Abstract

We experimentally demonstrate simultaneous phase and group velocity locking of fundamental and generated second harmonic pulses in Lithium Niobate, under conditions of material phase mismatch. In phase-mismatched, pulsed second harmonic generation in addition to a reflected signal two forward-propagating pulses are also generated at the interface between a linear and a second order nonlinear material: the first pulse results from the solution of the homogeneous wave equation, and propagates at the group velocity expected from material dispersion; the second pulse is the solution of the inhomogeneous wave equation, is phase-locked and trapped by the pump pulse, and follows the pump trajectory. At normal incidence, the normal and phase locked pulses simply trail each other. At oblique incidence, the consequences can be quite dramatic. The homogeneous pulse refracts as predicted by material dispersion and Snell’s law, yielding at least two spatially separate second harmonic spots at the medium’s exit. We thus report the first experimental results showing that, at oblique incidence, fundamental and phase-locked second harmonic pulses travel with the same group velocity and follow the same trajectory. This is direct evidence that, at least up to first order, the effective dispersion of the phase-locked pulse is similar to the dispersion of the pump pulse.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. A. Feisst and P. Koidl, "Current induced periodic ferroelectric domain structures in LiNbO3 applied for efficient nonlinear optical frequency mixing," Appl. Phys. Lett. 47, 1125 (1985).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. S. K. Kurtz and T. T. Perry, "A Powder Technique for the Evaluation of Nonlinear Optical Materials," J. Appl. Phys. 39, 3798 (1968).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  21. P. D. Marker, R. W. Terhune, M. Nisenoff and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics", Phys. Rev. Lett. 8, 21 (1962).
    [CrossRef]

2008

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

2007

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

2006

1999

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, "Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric", Phys. Rev. Lett. 83, 2934 (1999).
[CrossRef]

1998

1997

1996

K. Sakoda and K. Othaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742 (1996).
[CrossRef]

1992

1990

D. Noordam, H. J. Bakker, M. P. de Boer, H. B. van Linden van den Heuvell, "Second-harmonic generation of femtosecond pulses: observation of phase-mismatch effects," Opt. Lett. 15, 1464 (1990)
[CrossRef] [PubMed]

G. A. Magel, M. M. Fejer, and R. L. Byer, "Quasi-phase-matched second-harmonic generation of blue light in periodically poled LiNbO3," Appl. Phys. Lett. 56, 108 (1990).
[CrossRef]

1988

1986

1985

A. Feisst and P. Koidl, "Current induced periodic ferroelectric domain structures in LiNbO3 applied for efficient nonlinear optical frequency mixing," Appl. Phys. Lett. 47, 1125 (1985).
[CrossRef]

1976

J. P. van der Ziel, "Phase−matched harmonic generation in a laminar structure with wave propagation in the plane of the layers," Appl. Phys. Lett. 26, 60 (1976).
[CrossRef]

1969

N. Bloembergen, H. J. Simon, and C. H. Lee, "Total Reflection Phenomena in Second-Harmonic Generation of Light," Phys. Rev. 181, 1261 (1969).
[CrossRef]

1968

S. K. Kurtz and T. T. Perry, "A Powder Technique for the Evaluation of Nonlinear Optical Materials," J. Appl. Phys. 39, 3798 (1968).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
[CrossRef]

N. Bloembergen and P. S. Pershan, "Light Waves at the Boundary of Nonlinear Media," Phys. Rev. 128, 606 (1962).
[CrossRef]

P. D. Maker, R. W. Terhune, N. Nisenoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

P. D. Marker, R. W. Terhune, M. Nisenoff and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics", Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Akozbek, N.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
[CrossRef]

Bakker, H. J.

Beckwitt, K.

Bertolotti, M.

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

E. Fazio, M. Zitelli, S. Dominici, C. Sibilia, G. D'Aguanno, and M. Bertolotti, "Phase-driven pulse breaking during perfectly-matched second harmonic generation," Opt. Commun. 148, 427 (1998).
[CrossRef]

Bloembergen, N.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, "Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric", Phys. Rev. Lett. 83, 2934 (1999).
[CrossRef]

N. Bloembergen, H. J. Simon, and C. H. Lee, "Total Reflection Phenomena in Second-Harmonic Generation of Light," Phys. Rev. 181, 1261 (1969).
[CrossRef]

N. Bloembergen and P. S. Pershan, "Light Waves at the Boundary of Nonlinear Media," Phys. Rev. 128, 606 (1962).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
[CrossRef]

Bloemer, M. J.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Byer, R. L.

G. A. Magel, M. M. Fejer, and R. L. Byer, "Quasi-phase-matched second-harmonic generation of blue light in periodically poled LiNbO3," Appl. Phys. Lett. 56, 108 (1990).
[CrossRef]

Carlotti, X.

Centini, M.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Chen, Y.

Cussat-Blanc, S.

D'Aguanno, G.

E. Fazio, M. Zitelli, S. Dominici, C. Sibilia, G. D'Aguanno, and M. Bertolotti, "Phase-driven pulse breaking during perfectly-matched second harmonic generation," Opt. Commun. 148, 427 (1998).
[CrossRef]

de Boer, M. P.

de Ceglia, D.

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

DeSalvo, R. J.

Dominici, S.

E. Fazio, M. Zitelli, S. Dominici, C. Sibilia, G. D'Aguanno, and M. Bertolotti, "Phase-driven pulse breaking during perfectly-matched second harmonic generation," Opt. Commun. 148, 427 (1998).
[CrossRef]

Ducasse, A.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
[CrossRef]

Fazio, E.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

E. Fazio, M. Zitelli, S. Dominici, C. Sibilia, G. D'Aguanno, and M. Bertolotti, "Phase-driven pulse breaking during perfectly-matched second harmonic generation," Opt. Commun. 148, 427 (1998).
[CrossRef]

Feisst, A.

A. Feisst and P. Koidl, "Current induced periodic ferroelectric domain structures in LiNbO3 applied for efficient nonlinear optical frequency mixing," Appl. Phys. Lett. 47, 1125 (1985).
[CrossRef]

Fejer, M. M.

G. A. Magel, M. M. Fejer, and R. L. Byer, "Quasi-phase-matched second-harmonic generation of blue light in periodically poled LiNbO3," Appl. Phys. Lett. 56, 108 (1990).
[CrossRef]

Foreman, J. V.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

Freysz, E.

Fu, X.

Hache, F.

Hagan, D. J.

Haus, J. W.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Ivanov, A.

Kandidov, V. P.

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Koidl, P.

A. Feisst and P. Koidl, "Current induced periodic ferroelectric domain structures in LiNbO3 applied for efficient nonlinear optical frequency mixing," Appl. Phys. Lett. 47, 1125 (1985).
[CrossRef]

Kosareva, O. G.

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Kothari, N. C.

Kurtz, S. K.

S. K. Kurtz and T. T. Perry, "A Powder Technique for the Evaluation of Nonlinear Optical Materials," J. Appl. Phys. 39, 3798 (1968).
[CrossRef]

Lee, C. H.

N. Bloembergen, H. J. Simon, and C. H. Lee, "Total Reflection Phenomena in Second-Harmonic Generation of Light," Phys. Rev. 181, 1261 (1969).
[CrossRef]

Luo, H.

Magel, G. A.

G. A. Magel, M. M. Fejer, and R. L. Byer, "Quasi-phase-matched second-harmonic generation of blue light in periodically poled LiNbO3," Appl. Phys. Lett. 56, 108 (1990).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, N. Nisenoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Maleck Rassoul, R.

Maleck-Rassoul, R.

Margulis, W.

Marker, P. D.

P. D. Marker, R. W. Terhune, M. Nisenoff and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics", Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Mlejnek, M.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, "Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric", Phys. Rev. Lett. 83, 2934 (1999).
[CrossRef]

Moloney, J. V.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, "Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric", Phys. Rev. Lett. 83, 2934 (1999).
[CrossRef]

Nisenoff, M.

P. D. Marker, R. W. Terhune, M. Nisenoff and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics", Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Nisenoff, N.

P. D. Maker, R. W. Terhune, N. Nisenoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Noordam, D.

Osterberg, U.

Othaka, K.

K. Sakoda and K. Othaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742 (1996).
[CrossRef]

Perry, T. T.

S. K. Kurtz and T. T. Perry, "A Powder Technique for the Evaluation of Nonlinear Optical Materials," J. Appl. Phys. 39, 3798 (1968).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
[CrossRef]

N. Bloembergen and P. S. Pershan, "Light Waves at the Boundary of Nonlinear Media," Phys. Rev. 128, 606 (1962).
[CrossRef]

Pettazzi, F.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

Qian, L.

Roppo, V.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Sakoda, K.

K. Sakoda and K. Othaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742 (1996).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, N. Nisenoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

P. D. Marker, R. W. Terhune, M. Nisenoff and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics", Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Scalora, M.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

Sheik-Bahae, M.

Sibilia, C.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007)
[CrossRef]

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[CrossRef]

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M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008)
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Figures (4)

Fig. 1.
Fig. 1.

Numerical simulation of pulsed second-harmonic in a generic Lorentz medium, having γ=10-8, ωp=4, ω0=4, under phase-mismatched conditions. The small γ keeps absorption at negligible levels. The yellow box delineates the medium. (a) pump pulse. (b)SH pulses. Two SH signals are discernable, one that tracks the pumps pulse (yellow arrow), the other that refracts according to material dispersion (red arrow). These two components travel with different group velocities and thus tend to separate as distance is gained inside the sample. The leading pulse is phase-locked and trapped by the pump; the second pulse propagates freely and at a group velocity approximately three times smaller compared to the pump.

Fig. 2.
Fig. 2.

The experimental set-up. A prismatic lithium niobate crystal was used to generate the second harmonic 400 nm signal from a 800 nm pump pulse. The titled output face of the crystal forced the pulses to exit at different angles, according to their refractive indices. The locked arm was then refocused onto a thin BBO crystal at perfect phase-matching for ω+2ω=3ω interaction.

Fig. 3.
Fig. 3.

Experimental images of the fundamental (800 nm) and second harmonic (400 nm) signals exiting the tilted output surface. Each position corresponds to a different output angle according to the Snell’s law.

Fig. 4.
Fig. 4.

The generated 3ω signal as recorded by the photomultiplier tube (I), with a thin (II) or a thick (III) glass plate at the reference position of the experimental set-up in fig.2.

Equations (8)

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H = x ̂ = 1 ( 𝓗 x ω ( z , y , t ) + c . c ) = x ̂ = 1 ( 𝓗 x ω ( z , y , t ) e i ( kz ωt ) + c . c )
E = y ̂ = 1 ( E y ℓω ( z , y , t ) + c . c ) + z ̂ = 1 ( E z ℓω ( z , y , t ) + c . c ) = y ̂ = 1 ( y ℓω ( z , y , t ) + c . c ) + z ̂ = 1 ( z ℓω ( z , y , t ) + c . c )
𝓗 x ℓω τ = i β ( 𝓗 z ℓω + z ℓω sin θ i + y ℓω cos θ i ) z ℓω y ˜ + y ℓω ξ
y ℓω τ = i β ( y ℓω + 𝓗 x ℓω cos θ i ) 4 π ( 𝒥 y ℓω i β 𝒫 y ℓω ) + i 4 π β 𝒫 y , NL ℓω 4 π 𝒫 y , NL ℓω τ + 𝓗 x ℓω ξ
z ℓω τ = i β ( z ℓω + 𝓗 x ℓω sin θ i ) 4 π ( 𝒥 z ℓω i β 𝒫 z ℓω ) + i 4 π β 𝒫 z , NL ℓω 4 π 𝒫 z , NL ℓω τ 𝓗 x ℓω y ˜ .
𝒥 y ℓω τ = ( 2 i β γ ) 𝒥 y ℓω + ( β 2 + β β r 2 ) 𝒫 y ℓω + π ω 𝒫 2 ω 0 2 y ℓω
𝒥 z ℓω τ = ( 2 i β γ ) 𝒥 z ℓω + ( β 2 + β β r 2 ) 𝒫 z ℓω + π ω 𝒫 2 ω 0 2 z ℓω
𝒥 y ℓω = 𝒫 y ℓω τ ; 𝒥 z ℓω = 𝒫 z ℓω τ

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