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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  1. 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]
  2. D. Lugara and C. Letrou, “Alternative to Gabor’s representation of plane aperture radiation,” Elect. Lett. 34, 2286–2287 (1998).
    [Crossref]
  3. 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]
  4. C. Rieckman, M. R. Rayner, and C. G. Parini, “Diffracted Gaussian beam analysis of quasioptical multi-reflector systems,” Elect. Lett. 36, 1600–1601 (2000).
    [Crossref]
  5. J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1968).
  6. G. L. James, The geometrical theory of diffraction for electromagnetic waves (IEEE Electromagnetic Waves Series, vol. 1, 3d ed., Peter Peregrinus Ltd., 1986).
  7. J. M. Arnold, “Geometrical methods in the theory of wave propagation: a contemporary review,” IEE Proc. J 133, 165–188 (1986).
  8. 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]
  9. M. A. Alonso and G. W. Forbes, “Phase space distributions for high-frequency fields,” J. Opt. Soc. Am. A 17, 2288–2300 (2000).
    [Crossref]
  10. D. Gabor, “Theory of communication,” Jour. IEE (III) 93, 429–457 (1946).
  11. K. G. Wilson, “Generalised Wannier functions,” Cornell University preprint (1987).
  12. I. Daubechies, S. Jaffard, and J. L. Journe, “A simple Wilson orthonormal basis with exponential decay,” SIAM Jour. Math Anal. 22, 554–572 (1991).
    [Crossref]
  13. J. M. Arnold, “Phase space localisation in high-frequency scattering using elementary wavelets,” URSI General Assembly, Kyoto, Japan, Paper B8-2, p60, (1993).
  14. J. M. Arnold, “Phase-space localisation and discrete representations of wave fields,” Jour. Opt. Soc. Am. A 12, 111–123 (1995).
    [Crossref]
  15. J. F. Nye, Natural focusing and fine structure of light (IOP Publishing, Ltd., Bristol, 1999).

2001 (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]

2000 (2)

C. Rieckman, M. R. Rayner, and C. G. Parini, “Diffracted Gaussian beam analysis of quasioptical multi-reflector systems,” Elect. Lett. 36, 1600–1601 (2000).
[Crossref]

M. A. Alonso and G. W. Forbes, “Phase space distributions for high-frequency fields,” J. Opt. Soc. Am. A 17, 2288–2300 (2000).
[Crossref]

1998 (1)

D. Lugara and C. Letrou, “Alternative to Gabor’s representation of plane aperture radiation,” Elect. Lett. 34, 2286–2287 (1998).
[Crossref]

1995 (1)

J. M. Arnold, “Phase-space localisation and discrete representations of wave fields,” Jour. Opt. Soc. Am. A 12, 111–123 (1995).
[Crossref]

1991 (1)

I. Daubechies, S. Jaffard, and J. L. Journe, “A simple Wilson orthonormal basis with exponential decay,” SIAM Jour. Math Anal. 22, 554–572 (1991).
[Crossref]

1986 (1)

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

1946 (1)

D. Gabor, “Theory of communication,” Jour. IEE (III) 93, 429–457 (1946).

Alonso, M. A.

Arnold, J. M.

J. M. Arnold, “Phase-space localisation and discrete representations of wave fields,” Jour. Opt. Soc. Am. A 12, 111–123 (1995).
[Crossref]

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

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

Burkholder, R. J.

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]

Castanon, D.

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]

Chou, H. T.

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]

Daubechies, I.

I. Daubechies, S. Jaffard, and J. L. Journe, “A simple Wilson orthonormal basis with exponential decay,” SIAM Jour. Math Anal. 22, 554–572 (1991).
[Crossref]

Dragt, A. J.

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]

Felsen, L. B.

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]

Forbes, G. W.

Forest, E.

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]

Gabor, D.

D. Gabor, “Theory of communication,” Jour. IEE (III) 93, 429–457 (1946).

Galdi, V.

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]

Goodman, J. W.

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

Jaffard, S.

I. Daubechies, S. Jaffard, and J. L. Journe, “A simple Wilson orthonormal basis with exponential decay,” SIAM Jour. Math Anal. 22, 554–572 (1991).
[Crossref]

James, G. L.

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

Journe, J. L.

I. Daubechies, S. Jaffard, and J. L. Journe, “A simple Wilson orthonormal basis with exponential decay,” SIAM Jour. Math Anal. 22, 554–572 (1991).
[Crossref]

Letrou, C.

D. Lugara and C. Letrou, “Alternative to Gabor’s representation of plane aperture radiation,” Elect. Lett. 34, 2286–2287 (1998).
[Crossref]

Lugara, D.

D. Lugara and C. Letrou, “Alternative to Gabor’s representation of plane aperture radiation,” Elect. Lett. 34, 2286–2287 (1998).
[Crossref]

Nye, J. F.

J. F. Nye, Natural focusing and fine structure of light (IOP Publishing, Ltd., Bristol, 1999).

Parini, C. G.

C. Rieckman, M. R. Rayner, and C. G. Parini, “Diffracted Gaussian beam analysis of quasioptical multi-reflector systems,” Elect. Lett. 36, 1600–1601 (2000).
[Crossref]

Pathak, P. H.

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]

Rayner, M. R.

C. Rieckman, M. R. Rayner, and C. G. Parini, “Diffracted Gaussian beam analysis of quasioptical multi-reflector systems,” Elect. Lett. 36, 1600–1601 (2000).
[Crossref]

Rieckman, C.

C. Rieckman, M. R. Rayner, and C. G. Parini, “Diffracted Gaussian beam analysis of quasioptical multi-reflector systems,” Elect. Lett. 36, 1600–1601 (2000).
[Crossref]

Wilson, K. G.

K. G. Wilson, “Generalised Wannier functions,” Cornell University preprint (1987).

Wolf, K. B.

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]

Elect. Lett. (2)

D. Lugara and C. Letrou, “Alternative to Gabor’s representation of plane aperture radiation,” Elect. Lett. 34, 2286–2287 (1998).
[Crossref]

C. Rieckman, M. R. Rayner, and 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)

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]

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]

J. Opt. Soc. Am. A (1)

Jour. IEE (III) (1)

D. Gabor, “Theory of communication,” Jour. IEE (III) 93, 429–457 (1946).

Jour. Opt. Soc. Am. A (1)

J. M. Arnold, “Phase-space localisation and discrete representations of wave fields,” Jour. Opt. Soc. Am. A 12, 111–123 (1995).
[Crossref]

SIAM Jour. Math Anal. (1)

I. Daubechies, S. Jaffard, and J. L. Journe, “A simple Wilson orthonormal basis with exponential decay,” SIAM Jour. Math Anal. 22, 554–572 (1991).
[Crossref]

Other (6)

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

J. F. Nye, Natural focusing and fine structure of light (IOP Publishing, Ltd., Bristol, 1999).

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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