Abstract

The present work is concerned with applying a ray-centered non-orthogonal coordinate system which is a priori matched to linearly-phased localized aperture field distributions. The resulting beam-waveobjects serve as the building blocks for beam-type spectral expansions of aperture fields in 2D inhomogeneous media that are characterized by a generic wave-velocity profile. By applying a rigorous paraxial-asymptotic analysis, a novel parabolic wave equation is obtained and termed “Non-orthogonal domain parabolic equation”—NoDope. Tilted Gaussian beams, which are exact solutions to this equation, match Gaussian aperture distributions over a plane that is tilted with respect to the beam-axes initial directions. A numerical example, which demonstrates the enhanced accuracy of the tilted Gaussian beams over the conventional ones, is presented as well.

© 2010 Optical Society of America

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  1. V. Cerveny, M. M. Popov, and I. Pŝencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
    [CrossRef]
  2. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [CrossRef]
  3. T. Melamed, “Phase-space Green’s functions for modeling time-harmonic scattering from smooth inhomogeneous objects,” J. Math. Phys. 46, 2232–2246 (2004).
    [CrossRef]
  4. S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
    [CrossRef]
  5. R. Simon and N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–70 (1998).
    [CrossRef]
  6. E. Poli, G. V. Pereverzev, and A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
    [CrossRef]
  7. L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).
  8. I. Tinkelman and T. Melamed, “Gaussian beam propagation in generic anisotropic wavenumber profiles,” Opt. Lett. 28, 1081–1083 (2003).
    [CrossRef] [PubMed]
  9. I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in anisotropic media. I. Time-harmonic fields,” J. Opt. Soc. Am. A 22, 1200–1207 (2005).
    [CrossRef]
  10. I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in anisotropic media. II. Time-dependent fields,” J. Opt. Soc. Am. A 22, 1208–1215 (2005).
    [CrossRef]
  11. A. G. Khatkevich, “Propagation of pulses and wave packets in dispersive gyrotropic crystals,” J. Appl. Spectrosc. 46, 203–207 (1987).
    [CrossRef]
  12. T. Melamed and L. B. Felsen, “Pulsed beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
    [CrossRef]
  13. T. Melamed and L. B. Felsen, “Pulsed beam propagation in lossless dispersive media. 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. A. P. Kiselev, “Localized light waves: paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
    [CrossRef]
  16. 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]
  17. 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]
  18. T. Melamed, “Exact beam decomposition of time-harmonic electromagnetic waves,” J. Electromagn. Waves Appl. 23, 975–986 (2009).
  19. B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
    [CrossRef]
  20. B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase space beam summation for time dependent radiation from large apertures: Continuous parametrization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
    [CrossRef]
  21. T. Melamed, “Phase-space beam summation: A local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
    [CrossRef]
  22. M. A. Leontovich and V. A. Fock, “Solution of the problem of EM wave propagation along the earth surface by the parabolic equation method,” J. Phys. 10, 13 (1946).
  23. G. D. Malyuzhinets, “Progress in understanding diffraction phenomena (in Russian),” Sov. Phys. Usp. 69, 321–334 (1959).
  24. A. V. Popov, “Numerical solution of the wedge diffraction problem by the transversal diffusion method (in Russian),” Sov. Phys. Acoust. 15, 226–233 (1969).
  25. S. N. Vlasov and V. I. Talanov, “The parabolic equation in the theory of wave propagation,” Radiophys. Quantum Electron. 38, 1–12 (1995).
    [CrossRef]
  26. M. Levys, Parabolic Equation Methods for Electromagnetic Wave Propagation (The Institution of Electrical Engineers, 2000).
    [CrossRef]
  27. C. Chapman, Fundamentals of Seismic Wave Propagation (Cambridge Univ. Press, 2004).
    [CrossRef]
  28. J. A. Arnaud and H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
    [CrossRef] [PubMed]
  29. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  30. S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Lasers Eng. 47, 604–606 (2009).
    [CrossRef]
  31. Y. Hadad and T. Melamed, “Non-orthogonal domain parabolic equation and its Gaussian beam solutions,” IEEE Trans. Antennas Propag. 58, 1164–1172 (2010).
    [CrossRef]
  32. Y. Hadad and T. Melamed, “Parameterization of the tilted Gaussian beam waveobjects,” PIER 102, 65–80 (2010).
    [CrossRef]
  33. V. M. Babič and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer-Verlag, 1991).
  34. E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
    [CrossRef]
  35. E. Heyman and T. Melamed, “Space-time representation of ultra wideband signals,” in Advances in Imaging and Electron Physics, Vol. 103 (Elsevier, 1998), pp.1–63.
    [CrossRef]
  36. 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]
  37. A. P. Wills, Vector Analysis with an Introduction to Tensor Analysis (Dover, 1958).
  38. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
  39. E. Heyman and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
    [CrossRef]

2010 (2)

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,” PIER 102, 65–80 (2010).
[CrossRef]

2009 (2)

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Lasers Eng. 47, 604–606 (2009).
[CrossRef]

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

2007 (1)

A. P. Kiselev, “Localized light waves: paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
[CrossRef]

2005 (4)

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]

I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in anisotropic media. 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 anisotropic media. 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]

2004 (2)

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. 46, 2232–2246 (2004).
[CrossRef]

2003 (1)

2000 (2)

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

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]

1999 (1)

E. Poli, G. V. Pereverzev, and A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

1998 (3)

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]

1995 (1)

S. N. Vlasov and V. I. Talanov, “The parabolic equation in the theory of wave propagation,” Radiophys. Quantum Electron. 38, 1–12 (1995).
[CrossRef]

1994 (2)

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

1987 (2)

E. Heyman and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
[CrossRef]

A. G. Khatkevich, “Propagation of pulses and wave packets in dispersive gyrotropic crystals,” J. Appl. Spectrosc. 46, 203–207 (1987).
[CrossRef]

1982 (1)

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

1974 (1)

S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

1971 (1)

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

1969 (2)

A. V. Popov, “Numerical solution of the wedge diffraction problem by the transversal diffusion method (in Russian),” Sov. Phys. Acoust. 15, 226–233 (1969).

J. A. Arnaud and H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

1959 (1)

G. D. Malyuzhinets, “Progress in understanding diffraction phenomena (in Russian),” Sov. Phys. Usp. 69, 321–334 (1959).

1946 (1)

M. A. Leontovich and V. A. Fock, “Solution of the problem of EM wave propagation along the earth surface by the parabolic equation method,” J. Phys. 10, 13 (1946).

Arnaud, J. A.

Babic, V. M.

V. M. Babič and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer-Verlag, 1991).

Boag, A.

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]

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]

Buldyrev, V. S.

V. M. Babič and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer-Verlag, 1991).

Cerveny, V.

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

Chapman, C.

C. Chapman, Fundamentals of Seismic Wave Propagation (Cambridge Univ. Press, 2004).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

Deschamps, G. A.

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

Dongare, M. B.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Lasers Eng. 47, 604–606 (2009).
[CrossRef]

Felsen, L. B.

Fock, V. A.

M. A. Leontovich and V. A. Fock, “Solution of the problem of EM wave propagation along the earth surface by the parabolic equation method,” J. Phys. 10, 13 (1946).

Garea, M. T.

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

Hadad, Y.

Y. Hadad and T. Melamed, “Parameterization of the tilted Gaussian beam waveobjects,” 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]

Heyman, E.

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]

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. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase space beam summation for time dependent radiation from large apertures: Continuous parametrization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
[CrossRef]

B. Z. Steinberg, E. Heyman, and L. B. 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 L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
[CrossRef]

E. Heyman and T. Melamed, “Space-time representation of ultra wideband signals,” in Advances in Imaging and Electron Physics, Vol. 103 (Elsevier, 1998), pp.1–63.
[CrossRef]

Khatkevich, A. G.

A. G. Khatkevich, “Propagation of pulses and wave packets in dispersive gyrotropic crystals,” J. Appl. Spectrosc. 46, 203–207 (1987).
[CrossRef]

Kiselev, A. P.

A. P. Kiselev, “Localized light waves: paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
[CrossRef]

Kogelnik, H.

Leontovich, M. A.

M. A. Leontovich and V. A. Fock, “Solution of the problem of EM wave propagation along the earth surface by the parabolic equation method,” J. Phys. 10, 13 (1946).

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]

Levys, M.

M. Levys, Parabolic Equation Methods for Electromagnetic Wave Propagation (The Institution of Electrical Engineers, 2000).
[CrossRef]

Malyuzhinets, G. D.

G. D. Malyuzhinets, “Progress in understanding diffraction phenomena (in Russian),” Sov. Phys. Usp. 69, 321–334 (1959).

Melamed, T.

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,” PIER 102, 65–80 (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 anisotropic media. 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 anisotropic media. II. Time-dependent fields,” J. Opt. Soc. Am. A 22, 1208–1215 (2005).
[CrossRef]

T. Melamed, “Phase-space Green’s functions for modeling time-harmonic scattering from smooth inhomogeneous objects,” J. Math. Phys. 46, 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. B. Felsen, “Pulsed beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
[CrossRef]

T. Melamed and L. B. Felsen, “Pulsed beam propagation in lossless dispersive media. 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]

E. Heyman and T. Melamed, “Space-time representation of ultra wideband signals,” in Advances in Imaging and Electron Physics, Vol. 103 (Elsevier, 1998), pp.1–63.
[CrossRef]

Mukunda, N.

Navare, S. T.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Lasers Eng. 47, 604–606 (2009).
[CrossRef]

Patil, S. D.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Lasers Eng. 47, 604–606 (2009).
[CrossRef]

Peeters, A. G.

E. Poli, G. V. Pereverzev, and A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

Pereverzev, G. V.

E. Poli, G. V. Pereverzev, and A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

Perez, L. I.

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

Poli, E.

E. Poli, G. V. Pereverzev, and A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

Popov, A. V.

A. V. Popov, “Numerical solution of the wedge diffraction problem by the transversal diffusion method (in Russian),” Sov. Phys. Acoust. 15, 226–233 (1969).

Popov, M. M.

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

Psencik, I.

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

Shin, S. Y.

S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

Shlivinski, A.

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]

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]

Simon, R.

Steinberg, B. Z.

Takale, M. V.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Lasers Eng. 47, 604–606 (2009).
[CrossRef]

Talanov, V. I.

S. N. Vlasov and V. I. Talanov, “The parabolic equation in the theory of wave propagation,” Radiophys. Quantum Electron. 38, 1–12 (1995).
[CrossRef]

Tinkelman, I.

Vlasov, S. N.

S. N. Vlasov and V. I. Talanov, “The parabolic equation in the theory of wave propagation,” Radiophys. Quantum Electron. 38, 1–12 (1995).
[CrossRef]

Wills, A. P.

A. P. Wills, Vector Analysis with an Introduction to Tensor Analysis (Dover, 1958).

Appl. Opt. (1)

Appl. Phys. (1)

S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

Electron. Lett. (1)

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

Geophys. J. R. Astron. Soc. (2)

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

E. Heyman and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
[CrossRef]

IEEE Trans. Antennas Propag. (7)

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]

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]

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]

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

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

J. Appl. Spectrosc. (1)

A. G. Khatkevich, “Propagation of pulses and wave packets in dispersive gyrotropic crystals,” J. Appl. Spectrosc. 46, 203–207 (1987).
[CrossRef]

J. Electromagn. Waves Appl. (2)

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

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

J. Math. Phys. (1)

T. Melamed, “Phase-space Green’s functions for modeling time-harmonic scattering from smooth inhomogeneous objects,” J. Math. Phys. 46, 2232–2246 (2004).
[CrossRef]

J. Opt. Soc. Am. A (7)

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Figures (5)

Fig. 1
Fig. 1

Tilted Gaussian beam waveobject is propagating along a ray trajectory (beam-axis) in an inhomogeneous medium that is characterized by a generic wave-velocity profile V ( x , z ) . This waveobject carries Gaussian distributions over transverse lines that are tilted by angle ϑ with respect to the beam-axis.

Fig. 2
Fig. 2

Observation point r is described by a non-orthogonal coordinate system r = ( x b , z b ) , where x b is the distance along the axis that is tilted by ϑ with respect to the ray trajectory tangent, t ̂ o , and z b is the arclength along the trajectory up to the intersection point of the ray with the x b axis. The (positive) x ̂ b and n ̂ o directions remain constant with respect to the trajectory tangent, and therefore, the curvature K o ( z b ) is negative or positive in convex (left) or concave (right) regions, respectively.

Fig. 3
Fig. 3

Δ s approximation. The difference between the Eikonal s and the beam local coordinate z b , Δ s = s z b , is expressed in terms of on-axis point r o using a Taylor series. Unit-vectors n ̂ o and t ̂ o denote the normal and tangent to the trajectory at r o , respectively, and unit-vectors n ̂ ( Δ s ) and t ̂ ( Δ s ) denote the normal and tangent to the trajectory at point s.

Fig. 4
Fig. 4

Tilted GB (solid curve), the conventional GB (dashed curve), and the reference field (light gray) curves that are sampled over a line perpendicular to the beam-axis. (a) Absolute value, (b) phase in radians. The inhomogeneous parameters are V 0 = 1 , α = 0.01 , and the fields parameters are ω = 5000 , Γ 0 = i 3 , ϑ = 15 o , and s m = 32 λ 0 .

Fig. 5
Fig. 5

L 2 error norms of the tilted and approximated GBs in percent as a function of the normalized on-axis arclength s m F h are plotted for three different values of ϑ. The curves are arranged in pairs with continuous and dashed lines corresponding to errors of the tilted GB and the approximated one, respectively. The medium and field parameters are as in Fig. 4. The figure demonstrates the enhanced accuracy of tilted GBs within the collimated beam domain s m < 0.7 F h .

Equations (107)

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u 0 ( x ) = exp [ i ω ( V 0 1 x cos ϑ + 1 2 x 2 Γ 0 ) ] ,
[ 2 + ω 2 V 2 ( x , z ) ] u ( x , z ) = 0 , 2 = 2 z 2 + 2 x 2 ,
d r o d s = t ̂ o , d t ̂ o d s = K o n ̂ o , d n ̂ o d s = K o t ̂ o ,
x ̂ b = cos ϑ t ̂ o + sin ϑ n ̂ o , z ̂ b = t ̂ o .
r = r o ( z b ) + x b x ̂ b ,
u ( x b , z b ) = U ( x ¯ b , z b ) exp [ i Ψ ( x ¯ b , z b ) ] ,
x ¯ b = x b ω
Ψ ( x ¯ b , z b ) = ω 0 s ( x ¯ b , z b ) v 1 ( σ ) d σ
csc ϑ v 3 ( z b ) U x ¯ b x ¯ b + 2 i sin ϑ v 2 ( z b ) U z b [ x ¯ b 2 sin 3 ϑ v n n ( z b ) + i sin ϑ v ( z b ) v ( z b ) ] U = 0 ,
U ( x ¯ b , 0 ) = | u ( x , 0 ) | x = x ¯ b ω exp [ i Ψ ( x ¯ b , 0 ) ] .
U ( x ¯ b , z b ) = A ( z b ) exp [ i x ¯ b 2 Γ ( z b ) 2 ] ,
Γ ( z b ) = sin 2 ϑ p ( z b ) q ( z b )
d d z b [ q p ] = [ 0 v ( z b ) v n n ( z b ) v 2 ( z b ) 0 ] [ q p ] .
q ( 0 ) = 1 , p ( 0 ) = Γ 0 sin 2 ϑ + V 0 V 0 2 cot 2 ϑ 2 K 0 cot ϑ V 0 ,
A ( z b ) = q ( 0 ) q ( z b ) v ( z b ) v ( 0 ) .
u ( x b , z b ) = q ( 0 ) q ( z b ) v ( z b ) v ( 0 ) exp [ i ω ( 0 s ( x b , z b ) d σ v ( σ ) + 1 2 x b 2 Γ ( z b ) ) ] .
u ( x b , z b ) = Γ ( z b ) Γ 0 exp [ i ω ( v 1 z b + 1 2 x b 2 Γ ( z b ) ) ] ,
u ( s , n ) = q ( 0 ) q ( s ) v ( s ) v ( 0 ) exp [ i ω ( 0 s 1 v ( σ ) d σ + 1 2 n 2 Γ ( s ) ) ] .
W ( z b ) = 8 ω Im Γ ( z b ) ,
Γ ( z b ) Γ ( s ) Γ ( z b ) Δ s , Δ s = s z b .
Re [ 1 2 x b 2 Γ ( z b ) ] 1 2 n 2 Re [ Γ ( s ) ] sin 2 ϑ .
d r = r z b d z b + r x b d x b .
a 1 = r z b = t ̂ o + x b K o ( cos ϑ n ̂ o sin ϑ t ̂ o ) ,
a 2 = r x b = sin ϑ n ̂ o + cos ϑ t ̂ o .
G ( x b , z b ) = [ 1 2 x b sin ϑ K o + x b 2 K o 2 cos ϑ cos ϑ 1 ] .
G 1 ( x b , z b ) = [ g z b z b g z b x b g x b z b g x b x b ] = 1 h 2 [ 1 cos ϑ cos ϑ 1 2 x b sin ϑ K o + x b 2 K o 2 ] ,
h ( x b , z b ) = det ( G ) = sin ϑ x b K o ( z b ) .
2 = 1 h i = 1 2 j = 1 2 x i [ g i j h u x j ] ,
k = 1 6 T k ( x b , z b ) = 0 ,
T 1 ( x b , z b ) = M 1 ( x b , z b ) u z b ( x b , z b ) ,
T 2 ( x b , z b ) = M 2 ( x b , z b ) u x b ( x b , z b ) ,
T 3 ( x b , z b ) = M 3 ( x b , z b ) u z b x b ( x b , z b ) ,
T 4 ( x b , z b ) = M 4 ( x b , z b ) u z b z b ( x b , z b ) ,
T 5 ( x b , z b ) = M 5 ( x b , z b ) u x b x b ( x b , z b ) ,
T 6 ( x b , z b ) = M 6 ( x b , z b ) u ( x b , z b ) ,
M 1 ( x b , z b ) = ( h g z b z b ) z b + ( h g x b z b ) x b ,
M 2 ( x b , z b ) = ( h g x b x b ) x b + ( h g z b x b ) z b ,
M 3 ( x b , z b ) = h ( g z b x b + g x b z b ) ,
M 4 ( x b , z b ) = h g z b z b ,
M 5 ( x b , z b ) = h g x b x b ,
M 6 ( x b , z b ) = h ω 2 V 2 ( x b , z b ) ,
Δ r ( Δ s ) | d r d s | r o Δ s + 1 2 | d 2 r d s 2 | r o Δ s 2 .
| d 2 r d s 2 | r o = | d t ̂ d s | r o = K o n ̂ o ,
Δ r ( Δ s ) = Δ s t ̂ o + K o Δ s 2 n ̂ o 2 .
t ̂ ( Δ s ) = d r d s = d Δ r ( Δ s ) d Δ s = t ̂ o + K o Δ s n ̂ o .
n ̂ ( Δ s ) = n ̂ o K o Δ s t ̂ o .
x b x ̂ b = t ̂ o Δ s ( 1 K o n ) + n ̂ o ( K o Δ s 2 2 + n ) .
Δ s = x b cos ϑ ( 1 + K o x b sin ϑ ) + O ( ω 3 2 ) .
s x ¯ b = ω 1 2 cos ϑ + K o ω 1 x ¯ b sin 2 ϑ + C 3 2 ω 3 2 ,
s z b = 1 + 1 2 K o x ¯ b 2 ω 1 sin 2 ϑ ,
v [ s ( x b , z b ) ] = v o + | d v d s | s = z b Δ s + | d 2 v d s 2 | s = z b Δ s 2 + O ( Δ s 3 ) .
v app ( x ¯ b , z b ) = v o + ω 1 2 x ¯ b v o cos ϑ + ω 1 x ¯ b 2 [ v o K o sin 2 ϑ + v o cos 2 ϑ ] 2 .
u z b = U z b ( x ¯ b , z b ) exp [ i Ψ ( x ¯ b , z b ) ] ,
U z b = i ω v 1 ( z b ) U ( x ¯ b , z b ) + O ( ω 0 ) .
U x b = i ω cos ϑ v o U ( x ¯ b , z b ) + O ( ω 1 2 ) ,
U z b z b = [ ω 2 v app 2 + ω ( K o x ¯ b 2 sin 2 ϑ + i v o ) v o 2 ] U + 2 i ω v ( z b ) U z b + O ( ω 1 2 ) ,
U x b x b = [ ( s x ¯ b ) 2 ω 3 v app 2 + i ω ( v o cos 2 ϑ K o v o sin 2 ϑ ) v o 2 ] U + [ 2 i ω K o x ¯ b sin 2 ϑ v o + 2 i ω 3 2 cos ϑ v app ] U x ¯ b + ω U x ¯ b x ¯ b + O ( ω 1 2 ) ,
U z b x b = [ ω 5 2 v app 2 s z b s x ¯ b + i ω v o cos ϑ v o 2 ] U + i ω cos ϑ v o U z b + [ i ω 3 2 v app i ω v o x ¯ b cos ϑ v o 2 ] U x ¯ b + O ( ω 1 2 ) ,
M 1 = cot ϑ csc ϑ K o + O ( ω 1 2 ) ,
M 2 = ( csc 2 ϑ 2 ) K o + O ( ω 1 2 ) ,
M 3 = 2 cot ϑ ( 1 + csc ϑ K o x ¯ b ω 1 2 + csc 2 ϑ K o 2 x ¯ b 2 ω 1 ) + O ( ω 3 2 ) ,
M 4 = csc ϑ ( 1 + csc ϑ K o x ¯ b ω 1 2 + csc 2 ϑ K o 2 x ¯ b 2 ω 1 ) + O ( ω 3 2 ) ,
M 5 = csc ϑ + ( csc 2 ϑ 2 ) K o x ¯ b ω 1 2 + cot 2 ϑ csc ϑ K o 2 x ¯ b 2 ω 1 + O ( ω 3 2 ) .
T ¯ 1 T 1 exp [ i Ψ ] = M 1 ( x ¯ b , z b ) U z b ,
k = 1 6 T ¯ k ( x b , z b ) = 0 .
T ¯ 1 = i ω cot ϑ csc ϑ K o v o 1 U + O ( ω 1 2 ) .
V 2 = 1 v o 2 ω 1 2 2 x ¯ b v o 3 v x b ω 1 x ¯ b 2 ( v x b x b v o 3 3 v x b 2 v o 4 ) + O ( ω 3 2 ) ,
v x b | V x b | r o = sin ϑ v o K o + cos ϑ v o ,
v x b x b o = sin 2 ϑ v n n + cos 2 ϑ ( v o + K o 2 v o ) sin 2 ϑ v o K o .
T ¯ 6 = v o 3 { ω 2 sin ϑ v o ω 3 2 x ¯ b [ cos 2 ϑ K o v o + sin 2 ϑ v o ] ω x ¯ b 2 [ sin 3 ϑ v n n + sin 3 ϑ K o 2 v o 1 2 ( cos ϑ + 3 cos 3 ϑ ) K o v o 2 cos ϑ sin 2 ϑ K o v o 3 cos 2 ϑ sin ϑ v o 2 v o 1 + cos 2 ϑ sin ϑ v o ] } U + O ( ω 1 2 ) .
2 d A d z b + A ( z b ) [ csc 2 ϑ v ( z b ) Γ ( z b ) v ( z b ) v ( z b ) ] = 0 ,
Γ ( z b ) + csc 2 ϑ v ( z b ) Γ 2 ( z b ) + sin 2 ϑ v 2 ( z b ) v n n ( z b ) = 0 .
Γ ( z b ) = sin 2 ϑ q ( q v ) ,
v ( z b ) q ( z b ) v ( z b ) q ( z b ) + v n n ( z b ) q ( z b ) = 0 .
v 1 ( σ ) V 0 1 V 0 V 0 2 σ ,
0 s ( x ) v 1 ( σ ) d σ s ( x ) V 0 s 2 ( x ) V 0 2 V 0 2 .
u ( x , 0 ) = A ( 0 ) exp { i ω [ x cos ϑ V 0 + 1 2 x 2 ( Γ ( 0 ) + K 0 sin 2 ϑ V 0 V 0 cos 2 ϑ V 0 2 ) ] } .
Γ ( 0 ) = Γ 0 K 0 V 0 1 sin 2 ϑ + V 0 V 0 2 cos 2 ϑ .
V ( z ) = V 0 + α z , z 0 ,
( x c , z c ) = ( tan ϑ V 0 α , V 0 α ) , R o = V 0 α cos ϑ ,
z t = R o ( 1 sec ϑ ) .
x o = x c [ R o 2 ( z o z c ) 2 ] 1 2 ,
σ = R o [ sin 1 ( cos ϑ + z o R o ) + ϑ π 2 ] .
x n o = R o ( x x c d + sin ϑ ) , z n o = R o ( z z c d cos ϑ ) ,
d ( x , z ) = ( x x c ) 2 + ( z z c ) 2 .
n = R o d , s = R o [ sin 1 ( x x c d ) + ϑ ] .
v ( σ ) = V 0 cos ( ϑ α σ cos ϑ V 0 ) cos ϑ ,
p ( σ ) = p ( 0 ) ,
q ( σ ) = 1 + V 0 2 α p ( 0 ) cos 2 ϑ [ sin ( ϑ cos ϑ α σ V 0 ) sin ϑ ] ,
x b = ± R o [ ( 1 n R o ) 2 cos 2 ϑ sin ϑ ] , d R o ,
z b = s R o sin 1 [ x b cos ϑ ( R o n ) ] .
u ref ( x , z ) = 1 V 0 i ω 2 π Γ 0 d ξ C ( ξ ) ( 1 ξ 2 ζ 2 ( z , ξ ) ) 1 4 exp [ i ω Ψ ̃ ( x , z ; ξ ) ] ,
Ψ ̃ ( x , z ; ξ ) = [ ( cos ϑ ξ ) 2 2 V 0 2 Γ 0 + V 0 1 0 z ζ ( z , ξ ) d z + V 0 1 ξ x ] ,
C ( ξ ) = 1 i exp { 2 i ω V 0 [ 0 z t ( ξ ) ζ ( z , ξ ) d z 0 z ζ ( z , ξ ) d z ] } .
L 2 [ u , u ref ] = 1 L L 2 L 2 | u ( s m , n ) u ref ( s m , n ) | 2 d n ,
T ¯ 2 = i ω cot ϑ cos 2 ϑ csc ϑ K v o U + O ( ω 1 2 ) ,
T ¯ 3 = { ω 2 2 cos ϑ cot ϑ v o 2 + ω 3 2 x ¯ b cot ϑ ( 2 ( cot ϑ + sin 2 ϑ ) K v o 2 4 cos 2 ϑ v o v o 3 ) + ω [ 6 x ¯ b 2 cos 3 ϑ cot ϑ v o 2 v o 4 + 2 C 3 2 cot ϑ v o 2 + 2 x ¯ b 2 ( 2 cos ϑ cot ϑ + cot 2 ϑ csc ϑ ) K 2 v o 2 + 2 x ¯ b 2 cos 3 ϑ K v o 2 + 2 i cot ϑ cos ϑ v o v o 2 2 x ¯ b 2 cos ϑ cot ϑ ( ( 5 3 cos 2 ϑ ) cot ϑ K v o v o 3 + cos 2 ϑ v o v o 3 ) ] } U [ i ω 3 2 2 cot ϑ v o + i ω x ¯ b ( 2 cot ϑ csc ϑ K v o 2 cos ϑ cot ϑ v o v o 2 ) ] U x ¯ b i ω 2 cot ϑ cos ϑ v o U z b + O ( ω 1 2 ) ,
T ¯ 4 = [ ω 2 csc ϑ v o 2 + ω 3 2 x ¯ b ( 2 cot ϑ v o v o 3 csc 2 ϑ K v o 2 ) ω csc ϑ ( x ¯ b 2 csc 2 ϑ K 2 v o 2 + 1 2 x ¯ b 2 ( cos 3 ϑ 5 cos ϑ ) csc ϑ K v o v o 3 + x ¯ b 2 sin 2 ϑ K + i v o v o 2 + 3 x ¯ b 2 cos 2 ϑ v o 2 v o 4 x ¯ b 2 cos 2 ϑ v o v o 3 ) ] U + i ω 2 csc ϑ v o U z b + O ( ω 1 2 ) ,
T ¯ 5 = [ ω 2 cos ϑ cot ϑ v o 2 + ω 3 2 x ¯ b cot ϑ csc ϑ 2 ( ( cos 3 ϑ 3 cos ϑ ) K v o 2 + 4 sin ϑ cos 2 ϑ v o v o 3 ) + ω ( 2 i cos ϑ K v o 3 x ¯ b 2 cos 3 ϑ cot ϑ v o 2 v o 4 2 C 3 2 cot ϑ v o 2 + x ¯ b 2 cos ϑ cot 3 ϑ ( 2 cos 2 ϑ 3 ) K 2 v o 2 i cos ϑ cot ϑ v o v o 2 + x ¯ b 2 cos ϑ cot ϑ ( ( 5 3 cos 2 ϑ ) cot ϑ K v o v o 3 + cos 2 ϑ v o v o 3 ) ) ] U + [ i ω 3 2 2 cot ϑ v o + i ω x ¯ b ( 2 cot ϑ csc ϑ K v o 2 cos ϑ cot ϑ v o v o 2 ) ] U x ¯ b + ω csc ϑ U x ¯ b x ¯ b + O ( ω 1 2 ) .
= n ̂ n + t ̂ h 1 s ,
h = 1 K ( s ) n ,
x b = ( x ̂ b n ̂ ) n + h 1 ( x ̂ b t ̂ ) s .
x b = sin ϑ n + cos ϑ z b .
x b x b = x ̂ b ( x ̂ b ) = [ ( x ̂ b n ̂ ) n x ̂ b + h 1 ( x ̂ b t ̂ ) s x ̂ b ] .
n = n ̂ n n 2 + t ̂ [ h 1 s n 2 + K ( s ) h 2 s ] ,
s = n ̂ [ s n 2 + K ( s ) h 1 s ] + t ̂ [ h 1 s s 2 h s h 2 s K ( s ) n ] .
x b x b = cos 2 ϑ z b 2 + sin 2 ϑ n z b 2 + sin 2 ϑ n n 2 + sin 2 ϑ K o z b cos 2 ϑ K o n .

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