Abstract

Recent experimental results have shown that the highly directional frequency-upconverted backward-stimulated emission from a two-photon-pumped lasing medium possesses the property of optical phase conjugation. To explain this property, a Gabor’s quasi-collinear holographic model is proposed for a two-photon-pumped gain medium. According to this model, the input IR pump beam (after passing through a phase-distorting medium or an aberration plate) can be assumed to consist of two parts: an undisturbed regular portion and a phase-distorted irregular portion. These two portions interfere with each other and create a volume holographic grating within the pumped region inside the gain medium. While an initial regular backward-stimulated emission (as a reading beam) is passing through this holographic grating, a diffracted wave is created and amplified together with the reading beam. A rigorous mathematical analysis shows that under certain conditions the combination of these two parts (the reading beam plus the diffracted wave) of the backward-stimulated emission can be an approximate phase-conjugate field of the total input-pump field.

© 1998 Optical Society of America

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1998 (1)

1997 (2)

1986 (1)

G. S. He, D. Liu, and S. H. Liu, “New physical explanation of phase-conjugate wave generation in backward stimulated scattering,” Chin. Phys. Lasers 13, 713–717 (1986).

1985 (1)

G. S. He, D. Liu, and S. H. Liu, “A new physical explanation of producing phase-conjugate wave process in backward stimulated scattering,” Bull. Am. Phys. Soc. 30, 1800 (1985).

1980 (1)

N. B. Baranova and B. Ya. Zel’dovich, “Wavefront reversal of focused beams (theory of stimulated Brillouin backscattering),” Sov. J. Quantum Electron. 10, 555–560 (1980).
[CrossRef]

1979 (5)

V. I. Bespalov, V. G. Manishin, and G. A. Pasmanik, “Nonlinear selection of optical radiation on reflection from a stimulated Mandel’shtam-Brillouin scattering mirror,” Sov. Phys. JETP 50, 879–886 (1979).

V. G. Sidorovich and V. V. Shkunov, “Capture of a Stokes pump wave in a stimulated-Raman-scattering amplifier,” Sov. Phys. Tech. Phys. 24, 472–476 (1979).

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[CrossRef]

G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

H. Hsu, “Large-signal theory of phase-conjugate backscatterings,” Appl. Phys. Lett. 34, 855–857 (1979).
[CrossRef]

1978 (6)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. 14, 650–660 (1978).
[CrossRef]

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, “Light beam wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,” Opt. Commun. 24, 74–76 (1978).
[CrossRef]

A. D. Kudriavtseva, A. L. Sokolovskaia, J. Gazengel, N. P. Xuan, and G. Rivoire, “Reconstruction of the laser wave-front by stimulated scattering in the pico-second range,” Opt. Commun. 26, 446–448 (1978).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, and V. V. Shkunov, “Wavefront reversal in stimulated light scattering in a focused spatially inhomogeneous pump beam,” Sov. J. Quantum Electron. 8, 559–566 (1978).
[CrossRef]

V. Wang and C. R. Giuliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[CrossRef] [PubMed]

R. W. Hellwarth, “Theory of phase conjugation by stimulated scattering in a waveguide,” J. Opt. Soc. Am. 68, 1050–1056 (1978).
[CrossRef]

1977 (6)

R. W. Hellwarth, “Generation of time reversal wavefronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
[CrossRef]

A. Yariv and D. M. Pepper, “Amplified refraction, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[CrossRef]

D. M. Bloom and G. C. Bjorklund, “Conjugate wave front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

G. G. Kochemasov and V. D. Nikolaev, “Reproduction of the spatial amplitude and phase distributions of a pump beam in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 7, 60–63 (1977).
[CrossRef]

I. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, and V. I. Mandrosov, “Complex conjugation of field in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 6, 1349–1351 (1977).
[CrossRef]

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

1972 (2)

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam-Brillouin scattering,” JETP Lett. 15, 109–112 (1972).

O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror, ' ” JETP Lett. 16, 435–438 (1972).

Baranova, N. B.

N. B. Baranova and B. Ya. Zel’dovich, “Wavefront reversal of focused beams (theory of stimulated Brillouin backscattering),” Sov. J. Quantum Electron. 10, 555–560 (1980).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, and V. V. Shkunov, “Wavefront reversal in stimulated light scattering in a focused spatially inhomogeneous pump beam,” Sov. J. Quantum Electron. 8, 559–566 (1978).
[CrossRef]

Bel’dyugin, I. M.

I. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, and V. I. Mandrosov, “Complex conjugation of field in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 6, 1349–1351 (1977).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov, V. G. Manishin, and G. A. Pasmanik, “Nonlinear selection of optical radiation on reflection from a stimulated Mandel’shtam-Brillouin scattering mirror,” Sov. Phys. JETP 50, 879–886 (1979).

Bhawalkar, J. D.

Bjorklund, G. C.

D. M. Bloom and G. C. Bjorklund, “Conjugate wave front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

Bloom, D. M.

D. M. Bloom and G. C. Bjorklund, “Conjugate wave front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

Brekhovskikh, G. L.

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, “Light beam wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,” Opt. Commun. 24, 74–76 (1978).
[CrossRef]

Cui, Y.

Faizullov, F. S.

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam-Brillouin scattering,” JETP Lett. 15, 109–112 (1972).

O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror, ' ” JETP Lett. 16, 435–438 (1972).

Galushkin, M. G.

I. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, and V. I. Mandrosov, “Complex conjugation of field in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 6, 1349–1351 (1977).
[CrossRef]

Gazengel, J.

A. D. Kudriavtseva, A. L. Sokolovskaia, J. Gazengel, N. P. Xuan, and G. Rivoire, “Reconstruction of the laser wave-front by stimulated scattering in the pico-second range,” Opt. Commun. 26, 446–448 (1978).
[CrossRef]

Giuliano, C. R.

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[CrossRef]

V. Wang and C. R. Giuliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
[CrossRef] [PubMed]

He, G. S.

Hellwarth, R. W.

Hsu, H.

H. Hsu, “Large-signal theory of phase-conjugate backscatterings,” Appl. Phys. Lett. 34, 855–857 (1979).
[CrossRef]

Kochemasov, G. G.

G. G. Kochemasov and V. D. Nikolaev, “Reproduction of the spatial amplitude and phase distributions of a pump beam in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 7, 60–63 (1977).
[CrossRef]

Kudriavtseva, A. D.

A. D. Kudriavtseva, A. L. Sokolovskaia, J. Gazengel, N. P. Xuan, and G. Rivoire, “Reconstruction of the laser wave-front by stimulated scattering in the pico-second range,” Opt. Commun. 26, 446–448 (1978).
[CrossRef]

Kudryavtseva, A. D.

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, “Light beam wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,” Opt. Commun. 24, 74–76 (1978).
[CrossRef]

Lam, J. F.

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[CrossRef]

Lind, R. C.

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[CrossRef]

Liu, D.

G. S. He, D. Liu, and S. H. Liu, “New physical explanation of phase-conjugate wave generation in backward stimulated scattering,” Chin. Phys. Lasers 13, 713–717 (1986).

G. S. He, D. Liu, and S. H. Liu, “A new physical explanation of producing phase-conjugate wave process in backward stimulated scattering,” Bull. Am. Phys. Soc. 30, 1800 (1985).

Liu, S. H.

G. S. He, D. Liu, and S. H. Liu, “New physical explanation of phase-conjugate wave generation in backward stimulated scattering,” Chin. Phys. Lasers 13, 713–717 (1986).

G. S. He, D. Liu, and S. H. Liu, “A new physical explanation of producing phase-conjugate wave process in backward stimulated scattering,” Bull. Am. Phys. Soc. 30, 1800 (1985).

Mandrosov, V. I.

I. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, and V. I. Mandrosov, “Complex conjugation of field in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 6, 1349–1351 (1977).
[CrossRef]

Manishin, V. G.

V. I. Bespalov, V. G. Manishin, and G. A. Pasmanik, “Nonlinear selection of optical radiation on reflection from a stimulated Mandel’shtam-Brillouin scattering mirror,” Sov. Phys. JETP 50, 879–886 (1979).

Martin, G.

G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

Nikolaev, V. D.

G. G. Kochemasov and V. D. Nikolaev, “Reproduction of the spatial amplitude and phase distributions of a pump beam in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 7, 60–63 (1977).
[CrossRef]

Nosach, O. Yu.

O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror, ' ” JETP Lett. 16, 435–438 (1972).

Pasmanik, G. A.

V. I. Bespalov, V. G. Manishin, and G. A. Pasmanik, “Nonlinear selection of optical radiation on reflection from a stimulated Mandel’shtam-Brillouin scattering mirror,” Sov. Phys. JETP 50, 879–886 (1979).

Pepper, D. M.

Popovichev, V. I.

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam-Brillouin scattering,” JETP Lett. 15, 109–112 (1972).

O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror, ' ” JETP Lett. 16, 435–438 (1972).

Prasad, P. N.

Ragul’skii, V. V.

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam-Brillouin scattering,” JETP Lett. 15, 109–112 (1972).

O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror, ' ” JETP Lett. 16, 435–438 (1972).

Rivoire, G.

A. D. Kudriavtseva, A. L. Sokolovskaia, J. Gazengel, N. P. Xuan, and G. Rivoire, “Reconstruction of the laser wave-front by stimulated scattering in the pico-second range,” Opt. Commun. 26, 446–448 (1978).
[CrossRef]

Shkunov, V. V.

V. G. Sidorovich and V. V. Shkunov, “Capture of a Stokes pump wave in a stimulated-Raman-scattering amplifier,” Sov. Phys. Tech. Phys. 24, 472–476 (1979).

N. B. Baranova, B. Ya. Zel’dovich, and V. V. Shkunov, “Wavefront reversal in stimulated light scattering in a focused spatially inhomogeneous pump beam,” Sov. J. Quantum Electron. 8, 559–566 (1978).
[CrossRef]

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

Sidorovich, V. G.

V. G. Sidorovich and V. V. Shkunov, “Capture of a Stokes pump wave in a stimulated-Raman-scattering amplifier,” Sov. Phys. Tech. Phys. 24, 472–476 (1979).

Sokolovskaia, A. L.

A. D. Kudriavtseva, A. L. Sokolovskaia, J. Gazengel, N. P. Xuan, and G. Rivoire, “Reconstruction of the laser wave-front by stimulated scattering in the pico-second range,” Opt. Commun. 26, 446–448 (1978).
[CrossRef]

Sokolovskaya, A. I.

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, “Light beam wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,” Opt. Commun. 24, 74–76 (1978).
[CrossRef]

Steel, D. G.

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[CrossRef]

Wang, V.

Xuan, N. P.

A. D. Kudriavtseva, A. L. Sokolovskaia, J. Gazengel, N. P. Xuan, and G. Rivoire, “Reconstruction of the laser wave-front by stimulated scattering in the pico-second range,” Opt. Commun. 26, 446–448 (1978).
[CrossRef]

Yariv, A.

Yoshida, M.

Zel’dovich, B. Ya.

N. B. Baranova and B. Ya. Zel’dovich, “Wavefront reversal of focused beams (theory of stimulated Brillouin backscattering),” Sov. J. Quantum Electron. 10, 555–560 (1980).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, and V. V. Shkunov, “Wavefront reversal in stimulated light scattering in a focused spatially inhomogeneous pump beam,” Sov. J. Quantum Electron. 8, 559–566 (1978).
[CrossRef]

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam-Brillouin scattering,” JETP Lett. 15, 109–112 (1972).

Zemskov, E. M.

I. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, and V. I. Mandrosov, “Complex conjugation of field in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 6, 1349–1351 (1977).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

D. M. Bloom and G. C. Bjorklund, “Conjugate wave front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

H. Hsu, “Large-signal theory of phase-conjugate backscatterings,” Appl. Phys. Lett. 34, 855–857 (1979).
[CrossRef]

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[CrossRef]

G. Martin and R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index gratings,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

Bull. Am. Phys. Soc. (1)

G. S. He, D. Liu, and S. H. Liu, “A new physical explanation of producing phase-conjugate wave process in backward stimulated scattering,” Bull. Am. Phys. Soc. 30, 1800 (1985).

Chin. Phys. Lasers (1)

G. S. He, D. Liu, and S. H. Liu, “New physical explanation of phase-conjugate wave generation in backward stimulated scattering,” Chin. Phys. Lasers 13, 713–717 (1986).

IEEE J. Quantum Electron. (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. 14, 650–660 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

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

JETP Lett. (2)

B. Ya. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam-Brillouin scattering,” JETP Lett. 15, 109–112 (1972).

O. Yu. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faizullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror, ' ” JETP Lett. 16, 435–438 (1972).

Opt. Commun. (2)

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, “Light beam wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,” Opt. Commun. 24, 74–76 (1978).
[CrossRef]

A. D. Kudriavtseva, A. L. Sokolovskaia, J. Gazengel, N. P. Xuan, and G. Rivoire, “Reconstruction of the laser wave-front by stimulated scattering in the pico-second range,” Opt. Commun. 26, 446–448 (1978).
[CrossRef]

Opt. Lett. (3)

Sov. J. Quantum Electron. (5)

G. G. Kochemasov and V. D. Nikolaev, “Reproduction of the spatial amplitude and phase distributions of a pump beam in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 7, 60–63 (1977).
[CrossRef]

I. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, and V. I. Mandrosov, “Complex conjugation of field in stimulated Brillouin scattering,” Sov. J. Quantum Electron. 6, 1349–1351 (1977).
[CrossRef]

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, and V. V. Shkunov, “Wavefront reversal in stimulated light scattering in a focused spatially inhomogeneous pump beam,” Sov. J. Quantum Electron. 8, 559–566 (1978).
[CrossRef]

N. B. Baranova and B. Ya. Zel’dovich, “Wavefront reversal of focused beams (theory of stimulated Brillouin backscattering),” Sov. J. Quantum Electron. 10, 555–560 (1980).
[CrossRef]

Sov. Phys. JETP (1)

V. I. Bespalov, V. G. Manishin, and G. A. Pasmanik, “Nonlinear selection of optical radiation on reflection from a stimulated Mandel’shtam-Brillouin scattering mirror,” Sov. Phys. JETP 50, 879–886 (1979).

Sov. Phys. Tech. Phys. (1)

V. G. Sidorovich and V. V. Shkunov, “Capture of a Stokes pump wave in a stimulated-Raman-scattering amplifier,” Sov. Phys. Tech. Phys. 24, 472–476 (1979).

Other (6)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), p. 254.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1983), p. 453.

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985), pp. 28–34.

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), pp. 245–252, 343–347.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

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Figures (2)

Fig. 1
Fig. 1

Schematic illustration of Gabor’s holographic model for the phase-conjugation formation of a backward-stimulated emission.

Fig. 2
Fig. 2

Schematic illustration of the detailed optical-path geometry for formation of a phase-conjugated backward-stimulated emission from a TPP gain medium.

Equations (91)

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U=U(i)+U(s)=A(i) exp(iφi)+A(s) exp(iφs)=exp(iφi){A(i)+A(s) exp[i(φs-φi)]}.
Δn(ω)=12n0(ω) χe(3)(-ω; ω, -ω, ω)|E(ω)|2,
I(l, ω)=I(0, ω)exp[(G-α)l],
exp[(G-α)l]1.
Ip(z, ω)z=-βIp2(z, ω),
ΔNIp(z, ω)zIp2(z, ω).
G(z)=gIp2(z, ω).
E(z, x, y, ω)=E(z, x, y)exp(-iωt)=A0(z, x, y)exp{i[kz+φ(z, x, y)]}×exp(-iωt).
E(z, x, y, ω)=aE*(z, x, y)exp(-iωt)=aA0(z, x, y)exp{-i[kz+φ(z, x, y)]}×exp(-iωt),
E(z, x, y, ω)=aA0(z, x, y)×exp{-i[kz+φ(z, x, y)]}×exp(-iωt),
E(z, x, y, ω)=C w0w(z) exp-(x2+y2)×1w2(z)-ik2R(z)-i tan-1(z/δz)×exp[i(kz-ωt)],
w2(z)=w02[1+(z/δz)2],
R(z)=z[1+(δz/z)2],
δz=kw02/2.
E(z, x, y, ω)=E1(z, x, y, ω)+E2(z, x, y, ω)=[A1(z, x, y)+A2(z, x, y)]×exp[i(kz-ωt)]=C1 w0w(z) exp-(x2+y2)×1w2(z)-ik2R(z)-i tan-1(z/δz)+C2 exp[iθ(z, x, y)]×exp[i(kz-ωt)].
E(z, x, y, ω)=E1(z, x, y, ω)+E2(z, x, y, ω)=[A1(z, x, y)+A2(z, x, y)]×exp[i(-kz-ωt)],
P1(ω)=ε0iχe|A1|4A1 exp[i(-kz-ωt)]=ε0iχ C14w04w4(z) exp[-4(x2+y2)/w2(z)]A1×exp[i(-kz-ωt)],
2E1-n02(ω)c2 2E1t2=μ0 2P1(ω)t2,
A1z+i2k 2A1=gC142[1+(z/δz)2]2×exp[-4(x2+y2)/w2(z)]A1,
A1=C1 w0w(z) exp[gC14(z-z0)/2]exp-(x2+y2)×1w2(z)+ik2R(z)+i tan-1(z/δz).
δz=ηδz,
w02=(k/k)ηw02,
w2(z)=w02[1+(z/δz)2],
R(z)=z[1+(δz/z)2],
η=23 gC14δz[(z/δz)3-(z0/δz)3].
η=23 gC14δz[(z/δz)3-(z0/δz)3]1
E(z, x, y, ω)=E1(z, x, y, ω)=A1(z, x, y)exp[i(kz-ωt)].
E(z, x, y, ω)=E1(z, x, y, ω)=A1(z, x, y)exp[i(-kz-ωt)].
A1(z, x, y)A1*(z, x, y).
P2(ω)=ε0iχe(3)A1A2*A1 exp[i(-kz-ωt)],
P2(ω)=ε0iχe(3) C1C2C1w02w2 exp-2(x2+y2)w2(z)×exp(-iθ)exp[gC14(z-z0)/2]×exp[i(-kz-ωt)].
A2z+i2k 2A2=g2 C1C2C11+(z/δz)2 exp-2(x2+y2)w2(z)×exp(-iθ)exp[gC14(z-z0)/2],
A2=C2C1C1 exp-2(x2+y2)w2(z)A2,
A2z-i 2δz[1+(z/δz)]2 x A2x+y A2y+i2k 2A2x2+2A2y2-iδz[1+(z/δz)2] 1-4(x2+y2)w2(z)A2+4(x2+y2)δzw2(z) A2
=12 gC12 11+(z/δz)2 exp[gC14(z-z0)/2]
×exp(-iθ).
A2z+i2k 2A2=gC122[1+(z/δz)2]×exp[gC14(z-z0)/2]exp(-iθ).
A2(z, x, y)=A2(z)exp[-iθ(z, x, y)].
A2z+A22k 2θ
=gC122[1+(z/δz)2] exp[gC14(z-z0)/2]cos δθ,
θz+12k θx2+θy2
=gC122[1+(z/δz)2] exp[gC14(z-z0)/2] sin δθA2,
A2z=gC122[1+(z/δz)2] exp[gC14(z-z0)/2]cos δθ,
θz=gC122[1+(z/δz)2] exp[gC14(z-z0)/2]sin δθA2.
θ/z0.
A2z=gC122[1+(z/δz)2] exp[gC14(z-z0)/2]cos δθ,
(δθ)z=-gC122[1+(z/δz)2] exp[gC14(z-z0)/2] sin δθA2.
A2(δθ)=-A2 cos δθ/sin δθ.
A2(z)sin δθ(z)=A2(z0)sin δθ(z0)=B,
A2/(A2)2-B2 A2z=gC122[1+(z/δz)2]×exp[gC14(z-z0)/2].
A2z=gC122[1+(z/δz)2] exp[gC14(z-z0)/2].
A2z=gC142[1+(z/δz)2] exp[gC14(z-z0)/2].
A2(z)=exp12 gC14(z-z0).
A2(z0)=1.
sin δθ(z)=sin δθ(z0)A2(z0)/A2(z).
sin δθ(z)=sin[θ(z)-θ(z)]=sin δθ(z0)exp-12 gC14(z-z0).
12 gC14(z-z0)1,
sin[θ(z)-θ(z)]  0,
θ(z)  θ(z).
A2=C2C1C1 exp12 gC14(z-z0)exp-2(x2+y2)w2(z)×exp(-iθ).
A1(z, x, y)+A2(z, x, y)
=C1C1 exp[gC14(z-z0)/2]C1 w0w(z) exp-(x2+y2)×1w2(z)+ik2R(z)+i tan-1(z/δz)+C2 exp-2(x2+y2)w2(z)exp(-iθ).
η=23 gC14δz[(z/δz)3-(z0/δz)3]1,
12 gC14(z-z0)1.
A1(z, x, y)+A2(z, x, y)
=C1 w0w(z) exp-(x2+y2)1w2(z)-ik2R(z)-i tan-1(z/δz)+C2 exp(iθ).
[2(x2+y2)/w2(z)]<1,
[A1(z, x, y)+A2(z, x, y)]
[A1(z, x, y)+A2(z, x, y)]*.
gC14δz(612),
A1z+i2k 2A1=gC142[1+(z/δz)2]2×exp[-4(x2+y2)/w2(z)]A1.
A1(x, y, z)=k2A˜1(ϑx, ϑy, z)×exp[ik(ϑxx+ϑyy)]dϑxdϑy,
A˜1(ϑx, ϑy, z)=14π2 A1(x, y, z)×exp[-ik(ϑxx+ϑyy)]dxdy.
A˜1z-ik2 (ϑx2+ϑy2)A˜1=k2g8π2  C14[1+(z/δz)2]2×exp[-4(x2+y2)/w2(z)]A˜1 exp{ik[(ϑx-ϑx)x+(ϑy-ϑy)y]}dϑxdϑydxdy.
A˜1(ϑx, ϑy, z)=S(ϑx, ϑy, z)exp12 ik(ϑx2+ϑy2)z.
Sz=kG016π  S1+(z/δz)2×exp-kz8 [1+(z/δz)2][(ϑx-ϑx)2+(ϑy-ϑy)2-iβ(ϑx2-ϑx2)-iβ(ϑy2-ϑy2)]dϑxdϑy,
G0=gC14δz,
β=4z[1+(z/δz)2]δz.
Sz=S G02δz[1+(z/δz)2]2(1-iβ)×exp-kδz8 [1+(z/δz)2]×β21-iβ (ϑx2+ϑy2).
S=S0 expz0z G02δz[1+(z/δz)2]2(1-iβ)×exp-kδz8 [1+(z/δz)2]×β21-iβ (ϑx2+ϑy2)dz.
S=S0 expξ0ξ G02 exp[-2kδz(ϑx2+ϑy2)ξ2]dξ=S0 expξ0ξ G02 [1-2kδz(ϑx2+ϑy2)ξ2]dξ=S0 expG02 (ξ-ξ0)×exp-13 kG0δz(ϑx2+ϑy2)(ξ3-ξ03).
A1(x, y, z)=k2A˜1(ϑx, ϑy, z)×exp[ik(ϑxx+ϑyy)]dϑxdϑy=k2S exp12 ik(ϑx2+ϑy2)z×exp[ik(ϑxx+ϑyy)]dϑxdϑy=A10 expG02 (ξ-ξ0)× exp[-u(ϑx2+ϑy2)+ik(ϑxx+ϑyy)]dϑxdϑy,
u=13 kG0δz(ξ3-ξ03)-12 ikz.
A1=A10 πu expG02 (ξ-ξ0)exp-k24u (x2+y2)=πA10 13 kG0δz(ξ3-ξ03)+12 ikz13 kG0δz(ξ3-ξ03)2+14 k2z2×expG02 (ξ-ξ0)exp-k24 (x2+y2)×13 kG0δz(ξ3-ξ03)+12 ikz13 kG0δz(ξ3-ξ03)2+14 k2z2.
A1=C1 w0w(z) exp[gC14(z-z0)/2]exp-(x2+y2)×1w2(z)+ik2R(z)+i tan-1(z/δz).
δz=ηδz,
w02=(k/k)ηw02,
w2(z)=w02[1+(z/δz)2],
R(z)=z[1+(δz/z)2],
η=23 gC14δz[(z/δz)3-(z0/δz)3],
C1=3πA10kgC14(δz)2[(z/δz)3-(z0/δz)3].

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