APPLICATION OF DUAL INTEGRAL EQUATIONS TO DIFFRACTION PROBLEMS

by

Man Fong Yan,B.Eng.

Department of Electrical Engineering

APPLICATION OF DUAL INTEGRAL EQUATIONS

TO DIFFRACTION PROBLEMS

by

Man Fong Yan. B. Eng.

A thesis submitted to the Faculty of Graduate Studies

and Research in partial fulfillment of the

requirements for the degree of

Master of Engineering.

Department of Electrical Engineering,

McGill University,

Montreal, Quebec.

July, 1969.

1970

·e

i

ABSTRACT

A rigorous technique is proposed for the solution

of electromagnetic wave diffraction by circular aperture ,

multiple annular apertures, single strip and multiple strips

in terms of an electric Hertz vector and a magne tic Hertz

vector. It is shown that the mixed boundary conditions lead

to multiple integral equations, which can be solved by a

Fredholm integral equation of the second kind by using the

modified operator of tœHankel transform and the Erdelyi­

Kober fractional integration operator. Calculations are

presented and field maps shown for a circular aperture

12 wavelengths in diameter.

Man-Fong YAN

Electrical Engineering

Master of Engineering

ii

ACKNOWLEDGEMENTS

The author wishes to express his sincere gratitude

to Dr. T. J. F. Pavlasek for his continued guidance and

encouragement throughout the course of this research.

The author is deeply grateful to Dr. Lan Jen Chu

for giving the original impetus for this subject.

Special thanks are also due to Mr. J. P. Legendre

and Mr. 1. A. Cermak for their many helpful discussions

and excellent graph plotting subroutines.

Grateful acknowledgement is made to the National

Research Council of Canada for financial support of the

research.

111

CONTENTS

ABSTRACT 1

ACKNOWLEDGEMENTS 11

111 CONTENTS

CHAPTER

CHAPTER

2.1

2.2

CHAPTER

3.1

3.2

3.3

3.4

CHAPTER

4.1

4.2

4.3

l

2

3

4

INTRODUCTION

GENERAL TREATMENT OF MULTIPLE INTEGRAL

EQUATIONS WITH ARBITRARY WEIGHTING

FUNCTION AND HANKEL KERNEL

1

6

Solutlon Of Dual Integral Equatlons 6

Solutlon Of Multlple Integral Equations 8

DIFFRACTION BY STRIPS

Dlffraction By A Strlp

Two Identical Strips

An Array Of Strips

Dlscuss lon

DIFFRACTION BY AXIALLY SYKMETRIC

PLANAR APERTURE

Dual Integral Equations For The

Clrcular Aperture

Dlffraction By A Clrcular Aperture­

Normal Incidence

The Annular Aperture--- Normal

Incidence

10

11

12

14

15

17

17

23

26

4.4

CHAPTER

5.1

5.2

CHAPTER

5

6

APPENDIX A

A.l

A.2

A.3

APPENDIX B

The Boundary Cond~tions On The Rim

Of The Aperture

NUMERICAL ANALYSIS

Asymptotic Evaluation Of The Field

Numerical Calculation

CONCLUSION

The Hankel Inversion Theorem

The Modified Operator Of Hankel

Transform

The Erdelyi-Kober Operator

iv

30

33

33

36

46

50

50

51

B.l The Approximate Solution Of The Fred­

holm Integral Equation Of The Second

Kind : Replacement By A Finite System 53

REFERENCES 55

CHAPTER 1

INTRODUCTION

Planar diffraction problems are treated exten-

s1vely in the literature. Excellent bibliographies can be

1 found in a series of papers by J.P. Vasseur and in Wu's

thesis 2 . The case of diffraction of a plane wave incident

normally on an infinitely extended slit has been solved

3 rigorously by Morse and Rubenstein while the circular

aperture has been treated rigorously by Meixner and Andre-

456 jewski et al " . The solution for the slit is in the

form of an Infinite series of ellipsoidal functions,Meixner's

circular aperture solution is in the form of a series of

7 8 oblate spheroidal functions. Also, Copson' derived a set

of integro-differential equations for the electric field

components. In Copson's solution, singularities at the edge

introduce difficulties and the set of equations generally

9 admits a number of solutions. Bouwkamp studied diffraction

of a plane-polarized electromagnetic wave by a small con-

ducting circular disk. A set of integro-differential equations

for the currents in the disk was set up. The method was

confined to the case of normal incidence. The dual integral

10 equations approach was used by Hanl and Zimmer as weil as

2

• Tranterll

for the strip problern. lt seerns quite attractive

to ex tend the sarne scheme to axially symmetric diffracting

apertures. Usually, in the treatment of half planes and

strips, pure TM and TE waves arc 12 13 . assumcd ' ~n order to

make substantial simplifications when reducing the complex

formulations, but in general, both TM and TE waves should

be taken into account. Actually, electromagnetic fields

are described insufficiently by a single scalar wave func-

tion, as is frequcntly attempted and should be dcter~ined

by a set of such functions interrelated by Maxwell's equa-

tions.

This thesis presents a general survey of the

advantages of making use of the concept of TM and TE waves

for axially symmetric apertur'es. lt is .found that dual

integral equations, triple integral equations and, in general,

multiple integral equations with the Hankel kernel can be

formulated. By means of reeently developed theory of this

branch of classieal mathematics the problem can be solved

formally. lt is believed that this is the first time this

method is applied to the three dimensional diffraction problem.

The theory presented here is based on the con-

cept that any eleetromagnetic field, in a homogeneous,iso-

tropie medium, free from charges and eurrents, can be ex-

pressed in terms of an eleetrie Hertz vector ~ and a

3

13,14 magne tic Hertz vector M both being along an arbitrarily

chosen direction. Speclflcally,the N and M vector8 are

written as

if· N k M .. M k

where k i8 a unit vector along the z -ax 18. Then

TM TM · E-ft 'V x E - '\lx\} x N H N

TE -,bit'Vx M ,

TE - -\lx\lx M E - H

or explicitly

El

E2

E. z

Hl

H2

H z

in

-

•

where if and M must satlsfy the veCLor wave equations

( 1)

(2)

(3)

02-

-PE ~ ~~ = 0

4

(6)

n2ii -11~ ~2M = 0 V I~ a t 2 (7)

Since cy1indrica1 waves are of interest in this

work, a

+ e-

(8)

by the princip1e of superposition. It is easy to see that

particu1ar solutions of the Helmholtz's equations (6) and

(7) which are periodic in both the azimuthal angle 9 and

the time t,can be constructed from elementary waves of the

t 14,15 ype

General speaking, the propagation cons tant t assumes any

complex value. Explicit expression for tin terms of the

frequency w and the constant of the medium is determined

by prescribed boundary conditions on a plane z·constant

2 2 or r-constant. For the present case the term ~ jJ(k -u )z

ln the expression (8) must be so chosen that the field

vanishes as z tends to ~ infinity.

The next chapter deals with the mathematical

treatment of the appropriate integral equations. In

subsequent chapters, it will be shown that by means of

(4) ,(5) and (8) a set of multiple Integral equations

can be formulated which can be solved for the unknown

weighting functions and hence the diffracted fields.

Detailed discussion of the boundary condition at the

rim of the aperture is given and a comparison Is made

9 16 17 with the result given by Bouwkamp' ,

The method of steepest descent is used in

chapter 5 to evaluate asymptotic expansions for the E

and H field. The standard theory developed by Kantoro-

18 19 vicb and Krylov , and Milkhlin for the approximate

5

solution of tbe Fredholm Integral equat~on of the second

kind is invoked. Gauss-Legendre quadrature of order n a 20

is used to approximate the integration numerically. A

summary of the method is given in Appendix B.

CHAPTER 2

GENERAL TREATMENT OF MULTIPLE INTEGRAL EQUATIONS

WITH ARBITRARY WEIGHTING FUNCTION AND HANKEL KERNEL

In this chapter are presented the mathematical

techniques essential to the solution of the dual, triple

and sets of multiple integral equations of interest to

thls work. A complete collection of significant texts and

papers ls given in the book by Sneddon20 ,2l. All the opera-

tors lnvolved here are deflned in Appendix A.

2.1 Solution Of Dual Integral Equations

A useful solutlon ls sought to the dual integral

equatlons of the type

where

Si"-etJùl«l + w(u)] tp(u); r} - 9 1 (r) ,

S.J.. t1J( r) - 11.2 (r) • "j,u,O T T

w(r). 9 1 (r). P2(r) - known functlons

~ (u) - unknown function

Dl -{ r: 0 ~ r ~ 1 1 D2 -{ r: r ~ l J

( 1)

(2)

7

22 The Method described below is due to Erdélyi and Sneddon •

Put (3)

where heu) is a new unknown function defined in the range

DIU D2 " From (1) and (2) there is obtained

SJ.JJ~ Gol SJ. h(r) + S [w(r) S, h(r)} = 9 1 (r), r E Dl ~ ~ ?)J)-rJ.. iJ)~Jc( ~",.Dl

S.J..J 0 S h(r) = r2(r) z.y 1 iJJ,-Dl

After some manipulation it is found that

and

where

and

hl r ) = h(r) H(r-l) = K -tJJ-ot/~ <J>i r)

+J:h1<U)G<r.U)dU hl (r) = F(r), réDl

hl (r) = h(r) H(l-r)

H(u) = the Heaviside's unit function 00

-OC 1+01'1 G(r,u)= r u tw(t)JJI_ot (rt) J .J.I-ot(ut) dt

F(r) = l 9 l (r) - S~ w(r) S, #.r) !~,;l 1 !Jo';',,(/. yP" () 2-

2 -'" U 1+"'J t l+ oC JJ)-cI. (ut) u'f'<u) = hl (t)dt +

0 00

-" 1+1. 1+" K ~t) dt + 2 U 1 t J~<ut) iV-1ll"cI. 2.

lt may be shown that the above method is valid for a

convergent kernel (6).

(4)

(5)

(6)

(7)

(8)

8

2.2 Solution Of Multiple Integral Equations

The set of multiple integral equations of parti-

cular interest are defined as

S "Av, 0 cp< r) = 0, r E. Dl LJ D 3U ••• • • • LI D 2 n+ 1 ( 9 )

~ ([1 + w(u»),/,(u) j r} = 9 (r) , (10) ,",0 r f- D

2 U D4

U •••• UD 2n

where w(r) = known function

9 i (r) = 9(r) [ H(r-ai

_l

) - H(a i -r)]

Dl =[r o~r,al}

D2 ={r al r a 2 }

• • • • • • • • • • • • • • • • • • • • •

D2n+ 1 = {r: a 2n < r }

lt can be verified that (9) and (10) may be put in a more

convenient single equation by use of the Heaviside's unit

23 function, (the idea i8 similar to that of Noble ),namely,

n

S.J. rp<r) = r: f9 2m(r) - S, w(r) tf'(r)J[H(r-a2m_ l )-l.~O m= Il Z. v,o

(11)

9

where

-1 App1ying Si~oto equation (11) it may be found that the

system reduces to a Fredholm Integral equation of the second

kind, viz, 00

"'(r) • F(r) -J: G(r •• ) (s) ds (12) n a 0

where F(r) . .L J 2~ 9 2,.(t) J (rt) dt (13) m-1 a 2m-1

G(r,s) -w(s)L(s,r;a l ,a2 ,··· ,a2n )

(14) 2 2 .. - u

2n L(s,r;a l ,a2 ,···· , a 2n ) .~

m (-) lv(ams ,amr) (15)

l,,(x,y) • xJv .. ,(x) J Vey) - YJv(x) JJlt~Y) (16)

In princip1e the Fredholm equation of the second

kind (12) can be solved by various approximate 18 19 techniques '

such as Iteration methods, replacement by suitable quadrature

formu1ae, the method of moments, the method of Galerkin,

the method of 1east squares, and Ritz's method. Practically,

as in mapy other cases, the method of Galerkin is usually

used. Due to the Infinite range of Integration in equation

(12) there is difficulty in numerical integration.

CHAPTER 3

DIFFRACTION BY STRIPS

The first rigorous solution of the diffraction

of a plane electromagnetic wave by an infinitely long slit

24 in a plane screen was presented by Schwarzschild who used

a scheme of successive approximations based on the Sommerfeld

25 half-plane theory. Sieger obtained a solution by separation

of variables in the wave equation using elliptical coordi-

nates. An extensive numerical computation was performed by

. 3 Morse and Rubenstein • An integral equation treatment has

26 27 been given by Levine , while Millar derived an asymptotic

expansion based on Schwarzschild's approach.

In the following are presented some known but

interesting problems illustrating the application of the

above developed theory. Diffraction by a single strip was

28 considered early by Rayleigh in 1897 . It is shown below

how to apply the dual integral equations technique to two

dimensional problems and extend it to the case of multiple

strips. Note that the dual integral equations used here

12 are essentially those derived in the book by Born and Wolf •

The

and

same approach has been used by Tranterlt

,HBnl and Zimmer1? 29

Groschwitz and H8nl •

l'

Il

3.1 Diffraction By A Strip

Consider a strip occupying y=O, ~\<a, and a normally

incident H-po1arized plane wave. The dual integral equations

are 'then found to be

j[=(S)COS(kXS)dS = 1/2 ,

0()

i P(s)cos(kxs)ds

o j 1- s2

.. 0

Let r- Ixl , u-kas -a

Dual equations with a Hankel kernel 00

Ixl" a

\xl ~ a

are then obtained,

-1/2 l J (ka)2 -(ka)2 2

u yeu) J_ 1/2 (ru)du • r

2

00 r E: Dl

~r(U)J_1/2(rU)dU a 0

o 1

where yeu) .(1/27I u )2 peu)

) u2

1 -(ka)

D -1 {r 0 ~r~l} D -2 fr r >1}

( 1)

(2)

Mak1ng use of the result of the last chapter, 1t is found

tbat v-and F(r)

Benèe the zero order solyt1on for yeu) 1s

y&u) .(2U)~ ~ t~ JO(ut)h1(t)dt

•

= fif. (ka) 2 JI (u)

,}ï j 2 1/2

It follows that

P(s) =

~l _ s2

ka 2j

u

Jl(kas)

s

. 12

which agrees with the resu1t given in the above mentioned

texte Once· the weighting function P(s) is obtained the \

problem ls comp1etely determined since the induced current

can be evaluated from the following formula ll

c jClO P(s) jkstd Jx(t) = 271 e s

J1 - s2 -00

Substituting J into Helmholtz formulas the electromagnetic x

field can then be calculated by a suitable quadrature formula.

3.2 Two ldentical Strips

Consider a more complicated diffracting system of

conducting .: strips occupying y=O , al

</xl<:.a2

as shown in Fig.

2. By an argument similar to that above, a set of Integral

equations is obtained.

ra;, P (5) COS (kx 5) d s = 0

}0)1-s2

~a> P(s)cos(kxs)ds ~ 1/2

where ~ Dl = {x : 0 ~Ixl< aIt

~f'XI< a;}

I~ >a21

(3)

lxl € D2 (4)

13

lt may be put in form

100 P 1 (u) ( tJ J_ 1/2 xu)du = 0 , ~IEDt D3

o ,/1 -(u/k) 2

[

-112 o P'(u)J_ 1/2(xu)du = ~ x ,\x\é-D2

where ·ptu)- P(s) (!nks)1/2 2

u - ks

If zero order approximatioc ls sought, it cau be found that

by equation 2.2(13)

F(u) _~ k 2u-3/2

Bence, to this approximation

The current density ln either strip 18 thus given byl1

J ( x) _ ciC:Op(s) jkX8d x 2~ e s

J1- s2 -ct> k k

- ~ i~Sin-(a2-a1)S c08ï(a2+a 1)8 7I 2 _-..;;2;;..... ___________ c 0 s kxs d 8

o S

which ls exa1uated as in Reference 30, then

~ lt i8 interestlng to note that this zero order induced

current is identically equal to zero for 'xl ~al and Ixl ~a2·

14

A1though J(O)(x) is constant in the strip, yet the sharp x

edge condition as established by Meixner and Andrejewski,

Bouwkamp, and Jones is satisfied, viz,

J - 0 n

3.3 An A~ray Of Strips

at the edge.

Assume an array of strips occupying y=O, IXIED2U

D4U ••.. UD2n, where

D2 - {IXI

D4 - ~x\ a3<lx\ <.a4 }

D;:~ ·f~'· ; . ~;::~~·'~I ~a2n} Fo11owing the same reasoning as above, a set of multiple

lntegra1 equatlons ls obtalned, namely,

- J -11 2 x u ) du • 0 , Ixl E. Dl \J D 3 [

00 p' (u) (

o Jl-(u/k)2

~P'(U)J_1/2(XU)dU • ~

Ù D2n

_l

(5)

Thls set of integral equatlons can be solved by the scheme

described in 2.2. Therefore the whole problem is reduced

to solving a Fredholm lntegral equation of the second kind.

15

lt can be verified that the zero order induced currents are

mere constants, and that the edge condition mentioned ab ove

is obeyed.

3.4 Discussion

In conclusion,the basic ideas involved in the

formulation of planar strips diffraction problems developed

above may be summarized . lt will be sho~n later the same

ideas can be applied to the circular aperture as well. To

start with, it is required that the boundary conditions on

the strips and on the rest of the plane (free space) be

satisfied by a set of "discrete" integral equations, namely,

each integral equation is defined in a particular interval

only. Then, the specific problem is reduced to that of

solving a Fredholm integral equation of the second kind

for the weighting function, after suitable transformations.

Once the weighting functions is solved, the induced current

distribution in each strip can be evaluated. lt follows

that the electric and magne tic field can then be calculated

from the Helmholtz formula in straightforward manner.

Bowever, in practice it ia not a simple matter to solve

equation (12) in chapter 2 numerically. lt may be that Galerkin's

method will be useful,provided that an economic algorithm

can be found.

16

Fig. 1

S ing1e S tr ip

Fig. 2

Two Identical Strips

•

CllAPTER 4

DIFFRACTION BY AXIALLY SYMMETRIC PLANAR APERTURES

In the case of the circular disk, it is a straight-

forward matter to derive a set of integro-diffrential equations

Ë~ + ( \J V· + k2

) J J <fds =0 (1)

disc

by requiring that the tangential components of the electric

field vanish on the disk. The detail of solving (1) is given

in the book by Jones l2 • It will be of intrinsic interest to

attack the problem by splitting the diffracted field into an

electric wave-(TM) ,(transverse magnetic), and a magnetic

wave-(TE), (transverse electric). This chapter shows how it

is possible to formulate the dual integral equations appro-

priate to the circular aperture, and hence extends the idea

to multiple annular apertures.

4.1 Dual Integral Equations For The Circular Aperture

Consider a perfectly conducting infinite screen

with a circular aperture of unit radius (without loss of

generality the radius is normalized to unit y) with centre

at the origin of circular cylindrical coordinates (r,B,z).

lt will be assumed that the time depence i5 of the form

-jwt 1 . e in the subsequent derivation, thus al OW1ng +jkr

e r

describe an out-going spherica1 wave according to Stra­

tton's14 and Sommerfeld's15notation.

to

lt is convenient to suppose that the aperture

gives rise to a perturbation field ~, ~ which is super-

posed on the field that exists when the screen is completely

31 closed . Consequently then results

-i -r +~ for z~O E + E

~ for z ~O

(1) Etotal =

H = total

ifi + iir + uct for zc::::.O

~ for z >0

( 2)

where the superscripts i,r, and d denote incidence, reflec-

tion, and diffraction respective1y. By means of the Green's

function it has been shown that the tangential components

of the magne tic field and the normal component of the elec-

tric field in the aperture proper are the same as those of the

incident wave, provided that the condition of integrability

6 developed by Meixner is satisfied, viz,

r'E~ ds is finite surfac§

Bence the appropriate boundary conditions for

the problem are,

(3)

e·

•

•

=-cl -i E (r,e,O) = E (r,e,O) n n

and =<i E (r,S,O) = 0 tg

~ (r,e,O) = 0 n

1 . 1 14 or a ternat~ve y

dH: (r,e,O) = 0

dT

where Dland D2 are defined as

Dl = {x 0 .sr < 1 }

D2

= {x r ,. 1 }

, r éD l

rE:.D2

,

19

!4)

(5)

( 6)

(7)

(8)

(9)

(10)

Making use of the linear property of the Maxwell's equations

the perturbation field can be written in terms of TE and

TH wave components, namely,

E" = ~E +

r = ~E +

If we make the assumption that the incident wave at z=O

can be expressed in

1 f (r,e,O)

the following form

00

= L f n (r) ejnS

n=-cO

( 11)

( 12)

(13)

and that from Chapter 1 equations (8) and (5), it can be

assumed that

There is

If Z

H Z

Z

20

du , (14)

Z du , (15)

1 Y/k2 2 Q {u)J (ru)e -u zdu

n n

(16)

Mk2 2 Q (u)J (ru)e -u Zdu ,

n n

Where z is supposed to be positive. Let z·O in the above

expressions and make use of the recursion relations of

Bessel function of the first kind, viz,

J l(x)-n-

nJ (x) n· . x

nJ (x) n

x

It 18 a slmple matter to show that

JI (x) n

JI (x) n

(17)

e -jwt .f ejn9r:2Y(U)Jn_l (ru)du n--oo Jo

00 OC

and_ jH +H __ e- jwt ~ e jne[u2X(U)J l(ru)du . r e ~ ~

n--oO 0

where yeu) :& jw e p (u) _/(k 2 _u 2) Q (u) D n

21

, (18)

( 19)

(20)

(21)

Benceforth by (3), (1) and (8) two pairs of dual Integral

equations are obtained, Damely,

J:=2Y(U)Jn _ 1(rU)dU - H!n + jH!n

and

Let

~U2~(k2_U2) Y(u)Jn_1(ru)du - 0

~U2x(U)Jn+l(rU)dU - -H!n + jH!n

~ r 2 2 2 Jo u ;.I(k -u ) X(U)J D+1 (ru)du - 0

Y'(u) - u2J(k

2_u

2) yeu)

~he above equations can be put iD the form

00 la Y'(u) J l(ru)du - Bi + jH i

o Jk2_ u 2 D- Qn rD

eo l Y'(u) J l (ru) du - 0 n-

, réD1

(22)

, rc;D2

(23)

, r E. Dl (24)

, rED2

(25)

(26)

(21)

, reD1

(28)

1 rE:D2

(29)

•

•

•

22

and , rED

I (30)

= 0 ,r€D2

(31)

These two pairs of dual Integral equations will be used

later.

However, it is important to remark that the two

pairs of dual integral equations derived above are by no

means sufficient to determine the weighting functions P (u) n

and Q (u). Since the boundary conditions (5) and (6) on n

the screen with infinite conductivity imply that (7)

and (8) are satisfied. On the contrary, the mere conditions

(7) and (8), namely, that the normal derivatives of the

covariant components of magnetic field tangential to the

conductor surface vanishes, will not always provide suffi-

cient information from which the former can be inf~ed.

Bence in the present case it cannot be asserted that P (u) n

and Q (U) are completely determined without resort to n

equations (4), (5) and (6). lt is clear that in order to

avoid ambiguity either equations (28) and (29) or (30) and

(31) should be supplemented by equations (44) and (45)

established in the next section. In short, boundary condi­

(7) and (8)14 are convenient in certain cases, but in the pre-

sent case care should be taken in using them •

23

4.2 Diffraction By A tircular Aperture--Normal Incidence

Let a monochromatic plane-polarized electromagnetic

wave, jkz -.. ei x _ Re jkz l ~ y

travelling in the direction from z=-oo to z=oo, impinge

(32)

(33)

upon a circular aperture of unit radius ( The rationalized

MKS system of units is adopted throughout ) • Then the

magnetic field distribution in the aperture proper, can be

expressed in circular cylindrical coordinates as

-i (2~ J).. lH. 0 ) ejkz+ j9 (34) B - Ji.JA ' 1

-i (- h J,t ,y;: 0 ) jkz-j9 (35) B - e

-1

Whence by (28) , (29) , (30) , (31) the following sets of

dual integral equations are obtained, namely,

(i) For the case n-l

Y'(u) JO(ru)du .. ~ k 2 2 -u

(36)

Y'(u)Jo(ru)du • o J (37)

and o (38)

J (39)

24

( 11) For the case n=-l

J:.j Y'(U)JZ(rU)dU - 0 , r E Dl (40) Jk2 2 o - u

~Y'(U)J2(rU)dU - 0 , r E. D2 (41)

00 and i X'(U)Jo(ru)du = -/J. ' r E Dl (42)

'/k2_ u 2 0

10"" X'(u)JO(ru) du - 0 , r E: D2

(43)

The above equatlons for both cases should be supplemented

by the equatlons as mentloned ln

1:~3Pn(U)Jl(rU)dU - El zn

/,-0 3 o u Qn(u)Jl(ru)du - 0

(1) for n-l, ls obtalned

last section, vlz,

r ~ Dl

, r ~D2

(44)

(45)

(46)

(47)

(11) for n--l, lt can be shown that the same set of dual

lntegral equatlons (46) and (47) are obtalned except that Ql(u)

i. replaced by Q_l(u).

25

By means of the method described in 2.1, the

dual integra1 equations (36) and (37) are reduced to the

prob1em of solving the following Fredholm integra1 equation

of the second kind,

where

+(1 hl(u)G(r,u)du = F(r) ,r éD )0 1

00

-cx 1+rJ.( - r u Jo tw(t) JY-ol(r t) Jy-oa..(ut) dt G(r,u)

F(r) - 220( r-~l Jf. 1V,~P

and - 2-~ul+o( J: tl+"'JY-ol(ut)h l (t)dt

If ol, c and w(t) are chosen so that

y' (u)

c --j ".

and w(t) - j - 1 ,

Jê.k/t) 2_1

(48)

(49)

(50)

(51)

then they correspond to a similar case which has been treated

32 by Lebedev and Uf1yand in the discussion of certain problem~

in elasticity. lt i8 easily verified that the kernel can be

put in the form

G(r ,u) - r.J~1[ { Go ( Ir-u 1) + Go ( Ir+ul) } (Si)

where G (u} is the Fourier cosine transform of w(t) and the c

free term i8 given by

2 F(r) - ,./n. J(e~)

-1 r

(52)

26

S1m1larly, for (46) and (47), 1t 1s conven1ent to chotle

0( - -1/2 , c - j

w(t) .. ,J(k/t)2_ l - j

Whence G(r,u) 1s g1ven by

00

G(r ,u) - ,f\Xü}[tW( t)J J/2(xt)J J/2(ut) dt ,

Jx 2

F(r) 1 u f(u)du -r<1/2)x ,.J i 2

o x-u

where

3 - l/2~1 1/2 u Ql(u) .. "2 u t J 3 / 2 (ut)gl(t)dt

~ 0

i sY'(s) feu) - - Jl(us)ds

J 2 2 Ok -s

and gl(t) 1s the solut1on of the Fredholm equation (48).

The numer1cal evaluation of the field components

1a presented in the next chapter.

4.3 The Annular Aperture --- Normal Incidence

Top1cs on the diffraction by annular apertures

have received considerable attention in the past years.

Deap1te this, the rigorous solution of the problem still

rema1ns largely unsolved. In this section a set of triple

equat10ns for the annular aperture is given, which can be

21 solved by a method shown by Sneddon .

27

A perfectly conducting Infinite screen with an

annular aperture specified by the radius al and a2

' with

centre at the origin of circular cylindrical coordinates

(r,9,z) is assumed (see Fig. 2). Following the same rea-

soning as in the previous section, the triple equations

are obtained for normal incidence as follows

(i) for the case n-l

00 .

-{jr J' Y'(u)Jo(ru)dU , r ~ D2 (53)

J- 2 2 o k -u

~o~Y'(U)Jo(rU)dU - 0 , r€: DlUD3 (54)

~ and i X'(u) J

2(ru)du

- 0 , ré D2 (55)

o Jk2 _u 2

1000

x' (u)J 2 (ru) du - 0 , r6 DI U D

3 (56)

where y' (u) and X'(u) are as before, and ,

D -1 fr o ~r ~ al }

D -2 {r a 1< r < a 2 }

D -3 {r r >a 2 }

( ii) for the case n--1

•

•

•

and

00

1 :t'(u) J

2(ru)du = 0

01 k 2 _ u 2

r Y'(u)J2(ru)du = 0 o.

r; X'(u) J (ru)du = -~, J -2 2 0 j;iJ; Nk - u o

00

~o X'(u)JO(ru)du = 0

28

(57)

(58)

(59)

(60)

lt fo11ows that the main concern is to find a way to solve

the triple equations (53) and (54). Fortunate1y the same

type of equations have been investigated by Cooke 33 . The

~et of triple equations with the kerne1 JO(xu) arises in

the discussion of prob1ems in electrostatics, e1asticity,

and the conduction of heat. lt is interesting to encounter

equations of this type in diffraction problems.

Detai1ed derivation is avai1ab1e in the above

reference and therefore on1y the final result is given

be1ow. Cooke has shown that the problem can be reduced to

solving a Fredholm Integral equation of the second kind.

Name1y,

F(u) ( 61)

where u~ - a~ 1 u+a, t

t _ A. n a. u -a,

29

Jp, t +&1 (62) - u 2 2 ln t -a

u al 1

H(u. t) ~(X)I(U.X)I(t.X)dX (63)

o s

I(s.x) - L [ y Jo (xy) dy (64) ds J 2 2

:1 s - y

peu) - A~ x dx (65) )bdu ) 2 2

a u - x 1

where y' (u) ls given by the formula

y' (u) • ~!g2(')JO(U')d. (66)

al

sg2(s) -_ ~ L~2 yG2 (y)

dy (67) 1l da ,/ 2 2

s y - s

lu addltion to the set of equatlous mentloned. the followlng

set of triple equatlous must be considered, whlch for the

case n-l bas the form

CIO

~ u~ _u2

Ql(u)

rG. D2 (68)

co

1: u3Ql(U) J 1(ru)du - 0 (69)

If tbese cau be solved uumerlcally for Ql(u) then equatlons

•

.,

•

30

(53) and (54) can be solved explicitly for Pl(u) • The

task of solving the annular aperture problem thus depends

on the ability to evaluate Ql(u).

4.4 The Boundary Conditions On The Rim Of The Aperture

The behaviour of the electromagnetic fields

in the neighbourhood of a sharp edge has been investigated

b M i d A d . k· 5 B k 9,16,17 d b J 13 Y e xner an n reJews ~, ouw amp ,an y ones •

It has been shown that the admissible singularities of E - -1/2 and H at a sharp edge are at most of order d ,since

such singularities cannot magnifest themselves as real

sources of true electromagnetic energy. "d" measures the

distance from the observation point to the cdge. There-

fore in most cases in addition to writing the correct

equations governing the system additional care must be

taken to chose those solutions which are physically appro-

priate. For instance the integro-diffrentia1 equations

13 described in the book of Jones ,and in the paper of

17 7 8 Bouwkamp and Copson' admits several solutions.

It 18 lnteresting to find that in the present

method no such ambiguity arises. The set of multiple

integral equations will "automaticallyll provide the acceptable

31

conditions on the rim of the aperture. This fact can be seen

from the rigorous result for the charge distribution of a

disk charged to unit potential as depicted in reference

21. For the case of diffraction by a small hole i.e. ka«I.

the E component and the H component in the plane zcO z z 17 for r:7l and for r~1 respectively are given by Bouwkamp

as follows

B - ~ j ~ z 3 1T

cos 9

r R-l B z

4 rsin 9 - -jj -;Jî=-=r::;2~

+ • z-O • r;> 1

J z-O J r < 1

From equations (36), (37), (48), (50) and (46) ,(41), it

can be shown that (ka<:::< 1) 1

u'Q1(U) -)2 u1/

}[ .1/2 J,/2(ut)Sl(.)dt

3 u '1 (u) ~ clain u

QI Cu) - - Q-l (u)

'1 Cu) - , -1 (u)

vhere cl ia a constant. Bence it is eaaily verified that 30,34

r -1 z-O

E ~ ·z

j cos 9

rh2-l

o

r>l

r =1

r < 1

+ z=O

The above results are in agreement with those given by

Meixner and Andrejewski, Bouwkamp, and by Sommerfeld's

half-plane solution.

Fig. 1

Fig. 2

32

CHAPTER 5

NUHERICAL ANALYSIS

As in aIl diffraction problems, further substantial

difficulties arise when reducing the formaI solution to a

state which permits numerical computation. By use of the

13 34 35 standard method of steepest descent ' , asymptotic expan-

sions for the magne tic and electric field components are

obtained but the formulae are then valid only when the point

.of observation is several wavelengths away from the screen and

from the axis.

5.1 Asymptotic Evaluation Of The Field·

Prom Chapter 4 it can be shown that

N z

H z

(1)

(2)

34

«J

Hz DB ~n29 1.::,2 LJWEP 1 (u) +)k2 _u2 QI (u) ] H~l) (ru) e _.}iJ2 _k2

du

HyD0H~~~(ru> [JwePI(u) _A2_u 2 QI(U)]e-z)u2~:~ du

00

+frWêP 1 (:.! C0829 ..lé _u 2 QI (u) sin2e Ju2H~l) (ru)e _Ju

2 _k

2 du

J j 2 2 (4)

Hz. cos9 u3Ql(u)H~1)(ru)e-z u -k du (5) .

-0()

and similar expressions are obtained for the electric field

components. In fact, once N and Mare given all the E z z

and H field may be derived from equations (2) and (3) in

Chapter 1. Whence the main concern here is to evaluate an

infinite integral of the type

00 .

1 -z u -k J J 2 2

1. !(U) H~ )(ru) e du (6)

Notice that in order to facilitate the calculation

in the complex domain, the Bessel function in the integral

14 has been tactically transformed intoAHankel function of

the first kind which is the only pertinent form for an out-

going wave in this case. Furtherm9re the lower limit is ex-

tended to - 0() so that the ·contour may be closed at infinity.

For sufficiently large arguments the following

approximation can be employed, but it is this approximation

3S

which limits the validity of the region in which the field

can be calculated,

Then (6)

H( 1) (x) " n:-u --.["'ifX

j [x - (n+ 1/2) 1! 1 e 2

becomes 00

1 ~J f(u)e j [ru-(n+1/2)~1 - z)u2

_k2

du 7[ ,JrU

(7)

By meaus f h -00 h d f d 13,34,36 i ote met 0 0 steepest escent , t may

be shown that

-j(n-l)Z! I~2e 2f(kainoi.)

sin 0<

where a in a( - fi. •

It follows then that

jkR Hz ~ 2jsiu9 Pl (kainor-.) ~

e jkR

R

and that

jkR e kR

(8)

(9)

(10)

•

•

36

+ 2 e - j 1T /2 ksi nlX( j w t Pl (k sin 0<. ) cos 2

9 +J k 2

- u 2

Q 1 (k sin 0( ) sin 2 ~e ! k

(12) 2 2 jkR

Hz::::::2cos9 k sin rJ. Ql (ksinoe) e R (13)

From the above derivation, it is obvious that expressions

(9) to (13) will be valid ln the region such that equatlon (7)

ls satisfied. Thus, for points about a few wavelengths away

from the origin and those slowly approaching to the z-axls

37 as z increases the above formulae are valid • For a moderate

38 39 size aperture Bekefi's method or Miyamoto-Wolf formula

could be used to calculate the field distribtion ln the axial

region •

5.2 Numcrical Calculation

Throughout this work Gaussian quadrature of order

4 and 20 are used. The Fredholm integral equatlon of the

second klnd encountered in Chapter 4 is solved by the method

of replacement by a finite system of equatlons. For conve-

nience twenty points are used, that ls, twenty discrete

nolnts are calculated (sce Table 1). Detailed formulation

of the method 15 glven in appendix B. An lnterestlng example

can be found in reference 18.

37

Figure 2 is the photographic reduction (15 times)

of the phase map produced on the IBM 1403 line printer by

40 a program written by Cermak, Legendre and Silvester •

The map consista of 10 submaps joined together. Each submap

(outlined rectangle) is a contour plot of a matrix contàittlu&

21x81 equal1y spaced values, i.e. 2 wavelengths by 8 wave-

lengths.

Figures 1 to 6 are the calcu1ated field maps of

the three magne tic field components from formulas (11) to

(13) using an IBM 360/75 computer. The x and z components are

purely diffracted waves. Expressions (11) and (12) indicate

that these two purely diffracted waves will be zero along

the z-axis which is in agreement with the resu1t ca1cu1ated

from the Helmholtz representations as described in reference

41. It is interesting to point out that the phase distribution

for the H component ( Fig. 1 ) near the axis. and approximate1y y

a distance equal to the aperture radius from the aperture

is essential1y plane. A simi1ar phase distribution can be found

42 in Tanis work • Essentia11y the amplitude patterns are simi-

lar to the case of diffraction of a plane wave by a circu1ar

41 43 mirror or by a focusing 1ens .

The ahove comparisons confirm the va1idity of the

38

above asymptotic expansions and aloo suggests that the

choice of twenty discrete point values for the Helmholtz

integral equation (48) is good enough for this case.

e· e x

À

/2

9

b.

3 t ..tCZL a'Z1. 44IJL .It; 6.'.?L.u!1L~lll2i'l1.Z.f1!J...l2.'!LJ;.!U~ll..1

d ' 1 ~ 3 , 9 ~ ~ n A

FIGURE 1. THE PHASE STRUCTURE OF THE CALCULATED Hy COMPONENT.

E .. PLAN E • a • 6 À.. 1\.. 3. 20 3 cm.

e

z

~ \Q

40

• __ .... ' ••••.•• ;:.., • .... • • ':-~ ..... :or ... __ ...... • ::0' ....... _ •. _ .•• ;, ....

FIGURE 2. A PHOTOGRAPHIC REDUCTION OF THE PHASE

MAP (H COMPONENT, H-PLANE) AS PRODUCED y

ON THE IBM 1403 LINE PRIN TER. THE MAP

CONSISTS OF 10 SUB-MAPS JOINED TOGETHER.

EACH SUB-MAP (OUTLINED RECTANGLE) IS A

~ONTOUR PLOT OF A MATRIX CONTAINING

21x81 EQUALLY SPACED VALUES.

Â..

/3

/2

9

x

6

o 3

FIGURE 3. THE

1= .1433E-01 2=

5= .5063E-Ol 6=

9= .8693E-Ol 0=

E-PLANE , a = 6

9

CALCULATED IH 1 COHPONENT. y

.2341E-01 3= • 3248E -0 l

• 5971E-0 l 7= .6878E-Ol

.9601E-Ol A= .1051E 00

, . " = 3.203cm •

4=

8=

B=

41

, ~ Z Il À..,

.4156E-Ol

• 77 86E-0 l

.1142E 00

~

13

/2

y

6

o

1 1

~~ ~-- -

3

FIGURE 4.

9

THE PHASE STRUCTURE OF THE

CALCULATED il COMPONENT. z

42

1 ,... z 1/ /\.

H-Plane , a = 6 ï\, Â = 3.203cm.

1\

13

/2

9

y

6

,-

t o

FIGURE 5.

1= .4985E-02

5= .7058E-Ol

1

3

THE

2=

6=

H-PLANE , a = 6

6 1 9

CALCULATED IH 1 C

2

~ OMPONENT

• 138E-Ol - • .8698E-Ol 3= .3778E-Ol 4=

7- .1034E+OO

, 7\= 3.203cm •

43

, 11 >' Z

/~

.5418E-Ol

À..

13

/2

9

6

r

1 o

1 3

FIGURE 6.

,

------, ~

1

9

1

1

1

THE PHASE STRUCTURE OF THE

CALCULATED H COMPONENT. x

e = 45°, a = 6"-., Â= 3.203cm.

44

45

TABLE 1

The ca1cu1ated 20 discrete point values of h1

(xm

)

and gl(xm).

""T

ti m hl (xm) gl (xm)

.07652 1 -.209034 -.175019 1 1

11 -.18l902 -.166529 i .22778 1 2 -.284151 +.210429"'

12 -.214110 +.718288 1

.37371 3 -.985697 +.398065

13 -.213516 -.973806

.51087 4 -.251889 -.310033 1 14 -.231810 +.819069 1

.63605 5 -.196778 -.166027 1

i 15 -.248249 -.203050

1 .74633 6 +.840582 +.184890 i 16 -.283317 +.340470 1

.83911 7 -.341231 +.610218,

17 -.336396 -.469123; .91223 8 -.377140 -.473578 1

1 18 -.438247 +.864816. ,

.96397 9 -.901439 +.165690 i

19 -.632628 -.646038 1

.99312 10 +.879941 +.993700 : 1

20 -.154431 -.486233

•

•

CHAPTER 6

CONCLUSION

Dual integral equations with Hankel kernels have

been successfully applied to mixed boundary value problems

in potential theory. In the realm of electromagnetic diffrac­

tion problems only the two dimensional single strip ( or

slit ) case has been examined by this technique. The present

work shows that the method can be extended to the three

dimensional case as weIl, ndmely, diffraction by a circular

d1sc ( or aperture ). As far as i8 known to the author this

is the first instance of applying the method to three dimen­

s10nal diffraction problems. It is also pointed out in Chapter

4 that by mean? of thi8 method it is possible to obtain a

r1gorous solution of the hitherto unsolved problem of diffrac­

tion by an annular aperture. The fundamental formulation for

the problem is expounded clearly in this work. The problem

involves solving two sets of triple integral equations. One

set can be solved by the theory given by Cooke while the

other set is difficult to solve. The complete theory of

multi-integral equations with Hankel kernel 18 still not

47

fully developed and i5 beyound the scope of this thesis.

However, realizing that an arbitrarily shaped non-

planar aperture is completely unmanageable by rigorous theory,

an adequate general formulation to solve this diffraction

problem is required. From past investigations, there

1s no evidence that any particular formulation exists.

In fact the general objective of the future vork would

be to give a systematic way of calculating the mean square

error in any proposed approximation formula. Moreover, if

one intends to work with approximate scalar or vector

formulas a plan for improved approximations is needed.

For instance, the rigorous solution for the case of diffrac-

tion by a circular aperture givcn in Wu's thesis can be

used as a criterion for the calculation ~f the mean square

errors. The formulas to be considercd would be

(1)

(2)

(3)

(4)

(5)

(6)

(7)

the Helmholtz-Kirchhoff formula,

the edge-current method,

39 Miyamoto-Wolf formula,

Millar's formula,

Braunbek's method,

Bekefi's method~8

tbe Rayle1gh's first and second formulas.

•

•

48

The next step is the study of the behaviour of the mean

square errors from various formulas and search for a clue

to improve the approximation.

A difnrent approach in pursuing the subject i8

via a vector integral equation ( or integro-diffrential

equation) • The main concern in this approach is to seek

an efficient numerical technique to solve for the induced

current. Two dimensional cases have been investigated by

44 Harrington . The method of moments has been found to give

fruitful results. Three dimensional diffraction problems do

not seem as yet to have been systematically attempted by

this scheme. Following the same line of reasoning as for

the case of two dimensio~lproblems, the integral equations

obtained from a specifie boundary value problem May be

treated by the method of moments in exactly the same manner as

before, or, alternatively, it may be cast in such a form

that the problem can be reduced to minimize a suitable

functional. An interesting and important way is to make

use of the method of finite triangular elements, since an

obstacle of any confifuration can be approximated to

suffieient aceuracy by a number of sueh elements. The method

is especially elegant in the sense that once the convergence

,

49

of the various orders of approximation and an economic

algorithm have been worked out, all kinds of diffraction

problems can be synthesized from a single element, irres-

pective of the nature of the problems. As yet the only

non-planar problem that has been solved has been the slit

investigated by Tan~2. It is hoped that the ab ove approach

will prove useful for this class of problems.

50

APPENDIX A

1. The Hankel Inversion Theorem

If the function f(x) is integrable over (o,ooJ and its variation i8 bounded near the point x, then, for

))~ -1/2

fo"'(XU) 1/2 -!v (xu) du «ut) 1/2 JI' (ut) f(t) dt -~~(X+O)+f(X-O)} ( 1)

If f(x) i8 continou8 at the point x then (1) may be restated

in the following manner. Let

g(YilJ) .. . 1/2

Jp(xy ) (xy) dx

y ~ 0 (2)

then (3)

Where J{p[f(X); yt

f(x). It i8 obvious

is the Hankel transform of order V of

that fÇ.... i{,1 ·

2. The Modified Operator Of Hankel Transforms

,

51

In certain occasions it is convenient to use the

mod if ied oper a tor of Ranke 1 transforms, S ,,,Dl' in troduced

by Erde1yi and Kober (1940) which is defin.ed as fo110ws

J., (xt}dt .6o~"rI-

(4)

so that f(t)j.XJ (5)

From (1) it can be shown that

f(t) = S f S"I~f(x)j t L 'tri-" - DC. 1, l (6)

hence (7)

3. The Erde1yi-Kober Operator

The oper a tor l 7 I~ , K 7),,'" are def ined by the

equations

_1II.~'X ail 2 " 2 2"· '2.IW J I.,,,oc. f (x) ... 2- (x - u) u-r f (u) du

Fe-} o 00

2 '1, 1. 2 2J-' -1.,.-1, ... ' K ~ /0( f (x}... x ( u - x ) u

F(c:() x

, (S)

f(u)du ,(9)

I.e 0<. > 0

I.e ? " -1/2

52

1t can be shown that

1",,'" 1 (10)

( Il)

1 , J ~ 1 -,.-01. J F ... 1." CXt"f . (12)

K -,,'" K ?f-O"r ... K'/Ct"'f> (13)

1- 1 ... l (14) T)~ '1'0( ~-oc -1

K ... K (15) ,~CJ( ,? ... Qt~-()(

•

•

53

, APPENDIX B

1. The Approximate Solution Of The Fredholm Integral Equation

Of The Second Kind : Replacement Br A F1n1te System

Consider a Fredholm integral equation of the

second kind b

Y(x) -~ K(x,u)Y(u)du = f(x) ( 1)

Let it be assumed that it 1s just1fied to replace the

integral involved in (1) by an approximate linear formula

of the form

b

fa g(x)dx = + E (2)

where Ai and x k are the weights and abscissas for the

particular quadrature, respectively, and E is the error.

Equation (1), then becomes

n

Y(x) -À <;" A.K(x,x.)Y(x.) ~ f(x) Ll. 1. 1. (3)

i=1

It is then possible to solve for the unknown function

Y(x) at discrete points; for convenience, it 15 re~uired

"

that n

Y(x j ) - À ~AiK(Xj ,xi)Y(x i ) = f(xi

)

i= 1

(j a 1,2, •••• n )

54

(4)

Once this system of equations has been solved, it is obvious

that the unknown function Y(x) can be approximated by the

following formula :

n Y(x) = f(x) +À Z, AiK(x,xi)Y(x i ) (5)

i=l

In order to ensure good accuracy Gauss's quadrature

18 employed in this work. The error incurred in using Gauss's

quadrature is shown in reference ,go. The estimate reads

where

and

Error L (b-a) 2n+ 1

2n+l

B(S) = ( y(s) (x) J

2

{1.2.3 ••• n l (n+ 1) ••• 2n\ 1.2.3 •••• 2n

•

•

•

55

REFERENCES

1. Vasseur, J. P., "Diffraction of Electromagnetic Waves

by Openings in Blane Conducting Sheets", Onde Elect,

Vol.32, pp 1-10,55-71,97-111; 1952.

2. Wu, C. J. , Diffraction of An Electromagnetic Wave By

The Aperture, Ph.D. Thesis, McGill University,

1966.

3. Morse, p. M. ,and Rubenstein, P. , Phys. Rev. Vol. 54,

4.

895,1938.

Andrejewski, W. , "Die Beugung elektromagnetischer

Wellen an der leitenden Kreisscheibe und an der

kreisformigen Offnung im leitenden ebenen Schirm",

Dissertation, Univ. of Anchen, 1952; Z. Angew

Phys. , 5,178, 1953.

5. Mcixer, J. , and Andrejewski, W. , Ann. d. Physik, 7,

157, 1950; Andrejewski, W. , Naturwiss 38, 406,

1951.

6. Heixner, J. , "Die Kantenbedingung in der Theorieder

Bcugung elektromagnetischer Wellen an Vol1kmmen

lcitenden ebcnen Schirmen~ Ann. d. Physik,(6) 6,

1-9, 1949.

7. Copson, E. T. , "An Integral Equation Method Of Solving

Plane .,iffraction Problem~", Proc. Roy. Soc. ,

56

Vol. l86A, 100, June, 1946.

8. Copson, E. T. , "Diffraction By A Plane Screen ",

Proc. Roy. Soc. , Vol. 202A, 277, July, 1950.

9. Bouwkamp, C. J. , " On The Diffraction Of Electromag-

netic Waves By Small Circular Discs And Holes ",

Philips Res. Rep., Vol. S, 401, Dec. , 1950.

10. HSnl, H. , and Zimmer, E. , z.f. Phys., 135, p. 196,

1953 _

Il. Tranter, C. J. ,Quart. J. Mech. Appl. Maths.,Vol. 7,

317, 1954.

12. Born, M. , and Wolf, E. , Principle Of Optics, Perga­

men Press, London, 1954.

13. Jones, D. S. , The ~heory Of Electromagnetism, Perga­

men Press, London, 1964.

14. Stratton, J. A. , Electromagnetic Theory, McGraw-Hi11

Co. Inc., New York, 1941.

15. Sommerfeld, A. , Partial DifferentiaI Equations In

Physics, Academic Press, New York, 1949.

16. Bouwkamp, C. J. , " On Bethe's Theory Of Diffraction

By Sma1l Holes ", Philips Res. Rep., Vol. 5,

321, Oct., 1950.

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