0 - University of California, Berkeley

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 of Engi neeri ng
University of California, 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 ..........................................................
9.1 Introduction ........................................................................................... 7
3.2Results .................................................................................................... 7
3.3 Conclusion .............................................................................................. 10
4 2-D User's Manual ......................................................................................... 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
1
2
3
3
3
5
6
7
.

fist of F'igures
F'igure 1. Region of integration for equation (7)
Qure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
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).
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.
2

1. ~uction
#dditims bo the Imaging Capability of sAMpI;E
Mark D. Prouty
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, are being used to 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 will 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. if the light coming
from one of the features is delayed by a coating so that it arrives 180" out of
3

phase. the two Wacted beams will cancel, and the desired dark area will'be
obtained.
A mask using this technique, Plrst proposed by Levenson, et 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 will 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.
4

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 it
is shown that the transform of the image intensity may be given as
I
W)= 2 (%%mfp. a+ E %+rn

the region where KCf+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.
Therefore, a transformation of the form
Then the fhst integral becomes
= s cos z or f +f ' = cos z is made.
j I
sin2z cos-2np f *2-f"Z+2(cosz -f ')(f '-f** I] (9)
rn84fff + f ')
With the further changes sin'z + (1-cos 22)/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 gtl and gtli (gtl calculates the real
part and gtli calculates the imaginary part). The second integral becomes func-
tions gt2 and gt2i. The functions gtl and gtli are also used to integrate the
third expression.
6

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 hal 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 narkc!
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 to 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 in conventional
lithography now arrive in phase.
7

This improvement is quantified more Nly 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 Qure 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 lies and spaces. These curves show that for widths of less than 1
pm, 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
8

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 at that phase angle is 0.2 Fm.
In a 2-dimensional mask, it would be desirable for all adjacent features
to 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, to 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 Qure 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, this 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,
9

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 pm 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 to 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.
nus, 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
10

dear '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 pn , 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) =Xb)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
Y(z)
I
is dehed similarly. The parameter s is the partial coherence factor, and
0 -(zs+21)/2 c 2 s -zs/2
X(z> = [ 1 --2s/2 42s zs/2
0 zs/2 czs (zs+zL)/2
def is the defocus in pm. The program nodef.out ignores cf.!?f.
11
(11)
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

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 with 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 suma-
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.
12

and spec1.f.
The run time 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).
13

-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
input k. proj .6
lambda .5
trial 3 1 0 1
linespace .75 .?5
run 1
end
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
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 It 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,
produces :
I
trial 19 1 180 .75 0 0 .5 10 .75 0 0 5
I I

-15-
Appendix 2
The following is an input example for image.out The Ale lines contains the
me 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.
0. o.t.0 0.400 0.600 0.000 1 .m 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
*0.02.
The followkg figure plots the intensity at the center of small opaque and
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
ED-29 NO. 12. ~1828. 1982.
[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.

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
00, co + a Lo d rc) N--
- 0 0 O d d 0 0 0 0 0
AlISN3lNI
figure 5. Image intensity for a eriodic series of lines and spaces 0.75 pm
wide, for no phase shift fB) and l8Oo phase shift (A).
II
b

0 00 a w- CV 0
0 -
0
0
0
0 0 0 0 0
lSWtllNO3
Fuure 6. Contrast vs. Defocus for series of lines and spacps.
m

0
-
a3
0 0 0
1SWl N 03
Kgure 7. Contrast vs. Defocus for series of lines and spaces.
cu
0
rr)
0
II
b
0
0

0 -
I1 IJ
Q)
e
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
Figure 9. Contrast vs. Feature size for series of lines and spaces.
cv .
0
0
0
n
v)
w
LL

0 00 - 0
u3
0
AlISN31NI
E
=t
mLn
Flgure 10. Intensity profile for 0.2 pm wide phase shift line defect.
0
+
Q,
D
d cv
0 0
0
0

0
u)
Figure 11. Intensity minimum vs. size for phase shift line defects.
0
0
0

0
m
0
(D
I I I I I I 1 I 1
UUnUUINlUU AlISN31NI
Figure 12. Intensity minimum w size for phase shift line defects.

0
00
-
0
-
0 0 0
cu 6) (D
319NW 3SVHd
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 AlISN31N I
F'igure 14. Minimum intensity for clear field phase transition.
0
0
0
do

AlISN3lNI
a w
0 0
cu
0 0
Figure 15. Image proflle for clear field phase transition. Phase shifting material
is on the right hand side.

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

E
=t
mln
rn
b 3
0
Y-
Q)
c3
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 POI.
the vzrious ra~es. a maximum of 41 Frequencies is taken far the
linespace and 41 frbequencies zre taken for line and for spdce.
nota - each horizontal point is the center of a cell.
(pr?jEction zItC.tem only. 1
corn:, I ex Fcamsk ( 41 1, Fsaimg (1 1)
com:n.ir, /cbwilici/ window, edge, wndorg
cornrn~r~
com:ntn /harimqi del tx, mnhpts, nmhpts, horint (50)
/fouseri mxrmfr, nmfrcp, fsarnsk(411, fsaimg(41)
comm!~n Iirnq2pi.l rna, mrnwtf, nmwtfc, rrntfwt(41)
comin:~~ /img3pl-i iincoh, ipdrco~ ifulco, icoher, signa, dfdist corn~.'~ri /io1 i itei.;ni, ibulk, iprout, ircsvl, iin. iprint, ipunrh
corr.:nrln /ob,~nski mlinc, rnEpace, mlnspa, mirreg, mask ty
comrn:l,; 103 jn~;~/ r*lwl rsw, rjw2, rsw2
cnm:r7:::i /optic i cui.i,jl, VlJ vmax, thorin(50)
CQPQ-,~~~ /spectr/' mnlrnbd, nmlmbd, rlambd( 101, relint ( 10)
cnm:r.-tii fcnosk/ nurni.e:pa am(100),phi(100), dsize(1001, zperid
i.c st?t rn~xinum freqttcncy for coherent, partially coherent,
/it -11- inc ohwC.::t case.
ii: (iroher ~ 3 . iinc.oh) v:nar = 2. %rna/rlambd(ilmbd)
ii: (~roher . eq. iparco) vmax = (rna/rlarnbd(ilmbd))+(l. +sigma)
if orolrcr . 58 ifulco) vinax = rna/rlambd(ilmbd)

c
C
C
C
C
C
c
C
C
C
C
c
C
C
C
C
C
/* ca!cula?e t.3e fourier coeifficents for the pattern
450 call Foirrcf(. t.rue. , PSW~ rlw;'. T'SW~, rlw, fsamsk, mxnmfr, nmfrcp-1 1
q 0 t ; ; r,:r+r)

L
C
C
C
C
C
20
C
10
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 >=phi ( i )/360. *2*pi
. ...._.
do 1030 n=l, wmcof x 130.
x 24.
reor9.
a i mo-9.
do 40 i=l,numreg
~ 2 1 x rx 2 ( i 1
rel=2+rm~i~+~sin~n+u~x2~~si~~n*~xl~~/~n*pi~
reo=reo+rel+cos(ang(i) 1
rimo=aimo+rel+iin(.ng(i))
xl=xL?
continue
rcoeff (n+l)ncapIx(reo,.iIo)
continue
return
end
._ . .,'..,.T. .. . .- I .
. -.

c
c
subr;vtine foi~rcf(lext8 rsl,TI&', ro2,rlf, d1,nsnutncof)
c 5
C the t-ci~'rier coefficients for the mask amplitude transmission are
C tal- *:at€d.

C
C
c 2.
sub; -.:.tine p?~coZ(iIrnbd)
C this Eubr is a modified version of Mike O'loole's subr parcoh.
C It iicrs the values of rlw arid rsw calculated irl subr image
C
C
to calculate the Fourier coeFficients for the mask amplitude
tra:,r:~i~5ian (exterided as aw even periodic furrction if necessary)
C and places ?hfn in the arrau ai(41). It then calls subr crus5
C which calculates thF FouriFr coefficients for the image intensity
C pattcrn and lrcves them in thc array spec(U1)
C the cGrnmon blocks /trans/ contains ai(411, sprc(81), 5 (sigma),
C n (cht number of spD.tia1 Frequency components (at most 40)),
C and dl- (norrn.Jlised dcfocus).
C
(.
comsi cx f samsk ( 41 1, f5aimg (4 1)
comui~x ai(4i)s spec(U1)
c om:p urI / c 3 w i rr ci i w i 1-1 d ow, e d 9 e, WII d or g
corn;n-,fi-! /fouser/ mxnmfr, nmft-cp, fsamsk(41), fr;aimg(41)
com~qn /hariaqi deltx, mnhptr., nmhptr., horint(50)
com~:tn /ingf!o/' imgfl (5)
com:a.r!i /jmg2pi-/ rna, mxnwtf, rimwtfc, rmtfwt(41 1
comv:~*i /img3pt.i iincoh, iparco, ifulco, icoher, sigma, dfdist
cornn:;~.~ /io1 i itarmi, ibulk, iprout, iresvl, iin, iprint, ipunch
com;7,;:ri /optic i curt((l. +r!l'p)
if !I: . It. 4\11 goto 100
I ' -
'0

z
c:'
C
C
100
/* tw5y 15 thcre no check for. ni=O? march 25, 1981)
/it tht Fourier coefficients for the mask amplitude transmission
/+ i.1'- now caicu1atr.d.
if ta.:-skty. e?. 5) call cFour(ai,n)
if (n:..skty. e? 2.) golo 101
call
kour.cF(. Palse. , rsw, 1.1~12, i.si&J rlw# si, 41J n)
call crc155
or i '1 I I:= ( --r s w/d -e d 9 e 1 /r p er
i + ( m i = k t y . e q. m 5. p a c e 1 or i Q i n= ( 1. sw/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
aftn;=kty. eq. 5) ai-igin = (j~.ize(l)/2..rdge)/rp~~.
X=QI I .-, i ri
n2 et;
de 1 ti!- window/ i nmhp te,--l) *rper 1

C
t
C I
subcoutine cross
-_ .-- __- - --_ -
-
f
commn /iapQl / iapert, isquare icirc
common /trans/s, n, p, df, ii(4l)r spae(81)
comean /Qig/P, g
complex ai(41), rpec(81)
cornpiex fac
external gcl, gtl, gt2, ft2, ft3
external gcli, gtli, gt2ic ftZi, Qt3i
C
C /* set all elements oQ spec to zero
do 1 .iL-1,81
SP'C (j)=(O. 0.0.0)
C
c??
1 contirvuc
specc&)=ai(l)uconjg(ai(l))
p i 4 34159265558
p iZrp i 12.
/* ciFcular and square pupils me handled separately
/* the following line was present in a previous version.
c??
c??
C
/w whtt? (march 31, 1981).
if (n. eq. 0 ) return
if (ispert. eq. i squar) goto 66
i9 (r.spert. ne. icirc) return
/w set, amax, the mox. area of ovwlap
amaz L. s*s+pi
if (5. gt. 1. 1 aaax=pi
C /w for circular pupils 131 and s

.c?? /* whu doesn't the rtrnt for ac3 h8v8- the bias oP O.-OOOl 3
ac4 = .cos(d2~(s+O.0001))
Call g.USS(gt1, 8Cfr 8C218lt)
- call grUciS(gt2. 8C3, aC4,82t)
t (slt 482t)fi2./amax
calf g~ur,r(gtli~~cI,~~Z~~fti)
call gauss(gt2ir ac3, ac44 a2ti)
ti = (alti +a,3ti)*2./arnsx
5 spec (l+j)=con jg (ai( 1) )*ai( j+1)*2. *cmplx (t, ti)
3 contirue
C /* the nested do loops that follow make single sum series out
C /+ of the double sum series
do 1 ;=I,,
e=J*D
do 4 k=frJ
gr)r*,l
tf =-ZJ
ta =i:.
tli =-?.
t2i ~2.
C /w calculate tl and t2. in qeneral. integrate OYCP three
c /w circles. in some cases the limits UP integration are
C /* are the ssme Qor an integral. this io because the
c /* reqion of integration does not include that mea.
e /w this depends on the values of f and g.
C /* eo..rt tl
altt =-om
a2tl =o.
a3tl -0.
altli =O.
o2tli =O.
a3tli =Q.
C /* if (f+g).qc 2 C' and C" don't intersect and t14.
if((t'+g).ge.s !goto 25
C /w set the limits for integrating over C 'tC and C" for tl
a1 = r-1.
a2 =

C
C
C
C??
C
C
C
C
C
24
28
25
/* br zero - __.____
a2 = if-9)/2.
a3 = a2
/w transform limits
acl = rcoo(f-el)
ac2 = ecos(f-e2)
irc3 = acoo(a3ls)
/* s is incrmsed by 0.0001 to keep the 8rpument to COS .Xe. 1 .
/it (march 318 1981) why does the above stmt for re3 not have it?
ac4 = acos(a2/(s+O. 0001))
ac5 = acos(a4 +g)
acb = acos(a3 + g)
/* iqtegrate over C',C & C"
gtl celculates the integral over C'. The reel part of the
integral over C" uses the same function, but the
imaginary part uses the negation. Hence the switching
of the limits of integration in calculating the imaginary
part of t he 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
got:) 25
call oauss(gclI 0. I pi21 tl)
tl = tl/amax
call oauss(gcli,O. ,pi28 tli)
tli = tlj/amsx
/* heie calcglete t2
/* if (f-g).ge 2 C' 8t C" don't intersect.
if(if-g).ge.2 )goto 10
/* 2' is centred at f I Cy at g. g . le. f so if 9. la. (l.-s) C is
/* inside both C' & C". In all cases there's no integral over C".
if ! F. le. (1. -5) 1 goto 29
/* if 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.
a2 = ((.+.-1. )/f + f)/;t.
if(9. gt. (1. -s))goto 26
81 * -8
a2 = -8

!
(
26 a3 =s
I-p-__
c??
acl = scos(Q-et)
8C2 8COS(9-82)
/* (march 3 1 8 1981) .gain uhy no bi88 09 0.0001 to 8 9or oc3 3
c
c
c
ac3 =acos(r3/r)
ac4 = rcos(a2/(~+0.0001~) . I
/* the same Punctionr that wet* used 8s integrands +or tl 8re
/* used here. since C" is now centred et g and not at -g put g=-g
/* before integrating and change g back to it's previous value by
I C /* 9"-9
g =-.- 9
call aausr(gtlB acl, ac2, alt2)
I call osuss(gt2. ac3, ac4~ a2t2)
call qauss(gtli,aclJec2~alt2i)
call gauss(gt2rl ac3, ac4, a2t2i 1
9"'::
t2 = (aft2 + i?Zt2)/amax
t2i (rlt2i f a2t2i)/amax
goto 10
29 g=-g
call oauss(gcJ, 0. J p i 2 t2) ~
t2 = i2iamax
call causs(gcli,O. ,pi28 t2i)
t2i = t2ilamex
g '-2
10 fac=ai (J+1 )+con Jg (ai (k+I 1 )/2.
tl =z +tit
f-
ta = ;..*ta
tfi ~ 2 *tli .
t2i = ?.+t2i
ml= i+k
m2=1-Y
if !mL.eq.O) qoto 113
tis:? K t 1
C
t2fl E t 2
tli-? *tli
tzi--:t *t:Ji
1 3 spec(l+ml)==spec(l+ml)+fac*crnp x(,,, i)
spec (I+m;l)=spec (X+m2)+fac*cmplx (t2, t2i)
4 contivue
retu: :?
/* whst Follows is for a circular pupil with s>l
41 do 2 .1=1,n
f= J*P
g =o.
t =o.
tid.
C
if(f. oe.2. IgDto 2
/* integ~ate over C' int C"i C" centred at 0.
acl =acos(f/2 1
call gauss(gt1, 0. , ecl~ t)
t=4. *t/pi
call gaurs(gtlir0. # acl~ ti)
ti=4. *ti/pi
SpeC(~+J)fCOnJg(.i(i))+ri(J+1)~2.+Cmp1X(trti)
2 continue
do 14 J=l,n
frJ*P
do 14 k=lrJ
a=k*p
. .
. . .~ -,,. . .
. -. .
_ ..._.

C
C
C
C
C
C
c ??
11
12
13
100
--_ -_ -_ - -- __ . -
-- -
7* finding tl end t2
tl =o.
ti! -0.
.It2 so.
aZt2 -0.
r3t2 so.
tli e.
t2i 4.
d1tZi fO.
a2tTi =O.
a3t2i SO.
if ((F+g).ge.2. 1 goto 11- - .- - - - _ -
/* chccked to see if C' and C" intetsecti if not goto 11 and
/U c~.?culate t2
acl = acos((f+q)/2.)
C a l l GaUSS(gt1, 0. , aC1~ t1)
tl I'.*tl/pi
tli :- 0.
/* calculate t? here
if((f-9). ge.2 )goto 100
if(!q+l. 1. gt. slgoto 12
/* 17tt C' C" is inside C
acl=scos((f-q)/2. 1
call aauss(ft2.0. , acl, t2)
t2 = 2 . *t2/pi
t2i = 0.
goto 100
/* Int C' C" is not all inside C
a2 = (f49)/2.
a3 = ((s*s-l. )/g +g)/2.
if(a2 It. a3)ooto 13
a2=I!s*s-l. )/P +f)/2.
a3 2 22
acl = dcootf-02)
ac2 = acos(a2-g)
(march 31, 1931) again why no bias of 0.0001 to s for ac3 ?
ac3 =- acos(a3-9)
ac4 = acos(a3/( o+O. 0001 1)
call oauss(ft2.0. acl, alt2)
call cia~ss(ft2~ ec3, ac2, a2t2)
call oauss(ft3, 0. 8 ac4, a3t2)
t2 c(alt2 + a2t2 + a3t2)/pi
call gauss(ft2i.O. ,acl,alt2i)
call gruss(ft2i. ac2, ac3, a2t2i)
call aaurr(ft3i,O. ,acS,r3t2i)
t2i =;(alt2i + a2t2i + ;r3t2i)/pi
frc"ai(J+l)*coqjg(ai(&+~))/2.
tl =2 *ti
t2 = 2.+t2
tli =Z.+tli
tai = 2.*t21
ml=J+k
m2= J-- t
iP(mn2 eq.0)goto 114
t1-2. *tl
t2e. *t2
tli=2 +tli
1 -
t2i=2. *t2i
- - - ----
i+ ~ L- -
I .u; .* . \,-. __? , A- . __ . , .

I
t
-
114 ~pec~l+ml~=~pee~l+~l~+rlc+crplx~tl,~t~~~
~pecil+cn2~=~pcc~l+~~+fac*c~plx~t2,t2i~
14 continue
return
C /* the rest is ?or square pupils
66 s2 9s
if(?;. gt. 1. 1 s2 -1.
amax= 4. +s2*~2
do 82 J'1,n
faJ*P
t=O.
if(f. 9.. 2. )goto 51
C /+ set the limits 00 integration a1 8nd a2
al=f -1.
a2 L 52
if((5. It. 1. ).and. (9. lt. (l.-S)))al=-s
c /w if df is zero the integral for t is different.
if (d9. eq. 0. )goto 50
t = 2 *s2*,(sin(df+f*(o2-f/2.)) -sin(df*9*(ob-9/2. )))/(df+9+anrr)
gOtQ 51
50 t = 3 *s2+(a2-a 1) /amax
51 spec(l+~)=ai(t)+ai( j+1)*2. +t
8% continue
do YS J"I,n
f=J*P
do 84 b=t, J
g=k*p
t /* calculate tl and t2
tl = ?.
if((f+g). ge.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
53 if (df. ne. 0. 1 tl = 2. +s2+(rin(dQ+(f+g)~(2. *a2-f+g)/2. 1
* -sin (d 9+ f+g )*(Z. +al-f +g 1 /2. 1 1 / ( df+( f+g )+ami x
if (df. eq. 0. 1 tl = 2. fis2*(a2-al)/amax
E /W calculate t2
52 t24.
if((f-0). ga.2. )goto 54
a1 =f-1.
iQ((f. It. (1. 1. rnd. (8. It. 1. )).l=-s
=3+1.
if(a2 gt. .)a2 = s
if(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
-sintdP*(f-gl*(2.+al-f-g)/2. ))/(df*(P-g)*aa81)
4.
if((df. eq.0. 1. or. (f. ep. g))t2 = 1. *s2*(&-all/amax
if((f+g). ge.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
- ...a* ,-. .
t2e *t2 - - --
- i_- r , ._ __ - -
L. .

I
L
I
C P
C
c f
C
cf
C
CP
C
143 sp.c(i+mii-spec (1+n1)+~ac*2. +t1
specil+m2)=rpec (l+m2)+Qec*2. +t2
t14.
t2 = c.
84 continue
retutn
end
function ft2(x)
_ _
. . . - . . . . -
ft2 = 0. S*Xl. COS(^ *~))*~0~(df*(f-g)*(2. +cos(x)-$+~)/~.
1
return
end
funct:on ft34x)

C
cf
C
gtli = 0. 5*(1. -cos(2. *x)l*sin(O. 5*df*(f+p)+(2. *cos(x)-f-g))
return
end
function gt2i(x)
comman /trans/s,n#pI df,ai(41),sprc(81)
cornan /fig/f# g
complex si(41)# spar(81) _- --- 1
gt21 = 0. S+s*s*(l. -cos(2. *x))*rin(-O. S*df*(f+g)+(2. +r*cos(x)-f+g))
return
end