Abstract

The analytical properties and computational implications of the Gabor representation are investigated within the context of aperture theory. The radiation field in the pertinent half-space is represented by a discrete set of linearly shifted and spatially rotated elementary beams that fall into two distinct categories, the propagating (characterized by real rotation angles) and evanescent beams. The representation may be considered a generalization in the sense that both the classical plane wave and Kirchhoff’s spatial-convolution forms are directly recoverable as limiting cases. The choice of a specific window function [w(x)] and the corresponding characteristic width (L) are, expectedly, cardinal decisions affecting the analytical complexity and the convergence rate of the Gabor series. The significant spectral compression achievable by an appropriate selection of w(x) and L is demonstrated numerically, and simple selection guidelines are derived. Two specific window functions possessing opposite characteristics are considered, the uniformly pulsed and the Gaussian distributions. These are studied analytically and numerically, highlighting several outstanding advantages of the latter. Consequently, the primary attention is focused on Gaussian elementary beams in their paraxial and their far-field estimates. Although the main effort is devoted to aperture analysis, demonstrating the advantages and limitations of the proposed approach, reference is also made to its potential when applied to aperture-synthesis and spatial-filtering problems. The quantitative effects of basic filtering in the discrete Gabor space are depicted.

© 1986 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,”J. Inst. Elec. Eng. (London) 93III, 429–457 (1946).
  2. M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
    [CrossRef]
  3. M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Soc. Photo-Opt. Instrum. Eng. International Opt. Comput. Conf. 231, 274–279 (1980).
  4. M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–101 (1980).
  5. A. J. E. M. Janssen, “Gabor representation of generalized functions,”J. Math. Anal. Appl. 83, 377–394 (1981).
    [CrossRef]
  6. G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  7. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).
  8. P. D. Einziger, L. B. Felsen, “Evansecent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–605 (1982).
    [CrossRef]
  9. A. Papoulis, The Fourier Integral and Its Application (McGraw-Hill, New York, 1962), p. 62.
  10. M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 590.
  11. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, New York, 1975).
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. III.
  13. J. B. Keller, W. Streifer, “Complex ray with an application to Gaussian beams,”J. Opt. Soc. Am. 61, 40–43 (1971).
    [CrossRef]
  14. P. D. Einziger, Y. Haramaty, L. B. Felsen, “Radiation from planar aperture distributions by evanescent wave and complex ray analysis,” in Proceedings of the 13th Convention of IEEE in Israel (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 2.1.2.1–2.1.2.3.

1982 (1)

P. D. Einziger, L. B. Felsen, “Evansecent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–605 (1982).
[CrossRef]

1981 (1)

A. J. E. M. Janssen, “Gabor representation of generalized functions,”J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

1980 (3)

M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Soc. Photo-Opt. Instrum. Eng. International Opt. Comput. Conf. 231, 274–279 (1980).

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–101 (1980).

1971 (2)

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

J. B. Keller, W. Streifer, “Complex ray with an application to Gaussian beams,”J. Opt. Soc. Am. 61, 40–43 (1971).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,”J. Inst. Elec. Eng. (London) 93III, 429–457 (1946).

Abramovitz, M.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 590.

Bastiaans, M. J.

M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Soc. Photo-Opt. Instrum. Eng. International Opt. Comput. Conf. 231, 274–279 (1980).

M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–101 (1980).

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, New York, 1975).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. III.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

Deschamps, G. A.

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Einziger, P. D.

P. D. Einziger, L. B. Felsen, “Evansecent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–605 (1982).
[CrossRef]

P. D. Einziger, Y. Haramaty, L. B. Felsen, “Radiation from planar aperture distributions by evanescent wave and complex ray analysis,” in Proceedings of the 13th Convention of IEEE in Israel (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 2.1.2.1–2.1.2.3.

Felsen, L. B.

P. D. Einziger, L. B. Felsen, “Evansecent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–605 (1982).
[CrossRef]

P. D. Einziger, Y. Haramaty, L. B. Felsen, “Radiation from planar aperture distributions by evanescent wave and complex ray analysis,” in Proceedings of the 13th Convention of IEEE in Israel (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 2.1.2.1–2.1.2.3.

Gabor, D.

D. Gabor, “Theory of communication,”J. Inst. Elec. Eng. (London) 93III, 429–457 (1946).

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, New York, 1975).

Haramaty, Y.

P. D. Einziger, Y. Haramaty, L. B. Felsen, “Radiation from planar aperture distributions by evanescent wave and complex ray analysis,” in Proceedings of the 13th Convention of IEEE in Israel (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 2.1.2.1–2.1.2.3.

Janssen, A. J. E. M.

A. J. E. M. Janssen, “Gabor representation of generalized functions,”J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

Keller, J. B.

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Application (McGraw-Hill, New York, 1962), p. 62.

Stegun, I.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 590.

Streifer, W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. III.

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

P. D. Einziger, L. B. Felsen, “Evansecent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–605 (1982).
[CrossRef]

J. Inst. Elec. Eng. (London) (1)

D. Gabor, “Theory of communication,”J. Inst. Elec. Eng. (London) 93III, 429–457 (1946).

J. Math. Anal. Appl. (1)

A. J. E. M. Janssen, “Gabor representation of generalized functions,”J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Optik (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–101 (1980).

Proc. IEEE (1)

M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

Soc. Photo-Opt. Instrum. Eng. International Opt. Comput. Conf. (1)

M. J. Bastiaans, “Sampling theorem for the complex spectrogram, and Gabor’s expansion of a signal in Gaussian elementary signals,” Soc. Photo-Opt. Instrum. Eng. International Opt. Comput. Conf. 231, 274–279 (1980).

Other (6)

A. Papoulis, The Fourier Integral and Its Application (McGraw-Hill, New York, 1962), p. 62.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 590.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, New York, 1975).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. III.

P. D. Einziger, Y. Haramaty, L. B. Felsen, “Radiation from planar aperture distributions by evanescent wave and complex ray analysis,” in Proceedings of the 13th Convention of IEEE in Israel (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 2.1.2.1–2.1.2.3.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

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Figures (19)

Fig. 1
Fig. 1

Geometrical configuration.

Fig. 2
Fig. 2

Gabor coefficients (|Am,n|) for a pulsed aperture, L0/L = 0.2.

Fig. 3
Fig. 3

Gabor coefficients (|Am,n|) for a pulsed aperture, L0/L = 9.

Fig. 4
Fig. 4

Gabor coefficients (|Am,n|) for apulsed aperture, L0|L = 1.

Fig. 5
Fig. 5

The number of contributing Gabor coefficients |Am,n| ≥ 10−2 (pulsed, cosine, raised cosine aperture distributions, elementary Gaussian-window functions).

Fig. 6
Fig. 6

A Gabor representation of a pulsed aperture with L0/L = 0.6 and two choices (|m| ≤ 4, |n| ≤ 2), (|m| ≤ 4, |n| ≤ 60).

Fig. 7
Fig. 7

Shifted and rotated coordinate system for the elementary beam Bm,n(x, z).

Fig. 8
Fig. 8

Integration contours in the complex ϑ plane.

Fig. 9
Fig. 9

Field distribution of a pulsed aperture at z = 0.02L02/λ. The solid and the dashed lines represent reference (FFT) and Gabor solutions, respectively (|m| ≤ 3, |n| ≤ 5, L0/L = 2, L0/λ = 11).

Fig. 10
Fig. 10

Field distribution of a pulsed aperture at z = 0.22L02/λ. The solid and the dashed lines represent reference (FFT) and Gabor solutions, respectively (|m| ≤ 3, |n| ≤ 5, L0/L = 2, L0/λ = 11).

Fig. 11
Fig. 11

Field distribution of a pulsed aperture at z = 2L02/λ. The solid and the dashed lines represent reference (FFT) and Gabor solutions, respectively (|m| ≤ 3, |n| ≤ 5, L0/L = 2, L0/λ = 11).

Fig. 12
Fig. 12

The far-field pattern for a pulsed aperture. The solid and the dashed lines represent exact and Gabor solutions, respectively (|m| ≤ 2, |n| ≤ 4, L0/L = 0.6, L0/λ = 2.4).

Fig. 13
Fig. 13

The far-field pattern for a cosine aperture. The solid and the dashed lines represent exact and Gabor solutions, respectively (|m| ≤ 1, |n| 1, L0/L = 1.6, L0/λ = 2.4).

Fig. 14
Fig. 14

The spectral content in the transition from propagating to evanescent elementary beams.

Fig. 15
Fig. 15

Field distribution of a pulsed aperture at (L/λ = 2.5). The solid and the dashed lines represent reference (FFT) and Gabor solutions, respectively (L0/λ = 4, z = 0.026L02/λ, |m| ≤ 2, |n| ≤ 2).

Fig. 16
Fig. 16

Field distribution of a pulsed aperture at (L/λ = 2.9). The solid and the dashed lines represent reference (FFT) and Gabor solutions, respectively (L0/λ = 4, z = 0.026L02/λ, |m| ≤ 2, |n| ≤ 2).

Fig. 17
Fig. 17

The far field in normalized amplitude units. Solid line, a pulsed-aperture response (L0/λ = 2.4, L0/L = 0.6, |m| ≤ 2, |n| ≤ 4); dashed line, a constrained zero introduced at φ = 30° (Am,2 = 0).

Fig. 18
Fig. 18

The far field in normalized amplitude units. Solid line, a pulsed-aperture response (L0/λ = 2.4, L0/L = 0.6, |m| ≤ 2, |n| ≤ 4); dashed line, a constrained zero introduced in the interval 22.5° < φ < 37.5°.

Fig. 19
Fig. 19

The aperture field distributions. Dashed line, the unperturbed far field described by the solid lines in Figs. 17 and 18; solid line, the far field described by the dashed line in Fig. 17; dotted line, the far field described by the dashed line in Fig. 18.

Equations (119)

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E ( x , 0 ) = y ^ f ( x ) ,
E ( x , z ) = ( 2 π ) - 1 - F ( η ) exp [ i ( η x + κ z ) d η
F ( η ) = - f ( x ) exp ( - i η x ) d x
κ = ( k 2 - η 2 ) 1 / 2
E ( x , z ) = ( i k z / 2 ) - f ( x ) [ H 1 ( 1 ) ( k R ) / R ] d x ,
R [ ( x - x ) 2 + z 2 ] 1 / 2
f ( x ) = - f ( x ) δ ( x - x ) d x .
f ( x ) = m n A m , n w ( x - m L ) exp ( i n Ω x ) .
Ω L = 2 π ,
- w ( x ) 2 d x = 1.
w ( x ) = g ( x / L ) ( 2 1 / 2 / L ) 1 / 2 exp [ - π ( x / L ) 2 ] ,
F ( η ) = m n A m , n W ( η - n Ω ) exp ( - i m L η ) ,
W ( η ) = - w ( x ) exp ( - i η x ) d x
E ( x , z ) = m n A m , n B m , n ( x , z ) ,             z > 0 ,
B m , n ( x , z ) = ( 2 π ) - 1 - W ( η - n Ω ) exp { i [ η ( x - m L ) + κ z ] } d η
B m , n ( x , z ) = ( i k z / 2 ) - w ( x - m L ) exp ( i n Ω x ) [ H 1 ( 1 ) ( k R ) / R ] d x .
W ( η ) = G ( η / Ω ) = ( 2 π ) 1 / 2 g ( η / Ω ) = ( 2 3 / 2 π / Ω ) 1 / 2 exp [ - π ( π / Ω ) 2 ] .
D x D η ( π / 2 ) 1 / 2 ,
D x 2 - x 2 w ( x ) 2 d x ,             D η 2 - η 2 W ( η ) 2 d η .
m M L 0 / 2 L .
n N k / Ω = L / λ .
E ( x , z ) - M M - N N A m , n B m , n ( x , z ) ,             N k / Ω ,             M L 0 / 2 L .
w ( x ) = p ( x / L ) L - 1 / 2 rect ( x / L ) ,             rect ( x ) = { 1 x < 1 / 2 0 x > 1 / 2
W ( η ) = P ( η / Ω ) = ( 2 π / Ω ) 1 / 2 sinc ( η / Ω ) , sinc ( x ) = sin ( π x ) / π x
- w ( x ) γ * ( x - m L ) exp ( - i n Ω x ) d x = δ m δ n ,             δ m = { 1 m = 0 0 m 0
( 2 π ) - 1 - W ( η ) Γ * ( η - n Ω ) exp ( i m L η ) d η = δ m δ n .
Γ ( η ) = - γ ( x ) exp ( - i η x ) d x
A m , n = - f ( x ) γ * ( x - m L ) exp ( - i n Ω x ) d x
A m , n = ( 2 π ) - 1 - F ( η ) Γ * ( η - n Ω ) exp ( i m L η ) d η ,
m n w ( x - m L ) γ * ( x - m L ) exp [ i n Ω ( x - x ) ] = δ ( x - x )
( 2 π ) - 1 m n W ( η - n Ω ) Γ * ( η - n Ω ) exp [ - i m L ( η - η ) ] = δ ( η - η ) .
A m , - n = A * m , n             i f             Im [ f ( x ) γ * ( x - m L ) ] = 0 ,
A - m , n = A * m , n             if f ( - x ) γ * ( - x + m L ) = [ f ( x ) γ * ( x - m L ) ] *
A - m , n = - A * m , n if             f ( - x ) γ * ( - x + m L ) = - [ f ( x ) γ * ( x - m L ) ] * .
γ ( x ) = γ g ( x / L ) = C L exp [ π ( x / L ) 2 ] S ( x / L ) ,
S ( x / L ) = n ( x / L - 1 / 2 ) ( - 1 ) n exp [ - π ( n + 1 / 2 ) 2 ] ,
C L = [ π 3 / ( K 0 3 L 2 1 / 2 ) ] 1 / 2 ,
Γ ( η ) = Γ g ( η / Ω ) = ( 2 π ) 1 / 2 γ g ( η / Ω ) = ( 2 π ) 1 / 2 C Ω × exp [ π ( η / Ω ) 2 ] S ( η / Ω ) ,
γ ( x ) = γ p ( x / L ) = p ( x / L )
Γ ( η ) = Γ P ( η / Ω ) = P ( η / Ω ) .
f ( x ) = 0 ,             x > L 0 / 2
A m , n = A m , n ( 0 ) + ( L 0 / L ) m , n ,
A m , n ( 0 ) = γ * ( - m L ) - L 0 / 2 L 0 / 2 f ( x ) exp ( - i n Ω x ) d x = γ * ( - m L ) F ( n Ω )
m , n = ( L / L 0 ) - L 0 / 2 L 0 / 2 f ( x ) [ γ * ( x - m L ) - γ * ( - m L ) ] exp ( - i n Ω x ) d x .
A m , n = γ * ( - m L ) F ( n Ω ) [ 1 + 0 ( L 0 / L ) ] ,             L 0 / L 0 ,
F ( η ) = 0 ,             η > Ω 0 / 2
A m , n = Γ * ( - n Ω ) f ( m L ) [ 1 + 0 ( Ω 0 / Ω ) ] ,             Ω 0 / Ω 0 ,
F ( η ) = 0 [ η - ( q + 2 ) ] ,             η .
A m , n = { 0 [ ( n Ω ) - ( q + 2 ) ] , n Ω m = 0 , 0 m 0 .
A m , n = { 0 [ ( n Ω ) - 1 ] , n Ω , m < ( L 0 / L + 1 ) / 2 0 , m > ( L 0 / L + 1 ) / 2 .
A m , n = 0 { [ ( n - i m ) Ω ] - ( q + 2 ) exp [ - π ( 1 - L 0 / L ) ( 4 m + 1 + L 0 / L ) / 4 ] } ,             ( n - i m ) Ω ,
A m , n = 0 [ ( n - i m ) - 1 Ω - 1 ] ,             ( n - i m ) Ω ,
f ( x ) = ( 2 / L 0 ) 1 / 2 cos ( π x / L 0 ) rect ( x / L 0 )
f ( x ) = ( 2 / 3 L 0 ) 1 / 2 ( 1 + cos ( 2 π x / L 0 ) ) rect ( x / L 0 ) ,
x - m L = ρ t sin φ = x t cos φ n + z t sin φ n , z = ρ t cos φ = - x t sin φ n + z t cos φ n ,
η = k sin ϑ ,             κ = k cos ϑ
B m , n = ( 2 π ) - 1 k c W ( k sin ϑ - k sin φ n ) × exp [ i k ρ r cos ( ϑ - φ ) ] cos ϑ d ϑ ,
sin φ n n Ω / k = n λ / L ,             φ = φ t + φ n ,
B m , n = ( 2 1 / 2 π Ω ) - 1 / 2 k c exp [ i k l Φ ( ϑ ) ] cos ϑ d ϑ ,
Φ ( ϑ ) = ( i a n / 2 l ) [ ( sin ϑ - sin φ n ) sec φ n ] 2 + ( ρ t / l ) cos ( ϑ - φ )
a n ( L 2 cos 2 φ n ) / λ
( d Φ / d ϑ ) ϑ = ϑ s = 0
i a n ( sin ϑ s - sin φ n ) cos ϑ s sec 2 φ n = ρ t sin ( ϑ s - ϕ )
B m , n = [ 2 1 / 2 D ( ϑ s ) / L ] 1 / 2 exp [ i k l Φ ( ϑ s ) ] [ 1 + 0 ( l / k l ) ] ,             k l ,
D ( ϑ s ) = [ i a n / l Φ ( ϑ s ) ] cos 2 ϑ s sec 2 φ n .
ϑ s = φ n + x t / ( z t - i a n ) - ( 3 / 2 ) i a n x t 2 / ( z t - i a n ) 3 tan φ n + 0 [ x t 3 / ( z t - i a n ) 3 ] ,             x t / ( z t - i a n ) 0.
a n Φ ( ϑ s ) = ( z t - i a n ) { 1 + ( 1 / 2 ) x t 2 / ( z t - i a n ) 2 + 0 [ z t 3 / ( z t - i a n ) 3 ] } + i a n ,             x t / ( z t - i a n ) 0
D ( ϑ s ) = - i a n / ( z t - i a n ) { 1 + 0 [ x t / ( z t - i a n ) ] } ,             x t / ( z t - i a n ) 0.
B m , n ~ ( 2 1 / 2 / L ) 1 / 2 [ - i a n / ( z t - i a n ) ] 1 / 2 × exp { i k [ z t + x t 2 / 2 ( z t - i a n ) ] } ,             k a n ,             x t / ( z t - i a n ) 0.
[ ( z t - i a n ) 3 + x t 2 ] 1 / 2 = r t - r t = ( z t - i a n ) { 1 + ( 1 / 2 ) x t 2 ( z t - i a n ) 2 + 0 [ x t 4 / ( z t - i a n ) 4 ] } ,             x t / ( z t - i a n ) 0 ,
r t - r t = r - r ,
r = ( i a n sin φ n + m L , i a n cos φ n )
sin φ n = n Ω / k = n λ / L 1
sin φ n = n λ / L > 1.
ϑ s = φ + i ( a n / ρ t ) ( sin φ - sin φ n ) sec 2 φ n × cos φ + 0 [ ( a n / ρ t ) 2 ] ,
ρ t a n = ( L cos φ n ) 2 / λ ,
Φ ( ϑ s ) = 1 + ( i / 2 ) ( a n / ρ t ) ( sin φ - sin φ n ) 2 × sec 2 φ n + 0 [ ( a n / ρ t ) 2 ] ,             a n / ρ t 0
D ( ϑ s ) = - i ( a n / ρ t ) ( cos φ / cos φ n ) 2 × [ 1 + 0 ( a n / ρ t ) ] ,             a n / ρ t 0.
B m , n ~ ( 2 1 / 2 / L ) 1 / 2 ( - i a n / ρ t ) 1 / 2 ( cos φ / cos φ n ) × exp { i k [ ρ t + ( i a n / 2 ) ( sin φ - sin φ n ) 2 sec 2 φ n ] } , k ρ t ,             a n / ρ t 0
ρ t = [ ρ 2 - 2 m L x + ( m L ) 2 ] 1 / 2 = ρ { 1 + ( m L ) / ρ ) sin φ + 0 [ ( m L / ρ ) 2 cos 2 φ ] }             m L / ρ 0 ,
B m , n ~ ( λ ρ ) - 1 / 2 H m , n ( k sin φ ) cos φ exp [ i ( k ρ - π / 4 ) ] ,             k ρ ,             a n / ρ 0 ,             m L / ρ 0.
H m , n ( k sin φ ) = G [ ( sin φ - sin φ n ) k / Ω ] exp ( - i m L k sin φ ) ,
k - N Ω = ( N + 1 ) Ω - k
L / λ = N + 1 / 2.
k a n = 2 π ( N + 1 / 4 ) .
ψ ( φ ) { 1 - rect [ 2 ( φ - φ 2 ) / φ 2 ] } ,
E t ( x , y , 0 ) = f t ( x , y )
F t ( η , ξ ) = - f t ( x , y ) exp [ - i ( η x + ξ y ) ] d x d y
E ( r ) = ( 2 π ) - 2 - F ( η , ξ ) exp ( ik · r ) d η d ξ ,
F = F t - κ - 1 ( k · F t ) z ^ ,
r = x ^ x + y ^ y + z ^ z ,
k = x ^ η + y ^ ξ + z ^ κ ,
κ = [ k 2 - η 2 - ξ 2 ] 1 / 2 ,             Re ( κ ) 0 ,             Im ( κ ) 0 ,
E ( r ) = × ψ ( r ) ,
ψ ( r ) = i ( 2 π ) - 2 - z ^ × F t ( η , ξ ) κ - 1 exp ( ik · r ) d η d ξ ,
ψ ( r ) = ( 2 π ) - 1 - ( z ^ × f t ) R - 1 exp ( i k R ) d x d y ,
R = [ ( x - x ) 2 + ( y - y ) 2 + z 2 ] 1 / 2 .
f t ( x , y ) = m , n , l , j A m , n , l , j w ( x - m L 1 , y - l L 2 ) exp [ i ( n Ω 1 x + j Ω 2 y ) ]
F t ( η , ξ ) = m , n , l , j A m , n , l , j W ( η - n Ω 1 , ξ - j Ω 2 ) × exp [ - i ( m L 1 η + l L 2 ξ ) ]
W ( η , ξ ) = - w ( x , y ) exp [ - i ( η x + ξ y ) ] d x d y ,
ψ ( r ) = m , n , l , j z ^ × A m , n , l , j B m , n , l , j ,
B m , n , l , j = i ( 2 π ) - 2 - W ( η - n Ω 1 , ξ - j Ω 2 ) κ - 1 × exp [ i ( x - m L 1 ) η + i ( y - l L 2 ) ξ ] d η d ξ
B m , n , l , j = ( 2 π ) - 1 - w ( x - m L 1 , y - l L 2 ) × exp [ i ( n Ω 1 x + j Ω 2 y ) ] R - 1 exp ( i k R ) d x d y ,
- w ( x , y ) γ * ( x - m L 1 , y - l L 2 ) × exp [ - i ( n Ω 1 x + j Ω 2 y ) ] d x d y = δ m δ n δ l δ j
( 2 π ) - 2 - W ( η , ξ ) Γ * ( η - n Ω 1 , ξ - j Ω 2 ) × exp [ i ( m L 1 η + l L 2 ξ ) ] d η d ξ = δ m δ n δ l δ j .
A m , n , l , j = - f t ( x , y ) γ * ( x - m L 1 , y - l L 2 ) × exp [ - i ( n Ω 1 x + j Ω 2 y ) ] d x d y
A m , n , l , j = ( 2 π ) - 2 - F t ( η , ξ ) Γ * ( η - n Ω 1 , ξ - j Ω 2 ) × exp [ i ( m L 1 η + l L 2 ξ ) ] d η d ξ .
S n n exp ( i n Ω x 0 ) = L n δ ( x 0 + n L ) ,             x 0 x - x
S m = m w ( x - m L ) γ * ( x - x 0 - m L ) = L - 1 m I ( m Ω , x 0 ) exp ( i m Ω x ) .
I ( m Ω , x 0 ) - w ( x ) γ * ( x - x 0 ) exp ( - i m Ω x ) d x .
I ( m Ω , - n L ) = δ m δ n ;
S n S m = δ ( x 0 ) = δ ( x - x ) .
S ( x / L ) = ( - 1 ) m exp [ - π ( m + 1 / 2 ) 2 ] s ( m ) ,             m - 1 / 2 < x / L m + 1 / 2 ,
s ( m ) j = 0 ( - 1 ) j exp [ - π j ( 2 m + j + 1 ) ]
0 < 1 - s ( m ) < exp [ - 2 π ( m + 1 ) ] < 2 × 10 - 3 ,
γ g ( x / L ) = ( - 1 ) m C L s ( m ) exp { - π [ ( m + 1 / 2 ) 2 - ( x / L ) 2 ] } ,             m - 1 / 2 < x / L < m + 1 / 2 ,
s ( m ) 1.
γ g ( m ) = ( - 1 ) m C L s ( m ) exp [ - π ( 4 m + 1 ) / 4 ] ,
A m , n = ( - 1 ) m C L s ( m ) exp [ - π ( 4 m + 1 ) / 4 ] × - L 0 / 2 L 0 / 2 f ( x ) exp [ π ( x / L ) 2 ] exp [ - i ( n - i m ) Ω x ] d x .

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