Abstract

It is shown that the copropagating three-wave-mixing parametric process, with appropriate type-II extended phase matching and pumped with a short second-harmonic pulse, can perform spectral phase conjugation and parametric amplification, which shows a threshold behavior analogous to backward-wave oscillation. The process is also analyzed in the Heisenberg picture, which predicts a spontaneous parametric downconversion rate in agreement with the experimental result reported by Kuzucu et al. [Phys. Rev. Lett. 94, 083601 (2005)] . Applications in optical communications, signal processing, and quantum information processing can be envisaged.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  32. H.-Y. Fan and N.-Q. Jiang, "Special two-mode unitary transform and maximum entanglement state for four wave mixing," Phys. Scr. 71, 277-279 (2005).
    [CrossRef]
  33. A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, "Stimulated emission of polarization-entangled photons," Nature 412, 887-890 (2001).
    [CrossRef] [PubMed]
  34. G. Michaeli and A. Arie, "Optimization of quasi-phase-matched non-linear frequency conversion for diffusion bonding applications," Appl. Phys. B 77, 497-503 (2003).
    [CrossRef]
  35. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

2005

M. Tsang and D. Psaltis, "Spontaneous spectral phase conjugation for coincident frequency entanglement," Phys. Rev. A 71, 043806 (2005).
[CrossRef]

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kaertner, "Two-photon coincident frequency entanglement via extended phase matching," Phys. Rev. Lett. 94, 083601 (2005).
[CrossRef] [PubMed]

H.-Y. Fan and N.-Q. Jiang, "Special two-mode unitary transform and maximum entanglement state for four wave mixing," Phys. Scr. 71, 277-279 (2005).
[CrossRef]

2004

M. Tsang and D. Psaltis, "Spectral phase conjugation with cross-phase modulation compensation," Opt. Express 12, 2207-2219 (2004).
[CrossRef] [PubMed]

F. Konig and F. N. C. Wong, "Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch," Appl. Phys. Lett. 84, 1644-1646 (2004).
[CrossRef]

M. Tsang and D. Psaltis, "Spectral phase conjugation by quasi-phase-matched three-wave mixing," Opt. Commun. 242, 659-664 (2004).
[CrossRef]

2003

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion," Phys. Rev. A 67, 053810 (2003).
[CrossRef]

S. Fraigne, J. P. Galaup, J. L. Le Gouet, B. Bousquet, L. Canioni, M. Joffre, and J. P. Likforman, "Amplitude and phase measurements of femtosecond pulses shaped by use of spectral hole burning in free-base naphthalocyanine-doped films," J. Opt. Soc. Am. B 20, 1555-1558 (2003).
[CrossRef]

M. Tsang and D. Psaltis, "Dispersion and nonlinearitycompensation by spectral phase conjugation," Opt. Lett. 28, 1558-1560 (2003).
[CrossRef] [PubMed]

G. Michaeli and A. Arie, "Optimization of quasi-phase-matched non-linear frequency conversion for diffusion bonding applications," Appl. Phys. B 77, 497-503 (2003).
[CrossRef]

2002

N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, "Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band," Opt. Lett. 27, 1046-1048 (2002).
[CrossRef]

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, "Generating entangled two-photon states with coincident frequencies," Phys. Rev. Lett. 88, 183602 (2002).
[CrossRef] [PubMed]

G. A. Durkin, C. Simon, and D. Bouwmeester, "Multiphoton entanglement concentration and quantum cryptography," Phys. Rev. Lett. 88, 187902 (2002).
[CrossRef] [PubMed]

2001

V. Giovannetti, S. Lloyd, and L. Maccone, "Quantum-enhanced positioning and clock synchronization," Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

D. M. Marom, D. Panasenko, P.-C. Sun, Y. T. Mazurenko, and Y. Fainman, "Real-time spatial-temporal signal processing with optical nonlinearities," IEEE J. Sel. Top. Quantum Electron. 7, 683-693 (2001).

A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, "Stimulated emission of polarization-entangled photons," Nature 412, 887-890 (2001).
[CrossRef] [PubMed]

2000

1995

Y. J. Ding, S. J. Lee, and J. B. Khurgin, "Transversely pumped counterpropagating optical parametric oscillation and amplification," Phys. Rev. Lett. 75, 429-432 (1995).
[CrossRef] [PubMed]

1992

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251-2261 (1992).
[CrossRef]

1991

1989

C. Joubert, M. L. Roblin, and R. Grousson, "Temporal reversal of picosecond optical pulses by holographic phase conjugation," Appl. Opt. 28, 4604-4612 (1989).
[CrossRef] [PubMed]

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93-95 (1989).
[CrossRef]

1983

1981

R. A. Fisher, B. R. Suydam, and B. J. Feldman, "Transient analysis of Kerr-like phase conjugators using frequency-domain techniques," Phys. Rev. A 23, 3071-3083 (1981).
[CrossRef]

1980

1979

1978

1966

S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

1965

N. M. Kroll, "Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves," J. Appl. Phys. 36, 34-43 (1965).
[CrossRef]

D. Bobroff, "Coupled-mode analysis of phonon-photon parametric backward-wave oscillator," J. Appl. Phys. 36, 1760-1769 (1965).
[CrossRef]

1954

H. Heffner, "Analysis of the backward-wave traveling-wave tube," Proc. IRE 42, 930-937 (1954).
[CrossRef]

1953

R. Kompfner and N. T. Williams, "Backward-wave tubes," Proc. IRE 41, 1602-1611 (1953).
[CrossRef]

Aaviksoo, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93-95 (1989).
[CrossRef]

Abrams, R. L.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Albota, M. A.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kaertner, "Two-photon coincident frequency entanglement via extended phase matching," Phys. Rev. Lett. 94, 083601 (2005).
[CrossRef] [PubMed]

Andrejco, M. J.

Arie, A.

G. Michaeli and A. Arie, "Optimization of quasi-phase-matched non-linear frequency conversion for diffusion bonding applications," Appl. Phys. B 77, 497-503 (2003).
[CrossRef]

Babbitt, W. R.

Bobroff, D.

D. Bobroff, "Coupled-mode analysis of phonon-photon parametric backward-wave oscillator," J. Appl. Phys. 36, 1760-1769 (1965).
[CrossRef]

Booth, M. C.

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion," Phys. Rev. A 67, 053810 (2003).
[CrossRef]

Bousquet, B.

Bouwmeester, D.

G. A. Durkin, C. Simon, and D. Bouwmeester, "Multiphoton entanglement concentration and quantum cryptography," Phys. Rev. Lett. 88, 187902 (2002).
[CrossRef] [PubMed]

A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, "Stimulated emission of polarization-entangled photons," Nature 412, 887-890 (2001).
[CrossRef] [PubMed]

Bremmer, H.

B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-sided Laplace Integral (Cambridge U. Press, 1964).

Canioni, L.

Carlson, N. W.

Cha, M.

Chase, E. W.

da Silva, V. L.

Ding, Y. J.

Y. J. Ding, S. J. Lee, and J. B. Khurgin, "Transversely pumped counterpropagating optical parametric oscillation and amplification," Phys. Rev. Lett. 75, 429-432 (1995).
[CrossRef] [PubMed]

Durkin, G. A.

G. A. Durkin, C. Simon, and D. Bouwmeester, "Multiphoton entanglement concentration and quantum cryptography," Phys. Rev. Lett. 88, 187902 (2002).
[CrossRef] [PubMed]

Fainman, Y.

D. M. Marom, D. Panasenko, P.-C. Sun, Y. T. Mazurenko, and Y. Fainman, "Real-time spatial-temporal signal processing with optical nonlinearities," IEEE J. Sel. Top. Quantum Electron. 7, 683-693 (2001).

D. M. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, "Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves," Opt. Lett. 25, 132-134 (2000).

Fan, H.-Y.

H.-Y. Fan and N.-Q. Jiang, "Special two-mode unitary transform and maximum entanglement state for four wave mixing," Phys. Scr. 71, 277-279 (2005).
[CrossRef]

Fekete, D.

Feldman, B. J.

R. A. Fisher, B. R. Suydam, and B. J. Feldman, "Transient analysis of Kerr-like phase conjugators using frequency-domain techniques," Phys. Rev. A 23, 3071-3083 (1981).
[CrossRef]

Fiorentino, M.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kaertner, "Two-photon coincident frequency entanglement via extended phase matching," Phys. Rev. Lett. 94, 083601 (2005).
[CrossRef] [PubMed]

Fisher, R. A.

R. A. Fisher, B. R. Suydam, and B. J. Feldman, "Transient analysis of Kerr-like phase conjugators using frequency-domain techniques," Phys. Rev. A 23, 3071-3083 (1981).
[CrossRef]

Fraigne, S.

Galaup, J. P.

Giovannetti, V.

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, "Generating entangled two-photon states with coincident frequencies," Phys. Rev. Lett. 88, 183602 (2002).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, "Quantum-enhanced positioning and clock synchronization," Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

Grousson, R.

Harris, S. E.

S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

Heffner, H.

H. Heffner, "Analysis of the backward-wave traveling-wave tube," Proc. IRE 42, 930-937 (1954).
[CrossRef]

Heritage, J. P.

Howell, J. C.

A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, "Stimulated emission of polarization-entangled photons," Nature 412, 887-890 (2001).
[CrossRef] [PubMed]

Jiang, N.-Q.

H.-Y. Fan and N.-Q. Jiang, "Special two-mode unitary transform and maximum entanglement state for four wave mixing," Phys. Scr. 71, 277-279 (2005).
[CrossRef]

Joffre, M.

Joubert, C.

Kaertner, F. X.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kaertner, "Two-photon coincident frequency entanglement via extended phase matching," Phys. Rev. Lett. 94, 083601 (2005).
[CrossRef] [PubMed]

Khurgin, J. B.

Y. J. Ding, S. J. Lee, and J. B. Khurgin, "Transversely pumped counterpropagating optical parametric oscillation and amplification," Phys. Rev. Lett. 75, 429-432 (1995).
[CrossRef] [PubMed]

Kompfner, R.

R. Kompfner and N. T. Williams, "Backward-wave tubes," Proc. IRE 41, 1602-1611 (1953).
[CrossRef]

Konig, F.

F. Konig and F. N. C. Wong, "Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch," Appl. Phys. Lett. 84, 1644-1646 (2004).
[CrossRef]

Kroll, N. M.

N. M. Kroll, "Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves," J. Appl. Phys. 36, 34-43 (1965).
[CrossRef]

Kuhl, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93-95 (1989).
[CrossRef]

Kurimura, S.

Kuzucu, O.

O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kaertner, "Two-photon coincident frequency entanglement via extended phase matching," Phys. Rev. Lett. 94, 083601 (2005).
[CrossRef] [PubMed]

Lamas-Linares, A.

A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, "Stimulated emission of polarization-entangled photons," Nature 412, 887-890 (2001).
[CrossRef] [PubMed]

Le Gouet, J. L.

Leaird, D. E.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251-2261 (1992).
[CrossRef]

Lee, S. J.

Y. J. Ding, S. J. Lee, and J. B. Khurgin, "Transversely pumped counterpropagating optical parametric oscillation and amplification," Phys. Rev. Lett. 75, 429-432 (1995).
[CrossRef] [PubMed]

Likforman, J. P.

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, "Quantum-enhanced positioning and clock synchronization," Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

Maccone, L.

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, "Generating entangled two-photon states with coincident frequencies," Phys. Rev. Lett. 88, 183602 (2002).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, "Quantum-enhanced positioning and clock synchronization," Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

Marom, D. M.

D. M. Marom, D. Panasenko, P.-C. Sun, Y. T. Mazurenko, and Y. Fainman, "Real-time spatial-temporal signal processing with optical nonlinearities," IEEE J. Sel. Top. Quantum Electron. 7, 683-693 (2001).

D. M. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, "Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves," Opt. Lett. 25, 132-134 (2000).

Mazurenko, Y. T.

D. M. Marom, D. Panasenko, P.-C. Sun, Y. T. Mazurenko, and Y. Fainman, "Real-time spatial-temporal signal processing with optical nonlinearities," IEEE J. Sel. Top. Quantum Electron. 7, 683-693 (2001).

Michaeli, G.

G. Michaeli and A. Arie, "Optimization of quasi-phase-matched non-linear frequency conversion for diffusion bonding applications," Appl. Phys. B 77, 497-503 (2003).
[CrossRef]

Miller, D. A. B.

Mossberg, T. W.

Paek, E. G.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251-2261 (1992).
[CrossRef]

Panasenko, D.

D. M. Marom, D. Panasenko, P.-C. Sun, Y. T. Mazurenko, and Y. Fainman, "Real-time spatial-temporal signal processing with optical nonlinearities," IEEE J. Sel. Top. Quantum Electron. 7, 683-693 (2001).

D. M. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, "Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves," Opt. Lett. 25, 132-134 (2000).

Pepper, D. M.

Psaltis, D.

M. Tsang and D. Psaltis, "Spontaneous spectral phase conjugation for coincident frequency entanglement," Phys. Rev. A 71, 043806 (2005).
[CrossRef]

M. Tsang and D. Psaltis, "Spectral phase conjugation by quasi-phase-matched three-wave mixing," Opt. Commun. 242, 659-664 (2004).
[CrossRef]

M. Tsang and D. Psaltis, "Spectral phase conjugation with cross-phase modulation compensation," Opt. Express 12, 2207-2219 (2004).
[CrossRef] [PubMed]

M. Tsang and D. Psaltis, "Dispersion and nonlinearitycompensation by spectral phase conjugation," Opt. Lett. 28, 1558-1560 (2003).
[CrossRef] [PubMed]

Rebane, A.

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93-95 (1989).
[CrossRef]

Reitze, D. H.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251-2261 (1992).
[CrossRef]

Ro, J. H.

Roblin, M. L.

Rokitski, R.

Rothberg, L. J.

Saifi, M. A.

Saleh, B. E. A.

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion," Phys. Rev. A 67, 053810 (2003).
[CrossRef]

Sergienko, A. V.

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion," Phys. Rev. A 67, 053810 (2003).
[CrossRef]

Shapiro, J. H.

V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, "Generating entangled two-photon states with coincident frequencies," Phys. Rev. Lett. 88, 183602 (2002).
[CrossRef] [PubMed]

Silberberg, Y.

Simon, C.

G. A. Durkin, C. Simon, and D. Bouwmeester, "Multiphoton entanglement concentration and quantum cryptography," Phys. Rev. Lett. 88, 187902 (2002).
[CrossRef] [PubMed]

Sun, P.-C.

D. M. Marom, D. Panasenko, P.-C. Sun, Y. T. Mazurenko, and Y. Fainman, "Real-time spatial-temporal signal processing with optical nonlinearities," IEEE J. Sel. Top. Quantum Electron. 7, 683-693 (2001).

D. M. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, "Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves," Opt. Lett. 25, 132-134 (2000).

Suydam, B. R.

R. A. Fisher, B. R. Suydam, and B. J. Feldman, "Transient analysis of Kerr-like phase conjugators using frequency-domain techniques," Phys. Rev. A 23, 3071-3083 (1981).
[CrossRef]

Taira, T.

Teich, M. C.

Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion," Phys. Rev. A 67, 053810 (2003).
[CrossRef]

Tsang, M.

M. Tsang and D. Psaltis, "Spontaneous spectral phase conjugation for coincident frequency entanglement," Phys. Rev. A 71, 043806 (2005).
[CrossRef]

M. Tsang and D. Psaltis, "Spectral phase conjugation by quasi-phase-matched three-wave mixing," Opt. Commun. 242, 659-664 (2004).
[CrossRef]

M. Tsang and D. Psaltis, "Spectral phase conjugation with cross-phase modulation compensation," Opt. Express 12, 2207-2219 (2004).
[CrossRef] [PubMed]

M. Tsang and D. Psaltis, "Dispersion and nonlinearitycompensation by spectral phase conjugation," Opt. Lett. 28, 1558-1560 (2003).
[CrossRef] [PubMed]

van der Pol, B.

B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-sided Laplace Integral (Cambridge U. Press, 1964).

Walton, Z. D.

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Figures (5)

Fig. 1
Fig. 1

Schematic of SPC via type-II EPM. The signal and idler pulses, in orthogonal polarizations, have carrier frequencies of ω s and ω i , while the pump pulse has a carrier frequency of ω p = ω s + ω i . The EPM condition requires that the signal and the idler counterpropagate with respect to the pump, which should be much shorter than the input signal.

Fig. 2
Fig. 2

Normalized poles p ( χ A p 0 ) plotted against G, obtained by numerically solving Eq. (32), indicating the onset of spatial instability beyond the threshold G > π 2 . More poles appear as G is increased.

Fig. 3
Fig. 3

Plots of intensity and phase of the input signal, output signal, and output idler, from numerical analysis of Eqs. (5, 6). Parameters used are k p = 1 ( 1.5 × 10 8 ms 1 ) , k s = 1.025 k p , k i = 0.975 k p , T p = 100 fs , T s = 2 ps , L = 10 cm , t s = 4 T s , beam diameter = 200 μ m , A s 0 = 0.5 exp [ ( t 2 T s ) 2 ( 2 T s 2 ) ] exp [ ( 1 + 0.5 j ) ( t + 2 T s ) 2 ( 2 T s 2 ) ] , A p 0 = exp [ t 2 ( 2 T p 2 ) ] , and G = π 4 . The plots clearly show that the idler is the time-reversed and phase-conjugated replica, i.e., SPC, of the signal.

Fig. 4
Fig. 4

Signal gain η + 1 and idler gain η versus G from numerical analysis compared with theory. See the caption of Fig. 3 for parameters used.

Fig. 5
Fig. 5

Plot of numerical idler gain η in decibels against G for L = 10 cm (solid curve) and L = 1 cm (dashed–dotted curve), compared with the Fourier theory (dashed curve), tan 2 ( G ) in decibels. Three distinct regimes can be observed for the L = 10 cm case: the moderate-gain regime where the Fourier theory is accurate, the unstable regime where the gain increases exponentially, and the oscillation regime where significant pump depletion occurs. For L = 1 cm , the medium is not long enough for oscillation to occur in the parameter range of interest.

Equations (41)

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A p z + k p A p t = j χ p A s A i ,
A s z + k s A s t = j χ s A p A i * ,
A i * z + k i A i * t = j χ i A p * A s ,
A p z = j χ p A s A i ,
A s z + ( k s k p ) A s τ = j χ s A p A i * ,
A i * z + ( k i k p ) A i * τ = j χ i A p * A s .
L T s , i k s , i k p ,
A ̃ s ( κ , τ ) A s ( z , τ ) exp ( j κ z ) d z ,
A ̃ i * ( κ , τ ) A i * ( z , τ ) exp ( j κ z ) d z .
j κ A ̃ s + ( k s k p ) A ̃ s τ = j χ s A p 0 ( τ ) A ̃ i * ,
j κ A ̃ i * + ( k i k p ) A ̃ i * τ = j χ i A p 0 ( τ ) A ̃ s .
γ s k s k p , γ i k i k p , r γ s χ i γ i χ s .
A = r A ̃ s exp ( j κ γ s τ ) , B = A ̃ i * exp ( j κ γ i τ ) ,
A τ = j χ s χ i γ s γ i A p 0 ( τ ) B exp [ j κ ( 1 γ s 1 γ i ) τ ] ,
B τ = j χ s χ i γ s γ i A p 0 ( τ ) A exp [ j κ ( 1 γ s 1 γ i ) τ ] .
T p 2 π κ ( 1 γ s 1 γ i ) T s , i γ s , i ( 1 γ s 1 γ i ) ,
g ( τ ) χ s χ i γ s γ i A p 0 ( τ ) ,
A τ = j g ( τ ) B ,
B τ = j g ( τ ) A .
A ( κ , τ ) = sec ( G ) { A ( κ , T p 2 ) cos [ T p 2 τ g ( τ ) d τ ] + j B ( κ , T p 2 ) sin [ T p 2 τ g ( τ ) d τ ] } ,
B ( κ , τ ) = sec ( G ) { j A ( κ , T p 2 ) sin [ T p 2 τ g ( τ ) d τ ] + B ( κ , T p 2 ) cos [ T p 2 τ g ( τ ) d τ ] } ,
G T p 2 T p 2 g ( τ ) d τ g ( τ ) d τ .
A s ( L , t ) = A s 0 ( t k s L + t s ) sec ( G ) + j 1 r A i 0 * [ 1 r ( t k s L t i ) ] tan ( G ) ,
A i ( L , t ) = A i 0 ( t k i L + t i ) sec ( G ) + j r A s 0 * [ r ( t k i L + t s ) ] tan ( G ) .
k s + k i = 2 k p , k s k i ,
η A i ( L , t ) 2 d t A s ( 0 , t ) 2 d t = tan 2 ( G ) .
A ¯ s ( p , τ ) A s ( z , τ ) exp ( p z ) d z ,
A ¯ i * ( p , τ ) A i * ( z , τ ) exp ( p z ) d z .
A ¯ s ( p , T p 2 ) = 1 P 2 csc ( G 1 P 2 ) P + 1 P 2 cot ( G 1 P 2 ) A ¯ s ( p , T p 2 ) ,
A ¯ i * ( p , T p 2 ) = j P + 1 P 2 cot ( G 1 P 2 ) A ¯ s ( p , T p 2 ) ,
P p χ A p 0 , G χ A p 0 ( T p γ ) .
p + ( χ A p 0 ) 2 p 2 cot [ G 1 p 2 ( χ A p 0 ) 2 ] = 0 .
A s , i ( z , τ ) z = j χ A p 0 ( τ ) A i , s * ( z , τ ) ,
A s , i ( z , τ ) = A s , i ( 0 , τ ) cosh [ χ A p 0 ( τ ) z ] + j A i , s * ( 0 , τ ) sinh [ χ A p 0 ( τ ) z ] .
A ̂ s = A ̂ s 0 sec ( G ) + j A ̂ i 0 tan ( G ) ,
A ̂ i = j A ̂ s 0 tan ( G ) + A ̂ i 0 sec ( G ) .
n s , i A ̂ s , i A ̂ s , i = A ̂ s , i A ̂ s , i 1 ,
A ̂ s 0 A ̂ i 0 = A ̂ i 0 A ̂ s 0 = A ̂ s 0 A ̂ i 0 = A ̂ i 0 A ̂ s 0 = 0 .
n s = n s 0 sec 2 ( G ) + ( n i 0 + 1 ) tan 2 ( G ) ,
n i = n i 0 sec 2 ( G ) + ( n s 0 + 1 ) tan 2 ( G ) .
ψ = cos ( G ) n = 0 sin n ( G ) n s n i ,

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