Abstract

Diffraction of a plane wave by multiple slits is investigated by a method that uses a set of orthonormal functions and Fourier transformations. A solution satisfying the wave equation and the boundary conditions is obtained. Expressions for the transmission coefficient and the far field are derived from the solution, and some numerical results are given.

© 1990 Optical Society of America

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References

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  1. J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
    [CrossRef]
  2. I. Lazar, L. A. DeAcetis, “Experimental diffraction by two long, parallel strips in a plane. I: Vertical polarization,” Appl.Opt. 7, 1609–1612 (1968).
    [CrossRef] [PubMed]
  3. T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier-orthogonal functions transformation,” J.Phys. Soc. Jpn. 41, 2046–2051 (1976).
    [CrossRef]
  4. T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
    [CrossRef]
  5. The series expressions of Pmn(MN) are obtained by the same procedure in the appendix of Ref. 4. In Ref. 4 there are three errors, as follows: The factor Γ(ν − μ − λ − 3/2) in Q4(μ,ν) on p. 913 should read as Γ(ν − μ − λ − 1/2). The factor Γ(ν+ 1/2) in Q5(μ, ν) on p. 913 should read as Γ (ν+ 3/2). The factor ψ(ν+ 1/2) in ψ3(μ,ν) on p. 913 should read as ψ(ν+ 3/2).
  6. S. W. Lee, J. Boersama, “Ray-optical analysis of fields on shadow boundaries of two parallel plates,” J. Math. Phys. 16, 1746–1764 (1975).
    [CrossRef]

1978 (1)

T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
[CrossRef]

1976 (1)

T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier-orthogonal functions transformation,” J.Phys. Soc. Jpn. 41, 2046–2051 (1976).
[CrossRef]

1975 (1)

S. W. Lee, J. Boersama, “Ray-optical analysis of fields on shadow boundaries of two parallel plates,” J. Math. Phys. 16, 1746–1764 (1975).
[CrossRef]

1968 (1)

I. Lazar, L. A. DeAcetis, “Experimental diffraction by two long, parallel strips in a plane. I: Vertical polarization,” Appl.Opt. 7, 1609–1612 (1968).
[CrossRef] [PubMed]

1957 (1)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

Boersama, J.

S. W. Lee, J. Boersama, “Ray-optical analysis of fields on shadow boundaries of two parallel plates,” J. Math. Phys. 16, 1746–1764 (1975).
[CrossRef]

DeAcetis, L. A.

I. Lazar, L. A. DeAcetis, “Experimental diffraction by two long, parallel strips in a plane. I: Vertical polarization,” Appl.Opt. 7, 1609–1612 (1968).
[CrossRef] [PubMed]

Keller, J. B.

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

Lazar, I.

I. Lazar, L. A. DeAcetis, “Experimental diffraction by two long, parallel strips in a plane. I: Vertical polarization,” Appl.Opt. 7, 1609–1612 (1968).
[CrossRef] [PubMed]

Lee, S. W.

S. W. Lee, J. Boersama, “Ray-optical analysis of fields on shadow boundaries of two parallel plates,” J. Math. Phys. 16, 1746–1764 (1975).
[CrossRef]

Otsuki, T.

T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
[CrossRef]

T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier-orthogonal functions transformation,” J.Phys. Soc. Jpn. 41, 2046–2051 (1976).
[CrossRef]

Appl.Opt. (1)

I. Lazar, L. A. DeAcetis, “Experimental diffraction by two long, parallel strips in a plane. I: Vertical polarization,” Appl.Opt. 7, 1609–1612 (1968).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

J. Math. Phys. (2)

S. W. Lee, J. Boersama, “Ray-optical analysis of fields on shadow boundaries of two parallel plates,” J. Math. Phys. 16, 1746–1764 (1975).
[CrossRef]

T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
[CrossRef]

J.Phys. Soc. Jpn. (1)

T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier-orthogonal functions transformation,” J.Phys. Soc. Jpn. 41, 2046–2051 (1976).
[CrossRef]

Other (1)

The series expressions of Pmn(MN) are obtained by the same procedure in the appendix of Ref. 4. In Ref. 4 there are three errors, as follows: The factor Γ(ν − μ − λ − 3/2) in Q4(μ,ν) on p. 913 should read as Γ(ν − μ − λ − 1/2). The factor Γ(ν+ 1/2) in Q5(μ, ν) on p. 913 should read as Γ (ν+ 3/2). The factor ψ(ν+ 1/2) in ψ3(μ,ν) on p. 913 should read as ψ(ν+ 3/2).

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Figures (6)

Fig. 1
Fig. 1

Configuration of 2L slits and coordinate systems.

Fig. 2
Fig. 2

Diffraction patterns of two slits of width 2 and slit separation 2D = 4 due to a normally incident plane wave (incident angle θ0 = 0°) of different values of the wave number k.

Fig. 3
Fig. 3

Diffraction patterns of four slits of width 2 and slit separation 2D = 4 due to a normally incident plane wave (incident angle θ0 = 0°) of different values of the wave number k.

Fig. 4
Fig. 4

Diffraction patterns of six slits of width 2 and slit separation 2D = 4 due to a normally incident plane wave (incident angle θ0 = 0°) of different values of the wave number k.

Fig. 5
Fig. 5

Diffraction pattern of two slits of width 2 and slit separation 2D = 2.66 due to an obliquely incident plane wave (incident angle θ0 = −36°) of the wave number k = 7.30.

Fig. 6
Fig. 6

Diffraction pattern of two slits of width 2 and slit separation 2D = 2.85 due to an obliquely incident plane wave (incident angle θ0 = −46°) of the wave number k = 8.69.

Tables (1)

Tables Icon

Table 1 Plane-Wave Transmission Coefficients t of Multiple Slits for Normal Incidencea

Equations (48)

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( 2 x 2 + 2 z 2 + k 2 ) ϕ = 0 ( z > 0 )
ϕ = 0 on S ,
ϕ z = ϕ 0 z f ( x ) on A ,
S = [ ( x , z ) ; z = 0 ] A , A = [ ( x , z ) ; z = 0 , D 1 < x < D + 1 , 3 D 1 < x < 3 D + 1 , ; ( 2 L 1 ) D 1 < x < ( 2 L 1 ) D + 1 ] .
x M = x + ( 2 L 2 M + 1 ) D ( M = 1 , 2 , 3 , , 2 L )
f M ( x M ) = { f ( x ) ( | x M | < 1 ; M = 1 , 2 , 3 , , 2 L ) 0 ( | x M | > 1 ; M = 1 , 2 , 3 , , 2 L ) ,
ϕ z = f M ( x M ) ( z = 0 , | x M | < 1 ; M = 1 , 2 , 3 , , 2 L ) .
ϕ ( x , z ) = N = 1 2 L n = 0 A n ( N ) U n ( s ) exp ( is x N ) × exp [ ( s 2 k 2 ) 1 / 2 z ] d s ,
N = 1 2 L n = 0 A n ( N ) U n ( s ) exp ( is x N ) d s = 0 on S
N = 1 2 L n = 0 A n ( N ) ( s 2 k 2 ) 1 / 2 U n ( s ) exp ( is x N ) d s = f M ( x M ) ( | x M | < 1 ; M = 1 , 2 , 3 , , 2 L ) ,
υ n ( x ) = { 0 ( | x | > 1 ) w ( x ) u n ( x ) ( | x | < 1 ) .
U n ( s ) = υ n ( x ) exp ( isx ) d x = 1 1 w ( x ) u n ( x ) exp ( is x ) d x .
U n ( s ) exp ( isx ) d s = 2 π υ n ( x ) ,
N = 1 2 L n = 0 A n ( N ) ( s 2 k 2 ) 1 / 2 U n ( s ) exp ( is x M ) × exp [ 2 is ( M N ) D ] d s = f M ( x M ) ( | x M | < 1 ; M = 1 , 2 , 3 , , 2 L ) ,
exp ( is x M ) = m = 0 E m ( M ) ( s ) u m ( x M ) ,
E m ( M ) ( s ) = 1 1 w ( x ) u m ( x ) exp ( isx ) d x = U m ( s )
f M ( x M ) = m = 0 F m ( M ) u m ( x M ) ,
F m ( M ) = w ( x ) u m ( x ) f M ( x ) d x ,
N = 1 2 L n = 0 P mn ( MN ) A n ( N ) = F m ( M ) ( M = 1 , 2 , 3 , , 2 L ; m = 0 , 1 , 2 , ) ,
P mn ( MN ) = ( s 2 k 2 ) 1 / 2 U n ( s ) U m ( s ) exp [ 2 is ( M N ) D ] d s .
P mn ( MN ) = 0 ( D ; M N ) .
n = 0 P mn ( MM ) A n ( M ) = F m ( M ) ( M = 1 , 2 , 3 , , 2 L ; m = 0 , 1 , 2 , ) ,
u n ( x ) = Γ ( n + 2 ) 2 Γ ( n + 3 / 2 ) P n ( 1 / 2 , 1 / 2 ) ( x ) ( n = 0 , 1 , 2 , ) ,
w ( x ) = ( 1 x 2 ) 1 / 2 ,
P n ( 1 / 2 , 1 / 2 ) ( x ) = ( 1 ) n ( 1 x 2 ) 1 / 2 2 n Γ ( n + 1 ) d n d x n ( 1 x 2 ) n + 1 / 2 .
U n ( s ) = ( 2 π ) 1 / 2 ( i ) n ( n + 1 ) J n + 1 ( s ) / s ,
P mn ( MN ) = C mn 0 ( s 2 k 2 ) 1 / 2 cos [ 2 D ( M N ) s ] × [ J m + 1 ( s ) J n + 1 ( s ) / s 2 ] d s + S mn 0 ( s 2 k 2 ) 1 / 2 × sin [ 2 D ( M N ) s ] [ J m + 1 ( s ) J n + 1 ( s ) / s 2 ] d s ( M , N = 1 , 2 , 3 , , 2 L ; m , n = 0 , 1 , 2 , , ) ,
C mn = 2 π i m + n [ ( 1 ) m + ( 1 ) n ] ( m + 1 ) ( n + 1 ) , S mn = 2 π i m + n + 1 [ ( 1 ) n ( 1 ) m ] ( m + 1 ) ( n + 1 ) .
P mn ( MN ) = P nm ( NM )
ϕ 0 ( x , z ) = exp [ i ( k x x + k z z ) ] ,
F m ( M ) = i k z ( i k x ) m exp [ i k x ( 2 L 2 M + 1 ) D ] × m + 1 2 m + 1 / 2 Γ ( m + 3 / 2 ) 1 1 ( 1 x 2 ) m + 1 / 2 cos ( k x x ) d x .
t = Im A [ ϕ * ( x , 0 ) ϕ ( x , 0 ) / z ] d x Im A [ ϕ 0 * ( x , 0 ) ϕ 0 ( x , 0 ) / z ] d x ,
ϕ ( x , 0 ) = 2 π N = 1 2 L n = 0 A n ( N ) w ( x N ) u n ( x N ) ( | x N | < 1 ) ,
ϕ ( x , 0 ) z = ϕ 0 ( x , 0 ) z = f N ( x N ) = m = 0 F m ( N ) u m ( x N ) ( | x N | < 1 ; N = 1 , 2 , 3 , , 2 L ) .
t = 2 π Im N = 1 2 L n = 0 A n ( N ) * F n ( N ) Im A ϕ 0 * ( x , 0 ) f ( x ) d x .
F 0 ( N ) = i k ( π / 2 ) 1 / 2 ( N = 1 , 2 , 3 , , 2 L ) , F 0 ( N ) = 0 ( n 1 ; N = 1 , 2 , 3 , , 2 L ) ,
t = ( π / 2 ) 3 / 2 ( 1 / L ) N = 1 2 L Re A 0 ( N ) .
ϕ ( x , z ) = N = 1 2 L n = 0 A n ( N ) I n ,
I n = U n ( s ) exp ( is x N ) exp [ ( s 2 k 2 ) 1 / 2 z ] d s .
x = r sin θ , z = r cos θ ; x N = r N sin θ N , z = r N cos θ N
s = k sin t .
I n = k c U n ( k sin t ) cos t exp [ i k r N cos ( t θ N ) ] d t ,
C 1 = { t ; Re t = π / 2 , 0 < Im t < } , C 2 = { t ; π / 2 < Re t < π / 2 , Im t = 0 } , C 3 = { t ; Re t = π / 2 , < Im t < 0 } .
I n = ( 2 π i k / r N ) 1 / 2 exp ( ik r N ) U n ( k sin θ N ) cos θ N .
r N r ( 2 N 2 L 1 ) D sin θ , θ N θ .
ϕ ( x , z ) ϕ a ( θ ) exp ( ikr ) / ( kr ) 1 / 2 ( r ) ,
ϕ a ( θ ) = 2 π ( i ) 1 / 2 cot θ N = 1 2 L n = 0 exp [ ik ( 2 N 2 L 1 ) D sin θ ] × A n ( N ) ( i ) n ( n + 1 ) J n + 1 ( k sin θ ) .
N = 1 2 L n = 0 K P mn ( MN ) A n ( N ) = F m ( M ) ( M = 1 , 2 , 3 , , 2 L ; m = 0 , 1 , 2 , K )

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