A 1 1 1

2

71?bTS NBS

PUBLICATIONS

NBSIR 87-3618

Computer Software for the

Computation of the Scattered Field

and the Optical Microscope Image of

Line Objects Patterned in Thick

Layers

D. Nyyssonen

CD Metrology, Inc. Germantown, MD 20874

December 1987

Prepared for: U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards —,„i Engineering Laboratory

^ >r Manufacturing Engineering

100 i Engineering Division

. U56 aurg, MD 20899

87-3618

1987

C.2

NBSIR 87-3618

COMPUTER SOFTWARE FOR THE

COMPUTATION OF THE SCATTERED FIELD

AND THE OPTICAL MICROSCOPE IMAGE OF

LINE OBJECTS PATTERNED IN THICK

LAYERS

D. Nyyssonen

CD Metrology, Inc. Germantown, MD 20874

December 1987

Prepared for: U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Manufacturing Engineering

Precision Engineering Division

Gaithersburg, MD 20899

Research Information Center

National bureau of Standards

Gaithersburg, Maryland 20899

(JJ , / (Q O

U.S. DEPARTMENT OF COMMERCE, C. William Verity, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Table of Contents

Page

Abstract 1

Introduction 1

Software Structure . . 3

Implementation and Testing 5

Accuracy 6

Acknowledgments 7

References . ........... 7

Table .................... 9

Appendix I 10

Appendix II 30

Appendix III .52

iii

Computer Software for the Computation of

the Scattered Field and the Optical Microscope Image of

Line Objects Patterned in Thick Layers

Diana Nyyssonen

CD Metrology, Inc.

Germantown, MD 20874

ABSTRACT: This report contains computer software for calculating

optical microscope images of line objects patterned in thick

layers (>A/4 thick). The algorithms used are based on a monochromatic, waveguide model which can predict the images of line

objects with arbitrary edge geometry including multilayer structures with sloped, curved, asymmetric, and undercut edges. Along

with the computer software listing, the mathematics of the model,

a short description of its structure and use, and test cases for

help in implementation are given.

KEY WORDS: computer software; diffraction; dimensional metrology; linewidth; microscopy; optical imaging; optical metrology.

INTRODUCTION

The computer software described in this report was written in

conjunction with the NBS project to develop fundamentally accurate optical measurement techniques for the width of micrometer

and submicrometer lines patterned on integrated circuit wafers.

Accurate and precise measurement techniques for linewidth are

needed to improve yield, to ensure that lithographic and critical

dimension (CD) measurement systems meet specification, to establish control of fabrication processes, and as input to device

modeling and simulation programs.

In the course of research and development of the NBS laser linewidth measurement system [1], it was found that the microscope

1
image profiles for lines patterned in thick layers OA/4 thick)

were not properly predicted by the scalar theory conventionally

used to calculate images of line objects [2, 3]. Scalar imaging

theory assumes that the object is planar (thin compared to the

wavelength of the illumination) and can be represented by a complex transmittance or reflectance function. The scalar model

does not take into account the multiple reflections or standing

waves that may occur within the patterned layer or the edge effects that occur in thick layers.

A new model was developed by Nyyssonen [4, 5] based on a waveguide approach which characterizes the patterned layer by its

complex dielectric constant and calculates the waveguide modes

supported by the line structure. First, the mode expansion of

the electromagnetic fields within the patterned layer is found

and then the appropriate boundary conditions are used to calculate the scattered field.

Nyyssonen' s model assumes that the line structure is patterned in

a nonmagnetic layer which can be characterized by its complex index of refraction which is taken to be constant with depth within

the layer. Thus, this model can be used to represent homogeneous

line structures with vertical edge walls. The spatial function

representing the variation in the dielectric constant (square of

the complex index of refraction) in the layer is expanded in a Fourier series. This series is substituted into the wave equation and the eigenvalue solutions to this equation which represent the waveguide modes are found. Assuming a single incident

plane wave normal to the surface, the boundary value problem at

the layer interfaces is solved to determine the Fourier coefficients in the expansions for the transmitted and reflected

fields. This method allows the use of conventional scalar imaging theory to compute the image when no polarization effects are

present. In such a case, the E- and H-field components are

equivalent and either may be used in the scalar imaging

equations

.

2
To include the effects of partial coherence in the imaging system, it is possible to integrate over a finite illumination cone

angle as long as the variation of the scattered field coefficients is negligible for the cone angle used. Hence, for layers

approximately a few micrometers or less thick, finite coherence

may be included.

This report covers the extension of this model by Kirk and

Nyyssonen [6] to line structures of arbitrary edge geometry whose

index of refraction may vary with depth in the layer. This line

structure is approximated by subdividing it into a set of sublayers, each consisting of a line object with vertical edges and

having a constant index of refraction over its small interval of

depth. Each sublayer is treated in the same way as the single

layer of Nyyssonen' s earlier model. The waveguide modes and

solution for the electromagnetic fields are found for each sublayer. There are now n+1 boundary value equations where n is the

number of sublayers. These equations when solved allow for the

substitution of a single "equivalent" planar scattering layer for

the multilayer structure. Thereafter, the solution for the

transmitted and reflected field components and image are found in

the same manner as for the single layer case. This model therefore allows for the modeling of line structures which contain

different materials, have curved edges, asymmetry, and unpatterned (or patterned) sublayers. The mathematical details of the

model are given in the expanded version of ref. [6] reproduced in

Appendix III. A list of definitions of symbols used in the software and Appendix III is given in the attached table.

Software Structure

As input to the software, the line structure may be characterized

either by giving the index of refraction, thickness, and edge

locations for the individual sublayers, or by a single refractive

index, total thickness, and the polynomial coefficients used to

3
describe the edge geometry (See Appendix III.) In the latter

case, the number of layers used to approximate this structure in

the calculations must be designated. A good rule of thumb to use

in determining the number of layers is to divide the change in * linewidth (between the top and bottom interfaces) by 0.1 urn and

use a value larger than this. For most cases of interest in integrated circuit processing, 9 layers are sufficient. The program arrays, as written, allow up to 20 layers. The program

calculates both the scattered field and the microscope image for

TE-mode illumination. That is, the direction of polarization

(i.e. E-field) is assumed to be parallel to the length of the

line structure.

The scattered field is given in terms of the Fourier coefficients

of the resulting far field diffraction pattern. In the test cases shown, 45 coefficients are given corresponding to +22 diffracted orders at diffraction angles corresponding to nA/P where

n is the order number, A the wavelength, and P the period. The

line structure is assumed to be repeated at a period P for convenience in calculating the Fourier series coefficients. An

isolated line image is calculated by taking P large enough so

that there is no influence from adjacent edges. Near resonances,

this is not always possible due to time or storage limits on

array sizes. The largest period used has been 12 um corresponding to a 90x90 complex matrix for the boundary condition equations .

The microscopic image is calculated using conventional scalar imaging theory with the complex Fourier coefficients of the scattered field substituted for those of the planar object traditionally used. The software, as written, assumes a 1-D imaging system with a finite illumination cone angle. The Fourier coefficients of the scattered field are assumed to be constant over

this angle. The calculations are accurate for a 2-D system as

long as the condenser numerical aperture (N.A. ) is much smaller

4
than that of the imaging objective. Hence, the calculations will

accurately predict the image waveforms for the NBS laser system

which has an objective N . A. of 0.85 or 0.95 and an illumination

aperture 1/5 that of the objective at a wavelength of 514 nm.

The software also allows the calculation of the image for varying

focus positions. Best focus is the top surface of the patterned

layer. For all other focus positions, a quadratic phase factor

is introduced (just as in scalar image theory).

The input for focus position is given as the number of wavelengths of defocus m where m is a dimensionless quantity. The

relationship between the displacement of the measurement plane

from the Gaussian focal plane, z, is given by

tan 2 8

where A is the wavelength and tan© is substituted for N . A.

(=sin©) for the high numerical apertures conventionally used for

linewidth measurement.

Implementation and Testing

This software (See Appendix I) is written in ANSI FORTRAN 77 consisting of a main program and 9 modules. The software is portable except for the complex matrix subroutines used for calculation of the eigenvectors and eigenvalues and matrix inversions.

The software requires highly accurate routines because of the

cascading of matrix multiplications and inversions. Those used

if if at NBS on the Cyber 855 are from the NAG library [7]. If these

Certain commercial equipment, instruments, or materials are

identified in this paper in order to adequately specify experimental or computational procedure. Such identification

does not imply recommendation or endorsement by the National

Bureau of Standards, nor does it imply that the materials or

equipment identified are necessarily the best available for

the purpose.

5
routines are unavailable, the user may substitute others as noted

in the software. However, several test conditions should be met.

A symmetric line object should produce symmetric Fourier coefficients to at least six significant figures. Also, a single-layer,

thick line object with complex index of refraction should produce

the same image structure as for a single layer when divided into

20 sublayers (or the maximum number of sublayers to be used).

Repeatability to three or four significant figures in the image

is sufficient to ensure this. If the matrix routines are inaccurate, the errors will increase as the number of layers is increased. Test cases are given in Appendix II. These six test

cases are designed to test different portions of the software.

Successful execution of one test case does not guarantee accuracy

for the others.

All of the data calculated are output to files for use with

available graphics. Plotting routines are not included here

because they tend to be system dependent and user requirements

may vary.

Accuracy

Testing of this software poses a challenging problem in that

the only cases for which answers are known do not fully test the

algorithms. For example, in comparing this model with scalar

theory, agreement can be expected only in the limit as the thickness of the patterned layer approaches zero. Test case #2 for a patterned 0.09 uni thick chromium layer is one such case which has

been shown to agree with scalar theory. (See ref. 5.) Comparison has also been done for patterned thin layers of silicon dioxide showing excellent agreement. (See ref. 4.) In addition, it

is required that the ratio of reflectivities of the patterned

layer and substrate (or sublayers) away from the edges agrees with

the values calculated from the Fresnel equations. (See ref. 8.)

Otherwise, calculations can at this time only be compared with

experimental measurements. This has been done for some cases

6
[6, 9]. However, although the agreement is good in most cases,

the accuracy is suspect near resonances (where the line dimensions of width and/or thickness are equal to an integer multiple

of the wavelength) and for small line dimensions. In both cases,

the energy in the high spatial frequencies increases and errors

in the eigenvalues and eigenvectors are expected to increase due

to truncation errors.

Acknowledgments

A number of people have contributed to this software development.

* The original software was written for the Univac 1108 computer

by Chris Kirk while a research associate at NBS . The author

wishes to especially thank Ruth Varner, NBS, for her assistance

* in converting this software for use on the Cyber 855. The present software has been modified by the author to work for patterned metal layers. Other modifications to simplify its use and

provide additional documentation have also been made.

References

1. D. Nyyssonen, "Calibration of Optical Systems for Linewidth

Measurements on Wafers," Opt. Eng. 21, 882-887 (1982).

2. L. C. Martin, The Theory of the Microscope , (Blackie, London,

1966), Chapters V and VIII.

3. E. C. Kintner, "Method for the Calculation of Partially Coherent Imagery," Appl. Opt. V7, 2747-2753 (1978).

4. D. Nyyssonen, "Theory of Optical Edge Detection and Imaging

of Thick Layers," J. Opt. Soc. Am. 72^, 1425-1436 (1982).

*See previous footnote.

7
5. D. Nyyssonen, "Optical Linewidth Measurement on Patterned

Metal Layers , " Proc. SPIE Vol. 480, Integrated Circuit Metrology II, 65-70 (1984)

.

6. D. Nyyssonen and C. Kirk, "Modeling the Optical Microscope

Images of Thick Layers for the Purpose of Linewidth Measurement," Proc. SPIE Vol. 538, Optical Microlithography IV,

179-187 (1985).

7. These routines are available from: Numerical Algorithms

Group (NAG) Ltd., 7 Banbury Road, Oxford, Oxfordshire 0X2

6NN, England.

8. D. Nyyssonen, "Narrow- Angle Laser Scanning Microscope System

for Linewidth Measurement on Wafers," NBSIR (to be published) .

9. C. Kirk, "A Study of the Instrumental Errors in Linewidth and

Registration Measurements Made with an Optical Microscope,"

Presented at the SPIE 1987 Santa Clara Symposium on Microlithography, Vol. 775, Integrated Circuit Metrology, Inspection and Process Control.

8
Table: Definition of Symbols Used in Software

Math Software

Symbol Label

Free space wavelength in microns A WAVE

Total layer thickness T TL

The number of layers N NS

Period of the grating P PER

RI of the air layer ^o CRI ( 0

)

RI of the substrate ^s CRI( 21)

RI of the n-th layer % CRI ( N

)

Width polynomial coefficients x

j z

j

W( J)

Starting points for 1st, 2nd,

and 3rd order

P0( J)

The X position offset for each

layer

- - XP ( J

)

The 2 locations of the layer

interfaces

z n

ZP (N)

Spatial frequency of the grating 1/P SPAFRE

Free space wavenumber ( in units

of ym~

1

) k

o

RKO

Fourier coefficients of dielectric

constant

E

q,n

EQ ( I

)

Eigenvalue matrix D

i, j

D ( I , J

)

Eigenvalues of the layers a

m,n

VAL(M)

Eigenvectors of the layers B, m n 3 / m , n

VEC

Boundary equation matrices D

11

, 3 »

m

12 21 22

D ,D ,D

3 ,m j,m' j ,m

RM(L,M)

(A) coefficients A A

in, 1 ' m,

1

AV(M)

Fourier coefficients of the pseudoobject

FC( J)

1-D image of the line -- ELEC ( I

)

9
Appendix I

Computer Software Listing ( THKIMAG

)

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PROGRAM THKIMAG

THE ORIGINAL VERSION OF THIS PROGRAM WAS WRITTEN BY

CHRIS KIRK, WHILE AN EMPLOYEE OF VICKERS INSTRUMENTS,

( HAXBY ROAD, YORK, NORTH YORKSHIRE , ENGLAND ) WHILE A

RESEARCH ASSOCIATE AT THE NATIONAL BUREAU OF STANDARDS,

GAITHERSBURG, MARYLAND, USA. SEPTEMBER 1984.

THE CURRENT VERSION HAS BEEN MODIFIED BY D. NYYSSONEN

TO WORK FOR METAL LAYERS. JUNE 1986 *********************************************************

THIS PROGRAM COMPUTES THE OPTICAL IMAGE OF LINE OBJECTS

WITH ARBITRARY EDGE GEOMETRY, PATTERNED IN THICK LAYERS

INCLUDING MULTILAYER STRUCTURES WITH SLOPING, CURVED, AND

UNDERCUT EDGES, AS WELL AS ASYMMETRIC OBJECTS.

SEE REFERENCE: "MODELING OF THE OPTICAL IMAGING OF LINES

PATTERNED IN THICK LAYERS WITH VARIABLE EDGE GEOMETRY,"

BY D. NYYSSONEN AND C. P. KIRK

ALL EQUATION NUMBERS GIVEN REFER TO THIS MANUSCRIPT

FOR QUESTIONS CONCERNING THIS PROGRAM CONTACT:

R. D. LARRABEE, PRECISION ENGINEERING DIVISION, NATIONAL

BUREAU OF STANDARDS OR D. NYYSSONEN, CD METROLOGY, INC. **********************************************************

SYSTEM DEPENDENT FEATURES

.

ALTHOUGH THIS SOFTWARE HAS BEEN WRITTEN IN FORTRAN 77 IN

ORDER TO ALLOW FOR EASY PORTABILITY, THERE ARE A NUMBER

OF FEATURES WHICH MAY BE SYSTEM DEPENDENT.

1. A FORTRAN 77 COMPILER IS REQUIRED.

2. THE SYSTEM MUST SUPPORT CERTAIN NAG LIBRARY ROUTINES.

A NOTE CONCERNING THE NAG LIBRARY.

THIS PROGRAM USES THE FOLLOWING NAG ROUTINES.

F02AKF F04ADF

THESE ARE SINGLE PRECISION VERSIONS OF LIBRARY ROUTINES

FOR THE CYBER WHICH MAY NOT BE SUPPORTED ON ALL

MACHINES

.

THESE ROUTINES ARE AVAILABLE FROM:

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C

C

cc

cc

c

c

c

cc

NUMERICAL ALGORITHMS GROUP (NAG) LTD , 7 BANBURY ROAD,

OXFORD, OXFORDSHIRE. OX2 6NN . ENGLAND. **********************************************************

PARAMETER ( NLAY=20 , NLAY1=21)

PARAMETER (NLIM=- 22, LIM=22, KLIM=45, MLIM=90)

REAL PI , RKO , WAVE , SPAFRE , WI ( NLAY ) ,ZP(0:NLAY) ,NS,XP(NLAY) , TRA ( 2

COMPLEX D ( NLIM : LIM , NLIM : LIM ) , RM ( MLIM , MLIM ) , T ( MLIM , MLIM

)

, RN ( KLIM , MLIM ) ,AV(MLIM) ,FC(NLIM:LIM) , CRI ( 0 : NLAY1 ) ,Q0 ,VAL ( KLIM

PI = 4.0*ATAN( 1.0)

OPEN ( UNIT=9 , FILE= ' PARFIL ’

)

OPEN ( UNIT=10 , FILE= ' FCOFDC ’

)

OPEN ( UNIT=11 , FILE= ' DMATRX '

)

OPEN ( UNIT=12 , FILE= ' EIGVAL '

)

OPEN ( UNITTM 13 , FILE= ’ EIGVEC '

)

OPEN ( UNIT=14 , FILE= ' BOUNEQ '

)

OPEN ( UNIT=1 5 , FILE= ' BEMATX '

)

OPEN ( UNIT=1 6 , FILE= ' FCOFPO '

)

OPEN ( UNIT=17 , FILE= ’ IMAGE '

)

**********************************************************

SUMMARY OF THE INPUT/ OUTPUT STRUCTURE.

76 C INPUT INPUT DATA FILE.

77 C PARFIL UNIT9 PARAMETER FILE CREATED.

78 C FCOFDC UNIT10 FOURIER COEFFICIENTS OF

79 C DIELECTRIC CONSTANT CALCULATED

80 C FOR FIRST LAYER.

81 C DMATRX UNITll EIGENVALUE MATRIX D OF THE 1ST

82 C LAYER

83 C EIGVAL UNIT12 EIGENVALUES OF THE LAYERS.

84 C EIGVEC UNIT13 EIGENVECTORS OF THE LAYERS.

85 C BOUNEQ UNIT14 BOUNDARY EQUATION MATRICES.

86 C BEMATX UNIT15 (A) COEFFICIENTS.

87 C FCOFPO UNIT16 FOURIER COEFFICIENTS OF

88 C THE PSEUDO-OBJECT.

89 C IMAGE UNIT17 1-D IMAGE OF THE LINE. **********************************************************

SET UP THE PARAMETERS FOR THE SECTIONED LINE.

INPUT

WAVE

NS

PER

CRI ( 0

)

FREE SPACE WAVELENGTH IN MICRONS

THE NUMBER OF LAYERS.

THE PERIOD.

RI OF THE AIR LAYER.

12
101 c 102 C

103 C

104 C

105 C

106 C

107 C

108 C

109 C

110 C

111 C

112 C

113 C

114 C

115 C

116 C

117 C

118 C

119 C

120 C

121 C

122 C

123 C

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125 C

126 C

127 C

128 C

129 C

130 C

131 C

132 C

133 C

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135 C

136 C

137 C

138 C

139 C

140

141

142 C

143 C

144 C

145 C

146 C

147 C

148 C

149 C

150 C

CRI ( 21 ) = RI OF THE SUBSTRATE.

ID = DEFINE LINE: PARAMETRICALLY ( 0

)

= : LAYER BY LAYER ( 1 )

W(I) = WIDTH POLYNOMIAL COEFFICIENTS. I=ORDER

PO(I) = STARTING POINTS FOR 1ST, 2ND AND 3RD ORDER.

CRI(l) = REFRACTIVE INDEX OF PATTERNED LAYER.

TL = LAYER THICKNESS.

TRA(l) = PHOTOMULTIPLIER SLIT WIDTH.

TRA( 2 ) = TELEVISION CAMERA GAUSSIAN WIDTH PARAMETER.

LSKIP = 0: GENERATE A NEW FOURIER SERIES FOR

THE PSEUDO OBJECT.

1: USE PREVIOUSLY GENERATED FOURIER SERIES.

-1: GENERATE THE IMAGE.

DF = DEFOCUS IN WAVES WHERE DF IS A DIMENSIONLESS QUANTITY.

IF DEFOCUS IS KNOWN IN TERMS OF DISPLACEMENT OF THE

MEASUREMENT PLANE FROM THE FOCAL PLANE DZ , IT MUST BE

CONVERTED TO WAVES USING THE RELATIONSHIP:

DZ=( 2*WAVE/ (TAN( ANGLE) )**2)*DF

WHERE ANGLE IS THE SAME AS DEFINED BY N.A. = SIN (ANGLE)

INPUT IF ID = 1.

WI(N) = THE WIDTH OF THE LINES IN THE NTH LAYER.

ZP ( N ) = THE Z LOCATIONS OF THE LAYER INTERFACES.

CRI(N) = RI OF EACH LAYER.

XP(N) = THE X POSITION OFFSET FOR EACH LAYER.

COMPUTATION

SPAFRE = SPATIAL FREQUENCY OF THE GRATING (1/PERIOD).

COMPUTATIONS IF ID = 0

CRI(N) = RI OF EACH LAYER.

WI(N) = THE WIDTHS OF THE LINES IN EACH LAYER.

ZP ( N ) = THE Z LOCATIONS OF THE LAYER INTERFACES.

THE TOP SURFACE IS ASSUMED TO BE Z=0.

XP ( N ) = THE X POSITION OFFSET FOR EACH LAYER (FOR

ASYMMETRIC CROSS-SECTIONS).

COMPUTATION

RK0 = FREE SPACE WAVENUMBER (IN UNITS OF 1/UM)

CALL LINE ( PI , WAVE , RK0 , CRI , ZP , WI , NS , SPAFRE

,

TRA,XP,DF, LSKIP)

NOTE: CALCULATIONS ARE FOR LINE OBJECT, SPACES ARE

CALCULATED BY TAKING LINEWIDTH = PERIOD MINUS

DESIRED SPACE WIDTH. **********************************************************

IF LSKIP =1 THEN GO STRAIGHT TO THE IMAGE SUBROUTINE.

13
151

152 C

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159 C

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172 10

173 C

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190 C

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200

IF (LSKIP.EQ.l) GO TO 20 **********************************************************

FIND THE EIGENVALUES AND EIGENVECTORS FOR EACH LAYER.

D IS THE EIGENVALUE MATRIX FOR EACH LAYER.

THE EIGENVALUES AND EIGENVECTORS ARE FOUND FOR EACH LAYER

AND THEN STORED ON DISK, STARTING WITH THE TOP LAYER.

THE DIFFRACTION SERIES IS TRUNCATED FOR DIFFRACTION

ANGLES WHICH EXCEED 90 DEGREES IN AIR. THIS IS SET

BY THE VARIABLE ’ LIM ' . LIM AND ITS RELATED VARIABLES

ARE SET IN THE PARAMETER STATEMENTS.

L = (REFRACTIVE INDEX OF AIR) *( PERIOD )/( WAVELENGTH

)

NLIM = -L, LIM = L, KLIM = 2L+1, MLIM = 2 ( 2L+1

)

DO 10 N=1,NS

CALL SETUPD ( PI , WI ( N ) ,CRI ,D, SPAFRE, N,XP(N) , WAVE , Q0 , EQ2

)

CALL EIGD(D,N,Q0 , RK0 , WAVE , SPAFRE , EQ2

)

CONTINUE **********************************************************

SET UP THE MATRIX ELEMENTS FOR THE FIRST INTERFACE

BOUNDARY CONDITIONS.

THE ARRAY RM CONTAINS THE UPPER MATRIX ELEMENTS (SEE EQ. 21).

CALL SETRM ( RM , RK0 , WAVE , SPAFRE , CRI ( 0 )

)

**********************************************************

SET UP THE PRODUCT MATRICES FOR THE N-LAYERS INTERFACES

SET THE MATRIX T TO THE IDENTITY MATRIX.

THE T MATRIX IS MODIFIED BY ALL THE INTERMEDIATE INTERFACES.

STARTING WITH THE IDENTITY MATRIX, THE T MATRIX IS PASSED

TO THE SUBROUTINE FOR EACH INTERFACE AND RETURNED AFTER

MODIFICATION.

DO 40 L=1 , MLIM

DO 30 M=1 , MLIM

IF(L.EQ.M) THEN

T(L,M) = CMPLX( 1.0, 0.0)

ELSE

T(L,M) = CMPLX( 0.0, 0.0)

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201

202 30

203 40

204

205

206

207 50

208

209 C

210 C

211 C

212 C

213 C

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215 C

216 C

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218 C

219

220 C

221 C

222 C

223 C

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225

226 C

227 C

228 C

229 C

230 C

231 C

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233

234 20

235 C

236 C

237 C

238

239 C

240 C

241 C

242 C

243

244 C

245 60

246

247

248 C

249 C

250 C

END IF

CONTINUE

CONTINUE

IF(NS.GT.l) THEN

DO 50 N=1,NS-1

CALL SETPQN ( N , ZP ( N ) , T

)

CONTINUE

END IF **********************************************************

SET UP THE MATRIX ELEMENTS FOR THE LAST INTERFACE BOUNDARY

CONDITIONS

.

THE ARRAY RN CONTAINS THE LAST INTERFACE MATRIX

ELEMENTS

.

CALL SETRN(RN,NS,ZP(NS) ,VAL)

SOLVE THE MATRIX EQUATION FOR THE (A) COEFFICIENTS. (EQ. 21)

THE VECTOR AV IS RETURNED CONTAINING THE (A) COEFFICIENTS.

CALL SETAM( RM,T , RN , RK0 , AV , VAL ,NS ,WAVE , SPAFRE , CRI ( 21 )

)

SET UP THE FOURIER COEFFICIENTS.

THE FOURIER COEFFICIENTS ARE RETURNED IN VECTOR FC

.

T IS USED AS A WORKSPACE.

CALL FCOEFF ( AV , FC , T

)

CONTINUE

IF LSKIP = “1 THEN SKIP THE IMAGE ROUTINE.

IF ( LSKIP. EQ.-l) GOTO 60

COMPUTE THE IMAGE FROM THE FOURIER COEFFICIENTS.

CALL IMAGE ( FC , PI , SPAFRE , TRA , DF , LSKIP , WAVE

)

CONTINUE

STOP

END**********************************************************

**********************************************************

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300 C

SUBROUTINES TO SUPPORT THE MAIN PROGRAM. **********************************************************

SUBROUTINE LINE ( PI , WAVE , RK0 , CRI , ZP , WI , NS , SPAFRE

,

TRA , XP , DF , LSKIP

)

COMPLEX CRI (0:21)

REAL PI , RK0 , WAVE , SPAFRE

,

W ( 0 : 3 ) ,WI(20) ,PER,PO(3)

,

Z,ZP(0:20) ,NS,TL,XP(20) , TRA ( 2

)

SET THE ARRAYS TO ZERO.

DO 210 1=0,20

ZP(I) = 0.0

CRI ( I ) = CMPLX( 0.0, 0.0)

IF(I.GT.O) THEN

WI(I) = 0.0

XP(I) = 0.0

END IF

CONTINUE

READ IN THE WAVELENGTH, THE PERIOD, THE NUMBER OF LAYERS

READ*, WAVE, PER, NS

SPAFRE = 1.0 /PER

READ IN THE RI ' S OF THE AIR LAYER AND THE SUBSTRATE.

ID DETERMINES WHETHER THE LINE IS DEFINED BY PARAMETERS

OR LAYER BY LAYER.

READ* , CRI ( 0 ) , CRI ( 21 ) , ID

READ IN THE WIDTH POLYNOMIAL COEFFICIENTS.

READ* , W( 0 ) ,W( 1) ,W( 2) ,W( 3

)

READ IN THE Z POSITION STARTING POINTS FOR THE 1ST, 2ND

AND 3RD ORDERS.

READ* , PO( 1 ) ,PO( 2) ,PO( 3)

READ IN THE REFRACTIVE INDEX OF THE PATTERNED LAYER

AND THE LAYER THICKNESS.

READ*, CRI ( 1) , TL

READ IN THE PHOTOMULTIPLIER SLIT WIDTH AND THE VIDEO

CAMERA WIDTH PARAMETER.

16
301

302

303

304

305 C

306 C

307 C

308 C

309 C

310 C

311 C

312 C

313 C

314

315 C

316

317 C

318 C

319 C

320

321

322

323

324

325

326 220

327

328 C

329 C

330 C

331 C

332 C

333 C

334

335

336 230

337

338 C

339 C

340 C

341

342

343 C

344 C

345

346

347

348

349

350

READ* , TRA( 1 ) ,TRA(2)

IF(TRA(1) .GT.0.01) THEN

TRA ( 2 ) =0.0

END IF

THE CAMERA WIDTH PARAMETER TRA ( 2 ) IS SET TO ZERO WHENEVER

THERE IS A FINITE SLIT WIDTH TRA ( 1 )

.

READ IN THE SKIP PARAMETER (TO SKIP IMAGE CALCULATIONS)

AND THE AMOUNT OF DEFOCUS IN NUMBER OF WAVES.

READ* , LSKIP , DF

IF(ID.EQ.O) THEN

SET UP THE LINE DEFINED PARAMETRICALLY.

DO 220 1=1, NS

CRI(I) = CRI(l)

Z = TL*( 1-0.5) /NS

WI(I) = W ( 3

) * ( Z-PO( 3 )

) **3+W( 2) * ( Z-PO( 2 ) )**2

WI(I) = WI(I)+W(l)*(Z-PO(l) )+W(0)

ZP ( I ) = TL*I/NS

CONTINUE

ELSE IF(ID.EQ.l) THEN

SET UP THE LINE LAYER BY LAYER.

READ IN THE WIDTH , INTERFACE POSITION RI , AND X OFFSET

OF EACH LAYER

DO 230 1=1, NS

READ* , WI ( I ) ,ZP(I) , CRI ( I ) ,XP(I)

CONTINUE

END IF

CALCULATE THE WAVE NUMBER.

RK0 = 2 . 0*PI/WAVE

SEND PARAMETERS TO A FILE.

WRITE ( 9

, * ) ' RUN PARAMETERS .

WRITE ( 9

, *

)

' WAVELENGTH = ’ , WAVE

WRITE ( 9

, * ) ' NUMBER OF LAYERS=

' , NS

WRITE ( 9

, * )

' WAVE NUMBER =

' , RK0

WRITE ( 9

, * ) 'AIR LAYER =',CRI(0)

WRITE ( 9

, * ) 'SUBSTRATE =’, CRI (21)

17
351

352

353

354

355

356

357

358 240

359

360 299

361

362

363

364 C

365 C

366 C

367

368

369

370

371

372 C

373 C

374 C

375 C

376 C

377 C

378

379 C

380 C

381 C

382 C

383

384

385

386

387 310

388 320

389 C

390 C

391 C

392 C

393 C

394

395

396

397

398

399

400

WRITE (9,*) 'DEFOCUS =',DF

WRITE (9,*) 'SLIT WIDTH IN MICRONS =',TRA(1)

WRITE (9,*) 'CAMERA WIDTH PARAMETER =',TRA(2)

IF ( LSKIP . LE . 0 ) THEN

WRITE ( 9

, * ) 'LAYER WIDTH POSITION RI OFFSET’

DO 299 1=1, NS

WRITE (9,240) I ,WI ( I ) , 2P( I ) , CRI ( I ) ,XP(I)

FORMAT (1X,I2,6X,F6.3,6X,F5.3,4X,F5.3,1X,F5.3,

6X, F5 . 3

)

CONTINUE

END IF

RETURN

END **********************************************************

SUBROUTINE SETUPD ( PI , W , CRI , D , SPAFRE , N , XP , WAVE , Q 0

)

PARAMETER ( NLIM=- 2 2 , LIM=22, KLIM=45, MLIM=90)

REAL P I , GO 5 CAF , SPAFRE , RP , W , XP , RI P , WAVE

COMPLEX D ( NLIM : LIM , NLIM : LIM ) ,EQ( NLIM : LIM ) , El , Cl , Q0

,

CRI (0:21)

W = LINEWIDTH OF LAYER N.

XP = X OFFSET OF THE LINE IN LAYER N.

THE LAYER FOURIER COEFFI CENTS FOR THE EXPANSION OF THE

DIELECTRIC CONSTANT ARE STORED IN ARRAY EQ.

Cl = CMPLX ( 0 . 0 , 1 . 0

)

SET THE ARRAYS TO ZERO.

DO 320 I=NLIM, LIM

EQ ( I ) = CMPLX ( 0.0 ,0.0)

DO 310 J=NLIM, LIM

D ( I , J ) = CMPLX (0.0, 0.0)

CONTINUE

CONTINUE

ROUTINE TO GENERATE THE FOURIER COEFFICIENTS.

THE OFFSET XP ADDS A PHASE TERM TO THE FOURIER

COEFFICIENTS. SEE EQUATION 8 OF APPENDIX III.

El = CRI ( N

) **2 - CRI ( 0 ) **2

EQ ( 0 ) = CRI ( 0

) **2+El*W*SPAFRE

DO 330 I=NLIM, LIM

IF(I.EQ.O) GO TO 330

EQ ( I ) = El* SIN ( PI*I*SPAFRE*W ) /PI /I

EQ ( I ) = EQ( I

) *CEXP( 2 . 0*PI*CI*SPAFRE*XP*I

)

IF(N.EQ.l) THEN

18
401

402

403 330

404

405 C

406 C

407 C

408

409

410

411

412

413

414

415

416

417 340

418 350

419

420

421 C

422 C

423 C

424

425

426

427

428

429 C

430 C

431 C

432 C

433 C

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448 410

449 420

450 C

WRITE ( 10 , * ) EQ( I ) ,

I

END IF

CONTINUE

Q0 = EQ(0)

ROUTINE TO CONSTRUCT THE D MATRIX. SEE APPENDIX OF

APPENDIX III.

DO 350 L=NLIM,LIM

DO 340 M=NLIM, LIM

IF(L.EQ.M) THEN

D( L,M) = - ( WAVE*L*SPAFRE ) **2

ELSE IF( (IABS(L-M) ) .GT. 22) THEN

D(L,M) = CMPLX ( 0 . 0 , 0 .

0

) ELSE

D( L,M) = EQ(M-L)

END IF

CONTINUE

CONTINUE

RETURN

END **********************************************************

SUBROUTINE EIGD( D,N,Q0 ,RK0

)

PARAMETER ( NLIM=- 2 2 , LIM=22, KLIM=45, MLIM=90 ) COMPLEX D ( KLIM , KLIM ) ,VAL( KLIM) ,Q0

REAL AR ( KLIM , KLIM ) , AI ( KLIM , KLIM ) , RR ( KLIM )

,

RI (KLIM) ,VR( KLIM, KLIM) , VI (KLIM, KLIM) ,IWORK(KLIM) ,RK0

THE D MATRIX FOR THE FIRST LAYER IS WRITTEN ON TAPE11.

THE COMPLEX D MATRIX IS SPLIT INTO TWO REAL MATRICES

AND STORED IN ARRAYS AR(

)

(REAL) AND AI (

)

(IMAGINARY).

DO 420 1=1, KLIM

RR( I

)

= 0.0

RI ( I

)

= 0.0

IWORK(I) =0.0

DO 410 J=1,KLIM

VR( I , J ) = 0.0

VI ( I, J) = 0.0

AR( I , J ) = REAL ( D ( I , J )

)

IF ( ABS ( AR( I , J ) ) .LT.l.E-9) AR(I, J)=+0.0

AI ( I , J ) =AIMAG(D(I,J)

)

IF ( ABS ( AI ( I , J ) ) .LT.l.E-9) AI(I, J)=+0.0

IF(N.EQ.l) THEN

WRITE ( 11 , * ) D( I , J ) , I ,

J

END IF

CONTINUE

CONTINUE

19
451 C

452 C

453 C

454 C

455 C

456 C

457

458

459

460 C

461 C

462 C

463 C

464 C.

465 C

466

467

468

469

470

471

472

473

474

475

476

477 430

478 440

479

480

481

482

483

484

485

486 C

487 C

488 C

489

490

491

492

493 C

494 C

495 C

496 C

497 C

498 C

499

500

F02AKF SOLVES FOR THE EIGENVALUES AND EIGENVECTORS.

THE VECTORS ARE RETURNED IN VR AND VI AND THE EIGENVALUES

ARE RETURNED IN RR AND RI

THE SQUARE ROOTS OF THE EIGENVALUES ARE WRITTEN ON

TAPE12 AND THE EIGENVECTORS ON TAPE13

I FAIL = 0

CALL F0 2AKF ( AR, KLIM, AI, KLIM, KLIM, RR,RI,VR, KLIM, VI, KLIM,

IWORK, IFAIL

)

THE CYBER RESTRICTS PHASE ANGLES ON SQRT OF A COMPLEX

NO. TO -PI/2<0<PI/2 WHICH RESULTS IN INCORRECT

PHASE ANGLES. ANGLES ARE CORRECTED FOLLOWING

CALCULATION OF CSQRT

DO 440 1=1, KLIM

VAL(I) = CMPLX ( RR ( I ) , RI ( I )

)

VAL(I) = -VAL ( I

) -Q0

VAL(I) = CSQRT ( VAL ( I )

) *RK0

IF ( AIMAG( VAL ( I ) ) . LT .0.0) THEN

VAL ( I ) = -VAL(I)

END IF

WRITE (12,*) VAL ( I

)

PRINT* , VAL ( I ) ,1

DO 430 J=1 , KLIM

WRITE ( 13,*) VR ( I , J ) , VI ( I , J

)

CONTINUE

CONTINUE

IF ( IFAIL. EQ.0) THEN

PRINT* EIGENVALUE SUCCESS AT LAYER’ ,N

ELSE

PRINT* ,' EIGENVALUE FAILURE AT LAYER' ,N

END IF

RETURN

END **********************************************************

SUBROUTINE SETRM ( RM , RK0 , WAVE , SPAFRE , CR0

)

PARAMETER ( NLIM=- 2 2 , LIM=22, KLIM=45, MLIM=90)

COMPLEX RM ( MLIM , MLIM ) , VEC , VECCN , VAL ( KLIM ) , Cl , RKL , V , CR0

REAL VR, VI, RK0, WAVE, SPAFRE

CR0 = REFRACTIVE INDEX OF THE AIR LAYER.

THE EIGENVALUE AND EIGENVECTORS ARE RECOVERED FOR THE

FIRST LAYER FROM UNIT12 AND UNIT13.

REWIND 12

REWIND 13

20
501 Cl = CMPLX( 0 . 0

, -1 . 0 ) 502 DO 520 L=1 , KLIM

503 V = CMPLX (

( WAVE*SPAFRE* ( L-LIM-1 ) ) **2 , 0 . 0

)

504 RKL = CSQRT ( CRO * * 2 -V

)

505 DO 510 M=1,KLIM

506 IF(L.EQ.l) THEN

507 READ (12,*) VAL(M)

508 END IF

509 READ (13,*) VR, VI

510 VEC = CMPLX ( VR , VI

)

511 RM(L,M) = ( RKL-VAL ( M

) *CI /RKO ) *VEC

512 RM(L,M+KLIM) = ( RKL+VAL ( M) * Cl /RKO ) *VEC

513 RM(L+KLIM,M) = CMPLX ( 0 . 0 , 0 . 0

)

514 RM ( L+KLIM , M+KLIM ) = CMPLX ( 0 . 0 , 0 . 0

)

515 510 CONTINUE

516 520 CONTINUE

517 RETURN

518 END

519 C

520 C **********************************************************

521 C

522 SUBROUTINE SETPQN ( N , ZP 1 , T

)

523 PARAMETER ( NLI M=- 22, LIM=22, KLIM=45, MLIM=90)

524 COMPLEX PQ(MLIM,MLIM) ,T(MLIM,MLIM) , RIPQ ( MLIM , MLIM )

,

525 * VEC , VECCN , VAL ( KLIM

)

526 REAL WKSPCE ( MLIM ) , ZP1 , VR , VI

527 C

528 c ZP1 = THE Z LOCATION OF THE INTERFACE.

529 c 530 REWIND 12

531 REWIND 13

532 c 533 c READ PAST THE EIGENVALUES AND EIGENVECTORS RELATING TO

534 c THE PREVIOUS LAYERS.

535 c 536 IF(N.GT.l) THEN

537 DO 630 1=1, N-l

538 DO 620 J=1 , KLIM

539 READ (12,*) VAL(J)

540 DO 610 K=1 , KLIM

541 READ (13,*) VR , VI

542 610 CONTINUE

543 620 CONTINUE

544 630 CONTINUE

545 END IF

546 C

547 C SET UP THE UPPER LAYER MATRIX.

548 C

549 DO 650 L=1 , KLIM

550 DO 640 M=1 , KLIM

21
551

552

553

554

555

556

557

558

559

560 640

561 650

562 C

563 C

564 C

565

566

567

568

569

570 675

571 680

572 685

573 C

574 C

575 C

576 C

577

578

579

580

581

582

583

584

585

586

587 654

588 658

589 C

590 C

591 C

592

593

594

595

596

597

598

599

600

XF(L.EQ.l) THEN

READ (12,*) VAL(M)

END IF

READ (13,*) VR, VI

VEC = CMPLX ( VR , VI

)

PQ(L,M) = VEC*CEXP ( -VAL ( M) *ZP1

)

PQ(L, M+KLIM) = VEC*CEXP( VAL(M) *ZP1

)

PQ(L+KLIM,M) = VAL(M) *PQ( L,M)

PQ ( L+KLIM , M+KLIM ) = -VAL(M) *PQ( L, M+KLIM)

CONTINUE

CONTINUE

MULTIPLY THE PREVIOUS T MATRIX BY THE UPPER LAYER MATRIX.

DO 685 L=1 , MLIM

DO 680 M=1,MLIM

RIPQ( L,M) = CMPLX ( 0 . 0 , 0 . 0

)

DO 675 1=1, MLIM

RIPQ(L,M) = RIPQ(L,M)+PQ(L,I)*T(I,M)

CONTINUE

CONTINUE

CONTINUE

WRITE THE MATRIX PRODUCT ONTO UNIT14.

CLEAR THE MATRICES RIPQ AND T.

REWIND -14

DO 658 L=1,MLIM

DO 654 M=1,MLIM

WRITE ( 14,*) RIPQ(L,M)

RIPQ(L,M) = CMPLX ( 0 . 0 , 0 . 0

)

IF( L.EQ.M) THEN

T(L,M) = CMPLX(1. 0,0.0)

ELSE

T(L,M) = CMPLX (0.0, 0.0)

END IF

CONTINUE

CONTINUE

SET UP THE LOWER LAYER MATRIX.

DO 670 L=1 ,KLIM

DO 660 M=1 , KLIM

IF(L.EQ.l) THEN

READ (12,*) VAL(M)

END IF

READ (13,*) VR, VI

VEC = CMPLX (VR, VI)

PQ(L,M) = VEC*CEXP( -VAL(M) *ZP1)

PQ(L, M+KLIM) = VEC*CEXP ( VAL ( M

) *ZP1

)

22
601

602

603 660

604 670

605 C

606 C

607 C

608 C

609

610

611

612

613

614

615

616

617

618 C

619 C

620 C

621

622

623

624

625 604

626 608

627 C

628 C

629 C

630

631

632

633

634

635 690

636 694

637 698

638

639

640 C

641 C

642 C

643

644

645

646

647 C

648 C

649 C

650 C

PQ(L+KLIM,M) = VAL ( M

) *PQ ( L , M

)

PQ ( L+KLIM , M+KLIM ) = -VAL ( M

) *PQ ( L , M+KLIM

)

CONTINUE

CONTINUE

F04ADF INVERTS THE MATRIX PQ AND RETURNS THE INVERSE IN

THE MATRIX RIPQ.

IFAIL = 0

CALL F0 4 ADF ( PQ , MLIM , T , MLIM , MLIM , MLIM , RIPQ , MLIM

,

WKSPCE , IFAIL

)

IF ( IFAIL. EQ.O) THEN

PRINT* ,

' SUCCESSFUL MATRIX INVERSION AT INTERFACE' ,

ELSE

PRINT* ,' SINGULAR MATRIX AT INTERFACE' ,N

STOP

END IF

RECOVER THE PREVIOUS MATRIX FROM UNIT14.

REWIND 14

DO 608 L=1,MLIM

DO 604 M=1 , MLIM

READ(14,*) PQ ( L , M

)

CONTINUE

CONTINUE

MULTIPLY THE TWO MATRICES AND RETURN THE PRODUCT IN T.

DO 694 L=1,MLIM

DO 694 M=1 , MLIM

T( L,M) = CMPLX ( 0 . 0 , 0 . 0 ) DO 690 1=1, MLIM

T(L,M) = T ( L , M ) +RIPQ ( L , I

) *PQ ( I ,M

)

CONTINUE

CONTINUE

CONTINUE

RETURN

END **********************************************************

SUBROUTINE SETRN ( RN ,NS , ZS , VAL

)

PARAMETER (NLIM=-22, LIM=22, KLIM=45, MLIM=90)

COMPLEX RN ( KLIM , MLIM ) , VEC , VECCN , VAL ( KLIM

)

REAL VR, VI , ZS , NS

ZS = THE Z POSITION OF THE SUBSTRATE INTERFACE.

CRS = THE COMPLEX REFRACTIVE INDEX OF THE SUBSTRATE.

23
651

652

653 C

654 C

655 C

656 C

657

658

659

660

661

662

663 710

664 720

665 730

666

667 C

668 C

669

670

671

672

673

674

675

676

677

678 740

679 750

680

681

682 C

683 C

684 C

685

686

687

688

689

690 C

691 C

692 C

693 C

694 C

695 C

696 C

697 C

698

699

700

REWIND 12

REWIND 13

SKIP OVER THE ALPHA ( M ) AND B(L,M) TERMS OF THE UPPER

LAYERS

IF(NS.GT.l) THEN

DO 730 1=1 ,NS-1

DO 720 J=1 , KLIM

READ (12,*) VAL(J)

DO 710 K=1 , KLIM

READ (13,*) VR, VI

CONTINUE

CONTINUE

CONTINUE

END IF

SET UP THE RN MATRIX.

DO 750 L=1,KLIM

DO 740 M=1 , KLIM

IF(L.EQ.l) THEN

READ (12,*) VAL(M)

END IF

READ (13,*) VR, VI

VEC = CMPLX ( VR , VI

)

RN ( L ,M) = CEXP ( -VAL ( M

) * ZS ) *VEC

RN ( L,M+KLIM) = CEXP ( VAL( M) *ZS ) *VEC

CONTINUE

CONTINUE

RETURN

END

if*********************'***********************'********

SUBROUTINE SETAM ( RM , T , RN , RK0 , AV , VAL , NS , WAVE , SPAFRE , CRS

)

PARAMETER ( NLIM=- 2 2 , LIM=22, KLIM=45, MLIM=90)

COMPLEX RM ( MLIM , MLIM ) ,T( MLIM , MLIM ) , RN ( KLIM , MLIM ) , RKL

,

RV(MLIM) , AV ( MLIM ) , Cl , VAL ( KLIM) , CRS , CRSUB , S2 , S3 , A1 , ASUB

REAL RK0 , WKSPCE ( MLIM ) , WAVE , SPAFRE , NS

ASSEMBLE THE RM,RN AND T MATRICES INTO ONE MATRIX (RM).

THIS VERSION ALLOWS FOR THE INCLUSION OF AN UNPATTERNED

SUBLAYER (OXIDE). IF NOT WANTED, SET TSUB=0 .

TSUB IS THE THICKNESS IN UM OF THIS LAYER AND CRSUB IS

THE COMPLEX INDEX OF REFRACTION.

TSUB =0.0

CRSUB = CMPLX( 1.46,0.0)

Cl = CMPLX ( 0 . 0

, - 1 . 0

)

24
701

702

703

704

705

706

707

708

709

710

711

712 810

713 820

714

715

716

717

718

719

720

721

722

723

724

725 * 726 * 727

728 * 729 * 730 * 731 832

732 834

733 836

734

735 C

736 C

737 C

738

739

740

741 840

742

743 C

744 C

745 C

746 C

747 C

748

749 C

750

IF(NS.EQ.l) THEN

DO 820 L=1 , KLIM

V = CMPLX (

( WAVE*SPAFRE* ( L-LIM- 1 )

) ** 2 , 0 . 0

)

52 = CSQRT ( CRSUB * * 2 -V

)

53 = CSQRT ( CRS * * 2 -V

)

Al = CEXP ( CMPLX ( 0 . 0 , 2 . 0

) *RK0*TSUB*S2

)

ASUB = ( (S2+S3)-(S2-S3)*Al)/( ( S2+S3 ) +( S2-S3 ) *A1

)

RKL = ASUB* S

DO 810 M=1,KLIM

RM( L+KLIM, M) = RN ( L ,M) * ( RKL+VAL ( M) *CI /RKO

)

RM ( L+KLIM , M+KLIM ) = RN ( L , M+KLIM) * ( RKL-VAL ( M) *CI/RK0

)

CONTINUE

CONTINUE

ELSE

DO 836 L=1,KLIM

V = CMPLX (

( WAVE*SPAFRE* ( L-LIM- 1 ) ) **2,0.0)

52 = CSQRT ( CRSUB** 2 -V)

53 = CSQRT (CRS** 2 -V)

Al = CEXP (CMPLX (0.0, 2. 0

) *RK0*TSUB*S2

)

ASUB = ( (S2+S3)-(S2-S3)*Al)/( ( S2+S3 ) +( S2-S3 ) *A1

)

RKL = ASUB* S

DO 834 M=1,KLIM

DO 832 1=1 , KLIM

RM( L+KLIM, M) = RM( L+KLIM, M) + RKL* ( RN( L, I

) *T( I ,M) +

RN( L, I+KLIM) *T( I+KLIM,M) ) + ( RN( L, I

) *VAL( I) *CI/RK0*

T(I,M) - RN(L, I+KLIM) *VAL( I )*CI/RK0*T( I+KLIM, M)

)

RM( L+KLIM, M+KLIM) = RM( L+KLIM, M+KLIM) + RKL* ( RN ( L , I

)

T ( I , M+KLIM ) + RN(L, I+KLIM) *T( I+KLIM, M+KLIM ) ) +

( RN ( L , I

) *VAL ( I) *CI/RK0*T( I , M+KLIM) - RN( L , I+KLIM)

VAL( I

) *CI /RKO *T( I+KLIM, M+KLIM)

)

CONTINUE

CONTINUE

CONTINUE

END IF

SET UP THE RIGHT HAND SIDE VECTOR RV.

DO 840 L=1,MLIM

WKSPCE(L) =0.0

RV ( L) = CMPLX ( 0 . 0 , 0 . 0

)

CONTINUE

RV(LIM+1)=CMPLX( 2. 0,0.0)

F04ADF SOLVES THE SET OF SIMULTANEOUS EQUATIONS DEFINED

BY RM AND RV. SEE EQ. 21 OF APPENDIX III.

THE SOLUTIONS ARE RETURNED IN THE VECTOR AV.

IFAIL = 0

PRINT *, '4 SECTIONS OF RM MATRIX’

25
751 C

752

753 C

754

755 C

756

757

758 C

759

760

761 C

762

763

764 C

765 C

766

767

768

769

770

771

772

773

774 C

775

776

777 C

778 C

779 C

780

781

782

783

784

785 C

786 C

787 C

788 C

789 C

790

791

792

793

794

795 910

796 920

797 C

798 C

799 C

800 C

WRITE ( 1 ,*)' FIRST SECTION RM( 1 ,M) ,M=1, 45

PRINT *, (RM( 1,M) ,M=1, 45)

WRITE ( 1,*) (RM( 1,M) ,M=1,45)

PRINT*, ( RM( 1 ,M+45 ) ,M=1 , 45

)

WRITE ( 1

, * ) 'SECOND SECTION RM( 1 , M+45 ) ,M=1 , 45

WRITE ( 1

, * ) (RM( l,M+45) ,M=1,45)

PRINT*, (RM( 46, M) ,M=1,45)

WRITE(1,*) 'THIRD SECTION RM( 46 ,M) ,M=1 , 45

WRITE ( 1,*) (RM( 46, M) ,M=1,45)

PRINT*, (RM( 46,M+45) ,M=1,45)

WRITE ( 1 ,*)' FOURTH SECTION RM( 46 ,M+45 ) ,M=1 , 45

WRITE ( 1

, * ) (RM( 46 , M+45 ) ,M=1 , 45

)

IFAIL=0

CALL F0 4ADF (RM, MLIM, RV, MLIM, MLIM, 1,AV, MLIM,

WKSPCE , IFAIL

)

IF(IFAIL.EQ.O) THEN

PRINT* ,' SUCCESSFUL SOLUTION FOR (A) COEFFICIENTS.'

ELSE

PRINT* ,' FAILURE TO SOLVE FOR (A) COEFFICIENTS.’

END IF

PRINT*, ( AV ( L ) , L

,

' AV ' , L=1 , MLIM

)

RETURN

END **********************************************************

SUBROUTINE FCOEFF ( AV , FC , T

)

PARAMETER (NLIM=-22, LIM=22, KLIM=45, MLIM=90 ) COMPLEX AV ( MLIM ) , FC ( KLIM ) ,T( MLIM , MLIM

)

REAL VR, VI

REWIND 13

CALCULATE THE FOURIER COEFFICIENTS OF THE PSEUDO-OBJECT

AND WRITE THE RESULTS ON UNIT15.

STORE THE FIRST LAYER EIGENVECTORS IN MATRIX T.

DO 920 L=1 , KLIM

WRITE ( 15 , * ) AV ( L ) , AV ( L+KLIM ) ,

DO 910 M=1,KLIM

READ (13,*) VR, VI

T(L,M) = CMPLX( VR, VI

)

CONTINUE

CONTINUE

CLEAR THE FC VECTOR AND SET THE FC(0) VALUE TO -1.0.

CALCULATE FOURIER SERIES COEFFICIENTS FC USING THE ACOEFFICIENTS AND THE EIGENVECTORS. SEE EQ. 22 OF APPENDIX III.

26
801 C

802 DO 940 L=1 , KLIM

803 IF ( L . EQ . LXM+1 ) THEN

804 FC ( L ) = CMPLX ( - 1 . 0 f 0 . 0

805 ELSE

806 FC ( L ) = CMPLX ( 0 . 0 , 0 . 0 ) 807 END IF

808 DO 930 M=1 , KLIM

809 FC ( L ) = FC ( L ) + ( AV ( M ) +AV ( M+KLIM )

) *T ( L , M

810 930 CONTINUE

811 940 CONTINUE

812 C

813 C THE FOURIER SERIES FOR THE PSEUDO-OBJECT IS WRITTEN ON

814 C UNIT16, IN A FORMAT WHICH ALLOWS IT TO INTERFACE TO A

815 C PLANAR IMAGING OR OTHER PROGRAM.

816 C

817 DO 950 K=1,KLIM

818 WRITE (16,999) FC(K)

819 PRINT*, FC(K)

820 950 CONTINUE

821 999 FORMAT ( IX, 2F20 . 8

822 RETURN

823 END

824 C

825 C ********************************************************

826 C

827 SUBROUTINE IMAGE ( FC , PI , SPAFRE , TRA , DF , LSKIP , WAVE

828 PARAMETER (NLIM=- 22, LIM=22, KLIM=45, MLIM=90)

829 REAL PI , WAVE , SPAFRE , RR , RI , XPOS , TRA ( 2 )

,

830 2 RIM( -600:600) , IR,CAM( -100 : 100 )

, ELEC ( -500 : 500 ) ,X,YN,XI

831 COMPLEX FC ( NLIM : LIM ) , Cl , CCI , RT , FFC , FOC ( -KLIM : KLIM

832 C

833 c THIS SUBROUTINE COMPUTES THE IMAGE OF THE LINE ASSUMING

834 c ONE DIMENSIONAL OPTICS.

835 c 836 c THE IMAGE IS COMPUTED FOR 1000 POINTS OVER A RANGE

837 c OF +/- 0 . 5*THE PERIOD.

838 c 839 Cl = CMPLX (0.0, 1.0)

840 c 841 c IF LSKIP = 1 THEN READ IN THE PREVIOUS COEFFICIENTS.

842 c 843 IF ( LSKIP. EQ. 1) THEN

844 DO 105 I=NLIM, LIM

845 READ (16,*) RR, RI

846 FC ( I

= CMPLX (RR,RI)

847 105 CONTINUE

848 END IF

849 c 850 c OBJ = NUMERICAL APERTURE OF THE OBJECTIVE.

27
851 C

852 C

853 C

854 C

855

856

857

858

859 C

860 C

861 C

862 C

863

864

865

866 115

867 C

868 C

869 C

870 C

871 C

872

873

874

875

876

877

878

879

880

881

882

883

884 110

885

886 120

887 130

888 C

889 C

890 C

891 C

892 C

893 C

894 C

895 C

896 C

897 C

898 C

899 C

900 C

CON = NUMERICAL APERTURE OF THE "CONDENSER”

(ILLUMINATION APERTURE).

THESE PARAMETERS ARE PRESENTLY SET FOR THE NBS LASER SYSTEM

OBJ = 0.85

CON =0.17

FM = OBJ*LIM

FR = CON*LIM

THE DEFOCUS TERM DF ADDS A PHASE MODULATION TO THE

FOURIER SERIES.

CCI = CI*2 . 0*PI*SPAFRE

DO 115 I=-KLIM, KLIM

FOC(I) = CEXP( 2 . 0*PI*CI*DF* (WAVE*I*SPAFRE) **2

)

CONTINUE

THE IMAGE IS COMPUTED FOR AN ADDITIONAL 100 POINTS ON

EITHER SIDE IN ORDER TO ALLOW FOR CONVOLUTION WITH A

DETECTOR APERTURE OR IMPULSE RESPONSE.

DO 130 IX=-600,600

XPOS = IX/ (SPAFRE*1000.0)

RIM (IX) =0.0

DO 120 J=-FR , FR

RT = CMPLX ( 0 . 0 , 0 . 0

)

DO 110 K=J-FM, J+FM

IF( ABS(K) .GT.LIM) THEN

FFC = CMPLX ( 0 . 0 , 0 . 0

)

ELSE

FFC = FC ( K) *FOC ( K-J

)

END IF

RT = RT+FFC*CEXP ( CCI*XPOS*K)

CONTINUE

RIM (IX) = RIM( IX) +REAL( RT) **2+AIMAG( RT) **2

CONTINUE

CONTINUE

THE ARRAY RIM(

)

CONTAINS THE OPTICAL IMAGE PROFILE.

CONVOLVE THE IMAGE WITH THE DETECTOR LINE SPREAD

FUNCTION.

IF TRA(l) IS GREATER THAN 0.01 THEN A SLIT APERTURE IS

ASSUMED

OTHERWISE IF TRA( 2

)

IS GREATER THAN 0.05 THEN A VIDEO

CAMERA WITH A GAUSSIAN LINE SPREAD FUNCTION IS ASSUMED.

IF BOTH ARE ZERO THEN NO CONVOLUTION TAKES PLACE.

THE ARRAY CAM(

)

CONTAINS THE VIDEO CAMERA RESPONSE.

THE ARRAY ELEC(

)

CONTAINS THE ELECTRICAL OUTPUT SIGNAL

OF THE TRANSDUER.

28
901 C

902

903

904

905

906

907

908

909

910

911

912

913 140

914 150

915

916

917

918

919 160

920

921

922

923

924 170

925 180

926

927

928

929 190

930

931 C

932 C

933 C

934 C

935 C

936 C

937

938

939

940

941

942 195

943 C

944 C

945 C

946 C

947

948

949

IF(TRA(1) .GT.0.01) THEN

IR = TRA( 1

) *SPAFRE*500 .

IF(IR.GT.IOO) THEN

IR = 100

ELSE IF(IR.LT.l) THEN

IR = 1

END IF

DO 150 I=-500 ,500

ELEC ( I ) =0.0

DO 140 J=-IR,IR

ELEC ( I ) = ELEC ( I )/+RIM( J+I

CONTINUE

CONTINUE

ELSE IF(TRA( 2) .GT.0.05) THEN

DO 160 I=“100 , 100

X = 1/ ( SPAFRE*1000

CAM ( I ) = EXP(-0.5*(X**2)/(TRA(2)**2)

CONTINUE

DO 180 I=-500 , 500

ELEC ( I ) =0.0

DO 170 J=-100 / 100

ELEC ( I ) = ELEC ( I ) +CAM( J

) *RIM( I+J

CONTINUE

CONTINUE

ELSE

DO 190 I=-500 , 500

ELEC ( I ) = RIM(I)

CONTINUE

END IF

WRITE THE OUTPUT IMAGE DATA ON UNIT17

THE IMAGE IS NORMALISED WITH RESPECT TO THE INTENSITY AT

A DISTANCE OF 1/2 THE PERIOD AWAY FROM THE ORIGIN.

YN = AMAX1 ( ELEC ( 0 ) , ELEC (500)

DO 195 I=-500,500

XI = I / ( SPAFRE* 1000.0)

ELEC ( I ) = ELEC ( I ) /YN

WRITE ( 17 , * ) ELEC (I), XI

CONTINUE

THE FOLLOWING 'END OF DATA’ STATEMENT IS FOR THE

'DATAPLOT' GRAPHICS PACKAGE.

WRITE (17,*) 'END OF DATA'

RETURN

END

29
Appendix II

Test Cases for Thick Layer Imaging Program ( THKIMAG

)

30
Test

Case

#1:

THICK

DIELECTRIC

LAYER

WITH

VERTICAL

EDGES (6

um

wide

line

with

vertical

edges

patterned

m C/3 05 in w mH < 2 w J W u M M

fa

u 2 O M H

<d fa 05 Q fa 05 c Wh W fa 0 CQ O >« 2 S ' «< H u D — J Q o 2 fa J M H r<

<d ' < 2

rH

Q (H

H

N Pi < OHS fa 05

Q)

H C/3 o 2 >H

w

T3 OS 2 2 M < g W D £ 44 * fa c/3 ij m o c/i 0 — O H Q 2

°H ' H fa 2 W 'Q 2 05 fa 2 2 O H m S 2 H fa C O ' fa w Q W 0 2 — ' U H M Q U W 05 2 < H 3: •H t-J M H t-4 i< - rH fa < Q fa Oi H fa •H > M W —' IH M W < M 3 M M 2 fa£j • 5 05 Q 05 in m

44 • • U a) •H 4J o

(d o

4->

u • 4J o P t/i 3. n o o m w o o

VO • • • e <H O © O o oa

fd •H '****« rH U3 C •H O

01 • oo O r-I o O in

O

O • “ O O VO O O

(N • • • • • rH —> O O O O O oo oo O VO ^ O O 'T O

lo • o o • cm • rH • • tH • O — VO o o o i—1 OJ rO u") vo r—

31
Test Case #1: File FCOFPO: Fourier Coefficients of the

Pseudo-Object

1 -.09108273 .00519622

2 -.06079249 .01308535

3 .00478018 .01658667

4 .04455374 .00100212

5 .00810522 -.00558479

6 -.03495132 -.01667204

7 -.00672853 .00298502

8 .01767570 .02462243

9 .00840075 -.00504291

10 -.00193394 -.02052349

11 -.01237166 .00493059

12 -.00605733 .00957640

13 .01508445 -.00253726

14 .00599687 .00264588

15 -.01573390 -.00030316

16 -.00008822 -.01386078

17 .01502888 .00264068

18 -.01048221 .02557344

19 -.01386187 -.00424937

20 .02992448 -.04556850

21 .01290886 .00491631

22 -.10926479 .13891567

23 .39447293 .04366360

24 -.10926479 .13891567

25 .01290886 .00491631

26 .02992448 -.04556850

27 -.01386187 -.00424937

28 -.01048221 .02557344

29 .01502888 .00264068

30 -.00008822 -.01386078

31 -.01573390 -.00030316

32 .00599687 .00264588

33 .01508445 -.00253726

34 -.00605733 .00957640

35 -.01237166 .00493059

36 -.00193394 -.02052349

37 .00840075 -.00504291

38 .01767570 .02462243

39 -.00672853 .00298502

40 -.03495132 -.01667204

41 .00810522 -.00558479

42 .04455374 .00100212

43 .00478018 .01658667

44 -.06079249 .01308535

45 -.09108273 .00519622

32
CASE1

PROFILE

FOR

DN

THKIMAG

— ^r

— (M

•CD

-<S

CD

in LO

CD

(S

33

DISTANCE
TJ

0)cP

<L) -P

-P

tU

Coo

•H

r—

H

CO

(CJ

CO CO PS CO Ed CO fa

Eh Eh Z < Z fa tJ Ed cj M M

fa CJ z O M Eh

fa

PS

Q fa PS

Ed

H Ed Ed

m

O >n Z 2 -O < Eh D J Q Z Ed J M H*> i< Q PS O E-t Z

N3 PS

Ed

a

W CO

O Z Eh Ed

CO

C PS CQ Z h <C 2

Ed CO 0 Ed

D £ J <

cd <d fa CO

A CO CJ CO

Q O’ P —

O Eh

QD

fa *d CD * M fa Z Ed - CJ

CD >1 Z PS EdZZO PJ (CJ Eh co X 2 Eh fa

i< rH rH CD * Ed Ed

Q Ed

CJ aJ Z * CJ Eh

h

Q

M

U £ Ed PS Z «< Eh 15

E-* *«H JhEhJC PS -P •H Ed < Q fa fa Eh fa

Ed P E >w

H co-hh > CD 0 < H 5 H H iJ fa > P ^ ps a ps co co Z Z e • Eh A o m -P O

fa o 5 O 9 PS •H o Ed <D A

Eh c P

i4l -H o o J iH o o

P3 (D rH

O

OO < TJ O CD Eh -h P

Ed 5 • (d S O p VO £ -p o z 3. (CJ CO M A o Z VO c 3

Eh

— •H CO o O 1-H

OO <X>

OO . . o o o o o • • (N • • • • • fNJ rH — O O O O O

<DCO o i_n

(tJ

O lO

CJ • • O CM

-p

CO mo

O

CD 1/3 0 0 0^3*0

Eh . • O

O • (N OH • • <—I • — iflOwOO H (N m l/UO t'-

34
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

File FCOFPO : Fourier Coefficients of the

Pseudo-Object

.02598611 -.01092292

-.01353529 .00904450

-.01731358 .00311739

.01756685 -.00795909

.01356888 -.00057359

-.02047819 .00686085

-.01107900 -.00064297

.02327938 -.00598552

.00921602 .00126586

-.02648625 .00536360

-.00773381 -.00157148

.03061512 -.00501271

.00650267 .00169434

-.03646409 .00498365

-.00544231 -.00171108

.04564310 -.00541449

.00449766 .00167165

-.06227954 .00669869

-.00362787 -.00161367

.10149616 -.01032568

.00279663 .00156888

-.29965151 .02984478

-.14235833 -.46645156

-.29965151 .02984478

.00279663 .00156888

.10149616 -.01032568

-.00362787 -.00161367

-.06227954 .00669869

.00449766 .00167165

.04564310 -.00541449

-.00544231 -.00171108

-.03646409 .00498365

.00650267 .00169434

.03061512 -.00501271

-.00773381 -.00157148

-.02648625 .00536360

.00921602 .00126586

.02327938 -.00598552

-.01107900 -.00064297

-.02047819 .00686085

.01356888 -.00057359

.01756685 -.00795909

-.01731358 .00311739

-.01353529 .00904450

.02598611 -.01092292

35
CASE2,

ONE

LAYER,

3/20/87,

THKBEST

oiK-u<r j ujzcfl~H->-

36

DISTANCE
c

•H

(U <d / c

CO

u c w <u 0 o P Q -P p W <d 03 a Q <d M CO fH > 0) PS O' a D TJ TJ

CJ 03 •HX

as T3 0

Eh 0 •H M > 2 U3 c

PS CJ 0 w u

>H P •H < P iH • p •H H 5 CO 0) u -p M CD A! rd PS C U p

Eh °H •H +j O rH CO w P P p <u 3 w T3 £ CO M •H H Q 5 c

IS 0

ui P vs CJ cj 3. • •H M o rH

33 VO

H

Eh <d U) m

a)

ta

CJ

JJ

CO

<u

CO CO

OS CO

fa CO fa

>H EH Z < z 2 P W o M M

fa CJ as O H EH

fa

PSQ fa PS fa H W fa

CQO >h ac z - u < Eh

**“«*

p

p ZUJ IH

EH <

*< z

0^2

W Pi

H t/1 o z ^ u

tt! ffl Z H < S W p >-) J <

CU CO P CO

O CO

—

O EH P D

'H ft z w - cj as ccs uzso Eh mSKEHfa O v U w o w Z — - CJ Eh M P W OS 33 < Eh £ JhEhJC

fa < P fa fa Eh fa >-HI/]wHH

<C

H 5 HH

J ^ 5 OS Q OS U3 W • • oooo o o o• • CTi O M O

vsOOo O rH

OO tSOO • • O O VS O O

cm h*

rH

—

O

OOOO o o o o • • o o O VS <H O O H* O

IS • O

O • <N • rH • • rH • O VS O — O

O i—( OJ ro H* is vs t

3 7
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Case #3: File PARFIL: Parameter File Created

RUN PARAMETERS.

WAVELENGTH = .53

NUMBER OF LAYERS=9

WAVE NUMBER =11.85506661732

AIR LAYER =( 1. ,0.

)

SUBSTRATE =(4.1, .06)

DEFOCUS =0.

SLIT WIDTH IN MICRONS =.2

CAMERA WIDTH PARAMETER =0.

LAYER WIDTH POSITION RI OFFSET

1 6.003 .072 1.460 .000 .000

2 6.023 .144 1.460 .000 .000

3 6.065 .217 1.460 .000 .000

4 6.128 .289 1.460 .000 .000

5 6.211 .361 1.460 .000 .000

6 6.316 .433 1.460 .000 .000

7 6.441 .506 1.460 .000 .000

8 6.587 .578 1.460 .000 .000

9 6.754 .650 1.460 .000 .000

38
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

Case #3: File FCOFPO: Fourier Coefficients of the

Pseudo-Object

.03128047

-.00117774

-.04538472

.00285586

.04860053

-.01663746

-.02455180

.00370391

.01005389

.01510389

-.00775574

-.02538226

.01109421

.02493682

-.01569382

-.01660809

.01970374

.00248225

-.02276161

.02118083

.02524568

-.10579441

.38000129

-.10579441

.02524568

.02118083

-.02276161

.00248225

.01970374

-.01660809

-.01569382

.02493682

.01109421

-.02538226

-.00775574

.01510389

.01005389

.00370391

-.02455180

-.01663746

.04860053

.00285586

-.04538472

-.00117774

.03128047

-.04799119

-.02619416

.02843279

-.00315021

.01296409

-.01247821

-.02653567

.02884274

.01942759

-.02721254

-.00992680

.01395119

.00413423

.00192007

-.00211696

-.01617415

.00272111

.02957102

-.00472439

-.04960428

.00671538

.14143811

.04024071

.14143811

.00671538

-.04960428

-.00472439

.02957102

.00272111

-.01617415

-.00211696

.00192007

.00413423

.01395119

-.00992680

-.02721254

.01942759

.02884274

-.02653567

-.01247821

.01296409

-.00315021

.02843279

-.02619416

-.04799119

39
CASE3N,

2/22/87,

THKBEST

OQ»I” mUC 1 > 2 UJ 2 CO *•* h- ^

CD

CM

CM

I

CD

I CM LO LO ®

0“ OCLl— •— 2 LU 2CO •-•—>-

40

DISTANCE
TJ

d

03 0 C u P •rH 03 rH

ft

ft 03

fO

m ft fO w

ID 03

d

p 03 0 w P

TJ P p 03 03 < >i

cj rH o3 M 03 i—1

Eh C3 PS •H P

ft ft 3 > P •H

0)

g

a > 0

Ehp

H

a

a 5 p o

•H

OS 5 ft w CJ

03 •H < C A p •H P

rH

p P • < d) 3.

Eh T3 03 a •H VO P 2 • 03 o P

ft S P

CJ 3L 03 03 M ft a C d

Eh •H 03 aw

03 CJP0) 03 H

C/3 PS

ft C/3 >H EH C Z P ftM

ft

O o H

ft

a

p

p

ft

H

wo s - CJ

ft<Zso z w

Eh * i< P PS O eh H C/3 ps m

ft P >> — O - H ft a ps

Eh on O " Z '

ft PS Z

ft

H Eh

ft *=C P > — M

<C

M £ 5 PS

C/3 mwaMa

Eh

PSw

>•p

N PS

ft Z >* M *CP w ft

CJ Eh

*C Eh P <

ft ft

C/3

—

M

M P PS <

PS

ftO C/3 P- CJ a o

Eh ft P ft M P

Eh ft M M P ft

C/3 C/3

OOO

O

O• • i—I o o o

VOO

O rH

O

O

O

O

O . • O O VO o o

04 • • • • • rH O o o o o oo

Lnln

CN o o m o o o '3* o

in • O

O • <N • I—I • • «—t • o —— VO o —* o o h in m ^ i/i vo r41
Test Case #4: File FCOFPO: Fourier Coefficients of the

Pseudo-Object

1 .02870492 - . 00727836

2 -.03936288 -.01604885

3 .00858718 .02754465

4 .02527876 .00266374

5 -.01939032 -.03161583

6 -.00720966 .00412280

7 .01258914 .04300149

8 -.00154677 -.02651014

9 .00149106 -.03863546

10 -.00367560 .04706727

11 -.00975738 .02263821

12 .01454326 -.05862991

13 .01080683 -.00255415

14 -.02546317 .06130368

15 -.00753362 -.01701846

16 .03522425 -.05701120

17 .00344633 .03333270

18 -.04739825 . 04842201

19 -.00070328 -.04416899

20 .07514600 -.04152960

21 -.00182714 .04869017

22 -.21542281 .06822844

23 -.26974701 -.56211569

24 -.21542281 .06822866

25 -.00182724 .04869002

26 .07514611 -.04152962

27 -.00070319 -.04416899

28 -.04739853 .04842213

29 .00344651 .03333271

30 .03522438 -.05701149

31 -.00753393 -.01701814

32 -.02546294 .06130364

33 .01080669 -.00255426

34 .01454345 -.05863000

35 -.00975762 .02263852

36 -.00367547 .04706706

37 .00149104 -.03863546

38 -.00154668 -.02651021

39 .01258891 .04300184

40 -.00720944 .00412244

41 -.01939040 -.03161581

42 .02527873 .00266401

43 .00858720 .02754443

44 -.03936286 -.01604884

45 .02870492 -.00727828

42
CASE4,

2/22/87,

THKBEST

o cl t— •— cj <r „j ^Zh-ujzcn^*—

v

— ^

— CNJ

CD

-®

_C\J

_

CD

I L0 m cm

0“

o. t- *h u cj •—'Zi— uizrcn*—

43

DISTANCE
0

faoP

faPu>

OSD

cj

asH

osu

>«pp<

Ehw2

cjMX

Eh cH

T30GU0PP

HJa

co0

(CJ ao0

>i

<d o

vo co

cj

vo

O fH•H

(0 CO

C/3 C/3 OS C/3 fa 0 fa

>H EH Z c z 2 P w CJ M M

fa o 3S O w EH

fa OS P fa OS W H w fa

CQ o >c 3S 2 * CJ < Eh D ^ P P Z fa P M PoM

OSw

fa C/3

*CzH fa OS

—

O eh P ZH

C/3 Eh <

OS

fa

CJ C/3

v

D

T30

yif •H

»-

M fa Z W * CJ

as os wzKO Eh n 2 2 Eh fa T5 o 0 •* fa fa P fa 0s

•H Z ~ v CJ Eh

H P

Tl fa OS

X < EH 3 P H Eh P *• c fa < Q fa fa Eh fa

O 0u

•H > —

M 0 ' M

M < H 5 W H P fa Z 3 OS P OS 0 0 P rH • • • •H •H 5 CO 0p Oo 0 (CJ C u J-l o

•H •H

p

rH CO p ia o o 0 3 o o G CO •H 5. (Jl O (MO

voo

lD0

CO<0 CJP

CO0

EH

O rH

OO

OOO • • O O VO o o

CNt—I o o o o o oo

in

in<N o o m o o o ^ o

in • o o • cn • rH • • i-H • O VO O —' o o h (N n in vo t

44
Test Case #5: File PARFIL: Parameter File Created

1 RUN PARAMETERS

2 WAVELENGTH = .53

3 NUMBER OF LAYERS=9

4 WAVE NUMBER =11.85506661732

5 AIR LAYER =(1. ,0.

)

6 SUBSTRATE =( 4.1, .06)

7 DEFOCUS =0

8 SLIT WIDTH IN MICRONS =.2

9 CAMERA WIDTH PARAMETER =0.

10 LAYER WIDTH POSITION RI OFFSET

11 1 6.002 .067 1.400 2.550 .000

12 2 6.020 .133 1.400 2.550 .000

13 3 6.056 .200 1.400 2.550 .000

14 4 6.109 .267 1.400 2.550 .000

15 5 6.180 .333 1.400 2.550 .000

16 6 6.269 .400 1.400 2.550 .000

17 7 6.376 .467 1.400 2.550 .000

18 8 6.500 .533 1.400 2.550 .000

19 9 6.642 .600 1.400 2.550 .000

45
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

File FCOFPO : Fourier Coefficients of the

Pseudo-Object

02859869

01777494

01575921

01125702

03861030

05863673

00882784

04261764

02643598

01766390

01759161

00861215

00318939

01427936

00661590

02942126

00859497

05174989

00051795

08337158

01442410

20644118

27539556

20644114

01442399

08337133

00051788

05174964

00859462

02942143

00661589

01427944

00318921

00861184

01759189

01766396

02643602

04261751

00882814

05863689

03861020

01125717

01575919

01777511

02859880

01883469

05351227

07268917

04988998

00407903

02278771

01948456

00856627

00272235

00659771

01642617

00775205

01494420

01811821

00686578

02111680

00085589

02242236

00098279’ 03235164

00796601

07115434

53212378

07115399

00796627

03235159

00098292

02242235

00085587

02111668

00686571

01811798

01494381

00775220

01642605

00659770

00272207

00856600

01948459

02278783

00407919

04989043

07268954

05351234

01883462

46
CASE5,

2/22/87,

THKBEST

ocLH*^o<r_i

o- Q£Lh«(J<I J *-• Z I— LLl Z CO •-> I— >

47

DISTANCE
Test

Case

#6:

THICK

POLYSILICON

LAYER

WITH

CURVED

EDGES c

•H

r)ocu

<D V

4-> (0a co

C/3 a >h O' <u

a) TJ

<u>

Sh3oa4-J •H5

<DG

<u •H5

C/3

X

PS C/3 W 03 >4 Eh < Z W

.-3 fa CJ H H

fa

U z O M Eh c fa 0 os a fa PS o fa H fa fa •H 03 O >H S

rH 2 *U < Eh •H D —

J Q

CO Z W

J

H

Eh

C

fd z -— Q PS M N PS <

>i

fdGOO

•H

rH•H

CO>1 rHoao

•H

.c

4-»

VO

VO—r fd

(U V

fd

>H 4J

CO .Q3

CO 2O O Eh M CO

PS 03 fa D X CO

J

—

O

•>

H OU

X

X

Eh0 ' Z — - XXX i-3 h Eh W < Q > — M < H 3 5 PS

W OS

><

w < 2

GJ

m

C/3 EHZW2

fa fa O Eh <C Eh

tJ <

fa fa

C/3 H

M

Q

X

u

xD- U X o

Eh fa Q W H Q

Eh

X

wM

iJ fa X X

Ooo o o o

OlOfNO

VOO

O rH

O

O

OOO • • O O VO o o

fvj ^ • • • • • rH -—- O O O O O Ooo o m o o o oo o ln • o o • cm • r—t • • m • o — ‘O o o o h <n m in vo

48
Test Case #6: File PARFIL: Parameter File Created

1 RUN PARAMETERS

.

2 WAVELENGTH = .53

3 NUMBER OF LAYERS=9

.

4 WAVE NUMBER =11.85506661732

5 AIR LAYER = ( 1 . , 0 .

)

6 SUBSTRATE =( 4.1, .06)

7 DEFOCUS =0

.

8 SLIT WIDTH IN MICRONS =.2

9 CAMERA WIDTH PARAMETER =0.

10 LAYER WIDTH POSITION RI OFFSET

11 1 6.002 .067 3.800 .100 .000

12 2 6.020 .133 3.800 .100 .000

13 3 6.056 .200 3.800 .100 .000

14 4 6.109 .267 3.800 .100 .000

15 5 6.180 .333 3.800 .100 .000

16 6 6.269 .400 3.800 .100 .000

17 7 6.376 .467 3.800 .100 .000

18 8 6.500 .533 3.800 .100 .000

19 9 6.642 .600 3.800 .100 .000

49
Test Case #6: File FCOFPO : Fourier Coefficients of the

Pseudo-Object

1 -.06445069 .00175338

2 .10217661 -.01417309

3 -.06446184 .01696297

4 -.03134388 -.00120886

5 .07807896 -.01799169

6 -.01812634 .01902242

7 -.06195582 -.00139914

8 .04708281 -.01378403

9 .03549830 .01048018

10 -.05741779 .00419147

11 -.01086065 -.01166372

12 .05640716 .00465509

13 -.00874018 .00740007

14 -.04823824 -.01104616

15 .02114851 .00151754

16 .03683013 .01295479

17 -.02419279 -.01478962

18 -.02814956 -.00686661

19 .01508793 .03343811

20 .03801847 -.01927935

21 .00830037 -.06197198

22 -.16673161 .17059941

23 -.31875250 -.22803972

24 -.16673161 .17059941

25 .00830037 -.06197198

26 .03801847 -.01927935

27 .01508793 .03343811

28 -.02814956 -.00686661

29 -.02419279 -.01478962

30 .03683013 .01295479

31 .02114851 .00151754

32 -.04823824 -.01104616

33 -.00874018 .00740007

34 .05640716 .00465509

35 -.01086065 -.01166372

36 -.05741779 .00419147

37 .03549830 .01048018

38 .04708281 -.01378403

39 -.06195582 -.00139914

40 -.01812634 .01902242

41 .07807896 -.01799169

42 -.03134388 -.00120886

43 -.06446184 .01696297

44 .10217661 -.01417309

45 -.06445069 .00175338

50
CASE

6,

2/22/87,

THKBEST

O Q_ <1

J

51

DISTANCE
Appendix III

Reprint - "Modeling of the optical microscope image of lines

patterned in thick layers with variable edge geometry," by

D. Nyyssonen and C. P. Kirk (submitted to J. Opt. Soc. Am.).

52
Modeling of the optical microscope imaging of lines

patterned in thick layers with variable edge geometry*

* * Diana Nyyssonen

National Bureau of Standards, Gaithersburg, MD 20899

Chris P. Kirk+ Vickers Instruments, Haxby Road, York. Y03 7SD . England

and

Department of Electrical and Electronic Engineering,

Leeds University, Leeds. LS2 9JT . England

Abstract

A monochromatic, waveguide model is presented which can predict

the optical microscope images of line objects with arbitrary

edge geometry, patterned in thick-layers including multilayer

structures with sloping, curved, and undercut edges, granular

structures such as lines patterned in polysilicon, as well as

asymmetric objects. The model is used to illustrate the effects

of line edge structure on the optical image. Qualitative agreement with experimentally obtained optical image profiles is demonstrated. Application of the model to study the effects of

variations in layer thickness and edge geometry on linewidth

measurements made at different stages of manufacturing integratedcircuit devices is discussed.

* Contribution of the National Bureau of Standards; not subject

to copyright.

This work was performed under the Research Associate Agreement

(CN-454) between the National Bureau of Standards and Vickers

Instruments, July 1984.

** Currently with CD Metrology, Inc., Germantown, MD 20874.

* Currently with KLA Instruments, San Jose, CA 95131.

53
INTRODUCTION

During the manufacture of integrated circuit wafers, critical

dimensions of certain features must be measured at different

stages of the production process. The control of the critical

dimension of linewidth is usually monitored by using an optical

or scanning-electron microscope. The measurements made by a correctly aligned microscope fitted with an appropriate measurement attachment are usually limited in accuracy and precision not

by instrumental errors but by the lack of understanding of the

characteristics of the image profile and the lack of accurate

edge detection algorithms.

In the past, optical imaging in the microscope has been described

by the scalar theory of partially coherent imaging [1-3] which

characterizes the object by a planar complex transmittance or reflectance function. However, this approach does not accurately

predict either the scattered field or the image structure for

micrometer-sized line objects thicker than approximately onequarter of the illuminating wavelength [3]. Unfortunately, most

of the line features to be measured during the production of

integrated circuits fall into this category of thick objects.

Typical line objects are patterned in layers with thicknesses

ranging from 0.3 to over 1.0 um (e.g. photoresist).

Integrated-circuit features encompass a wide variety of materials

including dielectrics and absorbing materials of high and low

conductivity (e.g., semiconductors in crystalline and amorphous

states with varying amounts of dopant impurities, refractory

metals, etc.) and line geometries with edge shapes varying from

vertical to the complex shapes shown in Fig. 1. These lines may

be etched into layers and be situated on top of metal or dielectric layers. Because of this wide variation, most of the approaches found in the literature are of only limited usefulness.

54
Scattering from micrometer-sized thick objects with complex indices of refraction falls in the domain of scattering by objects

whose size is comparable to the illuminating wavelength such as

Mie scattering [4] and requires an electromagnetic field treatment to accurately predict the scattered field. The various approaches that have been used to treat scattering by objects on

the order of a wavelength can be found in the large volume of

papers primarily in the areas of scattering by particles [5],

scattering by surfaces with defects or protuberances [6], or

diffraction by gratings [7]. Many of these approaches are limited either to objects of specific shapes [4,8] or weakly scattering objects which satisfy either the Rayleigh [9] or Born

[10] approximations, or require infinite conductivity or specific

grating geometries. Many are mathematically cumbersome when applied to the wide range of object shapes and materials of present

interest.

Recently, the problem of imaging of line objects with vertical

edge walls patterned in thick layers of dielectrics and metals

was treated by Nyyssonen [3,11] who developed a model based

on the waveguide analysis for thick dielectric structures of

Burckhardt [12]. Burckhardt's model was originally developed

for diffraction from sinusoidal dielectric gratings and was

applied to the reconstruction of bleached holograms. Kaspar

[13,14] extended Burckhardt's approach to include absorbing

photographic emulsions and non-sinusoidal grating structures and

applied this approach to contact printing as well. Kaspar ' s work

related the dielectric constant of the grating to the photographic density of the film.

The waveguide model previously used by Nyyssonen [3,11] assumes

that the line structure is patterned in a nonmagnetic layer which

can be characterized by its complex index of refraction which is

taken to be constant with depth within the layer. Thus, this

model can be used to represent homogeneous line structures with

55
vertical edge walls. The spatial function representing the variation in the dielectric constant ( square of the complex index of

refraction) in the layer is expanded in a Fourier series. When

this is substituted into the wave equation. Hill's equation [15] * is obtained. The eigenvalue solutions to this equation represent

waveguide modes. Assuming a single incident plane wave normal to

the surface, the boundary value problem at the layer interfaces

is solved to determine the Fourier coefficients in the expansions

for the transmitted and reflected fields. This method allows the

use 'of conventional scalar imaging theory to compute the image

when no polarization effects are present. In such a case, the Eand H-field components are equivalent and either may be used in

the scalar imaging equations.

In this paper, the waveguide model is extended to line objects

whose index of refraction and geometry vary with depth in the

layer, thus enabling line objects with nonvertical edges and

multilayer structures to be considered. This extension of the

model allows the images of virtually all line structures commonly

encountered in IC fabrication to be modeled.

Thick-layer Model for Variable Edge Geometry

The method of solution for line structures of arbitrary edge geometry whose index of refraction may vary with depth in the layer

is an extension of Nyyssonen's earlier method used for vertical

edge walls discussed above. A line object of the type to be

treated in this paper is shown in Fig. 2a. This line structure

has been chosen because it contains different materials, curved

edges, asymmetry, and an unpatterned sub- layer. The line structure in the patterned layer varies in both width and dielectric

constant as a function of depth z within the layer. Figure 2b

shows how this line structure may be approximated by a multilayer

set of line objects each of which has a constant index of refraction over a small interval of depth z. The line may thus

56
be represented by a set of layers with the complex dielectric

constant e in each layer represented by the Fourier series

expansion of:

/ 1 , between the lines

where

:

n n

= refractive index of the nth layer of the line structure,

= width of the nth layer, and

= offset of the nth layer in the x-direction. (See Fig. 2b)

Nonperiodic structures are taken as periodic with a very large

repeating distance (period).

The eigenvalue solutions for the electromagnetic field components

are found independently for each layer.. The boundary value equations at each interface form a set of complex matrix equations

which when solved allow the substitution of a single "equivalent"

scattering layer for the multilayer structure. Thereafter, the

solution for the transmitted and reflected field components and

image are found in the same manner as for the single layer case

[3,11].

Following Burckhardt's method [12] and assuming monochromatic

illumination where E - we start with the inhomogeneous

wave equation for either the E- or H-field [16]:

/\ yiy 2 E + euk^E + grad ^

• grad e

e = 0

S H + euk^H + o—

grad e A

s

x curl H = 0

( 2 )

(3)

57
with k 2 eu - k 2 n

2 for nonmagnetic media,

o o

If we choose the (x,y ,z) -axes as we have done so that e is constant along the length of the line in the y-direction, for

TE-mode ( E - E

yy ) and TM-mode ( H = H^y ) respectively, these

equations simplify to

V

2 E + k 2 eE = 0

y O' y

( TE-mode

)

(4)

and

+ k 2 eH = 0

o y

( TM-mode

)

(5)

In the following analysis we consider only TE-mode because in the

experimental NBS laser linewidth measuring system, the direction

of polarization of the laser source is aligned parallel to the

y-direction in the sample plane and the x-direction is the image

scan direction. See Fig. 3. However, for any other system configuration and mode of illumination, the field can be considered

as a superposition of TE- and TM-modes [17] and the corresponding

analysis for TM-mode will be required. See ref. 12. Note that

these modes are determined with respect to the orientation of the

structure (not the incident wave) and are uncoupled (TE-mode: E^,

H , H , TM-mode H

x z

The solution for the TE-mode is taken in the form E (x,z) = X(x)

A A ^

Z(z) and since e=e(x), the method of separation of variables using a separation constant a results in

+ k 2 e ( x)X( x) + a2 X( x) = 0 ( 6 )

58
|if = a2 Z( z

)

(7)

These equations hold for any functional form of e(x). However,

A

following Burckhardt, we choose the Fourier approach where e (x)

from eq. (1) is represented by the Fourier series

e n

( x) “ 21 E

q n exp( 2niqx/P)

q

where

E

q,n

(W

n

/P) ^n

2 + d“VP) V

A A

( V"V } qn

exp ( 2ni A a/P ) sin( qnW /P

)

n n

( 8 )

q=0

q^O

where P is the period chosen arbitrarily large when the object

consists of a single line and q is the summation index. Substituting eq. (8) into eq. (6) yields the differential equation

known as Hill's equation

d

2 X(x) , ax2 + H E

q _ nexp ( 2n.igx/P

)

- q

X(x) + a2 X( x) 0 (9)

The solution for this general form of Hill's equation is given by

Kaspar and others [13,15]. The form of the eigenvalue matrix is

given in the Appendix. The solution of eq. (9) for the n-th

layer is of the form

Ey(X,Z) A exp (a z)

m, n m,

+ A’ exp (-a z

)

m , n m , n

m

Z B. m

exp( 2nijx/P)

j ,m,

j

( 10 )

59
where the a ' s are the eigenvalues and the B ' s are the

m,n j ,m,n

eigenvector solutions to Hill's equation. The A and A' are

m,n - m,n

weighting constants which must be determined from the boundary

conditions . Each of these terms represents an inhomogeneous

plane wave or waveguide mode which is supported by the line

structure. Note that, when there is no absorption, (n real) the

eigenvalues are purely imaginary and this form reduces to that of

Burckhardt

The solution for the transmitted and reflected fields is found by

equating the tangential components of the E- and H-fields at each

layer interface. The x-component of the H-field is

found from ik

Q

H = curl E or, here, ik

Q

H

x

= - BE^/dz:

H"(x,z) =

" exp ( -cl „

z ) m, n

B . m exp( 2ni jx/P )

:,m,n ( 11 )

With the tangential components of the incident field given by:

Ey = E^exp(ik

Q

z)

H

x

= *Ejexp(ik0

z),

the tangential components of the reflected field by:

= x E

j

exp - ik

o (f

1

)

x + K

?

z

_ j

(12a)

(12b)

(13a)

60
H*(x,z) (13b) = £ KjEj expj -ik

o [(fi) x + K*z]

where K' i -/Al 2

and the tangential components of the transmitted field by:

?-]| Ey(X,Z) = X j [(P

1

)

T

x + Kyz (14a)

H^(x,z) = X

" K

j

E

j

exp

j ik

c p

1

)

x + K

j

2 14b:

where K ft 2 - A1

S \P

. \ 2

n„~ and n is the complex index of the

substrate.

For each Fourier component j , the following boundary equations

must be satisfied:

1. At z=0 ( x=y=0

)

, the solution must match the tangential

components of the E- and H-fields at the top surface:

E

I

6 .

o]0 + E*T = S [

Am,l + A

m, ll

m

L J

B

j ,m,l' (15a)

B

j ,1X1,1'

where =

1 j = 0

0 j i* 0.

61
2. At the interface between the n-th and (n+l)-th layer,

z = -Z (x=y=0), the E- and H-fields must satisfy the boundary

conditions at each interface:

E

m u A exp (a Z ) + A' exp (-a Z ) m,n m, n n m,n m,n n

B . mri 3 , m , n

(16a)

= E A

»,n+lexp,am I n+ ln ) + A;,n,leXpl >,nnZ n> B j,m,n+1 ; m «- "* ? [Vn (^) eXp( VnV ' A

;,n (^)

=

[

Am,n+1 ( -ik

1

)

exp(am,n+l Z

n

} m L ' /

exp (-a Z ) m , n n

- Am,n+l

/ am, n+l\

\

- ik

o )

exp(a

m ,n+l Z

n'

B . 3 ,m,n+l 16b)

3. At z = -ZN

= -T ( x=y= 0 ) , the solution must satisfy the

boundary conditions imposed by the substrate (assumed infinite

in extent - see Fig. 3):

z

m

A Mexp (a MT ) + A ' Mexp ( -a T

)

m,N m,N m,N m,N

= E^exp(-ik

o

K^T) ;

B

j ,m,N (17a)

y~ \ A /Seln')

m 1-

x exp ( -cl MT ) - A

' m, N m

xt ( ) exp ( a vtT ) B . ,N y-ik

o J * m,N j,m,N

= -KjE^exp(-ik

oKjT) . (17b)

62
In matrix notation

where

B(Z) =

n n

B^Z ) B^ 2

(Z ) n n n n

B

Z1

(Z ) B

n

2

(Z n

} n n n n

and the matrix elements are given by

B

'!’ 1

(Z ) 3,m,n n exp (-a Z ) B

.

m,n n j ,m,

n

B

i 2m n

(Z n

: ^ f m /Fi n exp (a Z )B. m,n n m n

^ (2 ) = ( -?/

n

j exp ( -a Z )B. ,m,n n y-ik

J

F m,n n j,m,n

22

B . (Z ) = 3,m,n n -(^)exp (a Z ) B

.

m,n n 3 ,m,

n

(18a)

(18b)

(18c)

63
Depending upon whether the solution for the transmitted or reflec

ted field is desired, all of the A and A's are eliminated from

n n these equations except for either A^ and A^ or A^ and A^ , i.e.,

for the transmitted field:

B

1 (Z

1

} W b

2

(z

2

) -1

BN-1 (ZN-l } )

-1 VW

or, for the reflected field:

= (b

n

(zN-l } J BN-1 (ZN-1 BN-l (ZN-2 }

(19a)

(19b)

B

2

( V

-1

B

l

( z

l)

This operation replaces the multilayer structure with an equivalent layer characterized by the matrix B.

The equations are now solved as for the single layer case by

R T eliminating the unknown Ej amd Ej to solve for the A^s or A^s

again depending upon whether the Fourier expansion of the reflected or transmitted field is desired. In the present paper,

we compute the reflected field coefficients.

(

B

1 (0)

A

A

1

( 20a)

64
( 20b)

Using the format of eqs. (18) this reduces to the single matrix

equation:

where

11

?2

D 7

3

and

K

j

B

12

+ B

22

E

o

(1 + K*)6

( 21 )

where the desired coefficients E* are found from eq. (15a)

65
( 22 ) j,m,l " E

o

5 jO

These coefficients represent the Fourier coefficients of a pseudo

or equivalent (planar) object. The microscope image of the reflected field is then computed using scalar imaging theory (See

Ref. 3.) with the line object reflectance function given by

This method of computing both the reflected field and the corresponding microscope image requires no approximations of the type

usually found, such as limits on the conductivity or slope of the

surface, etc. Limitations may be imposed, however, by the computation capability available. First, the number of layers used to

approximate the structure increases the computing time linearly.

In most of the cases to be shown here, seven to nine layers were

sufficient to produce significant results and required approximately one minute of CPU time on a Univac 1108.

The second limitation is in the truncation of the series, i.e.,

the matrix sizes used in the computations. In the present case,

as for a single layer [3], all of the reflected plane waves are

included which have diffraction angles less than + n/2 in air.

With P = 12um and A = 0.53um, 22 diffracted orders are included

which requires a 45 x 45 complex eigenvalue matrix and a 90 x 90

*Certain commercial equipment, instruments, or materials are

identified in this paper in order to adequately specify

experimental or computational procedure. Such identification

does not imply recommendation or endorsement by the National

Bureau of Standards, nor does it imply that the materials or

equipment identified are necessarily the best available for the

purpose

t ( x

)

E? exp (ik

Q Ajx/P) (23)

j

*

66
complex matrix for inversion of the B matrices in eqs. (19).

This choice necessarily truncates the series which represents the

field in the layers with higher refractive index. In the cases

considered here, this truncation does not appear to significantly

affect the results.

Also, for grating objects with P £ 12|im, the computations are exact. However, for isolated line objects near resonances (where

the thickness times index of refraction of the patterned layer

is approximately equal to the wavelength) , P = 12um is not large

enough to eliminate the effect of the adjacent lines on the image.

Larger matrix sizes would have to be used where P > 12 urn is

required.

The Effect of Geometry on the Image Profile

The present method was developed primarily for the purpose of

modeling the effect of nonvertical edges on the image profile.

There is a very wide range of edge geometries which can be modeled, therefore, only a few key examples will be presented here

in order to demonstrate trends. Figure 1 shows six basic shapes

which are frequently encountered in integrated circuit processing. In order to simplify the definition of these shapes, they

will be represented by a polynominal expansion which defines the

width as a function of z.

J

W(z) = X X

j

(z-Z

j

)j ' (24)

j=0

where J = the polynominal order,

Xj = coefficient of the j-th order, and

Zj = offset of the j-th order.

67
The shapes in Fig. 1 have been restricted to fifth order polynomials (J<5). In these examples it will be assumed that the patterned layer is 0.6 um thick, and the nominal linewidth X

Q

is 6.0

urn. The line consists of loss-free silicon dioxide with a real * refractive index of 1.46. The substrate is taken to be silicon

with a complex refractive index of 4.1 + 0.06 i at an illumination

wavelength of 0.53 um. The diffraction-limited images have been

computed for a 0.14 NA illumination aperture and 0.85 NA objective aperture. Note that in this polynominal representation of

the line object, although X

Q

as shown in Fig. 1 is the nominal

linewidth, X

Q

is: the width at the top ( z=0 ) in cases (b) and

(g); the width at the bottom (z=d) in cases (c), (d), and (f),

and the mean width in cases (e) and (h)Figure 4 shows the theoretical image intensity profile of an ideal line with vertical

edge walls. The oxide line extends from -3.0 to +3.0 um as shown

by the dashed lines, and the rest of the object is bare silicon.

In the calculation of the Fourier series coefficients, this

structure is assumed to be repeated with a period of 12.0 um.

The line edge image is characterized by a dark fringe wider than

the interference fringe that would be calculated for a similar

thin layer.

The set of image profiles in Fig. 5 demonstrates the effect of

different edge geometries on the image. The line has been assumed to be symmetrical and so only half the image profile is

shown. The line objects are defined in terms of Eg. (2) and the

width of the line edge is getting progressively broader down each

of the three columns. There are two effects which are common to all three geometries. As the edge becomes broader, the

dark fringe associated with the edge also becomes broader with

the bright fringes on either side becoming brighter than occurs

with partially coherent imaging of planar objects. When the edge

becomes very broad, the dark fringe itself begins to broaden out

and small peaks form within it.

68
Modeling Granular Materials

According to the method described so far, the object is assumed

to consist of homogeneous slabs of materials. This assumption is

implicit in Eq. (1). In practice many materials such as polycrystalline silicon cannot be regarded as homogeneous as they

have a definite internal structure which will result in a variation in refractive index within the material. These structures

can be modeled by representing the random refractive index disturbances within the material by functions.

J

e'(x) = e ( x) + C . 6 ( x-X . ) , (25)

n n £—j j 3 j=l

where:

e n(x) = refractive index profile from Eg. (1),

Cj = amplitude of j-th refractive index disturbance, and

Xj = position of j-th disturbance.

The function e (x) now describes a noisy layer, and by restricting the range of values for Xj , the noise may be confined to different parts of the layer.

Figure 6 shows three image profiles for a noisy structure. The

object is the oxide structure used to produce the profile in Fig.

4 and noise has been added to the entire layer. In each of the

three cases shown in Fig. 6, different sets of noise data have

been used corresponding to different sets of Cj selected from a

random number table. The most striking feature of these curves

is that, although the field is perturbed within the area of the

object, the dark fringe at the edge of the line remains well defined, and relatively unperturbed.

69
Comparison with Experimental Results

In order to test the model against experimental image profiles,

a test specimen was prepared which consisted of lines patterned

in photoresist on a silicon substrate. The same test pattern was

put down repeatedly by projection printing but with a range of

focus positions. This produced a series of patterns with a range

of edge slopes. Two patterns were selected to illustrate different edge properties. One pattern corresponded to the in-focus

exposure and had near vertical edge walls; the second corresponded to a considerable defocus on exposure and had significantly

sloping edges.

The in-focus exposure was assumed to produce near vertical edge

walls, and when the structures were examined in a scanning electron microscope, this was found to be the case. The edge walls

had slopes of about 80°, which according to the theory may effectively be assumed to be vertical for films of submicrometer

thickness. The width of the window in the resist was taken as

2.5 um.

The out-of -focus pattern had significant edge slopes, and the

structure was modeled by the asymmetric third order window shown

in Fig. 7, based on SEM pictures of the cross section. The

photoresist thickness was not known exactly but was nominally

1 um. The refractive index of the resist is not known exactly,

but this is not serious as small changes in the refractive index

can be offset by small adjustments in the layer thickness without

significantly changing the image profile. The refractive index

of the photoresist was assumed to be 1.513. The theoretical

layer thickness was adjusted until it produced the same contrast

as the experimental data; this gave a thickness for the resist of

0.94 um.

It is very difficult to accurately determine the position corresponding to "best focus" when viewing these thick structures in

70
the optical linewidth measurement system. Therefore, it is necessary to generate a series of profiles as a function of focus

position and compare these with the theoretical data. Figure 8

shows a comparison of the experimental and theoretical image profiles produced by these structures. The experimental data were

generated using the coherent optical linewidth measurement system

which has been described in detail in the literature by

Nyyssonen. [ 3

]

The profiles for the sloping edge structure are quite different

from those for the vertical edge structure. However, in both

cases they show good agreement with the models. The theoretical

profiles display the same features as the experimental profiles,

but the agreement is not perfect. One significant source of differences is the assumption that the resist is homogeneous. This

is not strictly true. To get better agreement, it would have

been advantageous to use a material such as silicon dioxide which

is homogeneous and which would produce a stable and accurately

known refractive index.

A second source of differences may arise from the assumption that

the illumination is normally incident. As shown in the experimental data, the central peak of the sloping edge profile moves

as the object moves through focus and the line with nearly vertical edges generates slightly asymmetric profiles (See Fig. 8).

Both of these may be attributed to slight asymmetry in either the

illumination system or imaging optics. The NBS profiling system

used to generate these profiles was checked for illumination symmetry when profiling thin layers. However, it is expected that

these thick layers are significantly more sensitive to asymmetric

illumination than is a thin-layer object.

Case Study of the Production of an MOS Transistor

This section considers some typical structures which are encountered in microlithography and the effect of small variations in

71
these structures on the optical image profile. The two structures shown in Fig. 9 are considered. These structures represent

the key stages of patterning the polysilicon layer when making an

MOS device. The structure in Fig. 9(a) represents the stage between patterning the resist and etching the polysilicon. Controlling the resist dimensions at this stage will help control

the final etched polysilicon linewidth. However, measuring the

width of the resist line at this stage is difficult because the

image profile is a function of the geometry, refractive index,

and thickness of each of the three layers. The polysilicon was

assumed to have a refractive index of 3.8 + 0.1£.

The curves in Fig. 10 show how variations in the thickness of the

polysilicon affect the image profile of the resist line. Again,

only half the image profile is shown. The most striking feature

of these curves is that the image structure in the immediate vicinity of the line edge is sensitive to the thickness of the

polysilicon layer.

The structure in Fig. 11 represents a polysilicon line after

etching and stripping of the resist. The image profile is a function of the thickness of the silicon-dioxide layer. Figure

12 shows the effect of changing the thickness of the oxide sublayer on the image profile of the structure shown in Fig. 9(b).

The polysilicon is assumed in this case to have vertical edge

walls and the oxide thickness is varied over a range of 40 nm.

As the thickness changes, the image profile changes considerably.

This makes linewidth measurement difficult as there appears to be

no feature or threshold which locates the edge independently of

oxide thickness.

The image profile is also a function of edge geometry and in Fig.

9(b) it has been assumed that the edge shape can be defined by a

second order polynomial with the polynomial of Eg. (2) given by

72
W(z) = X ( 26 ) Q + X

2 .z

2

,

where, again, the nominal linewidth X

Q

is 6.0 urn. The curves in

Fig. 12 show the effect of variations in the edge curvature of

the polysilicon layer on the image profile. The edge image profile is clearly sensitive to the curvature of the physical edge

of a curved feature. In this case, three possible definitions of

linewidth are the minimum, maximum, or mean width of the layer.

For the examples shown in Fig. 12, these definitions all give

linewidths which do not correspond to that given by the position

of the minimum of the dark fringe. More importantly, however,

the offset between the true and measured linewidth varies with

the degree of curvature over a range of a few tenths of a micrometer regardless of the definition used for the line edge.

Summary

A waveguide model has been presented which enables the optical

images of line structures patterned in thick layers to be computed. The model has been shown to be applicable to a wide range of

structures, and qualitative agreement with experimental image

profiles has been demonstrated. From the results presented here,

it can be seen that the image profile of a thick line object, and

therefore the measured linewidth, is affected by the physical

shape of the edges.

The model enables the effects of process variations on the optical image profile to be determined and this, in turn, enables the

accuracy and sensitivity of different measurement techniques to

be investigated. Further work is in progress to improve existing

measurement methods in order to minimize the effects of process

variations on the accuracy and repeatability of linewidth measurements .

73
Acknowledgments

The authors wish to acknowledge the help of Marilyn J. Dodge who

collected the experimental image profiles for this work. This

work was carried out as part of one of the author's (Kirk) Ph.D.

research program supported by the Science and Engineering Research Council (UK).

74
Appendix

In matrix notation, the eigenvalue problem of eg. (9) is:

D • B = a2 1 • B

n n

where the matrix elements D. . for the n-th layer are given by i f 3 A ( E +K* ) E» . . . E. . 0 . . . 0

/ o,n j l,n i-D,

D =

1 / n

E. . i-3 /n

0 .

1 , n

E- n

- ( E +K

2 l,n o,n o

) En „ 1 ,n

E l,n *

0 E. . i-3 ,n

• • *l,n

i-3 /n

1 / n

- ( E +K?

o , n 3

and K . =

3 P

75
References

1. M. Born and E. Wolf, Principles of Optics , 5th ed. , (Pergamon Press, Oxford, 1975), pp. 522-532.

2. L. C. Martin, The Theory of the Microscope , (Blackie, London,

1966), Chapters V and VIII.

3. D. Nyyssonen, "Theory of optical edge detection and imaging

of thick layers", J. Opt. Soc. Am. 72, pp. 1425-1436 (1982).

4. Ref. 1, Section 13.5, pp. 633-664.

5. P. W. Barber, D. Y. Wang, and M. B. Long, "Scattering calculations using a microcomputer," App. Opt. 20., 1121-1123

(1981) .

6. J. P. Hugonin and R. Petit, "Theoretical and numerical study

of a locally deformed stratified media," J. Opt. Soc. Am.

71, 664-674 (1981).

7. R. Petit, ed. , Electromagnetic Theory of Gratings , (SpringerVerlag, Berlin, 1980).

8. K. A. Zaki and A. R. Neureuther, "Scattering from a perfectly conducting surface with sinusoidal height profile, TE

polarization," IEEE Trans. Antennas. Propagat. AP-19 , 208-

214 (1971).

9. J. P. Hugonin, R. Petit, and M. Cadilhac, "Plane-wave expansions used to describe the field diffracted by a grating,"

J. Opt. Soc. Am. 71, 593-598 (1981).

10.

Ref. 1, p. 453.

76
11. D. Nyyssonen, "Optical linewidth measurement on patterned

metal layers," Proc. SPIE 480 , Integrated Circuit Metrology

II, 65-70 (1984).

12. C. B. Burckhardt, "Diffraction of a plane wave at a sinusoidally stratified dielectric grating", J. Opt. Soc. Am.

56, pp. 1502-1509 (1966).

13. F. G. Kaspar, "Diffraction by thick, periodically stratified

gratings with complex dielectric constant," J. Opt. Soc. Am.

63 , 37-45 (1973). 14. F. G. Kaspar, "Computation of light

transmitted by a thick grating, for application to contact

printing," J. Opt. Soc. Am. 64., 1623-1630 (1974).

15. W. Magnus and S. Winkler, Hill’s Equation (John Wiley and

Sons, New York, 1966 and Dover, New York, 1979).

16. Ref. 1/ P- 612

17. Ref

.

1, P- 579

77
(a)

zero order

W(z)=X 0 vertical edges

(b)

first order

W(z)=X 0 +X 1 *z sloping edges

second order

W(z) = X 0 -X 2 (z-d) 2 baked resist

second order

W(z) = X 0

+ X 2 (z-d)

undercut

third order

W(z) = X 0

+ X 3 (z-d/2) 3 baked resist/

etched metal

fourth order

W(z) = X 0 ~X 4 (z-d) 4 baked resist

fourth order

W(z)= X 0 + X 4

z 4 etched metal

fifth order

W(z) = X 0

+ X 5 (z-d/2) 5 baked resist

Fig. 1. The geometry of most line objects encountered in integrated circuits may be approximated by low order polynomials.

X

Air ( n n)

Silicon substrate (ns)

(a)

"1

X

II o

JC r Xn n 2 j

\

\V

'V Jr

\

"3 1 (

V / V , X

^ n 4

ns

z

(b)

Fig. 2. Cross section of a hypothetical thick line object (a) and

the corresponding multi-layer representation (b).

78
INTENSITY

Fig. 3. Orientation of the line structure and incident fields in

the NBS linewidth measuring system. The y-direction is out of the

page and parallel to the length of the line patterns.

DISTANCE IN MICROMETERS

Fig. 4. Theoretical image intensity profile of a 6.0 um wide line

centered at zero, patterned in a 0.6 um thick SiO- layer on silicon. A = 530 nm.

z

79
First Order

W(z) = 6.0+ X i • z Second Order Third Order

W(z) = 6.0+ X 2 (z-0.6) 2 W(z) = 6.0+ X 3 (z -0.3) 3

DISTANCE IN MICROMETERS

DISTANCE IN MICROMETERS

DISTANCE IN MICROMETERS

Fig. 5. The effect of edge geometry on the calculated optical

image profile. The edge geometry has been superimposed on the

image profile for reference.

80
(a) (b) (c)

DISTANCE IN MICROMETERS DISTANCE IN MICROMETERS DISTANCE IN MICROMETERS

Fig. 6. Theoretical image intensity profiles of a line patterned

in a noisy dielectric layer on silicon. A different set of noise

data has been used for each curve.

0.94

i

i

Photoresist

Silicon

Fig. 7. Physical profile model of the asymmetric window structure

in resist used to compute the image profiles in Fig. 8. (Dimensions in micrometers.)

81
(a) (b)

e

INTENSITY

DISTANCE IN MICROMETERS

INTENSITY

DISTANCE IN MICROMETERS

Fig. 8. Comparison of experimental and theoretical image intensity profiles of a window in photoresist on silicon. The window

has either vertical edge walls (a) or sloping edges (b). The

experimental profiles are shown by the thick line and the theoretical profiles by the thin line. The photoresist was assumed to

have a thickness of 0.94 urn and the geometry shown in Fig. 7 was

used to model the window in the case of sloping edges.

82
(a) (b)

Fig. 9. Cross sections of the shapes used to model the polysilicon patterning stage of making an MOS transistor. The two shapes

are patterned resist in unetched polysilicon (a) and etched poly- silicon with the resist removed (b). (Dimensions in micrometers.)

DISTANCE

Fig. 10. Theoretical image profiles of a resist line on polysilicon (structure in Fig. 9a) for polysilicon thicknesses of a) 0.5, b) 0.6, and c) 0.7 urn. A vertical edge is assumed.

83
Fig. 11. Theoretical image profiles of a polysilicon line (structure in Fig. 9b) for oxide thicknesses of a) 85, b) 105, and c)

125 run. A vertical edge is assumed and 0.6 pm polysilicon thickness .

Fig. 12. Theoretical image profiles of a polysilicon line (structure in Fig. 9b) for a range of edge curvatures defined by Eg. 26.

Oxide thickness is 105 nm; polysilicon thickness is 0.6 um.

84
FEDERAL INFORMATION PROCESSING STANDARD SOFTWARE SUMMARY

01. Summary date

Yr. | Mo. Day

00

I 31— 0 |6

04. Software date

Yr. Mo. Day

1

02. Summary prepared by (Name and Phone)

Diana Nyyssonen (301)975-2099

8 7 10 1

06. Short title TIIKIMAG

05. Software title THKIMAG - Computer software for the computation of

optical microscope images of line objects patterned

in thick layers viewed in a coherent sysi-pm.

03. Summary action New Replacement Deletion

Previous noInternal Software ID

07. Internal Software ID

08. Software type

Automated Data

| | System

[X] Computer Program

| | Subroutine/Module

09. Processing mode

| | Interactive

(X] Batch

| | Combination

10. General Computer Systems

| | Support/Utility !'”j

[%] Scientific/Engineering Q

| | Bibliographic/Textual

Application area

Management/

Business

Process Control Other

Specific Optical imaging

I£. Submitting organization and address National Bureau of Standards

Microelectronics Dimensional Metrology Group

Precision Engineering Division, 731

Center for Manufacturing Engineering

Building 2? 5, Rm A3 U2

12. Technical contact(s) and phone

Robert D. Larrabee

(301)975-2298

Diana Nyyssonen

(301)975-2099

13. Narrative This computer software calculates the optical microscope images of line objects

patterned in thick layers (more than one-quarter of the illuminating wavelength

thick). The algorithms used are based on a monochromatic , waveguide model which

can predict the image of line objects with arbitrary cross-section geometry

including multilayer structures with sloped, curved, asymmetr ic , and undercut edges.

Along with the computer software, test cases for help in implementation are given.

As written, the program uses subroutines for complex matrix eigenvalues and

eigenvectors from the NAG library.

14. Keywords

diffraction; dimensional metrology; linewidth; microscopy; optical imaging; optical

metrology; waveguide

15. Computer manuf’r and model Cyber 855 (CDC)16. Computer o NOS 2 (aerating system

CDC)

17. Programing language(s)

ANSI FORTRAN77

18. Number of source program state- ments949

19. Computer memory requirements 20. Tape drives 21. Disk/Drum units 22. Terminals 91170 words . 120K words

23. Other operational requirements

24. Software availability

Available Limited In-house only

25. Documentation availability

Available Inadequate In-house only

m a.

26. FOR SUBMITTING ORGANIZATION USE

185-101 Standard Form 185

1974 July

U.S. Dept, of Commerce—NBS

(FIPS. Pub. 30)

NBS-114A (REV. 2 -aci U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See in struction s)

1. PUBLICATION OR

REPORT NO.

NBSIR 87 3 6 1

2. Performing Organ. Report NoJ 3. Publication Date

4. TITLE AND SUBTITLE

Computer Software for the Computation of the Scattered Field and the Optical

Microscope Image of Line Objects Patterned in Thick Layers

5. AUTHOR(S)

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City . Stote, ZIP)

10. SUPPLEMENTARY NOTES

|

I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant in formation. If document includes a significant

bibliography or literature survey, mention it here)

This report contains computer software for calculating optical microscope images of

line objects patterned in thick layers (>,\/4 thick). The algorithms used are based

on a monochromatic, waveguide model which can predict the images of line objects

with arbitrary edge geometry including multilayer structures with sloped, curved,

asymmetric, and undercut edges. Along with the computer software listing, the

mathematics of the model, a short description of its structure and use, and test

cases for help in implementation are given.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semi colon s) diffraction; dimensional metrology; linewidth; microscopy; optical imaging; optical

metrology computer software

13. availability

Eunl imited

I I For Official Distribution. Do Not Release to NTIS

H Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

ijj O f der From National Technical Information Service (NTIS). Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

86

IS. Price

$13.95

USCOMM-DC 6043-P80
'

•