ADDITIONS TO THE IMAGING CAPABILITY OF SAMPLE

by

Mark D. Prouty

Memorandum No. UCB/ERL M84/111

SAMPLE Report No. SAMD-8

10 December 1984

ADDITIONS TO THE IMAGING CAPABILITY OF SAMPLE

by

Mark D. Prouty

Memorandum No. UCB/ERL M84/111

10 December 1984

ELECTRONICS RESEARCH LABORATORY

Col 1 ege o f Engi neeri ng University of Cal i forn ia , Berkeley

94720

Table of Contents

.

Table of Contents ............................................................................................

list of lqplres .............................. ) ..................................................................

1 Introduction ..................................................................... ...........................

2 Calculating Images of Phase Shift Masks ....................................................

2.1 Introduction ........................................................................................... 2.2 Mathematical Methods ........................................................................... 2.3 Implementation ......................................................................................

3 Results using the Enhanced Program ..........................................................

1

2

3

3

3 5 6

7

9.1 Introduction ........................................................................................... 7

3.2Results .................................................................................................... 7 3.3 Conclusion .............................................................................................. 10

4 2-D User's M a n u a l ......................................................................................... 11

4.1 Introduction ........................................................................................... 11 4.2 Rwming the Image Program .................................................................. 11 4.3 How the Prograrn Works ......................................................................... 12

Appendix 1 Comparison of New and Old Versions of SAMPLE .......................... 14

Appendix 2 2-D Example ............................................................................... 15

Appendix 3 Defect Printability in 2-D Goodman vs . Subrammian ................ 17

References ....................................................................................................... 18

fist of F'igures

2

F'igure 1. Region of integration for equation ( 7 )

Qure 2.

Figure 3.

Figure 4.

Figure 5.

Region of integration for equation ( 7 )

Region of integration for equation ( 7 )

Region of integration for equation (7 )

Image intensity for a periodic series of lines and spaces 0.75 pm

wide, for no phase shift (B) and 180" phase shift (A).

Figure 6.

Figure 7.

Contrast vs. Defocus for series of lines and spaces.

Contrast vs. Defocus for series of lines and spaces.

Figure E. Contrast vs. Feature size for series of lines and spaces.

Frgure 9. Contrast vs. Feature size for series of lines and spaces.

Figure 10. Intensity proNe for 0.2 p m wide phase shift line defect.

Figure 11. Intensity minimum vs. size for phase shift line defects.

Figure 12. Intensity minimum vs size for phase shift line defects.

Figure 13. Phase angle for necessary feature contrast and defect minimum as a

function of size. o = 0.3.

Figure 14. Minimum intensity for clear field phase transition.

Figure 15. Image prome for clear field phase transition. Phase shifting material

is on the right hand side.

Fuure 16. Image profiles for smooth 180" transition over a distance of 0.6 (A),

0.9 (B), and 1.2 (C) pm.

Figure 17. Image proflles for 0.62 pm line with 0.3 pm (A), 0.2 p m (B), 0.1 +TI

(c), and no (D) 180" phase shifted proximity feature. Center to

center distance is 1.15 grn in all cases.

3

#dditims bo the Imaging Capability of sAMpI;E

Mark D. Prouty

1. ~ u c t i o n

As feature sizes decrease in optical lithography. becoming smaller than the

ratio of wavelength to numerical aperture, the faithfulness of imaging is

significantly reduced. This makes zomputer simulations of optical lithography

more difficult. flrst of all because more general techniques. such as phase shift

masks, a r e being used t o improve the image and secondly because the

differences between 1 dimensional and 2 dimensional patterns, such as line end

shortening and rounding of square apertures, are more important. Therefore,

the capabilities of imaging phase shifting masks and 2 dimensional masks have

been,added to the processing simulation program S.LMPLE.

The second chapter of this paper outlines the details of the changes this

author made in the program to calculate images of phase shift masks. The next

chapter outlines some results obtained using the new program. The fourth

chapter outlines the use of the'2-D code written by Shardcar Subramanian. Since

&is work was never documented or implemented in the SAMPLE program this

chapter wi l l serve as a user's manual for 2-D imaging.

2 Calculating Images of Wase Wt Masks.

21. lntrodllctioa

The resolution in uptical lithography is limited by proximity effects, that

is. the spillover of difkacted light between adjacent features. In conventional

lithography, 'light from each feature arrives in phase betMeen them, causing a

slightly lightened area where a dark one is-desired. However. i f the light coming

from one of the features is delayed by a coating so that it arrives 180" out of

4

phase. the two W a c t e d b e a m s wi l l cancel, and the desired dark area will'be

obtained.

A mask using this technique, Plrst proposed by Levenson, e t al.(l), has

been dubbed a phase shift mask Levenson, using the basic case of a 180" phase

shift. predicted theoretically and verified experimentally a much increased con-

trast. The success of this phase mask approach has raised many conceptual and

practical questions. Therefore, we decided to &enhance the capabilites of the

SAMPLE program.

The SAMPLE program calculates images using Hopkins(2) theory of partially

coherent imaging. Briefly, the image intensity is given as the convolution of the

Fourier transform of the mask intensity distribution with a function known as

the transmission cross-coefficient (TCC). The TCC is in turn the convolution of

two functions, one over the condenser aperture and the other over the objective

pupil. More details of this theory wil l be given later.

Previously, the SAMPLE program calculated images of real masks which

only required calculating the real part of the TCC. W-e have modified the pro-

gram to calculate complex coefficients for the Fourier series of the mask and to

also calculate the imaginary part of the TCC. This is a more complicated pro-

cedure, but many symmetries could be exploited to shorten the numerical

integration. In addition, the program will accept masks with up to 33 different

regions with arbitrary amplitude transmission and phase shift. This method

requires slightly more CPU time than the previous SAJfPLE version, up to twice

as long. This time is from 2 to 12 CPU seconds on a VAX 11/780 with UNIX,

depending on the size of the image. For an image period of 2 microns, for exam-

ple, 4 seconds are required.

5

22. &thematid Methods

Kintner(3) gives details of the partial coherence theory, and

Subrarnanian(4) shows how the theory is used in the SAMPLE program. There i t

is shown that the transform of the image intensity may be given as

I W ) = 2 (%%mfp. a+ E %+rn<T((n+m)fp, W ) + c h - , < T ( ( n - m ) f p . -fp) 6 b f p - f ) is1 m=1

where j p equals the reciprocal of the period.

T d f ' , f " ) is known as the transmission cross coefficient and is given by

W . f " ) = J J J ( f . g ) K(f+f',g)K'(f+f",g) df &7 (2)

where J, the Fourier transform of the mutual intensity function at the object is

where s is the partial coherence factor

f Z+g"<s f tg'2s

and K. the objective pupil function is

Here, p is related to the defocus in microns by

NA2 h

p = d -

(3)

( 5 )

The a,, ' s are the fourier coefficients of the mask transmission. so that F ( z )

, the intensity and phase of the lrght at the mask is

.. F ( 2 ) = q + 2 cos2nfp3:

n=l

Far masks without phase .shifts the a, 's are real. Here, however, they may be

complex.

These equations lead to a more specific expression for the TCC

(7) 1 TU', 1") = J J enp-znpdf *2-f"2+2f (1 '-f" )i df dg

n where n is the region of intersection of 3 circles, C, C', and C" . Cis the region

where J ( f , g ) is non-zero, viz. a circle of radius s centered at the origin. C'is

6

the region where K C f + f ' , 9 ) is non-zero, uiz. a circle of radius 1 centered at

(-f', 0) and .C" is the region where r(J +f". g ) is non-zero. viz. a circle of

radius 1 centered at (-I", 0) . Several.cases of these intersections are shown in

figures 14.

The speciAc formulas used in the program to integrate equation (7) are

dependent upon the geometry of the circles C, C', C" . For the case shown in

Figure 1

ReT(f ', I") = If cos -27rp[f'2-f''2+2( (f '-f" )} "If dg

ImT is identical, except cos is replaced by sin.

These integrands all contain radicals unsuitzbk for Gaussian quadrature.

= s cos z or f +f ' = cos z is made. Therefore, a transformation of the form

Then the fhs t integral becomes

j sin2z cos-2np f *2-f"Z+2(cosz -f ')(f '-f** I] ( 9 ) rn84fff + f ') I

With the further changes sin'z + (1-cos 2 2 ) / 2 , f ' + - f , and

cos x .* cos(n-z) ( which just changes the limits of integration), these integrals

are encoded in the program as functions g t l and gtl i ( g t l calculates the real

part and gtli calculates the imaginary part). The second integral becomes func-

tions gt2 and gt2i. The functions g t l and gtl i are also used to integrate the

third expression.

7

When the geometry is as in kure 2, gcl and gcli are used to inhegrate (7).

F’urther, for the case where s > 1. ft2(i) and ft3(i) are used forcases shown in

fjgure 3 and4, respectively.

9.1. Introduction

Here, we explore a number of the phase mask imaging issues raised by

Levenson’s work. The first question raised is how much the contrzst of periodic

features depends upon defocus and the amount of phase shift used, and how this

dependence changes with feature size. The second issue is what size defects in

the phase shifting material may be tolerated. We also explore the possibility of

making clear field transitions between phases without causing printable defects.

The h a l issue is whether unprintable prosimity features may be used to

enhance the image of isolated features.

3.2. Results

Image calculations of the new extension to SAMPLE indicate. tfit n a r k c !

improvement in image quality possible in using phase-shift masks. Fignre 5

shows the intensity profile for a periodic series of lines and spaces 0.75 prn in

width (in all simulations throughout this chapter the wavelength used i, 0.4355

prn the lens has a numerical aperture of 0.28 and the defocus is 1.5 w.). Curve

B is the image obtained with no phase shifting, and curve A is that obtained with

every other space phase shifted 180’. The large improvement in contrast is due

both to a decrease in the minimum intensity, as well as t o an increase in the

maximum intensity. This increase is due to the extra path difference!involved in

the light from both images. The beams that arrived out of phase i n conventional

lithography now arrive in phase.

8

This improvement is quantified more N l y in fgures 6 through 9. In

sure 8 we plot contrast ( ( I , - - f ~ / ( I - + f ~ - ) ) vs. defocus for a partial

roherence factor (sigma) of 0.7. Figure 7 shows the same thing for a sigma of

0.3. In both cases contrast is improved. The improvement is much greater for

the lower sigma values, which is what we would expect from the mutual coher-

ence function (MCF). This function gives the degree of correlation of the

incident light as a function of distance across the mask. The width of its central

lobe is equal to 1.22x/WAo. Since phase changes are significant only for phase

related beams, the latger the central lobe of the MCF (caussd by a smaller

sigma), the greater we would expect to be the effect of pha- .e shifts.

Examining Q u r e 7 more closely, we see how the phase shifting not only

increases the level of contrast, but also makes the contrast more focus tolerant.

This graph clearly shows the steady improvement possible by increasing the

phase shift up to 180". (However, we shall see later that we must pay a price for

this.

In figures '8 and 9 we now plot contrast vs. size, still for periodic

sequences of l i e s and spaces. These curves show that for widths of less than 1

p m , steady contrast improvement may be obtahed by increasing the phase

shift. Above 1 pin proximity effects are lessened, and the two features are

further apart than the diameter of the MCF. Therefore, little improvement is

possible for features larger than 1 &m.

Improvement in image quality, unfortunately, comes at the price of an

improvement in defect-printability. In figure 10 we plot the image intensity for

0.2 Fm wide 180" and 90" phase shift material defects on clear areas. This pro-

duces surprisingly large ripples in the intensity, which may leave resist remain-

ing on the wafer in what was supposed to be a clear area. For features this

amall, however, the assumption of uniform intensity across the opening in the

9

mas% made by the -image calculation algorithm, is not exactly true and these

results only give an approximate indication of the problem for a 1X system.

Defect printability may be reduced by using phase shifts of less than'

180". This is shown in f3gures 11 and 12 where we plot, for sigmas of 0.7 and 0.3

respectively, image intensity rninimum vs. defect size. Thus, by lowering the

phase angle, defect printability may also be lowered. For comparison. the con-

trast of an opaque defect is also shown in these Qures.

The tradeoff between defect printability and contrast improvement is

summarized in 'figure 13. There we have plotted, vs. defect size, the phase which

gives an intensity minimum of 0.5, which is the minimum tolerable intensity. We

have also plotted the phase angle which gives periodic features a contrast of

0.85, which we have taken as the minimum allowable feature contrast. As an

example, if we wanted periodic features 0.75 p m wide, we need at least a phase

shift of 90°. Moving to the left untii we.hit the defect line, then dropping down

we see that the largest allowable defect a t that phase angle is 0.2 Fm.

In a 2-dimensional mask, it would be desirable for all adjacent features

t o be out of phase with respect to each other. This is similar to the problem of

coloring a map with only 2 or 3 colors. Thus, it might be desirable to switch the

phase of different parts of the same opening, that is, t o color parts of the same

country different colors. This type of phase transition, however, will cause a

printable glitch as seen in figure 14. This Q u r e plots the phase shift depen-

dence of the image intensity minimum of a large clear region next to a large,

clear, phase shifted region. We see protlles for a 180" and 90" phase transition in

@re 15. Obviously, th is type of phase shift will cause a defect to print.

We may reduce the contrast of this type of transition be making a smooth

change in phase Over a small length. Figure 16 shows profiles for a 180" phase

shift made smoothly (by making 12 steps of 15" each) over a length of 0.6. 0.9,

10

and 1.2 pm. This reduces the contrast, but separating the Oo &d 180' regions

by more than 1.0 prn defeats the purpose of having the phase transition. Thus

even this smooth phase transition method is no good.

The discussion of printing features up to now has involved periodic

sequences of lines and spaces. Another use of the phase mask is to add non-

printable features next to an isolated feature to enhance its image. Figure 17

shows an attempt to improve the image of a .62 pm wide isolated line (this size

was chosen 'because the intensity peak of the unenhanced image drops to one

half the clear field value), Curves A through C are for t'ne line with a 0.3. 0.2, and

0.1 p m wide 180" phase shifted proximity feature on each side. respectively.

Curve I) is for the line by itself. The center to center distance between the line

and the proximity feature is 1.15 pn in all cases. The maximum intensity of the

line increases as the proximity linewidth increoses. and the peak becomes

slightly narrower, making this a potentially useful technique.

3.3. Conclusion

We see, then, that phase masks may improve both periodic and isolated

features. The size of periodic features may be reduced from 1.2 pm to 0.5 pm

end still have a contrast of 0.85 by using phase shifts. This improvement is

robust with respect t o phase shift angle; 120' gives nearly as good results as

180". Below 120", however, improvement does drop off. The peak intensity of

isolated features may be increased 30 percent by using unprintable proximity

features. This improvement decreases for phase shifts of less than 120 '.

We have also seen that phase transitions above 90n will produce defects.

n u s , clear held transitions are not viable even when the transition is relatively

smooth Another key problem is the printability of small defects. We have

shown that a 90" defect prints about the same as an opaque defect, and shifts of

120" and 180" cause defects twice as small to print. With the limitations of no

11

d e a r 'field transitions and proper defect control very significant improvements

in aerial irnage quality are possible with phase masks.

Not included within the SAMPLE program, but related to it, are a set of pro-

grams witken by Shankar Subrarnanian for calculating 2 dimensional images.

Programs for plotting contour plots and profiles of the images, written by this

author, are also included. This chapter will outline the use of these programs.

4.2. Running the Image Program

Two versions of Shankar's imaging program exist. The program image.out

calculates images with arbitrary defocus, while nodef.out calculates images with

no defocus. 930th programs assume X = .436 p n , NA = 0.28, and have the same

I/O format.

' Each program reads 6 parameters, zs,zf,ys,yl,s,def, in free format from a

file namedlines The &st four parameters specify the mask intznsity in the fol-

lowing way. The mask intensity is assumed to be a separable function of z and y,

Le.

J ( 2 . Y ) = X b ) Y ( y ) (10) X(Z) has period zs + ~ l , and Y(y) has period ys+yZ. The program defines X ( z )

in the first period as follows

(11) I 0 - (zs+21)/2 c 2 s -zs/2 X ( z > = [ 1 --2s/2 4 2 s z s / 2

0 zs/2 c z s (zs+zL)/2 Y(z ) is d e h e d similarly. The parameter s is the partial coherence factor, and

def is the defocus in pm. The program nodef.out ignores cf.!?f.

The output of the programs is written to an unforrnatted file called M t ,

which contains a 100 by 100 array of intensities, followed by the two parameters

12

dz and dy. These two parameters are the spacings between z and y points,

respectively, in'units af ,A/ NA

rint contains intensities for one .quadrant of one period of the image, which,

because of the z and symmetry. contains all the information of the entire

image. Tiht(1,l) is the intensity at the origin, &nt(1,100) is the intensity at

(0, 99dgUNA) . Tint(100.1) is the intensity at ( 99dyh/ArA, 0). The interactive

programs 'profile and contour sets up a file of points called f77punch7 suitable

for plotting w i t h the SAMPLE plotting programs.

4.3. How the Program Works

The 2-D program uses the same partial coherence formulas as the 1-D case,

except, of course. in two dimensions. This makes the geometry more compli-

cated, since the circles C: and C' are no longer centered on the z axis.

The main program calculates the Fourier coefficients of the mask.

Currently, the method used is a speciflc one for the separable fgnction of z and

I/ assumed. The rest of tbe program, however, does not require that

simplification. Therefore, a more general routine for calculating Fourier

coefficients could be added. The only requirement is that the intensity

transmissibn be real, and periodic and symmetric in both z and y directions.

After computing the Fourier coefficients, main calls spect, the subroutine

which calculates the transform of the image intensity, performing many s u m a -

tions similar to the 1-D case. 'The 'transmission cross-coefficients, now functions

of four variables, are calculated by the function t, two versions of which exist.

The file croas1.f contains the version for the no defocus case, while the general

case is located in crdd2o.f, each of which requires several functions. The

required modules are hain . f , crdss1.f (cross20.f). arc.f, rsect.l,inside.f, intf.

13

and spec1.f.

The run t ime for these programs varies roughly proportionally to the area

of the image period. The program image.out requires about 20 CPU seconds on a

VAX ,11/780 with UNIX per square micron, while nodef.out requires about half

that.

5. Acknowiedgement

This work was supported by the National Science Foundation under grant

ECS-8106234 and industrial contributions under the MICRO program (83-14).

-14-

Appendix 1

Comparison of old and new SAMPLE images.

The following table shows a comparison of the output for a periodic series of 0.75 pm lines and spaces calculated three different ways. First, using SAMPLE 1.5b and input A, second using SAMPLE 1 . 5 ~ and input A, and third using SAMPLE 1.5~ and input B. All images are identical, as expected.

distance

A 0.000 0.031 0.061 0.092 0.122 0.153 0.184 0.214 0.245 0.276 0.306 0.337 0.367 0.398 0.429 0.459 0.490 0.520 0.551 0.582 0.612 0.643 0.673 0.704 0.750

1.5b input A ,220 .221 .225 .231 .240 .251 .264 .280 .298 .319 .342 .367 .393 .420 .449 .4?7 .505 .533 .558 .582 .602 .619 .633 .642 .647

intensity 1.5~ input A

.220

.221

.225

.231

.240

.251

.264

.280

.298 -319 .342 .367 .393 .420 .449 .477 .505 .533 .558 .582 .602 .619 .633 .642 .647

input k. proj .6 lambda .5 trial 3 1 0 1 linespace .75 .?5 run 1 end

1.5~ inputB .220 .221 .225 .231 .240 .251 .264 -280 .295 .3 19 .342 .367 -393 .420 -449 .477 .505 .533 .558 .582 .602 .619 .633 .642 .647

input B: proj .6 lambda .5 trial 3 1 0 1 trial 19 0 0.75 1 0 .?5 run 1 end

- 14a-

Invoking the new SAMPLLE routines

The new routines are invoked through the 'trial 19' statement. Its form is:

trial 19 (ampl. phasel, distl). ... , (ampN. phaseN, distN )

where amp, phase, and dist specify the amplitude and phase (in degrees) transrqission for a region of length dist. Up to 33 regions may be specified.

The program then makes an even, periodic extension of this function I t does so by first halving the length of the first and last regions. This now forms one half-period of the function. The other half-period is the mirror image of the first.

For example, the statement,

trial 19 1 180 .75 0 0 .5 10 .75 0 0 5

produces :

I

I I

-15-

Appendix 2

The following is an input example for image.out The Ale lines contains the m e line: 1.0 1.0 1.0 1.0.3 1.5 . Running image.out then contour, and plotting the &e fT7punch7 produces:

- J

E & 9 d P

1 *.e

0 .eo0

0 06.0

0 A00

0 0200

0 .

1 .m 0 . o.t.0 0.400 0.600 0.000 Distance (urn)

These routines are currently located in the directory, /od/sample /#routy/Twodixu

16

Running profile and plotting f77punch7 produces:

1 .e0

0.808

8.200

e.

CPL' time for calculating this image was 20 seconds.

-17-

'. Appendix3

Comparison of Goodman and Subramanian

The following table compares the intensity at the center of small isolated squares of different sizes. Goodman's program was run by this author while working at IBM Watson Research Center. As shown, Subramanian's program gives identical results.

size A/ NA 1.0 0.8 0.7 0.6 0.5 0.4 0.3

Goodman 1.10 t .94 t .77 f .57 =f .34 t .16 7 .064

Subramanian 1.165 .954 .762 .547 .338 .171 .064

- t These numbers were extrapolated from a graph and are accurate only to

The followkg figure plots the intensity at the center of small opaque and *0.02.

small open squares and. for comparison, lines.

1.2

1.1

1 .e

.9

.8

.I

.6

.S

04

.3

.2

.1

0 .

-18 - References

[l] M.D. Levenson, N.S. Viswanathan, Rk Simpson, "Improving Resolution in Photolithography with a Phase-Shifting Mask," IEEE Trans. Elect. Dev., vol

[ Z ] H.H. Hopkins, Proc. R SOC. London Ser A, vol 217, ~405, 1953. [3] E.C. Kintner, "Method for the Calculation of Partially Coherent Imagery,"

Appl. Opt., vol 17, p2747, 1978. [4] S. Subramanian. "Rapid Calculation of Defocused Partially Coherent

Images," Appl. Opt., vol 20, p1854, 1951. [ 5 ] M. Prouty, A. Neureuther, "Optical Imaging With Phase Shift Masks,'' SPIE

conference on Microlithography, 470-26, March 1954.

ED-29 NO. 12. ~ 1 8 2 8 . 1982.

I

Figure 1. Region of integration for equation (7)

Figure 2 Region of integration for equation (7)

Figure 3. Region of integration for equation ( 7 )

Figure 4. Region of integration for equaLlull ;:)

m 0 II

b

00, co + a Lo d rc) N - - - 0 0 O d d 0 0 0 0 0

A l I S N 3 l N I

figure 5. Image intensity for a wide, for no phase shift fB) and l8Oo phase shift (A).

eriodic series of lines and spaces 0.75 pm

0 00 a w- CV 0 0 0 0 0 0

l S W t l l N O 3

0 0 0 0 m -

Fuure 6. Contrast vs. Defocus for series of lines and spacps.

0 - a3

0 0 0 1 S W l N 03

cu 0

rr)

0

b II

0 0

Kgure 7. Contrast vs. Defocus for series of lines and spaces.

I1 I J

Q) e

0 -

lSV~lN03

Fiure 8. Contrast vs. Feature size for series of lines and spaces.

n v9

w N v, -

W LL

E

8 - 11 11

rn 3 b o 0 Y-

u) Tt . 0 0

cv . 0

0 0

n

v)

w LL

Figure 9. Contrast vs. Feature size for series of lines and spaces.

0 - 00 0

u3 0

A l I S N 3 1 N I

E =t

m L n

0 Q, D +

d cv 0 0

Flgure 10. Intensity profile for 0.2 pm wide phase shift line defect.

0 0

0 0 u)

0 0 0

Figure 11. Intensity minimum vs. size for phase shift line defects.

0 0 m

0 0 (D

I I I I I I 1 I 1

UUnUUINlUU Al ISN31NI

Figure 12. Intensity minimum w size for phase shift line defects.

0 00 - 0 -

0 0 0 cu 6) (D

319NW 3 S V H d

Fuure 13. Phase angle 'for necessary feature contrast and defect minimum as a function of size. u = 0.3.

0 -

II / I

I I I I

a3 w d' cu 0 0 . o 0

Nnw I N I W Al ISN31N I

0 do

0 0

F'igure 14. Minimum intensity for clear field phase transition.

a w 0 0

A l I S N 3 l N I

cu 0 0

Figure 15. Image proflle for clear field phase transition. Phase shifting material is on the right hand side.

m Y L)

. m

mure 16. Image profiles for smooth 180° transition over a distance of 0.6 (A), 0.9 (B), and 1.2 (C) pm.

E =t

m l n

rn 0 0 Q) c3

b 3 Y-

Figure 17. Image' proflles for 0.62 Frn line with 0.3 pm (A), 0.2 prn (B), 0.1 pm (C). and no (I)) 180" phase shifted proximity feature. Center to center distance is 1.15 Frn in all cases.

C C

C

c E. C C C C C C

C

sub1 ..utine clwtf i (ilmbd 1

clmc.fi computes the diffraction limited fourier components P O I . the vzrious r a ~ e s . a maximum of 41 Frequencies is taken far the linespace and 41 frbequencies zre taken f o r line and for spdce. nota - each horizontal point is the center o f a cell. (pr?jEction zItC.tem only. 1

corn:, I e x F c a m s k ( 41 1 , Fsaimg (1 1 ) com:n.ir, / c b w i l i c i / window, edge, wndorg c o r n r n ~ r ~ /fouseri mxrmfr, nmfrcp, fsarnsk(411, fsaimg(41) com:ntn /harimqi d e l tx, mnhpts, nmhpts, horint (50) comm!~n I i r n q 2 p i . l rna, mrnwtf, nmwtfc, rrntfwt(41) c o m i n : ~ ~ /img3pl-i iincoh, ipdrco~ ifulco, icoher, signa, d f d i s t corn~.'~ri /io1 i itei.;ni, ibulk, iprout, ircsvl, iin. iprint, ipunrh corr.:nrln / o b , ~ n s k i mlinc, rnEpace, mlnspa, mirreg, mask ty comrn:l,; 1 0 3 j n ~ ; ~ / r*lwl r s w , rjw2, rsw2 cnm:r7:::i /optic i cui.i,jl, V l J vmax, thorin(50) C Q P Q - , ~ ~ ~ /spectr/' m n l r n b d , nmlmbd, rlambd( 101, relint ( 1 0 ) cnm:r.-tii f c n o s k / nurni.e:pa am(100),phi(100), dsize(1001, zperid

i . c st?t r n ~ x i n u m freqttcncy for coherent, partially coherent, /it -11- inc ohwC.::t c a s e . ii: (iroher ~ 3 . iinc.oh) v : n a r = 2. %rna/rlambd(ilmbd) ii: ( ~ r o h e r . eq. iparco) v m a x = (rna/rlarnbd(ilmbd))+(l. +sigma) i f o r o l r c r . 5 8 ifulco) v i n a x = rna/rlambd(ilmbd)

c

C

C

C

C

C

c C

C

C

C

c

C

C

C

C

C

/ * c a ! c u l a ? e t.3e f o u r i e r c o e i f f i c e n t s f o r t h e p a t t e r n 4 5 0 c a l l F o i r r c f ( . t . r u e . , P S W ~ rlw;'. T ' S W ~ , r l w , fsamsk, mxnmfr, n m f r c p - 1 1

q 0 t ; ; r , : r+r)

. ...._.

L

C

10 C

C C C

20 C

30

C C

40

C

1000

C

patperid p i-7. 14 1 5926 w=2*pi/p

convert degrees to radians

do 20 i=I,nuarreg ong i > = p h i ( i )/360. * 2 * p i

do 1030 n=l, w m c o f x 130. x 2 4 . reor9. a i mo-9. do 40 i = l , n u m r e g ~ 2 1 x 2 r x ( i 1 r e l = 2 + r m ~ i ~ + ~ s i n ~ n + u ~ x 2 ~ ~ s i ~ ~ n * ~ x l ~ ~ / ~ n * p i ~ reo=reo+rel+cos(ang(i) 1 rimo=aimo+rel+iin(.ng(i)) x l = x L ? continue rcoeff (n+l)ncapIx(reo,.iIo)

continue return end

. . . ._ . .,'..,.T. .- I . . - .

c c

s u b r ; v t i n e f o i ~ r c f ( l e x t 8 r s l , T I & ' , r o 2 , r l f , d 1 , n s n u t n c o f ) c 5

C t h e t - c i ~ ' r i e r c o e f f i c i e n t s f o r t h e mask a m p l i t u d e t r a n s m i s s i o n are C tal- * : a t € d .

C C

s u b ; - . : . t ine p ? ~ c o Z ( i I r n b d ) c 2.

C t h i s E u b r i s a m o d i f i e d v e r s i o n o f M i k e O ' l o o l e ' s s u b r p a r c o h . C I t i i c r s t h e v a l u e s o f r l w a r i d r s w c a l c u l a t e d i r l subr i m a g e C t o c a l c u l a t e t h e F o u r i e r c o e F f i c i e n t s f o r t h e m a s k a m p l i t u d e C t r a : , r : ~ i ~ 5 i a n ( e x t e r i d e d a s aw even periodic f u r r c t i o n i f n e c e s s a r y ) C and p l a c e s ?hfn i n t h e arrau a i ( 4 1 ) . I t t h e n c a l l s s u b r c r u s 5 C which c a l c u l a t e s t h F F o u r i F r c o e f f i c i e n t s for t h e i m a g e i n t e n s i t y C p a t t c r n and l r c v e s t h e m i n t h c array s p e c ( U 1 ) C t h e cGrnmon b l o c k s / t r a n s / c o n t a i n s a i ( 4 1 1 , s p r c ( 8 1 ) , 5 ( s i g m a ) , C n ( c h t number o f s p D . t i a 1 Frequency componen t s ( a t mos t 40)), C and d l - (norrn.Jl ised d c f o c u s ) . C

comsi cx f samsk ( 41 1, f5a img ( 4 1 ) c o m u i ~ x a i ( 4 i ) s s p e c ( U 1 ) c om:p urI / c 3 w i rr ci i w i 1-1 d ow, e d 9 e, W I I d or g corn;n-,fi-! / f o u s e r / mxnmfr, nmft-cp, f s a m s k ( 4 1 ) , f r ; a i m g ( 4 1 ) c o m ~ q n / h a r i a q i d e l t x , mnhptr., nmhptr., h o r i n t ( 5 0 ) c o m ~ : t n /ingf!o/' i m g f l ( 5 ) com:a.r!i / j m g 2 p i - / r n a , mxnwtf, rimwtfc, r m t f w t ( 4 1 1 c o m v : ~ * i / img3pt . i i i n c o h , i p a r c o , i f u l c o , i c o h e r , s i g m a , d f d i s t cornn:;~.~ / i o 1 i i t a r m i , i b u l k , i p r o u t , i r e s v l , i i n , i p r i n t , i punch com;7,;:ri / o p t i c i c u r < d l , ~ 1 , vmaxl t h o r i n ( 5 0 ) comncln / s p e c t r i nnirnbd, ntnlrnbd, r l a m b d ( 101, re1 i n t ( 10) com:n:trr / t r a n s i 5 , T i , p , d f , a i ('+I 1 , s p e c ( 3 1 ) corn:r!**ii / ob , in sk / m l i n e , mspace, m l n s p a . m i r r e g , m a s k t y cornx:::-I /ob jms;!/ rlw, r s w , rid, rsw2 comm?n / c rnask / n u m r e g , am(lOu), p h i ( 1 0 0 ) , d s i z e ( 1001, z p e r i d

(.

5=5 i .-.*,:a i f ( ? ~ g P l ( 4 ) . e ? . 1 ) w - i t e ( i p r j n t , 1 1 1 1 )

1111 f O r n a t ( / / / , 9x, 4Jhrlo d i a g n o s t i c s a r e a v a i l a b l e for shanka r - s , * 2 7 h p o r t i a l c n h r r e n c e r o u t i n e s . )

C

P I -- z:..l'?l!j9:?;;!*358 rpe l . =. r i w + r l w t 2. O * ( r s w 2 ~ v l i u ~ ! ) i f (n: s k t y . e ? . 5 ) r p e r = z p P 1 - i d p = 7 . 1 a m h d i 1n:h d 1 / ( rna*rprr 1 d f r L e . *p ic.d f 3 i s t* rna*rna / r l t l nbd ( i l m b d ) n = : . > t ( ( l . +r ! l 'p ) i f !I: . I t . 4\11 g o t o 100

I ' - '0

z c : ' C C

100

/* t w 5 y 15 thcre no check for. ni=O? march 25, 1981) / i t t h t Fourier coefficients for the mask amplitude transmission /+ i.1'- now caicu1atr.d. if t a . : - s k t y . e ? . 5 ) call cFour(ai,n) if ( n : . . s k t y . e ? 2 . ) g o l o 101 call c a l l crc155

or i '1 I I : = ( --r s w/d -e d 9 e 1 / r p e r i + ( m i = k t y . e q. m 5. p a c e 1 or i Q i n= ( 1. s w / 2 . -e d g e ) / r p er i f ( m.-, e k t y . e q. m i r r e 9 ) OP i g i n= ( T' p er +r l w 1 /2. 0-e d g e 1 / r p i.r a f t n ; = k t y . eq. 5 ) a i - i g i n = (j~.ize(l)/2..rdge)/rp~~. X = Q I I .-, i ri n 2 et; de 1 ti!- window/ i nmhp te,--l) *rper 1

k o u r . c F ( . Palse. , rsw, 1.1~12, i .si&J rlw# si, 41J n )

.-- __- - --_ - -_ - C

t subcoutine c r o s s

commn /iapQl / iapert, i square i c i r c common / t r a n s / s , n, p , df, ii(4l)r spae(81) comean /Qig/P, g

f complex ai(41), r p e c ( 8 1 ) cornpiex f a c external gcl, g t l , gt2, ft2, ft3 external gcli, g t l i , gt2ic f t Z i , Qt3i

C I

C C /* set all elements oQ spec to zero

d o 1 . i L - 1 , 8 1 SP'C ( j ) = ( O . 0 . 0 . 0 )

1 contirvuc specc&)=ai(l)uconjg(ai(l)) p i 4 34159265558 p i Z r p i 12.

C /* ciFcular and square pupils m e handled separately c?? /* t h e following line was present in a previous v e r s i o n . c?? / w whtt? (march 31, 1981). c?? if (n. eq. 0 ) return

if (ispert. eq. i squar) goto 66 i 9 (r.spert. ne. icirc) return

amaz L. s*s+pi if ( 5 . gt. 1. 1 aaax=pi

if (s . g t . 1. 1 g o t o 4 1

C / w set, amax, the m o x . area o f o v w l a p

C /w f o r c i r c u l a r pupils 131 and s<l are handled separately

C /u f o r circular pupils: C is a circle of ~ a d i u s s at the origin, C /u S ' and C " are c i r c l e s of unit radius. C ' is centred at 9. For C /u t C." i s centred at 0, for tl at -9 and Q o r t2 at +g. The C /* reqion OQ integration is t h e intersection o f the5e circles.

do 3 J 1,n f=Ji \J

t=O. t i 4

g 30.

a1 = f - 1 . a2 = ((s*s-l. I / + + f)/2. a3 = 5 if(f. gt. (l.-s))goto 21

C / # for t: c m t r e C" at 0 b y setting g=Q

C /* set t h e limits o f intrgpation over C and over C ' .

C t i ? C isn't inside C' goto 21, i f it is i n s i d e integrate over C only

call gaurs(gc11 0. , pi28 t) C /* norcnalise t b y dividing b y amax

tm2. *t /8n8 X Call g8USS(gClirO. api2,ti) t i e tti/amax gOtQ 5

c t h e integrand has been transforred b y u * cos(x).fransforr the limits 21 a C 1 = 8COS(f-.1)

.c?? /* whu d o e s n ' t t h e rtrnt f o r ac3 h8v8- t h e b i a s oP O.-OOOl 3 ac4 = . c o s ( d 2 ~ ( s + O . 0 0 0 1 ) ) C a l l g.USS(gt1, 8 C f r 8 C 2 1 8 l t ) c a l l grUc iS(g t2 . 8 C 3 , aC4,82t) t - (slt 482t)fi2./amax c a l f g ~ u r , r ( g t l i ~ ~ c I , ~ ~ Z ~ ~ f t i ) c a l l g a u s s ( g t 2 i r ac3, ac44 a 2 t i ) t i = ( a l t i +a,3ti)*2./arnsx

5 s p e c ( l + j ) = c o n j g ( a i ( 1) ) * a i ( j+1)*2. *cmplx (t, t i ) 3 c o n t i r u e

C /* t h e n e s t e d d o l o o p s t h a t follow m a k e s i n g l e s u m ser ies o u t C /+ o f t h e d o u b l e s u m s e r i e s

do 1 ; = I , , e = J * D

gr)r* , l t f =-ZJ ta =i:. t l i =-?. t2i ~ 2 .

do 4 k = f r J

C / w c a l c u l a t e t l and t2. i n q e n e r a l . i n t e g r a t e OYCP t h r e e c / w c i r c l e s . i n some c a s e s t h e l i m i t s UP i n t e g r a t i o n are C /* are t h e ssme Q o r an i n t e g r a l . t h i s i o because t h e c /* r e q i o n o f i n t e g r a t i o n d o e s n o t i n c l u d e t h a t m e a . e / w this d e p e n d s on t h e v a l u e s o f f and g. C /* e o . . r t tl

a l t t =-om a 2 t l =o. a 3 t l - 0 . a l t l i =O. o2tli =O. a3 t l i =Q.

i f ( ( t ' + g ) . g e . s ! g o t o 25

a 1 = r - 1 . a 2 = <(s*s-l. I / + + F)/2. i f ( f . qt. (1. -5))goto 22

C /* i f ( f + g ) . q c 2 C ' and C" d o n ' t i n t e r s e c t and t 1 4 .

C / w s e t t h e l i m i t s f o r i n t e g r a t i n g over C ' t C a n d C " f o r t l

C / w i f f . l e . (1-s) t h e r e ' s no i n t e g r a t i o n o v e r C ' , ~ h a n g e limits and C /* ? J u t al=a2 so t h a t t h e i n t e g r a l o v e r C ' is zero

a1 = - S

a2 = - S

a4 = 1. -g i f ( a a t . ( 1 . - 5 ) ) g o t o 23

22 a3 = - ( (s*s- l . ) / g +g)/2!.

C /+ i f g. le. ( l . - s ) t h e r e ' s no i n t e g r a t i o n over C*,xhangw l i m i t s C /u and p u t a3=r4 so t h a t t h e i n t e g ~ a l over C" w i l l . b e z e m

C /* i f C is i n s i d e C ' t C " i n t e g r a t e o v e r C o n 1 y . O o t o 28 POP t h s t .

C /* i f ( 1 - f * g ) . le..*. t h e i n t e r s e c t i o n 09 C ' L C" i s i n s i d e C. C /* No i n t e g r a t i o n O V ~ P C. Set 821.3 so t h a t integral 0 v e ~ 6 will

a 3 = S a4 = s

i f ( ( * . le. ( l . - % ) ) . a n d . ( 0 . le. (1:s)))goto 26 23 i f ( (1 . - f * g ) . g t . .*%)goto 24 -

C

C 24

C

C??

28

C C

25 C C

C

/* b r zero - __.____

a2 = if-9)/2. a3 = a 2 /w transform limits a c l = rcoo(f-el) ac2 = ecos(f-e2) irc3 = acoo(a3ls) /* s is incrmsed b y 0.0001 to keep the 8rpument to COS .Xe. 1 . / i t (march 318 1981) why does the above stmt for r e 3 not have it? ac4 = acos(a2/(s+O. 0001)) ac5 = acos(a4 + g ) acb = acos(a3 + g ) /* iqtegrate over C ' , C & C"

g t l celculates the integral over C ' . The reel part o f the integral o v e r C" uses the same function, but the imaginary part uses the negation. Hence the switching of the limits o f integration in calculating the imaginary part of t h e integral over C".

nw,

call causs(gt1, aclIac2,altl) call oou.ss(gt2. ac3, ac41 eat11 call G S U S S ( ~ ~ ~ ~ ac5,acb8a3tl) tl Firltl + a2tl + a3tl)/amax call oauss(gtii, aclI a628 altli) call qa~s5(gt2i,ac3~oc4, a2tli) call aauss(gtli,acbIac5,a3tli) tli =(altli f a2tfi + a3tli)/amax

call oauss(gclI 0. I p i 2 1 tl) tl = tl/amax call oauss(gcli,O. ,pi28 tli) tli = tlj/amsx /* h e i e calcglete t2 /* i f (f-g).ge 2 C ' 8t C" don't intersect. if(if-g).ge.2 ) g o t o 10 /* 2' is centred at f I C y at g. g . le. f so i f 9. la. (l.-s) C is /* inside both C' & C". In a l l cases t h e r e ' s no integral over C". if ! F. le. (1. - 5 ) 1 goto 29 /* i f f. le. (1. --SI C i5 inside C ' & em. a 1 tZ= 0. a2tZ- P. a3 td= 0. a1 t2i -0. aZt2i=O. a3t2i=Q. a1 = f-1. a 2 = ((.+.-1. ) / f + f)/;t. if(9. g t . (1. -s))goto 26 8 1 * -8 a2 = -8

got:) 25

!

I-p-__

26 a3 =s a c l = scos(Q-et) 8 C 2 8 C O S ( 9 - 8 2 )

c?? /* ( m a r c h 3 1 8 1981) .gain uhy no b i 8 8 09 0.0001 t o 8 9or o c 3 3 a c 3 =acos(r3/r) ac4 = r c o s ( a 2 / ( ~ + 0 . 0 0 0 1 ~ ) . I

c /* t h e same Punctionr t h a t w e t * used 8s integrands +or tl 8re c /* used h e r e . since C" is now centred e t g and not at -g p u t g=-g c /* b e f o r e integrating and c h a n g e g b a c k t o it's previous value b y

(

I C /* 9"-9 g =-.- call a a u s r ( g t l B acl, ac2, alt2)

I c a l l osuss(gt2. ac3, a c 4 ~ a2t2) call qauss(gtli,aclJec2~alt2i) c a l l gauss(gt2rl a c 3 , ac4, a2t2i 1 9"':: t2 = ( a f t 2 + i ? Z t 2 ) / a m a x t2i (r l t2 i f a2t2i)/amax

9

g o t o 10 29 g=-g

call oauss(gcJ, 0. J p i 2 ~ t2 ) t2 = i2iamax c a l l causs(gcli,O. , p i 2 8 t2 i ) t2i = t2ilamex g '-2

1 0 fac=ai (J+1 )+con J g ( a i ( k + I 1 )/2. t l =z +tit ta = ; . .*ta t f i ~ 2 . *tli t2i = ?.+t2i ml= i + k m 2 = 1 - Y if ! m L . e q . O ) qoto 113 tis:? K t 1 t2fl E t 2 tli-? *tli tzi--:t *t:Ji

f-

C

1 i ) 3 spec(l+ml)==spec(l+ml)+fac*crnp x ( , , , spec (I+m;l)=spec (X+m2)+fac*cmplx ( t2 , t 2 i )

4 contivue retu: :?

41 do 2 .1=1,n C /* w h s t Follows i s for a circular pupil w i t h s>l

f= J * P g =o. t =o. t i d . i f ( f . o e . 2 . I g D t o 2 /* integ~ate over C ' i n t C " i C" centred a t 0. a c l = a c o s ( f / 2 1 c a l l gauss(gt1, 0. , e c l ~ t ) t=4. * t / p i call gaurs(gtlir0. # a c l ~ ti) t i=4 . * t i /p i S p e C ( ~ + J ) f C O n J g ( . i ( i ) ) + r i ( J + 1 ) ~ 2 . + C m p 1 X ( t r t i )

do 14 J = l , n f rJ*P do 14 k = l r J

2 continue

a=k*p .~ -,,. . . _ . . ._ . . . . - . . . .

C

C C

C

C

C

11

12

13

c ??

100

-- - --_ -_ -_ - -- __ . - 7* f i n d i n g t l e n d t2 t l =o. ti! -0. .It2 so. a Z t 2 -0. r3t2 so. t l i e. t2i 4. d 1 t Z i fO. a 2 t T i =O. a3t2i SO. i f ( ( F + g ) . g e . 2 . 1 g o t o 11- - .- - - - _ - /* c h c c k e d t o see i f C ' and C " i n t e t s e c t i i f n o t g o t o 11 and /U c ~ . ? c u l a t e t 2 a c l = a c o s ( ( f + q ) / 2 . )

t l I ' . * t l / p i t l i :- 0. /* c a l c u l a t e t? h e r e i f ( ( f - 9 ) . g e . 2 ) g o t o 100 i f ( ! q + l . 1. g t . s l g o t o 12 /* 1 7 t t C ' C " i s i n s i d e C a c l = s c o s ( ( f - q ) / 2 . 1 c a l l a a u s s ( f t 2 . 0 . , a c l , t 2 ) t 2 = 2 . * t 2 / p i t 2 i = 0.

/* Int C ' C" is n o t a l l i n s i d e C

C a l l GaUSS(gt1, 0. , aC1~ t1)

g o t o 100

a2 = ( f 4 9 ) / 2 . a 3 = ((s*s-l. ) / g + g ) / 2 . i f ( a 2 I t . a 3 ) o o t o 13 a2=I!s*s-l. )/P +f)/2. a3 2 22 a c l = d c o o t f - 0 2 ) a c 2 = a c o s ( a 2 - g ) (march 31, 1 9 3 1 ) a g a i n w h y no b i a s o f 0.0001 t o s f o r a c 3 ? a c 3 =- acos(a3-9) a c 4 = acos(a3/( o+O. 0001 1 ) c a l l o a u s s ( f t 2 . 0 . a c l , a l t 2 ) c a l l c i a ~ s s ( f t 2 ~ e c 3 , ac2, a2t2) c a l l o a u s s ( f t 3 , 0. 8 a c 4 , a3t2) t2 c ( a l t 2 + a2t2 + a 3 t 2 ) / p i c a l l g a u s s ( f t 2 i . O . , a c l , a l t 2 i ) c a l l gruss(ft2i. ac2, a c 3 , a2t2i) c a l l aaurr(ft3i,O. , a c S , r 3 t 2 i ) t2i =;(alt2i + a2t2i + ;r3t2i)/pi f r c " a i ( J + l ) * c o q j g ( a i ( & + ~ ) ) / 2 . t l =2 *ti t 2 = 2.+t2 t l i = Z . + t l i t a i = 2.*t21 m l = J + k m 2 = J-- t iP(mn2 e q . 0 ) g o t o 114 t1-2. *tl t 2 e . *t2 tli=2 +tli t2i=2. *t2i

I .u; .* . \,-. __? , A - .. _ _ . , . 1 - i+ ~ L- -

---- - - ----

114 ~ p e c ~ l + m l ~ = ~ p e e ~ l + ~ l ~ + r l c + c r p l x ~ t l , ~ t ~ ~ ~ ~ p e c i l + c n 2 ~ = ~ p c c ~ l + ~ ~ + f a c * c ~ p l x ~ t 2 , t 2 i ~

14 continue return

-

C /* t h e rest is ?or square p u p i l s 66 s2 9s

if(?;. gt. 1. 1 s 2 -1. amax= 4. +s2*~2 do 82 J'1,n faJ*P t=O. if(f. 9.. 2. ) g o t o 51

al=f -1. a 2 L 5 2 if((5. I t . 1. ).and. (9 . lt. (l.-S)))al=-s

if ( d 9 . eq. 0. ) g o t o 50 t = 2 *s2*,(sin(df+f*(o2-f/2.)) -sin(df*9*(ob-9/2. )))/(df+9+anrr) gOtQ 51

I

C /+ set the l i m i t s 00 integration a1 8nd a2

t

c / w if df is z e r o the integral f o r t is different.

50 t = 3 *s2+(a2-a 1 ) /amax 51 spec(l+~)=ai(t)+ai( j+1)*2. +t 8% c o n t i n u e

do Y S J"I,n f=J*P do 8 4 b = t , J g=k*p

t /* calculate t l and t2 tl = ?. if((f+g). g e . 2 )goto 52

C /* at 52 find t2 81 = f-1.

- a2 =:I- -9 if ( 5 ge. 1. ) ooto 53 if ( f le. (1. --I) 1 al=-s if ! a le. (l.-s)) a2= s

- s i n ( d 9+ f+g )*(Z. +al-f +g 1 /2. 1 1 / ( df+( f+g ) + a m i x

E / W calculate t2

53 if ( d f . ne. 0. 1 t l = 2. +s2+(rin(dQ+(f+g)~(2. *a2-f+g)/2. 1 * if ( d f . eq. 0. 1 tl = 2. fis2*(a2-al)/amax

52 t 2 4 . if((f-0). ga.2 . )goto 54 a1 =f-1. i Q ( ( f . It. (1.

= 3 + 1 . i f ( a 2 g t . .)a2 = s i f ( s . It. 1. 1.2 s if((df. ne. 0. 1. and. (9. ne. g))t2=2. *s2*(sin(d~*(f-g)+(2. +a2-+-g)/2. 1

if((df. eq.0. 1. or. (f. ep. g))t2 = 1. *s2*(&-all/amax if((f+g). g e . 2 . )tl=O.

54 fac=ai (J+l )+ai ( k+1)/4. mi= J+k &= J-k if(n2 eq.O)goto 143 t1e *tl t 2 e *t2 - - - -

1. r n d . ( 8 . It. 1. )).l=-s

-sintdP*(f-gl*(2.+al-f-g)/2. ))/(df*(P-g)*aa81)

- ...a* ,,-... . 4 . L. . - i_- r , ._ __ - -

C P

C

c f

C

c f

C

I

L C P

I

C

_ _ 143 sp.c(i+mii-spec (1+n1)+~ac*2. +t1

specil+m2)=rpec (l+m2)+Qec*2. +t2 t 1 4 . t2 = c.

84 continue r e t u t n end function ft2(x)

.. . . - . . . . -

ft2 = 0. S*Xl. COS(^ *~))*~0~(df*(f-g)*(2. +cos(x)-$+~)/~. 1 return end funct:on ft34x)

C

g t l i = 0. 5*(1. -cos(2. *x)l*sin(O. 5*df*(f+p)+(2. *cos(x)-f-g)) r e t u r n end function gt2i(x)

comman /trans/s,n#pI df,ai(41),sprc(81) cornan /fig/f# g complex si(41)# s p a r ( 8 1 ) _- - - - 1

gt21 = 0. S+s*s*(l. -cos(2. *x))*rin(-O. S*df*(f+g)+(2. +r*cos(x)-f+g)) return end

c f

C