WAVEFRONT ERRORS PRODUCED BY MULTILAYER THIN-FILM OPTICAL COATINGS
Item Type text; Dissertation-Reproduction (electronic)
Authors Knowlden, Robert Edward
Publisher The University of Arizona.
Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Download date 05/10/2021 11:15:02
Link to Item http://hdl.handle.net/10150/281959 INFORMATION TO USERS
This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality--is-heavily dependent upon the quality of the material submitted.
The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction.
1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure you of complete continuity.
2. When an image on the film is obliterated with a round black mark it is an indication that the film inspector noticed either blurred copy because of movement during exposure, or duplicate copy. Unless we meant to delete copyrighted materials that should not have been filmed, you will find a good image of the page in the adjacent frame.
3. When a map, drawing or chart, etc., is part of the material being photo graphed the photographer has followed a definite method in "sectioning" the material. It is customary to begin filming at the upper left hand corner of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete.
4. For any illustrations that cannot be reproduced satisfactorily by xerography, photographic prints can be purchased at additional cost and tipped into your xerographic copy. Requests can be made to our Dissertations Customer Services Department.
5. Some pages in any document may have indistinct print. In all cases we have filmed the best available copy.
University Microfilms International 300 N. ZEEB ROAD. ANN ARBOR, Ml 48106 18 BEDFORD ROW, LONDON WC1R 4EJ, ENGLAND 8116708
Knowlden, Robert Edward
WAVEFRONT ERRORS PRODUCED BY MULTILAYER THIN-FILM OPTICAL COATINGS
The University of Arizona PH.D. 1981
University Microfilms International 300 N. Zeeb Road, Ann Arbor, MI 48106 IVAVEFRONT ERRORS PRODUCED BY MULTILAYER
THIN-FILM OPTICAL COATINGS
by
Robert Edward Knowlden
A Dissertation Submitted to the Faculty of the
COMMITTEE ON OPTICAL SCIENCES (GRADUATE)
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
198 1 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Robert Edward Knowlden entitled Wavefront Errors Produced by Multilayer Thin-Film Optical
Coatings
and recommend that it be accepted as fulfilling the dissertation requirement
for the Degree of Doctor of Philosophy . il ft — // fhd ft Date
IOM.
Date
Date
Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.
X hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.
Dissertation Director Date STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to bor rowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or re production of this manuscript in whole or in part may be granted by the head of the major department or the' Dean of the Graduate College when in his judgment the proposed use of the material is in the in terests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGNED ACKNOWLEDGMENTS
I wish to thank all of the people who have supported me through this work. Most especially I thank my parents, my advisor Bob Shannon, and Angus Macleod, who taught me most of the little I understand about optical thin films. I add that I will do my best to fail none of you.
Thanks also are due Sherrie Cornett, who put this in its present form.
iii TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS vi
LIST OF TABLES x
ABSTRACT xi
1. INTRODUCTION 1
Organization of the Dissertation 5 Use of the First Person 5
2. MULTILAYER THEORY AND ENHANCED REFLECTOR DESIGN 6
The Complex Plane Wave 6 The Matrix Formalism 7 The Optical Admittance 10 Reflection, Transmission and Optical Phase 11 Derivatives of the Phase 17 Basic Enhanced Reflectors 19
3. ANGLE OF INCIDENCE EFFECTS WITH ROTATIONALLY SYMMETRIC MIRRORS 24
Decomposition of Polarizations 24 Angles of Incidence for a Parabola 27 Computing Amplitudes and Phases 30 Extraction of the Focus Term 31 Examples 32
4. THE OPTICAL PHASE ON REFLECTION AS A FUNCTION OF STACK THICKNESS 46
An Infrared High-Reflectance Coating: Performance at IR and Visible Wavelengths 49 Measurements at Two Wavelengths 52 Inverting the Two-Wavelength Phase Difference Function . . 55 Base Thickness Measurements 57
5. A MODIFIED COATING DESIGN TO ALLOW ACCURATE COATING THICKNESS PROFILE ESTIMATION 61
iv V
TABLE OF CONTENTS--Continued
Page
An Error Function 61 An Optimization Technique: The Nonlinear Simplex Method 66 The Modified Design 70 Estimation of the Wavefront in the Infrared from Two Visible Measurements: Polynomials 73 Measurements Using Several Wavelengths 82
6. ROTATIONALLY SYMMETRIC COATING THICKNESS DISTRIBUTIONS .... 89
Vacuum Coating Geometries 89 The Use of Radial Polynomials 94 Determining a Radial Thickness Distribution from Measurements at Three Wavelengths 101 An Example 103
7. THE EFFECTS OF ERRORS ON THE ACCURACY OF ESTIMATION OF THE INFRARED WAVEFRONT FROM VISIBLE LIGHT INTERFEROGRAMS 109
Wavefront Measurements 114 The Need for a New Error Function 123 Choice of a Deposition Geometry 125 New Optimization Variables 130 An Example: Attainable Accuracy 131 The Initial Estimates 136 A Demonstration of the Effects of Wavefront Errors .... 138 The Effects of an Index Change 140
8. SUMMARY AND CONCLUSIONS 143
Additional Comments and Suggestions for Further Work . . . 147
APPENDIX A: A COATING DESIGN OF GUHA, SCOTT AND SOUTHWELL . . 150
APPENDIX B: AN INFRARED ENHANCED REFLECTOR WITH REDUCED WAVEFRONT ERRORS DUE TO PROPORTIONAL THICKNESS CHANGES 160
LIST OF REFERENCES 170 LIST OF ILLUSTRATIONS
Figure Page
1.1. Interferogram reproduced from Ramsay and Ciddor (1967) . . 3
2.1. A single layer, with forward and backward travelling waves 9
2.2. Intensity change due to refraction at an interface .... 13
2.3. Two dielectric stacks indicating phase changes 16
3.1. Circular aperture: Cartesian and polar coordinates .... 25
3.2. Section of a parabola being tested at its center of curvature 29
3.3. Contours for Example 1, an f/1.5 parabola with a 6 layer coating 34
3.4. Contours for Example 2, an f/1.5 parabola with a 35 layer coating after Heavens and Liddell (1966) 37
3.5. Contours for Example 3, a parabola tested at its center of curvature 40
3.6. Contours for Example 4, a 90 degree cone with a 6 layer coating 43
4.1. Optical path in microns (p) at 0.4p as a function of relative thickness change (k) for a 35 layer design by Heavens and Liddell (1966) 48
4.2. Optical path (in p) for a 6 layer enhanced reflector as a function of k at the design wavelength, 3.8p 50
4.3. Optical path (in p) for the 6 layer 3.8p design at 0.6328p 51
4.4. Optical path (in p) for the 6 layer 3.8p design at 0.5145p 53
4.5. Two wavelength optical path difference for the 6 layer design at 0.5145p and 0.6328p 54
vi vii
LIST OF ILLUSTRATIONS--Continued
Figure Page
4.6. Representation of displaced Fizeau fringes for measuring the stack thickness 59
4.7. Optical path change (in y) for the 6 layer design at 0.6328p; the stack thickness goes from zero (k=-l) to the nominal (k=0) 60
5.1. Flow chart for the nonlinear simplex search algorithm (Nelder and Mead 1965) 69
5.2. Two wavelength optical path difference (in y) as a function of relative stack thickness change (k) for the 6 layer modified design 72
5.3. Optical path (in y) at 3.8y for the modified design as a function of k 74
5.4. Optical path (in y) at 0.6328y for the modified design as a function of k 75
5.5. Optical path at 0.5145y for the modified design 76
5.6. Optical path at 0.6328y for the modified design, from zero stack thickness (k=-l) to l.lx the nominal design thickness (k=0.1) 81
5.7. Optical path at 3.8y for the six layer quarter wave design over a 10% range in k 83
5.8. Two wavelength path difference functions (in y) for 0.4880m - 0.6328y (lower curve) and 0.5145y - 0.6328y (upper) 85
5.9. Parametric curve of the two wavelength path difference for 0.5145y - 0.6328y (ordinate) vs. that for 0.4880y - 0.6328p (abscissa) 86
5.10. Point plot of the parametric curve in Fig. 5.9; spacing in k of 0.001 87
6.1. Offset source coating geometry 91
6.2. Radial coating thickness distribution from an offset source scheme, (h/A)=1.3, (ro/A)=0.7, as a function of normalized radius a 95 viii
LIST OF ILLUSTRATIONS--Continued
Figure Page
6.3. Two wavelength Optical Path Differences $2X1 anc* ^2X2 ^0 as a function of stack thickness (k) . . . . 97
6.4. Two wavelength radial O.P.D. for 0.4880p--0.6328y, for the coating thickness shown in Fig. 6.2 98
6.5. Two wavelength radial O.P.D., 0.5145p-0.6328y 99
6.6. Error (in p) as a function of k over the range in k of [-0.05, 0.05] 104
6.7. Error function over k in the range [-0.01, 0.01] 105
6.8. Error function over k in the range [0.01, 0.03] 107
7.1. A laser unequal path interferometer (LUPI) for use as a fringe scanning interferometer 116
7.2. Representation of a 32x32 detector array, masked to a circle , 117
7.3. Layer thickness uniformity for an offset source deposition scheme with a cos^ source distribution .... 126
7.4. Layer thickness uniformity with a cos^ distribution .... 127
7.5. Layer thickness uniformity with two source distributions . 128
7.6. Layer thickness uniformity for a cos2-3 and a cos1-1 source 129
7.7. Two wavelength radial O.P.D.'s for three simulations at (0.4880 - 0.6328 y) 133
. 7.8. Two wavelength radial O.P.D.'s for the same simulations as Fig. 7.7 but (0.5145 - 0.6328 u) 134
7.9. Radial optical paths at 3.8 y for the simulations in Fig. 7.7 134
7.10. RMS errors (in p) of estimations of the IR performance of a coating from measurements at three visible wave lengths as a function of errors in wavefront measurement 139 ix
LIST OF ILLUSTRATIONS--Continued
Figure Page
8.1. Two wavelength O.P.D. (0.5145 - 0.6328 y) vs. stack thickness (k) for the 35 layer coating of Heavens and Liddell (1966) 149
A.l. Optical paths (in y) vs. relative stack thickness for the design of Guha et al. (1980) 152
A.2. Absolute optical paths at 0.6471 y for s and p polarizations 153
A.3. Absolute optical paths at 3.8 y for s and p polarizations 155
A.4. Performance of the design of Guha et al. (1980) with approximately \% thickness errors 156
A.5. Performance of the design of Guha et al. (1980) with approximately 2% thickness errors 157
A.6. Radial optical paths vs. normalized radius (a) at 3.8 y and 0.6471 y 159
B.l. Optical path (in y) at 3.8 y vs. stack thickness (k) for a 14 layer quarter wave enhanced reflector and a modified design 163
B.2. Optical path at 0.6328 y vs. k for the 14 layer modified design 164
B.3. Performance of the 14 layer modified design with approximately 1% thickness errors 165
B.4. Optical path at 3.8 y vs. k for a 10 layer modified enhanced reflector design 168
B.5. Performance of the 10 layer modified design with approximately 1% thickness errors 169 LIST OF TABLES
Table Page
2.1. Reflectivities of silver and enhanced silver mirrors at 3.8p 25
5.1. Thickness prescription for the modified 6 layer design . . 71
5.2. Polynomial approximation of the phase at the IR wave length from the two wavelength path difference
6.1. Radial polynomial fits for two wavelength O.P.D.'s .... 100
6.2. Radial polynomial fits for two wavelength O.P.D.'s .... 108
7.1. Radial Zernike Polynomials (orthonormal) 124
7.2. Non-random parameters of the coating simulations 132
7.3. Errors in estimation caused by refractive index changes . . 142
A.l. Layer thicknesses for a design of Guha et al. (1980) . . . 151
B.l. Thickness for two 14 layer enhanced reflector coatings: a quarter wave design and a modified design ...... 162
B.2. Thicknesses for two 10 layer enhanced reflector coatings: a quarter wave design and a modified design 167
x ABSTRACT
The mirrors used in high energy laser systems have at least two requirements that are uncommon in optical engineering: the reflectance of such mirrors must be very high (> 0.999), and the level of aberra tions introduced by the mirrors is desired to be very low, typically
A/50 peak at 3.8 p. The first requirement can be met by using multi layer thin film coatings, but such coatings can themselves produce aberrations in an optical system.
One possible effect in multilayers is that such coatings produce an optical phase change on reflection that varies with angle of incidence and polarization of the illuminating beam. On a strongly curved mirror, such as an f/1.5 parabola used as a collimator, these effects may be appreciable for some coatings (e.g., A/15 for a broadband all-dielectric reflector), but for an enhanced silver coating the effects are small, typically A/400 of error that is almost entirely in the form of a small focus shift. If this same parabola is tested at its center of curvature, the coating-caused aberration due to angle of incidence effects are nearly zero (e.g., A/50,000 for the broadband reflector that gave A/13 when the parabola was used as a collimator). The wavefront errors due to coating nonuniformities are usually more important than angle of incidence effects.
The simplest type of coating nonuniformity to analyze is a proportional error, i.e., an error where the ratios of the thicknesses
xi xii
of the layers are fixed but the thin film stack varies in total thick
ness across a surface. For a six-layer enhanced reflector for use at
3.8 u, a 1% thickness error produces an approximate X/100 wavefront
error. At visible wavelengths, however, the aberration produced by such
a coating error can be very different because of the optical interference
nature of the coating.
Means may be developed to estimate the performance of such an
infrared reflector from measurements at visible wavelengths. If the
errors produced by the coating are to be distinguished from those exist
ing in the test due to misalignment or gravitational flexure of a large
mirror, two or more wavelengths must be chosen. There are ambiguities
in such a test that may be resolved by choice of an appropriate coating design or by using enough wavelengths in the visible, and both means have
been studied. A technique was found where the infrared wavefront can be determined for a coating with proportional thickness errors if the coating prescription is known: interferograms of the mirror are made at three visible wavelengths, and the IR wavefront error due to the coating error is determined in a way that is insensitive to any errors caused by distortion of the substrate or even fairly large misalignments in the optical test of a mirror's figure.
Simulations of some real coatings have determined that additional work needs to be done to improve the analysis procedures used in esti mating the infrared performance of an enhanced reflector from visible light measurements. However, initial results show that fairly accurate predictions of the IR wavefront errors can be made from measurements of infrared enhanced reflectors in visible light. CHAPTER 1
INTRODUCTION
The wavefront errors produced by multilayer thin film coatings
are small, typically a small fraction of the wavelength of use of the
coating. Some modern optical devices require wavefront error toler
ances in the range that can be easily produced by coatings, however;
such systems are the reason of existence for this study.
Some early users of multilayer coatings who were concerned with
the aberrations such coatings produce were researchers involved with
high-resolution spectroscopy using Fabry-Perot interferometers (cf.
Hecht and Zajac, 1974, pp. 509-311). Multilayer dielectric coatings are used when a very high finesse is required, which is to say a high reflectance of the mirrors of the etalon, while maintaining a low optical absorption, as highly reflecting silver films absorb too much light. If the finesse is large, then any departure from flatness of the etalon plates will reduce the effective finesse for spectroscopic work, as each area of the etalon will still only transmit a narrow band of frequencies, but a photoelectric detector that looks through many such areas will integrate over all the light transmitted so that the working bandwidth may be much broader than for perfectly flat plates. Because of the interferences of the light in a thin film stack, the effective lack of flatness of the plates caused by area nonuniformities of the
1 coating can be related to these nonuniformities but the relationship may
not be a simple one.
Giacomo (1958) has calculated these effects for small thickness
errors in several types of coatings used in Fabry-Perot work, and set
uniformity tolerances on these coatings. Pelletier, Chabbal, and
Giacomo (1964) presented some further sensitivity calculations and some
data from deposited films. The most beautiful results were obtained by
Ramsay and Ciddor (1967) who produced an interferogram showing a pair
of Fabrv-Perot plates that appeared slightly (A/60) concave in green
(0.546 y) light, almost flat in yellow (0.588 y) light, and A/10 convex
in red (0.644 y) light. Their interferogram is reproduced as Figure 1.1.
Another type of optical system that demands very low wavefront
aberrations is a laser weapon system where the highest possible target
irradiance is desired. Bennett (1980) describes many of the problems
with such systems, and specifically coating uniformity requirements.
The high energy flux levels on the optics of such a system require very
high (> 0.999) reflectance coatings for use at a typical 3.8 y wave
length, which implies that metallic coatings such as silver with a multi
layer dielectric stack to enhance the reflectance will be used. Bennett
(1980) gives A/50 at 3.8 y as a probable tolerance on the peak wavefront
error on any single optical component in a high-energy laser system, and
demonstrates that coating nonuniformities on the order of ls° can produce
an error of this magnitude. Because the largest optics are as large as several meters in diameter, a 1% coating uniformity is difficult to achieve. Vacuum chambers of this size exist to coat the primary mirrors Fig. 1.1. Interferogram reproduced from Ramsay and Ciddor (1967). 4
of large astronomical telescopes, but such mirrors are usually coated
with aluminum and the uniformity requirements for such a coating are
loose: if a 0.2 p thick coating is used, a 38% thickness variation
would be needed to give a X/50 error at 3.8 p. Bennett (1980) also
describes another problem: the wavefront error produced by an enhanced
reflector at its working wavelength can be much different from the
error given by that coating at a shorter wavelength. This means that
the errors produced by the coating cannot simply be determined by any of
the standard optical tests such as Twyman-Green interferometry in the
visible if the coating is designed for the infrared. Bennett (1980)
also remarks on p. 66 of his report:
If the nominal film design is known, it may be possible in principle to deduce the film-induced figure error in the in frared by measuring the apparent figure error at a variety of wavelengths in the visible region using a dye laser and de ducing from these results the infrared figure error. This approach could be tried, but several possible difficulties are apparent. Reliable values for the wavelength dispersion of index of refraction for materials in thin-film form will be required. Also, we have been assuming that the thickness of each film was in error by the same percentage, which will not necessarily be true in an actual situation. Finally, the film design must be known in detail and variations in thickness observed during the coating run should be available to make visual analysis successful. It may be difficult to obtain this data in many cases.
He suggests that a "more satisfying approach" would be direct qualifica tion of a component using interferometry at the working wavelength. The bulk of this study is to determine if, in fact, there exists a practical means of estimating the wavefront errors of a coated mirror for the in frared from a measurement or series of measurements at visible wavelengths.
Measurements in the visible would be simpler and cheaper to perform than 5
.,nes at 3.8 y, although the data reduction can be expected to be more
complex than for a simple interferogram.
Organization of the Dissertation
Chapter 2 develops the basic multilayer theory and the means to
design dielectric-enhanced metal reflectors. There is no need for the
reader to have a deep understanding of the material in this chapter, but notation is developed that will make the remainder of the study easier to understand. Chapter 3 consists of a series of calculations of wave- front errors due to angle of incidence effects in uniform coatings on curved mirrors. The remainder of the dissertation is mostly concerned with wavefront errors produced by coating thickness nonuniformities and finding ways to determine IR wavefronts from visible light measurements.
The results of this study are summarized in Chapter 8. The con clusion is reached that useful information can be determined about the wavefront errors that are produced by coating errors in an infrared en hanced reflector from measurements using visible light of only a few discrete wavelengths.
Use of the First Person
I use the first person singular voice in this dissertation as a means of making a distinction between statements of opinion and state ments of fact. Personal opinion which has little scientific necessity is marked by the first person (i.e., I, my), while supportable statements are not, for the most part. I have also tried to avoid the first person plural (i.e., we, our) and reserve it for kings and committees. CHAPTER 2
MULTILAYER THEORY AND ENHANCED REFLECTOR DESIGN
The purpose of this chapter is to set out the thin film theory
used in this work. It is not a comprehensive review of thin film theory,
nor are most of the basic calculational techniques new. The basic theory
used was developed by Abeles (1950). The complex notation and the Inter
national System (SI) of units are used unless otherwise specified.
The Complex Plane Wave
When using the complex notation, the signs used in complex
exponentials must be carefully kept consistent. For a plane wave propa
gating in a Cartesian positive x direction, the form of the electric
field used is
£(x,t) = Eq exp[i2ir(y t - y x)] (2.1)
nj where E(x,t) is the field amplitude, Eq a constant complex vector, c
the speed of light in vacuum, X the wavelength of the radiation in
vacuum, and N the complex refractive index of the medium (N=n-ik, n and k being real). The important feature is the sign in the exponent of the
x term, which fixes the geometrical part of the phase (as opposed to the
optical part, i.e., the phase on reflection from the stack).
6 The Matrix Formalism
The basic technique for thin film calculations involves recursively calculating the electric and magnetic fields in the stack. The approach used here is that of Abeles (1950), which has also been used by Macleod
(1969) and these works should be consulted for a more detailed under standing of the development. Basically, the development proceeds as follows: consider an incident plane wave. The plane-parallel inter faces in the thin film stack may only give rise to two waves: a forward and a backward running wave. Let the complex amplitudes of the tangential electric field components of the waves be written E+ and E . The tangen tial components are matched to take advantage of the boundary conditions on the fields. The corresponding magnetic fields are defined by
H+ = n E+ (2.2a)
H~ = -r, E~ (2.2b) where, for p or transverse magneti (TM) polarized radiation
Y N <2-3> m
where YQ is the admittance of free space (= , y being the permeability of free space, c the vacuum speed of light), N the complex refractive
index of the medium, and 6^ the angle of propagation in the medium. 0m is in general complex, and defined by
cose = /(I - sin^e.) (2.4) m \ N 1 where n. is the refractive index of the incident medium, and 0. the 1 1 angle of incidence, n^ is purely real, often 1. The appropriate branch of the complex square root function has the branch cut down the negative real axis, the branch point at the origin, and = -i (this is chosen to give a physically correct result for total internal reflection). For s or transverse electric (TE) polarized radiation,
n = Y N cos9 (2.5) o m
The total tangential fields at a given plane are defined
E = E + E (2.6a)
H = H + H (2.6b)
More interestingly,
E+ = y (E + ~j ) (2.7a)
E" = i (E - M ) (2.7b)
The electromagnetic boundary conditions are on the total fields; E and
H are the same on either side of a boundary. The total fields are affected by propagation, however, in the manner (see Fig. 2.1)
Ea"> = Ebl exP(i6) (2.8a)
Ea2 = Ebl e*P(-i6:> (2.8b) 9
l
m
Fig. 2.1. A single layer, with forward and backward travelling waves. 10 where
6 = Nd cos0 , (2.9) A m d being the mechanical thickness between planes a and b, and N being the complex index of the medium between the planes, the medium being assumed to be homogeneous and isotropic on a scale of the wavelength of the radiation. The value of 6 is complex in general. This may all be com bined in the final matrix form:
E cos6 — sin<5 a n ~Eb"
(2.10) X
H insin6 cos6 & a 1 1
The total field may be computed at any plane in the system by recursively applying Eq. (2.10), beginning at the substrate and proceeding backwards through the system.
The Optical Admittance
The optical admittance is defined
H (2.11) Ya = T-E a for any plane in the system. This may be seen to play the same role as n in Eqs. (2.2a) and (2.2b). The significance of Y is similar to that of the impedance concept in radio antenna theory. 11
Reflection, Transmission and Optical Phase
The amplitude reflection coefficient at the incident medium/film stack interface is
r = . (2.121
Eal
Let n be the admittance of the incident medium as determined o by Eq. (2.3) or Eq. (2.5). Then
H i n - Y o a The intensity reflection coefficient is 2 H = 1 ~ v" 11 ^. v: 1 (2.14) where the asterisk (*) indicates complex conjugate. The transmission of the stack is less straightforward to define, for a simple ratio of fields will not work if the media of entrance and exit for the stack have different indices of refraction. Worse, even if reflection is zero, the intensities of the entering and exiting beams will 12 be different if the media are not the same because of the different, inclination of the beams (Fig. 2.2). This is resolved by taking the transmittance as being the ratio of the normal components of the inten sity (i.e., the flux perpendicular to the interfaces). The intensity is further defined as the time average of the Poynting vector of the wave. This time average is developed on p. 30 of Knittl (1976), but is repro duced here in SI units. Let E(t) = i: exp(iwt) H(t) = Hq exp(iut) where w is some constant. The Poynting vector may be defined P = Re[E (t) ] x Re [H(t) ] = h{[E(t)+E*(t)] x [H(t)+H*(t)]} = HI[E q x H0] exp(i2wt) + [E* x H*] exp(-i2wt) + [E0 x H*] + [E* x H0]} P = Re[E 0 X H0 exp (i2wt)] + Re[E0 x H*]} . Taking the time average, the first term vanishes, leaving % i % ^ P = 4- Re[E x H ] . (2.15) av 2 1 o oJ Fig. 2.2. Intensity change due to refraction at an interface. Note change in beam cross-section. 14 The normal intensity at. surface a in Fig. 2.1 for the forward travelling wave may be written (2.16) Recall that E* and are the tangential components of the field for the forward wave, that they are perpendicular, and that for p or s polari zation one of the two field directions must lie in a plane parallel to the interfaces. Using Eq. (2.2a), the intensity transmittance is defined Re E T = 1 ^ b2 % *0 I *^1 "O E+J al or Re E M b2l (2.17) 2 te(n0)|E;ii where n is the admittance of the substrate. The amplitude transmission m r coefficient is defined indirectly by + + t = T b2_ + al whicli reduces to R e(nm) t = (2.18) Re(n0) This is not the standard form for amplitude transmittance, which is usually defined as the simple ratio of the fields, as on p. 39 of 15 Knittl (1976). The form given here I prefer because then t and T have the same relationship as r and R. The optical phase change on reflection is defined from r = p exp (i4>) (2.19) where p and ; are real valued, and p usually taken as being always non- negative (but this will be modified later). Eq. (2.19) gives [~ Im(r) = Arctan (2.20) [_Re(r) where the arc tangent function used is defined over a 2tt range by assigning a quadrant to the angle based on the signs of numerator and -1 5tt denominator (e.g., Arctan (—) = -j- ). The expression for the transmission phase is similar. In the study of optical wavefronts, geometrical factors must also be considered in obtaining the total optical phase. Fig. 2.3 shows two stacks on the same substrate; the stacks are identical except that one layer is increased in thickness a small amount d. Let be the phase change on reflection of one stack, and ^ be that of the stack with the thicker layer. The optical path difference (OPD) on reflection from the two stacks is 4irn.cos0. d *d = *2 + 1 A 1 - *1 [2'21) where n^ is the (real) index of the incident medium, 0^ the angle of incidence, and X the vacuum wavelength of the radiation. 16 Fig. 2.3. Two dielectric stacks indicating phase changes. 17 Derivatives of the Phase A variety of derivatives of the optical properties of a film stack may be computed. Dobrowolski (1965) mentions many of these. Con sider a matrix solution of the form B 1 M ... M M. (2.22) n 2 1 Mj, M^, ..., M being the 2 by 2 matrices of the form of Eq. (2.10). Eq. (2.15) takes the form noE-c r = (2.25) noB+C or (n B-C)(n B*+C*) o ' o 2 |n0B+c| allowing n =n0 (nQ is real), giving n2|B|2 - |C|2 + i2n Im(BC*) r = — _2 n B + C " 1 o Applying Eq. (2.20), 2n Im(BC*) Arctan (2.24) n2|B|2-|c|2 Now let a dot (•) over a variable indicate differentiation by some as yet unspecified variable, say x. Also define two \'ariables 6 and y such 18 that B = 2n0Im(BC*) (2.25a) 2 i i 2 Y = n0|B| (2.25b) Remember that d « 1 3— Arctan x = , dx 1 +x7 so that d . _ 6 1 / B By •j— Arctan — = , , ^ I dx y 1 + (8/y) \ y y or = 2n Im(BC* + BC*) (2.27a) o y = 2n~Re(BB*) - 2 Re(CC*) (2.27b) To calculate -5-^- , where d- is the mechanical thickness of layer j in Eq. ouj j (2.22), first form , as B M ... M.... Mn M, (2.28) n j 2 1 C m - with -sin6. cos6. M. J nj J J (2.29) iri.cosfi. -sin6. J J J 19 The d. derivative is found by 3 a 3 d6-i ~3d7 = "967 dd. (2. M)) J J J and d6. 2tiN.cos0. 33! = ^—1 • <2-» 3 X. The geometrical term in the total phase expression (Eq. 2.21) also has a derivative; if $ is that part of the phase, then 3d) 4im. cos0. £ = 1 L • (2 3^) 3d. . ' J >• higher derivatives are zero. The basic technique described above may be extended to higher and mixed derivatives, but the expressions become cumbersome. Dobrowolski (1980) has stated that in the previously mentioned program (Dobrowolski, 1965) he had originally used analytic expressions for the derivatives of order 2, but currently uses finite differences as approximations. Basic Enhanced Reflectors A simple means of obtaining a very high reflectance from a metal mirror is to apply a dielectric stack to boost the base reflectance. A simple design consists of a matching layer of low index, followed by the familiar quarter-wave stack of alternating high and low index layers. Metals are known to have a large imaginary part (k) of their complex indices of refraction. For a single layer of a dielectric (i.e., 20 having N purely real) film on a metal, the maximum reflectance is ob tained by having the admittance of the system real when the index of the incident medium is real, and that index will always be assumed to be real in this work. Let the admittance of the metal be a - iB, with both a and 6 being nonnegative real numbers, and let n be the (real) admittance of the film. Following Eq. (2.22), the admittance is computed so: 1 1 03 cos<5 — sin6 n n insin6 cos6 a-if I 1 giving C _ insin6 + acos5 - igcos5 Y = (2.33) B . . a . . 8 . coso + l — smo+ — sm6r Alternately CB* TBT7 ' so Im(Y)=0 if Im(CB*)=0 and |B|2^0. Expand CB* = insin<5cos6 + asin26 + if?sin26 a2 + acos26 - i — sin6cos6 n + — sinScosfi - i6cos26 n ct8 g2 - — sin5cos6 - i — sinficosfi n n so that Im(CB*) = nsin<5cos6 + Bsin2<5 a2 82 - — sin6cos6 - $cos26 - — sin6cos6, n n then use some trigonometric identities to reduce this to Im(CB*) = (n2-a2-62) sin26 - Bcos26. Set this equal to zero to find tan26 = 02 • (2.34) Because, for common materials, n2 < a2 + B2, the right side of Eq. (2.34) is negative. For a real layer, 6 must be positive, so the thinnest real layer that fulfills Eq. (2.34) is 6 = - i arctan —z (2.35) 2 2 or + Bz-n where this arc tangent is defined only on the interval (- j , j ). Eq. (2.34) may be resubstituted in Eq. (2.33) and solved for y _ 2 (X n ^ -i [az + 8z+T]z) - (a^B^rpTcoiTS + 2Bnsin26 by tedious but straightforward manipulation, or the result may be ob tained by finding 5 and solving Eq. (2.33) numerically. The admittance Y calculated above is a small real number in practice, and has a value to give a lower reflectance than the bare metal. Another quarter wave of material is required to raise the reflec tance to a higher value; if this is more of the matching layer material, the result is that of Park (1964) cited on p. 78 of Macleod (1969). A further increase in reflectance results from adding a quarter wave stack to produce a design m|L^HLH-•*LHLH|air, a prescription that reads left- to-right, m being the metal substrate, L and H quarter wave optical thickness (6 = j) layers of low and high index materials, and the sub script on Lx indicating that it is the fraction (x) of a quarter wave thick (for the matching layer, x i — B 0 1 n m. *< 1 1 1 1 0 or V (2.37) Y1 " Yo which may be seen to yield, for an odd number n of alternating high and low index layers of the form HLH---LHLH, the solution 2(n+1) 'H Y (2.38) n 2(n-l)v If Y is a small number, and the ratio n /n, is large, Y will be verv O Hu L n large and from Eq. (2.13), the reflectance will approach 1 closely. A high-reflectance mirror for use at the 3.8y laser wavelength is given as an example. The substrate (or, an opaque coating) is silver 23 with N=1.75 - i 27.3 by simple interpolation from data from Beattie (1957) as presented by Hass (1965). The low index layer is thorium fluoride (ThF^) with an assumed index of 1.50 (Guha, Scott and Southwell, 1980). The high index layer is zinc sulfide, with an index of 2.2535 from a Cauchy formula given by Pelletier (1970). Table 2.1 summarizes the reflectivities obtainable by adding layers of the above dielectrics to silver. The designs are all for normal incidence. Table 2.1. Reflectivities of silver and enhanced silver mirrors at 3.8p. Number of Layers Intensity of Dielectric Reflectance 0 0.990702 2 0.995877 4 0.998171 6 0.999189 8 0.999641 10 0.999841 12 0.999929 CHAPTER 3 ANGLE OF INCIDENCE EFFECTS WITH ROTATIONALLY SYMMETRIC MIRRORS Rotationally symmetric optical elements form a class that offers easy calculations. An example of such an element is a parabolic mirror used as a collimator. The analysis of coatings on curved surfaces gives some diffi culties; what, for example, is a uniform coating? I define a coating as uniform when its thickness along the local perpendicular to the surface is constant over the surface. A more fundamental problem is that the theory described in Chapter 2 is strictly valid only for plane waves and infinite plane-parallel layers of materials. In performing the calcula tions here, the wavefront is assumed to be nearly plane over a small area, and the layers are assumed to be flat slabs over that area. Angles of incidence are taken to be the ones given by geometrical optics. These approximations are not rigorous, but are hoped to give useful results. Decomposition of Polarizations The aperture of a circularly symmetric system is shov. in Figure 3.1. Let o and ip be polar coordinates, with the radius o normalized to 1 at the edge of the aperture, and with the azimuthal coordinate. The 24 25 y Fig. 3.1. Circular aperture: Cartesian and polar coordinates. 26 choice of origin of is one used in aberration theory, particularly by H. H. Hopkins (1950). The normalized cartesian coordinates are defined y = a cos iJj (3.1) x = a sin ijj (3.2) If rQ is the radius (semidiameter) of the aperture in physical units, then the physical position in units of length is the normalized quantity multiplied by rQ; mostly normalized quantities will be used here. Consider a plane wave entering the aperture in Fig. 3.1, the wave traveling perpendicular to the plane of the diagram. Any such monochromatic wave may be represented as the sum of 2 waves, with elec tric fields in the x and y directions and complex amplitudes E^ and E^r. Define the p and s components of the field for some ip as E (t/0 = E cost// + E sinsf< p J y x and Eg (iJO = -E sinij; + E^ cosiJj or, in matrix form • • • E cos sinijj E P y E -sinijI cosljj E (3.3) s x The matrix Eq. (3.3) has the inverse 27 cosljj -sinijj simp cost|) (3.4) for this is a simple coordinate rotation. The basic calculation proce dure for a given point in the aperture is: form E^ and Eg from E^ and Ey. The complex reflectances in the p and s directions will be functions of the angle of incidence, which itself is a function of o only. After multiplying E and E by these reflectances, Eq. (3.4) is used to give P s the new E and E . I neglect the effects caused by changing a plane x y wave into a converging beam; these may not be negligible for a mirror of numerical aperture 0.3, but they are beyond the scope of this study. Angles of Incidence for a Parabola For a parabola with f/number F (defined as aperture/focal length), the normalized sagitta h is defined as 2 (3.5) 8F having a slope dh a (3.6) da 4F If the parabola is used as a collimator (or receives a plane wave), this slope is the tangent of the angle of incidence; for an f/1.5 parabola, the angle of incidence at its edge is about 165 milliradians (9.46 degrees) and the tangent approximates the angle to within 1%. It may be desirable to test the parabola at its center of curva ture so that no other large optics will be required in the test. This is 28 shown in Figure 3.2; 0 is the angle the line joining some point P on the parabola with its center of curvature C makes with the axis (V - C). 0 is found as o tan6 = 4F-h or, using Eq. (3.5), 8Fo tan0 = 32F^ • Let 0^ be the angle of incidence on the parabola and 0g be the inclina tion of the surface defined as tan0 = 4^- . (3.8) s do Then tan0. = tan(0 -0), l s or ^ „ tan6c; - tan0 ^ tan9. = •:——f—r——- . (-5-9) 1 1 + tan0 tan0 s Performing substitutions for 0and 0 and simplifying the resulting ex pression gives tan8i ° 4f(32f*-3o*) ' (3'10) which, for F > a, may be approximated by 5 - i - 3 ' 128F 29 C 4F Fig. 3.2. Section of a parabola being tested at its center of curvature. 30 showing the form of classical third order spherical aberration. For F = 1.5, Eq. (3.10) gives an angle of incidence of about 2.42 milli- radians (8.30 arc minutes) at a=l; the approximation gives 2.31 mr, or about a 4% difference. Computing Amplitudes and Phases The result of the series of calculations that end with Eq. (3.4) is E and E^, both in the form of an ordered pair (real part, imaginary- part). The desired form is the polar form Ey = a exp (i<(0 , (3.12) where a and (J) are real numbers. The simplest way to define these is ct = |E | (3.13) and fIm(E )I = Arctan ' vi + 2im, (3.14) LRe(Ey) J with n some integer. If the angle of incidence is an infinitely differ- entiable (i.e., derivatives of all orders exist) function of position on the mirror's surface, and the p and s amplitude reflectances are infinitely differentiable functions of angle of incidence, then Ey is an infinitely differentiable function of position on the surface. The condition on the angle of incidence is met by choosing the right kind of surface; it is plainly true for a parabola. The condition on the amplitude reflec tance can be seen to be true by inspecting Eqs. (2.22) and (2.23). Observe that the only way that any of the derivatives of the reflectance r may be 31 singular is if n B + C = 0 . (3.15) o As nm in Eq. (2.22) must lie in the right half of the complex plane, then the equality (3.15) can never hold, and r and its derivatives are nonsingular. The point of all this discussion is that Eqs. (3.13) and (3.14) can give forms for a and Eq. (3.14) can be chosen for each surface point so that a and Extraction of the Focus Term For a rotationally symmetric element used on axis, the lowest order aberration present will be defocus of the form (like that of Hopkins 1950) °22 • • 0 2 where ck and <|k are data radii and phases, which has the least-squares solution Extending this to higher polynomial orders would be considerably more complicated. Examples The FORTRAN program CFILM was written to use the calculational procedures described above. The program performs calculations over an approximately circular part of a 32 by 32 array, with the (row, column) location of the center chosen as (17,17). A line-printer contour plot similar to that used by the FRINGE program (Loomis, 1976) was chosen to display both the modulus and phase (a and Figure 3.3(a) shows amplitude contours for example 1. The total range in amplitude is only 2.76x10 ^ (intensity range about twice this, or 5.5x10 ^) out of an amplitude of 0.9996, which corresponds to an in tensity of 0.9992. The contour interval is one tenth of the total range. Figure 3.3(b) shows the phase contours for the same example, with - 3 the focus aberration left in. The total range is + 2.46x10 waves (about A/407 more concave) at 3.8p; the contour interval is again 0.1 of the range. Figure 3.3(c) is the contour map for the phase with the best R.M.S. fit amount of defocus removed. The total range is 6.71x10 ^ (A/14,900) waves, which is likely to be very difficult to measure for any real optics. Coating thickness nonuniformities will be much more significant than this, as shall be seen in Chapter 4. Figure 3.3(d) shows the amplitude map of the small amount of elec tric field that appears in the x direction due to the differences between s and p complex reflectances. This radiation can be looked on as being lost from the plane wave, as orthogonally polarized light waves do not mutually interfere. To calculate the diffraction form of the image, the diffraction patterns for the two polarizations are computed, and then the 34 ,«•».,a,.,,,.,... 888S8BBRB»BBflBBBBBBBBB,,,. . CCC...... CCC *.CCCCCC,,.....,.,.,.,.,,,,,.CCCCCC,,. ^••••«CCCCCCC..,..,,,,,,,,,CCCCCCC,,,,,0 OPODO CCCCCCCCCCCCCCCCCCCCCCCC.....DOCO. ..0000,.,.,..cccccccccccecccccc,,.,,.00000,, E....00000.,..,.. CCCCCCCCCCCCC,.,.,... 00000,,,, E EE,,..OOOOD...... ,.,CC..,.00000.....EE •EFKE,,,,ODONOD.,,., .01000 EEE ••EEEE,.,.PPODOO,...... ,...... 0000000....EEEE >..FEE ononno ODOOOOO,,..EEEE. ...FEEE.'. ..DnDnono 0000000,,,,,EEEE, ...tEEE.,,,.nnDOOOO.,,,.,,,,,..,,0000000.,,,.EEE., FF...FEEE....,0000000,,„ ,...00000000....EEEE., ...EEEE 0000000. ..0000000 EEE. . ...EEEE,, .,0000000. DOOOOOO EEEE, ...TEEI,P,.DPOOOO, 0000000,,,,EEEE. ..EEEE....000000, .0000000,.,,EEEE •EEFE,.,.000000,,00000,,,,,EEE EE....ODDDD,,,,...... CC.#...... ,ODDDO,,,,,EG F,. ,.00000 CCCCCCCCCCCCC. ,,00000,., ,e ..rooD,,,..,.cccccccccccccccccc..,,,.ODOOD,, onopo..,..ccccccccccccccrccccccccc oooo, oo..,,,ccccccc,,,,,,,,,,,,,,ccccccc,,,,,o ..cccccc,,v .cccccc,,, CCC 8BB8BBBBBBBB,,,,,,,,CCC (a) - BB8BBBBBPBBB8BBBBBBBBB",,.. BBBB8..,,,,,,,,,,,BBBBBB EF,,.,.,...... E ....EEEEEEfEEEFEEEEE,..., F...EEEE,.EEEE,,. ,.EEE....OODOOOpDOOpOODOO..,,EEE,. ..EE...0000,,,,,,..,,,,,,,0000,,,EEC,, ,,.EE...oon....cccccccccpccc....,Dpo.,Eeeai ..FE,.OOP...CCCCC...,,,,,,,CCCCC,,,00*.,,EE,. EE..DOO.,.CCC...... ,,B,CCCC,,,OO,,EE, .EE..nn.,.CCC,.,.BBBBBB8fiBBBB8.,.,.CCC.,00.,Ee,, .E..OD,,.CC,...8BBBBB8.,..RBBBBBB..,CCC,,00,.EE, ,,EE.on.,.CC,,.BBBBB.8BBB,,,,CC«,0D,«£E,F ,EE,,DO..CC,,,BBBB,,,,,,.,.,,,,,,.BBBB,.CCC,,OO,,E,, ,EE,00.,CCC,,BBBB, AAAAAA BBB,,.CC...O,.EE, .F..O0.,CC...BBB,,...AAAAAAAAAA BBB,,,CC,,OD.EE, EEa.DP..CC.,BB8.,,.,AAAAAAAAAAAA,,.,B8B,,,CC,,0D,,CT F.EE..D..CCC', ,BBB AAAAAAAAAAAA,, , ,BBBB. ,CC,.00, ,E. FE.,00.,CC,,8BB,,...AAAAAAAAAAAA,,,,BBFI,,,CC,,0D,,E, .E..OD..CC...BBB AAAAAAAAAA,..,,BBB...CC,.00,EE, ,EE.OO.,CCC,,3BBB...,,,AAAAAA,,,,,,BBBP,,CC,,,0,.EE. .FE..DD.,CC...BBBB...,,,,,,,,,,,,,BBBB,,CCC,,0O,,E,, .,EF,00,p,CC.,,8B8BB...... ,,,,..BBB6...,CC,,0D,,EE,F •F,aOO,,.CC,,..BBBBBBB...gBBBBBBB..cCCC,,DO,,EE, ,FE..DO...CCC....BBBBBBBBBBBBB CCC,,00..EE., EE.,OOO...CCC,,,,.,a,.,a,.a.a.CCCC«,.OD,,EE, •,EE,.OOO...CCCCC...,,.....CCCCC.g,OOO..EE.A ,,EE...000,...CCCCCCCCCCCCC.,...ODD..EEE, ,.EE.aaODODa..aa...... ,,ODOO,..EEE., ..EEE,..0000000000000000. . . ,EEE,, F..,EEEE....•••••,...•.•EEFE...... EEEEEEEEEEFEFEEE,,., FF,,,,,,,,,...,,F Fig. 3.3. Contours for Example 1, an f/1.5 parabola with a 6 layer coating. (a) y amplitude. (b) y phase, defocus remaining. 35 ..FFmfFFFfFFf,, t.f..rfrrrrrrrrrrfr,,,.z ,EEE FFFFFFFFFFFFFF,,,,,EE. ..£EEr,....FFFFfrFFfFFF,1„,.!E!ll ,r>D..EEE..„ ,E!E..O. C.B..EEEE .....EEE...D.C ,nn..EEE',.,., EEEE..O.. iC«0(lEEEEi« i «•«•( iiEEFEEEFttd,^!VtfEEEE|iOH|Ci C.On,,EEEEE,....EEEEFEEEEEEEEEEE.t.,.EEEEE..O,.C B,C.DO,,EEEEEE.EEEEEEEEEEEEEEEEEEEEE..EEEEEE..DO.C.B B.C.O..EEEEEEEEEEEFEEEEEFEFEEEEEEEEEEEEEEEEEE..O.C.B BC.nt>,.EEEEEEEEEEEEEEEEEFEEEtrEEEEEEEEEFEEEEEE..O.C.B .C.nD..E*EEEEEEEEEEEEEEEE.EEEEFEFEEEEEEEEEEEE,.D...B .C.no,.EEEEEEEEEEEEEEEEE,.,.EFEEEEEEEEEEEEEEE..O,.CB A..C.no..EFEEEEEEEEEEEEE...,...EEEEEEEEEEEEEEEE,.D..CB .C.Dn,.EEEEEEEEEEEEEEEEE....FEEEEEEEEEEEEEEEE,.O..CB .C.PD,.EEEEEEEEEEEEEEEFEE.EEFFEEEEEEEEEEEEEEE,.D,t.B «C,no,.EFEEEEEEEEEEEEEEEFEEEFEEEEEEEEEEEEEEEE,.D.C,B R.C.O..EEEEEEEEEFEFEEEEEFEFEEEEEFEFEEEEEEEEEE..O.C.B B.C.OO..EEEEEE,EEEEEEEFFEEEEEEFEEEEE..EEEEEE.,OO.C.B C.OO.,EEEEE,,.,.EEEEEEEEFEEEEEEE,,,.,EEEEE.,O..C •CiDnEEEEfi^ti i tutEEEEEEEi ii^«^ii«^EEEE|iDQ|Ci iDDt(EEEMptfi|(t«^f^«tifitit*fiii|EEEE|iDfi C.6..EEEE,, ,.,EEE,RLM .DD,.EEE..... EEE..B. ^•*£EEt*(ti«*«iiiii«i(»iii*p*iiEEEEtOO , ..EEE...... FFFFFFFFFFFF FEE,, I.CJ ,EEE FFFFFFFFFFFFFF EE. E....FFFFFFFFFFFFFF....E ..FFFFFFFFFFFFF,, D..C,BB,AA,BB,C,0 EfO|CC|BR(^AiltB(iC||D|E F.E.O,.C..BB...AA..BBB.CC.nO.E, .E,DO.CC..BB,.,AAA.,.BBB..CC.0..E. .F..0.CC,.BBS....AAAA..,,8BB.;c..0,E,F r.E,D..C..B0IJ AAAAAA.,,.BBB..CC..D .E, .E.O..CC..BB,,.,.»AAAAAAAA.,.,,BBB..CC,DO.ET ,DO,CC..BBB,,,,.AAAAAAAAAAAA BBB.,CC.D,E D.,CC,.BB,,..,AAAAAAAAAAAAAAAAAAAA.,.,BB0.,Ce.eO B..CC,,BBB«..TAAAAAAAAAAAAAAAAAAAAAAAAt###0BBf.CCtOD .CC..BBR,,,,,AAAAAAAAAAAAAAAAAAAAAAAAAA,, ••.BBB...,.AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA#ft..BBBt,C BBB AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA BBBB AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA BBB ,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAtt.,lBBBB ...BBB.,,,.AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA,,,,,BBB,,C •CC,.BPB..,.PAAAAAAAAAAAAAAAAAAAAAAAAAA.p..,BBB#.CCt n..CC..BB8...,AAAAAAAAAAAAAAAAAAAAAAAA,.,,BBB,,CC.t>D n.,CC.,n8..,.,AAAAAAAAAAAAAAAAAAAA....BPBI.CC,D0 E.0..C.,BBB,.,',,A AAAAAAAAA AAAAAA BBB..C..D.E .(1n.CC.,BBB,,,.,AAAAAAAAAAAA BBB..CC.O.E .E.D..CC..BB AAAAAAAAA BAB..CC,DD,E, (d) F.E.D..C..BBB AAAAAA....BBB..CC..O.E, i^»«0iCC,lBRBl|i,A4AAjl,lflBBllCl(O,E|F ,E,I50.CC..BB...AAA,,.BBB..CC.D.,E. F.E,0..C..B8..,AA..BBB.CC.0D.E. £#0|CCiBB,p4A(tBMC|iO»E 0,,C.BB.AA,BB,C.D Fig. 3.3.--Continued (c) y phase, defocus removed. (d) x amplitude. intensities, not the amplitudes, are added. Such analysis has been used successfully to describe polarization microscopy by Kubota and Inoue (1959). The field amplitude is zero along the x and y axes; here all the field is either in the s or p direction (except at the center of the surface, where these directions are undefined). The x amplitude maximum -4 -8 is only 1.88x10 , corresponding to an intensity of 3.5x10 that of the incident wave. No phase map is given for the x field, as its amplitude passes through zero and the u phase jump occurs. Example 2 uses the same geometry as for Example 1 except the coating and substrate are taken from a design by Heavens and Liddell (1966), which was cited on p. 103 of Macleod (1969). The basic design is of 35 dielectric layers on glass, with alternating low and high index, the first and last layers being the low index material. The indices of the low and high index layers and the glass were taken to be 1.39, 2.36, and 1.53. Layer thicknesses formed a geometric series, i.e., the outer most layer was one quarter wave thick at 0.8p, the next layer 0.97 quarter waves, on to the layer in contact with the substrate, which was 34 (0.97) or about 0.355 quarter waves thick. The calculations were per formed at a 0.4y wavelength. This coating was chosen as an example because it has very rapid changes in the phase change on reflection with wavelength, but the reflectance change with wavelength is small; it was hoped that the coating would show large phase changes as the angle of incidence changed. Figure 3.4(a) gives the amplitude contours; the minimum amplitude reflectance is 0.9967 (0.9934 in intensity), with a total range of 1.01 x -3 -3 10 (or about 2x10 in intensity). 37 .3aBBBB.aaaaaa,aRBBB8a., A»A,,,BB8BRB..,,,,.BBBSBBB,,,AA ,,,,,,869BB88R.,,,.BBBRBB9B0.,.,.A BBf",BBBBBRBB,.BBBBBRBBBBB, R, ,,BH mBBBBBBBRBBBRBRBBBBBB,. aBB,... R flaRBRBBB8BBBBB8B8BBBRB BBRRBBRRRBBB8B8BBBB.B aaaaRBBBB8BBRBRBB8BBRBRBRBBBBBRBBBRB8BBBaaa, CCCCC,,..BBBBBBBRBBB.BRRBBBBBBBRBBBBSB,,,,,CCCCC ..CCCC..,,.888*988,,,88B8BRBB,,,,CCCCC., ni3DI)...CC BBBBB8 ,,..,BBBBBB,.,.CCC..,nDOD F..nn,.CCCC,,,BBBBBB,.,.,.B8BBB....CCCC..00,,E EaanODaa',CCCaaa8»8RBaBBBBB.,.CCC...0D0.aE EEaa0Daa,CC,,.,BBBP8,,..,,..,,..B3B88,.,CCC..D0D.,EE .EE..DO.,,CC...BBBB..,,..,.,,..,B8BBB,,.CC,.OOn,.EE, FF.aEEaa00,.CCC...RBB8....,.,.,...SBBB...CCC..0D.,EE,, .EE..DD...CC...BBBB BBBBB,,,CC..000..EE, EF.a00'.., CC,,,,BBBBB,.,, BBBBB., ,CCC,.000.. EE EaanDDaaaCCC,,.BB8RB,...,.,,,..,BBBBRaaaCCCaaa000aaE Faanna.CCCC,,,B8RBR8,,,,,,,a,a,,B8BBB.,.,CCCC..Dn,aE OOOOaaaCCaaaaaBBBBB6..p.,,,,,...BBRBB6..,.CCC..,ODOO ..CCCC.a.,,8RRBBBB,,,aa,,....B8B8BB88.,..CCCCC.. CCCCC.,,,8BRBBB8RBBB.BRB8BBBBBR8BBBBBB CCCCC •••.bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb,,,, BaRBRPflBBR8BB8RBBBRBBBB8RBRBR8BBBBBBBB8BBBa6 eaaaBBBBB8B8RRR0RBBRBBBRBBBa,aBBaa,. «.....BBBBBBBBBBBB..BBB8BRBBBB8,,,,,.. BBB8BBBB.,...RBflBBBR88,...,A AAAaaaRB8BBBaapaaaaBBBBBBBaaaAA aaaBBBBa,a,a,a,aBBBBBaaa BB,,,..,C..,....B EEEEEEEEEEEEEEEEE EEEE,,,,EEEE .EEEE..,.ODDnDnOnOOOOOO,,,,EEE. EEE,,.O00no,,... 00010.,..EE. .EE...ODOO CCCCCCC 0000,,EEE. .FEE..OOD,,.,CCCCCCCCCCCCCCCC....pD...Ef. «Er...00,,,CCCCC...... CCCC..,000..EE.. EE.,OQr>,,,CCC,,,,,,BBBnBB,,,,,,CCC,.,DO,,EE, .FE..OOO..CCC,...BBBBBBBeBBBBBB..,.CCC..OD..EE.. ,FE..DO,,CCC,,,BBBBBB .BBBBBB,,CC.,00,,EE, aaEFaanoaaCCCaaaBHBBaaa.,.,.,.,.R8B8..,CCC..00..E..F ..FF,0n,,,CC,,,BB8,,,,..,,,,,,,,,BBBB,p,CC.,00,,EE,, •FE,,D'"l,aCC,,aBRB,a,,,.AAAAAA,,, f,pBB8,.CCCa,D0,,E,, ,EE,.Dr<,,CC.,BBBB....AAAAAAAAA BBB. ., CC . .DO , .EE, .EE,,Oa,CCC,,dBB,,.«AAAAAAAAAAA,,,,,BBB,,CC,,On,,EE, F,,EE,00.,CCC.,BBB,,.,AAA«AAAAAAA RB8,,CC..00,,EE, .EE.,U..CCC..9BB...,AAAAAAAAAAA,,,a,8B8.,CC,,0n,,EE, ,EE,,00,,CC,,8BBB,,,aAAAAAAAAA,,,,,BB8,,,CC.,D0,,EE, •EE..OO,,CC,,,6BBa,.,,,AAAAAA..,a..8BB..CCC..no.,E.v ,,EF,0^,,,CC,,,BRB,,...BBB8,,,CC,,00,,EE,, a,EF,,ODa,CCC,..BBBB,,a,,,p.,.,,BBBB,.«CCC..OO..E..F .EE..DO..CCC.. jBBBBBB BBS8B8,, ,CCC, .00, .EE. (D) ,EE,.D0D.,CCC,.,.BBBBRRRRBRB«6B.,,.CCC.,n0.,EE,. EEa,pOD,,,CCC,..,,,BRBBBB,,,,.,CCC,,,OD,,ECa .EF,,,00,,,CCCCC,,.aaaa.,,«.CCCC...DOO,,EE.. .FEE,.000,.;,CCCCCCCCCCCCCCCC....nO,..EE, ,EE..,on00 CCCCCCC '.OOOO..EEE, EEEaaaODODO,aaaa.a,aaaOOOODaaaaEE, .EEEE.,.ODDnonooonoooo. , aaEEE. CEEE,,,,,,,,,aaaa,a,EEEE EEEEEEEEEEEEEEEEE Fig. 3.4. Contours for Example 2, an f/1.5 parabola with a 35 layer coating after Heavens and Liddell (1966). (a) y amplitude. (b) y phase, defocus remaining. 38 ....CCCCCCCCCCCCCCCCCCCCCCCC,,. ODOpOpOOOpOOOOOOODOnonDOOOOOOOOOOpOpOODOD ^•••••••^•^•,«oooonooo^oonnDOo»«^»ft»»«a**«OD • i|ODO^OOnDOD| f ,nCEEEE,t* •EEEFEEEEEEEE t,,##.#BOOOOOp,(, EEEEEEEEEEEE. £^£*«*f«EE£EE£aaaaaaaanr)papwacava£EEEEEEiaa«pECE EEta««ac^a^|EEEEEv«CcS|VfV|9^iytfC4EEEEEff»««»»»»*«EE •»iii«ii^*»i*EEEE|| ((nifi(if i^iyiECEEE|^niiiiiiiiC ...^PF.,m»EEEPE.,mm#Om, !EEE FFFFF,., f .." "FF,mmEEEEmmm,OOm aFFFFFFFFaf.,,EEEE,..,#,HOO.EEEE,,,,FFFFFFFFF, aaaFFFFFFFF#,.,#EEEE ODr50,.#.,.EEEE,,,,FFFFFFFFF, aFFFFFFFFa,a,#EEEEaa#a,aODnt,a,.,aeEEE,a,.FFFFFFFFF, .,FFFFFFF#,.,.EEEE,00,,., 0>EEEE,#,.,FFFFFFF## F ... "fFf„ttEEEEEMM,(iOM, EEEE..,a,tFFFFFa., •••••• ••?*p*a££££***t*p*p«pt*««*p*EEEEE a,*•••..••••£ EEiimif niEEEEE,fiii,i«iii|(f i|iyEEEEEat*(iii* M«*l|ODMIM|,^|.t|(f|r|.|f|tOM|aa|g fc*) ccc, •••••••.•••••••••••••ccc v J ••••ccccccccccccccpccccccccc... ,f, Mf|,6BB Cd) Fig. 3.4.--Continued (c) y phase, defocus removed. (d) x amplitude. 39 Figure 3.4(b) shows the phase, including the focus term, with a _2 total range of 7.59x10 (about A/13) waves at 0.4p. This is large m comparison to Example 1, and even when defocus is removed (Fig. 3.4c), _2 about 1.29x10 (X/78) waves of aberration remain. Figure 3.4(d) shows that the x field develops a maximum ampli- -2 -3 tude of 3.33x10 (intensity 1.1x10 ), with zero amplitude along the x and y axes. The angle of incidence effects with this coating are more severe than for Example 1, as expected. Example 3 is an f/1.5 parabola like that of Example 2, but computed at the center of curvature for the mirror using the angle of incidence given by Eq. (3.10). The total range for the y amplitude is extremely small: 2.55x10 ', _7 giving a range in intensity of only about 5.1x10 . Figure 3.5(a) gives the contour map for this, but it seems unlikely that such a small varia tion in intensity could be detected, much less measured, in practice. Figure 3.5(b) shows the phase contours for Example 3, with the defocus term not removed. The total range is only 1.88x10 ^ (X/53,000) waves, which should not be detectable optically. The amount of energy that appears in the x field is extremely small; the total amplitude range in Fig. 3.4(c) is only 8.38x10 6, for an intensity maximum of 7.0x10 ^ that of the incident radiation. Based on Example 3, I believe that the angle of incidence effects in testing an f/1.5 or slower parabola at its center of curvature are completely negligible, as they would not be detectable even for an impossibly ideal coating. 40 C BBBFLB, •• • • C C..,PB0B^B SB0 BBBBB,, .CC..BBBB...'. ,,.,BBB • Cf .C..B0B,., SB, C, •Mf# .BB..C.D O.C,.BB, i •• • ^ AAA,. I,6BB,CC,( ..AAA AAAAA •»• »BB,,C,« •Cf AAAAA AAAAA A,#MBB,CCO tCctBBtfac AAAAA AAAAA A*•p•fBB §C,0, 0.TC,BBJ M••* AAAAA AAAAA AAA,,,,0B,C,0 E..C:,BB.,,.»* AAAAA AAAAA AAAA,,,00,.CO., EO.C:, bb.,.»/U A AAA,,,,86,C,QE AAAAA AAAAA AAAAA,,,68,C«D, '"I- AAAAA AAAAA AAAAA,,,,0,,CD, o.c, AAAAA,,,,BB,C,, PCO,C.BB,.,AAAA AAAAA AAAAA &AAAAAV,,88,C», ^.C.BB, AAAAA AAAAA AAAAA,,,,BB,C,, .CO, .DC.•»®•0 •AAA AAAAA AAAAA AAAAA,,,8B,C,D# co.e2,BB.,,AAA AAJFTAA AAAAA AAAA,,,,BB,C.OE e..c:.BB #Ff;AA IAAAA AAAAA AAAA,.,00P,C0,, O.C.BB,,,.AAAAAAA AAAAA AAA,,,,80.C«D ,C,,00,.,, AAAAA AAAAA AA,^,,66,C,0, .CI.BB;,; AAAAA AAAAA A ^ % P, 8 B , C C 0 D.e.,B0., AAAAA i««• BB,,C, , 0,C,«68, AAA,. ..900.CC, .CC , BBB IIPI •MM .B6,,C,0 ,C,,BBB,,, • P •• • • •®B0,PC, ,CC.,0060 •«••••BBB • , C ..60BB6BB0 BBBBBBB,, c C•,,,•BBBBB,,,,,C (a) Fig. 3.5. Contours for Example 3, a parabola tested at its center of curvature. (a) y amplitude. 41 £...0000000000... ,00,,.CCCCCCCCCCCC,,00,. E.00ACCC.....B8BBBB....,CC,.D,E .D,iC...BBBBBB8J.,.BBBBB8B..Ce,.D. .p.CC..BBBB,...,.....,,,.,,BBBB..C..DE E'. ..C..BBB...... AAAA,...., ft ®?B..C.O. EDAC,.8B8.,..,AAAAAAAAAAAAAAA.....,B88.CC.,E 0,C,.BB,,...AAAAAAAAAAAAAAAAAAAA....,BB,,C,0 EO.C.BBB', ,.,AAAAAAAAAAAAAAAAAAAAAAAA,.,,BB.,C.,E O.C..BB.,.AAAAAAAAAAAAAAAAAAAAAAAAAAA.,..BB.,C,. ..DC,,BB',,AAAAAAAAAAAAAAAAAAAAAAAAAAAAA..,.BB.C.O,, ED.C,BB,,,.AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA.,,,B8,C,»E ,OC,,BB,,,AAAAAAAAAAAAAAAAAAAAAAAAAAAA»AAA,.,BB1C.OE ..C.BB.>,.AAAAAAAAAAAAAAAAAAAAAAAAAAAA»AAA,.,.B.,CO, O.C.OB. 'AAAAAAAAAAAAAAA'AAAAAAAAAAAAAAAAAA FE0,C.8B,,,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA.,,8B,C., D.C.BB,,,AAAAAAAAAAAAAAAAAA«AAAAAAAAAAAAAA,,,,B6.C., ,.C.BB..,,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA.,..8..CO, ,DC..BB.,,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA,,,BB,C»OE E0,C,BB,,.,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA,,.,BB.C,,E '..OC..BB..,AAAAAAAAAAAAAAAAAAAAAAAAAAAAA,.,.BB.C,O.. D.C.,BB.,,AAAAAAAAAAAAAAAAAAAAAAAAAAA.,..BB.»C,. EO.C.BBB..',,AAAAAAAAAAAAAAAAAAAAAAAA,,,.BB,.C..E 0.C.^BB.,.,.AAAAAAAAAAAAAAAAAAAA,.,..BB,.C,O 1C..BB8...', TAAAAAAAAAAAAAAA,,,...B88.CC,LL.E ...C'..BBB...*AAA;., BBB.-.C.O , .D,CC,,BB88.,.I..,»,,...... BBBB..C.,0E ,D..C,,.BBB9BBB....BBBBBBB..CC,.D. (D) EA00,CCC,....BBBB8B...,,CC,.0.E ,oo,,.cccccccccccc,,oo.. E...0000000000... D.CC.B.,AA,BB,C,0 ,0.CC..B,..AA.,BB..C,0,E ^.,O.CC..BB8,.AAAA^..BB.-C,D,E, ,,O..C..BB..,.AAAAA..,.BBB..C,D,E. •ED.,C..B6.,,.,AAAAAAAA..,,BB6,.C,D»E, FE,OTC..BB.F .,.AAAAAAAAAAA..,),,BBB..C.O.E ..0,C..BBB,,R,.AAAAAAAAAAAAAA,..,,BBB,CC.OE, , .,,C.BBB,,.,,AAAAAAAAAAAAAAAAAA,.,',BB..C.O, .O.C..BB,,,,^*«AAAAATAAAAAAAAAAAAAA, ,,,TBB,CC,OE 0,C.BBB,,,,AAAAAA£AAAAAAAAAAAAAAAAAA4.,,,B8.,C.D ?.C..BB,,,,.AAAAAAAAAA£AAAAAAAAAAAAAAAAA BB.CC.D ,C.BBB^,MAAAAAAAAAAAAAAA£AAAAAAAAAAAAAAAA.,..BB..C, ..6B,.,,,AAAAA«AAAAAAAAAAAAA£AAAAAAAAAAAA4J..,,,BB., 8B,.,..AAAAAAAAAAAAAAAAAAAAAAAATAAAAAAAAAAAAA,,.,ABB ....AAAAAAAAAAAAAAA/KAAAAAAAAAAAAAAAAAAAAAAAAAAAA,... AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ,.,.AAAAAAAAAAAAAA|AAAAAAAAAAAAAAAAAAAAA4AAAAAAA,... ?B«?»V4?444AA4444*4444444444444#44444444444*.|I...BB ,.BB,.,,FAAAAAAAAFAAAAAAAAAAAAAAAAAAAAAAAAA.,...BB.. .C, BBB. , , , A AA A A A AAAAAAIA AA A A A AA A A A A A AA AAA A, ...BB ..C, 0,C,,BB,LT,,AAAAAAAAAAAAAAAAAAAAAAAAAAAA,,,,,BB,CC.O D,C,6BBA,,,AAA4AAAAAAAAAAAAAAAAAAAAAA,,.,BB,.e.O .O.C,.BB,,,,,AAAAAAAAAAAAAAAAAAAAAA.' ..BB.CC.oe ...C.BBB...,,AAAAAAAAAAAAAAAAAA.,,..BB..C.O, ..0,CT.BBB.,,,,AAAAAAAAAAAAAA,,..,BBB>CC,OE, 'E.O.C..BB AAAAAAAAAAA,,,..BBB 'C.O.E F V ,E0,AC,.88....,AAAAAAAA,...BBBA,C,0,E, ..0,.C,.BB..,,AAAAA_.,.BBB.,C,0,E. F..D,CC,.8B.R,AAAAF,.B6.,C,0,E. .O.CC..6..FAA,.B8,,C.O.E 0.CC.B..AA.BB.C.0 Fig. 3.5.—Continued (b) y phase, defocus remaining. (c) x amplitude. 42 Another type of rotationally symmetric optical element is the 7T cone, with the enormous angle of incidence of radians (45 deg.). These are of interest in laser work for changing the amplitude profiles in annu lar beams. Using geometrical optics approximations here seems even more dubious than for fast parabolae, but the results may be of some interest. Using the methods of Chapter 2, a six-layer ThF^ and ZnS on silver coating was designed for high reflectance for s polarized light at 45 degree incidence. The angle of incidence was taken as constant over the surface. Figure 3.6(a) shows amplitude contours for the y field, as in Figs. 3.3 - 3.5. The contours appear like the spokes of a wheel, except near the center of the cone, where the output interpolation routine gives erroneous results. The maximum amplitude occurs where the field is locally completely s polarized, and has a value of 0.9998 (intensity -4 0.9996). The range in amplitude is 7.20x10 downwards from that (intensity 1.4x10 "5). Figure 3.6(b) shows the phase contours; the total range is 3.33 x 10"^ (X/300) waves at 3.8y. Figure 3.6(c) gives the amplitude of the x-field that is produced -2 -4 by the mirror; the amplitude ranges from 1.05x10 (intensity 1.1x10 ) to zero. This example failed to show the significance of the geometrical effects described by Fink (1979). However, it seems that even for this very simple reflector design, the goal of a small phase shift between the two polarizations is not too far from being met, as suggested by the fairly small amplitude of the field that develops in the perpendicular 43 ,BBBB AAAAAAA AAAAA,,,..BBBB ,,.BBBB' ,AAAAAAAAAAA,,.,.BBBB... CCC. 'BB0B,,.,AAAAAAAAAA,,..BBBB.,,CCC ,,,CCC...BBB.,.tAAAAAAAAA,,,•BBS# ,i«CCCf« 00. •1CCC.,BBB..,.AAAAAAAA,,.BBB,..CCC....OO 0000...CCC..',BB... AAA AA AAA,,.BB...CC,..0000,...... 0000..CCC.•BB,.,AAAAAA.,,BB,.CC,..0000,..,, EEE...,..OOO..CC.,BS.,AAAAA«..BB..CC..OOO,...EEEe ,.EEEEEEE,..ODD,'.C..BB,.AAAA,,BB.CC.IDO....EEEEEE... EEEEE»««OO,.CtB».AAAA.0,,C,,DO..EEEEEE,,,,,,, • • • •••••••••££££,.D.C.B,,AA,B,C,po,.£EEE,...... ,..,F FFFFFFFFF EEE.0,C.. A. . . .0. .EE FFFFFFFFF FFFFFFFFFFFFFFFFF.,BCOEE,,,FFFFFPFFFFFFPFFFPF FFFFFFFFFFFFFFFFFFFFFFFFFFEBCFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF....E..',.BCOEE,.,.FFFFFFFFFFFFFFFFFF FFFFFFFFF, J...•.EEE.O.C., A..'..0.. EE FFFFFFFFF ••••••••••••£EEE,,0,C,B.« AA FIB.C,00P ,EEEE.....,,,,.,F EEEEE...OO.,C.B..AAAA,B.,C..OO..EEEEEE,....., ..EEEEEEE...000..C.,BB.,AAAA..BB.CC..00....EEEEEE,,, EEE.'. ...DDO..CC..BB..AAAAAA,,BB.,CC,pOpO....EEEE •••••0000..CCC,.BB,..AAAAAA...BB.,CC,,,0000..... 0000J,,CCC...BB,,,AAAAAAAA..,BS,,.CC..,OODOT OD....CCC,.BBR.oe.AAAAAAAA.V,BBB..,CCC»...DD ...CCC,,TBBB,,.,AAAAAAAAA..,.BBB,..,CCC.. CCCL..8BBB,,,iAAAAAAAAAA,,.,BBB8,,,CCC .,.BBBB,,,,AAAAAAAAAAA,,,,,BBBB... .BBB8>r.,.AAAAAAAAAAAA..,.IBBBB ..•»,AAAAAAAAAAAAA,,,,,, (a) Fig. 3.6. Contours for Example 4, a 90 degree cone with a 6 layer coating. (a) y amplitude. 44 ,,,FFFFFFPFPFPF,, EE,....FFFFFFFFFFF.,.,.EE >,fEtE!,11,PFFPFRMM,,..EfEt1, OO.,,EEE, ,..PFFFFFFFFP,, .EEEE,,.DO .',DOO,.,EEE.,..FFFFFFFF,.,.;-'EE...ODO,, CC...DOO..,EEE.,.FFFFFFFF...EEE,.OOO....C .CCCC,^.OOO.'.EE...FFFFFFF..,EEE,.DOD,..CCCC, ..,CCCe.'.ODO.,EE.,,FFFFFP,,,EE,,OD,..CCC,.., BBB.. ' ,CCC.,00.,EE,,FFFFF,, .EE, .00, .CCC BBB BBBBBB.,,..CC.,'D..EE.,FFFF,.EE.OO,',Cpcj...BBBBBB B«BBBB,'.,CC..00. E ..FFFF,EF,DO. CCC..,BBBBB,.,.., ••••*»f»*#BB®B.,CC,,O,E,FFF,.E.0,CC,,,BBBB, AAA....a....,,BflB,.C,0,E.FP.E.O.C.,BB8,.,...... ,AAAA AAAAAAAAAAA .BB.C..EF.EO.C.BB., AAAAAAAAAAA AAA A AAA AAAAAAAAAAA,.,,BCO,EI'.B...AAAAAAA AAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAABEOAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAA...,BCO,E.'.B.,'.AAAAAAAAAAAAAAAAAAA AAAAAAAAAAA,,..,,,BB,C,,EF,E0.C,B8^..,j,j AAAAAAAAAAA AAA,,,,,,,.,.,BBBa,C.0.E.FF.E,0,C,,BBB.,.«.....,AAAA .».&BBB..CC..O.E.PFF.,E.D.CC...BSBB....,,,,,, 8BBBBB,.,CC,,00.E,,FPFP,EE,0D.CCC..FBBBB8 BBBB3Bl.i,CC'...O..EE..FFFF,,EE.OO.'.CCC...tBBpB0B B«B...,,CCC,.DO,.EET.FFFPP.,.eE.,DD,..CCC BBB ...CCCC,'.ODD:.EE.,.FFFFFF,.,EE..DO..,CCC..., .CCCC,..OOO..EE,..V PFFFFF.,,EEF.,DOD...CCCC, CC,;.OOD,,,EEE,..FFFFFFFF,,,EEE..noO....C ..000,.,EEE,...FFFFFFFF.,,.EEE.,.000,. ,, , 00...EEE,,..FFFFFFFFFF,,,EEEE..,00 (bJ ...EEEE....FFFFFFFFFF,,,,EEEE,. EE.,..FFFFFFFFFFF.,...EE ...FFFFFFFFFFFF,. 0«.C,,B,A,.B,.C.. E,,00,CC,B,A,,B,CC,DD,.E ••P£EE.,D,.C,B,A,,B,C,,0,,EE,,, FFL,.EEE.OO.C,B,A,B,.C1O..EE.,..FF T • FFFFAA,EE.,D.C.B.A,B,C,.0,EE,..FFFFFP FFFFFF?F,.,EE.,D.C.«A,B,C.D,.E,,.FFFFFFFP .FFFFFFFFFF.,EE. O.C.BA.B. COO. E'... FFFFFFFFFF, ..,FFFFFFPFFA,EE.O,.B.,BC.O.EEP.PFPFFFPFP,,, EE.,.,FFPFFFFFF,,E,O.CB,,,CO,EE..FPFPFFF,,,,,EEE .EEEE...,.FFFFFF..E, pel...'.O.E.FFFFFFF' ..EEEEE, 000....EEE,.,.FFFFF.,E CO.E.FFFFFF,..EEEE....000 ...nOOO.,,.EEE..FFFFF,EOC.B,.E.FFFP...eEE...00000.., CCCC,.,..D00,,EEE..FFF...,BDE.FFF.,Ee,;.000...,CCCCC ^••••••CCCC.,,pDO,EE,FF,PB.E,F,,E..OO,,,CCCCC,,...,a .,.BBBBBBBBB.,,.CCe,0.e,EBCF.E.D,CC,,.,.BBB8B8BBBB... AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ,,,BBB8BBBBB,...CCC.O,E,EBCF^E,0,CC,i,,BBB8BBBBBB,,, ••••fa,CCCC,^,OOD,EE,FF,pB,E.F,,E,,OD,,vCCCCC,,,,,,t CCCC,..,,Op0.iEEE,,FFF...,BOE.FFF,,EE,..DOO,,,.CCCCe ...0000,.,,EEE..FFFFF,EOC,8..E.FFFF...EEE..,00000... 000....EEE,...FFFFF..E..,.,CO. E,FFFFFF',.,EEEe,...000 .EEEE,,...FFFFFF O.E.FFFFFFF..,,EEEEE, EE FFFFFFFF1,E,D,CB,,,CD,EE.,FFFFFFF EEE ,,.FFFFFFFFF;,EE.D.,B..BC.O.EE,.FFFFFFFFF,,S .FFFFFFFFFF.,EE,0,C,BA,BJCD0,E..,FFFFFFFFFF, FFPFFFFF,;,EE,,0,C.8A,B,C,0..e.,,FFFFFFFF FFFFF,,,EE,,0,C,B,A,B,C,,D,EE,,^FFFFFF Ff,,.EEP,00,C.B,A,8..C,p.,EE,M;FF f r) o»*EEE,.0,,C,B,A,,B,C,,0,,EE,,, ' E,.OO,CC,B,A,,B„CC,OO,,E 0.,C,.B.A..8..C.. Fig. 3.6.--Continued (b) y phase. (c) x amplitude. (x) direction from y plane polarized light. The current computer code does not give the phase of the x field, so the ellipticity of the radii tion reflected from the surface is not known completely. This kind of coating would be useless for the sort of axicon systems that require a half-wave phase shift between s and p light. CHAPTER 4 THE OPTICAL PHASE ON REFLECTION AS A FUNCTION OF STACK THICKNESS While the angle of incidence effects on optical wavefronts des cribed in the preceding chapter may not always be trivial, the errors produced by thickness nonuniformities of the layers will be shown to be of more general importance. The departures from the nominal design for a coating are diverse. Each layer in a stack may vary in all its properties over a surface; such properties are thickness, index, and ability to scatter light. Any of the films may vary in index with thickness as well (i.e., be inhomogeneous). The complete description of a real thin film stack would contain many variables; in practice the number of variables must be limited. In this chapter, the variables will be restricted to one: the overall thickness of the stack. Giacomo (1958) calls this a "proportional" coating; the thickness ratio of any pair of layers is constant over the deposition surface, but the overall stack thickness varies. The thickness ratios are taken to be those in the nominal design; the thickness of each layer is described by tj(x,y) = [1 + k(x,y)]t0- , (4.1) where tj(x,y) is the thickness of layer (j) in the stack as a function of position (x,y), k is some dimensionless multiplier, and t^ is the design 46 47 thickness for the layer. The total optical phase 41 (Eq. 2.21) is a func tion of the layer thicknesses and therefore of k; or, let the wavenumber be written 0=i , (4.2) where X is the vacuum wavelength of the radiation. Let be the design wavenumber of a stack; then at some wavenumber Oq + £o, if wc define , Acr k = — , (4..>) °0 with the film materials assumed to be non-dispersing, the effect on the phase for a frequency shift Aa will be the same as a change of amount k in the stack thickness. Fig. (4.1) shows the optical path difference ^ as a function of k for the 35 layer design of Heavens and Liddell (1966) described in Chapter 3. A constant is subtracted from the phase to make k = 0; the absolute phase on reflection requires a special effort to measure, but this chapter is concerned mainly with phase changes. Over the range k = -0.025 to 0.025 (5% total thickness) the optical path varies over a 0.31 p range, or 0.775 wave at the 0.4y working wavelength. The path is in units of length on the graph to allow easier comparison among phase changes for widely different wavelengths. The total thickness of the stacks is 2.52p; for k = 0.025, a 0.126y (0.315A) path change would result from the geometrical term; the path change is a bit less than this, but in the opposite direction, with the function being nonlinear. Relative Thickness Change (k) -0.03 0.05 Cu •H 4-> Fig. 4.1. Optical path in microns (p) at 0.4y as a function of rela tive thickness change (k) for a 35 layer design by Heavens and Liddell (1966). An Infrared High-Reflectance Coating: Performance at IR and Visible Wavelengths The 6 layer design of Table 2.1 was chosen as a basis of study for enhanced-reflector performance; it has a reflectance of about 0.99919. The quarter-wave layers for use at 3.8y are thick, which implies deposi tion problems such as high scattering in the films, so the number of layers is as small as will allow a high reflectance. Fig. 4.2 shows the optical path difference (O.P.D.) behavior of this coating as a function of thickness change on the thickness range [-0.025, 0.025]. Over this 5% thickness range, the reflectance remains above 0.99917, and the O.P.D. may be represented as <)> = (5.97910 • k , (4.4) -4 where over the 5% thickness range. Another way of stating Eq. (4.4) is that a 1% thickness change for the stack produces about a A/100 path change. The slope in Eq. (4.4) that would result from the geometrical term alone would be 6.285y for this thick a coating; the value for the dielectric stack is about 0.65 of the purely geometrical value. This effect is related to Giacomo's (1958) discussion of the effects of coating non- uniformities on Fabry-Perot etalon mirrors; it will be studied in greater detail in Appendix B. While the coating is a high-reflector near the design wavelength, the optics may also be used at other wavelengths, such as in a high energy laser with a shared-aperture visible light tracker (Pitts, 1980). Figure 4.3 shows the O.P.D. behavior of the coating at 0.6328u, at the same scale as Fig. 4.2. The coating gives a useful reflectance -U.05 Relative Thickness Change (k) Fig. 4.2. Optical path (in y) for a 6 layer enhanced reflector as a function of k at the design wavelength, 3.8y. 51 Relative Thickness Change (k) -0.03 •px: n. Fig. 4.3. Optical path (in p) for the 6 layer 3.8p design at 0.6328p. 52 (above 0.93), but the path range increases to [-0.285, 0.033], totaling over A/2. The in the opposite direction to the geometrical term. Figure 4.4 shows the O.P.D. performance at 0.5145 (a green Argon ion laser line), also at the same scale as Fig. 4.2. The path range becomes [-0.108, 0.225], or 0.65A total; the reflectance is always at least 0.95 over the 5% thickness range. As described above, a 1 % thickness change in the stack produces about A/100 O.P.D. at the working wavelength. We would like to measure the thickness nonuniformities at some visible wavelength, for increased sensitivity and convenience. The nonlinearitv in the optical path/ thickness relation at the shorter wavelength makes this more complex. Worse, for large optics, the mirror would have to be tested, coated, and then tested again. Large mirrors are difficult to test; the familiar square-cube law of scaling tells that the resistance to flexure of an object is proportional to the square of its diameter, but the mass increases as the third power. Flexure in large mirrors under gravity means that it would be difficult to separate coating defects from deforma tion of the substrate with a simple interferogram. Measurements at Two Wavelengths Coating nonuniformities can be distinguished from substrate deformations by finding the apparent surface heights at two wavelengths, and subtracting; this can be seen in Figure 4.5, which may be regarded as the difference between the optical paths in Figure 4.4 and Figure 4.3; Figure 4.5 shows the difference in optical paths between 0.5145y and Relative Thickness Change (k) -0.03 0.03 C- Fig. 4.4. Optical path (in p) for the 6 layer 3.8p design at 0.5145p. 54 -0.03 Relative Thickness Change (k) Fig. 4.5. Two wavelength optical path difference for the 6 layer design at 0.5145p and 0.6328p. 0.6328p for the 6 layer coating (the vertical scale is changed). The optical phase is again set to be zero if k = 0, because the absolute phase difference between the reflections at the two wavelengths would not be known from measurements with a conventional interferometer (such as a Twyman-Green) used in optical testing. The measurements at the two wave lengths are made without changing the setup. This measurement technique is similar to two wavelength holography as described by Wyant (1978). Unlike two wavelength holography, no actual hologram is produced; the wavefronts are measured at two wavelengths, scaled to the physical opti cal paths involved, and subtracted. If a mirror produces no chromatic effects, such as in a conventional glass mirror with a nearly opaque metallic coating, this technique would give a phase difference of zero across any smooth surface. Ideally, both wavefront measurements would be made at once to eliminate turbulence effects, and the interferometer and any null optics would be perfectly achromatic, or have any chromatic effects completely known for computer removal. Assume that this has been done, and that a two wavelength path difference distribution has been found for the mirror under test. The two wavelength path difference as a function of k, which I write as Inverting the Two-Wavelength Phase Difference Function Taking the inverse of (^(k) can be visualized as laying Figure 4.5 on its side. The zero-slope points of become points of vertical 56 tangent, and k (<(>_,) becomes multiple-valued. The vertical tangent LA implies that a small error in measuring or even that the value of $ observed will not lie on the curve at all. A more general statement about the existence of such an inverse is the Inverse Function Theorem (Fleming 1977, p. 140). Consider a vec tor function £ that transforms an n-dimensional vector into another, as y = f(x) , (4.5) % where x may be represented x = (x , x2, . . ., x ) . (4.6) Then approximate y by y = F(XQ) + D (.X-Xq) , (4./) where D is an n by n matrix defined by 9yi C4 8) ' - J j % the determinant of D is the Jacobean of the transformation f (J=|D|). The approximation, Eq. (4.7), is the several-variable form for the Taylor series for y in x through the first order. If the Jacobean of f is not % zero, then x may be approximated by _1 x = xQ + D ty-f(xo)] , (4.9) which is a local approximation to the inverse of f. Unfortunately, finding J = 0 at some point is not necessarily a test for multi-valued behavior of 57 an inverse even for a single variable; consider the function y = x'\ Each real y corresponds to a unique real x, but J = 0 for x = 0. But if J = 0 for some value in the expected range of variables, the error in f\J- 1 % ^ finding f (y) will be large if any there is error in the value of y. The uniqueness problem would be less severe; if x lies on a continuous r\j curve, it should be easy to decide which of several values of x is the correct one. The computation of J may not be attractive in practice; consider a 6 layer coating, with the layer thicknesses as variables, which have 6 associated variables that may be phases or reflectances. There are 36 terms in J, and these must be computed over a suitably fine mesh in the 6 dimensional domain of the thicknesses to ensure that any zeros of J in the domain are found. The technique to compute the phase derivatives was developed in Chapter 2, but the amount of calculation involved for a realistic problem would be great; if only 10 points per variable were used, and there are n variables, then n^ x 10n derivatives 7 would have to be computed, or 5.6 x 10 for only six variables. This shows the utility of restricting calculation to proportional coatings, to reduce the number of variables to one. Base Thickness Measurements There is another difficulty in estimating thicknesses from the two wavelength phase difference: the absolute phases are not known. In Fig. 4.5, the path difference is arbitrarily set to zero for k=0, but if the deposition process is not calibrated with extreme accuracy, the value of k will be unknown for the point on the mirror's surface that serves as the phase reference. Because the function is nonlinear, an unknown shift in origin will cause systematic errors in estimating the thickness distribution. Let k be the relative thickness difference for the surface refer- o ence point described above. The value of kQ can be predicted if the absolute phase on reflection (including the geometrical term) is known at the reference point. A possible means of measuring this phase was described on pp. 105-111 of Heavens (1965), based on work by Tolansky (1948). The basic technique was originally intended for determining the optical properties of single films, but it could be used with stacks. A mask is placed over part of the silvered substrate so that an edge is produced where the stack goes from zero thickness (bare silver; thickness change k = -1) to the full thickness in an unbroken fashion. The phase on reflection is measured by displacement fringes at a single wavelength, as suggested in Figure 4.6. The layers need not maintain the proportional thicknesses or film optical constants in the wedged portion of the stack, as all that is needed are continuous fringes so that the total phase will be found with no doubts as to order number outside the region of the wedge. Figure 4.7 shows the total optical path excursion for the 6 layer enhanced reflector on a silver substrate as its relative thickness in creases from zero (k = -1) to 1 (k = 0). The total change is about 5.18y, or 8.18 waves for 0.6328p light. The accuracy in estimating kQ would de pend on the behavior of the path; if kQ had a value to give a local maxi mum or minimum of the O.P.D., the accuracy would be poor and a different wavelength should be used to make the measurements. 59 ^2 (a) (b) Fig. 4.6. Representation of displaced Fizeau fringes for measuring the stack thickness. (a) A wedged step in thickness. (b) Fizeau fringe centers for an 8.2A linear phase step (the phase is not linear in reality; see Fig. 4.7). Relative Thickness Change (k) Fig. 4.7. Optical path change (in p) for the 6 layer design at 0.6528p; the stack thickness goes from zero (k=-l) to the nominal (k=0). CHAPTER 5 A MODIFIED COATING DESIGN TO ALLOW ACCURATE COATING THICKNESS PROFILE ESTIMATION Some of the difficulties described in the preceding chapter might be circumvented by choosing a different pair of wavelengths than 0.6328p and 0.5145y. In practical interferometers used for testing large optics, unequal paths to the test and reference mirrors require a laser light source to be used for good fringe contrast. Lasers with tuneable output wavelengths (dye lasers) are expensive, and their wavelength range is limited; it is more attractive to modify the coating slightly to allow the use of standard wavelengths. An Error Function The 6 layer design will be modified through use of an optimiza tion algorithm; this requires an error function to be calculated to give a single number to describe the system performance. A goal of this study is to estimate the wavefront error at some infrared wavelength from measurements in the visible, by defining wavelength (3), {{>03 is a reference phase at wavelength 3, W1 is an optical path (in p) at a visible wavelength (1), W01 is a reference path at 61 62 wavelength 1, and and are optical paths at visible wavelength 2. G is defined from material in the preceding chapter; let [l 5 2 >2» • W - W ' - ' where • ^2 X = f If >3 = • g(k) , (5.5) then G is found as G(x) = g[fr-1_i(x)] . (£.4) For the 6 layer design, g is well approximated by Eq. (4.4), but neither g nor f will be used explicitly in the following discussion. The func tion G gives an estimate of This may not be true in practice, for the layer thicknesses will be in error from the original prescription. The layers will also have errors in their complex refractive indices, but these effects will not be in cluded in the error function, which is defined 1 (5 5) E • • - *03 - G[0»rWOi> " OVV ' ' 63 Let tj be the thickness of layer j in the stack; then d 3E ( 3W? 3Wn? \ (5.6) " V 3t. 3 t. / * 3 3 ' The derivative ^ is the derivative of G with respect to its argument. None of the terms with two subscripts C4>o3» w0l> w02) are constants; the phases at the reference point change if any of the film thicknesses change. If the stack has proportional thicknesses (i.e., if the layer thicknesses are only functions of k) and the reference thickness is fixed at k=0, Eq. (5.5) becomes 0 = E(k) = - [W2 Ck)-W2 CO)]} , (5.7) where <}>3(0), Wj(0) and W2 (0) are now regarded as constants. Then _ dE(k) _ d Suppose that the thicknesses of the layers are represented by a % vector t = (tl5 t2, ... t ), where n=6 for the present design. Then by the chain rule 64 d4>3(k) _ v Si}>3(k) 3tj ,r d k > 9 t. Tk ' 1 J J = 1 3 % Because t is of the form t = (l+k)(t0i> 1021 •••> tQn) , (5.10") t t ie vector where to= (toi»t"o2» • • • > on-' ' describing the prescribed coating thicknesses, then 9tH = ltn, , (5.111 7k - o 3 and dflg(k) y 8$3 (k) d k .L, °i 3 t. " j = l J The forms for the derivatives of Wj and W2 in k are similar, and the derivatives of the phase were developed in Chapter 2, so the necessary 9E information for calculating —— is complete. dt . 3E ^ Once —r— is known for each layer (i), the multivariate Taylor 3tj series through first order for E is written E * E(k) + (1+k) • I |^- Ato. , (5.13) j = l 3 th ^ where Atg^ is change in the j component of tg. This approximation is % assumed to be valid near the n-dimensional line for t that Eq. (5.10) describes. Let At . be a random variable with zero mean and variance oj 2 Oj, and let the several At0j be statistically independent; if A in brackets () represents the expected value of A, then having different At0j statistically independent (and zero mean) implies 65 (5.14) where 6.. is the Kronecker delta ij 0; i i j S. . ij 1; i = j The thickness errors Atg^ are not necessarily independent in practice; if ThF^ were to be deposited using optical monitoring, and the monitor glass chip had index 1.523, then if the ThF4 had an index of 1.50 the reflectance would have a maximum of 0.0430 and a minimum of 0.0371; a reflectance swing of 0.0058 is too small for accurate monitoring. If a quarter wave of ZnS (with an index near 2.35) was deposited first the reflectance would be increased to 0.3222, and a quarter wave of the ThF^ would reduce this to 0.0548. An error in the thickness of the ZnS layer would bias the monitoring of the ThF^ layer; in this case the layer thick ness errors would not be statistically independent, but they will be assumed to be so to simplify the analysis; they could be made independent by depositing a ZnS layer on the monitor chip (above) that is not a layer of the stack but present only for monitoring the low index layer; alter nately, nonoptical monitoring may be used. The variance of E is then o|(k) = This is the approximate variance of the estimate for the optical phase ( Atj = (l+k)AtQj , (5.16) c) the errors Ato^ are zero mean independent random variables with variances o_., and d) the errors AtQj are small enough so that the Taylor series approximation (Eq. 5.13) gives an adequate result. Note that assumption (c) does not require any specific form for the probability distributions At0j other than having zero mean and a known variance, and the thickness errors At0j are implicitly assumed to be the only errors present in the coating other than having overall stack thickness changes over the surface of the optic ( k(x,y) ). This fairly lengthy development has had the simple purpose of providing a single number that is an estimate of the probable error in determining a wavefront at an infrared wavelength from a pair of inter- ferograms taken of a coated mirror at two visible wavelengths, given that the only errors in depositing the coating are a fairly large overall total stack thickness error (k) that is produced by the deposition geo metry for the films, and basic small departures from the layer thickness prescriptions. The R.M.S. estimate of the error in $3 is and is hoped to be valid enough to be useful. An Optimization Technique: The Nonlinear Simplex Method A wide variety of optimization techniques for adjusting the performance of thin film stacks are available; review articles by Liddell (1976) and Bloom (1981) are concerned directly with multilayer 6" optimization, while the book by Adby and Dempster (1974) describes a wide variety of algorithms of general uses. The optimization technique chosen for use here is the nonlinear simplex method of Nelder and Mead (1965), which was used to modify broadband reflector designs by Pelletier (1970). The algorithm proceeds as follows: suppose there are n variables. 'Xi % ^ Construct (n + 1) n-element vectors x^, x^, x^ with each vector of the form 3f. = (x.r x.2, x.n) , (5.17) with the (n+1) vectors forming the vertices of a simplex (i.e., for n=2 ^ , a, forming a triangle, for n=5 a tetrahedron). Find the vectors x^ and x L that give maximum and minimum errors. The centroid of the hyperplane (or n-dimensional simplex) formed of all the points but x.. is n+1 \ ^ 1 x = — (5.18) c n .Ji \)-x» Form the reflection of x,. m x , H c xn1 = x - (x..-x ) R1 c He or xR1 = 2xc - xH . (5.19) Find the value of the error function for x , and write this as F(x ). KlD1 Hi If this is less than F(xH), try expanding the simplex: 68 y 'b % y . XR2 Xc " " H~Xc or % % XR2 = 3xc " 2xH ' (5.20J ^\j 'Xj If F(xno) is less than F(xT), replace xu by xDO; if not, then replace rw l h k<- ^ 'V % '"u xu bv xni . But if F(xn ) was not less than F(xu), then form a different h " hi k1 h x^ of the form 1 % XR2 = I (Xc + XhP ' (5-21) which tests to see if the error function minimum is inside the simplex. i\, <\, 'V If F(x^) is less than F(x^), replace x^ by x^?, which contracts the simplex in one direction. If F(x^?) is greater than F(x^), then contract the whole simplex about x^, o, 1 ^ A, Xi = 1 (xi + *0 ' (5"22) and then find a new x^ and xL and begin again. The algorithm is ex pressed in a flow chart as Figure 5.1. The nonlinear simplex method was chosen because of its advantages. It is a search algorithm, so it never diverges, it has no restrictions on the error function (which need not be continuous), and needs no error function derivatives. The algorithm is also very easy to turn into computer code. The final error function was of the form F(k,t0) = max {fi oF(k,t0) + |R0-R(k,t0)|} , (5.23) (k) p 69 Input Simplex no VI yes no no V2 ves ves Output New Simplex Fig. 5.1. Flow chart for the nonlinear simplex search algorithm (Nelder and Mead 1965). "0 which is shown as a function of k and the thickness prescription vector tq• Rg is a target reflectance; if a basic quarter-wave design is chosen as a starting point, its reflectance is a maximum for the number and indices of refraction of the films, so (RQ) is chosen to be that reflec tance (0.9992 for the six layer design), fi and fi are positive weights p K chosen to emphasize the optical phase or reflectance errors. Because F is defined as the maximum value of a function over a range in k (in practice, a set of values of k), F is nonlinear; a small change in the % prescription t0 can cause a large change in the value of k that gives the maximum error over the interval in k. The R.M.S. thickness variation of each layer ck was set to 5° of the thickness of a quarter wave of that material in the visible, or 0.002y for ZnS and O.OOSp for ThF^. The observations of Macleod (1981) suggest that this kind of accuracy should be attainable if the proper optical monitoring techniques are used when depositing the films. The Modified Design Use of the above optimization algorithm turned out to require a considerable amount of human intervention. The basic tactic used to modify the coating was to pick a range in k that had turning points of 4>2x(k) just beyond its endpoints; a few iterations increased the spacing of the turning points, and a new range was selected. The net effect is to maximize the range where cj>0 (k) has no turning points. A modified 2. A coating prescription is given in Table 5.1. Figure 5.2 gives the two wavelength path difference function <(>2^(k) over the range in k of [-0.04, 0.04]. This design should allow 71 Table 5.1. Thickness prescription for the modified 6 layer design. Layer Number Material Thicknesses (y) (from substrate) Quarter-Wave Modified Design Design 1 ThF, 0.6113 0.74215 4 2 ZnS 0.4216 0.45885 5 ThF, 0.6333 0.55863 4 4 ZnS 0.4216 0.45059 5 ThF, 0.6333 0.57588 4 6 ZnS 0.4216 0.45536 Relative Thickness Change (k) -0.04 n.04 rt o ^ . a. 5 O Fig. 5.2. Two wavelength optical path difference (in y) as a function of relative stack thickness change (k) for the 6 layer modi fied design. estimating k from over the range in k of [-0.025, 0.025]. Figure 5.3 shows the O.P.D. for the modified design at the 3.8p working wavelength, shown at the same scale as Figure 4.2, which repre sents the performance of the original quarterwave design. Figure 5.3 is also nearly a straight line over the 5% range in k, approximately (like Eq. 4.4) cj> = (4. 061 y) • k (5.24) with a peak error of O.OOlp (A/3800). The phase behavior is not materially different from the quarter-wave design, but the reflectance dips to 0.99894 for the modified design, which is slightly worse than the desired goal of always having R _> 0.999. Figure 5.4 and Figure 5.5 show the optical path behaviors of the modified design at 0.6328y and 0.5145y. The figures are at the same scale as Figure 5.3, and may be directly compared with Figures 4.3 and 4.4, which are curves at the same wavelengths for the quarter wave design Estimation of the V.'avefront in the Infrared from Two Visible Measurements: Polynomials The process of estimating the O.P.D. in the infrared from measure ments in visible light is described by Eq. (5.1) which is written slightly differently as $ = ( Rather than finding G analytically, it is approximated by a polynomial. The approximation technique is one described by Carnahan, Luther, and Wilkes (1969) and uses orthogonal polynomials of a general nature. 74 0.03 Relative Thickness Change (k) -0.3 Fig. 5.3. Optical path (in p) at 3.8y for the modified design as a function of k. 75 0.03 Relative Thickness Change (k) -0.3 Fig. 5.4. Optical path (in y) at 0.6328y for the modified design as a function of k. Relative Thickness Change (k) Fig. 5.5. Optical path at 0.5145y for the modified design. Suppose there are n points and function values for these, y\. Let P 1 and PQ be polynomials in x that are orthogonal over the set {x^}; orthogonality is defined n 0= I P (x.)P (x.) , (5.26) 1 1 i=l J R Typically, P = 0 and Pq e 1. Then higher order orthogonal polynomials may be constructed using the recursion relation P.+1(x) = (x-Y)Pj(x)-6Pj_1(x), (5.27) where n 7 x.[P.(x.)]2 1 1 i=1= 1 1 3J 1 y = — , (5.28a) ; I [Pi(xi)] i=l 3 and IX y x.[p.(x.)p. ,(x. )] ^ 1 J i j-i i (5.28b) 1=1 j except define 3=0 for j=0. The polynomial approximation of the y^ is defined m y- - I a. P.(x.) , (5.29) 1 j=o J J 1 where 7S 11 y y. p.(x." i=l 1 J 1- a. = — . (5.30) j n ; I [Pj(xi)] i=l J In practice, the various P are combined into a single polynomial of order m, which I will call P . If a polynomial of order j is defined i i P (x) = I C X1 , (5.51) J i=0 J then m C.M = I a. C.. (5.52) lM -j = o J l;l The useful feature of this approach is that it is general; any set {} may be used, and P^ gives the least-squares fit over that set of points for a polynomial of order m simply because the P^ are orthogonal. In practice, a set of points evenly spaced in the total stack thickness change k are chosen, the The coefficients of the polynomial P for the modified coating are given in Table 5.2. The peak error in the fit is 0.0008 waves (X/1250) if the zero order term is used; the error increases to 0.0015 waves (A/667) if the zero order term is omitted. This polynomial can be used to give an estimate of the accuracy that must be used to measure the base thickness R^ as described in Chapter 4. The derivative of P^ with gives an estimate of the error 79 Table 5.2. Polynomial approximation of the phase at the IR wavelength from the two wavelength path difference 4>2\' Order Coefficient. Value of term for 1 Number (j) (in waves/p" ) $2^=0.1 v, in p 0 6.8 x 10"4 6.8 x 10"4 1 -0.2814745 -2.814745 x 10 2 0.4575552 4.373532 x 10" 3 17.59589 1.739589 x 10" 4 -54.10548 -3.410548 x 10 5 - 768.7625 -7.687623 x 10 6 1011.042 1.011042 x 10" 7 17145.79 1.714379 x 10"' 8 -12995.20 -1.299320 x 10 9 -183154.3 -1.831543 x 10 10 60195.11 6.019311 x 10" 11 744919.5 7.449195 x 10"' 80 in the IR wavefront estimate of measuring the displacement fringes (Fig. 4.6) to give an accurate value for k . 0 Figure 5.6 shows the optical path across a wedged region of the stack, at the same scale as Fig. 4.7. Figure 5.4 gave the more detailed behavior of the stack near the prescribed Stack thickness value. If the maximum error allowed in the estimate of the performance at the IR wave length is 0.002 waves (A/500), by using the derivative of P^ as a guide and then using P itself with some trial values we find that the maximum allowable error in measuring the displacement fringes is 0.0062 waves (A/160) at 0.6328p. By reversing this reasoning and trying a more easily attained value of 0.0516P (A/20) at 0.6328y, the error in estimating 4> is found to be greater than 0.008 (A/125) over a restricted range in stack thick ness of +_ 2%; this error is 0.2 the total wavefront excursion for a +_ 2% thickness range. Measuring the total stack thickness at the reference point (kp) is seen to be difficult using simple interferometry. The value of kp must be kept within the 5% thickness range where estimates of k (and c(>) remain accurate. If the value of k is to be measured by interferometrv, ' 0 the simplest technique requires a spot near the reference point to have no dielectric stack coating, and for some kinds of optics, such as high- energy laser optics, an uncoated spot on a mirror may not be acceptable. 81 -1.0 o.o Relative Thickness Change (k) Fig. 5.6. Optical path at 0.6328y for the modified design, from zero stack thickness (k=-l) to l.lx the nominal design thickness (k=0.1). 82 These drawbacks make the use of a modified coating design unattractive; a more general approach that places fewer restrictions on the coating type is desired. Measurements Using Several Wavelengths The next step is to use three wavelengths; for convenience, 0.5145y and 0.6328p will be used again, with 0.4880p (a blue line from the Argon laser) chosen for the third wavelength. The Cauchy formula from Pelletier (1970) gives the index of ZnS at this wavelength as 2.4142, and Ag is assigned a complex index of 0.053-i2.76 from interpolation of a table by Hass (1965). ThF^ is assumed to have an index of 1.50 at all three wavelengths. An additional incentive for performing measurements at 3 wave lengths is given by Figure 5.7. The phase for the original 6 layer quarter wave design for the enhanced reflector is linear within O.OOlu (X/3800) over the range in k [-0.05, 0.05], or a 10% total thickness range; the reflectance remains above 0.9991 over that range. In order to keep the wavefront error less than A/50 at 3.8p, the excursion in k must be no more than 0.02; however, acceptable performance will be achieved if this 2% thickness range falls anywhere inside the 10% range above, giving a _+ 4% tolerance on the total stack thickness. For the modified 6 layer reflector with a 2% total excursion in k, the tolerance on kQ would be +_ 1.5%. The proposed 3 wavelength measurement procedure is to form two 2 wavelength optical path differences these functions are (Chapter 4) independent of substrate profile. Three different pairs of wavelengths 83 D. 4-> CL -0.05 0.05 Relative Thickness Change Fig. 5.7. Optical path at 3.8y for the six layer quarter wave design over a 10% range in k. 84 may be chosen; the two wavelength path difference functions in Figure 5.8 lire for 0.4880u - 0.6328p and 0.5145y - 0.6328p. Over the 10% thick ness range for the 6 layer quarter wave design, the derivatives of the (J> curves are never both zero for the same k, implying that k can always z a be estimated without large error; however, there are some values of k that have the slopes of both (^x-' sma11 at once. Let An alternate way of displaying the data of Figure 5.8 is to plot crosses itself; the value of k is ambiguous at these points, and which of the two values is correct is decided by physical arguments such as continuity of k over the surface of the optic under test. Figure 5.10 shows a version of Figure 5.9 that only plots points in increments of k; here the difference in k between points is 0.001. Wherever the points on the vs. curve are closely spaced, the first derivatives in k of and are both small; these are the locations of high sensitivity to measurement error. The estimation of the relative thickness of the stack over the surface (k(x,y) in Eq. 4.1) is made from cj^-y and by simultaneously finding the stack thickness at the reference point (kg) and the value of k for each surface point to best fit the observed a"d $2^7 • This corresponds to having a set of points on Figure 5,9 that in general do not lie on the curve because of some unknown shift of the origin; the curve is displaced (but not rotated) until all the points fall on the curve. 85 Relative Thickness Change (k) Fig. 5.8. Two wavelength path difference functions (in p) for 0.4880p - 0.6328p (lower curve) and 0.5145p - 0.6328p (upper). Fig. 5.9. Parametric curve of the two wavelength path difference for 0.5145p - 0.6328y (ordinate) vs. that for 0.4880p - 0.6328u (abscissa). 87 0.3 .. J -0. 2 (in p) 0.2 2X2 r-j -e- -0.1 Fig. 5.10. Point plot of the parametric curve in Fig. 5.9; spacing in k of 0.001. 88 In an electronic computer, this would be done through use of an optimization procedure. The means of putting this in practice are described in the following chapter. CHAPTER 6 ROTATIONALLY SYMMETRIC COATING THICKNESS DISTRIBUTIONS The use of numerical techniques for calculating thin film proper ties makes restricting the number of values in a calculation desirable; a flat mirror with a rotationally symmetric coating thickness profile is a two-dimensional surface that can be described by a one-dimensional radial variable. Vacuum Coating Geometries Holland and Steckelmacher (1952) describe the fundamental tech nique for predicting coating thicknes distributions. The basic approach uses the methods of radiometry; the streams of material from a vapor source are treated like the radiant flux from an incoherent light source. The molecules of evaporant are assumed to travel in straight trajectories from source to substrate, and to adhere where they strike. These assumptions may have to be modified in practice, but they will serve for an elementary analysis. Holland and Steckelmacher (1952) discussed sources with vapor distributions like that of point and cosine (Lambertian) light sources. Anastasio (1963), Keay and Lissberger (1967), Anastasio and Slattery (1967), and Graper (1973) have studied vapor distributions produced by several source types, using different analysis procedures. I conclude 89 90 from studying these papers that the practical source with the slowest fall off in rate/solid angle is the cosine source, which is achieved most nearly by crucible sources. The coating thickness distribution given by a cosine source is described by Holland and Steckelmacher (1952) and by Behrndt (1966). For 4 a flat substrate with the source on the axis, the cos law may be written 'W " 7^r"v • • t6-1' (h +r ) where t(r) is the relative thickness of a film, h is the distance from the center of the flat to the source, and r is the radius on the mirror. Alternately, t(E} = LTTT * (6-2) h [i + (£)] Eq. (6.2) may be solved for (^-) to give a specified value of t; for r - 2 t=0.99 (1% thickness range), the required value of (^) is 7.1x10 ; the flat must be displaced 7.04 times its diameter above the source. For a mirror several meters in diameter this is much too far to be convenient, and only 0.5% of the material from the source is deposited on the mirror. * A report by Bennett (1980) gives an overview of the coating problem for large optics; an offset source technique is recommended for obtaining better coating uniformity with a relatively compact coating chamber. The geometry of this technique is shown in Figure 6.1. The substrate rotates around an axis displaced from the source axis an amount (r,e) I I i ~ 3? _ik. Fig. 6.1. Offset source coating geometr 92 A; the plane of the substrate is spaced h above the source. If the sub strate performs many rotations during a film deposition, the effect is the same as averaging around a single 2n substrate rotation but should give better uniformity if the source is not constant. For the point on the substrate described by the polar coordinates (r,6) in Fig. 6.1, the deposition rate (normalized to the center of the substrate) is 2 L,h 2 +AA 2 D(r,6) = (6.3) 2 2 2 r +h +A +2rAcos0 Averaging this over a rotation of the substrate is represented by the integral r.2 ,2,2 (h + A ) de t(r) = (6.4) 2 2 7 2TT (r +h +A +2rAcos6)~ Fortunately this is a tabulated,integral, of the form (Dwight, 1961) IT d x (6.5) (1 ± a cos x)- (l-a2)2 After some manipulation the integral expression (6.4) becomes [(£>h 2 +i]2 u|rh ° + c£rr 2 + i] (6.6) t(r) = uc^)2-CJD2-I]2 - 4(J)2>3 The fraction of the material generated by the source that is deposited on the substrate is of some interest. For the substrate on the axis of a cosine source, this is 93 f»2h|2 »•') 0 which has the solution C^)2 f = ^T—2 ' t6'8) 1 • (,f)2 where r^ is the semidiameter of the circular substrate. For the offset source arrangement, the form similar to (6.7) is much more complex; it is r° 2tt ,,2 2. t t r dr de ,, f = (h +A ) [ ; T . (6.9) 7i JJ (h +r +A + 2r&cos0) 0 0 The 0 integral is performed first, using Eq. (6.5), to give ,r0 2 2 2 2.2. f (h+r +A )r d r f=2(h A) (6 10) J rJ 2 2.2 ,2.2,3 ' ' 0 [(h +r +A ) - 4r A J 2 2 2 2 the r integral is solved by the change of variable a = h +r +A (da=2rdr) and the integral has the form (Dwight 1961) x dx 2bx + 4c (6.11) 2 2 2 2 1 (ax +bx+c)2 (4ac-b )(ax +bx+c)2 Using Eq. (6.11), Eq. (6.10) may be solved as 94 / u£)2 • c^)2 • i] f [1 2 2 2 2 = 2 +$ l\u £) ^) +i] -<) }I " (6.12) which can be reduced to Eq. (6.8) in the limit as A goes to zero. Figure 6.2 shows the relative stack thickness, k, as a function h r of normalized radius, a, (a=l at the edge)' for (—)= 1.3 and (-^0 = 0.7 with a flat substrate. Alternately, (—)= 1.86 and the tank diameter is at least 1.22 times the mirror diameter. The total excursion in k is .019. _2 For these parameters, the flat should receive f=9.05x10 or slightly more than 9% of the total material emitted from the source. For the same range in k, using the on-axis configuration would require (—)=10.2, and the flat would receive less than 1°6 of the evaporated material. This apparent advantage of the offset source arrangement is somewhat mis leading, for the scheme gives an angle of incidence of the vapor stream on the flat as large as 52.6°, which may cause variations in the film properties. Such difficulties are inherent with a compact deposition scheme. The Use of Radial Polynomials Interferogram analysis programs such as FRINGE (Loomis, 1976) fit a set of polynomials to wavefront measurements both for numerical con venience and to make sense of data that consists of phase measurements over a set of points on an optical surface; the radial thickness distri bution shown in Figure 6.2 is for (^-)= 1.3 and (^0 = 0.7, and a sixth order 0.02 0. Normalized Radius (a) -0.02 Fig. 6.2. Radial coating thickness distribution from an offset source scheme, (h/A)=1.3, (rg/A)=0.7, as a function of normalized radius a. polynomial fits it to within better than 0.002% in thickness, or about 0.1% of the 1.9% thickness range. This would give about A/50,000 error at the working wavelength. Fitting through fourth order gives an 0.04% peak thickness error, corresponding to a A/2500 error at the infrared wavelength. The behaviors of the two wavelength path difference functions $2,1 and 0.002p or about A/250 at the shortest measuring wavelength, with an RMS error of a third of that. Taken with the fact that a fourth order fit to the thickness distribution of Figure 6.2 is adequate, it is plausible that radial polynomials of order 16 would suffice to represent the pair of two wavelength O.P.D. maps produced from three interferograms of a flat coated with the 6 layer quarterwave design, the coating having the radial thickness distribution shown by Figure 6.2. Figure 6.4 shows the computed radial two wavelength O.P.D. for such an optic, the wavelengths being 0.4880p and 0.6328y as in Chapter 5. Figure 6.5 is the radial function for 0.5145p - 0.6328y at the same scale. In practice both these curves may be fit to an R.M.S. error of less than 10 ^y by tenth order radial polynomials; the polynomials are in normalized radius (a), so the coefficients of the two polynomials are all in units of p. The coefficients are given in Table 6.1; the zero order term is included, although it would be discarded in ordinary use. 2A1 2A2 -o.oi o.oi Relative Thickness Change (k) and Fig. 6.3. Two wavelength Optical Path Differences 0 Normalized Radius (o) -0.00b 6.4. Two wavelength radial O.P.D. for 0.4880p-0.6328y, for the coating thickness shown in Fig. 6.2. 0.01 •!-> DO 0. Normalized Radius (a) -0.006 Fig. 6.5. Two wavelength radial O.P.D., 0.5145p-0.6328y. 100 Table 6.1. Radial polynomial fits for two wavelength O.P.D.'s. Coefficients in p Order (0.4880p-0.6328y) (0.5145p-0.6328p) 0 -0.00029 0.00042 2 0.37856 0.12840 4 0.15202 1.93902 6 -1.98514 -6.20348 8 1.77649 5.42029 10 -0.37044 -1.19039 101 One common feature of the fits to the radial polynomials that have been done so far is that the set of points used for the calculations are equally spaced in squared radius; this attempts to make each point of equal weight in area on the circular flat surface. Determining a Radial Thickness Distribution from Measurements at Three Wavelengths The information that is needed is the coating prescription with layer thicknesses and optical constants specified, an initial estimate of the radial thickness polynomial, and radial polynomials for the pair of two wavelength O.P.D. functions from measurements of the mirror. An optimization routine is used to find the radial thickness distribution for the coating that best fits the observed optical data, based on the assumption that the film stack varies only in its overall thickness proportionally (i.e., if to is the prescribed thickness vector for the stack, then the thickness vector at some radius r on the flat is [1+k(r)]t0). The use of an optimization procedure to determine the optical properties of a single film is not new: Hansen (1973) and Pelletier, Roche and Vidal (1976) have described techniques that employ optimization of layer index and thickness to match different types of physical data. The peculiar feature of the measurements here is that if the coating becomes uniform, then there is no information at all about the stack thickness. The basic procedure is as follows: take the initial estimate for the stack thickness distribution, and search for the base thickness 102 (i.e., zero order term in the radial thickness polynomial) that causes this distribution to give radial polynomials that best match the O.P.D. polynomials of the measured data. An optimization over all the terms in the estimated radial thickness polynomial follows to produce the final estimate. The error function used is described as follows: assume that the polynomial coefficients for the pair of two wavelength O.P.D. maps c are c: = [c12, c14, ..., and c2 = [c22, c21t, ..., 2^ ' and that the estimate of the thickness distribution leads to calculated estimates for these, which will be called ej and e2. The polynomials are each of the basic form n 0. P(o) = I c o'] ; (6.15) j=l J there are only even terms in a radial polynomial. Zero order terms are computed for e^ and e2 but are not used because the absolute phases of the wavefronts are not measured by ordinary interferometry. The error function in use is defined as E = max £2.. I e. . -c. . I , (6.14) i-1,2 13 13 13 j =2,4,...,2n where the are nonnegative weights. This error function is nonlinear, as was the one in Chapter 5, and the simplex search algorithm is again used to find an optimum fit. The basis for choice of this error function was that high order radial polynomials might have been needed to fit the two wavelength radial O.P.D. functions well, but that only relatively low 105 order polynomials would be available (order 12 for the above mentioned FRINGE program). The low order polynomials would be expected to be more accurate than high order ones given noisy data; this will be studied in the next chapter. The main advantage of the three wavelength procedure over the two wavelength method of Chapter 5 is that with three wavelengths a value for the base stack thickness (k^) can be found without modifying any area of the coating and without attaching a special importance to any single point on the mirror. This is achieved at the price of needing another measurement of a wavefront. An Example The coefficients presented in Table 6.1 may be used to estimate the thickness distribution that produced them. An initial search for a value of kg was made by plotting th- error function of Eq. (6.14) vs. k for a range in k of [-0.05, 0.05]. All the were set to 1, and the initial estimate for the radial thickness polynomial was chosen to be P(o) = -0.05a14 + 0.038a2 + k' , (6.15) with the coefficients of the second and fourth order terms being roughly those of the fourth order best fit to the known radial thickness distri bution. There would in general be random or systematic errors from the estimate, but these will not be introduced until the next chapter. Figure 6.6 shows the plot of the error E (Eq. 6.14) as a function of k for the tenth order fits of the pair of two wavelength radial O.P.D. functions. Figure 6.7 shows a plot of the same function over the restricted range 104 so. II 11.05 -0.05 Relative Thickness (k) function of k over the range in k of Fig. 6.6. Error (in u) as a [-0.05, 0.05]. u- W -o.oi o.oi Relative Thickness (k) Fig. 6.7. Error function over k in the range [-0.01, 0.01]. 106 in k of [-0.01, 0.01]. There seems to be a cusp near the origin; Figure 6.8 shows a plot of the function for the k range of [0.01, 0.03] at the same scale as Figure 6.7; the curve either has cusps or places of very small radius. Because of the presence of multiple minima, a plot such as Figure 6.6 would be the best way to find the appropriate value for kpj an automatic routine could choose the wrong region. If the data is noisy, it may be necessary to perform a complete optimization in several regions of the curve, and to choose the refined answer that gives the best fit to the data. The best sixth order fit to the actual thickness distribution was P (a) = 0. 006958a6-0.060725o l4+0. 042551a2-0. 000017 , (6.16) but the best fit using the terms through tenth order for the two wave length radial O.P.D. functions was P = -0.000507o6-0.043444crL,+ 0.035876a2-0.004152 . (6.17) e The fit of P to P_ is no better than 0.0015 or about 8% of the total e T 0.019 range in thickness even if the zero order term (true=0) is neglected. For this reason, the fit to the two wavelength O.P.D.'s was recalculated through twelfth order; the coefficients of this new fit are given in Table 6.2. The terms are quite different in the higher orders, and the optimum fit polynomial for the thickness goes to P „ = 0.007331a6-0.061343a'4+0.042353a2+0.000204 . (6.18) e2 3. •H +•> U. UJ o.oi o.o: Relative Thickness (k) Fig. 6.8. Error function over k in the range [0.01, 0.03]. 108 Table 6.2. Radial polynomial fits for two wavelength O.P.D.'s. Coefficients in y Order (0.4880y-0.6328y) (0.5145y-0.6328y) 0 -0.00147 -0.00082 2 0.40404 0.19305 4 -0.5051S 1.25415 6 0.69707 -3.39970 8 -3.30350 0.11006 10 4.11622 3.49962 12 -1.49555 -1.56334 With or without the zero order term, this produces an error of about 0.0002 in the thickness estimate or about 1% of the total thickness range; this is an adequate estimate but the effects of errors will not be tested until the next chapter. One last comparison of using the tenth and twelfth order fits to the O.P.D.'s: the final value of the error function using the tenth order fit was 0.40, but with the twelfth order fit it was 0.042. This would seem to imply that a good fit gives an error in the polynomial fits for the O.P.D.'s of approximately two orders of magnitude less than the value of the largest coefficient; how well this may be done in practice will be tested in Chapter 7. CHAPTER 7 THE EFFECTS OF ERRORS ON THE ACCURACY OF ESTIMATION OF THE INFRARED WAVEFRONT FROM VISIBLE LIGHT INTERFEROGRAMS Because of the complexity of analyzing the effects of errors on the previously described measuring process, the most straightforward means of studying this is to produce a simulation of a coating including some of these errors, then observing the effects on a measurement. The types of errors to be simulated are: a) a rotationally symmetric coating thickness distribution due to the evaporation source's vapor distribution, b) what I call the monitor ratio, c) random errors in overall layer thick ness, d) an overall change in the optical properties of a layer, and e) random errors in measuring the wavefronts. The rotationally symmetric coating thickness (a) is assumed to result from having an evaporation source with a vapor distribution of the g form cos 109 110 7T (¥) d 6 t(r). f2^)V2 (7.1) 7T [r^+hz+Az+2rAcos0] 0 which is seen to equal Eq. (6.4) when 6=1. For arbitrary val; s of 8 this integral may have no closed-form solution, but it is simple to integrate numerically; using Simpson's rule with 41 points on the inter val [0,tt] was found to give values for t(r) that were not significantly different from those of the exact formula for 6=1 (Eq. 6.6). The accu racy seemed unnecessarily high, but 41 points were retained as a pre caution; the calculations used little computer time. I define the monitor ratio (error b) as the ratio of the true thickness attained in the center of the substrate to the estimated value. This would be dependent on the source distribution and on the type and location of the film thickness monitor, but here the ratio can be specified for each layer arbitrarily. The monitor ratio is a multiplier for the overall thickness of a layer. Similarly, layer thickness multipliers due to random monitoring errors may be included (c); these are like the monitor ratios but the monitor ratios are systematic. I assume that what Macleod (1977) calls indirect monitoring will be used, i.e., each layer is monitored on a separate substrate, which implies that the errors are statistically in dependent. The layers are assumed to be homogeneous, but the index of a given layer will be allowed to vary from a nominal value (error e). The Ill optical thickness (index x thickness) of a layer will be taken as constant at some single visible wavelength, or alternately (as in monitoring with a quartz crystal monitor) the mass thickness will be preserved; the index will be assumed to vary due to a change in the packing fraction, according to the expression from Kinosita and Nishibori (1969) n n + f = P m (1-P)ny , (7.2) where is the apparent index of the film, is the index of the bulk film material, n is the index of the contents of the voids in the v film, and p is the packing fraction. This expression has no basis in electromagnetic theory but has been shown to give good results for (l-p)«l. The behavior of real thin films is not simple enough to be described completely by the variables given above. The cos vapor dis tribution was found by Graper (1973) to allow a good fit to the vapor distribution of an electron beam heated source at moderate evaporation rates, but measurements by Anastasio (1963) of boat and basket sources indicate more complicated distributions. Even if the source distribution is well modeled by the cos form, the source axis may be inclined with respect to the perpendicular to the substrate plane (Richmond 1976), which will alter the thickness distribution. Macleod (1977) further describes the effects of substrate temperature nonuniformities: the condensation coefficient of evaporant molecules (sometimes known as the sticking coefficient) is a function of substrate temperature, and Macleod 112 cites data by Giinther (1957), Ritter and Hoffman (1969), and Warren to demonstrate this; heating of the substrate by the condensing material and by thermal radiation from the source can create substrate temperature nonuniformities sufficient to have an appreciable effect. A more mysterious effect has been observed for silver films by Bennett and Ashley (1973); they found thickness fluctuations as large as ± 0.4% in silver films 2000S (0.2 p) thick, with the films being deposited on 38.6 mm diameter substrates located approximately 500 mm from a dimple boat source. The system was maintained at less than 10 - 8 torr during the evaporation, there were baffles to allow only those silver atoms directly evaporated from the source to reach the substrate, and the evaporation rate was about 50 R/sec. The lateral scale of the non- uniformities was a few millimeters across the substrate; 5 mm at 500 mm is only about 0.006 milliradians (about 20 arc minutes). The effect was felt to be due to either very small scale fluctuations in the vapor streams or nucleation effects on the substrate. Clearly the data on thickness distributions in real evaporation systems is incomplete; my Q use of the cos form for the source distribution and allotting all of the nonuniformity in a given layer to vapor distribution effects is a convenient means of simulating the thickness distributions. Specifying different values of 3 for different layers of the same material allows changes of the source distribution in time to be simulated. The monitor ratio for each film is dependent on source vapor distribution as well as the monitoring method (e.g., optical or quartz crystal monitoring) used, and the location of the monitor in the chamber, 113 heating effects, and residual gases; in brief, on any parameter that effects film deposition. Rather than making a detailed model of the behavior of monitoring systems, I will assign ad hoc values to the monitor ratio. The random errors that result from nonsystematic errors in the monitoring (e.g., electrical noise) will also be assigned values that are not rigorously computed for a specific monitoring process. These errors will be assumed to be independent and Gaussian with a specified standard deviation for each type of layer; I believe that this is a reasonable assumption if the layers are monitored separately or by a quartz crystal monitor. The final error I decided to include in the simulation of coating deposition errors is an error in a specified layer index due to a change in the packing fraction. There are many factors other than this that cause index changes: the chemical composition of a film may vary, especially for such materials as TiO^ and SiO^ prepared by reactive evaporation (Glang, 1970). Dirks and Leamy (1977) review the columnar properties of thin films; the changes in structure of the films should be accompanied by index changes and possibly by changes in the bi refringence of the layers. The layers may also be inhomogeneous; anti- reflection coatings are particuarly sensitive to errors of this type, as seen in some measurements by Borgogno et al. (1981). Even the silver coating I assume as the base of an enhanced reflector can cause diffi culties; especially for very thin layers, the optical constants of silver vary with film thickness and deposition rate (Philip, 1960). 114 The basic claim I wish to make is that the means I have chosen to simulate the behavior of a coating give an adequate representation of the behavior of a real coating. I believe that the effects I have omitted would not be readily distinguished from those I have included if the technique previously described is used to estimate the wavefront performance of an infrared reflector from visible light measurements. I am not, after all, attempting to completely specify a real coating from three interferogr.: >s, but I am using a simplified model for a stack that I know will be in error from the true nature of the coatings; the idea is to estimate the wavefront error in the infrared to a better accuracy than an initial estimate based on knowledge of the probable coating thickness distributions and indices. Wavefront Measurements Bennett (1980) suggests that a reasonable tolerance for a mirror in an IR laser system would be about X/50, including figuring error, thermal distortions, and aberrations introduced by coating nonuniformities. An error of X/50 at 3.8 y is less than A/8 at 0.6528 p, which is not easy to achieve for a figuring error alone on a mirror several meters in diameter, so that the error introduced by the coating must be smaller than X/50 to allow the optician part of the error budget. If the allow able error introduced by coating nonuniformities is restricted to about one third the total error budget, then the X/150 error so given at 3.8 p corresponds to a X/25 error at 0.6328 p. The error produced by the coating doesn't scale in this fashion, as shown earlier, but the idea that I am 115 suggesting here is that the visible light wavefronts must be measured to extreme accuracies even considering the increased sensitivity due to the use of short wavelengths. A modern method of measuring wavefronts accurately is to use what Bruning (1978) calls a fringe scanning interferometer. A laser un equal path interferometer (LUPI) to be used in the fringe scanning mode is shown in Figure 7.1. The medium for recording an interferogram (often photographic film) is replaced by an array of detectors. A phase shift is introduced into the reference beam; one way is to displace the reference mirror as shown; each detector sees a signal whose amplitude varies sinusoidally with the change in the optical path of the reference beam, and the phase of the A.C. signal from the detector gives the optical phase of the wavefront being measured (1 period = 1 wave). I assume that the form of the detector array is simply a square array as shown in Figure 7.2. The measurement error for the detectors will be represented by Gaussian random variables, identically distributed for each detector but having the errors all statistically independent. I will assume that if other random errors such as air turbulence are present, then a sufficiently long time or large number of measurements of the wavefront are made so that the errors become uncorrelated. System atic errors in measuring the wavefront will be assumed to be nonexistent, although reducing systematic errors to as little as A/1000 should be difficult. If a least squares polynomial fit is to be done on the data, a convenient means is using orthogonal polynomials as was described for 116 Reference (displacement Mirror >-Beamsplitter Laser Convergin Lens ' Mirror Imaging under Lens test Detector Array Fig. 7.1. A laser unequal path interferometer (LUPl) for use as a fringe scanning interferometer. aeaaasBB BaaaoDoagdosaa oaaaaoaaaaDaoaaaao DBaDOaaODOODODOflOBOO ooooaoaaDDooaaoaoaoooa oaaaoaaaaoooaoaaooaooaao oaooDDaoaoaDaoaooeaooooooo aaaBBsaoDDDOooBOSDogoBBaaoaD aoaBBDoaaaaaaoaDoaaaaaaaooaa oaaaooaooaaaooaaaaaoBaaaaaaaaa BBDBBBOBBBBBBBBBBBOBOBBBBBBBBB DBBBBBaBBBBBBBBBBBBBBBDBBBBBBB DOQDOBnBBBODBQODBOaBBBBBBOOBBBBB BBBBBBBBBBBBBOBBBBBBBaBOBBBBBOaO BBBBBBBBBBBBBBBBBOBBB9BOBBBBBBBB oaoooooaoaoaaooaooooaaoaaaoooooB BBBBBBBBBBBBBBBBBaaaBBBBBBBaBBBB BaBBOaBBBBBaaaBOeaBBBBBBOBaBBBao BBBBBBBBBBBBBBBfiBBBBaBBOBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBaBBBBBBB BBBBBOBBBBBBBBBBBBBBBBOBBOBBBB BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB OBBBBBBBBBBBBBBBBBBOBBBBOBOBBB BBOBBBBBOOODBBBBOBBBBBBBDBBB BBBBBBBBBflBBBBBBBBBBflBBBflBaB OBBOBBBOBBBBBBBBBBBBOBBBBO BBBBBBBBBBBBBBBBBBBBBBBB OBBBBBBBBBBBBBBBBBOBBB BBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBKOB BBBBBBOBBBBBOG BOBBBBOO 118 a one-dimensional problem in Chapter 5. Assume that the data is over a set of N points {p }, p = fx ,y ). Assume also that we have polynomials r rm rm m m r in (x,y) that are orthogonal on {p K that is, 1 N y p.(p )p.(P ) = 6.. , (7.3) N i rm j rm ij m where 6.. is the Kronecker delta. ij The coefficient for each polynomial fit to a data set is deter mined as (let j be the order of the polynomial term) 1 N a = ,v • v IL (pVJ )P. (p r ) , (7.4)v 1 h m=l, m l m where IV(Pm) is the value of the wavefront at the point p . The poly nomial estimate of the wavefront through order M is then M P.,(p ) = J" a. P. (p ) . (7.5) M rm j J m Let the measurement of W at each point p have an error z and let all rm m the e be identically distributed zero mean random variables, each with m ' 2 variance o ; then the variance of a. is defined E J a? = r = ({i • N m? I [WCp ) +e ][W(P ) + e ]P.(p )P-(p ) , m mJ1 rn nJ j rm j rn m= l ' n=l n —P J ('wCp )W(P )+ E W(p )+ £ W(p )+E E )P.(p )P.(p \ m mi m Kn n Mn m n / j Mn j r:J n=l - o .1 p a + 0 j- NM £ > where cu is the true (or mean) value of , so that o? = ^ a2 . (7.7) J N E p In Eq. (7.5), the zero order term ( 0=l) is included; in practice it is usually unknown. Define the statistical part of the wavefront variance as °»• * iiKUU v-']2) • (ji°s pitp-,v2 (7.8) M \ 2 r M -j: W> • r m 12 y a. P.(p ) (7.9) Lj=i j J m J 120 p a P (p DP, (p ) ([jl "i V »>] )" JI1< J°k> J um k rm y ' ' k=l = J" [a.a, + a?6., ]P.(p )P, (p ) •_2 J ^ j jk j m k rm k=l f" M T2 M y a. P.(p ) + I a? P? Lj=i 3 3 m J j=i j j (7.10) Eq. (7.8) then becomes N M °IV = I \ J. °i Pi (Pm} ; (7"11} m=l i=l if the (m) summation is performed first, then by Eq. (7.5): M o.f, = I o2 ; (7.12) J I W m=l but from Eq. (7.7), all the ck are identical, so that (7"13) This is the statistical variance of the wavefront given by the poly nomial fit; in practice, we would measure only the variance of the fit from the noisy data. The expected value of this is p 2 14 £ <[»CPu>-„ " j0«3 j The zero order term is included because it is needed to complete the fit to the measured wavefront; the zero order term is discarded in practice. Evaluate now M ([W(p )+e - I a. P . (p )] V m m JZQ J J ni J M W2(p ) - 2 W(p ) I a. P.(p ) rm vrm L i j rm j=0, n J N + a2 - 2 /£ y a.P.(p] e \ m j j m' <[1VJVT (7.15) E y a. P.(p )\= /E J y [W(p )+E ]P.(p )P.(p ) r n m j=0 j j m / N n=l ^ - i M i I a2 P2(p ) N e j=0 J a 2 M I p2(P ) (7.16) N >0 J 12: / r M "12 \ f M "12 M \ I a.P.(p) )= I £ a.P.(p)| + £ a? p2(p ). J J m j m J m \Lj=o J / Lj=o j J j=0 ^ (7.17) Then use Eqs. (7.15 - 7.17) to find that /fw(p )+ e - 7 a. P.(p )1 m m 1 3 m \L j=0 - J M W2(p ) - 2 W(p ) y a. P-. (p ) rm j 1 m 2 2 2a M r M *12 + o I P? (p ) + I a. P.(p ) £ N . , ... . . j=0ton - m LjLi=0 -1 -1 mJ M £ a? P? (p ) j =0 3 3 m [IV(p ) a P.(p )]' L r - Ly r J m ~\, =0n Ja l m ? M + [1 - 4- I P?(p )] a N . L i m e j=0n - M + I a? Pf CPm) • (7-18) j = 0 Now doing the summations and employing (7.3) and (7.7), Eq. (7.14) becomes 123 M °f = I"(V> - z s) v.'] 4 • (7'19) j=0 J J This is seen to consist of two parts: the deterministic and the random part. For real data, only the total variance expressed by (7.19) would be known; however, Eq. (7.14) shows that if M< A reasonable test for finding when the polynomial fit has the optimum number of terms is to stop when adding a term causes the total variance to decrease only about ^ a2, assuming o2 is known. In the simulation, the deterministic and random contributions to the total variance ap2 can be exactly determined; this might prove to be an acceptable means of estimating how many terms should be used in a polynomial fit to real data that is believed to be similar to the simulation. The Need for a New Error Function If the number of points used in an arrangement shown in Figure 7.2 becomes very large, the orthonormalitv condition (Eq. 7.3) can be seen to approach the defining integral for the Zernike polynomials (Bom and Wolf, 1975), which may be given 1 2 TT r 2tt6 P.(r,0) P (r,6)r dr d6 . (7.20) J * 0 0 The radial polynomials that fulfill this are given in Table 7.1; they differ from those cited above by the type of normalization. The main 124 Table 7.1. Radial Zernike Polynomials (orthonormal). Order Form 0 1 2 2/J (2r2-1) 4 2>/5 (6r4-6r2+l) 6 2/70 (20re-30r1|+ 12r2-l) 8 6/35 (70r8-140r2+ 90 l4-20r2 + l) 10 6/154 (252r1°-650r 8+560r6-210rL,+ 50r2-1) 12 2/6006 (924r12-2772r10+3150re-1680r6+420rl4-42r2 + l) feature of these is that their coefficients alternate in sign and achieve very large values; the error function defined in the preceding chapter would be very sensitive, to small errors in the a... A less pathological error function is a simple least squares one of the form Q / M j 2 E = ^ IL (Pr )-N-(Pr )1J " I C°t, ,-a_.)P.(p vt )> 1 m=l, 11 1 nr m .lji=l J i m )i Q ( M )2 + U^->(P -(P )] . -cx_ .)P. (p )> 2 I iL 2 *m)~N j> rm J - .I v(ot_ 2i 3j rm m=l" 3=1 • J J3 '| (7.21) where h' (p ), W_ (p ), and IV,(p ) are calculated wavefronts to go with a 1 rm 2 rm 3 rnr b set of parameters for wavelengths 1, 2, and 3 (e.g., 0.4880 p, 0.5145 y, and 0.6328 p), and a,., a„., and a,. are wavefront coefficients at those lj 2j 3j wavelengths; and are nonnegative weights. The use of such an error function would allow use of a more advanced sort of optimization routine than in the preceding chapters; for simplicity, the search algorithm will still be used. Choice of a Deposition Geometry The offset source arrangement (Figure 6.1) is examined again to see if it gives acceptable thickness uniformity. Calculations to test for uniformity are done with a modified form of Eq. (7.1): (¥) _A_ \2 [(^ ro d e t(a) a2+(— \ + 2a^— I cosf(¥) "(i)'' ro (7.22) where a = (—] • Figure 7.5 shows the radial thickness distributions for /h\ A I — 1 = 2.5,W the source having 6=2 (cos2 shows the curves for(—) = 2.5, 6=2 and /—] = 1.7, and for 6=1 and /4\ w (-)=,8S, the latter giving a peak error in thickness of about 0.0041 (0.41%). Figure 7.6 shows the changes in these curves caused by keeping the same source locations but changing to 6=2.3 and 6=1.1. I choose,to regard these as typical errors in the estimates of the source vapor dis tribution. 126 0.05 (1.9) tr, (1.8) (A o c AS o • r-l •e E— o.o (1.7) V 1.0 > o a. Normalized Radius (a) (1.6) -o.os (1.5) Fig. 7.3. Layer thickness uniformity for an offset source deposition scheme with a cos- source distribution. (h/rQ) = 2.5 and (A/rQ) = 1.5, 1.6, 1.7, 1.8, and 1.9. 127 0.05 in in G> C U •H (1.9) o > +-< Normalized Radius (a) (1.6) -0.05 Fig. 7.4. Layer thickness uniformity with a cos1 distribution. (h/r0) = 2.5 and (i/rQ) = 1.6, 1.7, 1.8, 1.9, and 2.0. 0.007 -o.ooi Normalized Radius (a) Fig. 7.5. Layer thickness uniformity with two source distributions. (h/r ) = 2.5 for both curves. The upper curve is for a 2 cos source and (A/r0) = 1.7, and the lower curve is for a cos1 source and (A/FQ) = 1-88. 129 (1.01 o.o l.o Normalized Radius (o) Fig. 7.6. Layer thickness uniformity for a cos2,3 and a cos1-1 source. Identical to Fig. 7.5 except for the source distributions. New Optimization Variables The optimization procedure described in Chapter 6 was suitable based on the assumption that the thickness distributions for the two kinds of materials are nearly the same. This is likely to be an un realistic assumption, and in fact the simulation will include radial distributions of the type shown in Fig. 7.5. The model I use for optimization here is one where radial polynomials are specified inde pendently for the two different layers; this is a step closer to what the real stack would be, and, unfortunately, increases the number of variables in the optimization. It would also be desirable to search for a base thickness for the layers of each type of material; call these relative thickness multipliers k^ and k^ for the high and low index materials. The plot for a single base thickness required many points (n) to be calculated; there are 101 points plotted in Fig. 6.6. Ex- 2 tending this to two variables would require n points to be calculated 4 (e.g., 10 ), which is generally an unattractively large number (expensive to compute). For this reason the values of the monitor ratios for the layers are restricted to lie within 1% of unity (i.e., in the range [0.99, 1.01]). With the offset source geometry and (—\ = 2.5,(—) = r r / \ 0/ \ o 1.7, with the monitor on the source axis, then assuming that the monitor ratio is 1.00 for a 6=2.0 source distribution gives a monitor ratio of 0.945 if B is changed to 2.3 for the true source distribution; this would be a poor location for placing the monitor, so it is likely that controlling the monitor ratio to within 1% is a reasonable hope if the source distribution changes as above. 151 An Example: Attainable Accuracy The number of variations of coating thicknesses that may be simulated using only the limited set of parameters described is great; I will choose one basic design in the hope that it will be representative, and run the tests with it. The coating to be examined is the six layer enhanced silver reflector with a low index matching layer on the silver followed by a quarter wave stack; the layer thickness prescription is given in Table 5.1. Table 7.2 lists the prescription again, and gives the values of the monitor ratios and 6 for each layer (the monitor ratios could have each been set to 1.0 and the changes incorporated in the prescription). The random thickness errors are taken to have an RMS of 0.05 of a quarter wave at 0.6528 y for each material used, or 0.002 y for the ZnS and 0.005 y for the ThF^. Different random parts in different runs can be produced by changing the seed value of the pseudorandom number generator used. Three runs with no wavefront measuring errors (a£=0) were made for the system described in Table 7.2; Figure 7.7 shows the two wavelength O.P.D. for (0.4880 - 0.6528 y) as a function of radius on the substrate for the three runs, and Figure 7.8 shows the similar curves for (0.5145 - 0.6528 y) at the same scale. Figure 7.9 shows the optical path produced by the mirror at 5.8 y for all three cases; the curves overlie each other. A basic difficulty was found with the use of the error function (Eq. 7.21); while the radial thickness distributions shown in Figure 7.5 and 7.6 can be fit well by fourth order polynomials, if these fourth order approximations are used to compute the pair of two-wavelength 132 Table 7.2. Non-random parameters of the coating simulations. Layer Number Material Nominal Monitor Cosine (From Sub Thickness Ratio Exponent strate) 00 (B) 1 ThF4 0..6115 1..000 1.,00 2 ZnS 0..4216 1,.000 2,,00 3 ThF4 0.6333 0.,995 1.,08 4 ZnS 0..4216 0.,998 2 10 5 ThF4 0.6355 0. 990 1. 15 6 ZnS 0. 4216 0. 992 2 ,20_ All layers deposited with (—)= 2.5; low index lavers with ( — VV ' \rn / A \ high index with I — 1 =1.7. 0.08 -A 3 QL. -C -o.oi Normalized Radius fol Fig. 7.7. Two wavelength radial O.P.D.'s for three simulations at (0.4880 - 0.6328 y). 0.08 A- -o.oi Normalized Radius (o) Fig. 7.8. Two wavelength radial O.P.D.'s for the same simulations as Fig. 7.7 but (0.5145 - 0.6328 0.03 (A, B,C) - 4-> W CL •H +J o 0 0.5 1.0 Normalized Radius (o) Fig. 7.9. Radial optical paths at 5.8 p for the simulations in Fig. 7.7. The three curves nearly coincide. 156 radial OPD functions, the error function will be large even if the only errors present are those radial thickness distributions that gave the fourth order polynomials; i.e., there is a poor relationship between the value of the error function and the coefficients of the fourth order polynomial descriptions of the radial thickness profiles of the layers. The error function can be made small by using an eighth or higher order approximation to the radial thickness profiles; however, attempts to extend the optimization to high order either did not converge or con verged too slowly to be of use. The optimization through fourth order gave the best results, which were achieved when the polynomial approxi mations to the pair of two wavelength O.P.D.'s were limited to eighth order. The Initial Estimates A simulation was run using the coating of Table 7.2 with all the random errors set to zero. The optical path at 3.8 p calculated for the simulation was compared to that for the best fourth order fits to the radial profiles for 6=1.0 (low index layers) and 6=2.0 (high index layers). For 6= 1.0,(—) = 2.5, and (—\ = 1.88 (Fig. 7.5) the fourth \r0 / \ro/ order fit to the radial profile was ' P (a) = -0.016779a4 + 0.016452a2 + 0.000055, (7.22) where a is the normalized radius, and all the coefficients are unitless (i.e., relative thickness). The similar approximation for the high index layers with 6 = 2.0,(—) = 2.5, and (—)= 1-7 was vr0' vV P (o) = -0.027858O4 + 0.027518a2 + 0.000137 (7.25) Hu The total RMS departure of the wavefront from the plane W=0 (i.e., the zero aberration plane, not the best-fit plane) is .0177 y (0.0047X or X/215 at 3.8 y); the RMS error in the estimate is .0048 y, or 0.27 of the total range. The optimization routine may give better results than this. Another bit of information is obtainable initially; the toleranc on measuring the wavefront. While not explicitly mentioned before, the simulation was run on an array like that pictured in Fig. 7.2, which rep sents a 32 by 32 square array masked to a circle, which contains 812 total points out of the 1024 in the original square. The RMS residuals of the fits with eighth order polynomials to the pair of two wavelength radial O.P.D. functions were 7.48x10 ^ y and 1.23x10 ^ y for (0.4880 - 0.6328 y) and (0.5145 - 0.6328 y). The desired wavefront error for the better of these is 5 a < 7.48xl0~ y (812)** ^ (? 24) E [(0.5145y)2+(0.6328y)2]^ which follows from the variance in optical path given by a sum or dif ference of two wavefronts at wavelengths Xj and X2, each with the same error in waves: °total = (X12 + ^ at ' (7"25} and from Eq. (7.7). The RMS wavefront. error should, by (7.24), be less than 0.0026X at the measuring wavelengths; by the optician's rule of 138 thumb that the peak-to-valley wavefront departure is 5 times the RMS, then this would be the same as measuring a wavefront to 0.013X, or a very tight specification. This gives the idea that using the eighth order fits to the two wavelength radial O.P.D.'s may be all that need be done even with a state-of-the-art fringe scanning interferometer. A Demonstration of the Effects of Wavefront Errors While the optimization has not worked as well as was hoped, 1 can still demonstrate some of the effects of random errors in measuring the wavefronts on the accuracy of the estimate for the wavefront at the IR wavelength. The basic procedure was as follows: the non-random parts of the description of the coating were taken from Table 7.2; the RMS base thick ness errors were given RMS values of 0.002 y and 0.003 p for the ZnS and ThF^ layers. Then three runs were,made using different pseudorandom number generator seeds for each of a range of wavefront errors )• The number of points in the assumed detector array (Fig. 7.2) was again 812. A fixed number of iterations of the optimization routine was used; the number chosen (90) should have been large enough to assure that little additional reduction in the error function would occur with more itera tions-using a fourth order approximation to the layer thickness profiles. The results of the set of trials is shown in Figure 7.10. The number of points is small, but it must be remembered that each one took a good deal of computer time (approximately 15 sec. on a CDC Cyber 175) to produce using the optimization procedure. A dotted line indicates 139 0.01 D *K X rx X • X rx (0.) 0.002 0.005 0.01 0.02 0.05 0.1 RMS Error in Wavefront Measurement at Each Detector for Each Wavelength (waves) 7.10. RMS errors (in p) of estimations of the IR performance of a coating from measurements at three visible wavelengths as a function of errors in wavefront measurement. The dotted line is the error given by the initial estimate, crosses (x) are data, and squares are RMS values from three data points. 140 the RMS error for the initial estimate; the crosses are the data points, and the squares are combined RMS values for the three data points for each value of RMS wavefront error. The three points for 0^=0.01 almost coincide. I believe that the dip at a^=0.01 results from this being the approximate location where the errors due to the wavefront measure ments are essentially equal to the errors caused by the error function (Eq. 7.21) that gives false results for the fourth order approximation. When the wavefront error is small, the ultimate value of the error function is small but the solution may be a worse fit to the actual thickness profiles than the fit that goes with a larger value of the error function. For all cases where a£ 0.02A, the optimization pro duced estimates of the thickness profiles that were better than the initial estimate, but not always as much better as might be hoped. The Effects of an Index Change The effects of an error in index in the layers on the accuracy of the estimation of the IR performance of the coating has been tested by running several simulations with index errors present. The non-random parts of the simulation are again described by the values of Table 7.2; the RMS errors in the high (ZnS) and low (ThF^) index layers' thicknesses were again chosen to be 0.002 y and 0.005 u, and the wavefront measurement RMS error was taken as 0.01X, which gave the most consistent results among a small set of data (3 points) in Fig. 7.10. The index error was assumed to arise because of a change in packing fraction from 1.0 to 0.90. The resulting voids in the layers were assumed to be filled with air of index 1.0; the voids would be 141 likely to be at least partially filled with water (cf. Macleod 1977) but I prefer to avoid discussion of the effects of moisture adsorption on a multilayer intended for use in the IR, where water is highly optically absorbing. The index change is greater if the voids contain only air. Under these assumptions, the index of the ThF^ layer changes from 1.50 to 1.45 at all wavelengths because the ThF^ is assumed to be nondis- persing. The original indices assumed for ZnS at 0.4880 y, 0.5145 y, 0.6328 y and 3.8 y were from Pelletier (1970): 2.4142, 2.3893, 2.3256, and 2.2535; these values become 2.2728, 2.2504, 2.1930, and 2.1282 with the packing fraction change. Three simulations were performed; (1) both high and low indices changed, (2) high index only changed, and (3) low index only changed. The optical thickness was kept constant for the prescription of each layer, which meant that the physical thicknesses changed, in an attempt to distinguish between the effects of layer thickness change and layer -3 index change. The RMS error in case (1) was 1.41x10 y, for case (2) -4 -3 the error was 7.67x10 y, and for case (3) the RMS was 3.16x10 y, to be _ 2 compared to the total RMS wavefront error of about 1.8x10 %. Another trial with only the low index changed, similar to case (3), was per- » formed; call it case (3b). The repeat of case (3) had an RMS error of _ 3 2.41x10 y. These results are presented in more concise form in Table 7.3. The results for cases (3) and (3b) suggest that the estimation procedure is sensitive to the index of the ThF^ layer to a much greater degree than to the index of the ZnS layer; however, the data is sparse and the values of the RMS errors observed were little larger than the 142 Table 7.3. Errors in estimation caused by refractive index changes. Trial Number Packing Fraction RMS Error (p) in Estimate Becomes 0.9 of TR Wavefront Performance ZnS ThF. 4 1 X X 1..41 X 10"3 1 o 2 X 7..67 X t—• 3 X 3..16 X lo"3 3b X 2.,41 X ID'3 _ ^ The total RMS of the wavefront was approximately 1.8x10 ~p in all cases. fluctuations in Fig. 7.10. It does seem that the phase behavior of the enhanced reflector would be most strongly dependent on the properties of the low index matching layer, so perhaps the sensitivity indicated above is a real effect. To be certain, a more effective means of estimating the IR performance than that described above should be developed; the errors that the current optimization allow are too large to make more detailed study of the errors worth while. CHAPTER 8 SUMMARY AND CONCLUSIONS The main driving force in this study has been the need to control the small aberrations that multilayer enhanced reflectors introduce in high energy laser optical systems. Chapter 3 involved calculations of coating induced aberrations produced with strongly curved mirrors that were entirely due to the changes in angle of incidence that the coating on such a mirror experiences. The aberrations resulting from the use of an enhanced reflector coating on an f/1.5 parabola used as a collimator are small, only A/400 at 3.8 p, and most of the error is in the form of a focus shift. The aberrations for a plane wave incident on a 90° cone are little larger, giving only about A/300 for an enhanced silver coating. Larger errors can appear with the use of more unusual coatings, but for high energy laser mirrors the aber rations due to the angle of incidence effects with multilayer coatings are small. Also, a large short radius mirror would typically undergo optical tests at its center of curvature. The departure of even a very fast f/1.5 parabola from a sphere was found to be too small to give angle of incidence effects that would be measurable on a mirror tested at its center of curvature, e.g., an f/1.5 parabolic mirror with a coating that gave A/13 of aberration when illuminated with a plane wave had less than A/50,000 error when calculated at the center of curvature. Because this 143 144 error was so tiny, it is safe to assume that the only measurable aberra tions on a mirror that is suitable for testing at its center of curvature would be due to errors in the coating rather than angle of incidence effects. For the entire class of mirrors that may be tested at small angles of incidence, calculations may be restricted to flat mirrors at normal incidence without loss of generality. An introduction to the effects of what Giacomo (1958) calls a proportional thickness change was made in Chapter 4. This sort of error occurs when the ratios of the thicknesses of the layers are fixed, but the total thickness of a thin film stack is allowed to vary over the coated surface. Taking the difference between wavefronts measured at two wavelengths was found to be a means of discriminating between errors due to misalignment, or deformation of the coating substrate, and aberrations produced by the coating. However, measurements at only two wavelengths were found to not allow unique determination of a coating thickness distribution from interferograms of the mirror at the two wavelengths because the two wavelength optical path difference function for a simple enhanced reflector coating was not monotonically increasing or decreasing, i.e., an inverse of this function was multi-valued. A modified coating design that did not suffer from this diffi culty over a 5% range in film thickness was described in Chapter 5. If the proportional thickness errors in the coating were the only errors present, then such a coating design would allow precise estimation of the wavefront errors produced by the coating of a mirror for 3.8 y from a pair of interferograms taken using the visible laser lines at 0.6528 y 145 and 0.5145 p. However, the actual stack thickness must be known some where on the mirror surface or the estimate of the IR wavefront will be in error; this difficulty makes the use of such a coating less attractive. Chapter 6 introduces the use of a third visible wavelength which would allow determination of an IR wavefront from three visible light interferograms of an IR enhanced reflector. If the wavefront errors are due only to proportional thickness changi in the coatings then, in principle, if the basic coating prescription is known, the IR wavefront can be precisely determined from the three interferograms. Even this determination has uniqueness problems at a few discrete points but the choice of the correct value of stack thickness should be straightforward for a continuous coating. The technique described in Chapter 6 for finding the film thickness distribution on the substrate (and therefore the IR wavefront) is one of finding, by means of an optimization proce dure, a radial polynomial description of the thickness distribution that gives a good match to the wavefronts measured in the visible. Only circularly symmetric thickness distributions were allowed, so that the computer time needed for the optimization runs would be fairly short. In the absence of errors other than proportional stack thickness errors this approach was found to allow accurate estimation of IR wavefronts from sets of computed wavefront polynomials in visible light for some simple coating thickness distributions. Real coatings are much more complex than those described in Chapters 3 through 6. Chapter 7 involves a more detailed but still crude representation of the coatings that might be expected to be 146 deposited in a real vacuum system, including the effects of different vapor source distributions for the two dielectric materials used to enhance the reflectance of a silver coating. Fringe scanning interfero- metrv at the three visible wavelengths was recommended to obtain the very high measurement precision needed, and the errors in that technique were discussed. An optimization technique was used again to find the radial thickness distributions for the layers of each dielectric material. Each layer of a given material was assumed for the purposes of the optimization to have the same sort of thickness distribution, but the two dielectric materials were allowed to have different distributions. The means of optimization was not completely satisfactory because the error function used was found to be inappropriate in judging when the desired values for the layer thicknesses were being approached. However, in the single type of coating studied at any length, the optimization technique did give estimates of the IR wavefront errors with an RMS error of not worse than 3.10 ^ y or about 1/6 of the total wavefront _ ~y RMS of about 1.8x10 % that resulted from the choice of coating errors. The difficulties with the optimization resulted in errors that were at least as large as the errors in the estimates of the IR wavefront given by errors in measuring the visible wavefronts if the RMS error for each point in an array of 812 detectors used to sample the fringes was <_ 0.02 (A/50) waves at each wavelength. The effect on the estimation procedure caused by errors in know ledge of the coating indices was also tested, if briefly, in Chapter 7. Index changes in either or both dielectric materials due to a reduction 147 of the packing fraction from 1.0 to 0.9 was found to have little effect, although there may be a higher sensitivity to changes in the low index material. Additional Comments and Suggestions for Further Work The most obvious need left unfulfilled by this study is that for physical data to support all these calculations. However, the facilities to perform these experiments do not exist at the time of this writing at the Optical Sciences Center. I am unaware of the existence of any apparatus to perform the precision IR interferometry the experiments would require. The final estimation procedure described in Chapter 7 was basi cally unsatisfactory; I find that an optimisation is not an attractive means of inferring data from other data. Numerical analysis is often useful, but I find it to be a poor substitute for insight. The fact that an optimization would be used limited the treat ment in Chapter 7 to one dimensional (radial) functions only; this was not too severe a restriction because the most practical means of obtain ing good layer thickness uniformity would produce nearly rotationally symmetric coatings. The approach used could be readily extended to general two dimensional coating thickness distributions, although I doubt that the problems with the current scheme would make it worth the computing expense. Regardless of what technique is used to qualify the optical performance of infrared mirrors, there must be a strong emphasis on 148 control of the deposition process for the coatings. Ramsay, Netterfield, and Mugridge (1974) have attained coating nonuniformities of 0.1% on 200 nm diameter substrates; it would be well if this could be extended to 2m or 4m substrates, but I expect that the degree of difficulty in creases for coating a piece as it does for optical figuring. In the course of early discussions of the effects in this study 1 received similar suggestions from Profs. R. V. Shack and H. A. Macleod involving the possible uses of deliberately nonuniform optical coatings in optical engineering. One such use would be as a coating for the primary mirror of a large Schmidt camera that compensates for the chromatic variation of spherical aberration (spherochromatism) caused by the corrector plate. I don't suggest that this will be easy or practical, but it would at least be a novel use of dielectric coatings. Figure 8.1 shows the two wavelength O.P.D. for 0.5145 - 0.6328 p for the 35 layer broadband coating of Heavens and Liddell (1966). The total O.P.D. be tween the two wavefronts is on the order of one wave in the visible for the 20% thickness range shown, which would be of the correct order for the aspherochromatism produced in a 50 cm aperture f/3 Schmidt. Thirty- five is an improbably large number of layers for a coating of a large mirror, but on the other hand, this coating was not designed to give large phase shifts. I think it may be possible to design a coating with a more reasonable number of layers that could be an element of the optical de sign, but perhaps it would be too odd to an attractive optical engineering tool. 149 Relative Thickness Change (k) o 4-> BO > Q rt • 2 Cl. o 6 Fig. 8.1. Two wavelength O.P.D. (0.5145 - 0.6328 p) vs. stack thick ness (k) for the 35 layer coating of Heavens and Liddell (1966). APPENDIX A A COATING DESIGN OF GUHA, SCOTT AND SOUTHWELL An alternate approach to the problem of estimating the IR per formance of a high reflectance coating from measurements in the visible was used by Guha, Scott, and Southwell (1980). The means they chose was to modify a quarter wave design such as the ones described in Chapter 2 to produce a phase behavior at a visible wavelength (the Krypton 0.6471 y line) identical to the performance in the IR (3.8 y) over some range of proportional stack thickness change (k in Chapter 4). Table A.l is a version of their eight-layer design, with the initial quarter wave design included as well, using my numbering convention. The coating is intended for use at a 16° angle of incidence; the substrate is silver with optical constants (n,k) = (0.07, -4.17) at 0.6471 y and (1.89, -28.7) at 3.8 y, and the low and high index layers are ThF^ and ZnS with indices of refrac tion assumed to be 1.50 and 2.25 at both wavelengths. Figure A.l shows that the desired effect is achieved over the ±1% thickness range as claimed; the curves shown are for s polarized light, and both curves have a constant phase subtracted so they both pass through the origin. Figure A.2 compares s and p polarizations for 0.6471 y light; the curves are fundamentally similar but the p polarization curve moves away from the line for smaller values of k near the origin than the s polarization curve. The optical paths shown in Figure A.2 are absolute with a constant (tt) subtracted before scaling to length units; the phase shift between s 151 Table A.l. Layer thicknesses for a design of Guha et al. (1980). Layer Number Layer Thickness, Layer Thickness, (From Substrate) Quarter Wave Modified Design Design (y) (y) 1 0 .644 0,.665 2 0.425 0.410 3 0,.644 0,.574 4 0,.425 0..481 5 0..644 0.,573 6 0,.425 0..482 7 0,.644 0..602 8 0..425 0.,495 152 c. c. -0.05 Relative 0.05 Thickness Change (k) Fig. A.l. Optical paths (in y) vs. relative stack thickness for the design of Guha et al. (1980). Both curves are for s polarization; the line is for 3.8 y, the curve is for 0.6471 p. 153 3. •P -o.os Relative .05 S- Thickness 00 Fig. A.2. Absolute optical paths at 0.6471 y for s and p polarizations. and p polarization for k=0 is only +0.0007 y (A/900) at 0.6471 y, with the (+) sign indicating that s polarization lags p following the sign convention of Eq. (2.1). This sign convention also gives the opposite slopes to these plots as compared to Guha et al. Figure A.3 gives the similar two curves for 3.8 y; the s-p phase shift at k=0 is +0.009 y (X/400), also a small value. The slopes of the lines that best fit the curves of Fig. A.3 are 5.586 y and 5.326 y for the s and p polarizations, which means that the phase shift between the two polarizations becomes no larger than +0.0115 y (X/330) over the ±1% range in k specified for the modified design. A crude tolerance analysis was performed in the manner of Ritchie as described on P. 231 ff. of Macleod (1969): independent random thick ness errors are included for each layer. As the layers' thicknesses have no simple relationship, the layers will be assumed to be monitored separately, so that independence is a reasonable assumption. Figure A.4 shows five curves, on the same plot, of the optical path as a function of k in the range ±1%; the RMS errors in the layer thicknesses were all about 0.06 quarter waves at 0.6471 y, or 1% of the layer thicknesses. With these random errors in thickness, the design still behaves acceptably. Figure A.5 is a similar plot when the RMS thickness errors are twice those in Figure A.4 or abour 2% of the layer thicknesses, with a different vertical scale than Figure A.4. A 1% thickness tolerance appears accepta ble; a 2% tolerance clearly is not. A final test was performed to check the practicality of the de sign of Guha et al. (1980); there are two dielectric materials used, and the source distributions for each would be at least slightly different. 155 a. G. s- -0.05 0.05 Relative Thickness (k) Fig. A.3. Absolute optical paths at 3.8 p for s and p polarizations. 156 Relative Thickness Change (k) Fig. A.4. Performance of the design of Guha et al. (1980) with approximately 1% thickness errors. o. 0.01 Relative Thickness "N^Change (k) A.5. Performance of the design of Guha et al. (1980) with approximately 2% thickness errors. 158 Using the methods of Chapter 7, I assume that the layers are deposited on a flat substrate using the offset source geometry (Fig. 6.1) with ^~^ = 2.5, = 1-88, and 3=1.1 for the ZnS and 3=1.0 for the ThF^. No other errors were assumed to be present. Figure A.6 shows the radial optical path functions for the modified coating with these parameters; the error in the wavefront at 0.6471 y compared to the true 3.8 y one is only -0.0018 y (X/2000 at 3.8 y) out of a total path excursion of 0.0265 y (X/140 at 3.8 y). This should be an acceptable error, but it is clear that if the coating is to fulfill its stated purpose, the coating techni cian must have excellent control over the coating process. 159 0.03 0.02 a. o.oi 0.5 i.o Normalized Radius (o) Fig. A.6. Radial optical paths vs. normalized radius (a) at 3.8 y and 0.6471 y. APPENDIX B AN INFRARED ENHANCED REFLECTOR WITH REDUCED IVAVEFRONT ERRORS DUE TO PROPORTIONAL THICKNESS CHANGES One means of reducing the uniformity requirements on an infrared enhanced reflector is to design a coating that gives a much smaller wave- front error for a given change in the proportional stack thickness (k). Ciddor (1968) attempted this for an all dielectric coating for Fabry- Perot interferometry; he faced a harder task than the one described here because the Fabry-Perot coating was to be broadband, while the enhanced reflector need only work near 3.8 y. The approach used here is to take a basic quarter wave design with a silver substrate, ThF4 matching layer, and ZnS and ThF4 quarter wave stack, and then to modify the design using an optimization technique. The optimization procedure was that of Nelder and Mead (1965) as des cribed in Chapter 5; the error function was of the form E = ft max |W(0) - W(k)| + fi max [1.0 - R(k)] , (B.l) k k where W is the wavefront as a function of the stack thickness multiplier k, R is the reflectance as a function of k, and and are nonnegative weights. 160 161 The initial design was chosen to have 14 layers because the modi fication can be expected to reduce the overall reflectance; the theoreti cal reflectance of such a coating is 0.999969, although deposition problems would limit this in practice. Table B.l gives the layer thick ness prescriptions for the initial quarter wave design and the modified design. All 14 layers were used in the optimization at once, and the range chosen for k was [-0.01, 0.01], or a 2% total thickness range. Figure B.l shows the performance of the quarter wave design and the modified design as a function of k. It would be more fair to compare the modified design to a quarter wave design of similar reflectance, or about 0.99976 minimum over the ±1% thickness range, such as a 10 layer coating; the slope of the line for the quarter wave design is 12.157 y; a six layer coating gives a slope (Eq. 4.4) of 3.979 y, with much of the difference due to the 2.3 ratio in total stack thickness. A maximum error of -0.0063 y (X/600) is reached over the ±1% thickness range for the modified coating, or roughly 1/6 that of even the six layer quarter wave design. Figure B.2 shows the optical path behavior of the modified design at 0.6328 y. The horizontal scale is the same as Fig. B.l, but the vertical scale includes larger values of the optical path. The advantages for alignment given by the design of Guha et al. (1980) are not present in this design, but the need for coating uniformity is much reduced. Figure B.3 indicates the sensitivity of the design to manufacturing tolerances similarly to the curves in Appendix A; random layer thickness fluctuations with RMS values of 0.004 y and 0.006 y for the ZnS and ThF^ 162 Table B.l. Thickness for two 14 layer enhanced reflector coatings: a quarter wave design and a modified design. Layer Number Q.W. Design Modified Design (From Substrate) Thicknesses (y) Thicknesses (n) 1 0.611 0.424 2 0.422 0 .435 3 0.633 0.743 4 0.422 0,.486 5 0.633 0.555 6 0.422 0 .:25 7 0.633 0,,380 8 0.422 0,.517 9 0.633 0,.498 10 0.422 0.447 11 0.633 0.,644 12 0.422 0..368 13 0.633 0. 768 14 0.422 0. 758 163 a. Relative Thickness Change (k) -0.05 Modified Fig. B.1. Optical path (in u) at 3.8 y vs. stack thickness (k) for a 14 layer quarter wave enhanced reflector and a modified design. 164 Relative Thickness Change (k) -0.05 0.05 Fig. B.2. Optical path at 0.6328 p vs. k for the 14 layer modified design. 165 & Relative Thickness Change (k) -o.o: Fig. B.3. Performance of the 14 layer modified design with approximately 1% thickness errors. 166 layers were used, or about 1% of the quarter wave thickness for these material at 3.8 y. This level of error is acceptable. A different design was sought that used a smaller number of layers, and that would perform over a larger thickness range if not as well. A 10 layer quarter wave design was chosen, and the optimization procedure was performed over the range of [-0.025, 0.025] in k. Table B.2 gives the layer thicknesses for the original and the modified designs. The performance of the coating is illustrated by Figure B.4, which shows the optical path at 3.8 y over the range [-0.05, 0.05] in k. The maximum error over the design range is -0.0310 y, or X/120, at 3.8 y; this is about 1/3 that of a 6 layer quarter wave design, and the minimum reflec tance is 0.99928 over the design range in k. This coating would work as well in terms of wavefront as the 14 layer design above over the [-0.01, 0.01] range in k as well, and the design reflectance should be more nearly reached. Figure B.5 shows that the same RMS coating thickness errors as above, or roughly 1%, will give acceptable performance for the coating. The reflectances are not shown, but they increase slightly with thick ness errors, while the sensitivity to changes in the proportional stack thickness k degrades. The design possibilities are far from exhausted] improved designs with few layers but good performance may be possible, and such designs would shift the burden of the coating technician from achieving high coating uniformity to obtaining good control over the absolute thickness and optical properties of the layers. 16" Table B.2. Thicknesses for two 10 layer enhanced reflector coatings: a quarter wave design and a modified design. Layer Number Q.W. Design Modified Design (From Substrate) Thicknesses (u) Thicknesses (p) 1 0.611 0.566 2 0 .422 0.409 3 0.633 0.586 4 0.422 0.404 5 0.633 0.653 6 0.422 0.358 7 0.633 0.649 8 0.422 0.380 9 0.633 0.621 10 0 .422 0.781 168 .e +j a. Relative Thickness Change (k) -0.05 0.05 Fig. B.4. Optical path at 3.8 p vs. k for a 10 layer modified enhanced reflector design. 169 0.05 n. Relative Thickness Change (k) -0.03 -0.05 Fig. B.5. Performance of the 10 layer modified design with approximately \% thickness errors. LIST OF REFERENCES Abeles, F. , "Rccherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces." Ann. Physique, 5_, 596-640 and 706-784 (1950). Adby, P. R., and M. A. H. Dempster, Introduction to Optimization Methods, Chapman and Hall, London, 1974. Anastasio, T. A., "Use of Optical Density Measurements of Thin Films to Determine Vapor Distributions," Rev. Sci. Inst. 54_, 740-741 (1965). Anastasio, T. A., and W. J. Slatterv, "Use of the Quartz-Crystal Rate Monitor to Determine Vapor Distributions," J. Vac. Sci. Tech. 4, 205-205 (1967). Beattie, J. R., "The Anomalous Skin Effect and the Infra-Red Properties of Silver and Aluminum," Phvsica 23_, 898-920 (1957). Behrndt, K. H. , "Film-Thickness and Deposition-Rate Monitoring Devices and Techniques for Producing Films of Uniform Thickness," in Physics of Thin Films, Vol. 5, G. Hass and R. E. Thun, eds., Academic Press, New York, 1966. Bennett, H. E., "Large Optics Coating and Evaluation Facility Study, Part 1: Technical Discussion," NWC TP 6177, Part 1, Naval Weapons Center, China Lake, CA, August 1980. Bennett, J. M., and E. J. Ashley, "Investigation of Film Thickness Uni formity on Substrates Located Close to the Source Axis," Appl. Opt. U, 758-765 (1975). Bloom, A. L. , "Refining and Optimization in Multilayers," Appl. Opt. 20_, 66-75 (1981). Borgogno, J. P., P. Bousquet, F. Flory, B. Lazarides, E. Pelletier, and B. Roche, "Inhomogeneity in Films: Limitation of the Accuracy of Optical Monitoring of Thin Films," Appl. Opt. 20, 90-94 (1981). Born, M., and E. Wolf, Principles of Optics, Pergamon Press, New York, 1975. Bruning, J. H., "Fringe Scanning Interferometers," in Optical Shop Test ing, D. Malacara, ed., Wiley, New York, 1978. 170 171 Carnahan, B., H. A. Luther, and J. 0. Wilkes, Applied Numerical Analysis, Wiley, New York, 1969. Ciddor, P. E., "Minimization of the Apparent Curvature of Multilayer Reflecting Surfaces," Appl. Opt. 7_, 2328-2329 (1968). Dirks, A. G., and H. J. Leamy, "Columnar Microstructure in Vapor-Deposited Thin Films," Thin Solid Films £7, 219-233 (1977). Dobrowolski, J. A., "Completely Automatic Synthesis of Optical Thin-Film Systems," Appl. Opt. 4_, 937-946 (1965). Dobrowolski, J. A., "Versatile Computer Program for Absorbing Optical Thin-Film Systems," paper presented at the OSA Topical Meeting on Optical Interference Coatings, Oakland, CA, June 1980. Dwight, H. B., Tables of Integrals and other Mathematical Data, fourth ed., Macmillan, New York, 1961. Fink, D., "Polarization Effects of Axicons," Appl. Opt. 1_8> 581-582 (1979). Fleming, W., Functions of Several Variables, second ed., Springer-Verlag, New York, 1977. Giacomo, P., "Proprietes Chromatiques des Couches Reflechissantes Multi- Dielectriques," J. de Phys. et Rad. IJ^, 307-310 (1958). Glang, R., "Vacuum Evaporation," in Handbook of Thin Film Technology, L. I. Maissel and R. Glang, eds., McGraw-Hill, New York, 1970. Graper, E. B., "Distribution and Apparent Source Geometry of Electron- Beam-Heated Evaporation Sources," J. Vac. Sci. Tech. 1_0, 100-103 (1973). Guha, J. K., P. W. Scott, and W. H. Southwell, "Effect of Infrared Coating Nonuniformity on Optical Systems," Appl. Opt. _1£, 1320-1322 (1980). Gunther, K. G., "Zum Kondensationsverhalten hochsiendender Substanzen," Z. Phys. 149, 538-549 (1957). Hansen, W. N., "Optical Characterization of Thin Films: Theory," JOSA 63, 793-802 (1973). Hass, G., "Mirror Coatings," in Applied Optics and Optical Engineering, Vol. Ill, R. Kingslake, ed., Academic Press, New York, 1965. Heavens, 0. S., Optical Properties of Thin Solid Films, Dover, New York, 1965. Heavens, 0. S., and H. Liddell, "Staggered Broad-Band Reflecting Multi layers," Appl. Opt. 5_, 373-376 (1966). 172 Hecht, E., and A. Zajac, Optics, Addison-Wesley, Reading, MA, 1974. Holland, L., and W. Steckelmacher, "The Distribution of Thin Films Con densed on Surfaces," Vacuum H_, 346-364 (1952). Hopkins, H. H., Wave Theory of Aberrations, Oxford Univ. Press, London 1950. Keav, D. and P. H. Lissberger, "Application of the Concept of Effective Refractive Index to the Measurement of Thickness Distributions of Dielectric Films," Appl. Opt. 6^ 727-730 (1967). Kinosita, K., and M. Nishibori, "Porosity of MgF2 Films - Evaluation Based on Changes in Refractive Index due to Adsorption of Vapors," J. Vac. Sci. Tech. 6, 730-733 (1969). Knittl, Z., Optics of Thin Films, IVilev, New York, 1976. Kubota, H., and S. Inoue, "Diffraction Images in the Polarizing Micro scope," JOSA 49, 191-198 (1959). Liddell, H. M., "Use of Optimization Techniques in Optical Filter Design," in Optimization in Action, L. C. IV. Dixon, ed., Academic Press, New York, 1976. Loomis, J. S., "FRINGE User's Manual Version 2," Optical Sciences Center, University of Arizona, Tucson, 1976. Macleod, H. A., Thin Film Optical Filters, Adam Hilger, London, 1969. Macleod, H. A., "The Monitoring of Thin Films for Optical Purposes," Vacuum 21_, 383-390 (1977). Macleod, H. A., "Monitoring of Optical Coatings," Appl. Opt. 2£, 82-89 (1981). Nelder, J. A., and R. Mead, "A Simplex Method for Function Minimization," Comp. J. 7_, 308-313 (1965). Park, K. C., "The Extreme Values of Reflectivity and the Conditions for Zero Reflection from Thin Dielectric Films on Metal," Appl. Opt. 3, 877-881 (1964). Pelletier, E., "Calcul et Realisation de Revetements Multidielectriques Presentant des Carateristiques Spectrales Imposees," Ph.D. Thesis, Orsay, France, 1970. Pelletier, E., R. Chabbal, and P. Giacomo, "Influence sur 1'Epaisseur Optique de 1'Interferometre de Fabry-Perot d'une Variation d'Epaisseur du Revetement Reflecteur," J. de Phys. 25_, 275-279 (1964). 173 Pelletier, E., P. Roche, and B. Vidal, "Determination Automatique des Constantes Optiques et de I'Epaisseur de Couches Minces: Appli cation aux Couches Dielectriques," Nouv. Rev. Optique 7_, 353- 362 (1976). Philip, R., "Proprietes Optiques et Structure des Couches Minces Metalli- ques Influence de la Vitesse de Formation," Ph.D. Thesis, L'Universite D'Aix-Marseille, France, 1960. Pitts, T., Reported by R. R. Shannon, personal communication, December 1980. Ramsay, J. V., and P. E. Ciddor, "Apparent Shape of Broad Band, Multi layer Reflecting Surfaces," Appl. Opt. 6_, 2003-2004 (1967). Ramsay, J. V., R. P. Netterfield, and E. G. V. Mugridge, "Large-Area Uniform Evaporated Thin Films," Vacuum 24_, 337-340 (1974). Richmond, D., "Thin Film Narrow Band Optical Filters," Ph.D. Thesis, Newcastle-upon-Tyne Polytechnic, England, 1976. Ritter, E., and R. Hoffman, "Influence of Substrate Temperature on the Condensation of Vacuum Evaporated Films of MgF2 and ZnS," J. Vac. Sci. Tech. IJD, 733-736 !969). Tolansky, S., Multiple Beam Interferometry, Oxford Univ. Press, London, 1948. Wyant, J. C., "Holographic and Moire Techniques," in Optical Shop Testing, D. Malacara, ed., Wiley, New York, 1978.