Beam Deviation Errors in Ellipsometric Measurements; an Analysis

Transcription

1 Beam Deviation Errors in Ellipsometric Measurements; an Analysis James R. Zeidler, R. B. Kohles, and N. M. Bashara The types of error produced by beam deviation in the optical elements of an ellipsometer are examined. It is shown that there are two types of error that may be significant-systematic errors due to a variation in the plane of incidence and in the angle of incidence at the specimen and errors due to the combined effects of beam displacement and nonuniformities in either the detector response or the optical properties of the specimen, the compensator, the polarizer, the analyzer, or the specimen cell. Analytic expressions for the variation in the plane of incidence and in the angle of incidence are given in terms of parameters that can be determined experimentally. A method by which these parameters can be measured is described. It is shown that the azimuthal variation in the angle of incidence produces fundamental errors in conventional zone averaging techniques because the values of lb and A are averaged at different angles of incidence in different zones. Methods of experimentally predetermining when such errors are likely to be significant are discussed, and a procedure that cancels most systematic errors due to beam deviation in each zone is described. The combined effects of beam deviation in the polarizer, the compensator, the cell windows, and the analyzer are considered in several commonly used configurations, and the configurations that minimize beam deviation errors are described. 1. Introduction In recent years there has been considerable interest in improving the absolute accuracy of ellipsometric measurements. It has been shown that most of the systematic errors produced by component imperfections and instrumental misalignment can either be canceled to first order by zone averaging or can be eliminated by accurate calibration techniques.1-7 Another important source of error that has not previously been analyzed and that does not cancel in the zone averaging techniques now used results from the angular deviation of the beam produced by imperfect components. Beam deviation errors must be considered if the absolute accuracy of zone averaged measurements is to be determined since, as previously noted, 8 such errors vary in magnitude as a function of the azimuthal settings of the components. It is often necessary to determine the azimuthal angles of the components of the ellipsometer to an accuracy of at least 0.01 in high precision measurements. The polarizing prisms supplied with commercial ellipsometers, for use in the visible and the near ir and uv spectral regions, typically have angu- Both authors were with the Electrical Materials Laboratory, College of Engineering, University of Nebraska, Lincoln, Nebraska 68508, when this work was done; J. R. Zeidler is now with the U.S. Naval Underseas Center, San Diogo, California Received 26 December lar deviations between 0.01 and Unless high tolerances for the wedge angles of the cell windows of the specimen cell are specified, each window may introduce an additional angular deviation of 0.01 to The deviation of the beam may be further increased by defects in the compensator. Since the combined effects of beam deviation in all the elements of the ellipsometer can produce an angular deviation of the beam which greatly exceeds the desired angular precision in the measured parameters, g and A, and since calculations have shown that in many materials ai/db and aa/ao are greater than unity at the angles of incidence at which ellipsometry measurements are often made,1 0 a detailed study of the effects of beam deviation in ellipsometric measurements is required. Beam deviation errors have previously been detected by several authors. Lukeg" found that beam deviation may produce variations in the angle of incidence, X, of as much as 0.20 and suggests that the angle of incidence be chosen as the mean of the two extreme values obtained in the measurement. Smith1 2 found that a prism with a deviation angle of 0.02 produced a maximum error in of This corresponds to an error in A of for an aluminum film at an angle of incidence of at 5461 A.12 He noted that the error in A due to a given angular deviation would be even greater for a low absorption substrate since the derivative a/0ka increases near the pseudo-brewster angle as k 0.12 The discrepancies noted in zone-averaged calibration 1938 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

2 measurements have been attributed primarily to the effect of beam deviation in the polarizing prisms. 4 ' 5 It has also been shown that beam deviation errors significantly limit the accuracy of conventional alignment techniques. 7 It is not sufficient simply to use an average value of X, as previously suggested," since 0 varies continuously as the polarizer azimuth is varied. Further, beam deviation not only changes the angle of incidence at the sample, but it also tilts the plane of incidence relative to the horizontal plane of the ellipsometer thereby varying the angle of incidence of the beam at the compensator, analyzer, and detector. An additional source of error results from the variation in the position of the beam on the cell windows, the specimen, the compensator, the analyzer, and the detector. Hunter 5 attributes the beam deviation errors that he detects in his measurements to the variation in photomultiplier response as the beam position is changed by beam deviation but gives no indication as to how he isolated the variation in photomultiplier response from the other types of beam deviation error that may be present. The goal of this paper is to analyze the effects of beam deviation in detail so that its significance can be accurately assessed and so that suitable corrections can be made where necessary. It will be assumed that the ellipsometer has been aligned as previously described 7 so that beam deviation does not reduce the accuracy of the alignment. Initially we will assume that the axis of rotation of the polarizer is parallel to the beam and that one face of the polarizer is normal to the beam. It will be shown that the major results of this idealized case also apply in the general case. Initially it will be assumed that the beam deviation is produced by the polarizer alone. The combined effects of beam deviation in the polarizer, compensator, analyzer, and cell windows are considered in several commonly used experimental configurations of elements in Sec. VI, and the configurations that minimize beam deviation errors are described. II. Variation in the Plane of Incidence and the Angle of Incidence at the Specimen If the polarizer has nonparallel end faces or a nonparallel air gap, the transmitted beam direction is deviated by an angle from the incident beam direction. As the polarizer is rotated through 360, the transmitted beam direction describes a right circular cone of apex angle, which is centered about the incident beam direction. As shown in Fig. 1, the intersection of this cone with the specimen is an ellipse and both the plane of incidence and the angle of incidence at the specimen vary for each point on this ellipse. The horizontal plane of the ellipsometer lies along the x axis in Fig. 1 and, in a properly aligned 7 ellipsometer, is perpendicular to the instrumental axis of rotation. The horizontal plane is the plane formed by the undeviated beam and the normal to the specimen at the point of incidence of the undeviated beam (the point 0 in Fig. 1). The vertical plane of the ellipsometer lies along the y axis of Fig. 1 and contains the instrumental axis of rotation. (The vertical plane is perpendicular to the horizontal plane, intersecting it at point 0). The ellipse formed by the deviated beam intersects the vertical plane at points A and B and the horizontal plane at points C and D, as shown in Fig. 1. The angles of incidence in the horizontal plane are given by aind rc = qo + 3, (1) where ko is the angle of incidence of the undeviated beam and is also the angle of incidence read from the ellipsometer scale. The angles of incidence in the vertical plane may be obtained directly from the law of cosines and are given by (2) COSq 5 A = COScB = COSOOCOS6. (3) Since in ellipsometric measurements, co >> 0 and < 0.050, Eq. (3) reduces to O'A = OB (4) The plane of incidence at points C and D coincides with the plane of the undeviated beam but is tilted relative to that of the undeviated beam by an angle of +6 and - at points A and B, respectively. As shown in Fig. 2, for an arbitrary azimuthal setting of the polarizer, P, the tilt of the plane of incidence and the variation in the angle of incidence can be determined by projecting the angle of incidence 0(P) and the deviation angle into the horizontal and vertical planes of the ellipsometer and applying the results given by Eqs. (1)-(4). From the law of cosines, the tilt in the plane of incidence, v, is obtained from l ~ OD = 00 6, ~~~~~ A y Fig. 1. The ellipse of incidence at the specimen formed by rotating the polarizer 360. I l x August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1939

3 x // \ /'AJ~~~~~~~~~~~~~~~ PH / ~~~NO Fig.2. Determination of the angle of incidence at the sample for an arbitrary polarized setting, (P). v and 6 are the projections of the deviation angle of the polarizer 6 into the vertical and horizontal planes of the ellipsometer, respectively. SbH and PH are the projections of (P) and P into the horizontal plane of the ellipsometer. N 0, NPH' and N, are the normals to the surface at points, 0, PH, and P, respectively. cos6 = cos6vcos6h, (5) where v and 3 H are the projections of 6 into the vertical and horizontal planes of the ellipsometer, respectively. Likewise, the angle of incidence may be determined from oos/j(p) = cos6vcosoh(p), (6) where 0/H(P) is the projection of 0(P) into the horizontal plane. Since 6v 6 and GO >> 0, Eq. (6) reduces to ()-H(P). (7) From Eqs. (1) and (2), Eq (7) reduces to OM(P) - 00 : H- (8) In order to determine H at an arbitrary setting of the polarizer, an analytic expression for the ellipse of incidence at the specimen must be determined. It may be shown for /o < 890 and 6 < 0.1 that the ellipse of incidence is symmetric about 0 and is described by the equation X 2 COS 2 /00 + Y 2 = r 2. (9) These coordinates may be determined experimentally for an arbitrary azimuthal setting of the polarizer in terms of the reference angle 0 defined in Fig. 3. As shown in Fig. 3, at an arbitrary azimuthal setting, P, of the polarizer, y = x tano, (10) where 0 is the angle between the horizontal plane and the radial vector to the point P on the ellipse of incidence at the specimen. From Eqs. (8), (9), and (10) it may be shown that (P) = 0/j A sin-'[sinbcosjo/(cos 2 /o + tan 2 0) 112 ], (11) where the sign is negative for 37r/2 < 0 < r/2 and positive for 7r/2 < 0 < 3r/ 2. The tilt angle of the plane of incidence at an arbitrary setting of the polarizer can likewise be determined from Eqs. (5) and (11) in terms of /o, 6, and 0. It has been shown that a tilt in the specimen produces direct errors in the azimuthal angles of all the components. 4 However, since beam deviation tilts both the electric vector of the transmitted beam and the plane of incidence, the magnitude of the error is less than that produced by a tilted specimen. A method by which the azimuthal correction term for a, given value of 6v, 0, and 0 has been derived,1 3 and these results may be used in conjunction with the results of Ref. 13 to calculate the expected error in specific cases. 11. Determination of b and 0 The polarizer reference angle 0 can be determined at an arbitrary azimuthal setting of the polarizer by 1940 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

4 Pi Fig. 3. Determination of the coordinates of the point of incidence, P. at an arbitrary setting of the polarizer. r(o) is a distance between P and the center of the cone of the deviated beam. 0 is the angle between the horizontal plane of the allipsometer and r(o). x(0) and y(o) are components of r(o) in the horizontal and vertical planes of the ellipsometer, respectively. do and d 1 are the distances between the apex of the cone of the deviated beam and the points 0 and Px, respectively. measuring the azimuthal settings at which the cone of the deviated beam intersects the horizontal plane of the ellipsometer. This is done by aligning a laser parallel to the polarizer telescope axis using variable aperture pinholes. With the polarizer removed, the diameters of the pinholes are adjusted to the same diameter as the laser beam and are rotated in their mountings to ensure that they are accurately centered in the telescope arm. The beam is reflected from a specimen that has been aligned so that its surface is parallel to the axis of rotation of the ellipsometer. 7 The reflected beam is observed on a target mounted normal to the beam at a point that is sufficiently far from the ellipsometer that the radius of the cone of the deviated beam can be resolved (a distance of 3.0 m to 6.0 m was sufficient in our measurements). The sample table is then rotated through a small angle and several points of incidence are marked on the target as shown in Fig. 4. The line formed by these points determines the horizontal plane of the ellipsometer. The sample table is then locked in place and the point of incidence on the target is marked. The polarizer is then reinserted and the beam position is marked as the polarizer is rotated through 3600, as indicated in Fig. 4. The azimuthal scale readings of the polarizer at which the cone of the deviated beam intersects the horizontal plane are then determined, and the azimuthal angle at which the deviated beam intersects the horizontal plane at point D is denoted as Po. Therefore, at an arbitrary azimuthal setting P of the polarizer, the reference angle is given by 0(P) = P - P (12) The deviation angle of the polarizer can also be determined if the radius of the cone r and the distance between the target and an arbitrary point of reference along the beam path d are measured. If the target is then moved and the distance between the target and the reference point d 2 and the radius of the cone r 2 are measured, 3 = tan 1 '[(r 2 - r)/(d 2 - d]. (13) It is necessary to use at least two different distances because the polarizer is not in general normal to the beam, and lateral, as well as angular, deviation of the beam will occur. At large distances from the ellipsometer, the angular deviation dominates and the surface of revolution may be approximated by a right circular cone, but the location of the apex of this cone is unknown. Conventional techniques' 4 "1 5 for measuring prism deviation angles thus give higher precision, but this method gives a convenient estimate of that requires no additional equipment and allows and Po to be measured simultaneously. The accuracy is limited primarily by the beam divergence of the laser, and the measurements of are not reliable if is less than the divergence angle of the laser. (Our lasers have divergence angles between 0.01 and 0.02.) This procedure works equally well for the compensator, analyzer, and cell windows, and its application allows a quick determination of the expected magnitude of beam deviation errors in a given experimental configuration. The analysis thus far has assumed that the polarizer was at normal incidence to the beam. This condition is not usually satisfied, since it is necessary to tilt the components relative to the incident beam to eliminate errors that would arise from multiple reflections between the components. In this case, lateral, as well as angular, deviation of the beam must Fig. 4. Measurement of the radius of the cone of the deviated beam and the reference angle 0. The X's are the points obtained on an arbitrarily positioned target by varying the angle of incidence at the sample with the polarizer removed. The polarizer is then reinserted. The *'s are the points of incidence on the target for various azimuthal settings of the polarizer. If the target is normal to the beam, these points should form a circle of radius r. Po is the azimuthal setting of the polarizer at which the cone of the deviated beam intersects the horizontal plane of the ellipsometer at point D. P August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1941

5 be considered, and the intersection between the beam and the specimen is no longer simply described by an ellipse. This does not change the results of the preceding analysis, however, since a parallel displacement of the beam merely changes the point at which the beam intersects the sample. Although the reference angle 0 at the specimen may not be described by Eq. (10) when the effects of lateral deviation are included, the correlation between 0 and P given by Eq. (12) remains valid for oblique incidence at the polarizer because the calibration measurements are made at a sufficiently large distance from the polarizer that the effects of lateral deviation are negligibly small compared with the effects of angular deviation. Since 0(P) and the tilt in the plane of incidence are affected only by the angular deviation of the beam, the methods described for determining 0(P),, and 0 are still valid when the polarizer is tilted at a slight angle to the incident beam. IV. Cancellation of Errors Due to the Angle of Incidence Variation in Zone-Averaged Measurements eraged measurements.cancel several types of systematic errors present in ellipsometric measurements to first order.1-3 In the fixed compensator nulling scheme, the compensator is set at +450 in zones 2 and 4 and nulls are obtained for an azimuthal setting of the polarizer P 2 in zone 2 and P 4 P 2 i 900 in zone 4. The compensator is set at -45 in zones 1 and 3 and a null is obtained at Pi in zone 1 and at P 3 P in zone 3. As shown by Eq. (11), a change in the polarizer azimuth of 90 produces a variation in 0 between zones 2 and 4 and between zones 1 and 3. Since the magnitudes of 4 and A depend on X, fit and A will vary between zones in a way that is correlated to the azimuthal variation in 0 (i.e., to the magnitude of Oo, 6, P, and 0) and to the magnitudes of OA/aq and aq/0a for 0 cj co. It has been shown that the magnitudes of ca/dla/ and ao4/d are often large,' 0 "L 6 and beam deviation errors of 0.0i0 to would therefore be significant in many cases. A method of canceling these errors to first order is described below. A rotation of the polarizer by 1800 varies the angle of incidence from 0 = o + H to 0 = o - H, as predicted by Eqs. (8) and (11). Since the polarizer azimuth is unchanged by a 180 rotation, the angle of incidence variations in Vt and A are canceled to first order (i.e., to the extent that 4i and A depend linearly on 0) if the nulls obtained at Pi and Pi P are averaged in each zone. The 3P term in the second null arises from two factors. First, the variation in 0 produced by beam deviation may alter the azimuthal angles of both the polarizer and analyzer at null; second, due to a misalignment of the axis of rotation with the geometric center of the azimuthal scale of the polarizer, a rotation of the scale by 180 may not actually rotate the azimuth of the polarizer by The magnitude of the latter source of error varies with polarizer azimuth, 5 and if the distance between the rotational axis and the geometric center is Ax, the maximum eccentricity error is given by (0'/)max = (1 + Ax/r), (14) where 0' is the angle through which the scale is rotated and 0 is the angle indicated by the scale. This error can be significant. For example, if Ax = sec of arc and r = A sec of arc, the maximum difference between 0' and 0 is 0.07 (we found that the maximum eccentricity was between and on our ellipsometers). These errors may be eliminated by determining a correction curve by simultaneously measuring 0 and its supplement. Alternatively, as shown by Hunter, 5 correction terms may be obtained on the basis of polarizer-analyzer extinction measurements. In many cases, a predetermination of the polarizer reference angle 0 will allow the cancellation of beam deviation errors without an additional nulling measurement. For example, if a null is obtained in zone 1 or 2, there are two equivalent 900 rotations by which the nulls in zone 3 or 4, respectively, can be obtained. For example, if PO is determined to be 420 and a null is obtained in zone 1 at Pi = 870, 01 = 450 and equivalent nulls can be obtained in zone 3 for P or P If P , , while if P , If P , /j = 00 - H and 03 = 00 + H, from Eqs. (8) and (11), and the two-zone average gives the values of 4 and A at 0 = co to first order in 0. However, if P 3-357, 0c1 = 3 = 'PO - H and there would be no variation in the angle of incidence due to beam deviation, but 4 and A are not determined at co. Either technique can be advantageous in certain circumstances, and the predetermination of 0 allows the optimum procedure to be determined a priori. As shown in the preceding example, whenever 0 450, it is not necessary to rotate the polarizer by 1800 in each zone of a two-zone measurement to cancel beam deviation errors since the same cancellation automatically occurs between the two zones if the polarizer is rotated in the proper sense. A two-zone average will allow partial cancellation of the errors due to the angle of incidence variation for most settings of the polarizer at null. The magnitude of the variation in 4' and A between the two zones will usually depend on the sense in which the polarizer is rotated between zones. The errors in the measured azimuthal angles produced by the azimuthal-dependent tilt in the plane of incidence also tend to be canceled by this procedure. It has been shown that the corrected value of 4' is given by' 3 "(0, y) = '(0o, 0) + 0(, Y). (29) Averaging ' for Pi and Pi gives the average of O(/o + ay) and O(/ - c,-y). Since the tilt of the plane of incidence is equal and opposite for the two settings, only the error resulting from 3 / remains. The magnitude of this error is further re APPLIED OPTICS / Vol. 13, No. 8 / August 1974

6 duced by the fact that y and 3/ are inversely related (i.e., y is a maximum when Hi = 0 and vice versa). V. Errors Produced by a Variation of the Angle of Incidence and the Point of Incidence at the Compensator, the Analyzer, the Cell Windows, and the Detector If the angle between the normal to the ith component of the ellipsometer and the direction of the undeviated beam is given by Ei, the angle at which the beam strikes that element will vary between 'e + 1 as a function of the azimuthal angles of the preceding components, where 6i is the maximum deviation angle produced by summing the angular deviations resulting from all the preceding components. It has previously been shown that the phase retardation and the relative transmittance ratio of the compensator depend on the angle of incidence.9,15,1 6 However, the analysis of Bennett and Bennett 9 for a thick waveplate or compensator in which coherent multiple reflections can be ignored and the analyses of Holmes1 5 and of Oldham1 6 for a waveplate in which coherent multiple reflections occur indicate that for a variation in the angle of incidence of less than 0.100, the variations in the phase retardation and the transmittance ratio will be negligible. For angles of incidence less than the field angle of the analyzer, there is no variation in the polarization state of the beam transmitted by the analyzer. The transmitted intensity does depend on the angle between the electric vector of the incident beam and the polarization axis of the analyzer, but, since E is small, the variations in the transmitted intensity due to beam deviation are negligible. The variation in the sensitivity of the detector due to small variations in the angle of incidence is likewise negligible at angles close to normal incidence. Since the polarization state of the beam transmitted by the cell windows is insensitive to the angle of incidence at angles near normal incidence, the variation in the angle of incidence at the cell windows produced by beam deviation will not produce any significant errors unless the specimen is mounted in a liquid cell. In this case, the variation in the angle of incidence at the cell window is magnified by refractive properties of the liquid, and significant variations in the angle of incidence at the specimen may occur.16 Beam deviation in the components produces an azimuthal-dependent displacement of the beam on the cell windows, the specimen, the compensator, the analyzer, and the detector. Since the specimen is mounted obliquely to the beam, the displacement of the beam on the specimen increases as o 900, as shown in Section II; and, for angles of incidence near 90, large samples are required to ensure that the beam is reflected from the sample for all settings of the polarizer. Errors will thus arise unless the optical properties of the specimen are uniform over the area sampled by the cone of the deviated beam. The displacement of the beam on the compensator, analyzer, and detector in the PSCA (polarizer, specimen, compensator, analyzer) arrangement and on the analyzer and the detector in the PCSA sequence thus depends on the angle of incidence at the specimen and increases as It is expected that the optical properties of the compensator and analyzer will usually be uniform across their apertures, but such variations cannot be summarily dismissed. For example, Smith1 2 has detected shifts of 0.04 in the position of the intensity minima due to grain structure in mica quarter-wave plates. The variation in the recorded intensity due to variations in the sensitivity of the detector cathode across its surface is a common source of error in many optical systems. A technique that has been used to minimize such errors is to place a scatterplate in front of the detector to diffuse the light across a large area of the cathode.1 8 Such techniques are not always necessary in conventional static ellipsometry measurements, however, because the experimentally determined quantities are the azimuthal angles of the components at null. The effect of a variation in the intensity due to nonlinearities in the response or nonuniformities in the sensitivity of the detector is analogous to the effect of the variations in the input intensity that occur when the beam incident on the polarizer is elliptically polarized. It has been shown' 9 that, to first order, the ellipticity of the incident beam does not affect the accuracy of ellipsometric measurements since the optical properties of the sample define the azimuthal angles at which a null will occur and the location of the null cannot be shifted by varying the input polarization. Variations in the transmitted intensity do occur at angles away from the null, and errors may arise if the null is determined by averaging points of equal intensity on each side of the null. However, since the intensity variations at null are large and the azimuthal variations necessary to determine the null are small (usually less than 1), the additional variations due to the ellipticity of the incident beam are usually ignorable. The same conclusions also apply to the variations in intensity due to beam displacement at the detector if the nulls are averaged over small azimuthal swings of the components and if the response function of the photomultiplier does not vary rapidly for small displacements of the beam. There are other ellipsometric configurations, e.g., those that do not use a compensator, 2 0 ' 2 in which the optical constants are calculated directly from the recorded intensity. In these systems, the combined errors of beam deviation and detector nonuniformity are more serious, and some method of eliminating these errors (such as using a scatterplate) should be considered. The magnitude of the errors due to beam displacement are difficult to determine since they may vary erratically as a function of the azimuthal angles of the components and are not canceled by the systematic procedures outlined in Sec. IV. August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1943

7 VI. Effect of Beam Deviation in the Compensator, Analyzer, and Cell Windows The analysis thus far has been concerned primarily with the effects of beam deviation in the polarizer. When the specimen is contained in a sample cell, the angle of incidence variation at the specimen is determined by the combined effects of one cell window and the polarizer. The window deviates the beam in a specific direction for all settings of the polarizer. Along the direction that the beam is deviated by the cell window the net deviation is 3, 4 bp and the length of the axis of the cone of the deviated beam is 23p. Perpendicular to this direction the net deviation is p and the length of the axis of the cone is also 2 p. The cone of the beam transmitted by the polarizer thus remains a right circular cone after it passes through the cell window, but the apex of the cone is shifted and 0 is symmetric about a new angle /', where O' Oo. The difference between ' and Go depends on the direction and magnitude of 3c. Except for this shift in X, the results of Secs. II and III are unchanged, and, in most cases, the differences between ' and c 0 can be ignored. The magnitude and direction of the angular deviation produced by the cell window can be measured by an extension of the techniques described in Sec. III, and /' can be determined in terms of the measured parameters. If a precise determination of is required, however, it is advisable to circumvent these difficulties by specifying cell windows with negligibly small wedge angles. (Cell windows typically have wedge angles between 0.05 and 0.10, although windows with much smaller wedge angles are usually available.) In the PSCA arrangement, any additional beam deviation produced by the compensator further alters both the point of incidence and the angle of incidence at the analyzer and the detector. The beam deviation produced by the analyzer also changes the position and angle of intersection between the beam and the detector. In the PCSA arrangement of components, the angle of incidence at the specimen depends on the azimuthal settings of both the polarizer and compensator. For a fixed setting of the compensator the results are analogous to the combined effects of the polarizer and a cell window. As the compensator azimuth is varied, the angle ' also varies. Precise corrections are difficult to obtain in this case, and if the angular deviation of the compensator is found to be significant, the ellipsometer should be operated in the PSCA arrangement to minimize beam deviation errors and to make the errors more amenable to analytic corrections. Beam deviation in the polarizer thus produces more significant errors than would be produced by an equivalent amount of beam deviation in the other components of the ellipsometer. The magnitude of the beam deviation errors can thus be decreased if the wedge angles of the two polarizing elements are measured and the element with the largest wedge angle is used as the analyzer. Vl. Conclusions The various types of errors produced by beam deviation have been examined. It has been shown that the variations in the angles of incidence at the compensator, cell windows (except possibly in a liquid cell), analyzer and detector produce no significant errors in the measurements. The variation in the angle of incidence at the specimen can in some cases produce significant errors, the magnitude of which is dependent upon the optical properties of the specimen and the experimental parameters. An analytic expression was derived that allows the angle of incidence and the tilt angle of the plane of incidence to be determined in terms of experimentally measured quantities at an arbitrary setting of the polarizer. The results indicate that fundamental errors occur in zone-averaged measurements because 4 and A are obtained at different angles of incidence and in different planes of incidence in each zone. An averaging procedure by which most of these systematic errors can be eliminated is described. The equations derived in Sec. II are strictly valid only when the deviation of the beam at the specimen is produced by the polarizer alone. It is shown, however, that, except for a slight shift in the angle of incidence, the basic results of Secs. II-IV remain true when the deviation of the specimen cell windows and that of the compensator at a fixed azimuthal position are included. A method for measuring the magnitude and direction of the angular deviation produced by each element of the ellipsometer is described. If the angular deviation of the compensator is found to be significant and the azimuthal position of the compensator is to be an experimental variable, the validity of the equations in Sec. II can be maintained by operating the ellipsometer in the PSCA arrangement. This also minimizes the magnitude of the beam deviation errors. The errors are further minimized by measuring the angular deviation of the two polarizers and using the polarizer with the largest angular deviation as the analyzer. An additional source of error results from the combined effects of beam displacement and nonuniformities in the specimen, the detector, and the components of the ellipsometer. The variation in the detector response as a function of beam position is of particular importance since such variations are frequently encountered in optical measurements. It was shown, however, that the variations in detector response will be less important in a nulling measurement than in an intensity measurement and can often be neglected. The errors resulting from beam displacement can vary erratically as a function of the azimuthal angles of the components and are not eliminated by the systematic procedures suggested in Sec. IV. The procedures suggested in Sec. IV do allow an important class of beam deviation errors to be eliminated to first order. However, since this procedure increases the measurement time and still does not eliminate the errors associated with beam displace APPLIED OPTICS / Vol. 13, No. 8 / August 1974

8 ment, an alternate method of eliminating all errors due to beam deviation should be considered. It is generally possible to obtain components with smaller wedge angles than are typically supplied in commercial ellipsometers. If close tolerances are specified for the wedge angles of the components of the ellipsometer (particularly for the polarizer), the errors due to beam deviation could be substantially reduced and several other simplifications in the alignment of the instrument could be realized. 7 We wish to thank R. M. A. Azzam for many interesting and informative discussions concerning this work. J. R. Zeidler's work was sponsored by Air Force Office of Scientific Research Grant AFOSR N. M. Bashara's work was sponsored in part by the National Scientific Foundation. References 1. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 61, 118, 600, 1236, 1380, (1971); 62, 700 (1972). 2. D. E. Aspnes, J. Opt. Soc. Am. 61, 1077 (1971). 3. D. E. Aspnes and A. A. Studna, Appl. Opt. 10, 1024 (1971). 4. F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970). 5. W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355 (1970). 6. W. R. Hunter, J. Opt. Soc. Am. 63, 951 (1973). 7. J. R. Zeidler, R. B. Kohles, and N. M. Bashara, Appl. Opt. 13, 1115 (1974). 8. A. B. Winterbottom, Roy. Norwegian Sci. Soc. Rep. No. 1 (F, Trondheim, 1955), pp J. M. Bennett and H. E. Bennett, in Handbook of Optics (W. G. Driscoll and W. Vaughan, Eds. (McGraw-Hill, New York, to be published). 10. J. R. Zeidler, R. B. Kohles, and N. M. Bashara, Appl. Opt. 13, 1591 (1974). 11. F. Lukeg, Surf. Sci. 16, 74 (i969). 12. P. H. Smith, Surf. Sci. 16, 34 (1969). 13. K. K. Svitashev, A. I. Semenenko, L. V. Semenenko, and V. K. Sokolov, Opt. Spectrosc. 34, 542 (1973). 14. J. F. Archard, J. Sci. Instrum. 26, 188 (1949). 15. D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964); D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 57, 466 (1967). 16. W. G. Oldham, J. Opt. Soc. Am. 57, 617 (1967). 17. S. S. So., W. H. Knausenberger, and K. Vedam, J. Opt. Soc. Am. 61, 124 (1971) Hunderi, Appl. Opt. 11, 1572 (1972). 19. Ref. 1, Paper J. N. Hodgson, Proc. Phys. Soc. (London) B68, 593 (1955). 21. D. E. Aspnes, Opt. Commun. 8, 222 (1973). The International Council of the oeronautical Sciences (ICAS) will hold its next biannual Congreas, the 9th, at the Techrion, Israel Institute of Technology, Haifa, August 25-30, Eminent scientists and engineers from the twenty eight ICAS member countries will present and discuss problems in aerodynamics, flight mechanics, structures, aeroelasticity, materials, propulsion, design, sim'ldtion, noise technology, operations and other topics of aeronauical,epience and technology. Preprints of papers will be available during the Conference and will later be published with discussions as the Conference Proceedings. Preceeding the ICAS Congress, on Slmday, 25th :.ugust 1974, the 16th Israel Annual Conference on Aviation and Astronautics will take place in Tel-Aviv. For further particulars please write to: ICAS Secretariat AIAA 1290 Avenue of the Americas New York, N.Y U.S.A. or to: Israel Organizing Committee 9th ICAS Congress P.0.B Tel-Aviv, Israel August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1945