uuieuu or c library, N.W. B

NOV 5 1964

^ecUnlcciL ^iote

COMPUTATION OF A MODIFIED FRESNEL

INTEGRAL ARISING IN THE THEORY OF

DIFFRACTION BY A VARIABLE SCREEN

L. C. Walters and J. R. Wait

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

I
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NATIONAL BUREAU OF STANDARDS

technical ^Ttote 224

Issued October 14, 1964

COMPUTATION OF A MODIFIED FRESNEL

INTEGRAL ARISING IN THE THEORY OF

DIFFRACTION GY A VARIA6LE SCREEN

L. C. Walters and J. R. Wait

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price: 20 cents 1

CONTENTS Page

Abstract 1

1. Introduction 1

2. A Simple Physical Problem 1

3. Various Aperture Functions and the

Resultant Fields 3

4. The Method of Evaluation 5

5. Concluding Remarks 7

6. References 8

Figures 9

Tables 13

Appendix 19

Appendix References 20

-in-

COMPUTATION OF A MODIFIED FRESNEL INTEGRAL

ARISING IN THE THEORY OF DIFFRACTION

BY A VARIABLE SCREEN

L. C. Walters and J. R. Wait

The integral of the form

+ 00

\ f(Z) exp (-in Z

2

/2) dZ

_ 00

is considered in this note. The function f(Z) varies

from to 1 in a monotonic fashion over the range

of Z. Some numerical results are given which should

be useful in certain diffraction and scattering problems

involving objects of variable density.

1. Introduction

In numerous problems in the wave theory of diffraction, the

solution is expressed approximately by expressions involving Fresnel

integrals. These integrals, which have the form \ exp (-in Z /Z) dZ,

are extensively tabulated for real limits. However, there is a class

of problems involving non-opaque or partially transparent objects

where functions of the form \ f (Z) exp (-in Z /Z) dZ occur where

f(Z) is some non-slowly varying function. It is the purpose of this

note to discuss this modified Fresnel integral for a class of functions

f(Z) which are monotonic in Z over the range - °° to + °° .

2. A Simple Physical Problem

To give the reader some insight, a simple physical model of

diffraction by a variable screen is described to illustrate how such

a modified Fresnel integral may arise. Thus, consider a thin planar

slab (in the plane x = of a Cartesian system x, y, z) of material

whose permittivity differs from the surrounding medium. The
permittivity ( or the density in the equivalent acoustic problem) is

assumed to be just a function of y, denoted K(y). The slab is now

illuminated by a plane wave from the negative x direction. The

observer, who is on the other side of the slab (i. e. , positive x),

will see a variation with x of the field intensity E(x) which bears

some relation to the permittivity function K(y). Letting the observer

be located at y = 0, the application of first-order Born scattering

theory leads readily to the result that [Born and Wolf, 1959]

f K<y) E(x) ss const X \ —Hr- e" v dy + E^ , (1) P

2 °

where p = (xs + y

2

) , k = 2 n/wavelength, and E is the field o

in the absence of the slab. Within this approximation, each line

element of the screen is a re-radiating source for a cylindrical

wave emanating from all points on the screen. The strength of

these secondary source functions varies according to K(y).

The scattered field, E , which is the difference E(x) - E , s o may be approximated in most cases by using the usual second-order

phase approximation for the exponential factor exp (-ikp ). Thus,

i

-ikp = -ik(xs +y2

) — ikx - iayp

where a = k/(2x). At the same time, p in the denominator

is replaced by x

2

. It now follows that

-ikx p

E

S

== const X e L \ K(y) exp (-id y

s

) dy . (2)

X 2 X 00

On making a simple change of variable via Z = (2a/n)y, it is seen

that

-2-
where

-i kx

E St const X e—j— F , (3) v 2

+ 00 F = \ f(Z) exp (-in Z

2

/2) dZ , (4)

and

f(Z) = const X K(y)

In this normalization, the function F, which is the modified Fresnel

integral, is proportional to the secondary field diffracted by the

screen whose scattering properties vary according to the function

f(Z). The modified Fresnel integral, as given by (4), also arises

in related diffraction problems such as occur in VLF propagation

in the earth-ionosphere waveguide of variable width [Wait, 1964].

3. Various Aperture Functions and the Resultant Fields

The aperture function f(Z) in the simplest case could be replaced by a step function. In other words, if

f(Z) = u(Z-Z)(= l f° r Z>Zo , ° 1=0 for Z < ZQ

it is seen that

00

= \ exp (-i rr Z

2

/2) dZ (5)

Z

o which is the standard Fresnel integral. Physically, this would be

applicable to the screen whose properties changed abruptly at

Z = Z . Of more interest is the case when the aperture function

o r

f(Z) changes gradually from to 1 about the central value Z = Z .

-3-
Three aperture functions are considered. They are defined as

follows, where g is a constant:

I Linear Tapered Aperture

f(Z) x

' = -\

2 + -|~(

2

x z - z ) fo r z -— <Z<Z + — o' op op

= for-°°<Z<Z- —

o 3

1 for Z + i < Z < «

o p

(6)

II Cubic Tapered Aperture

f(Z) = — + -T ( z " z ) " H-( z" z ) 3 for Z -,-^<Z<Z + ^-

2 2 o 27 o - o 2 6 o 2 8

3 = for -»<Z<Z - -

3

1 for Z + ~ < Z < «

o 2p

b ' 2 3

*

(7)

III Exponential Tapered Aperture

f(Z) = =—z—: rr- for - °° ^ Z <. « V

' 1 + exp [-2 3(Z - Z

q

)]

(8)

The three forms of the function f(Z) are shown in figure 1 where the ordinate is described as the "aperture blocking function. " The term blocking, in this sense, refers to the partial filling of the

aperture plane with scattering material.

The resulting behavior of the function | F(z) |, which is described

as the "magnitude of the field", is illustrated in figures 2,3, and 4

for the three respective forms of the aperture functions shown in

figure 1. For large negative values of Z the aperture is totally

-4-
effective, while for large positive values, the observer does not

"see" the scattering material in the aperture. The case Z = could be described as a grazing condition. It is interesting to observe that certain intermediate values of 3 result in a rapid decay

of the field as a function of Z . o

4. The Method of Evaluation

The evaluation of the function F(Z) for the aperture with a

linear taper is quite straightforward. By a suitable change of variable and after integrating parts, it is readily found that, for case I,

vr l p Zo\ „ .„. iB r . tt „2m 2

, i-i

zi

where

(9)

Z, = Z + (l/p), Z = Z - (l/p) and F (Z) = \ exp(-inX2 CQ /2) dX . 1 O O o

o

In a similar fashion, it is found for the cubic taper, namely, case II,

that

F<*> {(t- -rO F

o

(z) + f^ exp'-inz2/2)

. L£Zfl.(ii_ Z

2>(Z) - |/(iZ2 + -i -i3ZZ + i3Z2

27 V n oy o 27n V n o o .Z . x exp(-inZ 2/2)l + . ~- - F(Z_) , (10) 2 & V

z

i where

Z

3 - -J - - = r exp (-inX2

Z. = Z + /2)« -T— , Z1 = Z - ^-— , and F (Z) = \ exp(-iTTX /2)dX

2 o2p 1 o 2 B o J

-5-
The case for the exponential taper cannot be expressed in

simple form. Therefore, a numerical procedure was adopted.

First, a lower limit Z was chosen such that for Z < Z , there

was a negligible contribution to the integral. Then, a corresponding

upper limit was chosen such that for Z > Z,., f(Z) could be considered unity. For a specified accuracy, the ultimate choice of

Z and Z was a function of g and Z . For five -digit accuracy,

it was found that Z_CO= Z + (7/e) and Z, = Z -(7/p). Thus,

1 o

F(z) ~ \ f(Z) exp (-in Z

2/2) dZ + i-^-2- - F (Z_) , (11)

J ^ o c. z

l

where f(Z) = {l + exp [-2 g (Z - Z

Q

)] j" 1

.

The significance of the neglected portions of the integral could be

verified by using the following asymptotic estimates:

Z

.2 r«

- 00 and

f(Z)

Z) exp (-in Z /2) dZ ~ -—±- exp (-in zf/2) , (12)

TT Z 1

oo r 2

f(z

2

}

\ f(Z) exp (-irr Z /2) dZ _ 7rT- exp (-in Z^/2)

Z

2

" Z

2

(13)

which are strictly valid only when Z - °° and Z -• - °° .

r

2

The integral \ f(Z) exp (-in Z /2) dZ was evaluated using a

z

i Gaussian quadrature method. Following this standard technique

(Kopal, 1955), the integral was approximated by a weighted sum of

values of the ordinate. The required abscissas are roots of the

-6-
Legendre polynomials. The number of intervals required for a

given accuracy depend on (3 and Z . For the values of Z chosen

here, n =96 was sufficient to obtain five-figure accuracy for

8 = 10, 5; while n = 192 was required for g = 1, 2, 2. 5, 3, and 4;

and finally, n = 384 was required for 8 = 0. 5. In the

case 8 = 0. 2, it was necessary to modify the method because of

poor convergence. Here, n was fixed at 3 84 but Z and Z

were selected so that the quadrature could be employed with the

desired accuracy. For the regions Z < Z and Z > Z , equations

(12) and (13) were used to obtain the residual contributions. As a

check, Z and Z were shifted by small amounts in order to detect

any change in the final results.

The results of the calculations for all three aperture distributions

are given in tables la, lb, 2a, 2b, 3a, and 3b for Z ranging

from to + 5. 0. Results for negative values of Z can be obtained

readily by the relation

I(Z ) + I(-Z ) = 1 - i . o o

The accuracy of these tables to the number of digits shown was

checked by the method outlined in the appendix.

5. Concluding Remarks

It is believed that the results given here will be useful in various

types of diffraction or scattering problems which involve objects

with tapered distributions of density. A characteristic feature of the

tapering is to remove some of the ripples associated with diffraction

patterns from sharply bounded objects such as occur with the classical

half -plane screen. Furthermore, the tapering of the density near the

diffracting (or scattering) edge has the consequence of giving a

-7-
relatively rapid transition in the diffraction from the illuminated

region to the shadow region (with an analogous effect present in

the scattering problem).

6 . References

Born, M., and E. Wolf (1959), Principles of optics (Pergamon Press,

Oxford).

Kopal, Z. (1955), Numerical analysis (John Wiley and Sons, Inc. , New York, N. Y. ). Wait, J. R. (Feb. 1, 1964), On phase changes in very-low frequency

propagation induced by an ionospheric depression of finite extent, J. Geophys. Res. vol. 68, No. 3, 441-445.

-8-
N

g

O

o

o

3

m

UJ

QC

irUJ

Q.

<

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

LINEAR TAPER

CUBIC TAPER

EXPONENTIAL TAPER

-5 -3-2-10123

APERTURE COORDINATE, Z-Z

-9-

Fig.l
10

I

I I I I E

0-0.2

_ LINEAR TAPER

r«l I I L -5-4-3-2-1012345

FRESNEL PARAMETER, Z

Q

Fig. 2 -10-
uj 1.0

UJa3

-5-4-3-2-1012345

FRESNEL PARAMETER, Z

Fig. 3 -11-
" r=

ID —

i.o —

i.o

y i.o o _

i.o —

1.0

10 =

.:i

= 1 1 1 1

1 1

1

1

MM

_

__ ff'0.2 ^ Z^SSN^, 0'O ~~E

fl«0.5 ^S.

v

/9 = l 1II \ —

CI11

0*2

\\ " \

^5 ^~ \ \

\

\ \ \

'3l|° >^"\. \\ \

= \

\, vv \

~

-CO \ \ \ \

\

\ \ >r

EXPONENTIAL TAPER ^^\ —

— \—

1 1 1 1 1 i i i i \ -5-4-3-2-101234

FRESNEL PARAMETER, 2

Fig.

4

-12-
Table la

MODIFIED FRESNEL INTEGRAL

LINEAR TAPERED APERTURE

F(REAL)

z

o 6= 0.20 0= 0. 50 (3= 1.00 0= 2.00 3= 5.00 (3= 10.00

0. 0.50000 0.500C0 0.5000C 0.50000 0.50000 0.50000

0.1 0.48919 0.47559 0.42253 0.40173 0.40007 0.40001

0.2 0.48000 0.45124 0.34813 0.30473 0.3C042 0.30015

0.3 0.47081 0.42721 0.27954 0.21048 0.2016C 0.20083

0.4 0.46002 0.40382 0.21888 0. 12090 0.10475 0.10305

0.5 0.44922 0.38109 0. 16733 0.03842 0.01182 0.00867

0.6 0.43991 0.35829 0. 12492 -0.03404 -0.07421 -0.07940

0.7 0.43064 0.33390 0.09045 -0.09327 -0. 14936 -0.15710

0.8 0.42022 0.30628 0.06175 -0.13620 -0.20871 -0.21931

0.9 0.40970 0.27482 0.03624 -0. 16052 -0.24700 -0.26033

1.0 0.39965 0.24068 0.01175 -0. 16534 -0.25936 -0.27468

l.l 0.38974 0.20655 -0.01258 -0.15190 -0.24267 -0.25840

1.2 0.38036 0. 17503 -0.03576 -0.12384 -0.19691 -0.21065

1.3 0.37085 0.14697 -0.05514 -0.08698 -0.12666 -0.13542

1.4 0. 35992 0. 12108 -0.06745 -0.04842 -0.04184 -0.04267

1.5 0. 34900 0.09554 -0.07049 -0.01492 0.04273 0.05173

1.6 0.33955 0.07022 -0.06455 0.00887 0. 10963 0. 12814

1.7 0. 33028 0.04718 -0.05267 0.02174 0.14295 0.16752

1.8 0.32074 0.02858 -0.03931 0.02593 0.13373 0.15776

1.9 0.31096 0.01419 -0.02783 0.02562 0.08478 0.09995

2.0 0.29989 0.00157 -0.01863 0.02436 0.01244 0.01179

2.1 0.28881 -0.01073 -0.00944 0.02281 -0.05679 -0.07506

2.2 0.27900 -0.02117 0.00193 0.01849 -0.09556 -0. 12538

2.3 0.26948 -0.02729 0.01440 0.00836 -0.08795 -0. 11605

2.4 0.26065 -0.02940 0.02329 -0.00725 -0.03919 -0.05052

2.5 0.25170 -0.03024 0.02376 -0.02235 0.02414 0.03768

2.6 0.24131 -0.03072 0.01553 -0.02775 0.06648 0.09880

2.7 0.23060 -0.02858 0.00391 -0.01788 0.06393 0.09529

2.8 0.21920 -0.02260 -0.00458 0.00288 0.02086 0.02906

2.9 0.20789 -0.01542 -0.00786 0.02096 -0.03082 -0.05358

3.0 0. 1978L -0.C0957 -0.00860 0.02315 -0.05316 -0.09070

3.1 0. 18813 -0.00371 -0.00887 0.00824 -0.03147 -0.05275

3.2 0. 17970 0.00334 -0.00657 -0.01067 0.01271 0.02775

3.3 0. 17143 0.00869 0.00056 -0.01752 0.03930 0.07816

3.4 0.16291 0.00990 0.00865 -0.00847 0.02613 0.05085

3.5 0. 15410 0.00914 0.01002 0.00515 -0.00973 -0.02590

3.6 0. 14390 0.00791 0.00311 0.01000 -0.0297? -0.07030

3.7 0.13323 0.00409 -0.00448 0.00474 -0.01514 -0.03355

3.8 0. 12124 -0.00150 -0.00588 -0.00175 0.01311 0.04003

3.9 0. 10894 -0.00460 -0.00301 -0.00365 0.02085 0.05978

4.0 0.09616 -0.00517 -0.0005C -0.00311 0.00249 0.00166

4.1 0.08337 -0.00516 0.00201 -0.00203 -0.01467 -0.05479

4.2 0.07102 -0.0C264 0.00475 0.00170 -0.00930 -0.03100

4.3 0.05894 0.00187 0.00297 0.00575 0.00652 0.03619

4.4 0.04791 0.C0374 -0.00349 0.00246 0.00921 0.04404

4.5 0.03733 0.00328 -0.00547 -0.00614 -0.00110 -0.01778

4.6 0.02809 0.00224 0.00030 -0.00597 -0.00621 -0.04625

4.7 0.01939 -0.00100 0.00428 0.00476 -0.00121 0.00537

4.8 0.01211 -0.00348 0.00158 0.00769 0.00283 0.04383

4.9 0.00540 -0.00213 -0.00134 -0.00333 0.00167 0.00030

5.0 0.0001C -0.00028 -0.00174 -0.00782 0.00002 -0.04050

-13-
Table lb

MODIFIED FRESNEL INTFGRAL

LINEAR TAPFRED APERTURE

F (IMAG)

z o P= 0.20 (3= 0. 50 6 = 1. 00 6= 2.00 P= 5.00 6= 10.00

0. -0.50000 -0.50000 -0.50000 -0.50000 -0.50000 -0.50000

0.1 -0.49002 -0.48232 -0.45618 -0.48657 -0.49738 -0.49895

0.2 -0.48006 -0.46179 -0.41242 -0.47029 -0.49164 -0.49477

0.3 -0.47008 -0.43667 -0.36908 -0.44849 -0.47969 -0.48433

0.4 -0.45987 -0.40701 -0.32691 -0.41893 -0.45859 -0.46462

0.5 -0.44971 -0.37458 -0.28713 -0.38005 -0.42572 -0.43286

0.6 -0.44018 -0.34211 -0.25117 -0.33128 -0.37905 -0.38682

0.7 -0.43057 -0.31200 -0.22023 -0.27334 -0.31763 -0.32525

0.8 -0.41984 -0.28515 -0.19483 -0.20845 -0.24206 -0.24844

0.9 -0.40916 -0.26077 -0.17427 -0.14040 -0.15510 -0.15886

1.0 -0.40003 -0.23733 -0.15658 -0.07427 -0.06212 -0.06177

1.1 -0. 39090 -0.21405 -0.13890 -0.01576 0.02882 0.03462

1.2 -0.48025 -0.19182 -0.11840 0.02982 0.10749 0.11940

1.3 -0.36949 -0.17254 -0.09358 0.05884 0.16284 0.18035

1.4 -0.35948 -0.15725 -0.06518 0.07039 0.18535 0.20628

1.5 -0.34966 -0.14475 -0.03629 0.06688 0.16996 0.19034

1.6 -0.34045 -0.13234 -0.01114 0.05348 0.11904 0.13365

1.7 -0.33111 -0.11831 0.00692 0.03653 0.04402 0.04773

1.8 -0.32020 -0.10361 0.01728 0.02129 -0.03545 -0.04561

1.9 -0.30919 -0.09048 0.02228 0.01005 -0.09596 -0.11863

2.0 -0.29910 -0.07948 0.02534 0.00170 -0.11789 -0.14632

2.1 -0.28928 -0.06849 0.02808 -0.00653 -0.09393 -0.11715

2.2 -0.28041 -0.05546 0.02879 -0.01599 -0.03461 -0.04155

2.3 -0.27146 -0.04144 0.02400 -0.02428 0.03338 0.04794

2.4 -0.26111 -0.02928 0.01241 -0.02583 0.07771 0.10831

2.5 -0.25045 -0.01953 -0.00236 -0.01620 0.07676 0.10748

2.6 -0.23920 -0.00994 -0.01351 0.00268 0.03323 0.04520

2.7 -0.22807 0.00034 -0.01640 0.02105 -0.02480 -0.04103

2.8 -0.21833 0.00859 -0.01239 0.02671 -0.05964 -0.09479

2.9 -0.20896 0.01268 -0.00669 0.01477 -0.04928 -0.07825

3.0 -0.20051 0.01442 -0.00238 -0.00643 -0.00457 -0.00376

3.1 -0.19207 0.01572 0.00192 -0.02049 0.03741 0.06911

3.2 -0.18275 0.01464 0.00748 -0.01680 0.04283 0.07857

3.3 -0.17304 0.00967 0.01069 -0.00050 0.01061 0.01654

3.4 -0.16182 0.00384 0.00662 0.01244 -0.02615 -0.05735

3.5 -0.15027 -0.00025 -0.00278 0.01143 -0.03200 -0.06885

3.6 -0.13814 -0.00414 -0.00874 0.00116 -0.00495 -0.00666

3.7 -0.12603 -0.00775 -0.00623 -0.00608 0.02237 0.05894

3.8 -0.11462 -0.00764 -0.00010 -0.00532 0.02018 0.05140

3.9 -0.10358 -0.00426 0.00307 -0.00153 -0.00437 -0.01858

4.0 -0.09395 -0.00127 0.00351 0.00086 -0.01853 -0.06013

4.1 -0.08482 0.00172 0.00336 0.00302 -0.00700 -0.01848

4.2 -0.07710 0.00498 0.00049 0.00433 0.01041 0.04613

4.3 -0.06983 0.00476 -0.00465 0.00030 0.00985 0.03930

4.4 -0.06357 0.00156 -0.00473 -0.00629 -0.00338 -0.02644

4.5 -0.05760 -0.00075 0.00187 -0.00442 -0.00787 -0.04607

4.6 -0.05219 -0.00282 0.00524 0.00555 -0.00038 0.01070

4.7 -0.04692 -0.00378 0.00089 0.00706 0.00448 0.04532

4.8 -0.04186 -0.C0096 -0.00289 -0.00398 0.00158 -0.00174

4.9 -0.03683 0.00183 -0.00182 -0.00793 -0.00133 -0.04215

5.0 -0.03183 0.00225 -0.00015 0.00287 -0.00162 -0.00091

-14-
z o

(3 = 0.20

0. 0.50000

0.1 0.48995

0.2 0.4 f989

0.3 0.46985

0.4 0.45981

0.5 0.44979

0.6 0.43979

0.7 0.42981

0.8 0.41985

0.9 0.40992

1.0 0.40003

1.1 0. 39016

1.2 0. 38035

1.3 0. 37057

1.4 0.36083

1.5 0. 35115

1.6 0. 34152

1.7 0.33195

1.8 0. 32243

1.9 C. 31299

2.0 0. 30361

2.1 0. 29429

2.2 0.2H5C6

2.3 0.27591

2.4 0.26683

2.5 0.25784

2.6 0. 24894

2.7 0.24014

2.8 0.23142

2.9 0.22280

3.0 0.21430

3.1 0.20 591

3.2 0. L9761

3.3 0. 1894?

3.4 0. 18136

3.5 0. 17343

3.6 0. 16562

3.7 0. 15793

3.8 0. 150 36

3.9 0. 14293

4.0 0. 13566

4.1 0. 12854

4.2 0. 12155

4.3 0. 11469

4.4 0. 10797

4.5 0. 10142

4.6 0.C9506

4.7 0.08888

4.8 0.08286

4.9 0.07698

5.0 0.07125

Table 2 a

MODIFIED FRESNEL INTEGRAL

CUBIC TAPERED APERTURE

F(REAL)

3=0.50 6=1.00 8=2.00 0=5. 00 (3=10.00

0.5C000 0. r <000t 0.50000 0.50000 0.50000

0.47415 0.43775 0.40353 0.40013 0.40002.

0.44837 0.)7707 0.30865 0.30060 0.30018

0.42271 0. U926 0.21714 0.20204 0.20091

0.39720 0.26521 0.13105 0.10564 0.10324

0.37182 0.21534 0.05278 0.01341 0.00903

0.34659 0.16969 -0.01502 -0.07169 -0.07880

0.32164 0.12815 -0.06974 -0.14568 -0.15620

0.2972? 0.09C6fc -0.10916 -0.20379 -0.21807

0.27354 0.05751 -0.13195 -0.24092 -0.25877

0.25063 0.02914 -0.13818 -0.25252 -0.27288

0.22825 0.006C4 -0.12963 -0.2358C -0.25653

0.20622 -0.01160 -0.10972 -0.19108 -0.20901

0.18453 -0.02412 -0.08312 -0.12315 -0.13436

0.16335 -0.03235 -0.05483 -0.04183 -0.04256

0.14279 -0.03727 -0.02911 0.03859 0.05069

0.12291 -0.03951 -0.00859 0.10166 0.12595

0.10389 -0.03<>24 0.00609 0.13278 0.16461

0.08603 -0.03637 0.01570 0.12416 0.15486

0.0696C -0.0311) 0.02143 0.07908 0.09809

0.05466 -0.02430 0.02375 0.01311 0.01183

0.^4126 -0.0171C 0.02214 -0.04945 -0.07288

0.02949 -0.0104? 0.01599 -0.08426 -0.12176

0.01943 -0.90454 0.00602 -0.07783 -0.11261

0.01C97 0.00C59 -0.00485 -0.03552 -0.04908

0.00397 0.00471 -0.01237 0.01901 0.03606

-0.00156 0.00714 -0.01344 0.05532 0.09488

-0.00568 0.00745 -0.00833 0.05373 0.09137

-0.0C864 0.00605 -0.00076 0.01861 0.02799

-0.01053 0.00391 0.00477 -0.02341 -0.05084

-0.01139 0.00165 0.00610 -0.04187 -0.08597

-0.01141 -0.00059 0.00427 -0.02560 -0.04999

-0.01081 -0.00246 0.00153 0.00816 0.02595

-0.C0962 -0.00318 -0.00094 0.02881 0.07326

-0.00801 -0.00244 -0.00291 0.02005 0.04762

-0.00622 -0.00096 -0.00355 -0.00547 -0.02388

-0.00432 0.00035 -0.00166 -0.02024 -0.06509

•0.00246 0.00125 0.00190 -0.01131 -0.03105

-0.00085 0.00163 0.00371 0.00730 0.03652

0.00045 0.00113 0.00161 0.01327 0.05460

0.0C139 -0.00008 -0.00196 0.00285 0.00169

0.00183 -0.00097 -0.00274 -0.00778 -0.04932

0.00188 -0.00095 -0.00035 -0.00602 -0.02791

0.0C157 -0.00043 0.00165 0.00217 0.03205

0.00098 0.0002C 0.00122 0.00475 0.03903

0.C0034 0.00074 -0.00014 0.00091 -0.01555

-0.00025 0.00062 -0.00082 -0.00211 -0.04047

-0.00066 -0.00014 -0.00080 -0.00181 0.00451

-0.00075 -O.OOCSd 0.00003 -0.00021 0.03782

-0.0C061 -o!o0034 0.00113 0.00172 0.00039

-0.00026 0.00006 0.00057 0.00185 -0.03460

-15-
Table 2 b

MODIFIED FRESNEL INTEGRAL

CUBIC TAPERED APERTURE

F(IMAG)

z o

(3= 0.20 (3= 0. 50 (3= 1.00 P= 2.00 0= 5.00 (3= 10.00

0. -0.500C0 -0.50000 -0.500C0 -0.50000 -0.50000 -0.50000

0.1 -0.49006 -0.47621 -0.4 56 34 -0.48243 -0.49665 -0.49877

0.2 -0.48012 -0.45221 -0.41378 -0.46228 -0.49019 -0.49440

0.3 -0.47019 -0.42801 -0.37331 -0.43720 -0.47755 -0.48379

0.4 -0.46026 -0.40387 -0. 33566 -0.40533 -0.45583 -0.46391

0.5 -0.45036 -0.38014 -0.30121 -0.36555 -0.42248 -0.43202

0.6 -0.44047 -0.35699 -0.269 7 -0.31770 -0.37558 -0.38591

0.7 -0.43060 -0.33438 -0.24 112 -0.26285 -0.31429 -0.32435

0.8 -0.42076 -0.31214 -0.21416 -0.20331 -0.23935 -0.24768

0.9 -0.41094 -0.29019 -0. 18H17 -0.14253 -0.15364 -0.15841

1.0 -0.40115 -0.26858 -3. 16263 -0.08471 -0.06255 -0.06179

1.1 -0.39141 -0.24739 -0.13 757 0.03415 0.02598 0.03396

1.2 -0.38170 -0.22662 -0. 11348 0.00562 0.10202 0.U8C1

1.3 -0.37204 -0.20634 -0.09109 0.03258 0.155C9 0.17828

1.4 -0.36242 -0. 18676 -0.07090 0.04675 0.17635 0.20378

1.5 -0.35285 -0. 16824 -0.05295 0.05013 0. 16147 0.18789

1.6 -0.34333 -0. 15096 -0.03689 0.04600 0.11323 0.13187

1.7 -0.33387 -0.13491 -0.02244 0.03778 0.04288 0.04723

1.8 -0.32448 -0.12004 -0.0C975 0.02795 -0.03097 -0.04442

1.9 -0.31514 -0. 10640 0.00053 0.01756 -0.08671 -0.11593

2.0 -0.30587 -C. 09401 0.00780 0.00678 -0. 10679 -0.14289

2.1 -0.29667 -0.08265 0.012C2 -0.00399 -0.08529 -0.11432

2.2 -0.28756 -0.07214 o.o

i

*ao -0.01312 -0.03241 -0.04064

2.3 -0.27851 -0.C6246 0.013 85 -0.01799 0.02766 0.04621

2.4 -0.26954 -0.C5359 0.0 1249 -0.01652 0.06658 0.10458

2.5 -0.26067 -0.04537 0.0096 7 -0.00913 0.06618 0.10369

2.6 -0.25189 -0.03770 3.005 70 0.00065 0.02954 0.04368

2.7 -0.24319 -0.03067 0.00157 0.00792 -0.01905 -0.03908

2.8 -0.23459 -0.02429 -0. CO 161 0.00955 -0.04824 -0.09042

2.9 -0.22609 -0.01846 -0.00 345 0.00616 -0.04050 -0.07461

3.0 -0.21771 -0.01324 -0.00420 0.00113 -0.00521 -0.00376

3.1 -0.20941 -0.00874 -J. 004 CI -0.00244 0.02799 0.06515

3.2 -0.20122 -0.C0490 -0.00273 -0.00373 0.03298 0.07404

3.3 -0. 19 315 -0.00172 -0.00062 -0.00348 0.00936 0.01570

3.4 -0.18522 0.00069 0.0012 5 -0.00199 -0.01792 -0.05342

3.5 -0. 17740 0.00239 0.00205 0.00074 -0.02308 -0.06398

3.6 -0. 16969 0.00 346 0.001H6 0.00336 -0.00485 -0.00632

3.7 -0.16210 0.00386 0.001 17 0.00328 0.01415 0.05420

3.8 -0. 15465 0.00371 0.00007 -0.00000 0.01381 0.04711

3.9 -0. 14 7 36 0.00318 -0.00107 -0.00312 -0.00137 -0.01675

4.0 -0.14021 0.00231 -0.0^140 -0.00254 -0.01089 -0.05458

4.1 -0. 13318 0.00130 -0.00069 0.00069 -0.00521 -0.01686

4.2 -0. 12627 0.0C0 36 0.00022 0.00236 0.00472 0.04130

4.3 -0.11952 -0.00046 0.00070 0.00098 0.00571 0.03513

4.4 -0. 11295 -0.00097 0.00076 -0.00086 -0.00029 -0.02335

4.5 -0. 10655 -0.001 13 0.000 31 -0.00112 -0.00345 -0.04077

4.6 -0. 10031 -0.00100 -0.0004 8 -0.00042 -0.00157 0.00931

4.7 -0.09422 -0.00060 -0.00070 0.00043 0.00088 0.03957

4.8 -0.08826 -0.00012 -0.00015 0.00102 0.00182 -0.00113

4.9 -0.08246 0.00029 0.000 3 3 0.00031 0.00114 -0.03629

5.0 -0.07685 0.00054 0.00041 -0.00116 -0.00158 -0.00088

-16-
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-17-
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-18-
Appendix

The purpose of this appendix is to describe the methods used to

check the table entries. The Fresnel integral which was used throughout the calculations was evaluated by the method of Boersma [I960].

Boersma states that this method is suitable to evaluate the integral

to eight decimal places over the whole range from to °° with a

small restriction which did not apply in these calculations.

Entries in tables la, lb, 2a, and 2b were evaluated using (6)

with (4) for the linear aperture and (7) with (4) for the cubic aperture.

The quantities used involved only Fresnel integrals, sines, and cosines in various combinations. The sines and cosines were evaluated

-7

by subroutines with a maximum error of ± 1(10 ) and the Fresnel

integral as described above. An investigation into the possible precision loss for both types of aperture tapers indicated that for the

severest case this would be four digits at the very most. This loss

occurred principally in the linear taper case for 3^1. This was

verified by spot checking the tables for both types of apertures by

differencing techniques.

The accuracy of the entires in tables 3a and 3b for the exponential

taper was checked by increasing the number of Gaussian abscissas

until the values for two arbitrarily chosen n were essentially the

same. For n =96 and 192, the values for 8 = 5, 10 were exactly

the same to five decimals. For 3 = 1, 2, 2. 5, 3, and 4, values of

n = 192 and 384 gave the same results with a maximum difference of

-5

± 1(10 ). For g = 0. 5, the maximum difference for n = 192 and

_5 384 was ±1(10 ) for Z ^4.1. With the same n values for

° -3

4. 1 < Z ^5, the maximum difference was ± 4(10 ). However, for

o these few values, it did not seem feasible to repeat the calculations

for larger n. The application of differencing techniques verified the

above conclusions.

-19-
For all three aperture types, if |3 = °° ,

oo in 2 - r e dZ = -rr- - F (Z ) J £00

Z

o

where F (Z ) is the Fresnel integral (9, 10, 11). The entries for

o o

g = °° in tables 3a and 3b were checked using the tables of

Winjgaarden and Scheen [1949]. The values checked to the five

decimals for all entires.

Appendix References

Boersma, J. (I960), On a numerical method for the computation of

Fresnel integrals, Report TW2, Mathematics Institute, Univ.

of Groningen.

Winjgaarden, Van A. , and W. L. Scheen (1949], Table of Fresnel

Integrals, Report R 49 of the computation department of the

Mathematical Centre of Amsterdam.

_20 - GPO 844-661