Method and apparatus for measuring film thickness in multilayer thin film stack by comparison to a reference library of theoretical signatures

The thicknesses of a first layer and of a second layer on a semiconductor wafer can be measured together by assuming that the second layer has a substantially uniform thickness. The thicknesses are measured by measuring reflectivity as a function of wavelength at a plurality of points on the wafer to provide a plurality of signatures, comparing each signature with signatures from libraries of theoretical signatures by calculating an error value associated with each signature; and determining the minimum error value. Each library is based upon a unique assumed thickness of the second layer. Thus, the thickness of the second layer is determined by identifying the library associated with the minimum error value.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to apparatus and method for fabricating thin 
films. More specifically, the present invention relates to a method and 
apparatus for measuring the thickness of thin films. 
While the present invention is described herein with reference to 
illustrative embodiments for particular applications, it should be 
understood that the invention is not limited thereto. Those having 
ordinary skill in the art and access to the teachings provided herein will 
recognize additional modifications, applications, and embodiments within 
the scope thereof and additional fields in which the present invention 
would be of significant utility. 
2. Description of the Related Art 
A silicon-on-insulator (SOI) semiconductor wafer typically includes a 
sandwich structure fabricated by growing a silicon dioxide film on one 
surface of each of two silicon wafers and bonding the two silicon dioxide 
film surfaces together at high temperature. Other materials such as, for 
example, silicon nitride, may be used for the insulator material and other 
materials may be used for the wafer material. One of the two outer silicon 
surfaces of the sandwich structure is typically mechanically ground and 
polished to an average thickness of several microns. This mechanical 
process unfortunately results in large spatial variations in the thickness 
of this outer silicon layer over the surface of the wafer. To reduce these 
spatial variations, a thickness error map that indicates thickness 
non-uniformities of this outer silicon layer over the entire wafer 
surface, is typically required for subsequent manufacturing operations, 
e.g. micro-polishing. The steps of measuring the spatial variations in the 
thickness of the outer silicon layer followed by thinning and smoothing 
the surface by micro-polishing may need to be performed several times 
before the entire outer silicon layer achieves the desired thickness. In 
order to reduce costs and increase production, a measurement of at least 
400 points on a wafer surface in 60 seconds is desirable. 
Current commercial instruments, however, typically provide film thickness 
measurements at only a single point on a surface. These instruments use a 
focused lens or a fiber bundle to locally illuminate the film surface with 
a beam of monochromatic light. A grating or prism spectrograph is 
typically used to measure the surface spectral reflectance at each point. 
This surface spectral reflectance data must be numerically corrected due 
to variations in the angle of incidence caused by the illuminating beam 
f-number. 
These commercial instruments may be extended to cover the entire wafer 
surface by moving either the measuring instrument or the wafer in a 
controlled manner. However, the time required for these instruments to 
determine the thin film layer thickness at a single point is on the order 
of one minute. Further, characterizing an entire film surface of at least 
400 measurement points far exceeds the time typically afforded for 
efficient wafer production. 
The need in the art for a faster system and technique for measuring the 
thickness of thin films is addressed by the invention of two copending 
U.S. Patent Applications entitled, APATUS AND METHOD FOR MEASURING THE 
THICKNESS OF THIN FILMS, Ser. No. 07/804,872 filed Dec. 6, 1991 and Ser. 
No. 07/987,926 filed Dec. 10, 1992, by Anthony M. Ledger and assigned to 
the present Assignee. This application discloses an electro-optical 
imaging system for efficiently determining a thin film layer thickness of, 
for example, a wafer over a full aperture. Non-uniformities in this layer 
thickness are obtained by measuring the reflectance characteristics for a 
full aperture of a wafer surface and comparing this measured reflectance 
data to reference reflectance data by using numerical iteration or by 
using a calibration wafer having known layer thicknesses. 
To efficiently measure the reflectance characteristics of a wafer layer, a 
filtered white light source is used to produce a sequence of collimated 
monochromatic light beams at Several different wavelengths. These 
collimated monochromatic beams are individually projected onto the entire 
surface of the wafer, and coherent interactions occur between this light 
as it is reflected from the physical boundaries in the wafer structure. As 
a result of these interactions an interference fringe pattern is formed on 
the surface of the wafer for each projected beam and, consequently, for 
each wavelength. A reflected image of each fringe pattern is projected 
onto a detector array of, for example, a charge coupled device (CCD) 
camera, where the full aperture of this image is then captured. The fringe 
pattern image is captured by digitizing pixels in the CCD camera detector 
array corresponding to the image present. A reflectance map of the entire 
wafer surface is generated from this captured fringe pattern image. 
Several reflectance maps are generated from each measured wafer to 
eliminate thickness ambiguities which may result from outer layers having 
phase thicknesses greater that 2.pi.. 
The reference reflectance data for a wafer may be obtained by a theoretical 
calculation or through the use of a calibration wafer. The theoretical 
method consists of numerically computing reference reflectance 
characteristics based on assumed values for the intrinsic optical 
properties of the wafer materials. Alternatively, a calibration wafer, 
having a known thickness profile, may be constructed from the same batch 
of materials used to construct the wafer to be measured. By subjecting 
this calibration wafer to the measuring method of the present invention, 
reference reflectance data is obtained for the known wafer. 
The comparison between the measured reflectance data and the reference 
reflectance data can then be performed by a computer. The computer can 
then provide a mapping of layer thickness or a mapping of layer thickness 
non-uniformities over a full aperture of the wafer. 
While this invention represents a substantial advance in the state of the 
art, it is limited in the measurement of multi-layer film stacks. For 
multi-layer film stacks, film stack reflectivity is measured as a function 
of incident wavelength for a large number of physical locations. The 
"signature" of reflectivity vs. wavelength is then compared to a "library" 
of such signatures, each generated from theoretical predictions for a 
slightly different thickness of one layer in the stack. The library is 
essentially a column of data based on a theoretical prediction of the 
thickness of one layer. The library is normally generated with a fixed 
thickness for all other films in the stack. The best match to this library 
is determined by minimizing to the least squares error. 
Using this technique, measurement of the thickness of two layers of thin 
films requires that the thickness of one layer be known along with the 
index of refraction `n` and the imaginary index of refraction `k` for both 
layers. Unfortunately, in many applications, there is a need in the art 
for a system and technique for measuring the thickness of two adjacent 
thin film layers in a multi-layer film stack where the thickness of both 
layers is unknown. 
SUMMARY OF THE INVENTION 
The need in the art is addressed by the system and technique of the present 
invention for measuring the thickness of two layers of material where one 
of the layers may be assumed to have substantially uniform thickness. The 
novel method includes the steps of: 
a) providing a first reference library of theoretical signatures of 
reflectivity as a function of wavelength for a range of thicknesses of the 
first layer on the basis of a first assumed thickness of the second layer; 
b) measuring reflectivity as a function of wavelength for the first layer 
at plural points on the surface of the first layer to provide a plurality 
of signatures representing the measured value of reflectivity as a 
function of wavelength at each point measured; 
c) comparing each of the measured signatures to each of the theoretical 
signatures in the first reference library to identify an associated 
optimal correlation for each measured signature; 
d) calculating an error value associated with each optimal correlation of 
each signature with one of the theoretical signatures in the reference 
library; and 
e) analyzing the error value associated with each optimal correlation of 
each signature with one of the theoretical signatures in the reference 
library to provide a first measure of a quality of the correlation between 
the actual thickness of the second layer and the first assumed thickness 
of the second layer; 
f) providing plural reference libraries of theoretical signatures of 
reflectivity as a function of wavelength for a range of thicknesses of the 
first layer on the basis of plural associated assumed thicknesses of the 
second layer; 
g) comparing each of the measured signatures to each of the theoretical 
signatures in the reference library associated with each assumed thickness 
of the second layer to identify an associated optimal correlation for each 
measured signature in relation to each reference library; 
h) calculating an error value associated with each optimal correlation of 
each signature with one of the theoretical signatures in the each 
reference library; 
i) analyzing the error value associated with each optimal correlation of 
each signature with one of the theoretical signatures in each reference 
library to provide a measure of quality of a correlation between the 
actual thickness of the second layer and each assumed thickness of the 
second layer; 
j) comparing the measures of the quality of correlation between the actual 
thickness of the second layer and each assumed thickness of the second 
layer to identify an optimum reference library which represents the most 
likely estimate of the thickness of the second layer; and 
k) determining the thickness of the first layer based on the optimal 
correlations of each measured signature to the signatures of the optimum 
reference library. 
In the illustrative embodiment, the step of calculating an error value 
associated with each optimal correlation of each signature with one of the 
theoretical signatures in a reference library includes the step of 
calculating a least squares error. In addition, in the illustrative 
embodiment, the step of analyzing the error value associated with each 
optimal correlation of each signature with one of the theoretical 
signatures in the reference library to provide a measure of a quality of a 
correlation between the actual thickness of the second layer and the 
assumed thickness of the second layer includes the step of computing the 
average of the least squares error. Further, in the illustrative 
embodiment, the step of comparing the measures of the quality of 
correlation between the actual thickness of the second layer and each 
assumed thickness of the second layer to identify an optimum reference 
library which represents the most likely estimate of the thickness of the 
second layer includes the step of determining when the average least 
squares error approaches a minimum over the values thereof associated with 
the plural reference libraries of theoretical signatures of reflectivity 
as a function of wavelength for a range of thicknesses of the first layer 
on the basis of plural associated assumed thicknesses of the second layer.

DESCRIPTION OF THE INVENTION 
An electro-optical system for measuring a layer thickness of a wafer 24 is 
shown in FIG. 1. For the purposes of this description, the measurement of 
an outer silicon layer of a SOI semiconductor wafer 24 is described. 
A white light source is provided consisting of a circular aperture 14 
illuminated by a halogen lamp 10 and a condensing lens 12. Light passing 
through aperture 14 impinges on a collimator lens 16 to form a beam 15 of 
collimated light. The size of the aperture 14 determines the field angles 
in the collimated light sections of the optical system and the orientation 
is chosen to allow an aperture image to be projected onto the SOI wafer 
24. It should be noted that the condensing lens 12 may be replaced by the 
fiber optic light guide. 
The white light source is spectrally filtered by a series of narrow band 
filters 17, nominally of 30 to 50 angstroms half-bandwidth, placed in the 
collimated beam 15. The series of filters 17 are placed around the 
periphery of a rotating filter wheel assembly 18, whereby a corresponding 
series of collimated monochromatic light beams are produced. The 
wavelengths of these collimated monochromatic light beams 19 may typically 
range from 550 nm to 950 nm. Locating the filter wheel assembly 18 in a 
collimated light section 15 minimizes the spectral broadening of the 
filtered beam 19 caused by the field angle defined by the size of the 
aperture 14. A pair of electronic signals 32 are generated by the filter 
wheel assembly 18 to serve as a timing reference 33 for a digitizing 
circuit 34. One of these signals indicates the beginning of a filter wheel 
revolution, whereas the other signal indicates the beginning of each 
filter period. 
A second collimator lens 20 forms a monochromatic image of the aperture 14 
about a point 21 in a focal plane of a third collimator lens 22. This 
third collimator lens 22 produces a collimated beam 23 which illuminates 
the full aperture of the 100 millimeter diameter SOI wafer 24. Also, an 
extension of this wafer illumination technique to wafers of 150 
millimeters or 200 millimeters in diameter requires that the size of the 
third collimator lens 22 match the wafer size. It should be noted that a 
monochromator can replace the halogen lamp 10, the condensing lens 12, the 
first two collimator lenses 16, 20, and the narrow band filter wheel 18, 
provided that the slewing rate of the monochromator between different 
wavelengths is sufficiently high, up to twenty different wavelengths in 
less than one second per wavelength. 
Referring to FIG. 2, a cross-sectional view of a SOI semiconductor wafer 24 
is shown. This wafer 24 is constructed in a sandwich structure consisting 
of a mechanically polished outer silicon layer 40, an internal silicon 
dioxide film 42, and a silicon wafer substrate 44. This sandwich structure 
creates three interfaces 46, 48, 50 from which light, incident upon the 
outer silicon layer 40 may be reflected. The reflectance characteristics 
of these interfaces 46, 48, 50 are based upon the intrinsic optical and 
physical properties of the semiconductor materials in each layer 40, 42, 
44 of the SOI wafer 24. These properties consist of the absorption 
coefficient, .alpha., the index of refraction, n, and the thickness, t, of 
the material layers 40, 42, 44. For an SOI wafer, it is assumed that the 
absorption coefficient, .alpha., of the SiO.sub.2 layer 42 is zero. 
However, in general, it is permissible that the absorption coefficient be 
non-zero, provided that it is known. 
When the surface of the SOI wafer 46 is illuminated with collimated 
monochromatic light from beam 23, a series of coherent interactions occur 
as this light is reflected between the three material interfaces 46, 48 50 
of the SOI structure 24. These interactions produce a wavelength dependent 
interference pattern that is visible upon the surface of the wafer. The 
reflectance at any point on the wafer is determined by the multiple 
reflections between the three surfaces and by the magnitudes of their 
physical properties, n.sub.1, .alpha..sub.1, t.sub.1 and n.sub.2, 
.alpha..sub.2, t.sub.2, as well as properties of the substrate ns, 
.alpha..sub.s, where `.alpha.` represents the absorption coefficient of 
each respective layer. In the unique case of an SOI wafer structure, the 
substrate indices are identical to those of the outer film indices 
(n.sub.s =n.sub.2, .alpha..sub.s =.alpha..sub.2) since both are fabricated 
from single crystal silicon. The wafer reflectance at any wavelength can 
be calculated explicitly as a function of the outer film thickness if all 
other parameters are known, however, the reverse problem of computing the 
thickness from a single measured reflectance is ambiguous. This ambiguity 
is created by the fact that as the outer film thickness is increased, the 
measured reflectance cycles between maximum and minimum values as the 
phase thickness (n.sub.2 t.sub.2) increases by multiples of .pi./4. This 
multi-valued problem clearly makes the computation of the value of t.sub.2 
from a single reflectance measurement impossible. The use of multiple 
wavelength measurements can in principle overcome the multiple value 
problem but the wavelength dependent behavior of the material properties 
must be very accurately known otherwise large errors occur in the 
thickness computations. 
An alternate approach is a statistical one where measured reflectance data 
at several wavelengths is compared on a least squares best fit basis with 
a library of computed spectral data at the same wavelengths. In the case 
of an SOI wafer, the library of spectra is computed for all values of the 
outer film thickness and the selection is made by choosing that outer film 
thickness which minimizes the least squares fit. This is for the case 
where all layer thicknesses are known except for one layer, typically the 
top layer. 
Referring back to FIG. 1, a collimated light image of the interference 
fringe pattern is reflected off the surface of the SOI wafer 24 and 
returned through the third collimator lens 22. This third collimator lens 
22 projects a condensed image of the reflected fringe pattern upon an 
off-axis mirror 26. This mirror 26 is positioned at a point 25 in the 
focal plane of the third collimator lens 22, alongside the position of the 
aperture image at focal point 21. The separation of these two focal points 
21, 25 may be controlled with a slight lateral shift in the optical axis 
of the third collimator lens 22 with respect to the optical axis of the 
condensing lens 12 and the first two collimator lenses 16, 20. 
Equivalently, the wafer 24 may be tilted through a small angle, less than 
one degree, to achieve this same effect. This image separation scheme 
avoids the use of a beamsplitter which contains metallic coating with 
attendant optical losses. 
The off-axis mirror 26 is used to redirect the reflected fringe pattern 
image from the wafer 24 to a final collimator lens 28. This final 
collimator lens 28 projects a collimated beam 29 containing an image of 
the fringe pattern onto a CCD camera detector array 31. It should be noted 
that the filter wheel assembly 18 may also be placed in this collimated 
beam 29 provided that the field angle, which is approximately fifteen 
times larger than the field angle in the collimated beam 23 illuminating 
the wafer 24, can be tolerated by the narrow band filters. 
An alternate method of providing the reflected fringe pattern image to the 
CCD camera detector array 31 is shown in a dashed line block 61 in FIG. 1. 
An on-axis beamsplitter 60 is placed in the collimated light beam section 
19 where the filter wheel assembly 18 is positioned. The beamsplitter 60 
receives a collimated fringe pattern image from the second collimator lens 
20 and reflects a portion 62 of this collimated beam to a final collimator 
lens 64. This final collimator lens 64 converges the fringe pattern image 
onto the CCD camera detector array 31. Although this alternate method 
results in optical losses which are inherent in beamsplitters, it does not 
require an image separation scheme which can introduce field angle errors 
in the collimated light beam 23 reflected from an off-axis SOI wafer 24. 
As with the previous method, the filter wheel assembly 18 may be placed in 
the collimated beam 62 reflected by the beamsplitter 60, provided that the 
field angle can be tolerated by the narrow band filters 17. 
The determination of the method used to provide the reflected fringe 
pattern image to the CCD camera 30 is critically dependent upon the 
optical performance of the third collimator lens 22. When using the 
off-axis mirror method, the optical design of the third collimator lens 22 
must possess an optimal off-axis performance quality and provide a minimal 
radial color distortion effect. Optimal off-axis performance minimizes the 
distortion effects associated with field angles that are created when the 
collimated light beam 23 is reflected from an off-axis non-uniform surface 
of a SOI wafer 24. Also, the need for a consistent fringe pattern image 
size at the CCD camera detector array 31 requires radial color distortion 
correction over the wavelength region of the incident monochromatic light. 
When the on-axis beamsplitter method is used, however, only the radial 
color distortion correction requirement applies since the field angles 
produced in the collimated light beam 23 reflected from an on-axis SOI 
wafer surface 46 are negligible. Therefore, if the returned fringe pattern 
image is distorted due to sub-optimal off-axis performance by the third 
collimator lens 22, then the off-axis mirror method is unsuitable and an 
on-axis beamsplitter 60 must be used. 
Referring to FIG. 3, the CCD camera detector array 31 is shown with an 
image of a scaled SOI wafer outline 52, a pair of reference alignment 
images 54, and a pair of reference reflecting images 56, projected upon 
its surface. These reference images are formed by placing reference 
alignment marks and reference reflecting surfaces along the same plane as 
the surface of the SOI wafer 24. When illuminated with a collimated light 
beam 23 from the third collimator lens 22, these references provide 
reflections from their surfaces. Similar to the SOI wafer fringe pattern, 
images of these reflected references are returned through the third 
collimator lens 22 and are eventually projected upon the CCD camera 
detector array 31. The reference alignment marks provide aid in wafer 
alignment whereas the reference reflecting surfaces serve to normalize the 
CCD signals so that actual wafer reflectances can be calculated. Other 
wafer alignment techniques may be used without departing from the scope of 
the present teachings. 
Referring back to FIG. 1, the collimated beam 29 formed by the final 
collimator lens 28 contains an image of the reflected fringe pattern. This 
image is projected upon the CCD camera detector array 31 and captured by 
the CCD camera 30. A reflectance map is generated by digitizing the CCD 
pixel signals corresponding to the projected fringe pattern image with a 
digitizing circuit 34. This raw reflectance data may be normalized to 
eliminate variations in pixel sensitivity and may be reduced in size by 
averaging signals over blocks of pixels to match the spatial limitations 
of the subsequent chemical micro-polishing process. In determining the 
thickness, t.sub.2, of the outer silicon layer of the SOI wafer 24, either 
a numerical computation method or a SOI calibration wafer may be used. 
Both of these methods require the use of a computer 36. 
The numerical method of determining outer silicon layer thickness, t.sub.2 
consists of assuming values for the thin film constants n.sub.1, 
.alpha..sub.1, t.sub.1, n.sub.2, .alpha..sub.2, n.sub.3, and .alpha..sub.3 
and calculating spectral reflectances for a set of wavelengths 
corresponding to the monochromatic light produced by the filtered white 
light source. This calculation is done for a number of different outer 
layer thicknesses, t.sub.2, and provided that the initial thin film 
constant assumptions are correct should only need to be computed once. 
This calculation provides sets of reflectance values, R.sub.c 
(.lambda..sub.1, .lambda..sub.2, . . . .lambda..sub.n,t.sub.2) for 
thicknesses ranging from the thinnest to the thickest estimated value of 
the outer silicon layer. These computed spectral reflectances are stored 
in a reference library and then compared with the measured reflectance 
data, R.sub.m (.lambda..sub.1, .lambda..sub.2, . . . 
.lambda..sub.n,t.sub.2), at specific points on the wafer using a root mean 
square (rms) merit function of the form 
##EQU1## 
This merit function is evaluated for different values of t.sub.2 until a 
minimum or best match is found, which in turn indicates the most likely 
thickness. It is, of course, apparent that other pattern matching merit 
functions can be used, if desired. 
Unknown variations in any of the assumed thin film constants may cause 
errors to propagate through the computation process as outer layer 
thickness errors. Such first order error sources include the lack of 
knowledge of the SiO.sub.2 inner film thickness, t.sub.1, over the wafer 
aperture and the dispersive effects of the silicon index of refraction, 
n.sub.1. If the value of the merit function is too large, indicating a 
poor match, then new computed spectral reflectances will have to be 
generated for a closer set of t.sub.2 thicknesses, iterated with the 
absorption coefficients .alpha..sub.2, .alpha..sub.3, and the indices of 
refraction n.sub.2, n.sub.3, of the outer silicon layer 40 and the silicon 
substrate 44 respectively, or the index of refraction, n.sub.1, and the 
thickness, t.sub.1 of the SiO.sub.2 layer 42. 
In short, the above-described technique for determining the thickness of a 
layer of material in a stack where the thickness of the other layers is 
known is as follows. First, the film stack reflectivity is measured as a 
function of incident wavelength for a large number of physical locations. 
The "signature" of reflectivity vs wavelength is then compared to a 
library of such signatures, each generated from theoretical predictions 
for a slightly different thickness of one layer in the stack. A 
"signature" is simply a set of numbers which represents the reflectivity 
of one physical point on the wafer for a number of discrete wavelengths of 
incident light. The library is normally generated with a fixed thickness 
for all other films in the stack. The best match to this library is 
determined by minimizing the least squares error therebetween. The library 
is structured as a sequence of signatures representative of film thickness 
stored end-to-end, from smallest to longest thickness. In the illustrative 
embodiment, each signature is the same length, hence we can access any 
signature by calculating its position or offset in the file. However, in 
accordance with the present teachings, the signatures could be different 
lengths. 
In any event, the present invention extends this technique by providing a 
method for simultaneously determining the thickness of two layers of thin 
film material. In accordance with the present invention, one of the layers 
is assumed to be of substantially uniform thickness relative to the other. 
The other layer is permitted to vary in thickness considerably. 
Thus, the technique of the present invention is as follows. First a 
reference library of reflectivity vs wavelength is generated for a range 
of say 1000 thicknesses of one layer of film in the stack 40. The 
thickness of the second layer 42 is assumed to be uniform and is fixed at 
some arbitrary but plausible value for the sake of calculating the 
reflectivity of the stack theoretically. The experimental data of 
reflectivity versus wavelength is then acquired at a large number (e.g., 
1000) of points on the stack and each signature is matched to the library. 
A least square error is provided between each entry in the library and the 
measured signatures. Thus, the average least squares error is calculated 
from the best match of each signature to the library. (Those skilled in 
the art will appreciate that techniques other than the average least 
squares error may be used to evaluate the match of the signature to the 
library. For example, the kth-nearest neighbor and maximum likelihood 
techniques may be used for this purpose.) This average least squares error 
is intrinsically a measure of how well the actual thickness of the second 
layer 42 matches the arbitrary value selected for generating the reference 
library. 
A slightly different thickness is now selected for the second layer 42 
(e.g., .DELTA.t=0.01 .mu.m) and a new theoretical library is generated. 
The average least squares error is once again determined. As the estimated 
thickness of the second layer 42 approaches the actual thickness of the 
second layer 42 in the film stack, the average least squares error 
approaches a minimum. When the minimum has been found through iteration, 
the optimum library represents the most likely estimate of the thickness 
of the second layer 42. Thereafter, the best matches of each experimental 
signature yields the thickness of the first layer 40 at each point. 
FIG. 4 is a plot of average least squares error as a function of estimated 
oxide thickness in an SOI wafer. The minimum occurs for an oxide (second 
layer) thickness of 1.0 .mu.m. 
The procedure can be thought of as a two-dimensional optimization. It 
relies on the fact that each parameter (e.g., film thickness of the two 
layers) can be varied in the reference library independently. 
In the case of the optically transparent film stack, an ambiguity can exist 
in the determination of the best library match for a given experimental 
signature. This ambiguity derives in part from the cyclical nature of net 
reflectance from a multilayer film stack as the thickness of one of the 
layers (e.g., the first layer 40) changes. The present invention appears 
operative in spite of the ambiguity because the consistent uniformity of 
the second layer 42 yields the same contribution to the signature for each 
location on the stack, whereas the varying thickness of the first layer 40 
(and concomitant variation in reflectivity for a given wavelength) tends 
to "wash out" the ambiguity. The result is that, statistically, the 
average least squares error tends toward a minimum as the estimate of 
thickness of the second layer 42 approaches the actual thickness thereof, 
even though the individual experimental signatures are not precisely 
matched to the candidates in the reference library. 
Thus, the present invention has been described herein with reference to a 
particular embodiment for a particular application. Those having ordinary 
skill in the art and access to the present teachings will recognize 
additional modifications applications and embodiments within the scope 
thereof. For example, the invention is not limited to use to measure 
thickness. The technique of the present invention may be extended and used 
to determine the index of refraction of a film by iterating over the index 
in a narrow range as the thickness is held constant in the reference 
library. 
It is therefore intended by the appended claims to cover any and all such 
applications, modifications and embodiments within the scope of the 
present invention.