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f^^ -^x: T^. NEW YORK UNIVERSITY

CO

'f. I ^1 lY Â» Institute of Mathematical Sdences

m

'^ * ' ,< Division of Electromagnetic Research

'^occcxx'^

RESEARCH REPORT NO. EM-68

Diffraction of Electromagnetic Waves by a Slit

in a Conducting Plane Between Different Media

JOSEF MEIXNER

"7 CONTRACT NO. A F â– 1 9 ( 1 22)-42

OCTOBER, 1954

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM - 68

DIFFRACTION OF ELECTROMAGNETIC WAVES BY A SLIT

IN A CONDUCTING PLANE BETWEEN DIFFERENT MEDIA

/^osef Meixner

Morris Kline

Project Director

The research reported in this document has been made possible

through support and sponsorship extended by the Air Force Cam-

bridge Research Center, under Contract No. AF-19(122)-42. It

is published for technical information only and does not neces-

sarily represent recommendations or conclusions of the sponsor-

ing agency.

New York, 1954

Abstract

The problem of diffraction of electromagnetic

w?. /es by an i-nfinite slit in a perfect conductor at the

interface between two different dielectric media is solved

by the use of expansions of the fields in each mediijm in

terins of appropriate Mathieu functions. 3y iratcliing these

expansions across the slit an infinite set of linear equa-

tions is obtained for the coefficients in the expansion of

the diffracted fields. The solution is put in such a form

that numerical results can be readily obtained.

Table of Contents

1. Introduction

2, Formulation of the problem

3Â» Solution for electric vector parallel to 3

the slit

hÂ» Magnetic vector parallel to the slit 6

5. Numerical evaluation 8

References 11

!â™¦ Introduction

One of the classical diffraction problems is the problem of diffraction of

electromagnetic waves by an infinitely long parallel slit in a perfectly conducting

plane embedded in a homogeneous medium. The problem was first treated by Sieger '- -^ ,

r2i

and extensive numerical results are due especially to Morse and Rubenstein*- -â€¢ and

Skavlem L-'-' â€¢ The general procedure is to expand in terms of Mathieu fimctions the

incident, reflected, and diffracted fields, and then to evaluate explicitly the coef-

ficients of the series using the boundaiy conditions.

In the present paper we shall consider a modification of this problem: we

assume that the medium on one side of the conducting plane differs in dielectric con-

stant and magnetic permeability from the medium on the other side. The general pro-

cedure of expanding into a series of Mathieu functions is also appl-^able in this case,

but the coefficients in the expansion of tlie diffracted field can no longer be evaluated

explicitly} they must now be determined nijmerically for each wavelength by solving an

infinite set of linear equations. Fortunately this infinite set reduces, in practice,

to a finite set. (The number of equations increases with decreasing wavelength.) For

wavelengths not smaller than one-third of the slit width, it does not seem too diffi-

cult to vjork out the numerical solution.

- 2 -

2, Formulation of the prob lem

Let z > be a half-space with dielectric constant e^ and magnetic permeabil-

ity IX., and let z < be a half -space with corresponding quantities e, and ii^* At the

interface z â– there is a perfectly conducting plane with a slitÂ» V^e introduce an

elliptic cylinder coordinate system by the transformation

(1) X = a cosh E, cos "ii , y = y, z=a sinh 4 sin -9^ , (0 < ^ a, i.e., for '>^ - and '\ ' + ^ ,

E , E , H , H are continuous across the slit z Â«â– 0, |x| < a, i.e., 4=0

- 3 -

3, Solution for electric vector parallel to the slit

Let a plane wave be incident from the direction designated in Figure 1 by

the angle p. Let the electric vector of this plane wave have only one component, E . ,

which is parallel to the slit; if we denote the amplitude of E. by E then

(5) E = E exp ik, (x cos fi + z sin j3)| , z > .

If the slit were absent, we would have a reflected wave with an electric

vector p>arallel to the incident wave:

(6) E = - E expMJsLfT cos p - z sin p) | , z > .

Vie now expand each of these waves into a series of Mathieu functions;'^ we

have , for < lo < n,

JT - 2^ i'^ce (p; hj) Mc;l^ (4} h^) ce (^ j h^)

(7)

00

+

and

(8)

iii'*l

F m

o m^O

^ 2r i^e (pj h^) Ms^^^ (Cj h^)sej7^} h^) ,

ra=l

where

(9) 2h^ = k^a, k^ Â« (^-/e^l"!^ .

For the definitions and notations for the ^thieu functions used in this paper, see

Meixner-Schaef ke ^ -^ , For the expansion of a plane wave see Section 2,86 of this bookÂ©

- u -

To meet all tiie conditions of the problem it will be ?een that it is suffi-

cient to assume that the electric vector of the diffracted field also has only a

single component, E,, which is parallel to the slit* Now we expand also this dif-

fracted field E, into a series of Mathieu functions:

a

(10^ !r " ZI \ti ^^^ (5J \^^Â®m^^ Â» ^^ ^ - Â°Â» ^'^'' Â° - '^ - " '

o ni=l

o mÂ«=l

i'he \xse of the Mathieu fiinctions of the fourth kind arises from the radiation con-

dition, nanely that the diffracted wave must be an outgoing wave. The omission of

terms with Mathieu functions ce (''j, ; h. ) and ce (''?: h.,"* in (10) and (ll) respective-

m J. m ' ^

ly is again justifie-" by the fact that these terras are not necessary for satisfying

the reqixLreraents for the solution of our diffraction problem. The parameter hg is

given by

(12) 2h2 - k^a , k^ = a^/e^T^ .

The coefficient of the expansions , V. (i =1.2) are to be determined.

' xm ' '

On the perfectly conducting screen ( '^ Â° 0, \ = ^ ^) Â» E.+ E vardshesj

therefore E^ must also vanish 'there. This is indeed the case in e^juations (10) and

d

(11), since the functions se vanish for '^ = ai.d '^ = i "â€¢

In the slit 4 = tnt coriuitiuns (U) m\xst be satisfied; that is, for each

value of V we must Pave

E^ + E + E , - E,

^ ' o,i13^'^'(0; h^)3e^(^j h^) .Z:^ V^Â«s^^^'(0; h^)se^(^ , h^)

'J

These two equations imist hold identically in "^ â€¢ We multiply them both by se {\\ h- )

(n = 1,2,3, Â»â€¢â€¢) and introduce the abbreviations

Pnm- l/W^J^^ seJOijh2)d^

I'aking use of the orthonormalization properties

/ se^(^i hj)se^(9^j h^)d^

^ , n - m

, n / m

we obtain

(13)

and

^in^n^^ (Â°' \^ " -f , Pnm^2m 4^^^Â°' ^> '

2 ni=l

Eliminating V^ from (13) and (lii), we have finally the infinite set of linear equa-

tions for the V__

(15) mVjPi h^tef' CO, t^) -C^ p^v^ ^ Ms^^> (0,.^). ^tu (â€ž,^) Â»

m^

and

- 7 -

/ - exp[ik^(x cos p - z sin p)] - 2 ^ iâ„¢cejp} h^)Mc^^^e; hj^)cej^ jhj)

(17) ''^ m 2 (l) 9

-2r" i'^se^CB; h,) Ms^M'^; i^)seâ€ž(9? I hf), z>0

m=l

The difft*acted wave, H,, can be expanded as follows:

raÂ«^

or

(19) j^ Â« Â£3 U^^^""^ (^i h )ce (^ , h^), z < .

o in=0

2 2

In this e>:pansion terms with fimctions se ( i^ j h- ) and se (^ j h^) respectively can

be omitted, as in the preceeding section. Consequently, the electric field component

parallel to the plane z â– vanishes on the screen C â– 0, ^ â– + n.

The bovmdary conditions in the slit are now

(hb)

^ h^\^ "r^ V|c.O,^>0Â° '^^"dk = 0,^rerÂ«

- 10 -

The numerical evaluation of the coefficients in (25) can be done conveniently

M 2

using the 'Tables Relating to Mathieu Functions''- -'â€¢ There the parameter is s = Uh

and certain quantities f , g are given as functions of s from wMch we can compute

Ms^^^'(oj h) 2 r 1

-^^T-n = - i (g )^ â€¢ 1 - if

n '

M^l'Voih) ./T. J^ .

The p are evaluated by using the expansion

? 2t+p ? r 1

^^2t^j>^1i ^ ) = 11^ Vp ^^^ 3in[(2r+p)^J , P - 0,1 .

One gets

t2B 2t+p 2^ 2n+p , 2x

2t+p

The coefficients B can be derived from the coefficients Bo^ in the above -

2r+p 'â– r-ip

mentioned tables; LJ we obtain

,-^ , .â€ž,,.â€ž [t ,..â€žj^-^''^

Â«2^ = ^Â°2r.p!A,, '"-Wj P'0.1

- 13 -

References

[1]

BÂ« ^ieger. Die Beugung einer ebenen elektrischen Welle an einere Schirm von

elliptischen Querschnitt, Ann. Physik b, 27, 626-66h (1908)o

[2] P.M. Morse and P.J. Rubenstein, The Diffraction of Waves by Ribbons and

by Slits, Phys. Re v., 5u> â€¢89^-898 (1938).

[3]

M

K

[6]

[7]

St. Skavlem, On the Diffraction of Scalar Plane Waves by a Slit of Infinite

Length, Arch. Math. Naturvld, B ^, 61-SO (19$0).

J. Keixner and i'\uc SclSfke, I'lathieusche Functionen und SpHaroidfvinctionen,

Springp^r, Berlin-Gottincen-Heidelberg, 195h.

Tables relating to i'iathieu functions: Characteristic valuec, coefficients,

and join3.ni3 factors. Nev; iork: Columbia University Press, 19$1.

S.No Karp, Some resxilts in l^lectrostatics with Applications to i^dge oinfjulari-

ties, ^''C\'i York yr-? varsity. Institute of Mathematical Sciences,

Report No. EM-71.

J. Herxner, The Behavior of Klectronagnetic lields at Bdgcs;. Nev York University,

Institute of Mathem tical Sciences, Report No. EW-72,

CO

'f. I ^1 lY Â» Institute of Mathematical Sdences

m

'^ * ' ,< Division of Electromagnetic Research

'^occcxx'^

RESEARCH REPORT NO. EM-68

Diffraction of Electromagnetic Waves by a Slit

in a Conducting Plane Between Different Media

JOSEF MEIXNER

"7 CONTRACT NO. A F â– 1 9 ( 1 22)-42

OCTOBER, 1954

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM - 68

DIFFRACTION OF ELECTROMAGNETIC WAVES BY A SLIT

IN A CONDUCTING PLANE BETWEEN DIFFERENT MEDIA

/^osef Meixner

Morris Kline

Project Director

The research reported in this document has been made possible

through support and sponsorship extended by the Air Force Cam-

bridge Research Center, under Contract No. AF-19(122)-42. It

is published for technical information only and does not neces-

sarily represent recommendations or conclusions of the sponsor-

ing agency.

New York, 1954

Abstract

The problem of diffraction of electromagnetic

w?. /es by an i-nfinite slit in a perfect conductor at the

interface between two different dielectric media is solved

by the use of expansions of the fields in each mediijm in

terins of appropriate Mathieu functions. 3y iratcliing these

expansions across the slit an infinite set of linear equa-

tions is obtained for the coefficients in the expansion of

the diffracted fields. The solution is put in such a form

that numerical results can be readily obtained.

Table of Contents

1. Introduction

2, Formulation of the problem

3Â» Solution for electric vector parallel to 3

the slit

hÂ» Magnetic vector parallel to the slit 6

5. Numerical evaluation 8

References 11

!â™¦ Introduction

One of the classical diffraction problems is the problem of diffraction of

electromagnetic waves by an infinitely long parallel slit in a perfectly conducting

plane embedded in a homogeneous medium. The problem was first treated by Sieger '- -^ ,

r2i

and extensive numerical results are due especially to Morse and Rubenstein*- -â€¢ and

Skavlem L-'-' â€¢ The general procedure is to expand in terms of Mathieu fimctions the

incident, reflected, and diffracted fields, and then to evaluate explicitly the coef-

ficients of the series using the boundaiy conditions.

In the present paper we shall consider a modification of this problem: we

assume that the medium on one side of the conducting plane differs in dielectric con-

stant and magnetic permeability from the medium on the other side. The general pro-

cedure of expanding into a series of Mathieu functions is also appl-^able in this case,

but the coefficients in the expansion of tlie diffracted field can no longer be evaluated

explicitly} they must now be determined nijmerically for each wavelength by solving an

infinite set of linear equations. Fortunately this infinite set reduces, in practice,

to a finite set. (The number of equations increases with decreasing wavelength.) For

wavelengths not smaller than one-third of the slit width, it does not seem too diffi-

cult to vjork out the numerical solution.

- 2 -

2, Formulation of the prob lem

Let z > be a half-space with dielectric constant e^ and magnetic permeabil-

ity IX., and let z < be a half -space with corresponding quantities e, and ii^* At the

interface z â– there is a perfectly conducting plane with a slitÂ» V^e introduce an

elliptic cylinder coordinate system by the transformation

(1) X = a cosh E, cos "ii , y = y, z=a sinh 4 sin -9^ , (0 < ^ a, i.e., for '>^ - and '\ ' + ^ ,

E , E , H , H are continuous across the slit z Â«â– 0, |x| < a, i.e., 4=0

- 3 -

3, Solution for electric vector parallel to the slit

Let a plane wave be incident from the direction designated in Figure 1 by

the angle p. Let the electric vector of this plane wave have only one component, E . ,

which is parallel to the slit; if we denote the amplitude of E. by E then

(5) E = E exp ik, (x cos fi + z sin j3)| , z > .

If the slit were absent, we would have a reflected wave with an electric

vector p>arallel to the incident wave:

(6) E = - E expMJsLfT cos p - z sin p) | , z > .

Vie now expand each of these waves into a series of Mathieu functions;'^ we

have , for < lo < n,

JT - 2^ i'^ce (p; hj) Mc;l^ (4} h^) ce (^ j h^)

(7)

00

+

and

(8)

iii'*l

F m

o m^O

^ 2r i^e (pj h^) Ms^^^ (Cj h^)sej7^} h^) ,

ra=l

where

(9) 2h^ = k^a, k^ Â« (^-/e^l"!^ .

For the definitions and notations for the ^thieu functions used in this paper, see

Meixner-Schaef ke ^ -^ , For the expansion of a plane wave see Section 2,86 of this bookÂ©

- u -

To meet all tiie conditions of the problem it will be ?een that it is suffi-

cient to assume that the electric vector of the diffracted field also has only a

single component, E,, which is parallel to the slit* Now we expand also this dif-

fracted field E, into a series of Mathieu functions:

a

(10^ !r " ZI \ti ^^^ (5J \^^Â®m^^ Â» ^^ ^ - Â°Â» ^'^'' Â° - '^ - " '

o ni=l

o mÂ«=l

i'he \xse of the Mathieu fiinctions of the fourth kind arises from the radiation con-

dition, nanely that the diffracted wave must be an outgoing wave. The omission of

terms with Mathieu functions ce (''j, ; h. ) and ce (''?: h.,"* in (10) and (ll) respective-

m J. m ' ^

ly is again justifie-" by the fact that these terras are not necessary for satisfying

the reqixLreraents for the solution of our diffraction problem. The parameter hg is

given by

(12) 2h2 - k^a , k^ = a^/e^T^ .

The coefficient of the expansions , V. (i =1.2) are to be determined.

' xm ' '

On the perfectly conducting screen ( '^ Â° 0, \ = ^ ^) Â» E.+ E vardshesj

therefore E^ must also vanish 'there. This is indeed the case in e^juations (10) and

d

(11), since the functions se vanish for '^ = ai.d '^ = i "â€¢

In the slit 4 = tnt coriuitiuns (U) m\xst be satisfied; that is, for each

value of V we must Pave

E^ + E + E , - E,

^ ' o,i13^'^'(0; h^)3e^(^j h^) .Z:^ V^Â«s^^^'(0; h^)se^(^ , h^)

'J

These two equations imist hold identically in "^ â€¢ We multiply them both by se {\\ h- )

(n = 1,2,3, Â»â€¢â€¢) and introduce the abbreviations

Pnm- l/W^J^^ seJOijh2)d^

I'aking use of the orthonormalization properties

/ se^(^i hj)se^(9^j h^)d^

^ , n - m

, n / m

we obtain

(13)

and

^in^n^^ (Â°' \^ " -f , Pnm^2m 4^^^Â°' ^> '

2 ni=l

Eliminating V^ from (13) and (lii), we have finally the infinite set of linear equa-

tions for the V__

(15) mVjPi h^tef' CO, t^) -C^ p^v^ ^ Ms^^> (0,.^). ^tu (â€ž,^) Â»

m^

and

- 7 -

/ - exp[ik^(x cos p - z sin p)] - 2 ^ iâ„¢cejp} h^)Mc^^^e; hj^)cej^ jhj)

(17) ''^ m 2 (l) 9

-2r" i'^se^CB; h,) Ms^M'^; i^)seâ€ž(9? I hf), z>0

m=l

The difft*acted wave, H,, can be expanded as follows:

raÂ«^

or

(19) j^ Â« Â£3 U^^^""^ (^i h )ce (^ , h^), z < .

o in=0

2 2

In this e>:pansion terms with fimctions se ( i^ j h- ) and se (^ j h^) respectively can

be omitted, as in the preceeding section. Consequently, the electric field component

parallel to the plane z â– vanishes on the screen C â– 0, ^ â– + n.

The bovmdary conditions in the slit are now

(hb)

^ h^\^ "r^ V|c.O,^>0Â° '^^"dk = 0,^rerÂ«

- 10 -

The numerical evaluation of the coefficients in (25) can be done conveniently

M 2

using the 'Tables Relating to Mathieu Functions''- -'â€¢ There the parameter is s = Uh

and certain quantities f , g are given as functions of s from wMch we can compute

Ms^^^'(oj h) 2 r 1

-^^T-n = - i (g )^ â€¢ 1 - if

n '

M^l'Voih) ./T. J^ .

The p are evaluated by using the expansion

? 2t+p ? r 1

^^2t^j>^1i ^ ) = 11^ Vp ^^^ 3in[(2r+p)^J , P - 0,1 .

One gets

t2B 2t+p 2^ 2n+p , 2x

2t+p

The coefficients B can be derived from the coefficients Bo^ in the above -

2r+p 'â– r-ip

mentioned tables; LJ we obtain

,-^ , .â€ž,,.â€ž [t ,..â€žj^-^''^

Â«2^ = ^Â°2r.p!A,, '"-Wj P'0.1

- 13 -

References

[1]

BÂ« ^ieger. Die Beugung einer ebenen elektrischen Welle an einere Schirm von

elliptischen Querschnitt, Ann. Physik b, 27, 626-66h (1908)o

[2] P.M. Morse and P.J. Rubenstein, The Diffraction of Waves by Ribbons and

by Slits, Phys. Re v., 5u> â€¢89^-898 (1938).

[3]

M

K

[6]

[7]

St. Skavlem, On the Diffraction of Scalar Plane Waves by a Slit of Infinite

Length, Arch. Math. Naturvld, B ^, 61-SO (19$0).

J. Keixner and i'\uc SclSfke, I'lathieusche Functionen und SpHaroidfvinctionen,

Springp^r, Berlin-Gottincen-Heidelberg, 195h.

Tables relating to i'iathieu functions: Characteristic valuec, coefficients,

and join3.ni3 factors. Nev; iork: Columbia University Press, 19$1.

S.No Karp, Some resxilts in l^lectrostatics with Applications to i^dge oinfjulari-

ties, ^''C\'i York yr-? varsity. Institute of Mathematical Sciences,

Report No. EM-71.

J. Herxner, The Behavior of Klectronagnetic lields at Bdgcs;. Nev York University,

Institute of Mathem tical Sciences, Report No. EW-72,

1

Online Library → Josef Meixner → Diffraction of electromagnetic waves by a slit in a conducting plan between different media → online text (page 1 of 1)