Abstract

By use of the envelope-function approach, the equation governing propagation of the TE plane wave in a one-dimensional periodic structure is reduced to a set of coupled-mode dynamical equations for slowly varying amplitudes. We applied this method to layered media possessing both χ(2) and χ(3) nonlinearities, to study the possibility of simultaneous second- and third-harmonic generation. The phenomenon is based on the geometry of the structure and is observed in a wide class of photonic crystals of different natures, provided that the thickness and refractive indices of alternating dielectric layers are appropriately chosen. By imposing various initial distributions of the energy among the individual modes, we studied the evolution of the intensities. We found that the presence of two channels for the energy transfer from the fundamental mode prevents concentration of the total energy in either of the higher modes. In all the cases considered the energy exchange among the modes is observed. We present a particular solution for the energy transfer from the fundamental and the third harmonic to the second harmonic, obtained when the cubic nonlinearity is negligible.

© 1999 Optical Society of America

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References

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  1. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379 (1979); H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985); F. Deylon, Y. F. Lévy, and B. Souillard, “Nonperturbative bistability in nonlinear media,” Phys. Rev. Lett. PRLTAO 57, 2010–2013 (1980); L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of superlattices. Multistability and soliton trains,” Phys. Rev. B PRBMDO 37, 8072–8081 (1988); V. M. Agranovich, S. A. Kiselev, and D. L. Mills, “Optical multistability in nonlinear superlattices with very thin layers,” Phys. Rev. B PRBMDO 44, 10917–10920 (1991).
    [CrossRef]
  2. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [CrossRef] [PubMed]
  3. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrafast pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994); A. Kozhekin and G. Kurizki, “Self-induced transparency in Bragg reflectors,” Phys. Rev. Lett. 74, 5020–5023 (1995); M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. JAPIAU 76, 2023–2026 (1994); M. Scalora, R. L. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, J. Bendikson, H. Ledbetter, C. M. Bowden, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E PLEEE8 76, R1078–R1081 (1996).
    [CrossRef] [PubMed]
  4. W. Chen and D. L. Mills, “Gap solitons in nonlinear periodic structures,” Phys. Rev. Lett. 58, 160–163 (1987); D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947–952 (1987).
    [CrossRef] [PubMed]
  5. 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); “Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
    [CrossRef] [PubMed]
  6. S. Lee and S.-T. Ho, “Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic media,” Opt. Lett. 18, 962 (1993); V. V. Konotop, “Vector gap solitons,” Phys. Rev. A 51, R3422–R3425 (1995); V. V. Konotop and G. P. Tsironis, “Dynamics of coupled gap solitons,” Phys. Rev. E PLEEE8 53, 5393–5398 (1996); C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E PLEEE8 53, 1969–1989 (1996).
    [CrossRef] [PubMed]
  7. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
    [CrossRef]
  8. J. P. van Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs—Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
    [CrossRef]
  9. J. Martorell and R. Corbalan, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994); J. Trull, R. Vilaseca, J. Martorell, and R. Corbalan, “Second harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
    [CrossRef]
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    [CrossRef]
  11. M. J. Steel and C. M. de Sterke, “Second-harmonic generation in second-harmonic fiber Bragg gratings,” Appl. Opt. 35, 3211–3222 (1996); “Bragg-assisted parametric amplification of short optical pulses,” Opt. Lett. 21, 420–422 (1996); M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulse second-harmonic generation in nonlinear one-dimensional, periodic structures,” Phys. Rev. A PLRAAN 56, 3166–3174 (1997).
    [CrossRef] [PubMed]
  12. V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
    [CrossRef]
  13. V. Pruneri, G. Bonfrate, P. G. Kazansky, G. J. Richardson, N. G. Broderick, J. P. Sandro, C. Simonneau, P. Vidakovich, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
    [CrossRef]
  14. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” J. Opt. Quantum Electron. 28, 1691–1740 (1996).
    [CrossRef]
  15. X. Gu, M. Makarov, Y. Ding, J. B. Khurgin, and W. P. Risk, “Backward second-harmonic and third-harmonic generation in periodically poled potassium titanyl phosphate waveguide,” Opt. Lett. 24, 127–129 (1999).
    [CrossRef]
  16. R. A. Myers, N. Mukherjee, and S. R. J. Brueck, “Large second-order nonlinearity in poled fused silica,” Opt. Lett. 16, 1732–1734 (1991).
    [CrossRef] [PubMed]
  17. B. A. Saleh and T. M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  18. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

1999 (2)

1998 (1)

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

1996 (2)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” J. Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1991 (1)

1976 (1)

J. P. van Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs—Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

1970 (1)

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Bonfrate, G.

V. Pruneri, G. Bonfrate, P. G. Kazansky, G. J. Richardson, N. G. Broderick, J. P. Sandro, C. Simonneau, P. Vidakovich, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
[CrossRef]

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

Broderick, N. G.

Brueck, S. R. J.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Ding, Y.

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Gu, X.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” J. Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Ilegems, M.

J. P. van Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs—Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kazansky, P. G.

V. Pruneri, G. Bonfrate, P. G. Kazansky, G. J. Richardson, N. G. Broderick, J. P. Sandro, C. Simonneau, P. Vidakovich, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
[CrossRef]

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

Khurgin, J. B.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Levenson, J. A.

V. Pruneri, G. Bonfrate, P. G. Kazansky, G. J. Richardson, N. G. Broderick, J. P. Sandro, C. Simonneau, P. Vidakovich, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
[CrossRef]

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Makarov, M.

Mukherjee, N.

Myers, R. A.

Pruneri, V.

V. Pruneri, G. Bonfrate, P. G. Kazansky, G. J. Richardson, N. G. Broderick, J. P. Sandro, C. Simonneau, P. Vidakovich, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
[CrossRef]

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

Richardson, G. J.

Risk, W. P.

Sandro, J. P.

Sievers, A. J.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Simonneau, C.

V. Pruneri, G. Bonfrate, P. G. Kazansky, G. J. Richardson, N. G. Broderick, J. P. Sandro, C. Simonneau, P. Vidakovich, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
[CrossRef]

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” J. Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” J. Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

van Ziel, J. P.

J. P. van Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs—Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

Vidakovic, P.

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

Vidakovich, P.

Appl. Phys. Lett. (3)

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

J. P. van Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs—Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

V. Pruneri, G. Bonfrate, P. G. Kazansky, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Efficient frequency doubling of 1.5 μm femtosecond laser pulses in quasi-phase-matched optical fibers,” Appl. Phys. Lett. 72, 1007–1009 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

J. Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” J. Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Other (9)

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrafast pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994); A. Kozhekin and G. Kurizki, “Self-induced transparency in Bragg reflectors,” Phys. Rev. Lett. 74, 5020–5023 (1995); M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. JAPIAU 76, 2023–2026 (1994); M. Scalora, R. L. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, J. Bendikson, H. Ledbetter, C. M. Bowden, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E PLEEE8 76, R1078–R1081 (1996).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, “Gap solitons in nonlinear periodic structures,” Phys. Rev. Lett. 58, 160–163 (1987); D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947–952 (1987).
[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); “Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[CrossRef] [PubMed]

S. Lee and S.-T. Ho, “Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic media,” Opt. Lett. 18, 962 (1993); V. V. Konotop, “Vector gap solitons,” Phys. Rev. A 51, R3422–R3425 (1995); V. V. Konotop and G. P. Tsironis, “Dynamics of coupled gap solitons,” Phys. Rev. E PLEEE8 53, 5393–5398 (1996); C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E PLEEE8 53, 1969–1989 (1996).
[CrossRef] [PubMed]

M. J. Steel and C. M. de Sterke, “Second-harmonic generation in second-harmonic fiber Bragg gratings,” Appl. Opt. 35, 3211–3222 (1996); “Bragg-assisted parametric amplification of short optical pulses,” Opt. Lett. 21, 420–422 (1996); M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulse second-harmonic generation in nonlinear one-dimensional, periodic structures,” Phys. Rev. A PLRAAN 56, 3166–3174 (1997).
[CrossRef] [PubMed]

J. Martorell and R. Corbalan, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994); J. Trull, R. Vilaseca, J. Martorell, and R. Corbalan, “Second harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379 (1979); H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985); F. Deylon, Y. F. Lévy, and B. Souillard, “Nonperturbative bistability in nonlinear media,” Phys. Rev. Lett. PRLTAO 57, 2010–2013 (1980); L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of superlattices. Multistability and soliton trains,” Phys. Rev. B PRBMDO 37, 8072–8081 (1988); V. M. Agranovich, S. A. Kiselev, and D. L. Mills, “Optical multistability in nonlinear superlattices with very thin layers,” Phys. Rev. B PRBMDO 44, 10917–10920 (1991).
[CrossRef]

B. A. Saleh and T. M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

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Figures (3)

Fig. 1
Fig. 1

Photonic band structure of layered dielectric structures consisting of alternating slabs of GaAs with a=10.9 and β-BaB2O4 with b=2.74 in which simultaneous SHG and THG occur. q is measured in the units 2π/(a+b).

Fig. 2
Fig. 2

Evolution of the amplitudes of the fundamental (solid curve), the second- (long-dashed curve) and third-harmonic (short-dashed curve) signals in GaAs/β-BaB2O4 when the initial conditions are A1(0)=3.57×109 V/m. (a) A2(0)=0, A3(0)=0; (b) A2(0)=1.78×109 V/m, A3(0)=0; (c) A2(0)=0, A3(0)=1.78×109 V/m; (d) A2(0)=1.07×109=V/m, A3(0)=1.78×109 V/m. The intensity represented is normalized to (χ(2))-2.

Fig. 3
Fig. 3

Evolution of the fundamental signal (solid curve) and the second- (long-dashed curve) and third-harmonic (short-dashed curve) signals characterized with the same coefficients as in Fig. 2 when the initial conditions are A1(0)=0, A2(0)=3.93×109 V/m, A3(0)=3.12×109 V/m. The intensity represented is normalized to (χ(2))-2.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

q2=2q1+Q1,ω2=2ω1,
q3=3q1+Q2,ω3=3ω1,
-c2 2x2+0(x) 2t2E(x, t)
=-4π 2t2[χ(2)(x)E2(x, t)+χ(3)(x)E3(x, t)].
χ(2)E=o(1),
O(|χ(2)|/|χ(3)E|)=O(|χ(3)|/0|E|)=o(1).
E=j=13Ajϕj(x)exp(iωjt)+c.c.,
[c2(d2/dx2)+0(x)ωj2]ϕj(x)=0,
iω1dA1dt+γ3A¯2A3+2γ1 A¯1 A2+(λ1111|A1|2
+λ1122|A2|2+λ1133|A3|2)A1+λ1223 A22A¯3
+3λ1113 A¯12A3=0,
iω2dA2dt+γ3 A¯1 A3+γ1 A12+(λ1122|A1|2+λ2222|A2|2
+λ2233|A3|2)A2+2λ1223 A1A¯2A3=0,
iω3dA3dt+γ3 A1 A2+(λ1133|A1|2+λ2233|A2|2
+λ3333|A3|2)A3+λ1113 A13+λ1223 A¯1 A22=0,
γj=2π0Lχ(2)(x)ϕ1(x)ϕ2(x)ϕj(x)dx,
λijkl=2π0Lχ(3)(x)ϕi(x)ϕj(x)ϕk(x)ϕl(x)dx
N=|A1|2+2 ω1ω2|A2|2+3 ω1ω3|A3|2,
H=γ1(A12A¯2+A¯12A2)+γ3(A1 A2 A¯3+A¯1 A¯2 A3)+λ1113(A¯13 A3+A13 A¯3)+λ1223(A¯1 A22 A¯3+A1 A¯22A3)+12(λ1111|A1|4+λ2222|A2|4+λ3333|A3|4+2λ1122|A1|2|A2|2+2λ1133|A1|2|A3|2+2λ2233|A2|2|A3|2),
A1=iα1a sech(ηt),
A2=iα2a tanh(ηt),
A3=ia sech(ηt).
α1=γ3+γ32-3γ123γ1,
α2=2(3γ12+γ32±γ3γ32-3γ12)1/23γ1,
E1=j=13A˜j(t1,,x1,)ϕj(x0)exp(iωjt)+c.c.,
(ϕ(x), ψ(x))0Lϕ¯(x)(x)ψ(x)dx.
Ajt1+vj Ajx1=0,
vj=-i0Lϕj(x) xϕj(x)dx

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