U.S. patent application number 11/875273 was filed with the patent office on 2008-09-25 for broadband cavity spectrometer apparatus and method for determining the path length of an optical structure.
This patent application is currently assigned to Board of Regents, The University of Texas System. Invention is credited to Edward Bradstreet Flagg, Chih-Kang Shih, Christopher Frank Wieland.
Application Number | 20080231852 11/875273 |
Document ID | / |
Family ID | 37947860 |
Filed Date | 2008-09-25 |
United States Patent
Application |
20080231852 |
Kind Code |
A1 |
Shih; Chih-Kang ; et
al. |
September 25, 2008 |
BROADBAND CAVITY SPECTROMETER APPARATUS AND METHOD FOR DETERMINING
THE PATH LENGTH OF AN OPTICAL STRUCTURE
Abstract
A broadband light source with a sufficiently long coherence
length is impinged on the optical cavity. The broadband laser light
reflects from the first and second surfaces of the cavity. The two
light beams, either reflected or transmitted, are phase shifted
from one another by an amount proportional to the optical path
length of the cavity and inversely proportional to the wavelength
of the light (4.pi.nd/.lamda.) The two light beams interfere with
each other and form a modulated light beam that has a spectrum
approximately like the laser's broadband spectrum multiplied by a
cosine with a frequency 4.pi.nd/.lamda.. The modulated light beam
is coupled to a spectrometer that measures the intensity of the
light as a function of wavelength over a range of wavelengths. The
Fourier transform of the spectrum contains a peak that is related
to the OPL and is located at 2*n*d where n*d is the OPL.
Inventors: |
Shih; Chih-Kang; (Austin,
TX) ; Wieland; Christopher Frank; (Austin, TX)
; Flagg; Edward Bradstreet; (Austin, TX) |
Correspondence
Address: |
SCHWABE, WILLIAMSON & WYATT, P.C.;PACWEST CENTER, SUITE 1900
1211 SW FIFTH AVENUE
PORTLAND
OR
97204
US
|
Assignee: |
Board of Regents, The University of
Texas System
Austin
TX
|
Family ID: |
37947860 |
Appl. No.: |
11/875273 |
Filed: |
October 19, 2007 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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11250954 |
Oct 14, 2005 |
7289220 |
|
|
11875273 |
|
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Current U.S.
Class: |
356/326 |
Current CPC
Class: |
G01B 9/02057 20130101;
G01B 9/02044 20130101; G01B 9/0209 20130101 |
Class at
Publication: |
356/326 |
International
Class: |
G01J 3/28 20060101
G01J003/28 |
Claims
1. A method for determining an optical path length (OPL) of an
optical cavity comprising the steps of: a) impinging a light beam
from a broadband coherent light source on the optical cavity
thereby forming a modulated light beam by optical interference; b)
directing the modulated light beam from the optical cavity to a
light dispersive element thereby splitting the modulated light beam
into a spectrum of component light frequencies each having a light
intensity and a corresponding light wavelength; c) detecting the
component light frequencies in a light detector thereby generating
spectral data with dimensions of light intensity of the component
light frequencies as a function of the wavelength of the component
light frequencies; and d) processing the spectral data by
performing a mathematical Fourier transform of the spectral data
using a Fourier transform algorithm, thereby generating a transform
spectrum with a Fourier amplitude as a function of an independent
length variable, wherein the transform spectrum has a peak
corresponding to a non-changing (DC) component of the spectral
data, and at least one modulation peak located at a coordinate of
the length variable corresponding to twice the OPL of the optical
cavity.
2. The method of claim 1, wherein the broadband coherent light
source is specified with a set of parameters comprising: a
coherence length greater than twice the OPL of the optical cavity;
and a frequency bandwidth sufficient to generate a number of
interference fringes necessary to resolve the OPL to a
predetermined resolution when light from the broadband coherent
light source is modulated by the optical cavity.
3. The method of claim 1, wherein the broadband light source is
directed to the optical light cavity through a beam splitter.
4. The method of claim 1, wherein the modulated light beam results
from optical interference between a first light beam transmitted
through first and second surfaces of the optical cavity and a
second light beam transmitted through the second surface of the
optical cavity, wherein the second light beam is generated when
light from the broadband light source is transmitted through the
first surface, reflected off the second surface, reflected off the
first surface, and is then transmitted through the second surface
of the optical cavity.
5. The method of claim 1, wherein the modulated light beam results
from optical interference between a first light beam reflected from
a first surface of the optical cavity and a second light beam
reflected from a second surface of the optical cavity, wherein the
second light beam is generated when light from the broadband light
source is transmitted through the first surface, reflected from the
second surface and then transmitted through the first surface of
the optical cavity.
6. The method of claim 1, wherein the light detector is a linear
array light detector having a plurality of discrete light detecting
elements each corresponding to one of the component light
frequencies and generating an electrical signal proportional to an
intensity of received wavelength of light.
7. The method of claim 6, wherein the electrical signals from the
plurality of discrete light detecting elements are coupled to a
computing device that provides the processing of the spectral
data.
8. The method of claim 1, wherein the dispersive element is a
device selected from a set of devices consisting of a prism, a
reflection grating, and a transmission grating.
9. The method of claim 1, wherein the Fourier transform is
calculated over a first set of coordinate points of the length
variable in a Fourier domain with a coarse separation between the
first set of coordinate points to determine a first location of a
positive peak adjacent to the DC peak.
10. The method of claim 9, wherein the Fourier transform is
recalculated over a second set of coordinate points of the length
variable in the Fourier domain spanning the first location with a
fine separation between the second set of coordinate points to
increase the resolution of measuring the OPL while keeping
calculation time low.
11. The method of claim 9, wherein the Fourier transform is
performed using an efficient Fourier transform algorithm configured
to reduce processing time.
12. The method of claim 10, wherein the Fourier transform is
performed using an efficient Fourier transform algorithm configured
to reduce processing time.
13. A system for determining an optical path length (OPL) of an
optical cavity comprising: a broadband coherent light source
providing an incident light beam for impinging on the optical
cavity; a light dispersive element for receiving a light beam from
the optical cavity and splitting the light beam into a spectrum of
component light frequencies each having a light intensity and a
corresponding light wavelength; a detector for detecting the
component light frequencies thereby generating spectral data that
is light intensity of the component light frequencies as a function
of wavelength of the component light frequencies; and a computing
device for processing the spectral data by performing a
mathematical Fourier transform of the spectral data using a Fourier
transform algorithm, thereby generating a transform spectrum with a
Fourier amplitude as a function an independent length variable,
wherein the transform spectrum has a peak corresponding to a
non-changing (DC) component of the spectral data, and at least one
modulation peak located at a coordinate of the length variable
corresponding to twice the OPL of the optical cavity.
14. The system of claim 13, wherein the broadband coherent light
source is specified with a set of parameters comprising: a
coherence length greater than twice the OPL of the optical cavity;
and a frequency bandwidth sufficient to generate a number of
interference fringes necessary to resolve the OPL to a
predetermined resolution when light from the broadband coherent
light source is modulated by the optical cavity.
15. The system of claim 13, wherein the broadband light source is
directed to the optical light cavity through a beam splitter.
16. The system of claim 13, wherein the modulated light beam
results from optical interference between a first light beam
transmitted through first and second surfaces of the optical cavity
and a second light beam transmitted through the second surface of
the optical cavity, wherein the second light beam is generated when
light from the broadband light source is transmitted through the
first surface, reflected off the second surface, reflected off the
first surface, and is then transmitted through the second surface
of the optical cavity.
17. The system of claim 13, wherein the modulated light beam
results from optical interference between a first light beam
reflected from a first surface of the optical cavity and a second
light beam reflected from a second surface of the optical cavity,
wherein the second light beam is generated when light from the
broadband light source is transmitted through the first surface,
reflected from the second surface and then transmitted through the
first surface of the optical cavity.
18. The system of claim 13, wherein the light detector is a linear
array light detector having a plurality of discrete light detecting
elements each corresponding to one of the component light
frequencies and generating an electrical signal proportional to an
intensity of received wavelength of light.
19. The system of claim 18, wherein the electrical signals from the
plurality of discrete light detecting elements are coupled to a
computing device that provides the processing of the spectral
data.
20. The system of claim 13, wherein the dispersive element is a
device selected from a set of devices consisting of a prism, a
reflection grating, and a transmission grating.
21. The system of claim 13, wherein the Fourier transform is
calculated over a first set of coordinate points of the length
variable in a Fourier domain with a coarse separation between the
first set of coordinate points to determine a first location of a
positive peak adjacent to the DC peak.
22. The system of claim 21, wherein the Fourier transform is
recalculated over a second set of coordinate points of the length
variable in the Fourier domain spanning the first location with a
fine separation between the second set of coordinate points to
increase the resolution of measuring the OPL while keeping
calculation time low.
23. The system of claim 21, wherein the Fourier transform is
performed using an efficient Fourier transform algorithm configured
to reduce processing time.
24. The system of claim 22, wherein the Fourier transform is
performed using an efficient Fourier transform algorithm configured
to reduce processing time.
Description
TECHNICAL FIELD
[0001] The present invention relates to apparatus and methods for
measuring the optical path length in optical cavities.
BACKGROUND INFORMATION
[0002] An optical cavity may be any region bounded by two or more
reflective interfaces that are aligned to provide multiple
reflections of light waves. Optical cavities have been monitored or
measured using a single wavelength illumination source such as a
helium/neon (HeNe) laser. A change in the cavity size is detected
by observing the change in reflected or transmitted intensity at
the single wavelength. Monitoring a single wavelength reflected
intensity requires a much larger signal-to-noise ratio (S/N) than a
broadband technique. In certain applications, the cavity is filled
with a fluid that is designed to have an index of refraction as
close as possible to that of the bounding surfaces, making the
reflectivity very small. In such cases, the S/N will be small and
thus may not be sufficient to use a single wavelength technique. In
a single wavelength system, there is not a one-to-one
correspondence between the measured intensity to the optical path
length; in other words, a measured intensity may correspond to any
number of optical path lengths. Therefore, a single-wavelength
system cannot determine the absolute value of the optical path
length, it can only detect changes. The change in optical path
length as measured by a single-wavelength system is sufficient for
some servo applications where the path length is to be held
constant; however, this allows the possibility of mode-hopping
where the servo unintentionally and undesirably locks onto a
different spectral mode.
[0003] Therefore, there is a need for an apparatus and method for
measuring or monitoring an optical cavity path length with an
output that provides the absolute value of the optical path length,
has better S/N tolerance, is free of mode-hopping limitations, and
offers near real time operation.
SUMMARY OF THE INVENTION
[0004] A broadband light source with a sufficiently long coherence
length is impinged upon an optical cavity. In one embodiment, the
broadband laser light reflects from the first and second surfaces
of the optical cavity generating multiple reflected light beams.
Generally, the two most intense beams will be those that are only
reflected once: one from the first surface and one from the second.
These two beams are sufficient to produce the necessary
interference signal. Therefore, ignoring the weaker reflected beams
will not change the functionality of the invention and for
simplicity the discussion is limited to the first two beams.
[0005] In another embodiment, the modulated light beam transmitted
through the optical cavity is used to produce the necessary
interference signal. In general, when light is incident on a
reflective surface, it is partially transmitted and partially
reflected. The transmitted light beam is primarily comprised of the
following two beams: that which is transmitted through both the
first and second surfaces of the optical cavity; and that which is
transmitted through the first surface, reflected from the second
surface, reflected again from the first, and finally transmitted
through the second surface. These two beams are sufficient to
produce the necessary interference signal. Therefore, ignoring the
weaker transmitted beams will not change the functionality of the
invention and for simplicity the discussion is limited to these two
beams.
[0006] The two broadband reflected (or transmitted) light beams are
phase shifted from one another by an amount proportional to the
optical path length (OPL) of the optical cavity. They interfere
with each other and produce a modulated light beam that has a
spectrum that looks approximately like the broadband laser's
spectrum multiplied by a cosine with a frequency
(4*.pi.*n*d/.lamda.), where n is the index of refraction of the
medium within the cavity, d is the physical separation between the
cavity surfaces, and .lamda. is the wavelength of the light. The
combined modulated light is coupled to a spectrometer which outputs
a spectrum that is a measure of the intensity of the light as a
function of wavelength, over a range determined by the spectrometer
specifications. The spectrum is changed to a function of
wavenumber, where for our purposes the wavenumber is the reciprocal
of the wavelength. The spectrum is Fourier transformed, resulting
in the Fourier amplitude as a function of a variable that is the
reciprocal of the wavenumber. The Fourier transform contains at
least one peak that is located at the independent variable
coordinate 2*n*d, where n*d is the OPL.
[0007] The Fourier transform of the spectrum will contain a DC peak
and at least one other peak that is related to the OPL. The
certainty that the Fourier transform of the spectrum of the light
reflected from or transmitted through the optical cavity has a peak
located at a coordinate equal to twice the OPL is a key feature of
the present invention. By locating and tracking the position of
this peak, one is able to measure and track the OPL directly. Due
to the nature of the spectrum of the light reflected from the
optical cavity, this OPL peak will be the most prominent one aside
from the DC peak (whose location is known). This feature adds to
the ease of tracking the OPL with the Fourier transform. If
efficient Fourier algorithms are used with sufficiently fast
computer technology, the OPL may be tracked in real-time, for
example, during a manufacturing process.
[0008] The foregoing has outlined rather broadly the features and
technical advantages of the present invention in order that the
detailed description of the invention that follows may be better
understood. Additional features and advantages of the invention
will be described hereinafter which form the subject of the claims
of the invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0009] For a more complete understanding of the present invention,
and the advantages thereof, reference is now made to the following
descriptions taken in conjunction with the accompanying drawings in
which:
[0010] FIG. 1 is a diagram illustrating an apparatus for practicing
embodiments of the present invention;
[0011] FIG. 2A is a diagram of the broad band input spectrum and
the spectrum of the reflected light from the optical cavity;
[0012] FIG. 2B is a diagram illustrating which wavelengths of
incident light have constructive interference for a given cavity
and which have destructive interference;
[0013] FIG. 2C is a diagram illustrating a graph of a Fourier
transform of a reflected spectrum from an optical cavity according
to embodiments of the present invention; and
[0014] FIG. 3 is a flow diagram of method steps according to
embodiments of the present invention.
DETAILED DESCRIPTION
[0015] In the following description, numerous specific details are
set forth to provide a thorough understanding of the present
invention. For example, specific subsystems and functions may be
described; however, it would be recognized by those of ordinary
skill in the art that the present invention may be practiced
without such specific details. In other instances, well-known units
or systems have been shown in block diagram form in order not to
obscure the present invention in unnecessary detail. Refer now to
the drawings wherein depicted elements are not necessarily shown to
scale and wherein like or similar elements are designated by the
same reference numeral by the several views.
[0016] In the following detailed description, a variable may be
referred to as having a DC (direct current) component, wherein the
variable may not be electrical current. In these cases, DC means
that the variable has a static non-changing component. The variable
may also have a sinusoidal component which may be referred to as an
AC (alternating current) wherein again the variable is not related
to electrical current.
[0017] The term "spectrometer" (for light) is used to describe a
wide range of instruments that are used to determine frequency
content of a light signal as a function of a variable and may be a
device that incorporates a dispersive element (e.g., prism,
reflection grating, or transmission grating, etc.), a recording
element (e.g., charge coupled device (CCD) camera, CMOS camera,
photodiode array, etc.), and possibly other guiding elements (e.g.,
mirrors and lenses).
[0018] A Fourier transform is a mathematical operation that
re-expresses a function in terms of a sum or integral of sinusoidal
basis functions. Often it is associated with converting a periodic
signal as a function of time into an amplitude as a function of
frequency (the reciprocal of time). However, in this disclosure the
Fourier transform is applied to spectrum amplitudes (intensity) as
a function of wavenumber, wherein the Fourier transform again
generates a Fourier amplitude as a function of the reciprocal of
that variable (length). There exist many efficient Fourier
transform algorithms including so-called Fast Fourier Transform
(FFT) variants, Fourier transforms may be performed using such
techniques as streamlined Fourier transform algorithms, fast
non-equal-spaced discrete Fourier transform (FNDFT), or other fast
Fourier transform algorithms suitable for the desired speed of
determining the OPL. These algorithms may be performed with
specialized digital signal processing (DSP) hardware or computers
receiving data from a fast-recording charge coupled digital (CCD)
camera to facilitate real time operation.
[0019] In the following description, a "suitable" broadband light
source is used as an integral part of the measurement technique.
The requirements for the "suitable" broadband light source, for use
in embodiments of the present invention, depend somewhat on the
optical path length (OPL) to be measured. An example of a suitable
broadband light source is a mode-locked Titanium:Sapphire laser
(Ti:S). Specifications for a suitable broadband light source
comprise parameters as the following:
[0020] (1) The coherence length of a light source is the distance
over which the phase of the light wave remains in phase with light
just emerging from the source. A suitable broadband light source,
for the present invention, must have a coherence length longer than
twice the OPL to be measured. Precisely how much longer depends in
a complicated way on the reflectivities of the two cavity surfaces.
Any broadband laser has a coherence length of many tens of meters
and so would definitely be suitable for typical optical cavities.
For very short OPLs (e.g., several hundred nanometers), a light
source normally considered incoherent, like a halogen lamp, would
work (it is still "coherent" but only over much shorter
distances).
[0021] (2) The broadband light source has measurable bandwidth
large enough so that when its light is modulated by the optical
cavity the modulated light's spectrum contains the number of
interference fringes necessary to determine the OPL to a desired
resolution. For a given spectrometer resolution, more fringes
results in a more precise determination of the OPL.
[0022] FIG. 1 is an apparatus suitable for practicing embodiments
of the present invention. Ti:S Laser 101 is used as a broadband
light source suitable for practicing embodiments of the present
invention. The incident light beam 120 impinges on beam splitter
102 which transmits a portion of the light beam 122 and diverts a
portion 121 which is not used. Light beam 122 is reflected from
mirror 103 and to an exemplary optical cavity formed between silica
window 104 and silicon wafer 105. Light beam 106 represents the
combination of the light beams reflected from the two surfaces of
the optical cavity. Mirror 103 directs light beam 106 to beam
splitter 102 and a portion of light beam 106 is transmitted (107)
and a portion is directed to curved mirror 123 as light beam 108.
Light beam 108 is directed to reflection grating 118 by mirror 123.
Reflection grating 118 breaks light beam 108 into light beams 109
distributed uniformly by wavelength. Light beams 109 are each
directed to parabolic mirror 124 which then directs each light beam
to a light detector element in detector 110. The output of the
detector elements 110 are coupled in parallel as signals 111 to
computer 112 which has digital signal processing (DSP) hardware for
digitizing and processing the individual signals 111 produced by
detecting the individual light beams 109. The signals 111 form the
spectrum of the reflected light 106. The DSP hardware in computer
112 may be programmed to perform a Fourier transform on the signals
111 (spectrum of reflected light 106) according to embodiments of
the present invention.
[0023] FIG. 2A is a diagram illustrating spectrum of the incident
and reflected light according to embodiments of the present
invention. Optical cavity 250 has a first surface 204 and a second
surface 205 defining its optical path length. Broadband input light
beam 202 has a continuous distribution of many light frequencies
centered around frequency F1. Input light beam 202 is directed by
beam splitter 207 to optical cavity 250 as light beam 203. The
combination of the two beams reflected from surfaces 204 and 205 is
reflected light beam 206 which has a spectrum 208 which looks like
the input spectrum 201 modulated by the cosine of a frequency that
is dependent on the dimensions of the optical cavity and the index
of refraction of the material 210 within the cavity.
[0024] FIG. 2B illustrates constructive and destructive
interference of the light reflected from the surfaces 222 and 224
as a function of dimensions of the optical cavity 250. Exemplary
incident wave 220 is partially reflected from surface 222 as
reflected wave 221. A portion of incident wave 220 continues as
transmitted wave 226 and is reflected from second surface 224 of
optical cavity 250 as reflected wave 223. Incident wave 220 has a
wavelength whose relationship with the cavity OPL is such that
reflected wave 221 is in phase with reflected wave 223 and they
constructively interfere. The relationship between the light
wavelength and the optical cavity dimensions for constructive
interference are given by the formula .lamda.=(2*n*d)/m where (m=1,
2, 3 . . . ). In this example, 2*n*d (twice the optical cavity
length) is equal to 3 wave lengths of incident wave 220 therefore m
is equal to 3.
[0025] Exemplary incident wave 230 represents a different
wavelength of light. Incident wave 230 is partially reflected from
surface 222 as wave 231. A portion of incident wave 230 continues
as transmitted wave 232 and is reflected from second surface 224 of
optical cavity 250 as reflected wave 233. This time incident wave
230 has a wavelength whose relationship with the cavity OPL is such
that reflected wave 221 is out of phase with reflected wave 223,
resulting in destructive interference. The relationship between the
light wavelength and the optical cavity dimensions for destructive
interference are given by the formula (.lamda.=(4*n*d)/(2m+1))
where (m=0, 1, 2, 3 . . . ). In this example, 2*n*d (twice the
optical cavity length) is equal to 21/4 wavelengths of incident
wave 230 therefore m is equal to 2.
[0026] FIG. 2C illustrates the resulting graph 240 when the Fourier
transform is applied to the spectrum 208 according to embodiments
of the present invention. Since the spectrum 208 is the intensity
as a function of the wavenumber (inverse of wavelength), its
independent axis has the dimension of (1/length). The Fourier
transform of the spectrum 208 transforms this dimension so the
independent axis of graph 240 has the dimension of (length). Graph
240 will have a DC peak corresponding to the DC component of the
spectrum 208, and another significant peak located at a coordinate
corresponding to 2*n*d, or twice the optical length of the optical
cavity. This allows the optical length of the optical cavity to be
determined directly by locating this peak in the Fourier
transform.
[0027] Mathematically the Fourier transform of spectrum 208 has at
least three peaks; a center peak that corresponds to the
non-changing or DC component located at the coordinate F.sub.M=0.
There are two resulting sidebands that represent the modulation
frequency due to the optical cavity. The sidebands are located at
positive and negative values of the modulation frequency
F.sub.M=2*n*d. The method of the present invention comprises
calculating only the positive half of Fourier transform function,
and locating the position of the corresponding sideband. The
negative half of the Fourier transform function is a mirror image
of the positive half and is therefore redundant. While one may use
either the positive or negative half of the Fourier transform
function, according to embodiments of the present invention, the
negative half of the Fourier transform function is normally not
calculated to speed up the process of determining the OPL.
[0028] To obtain better precision, the Fourier transform may be
first coarsely calculated to locate the positive sideband. The
Fourier transform is then recalculated only in the region of the
sideband, resulting in a savings in processing time and better
resolution. This process is repeated until a satisfactory
resolution is achieved.
[0029] Processing or calculation time is reduced by the following:
[0030] (1) only dealing with the positive half of the Fourier
transform. [0031] (2) coarsely calculating the Fourier transform to
locate the sideband peak. [0032] (3) only calculating a small range
around the sideband or the very peak of the sideband both of which
also result in better resolution. [0033] (4) using more powerful
Fourier transform algorithms that are configured to do "fast"
transforms and thus speed up the actual calculation.
[0034] FIG. 3 is a flow diagram of method steps according to
embodiments of the present invention. In step 301, a suitable
broadband light source is impinged on the optical cavity with an
optical path length to be measured. In step 302, the reflected
light beams from the two surfaces of the optical cavity are
directed to a dispersive element that spatially splits the
reflected light beam into its component light frequencies uniformly
distributed by wavelength. In step 303, the spectrum of the
component light frequencies of the reflected light beam is
converted to electrical signals in a light detector array, wherein
the electrical signals have amplitudes proportional to the
intensities of the component light frequencies. In step 304, the
spectrum is processed to generate a mathematical Fourier transform
of the detected spectrum using efficient Fourier transform
algorithms, thereby generating a transform spectrum with a peak
corresponding to a DC, or non-changing component of the spectrum,
and a peak located at a coordinate corresponding to twice the OPL
of the optical cavity.
[0035] Calibrating the system involves primarily the spectrometer
which is the device that records the light intensities for
different wavelengths. Spectrometers have standard calibration
procedures. Usually they involve a light source that emits light
composed of only several known wavelengths. The spectrometer
records upon which pixel those lines fall and those pixels are
thereby defined using the known wavelength of light. The reason for
this is that there may not be, in general, a linear relationship
between pixel location and the corresponding wavelength that falls
on it. The spectrometer hardware or software takes this into
account when collecting data. There is, of course, a limit to how
well a spectrometer may be calibrated and any error in the
calibration will bias measurement of the OPL either high or low,
depending on the error.
[0036] Some embodiments of the present invention may allow for the
OPL to be measured with a 95% confidence range that the true value
of the OPL is within +/-32 nm of the measured value. Other
embodiments may have different measurement accuracies.
[0037] When the broadband light source with the spectrum 201 is
impinged onto an optical cavity (e.g., 250, FIG. 2A) this spectrum
is modulated. Let B(.lamda.) denote the spectral intensity of the
broadband light as a function of wavelength. The optical cavity
basically multiplies the spectrum (201) by a cosine function;
cos(4*.pi.*n*d/.lamda.). Here, .pi..apprxeq.3.14, n is the index of
refraction within the cavity (for air n=1), d is the distance
between cavity surfaces, and .lamda. is the wavelength of the light
as illustrated in spectrum 201. Now the spectrum of the light beam
that is reflected or transmitted from the cavity is described by a
new function: B.sub.M(.lamda.)=B(.lamda.)*cos(4*.pi.*n*d/.lamda.).
The variable B.sub.M of this spectrum is what is measured by the
spectrometer. Each pixel of the spectrometer records the value
B.sub.M of this spectrum at a different wavelength of light. The
result is a set of data B.sub.M(.lamda.), stored digitally, that if
plotted versus (1/.lamda.), yields the "spectrum" 208 as shown in
FIG. 2A. This stored data B.sub.M(.lamda.) may now be processed to
yield desired results.
[0038] Note that the function cos(4*.pi.*n*d/.lamda.) may be
rewritten as cos(2.pi.*(2*n*d)*(1/.lamda.)) which shows that it is
not periodic in wavelength, .lamda., but is periodic in 1/.lamda..
Thus, it becomes convenient to define the variable "wavenumber" as
1/.lamda.. This is a common variable when dealing with waves such
as light and is normally written as k=1/.lamda.. Now the function
cos(2.pi.*(2*n*d)*(1/.lamda.)) may be written as
cos(2.pi.*(2*n*d)*k). Therefore the modulation of the spectrum may
be described as periodic in k with a modulation frequency
f.sub.M=2*n*d, wherein f.sub.M has dimensions of length. The
modulation frequency is not to be confused with the frequency of
the light in the light beams. Spectrum 208 in FIG. 2A has peaks
spaced at 1/f.sub.M and refers to the "wavelength" of the cosine
function cos(2.pi.*(2*n*d)*k). Note that this wavelength
(1/f.sub.M) is not the wavelength of a light frequency and has the
dimension of "1/length". Since the optical path length (OPL) is
(n*d), extracting (by digital processing) this frequency f.sub.M is
how embodiments of the present invention determine the OPL of an
optical cavity.
[0039] Although the present invention and its advantages have been
described in detail, it should be understood that various changes,
substitutions and alterations can be made herein without departing
from the spirit and scope of the invention as defined by the
appended claims.
* * * * *