Method and system for calibrating an ellipsometer

A method for calibrating an ellipsometer, and an ellipsometer including a processor programmed to control the analyzer, polarizer, and other ellipsometer components, and to process the data measured by the ellipsometer to perform the calibration method automatically. Where the ellipsometer's polarizer rotates and the analyzer remains fixed during measurement, the method determines coarse approximations of values A.sub.0 and P.sub.0, and then processes reflectivity data obtained at two or more analyzer angles to determine refined approximations of the values A.sub.0 and P.sub.0, where P.sub.0 is the angle of the polarizer's optical axis at an initial time, and A.sub.0 is the offset of the actual orientation angle of the analyzer from a nominal analyzer angle. Preferably the ellipsometer is a spectroscopic ellipsometer, the reflectivity data determine a tan.psi. spectrum and a cos.DELTA. spectrum for each of the analyzer angles, and the coarse approximations of A.sub.0 and P.sub.0 are refined by processing the reflectivity data by performing regression on A.sub.0 and P.sub.0 until the differences among the tan.psi. and cos.DELTA. spectra for several analyzer angles are minimized. Where the ellipsometer's analyzer rotates and the polarizer remains fixed during measurement, the method coarsely determines values A'.sub.0 and P'.sub.0, and then processes reflectivity data obtained at two or more polarizer angles to determine refined approximations of the values A'.sub.0 and P'.sub.0, where P'.sub.0 is the angle of the analyzer's optical axis at an initial time, and A'.sub.0 is the offset of the actual orientation angle of the polarizer from a nominal polarizer angle.

FIELD OF THE INVENTION 
The invention relates to a method for calibrating an ellipsometer (such as 
a spectroscopic ellipsometer), and to an ellipsometer (such as a 
spectroscopic ellipsometer) including a means for automatically performing 
such calibration method. 
BACKGROUND OF THE INVENTION 
Among the well known nondestructive testing techniques is the technique of 
spectroscopic ellipsometry, which measures reflectance data by reflecting 
electromagnetic radiation from a sample (typically to measure the 
thickness of a very thin film on a substrate). In spectroscopic 
ellipsometry, an incident radiation beam having a known polarization state 
reflects from a sample, and the polarization of the reflected radiation is 
analyzed to determine properties of the sample. Since the incident 
radiation includes multiple frequency components, a spectrum of measured 
data (including data for incident radiation of each of at least two 
frequencies) can be measured. Typically, the polarization of the incident 
beam has a time-varying characteristic (produced, for example, by passing 
the incident beam through a mechanically rotating polarizer), and/or the 
means for analyzing the reflected radiation has a time-varying 
characteristic (for example, it may include a mechanically rotating 
analyzer). Examples of spectroscopic ellipsometry systems are described in 
U.S. Pat. No. 5,329,357, issued Jul. 12, 1994 to Bernoux, et al., and U.S. 
Pat. No. 5,166,752, issued Nov. 24, 1992 to Spanier, et al. Spectroscopic 
ellipsometry theory is described in F. Bernoux, et al., "Ellipsometrie," 
Techniques de l'Ingenieur, R6490, pp. 1-16 (1990). 
Reflectance data (measured by spectroscopic ellipsometry or other 
reflection techniques) are useful for a variety of industrial 
applications. The thickness of various coatings (either single layer or 
multiple layer) on a substrate can be determined from spectroscopic 
ellipsometry data (indicative of the polarization of radiation reflected 
from the sample in response to incident radiation having known 
polarization state), or a reflectance spectrum or relative reflectance 
spectrum. 
The reflectance of a sample (or sample layer) at a single wavelength can be 
determined by analyzing spectroscopic ellipsometry data (indicative of the 
polarization changes of radiation reflected from the sample, in response 
to incident radiation having known polarization state) or extracted from 
an accurately measured reflectance or relative reflectance spectrum. It is 
useful to determine sample reflectance in this way where the reflectance 
of photoresist coated wafers at the wavelength of a lithographic exposure 
tool must be found, to determine proper exposure levels for the wafers or 
to optimize the thickness of the resist to minimize reflectance of the 
entire coating stack. 
The present invention pertains to calibration of an ellipsometer (such as a 
spectroscopic ellipsometer). To appreciate the difference between the 
inventive calibration method, and conventional calibration methods, it is 
helpful to consider the method of operation of a spectroscopic 
ellipsometer. 
FIG. 1 is a schematic diagram of a typical spectroscopic ellipsometer. In 
operation of this ellipsometer, a beam of broadband radiation from 
broadband radiation source 150 is linearly polarized in polarizer 152, and 
the linearly polarized beam is then incident on sample 154o After 
reflection from sample 154, the beam propagates toward analyzer 156 with a 
changed polarization state (typically, the reflected beam has elliptical 
polarization, where the polarized beam emerging from polarizer 152 had 
linear polarization). The reflected beam propagates through analyzer 156 
into dispersion element (spectrometer) 158. In dispersion element 158, the 
beam components having different wavelengths are refracted in different 
directions to different detectors of detector array 160. Processor 162 
receives the measured data from each detector of array 160, and is 
programmed with software for processing the data it receives in an 
appropriate manner. Detector array 160 can be a linear array of 
photodiodes, with each photodiode measuring radiation in a different 
wavelength range. 
Either polarizer 152 or analyzer 156 is rotatably mounted for rotation 
about the optical axis during a measurement operation (or both of them are 
so rotatably mounted). During a typical measurement operation, polarizer 
152 is rotated and analyzer 156 remains in a fixed orientation, or 
analyzer 156 is rotated and polarizer 152 remains fixed. 
Processor 162 can be programmed to generate control signals for controlling 
the rotation (or angular orientation) of polarizer 152 and/or analyzer 
156, or for controlling other operating parameters of elements of the FIG. 
1 system (such as the position of a movable sample stage on which sample 
154 rests). Processor 162 can also receive data (indicative of the angular 
orientation of analyzer 156) from an analyzer position sensor associated 
with analyzer 156 and data (indicative of the angular orientation of 
polarizer 152) from a polarizer position sensor associated with polarizer 
152, and can be programmed with software for processing such orientation 
data in an appropriate manner. 
If polarizer 152 is controlled so that it rotates at a constant speed, the 
signal received at each detector of array 160 will be a time-varying 
intensity given by: 
##EQU1## 
where I.sub.0 is a constant that depends on the intensity of radiation 
emitted by source 150, .omega. is the angular velocity of polarizer 152, 
P.sub.0 is the angle between the optical axis of polarizer 152 and the 
plane of incidence (e.g., the plane of FIG. 1) at an initial time (t=0), 
and .alpha. and .beta. are sample related values defined as follows: 
EQU .alpha.=[tan.sup.2 .psi.-tan.sup.2 (A-A.sub.0)]/[tan.sup.2 .psi.+tan.sup.2 
(A-A.sub.0)] (2) 
and 
EQU .beta.=2(tan.psi.)(cos.DELTA.)(tan(A-A.sub.0)/[tan.sup.2 .psi.+tan.sup.2 
(A-A.sub.0)] (3) 
where tan.psi. is the amplitude of the complex ratio of the p and s 
components of the reflectivity of the sample, .DELTA. is the phase of the 
complex ratio of the p and s components of the reflectivity of the sample 
(where "p" denotes the component for polarized radiation whose electrical 
field is in the plane of FIG. 1, and "s" denotes the component for 
polarized radiation whose electrical field is perpendicular to the plane 
of FIG. 1), A is the nominal analyzer angle (a reading of analyzer 156's 
orientation angle, supplied for example from the above-mentioned analyzer 
position sensor associated with analyzer 156), and A.sub.0 is the offset 
of the actual orientation angle of analyzer 156 from the reading "A" (due 
to mechanical misalignment, A.sub.0 can be non-zero). 
The values .alpha.' and .beta.' are also sample related values, defined as 
follows: 
EQU .alpha.'=.alpha.cos (2P.sub.0)+.beta.sin (2P.sub.0) (4) 
and 
EQU .beta.'=.alpha.sin (2P.sub.0)-.beta.cos (2P.sub.0) (5) 
where .alpha., .beta., and P.sub.0 are defined above. 
To achieve measurement accuracy, it is crucial to determine P.sub.0 and 
A.sub.0 very precisely. 
Conventionally, P.sub.0 and A.sub.0 are calibrated simultaneously by a 
method known as the "minimum residual method," first proposed in David E. 
Aspnes and A. A. Studna, "High Precision Scanning Ellipsometer," Applied 
Optics, Vol. 14, No. 1, pp. 220-228 (1975). The minimum residual method is 
still widely used by ellipsometer users and manufacturers as of the filing 
date of this specification. 
The conventional minimum residual method determines (from measured data) a 
quantity known as the "residual" (R), which is: 
EQU R=1-.alpha..sup.2 -.beta..sup.2 ( 6) 
where .alpha. and .beta. are defined in equations (2) and (3). 
An equivalent quantity is R'=1-.alpha.'.sup.2 -.beta.'.sup.2 where .alpha.' 
and .beta.' are defined in equations (4) and (5). Of course, it follows 
from equations (2) through (5) that R=R', and both R and R' are denoted 
herein as the "residual." 
Using equations (2) and (3), it is apparent that the residual, R, can also 
be expressed as: 
EQU R=(4-cos.sup.2 .DELTA.) tan.sup.2 .psi. tan.sup.2 (A-A.sub.0)/[tan.sup.2 
.psi.+ tan.sup.2 (A-A.sub.0)].sup.2 ( 7) 
By orienting the analyzer so that A-A.sub.0 =.delta.A is very small, it can 
be assumed that tan.delta.A is approximately equal to .delta.A. Under this 
condition, equation (7) can be approximated by: 
EQU R=(4-cos.sup.2 .DELTA.) (.delta.A/tan.psi.).sup.2 
[1-2(.delta.A/tan.psi.).sup.2 ] (8) 
The "phase" of the residual R is defined by: 
##EQU2## 
where .alpha.' and .beta.' are defined by equations (4) and (5). 
To perform calibration (i.e., determine the values A.sub.0 and P.sub.0) in 
accordance with the conventional minimum residual method, the orientation 
of analyzer 156 is first scanned around the zero position, the values R 
and "Phase" are determined from the measured data at each measured value 
(A) of the analyzer's orientation, and the values R and "Phase" are 
plotted as a function of A, to generate a graph such as that shown in FIG. 
2. 
Then, the value A.sub.0 (the offset between analyzer 156's actual 
orientation angle and each reading "A") is identified as the minimum of 
the "R v. A" curve. When A=A.sub.0, it is true that .alpha.=1 and 
.beta.=0, so that Phase=2P.sub.0. Thus, having identified the value 
A.sub.0, the minimum residual method identifies 2P.sub.0 as the value of 
the "Phase v. A" curve at A=A.sub.0. 
To accurately determine the minimum of the "R v. A" curve, it is necessary 
to fit the bottom part of this curve with a parabolic curve. However, this 
cannot be done accurately under all conditions, for the reasons explained 
with reference to FIG. 3. 
FIG. 3 is a graph of three "R v. A" curves, each generated from 
measurements at cos (.DELTA.)=1 a different value of tan(.psi.), namely 
tan(.psi.)=10, tan(.psi.)=0.1 and tan(.psi.)=1. From FIG. 3, it is 
apparent that the shape of the "R v, A" curve strongly depends on the 
value of tan(.psi.). If tan(.psi.) is too small or too large, the bottom 
part of the curve is very flat, in which case a small perturbation caused 
by noise can cause a large shift in the estimated value of the curve's 
minimum position. Thus, the conventional "minimum residual method" is 
reliable and accurate only for samples for which the value of tan(.psi.) 
is in a very limited range. 
There are several other important limitations of prior art calibration 
methods (including the "minimum residual method"), including that they can 
be performed accurately only on very thick samples. Until the present 
invention, it had not been known how to avoid these limitations of prior 
art calibration methods. 
SUMMARY OF THE INVENTION 
The inventive method for calibrating an ellipsometer includes the steps of 
determining coarse approximations of the above-defined values A.sub.0 and 
P.sub.0 ; and then processing reflectivity data obtained at two or more 
analyzer angles to determine refined approximations of the values A.sub.0 
and P.sub.0. Where the ellipsometer is a spectroscopic ellipsometer, the 
reflectivity data determine a tan.psi. spectrum and a cos.DELTA. spectrum 
for each analyzer angle, each set of reflectivity data measured by 
operating the ellipsometer (with a rotating polarizer) at a different 
analyzer angle. The tan.psi. and cos.DELTA. spectra for each analyzer 
angle determine the amplitude and phase of the complex ratio of p and s 
components of sample reflectivity (at each of at least two incident 
radiation frequencies). Where the ellipsometer is not a spectroscopic 
ellipsometer, the reflectivity data determine a tan.psi. value and a 
cos.DELTA. value (for only single frequency or frequency range of incident 
radiation) for each analyzer angle. 
Preferably, the coarse approximations of A.sub.0 and P.sub.0 are refined by 
processing the reflectivity data in the following manner. The reflectivity 
data are processed to determine tan.psi. and cos.DELTA. values (e.g., 
tan.psi. and cos.DELTA. spectra) and regression is performed on A.sub.0 
and P.sub.0 until the differences among the determined tan.psi. and 
cos.DELTA. values for the different analyzer angles are minimized. In a 
class of preferred embodiments, the regression is performed using the 
well-known least square fit algorithm or another function minimization 
technique. 
Preferably, the invention generates data determining the coarse 
approximation of A.sub.0 as follows. A first nominal residual value is 
acquired with the fixed analyzer at a first nominal analyzer angle 
A.sub.1. Then, nominal residual values are acquired at several nominal 
analyzer angles A.sub.i all approximately equal to -A.sub.1, and a second 
nominal analyzer angle (A.sub.2) is identified, where -A.sub.2 is the one 
of the several nominal angles at which the nominal residual value is 
closest to the first nominal residual value. By averaging the first and 
second nominal analyzer angles, the coarse approximation of A.sub.0 is 
determined. 
Preferred embodiments of the inventive ellipsometer include a processor 
programmed to generate control signals for controlling the analyzer, 
polarizer, and other components of the ellipsometer appropriately, and to 
perform appropriate processing of the measured data received from the 
detector (or each detector of the detector array) of the ellipsometer, to 
perform the inventive calibration method automatically. 
Alternative embodiments of the invention are a method for calibrating an 
ellipsometer (where the ellipsometer has an analyzer that rotates during 
measurement of a sample and a fixedly mounted polarizer), and an 
ellipsometer of this type including a means for performing such method 
automatically. These embodiments implement a modified version of the 
above-described calibration method which determines coarse approximations 
(and then refined approximations) of values P'.sub.0 and A'.sub.0, where 
A'.sub.0 is the angle (at an initial time, t=0) of the analyzer optical 
axis, and P'.sub.0 is the offset of the actual orientation angle of the 
fixed polarizer from a nominal angle P' of the fixed polarizer.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Throughout the specification, the phrase "incidence angle" of radiation at 
a surface denotes the angle between the normal to the surface and the 
propagation direction of the radiation. Thus, radiation with normal 
incidence at a sample surface has an incidence angle of zero degrees, and 
radiation with grazing incidence at such surface has an incidence angle 
substantially equal to 90.degree.. Throughout the specification, the 
phrase "high incidence angle" denotes an incidence angle greater than 
30.degree.. 
Throughout the specification, the phrase "broadband radiation" denotes 
radiation whose frequency-amplitude spectrum includes two or more 
different frequency components. For example, broadband radiation may 
comprise a plurality of frequency components in the range from 230 nm to 
850 nm. 
Throughout the specification, including in the claims, the term "software" 
is employed in a broad sense to denote code which programs (instructions 
which program) a general purpose data processing apparatus, or firmware 
(microcode which resides in a read-only-memory apparatus). For example, 
when processor 101 of FIG. 4 is said to be programmed with software for 
generating control signals for controlling the orientation of rotatably 
mounted analyzer 8, such software can be implemented as firmware in 
processor 101 (e.g., when processor 101 is not a general purpose computer, 
but is instead another data processing apparatus). 
The inventive method for calibrating a spectroscopic ellipsometer has two 
steps: determining coarse approximations of the above-defined values 
A.sub.0 and P.sub.0 ; and then processing reflectivity data (tan.psi. and 
cos.DELTA. spectra) obtained at two or more analyzer angles to determine 
refined approximations of the values A.sub.0 and P.sub.0. The reflectivity 
data determine a tan.psi. spectrum and a cos.DELTA. spectrum for each of 
at least two analyzer angles, each set of reflectivity data measured by 
operating a spectroscopic ellipsometer (with a rotating polarizer) at a 
different analyzer angle. The tan.psi. and cos.DELTA. spectra for each 
analyzer angle determine the amplitude and phase of the complex ratio of p 
and s components of sample reflectivity (at each of at least two incident 
radiation frequencies). 
Preferably, the coarse approximations of A.sub.0 and P.sub.0 are refined by 
processing the reflectivity data in the following manner. Regression is 
performed on A.sub.0 and P.sub.0 (preferably using the well-known least 
square fit algorithm or another function minimization technique), until 
the differences among the tan.psi. and cos.DELTA. spectra for several 
analyzer angles are minimized. More specifically, to implement such 
regression, a tan.psi. spectrum and a cos.DELTA. spectrum for each of the 
analyzer angles is computed from the reflectivity data using the coarse 
approximations of A.sub.0 and P.sub.0. Then, regression is performed on 
A.sub.0 and P.sub.0 to generate a revised tan.psi. spectrum and a revised 
cos.DELTA. spectrum (for each of the analyzer angles) for each of a number 
of different value pairs A.sub.0 and P.sub.0, and computing (for each of 
the value pairs A.sub.0 and P.sub.0) the differences between the revised 
tan.psi. and cos.DELTA. spectra for pairs of the analyzer angles, until 
such differences are minimized. The invention identifies as the refined 
approximations of A.sub.0 and P.sub.0 those values of A.sub.0 and P.sub.0 
which result in minimization of such differences. 
The invention is based on recognition that when an ellipsometer (with 
rotating polarizer and fixed analyzer) is used to measure reflectivity 
data (e.g., when a spectroscopic ellipsometer with rotating polarizer and 
fixed analyzer is used to measure reflectivity data determining tan.psi. 
and cos.DELTA. spectra), the tan.psi. and cos.DELTA. spectra are 
independent of the analyzer angle. The invention is also based on 
recognition that this phenomenon can be exploited to estimate A.sub.0 and 
P.sub.0 in efficiently and accurately for virtually any sample. 
We next describe preferred embodiments of the calibration method of the 
invention which determine (from measured data obtained by operating a 
spectroscopic ellipsometer or other ellipsometer) the same "residual" 
value defined above in equation (6), namely R=1-.alpha..sup.2 
-.beta..sup.2 =1-.alpha.'.sup.2 -.beta.'.sup.2 =R'. 
When the ellipsometer is operated (to measure reflectivity data) with a 
rotating polarizer and fixed analyzer, the residual depends on the actual 
analyzer angle (a reading of the analyzer angle adjusted by the offset of 
the reading from the actual angle). 
In calibrating an ellipsometer with a rotating polarizer and fixed 
analyzer, the method of the invention determines coarse approximations of 
A.sub.0 and P.sub.0 as follows. First, data determining a nominal residual 
value R'(A.sub.1) are acquired with the fixed analyzer at a first nominal 
analyzer angle A.sub.1. This is done without knowledge of values A.sub.0 
and P.sub.0 by assuming (for the purpose of determining the nominal 
residual value R'(A.sub.1)) that A.sub.0 =0, and employing equation (7) to 
process data (indicative of A.sub.1 and cos.DELTA. and tan.psi. values for 
analyzer angle A.sub.1) to generate data indicative of the value 
R'(A.sub.1)=(4-cos.sup.2 .DELTA.)tan.sup.2 .psi.tan.sup.2 
(A.sub.1)/[tan.sup.2 .psi.+tan.sup.2 (A.sub.1)].sup.2. 
Similarly, data determining nominal residual values R'(A.sub.i) are 
acquired at several nominal analyzer angles A.sub.i, all approximately 
equal to -A.sub.1. 
Preferably, nominal angle A.sub.1 is chosen so that tan(A.sub.1)&gt;tan.psi. 
for all wavelengths. This can be achieved easily by using a sample 
consisting of bare silicon or glass, and setting A.sub.1 =45 degrees. If 
tan(A.sub.1) is less than or equal to tan.psi. for at least one relevant 
wavelength, then the coarse approximation of P.sub.0 (determined in the 
manner described below) can differ by 90 degrees from the correct value. 
Then, the inventive method exploits the fact that the actual value R' 
satisfies the symmetry relation R'(A-A.sub.0)=R'[-(A-A.sub.0)], by 
determining a second nominal analyzer angle A.sub.2 at which the nominal 
value R'(-A.sub.2) is closest to the nominal value R'(A.sub.1). The second 
nominal angle A.sub.2, which has the same sign as the first nominal angle 
A.sub.1, is equal to A.sub.2 =-A.sub.i1, where +A.sub.i1 is one of the 
nominal analyzer angles A.sub.i that are approximately equal to -A.sub.1. 
Then, the invention averages the two nominal angles A.sub.2 and A.sub.1, 
and identifies the resulting averaged value (shown in equation (10)) as 
the coarse approximation of A.sub.0 : 
EQU A.sub.0 =(A.sub.1 +A.sub.2)/2 (10) 
Using this coarse approximation of A.sub.0, the invention determines the 
coarse approximation of P.sub.0 as follows. In order readily to understand 
the preferred method for coarsely approximating P.sub.0, it is helpful to 
employ equations (2) and (3) to define the following quantities: 
EQU .alpha..sub.+ =.alpha.(A-A.sub.0)=.alpha.[-(A-A.sub.0)]=.alpha..sub.-(11) 
and 
EQU .beta..sub.+ =.beta.(A-A.sub.0)=-.beta.[-(A-A.sub.0)]=-.beta..sub.-(12) 
With the notation .alpha.'(A.sub.1 -A.sub.0)=.alpha..sub.+ ', 
.alpha.'[-(A.sub.2 -A.sub.0) =.alpha..sub.- ', .beta.'(A.sub.1 
-A.sub.0)=.beta..sub.+ ', and .beta.'[-(A.sub.2 -A.sub.0)]=.beta..sub.- ', 
and using equations (11) , (12) , (4) , and (5) , it is seen that: 
EQU .alpha..sub.+ '+.alpha..sub.- '=2.alpha.cos (2P.sub.0) (13) 
and 
EQU .beta..sub.+ '+.beta..sub.- '=2.alpha.sin (2P.sub.0) (14) 
The invention thus determines the coarse approximation of P.sub.0 to be: 
EQU P.sub.0 =tan.sup.-1 [(.beta..sub.+ '+.beta..sub.- ')/(.alpha..sub.+ 
'+.alpha..sub.- ')]/2 (15) 
The values .alpha.'(A.sub.1 -A.sub.0)=.alpha..sub.+ ', .alpha.'[-(A.sub.2 
-A.sub.0)=.alpha..sub.- ', .beta.'(A.sub.1 -A.sub.0)=.beta..sub.+ ', and 
.beta.'[-(A.sub.2 -A.sub.0)]=.beta..sub.- ' in equation (15) are 
preferably determined (using the well known Hadamard algorithm) as 
follows, where each detector is a charge integrating detector (e.g., where 
array 173 is an intensified photodiode array). Using an array of such 
charge integrating detectors, the output of each detector is proportional 
to the charge it collects during a period, T, known as the integration 
time. The output of each detector is thus given by: 
##EQU3## 
where I(t) is defined above, T is the integration time, P.sub.s is the 
starting polarizer angular position, P.sub.e is the ending polarizer 
angular position, and P=.omega.t. By dividing one polarizer revolution 
into eight segments and measuring (integrating) the signal I(t) over each 
of those segments, we measure the following eight signals: 
##EQU4## 
Each of the eight signals S.sub.i is known as a "sum." By processing the 
sums (and using equation (2), we determine .alpha.' and .beta.' to be: 
EQU .alpha.'=(1/4I.sub.0) (S.sub.1 -S.sub.2 -S.sub.3 +S.sub.4 +S.sub.5 -S.sub.6 
-S.sub.7 +S.sub.8) 
and 
EQU .beta.'=(1/4I.sub.0) (S.sub.1 +S.sub.2 -S.sub.3 -S.sub.4 +S.sub.5 +S.sub.6 
-S.sub.7 -S.sub.8) 
where I.sub.0 =(1/2.pi.) (S.sub.1 +S.sub.2 +S.sub.3 +S.sub.4 +S.sub.5 
+S.sub.6 +S.sub.7 +S.sub.8). 
By processing sums (S.sub.i) measured with the analyzer at a nominal angle 
A, the values .alpha..sub.+ ' and .beta..sub.+ ' are determined by the 
equations in the previous paragraph. Then, by processing sums (S.sub.i) 
measured with the analyzer at the opposite nominal angle (-A), the values 
.alpha..sub.- ' and .beta..sub.- ' are determined by the equations in the 
previous paragraph. Having thus determined .beta..sub.+ ', .beta..sub.- 
',.alpha..sub.+ ', and .alpha..sub.- ', equation (15) determines the 
coarse approximation of P.sub.0. 
If a photodetector array (such as a photodiode array or a linear CCD) is 
used to implement detector array 173, the coarse approximation of P.sub.0 
for each element of the array can be determined as follows. Typically, 
when the array is used to record reflectivity data at multiple 
wavelengths, each detector element is initiated at a different polarizer 
angle. Accordingly, P.sub.0 is different for each element of the array. In 
most cases, P.sub.0 for a particular one of the detector elements 
(P.sub.0N) can be assumed to be a linear function of the element's 
position: 
EQU P.sub.0N =aN+b, 
where N is a non-negative integer identifying each of the elements (the 
"first" detector element is identified by N=0), "b" is P.sub.0 for the 
first detector element, and "a" is a constant known as "P.sub.0 -slope". 
The P.sub.0 -slope (the value of "a") depends only on the integration time 
and readout time of the array, and hence can be precisely calculated. 
Having determined coarse approximations of the values A.sub.0 and P.sub.0 
(for each detector), the invention then determines refined approximations 
of the coarsely approximated values A.sub.0 and P.sub.0, in the following 
manner. First, the ellipsometer undergoing calibration is operated to 
measure reflectivity data at each of two or more analyzer angles (with a 
rotating polarizer and fixed analyzer at each of the analyzer angles). If 
the ellipsometer is a not a spectroscopic ellipsometer, the reflectivity 
data (for each analyzer angle) determine both a tan.psi. value and a 
cos.DELTA. value for one incident radiation frequency (or frequency 
range). 
If the ellipsometer being calibrated is a spectroscopic ellipsometer, the 
reflectivity data (for each analyzer angle) determine both a tan.psi. 
value and a cos.DELTA. value for each of two or more incident radiation 
frequencies (or frequency ranges). In this case, the reflectivity data 
determine a tan.psi. spectrum and a cos.DELTA. spectrum for each analyzer 
angle. The tan.psi. and cos.DELTA. spectra for each analyzer angle 
determine the amplitude and phase of the complex ratio of p and s 
components of sample reflectivity (at each of two or more incident 
radiation frequencies). 
For example, when calibrating the spectroscopic ellipsometer of FIG. 1, the 
reflectivity data (for each angle of analyzer 156) typically consist of a 
sequence of readings of detector array 160 taken while polarizer 152 
rotates, with each read-out of detector array 160 initiated at a different 
angular position of polarizer 152 (and integrated over forty-five degrees 
of polarizer rotation). 
In accordance with the invention, the reflectivity data (which determine 
tan.psi. and cos.DELTA. values or tan.psi. and cos.DELTA. spectra) 
obtained at the multiple analyzer angles are processed to determine 
refined approximations of the values A.sub.0 and P.sub.0. In preferred 
embodiments, this processing is accomplished by performing regression on 
A.sub.0 and P.sub.0 (preferably using the well-known least square fit 
algorithm) as follows. A tan.psi. and cos.DELTA. value pair (or tan.psi. 
spectrum and a cos.DELTA. spectrum) for each of the analyzer angles is 
computed from the reflectivity data using the coarse approximations of 
A.sub.0 and P.sub.0. Then, regression is performed on A.sub.0 and P.sub.0 
to generate a revised tan.psi. and cos.DELTA. value pair (or a revised 
tan.psi. spectrum and a revised cos.DELTA. spectrum) for each of the 
analyzer angles (and for each of a number of different value pairs A.sub.0 
and P.sub.0), and computing (for each of the value pairs A.sub.0 and 
P.sub.0) an error term for (e.g., differences between) revised tan.psi. 
and cos.DELTA. values (or spectra) for different analyzer angles, until 
such error term is minimized. As a result of such regression, preferred 
embodiments of the invention identify the values of A.sub.0 and P.sub.0 
which result in the minimum total difference between the tan.psi. and 
cos.DELTA. values (or tan.psi. and cos.DELTA. spectra) for each pair of 
analyzer angles, as the refined approximations of A.sub.0 and P.sub.0. 
The first phase of the inventive method (determining coarse approximations 
of A.sub.0 and P.sub.0) is necessary to ensure that the regression 
operation (the second phase of the inventive method) converges rapidly to 
the correct values (refined approximations) of A.sub.0 and P.sub.0. 
The regression step of the inventive method exploits the fact that when an 
ellipsometer is operated with a rotating polarizer and fixed analyzer to 
measure data which determine tan.psi. and cos.DELTA. values (or a tan.psi. 
spectrum and a cos.DELTA. spectrum), such values (or spectra) are 
independent of analyzer angle (for any given sample). 
Next, we describe in greater detail a class of embodiments in which the 
regression step (for calibrating a spectroscopic ellipsometer with a fixed 
analyzer and rotating polarizer) is performed using a standard function 
minimization technique. In describing these embodiments, we assume that: 
1. a set of n spectra have been measured, each at a different analyzer 
angle A.sub.n (where the absolute value of each angle A.sub.n is greater 
than 10 degrees); 
2. the coarsely approximated value of A.sub.0 is in the range [-10 degrees, 
+10 degrees]; 
3. "b" (the above-discussed coarsely approximated value of P.sub.0 for the 
first detector element) is known to within plus or minus 10 degrees; and 
4. "a" (the above-discussed "P.sub.0 -slope") is known to within plus or 
minus 0.02 degrees. 
These assumptions ensure that the sign of the true value of each analyzer 
angle A.sub.n is known, and reduce the chance of multiple possible 
solutions for "b" and "a." 
We use the notation cos.DELTA..sub.ij to denote the measured value of 
cos.DELTA. at frequency "i" and analyzer position A.sub.j, and 
tan.psi..sub.ij to denote the measured value of tan.psi. at frequency "i" 
and analyzer position A.sub.j. Cos.DELTA..sub.ij is a function of "a," 
"b," the eight sums defined above (S.sub.1 through S.sub.8), and the sign 
of A.sub.j -A.sub.0. Tan.psi..sub.ij is a function of "a," "b," the raw 
signal T.sub.kij, and tan(A.sub.j -A.sub.0). 
Let 
##EQU5## 
be the mean value of cos.DELTA..sub.ij (over all analyzer positions) for a 
single frequency. 
Let 
##EQU6## 
be the mean value of tan.psi..sub.ij (over all analyzer positions) for a 
single frequency. 
We define an error term to be one of 
##EQU7## 
In these expressions, E.sub.MSE represents minimum square error from mean, 
and E.sub.MINMAX represents minimum maximum deviation from mean. Each of 
E.sub.MSE and E.sub.MINMAX depends on "a" and "b" but not on A.sub.0 (if 
the assumptions enumerated above are true). 
The refined approximations of the values "a" and "b" (which determine the 
refined approximation of P.sub.0) are those that minimize the selected 
error term (either E.sub.EMS or E.sub.MINMAX). Since the characteristics 
of the error surface are unknown, the minimization procedure must allow 
for multiple local minima in the search region. The preferred technique is 
a global least squares function minimization. The straightforward 
procedure of locating the minima on a fine mesh and repeatedly refining 
each minimum on finer and finer meshes should suffice. 
Having thus determined the refined approximation of P.sub.0, the refined 
approximation of A.sub.0 is determined as follows. 
We define an error term to be one of 
##EQU8## 
In these expressions, E.sub.MSE represents minimum square error from mean, 
and E.sub.MINMAX represents minimum maximum deviation from mean. 
Then, employing the refined approximations of the "a" and "b" values, the 
value of A.sub.0 which minimizes the selected error term is identified as 
the refined approximation of A.sub.0. 
Another aspect of the invention is a spectroscopic ellipsometer including a 
means for automatically performing the calibration method of the 
invention. A preferred embodiment of such a spectroscopic ellipsometer 
will be described with reference to FIG. 4. 
The spectroscopic ellipsometer of FIG. 4 includes processor 101 which is 
programmed to control analyzer 8 and polarizer 5 (and other components of 
the ellipsometer) appropriately, and perform the appropriate processing of 
the measured data received from detector array 173, to perform the 
inventive calibration method automatically. 
The focused beam spectroscopic ellipsometer of FIG. 4 is identical to the 
ellipsometer described with reference to FIG. 1 of pending U.S. patent 
application Ser. No. 08/375,353 (filed Jan. 19, 1995), except in the 
following respect. The ellipsometer of FIG. 4 of the present disclosure 
includes programmed processor 101 rather than programmed processor 100 of 
the ellipsometer of U.S. application Ser. No. 08/375,353. The description 
in U.S. Ser. No. 08/375,353 of the FIG. 1 ellipsometer (and the details 
thereof and variations thereon described with reference to FIGS. 2-15 in 
U.S. Ser. No. 08/375,353) is incorporated herein by reference, since such 
description applies to the common features of the FIG. 1 ellipsometer of 
U.S. Ser. No. 08/375,353 (and the details thereof and variations thereon) 
and the FIG. 4 ellipsometer of the present disclosure (and the details 
thereof and variations thereon described in the present disclosure with 
reference to FIGS. 5-11). 
In preferred implementations, processor 101 of FIG. 4 is programmed with 
calibration software in accordance with the invention, in addition to 
being programmed to perform all the functions performed by processor 100 
in the FIG. 1 ellipsometer of U.S. application Ser. No. 08/375,353. 
The spectroscopic ellipsometer of FIG. 4 includes several subsystems: 
optical and signal processing components (components 1, 4-6, 6A, 7, 8, 10, 
10A, 14, 16, 17, spectrometer components 69, 170, 171, 172, and 173, and 
processor 101) for measuring polarized radiation of beam 9 which has 
reflected from a small spot on sample 3, and for processing the measured 
data; 
focusing and pattern recognition components (including objective 40 and 
subsystem 80) for controlling the focusing of beam 9 onto a desired small 
spot on sample 3, and optionally also for imaging sample 3 (or a selected 
portion of sample 3) and recognizing a pattern in such image; and 
sample stage 63 (for moving sample 3 relative to the ellipsometer's optical 
components and relative to objective 40). 
Beam 9 (radiation emitted from lamp 10 and then polarized in polarizer 5) 
is reflected from sample 3 through a slit in aperture plate 6A to 
collection mirror 6, is then reflected from mirror 6 to mirror 7, and is 
then directed by mirror 7 through analyzer 8 into a spectrometer. The 
spectrometer (to be described in detail below) comprises entrance slit 
member 69, folding mirror 170, Ebert spherical mirror 171, prism 172, and 
detector array 173. Alternatively, an Ebert-Fastie or Czerny-Turner 
spectrometer can be employed. 
Radiation (e.g., from lamp 10) is reflected from sample 3 back to objective 
40, and is focused by objective 40 onto optical elements or sensors within 
subsystem 80 (for use in performing pattern recognition, controlling the 
focusing of beam 9 onto sample 3, and optionally displaying an image of 
all or part of the sample). Sample 3 is typically a semiconductor wafer 
with at least one thin layer 3a on a substrate. 
The illumination subsystem of FIG. 4 includes lamp 10 (preferably a xenon 
arc lamp including heatsink window 10A) which emits radiation beam 12 
having a broad range of frequency components in the UV, visible, and near 
infrared wavelength bands, a lamp housing including lamp housing window 
14, off-axis paraboloid mirror 16, UV cutoff filter 18 and color filter 
20, paraboloid mirror 17, and optical fiber 1. Fiber 1 has an inlet end 
for receiving beam 12, after beam 12 has reflected from mirror 16, passed 
through UV cutoff filter 18 and color filter 20, and then reflected from 
mirror 17. Beam 12 propagates through fiber 1 to entrance slit member 2 
and then through the entrance slit in member 2. Mirrors 16 and 17 
preferably have identical design. 
Lamp 10 emits beam 12 through heatsink window 10A and then through lamp 
housing window 14, to mirror 16. Windows 10A and 14 are unnecessary for 
optical reasons, but function to keep lamp cooling air from being drawn 
through the optical path, thereby avoiding noise due to shimmering of the 
arc image. A xenon arc lamp is preferred over other lamps such as tungsten 
or deuterium lamps, because a xenon lamp will produce radiation having a 
flatter spectrum in the wavelength range from UV to near infrared. 
Alternatively, a tungsten lamp and a deuterium lamp can be used in 
combination to cover the substantially the same spectrum covered by a 
xenon lamp. Brightness of the spectrum is important, because with less 
intensity, reflected radiation must be collected for longer periods. The 
lower intensities slow the measurement process. In alternative 
embodiments, a lamp is chosen which emits broadband UV radiation without 
emitting significant visible or near infrared radiation. 
Preferably, optical fiber 1 is made of fused silica, a UV transmitting 
material, and has a core diameter of 365 microns. 
The illumination subsystem optionally includes actuator 17A connected to 
mirror 17. Actuator 17A operates to move mirror 17 between a first 
position (shown in FIG. 4) in which it reflects beam 12 from mirror 16 
toward the inlet end of fiber 1, and a second position (not shown in FIG. 
4). In such second position, mirror 17 is outside the optical path of beam 
12 and thus does not impede propagation of beam 12 from mirror 16 to a 
spectrophotometer. Such spectrometer is not shown in FIG. 4, but is 
preferably integrated with the ellipsometer. 
With reference again to FIG. 4, the sample illuminating radiation enters 
polarizer 5 after propagating from fiber 1 through the entrance slit in 
member 2. The portion of this radiation which propagates through polarizer 
5 emerges from polarizer 5 as polarized beam 9. Polarized beam 9 is a 
measurement beam having a known polarization state. Polarizer 5 preferably 
has apertured plate 5A, with a circular aperture therethrough, positioned 
at its input face to limit the size of the polarized beams so that the two 
polarizations do not overlap. The diameter of this circular aperture is 
about 1 mm in one preferred embodiment in which the distance between 
entrance slit member 2 and polarizer 5 is about three inches. 
Entrance slit member 2 is a substrate (preferably made of stainless steel) 
through which an elongated, rectangular entrance slit (60 
microns.times.500 microns) has been etched. Because of the elongated shape 
of the entrance slit, elliptical focusing mirror 4 images the entrance 
slit as a small (25 micron.times.25 micron), compact (square-shaped) spot 
on sample 3, by reflectively focusing the beam 9 onto sample 3 at high 
incidence angle. Polarized beam 9 is incident at mirror 4 with a low 
incidence angle. Due to its orientation and the shape of its elliptical 
focusing surface, mirror 4 images the entrance slit. Mirror 4 has a 
numerical aperture (0.15 or greater, in preferred implementations of FIG. 
4) selected so that the rays of beam 9 reflected from mirror 4 will be 
incident at sample 3 with a desired range (preferably, a substantial 
range) of high incidence angles. In preferred implementations of FIG. 4 in 
which the numerical aperture of mirror 4 is 0.15, the range of high 
incidence angles (at which beam 9 strikes sample 3) is the range from 
about 63.5 degrees to about 80.5 degrees (from the normal to the surface 
of sample 3). This range desirably includes incidence angles near 
Brewster's angle for crystalline silicon (about 75.degree. at 630 nm 
wavelength) so that the instrument displays a high degree of sensitivity 
for film variations on silicon substrates. 
The preferred shape of focusing mirror 4's reflective surface is 
elliptical. As is well known, an elliptical mirror has two foci. In 
embodiments in which mirror 4 is an elliptical mirror, sample 3 should be 
positioned at one focus of the mirror and the entrance slit (through 
member 2) should be positioned at the other focus of the mirror. 
The elongated shape of the entrance slit in member 2, with the described 
design and orientation of mirror 4, results in focusing of beam 9 onto a 
small, compact (preferably square-shaped) spot on sample 3 with high 
incidence angle. 
In alternative embodiments, other combinations of an entrance slit and a 
focusing mirror are employed (in place of elements 2 and 4 of FIG. 4) to 
focus a beam onto a small (but not compact) spot on sample 3 with a 
substantial range of high incidence angles. 
Designing the reflective surface of mirror 4 to have its preferred 
elliptical shape (rather than a spherical shape, for example) reduces 
off-axis aberrations (such as the aberration known as "coma") in the 
focused beam incident on the sample. Use of a reflective elements (mirrors 
4, 6, and 7) between the polarizer and analyzer, rather than transmissive 
lenses, minimizes chromatic aberration in the analyzed beam which reaches 
spectrometer entrance slit member 69. 
Collection mirror 6 receives that portion of the diverging beam reflected 
from sample 3 which passes through an aperture in apertured plate 6A. 
Mirror 6 preferably has a focal length of 70 mm and a diameter of 20 mm. 
Mirror 6, because it is spherical, introduces coma into the beam. However, 
the aberration spreads the beam in a direction parallel to the long axis 
of the spectrometer entrance slit so it does not affect the light 
transmission properties of the instrument. In addition the spectrometer 
entrance slit is preferably rotated by approximately 5 degrees in a plane 
perpendicular to the surface normal in order to better pass the aberrated 
beam. 
The aperture in plate 6A is preferably elongated, and oriented to pass only 
the radiation which has reflected from sample 3 after reaching the sample 
at a single incidence angle (or narrow range of incidence angles). The 
aperture is preferably about 2 mm tall (in the Z-direction shown in FIG. 
4) and 20 mm wide, and oriented so as to pass the radiation reflected from 
sample 3 at an angle in the range from 75.degree. to 77.degree., while 
plate 6A blocks all other radiation reflected from sample 3. Thus, where 
beam 9 strikes sample 3 with a substantial range of high incidence angles, 
apertured plate 6A passes (for propagation to analyzer 8 and then 
measurement by detector 173) only the radiation reflected from sample 3 
after striking the sample at a narrow subset of the substantial range of 
high incidence angles. 
Actuator 62 can position plate 6A at any selected one of a range of 
positions in the optical path of reflected beam 9, so that the slit 
(aperture) through plate 6A will pass only those rays of the reflected 
beam which have reflected from sample 3 at incidence angles in a selected 
narrow range. For example, actuator 62 can be operated to move plate 6A 
(downward along the Z-axis in FIGS. 4 and 9) from the position shown in 
FIG. 4 (and FIG. 9) to a position in which the slit through plate 6A 
passes radiation reflected from sample 3 at an angle in the range from 
77.degree. to 79.degree. (and in which plate 6A blocks all other radiation 
reflected from the sample). Plate 6A and actuator 62 are shown in both 
FIGS. 4 and 9, but the manner in which plate 6A blocks some of the 
radiation reflected from sample 3 is shown more clearly in FIG. 9. 
To measure a complicated film stack, it is necessary to perform multiple 
independent measurements at different settings of one or more measurement 
parameters (such as wavelength or incidence angle). Spectroscopic 
ellipsometric measurement (at a fixed incidence angle) simultaneously 
provides data for multiple wavelengths of radiation reflected from the 
sample. Varying incidence angle in a sequence of spectroscopic 
ellipsometric measurements provides data about the sample which usefully 
supplements the data obtained at one fixed incidence angle. 
The width of the slit through apertured plate 6A determines the spreading 
of the incidence angles associated with the measured portion of the 
radiation reflected from sample 3, and the location of the slit's center 
determines the average incidence angle associated with the measured 
portion of such reflected radiation. Preferably, actuator 62 includes 
means for controlling both the slit width and the location of the slit's 
center. However, in some embodiments of the invention, the slit width 
and/or the location of the slit center are fixed. In embodiments in which 
the location of the slit center can be controlled, such location will 
typically be chosen to be close to Brewster's angle for the sample being 
measured. For example, when the sample is a flat panel display comprising 
films deposited on a glass substrate, it is useful to locate the slit 
center so that plate 6A passes only rays reflected from the flat panel 
display after being incident at angles in a narrow range centered at 
57.degree. (since Brewster's angle for glass is about 57.degree. at 
visible wavelengths). The latter embodiment would require substitution of 
a differently shaped focusing mirror for above-described elliptical 
focusing mirror 4 (since above-described mirror 4 could not focus 
radiation to sample 3 at incidence angles close to 57 degrees). 
Apertured plate 6A functions as an incidence angle selection element. An 
alternative position for the incidence angle selection element is shown in 
FIG. 7, and another such alternative position is between mirror 6 and 
mirror 7. In FIG. 7, the incidence angle selection element is movable 
apertured plate 6B, which is located between folding mirror 7 and analyzer 
8. Actuator 6C of FIG. 7 controls the location of the center of the slit 
through plate 6B, so that when radiation reflected from sample 3 at a 
substantial range of angles reaches plate 6B, only a portion of such 
radiation (i.e., the radiation reflected from sample 3 at a selected, 
narrow subrange of the "substantial range") will pass through plate 6B's 
slit. The dimensions of apertured plate 6B and the slit therethrough can 
(but need not) be identical to those of apertured plate 6A. Actuator 6C 
can (but need not) be identical to actuator 62 of FIG. 4. 
It should be understood that in each of FIGS. 7, 8, and 9, the polarized 
beam emitted from polarizer 5 propagates directly to mirror 4 without 
interacting with mirror 7. Mirror 7 is positioned so as to reflect (toward 
analyzer 8) only radiation that has already reflected from mirror 6. 
With reference again to FIG. 4, either polarizer 5 or analyzer 8 is rotated 
(about the optical axis) during measurement operations (including 
calibration operations). When polarizer 5 is to be rotated and analyzer 8 
to remain fixed, each of polarizer 5 and analyzer 8 is preferably a 
minimal-length Rochon prism of the type shown in FIGS. 5 and 6. The Rochon 
prism consists of two pieces separated by interface 5C, and splits the 
incident beam into two components: an ordinary polarized through beam, and 
an extraordinary polarized beam that is deflected by an angle of 1.6 
degrees (a deflection of at least 1.5 degrees is preferred for 
implementing the invention). The ordinary polarized beam is employed as 
beam 9 (which is focused by mirror 4 on the sample). Since it is desired 
to focus beam 9 on a small spot on the sample (e.g., to measure film 
thickness at such spot), no motion in the ordinary polarized "through" 
beam emitted from prism 5 can be tolerated. Typically, the rotation of 
prism 5 must be controlled so that the through beam deviation is 
constrained to be less than 30 seconds of arc, in order to restrict the 
motion of the spot on the sample to less than 1 micron. In some cases, 
through beam deviation of up to one minute of arc can be tolerated. 
With reference to FIGS. 5 and 6, the preferred Rochon prism embodiment of 
polarizer 5 (and analyzer 8) has only the minimum length (along the axis 
of "through beam" propagation) needed to enable the beam to pass through 
its clear aperture, because the prism's length is proportional to the 
amount of chromatic aberrations introduced by the prism. 
The area within polarizer 5 bounded by rectangular perimeter 5D in FIG. 5 
is the projection of interface 5C onto the plane of FIG. 5, and is what is 
referred to as the "clear aperture" of polarizer 5. As shown in FIG. 4, 
plate 5A having an entrance aperture therethrough should be positioned 
along the optical path between entrance slit element 2 and polarizer 5, so 
that the aperture through plate 5A determines the diameter of the beam 
(which has passed through the entrance slit through element 2) which 
passes through polarizer 5. The length of polarizer 5 should be the 
minimum length (assuming a fixed angle between interface 5C and the right 
face of polarizer 5 in FIG. 6) which causes the clear aperture to be as 
large as the cross-section of the beam incident on polarizer 5. It will be 
apparent to those of ordinary skill that the mechanical constraints 
inherently faced in designing and mounting a polarizer will also affect 
the minimum practical length for polarizer 5, and that varying the 
position of prism 5 (in the FIG. 4 system) will affect the preferred size 
of the aperture through plate 5A. 
Rochon prism 5 of FIG. 6 (which is preferred for use as polarizer 5 and 
analyzer 8 in FIG. 4) has a length (along the axis of through beam 
propagation) equal to 12 mm, with a tolerance of plus or minus 0.25 mm. In 
contrast, the length of a conventional, commercially available Rochon 
prism 5' (shown in phantom view in FIG. 6) is approximately 25 mm. As 
shown in FIG. 5, the Rochon prism 5 preferred for use as polarizer 5 in 
FIG. 4, has a square cross-section (in a plane perpendicular to the axis 
of through beam propagation) with sides of length 8 mm, with a tolerance 
of plus or minus 0.1 mm. The preferred Rochon prism of FIGS. 5 and 6 
preferably uses UV transmitting crystalline quartz, is optically contacted 
for enhanced UV transmission, introduces wavefront distortion of less than 
one quarter of a wavelength (at 633 nm), has transmittance in the UV of at 
least 40% (for two open polarizers at 230 nm) when used with an 
unpolarized light source, and has uncoated faces. 
To measure a sample, analyzer 8 typically remains fixed while polarizer 5 
rotates about the optical axis. Analyzer 8 is mounted so as to be free to 
rotate into a different angular orientation when a new sample is placed in 
the instrument (or when a new measurement is to be conducted on the same 
sample). This technique of "analyzer tracking" is well known in the field 
of ellipsometry. 
To implement the inventive calibration method, processor controls operation 
of the FIG. 4 spectroscopic ellipsometer as follows. Processor 101 
generates control signals, and supplies them to polarizer 5, analyzer 8, 
and detector array 173 to control acquisition by detector array 173 of 
data of the type described above with referende to equations (10)-(15), 
and then receives this data from array 173 and processes the data to 
determine coarse approximations of the above-defined values A.sub.0 and 
P.sub.0. For example, in some embodiments processor 101 is programmed to 
generate a sequence of the control signals for rotating analyzer 8 into a 
sequence of different fixed positions at appropriate times during 
measurement of the data, and for causing polarizer 5 to rotate with 
appropriate substantially constant speed during measurement of the data. 
In the FIG. 4 embodiment, P.sub.0 is the angle between the optical axis of 
polarizer 5 and the plane of incidence at t=0, and A.sub.0 is the offset 
of the actual orientation angle of analyzer 8 from the nominal angle A of 
analyzer 8 (where "A" is a reading of analyzer 8's orientation angle, 
supplied to processor 101, for example, from an analyzer position sensor 
associated with analyzer 8). To control acquisition of this data, 
processor 101 generates control signals for controlling the orientation of 
analyzer 8 and the rotation of polarizer 5 (and the operation of the other 
system components) to cause detector array 173 to provide to processor 101 
the data to be processed in accordance with equations (10)-(15). 
After determining the coarse approximations of A.sub.0 and P.sub.0, 
processor 101 generates control signals for controlling the orientation of 
analyzer 8 and the rotation of polarizer 5 (and the operation of the other 
system components) to cause detector array 173 to provide to processor 101 
the necessary data to be processed to determine the refined approximations 
of A.sub.0 and P.sub.0 in accordance with the inventive method. More 
specifically, processor 101 generates control signals to cause detector 
array 173 to provide reflectivity data (tan.psi. and cos.DELTA. spectra) 
at each of two or more angles of analyzer 8, and processor 101 then 
performs regression on this data to determine the refined approximations 
of the values A.sub.0 and P.sub.0. The reflectivity data (for each angle 
of analyzer 8) consist of a sequence of readings of detector array 173 
taken while polarizer 5 rotates, with each read-out of detector array 173 
made at a different angular position of polarizer 5. The reflectivity data 
determine a tan.psi. spectrum and a cos.DELTA. spectrum for each of at 
least two analyzer angles, each set of reflectivity data measured by 
operating the ellipsometer (with rotating polarizer 5) at a different 
angle of analyzer 8. The tan.psi. and cos.DELTA. spectra for each analyzer 
angle determine the amplitude and phase of the complex ratio of p and s 
components of the reflectivity of sample 3 (at each of at least two 
incident radiation frequencies). 
Alternative embodiments of the FIG. 4 system employ an alternative type of 
polarizer (and analyzer), such as a Glan-Taylor polarizer. Other 
embodiments employ a phase modulator (such as a photoelastic modulator) in 
place of a rotating polarizer. 
In other embodiments, the invention is a method for calibrating a 
spectroscopic ellipsometer (or other ellipsometer) whose analyzer rotates 
during measurement of a sample, and whose polarizer (which can be a 
minimal-length polarizer or other polarizer) remains fixed during each 
measurement of a sample. Another aspect of the invention is an 
ellipsometer of this type which includes a means for performing the 
calibration method automatically (preferably including a processor 
programmed with software for performing the calibration method 
automatically). In these embodiments, the above-described calibration 
method of the invention is modified in a manner that will be apparent to 
those of ordinary skill in the art so that it determines coarse (and then 
refined) approximations of values P'.sub.0 and A'.sub.0, where A'.sub.0 is 
the angle (at t=0) of the optical axis of the analyzer, and P'.sub.0 is 
the offset of the actual orientation angle of the fixed polarizer from the 
nominal (measured) angle P' of the fixed polarizer. 
To understand this modified method, it is should be appreciated that when 
the analyzer is controlled so that it rotates at a constant speed, the 
signal received at the detector (or each detector of the detector array of 
a spectroscopic ellipsometer) will be a time-varying intensity given by: 
##EQU9## 
where I.sub.0 is a constant that depends on the intensity of radiation 
emitted by the source, .omega. is the angular velocity of the analyzer, 
A'.sub.0 is the angle of the optical axis of the analyzer at an initial 
time (t=0), and .alpha. and .beta. are sample related values defined as 
follows: 
EQU .alpha.=[tan.sup.2 .psi.-tan.sup.2 (P'-P'.sub.0)]/[tan.sup.2 
.psi.+tan.sup.2 (P'-P'.sub.0)] (2') 
and 
EQU .beta.=2(tan.psi. )(cos.DELTA. )tan(P'-P'.sub.0)/[tan.sup.2 .psi.+tan.sup.2 
(P'-P'.sub.0)] (3') 
where tan.psi. is the amplitude of the complex ratio of the p and s 
components of the reflectivity of the sample, .DELTA. is the phase of the 
complex ratio of the p and s components of the reflectivity of the sample, 
P' is the nominal polarizer angle (a reading of the polarizer's 
orientation angle, supplied for example from a sensor which senses the 
position of the polarizer), and P'.sub.0 is the offset of the actual 
orientation angle of the polarizer from the nominal polarizer angle "P'." 
The values .alpha.' and .beta.' are also sample related values, defined as 
follows: 
EQU .alpha.'=.alpha.cos (2A'.sub.0)+.beta.sin (2A'.sub.0) (4') 
and 
EQU .beta.'=.alpha.sin (2A'.sub.0)-.beta.cos (2A'.sub.0) (5') 
where .alpha., .beta., and A'.sub.0 are defined above. 
In embodiments in which ellipsometer measures reflectivity data with a 
rotating analyzer and fixed polarizer, the inventive calibration method 
determines (from reflectivity data measured using the ellipsometer) the 
same "residual" (R) defined above in equation (6): R=1-.alpha..sup.2 
-.beta..sup.2 =1-.alpha.'.sup.2 -.beta.'.sup.2 =R'. To calibrate such an 
ellipsometer, the inventive method determines the coarse approximations of 
A'.sub.0 and P'.sub.0 as follows. 
First, data determining nominal residual value R'(P'.sub.1) are acquired 
with the fixed polarizer at a first nominal polarizer angle P'.sub.1. This 
is done by exploiting symmetry (without knowledge of values A'.sub.0 and 
P'.sub.0), assuming for purposes of determining the nominal value 
R'(P'.sub.1) that P.sub.0 =0. 
Also, data determining nominal values R'(P'.sub.i) are acquired at several 
nominal polarizer angles P'.sub.i, all approximately equal to -P'.sub.1. 
The next step exploits the fact that the actual value R' satisfies the 
symmetry relation R'(P'-P'.sub.0) =R'[-(P'-P'.sub.0)]. In this step, the 
invention determines a second nominal polarizer angle P'.sub.2 at which 
the nominal value R'(-P'A.sub.2) is closest to the nominal value 
R'(P'.sub.1). The second nominal angle P'.sub.2, which has the same sign 
as the first nominal angle P'.sub.1, is equal to P'.sub.2 =-P'.sub.i1, 
where +P'.sub.i1 is one of the nominal polarizer angles P'.sub.i that are 
approximately equal to -P'.sub.1. 
Then, the invention averages the two nominal angles P'.sub.2 and P'.sub.1, 
and identifies the resulting averaged value (shown in equation (10')) as 
the coarse approximation of P'.sub.0 : 
EQU P'.sub.0 =(P'.sub.1 +P'.sub.2)/2 (10') 
Using this coarse approximation of P'.sub.0, the coarse approximation of 
A'.sub.0 is determined as follows. From equations (2') and (3'), it 
follows that: 
EQU .alpha..sub.+ =.alpha.(P'-P'.sub.0)]=.alpha.[-(P'-P'.sub.0)]=.alpha..sub.-( 
11') 
and 
EQU .beta..sub.+ =.beta.(P'-P'.sub.0)=-.beta.[-(P'-P'.sub.0)]=-.beta..sub.-(12' 
) 
We denote .alpha.'(P'-P'.sub.0) as .alpha..sub.+ ', .alpha.'[-P'-P'.sub.0) 
as .alpha..sub.- ', .beta.'(P'-P'.sub.0) as .beta..sub.+ ', and 
.beta.'[-(P'-P'.sub.0) as .beta..sub.- '. From equations (11'), (12'), 
(4'), and (5'), it follows that: 
EQU .alpha..sub.+ '+.alpha..sub.- '=2.alpha.cos (2A'.sub.0) (13') 
and 
EQU .beta..sub.+ '+.beta..sub.- '=2.alpha.sin (AP'.sub.0 ) (14') 
Thus, the coarse approximation of A'.sub.0 is determined to be 
EQU A'.sub.0 =tan.sup.-1 [(.beta..sub.+ '+.beta..sub.- ')/(.alpha..sub.+ 
'+.alpha..sub.- ')]/2 (15') 
Having determined coarse approximations of the values A'.sub.0 and 
P'.sub.0, the invention then determines refines approximations of the 
coarsely approximated values A'.sub.0 and P'.sub.0, in the following 
manner. First, the ellipsometer (undergoing calibration) is operated to 
measure reflectivity data at each of two or more polarizer angles (with a 
rotating analyzer and fixed polarizer at each of the polarizer angles). 
The reflectivity data (for each polarizer angle) determine both a tan.psi. 
value and a cos.DELTA. value for each incident radiation frequency (data 
for two or more frequency components are measured where the ellipsometer 
is a spectroscopic ellipsometer). In the case that the ellipsometer is a 
spectroscopic ellipsometer, the reflectivity data for each polarizer angle 
determine tan.psi. and cos.DELTA. spectra, which in turn determine the 
amplitude and phase of the complex ratio of p and s components of sample 
reflectivity (at each incident radiation frequency). 
The reflectivity data (e.g., tan.psi. and cos.DELTA. spectra) obtained at 
multiple polarizer angles are then processed to determine refined 
approximations of the values A'.sub.0 and P'.sub.0. Preferably, the 
reflectivity data are processed by performing regression on A'.sub.0 and 
P'.sub.0 (using the well-known least square fit algorithm or another 
function minimization technique, in a class of preferred embodiments) 
until the differences among the reflectivity data for different polarizer 
angles are minimized. 
With reference again to FIG. 4, the spectrometer subsystem of the FIG. 4 
embodiment of the inventive spectroscopic ellipsometer will next be 
described. This subsystem comprises entrance slit member 69, folding 
mirror 170, Ebert spherical mirror 171, prism 172, and detector 173. Slit 
member 69 is made of the same material as above-described entrance slit 
member 2. The spectrometer entrance slit through member 69 is preferably 
an elongated slit of size 230 microns by 1200 microns (the beam is focused 
to a spot on sample 3 which is smaller than this entrance slit, and so the 
beam passes through the entrance slit unobstructed). The spectrometer is 
of a standard Ebert design, in which the broadband beam passed through 
member 69 (from analyzer 8) reflects from mirror 170 to mirror 171, and 
from mirror 171 to prism 172. The beam components having different 
wavelengths are refracted in different directions from prism 172 to mirror 
171, and from mirror 171 to detector 173. Mirror 171 images the entrance 
slit (through member 69) to detector 173, and mirror 171 preferably has a 
focal length of 250 mm. In preferred embodiments, detector 173 is 
essentially a linear array of photodiodes, with each photodiode measuring 
radiation in a different wavelength range. Preferably the radiation 
includes components with wavelength in the range from 230 nm to 850 nm, 
detector array 173 includes 512 photodiodes, and each photodiode (or set 
of adjacent photodiodes) receives radiation in a different segment of the 
230-850 nm wavelength range. For example, the resolution of the photodiode 
array may be limited to groups of three to five adjacent photodiodes, in 
the sense that each resolvable radiation element has a width of three to 
five photodiodes. 
Preferably, detector array 173 is an intensified photodiode array. For 
example, it can be a photodiode array available from the Japanese company 
Hammamatsu, to which an intensifier, known as Part Number BV2532QZ-15 
available from Proxitronics, is mated. The photodiode array of this 
commercially available product has 512 photodiodes, which independently 
measure 512 different wavelengths. An alternative embodiment of detector 
array 173 is a UV enhanced CCD array detector. 
We next describe two embodiments of an autofocus assembly for the inventive 
ellipsometer. One such assembly is shown in FIG. 8, and the other will be 
described with reference to FIGS. 9 and 10. 
The autofocus assembly of FIGS. 9 and 10 includes split photodiode detector 
94, which receives a substantially focused image of the spot to which beam 
9 is focused on sample 3. This image is provided by positioning 
beamsplitting mirror 95 along the optical path between analyzer 8 and 
spectrometer entrance slit element 69 (of FIG. 4) to divert a portion of 
the beam transmitted through analyzer 8 to detector 94. Detector 94 has 
two photodiodes, 94A and 94B, which are best shown in FIG. 10. Each of 
photodiodes 94A and 94B provides a measured intensity signal to processor 
101. Processor 101 is programmed to be capable of processing these signals 
in the same manner (described in U.S. patent application Ser. No. 
08/375,353) as does processor 100 of U.S. application Ser. No. 08/375,353. 
Detector 94 is positioned so that an entire substantially focused image 
(94C) of the spot can be projected onto photodiodes 94A and 94B, with 
approximately half of image 94C projected onto each of photodiodes 94A and 
94B as shown in FIG. 10. 
The auto focus system of FIGS. 9-10 (and its method of operation) is fast 
(i.e., processor 101 determines the necessary values very quickly); the 
algorithm implemented by processor 101 is simple; and this auto focus 
system gives directional information (in the sense that it enables the 
operator to tell whether the sample is above or below the best focus 
position). 
In designing the autofocus assembly of FIGS. 9-10, it is important to 
consider that the image intensity seen by the camera is time-varying, and 
that the speed at which the video image can be digitized and processed 
should be sufficiently high to enable autofocus, 
The alternative autofocus assembly of FIG. 8 includes source 92 of off-axis 
illuminating radiation, apertured mirror 93, apertured mirror 90, and 
camera 91. Apertured mirror 90 has a slit extending through it, and 
functions as an incidence angle selection element similar to the way 
apertured plate 6B of FIG. 7 functions as an incidence angle selection 
element. Indeed mirror 90 can be of identical design as apertured plate 6B 
(but the planar surface of mirror 90 which faces away from mirror 7 is 
highly reflective, while the corresponding planar surface of plate 6B need 
not be highly reflective). A first portion of the radiation from 
collection mirror 6 passes through the slit in mirror 90, and then 
reflects from mirror 7 toward analyzer 8 (just as in FIG. 4 and FIG. 7). 
However, because mirror 90 is tilted at a small angle with respect to 
folding mirror 7 (and is positioned along the optical path), mirror 90 
reflects a second portion of the radiation that it receives from 
collection mirror 6 toward camera 91 (this second portion does not pass 
through the slit in mirror 90, and does not propagate to analyzer 8). The 
radiation reflected from mirror 90 is focused to camera 91, and camera 91 
thus observes the position and size of the spot on sample 3. 
Signals indicative of the position and size of the spot are supplied from 
camera 91 to processor 101. In response to these signals, processor 101 
generates focus control signals (in the same manner as does processor 100 
of U.S. Ser. No. 08/375,353) that are used for focusing the sample (e.g., 
the focus control signals are used for controlling the position of sample 
stage 63). Where camera 91 is part of focusing and pattern recognition 
subsystem 80 of FIG. 4, the signals output from camera 91 are used for 
pattern recognition as well as for the auto focus function described with 
reference to FIG. 8. 
Apertured mirror 93 has an aperture therethrough which allows polarized 
beam 9 from polarizer 5 to pass unimpeded to mirror 4. Apertured mirror 93 
also reflects off-axis illuminating radiation from source 92 toward mirror 
4. This off-axis illuminating radiation is reflected to camera 91, where 
it enables camera 91 to "see" the position of the spot to which beam 9 is 
focused on the sample (and to enable pattern recognition and auto focus 
operations). 
Next, with reference to FIG. 11, we describe embodiments in which the 
inventive ellipsometer includes a reference channel (in addition to a 
sample channel which detects radiation reflected from the sample). The 
ellipsometer of FIG. 11 has both a reference channel (including detector 
array 273) and a sample channel (including detector array 173). 
Illuminating radiation from lamp 10 reflects from mirror 16 to mirror 17, 
and then from mirror 17 to entrance end 102 of bifurcated optical fiber 
101A. As the radiation propagates within fiber 101A away from end 102, it 
is split into two portions: a reference beam 109 emitted from end 104 of 
fiber 101A; and sample beam 9 (identical to beam 9 of FIG. 4) emitted from 
end 103 of fiber 101A. Sample beam 9 is polarized in rotating polarizer 5, 
then is reflectively focused by mirror 4 to sample 3, then reflects from 
the sample surface to mirror 6 and then mirror 7, and then reflects from 
mirror 7 through analyzer 8 to the entrance slit in spectrometer entrance 
slit member 69. In the spectrometer, the portion of sample beam 9 passed 
through member 69 reflects from mirror 170 to mirror 171, and from mirror 
171 to prism 172. The beam components having different wavelengths are 
refracted in different directions from prism 172 to mirror 171, and from 
mirror 171 to sample channel detector array 173. 
Reference beam 109 does not reflect from sample 3, but is directed directly 
to the spectrometer. Specifically, beam 109 reflects from mirror 171 
(i.e., from a slightly different spot on mirror 171 than the spot from 
which beam 9 reflects) to prism 172. The components of beam 109 having 
different wavelengths are refracted in different directions from prism 172 
to mirror 171, and from mirror 171 to reference channel detector array 
273. Detector arrays 173 and 273 are identical, but have slightly offset 
positions, so that the former receives only radiation of beam 9 reflected 
from mirror 171, and the latter receives only radiation of beam 109 
reflected from mirror 171. 
Alternatively, a plate with a double entrance slit is substituted for plate 
69 of FIG. 11. In such embodiments, the sample beam passes through one 
entrance slot into the spectrometer and the reference beam passes through 
the other entrance slot into the spectrometer. 
By processing reference signals from reference channel detector array 273 
with signals from sample channel detector array 173, the thickness (or 
refractive index) of a thin film on sample 3 can (under some conditions) 
be more accurately determined than with the FIG. 4 ellipsometer (which has 
no reference channel). In the In the FIG. 11 system, processor 101 is 
programmed (in the same manner as is processor 100 of U.S. application 
Ser. No. 08/375,353) to normalize the reflectivity measured by sample beam 
9 using the reference beam measurements from detector array 273, to 
compensate for such effects as lamp intensity fluctuations and air 
currents. 
An alternative technique for obtaining a reference beam is to modify the 
FIG. 4 apparatus so that it splits beam 9 at the location of focus mirror 
4. This can be done by designing mirror 4 to have a more complicated shape 
which focuses a portion of beam 9 (which functions as the sample beam) to 
sample 3 and directs the remaining portion of beam 9 (the reference beam) 
directly to collection mirror 6. In this case, the shape of collection 
mirror 6 would also be modified to reflect the reference beam to a 
separate channel in the spectrometer, while directing the sample beam to 
mirror 7. 
Other variations on the FIG. 4 ellipsometer will include a second optical 
fiber, identical to fiber 1, for directing the radiation propagating out 
from analyzer 8 to the spectrometer entrance slit through member 69. 
Alternatively, the inventive ellipsometer can omit fiber 1, and include 
only one optical fiber which directs radiation from analyzer 8 to the 
spectrometer entrance slit. 
Several variations on the spectroscopic ellipsometer of FIG. 4 have been 
described with reference to FIGS. 5-11. In other alternative embodiments 
of the inventive ellipsometer, polarized radiation having only one 
wavelength (rather than broadband radiation) is reflected from the sample. 
These embodiments can include a spectrometer as in FIG. 4, or 
alternatively a simple photodiode detector which detects the radiation 
output from the analyzer. In the alternative embodiments which employ a 
simple photodiode detector (rather than a spectrometer including a 
detector array), the ellipsometer is not a "spectroscopic ellipsometer." 
FIG. 12 is a schematic diagram of an ellipsometer (not a spectroscopic 
ellipsometer) which includes a means for performing an embodiment of the 
inventive calibration method automatically. In operation of the FIG. 12 
ellipsometer, a beam of radiation (which can be monochromatic radiation) 
from radiation source 10' is linearly polarized in polarizer 5, and the 
linearly polarized beam is then incident on sample 3. After reflection 
from sample 3, the beam propagates toward analyzer 8 with a changed 
polarization state (typically, the reflected beam has elliptical 
polarization, where the polarized beam emerging from polarizer 5 had 
linear polarization). The reflected beam propagates through analyzer 8 to 
photodiode detector 73. Detector 73 outputs a signal indicative of the 
intensity of a single frequency component (or frequency components in a 
single frequency range) of the radiation incident thereon. Processor 101' 
receives the measured data from detector 73, and is programmed with 
software for processing the data it receives in an appropriate manner. 
Either polarizer 5 or analyzer 8 is rotatably mounted for rotation about 
the optical axis during a measurement operation (or both of them are so 
rotatably mounted). During a typical measurement operation, polarizer 5 is 
rotated and analyzer 8 remains in a fixed orientation, or analyzer 8 is 
rotated and polarizer 5 remains fixed. 
Processor 101' is programmed to generate control signals for controlling 
the rotation (or angular orientation) of polarizer 5 and/or analyzer 8, or 
for controlling other operating parameters of elements of the FIG. 12 
system (such as the position of a movable sample stage on which sample 3 
rests). Processor 101' also receives data (indicative of the angular 
orientation of analyzer 8) from an analyzer position sensor associated 
with analyzer 8 and data (indicative of the angular orientation of 
polarizer 5) from a polarizer position sensor associated with polarizer 5, 
and is programmed with software for processing such orientation data in an 
appropriate manner. 
To calibrate an ellipsometer (which is not a spectroscopic ellipsometer) 
such as that of FIG. 12 in accordance with the invention, for operation 
with a fixed analyzer and a rotating polarizer during measurement of a 
sample, the above-described calibration method of the invention (whose 
coarse approximation step is described with reference to equations 
(10)-(15)) is slightly modified in a manner that will be apparent to those 
of ordinary skill in the art so that the method determines coarse (and 
then refined) approximations of values P.sub.0 and A.sub.0 by processing 
reflectivity data measured at only a single frequency (or frequency range) 
of incident radiation (rather than tan.psi. and cos.DELTA. spectra 
comprising reflectivity data measured at each of distinguishable multiple 
frequencies of incident radiation). Processor 101' of FIG. 12 is 
programmed to implement this slightly modified version of the 
above-described calibration method of the invention. Processor 101' can 
comprise hardware identical to that of processor 101 (of FIG. 4). However, 
processor 101' but is programmed in a slightly different manner than is 
processor 101, so that processor 101' processes data (from detector 73) 
corresponding to a subset of the data processed by processor 101, with 
processor 101' processing this reduced set of data in the same manner that 
processor 101 processes data from only one detector of array 173. 
Other embodiments of the inventive apparatus comprise not only an 
ellipsometer, but a spectrophotometer integrated together with an 
ellipsometer (preferably any of the above-described spectroscopic 
ellipsometers) as a single instrument. An example of such a 
spectrophotometer integrated with a spectroscopic ellipsometer is the 
system described with reference to FIG. 14 of above-referenced U.S. Ser. 
No. 08/375,353 (modified only by addition of a processor 101 identical to 
that of FIG. 4 of the present disclosure, programmed for receiving and 
processing data output from its detectors and for generating control 
signals for controlling the polarizer, analyzer, and other components 
thereof to implement the inventive calibration method). 
Several embodiments of methods and optical systems according to the present 
invention have been described. The description is illustrative and not 
restrictive. Many other variations on the invention will become apparent 
to those of skill in the art upon review of this disclosure. Merely by way 
of example, the sample measured by the invention need not be a wafer (it 
can be any other reflective object), and fold mirrors can be removed where 
space allows and additional fold mirrors provided where space is limited. 
The scope of the invention should be determined not merely with reference 
to the above description, but should be determined with reference to the 
appended claims along with their full scope of equivalents.