Abstract

In a one-dimensional periodic nonlinear χ(2) medium, by choice of a proper material and geometrical parameters of the structure, it is possible to obtain two matching conditions for simultaneous generation of second and third harmonics. This leads to new diversity of the processes of the resonant three-wave interactions, which are discussed within the framework of the slowly varying envelope approach. In particular we concentrate on the fractional conversion of the frequency ω(2/3)ω. This phenomenon occurs by means of intermediate energy transfer to the first harmonic at the frequency ω/3 and can be controlled by this mode. By analogy the same medium allows nondirect second-harmonic generation, controlled by the cubic harmonic. Propagation of localized pulses in the form of two coupled bright solitons on first and third harmonics and a dark soliton on the second harmonic is possible.

© 2000 Optical Society of America

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References

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  1. For review, see C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993); J. Opt. Soc. Am. B 10, 280–413 (1993); Development and applications of materials exhibiting photonic band gaps feature, E. Burstein and C. Weisbuch, eds., Confined Electrons and Photons, NATO ASI Ser., Ser. B 340 (1995); C. M. Soukoulis, ed., Photonic Band Gap Materials, NATO ASI Ser., Ser. E 315 (1996); and in J. Rarity and C. Weisbuch, eds., Microcavities and Photonic Bandgaps: Physics and Applications, NATO ASI Ser., Ser. E 324 (1996).
  2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton University, Princeton, N.J., 1995).
  3. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–382 (1979); L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of superlattices. Multistability and soliton trains,” Phys. Rev. B 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]
  4. 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]
  5. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985); 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 PRBMDO 36, 947–952 (1987).
    [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. A. V. Buryak, I. Towers, and S. Trillo, “Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems,” Phys. Rev. A 267, 319–325 (2000).
  9. E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972); C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 7, 271–300 (1974).
    [CrossRef]
  10. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
    [CrossRef]
  11. J. P. van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs–Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
    [CrossRef]
  12. J. Martorell and R. Corbalan, “Enhancement of second har-monic 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]
  13. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quazi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  14. 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]
  15. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994); M. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge ofsemiconductor heterostructures,” Phys. Rev. A 53, 2799–1783 (1996).
    [CrossRef] [PubMed]
  16. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–686 (1946).
  17. V. V. Konotop and V. Kuzmiak, “Simultaneous second- and third-harmonic generation in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 16, 1370–1376 (1999).
    [CrossRef]
  18. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  19. E. D. Palick, ed., Handbook of Optical Constants (Academic, New York, 1985).
  20. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1984).
  21. S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Theory of Solitons: Inverse Scattering method (Consultants Bureau, New York, 1980).
  22. I. Towers, R. A. Sammut, A. V. Buryak, and B. A. Malomed, “Soliton multistability as a result of double-resonance wave mixing in χ(2) media,” Opt. Lett. 24, 1738–1740 (1999).
    [CrossRef]

2000 (1)

A. V. Buryak, I. Towers, and S. Trillo, “Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems,” Phys. Rev. A 267, 319–325 (2000).

1999 (2)

1996 (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]

1992 (1)

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

1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

1988 (1)

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]

1976 (1)

J. P. van der 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]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–686 (1946).

Bloembergen, N.

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

Buryak, A. V.

A. V. Buryak, I. Towers, and S. Trillo, “Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems,” Phys. Rev. A 267, 319–325 (2000).

I. Towers, R. A. Sammut, A. V. Buryak, and B. A. Malomed, “Soliton multistability as a result of double-resonance wave mixing in χ(2) media,” Opt. Lett. 24, 1738–1740 (1999).
[CrossRef]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quazi-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]

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]

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, “Quazi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Ilegems, M.

J. P. van der 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, “Quazi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Konotop, V. V.

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]

Kuzmiak, V.

Magel, G. A.

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

Malomed, B. A.

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–686 (1946).

Sammut, R. A.

Sievers, A. J.

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

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]

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]

Towers, I.

A. V. Buryak, I. Towers, and S. Trillo, “Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems,” Phys. Rev. A 267, 319–325 (2000).

I. Towers, R. A. Sammut, A. V. Buryak, and B. A. Malomed, “Soliton multistability as a result of double-resonance wave mixing in χ(2) media,” Opt. Lett. 24, 1738–1740 (1999).
[CrossRef]

Trillo, S.

A. V. Buryak, I. Towers, and S. Trillo, “Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems,” Phys. Rev. A 267, 319–325 (2000).

van der Ziel, J. P.

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

Appl. Phys. Lett. (2)

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

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

IEEE J. Quantum Electron. (1)

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

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

Opt. Lett. (1)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–686 (1946).

Phys. Rev. A (2)

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]

A. V. Buryak, I. Towers, and S. Trillo, “Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems,” Phys. Rev. A 267, 319–325 (2000).

Phys. Rev. B (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

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 (12)

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985); 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 PRBMDO 36, 947–952 (1987).
[CrossRef] [PubMed]

For review, see C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993); J. Opt. Soc. Am. B 10, 280–413 (1993); Development and applications of materials exhibiting photonic band gaps feature, E. Burstein and C. Weisbuch, eds., Confined Electrons and Photons, NATO ASI Ser., Ser. B 340 (1995); C. M. Soukoulis, ed., Photonic Band Gap Materials, NATO ASI Ser., Ser. E 315 (1996); and in J. Rarity and C. Weisbuch, eds., Microcavities and Photonic Bandgaps: Physics and Applications, NATO ASI Ser., Ser. E 324 (1996).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton University, Princeton, N.J., 1995).

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–382 (1979); L. Kahn, N. S. Almeida, and D. L. Mills, “Nonlinear optical response of superlattices. Multistability and soliton trains,” Phys. Rev. B 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]

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972); C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 7, 271–300 (1974).
[CrossRef]

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]

E. D. Palick, ed., Handbook of Optical Constants (Academic, New York, 1985).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1984).

S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Theory of Solitons: Inverse Scattering method (Consultants Bureau, New York, 1980).

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. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994); M. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge ofsemiconductor heterostructures,” Phys. Rev. A 53, 2799–1783 (1996).
[CrossRef] [PubMed]

J. Martorell and R. Corbalan, “Enhancement of second har-monic 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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Figures (4)

Fig. 1
Fig. 1

(a) Photonic band structure of a 1D periodic structure consisting of alternating slabs of Al0.1Ga0.9As with a=10.97 and InSb with b=16.4 at λ=2 µm (solid curves) in which fractional frequency conversion takes place. The dashed curves refer to the seventh-lowest band at λ=1 µm, which corresponds to the second harmonic, and the tenth-lowest band at λ=0.667 µm, which corresponds to the third-harmonic signal. q is measured in units of 2π/(a+b). (b) Detailed picture of the region of the photonic band structure shown in (a) in an extended zone scheme, in which both frequencies and the wave vectors satisfy the resonant conditions for simultaneous SHG and THG. The solid curve indicates the calculated dispersion curve near 2 µm, the dashed curve refers to the region near 1 µm with values divided by a factor of 2, and the dash-dotted curve refers to the region near 0.667 µm with values divided by a factor of 3. The exact phase matching between the forward-traveling fundamental and the oppositely traveling second harmonic occurs when q=qSHG=0.468, whereas exact phase matching between the fundamental wave and the third harmonic is possible for oppositely propagating waves when q=qTHG=0.471. q is measured in units of 2π/(a+b).

Fig. 2
Fig. 2

Examples of the effective potential U(Xp) for the structure parameters and the initial amplitudes of waves as follows: γ1=82.14+78.48i,γ3=-14.57-5.07i; (a) v1=0.5382, v2=0.2808, v3=0.2433, A1=0.13×109 V/m, A3=0.31×109 V/m, A2=0, and μ0.1225; (b) v1=2.382, v2=0.2808, v3=0.06433, A1=0.1×109 V/m, A3=-0.9×109 V/m, A2=0, and μ2.051; (c) v1=0.5382, v2=0.2808, v3=-0.2433, A1=0.13×109 V/m, A3=0.31×109 V/m, A2=0, and μ0.1225.

Fig. 3
Fig. 3

Evolution of intensities of the first- (solid curve), the second- (long-dashed curve), and the third-harmonic (short-dashed curve) signals: (a) the same structure parameters and input amplitudes as in Fig. 2(a); (b) the same structure parameters and input amplitudes as in Fig. 2(b); (c) the same structure parameters as in Fig. 2(b) and the input amplitudes A1=0.1×109 V/m, A3=0.4×109 V/m, and A2=0; (d) the same structure parameters and input amplitudes as in Fig. 2(a) except that the phase mismatches Δω2-0.002 and Δω30.0009 included. The intensities represented are normalized to [χ(2)]-2, and time is measured in ω1-1 units.

Fig. 4
Fig. 4

Evolution of intensities of the first- (solid curve), the second- (long-dashed curve), the third-harmonic (short-dashed curve) signals in a structure depicted in Fig. 1, where the input amplitudes A1=0.255×109 V/m, A3=-1.996×109 V/m, and A2=0.

Equations (60)

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

PNL(x, t)=χ(2)(x)E2(x, t).
-c2 2E(x, t)x2+2t2 -(x, t-t)E(x, t)dt
=-4π 2t2PNL(x, t),
(x, t-t)=δ(t-t)+4πχ(x, t-t)
ω3=3ω1+Δω3,ω2=2ω1+Δω2,
q3=3q1+Q1,q2=2q1+Q2,
E=j=13Ajϕj(x)exp(iωjt)+c.c.,
[c2(d2/dx2)+ˆ0(x; ωj)ωj2]ϕj(x)=0.
iω1 A1t+v1 A1x+γ3A¯2A3 exp[i(Δω3-Δω2)t]
+2γ1A¯1A2 exp(iΔω2t)=0, 
iω2 A2t+v2 A2x+γ3A¯1A3 exp[i(Δω3-Δω2)t]
+γ¯1A12 exp(-iΔω2t)=0, 
iω3 A3t+v3 A3x+γ¯3A1A2 exp[-i(Δω3-Δω2)t]
=0,
γ1=2π0Lχ(2)(x;-ω, 2ω)ϕ¯1(x)ϕ¯1(x)ϕ2(x)dx,
γ3=2π0Lχ(2)(x;-ω, 3ω)ϕ¯1(x)ϕ¯2(x)ϕ3(x)dx.
χ(2)(x; ω1, ω2)=χ(2)(x;-ω3, ω2)=χ(2)(x; ω1,-ω3)
v1W1(x)+v2W2(x)+v3W3(x)=0,
Wj(x)=-|Aj(x, t)|2dt.
a1=2w2 A1 exp(-iδω1t),a2=A2 exp(iφ1),
a3=Γ3Γ1 2w2 A3 exp[i(-δω3t+φ3+φ1)].
ia1τ+a1ξ+a¯2a3+2a¯1a2=0,
i1w2 a2τ+σ2 a2ξ+a¯1a3+a12=0,
i1w3 a3τ+σ3 a3ξ+μa1a2=0.
k21-v2v12>4,
3k1-v2v1+2k1-v2v3<μ,
(v1-v2)(v2-v3)>0.
limξ a1,3=0,limξ a2=ρ exp[i(kξ-w2τ)].
ia˜1τ+a˜1ξ+a˜¯2a˜3+2a˜¯1a2+ν˜1a˜1=0,
i1w2 a˜2τ+σ2 a˜2ξ+a˜¯1a˜3+a˜12=0,
i1w3 a˜3τ+σ3 a˜3ξ+μa˜1a˜2+ν˜3a˜3=0,
νj=Δωjω1 σjΓ1wj,ν˜j=δj-νj,δ1=12k(1-w2),
δ3=3k(w3-w2)2w3.
i da1dξ+a¯2a3+2a¯1a2-ν1a1=0,
iσ2 da2dξ+a¯1a3+a12=0,
iσ3 da3dξ+μa1a2-ν3a3=0.
N=|a1|2+2σ2|a2|2+3σ3μ|a3|2,
H=a12a¯2+a¯12a2+a1a2a¯3+a¯1a¯2a3,
H=σ2σ3μq1p2p3+2σ2q1p1p2-σ2p1p2q3+q1q2q3+σ3μp1q2p3+q12q2-p12q2,
H=q2 dp2dξ-p2 dq2dξ,
cot θ2=p1q3-μq1p3-2q1p1q1q3+μp1p3+q12-p12.
θ2=±π/2.
Xp=σ20ξp2(ξ)dξ,
dq1dXp-q3+2q1=0,
dq3dXp+σ3μq1=0,
dp1dXp-q3p1q1=0.
q1=C1 exp(ω+Xp)+C2 exp(ω-Xp),
q3=-C1ω- exp(ω+Xp)-C2ω+ exp(ω-Xp),
ω±=-1±1-σ3μ ifσ3μ<1-1±iσ3μ-1 ifσ3μ>1.
p1=p01 exp(2Xp)[C1 exp(ω+Xp)+C2 exp(ω-Xp)].
d2Xpdξ2=-U(Xp)Xp.
a˜1=α±βu1cosh(βζ),
a˜2=i α±2βu121-μα±2u1 i ν˜3βu1+tanh(βζ),
a˜3=μα±3βu121-μα±2u1 1cosh(βζ).
ζ=(ξ-vτ)v2v2-vv1,
u=v3σ3-vv1(1-v)v3,u1=σ3-vv1v3,
α±2=μ(μ+u)±u(u-μ)μu1(μ+3u),
ν˜3ν˜1=(1-μα±2u1)2α±2u1(μα±2u1-2).
ω3=2ω2+Δω3,ω2=2ω1+Δω2,
q3=2q2+Q1,q2=2q1+Q2,

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