Diffraction of an obliquely incident wave beam by a rectangular aperture

Volume 12, number 2

OPTICS COMMUNICATIONS

DIFFRACTION

OF AN OBLIQUELY

BY A RECTANGULAR

October 1974

INCIDENT WAVE BEAM APERTURE

K. TANAKA

Department of Electrical Engineering, Nagasaki University, Nagasaki, Japan Received 17 June 1974 The diffraction problem of an obliquely incident wave beam on a rectangular aperture can be reduced to that of normal incidence by expressing the incident wave beam in a form convenient for the discussion of the problem. In this paper the diffraction field is obtained by using this expression. The field is then expanded into a series of beam mode functions. The expansion coefficient of the series can be expressed in a simple form. A numerical example of the coefficient is shown for the diffraction by a half plane. 1. Introduction The diffraction problem of a wave beam by a plane aperture has been discussed. Two methods have mainly been used for the analysis o f this problem. They are (1) the investigation o f the diffraction field distribution [ 1,2] and (2) the beam mode expansion method [3, 4]. They both have some merits and demerits. But when optical systems which have more than two apertures must be investigated, only the second m e t h o d will be useful. Very simple expressions for the coefficients o f this mode expansion are obtained in the case o f the normal incidence [5]. And this result is applied to the analysis o f a system of two aperture stops [6]. In this paper the diffraction problem o f an obliquely incident wave beam by a rectangular aperture is discussed. The incident wave beam is expressed in a form convenient for the discussion o f this problem b y using the beam wave approximation. The diffraction field is obtained by using the Kirchhoff-Huyghens diffraction formula. This field is then expressed as a sum of beam mode functions which have arbitrary beam mode parameters; the smallest spot size and its position. It will be shown that the oblique incidence problem can be reduced to that of normal incidence. As a numerical example some expansion coefficients in the case o f a fundamental mode incidence upon a half-plane are shown.

2. Expression o f an obliquely iacident wave beam The wave beam which propagates along the z ' axis is, in general, given by [7]

~mn(X,,y,,z,)=

r

r

!

t

77 . e x p [ _ j k ( z , + Z s ) _ ½ ~ l , 2 o , 2 ( x , 2 + y , 2 ) + j ( m + n + l ) t a n _ l ~ ]Hm(•X X/ rr2m+nm!n!

t

t

)Hn(~y,), (1)

where k = 2 7r/X, X = wavelength, and ~ ' = 2(z '+ Zs)/(kw ' s' 2 ) ,

'7 ' = x / ~ ( w ' s I + x / F / ~ ) ,

o '2

=

l+j~'.

(2)

H m ( X ) are the Hermite polynomials defined by H m ( X ) = ( - l ) m exp (X 2) d m exp ( - X 2 ) / d X m .

(3)

This wave beam has the smallest spot size w~ at z ' = - z ~ as shown in fig. 1. The set o f these functions (C~mn} is orthogonal on a z = constant plane. This beam mode function can also be represented in the following form. 168

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OPTICS COMMUNICATIONS

October 1974

X

t: i, I js

Y ia, zo

I

~ss

~'Z

s~S

Fig. 1. Coordinate system, incident wave beam, and aperture.

~brnn(X',Y',Z') =

f f/mn(~',[3')exp[-j~'x'-j[3'y'-jYz']d~'d[3',

(4)

where a ' , [3', and 3" are the components of the wave vector in the x ' , y ', and z' directions respectively and they are related to k by a ' 2 + [3'2 + 7'2 = k 2 .

(5)

The fact that the propagation axis of the wave beam is the z ' axis means that the spectral function a finite value only for

la'lk'l "~ I,

113'lk'l '~ 1,

(6)

fmn(a', [3') is negligibly

and otherwise

small [8]. In this region, (5) can be approximated by

Y = ~ - 1 ( a ' 2 + [3'2)/k •

The spectral function expression

(7)

fmn(a', [3') can be

obtained by using the inverse Fourier formula which gives the following i

r

fmn
fmn(a', (3') has

(8)

is given by

t 4 S \~ Cbmn(X ,y z') _ "97r~x/~n2m+n+Im.n.1 v _~f f-~ expr--±w'2r~"2+[3'2)--J°t'x'--J[3'Y'-J3"(z' +Zs)] ,

,

t

z-,, P-- /rWs,1

(9)

t

dt~' d ~ ' .

J "L~/213j

In the (x, y, z) coordinate system this can be transformed into the following form.

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OPTICS COMMUNICATIONS

October 1974

• mn(X,y, z) = ~mn(X ',y ', z') =

f f exp[--¼w~2{(acos0-3,sin0)2+/32} 2rrx/Tr2 m+n+lm!n! _ ~ _ ¢

jol(X+Xs)-j/3y

(10)

j3`(Z+Zs) ]

r

ws

×HmI~/~(°~c°sO-"rsinO)l

Iln[~2/3](cosO-~sinO)dad[3,

where a,/3 and 3' represent the x , y and z components of the wave vector respectively, given by a = a ' c o s 0 + 3,'sin 0 ,

/3=/3',

3`= -c~' sin0 + 3`'cos0 .

(11)

The region described by (6) corresponds in the (a,/3, 3`) system to a neighborhood of (k sin 0,0, k cos 0); 7 can be approximated, except when 0 is very close to 7r/2, by [9] 3'=3'0

( a - a 0 ) tan 0

(a

aO)2

2k cos 30

/32

(12)

2k cos0 '

where a 0 = k sin 0, and 3'0 = k cos 0. Then XPmn(X,y,z ) can be represented as follows: ./

~mn(X,Y,~Z)

WlsW2s 2rr2 m ~

j m+n

cos 0 2rr

=

× f .

fexp[-¼W~s(a-a0)2-¼W~s/32-ja(x+Xs) .

.

j/3y-j3`(Z+Zs)]Hrn ~ w l s ( a - % )

L~

.

t

]

(13)

Hn

dad

t

where Wls = Ws/COS0, W2s = w s. Integrating the above, we obtain

qimn (X,Y, z)

r/1 r/2 1// ~ ~2 m+nm!n! cos 0

x exp E-jao(X + X s ) - , o ( ~ + ~s)1,7~o1~(x - z tan 0 ) ~ - ~ ~o~y~+j(m +~)tan-l~x + > + l ) t a n % l ×H m [~l(X - z t a n 0 ) ] H n [r~2y ] ,

(14)

where 2 (z + Zs)

~1 - kW2sCOS30 '

2 (z + Zs) ~2 = kW2s cosO

r/i-

~

Wis,,a; ,~

(i:1,2),

0 2= l + j ~ i

(i=1,2).

(15)

This expression coincides with that of a wave beam transmitted by a series of periodically and obliquely set lenses [ 10].

3. Diffraction field by a rectangular aperture The diffraction field by a rectangular aperture as shown in fig. 1, can be obtained for an obliquely incident wave beam by using the Kirchhoff approximation. In the Fresnel region this is given by [ 11 ]

170

Volume 12, number 2

Vmn(X,y, z) -

October 1974

OPTICS COMMUNICATIONS

Jkc°s20 bl b2 I { 2rrZ f f ~lmn(XO,YO, Zo) exp - j k ~ -Z

+ (X - X 0) sin 0

- a l -a2 + c°s30 2Z

(X_Xo)2+c°sO 2Z-

(16)

( Y - Y0)2} t d X 0 d Y 0

'

where X = x - z tan0 ,

Y= y,

Z = z - z0 ,

X0 = x 0- z0 tan0,

Y0 = y o ,

and (x 0, YO, z0) represents the coordinates of the aperture. Substituting (14) for diffraction field, given by

Vmn(X, Y, z) ×exp

_ jk cos20 1 /

27rZ

~mn

(17) in (16) we obtain the

r/10r/20

~/ 7r2m+nm!n!cos0

[- Jao(X+Xs)-JTo(Z+Zs)+j(m+½)tan-t~lO+j(n+½)tan-l~20- jkc°s3O x2-jkc°sO y2] 2~ 2Z

X

~_~ p!q!(m_2q)!Hp(A1X)!,[ jkc°s30

V ~

(2rllo)m-2q {Gl.(P+m-2q,bl) +( - 1 ) p+mGl(P+m-2q,al)}

[n/21

X Lp--~0q--~0""QY)"'3 '

(18)

where

(2u-1)t~

IV~

G1 (p + m - 2q, b 1) = (r/10B1)2 u+l

u - 1 (r/10Blbl)2S+ 11

~(~1081bl)-(2-eu)exp(-½~@~b~)~o=

(2s,l)!!

3

for p + m - 2q = 2u (even),

(19)

U

(2u)!i (r/10B1)2u+2

-exp(-½rhnB,b,)~ .... s=0

E 1

(rlloBlbl)2S (2s)l!

1 forp+m-2q

= 2 u + 1 (odd)

and

~, -- ,,~ + jk cos30/(~a~0Z),

~ -- O~o + j~ cos0/(.~oZ),

(20)

:1::1 +: cos60/(2~:~1o:),

..~: ~-~+ :~o:Ol(2A~,~o:),

(21)

W

@(w)=(1/x/~)fexp(-½v2)dv,

eu:2

foru:0,

eu:l

forum0.

(22,23)

0 The subscript " 0 " represents the values of the parameters at the position of the aperture. In obtaining the above expression the following formula is used: exp [-- V2 + 2uo] The functions

= ~ Hp(U) vP/p!.

(24)

p=0

G1(p +m - 2q, a 1), G2(p +n - 2q, b2)

and

G2(p +n - 2q, a2)

are defined by (19) if we substitute 171

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OPTICS COMMUNICATIONS

October 1974

a 1 in place of b 1 , or n in place of m, etc. A 1 and A 2 are arbitrary parameters of which the field is independent. They will be determined when we discuss the beam mode expansion. The diffraction field in the Fraunhofer region is obtained if we neglect the second terms in the expressions for

and The expression (18)is used for numerical computations of the field [t2].

4. Beam mode expansion The diffraction field obtained above can be represented as a sum of beam mode functions that have the same propagation axis z' as the incident wave beam. Let Vmn(X,y,z ) be represented as follows,

Vmn(X,y,z ) = ~ Cmn mn ~-~E(x,y,z). m,n

(25)

The beam parameters of ~mff are designated as Wis,z s, ~i, -~i,etc. These are not necessarily coincident with those of the incident wave beam. The coefficients {Cmmntr},which are called "mode coupling coefficients" by Kogelnik [13], can be obtained by using the orthogonality of {qJ~n).

-Ng'_- f

f gmn(X,y,z ) ~mf(xy,z,)dx-*

Cmn

Substituting (18) for

dy

I~ff(x,y,z)12dx dy.

Vmn, this can be written as follows

C- ~mn = exp [ - ja0(x s - 2 s ) - j'y0(Zs- Zs)] Cm Cnn ' __

, /r

(26)

~1o~1o

cm = V 7r2m+-~m!~]

(27)

exp [j (m + 1) tan-1 ~10 - j (r~ +1) tan-1 ~-10]

( - 1)q+tm !~ !

X q=0 ~ t=0 ~ q!t!(m-2q)!(Tn-2t)Z X [Gl(m +t~-2q-2t,

(2 rtlo)m

b l ) + ( - 1 ) m+N

_2q(2~lO)m-2t

(28)

Gl(m +Un-2q-2t, al) ] ,

and C n is defined in the same way. The arbitrary parameter A 1 has been chosen as

rll [_62,+jkcos3 0 i

1/2

(29)

)

In obtaining this the orthogonality of the Hermite polynomials and the formula m!

(30)

Hm(vW) = ~ t!(m - 2 0 ! vm -2t(v2 - 1)tHin - 2 t ( w ) t=0 have been used. The parameter B 1 defined by (21) is, in this case, given by

2

B2 -

r/120

-

-2

~JlOnlO -- ~ 10r/10 +j 7720 '

which is independent ofz. Therefore the coefficient 172

(31) -Ng

Cmn is also independent ofz.

Volume 12, number 2

OPTICS COMMUNICATIONS

October 1974

When the parameters of the wavebeam functions{C~E} are coincident with those of the incident wave beam, mn Cram' and Cnn) represent the mode transmission (N = m and the coefficients (in this case, they are written as Cmn, = n) and mode conversion ( N 4 : m or ~ :/:n) coefficients by the aperture. In this case,

Cm ~ =I ~ V 2rr

X ~

~

q=0 t=0

exp [j(m - if0 tan-l~10]

(32) (-1)q+t(m+m-2q-2t)!! [Fl(m+~-2q-2t, bl)+(-1)m+ff~Fl(m+ff*-2q-2t, al)] (2q)l[(2t)!l(m-2q)!(tfi-2t)!

where F 1 (m + fit - 2 ( / - 2t, b 1 ) = x / r ~ d P (X/~-r/10b 1 ) - (2 -

eu ) exp [ - r/20b 2 ] ,,,~1 (x/2r/10b~ 1 s=0

)2s+1

(2s+l)!?

for m + ~ - 2q - 2t = 2u (even). =1-

exp [-n2ob~] s=0

(X/2~lobl) 2s (2s)!! for m + ~ -

(33)

2q - 2t = 2u + 1 (odd).

This shows that the angle 0 does not appear explicitly in this expression but is included in ~ and 77. Therefore if we represent the wave beam by using these parameters, the discussion of the mode expansion coefficients can be done in the same way as in the case of normal incidence. As a numerical example some expansion coefficients C~ for fundamental mode incidence on a half-plane are shown in fig. 2.

5. Conclusions The diffraction problem o f an obliquely incident wave beam by a rectangular aperture can be discussed almost in the same way as in the case of normal incidence if we represent the incident wave beam in an appropriate form.

1.0 ~

-Ic:l

0.5-

rl.i n.3

-210

-I~0

q

2'.0 :

Fig. 2. Transmission and conversion coefficients for fundamental mode incidence upon a half plane. 173

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October 1974

The magnitude of the mode transmission and mode conversion coefficients of the diffraction field is independent o f z and depends only upon the ratio of the spot size to the length of the side of the aperture.

Acknowledgement The author wishes to thank Prof. Fukumitsu for his valuable suggestions.

References [1] [2] [3] [41 [5] [6] [7] [8] [91 [10] [11] [121 [13]

174

J.E. Pearson, T.C. McGill, S. Kurtin and A. Yariv, J. Opt. Soc. Amer. 59 (1969) 11. R.G. Schell and G. Tyras, J. Opt. Soc. Amer. 61 (1971) 1. S. Nemoto and T. Makimoto, Trans. Inst. Electron. Commun. Eng. Japan, 52-B (June 1969). O.O. Andrade and G.C. Thomas, Joint Conf. Lasers and Opto-Electronics, Univ. of Southampton, Mar. (1969). K. Tanaka, M. Shibukawa and O. Fukumitsu, IEEE Trans. MTT, MTT-20 (1972) 11. K. Tanaka and O. Fukumitsu, IEEE Trans. MTT, MTT-22 (1974) 2. G.D. Boyd and J.P. Gordon, Bell Syst. Techn. J. 40 (Mar. 1961). G. Goubau and F. Schwering, IRE Trans. AP, AP-9 (May 1961). B.R. Horowits and T. Tamir, J. Opt. Soc. Amer. 61 (1971) 5. K. Tanaka and K. Yasuura, Pep. Fac. Eng. Nagasaki Univ., 1 (1971). S. Silver, Microwave Antenna Theory and Design (New York, McGraw-Hill, 1941). K. Tanaka, Opt. Commun. 6 (1972) 3. H. Kogelnik, Proc. Syrup. Quasi-Optics (New York: Polytechnic Press, 1964).