U.S. patent application number 11/388205 was filed with the patent office on 2006-10-26 for spectroscopic polarimetry.
This patent application is currently assigned to OMRON CORPORATION. Invention is credited to Kenichi Matoba, Kazuhiko Oka, Hiroshi Okabe.
Application Number | 20060238759 11/388205 |
Document ID | / |
Family ID | 36617143 |
Filed Date | 2006-10-26 |
United States Patent
Application |
20060238759 |
Kind Code |
A1 |
Okabe; Hiroshi ; et
al. |
October 26, 2006 |
Spectroscopic polarimetry
Abstract
In the channeled spectroscopic polarimetry, a measurement error
of a parameter showing a spectropolarization characteristic of a
sample is effectively removed, the error being generated by various
variations in retardation of a retarder depending upon the state of
the sample. With attention being focused on that the retardation of
the retarder may be kept constant by stabilization of an incident
direction of light that transmits through the retarder, the
retarder was arranged on the light source side with respect to the
sample so as to effectively remove an influence relative to a
measurement error, such as variations in direction of a light ray
due to the sample.
Inventors: |
Okabe; Hiroshi; (Kyoto-shi,
JP) ; Matoba; Kenichi; (Kyoto-shi, JP) ; Oka;
Kazuhiko; (Sapporo-shi, JP) |
Correspondence
Address: |
FOLEY AND LARDNER LLP;SUITE 500
3000 K STREET NW
WASHINGTON
DC
20007
US
|
Assignee: |
OMRON CORPORATION
National University Corporation Hokkaido University
|
Family ID: |
36617143 |
Appl. No.: |
11/388205 |
Filed: |
March 24, 2006 |
Current U.S.
Class: |
356/369 |
Current CPC
Class: |
G01J 3/447 20130101;
G01J 4/04 20130101 |
Class at
Publication: |
356/369 |
International
Class: |
G01J 4/00 20060101
G01J004/00 |
Foreign Application Data
Date |
Code |
Application Number |
Mar 28, 2005 |
JP |
P2005-091763 |
Dec 15, 2005 |
JP |
P2005-362047 |
Claims
1. A spectroscopic polarimetry, comprising the steps of: preparing
an object to be measured; preparing a polarimetric spectroscope
which includes a projection optical system, comprising a light
source, a polarizer and a plurality of retarders, where the light
source, the polarizer and the plurality of retarders are arranged
such that light emitted from the light source is irradiated on the
object to be measured after passing through the polarizer and the
plurality of retarders in this order, an analyzer for allowing
light to transmit therethrough, the light having been emitted from
the projection optical system and reflected on or transmitted
through the object to be measured, and a device for obtaining the
spectral intensity of the light having transmitted through the
analyzer; and obtaining the spectral intensity of the object to be
measured by use of the polarimetric spectroscope.
2. The spectroscopic polarimetry according to claim 1, further
comprising a step of obtaining at least one of spectropolarization
parameters of the object to be measured by use of the obtained
spectral intensity.
3. The spectroscopic polarimetry according to claim 2, wherein the
plurality of retarders that the projection optical system comprises
are a first retarder and a second retarder, and the light source,
the polarizer, the first retarder and the second retarder are
arranged such that light emitted from the light source transmits
through the polarizer, the second retarder and the first retarder
in this order, the orientation of a transmission axis of the
polarizer disagrees with the orientation of a principal axis of the
second retarder, and the orientation of the principal axis of the
second retarder disagrees with the orientation of a principal axis
of the first retarder.
4. The spectroscopic polarimetry according to claim 3, wherein the
step of obtaining at least one of spectropolarization parameters is
a step which comprises: obtaining, from the spectral intensity, a
spectral intensity component (first spectral intensity component)
which nonperiodically vibrates with wavenumber and a spectral
intensity component (third spectral intensity component) which
vibrates at a frequency depending upon a retardation
(.phi..sub.2(.sigma.)) of the second retarder and not depending
upon a retardation (.phi..sub.1(.sigma.)) of the first retarder,
with wavenumber; and obtaining at least one of spectropolarization
parameters by use of each of the spectral intensity components.
5. The spectroscopic polarimetry according to claim 3, wherein the
step of obtaining at least one of spectropolarization parameters is
a step which comprises: obtaining, from the spectral intensity, at
least one of a spectral intensity component (second spectral
intensity component) which vibrates at a frequency depending upon a
variation between the retardation (.phi..sub.1 (.sigma.)) of the
first retarder and the retardation (.phi..sub.2(.sigma.)) of the
second retarder with wavenumber, a spectral intensity component
(fourth spectral intensity component) which vibrates at a frequency
depending upon the sum of the retardation (.phi..sub.1(.sigma.)) of
the first retarder and the retardation (.phi..sub.2(.sigma.)) of
the second retarder with wavenumber, and a spectral intensity
component (fifth spectral intensity component) which vibrates at a
frequency depending upon the retardation (.phi..sub.1(.sigma.)) of
the first retarder and not depending upon the retardation
(.phi..sub.2(.sigma.)) of the second retarder, with wavenumber; and
obtaining at least one of the spectropolarization parameters of the
object to be measured by use of the obtained spectral intensity
component.
6. The spectroscopic polarimetry according to claim 3, wherein the
step of obtaining at least one of spectropolarization parameters is
a step which comprises: obtaining, from the spectral intensity, at
least one of the spectral intensity component (first spectral
intensity component) which nonperiodically vibrates with wavenumber
and the spectral intensity component (third spectral intensity
component) which vibrates at a frequency depending upon the
retardation (.phi..sub.2(.sigma.)) of the second retarder and not
depending upon the retardation (.phi..sub.1(.sigma.)) of the first
retarder, with wavenumber, and at least one of the spectral
intensity component (second spectral intensity component) which
vibrates at a frequency depending upon the difference between the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder with
wavenumber, the spectral intensity component (fourth spectral
intensity component) which vibrates at a frequency depending upon
the sum of the retardation (.phi..sub.1(.sigma.)) of the first
retarder and the retardation (.phi..sub.2(.sigma.)) of the second
retarder with wavenumber, and the spectral intensity component
(fifth spectral intensity component) which vibrates at a frequency
depending upon the retardation (.phi..sub.1(.sigma.)) of the first
retarder and not depending upon the retardation
(.phi..sub.2(.sigma.)) of the second retarder, with wavenumber; and
obtaining at least one of the spectropolarization parameters of the
object to be measured by use of each of the obtained spectral
intensity components.
7. The spectroscopic polarimetry according to claim 3, wherein the
step of obtaining at least one of spectropolarization parameters is
a step which comprises: obtaining the retardation
(.phi..sub.2(.sigma.)) of the second retarder from the spectral
intensity; and obtaining at least one of the spectropolarization
parameters of the object to be measured by use of the spectral
intensity and the retardation (.phi..sub.2(.sigma.)) of the second
retarder.
8. The spectroscopic polarimetry according to claim 3, further
comprising a step of acquiring data showing the relation between
the retardation (.phi..sub.1(.sigma.)) of the first retarder and
the retardation (.phi..sub.2(.sigma.)) of the second retarder,
wherein the step of obtaining at least one of spectropolarization
parameters is a step which comprises: obtaining the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder from the spectral
intensity and the data showing the relation between the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder; and obtaining at
least one of the spectropolarization parameters of the object to be
measured by use of the spectral intensity, the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder.
9. The spectroscopic polarimetry according to claim 3, further
comprising the steps of: acquiring data showing the relation
between the retardation variation (.DELTA..phi..sub.1(.sigma.)) of
the first retarder and the retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder; and acquiring
a reference value (.phi..sub.1.sup.(i)(.sigma.)) for calibration of
retardation of the first retarder and a reference value
(.phi..sub.2.sup.(i)(.sigma.)) for calibration of retardation of
the second retarder, wherein the step of obtaining at least one of
spectropolarization parameters is a step which comprises:
obtaining, from the spectral intensity, the retardation
(.phi..sub.2(.sigma.)) of the second retarder and the retardation
variation (.DELTA..phi..sub.2(.sigma.)) of the second retarder from
the reference value (.phi..sub.2.sup.(i)(.sigma.)) for calibration;
obtaining the retardation variation (.DELTA..phi..sub.1(.sigma.))
of the first retarder by use of the obtained retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder and data
showing the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder; obtaining the retardation (.phi..sub.1(.sigma.)) of the
first retarder from a reference value
(.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation of
the first retarder and the obtained retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder; and obtaining
at least one of the spectropolarization parameters of the object to
be measured by use of the spectral intensity, the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder.
10. The spectroscopic polarimetry according to claim 3, wherein the
polarizer and the second retarder are arranged such that an angle
between the orientation of the transmission axis of the polarizer
and the orientation of a fast axis of the second retarder is
45.degree..
11. The spectroscopic polarimetry according to claim 2, further
comprising a step of obtaining a spectral intensity for calibration
by use of the polarimetric spectroscope in a state where an object
to be measured having an unknown spectropolarization characteristic
does not exist in a light path between the projection optical
system and the analyzer, wherein the step of obtaining at least one
of spectropolarization parameters is a step of obtaining at least
one of the spectropolarization parameters of the object to be
measured by use of the spectral intensity regarding the object to
be measured and the spectral intensity for calibration or data
based upon the spectral intensity for calibration.
12. The spectroscopic polarimetry according to claim 11, wherein
the step of obtaining the spectral intensity for calibration is a
step of preparing an analyzer for calibration in a position in
which light emitted from the projection optical system is received
in a state where the object to be measured having an unknown
spectropolarization characteristic does not exist in the light path
between the projection optical system and the analyzer.
13. The spectroscopic polarimetry according to claim 11, further
comprising a step of obtaining the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder by use of the
spectral intensity for calibration, wherein the step of obtaining
at least one of spectropolarization parameters is a step of
obtaining at least one of the spectropolarization parameters of the
object to be measured by use of the spectral intensity regarding
the object to be measured, the retardation (.phi..sub.1(.sigma.))
of the first retarder, and the retardation (.phi..sub.2(.sigma.))
of the second retarder, which are obtained by use of the spectral
intensity for calibration.
14. The spectroscopic polarimetry according to claim 8, wherein the
step of acquiring data showing the relation between the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder is a step which
comprises: obtaining the spectral intensity for calibration by use
of the polarimetric spectroscope in a state where the object to be
measured having an unknown spectropolarization characteristic does
not exist in the light path between the projection optical system
and the analyzer; and obtaining the data showing the relation
between the retardation (.phi..sub.1(.sigma.)) of the first
retarder and the retardation (.phi..sub.2(.sigma.)) of the second
retarder by use of the obtained spectral intensity for
calibration.
15. The spectroscopic polarimetry according to claim 9, wherein the
step of acquiring data showing the relation between the retardation
variation (.DELTA..phi..sub.1(.sigma.)) of the first retarder and
the retardation variation (.DELTA..phi..sub.2(.sigma.)) of the
second retarder is a step which comprises: obtaining the spectral
intensity for calibration by use of the polarimetric spectroscope
in a state where the object to be measured having an unknown
spectropolarization characteristic does not exist in the light path
between the projection optical system and the analyzer; and
obtaining the data showing the relation between the retardation
variation (.DELTA..phi..sub.1(.sigma.)) of the first retarder and
the retardation variation (.DELTA..phi..sub.2(.sigma.)) of the
second retarder by use of the obtained spectral intensity for
calibration.
16. The spectroscopic polarimetry according to claim 1, comprising
a step of obtaining the spectroscopic quasi-tokes parameter of the
object to be measured by use of the obtained spectral
intensity.
17. The spectroscopic polarimetry according to claim 16, wherein
the plurality of retarders that the projection optical system
comprises are a first retarder and a second retarder, the light
source, the polarizer, the first retarder and the second retarder
are arranged such that light emitted from the light source
transmits through the polarizer, the second retarder and the first
retarder in this order, the orientation of the transmission axis of
the polarizer disagrees with the orientation of the principal axis
of the second retarder, and the orientation of the principal axis
of the second retarder disagrees with the orientation of the
principal axis of the first retarder, the spectroscopic polarimetry
further comprises a step of acquiring data showing the relation
between the retardation (.phi..sub.1(.sigma.)) of the first
retarder and the retardation (.phi..sub.2(.sigma.)) of the second
retarder, and the step of obtaining the spectroscopic quasi-tokes
parameter comprises: obtaining, from the obtained spectral
intensity, at least one of the spectral intensity component (first
spectral intensity component) which nonperiodically vibrates with
wavenumber and the spectral intensity component (third spectral
intensity component) which vibrates at a frequency depending upon
the retardation (.phi..sub.2(.sigma.)) of the second retarder and
not depending upon the retardation (.phi..sub.1(.sigma.)) of the
first retarder, with wavenumber, and at least one of the spectral
intensity component (second spectral intensity component) which
vibrates at a frequency depending upon the difference between the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder with
wavenumber, the spectral intensity component (fourth spectral
intensity component) which vibrates at a frequency depending upon
the sum of the retardation (.phi..sub.1(.sigma.) of the first
retarder and the retardation (.phi..sub.2(.sigma.)) of the second
retarder with wavenumber, and the spectral intensity component
(fifth spectral intensity component) which vibrates at a frequency
depending upon the retardation (.phi..sub.1(.sigma.)) of the first
retarder and not depending upon the retardation
(.phi..sub.2(.sigma.)) of the second retarder, with wavenumber; and
obtaining the retardation (.phi..sub.1(.sigma.)) of the first
retarder, the retardation (.phi..sub.2(.sigma.)) of the second
retarder and the spectroscopic quasi-Stokes parameter by use of the
data showing the relation between the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder, and each of the
obtained spectral intensity components.
18. The spectroscopic polarimetry according to claim 16, wherein
the plurality of retarders that the projection optical system
comprises are a first retarder and a second retarder, the light
source, the polarizer, the first retarder and the second retarder
are arranged such that light emitted from the light source
transmits through the polarizer, the second retarder and the first
retarder in this order, the orientation of a transmission axis of
the polarizer disagrees with the orientation of a principal axis of
the second retarder, and the orientation of the principal axis of
the second retarder disagrees with the orientation of a principal
axis of the first retarder, the spectroscopic polarimetry further
comprises the steps of: acquiring data showing the relation between
the retardation variation (.DELTA..phi..sub.1(.sigma.)) of the
first retarder and the retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder; and acquiring
a reference value (.phi..sub.1.sup.(i)(.sigma.)) for calibration of
retardation of the first retarder and a reference value
(.phi..sub.2.sup.(i)(.sigma.)) for calibration of retardation of
the second retarder, and the step of obtaining the spectroscopic
quasi-Stokes parameter comprises: obtaining, from the obtained
spectral intensity, at least one of the spectral intensity
component (first spectral intensity component) which
nonperiodically vibrates with wavenumber and the spectral intensity
component (third spectral intensity component) which vibrates at a
frequency depending upon the retardation (.phi..sub.2(.sigma.)) of
the second retarder and not depending upon the retardation
(.phi..sub.1(.sigma.)) of the first retarder, with wavenumber, and
at least one of the spectral intensity component (second spectral
intensity component) which vibrates at a frequency depending upon
the difference between the retardation (.phi..sub.1(.sigma.)) of
the first retarder and the retardation (.phi..sub.2(.sigma.)) of
the second retarder with wavenumber, the spectral intensity
component (fourth spectral intensity component) which vibrates at a
frequency depending upon the sum of the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder with wavenumber, and
the spectral intensity component (fifth spectral intensity
component) which vibrates at a frequency depending upon the
retardation (.phi..sub.1(.sigma.)) of the first retarder and not
depending upon the retardation (.phi..sub.2(.sigma.)) of the second
retarder, with wavenumber; obtaining the retardation
(.phi..sub.2(.sigma.)) of the second retarder and the retardation
variation (.DELTA..phi..sub.2(.sigma.)) of the second retarder from
the reference value (.phi..sub.2.sup.(i)(.sigma.)) for calibration
by use of the obtained spectral intensity; obtaining the
retardation variation (.DELTA..phi..sub.1(.sigma.)) of the first
retarder by use of the obtained retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder and data
showing the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder; obtaining the retardation (.phi..sub.1(.sigma.)) of the
first retarder from the reference value
(.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation of
the first retarder and the obtained retardation variation
(.DELTA..phi.1(.sigma.)) of the first retarder; and obtaining the
spectroscopic quasi-Stokes parameter by use of each of the obtained
spectral intensity components, the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder.
19. A polarimetric spectroscope, comprising: a projection optical
system, comprising a light source, a polarizer and a plurality of
retarders, where the light source, the polarizer and the plurality
of retarders are arranged such that light emitted from the light
source is irradiated on the object to be measured after passing
through the polarizer and the plurality of retarders in this order;
an analyzer for allowing light to transmit therethrough, the light
having been emitted from the projection optical system and
reflected on or transmitted through the object to be measured; and
a device for obtaining the spectral intensity of the light having
transmitted through the analyzer.
20. The polarimetric spectroscope according to claim 19, wherein
the plurality of retarders that the projection optical system
comprises are a first retarder and a second retarder, and the light
source, the polarizer, the first retarder and the second retarder
are arranged such that light emitted from the light source
transmits through the polarizer, the second retarder and the first
retarder in this order, the orientation of a transmission axis of
the polarizer disagrees with the orientation of a principal axis of
the second retarder, and the orientation of the principal axis of
the second retarder disagrees with the orientation of a principal
axis of the first retarder.
21. The polarimetric spectroscope according to claim 20, wherein
the polarizer and the second retarder are arranged such that an
angle between the orientation of the transmission axis of the
polarizer and the orientation of a fast axis of the second retarder
is 45.degree..
22. The polarimetric spectroscope according to claim 19, further
comprising: an analyzer for calibration, detachably provided in a
position in which light emitted from the projection optical system
is received in a state where an object to be measured having an
unknown spectropolarization characteristic does not exist in a
light path between the projection optical system and the analyzer;
and a device for obtaining the spectral intensity of the light
having transmitted through the analyzer for calibration.
23. The polarimetric spectroscope according to claim 19, further
comprising an optical fiber for projecting light which guides the
light emitted from the light source to the polarizer.
24. The polarimetric spectroscope according to claim 23, wherein
the device for obtaining a spectral intensity comprises a
light-reception element or a spectrometer, and further comprises an
optical fiber for receiving light which guides the light having
transmitted through the analyzer to the light-reception element or
the spectrometer.
25. A spectroscopic polarimeter, comprising: a polarimetric
spectroscope, which comprises a projection optical system,
comprising a light source, a polarizer and a plurality of
retarders, where the light source, the polarizer and the plurality
of retarders are arranged such that light emitted from the light
source is irradiated on the object to be measured after passing
through the polarizer and the plurality of retarders in this order,
an analyzer for allowing light to transmit therethrough, the light
having been emitted from the projection optical system and
reflected on or transmitted through the object to be measured, and
a device for obtaining the spectral intensity of the light having
transmitted through the analyzer; and an arithmetic unit for
obtaining at least one of spectropolarization parameters of an
object to be measured, by use of the spectral intensity.
26. The spectroscopic polarimeter according to claim 25, wherein
the plurality of retarders that the projection optical system
comprises are a first retarder and a second retarder, the light
source, the polarizer, the first retarder and the second retarder
are arranged such that light emitted from the light source
transmits through the polarizer, the second retarder and the first
retarder in this order, the orientation of a transmission axis of
the polarizer disagrees with the orientation of a principal axis of
the second retarder, and the orientation of the principal axis of
the second retarder disagrees with the orientation of a principal
axis of the first retarder, the arithmetic unit is made capable of
using data showing the relation between the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder, the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder are obtained from the
spectral intensity and the data showing the relation between the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder; and at
least one of the spectropolarization parameters of the object to be
measured is obtained by use of the spectral intensity, the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder.
27. The spectroscopic polarimeter according to claim 25, wherein
the plurality of retarders that the projection optical system
comprises are a first retarder and a second retarder, the light
source, the polarizer, the first retarder and the second retarder
are arranged such that light emitted from the light source
transmits through the polarizer, the second retarder and the first
retarder in this order, the orientation of the transmission axis of
the polarizer disagrees with the orientation of the principal axis
of the second retarder, and the orientation of the principal axis
of the second retarder disagrees with the orientation of the
principal axis of the first retarder, the arithmetic unit is made
capable of using data showing the relation between the retardation
variation (.DELTA..phi..sub.1(.sigma.)) of the first retarder and
the retardation variation (.DELTA..phi..sub.2(.sigma.)) of the
second retarder, a reference value (.phi..sub.1.sup.(i)(.sigma.))
for calibration of retardation of the first retarder and a
reference value (.phi..sub.2.sup.(i)(.sigma.)) for calibration of
retardation of the second retarder, the retardation
(.phi..sub.2(.sigma.)) of the second retarder and the retardation
variation (.DELTA..phi..sub.2(.sigma.)) of the second retarder from
the reference value (.phi..sub.2.sup.(i)(.sigma.)) for calibration
are obtained from the spectral intensity, the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder is obtained by
use of the obtained retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder and data
showing the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder, the retardation (.phi..sub.1(.sigma.)) of the first
retarder is obtained from the reference value
(.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation of
the first retarder and the obtained retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder, and at least
one of the spectropolarization parameters of the object to be
measured is obtained by use of the spectral intensity, the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder.
28. An optical device, comprising: a projection optical system,
comprising a polarizer and a plurality of retarders, where the
polarizer and the plurality of retarders are arranged such that
light incident on the polarizer is irradiated on the object to be
measured after passing through the polarizer and the plurality of
retarders in this order; and an analyzer for allowing light to
transmit therethrough, the light having been emitted from the
projection optical system and reflected on or transmitted through
the object to be measured.
29. A light-projection device, comprising a polarizer and a
plurality of retarders, wherein the polarizer and the plurality of
retarders are arranged such that light incident on the polarizer is
irradiated on the object to be measured after passing through the
polarizer and the plurality of retarders in this order.
30. The light-projection device according to claim 29, wherein the
plurality of retarders are a first retarder and a second retarder,
and the polarizer, the first retarder and the second retarder are
arranged such that light incident on the polarizer transmits
through the polarizer, the second retarder and the first retarder
in this order, the orientation of a transmission axis of the
polarizer disagrees with the orientation of a principal axis of the
second retarder, and the orientation of the principal axis of the
second retarder disagrees with the orientation of a principal axis
of the first retarder.
31. The spectroscopic polarimetry according to claim 1, wherein the
polarimetric spectroscope prepared in the step of preparing a
polarimetric spectroscope further comprises a device for changing
the azimuth angle of the analyzer, the step of obtaining a spectral
intensity is a step of obtaining a spectral intensity regarding the
object to be measured in a plurality of states where azimuth angles
of the analyzer are made different from one another, by use of the
spectroscopic polarimetry, and the spectroscopic polarimetry
further comprises a step of obtaining at least one of the
spectropolarization parameters of the object to be measured by use
of the spectral intensity obtained in the plurality of states.
32. The spectroscopic polarimetry according to claim 1, wherein the
step of preparing an object to be measured is a step of preparing
an object to be measured which includes a sample and a polarization
element on which light emitted from the sample is incident, the
polarimetric spectroscope to be prepared in the step of preparing a
polarimetric spectroscope further comprises a device for changing
the characteristic of the polarization element, the step of
obtaining a spectral intensity is a step of obtaining a spectral
intensity regarding the object to be measured in a plurality of
states where characteristics of the polarization element are made
different from one another, by use of the spectroscopic
polarimetry, and the spectroscopic polarimetry further comprises a
step of obtaining at least one of spectropolarization parameters of
the sample by use of the spectral intensity obtained in the
plurality of states.
33. The spectroscopic polarimetry according to claim 1, wherein the
step of preparing an object to be measured is a step of preparing
an object to be measured which includes a sample and a polarization
element on which light emitted from the sample is incident, the
polarimetric spectroscope to be prepared in the step of preparing a
spectroscopic polarimetry further comprises a device for changing
the characteristic of the polarization element and a device for
changing the azimuth angle of the analyzer, the step of obtaining a
spectral intensity is a step of obtaining a spectral intensity
regarding the object to be measured in a plurality of states where
characteristics of the polarization element, or azimuth angles of
the analyzer, are made different from one another, by use of the
spectroscopic polarimetry, and the spectroscopic polarimetry
further comprises a step of obtaining at least one of
spectropolarization parameters of the sample by use of the spectral
intensity obtained in the plurality of states.
34. The polarimetric spectroscope according to claim 19, further
comprising a device for changing the azimuth angle of the
analyzer.
35. The spectroscopic polarimetry according to claim 19, further
comprising a device for changing the characteristic of the
polarization element in a case where the object to be measured
includes a sample and a polarization element on which light emitted
from the sample is incident.
36. The spectroscopic polarimetry according to claim 19, further
comprising a device for changing the characteristic of the
polarization element, and a device for changing the azimuth angle
of the analyzer, in a case where the object to be measured includes
a sample and a polarization element on which light emitted from the
sample is incident.
37. The optical device according to claim 28, further comprising a
device for changing the azimuth angle of the analyzer.
38. The optical device according to clam 28, further comprising a
device for changing the characteristic of the polarization element
in a case where the object to be measured includes a sample and a
polarization element on which light emitted from the sample is
incident.
39. The optical device according to claim 28, further comprising a
device for changing the characteristic of the polarization element,
and a device for changing the azimuth angle of the analyzer, in a
case where the object to be measured includes a sample and a
polarization element on which light emitted from the sample is
incident.
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The present invention relates to a technique for stabilizing
measurement of spectropolarization characteristics of an object to
be measured by use of a channeled spectrum.
[0003] 2. Description of the Related Art
[0004] Light has properties of a "transverse wave". Based upon the
premise of three mutually orthogonal axes (x, y, z), when a
propagation direction of light is assumed to be the z-axis
direction, a vibration direction of the light is a direction along
the x-y plane. The vibration direction of the light within the x-y
plane has a bias. This bias of light is referred to as
"polarization". A biased state of light is referred to as a "state
of polarization (SOP)" in this specification. Typically, the SOP
varies depending upon wavelengths (colors) of light.
[0005] When light in some state of polarization is incident on an
object to be measured to acquire outgoing light such as transparent
light or reflected light and the object to be measured has optical
anisotropy, a change in SOP is observed between the incident light
and the outgoing light. Acquiring information on anisotropy of the
object to be measured from the change in SOP is referred to as
"polarimetry". It is to be noted that causes of such anisotropy may
include anisotropy of a molecular structure, presence of stress
(pressure), and presence of a local field and a magnetic field.
[0006] A measurement in which a change in SOP between the incident
light and the outgoing light is obtained with respect to each
wavelength and information on anisotropy of an object to be
measured is then acquired is especially referred to as
"spectroscopic polarimetry". This spectroscopic polarimetry has an
advantage of acquiring a great amount of information as compared to
the case of measurement by use of a single wavelength (single
color). In the spectroscopic polarimetry, a device for measuring
the change in SOP between the incident light and the outgoing
light, namely a spectroscopic polarimeter, serves as a key
device.
[0007] As fields of application of the spectroscopic polarimetry,
there are known the field of spectroscopic ellipsometry, the
medical field, and the like. In the field of spectroscopic
ellipsometry, for example, since thickness as well as a complex
refractive index of a thin film can be measured in a nondestructive
and non-contact manner, spectroscopic polarimetry has been applied
to optical electronic devices, examination/study of semiconductors,
and the like. In the medical field, attempts have been made for
early detection of glaucoma and a cancer cell since several kinds
of cells have polarization characteristics.
[0008] As conventional typical spectroscopic polarimetry,
rotating-retarder polarimetry and polarization-modulation
polarimetry are known.
[0009] In these methods, a mechanical or electric polarization
control element is used to modulate light to be measured, and from
a change in spectral with the modulation, a Stokes parameter or the
like is obtained.
[0010] However, the following problems and some other problems with
the above methods have been pointed out: [1] a mechanical or
electric drive unit is required; and [2] it is necessary to
repeatedly change a plurality of spectrums while changing
conditions of the spectroscopic polarimetry.
[0011] In order to solve these problems, channeled
spectropolarimetry was previously contrived (cf. "Measurement of
spectral distribution of polarized light based on frequency region
interference method", written by Takayuki Katoh, Kazuhiko Oka,
Tetsu Tanaka, and Yoshihiro Ohtsuka, preliminary manuscript
collection for 34th Academic Lecture Meeting of Hokkaido Branch of
Japan Society of Applied Physics, (Hokkaido Branch of Japan Society
of Applied Physics, Sapporo, 1998) p. 41).
[0012] Further, spectroscopic ellipsometry utilizing the channeled
spectropolarimetry has also been reported (cf. "Spectroscopic
ellipsometry using channeled spectrum", written by Kazuhiko Oka and
Takayuki Katoh, collected papers of lectures in 26th Study Session
on Light Wave Sensing Technology (Light wave Sensing Technology
Study Session held by Japan Society of Applied Physics, Dec. 19-20,
2000) pp. 107-114).
[0013] A configuration view of an experiment system for explaining
the channeled spectroscopic polarimetry is shown in FIG. 26. As
apparent from this figure, white light emitted from a xenon lamp 1
is transmitted through a polarizer 2 and a Babinet-Soleil
compensator 3, to obtain a light wave having an SOP depending upon
a frequency .nu.. Spectral distributions S.sub.0 (.nu.), S.sub.1
(.nu.), S.sub.2 (.nu.) and S.sub.3 (.nu.) of the Stokes parameters
of the light wave are obtained by a measurement system 4 surrounded
by a dashed line in the figure.
[0014] Light under measurement is first transmitted through two
retarders R1 and R2 having different thicknesses (d1, d2) and an
analyzer A, and then incident on a spectrometer 5. Here, a slow
axis of the retarder R2 is inclined at an angle of 45.degree. with
respect to a slow axis of the retarder R1, while a transmission
axis of the analyzer A is arranged in parallel to the slow axis of
the retarder R1.
[0015] In each of the two retarders R1 and R2, a phase difference
created between the orthogonal polarization components depends upon
a frequency. Hence, as shown in FIG. 27, a channeled spectrum
including three carrier components is obtained from the
spectrometer 5 which functions as an optical spectrum analyzer. An
amplitude and a phase of each of the carrier components are
modulated by the spectrum distribution of the Stokes Parameters of
the light under measurement. It is therefore possible to obtain
each of the Stokes Parameters by execution of a signal processing
with a computer 6 by use of Fourier transformation.
[0016] One example of results of an experiment is shown in FIG. 28.
This is a result obtained in the case of inclining the
Babinet-Soleil compensator 3 at an angle of 30.degree. with respect
to the slow axis of the retarder R1. Three solid lines respectively
show spectral distributions S.sub.1(.nu.)/S.sub.0(.nu.),
S.sub.2(.nu.)/S.sub.0(.nu.), S.sub.3(.nu.)/S.sub.0(.nu.) of the
standardized Stokes parameters. It is thereby understood that the
SOP depends upon the frequency.
[0017] As thus described, according to the channeled spectroscopic
polarimetry, it is possible to obtain each spectrally-resolved
Stokes Parameter by a frequency analysis (or wavenumber analysis)
of characteristics of a spectral intensity. It is reasonably
necessary to obtain respective retardations of the two retarders R1
and R2 prior to the frequency analysis. Here, the retardation means
a phase difference created between a fast axis component and a slow
axis component.
[0018] According to the foregoing channeled spectroscopic
polarimetry, advantages can be obtained including that: [1] a
mechanically movable element such as a rotating retarder is
unnecessary; [2] an active element such as an electro-optic
modulator is unnecessary; [3] four Stokes Parameters are obtained
at once with one spectrum so that a so-called snap shot measurement
can be performed; and [4] the configuration is simple, and thus
suitable for size reduction.
SUMMARY OF THE INVENTION
[0019] However, concerning the foregoing channeled spectroscopic
polarimetry, a problem of a relatively large measurement error has
been pointed out for the following reasons.
[0020] In measurement of a spectropolarization characteristic by
the channeled spectropolarimetry, it is necessary to previously
calibrate retardation of a retarder. However, when the incident
direction of light to be incident on the retarder changes between
the time of calibration and the time of sample measurement, a
distance of light to pass through the retarder changes, and thereby
the retardation changes. This change in retardation between the
time of calibration and the time of measurement has caused a
measurement error. Further, it has been pointed out that,
especially in cases including the case of using a higher-order
retarder, retardation widely varies due to variations in light ray
direction, or wave surface fluctuation, of light to pass through
the retarder, or the like.
[0021] Moreover, there are mainly two kinds of methods as follows
for investigating properties of an unknown sample by use of the
channeled spectropolarimetry: [A] light is reflected on the sample,
and by use of an SOP of light acquired from the reflected light,
the properties of the sample is investigated; and [B] light is
transmitted through the sample, and by use of an SOP of light
acquired from the transmitted light, the properties of the light
are investigated. The foregoing retardation variation can also be
seen in each of those cases. In the following, each of those cases
is described.
[0022] [A] When light is reflected on the sample so that a
spectropolarization characteristic of the sample is measured, it is
necessary to keep the incident direction of the wave surface of
light to be incident on the retarder constant between the time of
pre-calibration and the time of measurement. However, as also
apparent from FIG. 24, an incident angle of the light to be
incident on the sample varies in many ways due to variations in
state of the surface of each sample or arrangement position of the
sample, or the like, resulting in variations in incident direction
of the wave surface of the light to be incident on the sample,
whereby it becomes difficult to keep the retardation of the
retarder constant from the time of pre-calibration. It should be
noted that in FIG. 24, reference symbol B denotes a sample,
reference symbols R1 and R2 respectively denote a first retarder
and a second retarder, reference symbol A denotes an analyzer, and
an indicator indicates the traveling direction of the light.
[0023] [B] Also when light is transmitted through the sample so
that a spectropolarization characteristic of the sample is
measured, it is necessary to keep the incident direction of the
wave surface of light to be incident on the retarder constant
between the time of precalibration and the time of measurement.
However, the incident direction of the wave surface of the light to
be incident on the retarder varies in many waves caused by the
variations in light ray direction (cf. FIG. 25A) due to the
inclination characteristic of the sample (inclination of the sample
surface), scattering of the light ray (cf. FIG. 25B) attributed to
a physical characteristic such as a rough surface of the sample, or
the like, whereby it becomes difficult to keep the retardation of
the retarder constant from the time of pre-calibration. It should
be noted that in FIG. 25, reference symbol C denotes a birefringent
medium as the sample, reference symbols R1 and R2 respectively
denote the first retarder and the second retarder, reference symbol
A denotes the analyzer, and an indicator indicates the traveling
direction of the light.
[0024] The present invention was made by noting the problems as
described above, and has an object to solve the problem of
variations in incident angle of a retarder seen in the conventional
channeled spectroscopic polarimetry, to provide a channeled
spectroscopic polarimetry and spectroscopic polarimeter which are
capable of measurement with even higher accuracy.
[0025] Further objects and action effects of the present invention
are readily understood by the skilled in the art by referring to
the following description of the specification.
[0026] (1) A spectroscopic polarimetry of the present invention
includes the steps of: preparing an object to be measured;
preparing a polarimetric spectroscope; and obtaining the spectral
intensity of the object to be measured by use of the polarimetric
spectroscope.
[0027] Here, the polarimetric spectroscope includes: a projection
optical system; an analyzer for allowing light to transmit
therethrough, the light having been emitted from the projection
optical system and reflected on or transmitted through the object
to be measured; and a means of obtaining the spectral intensity of
the light having transmitted through the analyzer. The projection
optical system comprises a light source, a polarizer and a
plurality of retarders, where the light source, the polarizer and
the plurality of retarders are arranged such that light emitted
from the light source is irradiated on the object to be measured
after passing through the polarizer and the plurality of retarders
in this order.
[0028] Here, the "object to be measured" is a generic term of an
object placed in a light path between the projection optical system
and the analyzer. Namely, other than a sample having an unknown
spectropolarization characteristic that is intended to be an object
of the spectroscopic polarimetry, a polarization element having a
known spectropolarization characteristic such as a phase
compensator is also included in the "object to be measured" when
placed in the light path between the projection optical system and
the analyzer.
[0029] "A plurality of retarders" include: a retarder which is
arranged behind the polarizer with respect to the traveling
direction of light and whose principal axis is oriented differently
from the transmission axis of the polarizer; and another retarder
whose principal axis is oriented differently from the principal
axis of the above-mentioned retarder. The "analyzer is an optical
element showing a different transmittance from that of the
polarization component located in an orthogonal direction to the
optical element. The "analyzer" is not restricted to have a plate
shape or a film shape. For example, a polarization beam splitter is
usable as the "analyzer".
[0030] The "means of obtaining the spectral intensity" may be using
a spectrometer or using the light source where a wavelength is
scanned. The spectrometer in the case of using the light source
where the wavelength is scanned may be one capable of detecting an
amount of light received, and a timing for detecting the amount of
light received is corresponded to the wavelength of light.
[0031] According to the spectroscopic polarimetry of the present
invention, since the object to be measured has no influence on the
orientation of light to transmit through the retarder, it is
possible to perform the spectroscopic polarimetry with high
stability.
[0032] (2) The spectroscopic polarimetry of the present invention
may include a step of obtaining at least one of spectropolarization
parameters of the object to be measured by use of the obtained
spectral intensity.
[0033] In the specification, the "spectropolarization parameter" is
used in the meaning of a parameter that expresses the
spectropolarization characteristic of the object to be
measured.
[0034] (3) The plurality of retarders that the projection optical
system comprises may be a first retarder and a second retarder. In
this case, each of the elements of the projection optical system is
arranged such that light emitted from the light source transmits
through the polarizer, the second retarder and the first retarder
in this order, the orientation of a transmission axis of the
polarizer disagrees with the orientation of a principal axis of the
second retarder, and the orientation of the principal axis of the
second retarder disagrees with the orientation of a principal axis
of the first retarder.
[0035] (4) In the following, three techniques for obtaining the
spectropolarization parameter in the case of using two retarders
are described. A first technique is one comprising: obtaining, from
the obtained spectral intensity, a spectral intensity component
(first spectral intensity component) which nonperiodically vibrates
with wavenumber and a spectral intensity component (third spectral
intensity component) which vibrates at a frequency depending upon
the retardation (.phi..sub.2(.sigma.)) of the second retarder and
not depending upon the retardation (.phi..sub.1(.sigma.)) of the
first retarder, with wavenumber; and obtaining at least one of
spectropolarization parameters by use of each of the spectral
intensity components.
[0036] According to this method, it is possible to obtain a rate of
change in amplitude rate between linearly polarized light
components which are orthogonal to each other along the orientation
of the principal axis of the first retarder, an intensity
attenuation rate of each component, and the like. Further, the
optical disposition in this case may be in either a reflection mode
or a transmission mode. Namely, light allowed to transmit through
the analyzer may be light emitted from the projection optical
system and reflected on the object to be measured, light that
transmitted through the object to be measured, or light scattered
by the object to be measured. Examples of the rate of change in
amplitude rate which can be obtained by this method include an
arctangent .psi.(.sigma.) of a rate of change in amplitude rate
which is one of the ellipsometric parameters, and a rate of change
in amplitude rate caused by scattering of light due to
particles.
[0037] (5) A second technique is one comprising: obtaining, from
the obtained spectral intensity, at least one of a spectral
intensity component (second spectral intensity component) which
vibrates at a frequency depending upon a difference between a
retardation (.phi..sub.1(.sigma.)) of the first retarder and a
retardation (.phi..sub.2(.sigma.)) of the second retarder with
wavenumber, a spectral intensity component (fourth spectral
intensity component) which vibrates at a frequency depending upon
the sum of the retardation (.phi..sub.1(.sigma.)) of the first
retarder and the retardation (.phi..sub.2(.sigma.)) of the second
retarder with wavenumber, and a spectral intensity component (fifth
spectral intensity component) which vibrates at a frequency
depending upon the retardation (.phi..sub.1(.sigma.)) of the first
retarder and not depending upon the retardation
(.phi..sub.2(.sigma.)) of the second retarder, with wavenumber; and
obtaining at least one of the spectropolarization parameters of the
object to be measured by use of the obtained spectral intensity
component.
[0038] According to this method, it is possible to obtain a
variation in phase difference between linearly polarized light
components which are orthogonal to each other along the orientation
of the principal axis of the first retarder, and the like. Further,
the optical disposition in this case may be in either a reflection
mode or a transmission mode. Namely, light allowed to transmit
through the analyzer may be light emitted from the projection
optical system and reflected on the object to be measured, light
that transmitted through the object to be measured, or light
scattered by the object to be measured. Examples of the rate of
change in phase difference which can be obtained by this method
include a phase difference variation .DELTA.(.sigma.) which is one
of the ellipsometric parameters, and a variation in phase
difference caused by scattering of light due to particles.
[0039] (6) A third technique is one comprising: obtaining, from the
obtained spectral intensity, at least one of the spectral intensity
component (first spectral intensity component) which
nonperiodically vibrates with wavenumber and the spectral intensity
component (third spectral intensity component) which vibrates at a
frequency depending upon the retardation (.phi..sub.2(.sigma.)) of
the second retarder and not depending upon the retardation
(.phi..sub.1(.sigma.)) of the first retarder, with wavenumber, and
at least one of the spectral intensity component (second spectral
intensity component) which vibrates at a frequency depending upon
the difference between the retardation (.phi..sub.1(.sigma.)) of
the first retarder and the retardation (.phi..sub.2(.sigma.)) of
the second retarder with wavenumber, the spectral intensity
component (fourth spectral intensity component) which vibrates at a
frequency depending upon the sum of the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder with wavenumber, and
the spectral intensity component (fifth spectral intensity
component) which vibrates at a frequency depending upon the
retardation (.phi..sub.1(.sigma.)) of the first retarder and not
depending upon the retardation (.phi..sub.2(.sigma.)) of the second
retarder, with wavenumber; and obtaining at least one of the
spectropolarization parameters of the object to be measured by use
of each of the obtained spectral intensity components.
[0040] Examples of the spectropolarization parameter that can be
obtained by this method include the rate of change in amplitude
rate between linearly polarized light components which are
orthogonal to each other along the orientation of the principal
axis of the first retarder, such as an arctangent .psi.(.sigma.) of
the rate of change in amplitude rate which is one of the
ellipsometric parameters and a rate of change in amplitude rate
caused by scattering of light due to particles. Further, the
optical disposition in this case can be in either a reflection mode
or a transmission mode. Namely, light allowed to transmit through
the analyzer may be light emitted from the projection optical
system and reflected on the object to be measured, light that
transmitted through the object to be measured, or light scattered
by the object to be measured. scattering of light due to particles.
Other examples of the spectropolarization parameter that can be
obtained by this method include an azimuth R of a birefringent
medium and a retardation .delta.(.sigma.).
[0041] (7) It is possible to calibrate the retardation of the
retarder by use of light itself which is applied to the object to
be measured and used for calibration. Meanwhile, it is also
possible to calibrate the retardation of the retarder by use of the
foregoing polarimetric spectroscope by means of light not applied
to the object to be measured, or to separately calibrate the
retardation of the retarder without use of the polarimetric
spectroscope. In one case of calibrating the retardation of the
retarder by use of light for use in measurement, the foregoing
measurement method using the two retarders comprises: obtaining the
retardation (.phi..sub.2(.sigma.)) of the second retarder from the
spectral intensity; and obtaining at least one of the
spectropolarization parameters of the object to be measured by use
of the obtained spectral intensity and the retardation
(.phi..sub.2(.sigma.)) of the second retarder.
[0042] Examples of the spectropolarization parameter that can be
obtained by this method include the rate of change in amplitude
rate between linearly polarized light components which are
orthogonal to each other along the orientation of the principal
axis of the first retarder, such as an arctangent .psi.(.sigma.) of
a rate of a change in amplitude rate as one of the ellipsometric
parameters, and a rate of change in amplitude rate caused by
scattering of light due to particles.
[0043] (8) In another case of calibrating the retardation of the
retarder by use of light for use in measurement, the foregoing
measurement method using the two retarders includes a step of
acquiring data showing the relation between the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder, wherein the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder are
obtained from the spectral intensity and the data showing the
relation between the retardation (.phi..sub.1(.sigma.)) of the
first retarder and the retardation (.phi..sub.2(.sigma.)) of the
second retarder, and at least one of the spectropolarization
parameters of the object to be measured is then obtained by use of
the obtained spectral intensity, the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder.
[0044] The "data showing the relation between the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder" is for example a
rate between the retardation (.phi..sub.1(.sigma.)) of the first
retarder and the retardation (.phi..sub.2(.sigma.)) of the second
retarder for each wavelength.
[0045] "Obtaining the retardation" includes the case of obtaining a
parameter equivalent to the retardation. In particular, obtaining a
complex function including information as to the retardation
corresponds to obtaining a parameter equivalent to the
retardation.
[0046] According to this spectroscopic polarimetry, it is possible
to effectively reduce a measurement error of the
spectropolarization parameter caused by variations in retardation
of the retarder due to a temperature change or other factors.
[0047] (9) The retardation of the retarder may be calibrated by use
of a reference value for calibration of the retardation as well as
light for use in measurement. In one case of calibrating the
retardation in such a manner, the measurement method using the two
retarders includes the steps of: acquiring data showing the
relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder; and acquiring a reference value
(.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation of
the first retarder and a reference value
(.phi..sub.2.sup.(i)(.sigma.)) for calibration of retardation of
the second retarder, wherein the retardation
(.phi..sub.2.sup.(i)(.sigma.)) of the second retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder from the reference value (.phi..sub.2.sup.(i)(.sigma.))
for calibration of the retardation are obtained from the obtained
spectral intensity; the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder is obtained by
use of the obtained retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder and data
showing the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder; the retardation (.phi..sub.1(.sigma.)) of the first
retarder is obtained from the reference value
(.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation of
the first retarder and the obtained retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder, and at least
one of the spectropolarization parameters of the object to be
measured is obtained by use of the obtained spectral intensity, the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder.
[0048] The "data showing the relation between the retardation
variation (.DELTA..phi..sub.1(.sigma.)) of the first retarder and
the retardation variation (.DELTA..phi..sub.2(.sigma.)) of the
second retarder" is for example a rate between the retardation
variation (.DELTA..phi..sub.1(.sigma.)) of the first retarder and
the retardation variation (.DELTA..phi..sub.2(.sigma.)) of the
second retarder for each wavelength. To this rate for each
wavelength, it is possible to apply the rate between the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder for each
wavelength so long as a medium of the first retarder is the same as
that of the second retarder.
[0049] The retardation of the second retarder obtained by use of
light for use in measurement is accompanied with phase ambiguity by
an integer multiple of 2.pi.. Although this does not have an
influence on a calculation error of the spectropolarization
parameter by itself, phase unwrapping, which is performed in
obtaining the retardation of the first retarder from the
retardation of the second retarder, may cause a calculation error
of retardation of the first retarder, thereby leading to generation
of the calculation error of the spectropolarization parameter.
Phase unwrapping is a process of determining a value of retardation
of the second retarder such that the value of retardation of the
second retarder continuously changes over the range of 2.pi. with
respect to a wavelength change. In the case of not using the
retardation variation of the second retarder, the retardation of
the first retarder is obtained by applying the "data showing the
relation between the retardation of the first retarder and the
retardation of the second retarder" to the retardation of the
second retarder after phase unwrapping. When wavenumber intervals
are not sufficiently large during change in value of retardation of
the second retarder by 2.pi. as compared to sampling wavenumber
intervals, or when noise is included in the measured value of
retardation of the second retarder, the retardation of the second
retarder after phase unwrapping could be calculated by a wrong
unit, 2.pi.. If the retardation of the first retarder is obtained
from the retardation of the second retarder including the error by
the unit of 2.pi. as thus described, the retardation of the first
retarder typically includes an error not by the unit of 2.pi., and
the included error becomes a large error when the
spectropolarization parameter is calculated. According to the
method for obtaining the retardation variation of the first
retarder from the retardation variation of the second retarder, and
then obtaining the retarder of the first retardation from the
retardation variation of the first retarder and the reference value
for calibration of the first retarder, since the retardation
variation of the second retarder changes modestly with wavenumber,
performing phase unwrapping on the retardation variation of the
second retarder is unnecessary, or necessary only in a small
frequency, which enables elimination of, or extreme reduction in,
the possibility for generation of an error in the retardation
variation of the first retarder due to phase unwrapping.
[0050] (10) In the spectroscopic polarimetry using the two
retarders, the polarizer and the second retarder may be arranged
such that an angle between the orientation of the transmission axis
of the polarizer and the orientation of a fast axis of the second
retarder is 45.degree..
[0051] In the case of arranging the polarizer and the second
retarder such that the angle between the orientation of the
transmission axis of the polarizer and the orientation of the fast
axis of the second retarder is 45.degree., there is an advantage of
simplifying calculation for obtaining the spectropolarization
parameter. Meanwhile, in the case of not limiting the angle between
the orientation of the transmission axis of the polarizer and the
orientation of the fast axis of the second retarder to 45.degree.,
there is an advantage of facilitating manufacturing of the optical
system since a limit on an assembly error of the optical system is
alleviated.
[0052] (11) The spectroscopic polarimetry for obtaining at least
one of the spectropolarization parameters further includes a step
of obtaining a spectral intensity for calibration by use of the
polarimetric spectroscope in a state where an object to be measured
having an unknown spectropolarization characteristic does not exist
in a light path between the projection optical system and the
analyzer, wherein at least one of the spectropolarization
parameters of the object to be measured is obtained by use of the
obtained spectral intensity regarding the object to be measured and
the spectral intensity for calibration or data based upon the
spectral intensity for calibration.
[0053] In obtaining the spectral intensity for calibration, an
object to change the spectropolarization state of light may be
prevented from existing in the light path between the projection
optical system and the analyzer, or an object having a known
spectropolarization characteristic may exist in the light path.
[0054] (12) In obtaining the spectral intensity for calibration, an
analyzer for calibration may be prepared in a position in which
light emitted from the projection optical system is received in a
state where the object to be measured having an unknown
spectropolarization characteristic does not exist in the light path
between the projection optical system and the analyzer, and a
spectral intensity of light having passed through the analyzer for
calibration may be obtained.
[0055] When the light path between the projection optical system
and the analyzer is bent due to reflection on or inflection in the
object to be measured, it is possible to obtain the spectral
intensity for calibration by arrangement of the analyzer for
calibration in a position in which light emitted from the
projection optical system is received in a state where an object to
bend the light path as in the case where the object to be measured
exists in the light path does not exist. In this case, an object
having a known spectropolarization characteristic may also exist in
the light path. Further, the analyzer for calibration may be
prepared separately from the analyzer for measurement, or the
position of the analyzer for measurement may be temporarily changed
so that the analyzer for measurement may be used as the analyzer
for calibration.
[0056] (13) The spectroscopic polarimetry using the spectral
intensity for calibration may include a step of obtaining the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder by use of
the spectral intensity for calibration, wherein at least one of the
spectropolarization parameters of the object to be measured may be
obtained by use of the obtained spectral intensity regarding the
object to be measured, the retardation (.phi..sub.1(.sigma.)) of
the first retarder, and the retardation (.phi..sub.2(.sigma.)) of
the second retarder, which are obtained by means of the spectral
intensity for calibration.
[0057] (14) In the foregoing measurement method in which data
showing the relation between the retardation (.phi..sub.1(.sigma.))
of the first retarder and the retardation (.phi..sub.2(.sigma.)) of
the second retarder is acquired by use of the two retarders and the
retardation of the retarder is then calibrated by use of light for
use in measurement, the spectral intensity for calibration may be
obtained by use of the polarimetric spectroscope in a state where
the object to be measured having an unknown spectropolarization
characteristic does not exist in the light path between the
projection optical system and the analyzer, and the data showing
the relation between the retardation (.phi..sub.1(.sigma.)) of the
first retarder and the retardation (.phi..sub.2(.sigma.)) of the
second retarder may be obtained by use of the obtained spectral
intensity for calibration.
[0058] (15) In the foregoing measurement method in which data
showing the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder is acquired by use of the two retarders and the
retardation of the retarder is then calibrated by use of a
reference value for calibration of the retardation and light for
use in measurement, the spectral intensity for calibration may be
obtained by use of the polarimetric spectroscope in a state where
the object to be measured having an unknown spectropolarization
characteristic does not exist in the light path between the
projection optical system and the analyzer, and the data showing
the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder may be obtained by use of the obtained spectral intensity
for calibration.
[0059] (16) In the spectroscopic polarimetry of the present
invention, a spectroscopic quasi-Stokes parameter of the object to
be measured may be obtained by use of the obtained spectral
intensity.
[0060] (17) In the spectroscopic polarimetry for obtaining the
spectroscopic quasi-Stokes parameter, the plurality of retarders
that the projection optical system comprises may be a first
retarder and a second retarder. In this case, each of the elements
of the projection optical system is arranged such that light
emitted from the light source transmits through the polarizer, the
second retarder and the first retarder in this order, the
orientation of the transmission axis of the polarizer disagrees
with the orientation of the principal axis of the second retarder,
and the orientation of the principal axis of the second retarder
disagrees with the orientation of the principal axis of the first
retarder. Further, the spectroscopic polarimetry may include a step
of acquiring data showing the relation between the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder. From the obtained
spectral intensity, at least one of the spectral intensity
component (first spectral intensity component) which
nonperiodically vibrates with wavenumber and the spectral intensity
component (third spectral intensity component) which vibrates at a
frequency depending upon the retardation (.phi..sub.2(.sigma.)) of
the second retarder and not depending upon the retardation
(.phi..sub.1(.sigma.)) of the first retarder, with wavenumber, and
at least one of the spectral intensity component (second spectral
intensity component) which vibrates at a frequency depending upon
the difference between the retardation (.phi..sub.1(.sigma.)) of
the first retarder and the retardation (.phi..sub.2(.sigma.)) of
the second retarder with wavenumber, the spectral intensity
component (fourth spectral intensity component) which vibrates at a
frequency depending upon the sum of the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder with wavenumber, and
the spectral intensity component (fifth spectral intensity
component) which vibrates at a frequency depending upon the
retardation (.phi..sub.1(.sigma.)) of the first retarder and not
depending upon the retardation (.phi..sub.2(.sigma.)) of the second
retarder, with wavenumber, may be obtained. The retardation
(.phi..sub.1(.sigma.)) of the first retarder, the retardation
(.phi..sub.2(.sigma.)) of the second retarder and the spectroscopic
quasi-Stokes parameter may then be obtained by use of the data
showing the relation between the retardation (.phi..sub.1(.sigma.))
of the first retarder and the retardation (.phi..sub.2(.sigma.)) of
the second retarder, and each of the obtained spectral intensity
components.
[0061] The meaning of "obtaining the spectroscopic quasi-Stokes
parameter" includes obtaining all or part of four spectroscopic
quasi-Stokes parameters M.sub.0, M.sub.1, M.sub.2, M.sub.3
(respective definitonal equations for these parameters are provided
in DETAILED DESCRIPTION OF THE INVENTION). Whether or not all the
spectroscopic quasi-Stokes parameters are practically obtained is
referred to a choice of a person implementing the spectroscopic
polarimetry. However, according to the spectroscopic polarimetry of
the present invention, it is possible in theory to obtain all the
spectroscopic quasi-Stokes parameters.
[0062] For obtaining the spectroscopic quasi-Stokes parameter
M.sub.0(.sigma.), the first spectral intensity component and a
reference amplitude function m.sub.0(.sigma.) are needed. For
obtaining the spectroscopic quasi-Stokes parameter
M.sub.1(.sigma.), the third spectral intensity component, the
retardation of the second retarder and a reference amplitude
function m.sub.2(.sigma.) are needed.
[0063] For obtaining the spectroscopic quasi-Stokes parameter
M.sub.2(.sigma.) and M.sub.3(.sigma.), at least any one of a group
of the second spectral intensity component, the retardations of the
first and second retarders and a reference amplitude function
m.sub.-(.sigma.), a group of the forth spectral intensity
component, the retardations of the first and second retarders, and
a reference amplitude function m.sub.+(.sigma.), and a group of the
fifth spectral intensity component, the retardation of the first
retarder and a reference amplitude function m.sub.1(.sigma.) is
needed.
[0064] Further, the reference amplitude functions needed for
obtaining the spectroscopic quasi-Stokes parameter need to be made
usable when the spectroscopic quasi-Stokes parameter is
obtained.
[0065] According to this spectroscopic polarimetry for obtaining
the spectroscopic quasi-Stokes parameter, an active element such as
a mechanically moving part or an electric optical modulator for
polarization control is not necessary. In the spectroscopic
polarimetry, by acquiring a one-time spectral, it is possible in
theory to obtain all spectroscopic quasi-Stokes parameter of the
object to be measured, and further possible to effectively reduce a
measurement error of the spectroscopic quasi-Stokes parameter
generated caused by variations in retardation of the retarder due
to a temperature change or other factors. By performing further
calculation by use of the spectroscopic quasi-Stokes parameters, it
is possible to obtain a variety of spectropolarization parameters
regarding the object to be measured. Especially in a case where a
Mueller matrix of the object to be measured is determined from only
two to three parameters at most, an arbitrary spectropolarization
parameter can be obtained from the spectroscopic quasi-Stokes
parameters.
[0066] (18) In the spectroscopic polarimetry for obtaining the
spectroscopic quasi-Stokes parameter includes the steps of
acquiring data showing the relation between the retardation
variation (.DELTA..phi..sub.1(.sigma.)) of the first retarder and
the retardation variation (.DELTA..phi..sub.2(.sigma.)) of the
second retarder by use of the two retarders; and acquiring a
reference value (.phi..sub.1.sup.(i)(.sigma.)) for calibration of
retardation of the first retarder and a reference value
(.phi..sub.2.sup.(i)(.sigma.)) for calibration of retardation of
the second retarder. In the spectroscopic polarimetry, from the
obtained spectral intensity, at least one of the spectral intensity
component (first spectral intensity component) which
nonperiodically vibrates with wavenumber and the spectral intensity
component (third spectral intensity component) which vibrates at a
frequency depending upon the retardation (.phi..sub.2(.sigma.)) of
the second retarder and not depending upon the retardation
(.phi..sub.1(.sigma.)) of the first retarder, with wavenumber, and
at least one of the spectral intensity component (second spectral
intensity component) which vibrates at a frequency depending upon
the difference between the retardation (.phi..sub.1(.sigma.)) of
the first retarder and the retardation (.phi..sub.2(.sigma.)) of
the second retarder with wavenumber, the spectral intensity
component (fourth spectral intensity component) which vibrates at a
frequency depending upon the sum of the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder with wavenumber, and
the spectral intensity component (fifth spectral intensity
component) which vibrates at a frequency depending upon the
retardation (.phi..sub.1(.sigma.)) of the first retarder and not
depending upon the retardation (+2(.sigma.)) of the second
retarder, with wavenumber, may be obtained. The retardation
(.phi..sub.2(.sigma.)) of the second retarder and the retardation
variation (.DELTA..phi..sub.2(.sigma.)) of the second retarder from
the reference value (.phi..sub.2.sup.(i)(.sigma.) for calibration
may be obtained by use of the obtained spectral intensity. The
retardation variation (.DELTA..phi..sub.1(.sigma.)) of the first
retarder may be obtained by use of the obtained retardation
variation (.DELTA..phi..sub.2(.sigma.)) of the second retarder and
data showing the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder. The retardation (.phi..sub.1(.sigma.)) of the first
retarder may be obtained from the reference value
(.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation of
the first retarder and the obtained retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder. The
spectroscopic quasi-Stokes parameter may be obtained by use of each
of the obtained spectral intensity components, the retardation
(.phi..sub.1(.sigma.)) of the first retarder and the retardation
(.phi..sub.2(.sigma.)) of the second retarder.
[0067] (19) A polarimetric spectroscope of the present invention
comprises: a projection optical system, comprising a light source,
a polarizer and a plurality of retarders, where the light source,
the polarizer and the plurality of retarders are arranged such that
light emitted from the light source is irradiated on the object to
be measured after passing through the polarizer and the plurality
of retarders in this order; an analyzer for allowing light to
transmit therethrough, the light having been emitted from the
projection optical system and reflected on or transmitted through
the object to be measured; and a means of obtaining the spectral
intensity of the light having transmitted through the analyzer.
[0068] According to this polarimetric spectroscope, the direction
of light that transmits through the retarder is unsusceptible to
the object to be measured, enabling spectroscopic polarimetry with
high stability.
[0069] (20) In this polarimetric spectroscope, the plurality of
retarders that the projection optical system comprises may be a
first retarder and a second retarder. In this case, each of the
elements of the projection optical system is arranged such that
light emitted from the light source transmits through the
polarizer, the second retarder and the first retarder in this
order, the orientation of a transmission axis of the polarizer
disagrees with the orientation of a principal axis of the second
retarder, and the orientation of the principal axis of the second
retarder disagrees with the orientation of a principal axis of the
first retarder.
[0070] (21) In the polarimetric spectroscope using the two
retarders, the polarizer and the second retarder may be arranged
such that an angle between the orientation of the transmission axis
of the polarizer and the orientation of a fast axis of the second
retarder is 45.degree..
[0071] (22) The spectroscopic polarimetry of the present invention
may comprise: an analyzer for calibration, detachably provided in a
position in which light emitted from the projection optical system
is received in a state where an object to be measured having an
unknown spectropolarization characteristic does not exist in the
light path between the projection optical system and the analyzer;
and a means of obtaining the spectral intensity of the light having
transmitted through the analyzer for calibration.
[0072] Here, the whole or part of the "means of obtaining the
spectral intensity of the light having transmitted through the
analyzer" may double as the "means of obtaining the spectral
intensity of the light having transmitted through the analyzer for
calibration".
[0073] Also when the light path between the projection optical
system and the analyzer is bent due to reflection on or inflection
in the object to be measured, the use of this polarimetric
spectroscope enables calibration even in a state where an object to
bend the light path, in the same manner as in the case of existence
of the object to be measured, does not exist in the light path. In
this case, an object having a known spectropolarization
characteristic may exist in the light path. It is therefore
possible to perform calibration by singly using the device, prior
to installation of the object to be measured in the device, or
installation of the device on the object to be measured.
[0074] (23) The polarimetric spectroscope of the present invention
may further comprise an optical fiber for projecting light which
guides the light emitted from the light source to the
polarizer.
[0075] According to this spectroscopic polarimetry, the light
source can be installed in a position apart from the measurement
place, thereby facilitating size reduction in a portion of the
polarimetric spectroscope, the portion being used in the vicinity
of the measurement place.
[0076] (24) In the polarimetric spectroscope of the present
invention, the means of obtaining a spectral intensity may comprise
a light-reception element or a spectrometer, and may further
comprise an optical fiber for receiving light which guides the
light having transmitted through the analyzer to the
light-reception element or the spectrometer.
[0077] According to this polarimetric spectroscope, the light
source can be installed in a position apart from the measurement
place, thereby facilitating size reduction in a portion of the
polarimetric spectroscope, the portion being used in the vicinity
of the measurement place.
[0078] (25) A spectroscopic polarimeter of the present invention
comprises the foregoing polarimetric spectroscope of the present
invention and an arithmetic unit for obtaining at least one of
spectropolarization parameters of an object to be measured, by use
of a spectral intensity of light having transmitted through an
analyzer.
[0079] (26) In the spectroscopic polarimeter of the present
invention, the plurality of retarders that the projection optical
system comprises may be a first retarder and a second retarder. In
this case, each of the elements of the projection optical system is
arranged such that light emitted from the light source transmits
through the polarizer, the second retarder and the first retarder
in this order, the orientation of a transmission axis of the
polarizer disagrees with the orientation of a principal axis of the
second retarder, and the orientation of the principal axis of the
second retarder disagrees with the orientation of a principal axis
of the first retarder. Further, the arithmetic unit of this
spectroscopic polarimeter is made capable of using data showing the
relation between the retardation (.phi..sub.1(.sigma.)) of the
first retarder and the retardation (.phi..sub.2(.sigma.)) of the
second retarder. The retardation (.phi..sub.1(.sigma.)) of the
first retarder and the retardation (.phi..sub.2(.sigma.)) of the
second retarder may be obtained from the spectral intensity of the
light having transmitted through the analyzer and the data showing
the relation between the retardation (.phi..sub.1(.sigma.)) of the
first retarder and the retardation (.phi..sub.2(.sigma.)) of the
second retarder. At least one of the spectropolarization parameters
of the object to be measured may then be obtained by use of the
spectral intensity of the light having transmitted through the
analyzer, the retardation (.phi..sub.1(.sigma.)) of the first
retarder and the retardation (.phi..sub.2(.sigma.)) of the second
retarder.
[0080] (27) In the spectroscopic polarimeter of the present
invention, the plurality of retarders that the projection optical
system comprises may be a first retarder and a second retarder. The
arithmetic unit is made capable of using data showing the relation
between the retardation variation (.DELTA..phi..sub.1(.sigma.)) of
the first retarder and the retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder, a reference
value (.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation
of the first retarder and a reference value
(.phi..sub.2.sup.(i)(.sigma.)) for calibration of retardation of
the second retarder. The retardation (.phi..sub.2(.sigma.)) of the
second retarder and the retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder from the
reference value (.phi..sub.2.sup.(i)(.sigma.)) for calibration may
be obtained from the spectral intensity of the light having
transmitted through the analyzer, the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder may be obtained
by use of the obtained retardation variation
(.DELTA..phi..sub.2(.sigma.)) of the second retarder and data
showing the relation between the retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder and the
retardation variation (.DELTA..phi..sub.2(.sigma.)) of the second
retarder, the retardation (.phi..sub.1(.sigma.)) of the first
retarder may be obtained from the reference value
(.phi..sub.1.sup.(i)(.sigma.)) for calibration of retardation of
the first retarder and the obtained retardation variation
(.DELTA..phi..sub.1(.sigma.)) of the first retarder. At least one
of the spectropolarization parameters of the object to be measured
may then be obtained by use of the obtained spectral intensity, the
retardation (.phi..sub.1(.sigma.)) of the first retarder and the
retardation (.phi..sub.2(.sigma.)) of the second retarder.
[0081] (28) An optical device of the present invention comprises: a
projection optical system, comprising a polarizer and a plurality
of retarders, where the polarizer and the plurality of retarders
are arranged such that light incident on the polarizer is
irradiated on the object to be measured after passing through the
polarizer and the plurality of retarders in this order; and an
analyzer for allowing light to transmit therethrough, the light
having been emitted from the projection optical system and
reflected on or transmitted through the object to be measured.
[0082] Such an optical device can be used for the foregoing
polarimetric spectroscope.
[0083] (29) A light-projection device of the present invention
comprises a polarizer and a plurality of retarders, wherein the
polarizer and the plurality of retarders are arranged such that
light incident on the polarizer is irradiated on the object to be
measured after passing through the polarizer and the plurality of
retarders in this order.
[0084] Such a light-projection device can be used for the foregoing
polarimetric spectroscope.
[0085] (30) In this light-projection device, the plurality of
retarders may be a first retarder and a second retarder. Each of
the elements of the light-projection device may be arranged such
that light incident on the polarizer transmits through the
polarizer, the second retarder and the first retarder in this
order, the orientation of a transmission axis of the polarizer
disagrees with the orientation of a principal axis of the second
retarder, and the orientation of the principal axis of the second
retarder disagrees with the orientation of a principal axis of the
first retarder.
[0086] Next described are a spectroscopic polarimetry, a
polarimetric spectroscope and an optical device which make a
characteristic of a polarization element or an azimuth angle of an
analyzer changeable. The polarization element here refers to a
polarization element in a case where the object to be measured is
composed of a sample and a polarization element on which light,
having transmitted through or been reflected on the sample, is
incident. The polarization element is an optical element where
incident light and outgoing light have a relation depending upon
polarization. For changing the characteristic of the polarization
element, it is possible to use, for example, a means of changing
the azimuth angle of the polarization element, a means of changing
the retardation of the polarization element, and some other
means.
[0087] (31) In the spectroscopic polarimetry of the present
invention, a polarimetric spectroscope further comprising a means
of changing the azimuth angle of the analyzer may be used. By use
of this polarimetric spectroscope, a spectral intensity regarding
the object to be measured may be obtained in a plurality of states
where azimuth angles of the analyzer are made different from one
another, and at least one of the spectropolarization parameters of
the object to be measured may be obtained by use of the obtained
spectral intensity.
[0088] (32) In the spectroscopic polarimetry of the present
invention, an object to be measured including a sample and a
polarization element is prepared. The polarimetric spectroscope,
further comprising a means of changing the characteristic of the
polarization element, is used. By use of this spectroscopic
polarimetry, a spectral intensity regarding the object to be
measured may be obtained in a plurality of states where
characteristics of the polarization element are made different from
one another. At least one of spectropolarization parameters of the
sample may then be obtained by use of the obtained spectral
intensity.
[0089] (33) Here, the spectroscopic polarimetry may be used which
further comprises a means of changing the azimuth angle of the
analyzer in addition to the means of changing the characteristic of
the polarization element. By use of this spectroscopic polarimetry,
a spectral intensity regarding the object to be measured may be
obtained in a plurality of states where characteristics of the
polarization element or azimuth angles of the analyzer are made
different from one another. At least one of spectropolarization
parameters of the sample may then be obtained by use of the
obtained spectral intensity.
[0090] (34) The polarimetric spectroscope of the present invention
may further comprise a means of changing the azimuth angle of the
analyzer. This polarimetric spectroscope may be combined with an
arithmetic unit for obtaining at least one of spectropolarization
parameters of the object to be measured by use of this
spectroscopic polarimetry a spectral intensity regarding the object
to be measured obtained in a plurality of states where azimuth
angles of the analyzer are made different from one another, to give
a spectroscopic polarimeter.
[0091] (35) In a case where the object to be measured includes a
sample and a polarization element, the polarimetric spectroscope of
the present invention may further comprise a means of changing the
characteristic of the polarization element. This polarimetric
spectroscope may be combined with an arithmetic unit for obtaining
at least one of spectropolarization parameters of the sample by use
of a spectral intensity regarding the object to be measured
obtained in a plurality of states where characteristics of the
polarization element are made different from one another, to give a
spectroscopic polarimeter.
[0092] (36) Here, the spectroscopic polarimetry may further
comprise a means of changing the azimuth angle of the analyzer in
addition to the means of changing the characteristic of the
polarization element. This polarimetric spectroscope may be
combined with an arithmetic unit for obtaining at least one of
spectropolarization parameters of the sample by use of a spectral
intensity regarding the object to be measured obtained in a
plurality of states where characteristics of the polarization
element or azimuth angles of the analyzers are made different from
one another, to give a spectroscopic polarimeter.
[0093] (37) The optical device of the present invention may further
comprise a means of changing the azimuth angle of the analyzer.
[0094] (38) In a case where the object to be measured includes a
sample and a polarization element, the optical device of the
present invention may further comprise a means of changing the
characteristic of the polarization element.
[0095] (39) Here, the optical device may further comprise a means
of changing the azimuth angle of the analyzer in addition to the
means of changing the characteristic of the polarization
element.
[0096] As thus described, by making the characteristic of the
polarization element or the azimuth angle of the analyzer
changeable, it is possible to obtain a spectral intensity in a
plurality of states where characteristics of the polarization
element or azimuth angles of the analyzers are made different from
one another. Thereby, from the spectral intensity obtained in a
state where the number of characteristics of the polarizer or the
number of azimuth angles of the analyzer is relatively small, a
relatively many kinds of spectropolarization parameters can be
obtained, and an error due to an influence such as noise included
in the obtained value of the spectropolarization parameter can be
reduced. Or, the characteristic of the polarizer or the azimuth
angle of the analyzer can be selected such that a specific one or
more than one kinds of spectropolarization parameters of the object
to be measured or the sample can be obtained with high
sensitivity.
[0097] According to the present invention, it is possible to
perform the spectroscopic polarimetry with high stability since the
direction of light that transmits through the retarder is not
susceptible to the object to be measured.
BRIEF DESCRIPTION OF THE DPRWINGS
[0098] FIG. 1 shows a view for explaining a principle of solving a
problem.
[0099] FIG. 2 shows a view for explaining a configuration of an
optical system device and an azimuth angle of each optical
element.
[0100] FIG. 3 shows a view for explaining spectroscopic
polarimetry.
[0101] FIG. 4 shows a view for explaining a relation between a
channeled spectrum obtained from a spectrometer and its four
components.
[0102] FIG. 5 shows a view for explaining a procedure (flows of
signal processing) for demodulating a spectrometric quasi-Stokes
parameter.
[0103] FIG. 6 shows a view for explaining one example of Step
2.
[0104] FIG. 7 shows a view for explaining Fourier
transformation.
[0105] FIG. 8 shows a view for explaining flows of signals for
calibration during measurement.
[0106] FIG. 9 shows a view for explaining flows of signals in
combination of "calibration during measurement" and "measurement of
a spectrometric quasi-Stokes parameter".
[0107] FIG. 10 shows a comparative view for explaining methods (No.
1, 2) for calibrating a reference phase function during
measurement.
[0108] FIG. 11 shows a device configuration view (No. 1) in
ellipsometry.
[0109] FIG. 12 shows a device configuration view (No. 2) in
ellipsometry.
[0110] FIG. 13 shows a view for explaining a principle of solving a
problem in the case of reflecting light on a sample.
[0111] FIG. 14 shows a view for explaining a device configuration
in the case of separately installing an optical system for
calibration.
[0112] FIG. 15 shows a view for explaining a polarization state of
light reflected on the sample.
[0113] FIG. 16 shows a view for explaining a device configuration
(No. 1) in a double refraction measurement.
[0114] FIG. 17 shows a view for explaining a device configuration
(No. 2) in the double refraction measurement.
[0115] FIG. 18 shows a view for explaining a principle of solving
the problem in the case of allowing light to transmit through the
sample.
[0116] FIG. 19 shows a view for explaining a device configuration
in the case of arranging a polarization element having a known
spectropolarization characteristic in front of or behind the
sample.
[0117] FIG. 20 shows a configuration view (No. 1) of one example of
a spectroscopic polarimeter.
[0118] FIG. 21 shows a configuration view (No. 2) of one example of
the spectroscopic polarimeter.
[0119] FIG. 22 shows a flowchart of a pre-calibration
procedure.
[0120] FIG. 23 shows a flowchart of a measurement procedure.
[0121] FIG. 24 shows a view (No. 1) for explaining variations in
wave surface of light that passes through a retarder in the
incident direction.
[0122] FIG. 25 shows a view (No. 2) for explaining variations in
wave surface of the light that passes through the retarder in the
incident direction.
[0123] FIG. 26 shows a configuration view of an experimental system
of channeled spectroscopic polarimetry.
[0124] FIG. 27 shows a graph of a Stokes parameter in the
experimental system.
[0125] FIG. 28 shows a graph of a standardized Stokes parameter in
the experimental system.
[0126] FIG. 29 shows a device configuration view in the case of
measuring a spectropolarization parameter of the sample.
[0127] FIG. 30 shows a view of a configuration of a channeled
spectroscopic polarization state generator (CSPSG).
DETAILED DESCRIPTION OF THE INVENTION
[0128] In the following, one preferred embodiment of the present
invention is specifically described with reference to attached
drawings (FIGS. 1 to 19).
[0129] Chapter 1: Principle of Channeled Spectroscopic
Polarimetry
1.1 Constitution of Optical System in Present Invention
[0130] FIG. 1 shows a view for explanation comparing a
configuration of an optical system in a conventional channeled
spectroscopic polarimetry and a configuration of an optical system
in a channeled spectroscopic polarimetry of the embodiment of the
present invention. The optical system (cf. FIG. 1B) in the
conventional channeled spectroscopic polarimetry is comprised of a
light source 7, a polarizer P, and a polarimeter. The polarimeter
is comprised of two thick retarders R1 and R2, an analyzer A and a
spectrometer 8. It is to be noted that reference symbol D denotes a
sample through which light is transmitted or on which light is
reflected. Here, fast axes of the retarder R1 and the retarder R2
are inclined at an angle of 45.degree. from each other. Meanwhile,
a transmission axis of the analyzer A agrees with the fast axis of
the retarder R1.
[0131] It is to be noted that crossing angles among these three
elements may not necessarily be 45.degree.. Measurement is possible
even with a different crossing angle, although less efficient to
some extent. In short, any crossing angle can be applied so long as
principal axes of the adjacent elements are not superposed on each
other. A description in this respect is given later. What is
important is that each element is fixed and thus not required to be
rotated or modulated as in the conventional method.
[0132] Light having a broad spectrum is emitted from the light
source 7 on the left side of the figure, transmits through the
polarizer and is reflected on or transmits through the sample D
before being incident on the polarimeter. Thereafter, the light is
incident on the polarimeter. A spectral distribution of state of
polarization (SOP) of the light having been emitted from the sample
D can be expressed by spectrometric Stokes parameters
S.sub.0(.sigma.), S.sub.1(.sigma.), S.sub.2(.sigma.), and
S.sub.3(.sigma.). Here, .sigma. is a "wavenumber" defined by an
inverse number of a wavelength .lamda.. Further, coordinate axes x
and y for determining the spectrometric Stokes parameters is taken
so as to agree with the fast and late axes of the retarder R1.
[0133] The light incident on the spectrometer passes in this order
through the retarders R1 and R2 and the analyzer A, and is incident
on the polarimeter 8. The Stokes parameters depending upon the
wavenumber a are obtained from a spectrum acquired from the
polarimeter 8.
[0134] However, the optical system shown in FIG. 1B has caused a
problem of a variety of changes in incident direction of the wave
surface of the light that transmits through the retarder under the
influence of the sample to result in generation of an error in
spectroscopic polarimetry. The present invention is provided to
solve such a problem.
[0135] The optical system of the embodiment of the present
invention shown in FIG. 1A is comprised of the light source 7, the
polarizer P, the retarders R2 and R1, the analyzer A and the
spectroscope 8. The light emitted from the light source 7 transmits
through the polarizer P, the retarder R2 and the retarder R1 in
this order, and is reflected on or transmits through the sample D.
The light then transmits through the analyzer A, and is incident on
the spectroscope 8. Thereafter, a spectrum of the incident light is
acquired in the spectroscope 8, and spectropolarization parameters
of the sample and the like are calculated according to a
later-described procedure.
[0136] As described above, in the present specification, the
"spectropolarization parameter" is used in the meaning of a
parameter for expressing the spectropolarization characteristic of
the object to be measured. This is a generic name of parameters for
use in quantitatively expressing a polarization change caused by
reflection or transmission of light on or through the object to be
measured. Examples of the spectropolarization parameter include
ellipsometric parameters .psi.(.sigma.), .DELTA.(.sigma.), and a
retardation .delta.(.sigma.) of a double refraction medium. It
should be noted that, although the spectropolarization
characteristic of the object to be measured is typically expressed
completely by 16 elements of a 4.times.4 Mueller matrix, there are
few cases where all of those 16 elements are independent variables.
In the spectroscopic polarimetry, all of those elements are often
determined from only two to three parameters at most. In practice,
those independent parameters may be obtained as the
spectropolarization parameters. Moreover, there are applications in
which simply obtaining part of the spectropolarization parameters
satisfies the case regardless of whether the parameters are
independent or non-independent.
[0137] Here, it is important that the retarders R2 and R1 have been
arranged on the light-source side with respect to the sample D.
This makes it possible to keep the incident direction of the wave
surface of light to be incident on the retarder constant so as to
realize highly-accurate, stable spectroscopic polarimetry. It
should be noted that variations in incident direction of the wave
surface of light to be incident on the analyzer A has almost no
influence on the measurement result. This leads to solving the
foregoing problem of the retardation change due to changes in
distance and direction of a light ray that passes through the
retarder during calibration of the retardation and during sample
measurement.
[0138] Next, the embodiment of the present invention is
specifically described with reference to FIG. 2. This optical
system is comprised of the light source 7, the polarizer P, the
retarders R2 and R1, the analyzer A and the spectroscope 8. It is
to be noted that reference symbol D denotes a sample. Here, the
orientations of the fast axes of the retarders R1 and R2 are
inclined at an angle of 45.degree. from each other. Meanwhile, the
orientation of a transmission axis of the polarizer P agrees with
the orientation of the fast axis of the retarder R1. In the figure,
the fast axes of the retarders are denoted by "fast" and the slow
axes thereof are denoted by "slow". Further, .theta. is the azimuth
angle of the transmission axis of the analyzer with respect to the
fast axis of the retarder R1.
[0139] The Mueller matrix of the sample at this time is described
as follows. [Mathematical Expression 1] M .function. ( .sigma. ) =
[ m ^ 00 .function. ( .sigma. ) m ^ 01 .function. ( .sigma. ) m ^
02 .function. ( .sigma. ) m ^ 03 .function. ( .sigma. ) m ^ 10
.function. ( .sigma. ) m ^ 11 .function. ( .sigma. ) m ^ 12
.function. ( .sigma. ) m ^ 13 .function. ( .sigma. ) m ^ 20
.function. ( .sigma. ) m ^ 21 .function. ( .sigma. ) m ^ 22
.function. ( .sigma. ) m ^ 23 .function. ( .sigma. ) m ^ 30
.function. ( .sigma. ) m ^ 31 .function. ( .sigma. ) m ^ 32
.function. ( .sigma. ) m ^ 33 .function. ( .sigma. ) ] ( 1.1 )
##EQU1##
[0140] Further, as parameters for effectively expressing a degree
of polarization of light, an ellipticity angle, an azimuth angle,
and the like, Stokes Parameters are used. The Stokes Parameters are
composed of four parameters having definitions as follows:
[0141] S.sub.0: total intensity
[0142] S.sub.1: difference between intensities of linearly
polarized light components with angles of 0.degree. and
90.degree..
[0143] S.sub.2: difference between intensities of linearly
polarized light components with angles .+-.45.degree..
[0144] S.sub.3: difference between intensities of left and right
circularly polarized light components.
[0145] In a third-dimensional space where the three mutually
orthogonal axes are taken as S.sub.1, S.sub.2 and S.sub.3, assuming
a sphere with a radius S.sub.0 taking an original point of the axes
as a center, an SOP of arbitrary light is expressed as one point in
this third-dimensional space and a degree of polarization is
expressed by the following expression: Degree of
polarization=(distance from original point to point (S.sub.1,
S.sub.2,
S.sub.3))/S.sub.0=(S.sub.1.sup.2+S.sub.2.sup.2+S.sub.3.sup.2).sup.1/2/S.s-
ub.0
[0146] Here, the Mueller matrix is described with reference to FIG.
3. The Mueller matrix is a matrix for expressing an interaction of
light of reflection, transmission, etc. on or through a sample as
an object to be measured. As an example, the following case is
considered. Light whose SOP is expressed by S(.sigma.) as a state
of polarization 1 is incident on the sample, and comes under the
influence of a polarization element and an object to be measured
such as the sample. Light is then emitted whose SOP is expressed by
S'(.sigma.) as a state of polarization 2 (cf. FIG. 3A). Here, the
Mueller matrix of the object to be measured is expressed by a
relational expression as a 4.times.4 matrix shown in FIG. 3B (cf.
FIG. 3B).
[0147] In the following described is a procedure for obtaining a
spectropolarization parameter of the sample from the foregoing
Mueller matrix or the like.
[0148] Before the procedure for obtaining a spectropolarization
parameter of the sample is described, characteristics of the
retarders R1 and R2 are formulated as preparation for the
description. A retarder is an element having the property of
changing a phase difference between mutually orthogonal linearly
polarized light components before and after transmission of light
through the element. An amount of such a change in phase difference
is referred to as retardation. Further, coordinate axes taken along
the two linearly polarized light directions are referred to as
principal axes. Among them, the axis along the linearly polarized
light whose phase relatively moves fast is referred to as a fast
axis, and the other axis is referred to as a slow axis.
[0149] The retardation of a retarder R.sub.j (j=1,2) made of a
double refraction medium changes with wavenumber a as expressed in
the following expression:
.phi..sub.j(.sigma.)=2.pi.d.sub.jB(.sigma.).sigma.=2.pi.L.sub.j.sigma..PH-
I..sub.j(.sigma.) (1.2) [Mathematical Expression 2] L j = 1 2
.times. .pi. .times. d .PHI. j d .sigma. .times. .sigma. 0 = d j
.function. ( B .function. ( .sigma. 0 ) + d B d .sigma. .times.
.sigma. 0 .times. .sigma. 0 ) ( 1.3 .times. a ) .PHI. j .function.
( .sigma. ) = { .PHI. j .function. ( .sigma. 0 ) - 2 .times. .pi.
.times. .times. L j .times. .sigma. 0 } + 1 2 .times. d 2 .times.
.PHI. j d .sigma. 2 .times. .sigma. 0 .times. ( .sigma. - .sigma. 0
) 2 + .times. ( 1.3 .times. b ) ##EQU2## where d.sub.j is a
thickness of R.sub.j, and B(.sigma.) is its double refraction.
Further, .sigma..sub.0 indicates a center wavenumber of light under
measurement. Hereinafter, the retardation .phi..sub.j(.sigma.)) of
the retarder is referred to as a reference phase function.
[0150] Assuming now that dispersion (change rate with wavenumber)
of B(.sigma.) is not so large, as seen from the expression (1.2),
.phi.(.sigma.) increases almost linearly with respect to wavenumber
.sigma.. Such a property serves as a basis of measurement of the
spectropolarization parameter of the sample in a later-described
procedure.
1.2 Channeled Spectrum Acquired in Spectrometer
[0151] In the "channeled spectroscopic polarimeter" (polarimetric
spectroscope) shown in FIG. 2, a spectrum (spectral intensity)
acquired in the spectrometer 8 is expressed by the following
expression. [Mathematical Expression 3] P .function. ( .sigma. ) =
.times. 1 2 .times. m 0 .function. ( .sigma. ) .times. M 0
.function. ( .sigma. ) + 1 4 .times. m_ .times. ( .sigma. ) .times.
M 23 .function. ( .sigma. ) .times. cos .times. { .PHI. 2
.function. ( .sigma. ) - .PHI. 1 .function. ( .sigma. ) + arg
.function. ( M 23 .function. ( .sigma. ) ) } + .times. 1 2 .times.
m 2 .function. ( .sigma. ) .times. M 1 .function. ( .sigma. )
.times. cos .times. .times. .PHI. 2 .function. ( .sigma. ) -
.times. 1 4 .times. m + .function. ( .sigma. ) .times. M 23
.function. ( .sigma. ) .times. cos .times. { .PHI. 2 .function. (
.sigma. ) + .PHI. 1 .function. ( .sigma. ) - arg .function. ( M 23
.function. ( .sigma. ) ) } ( 1.4 ) ##EQU3## where
M.sub.23(.sigma.)=M.sub.2(.sigma.)+iM.sub.3(.sigma.) (1.5)
[Mathematical Expression 4] M 0 .function. ( .sigma. ) = 1 2
.times. P 0 .function. ( .sigma. ) .function. [ m ^ 00 .function. (
.sigma. ) + m ^ 10 .function. ( .sigma. ) .times. cos .times.
.times. 2 .times. .theta. + m ^ 20 .function. ( .sigma. ) .times.
sin .times. .times. 2 .times. .theta. ] ( 1.6 .times. a ) M 1
.function. ( .sigma. ) = 1 2 .times. P 0 .function. ( .sigma. )
.function. [ m ^ 01 .function. ( .sigma. ) + m ^ 11 .function. (
.sigma. ) .times. cos .times. .times. 2 .times. .theta. + m ^ 21
.function. ( .sigma. ) .times. sin .times. .times. 2 .times.
.theta. ] ( 1.6 .times. b ) M 2 .function. ( .sigma. ) = 1 2
.times. P 0 .function. ( .sigma. ) .function. [ m ^ 02 .function. (
.sigma. ) + m ^ 12 .function. ( .sigma. ) .times. cos .times.
.times. 2 .times. .theta. + m ^ 22 .function. ( .sigma. ) .times.
sin .times. .times. 2 .times. .theta. ] ( 1.6 .times. c ) M 3
.function. ( .sigma. ) = 1 2 .times. P 0 .function. ( .sigma. )
.function. [ m ^ 03 .function. ( .sigma. ) + m ^ 13 .function. (
.sigma. ) .times. cos .times. .times. 2 .times. .theta. + m ^ 23
.function. ( .sigma. ) .times. sin .times. .times. 2 .times.
.theta. ] ( 1.6 .times. d ) ##EQU4## Here, M.sub.0(.sigma.) to
M.sub.3(.sigma.) are referred to as spectroscopic quasi-Stokes
parameters of the sample. As thus described, the spectroscopic
quasi-Stokes parameter is expressed by the sum of values each
obtained by multiplying each element of each column of the Mueller
matrix of the sample by a coefficient determined by an azimuth of
the analyzer. The spectropolarization parameter of the sample can
be obtained by simultaneously solving the equations of 1.6a to
1.6d. m.sub.0(.sigma.), m.sub.-(.sigma.), m.sub.2(.sigma.), and
m.sub.+(.sigma.) each shows a ratio of amplitude attenuation due to
failure of the spectrometer to follow a fine vibration component.
P.sub.0(.sigma.) shows a "spectrum of the light source". However,
in the optical system, attenuation exists caused by the retarder,
the polarizer, a lens, a fiber or the like. Therefore, in the
present specification, the "spectrum of the light source"
P.sub.0(.sigma.) includes the attenuated portion. Further,
.phi..sub.1 and .phi..sub.2 are retardations of the retarder R1 and
R2.
[0152] The elements of the Mueller matrix of the sample included in
M.sub.0(.sigma.) to M.sub.3(.sigma.) are each relative to "each
column" of the Mueller matrix M(.sigma.). [Mathematical Expression
5] M .function. ( .sigma. ) = [ m ^ 00 .function. ( .sigma. ) m ^
10 .function. ( .sigma. ) m ^ 20 .function. ( .sigma. ) m ^ 01
.function. ( .sigma. ) m ^ 11 .function. ( .sigma. ) m ^ 21
.function. ( .sigma. ) m ^ 02 .function. ( .sigma. ) m ^ 12
.function. ( .sigma. ) m ^ 22 .function. ( .sigma. ) m ^ 03
.function. ( .sigma. ) m ^ 13 .function. ( .sigma. ) m ^ 23
.function. ( .sigma. ) m ^ 30 .function. ( .sigma. ) m ^ 31
.function. ( .sigma. ) m ^ 32 .function. ( .sigma. ) m ^ 33
.function. ( .sigma. ) ] ( 1.7 ) ##EQU5##
[0153] Information inside the frames (sum of values each obtained
by multiplying each element of each column by a coefficient
determined by an azimuth angle (.theta.) of the analyzer A can be
demodulated.
[0154] While 16 elements exist in the 4.times.4 Mueller matrix, it
is extremely rare that all those elements are independent. In many
cases, in polarimetry, only two to three independent parameters at
most are included in the Mueller matrix of the sample. Even with a
spectrum intensity of the light source included, the total number
of parameters required to be measured is often as small as four at
most. It is therefore possible by simultaneously solving obtained
four equations to obtain up to four parameters, which are
independent from each other and show the polarization
characteristic of the sample.
[0155] For the sake of understanding the property of this
expression, Expression (1.2) is substituted therein as follows.
[Mathematical Expression 6] P .function. ( .sigma. ) = .times. 1 2
.times. m 0 .function. ( .sigma. ) .times. M 0 .function. ( .sigma.
) + .times. 1 4 .times. m_ .times. ( .sigma. ) .times. M 23
.function. ( .sigma. ) .times. cos .function. [ 2 .times. .pi.
.times. .times. L - .times. .sigma. + .PHI. - .function. ( .sigma.
) + arg .times. { M 23 .function. ( .sigma. ) } ] + .times. 1 2
.times. m 2 .function. ( .sigma. ) .times. M 1 .function. ( .sigma.
) .times. cos .function. [ 2 .times. .pi. .times. .times. L 2
.times. .sigma. + .PHI. 2 .function. ( .sigma. ) ] - .times.
.times. 1 4 .times. m + .function. ( .sigma. ) .times. M 23
.function. ( .sigma. ) .times. cos .times. [ 2 .times. .pi. .times.
.times. L + .times. .sigma. + .PHI. + .function. ( .sigma. ) - arg
.times. { M 23 .function. ( .sigma. ) } ] ( 1.8 ) ##EQU6## where it
is found that the following expressions are satisfied.
L.sub.-=L.sub.2-L.sub.1 (1.9a) L.sub.+=L.sub.2+L.sub.1 (1.9b)
.PHI..sub.-(.sigma.)=.PHI..sub.2(.sigma.)-.PHI..sub.1 (.sigma.)
(1.9c) .PHI..sub.+(.sigma.)=.PHI..sub.2(.sigma.)+.PHI..sub.1
(.sigma.) (1.9d)
[0156] As seen from Expression (1.8), the spectrum P(.sigma.)
obtained from the spectrometer contains four components. One of
them is a component gently varies with wavenumber .sigma., and the
other three components are quasi-sinusoidal components that vibrate
with wavenumber .sigma.. These are schematically shown in FIG.
4.
[0157] Here, the central periods of each of the three vibration
components are respectively 1/L, 1/L.sub.2 and 1/L.sub.+. The
spectrum containing components that periodically finely vibrate
with wavenumber (wavelength) as in the figure is referred to as a
channeled spectrum.
[0158] What needs to be concerned here is that these four
components have information of any one of M.sub.0(.sigma.)
M.sub.1(.sigma.) and M.sub.23(.sigma.). When each component can be
separated, it is possible to determine all the spectroscopic
quasi-Stokes parameters M.sub.0(.sigma.), M.sub.1(.sigma.),
M.sub.2(.sigma.) and M.sub.3 from one spectrum P(.sigma.).
1.3 When Crossing Angle Between Elements is not 45.degree.
[0159] Next described is a spectrum acquired in the spectrometer 5
when a crossing angle between the elements is not 45.degree..
[0160] Here also described as a supplemental explanation is a
spectrum obtained when a crossing angle between each element in the
optical system is not 45.degree..
[0161] It is assumed now that, in the optical system in FIG. 2, the
angle formed between the fast axes of the retarders R1 and R2 is
.theta..sub.RR and the angle formed between the fast axis of the
retarder R2 and the transmission axis of the polarizer P is
.theta..sub.PR. So far, the calculation has been made only in a
limited case of .theta..sub.RR=45.degree. and
.theta..sub.PR=45.degree.. Below, a case where those angles are
more common ones is shown.
[0162] An expression for the obtained channeled spectrum P(.sigma.)
is given as follows. [Mathematical Expression 7] P .function. (
.sigma. ) = .times. 1 2 .times. m 0 .function. ( .sigma. )
.function. [ M 0 .function. ( .sigma. ) + cos .times. .times. 2
.times. .times. .theta. PR .times. cos .times. .times. 2 .times.
.times. .theta. RR .times. M 1 .function. ( .sigma. ) _ ] - .times.
1 2 .times. ( sin .times. .times. 2 .times. .times. .theta. PR
.times. sin 2 .times. .theta. RR ) .times. m - .function. ( .sigma.
) .times. M 23 .function. ( .sigma. ) .times. cos .times. [ .PHI. 2
.function. ( .sigma. ) - .PHI. 1 .function. ( .sigma. ) + arg
.times. { M 23 .function. ( .sigma. ) } ] - .times. 1 2 .times. (
sin .times. .times. 2 .times. .times. .theta. PR .times. sin
.times. .times. 2 .times. .theta. RR ) .times. m 2 .function. (
.sigma. ) .times. M 1 .function. ( .sigma. ) .times. cos .function.
[ .PHI. 2 .function. ( .sigma. ) ] + .times. 1 2 .times. ( sin
.times. .times. 2 .times. .times. .theta. PR .times. cos 2 .times.
.theta. RR ) .times. m + .function. ( .sigma. ) .times. M 23
.function. ( .sigma. ) .times. cos .times. [ .PHI. 2 .function. (
.sigma. ) - .PHI. 1 .function. ( .sigma. ) - arg .times. { M 23
.function. ( .sigma. ) } ] + .times. 1 2 .times. ( cos .times.
.times. 2 .times. .times. .theta. PR .times. sin .times. .times. 2
.times. .theta. RR ) .times. m 1 .function. ( .sigma. ) .times. M
23 .function. ( .sigma. ) .times. cos _ .times. [ .PHI. 1
.function. ( .sigma. ) - arg .times. { M 23 .function. ( .sigma. )
} _ ] ( 1.10 ) ##EQU7##
[0163] When this expression is compared with the spectrum in the
previous expression (1.4), namely when the angles .theta..sub.RR
and .theta..sub.PR are respectively limited to 45.degree. and
45.degree., the following differences are found in addition to a
mere difference in constant multiple of a coefficient. It is to be
noted that the different part is indicated by an underline in
Expression (1.10).
[0164] The component that gently varies with wavenumber a depends
not only upon M.sub.0(.sigma.) but additionally upon
M.sub.1(.sigma.).
[0165] A component that quansi-sinusoidally vibrates according to
the phase .phi..sub.1(.sigma.), namely a component that vibrates at
a central period 1/L.sub.1, is added. It should be noted that this
component has information of M.sub.23(.sigma.), as in the cases of
the two respective components that vibrate according to
(.phi..sub.2(.sigma.)-.phi..sub.1(.sigma.)) and
(.phi..sub.2(.sigma.)+.phi..sub.1(.sigma.)). It means that this
term can be treated in the same manner as the other two terms
including M.sub.23(.sigma.).
[0166] Here, conditions for nonappearance of the above two
components are considered.
[0167] The former term appears in a limited case "where both
.theta..sub.RR.noteq..+-.45.degree. and
.theta..sub.PR.noteq..+-.45.degree. are satisfied". In the
meantime, the latter term appears in a case "where
.theta..sub.PR.noteq..+-.45.degree. "(regardless of whether
.theta..sub.RR agrees with .+-.45.degree. or not)". From these, the
following fact can be mentioned.
[0168] When the fast axis of the retarder R2 and the transmission
axis of the polarizer P cross each other at an angle of 45.degree.
(i.e. .theta..sub.PR=.+-.45.degree.), the channeled spectrum is
given by Expression (1.4) except for the difference in constant
multiple of the coefficient of each term. Here, whether the angle
.theta..sub.RR formed between the principle axes of the retarders
R1 and R2 agrees with .+-.45.degree. or not is irrelevant.
[0169] In other words, the channeled spectrum can take the form of
Expression (1.4) under a condition that the fast axis of the
retarder R2 and the transmission axis of the polarizer P cross each
other at an angle of .+-.45.degree.. Meanwhile, whether the angle
formed between the fast axes of the retarders R1 and R2 agrees with
.+-.45.degree. or not is irrelevant.
1.4 When the Number of Retarders is Three or More
[0170] The case where the number of retarders is two was described
above regarding the spectrum acquired in the spectrometer 5. In the
same manner as this case, a spectrum having information of a
spectroscopic quasi-Stokes parameter particular to each component
is obtained in a case where the number of retarders is three or
more. As in the case of the number of retarders is two, each
element is separated so as to demodulate all spectroscopic
quasi-Stokes parameters from one spectrum P(.sigma.). By
simultaneously solving the obtained expressions, it is possible to
obtain a spectropolarization parameter of the sample.
1.5 Procedure for Demodulating Spectrometric Stokes Parameter
[0171] A specific procedure for demodulating a spectrometric
quasi-Stokes parameter is described below with reference to FIG. 5.
A brief description of the flow of the procedure is as follows.
[0172] Step 1: Each term is separated from Spectrum P(ca).
[0173] Step 2: An amplitude and a phase of each component are
obtained. (Or equivalent quantities, e.g. a real part and an
imaginary part in complex representation are obtained).
[0174] Step 3: Reference functions (below) included in the
amplitude and phase of each vibration component are removed.
[Mathematical Expression 8] Reference .times. .times. amplitude
.times. .times. function .times. { m 0 .function. ( .sigma. ) m -
.function. ( .sigma. ) m 2 .function. ( .sigma. ) m + .function. (
.sigma. ) } ##EQU8## Reference .times. .times. phase .times.
.times. function .times. { .PHI. 1 .function. ( .sigma. ) .PHI. 2
.function. ( .sigma. ) } ##EQU8.2## Spectrometric Stokes parameters
M.sub.0(.sigma.), M.sub.1(.sigma.), M.sub.2(.sigma.), and
M.sub.3(.sigma.) are then obtained (These reference functions are
peculiar to the polarimeter, depending not upon the sample but only
upon parameters of the polarimeter.
[0175] Each of the steps is described as follows.
[Step 1]
[0176] As described in the previous section, the spectrum
P(.sigma.) contains four components. An operation for taking out
each component by a signal processing is performed. What is applied
to this operation is that each component vibrates at a different
period (frequency). With the use of (any one of) a variety of
frequency filtering techniques being broadly used in fields of
communication engineering, signal analysis and the like, it is
possible to separate each component. [Mathematical Expression 9]
Component .times. [ 1 ] .times. ( low .times. .times. frequency
.times. .times. component ) 1 2 .times. m 0 .function. ( .sigma. )
.times. M 0 .function. ( .sigma. ) ( 1.11 .times. a ) Component
.times. [ 2 ] .times. ( central .times. .times. period .times.
.times. 1 / L - ) 1 4 .times. m - .function. ( .sigma. ) .times. M
23 .function. ( .sigma. ) .times. cos .function. [ .PHI. 2
.function. ( .sigma. ) - .PHI. 1 .function. ( .sigma. ) + arg
.times. { M 23 .function. ( .sigma. ) } ] ( 1.11 .times. b )
Component .times. [ 3 ] .times. ( central .times. .times. period
.times. .times. 1 / L 2 ) 1 2 .times. m 2 .function. ( .sigma. )
.times. M 1 .function. ( .sigma. ) .times. cos .function. [ .PHI. 2
.function. ( .sigma. ) ] ( 1.11 .times. c ) Component .times. [ 4 ]
.times. ( central .times. .times. period .times. .times. 1 / L + )
- 1 4 .times. m + .function. ( .sigma. ) .times. M 23 .function. (
.sigma. ) .times. cos .function. [ .PHI. 2 .function. ( .sigma. ) +
.PHI. 1 .function. ( .sigma. ) - arg .times. { M 23 .function. (
.sigma. ) } ] ( 1.11 .times. d ) ##EQU9##
[0177] Component [1] above is a first spectral intensity component
which nonperiodically vibrates with wavenumber. Component [2] is a
second spectral intensity component which vibrates at a frequency
depending upon a difference between a reference phase function
(retardation) .phi..sub.1(.sigma.) of the first retarder R1 and a
reference phase function (retardation) .phi..sub.2(.sigma.) of the
second retarder R2 with wavenumber. Component [3] is a third
spectral intensity component which vibrates at a frequency
depending upon the reference phase function .phi..sub.2(.sigma.) of
the second retarder and not depending upon the reference phase
function .phi..sub.1(.sigma.) of the first retarder, with
wavenumber. Component [4] is a fourth spectral intensity component
which vibrates at a frequency depending upon the sum of the
reference phase function .phi..sub.1(.sigma.) of the first retarder
and the reference phase function .phi..sub.2(.sigma.) of the second
retarder with wavenumber. It is to be noted that, when the crossing
angle between the elements is not 45.degree., Component [5] is
generated. Component [5] vibrates at a frequency depending upon the
reference phase function .phi..sub.1(.sigma.) of the first retarder
and not depending upon the reference phase function
.phi..sub.2(.sigma.) of the second retarder, with wavenumber.
[0178] With reference to Expressions (1.11a) to (1.11d) and
Expression (1.5), it is found that M.sub.0 is obtained from
Component [1], M.sub.1 is from Component [3], M.sub.2 and M.sub.3
are from Component [2] or Component [4]. It should be noted that,
when the crossing angle between the elements is not 45.degree.,
M.sub.2 and M.sub.3 are obtained at least one of Component [2],
Component [4] and Component [5].
[Step 2]
[0179] As for each component separated in Step 1, a "paired
amplitude and phase" and a "complex representation" are obtained,
as shown in FIG. 6. This can be readily realized by using a variety
of demodulation methods in an operation which are common in fields
of communication engineering, signal analysis and the like, as
in
Step 1. Examples of those methods include:
[0180] Amplitude demodulation: rectifying demodulation, envelope
demodulation, etc.
[0181] Phase demodulation: frequency discrimination, zero-crossing
method, etc.
[0182] Complex representation demodulation: Fourier transform
method (later described), synchronous demodulation, etc.
[0183] Here, definitions and basic properties of the "amplitude",
"phase" and "complex representation" of a vibration component are
summarized below. As seen from Expressions (1.11 a) to (1.11 d),
each of the separated components except for Component [1] takes the
form of: a(.sigma.)cos .delta.(.sigma.). a(.sigma.) and
.delta.(.sigma.) here are respectively referred to as the
"amplitude" and "phase" of the vibration component. It is to be
noted that, if assuming that the phase .DELTA..sub.0(.sigma.)=0
(i.e. cos .delta..sub.0(.sigma.)=0) also in Component [1], the
amplitude of Component [1] can also be defined.
[0184] Further, F(.sigma.) having the following relation with the
amplitude and the phase is called a complex representation.
[Mathematical Expression 10] F .function. ( .sigma. ) = 1 2 .times.
a .function. ( .sigma. ) .times. exp .function. [ i .times. .times.
.delta. .function. ( .sigma. ) ] ( 1.12 .times. a ) .times. = [ 1 2
.times. a .function. ( .sigma. ) .times. cos .times. .times.
.delta. .function. ( .sigma. ) ] + i .function. [ 1 2 .times. a
.function. ( .sigma. ) .times. sin .times. .times. .delta.
.function. ( .sigma. ) ] ( 1.12 .times. b ) ##EQU10## The real part
of F(.sigma.) is formed by dividing the amplitude of the vibration
component into halves, and the imaginary part thereof is displaced
from the real part at the angle of 90.degree.. It should be noted
that in Component [1], the amplitude is not divided into halves
since .delta.(.sigma.)=0, i.e. no imaginary part exists.
[0185] What needs to be concerned here is that when either the
"paired amplitude and phase" or the "complex representation" can be
demodulated, the other one can be immediately calculated by use of
the following relational expression. [Mathematical Expression 8] "
amplitude .times. .times. a .times. .times. ( .sigma. ) , phase
.times. .times. .delta. .times. .times. ( .sigma. ) " .fwdarw. "
complex .times. .times. representation .times. .times. F .times.
.times. ( .sigma. ) " .times. .times. F .function. ( .sigma. ) = 1
2 .times. a .function. ( .sigma. ) .times. e i .times. .times.
.delta. .function. ( .sigma. ) ( 1.13 ) " complex .times. .times.
representation .times. .times. F .times. .times. ( .sigma. ) "
.fwdarw. " amplitude .times. .times. a .times. .times. ( .sigma. )
, phase .times. .times. .delta. .times. .times. ( .sigma. ) "
.times. .times. a .function. ( .sigma. ) = 2 .times. F .function. (
.sigma. ) ( 1.14 .times. a ) .delta. .function. ( .sigma. ) = arg
.function. [ F .function. ( .sigma. ) ] ( 1.14 .times. b )
##EQU11##
[0186] Namely, demodulation of one of the paired amplitude and
phase and the complex representation enables immediate calculation
of the other according to need.
[0187] When the "amplitude" and "phase" of each component are
demodulated, the following results are obtained. [Mathematical
Expression 12] .cndot. .times. .times. Component .times. [ 1 ]
.times. .times. ( low .times. .times. frequency .times. .times.
component ) .times. : .times. .times. .times. [ Amplitude ] a 0
.function. ( .sigma. ) = 1 2 .times. m 0 .function. ( .sigma. )
.times. M 0 .function. ( .sigma. ) [ Phase ] .delta. 0 .function. (
.sigma. ) = 0 ( 1.15 .times. a ) .cndot. .times. .times. Component
.times. [ 2 ] .times. .times. ( central .times. .times. period
.times. .times. 1 / L - ) .times. : .times. .times. .times. [
Amplitude ] a - .function. ( .sigma. ) = 1 4 .times. m - .function.
( .sigma. ) .times. M 23 .function. ( .sigma. ) [ Phase ] .delta. -
.function. ( .sigma. ) = .PHI. 2 .function. ( .sigma. ) - .PHI. 1
.function. ( .sigma. ) + arg .times. { M 23 .function. ( .sigma. )
} ( 1.15 .times. b ) .cndot. .times. .times. Component .times. [ 3
] .times. .times. ( central .times. .times. period .times. .times.
1 / L 2 ) .times. : .times. .times. .times. [ Amplitude ] a 2
.function. ( .sigma. ) = 1 2 .times. m 2 .function. ( .sigma. )
.times. M 1 .function. ( .sigma. ) [ Phase ] .delta. 2 .function. (
.sigma. ) = .PHI. 2 .function. ( .sigma. ) ( 1.15 .times. c )
.cndot. .times. .times. Component .times. [ 4 ] .times. .times. (
central .times. .times. period .times. .times. 1 / L + ) .times. :
.times. .times. .times. .times. [ Amplitude ] a + .function. (
.sigma. ) = 1 4 .times. m + .function. ( .sigma. ) .times. M 23
.function. ( .sigma. ) [ Phase ] .times. .delta. + .function. (
.sigma. ) = .PHI. 2 .function. ( .sigma. ) + .PHI. 1 .function. (
.sigma. ) - arg .times. { M 23 .function. ( .sigma. ) } + .pi. (
1.15 .times. d ) ##EQU12##
[0188] On the other hand, when the "complex representation" of each
component is demodulated, the following results are obtained.
[Mathematical Expression 13] .cndot. .times. .times. Component
.times. [ 1 ] .times. .times. ( low .times. .times. frequency
.times. .times. component ) .times. : .times. [ Complex .times.
.times. representation ] .times. .times. F 0 .function. ( .sigma. )
= 1 2 .times. m 0 .function. ( .sigma. ) .times. M 0 .function. (
.sigma. ) ( 1.16 .times. a ) .cndot. .times. .times. Component
.times. [ 2 ] .times. .times. ( central .times. .times. period
.times. .times. 1 / L - ) .times. : .times. [ Complex .times.
.times. representation ] .times. .times. F - .function. ( .sigma. )
= 1 8 .times. m - .function. ( .sigma. ) .times. M 23 .function. (
.sigma. ) .times. exp .times. .times. I .function. [ .PHI. 2
.function. ( .sigma. ) - .PHI. 1 .function. ( .sigma. ) ] ( 1.16
.times. b ) .cndot. .times. .times. Component .times. [ 3 ] .times.
.times. ( central .times. .times. period .times. .times. 1 / L 2 )
.times. : .times. [ Complex .times. .times. representation ]
.times. .times. F 2 .function. ( .sigma. ) = 1 4 .times. m 2
.function. ( .sigma. ) .times. M 1 .function. ( .sigma. ) .times.
exp .times. .times. I .times. .times. .PHI. 2 .function. ( .sigma.
) ( 1.16 .times. c ) .cndot. .times. .times. Component .times. [ 4
] .times. .times. ( central .times. .times. period .times. .times.
1 / L + ) .times. : .times. [ Complex .times. .times.
representation ] .times. .times. F + .function. ( .sigma. ) = - 1 8
.times. m + .function. ( .sigma. ) .times. M 23 * .function. (
.sigma. ) .times. exp .times. .times. I .function. [ .PHI. 2
.function. ( .sigma. ) + .PHI. 1 .function. ( .sigma. ) ] ( 1.16
.times. c ) ##EQU13##
[0189] Here, * indicates a complex conjugation. It is to be noted
that, for the sake of what is described below, the expressions of
the complex representations are rewritten as follows. [Mathematical
Expression 14] .cndot. .times. .times. Component .times. [ 1 ]
.times. .times. ( low .times. .times. frequency .times. .times.
component ) .times. : .times. .times. [ Complex .times. .times.
representation ] .times. .times. F 0 .function. ( .sigma. ) = K 0
.function. ( .sigma. ) .times. M 0 .function. ( .sigma. ) ( 1.17
.times. a ) .cndot. .times. .times. Component .times. [ 2 ] .times.
.times. ( central .times. .times. period .times. .times. 1 / L - )
.times. : .times. .times. [ Complex .times. .times. representation
] .times. .times. F - .function. ( .sigma. ) = K - .function. (
.sigma. ) .times. M 23 .function. ( .sigma. ) ( 1.17 .times. b )
.cndot. .times. .times. Component .times. [ 3 ] .times. .times. (
central .times. .times. period .times. .times. 1 / L 2 ) .times. :
.times. .times. [ Complex .times. .times. representation ] .times.
.times. F 2 .function. ( .sigma. ) = K 2 .function. ( .sigma. )
.times. M 1 .function. ( .sigma. ) ( 1.17 .times. c ) .cndot.
.times. .times. Component .times. [ 4 ] .times. .times. ( central
.times. .times. period .times. .times. 1 / L + ) .times. : .times.
.times. [ Complex .times. .times. representation ] .times. .times.
F + .function. ( .sigma. ) = K + .function. ( .sigma. ) .times. M
23 * .function. ( .sigma. ) .times. .times. where ( 1.17 .times. d
) K 0 .function. ( .sigma. ) = 1 2 .times. m 0 .function. ( .sigma.
) ( 1.18 .times. a ) K - .function. ( .sigma. ) = 1 8 .times. m -
.function. ( .sigma. ) .times. exp .times. .times. I .function. [
.PHI. 2 .function. ( .sigma. ) - .PHI. 1 .function. ( .sigma. ) ] (
1.18 .times. b ) K 2 .function. ( .sigma. ) = 1 4 .times. m 2
.function. ( .sigma. ) .times. exp .times. .times. I .times.
.times. .PHI. 2 .function. ( .sigma. ) ( 1.18 .times. c ) K +
.function. ( .sigma. ) = - 1 8 .times. m + .function. ( .sigma. )
.times. exp .times. .times. I .function. [ .PHI. 2 .function. (
.sigma. ) + .PHI. 1 .function. ( .sigma. ) ] ( 1.18 .times. d )
##EQU14## [Step 3]
[0190] Finally, from the "amplitude" and the "phase" or the
"complex representation" obtained in Step 2 above, the spectrometer
parameters M.sub.0(.sigma.), M.sub.1(.sigma.), M.sub.2(.sigma.),
and M.sub.3(.sigma.) are determined as functions of the wavenumber
.sigma..
[0191] The "amplitude" and the "phase" obtained in Step 2 above
include, other than the spectrometric Stokes parameters to be
obtained, parameters shown below. [Mathematical Expression 15]
Parameter .times. .times. ( function ) .times. .times. determined
.times. .times. based .times. .times. only .times. .times. upon
.times. .times. characteristic ##EQU15## of .times. .times.
polarimeter .times. .times. itself ##EQU15.2## { m 0 .function. (
.sigma. ) m - .function. ( .sigma. ) m 2 .function. ( .sigma. ) m +
.function. ( .sigma. ) } ##EQU15.3## and ##EQU15.4## { .PHI. -
.function. ( .sigma. ) = .PHI. 2 .function. ( .sigma. ) - .PHI. 1
.function. ( .sigma. ) .PHI. 2 .function. ( .sigma. ) .PHI. +
.function. ( .sigma. ) = .PHI. 2 .function. ( .sigma. ) + .PHI. 1
.function. ( .sigma. ) } ##EQU15.5##
[0192] The former are included in the amplitude while the latter
are included in the phase. These parameters provide references in
determining spectrometric quasi-Stokes parameters from the
amplitude and the phase of each vibration component. Thus, each of
these parameters is hereinafter referred to as a "reference
amplitude function" or a "reference phase function". Since these
parameters do not depend upon the sample, each of the parameters is
subjected to division or subtraction, to be determined as follows.
[0193] M.sub.0(.sigma.) can be determined from [Component [1]].
[0194] M.sub.2(.sigma.) and M.sub.3(.sigma.) can be determined from
(either) [Component [2]] or [Component [4]]. [0195]
M.sub.1(.sigma.) can be determined from [Component [3]].
[0196] Meanwhile, in the case of the "complex representation",
parameters (functions) determined only by the characteristic of the
polarimeter itself are K.sub.0(.sigma.), K.sub.-(.sigma.),
K.sub.2(.sigma.), and K.sub.+(.sigma.) which are defined by
Expressions (1.18a) to (1.18d). These are, so to speak, "reference
complex functions".
[0197] As revealed from Expressions (1.17a) to (1.17d), if the
above reference complex functions have been obtained, by division
of the complex representation of each vibration component
demodulated in Step 2, the parameters can be determined as follows.
[0198] M.sub.0(.sigma.) can be determined from [Component [1]]
[0199] M.sub.2(.sigma.) and M.sub.3(.sigma.) can be determined from
(either) [Component [2]] or [Component [4]]. [0200]
M.sub.1(.sigma.) can be determined from [Component [3]].
[0201] When the angle formed between the retarder R2 and the
polarizer P is not 45.degree., a fifth term that appears can be
used in place of [component [2]] and [component [4]]. Namely, the
description on lines 2-3 above can be rewritten to: [0202]
M.sub.2(.sigma.) and M.sub.3(.sigma.) can be determined from any
one of [component [2]], [component [4]] and [component [5]].
[0203] Next, as one of signal processing methods for demodulating
spectrometric quasi-Stokes parameters, a "Fourier transform method"
is described with reference to FIG. 7. The use of this method
allows efficient concurrent performance of Steps 1 and 2, leading
to immediate determination of all complex representations of each
vibration component.
[0204] In this method, first, the spectrum P(.sigma.) measured with
the spectrometer in the channeled spectroscopic polarimeter is
subjected to inverse Fourier transformation, to obtain the
following correlation function of light incident on the
spectrometer. [Mathematical Expression 16] C .function. ( h ) = A 0
.function. ( h ) + A - .function. ( h - L - ) + A - * .function. (
- h - L - ) + A 2 .function. ( h - L 2 ) + A 2 * .function. ( - h -
L 2 ) + A + .function. ( h - L + ) + A + * .function. ( - h - L + )
.times. .times. where ( 1.19 ) A 0 .function. ( h ) = F - 1
.function. [ 1 2 .times. m 0 .function. ( .sigma. ) .times. M 0
.function. ( .sigma. ) ] ( 1.20 .times. a ) A - .function. ( h ) =
F - 1 .function. [ 1 8 .times. m - .function. ( .sigma. ) .times. M
23 .function. ( .sigma. ) .times. exp .times. .times. I .times.
.times. .PHI. - .function. ( .sigma. ) ] ( 1.20 .times. b ) A 2
.function. ( h ) = F - 1 .function. [ 1 4 .times. m 2 .function. (
.sigma. ) .times. M 1 .function. ( .sigma. ) .times. exp .times.
.times. I .times. .times. .PHI. 2 .function. ( .sigma. ) ] ( 1.20
.times. c ) A + .function. ( h ) = F - 1 .function. [ - 1 8 .times.
m + .function. ( .sigma. ) .times. M 23 * .function. ( .sigma. )
.times. exp .times. .times. I .times. .times. .PHI. + .function. (
.sigma. ) ] ( 1.20 .times. d ) ##EQU16## As shown in the upper
right part of FIG. 7, this correlation function C(h) contains seven
components including inverse numbers of a period of each vibration
component, 0, .+-.L.sub.-, .+-.L.sub.2, .+-.L.sub.+ as main
components.
[0205] Here, appropriate selection of these inverse numbers of the
period enables separation of each component, contained in C(h),
from each other on the h-axis. When four components as the main
components, h=0, L.sub.-, L.sub.2, and L.sub.+, are taken out and
then subjected to the Fourier transformation, the following
expressions are satisfied. [Mathematical Expression 17] F
.function. [ A 0 .function. ( h ) ] = 1 2 .times. m 0 .function. (
.sigma. ) .times. M 0 .function. ( .sigma. ) = F 0 .function. (
.sigma. ) ( 1.21 .times. a ) F .function. [ A - .function. ( h - L
- ) ] = 1 8 .times. m - .function. ( .sigma. ) .times. M 23
.function. ( .sigma. ) .times. exp .times. .times. I .function. [
.PHI. 2 .function. ( .sigma. ) - .PHI. 1 .function. ( .sigma. ) ] =
F - .function. ( .sigma. ) ( 1.21 .times. b ) F .function. [ A 2
.function. ( h - L 2 ) ] = 1 4 .times. m 2 .function. ( .sigma. )
.times. M 1 .function. ( .sigma. ) .times. exp .times. .times. I
.times. .times. .PHI. 2 .function. ( .sigma. ) = F 2 .function. (
.sigma. ) ( 1.21 .times. c ) F .function. [ A + .function. ( h - L
+ ) ] = - 1 8 .times. m + .function. ( .sigma. ) .times. M 23 *
.function. ( .sigma. ) .times. exp .times. .times. I .function. [
.PHI. 2 .function. ( .sigma. ) + .PHI. 1 .function. ( .sigma. ) ] =
F + .function. ( .sigma. ) ( 1.21 .times. d ) ##EQU17##
[0206] As seen from the expressions above, what are obtained in the
above operation are just the complex representations of the
components [1] to [4] to be obtained in foregoing Step 2. Namely,
in the above operation, Steps 1 and 2 are concurrently realized.
Hence, when Step 3 is performed using the results of Steps 1 and 2,
spectrometric quasi-Stokes parameters are obtained all at once.
1.6 Pre-Calibration: Calibration of Reference Amplitude Function,
Reference Phase Function, Reference Complex Function "Prior to
Measurement"
[0207] As described in the previous section, when a spectrometric
quasi-Stokes parameter is determined from a channeled spectrum, it
is necessary in Step 3 to in advance determine parameters to be
obtained based only upon a characteristic of a polarimeter itself
parameters, namely:
[0208] "reference amplitude function" m.sub.0(.sigma.),
m.sub.-(.sigma.), m.sub.2(.sigma.), m.sub.+(.sigma.), and
"reference phase function" .phi..sub.2(.sigma.) and
.phi..sub.1(.sigma.), or
[0209] "reference complex function" K.sub.0(.sigma.),
K.sub.-(.sigma.), K.sub.2(.sigma.), K.sub.+(.sigma.)
[0210] The former ("reference amplitude function" and "reference
phase function") and the latter ("reference complex function") are
required in the respective cases of obtaining spectrometric
quasi-Stokes parameters from the "amplifier and phase" and the
"complex representation" of each vibration component. Since these
are functions not depending upon the sample, it is desirable to
calibrate the functions at least prior to measurement.
[0211] In this section, a procedure for calibrating these reference
functions "prior to measurement, i.e. in advance" is described.
There are two typical methods as follows. [0212] [Method 1]: a
method for calibrating reference phase functions and reference
amplitude functions from a characteristic of each element for use
in the optical system. [0213] [Method 2]: a method for calibrating
reference phase functions and reference amplitude functions by use
of a sample having a known polarization characteristic.
1.6.1 [Method 1]
[0214] Method for Calibrating Reference Phase Function and
Reference Amplitude Function from Characteristic of Each Element
for Use in Optical System
[0215] Characteristics of a reference phase function and a
reference amplitude function are essentially determined based upon
elements for use in a channeled spectroscopic polarimeter.
Therefore, optical characteristics of individual elements are
repeatedly examined by experiment or calculation to perform
calibration of parameters.
1.6.2 [Method 2]
[0216] Method for Calibrating Reference Phase Function and
Reference Amplitude Function by Use of Sample Having Known
Polarization Characteristic.
[0217] The reference phase function and the reference amplitude
function are in amount determined based not upon the "polarization
characteristic of the object to be measured (sample)", but only
upon the characteristic of the "channeled spectroscopic
polarimeter". Accordingly, a "sample having a known polarization
characteristic" (light whose measurement result is known)" is
inputted into the polarimeter, and using the result of the input,
it is possible to calculate backward the reference phase function
and the reference amplitude function.
[0218] Below, the procedure for calibration is shown. As described
at the beginning of this section, the following should be noted.
[0219] When the SOP is obtained from the "amplitude and phase" of
each vibration component, the "reference amplitude function" and
the "reference phase function" are required. [0220] When the SOP is
obtained from the "complex representation" of each vibration
component, the "reference complex function" is required.
[0221] In the following, the procedures for calibration in the
above respective cases are described. Although these procedures are
essentially equivalent and different only in calculation method,
they are separately put down for the sake of convenience.
A. Calibration Procedure for Separately Obtaining Reference
Amplitude Function and Reference Phase Function
[0222] In this calibration, first, a "sample having a known
polarization characteristic" is prepared, and then inputted into a
channeled spectroscopic polarimeter. In this case, spectrometric
quasi-Stokes parameters of light are referred to as
M.sub.0.sup.(0)(.sigma.) M.sub.1.sup.(0)(.sigma.),
M.sub.2.sup.(0)(.sigma.), and M.sub.3.sup.(0)(.sigma.). When the
sampler is subjected to the above-mentioned demodulation means, the
amplitude and the phase obtained in Step 2 are expressed as follows
according to Expressions (1.15a) to (1.15d). [Mathematical
Expression 18] .cndot. .times. .times. Component .times. [ 1 ]
.times. .times. ( low .times. .times. frequency .times. .times.
component ) .times. : .times. .times. .times. [ Amplitude ] a 0 ( 0
) .function. ( .sigma. ) = 1 2 .times. m 0 .function. ( .sigma. )
.times. M 0 ( 0 ) .function. ( .sigma. ) [ Phase ] .delta. 0 ( 0 )
.function. ( .sigma. ) = 0 ( 1.22 .times. a ) .cndot. .times.
.times. Component .times. [ 2 ] .times. .times. ( central .times.
.times. period .times. .times. 1 / L - ) .times. : .times. .times.
.times. [ Amplitude ] a - ( 0 ) .function. ( .sigma. ) .times. =
.times. 1 4 .times. .times. m - .function. ( .sigma. ) .times.
.times. M 23 ( 0 ) .function. ( .sigma. ) [ Phase ] .delta. - ( 0 )
.function. ( .sigma. ) .times. = .times. .PHI. 2 .function. (
.sigma. ) - .PHI. 1 .function. ( .sigma. ) + arg .times. { M 23 ( 0
) .function. ( .sigma. ) } ( 1.22 .times. b ) .cndot. .times.
.times. Component .times. [ 3 ] .times. .times. ( central .times.
.times. period .times. .times. 1 / L 2 ) .times. : .times. .times.
.times. [ Amplitude ] a 2 ( 0 ) .function. ( .sigma. ) .times. =
.times. 1 2 .times. .times. m 2 .function. ( .sigma. ) .times.
.times. M 1 ( 0 ) .function. ( .sigma. ) [ Phase ] .delta. 2 ( 0 )
.function. ( .sigma. ) .times. = .times. .PHI. 2 .function. (
.sigma. ) ( 1.22 .times. c ) .cndot. .times. .times. Component
.times. [ 4 ] .times. .times. ( central .times. .times. period
.times. .times. 1 / L + ) .times. : .times. .times. .times. [
Amplitude ] a + ( 0 ) .function. ( .sigma. ) .times. = .times. 1 4
.times. .times. m + .function. ( .sigma. ) .times. .times. M 23 ( 0
) .function. ( .sigma. ) [ Phase ] .delta. + ( 0 ) .function. (
.sigma. ) .times. = .times. .PHI. 2 .function. ( .sigma. ) + .PHI.
1 .function. ( .sigma. ) - arg .times. { M 23 ( 0 ) .function. (
.sigma. ) } + .pi. .times. .times. where ( 1.22 .times. d ) M 23 (
0 ) .function. ( .sigma. ) = M 2 ( 0 ) .function. ( .sigma. ) - I
.times. .times. M 3 ( 0 ) .function. ( .sigma. ) ( 1.23 ) ##EQU18##
It is to be noted that this is mere replacement of M.sub.0(.sigma.)
to M.sub.3(.sigma.) with M.sub.0.sup.(0)(.sigma.) to
M.sub.3.sup.(0)(.sigma.).
[0223] The amplitude and the phase of each vibration component are
determined only by the spectrometric quasi-Stokes parameters, the
reference phase functions and the reference amplitude functions.
Here, since the spectrometric quasi-Stokes parameters are known in
a "case where the sample having a known polarization
characteristic", the remaining reference amplitude functions
m.sub.0(.sigma.), m.sub.-(.sigma.), m.sub.2(.sigma.),
m.sub.+(.sigma.), and the remaining reference phase functions
.phi..sub.1(.sigma.) and .phi..sub.2(.sigma.) are determined from
the demodulated amplitude and phase. Specifically, these functions
are given according to the following expressions: [Mathematical
Expression 19] m 0 .function. ( .sigma. ) = 2 .times. a 0 ( 0 )
.function. ( .sigma. ) M 0 ( 0 ) .function. ( .sigma. ) ( 1.24
.times. a ) m - .function. ( .sigma. ) = 4 .times. a - ( 0 )
.function. ( .sigma. ) M 23 ( 0 ) .function. ( .sigma. ) ( 1.24
.times. b ) m 2 .function. ( .sigma. ) = 2 .times. a 2 ( 0 )
.function. ( .sigma. ) M 1 ( 0 ) .function. ( .sigma. ) ( 1.24
.times. c ) m + .function. ( .sigma. ) = 4 .times. a + ( 0 )
.function. ( .sigma. ) M 23 ( 0 ) .function. ( .sigma. ) ( 1.24
.times. d ) .PHI. - .times. .times. ( .sigma. ) = .times. .PHI.
.times. 2 .times. .times. ( .sigma. ) .times. - .times. .PHI.
.times. 1 .times. .times. ( .sigma. ) = .times. .delta. - ( 0 )
.times. .times. ( .sigma. ) - arg .times. { M .times. 23 ( 0 )
.times. .times. ( .sigma. ) } ( 1.24 .times. e ) .PHI. .times. 2
.function. ( .sigma. ) = .delta. .times. 2 ( 0 ) .times. .times. (
.sigma. ) ( 1.24 .times. f ) .PHI. + .function. ( .sigma. ) =
.times. .PHI. 2 .function. ( .sigma. ) + .PHI. 1 .function. (
.sigma. ) = .times. .delta. + 90 ) .function. ( .sigma. ) + arg
.times. { M 23 ( 0 ) .function. ( .sigma. ) } - .pi. ( 1.24 .times.
g ) ##EQU19## Once these reference functions are obtained (can be
calibrated), spectrometric quasi-Stokes parameters of a sample
having an unknown spectropolarization characteristic can be
determined.
[0224] When the case of leaving the analyzer without using the
sample is considered as an example, the following expressions are
formed where the azimuth angle of the analyzer with respect to the
fast axis of the retarder R1 is .theta..
M.sub.0.sup.(0)(.sigma.)=P.sub.0.sup.(0)(.sigma.)/2 (1.25a)
M.sub.1.sup.(0)(.sigma.)=P.sub.0.sup.(0)(.sigma.)cos 2.theta./2
(1.25b) M.sub.2.sup.(0)(.sigma.)=P.sub.0.sup.(0)(.sigma.)sin
2.theta./2 (1.25c) M.sub.3.sup.(0)(.sigma.)=0 (1.25d) Here,
P.sub.0.sup.(0)(.sigma.) is a spectrum of the light source. In this
case, the above expressions (1.24a) to (1.24g) are expressed as
follows. [Mathematical Expression 20] m 0 .function. ( .sigma. ) =
2 .times. a 0 ( 0 ) .function. ( .sigma. ) 1 2 .times. P 0 ( 0 )
.function. ( .sigma. ) ( 1.26 .times. a ) m - .function. ( .sigma.
) = 4 .times. a - ( 0 ) .function. ( .sigma. ) 1 2 .times. P 0 ( 0
) .function. ( .sigma. ) .times. sin .times. .times. 2 .times.
.times. .theta. ( 1.26 .times. b ) m 2 .function. ( .sigma. ) = 2
.times. a 2 ( 0 ) .function. ( .sigma. ) 1 2 .times. P 0 ( 0 )
.function. ( .sigma. ) .times. cos .times. .times. 2 .times.
.times. .theta. ( 1.26 .times. c ) m + .function. ( .sigma. ) = 4
.times. a + ( 0 ) .function. ( .sigma. ) 1 2 .times. P 0 ( 0 )
.function. ( .sigma. ) .times. sin .times. .times. 2 .times.
.times. .theta. ( 1.26 .times. d ) .PHI. - .function. ( .sigma. ) =
.PHI. 2 .function. ( .sigma. ) - .PHI. 1 .function. ( .sigma. ) =
.delta. - ( 0 ) .function. ( .sigma. ) ( 1.26 .times. e ) .PHI. 2
.function. ( .sigma. ) = .delta. 2 ( 0 ) .function. ( .sigma. ) (
1.26 .times. f ) .PHI. + .function. ( .sigma. ) = .PHI. 2
.function. ( .sigma. ) + .PHI. 1 .function. ( .sigma. ) = .delta. +
( 0 ) .function. ( .sigma. ) - .pi. ( 1.26 .times. g )
##EQU20##
[0225] It is revealed from the above that the reference amplitude
function and the reference phase function can be obtained so long
as the direction .theta. and the spectrum P.sub.0.sup.(0)(.sigma.)
of the light source are known in advance. Further, even with
P.sub.0.sup.(0)(.sigma.) unknown, if only the direction .theta. is
known, it serves sufficiently for obtaining part of (essential)
polarization parameters.
[0226] B. Calibration Procedure for Obtaining Amplitude and Phase
Altogether (by Regarding Both as Reference Complex Function) at
Once
[0227] The above-mentioned method was a method for calculating the
"amplitude" and the "phase" of each vibration component separately.
However, it may be more convenient (efficient) in some cases to
calculate them as the "complex representation" of each vibration
component. One example of such calculation may be the case of
directly obtaining the "complex representation" (Expressions
(1.17a) to (1.17d)), as in the Fourier transform method shown in
FIG. 7 above. In such a case, calibration is efficiently performed
by calibration of the "complex representation" as it is without
separation into the "amplitude" and "phase".
[0228] In the following, mathematical expressions for the
above-mentioned case are shown. What needs to be concerned here is
that the physical properties of the cases of using "amplitude and
phase" and the "complex representation" are completely the same. It
is just that in the latter case, a calculation is made using
complex numbers, and thus more efficient.
[0229] Similarly to the previous section, a case is considered
where the sample having a known polarization characteristic is
inserted into the channeled spectroscopic polarimeter. A complex
representation of each vibration component is obtained according to
Expressions (1.17a) to (1.17d) as follows.
F.sub.0.sup.(0)(.sigma.)=K.sub.0(.sigma.)M.sub.0.sup.(0)(.sigma.)
(1.27a)
F.sub.-.sup.(0)(.sigma.)=K.sub.-(.sigma.)M.sub.23.sup.(0)(.sigma-
.) (1.27b)
F.sub.2.sup.(0)(.sigma.)=K.sub.2(.sigma.)M.sub.1.sup.(0)(.sigma.)
(1.27c)
F.sub.+.sup.(0)(.sigma.)=K.sub.+(.sigma.)M.sub.23.sup.(0)+(.sigm-
a.) (1.27d)
[0230] Here, the complex functions K.sub.0(.sigma.),
K.sub.-(.sigma.), K.sub.2(.sigma.), and K.sub.+(.sigma.) are in
amount (reference complex function) determined based not upon the
sample, but only upon the reference amplitude function and the
reference phase function, as seen from Expressions (1.18a) to
(1.18d). Accordingly, these can be calculated backward as follows.
[Mathematical Expression 21] K 0 .function. ( .sigma. ) = F 0 ( 0 )
.function. ( .sigma. ) M 0 ( 0 ) .function. ( .sigma. ) ( 1.28
.times. a ) K - .function. ( .sigma. ) = F - ( 0 ) .function. (
.sigma. ) M 23 ( 0 ) .function. ( .sigma. ) ( 1.28 .times. b ) K 2
.function. ( .sigma. ) = F 2 ( 0 ) .function. ( .sigma. ) M 1 ( 0 )
.function. ( .sigma. ) ( 1.28 .times. c ) K + .function. ( .sigma.
) = F + ( 0 ) .function. ( .sigma. ) M 23 ( 0 ) * ( .sigma. ) (
1.28 .times. d ) ##EQU21##
[0231] Similar to the case of calculating the amplitude and the
phase separately, once the above reference complex function is
obtained (can be calibrated), spectrometric quasi-Stokes parameters
of the sample having an unknown polarization characteristic can be
obtained.
[0232] It is to be noted that, just for reference, mathematical
expressions in the case of leaving the analyzer without using the
sample are shown below. [Mathematical Expression 22] K 0 .function.
( .sigma. ) = F 0 ( 0 ) .function. ( .sigma. ) 1 2 .times. P 0 ( 0
) .function. ( .sigma. ) ( 1.29 .times. a ) K - .function. (
.sigma. ) = F - ( 0 ) .function. ( .sigma. ) 1 2 .times. P 0 ( 0 )
.function. ( .sigma. ) .times. sin .times. .times. 2 .times.
.theta. ( 1.29 .times. b ) K 2 .function. ( .sigma. ) = F 2 ( 0 )
.function. ( .sigma. ) 1 2 .times. P 0 ( 0 ) .function. ( .sigma. )
.times. cos .times. .times. 2 .times. .theta. ( 1.29 .times. c ) K
+ .function. ( .sigma. ) = F + ( 0 ) .function. ( .sigma. ) 1 2
.times. P 0 ( 0 ) .function. ( .sigma. ) .times. sin .times.
.times. 2 .times. .theta. ( 1.29 .times. d ) ##EQU22##
[0233] Chapter 2: Problems of Channeled Spectroscopic
Polarimeter
[0234] As described in Step 3 in Section 1.5, for demodulation of
spectrometric quasi-Stokes parameters M.sub.0(.sigma.), M.sub.1
(.sigma.), M.sub.2(.sigma.), and M.sub.3(.sigma.) from a measured
channeled spectrum P(.sigma.), it is necessary to obtain
(calibrate) the following functions in advance. [Mathematical
Expression 23] Reference amplitude function , .times. m 0
.function. ( .sigma. ) m - .function. ( .sigma. ) m 2 .function. (
.sigma. ) m + .function. ( .sigma. ) } .times. .times. Reference
phase function , or .times. .PHI. 1 .function. ( .sigma. ) .PHI. 2
.function. ( .sigma. ) } .times. .times. Reference complex function
.times. K 0 .function. ( .sigma. ) K - .function. ( .sigma. ) K 2
.function. ( .sigma. ) K + .function. ( .sigma. ) } ##EQU23##
[0235] However, the reference phase functions .phi..sub.1(.sigma.)
and .phi..sub.2(.sigma.) have the property of varying for a variety
of reasons.
When these functions vary, there occurs a problem in that a large
error occurs in measured values of the spectropolarization
parameters of the sample.
2.1 Cause of Variations in Reference Phase Function
2.1.1 Temperature Change
[0236] The reference phase functions .phi..sub.1(.sigma.) and
.phi..sub.2(.sigma.) are amounts (retardation) determined by the
retarders R1 and R2 in the spectroscopic polarimeter. This
retardation has the property of changing sensitively with respect
to a temperature. Hence the phase of the channeled spectrum is
displaced due to the temperature change. As a result, due to a
temperature rise, a measured value is displaced to cause occurrence
of an error therein. Moreover, a similar change occurs with
pressure change.
2.1.2 Variations in Wavelength Axis of Spectrometer
[0237] When a wavelength to be sampled with the spectrometer is
displaced, a problem that is "equivalent" to fluctuations in the
reference phase function occurs. When the wavelength to be sampled
is displaced during measurement, a similar effect to an effect in
lateral displacement of the spectrum is produced. This is an
equivalent phase displacement. In particular, in an ordinary
spectrometer (type of rotating a diffraction grating with a motor),
a wavelength to be sampled is displaced by small degrees (at
random) in every measurement, due to backlash of a motor or the
like.
2.1.3 Solution Easily Found
[0238] For preventing variations in reference phase function of
each vibration component, stabilizing a cause of the fluctuations
is considered. However, this is very hard to realize. For example,
when noting the temperature variation, the accuracy required for
wavenumber-distribution of an ellipsometric parameter in
spectrometric ellipsometry is in the order of 0.1.degree. or
smaller, and for satisfying this, it is necessary to keep the
temperature variation within the order of 0.5.degree. C. This
necessitates large equipment for temperature stabilization,
unfavorably leading to a loss of a variety of advantages (size
reduction, non-inclusion of an active element, etc.) of the
channeled spectroscopic polarimetry.
[0239] Chapter 3: Solution Against Variations in Reference Phase
Function
[0240] The reference phase functions .phi..sub.1(.sigma.) and
.phi..sub.2(.sigma.) (depending not upon the sample but only upon
parameters of the polarimeter) included in the channeled spectrum
vary by a variety of factors, which becomes a major contributor to
an error. In consideration of this respect, in one embodiment of
the present invention, the channeled spectroscopic polarimeter is
provided with a function capable of calibrating the reference phase
functions .phi..sub.1(.sigma.) and .phi..sub.2(.sigma.) of each
vibration component during measurement (concurrently with
measurement) (cf. FIGS. 8 to 10).
3.1 Method for Calibration "During Measurement" (No. 1)
[0241] The calibration method described in Section 1.6 was a method
for calibration "prior to measurement". As opposed to this, in the
following section, a method for calibration "during measurement" is
shown.
3.1.1 Basic Idea
[0242] The amplitude and the phase obtained in Step 2 in Chapter 1
during measurement (when light in an unknown SOP is incident on the
channeled spectroscopic polarimeter) is shown again below.
[Mathematical Expression 24] .cndot.Component .function. [ 1 ]
.times. ( low .times. .times. frequency .times. .times. component )
.times. : .times. .times. [ Amplitude ] .times. .times. a 0
.function. ( .sigma. ) = 1 2 .times. m 0 .function. ( .sigma. )
.times. M 0 .function. ( .sigma. ) .times. .times. [ Phase ]
.times. .times. .delta. 0 .function. ( .sigma. ) = 0 ( 3.1 .times.
a ) .cndot.Component .function. [ 2 ] .times. ( central .times.
.times. period .times. .times. 1 / L - ) .times. : .times. .times.
[ Amplitude ] .times. .times. a - .function. ( .sigma. ) = 1 4
.times. m - .function. ( .sigma. ) .times. M 23 .function. (
.sigma. ) .times. .times. [ Phase ] .times. .times. .delta. -
.function. ( .sigma. ) = .PHI. 2 .function. ( .sigma. ) - .PHI. 1
.function. ( .sigma. ) + arg .times. { M 23 .function. ( .sigma. )
} ( 3.1 .times. b ) .cndot.Component .function. [ 3 ] .times. (
central .times. .times. period .times. .times. 1 / L 2 ) .times. :
.times. .times. [ Amplitude ] .times. .times. a 2 .function. (
.sigma. ) = 1 2 .times. m 2 .function. ( .sigma. ) .times. M 1
.function. ( .sigma. ) .times. .times. [ Phase ] .times. .times.
.delta. 2 .function. ( .sigma. ) = .PHI. 2 .function. ( .sigma. ) (
3.1 .times. c ) .cndot.Component .function. [ 4 ] .times. ( central
.times. .times. period .times. .times. 1 / L + ) .times. : .times.
.times. [ Amplitude ] .times. .times. a + .function. ( .sigma. ) =
1 4 .times. m + .function. ( .sigma. ) .times. M 23 .function. (
.sigma. ) .times. .times. [ Phase ] .times. .times. .delta. +
.function. ( .sigma. ) = .PHI. 2 .function. ( .sigma. ) + .PHI. 1
.function. ( .sigma. ) - arg .times. { M 23 .function. ( .sigma. )
} + .pi. ( 3.1 .times. d ) ##EQU24##
[0243] Here, all needed for obtaining the four spectrometric
quasi-Stokes parameters are found to be: [0244] "amplitude" of
Component [1].fwdarw.M.sub.0(.sigma.) [0245] "amplitude" and
"phase" of one of Component [2] and Component
[4].fwdarw.M.sub.2(.sigma.) and M.sub.3(G) [0246] "amplitude" of
Component [3].fwdarw.M.sub.1(.sigma.) It is found that the
remaining ones as follows are not used for demodulation of the
spectrometric quasi Stokes parameters. [0247] "phase" of Component
[3] [0248] "amplitude" and "phase" of the remaining one of
Component [2] and component [4]
[0249] The present inventors and the like found it possible to
obtain not only the four spectrometric quasi-Stokes parameters but
also the "reference phase functions (.phi..sub.1(.sigma.),
.phi..sub.2(.sigma.), etc.)" all at once through use of the
remaining component. In this method, it is meant that calibration
can be concurrently performed in the midst of measurement without
particular input of light in a known SOP.
3.1.2 Preparation
[0250] In order to use the "calibration method during measurement",
the following prior preparation is necessary. [0251] The reference
amplitude functions m.sub.0(.sigma.), m.sub.-(.sigma.),
m.sub.2(.sigma.), and m.sub.+(.sigma.) are subjected to
pre-calibration (cf. FIG. 9)
[0252] Since the following method is effective only on the
reference phase function, any one of the methods descried in
Section 1.6 is to be performed as for the reference amplitude
function. It is to be noted that the fluctuations in the reference
amplitude function during measurement typically have considerably
small magnitude, and are ignorable in many cases. Namely, in
contrast to the reference phase function, there is generally almost
no need for recalibration of the reference amplitude function
during measurement. [0253] As for the reference phase function, the
pre-calibration is not necessarily required. However, a ratio
between .phi..sub.1(.sigma.) and .phi..sub.2(.sigma.) needs to be
obtained in advance.
Example 1
[0254] when the retarders R.sub.1 and R.sub.2 are made of the same
medium, the ratio between .phi..sub.1(.sigma.) and
.phi..sub.2(.sigma.) is determined from a ratio between thicknesses
of the two retarders.
Example 2
[0255] By further pre-calibration of the reference phase function,
the ratio between .phi..sub.1(.sigma.) and .phi..sub.2(.sigma.) is
determined (this ratio may be considered not to change during
measurement).
[0256] Note here that, in cases including a case where the ratio
between the retarders R.sub.1 and R.sub.2 changes (e.g.
temperatures of the two retarders are different) during
measurement, a method described below cannot be used.
3.1.3 Actual Calibration Method
[0257] Based upon the above-mentioned idea, a method for actual
calibration is described below.
[0258] A. Method for Obtaining Reference Phase Function (.sigma.)
from Vibration Component [3]
[0259] By noting only Vibration component [3], the amplitude and
the phase thereof are shown again as follows. [Mathematical
Expression 25] { [ Amplitude ] a 2 .function. ( .sigma. ) = 1 2
.times. m 2 .function. ( .sigma. ) .times. M 1 .function. ( .sigma.
) [ Phase ] .delta. 2 .function. ( .sigma. ) = .PHI. 2 .function. (
.sigma. ) ( 3.2 ) ##EQU25## What needs to be noted here is that the
phase .delta..sub.2(.sigma.) of this component is one
(.phi..sub.2(.sigma.)) of the reference phase functions (itself).
Namely, when the phase .delta..sub.2(.sigma.) of Component [3] is
measured, one (.phi..sub.2(.sigma.)) of the reference phase
functions is immediately determined according to the following
expression. .phi..sub.2(.sigma.)=.delta..sub.2(.sigma.) (3.3)
[0260] This relational expression is constantly satisfied
regardless of a polarization characteristic of a measurement
sample, meaning that one of the reference phase functions can be
immediately obtained from a measured value, even from a channeled
spectrum by any kind of sample. This is a calibration method that
can be performed utterly concurrently during measurement, and in
the case of "using a sample having a known polarization
characteristic", there is no need for performing calibration "prior
to measurement or after discontinuation of measurement" as in
(Section 1.6). However, it should be noted that at this time, the
condition of observing Component [3] at a sufficient SN ratio needs
to be satisfied (cf. later-described C).
[0261] It is to be noted that, when the "complex representation" is
obtained in place of the "paired amplitude and phase" in Step 2 of
the "procedure for demodulating spectrometric quasi-Stokes
parameters" in Section 1.5, a calculation method, rewritten from
the above and described below, may be applied.
[0262] From Expression (1.14)b, .delta..sub.2(.sigma.) has the
following relation with the complex representation F.sub.2(.sigma.)
of Component [3]. .delta..sub.2(.sigma.)=arg [F.sub.2(.sigma.(]
(3.4) Therefore, the reference phase function .phi..sub.2(.sigma.)
can be obtained from the complex representation of Component [3]
according to the following expression. .phi..sub.2(.sigma.)=arg
[F.sub.2(.sigma.)] (3.5) It should be noted that what is needed at
the time of complex representation is not the reference phase
function .phi..sub.2(.sigma.) but the reference complex function
K.sub.2(.sigma.). Since there is a relation between these two
functions as expressed by Expression (1.18c), once
.phi..sub.2(.sigma.) is determined, K.sub.2(.sigma.) can also be
determined (this will be later described in details in F).
[0263] B. Method for Obtaining Reference Phase Function (.sigma.)
from a Plurality of Vibration Components (Paired [2] and [4],
etc.)
[0264] The respective phases of Vibration Components [2] and [4]
are again shown as follows.
[0265] Phase of Component [2]:
.delta..sub.-(.sigma.)=.phi..sub.2(.sigma.)-.phi..sub.1(.sigma.)+arg
{M.sub.23(.sigma.)} (3.6a)
[0266] Phase of Component [4]:
.delta..sub.+(.sigma.)=.phi..sub.2(.sigma.)+.phi..sub.1(.sigma.)-arg
{M.sub.23(.sigma.)}+.pi. (3.6b) When the one phase is added to the
other, .phi..sub.1(.sigma.) and arg {M.sub.23(.sigma.)} are
canceled out, whereby it is found that the following expression is
satisfied. [Mathematical Expression 26] .PHI. 2 .function. (
.sigma. ) = 1 2 .times. { .delta. - .function. ( .sigma. ) +
.delta. + .function. ( .sigma. ) } - .pi. 2 ( 3.7 ) ##EQU26##
[0267] The right side of the above expression means that one
(.phi..sub.2(.sigma.)) of the reference phase functions can be
obtained by taking an average of the phases of Vibration components
[2] and [4]. Similarly to Method A, this relational expression can
also be constantly satisfied regardless of the SOP of the sample,
meaning that one of the reference phase functions can be
immediately obtained from a measured value, even from a channeled
spectrum by any kind of sample.
[0268] Namely, similarly to the case of Method A, this is a
"calibration method that can be performed utterly concurrently
during measurement", and in the case of "using a sample having a
known polarization characteristic", there is no need for performing
calibration "prior to measurement or after discontinuation of
measurement" as in (Section 1.6). However, it should be noted that
the condition of observing Components [2] and [4] at a sufficient
SN ratio needs to be satisfied (cf. later-described C).
[0269] Here, similarly to the case of Method A, a calculation
method is described which is used in the case of obtaining the
"complex representation" in place of the "paired amplitude and
phase" in Step 2 of Section 1.5.
[0270] From Expression (1.14b), .delta..sub.-(.sigma.) and
.delta..sub.+(.sigma.) have the following relations with the
complex representations F.sub.-(.sigma.) and F.sub.+(.sigma.) of
Components [2] and [4]. .delta..sub.-(.sigma.)=arg
[F.sub.-(.sigma.)] (3.8a) .delta..sub.+(.sigma.)=arg
[F.sub.+(.sigma.)] (3.8b)
[0271] Therefore, the reference phase function .phi..sub.2(.sigma.)
can be obtained from the complex representations of the two
components as follows. [Mathematical Expression 27] .PHI. 2
.function. ( .sigma. ) = 1 2 .times. { arg .function. [ F -
.function. ( .sigma. ) ] + arg .function. [ F + .function. (
.sigma. ) ] } - .pi. 2 ( 3.9 ) ##EQU27## Or, the following
expression obtained by rewriting the above expression to a simple
formula of the complex function may be applied. [Mathematical
Expression 28] .PHI. 2 .function. ( .sigma. ) = 1 2 .times. arg
.function. [ - F - .function. ( .sigma. ) .times. F + .function. (
.sigma. ) ] ( 3.10 ) ##EQU28##
[0272] In the optically system (channeled spectroscopic
polarimeter) in FIG. 2, an obtained spectrum includes another
component having a different period as described above except for
the case where the angle formed between the retarder R.sub.2 and
the polarizer P is not 45.degree..
[0273] As seen from Expression (1.10), the phase of this component
is .delta..sub.1(.sigma.)=.phi..sub.1(.sigma.) -arg
{M.sub.23(.sigma.)}", and similar to the phase terms of above
Vibration components [2] and [4]. Hence, even when the above
component is combined with Components [2] and [4] (or replaced by
one of them), it is possible to calibrate .phi..sub.2(.sigma.) in
the same manner.
[0274] C. Combination of A and B
[0275] The two methods (Method A and Method B) described above are
methods in which one (.phi..sub.2(.sigma.)) of the reference phase
functions can be calibrated utterly concurrently during
measurement. However, the vibration components used are different
between the two methods. What should be concerned here is that the
amplitude of Vibration component [3] used in Method A is
proportional to M.sub.1(.sigma.), while the amplitudes of Vibration
Components [2] and [4] used in Method B are proportional to the
following. |M.sub.23(.sigma.)|= {square root over
(M.sub.2.sup.2(.sigma.)+M.sub.3.sup.2(.sigma.))} [Mathematical
Expression 29]
[0276] Since the polarization characteristic of the sample is
unknown, there is no guarantee that the spectrometric quasi-Stokes
parameters are constantly sufficiently large for phase measurement
for each component. For example, when M.sub.1(.sigma.) is small,
obtaining .phi..sub.2(.sigma.) by Method A using the phase of this
component might result in occurrence of a large error. For solving
this problem, adaptive combination of Methods A and B is desired.
Specifically, a value of .phi..sub.2(.sigma.) with more certainty
can be obtained by selecting, or weighting up and balancing,
results of the two methods.
[0277] D. Combination of A and B (No. 2)
[0278] One idea for efficiently combining A and B is shown below.
This is a method in which direct calculation is possible without
particular separation by case. It should be noted that, in this
part (Method D), three complex representation functions
F.sub.-(.sigma.) F.sub.2(.sigma.) and F.sub.-(.sigma.) of
Components [2] to [4] are used for calculation. When a calculation
is to be made from the "paired amplitude and phase" of each
vibration component, this pair may once be changed to the "complex
representation" according to Expression (1.13), and then the
following calculation procedure may be performed.
[0279] As a preparation for explaining this method, first, the
following two expressions are derived and the properties thereof
are described. By transforming Expression (3.5), the following
expression can be obtained.
2.phi..sub.2(.sigma.)=arg[F.sub.2.sup.2(.sigma.)] (3.11) Meanwhile,
by doubling both sides of Expression (3.10), the following
expression can be obtained.
2.phi..sub.2(.sigma.)=arg[-F.sub.-(.sigma.)F.sub.+(.sigma.)] (3.12)
It is found from the comparison between the above two expressions
that the complex function in the brackets on the right side of each
of the expressions has the same argument 2.phi..sub.2(.sigma.).
[0280] "Appropriate weighting functions .alpha.(.sigma.) and
.beta.(.sigma.) which have the same argument" were respectively
multiplied by the above two complex functions, and then the
obtained two terms were added together.
[Mathematical Expression 30]
.alpha.(.sigma.)[F.sub.2.sup.2(.sigma.)].beta.(.sigma.)
[-F.sub.-(.sigma.)F.sub.+(.sigma.)] (3.13) The argument of this
expression is constantly equivalent to
2.phi..sub.2(.sigma.)+arg.alpha.(.sigma.). It is found that,
through use of the properties thus described, .phi..sub.2(.sigma.)
can be obtained according to the following expression even if one
of M.sub.1 and M.sub.23 decreases. [Mathematical Expression 31]
.PHI. 2 .function. ( .sigma. ) = .times. 1 2 .times. arg .times. {
.alpha. .function. ( .sigma. ) .function. [ F 2 2 .function. (
.sigma. ) ] + .beta. .function. ( .sigma. ) .function. [ - F -
.function. ( .sigma. ) .times. F + .function. ( .sigma. ) ] } -
.times. 1 2 .times. arg .function. [ .alpha. .function. ( .sigma. )
] ( 3.14 ) ##EQU29##
[0281] There are a variety of specific ways to select
.alpha.(.sigma.) and .beta.(.sigma.).
[0282] The simplest way to select .alpha.(.sigma.) and
.beta.(.sigma.) is making the two functions the same constant (1).
In this case, an expression for obtaining the reference phase
function .phi..sub.2(.sigma.) is shown below. [Mathematical
Expression 32] .PHI. 2 .function. ( .sigma. ) = 1 2 .times. arg
.times. { [ F 2 2 .function. ( .sigma. ) ] + [ - F - .function. (
.sigma. ) .times. F + .function. ( .sigma. ) ] } ( 3.15 )
##EQU30##
[0283] E. Calculation of .phi..sub.1(.sigma.)
[0284] Since fluctuations in .phi..sub.1(.sigma.) are considered to
be similar to those in .phi..sub.2(.sigma.), it is possible to
obtain .phi..sub.1(.sigma.) from a measured value of
.phi..sub.2(.sigma.) by proportional calculation (e.g. by using a
thickness ratio).
[0285] F. Calculation of Reference Complex Function
[0286] In the demodulation in Step 2 of the "procedure for
demodulating spectrometric quasi-Stokes parameters" in Section 1.5,
when (not the "paired amplitude and phase" but) the "complex
representation" is obtained, what are needed ultimately in the
operation of Step 3 for obtaining the spectrometric quasi-Stokes
parameters are not the reference phase functions
.phi..sub.1(.sigma.) and .phi..sub.2(.sigma.) but the reference
complex functions K.sub.0(.sigma.), K.sub.-(.sigma.),
K.sub.2(.sigma.), and K.sub.+(u). However, these can also be
immediately obtained through use of the relations of Expressions
(1.18a) to (1.18d) if the reference phase functions
.phi..sub.1(.sigma.) and .phi..sub.2(.sigma.) have been obtained by
the procedures up to E above.
3.2 Method for Calibrating Reference Phase Function "During
Measurement" (No. 2)
3.2.1 Basic Idea
[0287] In the same idea as described in the previous section 3.1,
"only a variation" in reference phase function can be obtained.
[0288] In the previous method (in the previous section 3.1), the
"reference amplitude function" was obtained in the pre-calibration,
and it was not particularly necessary to obtain the "reference
phase function". However, as appeared from Section 3.2, those two
functions can be calibrated almost concurrently. It is thus
possible to obtain an "initial value of the reference phase
function in pre-calibration" so as to only track the variation
thereof during measurement.
[0289] Advantages in this case include the following. [0290]
Slightly additional phase displacement part which might be
generated due to characteristics of the spectrometer or the signal
processing system can be removed. [0291] Burdensome phase
unwrapping is not necessary. [0292] Since a variation in phase
difference itself is small, a dynamic range in calculation can be
made small. Further, as a result of this, a calculation error can
be relatively made small in many cases.
[0293] Accordingly, "obtaining only the variation in reference
phase function" has its own meaning.
[0294] The following is described as supplement for the foregoing
explanation. As shown in FIG. 10, the two methods have different
factors of an error in the case of obtaining .phi..sub.1(c from
.phi..sub.2(.sigma.). Namely, as shown in FIG. 10A, it is necessary
to perform phase unwrapping in the case of obtaining
.phi..sub.1(.sigma.) from .phi..sub.2(.sigma.). This phase
unwrapping is a major factor of the error. Especially when a period
is at high frequency as compared with sampling, noise is included
in the period, or the like, wrong phase unwrapping might be
performed. With wrong phase wrapping performed, an error becomes an
integer multiple of 2.pi., leading to calculation of a wrong phase.
Further, this error affects a broad wavenumber region. This
difference is essentially caused by that a solution of an arg
operator (or an arctan operator) for obtaining an argument has
ambiguity by the integer multiple of 2.pi.. As opposed to this, as
shown in FIG. 10B, it is not necessary in obtaining
.DELTA..phi..sub.1(.sigma.) from .DELTA..phi..sub.2(.sigma.) to
perform phase unwrapping since the variation
.DELTA..phi..sub.2(.sigma.) from the initial value of the reference
phase function is small. This allows the measurement error to be
relatively small.
3.2.2 Preparation
[0295] The use of the "calibration method during measurement" is
based upon the premise of pre-calibration of both the "reference
amplitude function" and the "reference phase function" prior to
measurement. It is to be noted that, as for the phase, highly
accurately obtaining the reference phase function is not necessary
since a phase value can be corrected later by the variance and the
measurement error.
3.2.3 Actual Calibration Method
[0296] The basic idea on the calibration method is completely the
same as in Section 3.1. There thus exists a calculation method
corresponding to all A to E described in Section 3.1.3. Hence, in
this section, an idea different from that in the previous section
is shown and mathematical expressions are mainly cited in the
following description.
[0297] First, a couple of symbols are defined. The reference phase
functions obtained by the pre-calibration are defined as
.phi..sub.1.sup.(i)(.sigma.) and .phi..sub.2.sup.(i)(.sigma.).
Reference complex functions corresponding to these reference phase
functions are expressed as follows according to Expressions (1.18a)
to (1.18d). [Mathematical Expression 33] K 0 ( i ) .function. (
.sigma. ) = 1 2 .times. m 0 .function. ( .sigma. ) ( 3.16 .times. a
) K - ( i ) .function. ( .sigma. ) = 1 8 .times. m - .function. (
.sigma. ) .times. exp .times. .times. I .function. [ .PHI. 2 ( i )
.function. ( .sigma. ) - .PHI. 1 ( i ) .function. ( .sigma. ) ] (
3.16 .times. b ) K 2 ( i ) .function. ( .sigma. ) = 1 4 .times. m 2
.function. ( .sigma. ) .times. exp .times. .times. I .times.
.times. .PHI. 2 ( i ) .function. ( .sigma. ) ( 3.16 .times. c ) K +
( i ) .function. ( .sigma. ) = - 1 8 .times. m + .function. (
.sigma. ) .times. exp .times. .times. I .function. [ .PHI. 2 ( i )
.function. ( .sigma. ) + .PHI. 1 ( i ) .function. ( .sigma. ) ] (
3.16 .times. d ) ##EQU31## Assuming that the reference phase
functions changed during measurement as follows.
.phi..sub.1(.sigma.)=.phi..sub.1.sup.(i)(.sigma.)+.DELTA..phi..-
sub.1(.sigma.) (3.17a)
.phi..sub.2(.sigma.)=.phi..sub.2.sup.(i)(.sigma.)+.DELTA..phi..sub.2(.sig-
ma.) (3.17b) Below described are methods for obtaining the
variations .DELTA..phi..sub.1(.sigma.) and
.DELTA..phi..sub.2(.sigma.) of the reference phase functions or
changes in the reference complex functions corresponding to those
variations.
[0298] A. Method for Obtaining Reference Phase Function .phi..sub.2
from Vibration Component [3]
[0299] As described in Method A in the previous section, the phase
of Component [3] is expressed as follows.
.delta..sub.2(.sigma.)=.phi..sub.2(.sigma.)=.phi..sub.2.sup.(i)(.sigma.)+-
.DELTA..phi..sub.2(.sigma.) (3.18) Here, the variation in
.phi..sub.2(.sigma.) can be obtained as
.DELTA..phi..sub.2(.sigma.)=.delta..sub.2(.sigma.)-.phi..sub.2.sup.(i)(.s-
igma.) (3.19) Namely, this means that, once the phase .delta..sub.2
of Component [3] is measured, one of the variations
.DELTA..phi..sub.2(.sigma.) in reference phase functions can be
immediately determined.
[0300] It is to be noted that in Step 2, when not the "paired
amplitude and phase" but the "complex representation" is to be
obtained, the variation is obtained according to the following
expressions. [Mathematical Expression 34] .delta. 2 .function. (
.sigma. ) = arg .function. [ F 2 .function. ( .sigma. ) ] ( 3.20
.times. a ) .PHI. 2 ( i ) .function. ( .sigma. ) = arg .function. [
K 2 ( i ) .function. ( .sigma. ) ] ( 3.20 .times. b ) From .DELTA.
.times. .times. .PHI. 2 .function. ( .sigma. ) = arg .function. [ F
2 .function. ( .sigma. ) ] - arg .function. [ K 2 ( i ) .function.
( .sigma. ) ] ( 3.21 ) , or .DELTA. .times. .times. .PHI. 2
.function. ( .sigma. ) = arg [ F 2 .function. ( .sigma. ) K 2 ( i )
.function. ( .sigma. ) ] ( 3.22 ) ##EQU32##
[0301] B. Method for Obtaining Reference Phase Function
.phi..sub.2(.sigma.) from a Plurality of Vibration Components
(Paired [2] and [4], etc.)
[0302] In the method for obtaining the variation in
.phi..sub.2(.sigma.) from the phase of each of Vibration components
[2] and [4], the variation is obtained according to the following
expression. [Mathematical Expression 35] .DELTA. .times. .times.
.PHI. 2 .function. ( .sigma. ) = [ 1 2 .times. { .delta. -
.function. ( .sigma. ) + .delta. + .function. ( .sigma. ) } - .pi.
2 ] - .PHI. 2 ( i ) .function. ( .sigma. ) ( 3.23 ) ##EQU33##
[0303] When not the "paird amplitude and phase" but the "complex
representation" is to be obtained, the variation is obtained
according to the following expression. [Mathematical Expression 36]
.DELTA. .times. .times. .PHI. 2 .function. ( .sigma. ) = 1 2
.times. { arg .function. [ F - .function. ( .sigma. ) ] + arg
.function. [ F + .function. ( .sigma. ) ] - arg .function. [ K - (
i ) .function. ( .sigma. ) ] - arg .function. [ K + ( i )
.function. ( .sigma. ) ] } ( 3.24 ) ##EQU34## Or, the following
expressions obtained by rewriting the above expression using a
simple formula of the complex function may be applied.
[Mathematical Expression 37] .DELTA. .times. .times. .PHI. 2
.function. ( .sigma. ) = 1 2 .times. { arg [ F - .function. (
.sigma. ) K - ( i ) .function. ( .sigma. ) ] + arg [ F + .function.
( .sigma. ) K + ( i ) .function. ( .sigma. ) ] } ( 3.25 ) , or
.DELTA. .times. .times. .PHI. 2 .function. ( .sigma. ) = 1 2
.times. arg [ F - .function. ( .sigma. ) K - ( i ) .function. (
.sigma. ) .times. F + .function. ( .sigma. ) K + ( i ) .function. (
.sigma. ) ] ( 3.26 ) ##EQU35## It is to be noted that, as noted at
the end of Section 3.1.3, the same idea as shown above can be
applied to the case of using another term.
[0304] C. Combination of A and B
[0305] As in the case described in the previous section, adaptive
combination of Methods A and B is effective also in the case of
obtaining only the "variation" in reference phase functions. It
should be noted that a description of the combination is completely
the same as that in the previous section and it is thus
omitted.
D. Combination of A and B (No. 2)
[0306] One of desired mathematical expressions in the case of
obtaining only the variation is as follows. [Mathematical
Expression 38] .alpha. .function. ( .sigma. ) = [ 1 K 2 ( i )
.function. ( .sigma. ) ] 2 ( 3.27 .times. a ) .beta. .function. (
.sigma. ) = - 1 K - ( i ) .function. ( .sigma. ) .times. K + ( i )
.function. ( .sigma. ) ( 3.27 .times. b ) ##EQU36## Since
arg[.alpha.(.sigma.)]=arg[.beta.(.sigma.)]=2.phi..sub.2(.sigma.) in
the above expressions, the variation can be obtained as follows.
[Mathematical Expression 39] .DELTA. .times. .times. .PHI. 2
.function. ( .sigma. ) = 1 2 .times. arg .times. { [ F 2 .function.
( .sigma. ) K 2 ( i ) .function. ( .sigma. ) ] 2 + F - .function. (
.sigma. ) K - ( i ) .function. ( .sigma. ) .times. F + .function. (
.sigma. ) K + ( i ) .function. ( .sigma. ) } ( 3.28 ) ##EQU37##
[0307] E. Calculation of .DELTA..phi..sub.1(.sigma.)
[0308] Fluctuations in .DELTA..phi..sub.1(a are considered similar
to those in .DELTA..sub.2(.sigma.). It is thus possible to obtain
.DELTA..sub.1 (.sigma.) from a measured value of
.DELTA..sub.2(.sigma.) by a comparative calculation using, for
example, a thickness ratio.
[0309] F. Calculation of Reference Complex Function
[0310] In the demodulation of each vibration component in Step 2,
when not the "paired amplitude and phase" but the "complex
representation" is obtained, what are needed ultimately in
obtaining the spectrometric quasi-Stokes parameters (operation of
Step 3) are not the reference phase functions .phi..sub.1(.sigma.)
and .phi..sub.2(.sigma.) but the reference complex functions
K.sub.0(.sigma.), K.sub.-(.sigma.), K.sub.2(.sigma.), and
K.sub.+(.sigma.).
[0311] If the reference phase function variations
.DELTA..phi..sub.1(.sigma.) and .DELTA..phi..sub.2(.sigma.) have
been obtained by the procedures up to E above, the reference
complex functions can be immediately obtained as follows.
[Mathematical Expression 40]
K.sub.0(.sigma.)=K.sub.0.sup.(i)(.sigma.) (3.29a)
K.sub.-(.sigma.)=K.sub.-.sup.(i)(.sigma.)e.sup.i[.DELTA..phi..su-
p.2.sup.(.sigma.)-.DELTA..phi..sup.1.sup.(.sigma.)] (3.29b)
K.sub.2(.sigma.)=K.sub.2.sup.(i)(.sigma.)e.sup.i.DELTA..phi..sup.2.sup.(.-
sigma.) (3.29c)
K.sub.+(.sigma.)=K.sub.+.sup.(i)(.sigma.)e.sup.i[.DELTA..phi..sup.2.sup.(-
.sigma.)+.DELTA..phi..sup.1.sup.(.sigma.)] (3.29d)
[0312] Chapter 4: Specific Embodiment of Present Invention
4.1 Case of Spectroscopic Polarimetry Performed by Reflecting Light
on Sample
[0313] An embodiment of the optical system in the case of measuring
a spectropolarization characteristic of a sample by reflecting
light on the sample is described in detail with reference to FIGS.
11 to 13. In this case, as shown in FIG. 11 and FIG. 12, the
optical system includes the light source 7, the polarizer P, the
retarder R2, the retarder R1, the analyzer A, and the spectroscope
8. It is to be noted that reference symbol B denotes a sample on
which light is reflected. Further, light emitted from the light
source 7 is transmitted through the polarizer P, the retarder R2
and the retarder R1 in this order. The light is then incident on
the sample B in a slanting direction, to be reflected thereon.
Thereafter, the light is transmitted through the analyzer A and
then received in the spectroscope 8. Attention should be paid that
this device is configured by arranging an optical element on the
light source side with respect to the sample in the configuration
of the conventional optical system as shown in FIG. 1B. Further,
the orientation of the transmission axis of the polarizer P agrees
with the orientation of the principal axis of the retarder R1, and
the orientations of the fast axes of the retarder R1 and the
retarder R2 are inclined at an angle of -45.degree. from each
other. .theta. shows the azimuth angle of the transmission axis of
the analyzer with respect to the fast axis of the retarder R1.
Further, the incident plane of the light agrees with the
orientation of the fast axis of the retarder R1. It is to be noted
that the device including the polarizer P, the retarder R2, the
retarder R1 and the analyzer A is referred to as a channeled
spectrometer unit.
[0314] At the same time, according to the optical system as shown
in FIG. 11 and FIG. 12, a stable measurement can be realized since
the incident direction of the wave surface of light that transmits
through the retarders R1 and R2 is not susceptible to the sample,
as shown in FIG. 13. That is, it is possible to solve the problem
of the retardation change between the time of calibrating the
retardation of the retarder and the time of measuring the sample
due to changes in distance and direction of a light ray that passes
through the retarder.
[0315] Further, an ellipsometric parameter and the like are
determined from the light incident on the spectroscope 8. A
procedure for such determination is described below.
[0316] Here, an arctangent of a rate of change in amplitude rate of
a p-polarized light (light having an SOP where the polarizing
direction runs parallel with the incident plane) and an s-polarized
light (light having an SOP where the polarizing direction runs
vertical to the incident plane) is expressed as .psi.(.sigma.), and
a phase difference is expressed as .DELTA.(.sigma.). Then, a
Mueller matrix of an isotropic medium can be described as follows.
[Mathematical Expression 41] M .function. ( .sigma. ) = .function.
[ 1 - cos .times. .times. 2 .times. .PSI. .function. ( .sigma. ) 0
0 - cos .times. .times. 2 .times. .PSI. .function. ( .sigma. ) 1 0
0 0 0 sin .times. .times. 2 .times. .PSI. .function. ( .sigma. )
.times. cos .times. .times. .DELTA. .function. ( .sigma. ) sin
.times. .times. 2 .times. .PSI. .function. ( .sigma. ) .times. sin
.times. .times. .DELTA. .function. ( .sigma. ) 0 0 - sin .times.
.times. 2 .times. .PSI. .function. ( .sigma. ) .times. sin .times.
.times. .DELTA. .function. ( .sigma. ) sin .times. .times. 2
.times. .PSI. .function. ( .sigma. ) .times. cos .times. .times.
.DELTA. .function. ( .sigma. ) ] ( 4.1 ) ##EQU38## It is thereby
considered that .psi.(.sigma.) and .DELTA.(.sigma.) are obtained as
ellipsometric parameters.
[0317] Here, the following expressions are satisfied according to
Expressions (1.5a) to (1.5d). [Mathematical Expression 42] M 0
.function. ( .sigma. ) = 1 2 .times. P 0 .function. ( .sigma. )
.function. [ 1 - cos .times. .times. 2 .times. .PSI. .function. (
.sigma. ) .times. cos .times. .times. 2 .times. .times. .theta. ] (
4.2 .times. a ) M 1 .function. ( .sigma. ) = - 1 2 .times. P 0
.function. ( .sigma. ) .function. [ - cos .times. .times. 2 .times.
.PSI. .function. ( .sigma. ) + cos .times. .times. 2 .times.
.times. .theta. ] ( 4.2 .times. b ) M 2 .function. ( .sigma. ) = 1
2 .times. P 0 .function. ( .sigma. ) .times. sin .times. .times. 2
.times. .PSI. .function. ( .sigma. ) .times. cos .times. .times.
.DELTA. .function. ( .sigma. ) .times. sin .times. .times. 2
.times. .theta. ( 4.2 .times. c ) M 3 .function. ( .sigma. ) = 1 2
.times. P 0 .function. ( .sigma. ) .times. sin .times. .times. 2
.times. .PSI. .function. ( .sigma. ) .times. sin .times. .times.
.DELTA. .function. ( .sigma. ) .times. sin .times. .times. 2
.times. .theta. ( 4.2 .times. d ) ##EQU39## Here, assuming that
.theta.=45.degree., the following expressions are satsfied.
[Mathematical Expression 43] M 0 .function. ( .sigma. ) = 1 2
.times. P 0 .function. ( .sigma. ) M 1 .function. ( .sigma. ) = - 1
2 .times. P 0 .function. ( .sigma. ) .times. cos .times. .times. 2
.times. .PSI. .function. ( .sigma. ) M 2 .function. ( .sigma. ) = 1
2 .times. P 0 .function. ( .sigma. ) .times. sin .times. .times. 2
.times. .PSI. .function. ( .sigma. ) .times. cos .times. .times.
.DELTA. .function. ( .sigma. ) M 3 .function. ( .sigma. ) = 1 2
.times. P 0 .function. ( .sigma. ) .times. sin .times. .times. 2
.times. .PSI. .function. ( .sigma. ) .times. sin .times. .times.
.DELTA. .function. ( .sigma. ) ( 4.3 ) ##EQU40##
[0318] Since M.sub.0(.sigma.) to M.sub.3(.sigma.) which are
obtained by demodulation processing include three unknowns; the
spectrum P.sub.0(.sigma.) and the ellipsometric parameters
.psi.(.sigma.) and .DELTA.(.sigma.), as expressed in the above
expressions, it is possible to determine the ellipsometric
parameters .psi.(.sigma.) and .DELTA.(.sigma.).
[0319] Moreover, one example of a case where modulating at least
one "component" shown in Expressions (1.11a) to (1.11d) gives a
useful application is described here.
[0320] For example, it is assumed that there is a sample, only a
value of .DELTA.(.sigma.) of which is desired to be measured. In
such a case, by obtaining only M.sub.23(.sigma.), two equations of
M.sub.2(.sigma.) and M.sub.3(.sigma.) are obtained. Solving these
equations can lead to calculation of the ellipsometric parameter
.DELTA.(.sigma.) as follows. [Mathematical Expression 44] .DELTA.
.times. .times. ( .sigma. ) = tan - 1 .times. { M 3 .function. (
.sigma. ) M 2 .function. ( .sigma. ) } ( 4.4 ) ##EQU41##
[0321] As described before, M.sub.0(.sigma.) is obtained from the
first spectral intensity, M.sub.1(.sigma.) is from the third
spectral intensity, and M.sub.2(.sigma.) and M.sub.3(.sigma.) are
from at least one of the second spectral intensity, the fourth
spectral intensity, and the fifth spectral intensity. Therefore,
with reference to Expression (4.3), .psi.(.sigma.) which is
obtained from the expressions of M.sub.0(.sigma.) and
M.sub.1(.sigma.) can be obtained from the first spectral intensity
and the third spectral intensity. In this case, even if the
reference phase function is to be calibrated during measurement,
calibration of only .phi..sub.2(.sigma.) sufficiently meets the
need. .psi.(.sigma.) can also be obtained from at least one of the
first spectral intensity and the third spectral intensity and at
least one of the second spectral intensity, the fourth spectral
intensity and the fifth spectral intensity. Since .DELTA.(.sigma.)
can be obtained from M.sub.2(.sigma.) and M.sub.3(.sigma.), at
least one of the second spectral intensity, the fourth spectral
intensity and the fifth spectral intensity can be obtained.
4.2 Case of Spectroscopic Polarimetry Performed by Transmitting
Light Through Sample
[0322] An embodiment of the optical system in the case of measuring
a spectropolarization characteristic of a sample by transmitting
light through the sample is described in detail with reference to
FIGS. 16 to 18. In this case, as shown in FIG. 16 and FIG. 17, the
optical system includes the light source 7, the polarizer P, the
retarder R2, the retarder R1, the analyzer A, and the spectroscope
8. It is to be noted that reference symbol C denotes a sample
through which light is transmitted. Further, light emitted from the
light source 7 is transmitted through the polarizer P, the retarder
R2 and the retarder R1 in this order. The light is then incident on
the sample C in a vertical direction, to be transmitted
therethrough. Thereafter, the light is transmitted through the
analyzer A and then received in the spectroscope 8. Note should be
taken that this device is configured by arranging an optical
element on the light source side with respect to the sample in the
configuration of the conventional optical system as shown in FIG.
1B. Here, the orientation of the transmission axis of the polarizer
P agrees with the orientation of the fast axis of the retarder R1,
while the orientations of the fast axes of the retarder R1 and the
retarder R2 are inclined at an angle of 45.degree. from each other.
.theta. shows the azimuth angle of the transmission axis of the
analyzer A with respect to the fast axis of the retarder R1.
[0323] Moreover, according to the optical system as shown in FIG.
16 and FIG. 17, since the incident direction of the wave surface of
light that transmits through the retarders R1 and R2 is not
susceptible to the inclination characteristic (cf. FIG. 18A) and
the surface state (cf. FIG. 18B) of the sample, a stable
measurement can be realized without restrictions of being unable to
measure the surface shape of the sample and a disturbing substance
(living body, etc.). Namely, it is possible to solve the problem of
the retardation change between the time of calibrating the
retardation of the retarder and the time of measuring the sample
due to changes in distance and direction of a light ray that passes
through the retarder.
[0324] Below described is a procedure for obtaining an azimuth
angle R of a birefringent axis of a sample (birefringent medium)
with respect to the fast axis of the retarder R1, as well as a
retardation .delta.(.sigma.), from light incident on the
spectroscope 8.
[0325] When the azimuth angle of the birefringent axis of the
sample (birefringent medium) with respect to the fast axis of the
retarder R1 is R, and the retardation of the sample (birefringent
medium) is .delta.(.sigma.), a Mueller matrix for expressing the
sample (birefringent medium) is described as follows. [Mathematical
Expression 45] M .function. ( .sigma. ) = [ 1 0 0 0 0 cos 2 .times.
2 .times. R + .times. cos .times. .times. .delta. .function. (
.sigma. ) .times. sin 2 .times. 2 .times. R cos .times. .times. 2
.times. R .times. .times. sin .times. .times. 2 .times. R
.function. ( 1 - cos .times. .times. .delta. .function. ( .sigma. )
) - sin .times. .times. .delta. .function. ( .sigma. ) .times. sin
.times. .times. 2 .times. R 0 cos .times. .times. 2 .times. R
.times. .times. sin .times. .times. 2 .times. R .function. ( 1 -
cos .times. .times. .delta. .function. ( .sigma. ) ) sin .times. 2
.times. 2 .times. R + cos .times. .times. .delta. .function. (
.sigma. ) .times. cos .times. 2 .times. 2 .times. R cos .times.
.times. 2 .times. R .times. .times. sin .times. .times. .delta.
.function. ( .sigma. ) 0 sin .times. .times. .delta. .function. (
.sigma. ) .times. sin .times. .times. 2 .times. R - cos .times.
.times. 2 .times. R .times. .times. sin .times. .times. .delta.
.function. ( .sigma. ) cos .times. .times. .delta. .function. (
.sigma. ) ] ( 5.1 ) ##EQU42## Here, the following expressions are
satisfied according to Expressions (1.5a) to (1.5d). [Mathematical
Expression 46] M 0 .function. ( .sigma. ) = 1 2 .times. P 0
.function. ( .sigma. ) ( 5.2 .times. a ) M 1 .function. ( .sigma. )
= .times. 1 2 .times. P 0 .function. ( .sigma. ) .times. { cos
.times. .times. 2 .times. R .times. .times. cos .times. ( 2 .times.
R - 2 .times. .times. .theta. ) + .times. cos .times. .times.
.delta. .times. ( .sigma. ) .times. sin .times. .times. 2 .times. R
.times. .times. sin .function. ( 2 .times. R - 2 .times. .times.
.theta. ) } ( 5.2 .times. b ) M 2 .function. ( .sigma. ) = .times.
1 2 .times. P 0 .function. ( .sigma. ) .times. { sin .times.
.times. 2 .times. R .times. .times. cos .function. ( 2 .times. R -
2 .times. .times. .theta. ) - .times. cos .times. .times. .delta.
.function. ( .sigma. ) .times. cos .times. .times. 2 .times. R
.times. .times. sin .function. ( 2 .times. R - 2 .times. .times.
.theta. ) } ( 5.2 .times. c ) M 3 .function. ( .sigma. ) = - 1 2
.times. P 0 .function. ( .sigma. ) .times. .times. sin .times.
.times. .delta. .function. ( .sigma. ) .times. .times. sin .times.
.times. ( 2 .times. R - 2 .times. .times. .theta. ) ( 5.2 .times. d
) ##EQU43## Here, assuming that 0=45.degree., the following
expressions are satisfied. [Mathematical Expression 47] M 0
.function. ( .sigma. ) = 1 2 .times. P 0 .function. ( .sigma. ) (
5.3 .times. a ) M 1 .function. ( .sigma. ) = 1 2 .times. P 0
.function. ( .sigma. ) .times. cos .times. .times. 2 .times. R
.times. .times. sin .times. .times. 2 .times. R .function. ( 1 -
cos .times. .times. .delta. .times. .times. ( .sigma. ) ) ( 5.3
.times. b ) M 2 .function. ( .sigma. ) = 1 2 .times. P 0 .function.
( .sigma. ) .times. ( sin 2 .times. 2 .times. R + cos .times.
.times. .delta. .function. ( .sigma. ) .times. cos 2 .times. 2
.times. R ) ( 5.3 .times. c ) M 3 .function. ( .sigma. ) = 1 2
.times. P 0 .function. ( .sigma. ) .times. cos .times. .times. 2
.times. R .times. .times. sin .times. .times. .delta. .function. (
.sigma. ) ( 5.3 .times. d ) ##EQU44##
[0326] Since M.sub.0(.sigma.) to M.sub.3(.sigma.) which are
obtained by demodulation processing include three unknowns: the
spectrum P.sub.0(.sigma.) of the light ray, the azimuth angle R of
the birefringent axis, and the retardation .delta.(.sigma.) of the
sample (birefringent medium), it is possible to determine the
azimuth angle R of the birefringent axis and the retardation
.delta.(.sigma.) of the sample (birefringent medium).
[0327] Moreover, there exists a case where what needs to be
obtained may be either the azimuth angle R of the birefringent axis
or the retardation .delta.(.sigma.) of the sample (birefringent
medium). For example, in the cases of a liquid crystal, a polymer
film, and the like, the orientation of the birefringent axis can be
determined when the azimuth angle thereof is obtained. Further,
since the azimuth angle R can be obtained only from
M.sub.1(.sigma.) and M.sub.2(.sigma.) while M.sub.2(.sigma.) is
obtained from the second, fourth or fifth spectral intensity,
M.sub.3(.sigma.) can be obtained concurrently with
M.sub.2(.sigma.). Further, if M.sub.2(.sigma.) and M.sub.3(.sigma.)
are known, M.sub.0(.sigma.) (obtained from the first spectral
intensity) may be used in place of M.sub.1(.sigma.) (obtained from
the third spectral intensity). Eventually, the necessary spectral
intensities are either the first or third spectral intensity and
any one of the second, fourth and fifth spectral intensities. This
also applies to necessary spectral intensities for obtaining
.delta.(.sigma.).
4.3 Pre-Calibration
[0328] Next, pre-calibration of the device configuration which was
described in Sections 4.1, 4.2 and the like is described with
reference to FIG. 14 and FIG. 15.
[0329] FIG. 14 shows a device configuration necessary for
pre-calibration. This device is comprised of the light source 7,
the spectroscope 8, a unit 9 for pre-calibration, a unit 12 for
measurement, an optical fiber 10 for use in pre-calibration, and an
optical fiber 11 for use in measurement. The unit 12 for
measurement includes the polarizer P, the retarder R2, the retarder
R1, and an analyzer A2 for measurement. The unit 9 for
pre-calibration includes an analyzer A1 for pre-calibration. It is
to be noted that the analyzer A1 for pre-calibration is set to have
a known polarization angle.
[0330] According to this device, at the time of pre-calibration,
light is emitted from the light source 7, and transmitted through
the polarizer P, the retarder R2 and the retarder R1 in this order.
The light is then transmitted through the analyzer A1 for
pre-calibration included in the unit 9 for pre-calibration, to be
incident on the spectroscope 8 through the optical fiber 10. On the
other hand, when an object to be measured such as a sample exists,
a measurement is performed by the methods described in Sections 4.1
and 4.2, using the unit 12 for measurement.
[0331] What is of importance here is that there is no need for
using the same one analyzer "for pre-calibration" and `for
measurement`. This is because the analyzer has the (disappearing)
characteristic of being susceptible to fluctuations due to
variations in incident angle of a transmitted light ray, as above
described.
[0332] Therefore, the unit 9 for pre-calibration (light-reception
part) including the analyzer A1 for pre-calibration set to have a
known polarization angle becomes movable to place where calibration
is readily performed. Thereby, the advantage of being able to
perform precalibration in a place which is not a measurement place
is obtained. Further, the advantages of time reduction and the like
are also obtained simultaneously.
[0333] Further, as for the pre-calibration, a problem with the
channeled spectroscopic polarimetry described in "Spectroscopic
ellipsometry using channeled spectrum", written by Kazuhiko Oka and
Takayuki Katoh, collected papers of lectures in 26th Study Session
on Light Wave Sensing Technology (Light wave Sensing Technology
Study Session held by Japan Society of Applied Physics, Dec. 19-20,
2000) pp. 107-114 is shown in FIG. 15. In FIG. 15, light is
incident from the lower left of the figure. The light is
transmitted through the polarizer P, reflected on the sample B, and
then transmitted through the retarder R.
[0334] In this calibration, it is necessary to apply known linearly
polarized light to the channeled spectroscopic polarimeter at the
time of calibration. Namely, it is necessary to adjust an SOP of
light to be incident on the sample so that the light has the SOP of
the known linearly polarized light after reflection on the sample.
This has required changing the light to be incident on the sample
to the p-polarized light (light having an SOP where the polarizing
direction runs parallel with the incident plane) and an s-polarized
light (light having an SOP where the polarizing direction runs
vertical to the incident plane), limited to the time of
calibration. (Light other than the p-polarized light and the
s-polarized light undesirably becomes elliptically polarized light
after reflection on the sample. Since the SOP of the elliptically
polarized light depends upon a refraction index of the sample,
surface roughness of the sample, or the like, treating the
elliptically polarized light as calibration light might cause
generation of a measurement error, which is inconvenient in
calibration.) Further, it has been necessary to adjust the
polarizer P again so as to have a known rotation angle other than
the p-polarized light and the s-polarized light at the time of
measurement. This has required a system for adjusting the rotation
angle of the polarizer, such as a stage, preventing size reduction
in the channeled spectropolarimetric unit.
[0335] However, according to the calibration method of the present
embodiment, pre-calibration is not accompanied by reflection on the
sample as apparent from FIG. 14, thereby giving the advantage of
eliminating the need for arranging a polarization angle adjustment
system on the light-projection side (retarder side), to permit size
reduction in light-projection side unit.
4.4 Case where Object to be Measured Includes Known Polarization
Element Other Than Sample
[0336] There are some cases where the object to be measured
includes a polarization element (e.g. quarter wave plate) having a
known polarization characteristic, and FIG. 19 shows a device
configuration of an optical system in such cases. In this figure,
the optical system includes the light source 7, the polarizer P,
the retarder R2, the retarder R1, the analyzer A and the
spectroscope 8. The object to be measured includes the sample D and
a known polarization element E. Light emitted from the light source
7 is transmitted through the polarizer P, the retarder R2 and the
retarder R1 in this order. The light is then reflected on or
transmitted through the object to be measured, and transmitted
through the analyzer, to be incident on the spectroscope 8.
[0337] In the arithmetic processing for obtaining a spectroscopic
quasi-Stokes parameter of the sample in this case, the sample and
the known sample are regarded and measured as one object to be
measured, and the effect of the known polarization element may be
eliminated from the obtained equation (spectroscopic quasi-Stokes
parameter). In the following, as an example, a case is considered
where a quarter wave plate is arranged in a position after the
sample such that the slow axis thereof runs parallel with the fast
axis of the first retarder. (In FIG. 19, the known polarization
element E is regarded as the quarter wave plate.)
[0338] The sample and quarter wave plate are regarded as one object
to be measured and the Mueller matrix thereof is described as
follows. [Mathematical Expression 48] M ' .function. ( .sigma. ) =
( m ^ 00 ' .function. ( .sigma. ) m ^ 10 ' .function. ( .sigma. ) m
^ 20 ' .function. ( .sigma. ) .times. m ^ 01 ' .function. ( .sigma.
) m ^ 11 ' .function. ( .sigma. ) m ^ 21 ' .function. ( .sigma. )
.times. m ^ 02 ' .function. ( .sigma. ) m ^ 12 ' .function. (
.sigma. ) m ^ 22 ' .function. ( .sigma. ) .times. m ^ 03 '
.function. ( .sigma. ) m ^ 13 ' .function. ( .sigma. ) m ^ 23 '
.function. ( .sigma. ) m ^ 30 ' .function. ( .sigma. ) m ^ 31 '
.function. ( .sigma. ) m ^ 32 ' .function. ( .sigma. ) m ^ 33 '
.function. ( .sigma. ) ) ( 6.1 ) ##EQU45##
[0339] Spectroscopic quasi-Stokes parameters are obtained which are
described using Mueller matrix elements in the frame of Expression
(6.1). Here, it is assumed that a Mueller matrix of the sample
essentially required to be known is as follows. [Mathematical
Expression 49] M .function. ( .sigma. ) = ( m ^ 00 .function. (
.sigma. ) m ^ 01 .function. ( .sigma. ) m ^ 02 .function. ( .sigma.
) m ^ 03 .function. ( .sigma. ) m ^ 10 .function. ( .sigma. ) m ^
11 .function. ( .sigma. ) m ^ 12 .function. ( .sigma. ) m ^ 13
.function. ( .sigma. ) m ^ 20 .function. ( .sigma. ) m ^ 21
.function. ( .sigma. ) m ^ 22 .function. ( .sigma. ) m ^ 23
.function. ( .sigma. ) m ^ 30 .function. ( .sigma. ) m ^ 31
.function. ( .sigma. ) m ^ 32 .function. ( .sigma. ) m ^ 33
.function. ( .sigma. ) ) ( 6.2 ) ##EQU46## Then, the relation
between this Mueller matrix and a Mueller matrix M'(.sigma.) of the
object to be measured is as follows. [Mathematical Expression 50] M
.function. ( .sigma. ) = ( m ^ 00 .function. ( .sigma. ) m ^ 01
.function. ( .sigma. ) m ^ 02 .function. ( .sigma. ) m ^ 03
.function. ( .sigma. ) m ^ 10 .function. ( .sigma. ) m ^ 11
.function. ( .sigma. ) m ^ 12 .function. ( .sigma. ) m ^ 13
.function. ( .sigma. ) m ^ 20 .function. ( .sigma. ) m ^ 21
.function. ( .sigma. ) m ^ 22 .function. ( .sigma. ) m ^ 23
.function. ( .sigma. ) m ^ 30 .function. ( .sigma. ) m ^ 31
.function. ( .sigma. ) m ^ 32 .function. ( .sigma. ) m ^ 33
.function. ( .sigma. ) ) = ( m ^ 00 ' .times. ( .sigma. ) m ^ 01 '
.function. ( .sigma. ) m ^ 02 ' .function. ( .sigma. ) m ^ 03 '
.function. ( .sigma. ) m ^ 10 ' .function. ( .sigma. ) m ^ 11 '
.function. ( .sigma. ) m ^ 12 ' .function. ( .sigma. ) m ^ 13 '
.function. ( .sigma. ) - m ^ 30 ' .function. ( .sigma. ) - m ^ 31 '
.times. ( .sigma. ) - m ^ 32 ' .times. ( .sigma. ) - m ^ 33 '
.times. ( .sigma. ) m ^ 20 ' .function. ( .sigma. ) m ^ 21 '
.function. ( .sigma. ) m ^ 22 ' .function. ( .sigma. ) m ^ 23 '
.function. ( .sigma. ) ) ( 6.3 ) ##EQU47## This means that the
spectroscopic quasi-tokes parameters obtained using this
polarimeter are determined by the elements on the first row, the
second row and the fourth row of Expression (6.2) of the Mueller
matrix of the sample which is essentially required to be known.
Namely, the elements associated with the spectroscopic quasi-Stokes
parameters are different from those in the case without the quarter
wave plate. However, the number of parameters obtained is four,
which is the same as in the case without the quarter wave plate. It
should thus be noted that some of the spectropolarization
parameters of the sample can be calculated also in the case with
the quarter wave plate. As apparent form the above, the use of the
quarter wave plate or the like permits for changing the relational
expression of the Mueller matrix of the sample and the
spectroscopic quasi-Stokes parameters to be measured. Actively
using this fact can for example lead to enhancement of measurement
sensitivity with respect to the specific spectropolarization
parameter. As an example of such use, the case of calculating the
retardation of the sample shown in Section 4.2 is considered. In
the case without the quarter wave plate, when the azimuth angle R
almost agrees with the azimuth angle .theta. of the analyzer,
sin(2R-2.theta.) in Expressions (5.2a) to (5.2d) become a value
close to zero, the spectroscopic quasi-Stokes parameter has almost
no sensitivity with respect to the retardation .delta.. This
results in degradation in measurement accuracy of .delta.. On the
other hand, in the case with the quarter wave plate, the
spectroscopic quasi-Stokes parameters are obtained from the first
row, the second row and the fourth row in Expression (5.1) of the
Mueller matrix of the sample. [Mathematical Expression 51] M 0
.function. ( .sigma. ) = 1 2 .times. P 0 .function. ( .sigma. )
##EQU48## M 1 .function. ( .sigma. ) = .times. 1 2 .times. P 0
.function. ( .sigma. ) .times. { ( cos 2 .times. .times. 2 .times.
R .times. + cos .times. .times. .delta. .function. ( .sigma. )
.times. sin 2 .times. 2 .times. .times. R ) .times. cos .times.
.times. 2 .times. .times. .theta. - .times. sin .times. .times.
.delta. .function. ( .sigma. ) .times. sin .times. .times. 2
.times. R .times. .times. sin .times. .times. 2 .times. .times.
.theta. } ##EQU48.2## M 2 .function. ( .sigma. ) = .times. 1 2
.times. P 0 .function. ( .sigma. ) .times. { cos .times. .times. 2
.times. R .times. .times. sin .times. .times. 2 .times. R .times.
.times. ( 1 - cos .times. .times. .delta. .function. ( .sigma. )
.times. cos .times. .times. 2 .times. .times. .theta. - .times. sin
.times. .times. .delta. .function. ( .sigma. ) .times. cos .times.
.times. 2 .times. R .times. .times. sin .times. .times. 2 .times.
.times. .theta. } ##EQU48.3## M 3 .function. ( .sigma. ) = 1 2
.times. P 0 .function. ( .sigma. ) .times. { sin .times. .times.
.delta. .function. ( .sigma. ) .times. sin .times. .times. 2
.times. R .times. .times. cos .times. .times. 2 .times. .times.
.theta. + cos .times. .times. .delta. .times. .times. ( .sigma. )
.times. sin .times. .times. 2 .times. .times. .theta. } ##EQU48.4##
Hence the sensitivity with respect to the retardation .delta. can
be obtained regardless of the azimuth angle R. It is further found
that the sensitivity becomes constant especially when
.theta.=45.degree..
5. Demodulation of Analyzer and Known Polarization Element and
Expansion of Number of Measurable Spectropolarization Parameters by
Modulation
[0340] As described in the previous chapters, the use of
measurement principle of the present invention allows concurrent
and independent measurement of four spectroscopic quasi-tokes
parameters in one spectrum measurement. This links to the
characteristic of being capable of concurrently determining a
plurality of spectropolarization parameters regarding an object to
be measured (i.e. sample included therein).
[0341] However, there are some cases where measurement is not
sufficiently performed due to insufficient information obtained
only from the above-mentioned four spectroscopic quasi-Stokes
parameters, depending upon properties of the sample. Examples of
such a case include a case where the polarization parameter
necessary to concurrently measure exceeds four.
[0342] As shown in Expression (1.1), the Mueller matrix for
expressing the SOP of the sample has sixteen elements, and
depending upon the sample, each of those sixteen element may be a
different value. For example, in the case of measuring the
spectropolarization characteristic of the sample by reflecting
light on the sample, the condition of the sample was being an
isotropic medium in Expression (4.1), but when the sample is an
anisotropic medium, the sixteen elements are expressed by up to
seven independent parameter equations. This was demonstrated by G.
E. Jellison, Jr, ("Handbook of ellipsometry", edited by H. G
Thompkins and E. A. Irene, William Andrew Publishing, P. 244).
Further, there is even a case where, when the sample is a
heterogeneous medium, the sixteen elements are all independent
parameters in measurement of transmitted light or reflected light
on or through the sample.
[0343] In the present chapter described is expansion of the
principle of the present invention for the above-mentioned case
where a large number of parameters are required to be obtained. It
is to be noted that, although applying this expansion technique
causes loss of the characteristic of the channeled spectroscopic
polarimetry of "needing no mechanical or active polarization
control element", another advantage of "needing to measure a
necessary spectrum only an extremely few times" is generated which
is not included in the characteristics of the corresponding
conventional method.
5.1 Relation Between Spectroscopic Quasi-Stokes Parameter and
Mueller Matrix of Sample
[0344] Prior to description on a method for expanding the
principle, as preparation for performing the method, a relational
expression between the spectroscopic quasi-Stokes parameters and
Mueller matrix of the sample is derived. A case is considered here
where the object to be measured is composed of the sample D and the
known polarization element E, as in FIG. 19. Mueller matrixes of
the sample D and the known polarization element E are described as
follows. [Mathematical Expression 52] M sample .function. ( .sigma.
) = [ m ^ s .times. .times. 00 .function. ( .sigma. ) m ^ s .times.
.times. 01 .function. ( .sigma. ) m ^ s .times. .times. 02
.function. ( .sigma. ) m ^ s .times. .times. 03 .function. (
.sigma. ) m ^ s .times. .times. 10 .function. ( .sigma. ) m ^ s
.times. .times. 11 .function. ( .sigma. ) m ^ s .times. .times. 12
.function. ( .sigma. ) m ^ s .times. .times. 13 .function. (
.sigma. ) m ^ s .times. .times. 20 .function. ( .sigma. ) m ^ s
.times. .times. 21 .function. ( .sigma. ) m ^ s .times. .times. 22
.function. ( .sigma. ) m ^ s .times. .times. 23 .function. (
.sigma. ) m ^ s .times. .times. 30 .function. ( .sigma. ) m ^ s
.times. .times. 31 .function. ( .sigma. ) m ^ s .times. .times. 32
.function. ( .sigma. ) m ^ s .times. .times. 33 .function. (
.sigma. ) ] ( 7.1 .times. a ) M pc .function. ( .sigma. ) = [ m ^ p
.times. .times. 00 .function. ( .sigma. ) m ^ p .times. .times. 01
.function. ( .sigma. ) m ^ p .times. .times. 02 .function. (
.sigma. ) m ^ p .times. .times. 03 .function. ( .sigma. ) m ^ p
.times. .times. 10 .function. ( .sigma. ) m ^ p .times. .times. 11
.function. ( .sigma. ) m ^ p .times. .times. 12 .function. (
.sigma. ) m ^ p .times. .times. 13 .function. ( .sigma. ) m ^ p
.times. .times. 20 .function. ( .sigma. ) m ^ p .times. .times. 21
.function. ( .sigma. ) m ^ p .times. .times. 22 .function. (
.sigma. ) m ^ p .times. .times. 23 .function. ( .sigma. ) m ^ p
.times. .times. 30 .function. ( .sigma. ) m ^ p .times. .times. 31
.function. ( .sigma. ) m ^ p .times. .times. 32 .function. (
.sigma. ) m ^ p .times. .times. 33 .function. ( .sigma. ) ] ( 7.1
.times. b ) ##EQU49## The Mueller matrix of the object to be
measured is given by the product of the above two Mueller matrixes,
i.e. M(.sigma.)=M.sub.pc(u)M.sub.sample(.sigma.) (7.2) Expression
(7.2) is written as follows using the element of each matrix. m ^
ij .function. ( .sigma. ) = k = 0 3 .times. m ^ pik .function. (
.sigma. ) .times. m ^ skj .function. ( .sigma. ) ( 7.3 ) ##EQU50##
where i and j are integers from 0 to 3.
[0345] When Expression (7.3) is substituted into Expressions (1.6a)
to (1.6d), the following expression is derived for relating the
spectroscopic quasi-Stokes parameter M.sub.1(.sigma.) of the object
to be measured (where I=0 . . . 3) to the Mueller matrix of the
sample. [Mathematical Expression 53] M l .function. ( .sigma. ) = 1
2 .times. P 0 .function. ( .sigma. ) [ k = 0 3 .times. m ^ p0k
.function. ( .sigma. ) .times. m ^ skl .function. ( .sigma. ) + k =
0 3 .times. m ^ plk ( .sigma. .times. | .times. m ^ skl .function.
( .sigma. ) .times. cos .times. .times. 2 .times. .times. .theta. +
k = 0 3 .times. m ^ p2k .function. ( .sigma. ) .times. m ^ skl
.function. ( .sigma. ) .times. sin .times. .times. 2 .times.
.times. .theta. ] = P 0 .function. ( .sigma. ) .times. k = 0 3
.times. a ^ k .function. ( .sigma. ) .times. m ^ skl .function. (
.sigma. ) ( 7.4 ) ##EQU51## where a ^ k .function. ( .sigma. ) = 1
2 .times. m ^ p0k .function. ( .sigma. ) + 1 2 .times. m ^ plk
.function. ( .sigma. ) .times. cos .times. .times. 2 .times.
.times. .theta. + 1 2 .times. m ^ p2k .function. ( .sigma. )
.times. sin .times. .times. 2 .times. .times. .theta. ( 7.5 )
##EQU52## The next relational expression using vectors and columns
is immediately derived from Expression (7.4). [ M 0 .function. (
.sigma. ) M 1 .function. ( .sigma. ) M 2 .function. ( .sigma. ) M 3
.function. ( .sigma. ) ] = P 0 .function. ( .sigma. ) .function. [
a ^ 0 .function. ( .sigma. ) a ^ 1 .function. ( .sigma. ) a ^ 2
.function. ( .sigma. ) a ^ 3 .function. ( .sigma. ) ] [ m ^ s
.times. .times. 00 .function. ( .sigma. ) m ^ s .times. .times. 01
.function. ( .sigma. ) m ^ s .times. .times. 02 .function. (
.sigma. ) m ^ s .times. .times. 03 .function. ( .sigma. ) m ^ s
.times. .times. 10 .function. ( .sigma. ) m ^ s .times. .times. 11
.function. ( .sigma. ) m ^ s .times. .times. 12 .function. (
.sigma. ) m ^ s .times. .times. 13 .function. ( .sigma. ) m ^ s
.times. .times. 20 .function. ( .sigma. ) m ^ s .times. .times. 21
.function. ( .sigma. ) m ^ s .times. .times. 22 .function. (
.sigma. ) m ^ s .times. .times. 23 .function. ( .sigma. ) m ^ s
.times. .times. 30 .function. ( .sigma. ) m ^ s .times. .times. 31
.function. ( .sigma. ) m ^ s .times. .times. 32 .function. (
.sigma. ) m ^ s .times. .times. 33 .function. ( .sigma. ) ] = P 0
.function. ( .sigma. ) .function. [ a ^ 0 .function. ( .sigma. ) a
^ 1 .function. ( .sigma. ) a ^ 2 .function. ( .sigma. ) a ^ 3
.function. ( .sigma. ) ] .times. M sample .function. ( .sigma. ) (
76 ) ##EQU53## The above expression relates the four spectroscopic
quasi-Stokes parameters M.sub.0(.sigma.), M.sub.1(.sigma.),
M.sub.2(.sigma.), and M.sub.3(.sigma.) directly to the Mueller
matrix M.sub.sample(.sigma.) of the sample. Here, the element
a.sub.k(.sigma.)(k=0 . . . 3) of the vector for relating the
spectroscopic quasi-Stokes parameters to the Mueller matrix is an
amount determined only by the characteristic of the known
polarization element E and the azimuth angle .theta. of the
analyzer A (azimuth angle of the transmission axis of the analyzer
A with respect to the fast axis of the retarder R1), as apparent
from the definition of Expression (7.5), and does not depend upon
the "Mueller matrix of the sample". It is to be noted that the
above argument can be applied as it is to the case without the
known polarization element E after the sample D in FIG. 19 if
Expression (7.1) is replaced by a 4.times.4 unit matrix.
5.2 (Method for Increasing Number of Measurable Parameters (and
Method for Completely Measuring Mueller Matrix)
[0346] As described in the previous chapters, it is possible to
concurrently and independently measure the spectroscopic
quasi-Stokes parameters M.sub.0(.sigma.), M.sub.1(.sigma.),
M.sub.2(.sigma.), and M.sub.3(.sigma.) by the method of the present
invention. By forming an equation based upon Expression (7.6) using
the obtained spectroscopic quasi-Stokes parameters, it is possible
to obtain up to four spectroscopic quasi-Stokes parameters
regarding the sample, as shown in the example of Chapter 4.
[{circumflex over (.alpha.)}.sub.0(.sigma.) {circumflex over
(.alpha.)}.sub.1(.sigma.) {circumflex over
(.alpha.)}.sub.2(.sigma.) {circumflex over
(.alpha.)}.sub.3(.sigma.)] [Mathematical Expression 54]
[0347] However, there could be cases where a large number of
spectropolarization parameters need to be obtained, or the above
equation is difficult to solve. In such cases, the number of
equations can be increased by changing a vector for giving a
coefficient in several times and repeating measurement at each of
the changes. Since this coefficient vector depends only upon the
characteristic of the known polarization element E and the azimuth
angle .theta. of the analyzer A, controlling either one of them
allows changing of the vector. With the number of equations
increased, it is possible to increase the number of independent
spectropolarization parameters which are concurrently obtained.
[0348] Here, especially the case of performing measurements under
four different conditions regarding the known polarization element
E and the analyzer A is considered. When spectroscopic quasi-Stokes
parameters obtained in the respective cases are differentiated by
means of superscript characters (p)=0 . . . 3, the following
relational expressions of matrixes are established, as seen from
Expression (7.6). [Mathematical Expression 55] [ M 0 ( 0 )
.function. ( .sigma. ) M 1 ( 0 ) .function. ( .sigma. ) M 2 ( 0 )
.function. ( .sigma. ) M 3 ( 0 ) .function. ( .sigma. ) M 0 ( 1 )
.function. ( .sigma. ) M 1 ( 1 ) .function. ( .sigma. ) M 2 ( 1 )
.function. ( .sigma. ) M 3 ( 1 ) .function. ( .sigma. ) M 0 ( 2 )
.function. ( .sigma. ) M 1 ( 2 ) .function. ( .sigma. ) M 2 ( 2 )
.function. ( .sigma. ) M 3 ( 2 ) .function. ( .sigma. ) M 0 ( 3 )
.function. ( .sigma. ) M 1 ( 3 ) .function. ( .sigma. ) M 2 ( 3 )
.function. ( .sigma. ) M 3 ( 3 ) .function. ( .sigma. ) ] = P 0
.function. ( .sigma. ) .function. [ a ^ 0 ( 0 ) .function. (
.sigma. ) a ^ 1 ( 0 ) .function. ( .sigma. ) a ^ 2 ( 0 ) .function.
( .sigma. ) a ^ 3 ( 0 ) .function. ( .sigma. ) a ^ 0 ( 1 )
.function. ( .sigma. ) a ^ 1 ( 1 ) .function. ( .sigma. ) a ^ 2 ( 1
) .function. ( .sigma. ) a ^ 3 ( 1 ) .function. ( .sigma. ) a ^ 0 (
2 ) .function. ( .sigma. ) a ^ 1 ( 2 ) .function. ( .sigma. ) a ^ 2
( 2 ) .function. ( .sigma. ) a ^ 3 ( 2 ) .function. ( .sigma. ) a ^
0 ( 3 ) .function. ( .sigma. ) a ^ 1 ( 3 ) .function. ( .sigma. ) a
^ 2 ( 3 ) .function. ( .sigma. ) a ^ 3 ( 3 ) .function. ( .sigma. )
] .times. M sample .function. ( .sigma. ) . ( 7.7 ) ##EQU54## In
this expression, the first matrix on the right side above is: N
.function. ( .sigma. ) = [ a ^ 0 ( 0 ) .function. ( .sigma. ) a ^ 1
( 0 ) .function. ( .sigma. ) a ^ 2 ( 0 ) .function. ( .sigma. ) a ^
3 ( 0 ) .function. ( .sigma. ) a ^ 0 ( 1 ) .function. ( .sigma. ) a
^ 1 ( 1 ) .function. ( .sigma. ) a ^ 2 ( 1 ) .function. ( .sigma. )
a ^ 3 ( 1 ) .function. ( .sigma. ) a ^ 0 ( 2 ) .function. ( .sigma.
) a ^ 1 ( 2 ) .function. ( .sigma. ) a ^ 2 ( 2 ) .function. (
.sigma. ) a ^ 3 ( 2 ) .function. ( .sigma. ) a ^ 0 ( 3 ) .function.
( .sigma. ) a ^ 1 ( 3 ) .function. ( .sigma. ) a ^ 2 ( 3 )
.function. ( .sigma. ) a ^ 3 ( 3 ) .function. ( .sigma. ) ] ( 7.8 )
##EQU55## When this matrix has an inverse matrix N.sup.-1(.sigma.),
it becomes possible to inverse transform Expression (7.7), so as to
obtain the Mueller matrix M.sub.sample(.sigma.) as: M sample
.function. ( .sigma. ) = 1 P 0 .function. ( .sigma. ) .times. N - 1
.function. ( .sigma. ) .function. [ M 0 ( 0 ) .function. ( .sigma.
) M 1 ( 0 ) .function. ( .sigma. ) M 2 ( 0 ) .function. ( .sigma. )
M 3 ( 0 ) .function. ( .sigma. ) M 0 ( 1 ) .function. ( .sigma. ) M
1 ( 1 ) .function. ( .sigma. ) M 2 ( 1 ) .function. ( .sigma. ) M 3
( 1 ) .function. ( .sigma. ) M 0 ( 2 ) .function. ( .sigma. ) M 1 (
2 ) .function. ( .sigma. ) M 2 ( 2 ) .function. ( .sigma. ) M 3 ( 2
) .function. ( .sigma. ) M 0 ( 3 ) .function. ( .sigma. ) M 1 ( 3 )
.function. ( .sigma. ) M 2 ( 3 ) .function. ( .sigma. ) M 3 ( 3 )
.function. ( .sigma. ) ] . ( 7.9 ) ##EQU56## This expression means
that, by changing either one or both of the characteristic of the
known polarization element E and the azimuth angle .theta. of the
analyzer A to perform measurements of at least four kinds of
spectroscopic quasi-Stokes parameter, all of the sixteen elements
of the Mueller matrix of the sample can be concurrently and
independently determined. It is however necessary at this time to
control the known polarization element E and the analyzer A so that
N(.sigma.) has an inverse matrix. It is to be noted that, if the
number of conditions for measurement increases more than four, the
number of equations can be increased. If the idea of least-squares
is employed, an error due to an influence of noise or the like can
be reduced. On the contrary, when the number of measurements is
less than four, all the sixteen elements cannot be independently
determined, but still a large number of spectroscopic quasi-Stokes
parameter expressions can be obtained as compared with the case of
not changing the characteristic of the known polarization element E
and the azimuth angle .theta. of the analyzer A, resulting in that
a larger number of spectropolarization parameters of the sample can
be obtained.
[0349] Here, attention should be given to that the present
measurement method has a large advantage as compared with the
conventional method even in the case of moving the known
polarization element E and the analyzer A. Although "mechanical or
active polarization control" is necessary for certain, the number
of necessary measurement steps is significantly different from that
of the conventional method. According to the present method, four
spectroscopic quasi-Stokes parameters are obtained in one
measurement, meaning that in terms of the number of parameters to
be obtained, the number of measurements can be reduced to the order
of a quarter of the number of measurements in the conventional
method. For example, when all of the 16 elements of the Mueller
matrix are to be measured, only four times of measurements are
required at the minimum in the present method, whereas in the
conventional method, the spectrum measurement needs to be repeated
sixteen times at the minimum, and usually 20 to 30 times. This
means that the present method has large advantages in terms of
reduction in measurement time or simplification of the measurement
system.
[0350] In addition, the characteristic of the known polarization
element E and the azimuth angle .theta. of the analyzer A can be
changed by a variety of methods. A first method is changing either
or both of azimuth angles .theta. of the two elements. Such a
change may be made by actually rotating the element, or replacing
the element by an element having a different azimuth angle .theta.,
or inserting a Faraday cell or the like in a position before the
element so as to magneto-optically rotate the azimuth angle .theta.
of the element from the installation orientation. A second method
is introducing a compensator capable of moduclation such as an
electro-optic modulator, a photoelastic modulator, or a
liquid-crystal optical modulator, to change a retardation of one of
parameters for determining a Mueller matrix of this element. A
third method is combining the above-mentioned methods. (It is to be
noted that the above-mentioned methods do not limit the method to
be performed.) Further, the known polarization element E is not
necessarily composed of a single element. For example, the known
polarization element E may be constituted by combination of a
plurality of compensators capable of modulation.
5.3 Example
[0351] The case of rotating a compensator (lower-order retarder or
zero-order retarder) as the known polarization element E is shown.
When the retardation of this compensator is .delta..sub.c(.sigma.),
and the azimuth thereof is .theta..sub.c,
[Mathematical Expression 56]
[0352] a.sub.k(.sigma.)(k=0 . . . 3) are expressed as follows. a ^
0 .function. ( .sigma. ) = 1 2 ( 7.10 .times. a ) a ^ 1 .function.
( .sigma. ) = .times. 1 2 .times. cos .times. .times. 2 .times.
.theta. C .times. cos .times. .times. 2 .times. ( .theta. - .theta.
C ) + .times. 1 2 .times. cos .times. .times. .delta. C .function.
( .sigma. ) .times. sin .times. .times. 2 .times. .times. .theta. C
.times. sin .times. .times. 2 .times. ( .theta. - .theta. C ) (
7.10 .times. b ) a ^ 2 .function. ( .sigma. ) = .times. 1 2 .times.
sin .times. .times. 2 .times. .theta. C .times. cos .times. .times.
2 .times. ( .theta. - .theta. C ) + .times. - 1 2 .times. cos
.times. .times. .delta. C .function. ( .sigma. ) .times. cos
.times. .times. 2 .times. .times. .theta. C .times. sin .times.
.times. 2 .times. ( .theta. - .theta. C ) ( 7.10 .times. c ) a ^ 3
.function. ( .sigma. ) = - 1 2 .times. sin .times. .times. .delta.
C .function. ( .sigma. ) .times. sin .times. .times. 2 .times. (
.theta. - .theta. C ) ( 7.10 .times. d ) ##EQU57## As apparent from
this expression, changing the azimuth angle .theta. of the analyzer
or the azimuth angle .theta..sub.c of the compensator enables
control of the following vectors which give coefficients.
[{circumflex over (.alpha.)}.sub.0(.sigma.) {circumflex over
(.alpha.)}.sub.1(.sigma.) {circumflex over
(.alpha.)}.sub.2(.sigma.) {circumflex over
(.alpha.)}.sub.3(.sigma.)] That is, channeled spectrums may be
repeatedly measured while either one of the azimuth angles of the
two elements is changed.
[0353] For example, when .theta..sub.c is changed to four different
angles: -45.degree., 0.degree., 30.degree., and 60.degree., on the
basis of .delta..sub.c=90.degree. and .theta.=45.degree., a matrix
N(.sigma.) given by Expression (7.8) is expressed as follows.
[Mathematical Expression 57] N = 1 2 .function. [ 1 0 1 0 1 0 0 1 1
3 4 3 4 1 2 1 - 3 4 3 4 - 1 2 ] ( 7.11 ) ##EQU58## Here, the
inverse matrix of this matrix is given as follows. N - 1 = 2
.function. [ - 1 0 1 1 - 4 3 - 2 3 5 3 1 3 2 0 - 1 - 1 1 1 - 1 - 1
] ( 7.12 ) ##EQU59## By substituting spectroscopic quasi-tokes
parameters obtained by four times of measurements and N.sup.-1
mentioned above into Expression (7.9), it is possible to determine
all the sixteen elements of the Mueller matrix of the sample for
each wavenumber.
[0354] It should be noted that in actual measurement, a retardation
.delta..sub.c(.sigma.) of a compensation is a function of a
wavenumber a and is not constant. However, even in such a case, the
measurement remains unsusceptible since Expression (7.9) is
calculated for each wavenumber. Further, according to demonstration
similar to that in the case of "spectroscopic polarization state
measurement by retarder rotating method" shown in a reference
document ("Polarized light" written by D. Goldstein, Mercel Dekker
Inc., p. 555), it is possible to demonstrate that 132.degree. is
the optimum value as the retardation .delta..sub.c(.sigma.). The
closer to this value the retardation is, the less susceptible to
measurement noise the measurement can be.
[0355] In the meantime, the number of equations can be increased
also by rotating the azimuth angle .theta. of the analyzer.
However, it is necessary to note that rotating only the analyzer
prevents the matrix N(.sigma.) from having an inverse matrix due to
the property of the analyzer. If all the sixteen Mueller matrix
elements are to be obtained, it is at least necessary to rotate the
compensator.
[0356] Chapter 6 Channeled Spectroscopic Polarization State
Generator
[0357] As the embodiment of the present invention, it was described
in Chapter 1 that the optical system is comprised of the light
source 7, the polarizer P, the retarders R2 and R1, the analyzer A
and the spectroscope 8, and spectropolarization parameters of the
sample and the like are calculated by analysis of a spectrum of
incident light acquired in the spectroscope 8 in the foregoing
procedure. Meanwhile, when the role of the light-projection part
(the light source 7, the polarizer P, and the retarders R2 and R1)
of the optical system is considered, this part can be defined as a
"spectroscopic polarization state generator" for emitting light
having a modulated SOP. This is especially named a "channeled
spectroscopic polarization state generator (hereinafter referred to
as CSPSG)". In this chapter, the optical implication of this
generator is described.
[0358] FIG. 30 shows the configuration of the channeled
spectroscopic polarization state generator (CSPSG). This optical
system is configured to allow light emitted from the light source 7
to transit through the polarizer P and the retarders R1 and R2. The
constituent elements in this configuration are the same as those
starting with the light source and ending with the component before
the sample in FIG. 2. Further, the azimuth angle of the element is
the same as in FIG. 2. At this time, light emitted from the CSPSG
is light having an SOP modulated along the wavenumber axis. A
stokes spectrum S.sub.PSG(.sigma.) emitted from the CSPSG is
expressed below by calculation using the Mueller matrix.
[Mathematical Expression 58] S PSG .function. ( .sigma. ) = ( S 0
.function. ( .sigma. ) S 1 .function. ( .sigma. ) S 2 .function. (
.sigma. ) S 3 .function. ( .sigma. ) ) = 1 2 .times. P o .function.
( .sigma. ) .times. ( 1 cos .function. ( .PHI. 1 .function. (
.sigma. ) ) sin .times. .times. .PHI. 1 .function. ( .sigma. )
.times. sin .times. .times. .PHI. 2 .function. ( .sigma. ) sin
.times. .times. .PHI. 1 .function. ( .sigma. ) .times. cos .times.
.times. .PHI. 2 .function. ( .sigma. ) ) ( 8.1 ) ##EQU60## Here,
.phi..sub.j(.sigma.) (j=1, 2) is the retardation of the retarder
formulated by Expression (1.2). Assuming that dispersion of a
birefringence B(.sigma.) of the retarder is not so large,
.phi..sub.j(.sigma.) increases almost linearly with respect to wave
number a, as seen from Expression (1.2). By substitution of
Expression (1.2) into Expression (8.1), the following expression is
formed. [Mathematical Expression 59] S PSG .function. ( .sigma. ) =
( S 0 .function. ( .sigma. ) S 1 .function. ( .sigma. ) S 2
.function. ( .sigma. ) S 3 .function. ( .sigma. ) ) = 1 2 .times. P
o .function. ( .sigma. ) .times. ( 1 cos .times. { 2 .times. .pi.
.times. .times. L 1 .times. .sigma. + .PHI. 1 .function. ( .sigma.
) } 1 2 .function. [ cos .times. { 2 .times. .pi. .times. .times. L
- .times. .sigma. + .PHI. - .function. ( .sigma. ) } - cos .times.
{ 2 .times. .pi. .times. .times. L + .times. .sigma. + .PHI. +
.function. ( .sigma. ) } ] 1 2 .function. [ sin .times. { 2 .times.
.pi. .times. .times. L - .times. .sigma. + .PHI. - .function. (
.sigma. ) } - sin .times. { 2 .times. .pi. .times. .times. L +
.times. .sigma. + .PHI. + .function. ( .sigma. ) } ] ) ( 8.2 )
##EQU61## It is found from Expression (8.2) that the light emitted
from the CSPSG is light having been modulated along a wavenumber
axis having three characteristics as follows. (a) S.sub.1(.sigma.)
is modulated in a quasi-sinusoidal manner at a period 1/L.sub.1.
(b) S.sub.2(.sigma.) and S.sub.3(.sigma.) are both composed of two
components modulated in a quasi-sinusoidal manner at a period 1/L
and a period 1/L. In (c) S.sub.2(.sigma.) and S.sub.3(.sigma.), the
quasi-sinusoidal components at the same period has initial phases
90.degree. different from each other. Therefore, the light emitted
from the CSPSG can be considered as light having four spectroscopic
Stokes parameters modulated at a period or phase independently
different from one another. It can thereby be said that this CSPSG
is a complete spectroscopic polarization state generator. The
present invention can be defined as having a configuration for
obtaining a spectropolarization parameter of an object to be
measured, formed by combination, of the light source of the
above-mentioned CSPSG as a complete spectroscopic polarization
state generator with the light source, the analyzer and the
spectrometer.
Example 1
[0359] In the following, a preferred example of the present
invention is specifically described with reference to FIGS. 20 to
23. FIG. 20 shows a configuration view of one example of a
spectroscopic polarimeter. As shown in this figure, this device
comprises a light-projection side unit 200 and a light-reception
side unit 300. It is to be noted that reference numeral 400 denotes
a sample.
[0360] The light-projection side unit 200 comprises: a power source
201; a light source 202 that is turned on by power feeding from the
power source 201; a pinhole plate 203 arranged on the front face
side of the light source 202 in the light emitting direction; a
collimator lens 204 for collimating light transmitting through the
pinhole of the 203; a shutter 205 which is arranged on the front
face side of the collimator lens 204 and opens and closes to
transmit or block the transmitted light; a polarizer 206 on which
the light having transmitted through the shutter is incident; and a
second retarder 207 and a first retarder 208, through which the
light having transmitted through the polarizer transmits in this
order.
[0361] The light after passage of the first retarder 208 is emitted
from the light-projection side unit 200 and applied to the sample
400. The light transmitted through or reflected on the sample 400
is incident on the light-reception side unit 300.
[0362] On an incident light channel in the light-reception side
unit 300, an analyzer 301, and a spectrometer 302 intervene in this
order. Here, a respective angle between the first retarder 208 and
the analyzer 301 is set to be a known angle.
[0363] The spectrometer 302 comprises: a diffraction grating 302a
for spatially dispersing the incident light; a CCD 302b with the
light-reception face on which light spatially dispersed by the
diffraction grating 302a is incident; and an A/D converter 302c for
converting light-reception output from the CCD 302b into a digital
signal. The digital light-reception output signal obtained from the
A/D converter 302c is taken out from the spectrometer 302, and then
processed in a computer 303 such as a personal computer (PC).
[0364] As widely known, the computer 303 comprises: an arithmetic
processing part 303a comprised of a microprocessor and the like; a
memory part 303b comprised of an ROM, an RAM, an HDD and the like;
and a measurement result output part 303c comprised of a display, a
printer, a variety of data output devices, a communication device,
and the like.
[0365] Next, FIG. 21 shows a more specific configuration view
regarding a sensor head part of the spectroscopic polarimeter. A
sensor head part 100 comprises: a light-projection part 110 for
emitting light; a light-reception part 120 for receiving light
having been reflected on or transmitted through a sample; and a
housing 130 for protecting the light-projection part 110 and the
light-reception part 120. It is to be noted that reference numeral
50 denotes a sample.
[0366] The light-projection part 110 includes: a fiber-optic cable
111 for allowing light emitted from a light source (not shown) to
transmit therethrough; a cable head 112 for allowing transmitted
light from the fiber-optic cable 111 to transmit therethrough; a
collimator lens (light-projection lens) 115 for collimating
transmitted light from the cable head 112; a polarizer 116 which is
arranged on the front face side of the collimator lens 115 and
allows incident light to transmit therethrough; a second retarder
117 and a first retarder 118, through which light emitted from the
polarizer transmits in this order; and an optical system holding
member 113 and a fixing member 114 which are used to install the
optical system in the housing 130. It should be noted that a solid
line 119 is a light-projection axis of the light that transmits
through the light-projection part 110.
[0367] The light-reception part 120 includes: an analyzer 122 for
allowing light reflected on or transmitted through the sample 50 to
transmit therethrough; a light-reception lens 123 for condensing
transmitted light from the analyzer 122; a cable head 126 for
allowing light having transmitted through the light-reception lens
123 to transmit therethrough; a fiber-optic cable 127 connected to
a spectrometer (not shown); and a fixing member 124 and an optical
system holding member 125 which are used to install the optical
system in the housing 130. It should be noted that a solid line 121
is a light-reception axis of the light that is reflected on or
transmits through the sample 50.
[0368] Next, FIG. 29 shows a device configuration view in the case
of installing a polarization element having a known polarization
parameter in a position after the sample and measuring the
polarization parameter of the sample. As compared with FIG. 21,
FIG. 29 is configured to install a compensator (known polarization
element) 140 held in a hollow motor 141 between the sample 50 and
the analyzer 122. Here, reference numeral 142 denotes electrical
wiring for motor drive. A spectroscopic quasi-Stokes parameter can
be measured under a plurality of conditions by rotating the hollow
motor 141 to control the azimuth angle of the compensator 140.
Further, the hollow motor 141 is fixed to the housing 130 to form
an integral configuration as an element of the light-reception part
120. The hollow motor 141 is controlled by the arithmetic
processing part 303a in FIG. 20. It is to be noted that in the case
of not rotating the compensator 140 but rotating the analyzer 122,
the hollow motor 141 may be replaced by a fixing hardware of the
compensator 140, to provide a hollow motor for rotating the
analyzer 122. Further, both the compensator 140 and the analyzer
122 may be made individually rotatable.
[0369] Next, FIG. 22 shows a flowchart of a pre-calibration
procedure. As shown in this figure, the precalibration procedure is
started with application of light to the device in Step 2201.
However, in this device, the relative angle between the first
retarder 208 and the analyzer 301 is a known angle, and an element
for changing an SOP of light is not arranged between the first
retarder 208 and the analyzer 301.
[0370] Next, in Step 2202, a spectral intensity of transmitted
light from the analyzer 301 is measured using the spectrometer.
Here, the shutter 205 may be utilized for reduction in influence of
unnecessary light, such as lost light. Specifically, a spectrum of
the unnecessary light can be canceled out by taking a difference in
spectrum between when measured with the shutter open and when
measured with the shutter closed.
[0371] Next, in Step 2203, the spectral intensity of the
transmitted light received is forwarded from the spectrometer to
the computer 303, to be provided to calculation in the arithmetic
processing part 303a.
[0372] Next, in Step 2204, reference phase functions and reference
amplitude functions are calculated by the action of the arithmetic
processing part 303a.
[0373] Next, in Step 2205, the calculated reference phase functions
and reference amplitude functions are stored in the memory part
303b, whereby the pre-calibration procedure is completed.
[0374] FIG. 23 shows a flowchart of a measurement procedure. As
shown in the figure, the measurement procedure is started with
application of light to the device in Step 2301.
[0375] Next, in Step 2302, light is reflected on or transmitted
through the sample 400 using the spectrometer 302, and thereafter,
a spectral intensity of transmitted light having transmitted
through the analyzer 301 is measured. Here, the shutter 205 can be
utilized for reduction in influence of unnecessary light, such as
lost light. Specifically, the spectrum of the unnecessary light can
be canceled out by taking a difference in spectrum between when
measured with the shutter open and when measured with the shutter
closed.
[0376] Next, in Step 2303, the spectral intensity of the
transmitted light is forwarded from the spectrometer 302 to the
computer 303, to be provided to processing in the arithmetic
processing part 303a. At this time, in the case of implementing the
procedure described in Chapter 5, the optical system described by
use of FIG. 29 is used and the azimuth angle of the compensator 140
or the analyzer 122 is changed to acquire a spectral intensity more
than once.
[0377] Next, in Step 2304, in the computer 303, the arithmetic
processing part 303a acquires reference phase functions and
reference amplitude functions from the memory part 303b.
[0378] Next, in Step 2305, in the computer 303, the arithmetic
processing part 303a calculates reference phase function variations
(.DELTA..phi..sub.1 and .DELTA..phi..sub.2) by use of the measured
spectral intensity, the reference phase functions and the reference
amplitude functions.
[0379] Next, in Step 2306, in the computer 303, the arithmetic
processing part 303a calculates spectrometric Stokes parameters by
use of the measured spectral intensity, the reference phase
functions, the reference amplitude functions and the reference
phase function variations.
[0380] Next, in Step 2307, in the computer 303, the arithmetic
processing part 303a outputs the spectrometric Stokes parameters of
the sample 400. Examples of the measurement result output part 303c
may include a memory, a hard disc, and other processing parts
(calculating part for film thickness, complex refractive index,
etc.).
[0381] As described above, in the spectroscopic polarimetry of the
present example, spectrometric Stokes parameters of the sample are
calculated through the pre-calibration procedure shown in FIG. 22
and the measurement procedure shown in FIG. 23 in the system
constitution shown in FIGS. 20, 21 and 29.
* * * * *