Abstract

Using a first-order multiple-scale expansion approach, we derive a set of coupled-mode equations that describe both forward and backward second-harmonic generation and amplification processes in nonlinear, one-dimensional, multilayered structures of finite length. The theory is valid for index modulation of arbitrary depth and profile. We derive analytical solutions in the undepleted pump regime under different pumping circumstances. The model shows excellent agreement with the numerical integration of Maxwell’s equations.

© 2002 Optical Society of America

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  1. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
    [CrossRef]
  2. M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
    [CrossRef]
  3. A. V. Balakin, D. Boucher, V. A. Bushev, N. I. Koroteev, B. I. Mantsyzov, P. Masselin, I. A. Ozheredov, and A. P. Shurinov, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
    [CrossRef]
  4. G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
    [CrossRef]
  5. Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
    [CrossRef]
  6. M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
    [CrossRef]
  7. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
    [CrossRef]
  8. J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
    [CrossRef]
  9. M. J. Steel and C. M. de Sterke, “Second-harmonic generation in second-harmonic fiber Bragg gratings,” Appl. Opt. 35, 3211–3222 (1996).
    [CrossRef] [PubMed]
  10. C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
    [CrossRef] [PubMed]
  11. A. Arraf and C. M. de Sterke, “Coupled-mode equations for quadratically nonlinear deep gratings,” Phys. Rev. E 58, 7951–7958 (1998), and references therein.
    [CrossRef]
  12. T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
    [CrossRef]
  13. O. Di Stefano, S. Savasta, and R. Girlanda, “Mode expansion and photon operators in dispersive and absorbing dielectrics,” J. Mod. Opt. 48, 67–84 (2001).
    [CrossRef]
  14. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  15. A. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1993).
  16. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
    [CrossRef]
  17. G. D’Aguanno, “Nonlinear χ(2) interactions in bulk and stratified materials,” Ph.D. Thesis (National Library of Italy, Rome, 1999) pp. 72–78.
  18. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  19. The numerical calculations were performed by use of a fast Fourier transform beam-propagation method that integrates the equations of motion in the time domain, as outlined in Ref. 1. Incident pulses are assumed to be 2 ps in duration and have the following intensity profile in time: Ĩ(pump)(t)=I(peak) exp[−(t2/2σ2)], where I(peak) is the peak intensity of the pump. A comparison with plane-wave results by use of pulses is possible because the spatial extension of these pulses is nearly 3 orders of magnitude greater than the structure length. As far as the structure is concerned, this incident pulse is nearly monochromatic, and the dynamics yield results that are nearly identical to the plane-wave results, provided the intensity of the plane wave is properly averaged over the pulse width. The average intensity of an incident plane-wave pump field is defined as I(pump)=I(peak)4σ −∞+∞ exp−t22dt=(π/8)1/2I(peak).
  20. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).
  21. S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
    [CrossRef] [PubMed]
  22. P. M. Lushnikov, P. Lodahl, and M. Saffman, “Transverse modulational instability of counterpropagating quasi-phase-matched beams in a quadratically nonlinear medium” Opt. Lett. 23, 1650–1652 (1999), and references therein.
    [CrossRef]

2001

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

O. Di Stefano, S. Savasta, and R. Girlanda, “Mode expansion and photon operators in dispersive and absorbing dielectrics,” J. Mod. Opt. 48, 67–84 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

2000

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
[CrossRef]

1999

1998

A. Arraf and C. M. de Sterke, “Coupled-mode equations for quadratically nonlinear deep gratings,” Phys. Rev. E 58, 7951–7958 (1998), and references therein.
[CrossRef]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

1997

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

1996

M. J. Steel and C. M. de Sterke, “Second-harmonic generation in second-harmonic fiber Bragg gratings,” Appl. Opt. 35, 3211–3222 (1996).
[CrossRef] [PubMed]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1988

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[CrossRef] [PubMed]

Arraf, A.

A. Arraf and C. M. de Sterke, “Coupled-mode equations for quadratically nonlinear deep gratings,” Phys. Rev. E 58, 7951–7958 (1998), and references therein.
[CrossRef]

Balakin, A. V.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bertolotti, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

Bloemer, M.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Bloemer, M. J.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Boucher, D.

Bowden, C. M.

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Bushev, V. A.

Centini, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

Cole, J. D.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

de Sterke, C. M.

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
[CrossRef]

A. Arraf and C. M. de Sterke, “Coupled-mode equations for quadratically nonlinear deep gratings,” Phys. Rev. E 58, 7951–7958 (1998), and references therein.
[CrossRef]

M. J. Steel and C. M. de Sterke, “Second-harmonic generation in second-harmonic fiber Bragg gratings,” Appl. Opt. 35, 3211–3222 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[CrossRef] [PubMed]

Di Stefano, O.

O. Di Stefano, S. Savasta, and R. Girlanda, “Mode expansion and photon operators in dispersive and absorbing dielectrics,” J. Mod. Opt. 48, 67–84 (2001).
[CrossRef]

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Dumeige, Y.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

Girlanda, R.

O. Di Stefano, S. Savasta, and R. Girlanda, “Mode expansion and photon operators in dispersive and absorbing dielectrics,” J. Mod. Opt. 48, 67–84 (2001).
[CrossRef]

Haus, J. W.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Iizuka, T.

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
[CrossRef]

Kalocsai, A. G.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

Kohl, I.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Koroteev, N. I.

Levenson, A.

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

Levenson, J. A.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Lodahl, P.

Lushnikov, P. M.

Madsen, J. B.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Mantsyzov, B. I.

Masselin, P.

Nefedov, I.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Ozheredov, I. A.

Peatross, J.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Saffman, M.

Sagnes, I.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

Sauvage, S.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Savasta, S.

O. Di Stefano, S. Savasta, and R. Girlanda, “Mode expansion and photon operators in dispersive and absorbing dielectrics,” J. Mod. Opt. 48, 67–84 (2001).
[CrossRef]

Scalora, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Shurinov, A. P.

Sibilia, C.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

Simmons, J.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[CrossRef] [PubMed]

Steel, M. J.

Terry, N.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Theimer, J.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

Titensor, J.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Vidakovic, P.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Viswanathan, R.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Voronov, S. L.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Wang, Q.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in a 1-D semiconductor photonic bandgap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

J. Mod. Opt.

O. Di Stefano, S. Savasta, and R. Girlanda, “Mode expansion and photon operators in dispersive and absorbing dielectrics,” J. Mod. Opt. 48, 67–84 (2001).
[CrossRef]

Opt. Lett.

Opt. Photon. News

M. Scalora, M. J. Bloemer, C. M. Bowden, G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, Y. Dumeige, I. Sagnes, P. Vidakovic, and A. Levenson, “Choose your color from the photonic band edge: nonlinear frequency conversion,” Opt. Photon. News 12, 36–40 (2001).
[CrossRef]

Phys. Rev. A

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structure,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[CrossRef] [PubMed]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998), and references therein.
[CrossRef]

Phys. Rev. E

A. Arraf and C. M. de Sterke, “Coupled-mode equations for quadratically nonlinear deep gratings,” Phys. Rev. E 58, 7951–7958 (1998), and references therein.
[CrossRef]

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609–016619 (2001), and references therein.
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610–036615 (2001).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Phys. Rev. Lett.

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902–133914 (2001), and references therein.
[CrossRef] [PubMed]

Other

The numerical calculations were performed by use of a fast Fourier transform beam-propagation method that integrates the equations of motion in the time domain, as outlined in Ref. 1. Incident pulses are assumed to be 2 ps in duration and have the following intensity profile in time: Ĩ(pump)(t)=I(peak) exp[−(t2/2σ2)], where I(peak) is the peak intensity of the pump. A comparison with plane-wave results by use of pulses is possible because the spatial extension of these pulses is nearly 3 orders of magnitude greater than the structure length. As far as the structure is concerned, this incident pulse is nearly monochromatic, and the dynamics yield results that are nearly identical to the plane-wave results, provided the intensity of the plane wave is properly averaged over the pulse width. The average intensity of an incident plane-wave pump field is defined as I(pump)=I(peak)4σ −∞+∞ exp−t22dt=(π/8)1/2I(peak).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

G. D’Aguanno, “Nonlinear χ(2) interactions in bulk and stratified materials,” Ph.D. Thesis (National Library of Italy, Rome, 1999) pp. 72–78.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

A. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1993).

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Figures (7)

Fig. 1
Fig. 1

(a) Transmittance plotted versus the normalized frequency for a typical structure made of a half–quarter-wave stack. The structure is composed of 39 alternating layers of high and low refractive index. The indices of refraction of the layers are, respectively: n1=1.4285 and n2=1. The layers have thicknesses of a=λ0/(2n1) and b=λ0/(4n2), where λ0=1 µm and ω0=2πc/λ0; the total length of the structure is L=11.75 µm. The arrow identifies the first transmission resonance near the first-order bandgap. (b) Magnification of the first transmission resonance (solid curve) near the first-order bandgap. Also shown is the power spectrum (dashed curve) of a Gaussian input pulse of 2 ps in duration with its carrier frequency tuned to the first transmission resonance. Note that the spectral bandwidth of the pulse is approximately 1 order of magnitude narrower than the spectral bandwidth of the transmission resonance. This is a typical situation in which the interaction can be described by monochromatic waves.

Fig. 2
Fig. 2

Schematic representation of the boundary conditions imposed on: (a) the LTR and (b) the RTL modes. The terms rjω(±) are the LTR and the RTL reflection coefficients, respectively, and tjω(±) is the transmission coefficient. We find that tjω(+)=tjω(-)=tjω for a nonabsorbing structure as a consequence of time-reversal symmetry.

Fig. 3
Fig. 3

Forward (solid curve) and backward (dashed curve) conversion efficiency, defined by η(±)=Iω(±)/Iω(+,pump), plotted versus the input pump intensity Iω(+,pump) for an undepleted pump, as given by Eqs. (22). The layer thickness is L=10 µm, the refractive index is n=2.5, the nonlinear coefficient is taken to be equal to d(2)=100 pm/V. The wavelength of the pump beam is λ=1 µm, and the pump is tuned to the m=50 transmission resonance.

Fig. 4
Fig. 4

Forward (solid curve) and backward (dashed curve) conversion efficiencies for η(±)=Iω(±)/(Iω(+,pump)+Iω(-,pump)) plotted versus the phase difference of the input fields δϕω=γω(-)-γω(+) for different pump-intensity ratios Q=Iω(-,pump)/Iω(+,pump). (a) Q=0.5, (b) Q=1, (c) Q=2. For Q=1 the forward and the backward conversion efficiency are equal. All the parameters are the same as reported in the caption of Fig. 3 and Iω(+,pump)=0.1 GW/cm2.

Fig. 5
Fig. 5

(a) Generic N-period stack composed of two-layer unit cells of thicknesses a and b and constant, real indices n1 and n2, respectively. (b) Symmetry is restored to the structure by removing the last layer.

Fig. 6
Fig. 6

Forward (solid curve) and backward (dashed curve) SH conversion efficiencies for η(±)=Iω(±)/Iω(+,pump) plotted versus the input pump intensity as calculated from Eqs. (31) in the undepleted pump regime. Forward (squares) and backward (circles) conversion efficiencies were calculated by use of the algorithm of the fast Fourier transform beam-propagation method. The structure is the same as that described in the caption of Figs. 1. The nonlinear material is dispersed in the high-index layer, with d(2)=80 pm/V. The high-index material is also assumed to be dispersive, and its index of refraction at the SH frequency is n1(2ω)=1.5179. The wavelength of the incident pump is λ=1.69 µm and is tuned to the first transmission resonance near the first-order bandgap.

Fig. 7
Fig. 7

Forward (solid curves) and backward (dashed curves) conversion efficiencies for η(±)=Iω(±)/(Iω(+,pump)+Iω(-,pump)) plotted versus the phase difference of the input fields δϕω=γω(-)-γω(+), for different pump intensity ratios Q=Iω(-,pump)/Iω(+,pump): (a) Q=0.5, (b) Q=1, (c) Q=2. For Q=1 the backward and the forward conversion efficiencies are equal. Forward (squares) and backward (circles) conversion efficiencies were calculated numerically. All the parameters are the same as those reported in the caption of Fig. 6, and Iω(+,pump)=0.019 GW/cm2.

Equations (79)

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d2Eωdz2+ω2ω(z)c2Eω=-2 ω2c2d(2)(z)Eω*E2ω,
d2E2ωdz2+4ω22ω(z)c2E2ω=-4 ω2c2d(2)(z)Eω2,
d2Ejωdz2+(jω)2jω(z)c2Ejω=0.
Ejω(z)=Aω(+)Φjω(+)(z)+Ajω(-)Φjω(-)(z),
f|g1L 0Lf*(z)g(z)dz.
Ejω(z)=Bjω(+)Φjω(+)(z)+Ajω(-)Ujω(-)(z),
Bjω(+)=Ajω(+)+βjωAjω(-),
Ujω(-)(z)=Φjω(-)(z)-βjωΦjω(+),
βjω=Φjω(+)|Φjω(-)Φjω(+)|Φjω(+).
ddz=z0+λ z1+λ2 z2+.
Ejω=λEjω(1)(z0, z1, z2, )+λ2Ejω(2)(z0, z1, z2, )+ ,j=1, 2.
2Ejω(1)z02=-(jω)2c2jω(z0)Ejω(1),j=1, 2.
Ejω(1)=Bjω(+)(z1, z2, )Φjω(+)(z0)+Ajω(-)(z1, z2, )Ujω(-)(z0),
2z02+ω2ω(z0)c2Eω(2)+2 z0 z1Eω(1)
=-2 ω2c2d(2)(z0)Eω(1)*E2ω(1),
2z02+4ω22ω(z0)c2E2ω(2)+2 z0 z1E2ω(1)
=-4ω2c2d(2)(z0)[Eω(1)]2.
Ejω(2)=Cjω(+)(z1, z2, )Φjω(+)(z0)+Cjω(-)(z1, z2 ,)Ujω(-)(z0),
2Φω(+)z0Bω(+)z1+2Uω(-)z0Aω(-)z1
=-2 ω2c2d(2)[B2ω(+)Bω(+)*Φ2ω(+)Φω(+)*+B2ω(+)Aω(-)*Φ2ω(+)Uω(-)*+A2ω(-)Bω(+)*U2ω(-)Φω(+)*+A2ω(-)Aω(-)*Uω(-)*U2ω(-)],
2Φ2ω(+)z0B2ω(+)z1+2U2ω(-)z0A2ω(-)z1
=-4 ω2c2d(2)[Bω(+)2Φω(+)2+Aω(-)2Uω(-)2+2Bω(+)Aω(-)Uω(-)Φω(+)].
l=+,-pω(+,l) dAω(l)dz=i ωc (k,l)=(+,-)Γ(ω,+)(k,l) A2ω(k) Aω(l)*,
l=+,-pω(-,l) dAω(l)dz=i ωc (k,l)=(+,-)Γ(ω,-)(k,l) A2ω(k) Aω(l)*,
l=+,-p2ω(+,l) dA2ω(l)dz=i ωc (k,l)=(+,-)Γ(2ω,+)(k,l) Aω(k) Aω(l),
l=+,-p2ω(-,l) dA2ω(l)dz=i ωc (k,l)=(+,-)Γ(2ω,-)(k,l) Aω(k) Aω(l),
pjω(k,l)=Φjω(k)|pˆjωΦjω(l),
forj=1, 2andk, l=+,-,
Γ(ω,n)(k,l)=Φω(n)|d(2)Φ2ω(k)Φω(l)*,forn, k, l=+,-,
Γ(2ω,n)(k,l)=Φ2ω(n)|d(2)Φω(k)Φω(l),forn, k, l=+,-.
p2ω(+,+) dA2ω(+)dz+p2ω(+,-) dA2ω(-)dz=i ωcΓ(2ω,+)(+,+)[Aω(+)(0)]2,
p2ω(-,+) dA2ω(+)dz+p2ω(-,-) dA2ω(-)dz=i ωcΓ(2ω,-)(+,+)[Aω(+)(0)]2.
A2ω(+)(z)=i ωc[Aω(+)(0)]2 Γ(2ω,+)(+,+)p2ω(-,-)-Γ(2ω,-)(+,+)p2ω(+,-)p2ω(+,+)p2ω(-,-)-p2ω(+,-)p2ω(-,+)z,
A2ω(-)(z)=i ωc[Aω(+)(0)]2 Γ(2ω,-)(+,+)p2ω(+,+)-Γ(2ω,+)(+,+)p2ω(-,+)p2ω(+,+)p2ω(-,-)-p2ω(+,-)p2ω(-,+)×(z-L).
p2ω(+,-) dA2ω(+)dz+p2ω(+,-) dA2ω(-)dz
=i ωc{Γ(2ω,+)(+,+)[Aω(+)(0)]2+Γ(2ω,+)(-,-)[Aω(-)(L)]2+2Γ(2ω,+)(+,-)Aω(+)(0)Aω(-)(L)},
p2ω(-,+) dA2ω(+)dz+p2ω(-,-) dA2ω(-)dz
=i ωc{Γ(2ω,-)(+,+)[Aω(+)(0)]2+Γ(2ω,-)(-,-)[Aω(-)(L)]2+2Γ(2ω,-)(+,-)Aω(+)(0)Aω(-)(L)}.
A2ω(+)(z)=i ωzc[p2ω(+,+)p2ω(-,-)-p2ω(+,-)p2ω(-,+)]{[Aω(+)×(0)]2[Γ(2ω,+)(+,+)p2ω(-,-)-Γ(2ω,-)(+,+)p2ω(+,-)]+[Aω(-)(L)]2[Γ(2ω,+)(-,-)p2ω(-,-)-Γ(2ω,-)(-,-)p2ω(+,-)]+2Aω(+)(0)Aω(-)(L)[Γ(2ω,+)(+,-)p2ω(-,-)-Γ(2ω,-)(+,-)p2ω(+,-)]},
Aω(-)(z)=i ω(z-L)c[p2ω(+,+)p2ω(-,-)-p2ω(+,-)p2ω(-,+)]{[Aω(+)×(0)]2[Γ(2ω,-)(+,+)p2ω(+,+)-Γ(2ω,+)(+,+)p2ω(-,+)]+[Aω(-)(L)]2[Γ(2ω,-)(-,-)p2ω(+,+)-Γ(2ω,+)(+,+)p2ω(-,+)]+2Aω(+)(0)Aω(-)(L)[Γ(2ω,-)(+,-)p2ω(+,+)-Γ(2ω,+)(+,-)p2ω(-,+)]}.
A2ω(+)(z)=i 2ωzIω(+,pump) exp[2iγω(+)]0c2[p2ω(+,+)p2ω(-,-)-p2ω(+,-)p2ω(-,+)]{[Γ(2ω,+)(+,+)p2ω(-,-)-Γ(2ω,-)(+,+)p2ω(+,-)]+Q exp(2iδϕω)[Γ(2ω,+)(-,-)p2ω(-,-)-Γ(2ω,-)(-,-)p2ω(+,-)]+2Q exp[iδϕω][Γ(2ω,+)(+,-)p2ω(-,-)-Γ(2ω,-)(+,-)p2ω(+,-)]},
A2ω(+)(z)=i 2ωzIω(+,pump) exp[2iγω(+)]0c2[p2ω(+,+)p2ω(-,-)-p2ω(+,-)p2ω(-,+)]×{[Γ(2ω,+)(+,+)p2ω(-,-)-Γ(2ω,-)(+,+)p2ω(+,-)]+Q exp(2iδϕω)[Γ(2ω,+)(-,-)p2ω(-,-)-Γ(2ω,-)(-,-)p2ω(+,-)]+2Q exp[iδϕω]×[Γ(2ω,+)(+,-)p2ω(-,-)-Γ(2ω,-)(+,-)p2ω(+,-)]},
A2ω(-)(z)=i 2ω(z-L)Iω(+,pump) exp[2iγω(+)]0c2[p2ω(+,+)p2ω(-,-)-p2ω(+,-)p2ω(-,+)]×{[Γ(2ω,-)(+,+)p2ω(+,+)-Γ(2ω,+)(+,+)p2ω(-,+)]+Q exp(2iδϕω)[Γ(2ω,-)(-,-)p2ω(+,+)-Γ(2ω,+)(+,+)p2ω(-,+)]+2Q exp(iδϕω)×[Γ(2ω,-)(+,-)p2ω(+,+)-Γ(2ω,+)(+,-)p2ω(-,+)]},
Φjω(+)=12 1+1nexpi jmπLz+12 1-1nexp-i jmπLz,j=1, 2,
Φjω(-)=12 1-1nexpijmπLz+jmπ+12 1+1n×exp-ijmπLz-jmπ,j=1, 2,
dAω(+)dz=i ωc d(2)4 1+3n2A2ω(+)Aω(+)*+1-1n2×[exp(-imπ)A2ω(+)Aω(-)*+A2ω(-)Aω(+)*+exp(-imπ)A2ω(-)Aω(-)*],
dAω(-)dz=-i ωc d(2)4 1-1n2exp(-imπ)A2ω(+)Aω(+)*+A2ω(+)Aω(-)*+exp(-imπ)A2ω(-)Aω(+)*+1+3n2A2ω(-)Aω(-)*,
dA2ω(+)dz=i ωc d(2)4 1+3n2Aω(+)2+1-1n2×[2 exp(imπ)Aω(+)Aω(-)+Aω(-)2],
dA2ω(-)dz=-i ωc d(2)4 1-1n2[Aω(+)2+2 exp(imπ)Aω(+)Aω(-)]+1+3n2Aω(-)2,
I2ω(+)=2ω2c30 d(2)4 1+3n22L2[Iω(+,pump)]2,
I2ω(-)=2ω2c30 d(2)4 1-1n22L2[Iω(+,pump)]2.
I2ω(+)I2ω(-)=n2+3n2-12.
I2ω(+)=2ω2c30 d(2)4 1+3n22L2[Iω(+,pump)]2×1+Qn2-1n2+3exp(2iδϕω)+2Qn2-1n2+3exp[i(δϕω+mπ)]2,
I2ω(-)=2ω2c30 d(2)4 1-1n22L2[Iω(+,pump)]2×1+Qn2+3n2-1exp(2iδϕω)+2Q exp[i(δϕω+mπ)]2,
|Φjω(±)|2 dϕjω(±)dz=±jωc.
Φjω(+)(0)=Φjω(-)(L)=1,
Φjω(-)(0)=Φjω(+)(L)=exp[iϕt(jω)],
pjω(+,+)=Φjω(+)|pˆjωΦjω(+)=1,
pjω(-,-)=Φjω(-)|pˆjωΦjω(-)=-1,
pjω(-,+)=[pjω(+,-)]*+2cjωL sin[ϕt(jω)].
Φjω(-)=Φjω(+)* exp[iϕt(jω)].
pjω(-,+)=-[pjω(+,-)]=cjωL sin[ϕt(jω)].
dAω(+)dz=i ωc (k,l)=(+,-)Γ(ω,+)(k,l)A2ω(k)Aω(l)*,
dAω(-)dz=-i ωc (k,l)=(+,-)Γ(ω,-)(k,l)A2ω(k)Aω(l)*,
dA2ω(+)dz=i ωc (k,l)=(+,-)Γ(2ω,+)(k,l)Aω(k)Aω(l),
dA2ω(-)dz=-i ωc (k,l)=(+,-)Γ(2ω,-)(k,l)Aω(k)Aω(l).
I2ω(+)=2ω2c30L2[Iω(+,pump)]2|Γ(2ω,+)(+,+)|2×1+Q Γ(2ω,+)(-,-)Γ(2ω,+)(+,+) exp(2iδϕω)+2Q exp(iδϕω) Γ(2ω,+)(+,-)Γ(2ω,+)(+,+)2,
I2ω(-)=2ω2c30L2[Iω(+,pump)]2|Γ(2ω,-)(+,+)|2×1+Q exp(2iδϕω) Γ(2ω,-)(-,-)Γ(2ω,-)(+,+)+2Q Γ(2ω,-)(+,-)Γ(2ω,-)(+,+) exp(iδϕω)2.
Γ(2ω,+)(+,+)=(1/L)0LΦ2ω(+)*d(2)(z)[Φω(+)]2dz,
Γ(2ω,+)(+,+)(1/L)0Ld(2)|Φ2ω(+)||Φω(+)|2dz.
ρω=(1/Lc)0Lω(z)|Φω(+)|2dz.
pjω(-,+)-[pjω(+,-)]cjωL sin[ϕt(jω)].
dAω(+)dz=i ωc (k,l)=(+,-)Γ(ω,+)(k,l)A2ω(k)Aω(l)*,
dAω(-)dz=-i ωc (k,l)=(+,-)Γ(ω,-)(k,l)A2ω(k)Aω(l)*,
dA2ω(+)dz=i ωc (k,l)=(+,-)Γ(2ω,+)(k,l)Aω(k)Aω(l),
dA2ω(-)dz=-i ωc (k,l)=(+,-)Γ(2ω,-)(k,l)Aω(k)Aω(l).
I2ω(+)=2ω2c30L2[Iω(+,pump)]2|Γ(2ω,+)(+,+)|2×1+Q Γ(2ω,+)(-,-)Γ(2ω,+)(+,+) exp(2iδϕω)+2Q exp(iδϕω) Γ(2ω,+)(+,-)Γ(2ω,+)(+,+)2,
I2ω(-)=2ω2c30L2[Iω(+,pump)2|Γ(2ω,-)(+,+)|2×1+Q exp(2iδϕω) Γ(2ω,-)(-,-)Γ(2ω,-)(+,+)+2Q Γ(2ω,-)(+,-)Γ(2ω,-)(+,+) exp(iδϕω)2.
I(pump)=I(peak)4σ -+ exp-t22σ2dt=(π/8)1/2I(peak).

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