Abstract

The phase-space beam summation is a general analytical framework for local analysis and modeling of radiation from extended source distributions. In this formulation, the field is expressed as a superposition of beam propagators that emanate from all points in the source domain and in all directions. In this Part I of a two-part investigation, the theory is extended to include propagation in anisotropic medium characterized by a generic wave-number profile for time-harmonic fields; in a companion paper [J. Opt. Soc. Am. A 22, 1208 (2005) ], the theory is extended to time-dependent fields. The propagation characteristics of the beam propagators in a homogeneous anisotropic medium are considered. With use of Gaussian windows for the local processing of either ordinary or extraordinary electromagnetic field distributions, the field is represented by a phase-space spectral distribution in which the propagating elements are Gaussian beams that are formulated by using Gaussian plane-wave spectral distributions over the extended source plane. By applying saddle-point asymptotics, we extract the Gaussian beam phenomenology in the anisotropic environment. The resulting field is parameterized in terms of the spatial evolution of the beam curvature, beam width, etc., which are mapped to local geometrical properties of the generic wave-number profile. The general results are applied to the special case of uniaxial crystal, and it is found that the asymptotics for the Gaussian beam propagators, as well as the physical phenomenology attached, perform remarkably well.

© 2005 Optical Society of America

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References

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  1. M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).
  2. B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase-space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
    [CrossRef]
  3. A. Dendane, J. M. Arnold, “Scattered field analysis of a focused reflector using the Gabor series,” IEE Proc. Part H Microwaves, Antennas Propag. 141, 216–222 (1994).
    [CrossRef]
  4. T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
    [CrossRef]
  5. J. M. Arnold, “Phase-space localization and discrete representations of wave fields,” J. Opt. Soc. Am. A 12, 111–123 (1995).
    [CrossRef]
  6. J. M. Arnold, L. B. Felsen, “Rays, beams and diffraction in a discrete phase space: Wilson bases,” Opt. Express 10, 716–727 (2002).
    [CrossRef] [PubMed]
  7. D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).
  8. I. Tinkelman, T. Melamed, “Local spectrum analysis of field propagation in anisotropic media. Part II. Time-dependent fields,” J. Opt. Soc. Am. A 221208–1215 (2005).
    [CrossRef]
  9. C. Yangjian, L. Qiang, G. Di, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19, 2036–2042 (2002).
    [CrossRef]
  10. L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik (Stuttgart) 111, 297–306 (2000).
  11. E. Poli, G. V. Pereverzev, A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
    [CrossRef]
  12. R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998).
    [CrossRef]
  13. X. B. Wu, R. Wei, “Scattering of a Gaussian beam by an anisotropic material coated conducting circular cylinder,” Radio Sci. 30, 403–411 (1995).
    [CrossRef]
  14. A. Hanyga, “Gaussian beams in anisotropic elastic media,” Geophys. J. R. Astron. Soc. 85, 473–503 (1986).
    [CrossRef]
  15. K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
    [CrossRef]
  16. M. Spies, “Modeling of transducer fields in inhomogeneous anisotropic materials using Gaussian beam superposition,” Nondestr. Test. Eval. Int. 33, 155–162 (2000).
  17. S. Y. Shin, L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
    [CrossRef]
  18. I. Tinkelman, T. Melamed, “Gaussian beam propagation in generic anisotropic wave-number profiles,” Opt. Lett. 28, 1081–1083 (2003).
    [CrossRef] [PubMed]
  19. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).
    [CrossRef]
  20. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (IEEE Press, Piscataway, N.J., 1996).
  21. M. M. Popov, V. Červeny, I. Pšenčik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
    [CrossRef]
  22. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [CrossRef]
  23. E. Heyman, T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
    [CrossRef]
  24. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

2005

2003

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

D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).

2002

2000

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

M. Spies, “Modeling of transducer fields in inhomogeneous anisotropic materials using Gaussian beam superposition,” Nondestr. Test. Eval. Int. 33, 155–162 (2000).

1999

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

1998

1997

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

1995

1994

A. Dendane, J. M. Arnold, “Scattered field analysis of a focused reflector using the Gabor series,” IEE Proc. Part H Microwaves, Antennas Propag. 141, 216–222 (1994).
[CrossRef]

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

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

1991

1986

A. Hanyga, “Gaussian beams in anisotropic elastic media,” Geophys. J. R. Astron. Soc. 85, 473–503 (1986).
[CrossRef]

1982

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

1980

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

1974

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

Arnold, J. M.

Bastiaans, M. J.

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

Boag, A.

D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Cerveny, V.

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

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (IEEE Press, Piscataway, N.J., 1996).

Dendane, A.

A. Dendane, J. M. Arnold, “Scattered field analysis of a focused reflector using the Gabor series,” IEE Proc. Part H Microwaves, Antennas Propag. 141, 216–222 (1994).
[CrossRef]

Di, G.

Felsen, L. B.

Garea, M. T.

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

Hanyga, A.

A. Hanyga, “Gaussian beams in anisotropic elastic media,” Geophys. J. R. Astron. Soc. 85, 473–503 (1986).
[CrossRef]

Heyman, E.

D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).

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

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

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

Letrou, C.

D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).

Lugaraand, D.

D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).
[CrossRef]

Melamed, T.

I. Tinkelman, T. Melamed, “Local spectrum analysis of field propagation in anisotropic media. Part II. Time-dependent fields,” J. Opt. Soc. Am. A 221208–1215 (2005).
[CrossRef]

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

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

Mukunda, N.

Peeters, A. G.

E. Poli, G. V. Pereverzev, 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, 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, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik (Stuttgart) 111, 297–306 (2000).

Poli, E.

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

Popov, M. M.

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

Pšencik, I.

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

Qiang, L.

Shin, S. Y.

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

Shlivinski, A.

D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).

Simon, R.

Spies, M.

M. Spies, “Modeling of transducer fields in inhomogeneous anisotropic materials using Gaussian beam superposition,” Nondestr. Test. Eval. Int. 33, 155–162 (2000).

Steinberg, B. Z.

Sundar, K.

Tinkelman, I.

Wei, R.

X. B. Wu, R. Wei, “Scattering of a Gaussian beam by an anisotropic material coated conducting circular cylinder,” Radio Sci. 30, 403–411 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Wu, X. B.

X. B. Wu, R. Wei, “Scattering of a Gaussian beam by an anisotropic material coated conducting circular cylinder,” Radio Sci. 30, 403–411 (1995).
[CrossRef]

Yangjian, C.

Appl. Phys.

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

Geophys. J. R. Astron. Soc.

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

A. Hanyga, “Gaussian beams in anisotropic elastic media,” Geophys. J. R. Astron. Soc. 85, 473–503 (1986).
[CrossRef]

IEE Proc. Part H Microwaves, Antennas Propag.

A. Dendane, J. M. Arnold, “Scattered field analysis of a focused reflector using the Gabor series,” IEE Proc. Part H Microwaves, Antennas Propag. 141, 216–222 (1994).
[CrossRef]

IEEE Trans. Antennas Propag.

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

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

J. Electromagn. Waves Appl.

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

J. Opt. Soc. Am. A

Nondestr. Test. Eval. Int.

M. Spies, “Modeling of transducer fields in inhomogeneous anisotropic materials using Gaussian beam superposition,” Nondestr. Test. Eval. Int. 33, 155–162 (2000).

Opt. Express

Opt. Lett.

Optik (Stuttgart)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

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

Phys. Plasmas

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

Radio Sci.

X. B. Wu, R. Wei, “Scattering of a Gaussian beam by an anisotropic material coated conducting circular cylinder,” Radio Sci. 30, 403–411 (1995).
[CrossRef]

D. Lugaraand, C. Letrou, A. Shlivinski, E. Heyman, A. Boag, “Frame-based Gaussian beam summation method: theory and applications,” Radio Sci. 38, 1–15 (2003).

Other

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).
[CrossRef]

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (IEEE Press, Piscataway, N.J., 1996).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

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Figures (6)

Fig. 1
Fig. 1

Local processing of the time-harmonic field distribution; the PS distribution U ̂ o ( X ¯ ) is obtained by integrating the field distribution u ̂ o ( x ) with a linearly phased window function shifted to point x ¯ . The linear phase extracts from u ̂ o its local directional properties and by that matches a single beam propagator emanating from the window center at x ¯ along the (ordinary) κ ¯ = ( ξ ¯ , ζ ξ ¯ ) direction identified by the spherical angles ( ϑ ¯ , φ ¯ ) .

Fig. 2
Fig. 2

Wave-number surface and parameterization. The parameterization of the beam propagators is matched to local geometrical properties of the wave-number surface ζ ( ξ ) at the the processing point ξ = ξ ¯ : the normal κ ¯ ̂ and the radii of curvature along constant ξ 1 or ξ 2 , ρ 1 , 2 c , of the surface.

Fig. 3
Fig. 3

Local beam coordinate frame for the extraordinary GB propagator. The beam axis is directed along the unit vector κ ¯ ̂ . The local transverse coordinates x b are given by transformation (36) so that the GB [Eq. (40)] exhibits quadratic Gaussian decay in the local x b coordinates.

Fig. 4
Fig. 4

Gaussian beam amplitude contour plots over z = 1 plane for the reference solution evaluated by using exact plane-wave spectral representation (13).

Fig. 5
Fig. 5

Astigmatism and waist location; the asymptotic beam exp ( 1 2 ) contours in the ( x b 1 , 2 , z ) planes are shown in (a) and (b), respectively, as well as the waist location (circle) and the beam width (vertical line) in each plot.

Fig. 6
Fig. 6

Beam width in the local transverse coordinates. Solid curves plot the asymptotic-beam widths (48), D 1 , 2 ( z b ) , and circles plot the reference field exp ( 1 2 ) contours in (a) and (b) for the ( x b 1 , 2 , z b ) planes, respectively.

Equations (60)

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ζ ( ξ ) = k z ( k x 1 , k x 2 ) k o , k o = ω c ,
ξ = ( k x 1 , k x 2 ) k o ,
u ̃ ̂ o ( ξ ) = d 2 x u ̂ o ( x ) exp ( i k o ξ x ) ,
u ̂ o ( x ) = ( k o 2 π ) 2 d 2 ξ u ̃ ̂ o ( ξ ) exp ( i k o ξ x ) .
u ̂ ( r ) = ( k o 2 π ) 2 d 2 ξ u ̃ ̂ o ( ξ ) exp [ i k o ( ξ x + ζ ( ξ ) z ) ] .
κ ̂ ( ξ ) = [ ξ , ζ ( ξ ) ] [ ξ 2 + ζ 2 ( ξ ) ] 1 2 , ξ 2 = ξ ξ .
u o ( x ) = A o ( x ) exp [ i k o Φ o ( x ) ] ,
Φ o ( x ) = ξ , at x s ( ξ ) .
U ̂ o ( X ¯ ) = d 2 x u ̂ o ( x ) W ̂ * ( x ; X ¯ ) ,
W ̂ ( x ; X ¯ ) = w ̂ ( x x ¯ ) exp [ i k o ξ ¯ ( x x ¯ ) ] ,
U ̂ o ( X ¯ ) = ( k o 2 π ) 2 d 2 ξ u ̃ ̂ o ( ξ ) W ̃ ̂ * ( ξ ; X ¯ ) ,
W ̃ ̂ ( ξ ; X ¯ ) = w ̃ ̂ ( ξ ξ ¯ ) exp ( i k o ξ x ¯ ) ,
u ̂ o ( x ) = ( k o 2 π N ̂ ) 2 d 4 X ¯ U ̂ o ( X ¯ ) W ̂ ( x ; X ¯ ) ,
N ̂ = [ d 2 x w ̂ ( x ) 2 ] 1 2
u ̂ ( r , ω ) = ( k o 2 π N ̂ ) 2 d 4 X ¯ U ̂ o ( X ¯ ) B ̂ ( r ; X ¯ ) ,
B ̂ ( r ; X ¯ ) = ( k o 2 π ) 2 d 2 ξ W ̃ ̂ ( ξ ; X ¯ ) exp [ i k o ( ξ x + ζ ( ξ ) z ) ] ,
w ̂ ( x ) = exp ( i 2 k o x Γ x T ) ,
w ̃ ̂ ( ξ ) = 2 π i k o Γ exp ( i 2 k o ξ Γ 1 ξ T ) ,
B ̂ ( r ; X ¯ ) = i k o 2 π Γ d 2 ξ exp [ i k o Φ ( r , ξ , ξ ¯ ) ] ,
Φ ( r , ξ , ξ ¯ ) = ξ ( x x ¯ ) + ζ ( ξ ) z ( ξ ξ ¯ ) 2 ( 2 Γ ) .
ζ ( ξ ) = ( 1 ξ 2 ) 1 2 , Im ζ 0 .
( x b 1 x b 2 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 1 x ¯ 1 x 2 x ¯ 2 z ) ,
B ̂ ( r ; X ̂ ) = [ det Γ ( z b ) det Γ ( 0 ) ] 1 2 exp [ i k o ( z b + 1 2 x b Γ ( z b ) x b T ) ] ,
Γ ( z b ) = [ ( z b + ζ ¯ 2 Γ ) 1 0 0 ( z b + 1 Γ ) 1 ] .
Z 1 = Γ r Γ 2 ζ ¯ 2 , Z 2 = Γ r Γ 2
F 1 = Γ i Γ 2 ζ ¯ 2 , F 2 = Γ i Γ 2 ,
D 1 , 2 = ( F 1 , 2 k o ) 1 2 [ 1 + ( z b Z 1 , 2 ) 2 F 1 , 2 2 ] 1 2 ,
R 1 , 2 = ( z b Z 1 , 2 ) + F 1 , 2 2 ( z b Z 1 , 2 ) .
B ̂ ( r ; X ¯ ) z b = 0 = exp ( i 2 k o x b Γ parax x b T ) ,
Γ parax = [ Γ ζ ¯ 2 0 0 Γ ] .
( x x ¯ ) R ¯ = ξ ¯ , R ¯ ( x x ¯ 2 + z 2 ) 1 2 .
Φ ξ = ξ s = x x ¯ + z ζ ξ = ξ s ξ s ξ ¯ Γ = 0 ,
x x ¯ + ζ ¯ z = 0 ,
κ ̃ ̂ ( ξ ¯ ) = ( cos ϑ ¯ 1 , cos ϑ ¯ 2 , cos ϑ ¯ ) ,
cos ϑ ¯ 1 , 2 = cos ϑ ¯ ξ 1 , 2 ζ ¯ , cos ϑ ¯ = ( ζ ¯ 2 + 1 ) 1 2 .
Φ ( ξ ) Φ 0 + Φ 1 ( ξ ξ ¯ ) T + 1 2 ( ξ ξ ¯ ) Φ 2 ( ξ ξ ¯ ) T ,
Φ 0 = Φ ( ξ ) = ξ ¯ ( x x ¯ ) + ζ ¯ z , Φ 1 = ( x x ¯ ) + ζ ¯ z ,
Φ 2 = [ 1 Γ + ξ 1 2 ζ ¯ z ξ 1 ξ 2 2 ζ ¯ z ξ 1 ξ 2 2 ζ ¯ z 1 Γ + ξ 2 2 ζ ¯ z ] .
B ̂ ( r ; X ¯ ) = i Γ det Φ 2 exp [ i k o S ( r ) ] ,
S ( r ) = Φ 0 1 2 Φ 1 Φ 2 1 Φ 1 T .
( x b 1 x b 2 z b ) = T ( x 1 x ¯ 1 x 2 x ¯ 2 z ) ,
T = [ cos φ ¯ sin φ ¯ ( cos ϑ ¯ 2 sin φ ¯ cos ϑ ¯ 1 cos φ ¯ ) cos ϑ ¯ sin φ ¯ cos φ ¯ ( cos ϑ ¯ 2 cos φ ¯ + cos ϑ ¯ 1 sin φ ¯ ) cos ϑ ¯ 0 0 1 cos ϑ ¯ ] ,
tan 2 φ ¯ = 2 ξ 1 ξ 2 2 ζ ¯ ( ξ 2 2 ζ ¯ ξ 1 2 ζ ¯ ) .
T 1 = [ cos φ ¯ sin φ ¯ cos ϑ ¯ 1 sin φ ¯ cos φ ¯ cos ϑ ¯ 2 0 0 cos ϑ ¯ ] .
B ̂ ( r ; X ¯ ) = [ det Γ ( z b ) det Γ ( 0 ) ] 1 2 exp [ i k o ( κ ¯ ̂ iso [ r ( x ¯ , 0 ) ] + 1 2 x b Γ ( z b ) x b T ) ] ,
Γ 1 , 2 ( z ) = ( 1 Γ z 1 2 { ξ 1 2 ζ ¯ + ξ 2 2 ζ ¯ [ ( ξ 1 2 ζ ¯ ξ 2 2 ζ ¯ ) 2 + 4 ( ξ 1 ξ 2 2 ζ ¯ ) 2 ] 1 2 } ) 1 .
a 1 , 2 = { + } cos ϑ ¯ cos ( 2 φ ¯ ) [ ξ 1 2 ζ ¯ { cos 2 φ ¯ sin 2 φ ¯ } ξ 2 2 ζ ¯ { sin 2 φ ¯ cos 2 φ ¯ } ] .
ρ 1 , 2 c = [ 1 + ( ξ 1 , 2 ζ ¯ ) 2 ] 3 2 ξ 1 , 2 2 ζ ¯ .
ξ 1 , 2 2 ζ ¯ = ( sin ϑ 2 , 1 cos ϑ 3 ) 3 1 ρ 1 , 2 c ,
a 1 , 2 = { + } cos 2 ϑ ¯ cos ( 2 φ ¯ ) [ sin 3 ϑ 2 ρ 1 c { cos 2 φ ¯ sin 2 φ ¯ } sin 3 ϑ 1 ρ 2 c { sin 2 φ ¯ cos 2 φ ¯ } ] .
Z 1 , 2 = Γ r ( Γ 2 a 1 , 2 ) , F 1 , 2 = Γ i ( Γ 2 a 1 , 2 )
Γ 1 , 2 ( z b ) = 1 R 1 , 2 ( z b ) + i k o D 1 , 2 2 ( z b ) ,
D 1 , 2 ( z b ) = ( F 1 , 2 a 1 , 2 k o ) 1 2 [ 1 + ( z b Z 1 , 2 ) 2 F 1 , 2 2 ] 1 2
R 1 , 2 ( z b ) = a 1 , 2 [ ( z b Z 1 , 2 ) + F 1 , 2 2 ( z b Z 1 , 2 ) ]
( x x ¯ ) R ¯ = cos ϑ ¯ 1 , 2 , R ¯ ( x x ¯ 2 + z 2 ) 1 2 ,
ϵ r = [ ϵ 0 0 0 ϵ 0 0 0 ϵ z ] .
k x 1 2 c 2 ( k ) c x 2 + k x 2 2 c 2 ( k ) c x 2 + k z 2 c 2 ( k ) c z 2 = 0 ,
c ( k ) = ω ( k x 1 2 + k x 2 2 + k z 2 ) 1 2 ,
ζ ( ξ ) = ( 1 ξ 2 2 ξ 1 2 ) 1 2 ,
ζ ( ξ ) = ( 1 ( ξ 2 2 + ξ 1 2 ) ϵ z ) 1 2 .

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