Abstract

The present contribution is concerned with applying beam-type expansion to planar aperture time-harmonic electromagnetic field distribution in which the propagating elements, the electromagnetic beam-type wave objects, are decomposed into transverse electric (TE) and transverse magnetic (TM) field constituents. This procedure is essential for applying Maxwell’s boundary conditions for solving different scattering problems. The propagating field is described as a discrete superposition of tilted and shifted TE and TM electromagnetic beams over the frame-based spatial–directional expansion lattice. These vector wave objects are evaluated either by applying differential operators to scalar beam propagators, or by using plane-wave spectral representations. Explicit asymptotic expressions for scalar, as well as for electromagnetic, Gaussian beam propagators are presented as well.

© 2011 Optical Society of America

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  1. B. Steinberg, E. Heyman, and L. Felsen, “Phase space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
    [CrossRef]
  2. T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
    [CrossRef]
  3. J. M. Arnold, “Rays, beams and diffraction in a discrete phase space: Wilson bases,” Opt. Express 10, 716–727 (2002).
    [PubMed]
  4. A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for ultra wideband radiation,” IEEE Trans. Antennas Propag. 52, 2042–2056 (2004).
    [CrossRef]
  5. G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J. Acoust. Soc. Am. 117, 1911–1921 (2005).
    [CrossRef] [PubMed]
  6. M. Katsav and E. Heyman, “Phase space Gaussian beam summation analysis of half plane diffraction,” IEEE Trans. Antennas Propag. 55, 1535–1545 (2007).
    [CrossRef]
  7. T. Melamed, “Phase-space Green’s functions for modeling time-harmonic scattering from smooth inhomogeneous objects,” J. Math. Phys. 45, 2232–2246 (2004).
    [CrossRef]
  8. T. Melamed, “Time-domain phase-space Green’s functions for inhomogeneous media,” in Ultrawideband/Short Pulse Electromagnetics 6, E.L.Mokole, M.Kragalott, K.R.Gerlach, M.Kragalott, and K.R.Gerlach, ed. (Springer-Verlag, 2007), pp. 56–63.
  9. I. Tinkelman and T. Melamed, “Gaussian beam propagation in generic anisotropic wavenumber profiles,” Opt. Lett. 28, 1081–1083 (2003).
    [CrossRef] [PubMed]
  10. I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in an anisotropic medium. Part I. Time-harmonic fields,” J. Opt. Soc. Am. A 22, 1200–1207 (2005).
    [CrossRef]
  11. I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in an anisotropic medium. Part II. Time-dependent fields,” J. Opt. Soc. Am. A 22, 1208–1215 (2005).
    [CrossRef]
  12. T. Melamed and L. Felsen, “Pulsed beam propagation in lossless dispersive media. Part I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
    [CrossRef]
  13. T. Melamed and L. Felsen, “Pulsed beam propagation in lossless dispersive media. Part II. A numerical example,” J. Opt. Soc. Am. A 15, 1277–1284 (1998).
    [CrossRef]
  14. T. Melamed and L. B. Felsen, “Pulsed beam propagation in dispersive media via pulsed plane wave spectral decomposition,” IEEE Trans. Antennas Propag. 48, 901–908 (2000).
    [CrossRef]
  15. V. Ĉerveny, M. M. Popov, and I. Pŝenĉik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
    [CrossRef]
  16. B. W. A. N. Norris and J. Schrieffer, “Gaussian wave packets in inhomogeneous media with curved interfaces,” Proc. R. Soc. London Ser. A 412, 93–123 (1987).
    [CrossRef]
  17. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [CrossRef]
  18. R. Collin, “Scattering of an incident Gaussian beam by a perfectly conducting rough surface,” IEEE Trans. Antennas Propag. 42, 70–4 (1994).
    [CrossRef]
  19. O. Kilic and R. Lang, “Scattering of a pulsed beam by a random medium over ground,” J. Electromagn. Waves Appl. 15, 481–516(2001).
    [CrossRef]
  20. O. Pascal, F. Lemaitre, and G. Soum, “Paraxial approximation effect on a dielectric interface analysis,” Ann. Telecommun. 51, 206–218 (1996).
  21. H. Anastassiu and P. Pathak, “Closed form solution for three-dimensional reflection of an arbitrary Gaussian beam by a smooth surface,” Radio Sci. 37, 1–8 (2002).
    [CrossRef]
  22. J. Hillairet, J. Sokoloff, S. Bolioli, and P. F. Combes, “Analytical physical optics scattering from a PEC finite plate illuminated by a vector Gaussian beam,” in 2007 International Conference on Electromagnetics in Advanced Applications (2007), pp. 170–173.
  23. F. Bass and L. Resnick, “Wave beam propagation in layered media,” Prog. Electromagn. Res. 38, 111–123 (2002).
    [CrossRef]
  24. J. Kong, “Electromagnetic wave interaction with stratified negative isotropic media,” J. Electromagn. Waves Appl. 15, 1319–1320 (2001).
    [CrossRef]
  25. Y. Hadad and T. Melamed, “Non-orthogonal domain parabolic equation and its Gaussian beam solutions,” IEEE Trans. Antennas Propag. 58, 1164–1172 (2010).
    [CrossRef]
  26. Y. Hadad and T. Melamed, “Parameterization of the tilted Gaussian beam waveobjects,” Prog. Electromagn. Res. PIER 102, 65–80 (2010).
    [CrossRef]
  27. Y. Hadad and T. Melamed, “Tilted Gaussian beam propagation in inhomogeneous media,” J. Opt. Soc. Am. A 27, 1840–1850(2010).
    [CrossRef]
  28. B. Steinberg and E. Heyman, “Phase space beam summation for time dependent radiation from large apertures: discretized parametrization,” J. Opt. Soc. Am. A 8, 959–966 (1991).
    [CrossRef]
  29. H.-T. Chou, P. Pathak, and R. Burkholder, “Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas,” IEE Proc. Microw. Antennas Propag. 150, 177–83 (2003).
    [CrossRef]
  30. H.-T. Chou, P. Pathak, and R. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag. 49, 880–93 (2001).
    [CrossRef]
  31. H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
    [CrossRef]
  32. T. Melamed, “Exact beam decomposition of time-harmonic electromagnetic waves,” J. Electromagn. Waves Appl. 23, 975–986(2009).
  33. R. Martínez-Herrero, P. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001).
    [CrossRef]
  34. H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14, 2095–2100 (2006).
    [CrossRef] [PubMed]
  35. D. Gabor, “A new microscopic principle,” Nature 161, 777(1948).
    [CrossRef] [PubMed]
  36. J. Wexler and S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–20 (1990).
    [CrossRef]
  37. E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
    [CrossRef]
  38. E. Heyman and T. Melamed, Space-Time Representation of Ultra Wideband Signals (Elsevier, 1998), pp. 1–63.
  39. A. Shlivinski, E. Heyman, and A. Boag, “A phase-space beam summation formulation for ultrawide-band radiation. Part II: A multiband scheme,” IEEE Trans. Antennas Propag. 53, 948–957 (2005).
    [CrossRef]
  40. A. Shlivinski, E. Heyman, and A. Boag, “A pulsed beam summation formulation for short pulse radiation based on windowed radon transform (WRT) frames,” IEEE Trans. Antennas Propag. 53, 3030–3048 (2005).
    [CrossRef]
  41. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Classic reissue (IEEE, 1994), Chap. 4.2.
    [CrossRef]

2010 (3)

Y. Hadad and T. Melamed, “Non-orthogonal domain parabolic equation and its Gaussian beam solutions,” IEEE Trans. Antennas Propag. 58, 1164–1172 (2010).
[CrossRef]

Y. Hadad and T. Melamed, “Parameterization of the tilted Gaussian beam waveobjects,” Prog. Electromagn. Res. PIER 102, 65–80 (2010).
[CrossRef]

Y. Hadad and T. Melamed, “Tilted Gaussian beam propagation in inhomogeneous media,” J. Opt. Soc. Am. A 27, 1840–1850(2010).
[CrossRef]

2009 (1)

T. Melamed, “Exact beam decomposition of time-harmonic electromagnetic waves,” J. Electromagn. Waves Appl. 23, 975–986(2009).

2007 (1)

M. Katsav and E. Heyman, “Phase space Gaussian beam summation analysis of half plane diffraction,” IEEE Trans. Antennas Propag. 55, 1535–1545 (2007).
[CrossRef]

2006 (1)

2005 (5)

I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in an anisotropic medium. Part I. Time-harmonic fields,” J. Opt. Soc. Am. A 22, 1200–1207 (2005).
[CrossRef]

I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in an anisotropic medium. Part II. Time-dependent fields,” J. Opt. Soc. Am. A 22, 1208–1215 (2005).
[CrossRef]

A. Shlivinski, E. Heyman, and A. Boag, “A phase-space beam summation formulation for ultrawide-band radiation. Part II: A multiband scheme,” IEEE Trans. Antennas Propag. 53, 948–957 (2005).
[CrossRef]

A. Shlivinski, E. Heyman, and A. Boag, “A pulsed beam summation formulation for short pulse radiation based on windowed radon transform (WRT) frames,” IEEE Trans. Antennas Propag. 53, 3030–3048 (2005).
[CrossRef]

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J. Acoust. Soc. Am. 117, 1911–1921 (2005).
[CrossRef] [PubMed]

2004 (3)

A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for ultra wideband radiation,” IEEE Trans. Antennas Propag. 52, 2042–2056 (2004).
[CrossRef]

T. Melamed, “Phase-space Green’s functions for modeling time-harmonic scattering from smooth inhomogeneous objects,” J. Math. Phys. 45, 2232–2246 (2004).
[CrossRef]

H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
[CrossRef]

2003 (2)

I. Tinkelman and T. Melamed, “Gaussian beam propagation in generic anisotropic wavenumber profiles,” Opt. Lett. 28, 1081–1083 (2003).
[CrossRef] [PubMed]

H.-T. Chou, P. Pathak, and R. Burkholder, “Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas,” IEE Proc. Microw. Antennas Propag. 150, 177–83 (2003).
[CrossRef]

2002 (3)

H. Anastassiu and P. Pathak, “Closed form solution for three-dimensional reflection of an arbitrary Gaussian beam by a smooth surface,” Radio Sci. 37, 1–8 (2002).
[CrossRef]

F. Bass and L. Resnick, “Wave beam propagation in layered media,” Prog. Electromagn. Res. 38, 111–123 (2002).
[CrossRef]

J. M. Arnold, “Rays, beams and diffraction in a discrete phase space: Wilson bases,” Opt. Express 10, 716–727 (2002).
[PubMed]

2001 (4)

R. Martínez-Herrero, P. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001).
[CrossRef]

J. Kong, “Electromagnetic wave interaction with stratified negative isotropic media,” J. Electromagn. Waves Appl. 15, 1319–1320 (2001).
[CrossRef]

O. Kilic and R. Lang, “Scattering of a pulsed beam by a random medium over ground,” J. Electromagn. Waves Appl. 15, 481–516(2001).
[CrossRef]

H.-T. Chou, P. Pathak, and R. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag. 49, 880–93 (2001).
[CrossRef]

2000 (1)

T. Melamed and L. B. Felsen, “Pulsed beam propagation in dispersive media via pulsed plane wave spectral decomposition,” IEEE Trans. Antennas Propag. 48, 901–908 (2000).
[CrossRef]

1998 (2)

1997 (1)

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

1996 (1)

O. Pascal, F. Lemaitre, and G. Soum, “Paraxial approximation effect on a dielectric interface analysis,” Ann. Telecommun. 51, 206–218 (1996).

1994 (3)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

R. Collin, “Scattering of an incident Gaussian beam by a perfectly conducting rough surface,” IEEE Trans. Antennas Propag. 42, 70–4 (1994).
[CrossRef]

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[CrossRef]

1991 (2)

1990 (1)

J. Wexler and S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–20 (1990).
[CrossRef]

1987 (1)

B. W. A. N. Norris and J. Schrieffer, “Gaussian wave packets in inhomogeneous media with curved interfaces,” Proc. R. Soc. London Ser. A 412, 93–123 (1987).
[CrossRef]

1982 (1)

V. Ĉerveny, M. M. Popov, and I. Pŝenĉik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777(1948).
[CrossRef] [PubMed]

Anastassiu, H.

H. Anastassiu and P. Pathak, “Closed form solution for three-dimensional reflection of an arbitrary Gaussian beam by a smooth surface,” Radio Sci. 37, 1–8 (2002).
[CrossRef]

Arnold, J. M.

Bass, F.

F. Bass and L. Resnick, “Wave beam propagation in layered media,” Prog. Electromagn. Res. 38, 111–123 (2002).
[CrossRef]

Boag, A.

A. Shlivinski, E. Heyman, and A. Boag, “A pulsed beam summation formulation for short pulse radiation based on windowed radon transform (WRT) frames,” IEEE Trans. Antennas Propag. 53, 3030–3048 (2005).
[CrossRef]

A. Shlivinski, E. Heyman, and A. Boag, “A phase-space beam summation formulation for ultrawide-band radiation. Part II: A multiband scheme,” IEEE Trans. Antennas Propag. 53, 948–957 (2005).
[CrossRef]

A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for ultra wideband radiation,” IEEE Trans. Antennas Propag. 52, 2042–2056 (2004).
[CrossRef]

Bolioli, S.

J. Hillairet, J. Sokoloff, S. Bolioli, and P. F. Combes, “Analytical physical optics scattering from a PEC finite plate illuminated by a vector Gaussian beam,” in 2007 International Conference on Electromagnetics in Advanced Applications (2007), pp. 170–173.

Bosch, S.

Burkholder, R.

H.-T. Chou, P. Pathak, and R. Burkholder, “Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas,” IEE Proc. Microw. Antennas Propag. 150, 177–83 (2003).
[CrossRef]

H.-T. Chou, P. Pathak, and R. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag. 49, 880–93 (2001).
[CrossRef]

Carnicer, A.

Cerveny, V.

V. Ĉerveny, M. M. Popov, and I. Pŝenĉik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Chen, J.

Chou, H.-T.

H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
[CrossRef]

H.-T. Chou, P. Pathak, and R. Burkholder, “Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas,” IEE Proc. Microw. Antennas Propag. 150, 177–83 (2003).
[CrossRef]

H.-T. Chou, P. Pathak, and R. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag. 49, 880–93 (2001).
[CrossRef]

Collin, R.

R. Collin, “Scattering of an incident Gaussian beam by a perfectly conducting rough surface,” IEEE Trans. Antennas Propag. 42, 70–4 (1994).
[CrossRef]

Combes, P. F.

J. Hillairet, J. Sokoloff, S. Bolioli, and P. F. Combes, “Analytical physical optics scattering from a PEC finite plate illuminated by a vector Gaussian beam,” in 2007 International Conference on Electromagnetics in Advanced Applications (2007), pp. 170–173.

Felsen, L.

Felsen, L. B.

T. Melamed and L. B. Felsen, “Pulsed beam propagation in dispersive media via pulsed plane wave spectral decomposition,” IEEE Trans. Antennas Propag. 48, 901–908 (2000).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777(1948).
[CrossRef] [PubMed]

Gordon, G.

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J. Acoust. Soc. Am. 117, 1911–1921 (2005).
[CrossRef] [PubMed]

Guo, H.

Hadad, Y.

Y. Hadad and T. Melamed, “Parameterization of the tilted Gaussian beam waveobjects,” Prog. Electromagn. Res. PIER 102, 65–80 (2010).
[CrossRef]

Y. Hadad and T. Melamed, “Non-orthogonal domain parabolic equation and its Gaussian beam solutions,” IEEE Trans. Antennas Propag. 58, 1164–1172 (2010).
[CrossRef]

Y. Hadad and T. Melamed, “Tilted Gaussian beam propagation in inhomogeneous media,” J. Opt. Soc. Am. A 27, 1840–1850(2010).
[CrossRef]

Heyman, E.

M. Katsav and E. Heyman, “Phase space Gaussian beam summation analysis of half plane diffraction,” IEEE Trans. Antennas Propag. 55, 1535–1545 (2007).
[CrossRef]

A. Shlivinski, E. Heyman, and A. Boag, “A phase-space beam summation formulation for ultrawide-band radiation. Part II: A multiband scheme,” IEEE Trans. Antennas Propag. 53, 948–957 (2005).
[CrossRef]

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J. Acoust. Soc. Am. 117, 1911–1921 (2005).
[CrossRef] [PubMed]

A. Shlivinski, E. Heyman, and A. Boag, “A pulsed beam summation formulation for short pulse radiation based on windowed radon transform (WRT) frames,” IEEE Trans. Antennas Propag. 53, 3030–3048 (2005).
[CrossRef]

A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for ultra wideband radiation,” IEEE Trans. Antennas Propag. 52, 2042–2056 (2004).
[CrossRef]

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[CrossRef]

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

B. Steinberg and E. Heyman, “Phase space beam summation for time dependent radiation from large apertures: discretized parametrization,” J. Opt. Soc. Am. A 8, 959–966 (1991).
[CrossRef]

B. Steinberg, E. Heyman, and L. Felsen, “Phase space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
[CrossRef]

E. Heyman and T. Melamed, Space-Time Representation of Ultra Wideband Signals (Elsevier, 1998), pp. 1–63.

Hillairet, J.

J. Hillairet, J. Sokoloff, S. Bolioli, and P. F. Combes, “Analytical physical optics scattering from a PEC finite plate illuminated by a vector Gaussian beam,” in 2007 International Conference on Electromagnetics in Advanced Applications (2007), pp. 170–173.

Katsav, M.

M. Katsav and E. Heyman, “Phase space Gaussian beam summation analysis of half plane diffraction,” IEEE Trans. Antennas Propag. 55, 1535–1545 (2007).
[CrossRef]

Kilic, O.

O. Kilic and R. Lang, “Scattering of a pulsed beam by a random medium over ground,” J. Electromagn. Waves Appl. 15, 481–516(2001).
[CrossRef]

Kong, J.

J. Kong, “Electromagnetic wave interaction with stratified negative isotropic media,” J. Electromagn. Waves Appl. 15, 1319–1320 (2001).
[CrossRef]

Lang, R.

O. Kilic and R. Lang, “Scattering of a pulsed beam by a random medium over ground,” J. Electromagn. Waves Appl. 15, 481–516(2001).
[CrossRef]

Lemaitre, F.

O. Pascal, F. Lemaitre, and G. Soum, “Paraxial approximation effect on a dielectric interface analysis,” Ann. Telecommun. 51, 206–218 (1996).

Letrou, C.

A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for ultra wideband radiation,” IEEE Trans. Antennas Propag. 52, 2042–2056 (2004).
[CrossRef]

Marcuvitz, N.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Classic reissue (IEEE, 1994), Chap. 4.2.
[CrossRef]

Martínez-Herrero, R.

Mazar, R.

G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J. Acoust. Soc. Am. 117, 1911–1921 (2005).
[CrossRef] [PubMed]

Mejías, P.

Melamed, T.

Y. Hadad and T. Melamed, “Parameterization of the tilted Gaussian beam waveobjects,” Prog. Electromagn. Res. PIER 102, 65–80 (2010).
[CrossRef]

Y. Hadad and T. Melamed, “Tilted Gaussian beam propagation in inhomogeneous media,” J. Opt. Soc. Am. A 27, 1840–1850(2010).
[CrossRef]

Y. Hadad and T. Melamed, “Non-orthogonal domain parabolic equation and its Gaussian beam solutions,” IEEE Trans. Antennas Propag. 58, 1164–1172 (2010).
[CrossRef]

T. Melamed, “Exact beam decomposition of time-harmonic electromagnetic waves,” J. Electromagn. Waves Appl. 23, 975–986(2009).

I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in an anisotropic medium. Part II. Time-dependent fields,” J. Opt. Soc. Am. A 22, 1208–1215 (2005).
[CrossRef]

I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in an anisotropic medium. Part I. Time-harmonic fields,” J. Opt. Soc. Am. A 22, 1200–1207 (2005).
[CrossRef]

T. Melamed, “Phase-space Green’s functions for modeling time-harmonic scattering from smooth inhomogeneous objects,” J. Math. Phys. 45, 2232–2246 (2004).
[CrossRef]

I. Tinkelman and T. Melamed, “Gaussian beam propagation in generic anisotropic wavenumber profiles,” Opt. Lett. 28, 1081–1083 (2003).
[CrossRef] [PubMed]

T. Melamed and L. B. Felsen, “Pulsed beam propagation in dispersive media via pulsed plane wave spectral decomposition,” IEEE Trans. Antennas Propag. 48, 901–908 (2000).
[CrossRef]

T. Melamed and L. Felsen, “Pulsed beam propagation in lossless dispersive media. Part I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
[CrossRef]

T. Melamed and L. Felsen, “Pulsed beam propagation in lossless dispersive media. Part II. A numerical example,” J. Opt. Soc. Am. A 15, 1277–1284 (1998).
[CrossRef]

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[CrossRef]

T. Melamed, “Time-domain phase-space Green’s functions for inhomogeneous media,” in Ultrawideband/Short Pulse Electromagnetics 6, E.L.Mokole, M.Kragalott, K.R.Gerlach, M.Kragalott, and K.R.Gerlach, ed. (Springer-Verlag, 2007), pp. 56–63.

E. Heyman and T. Melamed, Space-Time Representation of Ultra Wideband Signals (Elsevier, 1998), pp. 1–63.

Norris, B. W. A. N.

B. W. A. N. Norris and J. Schrieffer, “Gaussian wave packets in inhomogeneous media with curved interfaces,” Proc. R. Soc. London Ser. A 412, 93–123 (1987).
[CrossRef]

Pascal, O.

O. Pascal, F. Lemaitre, and G. Soum, “Paraxial approximation effect on a dielectric interface analysis,” Ann. Telecommun. 51, 206–218 (1996).

Pathak, P.

H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
[CrossRef]

H.-T. Chou, P. Pathak, and R. Burkholder, “Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas,” IEE Proc. Microw. Antennas Propag. 150, 177–83 (2003).
[CrossRef]

H. Anastassiu and P. Pathak, “Closed form solution for three-dimensional reflection of an arbitrary Gaussian beam by a smooth surface,” Radio Sci. 37, 1–8 (2002).
[CrossRef]

H.-T. Chou, P. Pathak, and R. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag. 49, 880–93 (2001).
[CrossRef]

Popov, M. M.

V. Ĉerveny, M. M. Popov, and I. Pŝenĉik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Psencik, I.

V. Ĉerveny, M. M. Popov, and I. Pŝenĉik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Raz, S.

J. Wexler and S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–20 (1990).
[CrossRef]

Resnick, L.

F. Bass and L. Resnick, “Wave beam propagation in layered media,” Prog. Electromagn. Res. 38, 111–123 (2002).
[CrossRef]

Schrieffer, J.

B. W. A. N. Norris and J. Schrieffer, “Gaussian wave packets in inhomogeneous media with curved interfaces,” Proc. R. Soc. London Ser. A 412, 93–123 (1987).
[CrossRef]

Shlivinski, A.

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Figures (7)

Fig. 1
Fig. 1

Discrete frame lattice. The fields in z 0 are evaluated by a superposition of tilted and shifted beams that originate from the aperture distribution plane over the discrete frame spatial–directional lattice in Eq. (12). Each beam propagator emanates from a lattice point ( x ¯ , y ¯ ) = ( N x Δ x ¯ , N y Δ y ¯ ) and in a direction of ( θ ¯ x , θ ¯ y ) = cos 1 [ ( k ¯ x , k ¯ y ) / k ] with respect to the corresponding axis.

Fig. 2
Fig. 2

Local beam coordinates. The asymptotic Gaussian beam propagator is described in terms of the local beam-coordinates, r b = ( x b , y b , z b ) , which are defined in Eq. (42). The beam propagates along the z b axis and exhibits a Gaussian decay in the transverse coordinates x b and y b .

Fig. 3
Fig. 3

Coefficients map for u 0 ( r t ) over the ( k ¯ x , x ¯ ) plane.

Fig. 4
Fig. 4

Real part of the (a) reference and (b) reconstructed scalar field over the z = 7 λ plane.

Fig. 5
Fig. 5

Error of reconstructed field over the z = 7 λ plane with respect to the reference field in decibels from the maximum value of the reference field for (a) the real part and (b) the imaginary part. The errors do not exceed 62 dB error.

Fig. 6
Fig. 6

Coefficients map for E 0 ( r t ) over the ( k ¯ x , x ¯ ) plane for (a) TE coefficients and (b) TM coefficients.

Fig. 7
Fig. 7

Error of reconstructed E x field over the z = 7 λ plane with respect to the reference field in decibels from the maximum value of the reference field for (a) the real part and (b) the imaginary part. The errors do not exceed 50 dB error.

Equations (65)

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E 0 ( r t ) = E x ( r t ) x ^ + E y ( r t ) y ^ ,
E ˜ 0 ( k t ) = E ˜ x ( k t ) x ^ + E ˜ y ( k t ) y ^ ,
E ˜ 0 ( k t ) = d 2 r t E 0 ( r t ) exp ( j k t · r t ) ,
E ˜ z ( k t ) = ( k x E ˜ x + k y E ˜ y ) / k z ,
E ( r , t ) = 1 ( 2 π ) 2 d 2 k t E ˜ ( k t ) exp ( j k · r ) ,
E ˜ ( k t ) = E ˜ 0 ( k t ) + z ^ E ˜ z ( k t ) ,
n ^ ( k t ) = 1 k t ( k y x ^ k x y ^ ) , t ^ ( k t ) = k z k k t ( k x x ^ + k y y ^ ) k t k z ^ ,
E ˜ ( k t ) = E ˜ TE ( k t ) n ^ ( k t ) + E ˜ TM ( k t ) t ^ ( k t ) ,
E ˜ TE ( k t ) = E ˜ ( k t ) · n ^ ( k t ) = 1 k t ( k y E ˜ x k x E ˜ y ) , E ˜ TM ( k t ) = E ˜ ( k t ) · t ^ ( k t ) = k k z k t ( k x E ˜ x + k y E ˜ y ) .
E ( r , t ) = E TE ( r , t ) + E TM ( r , t ) ,
E TE ( r , t ) = 1 ( 2 π ) 2 d 2 k t E ˜ TE ( k t ) n ^ ( k t ) exp ( j k · r ) , E TM ( r , t ) = 1 ( 2 π ) 2 d 2 k t E ˜ TM ( k t ) t ^ ( k t ) exp ( j k · r ) .
( x ¯ , y ¯ , k ¯ x , k ¯ y ) = ( N x Δ x ¯ , N y Δ y ¯ , N k x Δ k ¯ x , N k y Δ k ¯ y ) ,
Δ x ¯ Δ k ¯ x = 2 π ν x , Δ y ¯ Δ k ¯ y = 2 π ν y ,
u 0 ( r t ) = N a N ψ N ( r t ) ,
ψ N ( r t ) = ψ ( r t r ¯ t ) exp [ j k ¯ t · ( r t r ¯ t ) ] ,
a N = d 2 r t u 0 ( r t ) φ N * ( r t ) ,
φ N ( r t ) = φ ( r t r ¯ t ) exp [ j k ¯ t · ( r t r ¯ t ) ] .
u ( r , t ) = N a N P N ( r , t ) ,
( 2 + k 2 ) P N ( r , t ) = 0 .
P N ( r , t ) = 1 ( 2 π ) 2 d 2 k t ψ ˜ N ( k t ) exp ( j k · r ) ,
ψ ˜ N ( k t ) = ψ ˜ ( k t k ¯ t ) exp ( j k t · r ¯ t ) ,
u ˜ 0 ( k t ) = d 2 r t u 0 ( r t ) exp ( j k t · r t ) .
u ˜ 0 ( k t ) = N a N ψ ˜ N ( k t ) ,
a N = 1 ( 2 π ) 2 d 2 k t u ˜ 0 ( k t ) φ ˜ N * ( k t ) ,
φ ˜ N ( k t ) = φ ˜ ( k t k ¯ t ) exp ( j k t · r ¯ t ) ,
a N TE = 1 ( 2 π ) 2 d 2 k t k t 1 E ˜ TE ( k t ) φ ˜ N * ( k t ) , a N TM = 1 ( 2 π ) 2 d 2 k t k t 1 E ˜ TM ( k t ) φ ˜ N * ( k t ) ,
E ˜ TE ( k t ) = N a N TE k t ψ ˜ N ( k t ) , E ˜ TM ( k t ) = N a N TM k t ψ ˜ N ( k t ) ,
E TE ( r , t ) = N a N TE E N TE ( r , t ) , E TM ( r , t ) = N a N TM E N TM ( r , t ) ,
E N TE ( r , t ) = 1 ( 2 π ) 2 d 2 k t k t n ^ ( k t ) ψ ˜ N ( k t ) exp ( j k · r ) , E N TM ( r , t ) = 1 ( 2 π ) 2 d 2 k t k t t ^ ( k t ) ψ ˜ N ( k t ) exp ( j k · r ) ,
κ ¯ ^ = ( k ¯ t , k ¯ z ) / k , k ¯ z = k 2 k ¯ t 2 ,
E N TE ( r , t ) = 1 ( 2 π ) 2 d 2 k t ( k y x ^ k x y ^ ) ψ ˜ N ( k t ) exp ( j k · r ) , E N TM ( r , t ) = 1 k ( 2 π ) 2 d 2 k t [ k z ( k x x ^ + k y y ^ ) k t 2 z ^ ] ψ ˜ N ( k t ) exp ( j k · r ) .
E N TE ( r , t ) = j ( x ^ y y ^ x ) P N ( r , t ) , E N TM ( r , t ) = k 1 ( x ^ x z + y ^ y z z ^ t 2 ) P N ( r , t ) ,
H ( r , t ) = H TE ( r , t ) + H TM ( r , t ) ,
H TE ( r , t ) = N a N TE H N TE ( r , t ) , H TM ( r , t ) = N a N TM H N TM ( r , t ) ,
H N TE ( r , t ) = 1 k η 0 ( x ^ x z + y ^ y z z ^ t 2 ) P N ( r , t ) , H N TM ( r , t ) = 1 j η 0 ( x ^ y y ^ x ) P N ( r , t ) ,
H N TE ( r , t ) = 1 η 0 1 ( 2 π ) 2 d 2 k t k t t ^ ( k t ) ψ ˜ N ( k t ) exp ( j k · r ) , H N TM ( r , t ) = 1 η 0 1 ( 2 π ) 2 d 2 k t k t n ^ ( k t ) ψ ˜ N ( k t ) exp ( j k · r ) ,
ψ ( r t ) = exp ( j k Γ r t 2 / 2 ) ,
ψ ˜ ( k t ) = 2 π j ( k Γ ) 1 exp [ j k t 2 / ( 2 k Γ ) ] .
φ ( r t ) ν x ν y ψ 2 ψ ( r t ) ,
φ ( r t ) = ( ν 2 k Γ j / π ) exp ( j k Γ r t 2 / 2 ) ,
φ ˜ ( k t ) = j 2 ν 2 Γ j ( Γ ) 1 exp [ j k t 2 / ( 2 k Γ ) ] ,
[ x b y b z b ] = [ cos θ ¯ cos ϕ ¯ cos θ ¯ sin ϕ ¯ sin θ ¯ sin ϕ ¯ cos ϕ ¯ 0 sin θ ¯ cos ϕ ¯ sin θ ¯ sin ϕ ¯ cos θ ¯ ] [ x x ¯ x y x ¯ y z ] ,
cos θ ¯ = k ¯ z / k , cos ϕ ¯ = k ¯ x / k ¯ t , sin ϕ ¯ = k ¯ y / k ¯ t .
r t r ¯ t = z tan θ ¯ ( cos ϕ ¯ x ^ + sin ϕ ¯ y ^ ) ,
P N ( r , t ) Γ x ( z b ) Γ x ( 0 ) Γ y ( z b ) Γ y ( 0 ) exp [ j k Ψ ( r b ) ] , Ψ ( r b ) = z b + 1 2 [ Γ x ( z b ) x b 2 + Γ y ( z b ) y b 2 ] ,
Γ x ( z b ) = 1 / ( z b + cos 2 θ ¯ Γ 1 ) , Γ y ( z b ) = 1 / ( z b + Γ 1 ) ,
Γ 1 = Z + j F .
Γ x , y ( z b ) = 1 / ( z b Z x , y + j F x , y ) ,
Z x = Z cos 2 θ ¯ , Z y = Z , F x = F cos 2 θ ¯ , F y = F .
W x , y ( z b ) = D x , y 1 + ( z b Z x , y ) 2 F x , y 2 ,
R x , y = ( z b Z x , y ) + F x , y 2 / ( z b Z x , y ) .
E N TE ( r , t ) k ¯ t n ^ ( k ¯ t ) P N ( r , t ) ,
H N TE ( r , t ) 1 η 0 k ¯ t t ^ ( k ¯ t ) P N ( r , t ) .
E N TM ( r , t ) k ¯ t t ^ ( k ¯ t ) P N ( r , t ) , H N TM ( r , t ) η 0 1 k ¯ t n ^ ( k ¯ t ) P N ( r , t ) .
u ( r ) = A k R exp ( j k R ) , R = ( x x ) 2 + ( y y ) 2 + ( z z ) 2 ,
P N ( r , t ) = ( 2 π j k Γ ) 1 d 2 k t exp [ j q ( k t ) ] , q ( k t ) = k t T ( r t r ¯ t ) + k z z ( 2 k Γ ) 1 ( k t k ¯ t ) T ( k t k ¯ t ) ,
k t q = r t r ¯ t k t s z / k z s ( k Γ ) 1 ( k t s k ¯ t ) = 0 ,
q ( k t ) = q o + q 1 T ( k t k ¯ t ) + 1 2 ( k t k ¯ t ) T q 2 ( k t k ¯ t ) ,
q o = k ¯ t T ( r t r ¯ t ) + k ¯ z z , q 1 = r t r t z k ¯ t / k z ,
q 2 = [ 1 k Γ + z ( k ¯ x 2 + k ¯ z 2 ) k ¯ z 3 z k ¯ x k ¯ y k ¯ z 3 z k ¯ x k ¯ y k ¯ z 3 1 k Γ + z ( k ¯ y 2 + k ¯ z 2 ) k ¯ z 3 ] .
k t s = k ¯ t q 2 1 q 1 ,
P N ( r , t ) ( k Γ ) 1 det q 2 exp [ j ( q o 1 2 q 1 T q 2 1 q 1 ) ] .
q 1 = [ cos ϕ ¯ / cos θ ¯ sin ϕ ¯ sin ϕ ¯ / cos θ ¯ cos ϕ ¯ ] [ x b y b ] ,
q 1 T q 2 1 q 1 = x b 2 Γ x ( z ) + y b 2 Γ y ( z ) ,
Γ x ( z ) = ( z / cos θ ¯ + cos 2 θ ¯ Γ 1 ) 1 , Γ y ( z ) = ( z / cos θ ¯ + Γ 1 ) 1 .

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