Abstract

For high-frequency fields, which can be separated into superpositions of a few distinct components with rapidly varying phases and slowly varying amplitudes, phase-space representations exhibit a strong localisation in which the coefficients are negligible over most of the phase space. This leads, potentially, to a very large reduction in the computational cost of computing propagators. Using the Windowed Fourier Transform, a number of fundamental problems from diffraction theory are studied using a representation of continuous wavefields by superpositions of beams that are continuously parameterised in phase-space and which propagate along ray trajectories. The existence of noncanonical WFT coefficients is observed, due to the nonuniqeness of the WFT. Numerical evaluations require discrete finite bases. The discrete Wilson basis is generated by a discrete sampling of the windowed Fourier Transform in the phase-space. The sampling is optimal, in the sense that the smallest number of coefficients is generated in an orthogonal basis.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]

Elect. Lett. (2)

D. Lugara, C. Letrou, �??Alternative to Gabor�??s representation of plane aperture radiation,�?? Elect. Lett. 34, 2286�??2287 (1998).
[CrossRef]

C. Rieckman, M. R. Rayner, C. G. Parini, �??Diffracted Gaussian beam analysis of quasioptical multi-reflector systems,�?? Elect. Lett. 36, 1600�??1601 (2000).
[CrossRef]

IEE Proc. J. (1)

J. M. Arnold, �??Geometrical methods in the theory of wave propagation: a contemporary review,�?? IEE Proc. J 133, 165�??188 (1986).

IEEE Trans. Ant. Prop. (2)

H. T. Chou, P. H. Pathak and R. J. Burkholder, �??Novel guassian beam method for the rapid analysis of large reflector antennas,�?? IEEE Trans. Ant. Prop. 49, 880�??893 (2001).
[CrossRef]

V. Galdi, L. B. Felsen and D. Castanon, �??Quasi-ray Gaussian beam algorithm for time-harmonic two-dimensional scattering by moderately rough interfaces,�?? IEEE Trans. Ant. Prop. 49, 1305�??1314 (2001).
[CrossRef]

J. IEE (1)

D. Gabor, �??Theory of communication,�?? Jour. IEE (III) 93, 429-457 (1946).

J. Opt. Soc. Am. A (2)

SIAM Jour. Math Anal. (1)

I. Daubechies, S. Ja.ard, and J. L. Journe, �??A simple Wilson orthonormal basis with exponential decay,�?? SIAM Jour. Math Anal. 22, 554�??572 (1991).
[CrossRef]

Other (5)

J. M. Arnold, �??Phase space localisation in high-frequency scattering using elementary wavelets,�?? URSI General Assembly, Kyoto, Japan, Paper B8-2, p60, (1993).

K. G. Wilson, "Generalised Wannier functions,�?? Cornell University preprint (1987).

A. J. Dragt, E. Forest and K. B. Wolf, �??Foundations of a Lie-algebraic theory of geometrical optics,�?? Chap. 4 in Lie methods in optics, ed. J. Sanchez Mondragon and K. B. Wolf (Springer Lecture Notes in Physics, vol. 250, Springer-Verlag, Berlin, 1986).
[CrossRef]

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1968).

G. L. James, The geometrical theory of diffraction for electromagnetic waves (IEEE Electromagnetic Waves Series, vol. 1, 3d ed., Peter Peregrinus Ltd., 1986).

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Equations (56)

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u ( x , z ) = ( 2 π ) 1 u ˜ ( ξ ) e i β z e i ξ x d ξ ,
u ˜ ( ξ ) = u ( x , 0 ) e i ξ x d x ,
S 1 : x x , ξ ξ k x p ( x ) ,
S 2 : x x γ ( ξ ) z , ξ ξ ,
u ( x ) = ( 2 π ) 1 U ( x 0 , ξ 0 ) g ( x x 0 ) e i ξ 0 x d x 0 d ξ 0 ,
u ˜ ( ξ ) = ( 2 π ) 1 U ( x 0 , ξ 0 ) g ˜ ( ξ ξ 0 ) e i ( ξ ξ 0 ) x 0 d x 0 d ξ 0
U ( x 0 , ξ 0 ) = u ( x ) g * ( x x 0 ) e i ξ 0 x d x
= ( 2 π ) 1 u ˜ ( ξ ) g ˜ * ( ξ ξ 0 ) e i ξ x 0 d ξ
g ( x ) 2 d x = 1 .
u ( x ) ~ a ( x ) exp [ i k σ ( x ) ] ,
k σ ( x ) = k { σ ( x 0 ) + ( x x 0 ) σ ( x 0 ) + 1 2 ( x x 0 ) 2 σ ( x 0 ) + . . . } .
A exp [ i k x ( x 0 ) ( x x 0 ) + 1 2 i k σ ( x 0 ) ( x x 0 ) 2 ]
U ( x 0 , ξ 0 ) ~ u ( x 0 ) G ˜ ( x 0 , ξ 0 k x ( x 0 ) ) .
G ˜ ( x 0 , ξ ) = g ( x ) exp ( 1 2 i k σ ( x 0 ) x 2 ) ) exp ( i ξ x ) d x .
g ( x x 1 ) e i ξ 1 x = ( 2 π ) 1 K ( x 0 , ξ 0 ; x 1 , ξ 1 ) g ( x x 0 ) e i ξ 0 x d x 0 d ξ 0 ,
K ( x 0 , ξ 0 ; x 1 , ξ 1 ) = g ( x x 0 ) g ( x x 1 ) e i ( ξ 0 ξ 1 ) x d x .
U ( x 0 , ξ 0 ) ( 2 π ) 1 K ( x 0 , ξ 0 ; x 1 , ξ 1 ) U ( x 1 , ξ 1 ) d x 0 d ξ 0 .
U U = KU ,
K 2 = K .
g ( x ) = 2 1 4 L 1 2 exp ( π x 2 L 2 )
u ( x ) = exp ( i k x 2 2 f ) .
U ( m , n ) = 2 1 4 ( 1 i κ ) 1 2 exp [ i κπ m 2 2 i πmn π 1 i κ ( n κ m ) 2 ]
ξ 0 = d d x ( k x 2 2 f ) | x = x 0 = k x 0 f .
U ( m , n ) = 2 1 4 ( 1 + i κ ) 1 2 exp [ i κπ m 2 2 i πmn ] δ ( n κ m )
χ A ( x ) = { 1 , x A 0 , x A
U ( m , n ) = 1 2 π i 1 ( n b ) δ ( m a ) exp ( 2 π i ab )
U ( m , n ) = 1 2 π i 1 ( n b ) δ ( m a ) exp [ π i ( n b ) d ] exp ( 2 π i ab )
1 2 π i 1 ( n b ) δ ( m a ) exp [ π i ( n b ) d ] exp ( 2 π i ab )
U ( m , n ) = 1 2 π i 1 n b { exp [ π i ( n b ) d ] exp [ π i ( n b ) d ] }
U ( m , n ) = 1 2 π i 1 n κ m exp [ i κπ m 2 ] exp [ π i ( n κ m ) d ]
1 2 π i 1 n κ m exp [ i κπ m 2 ] exp [ π i ( n κ m ) d ]
U ( m , n ) = 1 2 π i 1 n b
U ( m , n ) = 2 1 4 exp [ π ( n b ) 2 + 2 π i m ( n b ) ] m i ( n b ) exp ( π s 2 ) d s .
U ( m , n ) ~ 2 1 4 exp [ π ( n b ) 2 + 2 π i m ( n b ) ] , m
U ( m , n ) ~ 0 , m
m i ( n b ) exp ( π s 2 ) d s ~ ( 2 π ( m + i ( n b ) ) ) 1 exp [ π ( m + i ( n b ) ) 2 ] ,
U ( m , n ) ~ 2 1 4 ( 2 π i ( n b ) ) 1 exp [ π m 2 ] , n ± .
U ( m , n ) ~ 2 1 4 ( 1 i κ ) 1 2 exp [ i κπ m 2 2 i πmn π 1 i κ ( n κm ) 2 ]
( 1 2 d m 1 2 d ) .
U ( m , n ) ~ 2 1 4 ( 2 π i ( n ± 1 2 κ d ) ) 1 exp [ π ( m 1 2 d ) 2 ] , ( m ~ ± 1 2 d ) .
u ( x ) = m Z n Z A mn w mn ( x )
w mn ( x ) = { ϕ ( x m ) , n = 0 2 ϕ ( x 1 2 m ) cos ( 2 πnx ) , m + n = 0 ( mod 2 ) , n 0 2 ϕ ( x 1 2 m ) sin ( 2 πnx ) , m + n = 1 ( mod 2 ) . n 0
m Ƶ ϕ * ( x k 1 2 m ) ϕ ( x 1 2 m ) = 2 δ 0 k , k Ƶ , x ,
m Ƶ ( 1 ) m ϕ * ( x k 1 2 m 1 2 ) ϕ ( x 1 2 m ) = 0 , k Ƶ , x ,
A mn = w mn * ( x ) u ( x ) d x .
A mn = { ϕ * ( x m ) u ( x ) d x , n = 0 2 ϕ * ( x 1 2 m ) cos ( 2 πnx ) u ( x ) d x , m + n = 0 ( mod 2 ) , n 0 2 ϕ * ( x 1 2 m ) sin ( 2 πnx ) u ( x ) d x , m + n = 1 ( mod 2 ) . n 0
ϕ ( x ) = ( r , s ) Ƶ 2 C r s g ( x 1 2 r ) exp ( 2 π i sx )
u ( x ) = B r s g ( x 1 2 r ) exp ( 2 π i sx )
w mn ( x ) = { r , s C r 2 m s g r s ( x ) , n = 0 2 1 2 r , s { C r m s n + C r m s + n } g rs ( x ) , n 0 , m + n = 0 ( mod 2 ) 2 1 2 r , s { C r m s n C r m s + n } g rs ( x ) , n 0 , m + n = 1 ( mod 2 )
u ( x ) = r , s , r , s U r s T r s r s g r s ( x )
g r s ( x ) = g ( x 1 2 r ) exp ( 2 π i sx )
U r s = u ( x ) g r s * ( x ) d x
T r s r s = m = n = 0 D mn r s * D mn r s
D mn r s = { C r 2 m s , n = 0 2 1 2 ( C r m s n + ( 1 ) m + n C r m s + n ) , n 0
w mn ( x , 0 ) w mn ( x , z )
w ˜ mn ( ξ , 0 ) w ˜ mn ( ξ , z ) = w ˜ mn ( ξ , 0 ) e 2 π i k 2 ξ 2 z

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