U.S. patent application number 11/684779 was filed with the patent office on 2007-08-30 for thin film interference filter and bootstrap method for interference filter thin film deposition process control.
This patent application is currently assigned to UNIVERSITY OF SOUTH CAROLINA. Invention is credited to Michael L. Myrick.
Application Number | 20070201136 11/684779 |
Document ID | / |
Family ID | 38443717 |
Filed Date | 2007-08-30 |
United States Patent
Application |
20070201136 |
Kind Code |
A1 |
Myrick; Michael L. |
August 30, 2007 |
Thin Film Interference Filter and Bootstrap Method for Interference
Filter Thin Film Deposition Process Control
Abstract
A thin film interference filter system includes a plurality of
stacked films having a determined reflectance; a modeled monitor
curve; and a topmost layer configured to exhibit a wavelength
corresponding to one of the determined reflectance or the modeled
monitor curve. The topmost layer is placed on the plurality of
stacked films and can be a low-index film such as silica or a high
index film such as niobia.
Inventors: |
Myrick; Michael L.; (Irmo,
SC) |
Correspondence
Address: |
DORITY & MANNING, P.A.
POST OFFICE BOX 1449
GREENVILLE
SC
29602-1449
US
|
Assignee: |
UNIVERSITY OF SOUTH
CAROLINA
1200 Catawba Street
Columbia
SC
29208
|
Family ID: |
38443717 |
Appl. No.: |
11/684779 |
Filed: |
March 12, 2007 |
Current U.S.
Class: |
359/578 |
Current CPC
Class: |
G01N 2021/8438 20130101;
G02B 5/28 20130101; G01N 21/55 20130101; G01N 21/8422 20130101 |
Class at
Publication: |
359/578 |
International
Class: |
G02B 27/00 20060101
G02B027/00 |
Goverment Interests
STATEMENT OF FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
[0002] Research described in this application was sponsored by the
United States Air Force Research Laboratory grant number
F22615-00-2-6059.
Foreign Application Data
Date |
Code |
Application Number |
Sep 13, 2004 |
WO |
US2005/032420 |
Claims
1. A method using experimental measurements to determine
reflectance phase and complex reflectance for arbitrary thin film
stacks, the method comprising the steps of: determining reflectance
of a stack of a plurality of films before depositing a topmost
layer; considering a modeled monitor curve for a wavelength of a
high-index layer; and discarding a plurality of monitor curves
without maxima in their reflectance during the topmost layer
deposition.
2. The method as in claim 1, wherein the topmost layer is a niobia
layer.
3. The method as in claim 2, further comprising the steps of
determining an anticipated standard deviation in .phi.k for a
plurality of monitor wavelengths in the niobia layer and discarding
any with .sigma. greater than 0.9 degrees.
4. The method as in claim 3, further comprising the step of
computing expected error in .delta. for wavelengths with .sigma.
less than 0.9 degrees at a target thickness of the niobia
layer.
5. The method as in claim 4, further comprising the step proceeding
with a full model deposition of the niobia layer when no
wavelengths have an error less than 0.9 degrees.
6. The method as in claim 1, further comprising the step of using
only the modeled monitor curve during the topmost layer deposition
when there is no maxima in the reflectance of each of the plurality
of monitor curves.
7. The method as in claim 1, further comprising the step of
computing a value of .delta. for all wavelengths based on a value
calculated for the monitor wavelength.
8. The method as in claim 1, further comprising the step of
computing two possible values of phase angle for each wavelength
other than the monitor wavelength.
9. The method as in claim 1, further comprising the steps of using
information extracted from the model for r.sub.k at each wavelength
and the computed best value of .delta., and computing an estimated
standard deviation of phase at all wavelengths except the
monitor.
10. The method as in claim 9, further comprising the steps of using
the computed phase closest to the model phase for r.sub.k at each
wavelength, measured R.sub.f and R.sub.k values and the computed
best value of .delta., and computing the estimated standard
deviation of phase at all wavelengths for which the magnitude of
r.sub.k was estimated other than the monitor.
11. The method as in claim 9, further comprising the steps of
determining if a phase error estimate is less than about 1.3
degrees and averaging calculated and modeled reflectance and phase
values to obtain a new value for use in subsequent modeling at that
wavelength.
12. The method as in claim 1, wherein the topmost layer is a silica
film.
13. The method as in claim 12, further comprising the step of
replacing the magnitude of the amplitude reflectance at each
wavelength with {square root over (R.sub.k)} whenever measuring a
latest depositing silica film having an intensity reflectance
greater than 9%.
14. The method as in claim 13, further comprising the step of
determining if a phase error estimate is less than about 1.3
degrees when the magnitude of the amplitude reflectance at each
wavelength has been replaced with {square root over (R.sub.k)} and
averaging calculated and modeled reflectance and phase values to
obtain a new value for use in subsequent modeling at that
wavelength.
15. A method for correcting thin film stack calculations for
accurate deposition of complex optical filters, the method
comprising the steps of: determining phase angle .phi.k at a
monitor wavelength from |r'.sub.k| and Rk using a first equation
expressed as: cos .function. ( .+-. .PHI. k ) = r k ' 2 .times. ( 1
+ R k .times. r 2 2 ) - r 2 2 - R k 2 .times. r 2 .times. R k
.times. ( 1 - r k ' 2 ) ; ##EQU37## and estimating r'.sub.k using a
second equation r k ' = r k - r 2 1 - r 2 .times. r k ; ##EQU38##
and obtaining a value for phase at a monitor wavelength.
16. A method for automated deposition of complex optical
interference filters, the method comprising the steps of:
determining from a measurement of intensity reflectance at a
topmost interface a phase angle .phi. at an interface k according
to an equation expressed as: .times. cos .function. ( .PHI. k ) = (
A .function. ( 1 + r 2 2 ) .times. sin .function. ( .delta. ) .+-.
B .times. .times. cos .function. ( .delta. ) C ) ##EQU39## .times.
A = R f + r 2 4 .function. ( R f - R k ) - R k + 2 .times. r 2 2
.function. ( ( 1 - R f ) .times. ( 1 + R k ) .times. cos .times. (
2 .times. .delta. ) - ( 1 - R f .times. R k ) ) ##EQU39.2## .times.
B = D .function. ( 1 + r 2 12 ) + F .function. ( r 2 2 + r 2 10 ) +
G .function. ( r 2 4 + r 8 2 ) + H .times. .times. r 6 2
##EQU39.3## .times. C = sin .function. ( .delta. ) .times. ( 4
.times. r 2 .function. ( 1 - R f ) .times. R k 1 / 2 .function. ( 2
.times. r 2 2 .times. cos .function. ( 2 .times. .delta. ) - 1 - r
2 4 ) ) ##EQU39.4## .times. D = - ( R f - R k ) 2 ##EQU39.5##
.times. F = 2 .times. ( R k .function. ( 2 + R k ) + R f 2
.function. ( 1 + 2 .times. R k ) + 2 .times. R f .function. ( 1 - 5
.times. R k + R k 2 ) - 2 .times. ( 1 - R f ) .times. ( 1 - R k )
.times. ( R f + R k ) .times. cos .function. ( 2 .times. .delta. )
) ##EQU39.6## G = - 6 - 4 .times. R f - 5 .times. R f 2 - 4 .times.
R k + 38 .times. R f .times. R k - 4 .times. R f 2 .times. R k - 5
.times. R k 2 - 4 .times. R f .times. R k 2 - 6 .times. R f 2
.times. R k 2 + 8 .times. ( 1 - R f 2 ) .times. ( 1 - R k 2 )
.times. cos .function. ( 2 .times. .delta. ) - 2 .times. ( 1 - R f
) 2 .times. ( 1 - R k ) 2 .times. cos .function. ( 4 .times.
.delta. ) ##EQU39.7## .times. H = 4 .times. ( 3 + 2 .times. R f 2 -
10 .times. R f .times. R k + 2 .times. R k 2 + 3 .times. R f 2
.times. R k 2 - 2 .times. ( 1 - R f ) .times. ( 1 - R k ) .times. (
2 + R f + R k + 2 .times. R f .times. R k ) .times. cos .function.
( 2 .times. .delta. ) + ( 1 - R f ) 2 .times. ( 1 - R k ) 2 .times.
cos .function. ( 4 .times. .delta. ) ) ##EQU39.8##
17. The method as in claim 16, wherein a process control for a
deposition system is bootstrapped by detaching the deposition
system from all but the topmost interface.
18. The method as in claim 16, further comprising the step of
validating two resultant solutions according to the expression:
.times. R f = 2 .times. r 2 2 + R k .function. ( 1 + r 2 4 ) + 2
.times. r 2 .times. Q 1 + r 2 4 + 2 .times. r 2 2 .times. R k + 2
.times. r 2 .times. Q ##EQU40## Q = r 2 2 .times. R k 1 / 2 .times.
cos .function. ( 2 .times. .delta. + .PHI. k ) + R k 1 / 2 .times.
cos .function. ( 2 .times. .delta. - .PHI. k ) - r 2 .function. ( 1
+ R k ) .times. cos .function. ( 2 .times. .delta. ) - ( 1 + r 2 2
) .times. R k 1 / 2 .times. cos ##EQU40.2##
19. The method as in claim 16, further comprising the step of
averaging calculated and modeled reflectance and phase values to
obtain a new value to be used in all future modeling at a given
wavelength.
20. A thin film interference filter system, comprising: a plurality
of stacked films having a determined reflectance; a modeled monitor
curve; and a topmost layer configured to exhibit a wavelength
corresponding to one of the determined reflectance or the modeled
monitor curve, the topmost layer being disposed on the plurality of
stacked films.
21. The thin film interference filter system as in claim 20,
wherein the topmost layer is a silica film.
22. The thin film interference filter system as in claim 20,
wherein the topmost layer is a niobia film.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims benefit of U.S. Provisional Patent
Application, Ser. No. 60/609,406, entitled "Bootstrap Method for
Interference Filter Thin Film Deposition Process Control," filed
Sep. 13, 2004.
FIELD OF THE INVENTION
[0003] The present invention relates to thin film optical devices.
More particularly, the invention relates to complex interference
filters.
BACKGROUND OF THE INVENTION
[0004] Highly accurate optical interference filters can be
manufactured using thin film deposition processes. These optical
interference filters are used for multivariate optical computing,
multiple-band-pass, and the like and can exhibit complex optical
spectra defined over a range of wavelengths. These filters are
typically constructed by depositing alternating layers of
transparent materials where one layer possesses a much larger
refractive index relative to the other layer. Theoretically, the
proper choice of composition, thickness and quantity of layers
could result in a device with any desired transmission
spectrum.
[0005] Among the simplest devices is the single cavity bandpass
filter; i.e., the thin-film form of an etalon. This device consists
of three sets of layers. The first stack is a dielectric mirror, a
next thicker layer forms a spacer, and a second stack forms another
dielectric mirror. The mirror stacks are typically fabricated by
depositing alternating transparent materials that have an optical
thickness that is one quarter of the optical wavelength of
light.
[0006] To achieve theoretical optical performance, each layer must
possess a precise and specific physical thickness and refractive
index. Any nonuniformity in the deposition of the layers can affect
the spectral placement and transmission or reflection
characteristics of the device. A design that requires very tight
manufacturing tolerances over large substrate areas could result in
the costly rejection of many devices. Given these manufacturing
limits, it would be desirable to analyze the devices after
construction and alter the devices that do not meet a predetermined
optical transmission or reflection specification by some electrical
or mechanical means. For example, if the peak transmission
wavelength of a manufactured optical bandpass cavity filter was
slightly out of tolerance, it would be desirable to have a
mechanism or process for shifting the peak back to the desired
spectral location. It is also desirable that the optical filters
have precise rejection bands and passbands that are electrically or
mechanically selectable.
[0007] Mechanical methods of achieving a variable transmission
spectrum device are well known. This includes changing a prism or
grating angle, or altering the optical spacing between mirrors of
an etalon. To overcome the performance, size and cost disadvantages
of using mechanical schemes, many have conceived of electrical
methods for varying a transmission spectrum. For example, U.S. Pat.
No. 5,150,236, issued Sep. 22, 1992 to Patel, discloses a tunable
liquid crystal etalon filter. The liquid crystal fills the space
between dielectric mirrors. Electrodes on the mirrors are used to
apply an electric field, which changes the orientation of the
liquid crystal that changes the optical length for tuning. The
change in the optical length corresponds to a change in the
location of the passband. U.S. Pat. No. 5,103,340, issued Apr. 7,
1992 to Dono et al., discloses piezoelectric elements placed
outside the optical path that are used to change the spacing
between cascaded cavity filters. Furthermore, U.S. Pat. No.
5,799,231, issued Aug. 25, 1998 to Gates et al., discloses a
variable index distributed mirror. This is a dielectric mirror with
half of the layers having a variable refractive index that is
matched to other layers. Changing the applied field increases the
index difference that increases the reflectance. The mathematics
that describes the transmission characteristics of multilayer films
composed of electro-optic and dielectric materials are well
known.
[0008] Another electrically actuated thin film optical filter uses
a series of crossed polarizers and liquid crystalline layers that
allow electrical controls to vary the amount of polarization
rotation in the liquid by applying an electric field in such a way
that some wavelengths are selectively transmitted. However, these
electrically actuated thin film optical filters have the
characteristic that the light must be polarized and that the
frequencies of light not passed are absorbed, not reflected.
Another electrically actuated thin film optical device is the
tunable liquid crystal etalon optical filter. The tunable liquid
crystal etalon optical filter uses a liquid crystal between two
dielectric mirrors.
[0009] The common cavity filter, such as the etalon optical filter,
is an optical filter with one or more spacer layers that are
deposited in the stack and define the wavelength of the rejection
and pass bands. The optical thickness of the film defines the
placement of the passband. U.S. Pat. No. 5,710,655, issued Jan. 20,
1998 to Rumbaugh et al., discloses a cavity thickness compensated
etalon filter.
[0010] In the tunable liquid crystal etalon optical filter, an
electric field is applied to the liquid crystal that changes the
optical length between the two mirrors so as to change the passband
of the etalon. Still another tunable optical filter device tunes
the passband by using piezoelectric elements to mechanically change
the physical spacing between mirrors of an etalon filter.
[0011] Bulk dielectrics are made by subtractive methods like
polishing from a larger piece; whereas thin film layer are made by
additive methods like vapor or liquid phase deposition. A bulk
optical dielectric, e.g., greater than ten microns, disposed
between metal or dielectric mirrors suffers from excessive
manufacturing tolerances and costs. Moreover, the bulk material
provides unpredictable, imprecise, irregular, or otherwise
undesirable passbands. These electrical and mechanical optical
filters disadvantageously do not provide precise rejection bands
and passbands that are repeatably manufactured.
[0012] In an attempt to avoid some of the foregoing problems,
modeling of interference filters can be conducted during on-line
fabrication with in-situ optical spectroscopy of the filter during
deposition. The current state of the art for on-line correction of
the deposition involves fitting the observed spectra to a
multilayer model composed of "ideal" films based on a model for
each film. The resulting model spectra are approximations of the
actual spectra. To use reflectance as an example: the measured
reflectance of a stack of films can be approximately matched to a
theoretical reflectance spectrum by modeling. Layers remaining to
be deposited can then be adjusted to compensate for errors in the
film stack already deposited, provided the film stack has been
accurately modeled. However, films vary in ways that cannot be
readily modelled using any fixed or simple physical model.
Heterogeneities in the films that cannot be predicted or
compensated by this method cause the observed spectra to deviate
more and more from the model. This makes continued automatic
deposition very difficult; complex film stacks are therefore very
operator-intensive and have a high failure rate. To improve
efficiency in fabrication, laboratories that fabricate these stacks
strive to make their films as perfectly as possible so the models
are as accurate as possible.
[0013] As outlined above, many thin films are usually designed in a
stack to produce complex spectra and small variations in deposition
conditions make it difficult to accurately model in situ film
spectra for feedback control of a continuous deposition process
because it is practically impossible to obtain full knowledge of
the detailed structure of the stack from reflectance,
transmittance, ellipsometry, mass balance or other methods. Thus, a
thin film interference filter is needed that is less difficult to
manufacture, which will address varying refractive indices of thin
films and varying absorptions with deposition parameters.
BRIEF SUMMARY OF INVENTION
[0014] In general, the present invention is directed to a layered,
thin film interference filter and related bootstrap methods. A
bootstrap method according to one aspect of the invention permits a
user to focus on a single layer of a film stack as the layer is
deposited to obtain an estimate of the properties of the stack.
Although the single layer model is a guideline and not a basis for
compensating errors, only the most-recently-deposited layer--and
not the already-deposited film stack--need be modeled according to
an aspect of the present invention. Thus, the user can neglect
deviations of the stack from ideality for all other layers. The
single-layer model can then be fit exactly to the observed spectra
of the film stack at each stage of deposition to allow accurate
updating of the remaining film stack for continued deposition.
[0015] The present invention works with any type of films, whether
absorbing or non-absorbing, and regardless of whether the control
of the deposition conditions are state-of-the-art or not. The
methods of the invention are relatively straightforward, and a
resultant thin film interference filter is economical to produce
and use.
[0016] According to a particular aspect of the invention, a method
using experimental measurements to determine reflectance phase and
complex reflectance for arbitrary thin film stacks includes the
steps of determining reflectance of a stack of a plurality of films
before depositing a topmost layer; considering a modeled monitor
curve for a wavelength of a high-index layer; and discarding a
plurality of monitor curves without maxima in their reflectance
during the topmost layer deposition. In this aspect of the
invention, the topmost layer can be a niobia layer.
[0017] The exemplary method can also include the steps of
determining an anticipated standard deviation in .phi..sub.k for a
plurality of monitor wavelengths in the niobia layer and discarding
any with .sigma. greater than 0.9 degrees. Another step in this
aspect is computing expected error in .delta. for wavelengths with
.sigma. less than 0.9 degrees at a target thickness of the niobia
layer. When no wavelengths have an error less than 0.9 degrees, a
further step according to this method is to proceed with a full
model deposition of the niobia layer. When there are no maxima in
the reflectance of each of the plurality of monitor curves, another
step according to the exemplary method is to use only the modeled
monitor curve during the topmost layer deposition.
[0018] The method according to this aspect of the invention can
also include the step of computing a value of .delta. for all
wavelengths based on a value calculated for the monitor
wavelength.
[0019] The method according to this aspect of the invention can
further include the step of computing two possible values of phase
angle for each wavelength other than the monitor wavelength.
[0020] Additional steps according to the exemplary method include
using information extracted from the model for r.sub.k at each
wavelength and the computed best value of .delta., and computing an
estimated standard deviation of phase at all wavelengths except the
monitor.
[0021] The method may further include the steps of using the
computed phase closest to the model phase for r.sub.k at each
wavelength, measured R.sub.f and R.sub.k values and the computed
best value of .delta., and computing the estimated standard
deviation of phase at all wavelengths for which the magnitude of
r.sub.k was estimated other than the monitor.
[0022] Further steps according to the exemplary method include the
steps of determining if a phase error estimate is less than about
1.3 degrees and averaging calculated and modeled reflectance and
phase values to obtain a new value for use in subsequent modeling
at that wavelength.
[0023] In yet another aspect according to the exemplary method, the
topmost layer can be a silica film. Accordingly, the method can
include the step of replacing the magnitude of the amplitude
reflectance at each wavelength with {square root over (R.sub.k)}
whenever measuring a latest depositing silica film having an
intensity reflectance greater than 9%. The method can also include
the step of determining if a phase error estimate is less than
about 1.3 degrees when the magnitude of the amplitude reflectance
at each wavelength has been replaced with {square root over
(R.sub.k)} and averaging calculated and modeled reflectance and
phase values to obtain a new value for use in subsequent modeling
at that wavelength.
[0024] In yet another aspect of the invention, a method for
correcting thin film stack calculations for accurate deposition of
complex optical filters can include the steps of determining phase
angle .phi..sub.k at a monitor wavelength from |r'.sub.k| and Rk
using a first equation expressed as: cos .times. .times. ( .+-.
.times. .PHI. k ) = r k ' 2 .times. ( 1 + R k .times. r 2 2 ) - r 2
2 - R k 2 .times. .times. r 2 .times. R k .times. ( 1 - r k ' 2 ) ;
##EQU1##
[0025] and estimating r'.sub.k using a second equation r k ' = r k
- r 2 1 - r 2 .times. r k ; ##EQU2## and obtaining a value for
phase at a monitor wavelength.
[0026] Still another aspect of the invention includes a method for
automated deposition of complex optical interference filters
including the steps of determining from a measurement of intensity
reflectance at a topmost interface a phase angle .phi. at an
interface k according to an equation expressed as: cos .times.
.times. ( .PHI. k ) = ( A .function. ( 1 + r 2 2 ) .times. sin
.times. .times. ( .delta. ) .+-. B .times. .times. cos .function. (
.delta. ) C ) ##EQU3## A = .times. R f + r 2 4 .function. ( R f - R
k ) - R k + .times. 2 .times. .times. r .times. 2 .times. 2
.function. ( ( 1 - R .times. f ) .times. ( 1 + R .times. k )
.times. cos .times. .times. ( 2 .times. .times. .delta. ) - ( 1 - R
.times. f .times. R .times. k ) ) ##EQU3.2## B = D .function. ( 1 +
r 2 12 ) + F .function. ( r 2 2 + r 2 10 ) + G .function. ( r 2 4 +
r 2 8 ) + Hr 2 6 ##EQU3.3## C = sin .times. .times. ( .delta. )
.times. ( 4 .times. .times. r 2 .function. ( 1 - R f ) .times. R k
1 / 2 .function. ( 2 .times. .times. r 2 2 .times. cos .times.
.times. ( 2 .times. .times. .delta. ) - 1 - r 2 4 ) ) ##EQU3.4## D
= - ( R f - R k ) 2 ##EQU3.5## F = 2 .times. ( R k .function. ( 2 +
R k ) + R f 2 .function. ( 1 + 2 .times. .times. R k ) + 2 .times.
.times. R f .function. ( 1 - 5 .times. .times. R k + R k 2 ) - 2
.times. ( 1 - R f ) .times. ( 1 - R k ) .times. ( R f + R k )
.times. cos .times. .times. ( 2 .times. .times. .delta. ) )
##EQU3.6## G = .times. - 6 - 4 .times. .times. R f - 5 .times.
.times. R f 2 - 4 .times. .times. R k + 38 .times. .times. R f
.times. R k - 4 .times. .times. R f 2 .times. R k - 5 .times.
.times. R k 2 - .times. 4 .times. .times. R f .times. R k 2 - 6
.times. .times. R f 2 .times. R k 2 + 8 .times. ( 1 - R f 2 )
.times. ( 1 - R k 2 ) .times. cos .times. .times. ( 2 .times.
.times. .delta. ) - .times. 2 .times. ( 1 - R f ) 2 .times. ( 1 - R
k ) 2 .times. cos .times. .times. ( 4 .times. .times. .delta. )
##EQU3.7## H = 4 .times. ( 3 + 2 .times. .times. R f 2 - 10 .times.
.times. R f .times. R k + 2 .times. .times. R k 2 + 3 .times.
.times. R f 2 .times. R k 2 - 2 .times. ( 1 - R .times. f ) .times.
( 1 - R .times. k ) .times. ( 2 + R f + R k + 2 .times. .times. R f
.times. R k ) .times. cos .times. .times. ( 2 .times. .times.
.delta. ) + ( 1 - R f ) 2 .times. ( 1 - R k ) 2 .times. cos .times.
.times. ( 4 .times. .times. .delta. ) ) .times. ##EQU3.8##
[0027] According to this exemplary method, a process control for a
deposition system is bootstrapped by detaching the deposition
system from all but the topmost interface.
[0028] The method can further include the step of validating two
resultant solutions according to the expression: R f = 2 .times.
.times. r 2 2 + R k .function. ( 1 + r 2 4 ) + 2 .times. .times. r
2 .times. Q 1 + r 2 4 + 2 .times. .times. r 2 2 .times. R k + 2
.times. .times. r 2 .times. Q ##EQU4## Q = .times. r 2 2 .times. R
k 1 / 2 .times. cos .times. .times. ( 2 .times. .times. .delta. +
.PHI. k ) + R k 1 / 2 .times. cos .times. .times. ( 2 .times.
.times. .delta. - .PHI. k ) - .times. r 2 .function. ( 1 + R k )
.times. cos .times. .times. ( 2 .times. .times. .delta. ) - ( 1 + r
2 2 ) .times. R k 1 / 2 .times. cos .times. .times. ( .PHI. k )
##EQU4.2##
[0029] The method can also include the step of averaging calculated
and modeled reflectance and phase values to obtain a new value to
be used in all future modeling at a given wavelength.
[0030] According to another aspect of the invention, a thin film
interference filter system includes a plurality of stacked films
having a determined reflectance; a modeled monitor curve; and a
topmost layer configured to exhibit a wavelength corresponding to
one of the determined reflectance or the modeled monitor curve, the
topmost layer being disposed on the plurality of stacked films. The
topmost layer according to this aspect can be a low-index film such
as silica or a high index film such as niobia.
[0031] Other aspects and advantages of the invention will be
apparent from the following description and the attached drawings,
or can be learned through practice of the invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0032] A full and enabling disclosure of the present invention,
including the best mode thereof to one of ordinary skill in the
art, is set forth more particularly in the remainder of the
specification, including reference to the accompanying figures in
which:
[0033] FIG. 1 is a schematic, cross sectional view of a stack of
films according to an aspect of the invention;
[0034] FIG. 2 is a schematic, cross sectional view of a stack of
films similar to FIG. 1 showing an internal interface in the stack
of films according to another aspect of the invention;
[0035] FIG. 3 is a schematic, cross sectional view of a stack of
films according to another aspect of the invention similar to FIG.
1 but showing amplitude reflectance of a buried interface;
[0036] FIG. 4 is a representation of regions of permissible values
of Rk and Rmax, particularly showing lowest values of
.sigma..sub..phi.k for silica according to an aspect of the
invention;
[0037] FIG. 5 is similar to FIG. 4 but for niobia according to
another aspect of the invention;
[0038] FIG. 6 is a histogram for silica as in FIG. 4;
[0039] FIG. 7 is a histogram for niobia as in FIG. 5;
[0040] FIG. 8 is a perspective plot showing an error logarithm
versus Rmax and .delta. in accordance with an aspect of the
invention; and
[0041] FIG. 9 is a histogram of FIG. 8 data.
DETAILED DESCRIPTION OF THE INVENTION
[0042] Detailed reference will now be made to the drawings in which
examples embodying the present invention are shown. Repeat use of
reference characters in the drawings and detailed description is
intended to represent like or analogous features or elements of the
present invention.
[0043] The drawings and detailed description provide a full and
detailed written description of the invention and the manner and
process of making and using it, so as to enable one skilled in the
pertinent art to make and use it. The drawings and detailed
description also provide the best mode of carrying out the
invention. However, the examples set forth herein are provided by
way of explanation of the invention and are not meant as
limitations of the invention. The present invention thus includes
modifications and variations of the following examples as come
within the scope of the appended claims and their equivalents.
[0044] Turning now to the figures, FIG. 1 shows a thin film
interference filter 10, which broadly includes a substrate 12 upon
which a stack of films 14A-X is deposited (where x represents a
theoretically infinite number of film layers). As shown, the last
(alternatively, final, top or topmost) deposited film is designated
by the alphanumeral 14A while previously deposited or lower level
films are designated 14B-X. An incoming ray 18 is shown in FIG. 1
being reflected at an interface 16 (also referred to herein as top
or top surface and when mathematically referenced as k). The
reflected ray is designated by the number 20. For simplicity, any
contribution from multiple incoherent reflections in the substrate
12 is ignored in the following discussion and only reflections with
respect to the film stack 14A-X are described.
[0045] Typically, reflectance of the top surface 16 is obtained
using a matrix calculation that is in turn built from the
characteristic matrices of each of the preceding films 14A-X. As
shown in FIG. 1, a computed value of an electric field amplitude
reflectance r.sub.k is obtained by optimizing the thickness of the
topmost film 14A to provide the best fit over the fall spectrum
consistent with an understanding of the existing film stack 14B-X.
The calculated value of r.sub.k can be considered an estimate of
the actual value of r.sub.k exhibited by the film stack 14A-X.
Since r.sub.k is a complex value, it cannot be measured
directly.
Normal Matrx-type Calculations:
[0046] For standard calculations, the complex reflectance of a
stack of films is computed using the admittances of the incident
medium (often air), and the first interface of the stack. For s and
p polarization, this reflectance is: r s ( f 1 ) = - .eta. s ( inc
) - .eta. s ( f 1 ) - .eta. s ( inc ) + .eta. s ( f 1 ) .times.
.times. r p ( f 1 ) = .eta. p ( inc ) - .eta. p ( f 1 ) .eta. p (
inc ) + .eta. p ( f 1 ) 6. ##EQU5## where the subscript s or p
indicates for s or p-polarized light, .eta. is a complex
admittance, the superscript "inc" indicates the incident medium and
(f.sub.1) indicates the first interface the light strikes when
coming from the incident medium.
[0047] For s and p polarized light, the admittances of the incident
media are written: .eta. s ( inc ) = inc - inc .times. sin .times.
2 .times. ( .theta. inc ) .times. .times. .eta. p ( inc ) = inc
.times. inc - inc .times. .times. sin 2 .times. .times. ( .times.
.theta. inc ) 7. ##EQU6## where .di-elect cons. indicates a complex
dielectric constant, and .theta..sub.inc is the angle of incidence
in the medium of incidence.
[0048] The problematic part of the calculation is how to express
the admittance of the initial interface. The matrix calculation
proceeds by relating the admittance of the initial interface to
that of the second interface, the admittance of the second to the
third, etcetera, through a series of 2.times.2 matrices, until the
calculation is related to the final interface. At the final
interface, the admittance (ratio of magnetic to electric fields) is
equal to the admittance of the exit medium, which is simple to
compute because there is only a single ray (the transmitted ray),
rather than rays propagating in two different directions.
[0049] For a single layer stack the admittance of the initial
interface is related to the exit medium admittance according to the
following equations: ( E x ( f 1 ) H y ( f 1 ) ) = .times. ( s 11 s
12 s 21 s 22 ) .times. ( E x ( f 2 ) H y ( f 2 ) ) = .times. ( s 11
s 12 s 21 s 22 ) .times. ( E x ( 3 ) H y ( 3 ) ) .thrfore. ( E x (
f 1 ) / E x ( 3 ) H y ( f 1 ) / E x ( 3 ) ) = .times. ( s 11 s 12 s
21 s 22 ) .times. ( 1 - .eta. s ( 3 ) ) .thrfore. .eta. s ( f 1 ) =
.times. H y ( f 1 ) E x ( f 1 ) = s 11 + s 12 .function. ( - .eta.
s ( 3 ) ) s 21 + s 22 .function. ( - .eta. s ( 3 ) ) 8.
##EQU7##
[0050] In these equations, the superscript (3) indicates the exit
medium. The admittance of the exit medium for s-polarized light is
given by .eta..sub.s.sup.(3)= {square root over (.di-elect
cons..sub.3-.di-elect cons..sub.inc sin.sup.2(.theta..sub.inc))}.
The negative sign in front of .eta..sub.s.sup.(3) in the second
line results from defining light as propagating in the negative z
direction. For s-polarized light, the magnetic and electric fields
are of opposite signs in this case. For p-polarized light, they are
of the same sign. The 2.times.2 matrix for the single film is
described below.
[0051] For p-polarized light, the admittance of the initial
interface is arrived at in the same way: ( E y ( f 1 ) H x ( f 1 )
) = .times. ( p 11 p 12 p 21 p 22 ) .times. ( E y ( f 2 ) H x ( f 2
) ) = .times. ( p 11 p 12 p 21 p 22 ) .times. ( E y ( 3 ) H x ( 3 )
) .thrfore. ( E y ( f 1 ) / E y ( 3 ) H x ( f 1 ) / E y ( 3 ) ) =
.times. ( p 11 p 12 p 21 p 22 ) .times. ( 1 .eta. p ( 3 ) )
.thrfore. .eta. p ( f 1 ) = .times. H x ( f 1 ) E y ( f 1 ) = p 11
+ p 12 .times. .eta. p ( 3 ) p 21 + p 22 .times. .eta. p ( 3 ) 9.
##EQU8##
[0052] For p-polarized light, the admittance of the exit medium is
written as .eta..sub.p.sup.(3)=.di-elect cons..sub.3/ {square root
over (.di-elect cons..sub.3-.di-elect cons..sub.inc
sin.sup.2(.theta..sub.inc))}. The 2.times.2 matrices for s and p
polarizations are defined by: s 12 = s 22 = p 11 = p 22 = cos
.times. .times. ( .delta. film ) .times. .times. s 12 = - i .eta. s
( film ) .times. sin .times. .times. ( .delta. film ) .times.
.times. s 21 = - i .times. .times. .eta. s ( film ) .times. sin
.times. .times. ( .delta. film ) .times. .times. p 12 = i .eta. p (
film ) .times. sin .times. .times. ( .delta. film ) .times. .times.
p 21 = i .times. .times. .eta. p ( film ) .times. sin .times.
.times. ( .delta. film ) 10. ##EQU9##
[0053] In these equations, the admittances of the film are written
in the same form as the admittances of the exit medium given above,
but with .di-elect cons.film replacing .di-elect cons.3. The value
.delta.film is the phase thickness of the film, given by .delta.
film = 2 .times. .times. .pi. .times. .times. d film .times. film -
inc .times. sin 2 .function. ( .theta. inc ) .lamda. 0 11.
##EQU10## where dfilm is the physical thickness of the film and
.lamda.0 is the free-space wavelength of the incident light.
[0054] If there are multiple films, the matrix for a stack of films
is obtained from S = ( s 11 s 12 s 21 s 22 ) = inc exit .times. S
films 12. ##EQU11## where the product is over the 2.times.2
matrices of each individual film from the entrance to the exit. The
final product matrix is used as though it described a single
equivalent layer.
[0055] Notably in the preceding calculation, the matrices
describing the film are used as "transfer" matrices. This permits
propagation of the calculation of the admittance of the initial
interface down through a stack of films. The downward propagation
is stopped at the substrate because, once there are no longer rays
propagating in both directions, a simple form (the admittance of
the exit medium) can be written. Thus, the matrices allow an
impossible calculation to be related to a simplified calculation
via a 2.times.2 matrix.
[0056] A critical piece of understanding results from the following
discussion. Referring to Equation 8 above for s polarization,
-.eta..sub.s.sup.(3), is the admittance of the final interface
(thus the basis for preserving the negative sign). In comparison, p
polarization, .eta..sub.p.sup.(3), in Equation 6 is the admittance
of the final interface. Thus, both the s and p calculation of
admittance for the initial interface can be written in a generic
form: .eta. ( f 1 ) = H ( f 1 ) E ( f 1 ) = m 11 + m 12 .times.
.eta. ( f .omega. ) m 21 + m 22 .times. .eta. ( f .omega. ) 13.
##EQU12## where the matrix elements are for either the s or p
matrices, and .eta..sup.(f.omega.) is the admittance of the final
interface. The final interface is always chosen because a simple
expression for its admittance can be written in terms of the
admittance of the exit medium.
[0057] A Bootstrap Method according to an aspect of the invention
depends on finding experimental values for the complex reflectance
at a given interface in a film stack. Thus, an initial matter of
using the amplitude reflectance of a film surface alone to complete
the matrix calculation for the films above the film surface in
question will be described.
Solving the Top of the Stack
[0058] Turning now to a problem illustrated in FIG. 2, reflectance
at some interface 160 (mathematically, k') below a film stack
115A-X is assumed. Earlier layers 114A-X are shown "grayed out" and
with diagonal lines to indicate a vague idea of what those layers
114A-X are. The reflectance of the "known" layer 114A differs in
value from the reflectance described above (thus, a "prime" symbol
on k at the interface 116 indicates the different value). However,
it is still desirable to be able to compute the reflectance
spectrum of the stack of films 115A-X layered on top of the "known"
layer 114A.
[0059] It is possible to compute the spectrum of a film stack when
the amplitude reflectance at one interface at the bottom is known.
To appreciate how this calculation is done, it is useful to review
the known method for calculating reflectance and understand what
the assumptions are.
[0060] If the amplitude reflectivity for an interface k (which can
be any interface, including the final interface) is known, an
equivalent expression to Equation 13 can be obtained in terms of
the admittance of that interface in lieu of carrying the
calculation all the way down to the substrate. The admittance for
the k.sup.th interface can be written as follows: for .times.
.times. s .times. : .times. { ( E x ( f k ) H y ( f k ) ) = ( ( r k
' + 1 ) .times. E x ( i 2 ) ( r k ' - 1 ) .times. .eta. s ( 2 )
.times. E ^ x ( i 2 ) ) .eta. s ( f k ) = .eta. s ( 2 ) .times. ( r
k ' - 1 ) ( r k ' + 1 ) } .times. .times. for .times. .times. p
.times. : .times. { ( E y ( f k ) H x ( f k ) ) = ( ( 1 + r k ' )
.times. E y ( i 2 ) ( 1 - r k ' ) .times. .eta. p ( 2 ) .times. E y
( i 2 ) ) .eta. p ( f k ) = .eta. p ( 2 ) .times. ( 1 + r k ' ) ( 1
- r k ' ) } 14. ##EQU13##
[0061] In this expression for the admittance of the known
interface, "2" indicates the admittance of the film deposited
directly on the known interface. This can then used as the starting
point for computing the reflectance of a stack of films above the
known interface. The admittance of the top interface can be written
as: .eta. s ( f 1 ) = H y ( f 1 ) E ^ x ( f 1 ) = s 21 ' + s 22 '
.times. .eta. s ( f k ) s 11 ' + s 12 ' .times. .eta. s ( f k ) = s
21 ' + s 22 ' .times. .eta. s ( 2 ) .function. ( r k ' - 1 r k ' +
1 ) s 12 ' + s 12 ' .times. .eta. s ( 2 ) .function. ( r k ' - 1 r
k ' + 1 ) .times. .times. .eta. p ( f 1 ) = H x ( f 1 ) E ^ y ( f 1
) = p 21 ' + p 22 ' .times. .eta. p ( f k ) p 11 ' + p 12 ' .times.
.eta. p ( f k ) = p 21 ' + p 22 ' .times. .eta. p ( 2 ) .function.
( 1 - r k ' 1 + r k ' ) p 11 ' + p 12 ' .times. .eta. p ( 2 )
.function. ( 1 - r k ' 1 + r k ' ) 15. ##EQU14##
[0062] In addition to the change in definition for the terminal
interface of the calculation, another difference is the matrix
elements come from a modified 2.times.2 matrix for the film stack.
The modified matrix is computed as: S ' = ( s 11 ' s 12 ' s 21 ' s
22 ' ) = inc k + .times. S films .times. .times. P ' = ( p 11 ' p
12 ' p 21 ' p 22 ' ) = inc k + .times. P films 16. ##EQU15## where
the product is taken in the order of incident light penetrating the
stack as before, but the calculation ends with the film deposited
directly onto the known interface. The symbol k+ in Equation 16 is
used to indicate that the product terminates with a layer 115A
directly above the known interface 114A. The change is a
significant one, in that the 2.times.2 matrices for any of the
films below the known interface 114A no longer have to be
computed.
[0063] Returning to Equation 6, the following can be expressed: r s
( f 1 ) = .eta. s ( inc ) .times. s 11 ' .function. ( 1 + r k ' ) -
.eta. s ( inc ) .times. .eta. s ( 2 ) .times. s 12 ' .times. ( 1 -
r k ' ) + s 21 ' .function. ( 1 + r k ' ) + .eta. s ( 2 ) .times. s
22 ' .function. ( 1 - r k ' ) .eta. s ( inc ) .times. s 11 '
.function. ( 1 + r k ' ) - .eta. s ( inc ) .times. .eta. s ( 2 )
.times. s 12 ' .times. ( 1 - r k ' ) - s 21 ' .function. ( 1 + r k
' ) + .eta. s ( 2 ) .times. s 22 ' .function. ( 1 - r k ' ) .times.
.times. r p ( f 1 ) = .eta. p ( inc ) .times. p 11 ' .function. ( 1
+ r k ' ) + .eta. p ( inc ) .times. .eta. p ( 2 ) .times. p 12 '
.function. ( 1 - r k ' ) - p 21 ' .function. ( 1 + r k ' ) - .eta.
p ( 2 ) .times. p 22 ' .function. ( 1 - r k ' ) .eta. p ( inc )
.times. p 11 ' .function. ( 1 + r k ' ) + .eta. p ( inc ) .times.
.eta. p ( 2 ) .times. p 12 ' .function. ( 1 - r k ' ) + p 21 '
.function. ( 1 + r k ' ) + .eta. p ( 2 ) .times. p 22 ' .function.
( 1 - r k ' ) 17. ##EQU16##
[0064] Note that these equations feature a complex quantity called
r'.sub.k, which, as mentioned above, is not the same as r.sub.k,
the amplitude reflectance of the top of the film stack before the
topmost layer was added. The two things are related to one another,
however, as evident in the following discussion.
Amplitude Reflectance of a Buried Interface
[0065] As discussed above, the magnitude of the reflectance of the
interface 116 (mathematically, k) in air can be learned by
measuring its intensity reflectance but not the phase of the
reflectance in the complex plane. With reference to FIG. 3, when
the interface 116 (k) is covered by another material 115, the known
reflectance is changed. If how the reflectance changes cannot be
computed in a simple way, then having learned anything about that
reflectance is of no use. Fortunately, there is a straightforward
way to relate this to the amplitude reflectance of the interface in
air.
[0066] In Optical Properties of Thin Solid Films (Dover
Publications, Inc., Mineola, USA, 1991), O. S. Heavens gives an
expression for the amplitude reflectance of a film in terms of the
reflectance of the two interfaces of the film: r film = r top + r
bot .times. e - I2.delta. 1 + r top .times. r bot .times. e -
I2.delta. 18. ##EQU17## where .delta. is the optical phase change
.delta. = 2 .times. .pi. .times. .times. d .times. 2 - inc .times.
sin 2 .function. ( .theta. inc ) .lamda. 0 , ##EQU18## directly
proportional to the physical thickness of the film. If rtop is
replaced with r2 (the Fresnel coefficient for reflectance off an
infinite slab of the film material with dielectric constant
.di-elect cons.2), and rbot is allowed to be r'.sub.k, the
reflectance of the multilayer stack when the entrance medium is an
infinite slab of film material, then the reflectance of the film's
top interface can be written as: r .function. ( f 1 ) = r 2 + r k '
.times. e - I2.delta. 2 1 + r 2 .times. r k ' .times. e - I2.delta.
2 19. ##EQU19##
[0067] When the thickness of the film goes to zero, the exponential
equals 1 and the film reflectance must be identical to r.sub.k.
This allows r'.sub.k to be solved in terms of r.sub.k as: r k ' = r
k - r 2 1 - r 2 .times. r k 20. ##EQU20##
[0068] This provides an estimate of r'.sub.k that is partially
independent of the preceding film stack, since r.sub.2 does not
depend on it at all and r.sub.k has been modified, keeping only the
phase determined by the film stack calculation.
The Bootstrap Method.
[0069] In light of the foregoing introduction, a Bootstrap method
for film deposition and refinement is described in the following
sections; more particularly, steps to perform Bootstrap refinement
of the optical model of a thin film stack are provided as
follows.
[0070] Step 1. Determine the reflectance of an existing film stack
prior to the deposition of a new layer.
[0071] The reflectance of a film stack provides some information
regarding the complex amplitude reflectance that can be used to
refine the model of the reflectance, and that is totally
independent of any modeling. If one does not measure reflectance
directly, it can be obtained by noting that transmission plus
reflectance for an absorption-free thin film stack is unity.
[0072] The relationship between the amplitude and intensity
reflectance is that the intensity reflectance is the absolute
square of the amplitude reflectance. Considering the amplitude
reflectance for a moment, it will be clear that it can be expressed
in standard Cartesian coordinates on a complex plane, or in complex
polar coordinates: r k = a + I .times. .times. b = r k .times. e
I.PHI. k .times. .times. r k = a 2 + b 2 .times. .times. .PHI. k =
tan - 1 .function. ( b a ) . 1 ##EQU21##
[0073] If the amplitude reflectance is expressed in polar
coordinates, it is the magnitude of the amplitude reflectance that
is provided by a measure of intensity reflectance, |r.sub.k|=
{square root over (R.sub.k)}.
[0074] It is possible at this point to replace the magnitude of the
amplitude reflectivity in Equation 1, |r.sub.k|, with the square
root of the intensity reflectance. This can be done whenever the
anticipated error in future reflectance values due to errors in
this initial measurement of R.sub.k is expected to be small. How to
obtain this relationship is shown in the following.
Calculating the Worst-case Future Reflectance
[0075] The standard deviation of the magnitude of the amplitude
reflectance is given by Equation 3: r k = R k .times. .times. r k =
1 R k .times. R k .times. .times. .sigma. r k = .sigma. R k R k . 3
##EQU22## In the worst-case scenario, Rk is a minimum (the
amplitude reflection is on the real axis nearest the origin). This
would make the error in magnitude relatively larger. Further, the
next layer could result in this vector being advanced by .pi.,
crossing the real axis at the furthest point from the origin,
producing a reflectance maximum. Again, in the worst case scenario,
the maximum reflectance that could be generated as a result of the
observed Rk is: R max .function. ( max ) = ( ( 1 + r 2 2 ) .times.
R k 1 / 2 - 2 .times. r 2 1 + r 2 2 - 2 .times. r 2 .times. R k 1 /
2 ) 2 . 4 ##EQU23##
[0076] To assure that replacing the reflectance amplitude with a
measured value does not affect future reflectance measurements by
more than the standard deviation of the reflectance measurement,
the worst-case scenario must be known. This is: .sigma. R max
.function. ( max ) = ( r 2 2 - 1 ) 2 .times. ( 1 + r 2 2 - 2
.times. r 2 .times. R k - 1 / 2 ) ( 1 + r 2 2 - 2 .times. r 2
.times. R k 1 / 2 ) 3 .times. .sigma. R k . 5 ##EQU24## Solving
Equation 5 for a factor of 1.sigma. is not easy, and the result is
very complicated. However, a numerical solution is straightforward.
For silica (r2=-0.2), this value is about Rk=0.19 or 19%
reflectance. For niobia (r2=-0.4), the value is about 9%
reflectance. In other words, when about to deposit a silica layer,
values of |r.sub.k| should not be adjusted when the measured
reflectance is less than 19%. When about to deposit a niobia layer,
values should not be replaced when the measured reflectance is less
than 9%. Instead, assume the modeled reflectance is more accurate
in these cases, although it is not necessary to do this often. In
the following sections, reasons are discussed to perform
bootstrapping only on high-index layers, so by extension, this step
is recommended only when a low-index layer is completed. [0077]
Step 2: Replace the magnitude of the amplitude reflectance at each
wavelength with {square root over (R.sub.k)} whenever measuring a
freshly completed silica film with an intensity reflectance greater
than 9% or a low-index film with an intensity reflectance greater
than the limiting value of the high-index material.
[0078] While the intensity measurement provides useful information
(most of the time) about the magnitude of reflectance, it
unfortunately provides no information about the phase angle in the
complex plane, .phi.. Much of the remainder of the present
description relates to how to obtain these phase angles in at least
some circumstances.
Monitor Curves Can Give Non-Redundant Calculations of Phase
[0079] In most cases, the magnitude of r.sub.k at the base of a
niobia layer can be obtained from the measured reflectance of the
film stack terminating in a fresh silica layer. The phase of the
amplitude reflectance is more difficult to ascertain, but there are
two general approaches. The first is to consider what values of
phase are consistent with the final value of reflectance after the
next layer is added. To use this information, the optical thickness
of the next layer must be known. This is sometimes a redundant
calculation since the estimation of optical thickness is usually
based on an understanding of the initial reflectance. This is, in
fact, a weakness of the usual matrix modeling approach--the
calculation is somewhat redundant.
[0080] Without additional information, redundant calculation would
normally be the only option. However, monitor curves are usually
recorded during deposition, and those curves contain all the
information necessary to compute the phase, .phi..sub.k, without
the need for redundant calculations. This involves the use of
reflectance maxima in the monitor curves. [0081] Step 3. Consider
the modeled monitor curve for each wavelength of a niobia
(high-index) layer. Discard any monitor curves without maxima in
their reflectance during the niobia layer deposition. If none meet
this criterion, deposit the layer using a pure model approach.
[0082] Based on Equation 14 above, the maxima and minima of a
monitor curve can be shown to depend solely on |r'.sub.k|, the
magnitude of the buried interface's reflectance, and not at all on
its phase. The maximum and minimum reflectance during the monitor
curve are given by Equation 21: R max = ( r k ' + r 2 1 + r 2
.times. r k ' ) 2 .times. .times. R min = ( r k ' - r 2 1 - r 2
.times. r k ' ) 2 . 21 ##EQU25##
[0083] Therefore, R.sub.max or R.sub.min can be used in the monitor
curve to convey the magnitude of the buried reflectance. This
magnitude can be related to the reflectance as follows. r k ' = R
max - r 2 1 - r 2 .times. R max .times. .times. or .times. .times.
r k ' = R min + r 2 1 + r 2 .times. R min ; r 2 - R min 1 - r 2
.times. R min . 22 ##EQU26## Thus, from a monitor curve covering at
least a quarter wave at the monitor curve wavelength, |r'.sub.k|
can be determined.
[0084] A caveat to using these equations is as follows. The
R.sub.min expression has two failings. First, there are two
possible solutions for |r'.sub.k| based on R.sub.min depending on
whether |r'.sub.k| is less than or greater than |r.sub.2|. If it is
less than |r.sub.2|, the right-hand solution is appropriate. If it
is greater than |r.sub.2|, the left-hand solution is appropriate.
The R.sub.max expression also has two solutions in principle but
can be discarded because it provides nonphysical results. The
second problem with the equation from R.sub.min is the issue of
experimental error. The error expected in the estimation of
|r'.sub.k| is related to the error in measurement of R.sub.min and
R.sub.max by Equation 23: r k ' = ( 1 - r 2 2 ) 2 .times. R min
.times. ( 1 + r 2 .times. R min ) 2 .times. R min .apprxeq. R min 2
.times. R min .times. .times. r k ' = ( 1 - r 2 2 ) 2 .times. R max
.times. ( 1 - r 2 .times. R max ) 2 .times. R max .apprxeq. R max 2
.times. R max . 23 ##EQU27## In other words, the error goes up as
the key reflectance diminishes. Since the minimum is, by
definition, smaller than the maximum, the error expected in
estimating |r'.sub.k| goes up accordingly. Thus, for both reasons,
the calculation of magnitude from a maximum reflectance (i.e., a
minimum in the transmission monitor curve) is preferred. Choosing
the Best Monitor Wavelength
[0085] The phase angle .phi.k at the monitor wavelength can be
determined from |r'.sub.k| and the Rk as: cos .function. ( .+-.
.PHI. k ) = r k ' 2 .times. ( 1 + R k .times. r 2 2 ) - r 2 2 - R k
2 .times. r 2 .times. R k .times. ( 1 - r k ' 2 ) . 25 ##EQU28##
This, of course, provides 2 solutions. Once the phase angle .phi.k
from Equation 25 is determined, r'.sub.k can be computed using
Equation 20 and a monitor curve can be generated if desired for
comparison with the actual to help determine which solution is
better. Afterwards, a value for phase at the monitor wavelength
should have been obtained that is as correct as possible. It
depends, of course, on accurately measuring the maximum reflectance
value and Rk. Thus, not all wavelengths are created equal as
potential monitor wavelengths. A full-spectrum monitor (acquiring
many monitor wavelengths) is the best solution, but if only a
single wavelength is available, then there is a systematic approach
to choosing the best.
[0086] Equation 25 depends, ultimately, on only two measurements:
the measurement of the initial reflectance and the measurement of
the maximum reflectance. For those wavelengths that exhibit a
maximum reflectance during deposition, these can be evaluated
quantitatively as possible monitor wavelengths.
[0087] It can be shown that the anticipated standard deviation of
the phase calculation can be written as: .sigma. .PHI. k = A
.times. .times. .sigma. R k 2 + B .times. .times. .sigma. R max 2
.times. .times. A = ( ( 1 + r 2 2 ) .times. R k + 2 .times. r 2
.function. ( 1 + R k ) .times. R max + ( 1 + r 2 2 ) .times. R max
) 2 4 .times. R k 2 .function. ( 4 .times. r 2 2 .times. R k
.function. ( 1 - R max ) 2 - ( 2 .times. r 2 .times. R max + ( 1 +
r 2 2 ) .times. R max - R k .function. ( 1 + r 2 2 + 2 .times. r 2
.times. R max ) ) 2 ) .times. .times. B = ( 1 - R k ) 2 .times. ( r
2 + ( 1 + r 2 2 ) .times. R max + r 2 .times. R max ) 2 R max
.function. ( 1 - R max ) 2 .times. ( 4 .times. r 2 2 .times. R k
.function. ( 1 - R max ) 2 - ( 2 .times. r 2 .times. R max + ( 1 +
r 2 2 ) .times. R max - R k .function. ( 1 + r 2 2 + 2 .times. r 2
.times. R max ) ) 2 ) . 27 ##EQU29##
[0088] This equation is helpful selecting the best monitor
wavelength for the purpose of determining the phase of the
amplitude reflectivity at the monitor wavelength.
[0089] FIG. 4 shows a representation of the regions of permissible
values of Rk and Rmax, with a color code for the lowest values of
.sigma..sub..phi.k for silica (assuming r2=-0.2 and the standard
deviation of the reflectance measurements is 0.003). The lowest
value possible under these conditions is 0.0262 radians (1.5
degrees). The lower axis, Rmax, represents possible values of Rmax,
while the left axis, Rk, gives possible values of Rk. Note that
large regions of reflectance are not possible--there are many
combinations of Rk and Rmax that cannot coexist. On the boundaries
of those disallowed regions, the error in estimating the phase
angle, .phi.k, becomes infinite.
[0090] The same plot for niobia, assuming r.sub.2=-0.4, is given in
FIG. 5. The minimum value of error here under the same conditions
is 0.00875 radians (0.5 degrees)--a much better phase
calculation.
[0091] It is possible to develop a histogram of the number of
combinations of allowed Rmax and Rk that provide a specific level
of error in .phi.k. This is accomplished first by considering the
range of possible Rmax values when depositing a layer: it cannot,
as FIGS. 4 and 5 illustrate, be less than the reflectance of the
thin film material being deposited itself (e.g., Rmax for niobia in
FIG. 5 cannot be less than -0.4.sup.2=0.2 as shown in a leftmost
portion of FIG. 5 in dark yellow). Possible values of Rk can be
evaluated for each value of Rmax, and the values of phase precision
those values provide can be determined using Equation 27. This is
accomplished by dividing the phase thickness of the top layer into
increments and computing the reflectance at each increment (this is
done because the reflectances at the turning points are more likely
than those in between the turning point values). In the end, a
histogram of the resulting precision values can be formed and a
determination made as to how likely each will appear for a given
film material. This has been done for silica (assuming
r.sub.2=-0.2, and reflectance standard deviations of 0.003) and for
niobia (assuming r.sub.2=-0.4) as shown respectively in FIGS. 6 and
7.
[0092] For silica, 30% of all observed combinations will have a
phase error given by Equation 24 that is less than 2.4 degrees. No
values less than about 1.5 degrees error in .phi.k is possible for
silica under these conditions. For niobia, the same fraction will
have .phi.k errors less than 0.9 degrees, as illustrated in the
FIG. 7, also derived from a full numeric simulation.
[0093] Thus, the calculated phase error of possible monitor
wavelengths will tend to be considerably better for niobia films
than for silica. If a limit of 0.9 degrees phase error is placed on
the monitors before this calculation is performed, only niobia will
give possible monitor wavelengths, and 30% of all wavelengths
(overall) will meet this criterion. On some layers, it is possible
that no wavelengths will meet this criterion, while on others many
may do so. [0094] Step 4. For the remaining possible monitor
wavelengths in a niobia layer deposition, determine the anticipated
standard deviation in .phi.k. Discard any with cy greater than 0.9
degrees (0.016 radians). If none remain, proceed with a pure model
deposition.
[0095] For low-index layers, bootstrapping is not recommended. For
layers with larger magnitudes of r.sub.2 (such as niobia), the
precision of the bootstrap is almost always better, but there is no
guarantee that a specific layer will include a set of R.sub.k and
R.sub.max values anticipated to provide excellent precision in
calculating the phase .phi.k. If no wavelength with an anticipated
reflectance maximum meets this criterion, modeling alone should be
relied upon to deposit the layer until a suitable bootstrap layer
is reached.
[0096] An ingenious characteristic about this calculation is that
it provides .phi.k--and thus also .phi.'.sub.k--that is consistent
with the monitor curve and is independent of .delta., the phase
thickness of the film. Once a valid solution for rk is determined,
the valid solution for r'.sub.k can be obtained. Thus, the final
transmission value at the monitor curve wavelength can be used to
determine what value of .delta. is most accurate for the monitor
wavelength.
[0097] If a wavelength meets the criterion specified above, then
consideration should be given as to whether the anticipated end of
the layer will have a reflectance, Rf, suitable for estimating
.delta., the phase thickness of the layer with some precision. This
calculation is performed with Equation 26, where .delta. depends on
Rf and rk'. .delta. = .PHI. k ' 2 .+-. 1 2 .times. cos - 1
.function. ( R f + R f .times. r 2 2 .times. r k ' 2 - r 2 2 - r k
' 2 2 .times. r 2 .times. r k ' .times. ( 1 - R f ) ) . 26
##EQU30##
[0098] This equation provides fairly unique solutions for .delta.
that can be used to correct the reflectance at all wavelengths. The
possible solutions for .delta. that are obtained can be tested
against the observed monitor curve to determine which is
correct.
[0099] For a given monitor curve with a given expectation of Rmax,
and with a known value of r2, the sensitivity of .delta. to errors
in Rf can be determined according to Equation 28: .sigma. .delta. =
( 1 - r 2 2 ) .times. ( 1 - R max ) 2 .times. ( R f - 1 ) .times. A
.times. .sigma. R f .times. .times. A = ( R max - R f ) .times. ( 2
.times. r 2 2 .times. ( R max .times. R f + R f - R max - 2 ) - ( 1
+ r 2 4 ) .times. ( R max - R f ) + 4 .times. ( r 2 + r 2 3 )
.times. ( 1 - R f ) .times. R max ) . 28 ##EQU31##
[0100] If the type of numerical analysis above using Equation 28 is
repeated, a plot as shown in FIG. 8 is obtained (shown as the
logarithm of error vs. Rmax and .delta. because otherwise the scale
would be difficult to see). This plot is made over the entire range
of possible values of Rmax between r.sub.2.sup.2 and 1 (on the
receding axis) and .delta. angle between 0 and 2.pi. (the front
axis).
[0101] The data in FIG. 8 can also be rendered as a histogram as
shown in FIG. 9. The histogram in FIG. 9 implies that the error in
phase thickness is usually satisfactory compared to the error in
.phi.k. At a cutoff of 0.9 degrees error in .delta., about 51% of
the remaining monitor wavelengths for niobia, for instance, should
be usable. Thus, given a good monitor wavelength for determining
the phase angle .phi.k, there is a good chance of having one that
also provides a good precision in .delta..
[0102] To conserve time, and when modeling is running fairly well,
it is reasonable to only select monitor wavelengths for niobia (in
a silica/niobia stack), and only when they meet these two criteria
(standard deviation of .phi.k<0.9 degrees, and standard
deviation of .delta.<0.9 degrees at the end of the layer). When
no monitor wavelengths meet these criteria, it is reasonable to
proceed with a pure model matrix approach to depositing the next
layer. [0103] Step 5. Compute the expected error in .delta. for the
remaining wavelengths at the target thickness of the niobia layer.
If no wavelengths have an error less than 0.9 degrees, proceed with
a full model deposition of the layer. If some do meet this
criterion, select the lowest error in this category. Reflectance of
a New Film and Determination of the Old Phase at Wavelengths Other
than the Monitor Wavelength.
[0104] Why bother with determining the .phi.k and .delta. from a
monitor curve as precisely as possible? First, if a monitor
wavelength for bootstrapping has been selected successfully, all
connection to the previous dependence on the matrix calculation for
the monitor wavelength may be avoided. For all other wavelengths,
there is at least the opportunity to determine the magnitude of rk,
but only the original modeled estimate of phase. The question
arises: how to "repair" the phases of all other wavelengths? Since
in a single-channel monitor there are no monitor curves at those
wavelengths, the phase at each wavelength cannot be directly
obtained. (If there was spectrograph recording all wavelengths all
of the time, such as with an FTIR system, the steps provided below
would not be needed). However, to reach this point in
bootstrapping, there must be a good value for .delta. for the
monitor wavelength, plus good values of Rmax and Rk. With this, at
least some of the other measurements can be "fixed" to accord with
measurements already taken. This will restrict errors to those of a
single layer at the worst for those wavelengths where the following
calculation is possible.
[0105] From .delta. for the monitor wavelength, the physical
thickness of the layer consistent with the modeled refractive index
of the film material can be estimated. This physical thickness and
the modeled refractive index of the film can be used to estimate
.delta. for all other wavelengths. [0106] Step 6. Compute the value
of .delta. for all wavelengths based on the value calculated for
the monitor wavelength.
[0107] Returning to Equation 16 and replacing rk' with the
definition in Equation 20, the following is obtained: r f = r 2
.function. ( 1 - r 2 .times. r k ) + ( r k - r 2 ) .times. ( cos
.function. ( 2 .times. .delta. ) - I .times. .times. sin .function.
( 2 .times. .delta. ) ) 1 - r 2 .times. r k + r k .function. ( r k
- r 2 ) .times. ( cos .function. ( 2 .times. .delta. ) - I .times.
.times. sin .function. ( 2 .times. .delta. ) ) . 29 ##EQU32##
[0108] In Equation 29, the exponential has been replaced with a
trigonometric expression using Euler's relation. If r.sub.k is
replaced with a+i b, a conventional expression for r.sub.f can be
obtained. The complex conjugate of r.sub.f can be formed and the
product taken of the two. This provides the intensity reflectance
of the top interface in terms of values from the new film, plus a
and b. The following can then replace a and b: a=cos (.phi..sub.k)
{square root over (R.sub.k)} b=sin (.phi..sub.k) {square root over
(R.sub.k)} (30.) where use is made of the intensity reflectance
measured in vacuum for interface k to represent the magnitude of
the amplitude reflectance vector at the interface k in vacuum.
[0109] The resulting expression can be simplified as Equation 31:
.times. R f = 2 .times. r 2 2 + R k .function. ( 1 + r 2 4 ) + 2
.times. r 2 .times. Q 1 + r 2 4 + 2 .times. r 2 2 .times. R k + 2
.times. r 2 .times. Q .times. .times. Q = r 2 2 .times. R k 1 / 2
.times. cos .function. ( 2 .times. .delta. + .PHI. k ) + R k 1 / 2
.times. cos .function. ( 2 .times. .delta. - .PHI. k ) - r 2
.function. ( 1 + R k ) .times. cos .function. ( 2 .times. .delta. )
- ( 1 + r 2 2 ) .times. R k 1 / 2 .times. cos .function. ( .PHI. k
) . 31 ##EQU33##
[0110] In this expression, everything is known EXCEPT the phase
angle .phi. at interface k, allowing it to be determined from the
measurement of intensity reflectance at the subsequent
interface.
[0111] Dispensing with numerical solution methods, this expression
can be solved for the Cosine of the angle: .times. cos .function. (
.PHI. k ) = ( A .function. ( 1 + r 2 2 ) .times. sin .function. (
.delta. ) .+-. B .times. .times. cos .function. ( .delta. ) C )
.times. .times. A = R f + r 2 4 .function. ( R f - R k ) - R k + 2
.times. r 2 2 .times. ( ( 1 - R f ) .times. ( 1 + R k ) .times. cos
.times. ( 2 .times. .delta. ) - ( 1 - R f .times. R k ) ) .times.
.times. .times. B = D .function. ( 1 + r 2 12 ) + F .function. ( r
2 2 + r 2 10 ) + G .function. ( r 2 4 + r 8 2 ) + H .times. .times.
r 6 2 .times. .times. .times. C = sin .function. ( .delta. )
.times. ( 4 .times. r 2 .function. ( 1 - R f ) .times. R k 1 / 2
.function. ( 2 .times. r 2 2 .times. cos .function. ( 2 .times.
.delta. ) - 1 - r 2 4 ) ) .times. .times. .times. D = - ( R f - R k
) 2 .times. .times. .times. F = 2 .times. ( R k .function. ( 2 + R
k ) + R f 2 .function. ( 1 + 2 .times. R k ) + 2 .times. R f
.function. ( 1 - 5 .times. R k + R k 2 ) - 2 .times. ( 1 - R f )
.times. ( 1 - R k ) .times. ( R f + R k ) .times. cos .function. (
2 .times. .delta. ) ) .times. .times. G = - 6 - 4 .times. R f - 5
.times. R f 2 - 4 .times. R k + 38 .times. R f .times. R k - 4
.times. R f 2 .times. R k - 5 .times. R k 2 - 4 .times. R f .times.
R k 2 - 6 .times. R f 2 .times. R k 2 + 8 .times. ( 1 - R f 2 )
.times. ( 1 - R k 2 ) .times. cos .function. ( 2 .times. .delta. )
- 2 .times. ( 1 - R f ) 2 .times. ( 1 - R k ) 2 .times. cos .times.
( 4 .times. .delta. ) .times. .times. .times. H = 4 .times. ( 3 + 2
.times. R f 2 - 10 .times. R f .times. R k + 2 .times. R k 2 + 3
.times. R f 2 .times. R k 2 - 2 .times. ( 1 - R f ) .times. ( 1 - R
k ) .times. ( 2 + R f + R k + 2 .times. R f .times. R k ) .times.
cos .function. ( 2 .times. .delta. ) + ( 1 - R f ) 2 .times. ( 1 -
R k ) 2 .times. cos .function. ( 4 .times. .delta. ) ) 32
##EQU34##
[0112] In principle, Equation 32 can be used to solve for the phase
angle. By doing so, the deposition system process control is
effectively "bootstrapped" by detaching the system completely from
everything that came before the last layer. Equation 32 provides
four (4) solutions for the phase angle; two come from the +/-
portion of the calculation; two more from the fact that cosine is
an even function, so positive and negative angles both work equally
well. However, only two of these solutions are consistent with the
measured value of R.sub.f. Thus, solutions should be checked via
Equation 31 for validity. Only two solutions should be left after
this process is complete. [0113] Step 7. Compute the two possible
values of phase angle for each wavelength other than the monitor
wavelength. Estimating Error for Non-Monitor Wavelengths
[0114] The phase angle is dependent on three reflectivity
measurements (R.sub.max, R.sub.f and R.sub.k), those being
scrambled together in Equation 32. While this works well for a
hypothetical system with no noise, a real spectrometer exhibits
errors in measurement of the intensity transmittance.
[0115] Based on work already done, the analysis of Equation 32 is
fairly straightforward for extending phase information to other
wavelengths. The following expression can be constructed from it:
.PHI. k = - 1 sin .function. ( .PHI. k ) .times. ( ( .differential.
cos .function. ( .PHI. k ) .differential. R k ) .times. R k + (
.differential. cos .function. ( .PHI. k ) .differential. R f )
.times. R f + ( .differential. cos .function. ( .PHI. k )
.differential. .delta. ) .times. .delta. ) .times. .times. .times.
.sigma. .PHI. k .apprxeq. .sigma. R sin .function. ( .PHI. k )
.times. ( ( .differential. cos .function. ( .PHI. k )
.differential. R k ) 2 + ( .differential. cos .function. ( .PHI. k
) .differential. R f ) 2 ) 1 / 2 . 33 ##EQU35##
[0116] In Equation 33 error in phase thickness has been omitted
from the calculation since it is of minor concern at this stage.
The phase thickness was settled previously for the sake of
argument; therefore, making that approximation, computing the sine
of the angle and the sensitivity of the cosine functions to errors
in R.sub.k and R.sub.f remain. There is a subtle issue to be
considered at this point. What angle should be focused on--the
modeled angle or the computed angle using Equations 31/32? In
principle, both values are available.
[0117] Thus, the conservative answer to the foregoing question is
"both". Assume, for example, that the model has a complex
reflectance rk at a given wavelength from which the phase angle
.phi.k and the predicted value of Rk can be obtained. Using a best
estimate of the phase thickness at the wavelength, Rf can be
computed using Equation 31. Next, compute the phase angle using
Equation 32 after varying Rk and Rf each by a small amount--e.g.,
by 10.sup.-5, 1/100.sup.th of a percent transmission. Since the
modeled phase is known exactly, it will be trivial to identify
which results are the ones closest to the modeled phase; therefore,
the sensitivities are computed as: .differential. cos .function. (
.PHI. k ) .differential. R k .apprxeq. cos .function. ( .PHI. k
.function. ( R k , m , R f , m , .delta. ) ) - cos .function. (
.PHI. k .function. ( R k , m - 10 - 5 , R f , m , .delta. ) ) 10 -
5 34 .differential. cos .function. ( .PHI. k ) .differential. R f
.apprxeq. cos .function. ( .PHI. k .function. ( R k , m , R f , m ,
.delta. ) ) - cos .function. ( .PHI. k .function. ( R k , m , R f ,
m - 10 - 5 , .delta. ) ) 10 - 5 . ##EQU36##
[0118] From the model, the sine of the phase angle that appears in
Equation 33 is trivial to obtain.
[0119] Thus, the standard deviation of the phase calculation can be
estimated from the model. If the model were absolutely trustworthy,
this would be sufficient; however, this is not the case. [0120]
Step 8. Using information extracted from the model for r.sub.k at
each wavelength and the computed best value of .delta., compute the
estimated standard deviation of phase at all wavelengths except the
monitor.
[0121] Since the model is not completely trustworthy, the process
is repeated using values taken from the calculated value of phase
closest to the model value of phase. In other words, for the two
possible values of .phi.k, the one closest numerically to the model
phase is selected. Now, keying on that value, compute the
sensitivities according to Equation 34 using the measured values of
R.sub.k, R.sub.f and the estimate of .delta. for the wavelength
being tested. [0122] Step 9. Using the computed phase closest to
the model phase for r.sub.k at each wavelength, the measured
R.sub.f and R.sub.k values and the computed best value of .delta.,
compute the estimated standard deviation of phase at all
wavelengths for which the magnitude of r.sub.k was estimated (Step
2) other than the monitor.
[0123] Now, compute the estimated error in phase as follows:
.sigma..sub..phi.k(estimated)=(.sigma..sub..phi.k.sup.2(model)+.sigma..su-
b..phi.k.sup.2(calculated)).sup.1/2 35.
[0124] What are the probable limits to this calculation? A bit of
experimentation suggests that the value of standard deviation in
phase angle could be between about that of the monitor measurement
(on the low end) to nearly infinite. A sufficient approach is to
replace the modeled values with calculated values only if a rather
conservative cutoff is obtained. If the error estimates are less
than about 1.3 degrees from Equation 35 (square root of two times
0.9), and if the reflectance was greater than 9 percent at the
start of the layer (Step 2), then the model and calculated values
are averaged together to obtain a new estimate. If either the phase
error OR the reflectance criteria are not met, the estimate is
returned alone to the model. [0125] Step 10. If the phase error
estimate is less than 1.3 degrees AND the criterion of Step 2 is
met, average the calculated and modeled reflectance and phase
values to obtain a new value that will be used in all future
modeling at that wavelength.
[0126] A further refinement of this approach is to fit the phase
values obtained above to a Kramers-Kronig model of the phase to
fill in values of the phase that have not been determined
previously.
[0127] While preferred embodiments of the invention have been shown
and described, those of ordinary skill in the art will recognize
that changes and modifications may be made to the foregoing
examples without departing from the scope and spirit of the
invention. Furthermore, those of ordinary skill in the art will
appreciate that the foregoing description is by way of example
only, and is not intended to limit the invention so further
described in such appended claims. It is intended to claim all such
changes and modifications as fall within the scope of the appended
claims and their equivalents.
* * * * *