Abstract

The system of four differential equations governing counterpropagating quasi-phase matching are recast in a Hamiltonian form that leads to immediate insight into the nonlinear mixing process by inspection for all possible boundary conditions. A reduced Hamiltonian is found using conservation relations that are dependent on only two normalized field efficiencies and two aggregate phases. Hamiltonian contours are plotted in a series of phase-space cross sections to provide insight into the generalized behavior. The nonlinear eigenmodes are found, and their stability is examined. Finally, two specific counterpropagating quasi-phase-matching configurations, i.e., mirrored and mirrorless, are analyzed using this general Hamiltonian with the appropriate boundary conditions.

© 2004 Optical Society of America

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2002 (5)

Y. Fukuchi, T. Sakamoto, K. Taira, K. Kikuchi, D. Kunimatsu, A. Suzuki, and H. Ito, “Speed limit of all-optical gate switched using cascaded second-order nonlinear effect in quasi-phase-matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 13, 1267–1269 (2002).
[CrossRef]

Y. Fukuchi and K. Kikuchi, “Novel design method for all-optical ultrafast gate switches using cascaded second-order nonlinear effect in quasi-phase matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 14, 1409–1411 (2002).
[CrossRef]

B. Chen, C. Q. Xu, B. Zhou, and X. H. Tang, “Analysis of cascaded second-order nonlinear interaction based on quasi-phase-matched optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 675–680 (2002).
[CrossRef]

H. Liu, N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, “Multiple-wavelength second-harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 81, 3326–3328 (2002).
[CrossRef]

D. C. Hutchings and T. C. Kleckner, “Quasi phase matching in semiconductor waveguides by intermixing: optimization considerations,” J. Opt. Soc. Am. B 19, 890–894 (2002).
[CrossRef]

2001 (3)

2000 (3)

1999 (5)

1998 (4)

1997 (12)

P. Vidakovic, D. J. Lovering, J. A. Levenson, J. Webjorn, and P. St. J. Russell, “Large nonlinear phase shift owing to cascaded χ(2) in quasi-phase-matched bulk LiNbO3,” Opt. Lett. 22, 277–279 (1997).
[CrossRef]

J. U. Kang, Y. J. Ding, W. K. Burns, and J. S. Melinger, “Backward second-harmonic generation in periodically poled bulk LiNbO3,” Opt. Lett. 22, 862–864 (1997).
[CrossRef] [PubMed]

G. D. Landry and T. A. Maldonado, “Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration,” Opt. Lett. 22, 1400–1402 (1997).
[CrossRef]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

G. D’Alessandro, P. S. J. Russell, and A. A. Wheeler, “Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator,” Phys. Rev. A 55, 3211–3218 (1997).
[CrossRef]

T. Gase and W. Karthe, “Quasi-phase matched cascaded second order processes in poled organic polymer waveguides,” Opt. Commun. 133, 549–556 (1997).
[CrossRef]

H. O. Wagner, M. Kühnelt, G. Wein, B. Hahn, W. Genhardt, D. Eisert, G. Bacher, and A. Forchel, “Phase matched second harmonic generation using a χ(2) modulated ZnTe/ZnSe optical waveguide,” J. Lumin. 72, 87–89 (1997).
[CrossRef]

Y. Paltiel, D. Mahalu, H. Shtrikman, G. Bunin, and U. Meirav, “Short-period surface superlattices formed by plasma etching,” Semicond. Sci. Technol. 12, 987–990 (1997).
[CrossRef]

K. Mizuuchi, K. Yamamoto, and M. Kato, “Generation of ultraviolet light by frequency doubling of a red laser diode in a first-order periodically poled bulk LiTaO3,” Appl. Phys. Lett. 70, 1201–1203 (1997).
[CrossRef]

Y. Shuto, T. Watanabe, S. Tomaru, I. Yokohama, M. Hikita, and M. Amano, “Quasi-phase-matched second-harmonic generation in diazo-dye- substituted polymer channel waveguides,” IEEE J. Quantum Electron. 33, 349–357 (1997).
[CrossRef]

N. Hashizume, T. Tsuruzono, T. Kondo, and R. Ito, “Fabrication of periodic waveguides using organic crystals and fluorinated polyimides for quasi-phase-matched second-harmonic generation,” Opt. Rev. 4, 316–320 (1997).
[CrossRef]

J. Pierce and D. Lowenthal, “Periodically poled materials and devices,” Laser Optoelektron. 16, 25–27 (1997).

1996 (4)

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,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

S. J. B. Yoo, “Wavelength conversion technology for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996).
[CrossRef]

W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-W continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 21, 1336–1338 (1996).
[CrossRef] [PubMed]

Y. J. Ding and J. B. Khurgin, “Second-harmonic generation based on quasi-phase matching: a novel configuration,” Opt. Lett. 21, 1445–1447 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Houe and P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D 28, 1747–1763 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (4)

1991 (1)

P. S. J. Russell, “Theoretical study of parametric frequency and wavefront conversion in nonlinear holograms,” IEEE J. Quantum Electron. 27, 830–835 (1991).
[CrossRef]

1990 (1)

R. Normandin, R. L. Williams, and F. Chatenoud, “Enhanced surface emitting waveguides for visible, monolithic semiconductor laser sources,” Electron. Lett. 26, 2088–2089 (1990).
[CrossRef]

1989 (1)

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. 66, 806–808 (1989).

1982 (1)

G. R. Meredith, “Second-order cascading in third-order nonlinear optical processes,” J. Chem. Phys. 77, 5863–5871 (1982).
[CrossRef]

1972 (1)

J. M. R. Thomas and J. P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–334 (1972).
[CrossRef]

1967 (1)

L. A. Ostrovskii, “Self-action of light in crystals,” JETP Lett. 5, 272–275 (1967).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

1959 (1)

J. M. Manley and H. E. Rowe, “General energy in nonlinear reactances,” Proc. IRE 47, 2115 (1959).

Aitchison, J. S.

Alexander, J. I.

Amano, M.

Y. Shuto, T. Watanabe, S. Tomaru, I. Yokohama, M. Hikita, and M. Amano, “Quasi-phase-matched second-harmonic generation in diazo-dye- substituted polymer channel waveguides,” IEEE J. Quantum Electron. 33, 349–357 (1997).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Arnold, J. M.

Assanto, G.

Bacher, G.

H. O. Wagner, M. Kühnelt, G. Wein, B. Hahn, W. Genhardt, D. Eisert, G. Bacher, and A. Forchel, “Phase matched second harmonic generation using a χ(2) modulated ZnTe/ZnSe optical waveguide,” J. Lumin. 72, 87–89 (1997).
[CrossRef]

Baldi, P.

Batchko, R.

V. Shur, E. Rumyantsev, R. Batchko, G. Miller, M. Fejer, and R. Byer, “Physical basis of the domain engineering in the bulk ferroelectrics,” Ferroelectrics 221, 157–167 (1999).
[CrossRef]

Batchko, R. G.

Belashenkov, N. R.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. 66, 806–808 (1989).

Bierlein, J. D.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bosenberg, W. R.

Bosshard, C.

Britton, P.

Brown, C. T. A.

Bryce, A. C.

Bunin, G.

Y. Paltiel, D. Mahalu, H. Shtrikman, G. Bunin, and U. Meirav, “Short-period surface superlattices formed by plasma etching,” Semicond. Sci. Technol. 12, 987–990 (1997).
[CrossRef]

Burak, A. V.

Burns, W. K.

Byer, R.

V. Shur, E. Rumyantsev, R. Batchko, G. Miller, M. Fejer, and R. Byer, “Physical basis of the domain engineering in the bulk ferroelectrics,” Ferroelectrics 221, 157–167 (1999).
[CrossRef]

Byer, R. L.

Cappellini, G.

Chatenoud, F.

S. Janz, F. Chatenoud, and R. Normandin, “Quasi-phase-matched second-harmonic generation from asymmetric coupled quantum wells,” Opt. Lett. 19, 622–624 (1994).
[CrossRef] [PubMed]

R. Normandin, R. L. Williams, and F. Chatenoud, “Enhanced surface emitting waveguides for visible, monolithic semiconductor laser sources,” Electron. Lett. 26, 2088–2089 (1990).
[CrossRef]

Chen, B.

B. Chen, C. Q. Xu, B. Zhou, and X. H. Tang, “Analysis of cascaded second-order nonlinear interaction based on quasi-phase-matched optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 675–680 (2002).
[CrossRef]

Chisari, R.

D’Alessandro, G.

G. D’Alessandro, P. S. J. Russell, and A. A. Wheeler, “Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator,” Phys. Rev. A 55, 3211–3218 (1997).
[CrossRef]

Daneshvar, K.

K. Daneshvar and D. H. Kang, “A novel method for laser-induced periodic domain reversal in LiNbO3,” IEEE J. Quantum Electron. 36, 85–88 (2000).
[CrossRef]

De Micheli, M.

DeSalvo, R.

Ding, Y. J.

Drobshoff, A.

Ducuino, J.

J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ebrahimzadeh, M.

Eisert, D.

H. O. Wagner, M. Kühnelt, G. Wein, B. Hahn, W. Genhardt, D. Eisert, G. Bacher, and A. Forchel, “Phase matched second harmonic generation using a χ(2) modulated ZnTe/ZnSe optical waveguide,” J. Lumin. 72, 87–89 (1997).
[CrossRef]

Fejer, M.

V. Shur, E. Rumyantsev, R. Batchko, G. Miller, M. Fejer, and R. Byer, “Physical basis of the domain engineering in the bulk ferroelectrics,” Ferroelectrics 221, 157–167 (1999).
[CrossRef]

Fejer, M. M.

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

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]

Forchel, A.

H. O. Wagner, M. Kühnelt, G. Wein, B. Hahn, W. Genhardt, D. Eisert, G. Bacher, and A. Forchel, “Phase matched second harmonic generation using a χ(2) modulated ZnTe/ZnSe optical waveguide,” J. Lumin. 72, 87–89 (1997).
[CrossRef]

Freysz, E.

Fukuchi, Y.

Y. Fukuchi and K. Kikuchi, “Novel design method for all-optical ultrafast gate switches using cascaded second-order nonlinear effect in quasi-phase matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 14, 1409–1411 (2002).
[CrossRef]

Y. Fukuchi, T. Sakamoto, K. Taira, K. Kikuchi, D. Kunimatsu, A. Suzuki, and H. Ito, “Speed limit of all-optical gate switched using cascaded second-order nonlinear effect in quasi-phase-matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 13, 1267–1269 (2002).
[CrossRef]

Gagarskii, S. V.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. 66, 806–808 (1989).

Gallo, K.

Gase, T.

T. Gase and W. Karthe, “Quasi-phase matched cascaded second order processes in poled organic polymer waveguides,” Opt. Commun. 133, 549–556 (1997).
[CrossRef]

Genhardt, W.

H. O. Wagner, M. Kühnelt, G. Wein, B. Hahn, W. Genhardt, D. Eisert, G. Bacher, and A. Forchel, “Phase matched second harmonic generation using a χ(2) modulated ZnTe/ZnSe optical waveguide,” J. Lumin. 72, 87–89 (1997).
[CrossRef]

Gu, X.

Guillet de Chatellus, H.

Hagan, D. J.

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N. Hashizume, T. Tsuruzono, T. Kondo, and R. Ito, “Fabrication of periodic waveguides using organic crystals and fluorinated polyimides for quasi-phase-matched second-harmonic generation,” Opt. Rev. 4, 316–320 (1997).
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K. Mizuuchi, K. Yamamoto, and M. Kato, “Generation of ultraviolet light by frequency doubling of a red laser diode in a first-order periodically poled bulk LiTaO3,” Appl. Phys. Lett. 70, 1201–1203 (1997).
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Kikuchi, K.

Y. Fukuchi and K. Kikuchi, “Novel design method for all-optical ultrafast gate switches using cascaded second-order nonlinear effect in quasi-phase matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 14, 1409–1411 (2002).
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Miller, G. D.

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H. Liu, N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, “Multiple-wavelength second-harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 81, 3326–3328 (2002).
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K. Mizuuchi, K. Yamamoto, and M. Kato, “Generation of ultraviolet light by frequency doubling of a red laser diode in a first-order periodically poled bulk LiTaO3,” Appl. Phys. Lett. 70, 1201–1203 (1997).
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L. A. Ostrovskii, “Self-action of light in crystals,” JETP Lett. 5, 272–275 (1967).

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Y. Paltiel, D. Mahalu, H. Shtrikman, G. Bunin, and U. Meirav, “Short-period surface superlattices formed by plasma etching,” Semicond. Sci. Technol. 12, 987–990 (1997).
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Y. Qin, S. N. Pietranlunga, and M. Martinelli, “Quasi-phase-matched (QPM) difference frequency generation in a mirrorless counterpropagating configuration,” J. Lightwave Technol. 19, 1298–1306 (2001).
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Y. Qin, S. N. Pietranlunga, and M. Martinelli, “Correction to quasi-phase-matched (QPM) difference frequency generation in a mirrorless counterpropagating configuration,” J. Lightwave Technol. 19, 1794 (2001).
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Y. Qin, S. N. Pietranlunga, and M. Martinelli, “Correction to quasi-phase-matched (QPM) difference frequency generation in a mirrorless counterpropagating configuration,” J. Lightwave Technol. 19, 1794 (2001).
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Y. Qin, S. N. Pietranlunga, and M. Martinelli, “Quasi-phase-matched (QPM) difference frequency generation in a mirrorless counterpropagating configuration,” J. Lightwave Technol. 19, 1298–1306 (2001).
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V. Shur, E. Rumyantsev, R. Batchko, G. Miller, M. Fejer, and R. Byer, “Physical basis of the domain engineering in the bulk ferroelectrics,” Ferroelectrics 221, 157–167 (1999).
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Y. Fukuchi, T. Sakamoto, K. Taira, K. Kikuchi, D. Kunimatsu, A. Suzuki, and H. Ito, “Speed limit of all-optical gate switched using cascaded second-order nonlinear effect in quasi-phase-matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 13, 1267–1269 (2002).
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Y. Paltiel, D. Mahalu, H. Shtrikman, G. Bunin, and U. Meirav, “Short-period surface superlattices formed by plasma etching,” Semicond. Sci. Technol. 12, 987–990 (1997).
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Y. Fukuchi, T. Sakamoto, K. Taira, K. Kikuchi, D. Kunimatsu, A. Suzuki, and H. Ito, “Speed limit of all-optical gate switched using cascaded second-order nonlinear effect in quasi-phase-matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 13, 1267–1269 (2002).
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B. Chen, C. Q. Xu, B. Zhou, and X. H. Tang, “Analysis of cascaded second-order nonlinear interaction based on quasi-phase-matched optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 675–680 (2002).
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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,” Opt. Quantum Electron. 28, 1691–1740 (1996).
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N. Hashizume, T. Tsuruzono, T. Kondo, and R. Ito, “Fabrication of periodic waveguides using organic crystals and fluorinated polyimides for quasi-phase-matched second-harmonic generation,” Opt. Rev. 4, 316–320 (1997).
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H. O. Wagner, M. Kühnelt, G. Wein, B. Hahn, W. Genhardt, D. Eisert, G. Bacher, and A. Forchel, “Phase matched second harmonic generation using a χ(2) modulated ZnTe/ZnSe optical waveguide,” J. Lumin. 72, 87–89 (1997).
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R. Normandin, R. L. Williams, and F. Chatenoud, “Enhanced surface emitting waveguides for visible, monolithic semiconductor laser sources,” Electron. Lett. 26, 2088–2089 (1990).
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B. Chen, C. Q. Xu, B. Zhou, and X. H. Tang, “Analysis of cascaded second-order nonlinear interaction based on quasi-phase-matched optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 675–680 (2002).
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H. Liu, N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, “Multiple-wavelength second-harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 81, 3326–3328 (2002).
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K. Mizuuchi, K. Yamamoto, and M. Kato, “Generation of ultraviolet light by frequency doubling of a red laser diode in a first-order periodically poled bulk LiTaO3,” Appl. Phys. Lett. 70, 1201–1203 (1997).
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H. Liu, N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, “Multiple-wavelength second-harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 81, 3326–3328 (2002).
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Y. Shuto, T. Watanabe, S. Tomaru, I. Yokohama, M. Hikita, and M. Amano, “Quasi-phase-matched second-harmonic generation in diazo-dye- substituted polymer channel waveguides,” IEEE J. Quantum Electron. 33, 349–357 (1997).
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B. Chen, C. Q. Xu, B. Zhou, and X. H. Tang, “Analysis of cascaded second-order nonlinear interaction based on quasi-phase-matched optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 675–680 (2002).
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H. Liu, N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, “Multiple-wavelength second-harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 81, 3326–3328 (2002).
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H. Liu, N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, “Multiple-wavelength second-harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 81, 3326–3328 (2002).
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Appl. Opt. (1)

Appl. Phys. Lett. (2)

K. Mizuuchi, K. Yamamoto, and M. Kato, “Generation of ultraviolet light by frequency doubling of a red laser diode in a first-order periodically poled bulk LiTaO3,” Appl. Phys. Lett. 70, 1201–1203 (1997).
[CrossRef]

H. Liu, N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, “Multiple-wavelength second-harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 81, 3326–3328 (2002).
[CrossRef]

Electron. Lett. (1)

R. Normandin, R. L. Williams, and F. Chatenoud, “Enhanced surface emitting waveguides for visible, monolithic semiconductor laser sources,” Electron. Lett. 26, 2088–2089 (1990).
[CrossRef]

Ferroelectrics (1)

V. Shur, E. Rumyantsev, R. Batchko, G. Miller, M. Fejer, and R. Byer, “Physical basis of the domain engineering in the bulk ferroelectrics,” Ferroelectrics 221, 157–167 (1999).
[CrossRef]

IEEE J. Quantum Electron. (4)

K. Daneshvar and D. H. Kang, “A novel method for laser-induced periodic domain reversal in LiNbO3,” IEEE J. Quantum Electron. 36, 85–88 (2000).
[CrossRef]

Y. Shuto, T. Watanabe, S. Tomaru, I. Yokohama, M. Hikita, and M. Amano, “Quasi-phase-matched second-harmonic generation in diazo-dye- substituted polymer channel waveguides,” IEEE J. Quantum Electron. 33, 349–357 (1997).
[CrossRef]

P. S. J. Russell, “Theoretical study of parametric frequency and wavefront conversion in nonlinear holograms,” IEEE J. Quantum Electron. 27, 830–835 (1991).
[CrossRef]

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).
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IEEE J. Sel. Top. Quantum Electron. (1)

B. Chen, C. Q. Xu, B. Zhou, and X. H. Tang, “Analysis of cascaded second-order nonlinear interaction based on quasi-phase-matched optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 675–680 (2002).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

Y. Fukuchi, T. Sakamoto, K. Taira, K. Kikuchi, D. Kunimatsu, A. Suzuki, and H. Ito, “Speed limit of all-optical gate switched using cascaded second-order nonlinear effect in quasi-phase-matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 13, 1267–1269 (2002).
[CrossRef]

Y. Fukuchi and K. Kikuchi, “Novel design method for all-optical ultrafast gate switches using cascaded second-order nonlinear effect in quasi-phase matched LiNbO3 devices,” IEEE Photonics Technol. Lett. 14, 1409–1411 (2002).
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J. Chem. Phys. (1)

G. R. Meredith, “Second-order cascading in third-order nonlinear optical processes,” J. Chem. Phys. 77, 5863–5871 (1982).
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J. Lightwave Technol. (4)

Y. Qin, S. N. Pietranlunga, and M. Martinelli, “Correction to quasi-phase-matched (QPM) difference frequency generation in a mirrorless counterpropagating configuration,” J. Lightwave Technol. 19, 1794 (2001).
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J. Lumin. (1)

H. O. Wagner, M. Kühnelt, G. Wein, B. Hahn, W. Genhardt, D. Eisert, G. Bacher, and A. Forchel, “Phase matched second harmonic generation using a χ(2) modulated ZnTe/ZnSe optical waveguide,” J. Lumin. 72, 87–89 (1997).
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J. Opt. Soc. Am. B (2)

J. Phys. D (1)

M. Houe and P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D 28, 1747–1763 (1995).
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JETP Lett. (1)

L. A. Ostrovskii, “Self-action of light in crystals,” JETP Lett. 5, 272–275 (1967).

Laser Optoelektron. (1)

J. Pierce and D. Lowenthal, “Periodically poled materials and devices,” Laser Optoelektron. 16, 25–27 (1997).

Opt. Commun. (2)

J. M. R. Thomas and J. P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–334 (1972).
[CrossRef]

T. Gase and W. Karthe, “Quasi-phase matched cascaded second order processes in poled organic polymer waveguides,” Opt. Commun. 133, 549–556 (1997).
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Opt. Express (1)

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Figures (6)

Fig. 1
Fig. 1

Wave-vector matching diagrams for (a) forward QPM (FQPM), (b) backward QPM (BQPM), (c) counterpropagating QPM (CQPM), and (d) surface-emitting QPM (SEQPM). There are two simultaneously phase-matchable processes for cases (a), (b), and (d), while there are six for case (c). All cases except case (d) are collinear. The fundamental and second-harmonic wave vectors are represented by thin black and thin gray arrows, respectively. The grating wave vector is represented by a thick black arrow and has a magnitude given by K=2π/Λ.

Fig. 2
Fig. 2

Schematics of the two CQPM devices under study with the distance normalized such that ξ=Ξz/L. (a) Mirrored CQPM consists of a common input and output interface at ξ=0 (i.e., z=0) and highly reflective mirror at ξ=Ξ (i.e., z=L). (b) Mirrorless CQPM has two FF counterpropagating waves of unequal amplitude with the stronger wave input at ξ=0. In both configurations, the two FF waves, A(ξ) and B(ξ), interact through the material’s nonlinear periodicity Λ=2π/K to produce two counterpropagating SH waves, C(ξ) and D(ξ). All fields are normalized as shown in Eq. (2) with Ξ given by Eq. (1).

Fig. 3
Fig. 3

Array of contour plots of η2 versus η1 with each plot representing a constant ϕ1 and ϕ2 given by the location in the array. The free parameters are F=3, G=0.25, and ϑ=2. The domain and range in all plots are 0η13 and 0η23, respectively. The nine contours in each plot are evenly spaced and range from H=0 (black) to H=8 (lightest gray). The domain restriction of Eq. (27) is visible in the upper-left and lower-right of each plot.

Fig. 4
Fig. 4

Array of contour plots of ϕ2 versus ϕ1 with each plot representing a constant η1 and η2 given by the location in the array. The free parameters are the same as in Fig. 3: F=3, G=0.25, and ϑ=2. The domain and range in all plots are 0ϕ1π, and 0ϕ2π, respectively. The nine contours in each plot are evenly spaced and range from H=0 (black) to H=8 (lightest gray). The domain restriction of Eq. (26) is visible by the absence of contours in the upper-left and lower-right sections of the array.

Fig. 5
Fig. 5

Schematic of the normalized intensity flow for the two CQPM configurations studied in this paper: (a) the mirrored configuration and (b) the mirrorless configuration.

Fig. 6
Fig. 6

Relationship of F to Ξ2 for (a) the mirrored configuration with G=0 and (b) the mirrorless configuration with FF input imbalances of ρ=1 (solid curve) and ρ=0.1 (dashed curve). Both cases assume perfect phase matching ϑ=0. For the mirrored case (a), only one value of F is found for Ξ2(π/4)2 and new pairs of solutions occur for every π above this point. The abscissa is not the same in these two figures.

Equations (106)

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Ξ=ω(2/π)d0Lcnω 2Z0n2ω I0=ΓI0,
A(ξ)=2 Eω+(ξ)2Z0/nωI0,
B(ξ)=2 [Eω-(ξ)]*2Z0/nωI0,
C(ξ)=E2ω+(ξ)2Z0/n2ωI0 exp(-jϑξ),
D(ξ)=[E2ω-(ξ)]*2Z0/n2ωI0 exp(-jϑξ),
ϑ=ΔkLΞ=(k2ω-K)LΞ=ΔκΞ,
dA(ξ)dξ=-jB(ξ)[C(ξ)+D*(ξ)],
dB(ξ)dξ=-jA(ξ)[C*(ξ)+D(ξ)],
dC(ξ)dξ=-jA(ξ)B*(ξ)-jϑC(ξ),
dD(ξ)dξ=-jA*(ξ)B(ξ)-jϑD(ξ),
|A(ξ)|2+|B(ξ)|2=F,
|A(ξ)|2-|B(ξ)|22+|C(ξ)|2-|D(ξ)|2=G.
0=d|A(ξ)|2dξ+d|B(ξ)|2dξ,
0=d|A(ξ)|2dξ-d|B(ξ)|2dξ+2d|C(ξ)|2dξ-d|D(ξ)|2dξ,
-d|A(ξ)|2dξ=d|B(ξ)|2dξ=d|C(ξ)|2dξ-d|D(ξ)|2dξ.
|A(Ξ)|2+|B(0)|22+|C(Ξ)|2+|D(0)|2=1,
|A(0)|2+|B(Ξ)|22+|C(0)|2+|D(Ξ)|2=1,
dXdξ=-j HCX*,
HC=A*B(C+D*)+AB*(C*+D)+ϑ(|C|2+|D|2).
A=a exp(jϕa),C=c exp(jϕc),
B=b exp(jϕb),D=d exp(jϕd).
ηα=x2,
dηadξ=-2ηaηb[ηc sin(ϕa-ϕb-ϕc)+ηd sin(ϕa-ϕb+ϕd)],
dηbdξ=+2ηaηb[ηc sin(ϕa-ϕb-ϕc)+ηd sin(ϕa-ϕb+ϕd)],
dηcdξ=+2ηaηbηc sin(ϕa-ϕb-ϕc),
dηddξ=-2ηaηbηd sin(ϕa-ϕb+ϕd).
dϕadξ=-ηbηa [ηc cos(ϕa-ϕb-ϕc)+ηd cos(ϕa-ϕb+ϕd)],
dϕbdξ=-ηaηb [ηc cos(ϕa-ϕb-ϕc)+ηd cos(ϕa-ϕb+ϕd)],
dϕcdξ=-ηaηbηc cos(ϕa-ϕb-ϕc)-ϑ,
dϕddξ=-ηaηbηd cos(ϕa-ϕb+ϕd)-ϑ.
pi=ϕi,
qi=ηi,
dqidξ=HRepi,
dpidξ=-HReqi,
dηidξ=HReϕi,
dϕidξ=-HReηi,
HRe=2ηaηb[ηc cos(ϕa-ϕb-ϕc)+ηd cos(ϕa-ϕb+ϕd)]+ϑ(ηc+ηd).
ϕ1=-(ϕa-ϕb)+ϕc,
ϕ2=+(ϕa-ϕb)+ϕd,
ηa+ηb=F,
ηa-ηb2+ηc-ηd=G.
ηa=F2+(G-η1+η2),
ηb=F2-(G-η1+η2),
η1=ηc,
η2=ηd.
H=F2-4(G-η1+η2)2[η1 cos(ϕ1)+η2 cos(ϕ2)]+ϑ(η1+η2).
dηidξ=Hϕi,
dϕidξ=-Hηi,
-F2+Gη1-η2F2+G.
dxdξ=f(x; α),
x=[η1η2ϕ1ϕ2]T,
f(x; α)=f1(x; α)f2(x; α)f3(x; α)f4(x; α)=Hϕ1Hϕ2-Hη1-Hη2T,
α=[FGϑ]T.
f(x; α)=-F2-4(G-η1+η2)2η1 sin(ϕ1)-F2-4(G-η1+η2)2η2 sin(ϕ2)-4(G-η1+η2)[η1 cos(ϕ1)+η2 cos(ϕ2)]F2-4(G-η1+η2)2-F2-4(G-η1+η2)2 cos(ϕ1)2η1-ϑ+4(G-η1+η2)[η1 cos(ϕ1)+η2 cos(ϕ2)]F2-4(G-η1+η2)2-F2-4(G-η1+η2)2 cos(ϕ2)2η2-ϑ.
F2-4(G-η1+η2)2cos(ϕ1)2η1+cos(ϕ2)2η2+2ϑ=0.
J(x; α)=dfdx=fmxn,
η1-η2=±F2+G.
η1 cos(ϕ1)+η2 cos(ϕ2)=0.
ϑ=F2-4(G+η2)24η2,
η2=-6G±5F2+16G210,
ϑ=±5F2+2G(-4G+5F2+16G2)10-6G-5F2+16G2,
ηa(0)=|A(0)|2=2,
ηc(0)=|C(0)|2=0.
ηb(0)=F-2,
ηd(0)=2-F2-G.
Eω-(Ξ)=|rω|exp(jϕrω)Eω+(Ξ),
E2ω-(Ξ)=|r2ω|exp(jϕr2ω)E2ω+(Ξ),
B*(Ξ)=|rω|exp(jϕrω)A(Ξ),
D*(Ξ)=|r2ω|exp(jϕr2ω)C(Ξ).
ηb(Ξ)=|rω|2ηa(Ξ),
ηd(Ξ)=|r2ω|2ηc(Ξ),
ϕa(Ξ)+ϕb(Ξ)=-ϕrω,
ϕc(Ξ)+ϕd(Ξ)=-ϕr2ω.
ηa(Ξ)=F11+|rω|2,
ηb(Ξ)=F|rω|21+|rω|2.
ηc(Ξ)=G-F2 1-|rω|21+|rω|211-|r2ω|2,
ηd(Ξ)=G-F2 1-|rω|21+|rω|2|r2ω|21-|r2ω|2.
C(ξ)+D*(ξ)=Ω,
ηd(0)=|1+r2ω|2ηc(Ξ).
ϕc(Ξ)=ϕΩ-arg(1+r2ω).
|Ω|2=ηc+ηd+2ηcηd cos(ϕc+ϕd),
ϕΩ=arcsinηc|Ω| sin(ϕc+ϕd)-ϕd,
2-F2-G=ηc+ηd+2ηcηd cos(ϕc+ϕd),
F=4-2[G+η1+η2+2η1η2 cos(ϕ1+ϕ2)].
G=2+F2 |1+r2ω|21+|r2ω|21-|rω|21+|rω|2-1|1+r2ω|21+|r2ω|2+1.
2F4,
0G1.
ηSHG=1-tan2π4-Ξ2ηSHG.
2-F2=1-tan2π4-Ξ22-F2.
ρ=|B(Ξ)|2|A(0)|2=ηb(Ξ)ηa(0).
ηa(0)+ηb(Ξ)=2.
ηa(0)=21+ρ,
ηb(Ξ)=2ρ1+ρ.
ηb(0)=F-21+ρ,
ηa(Ξ)=F-2ρ1+ρ.
ηc(0)=0,
ηd(0)=21+ρ-G-F2,
ηc(Ξ)=2ρ1+ρ+G-F2,
ηd(Ξ)=0.
G=1-ρ1+ρ.
ηd(0)=ηc(Ξ)=1-F2.
F=2-2[η1+η2+2η1η2 cos(ϕ1+ϕ2)].
21+ρF2,
0G1.
ηSHG=21+ρ ρ-ρ-sin ηSHGΞ2/2cos ηSHGΞ2/22.
1-F2=11+ρ ρ-ρ-sin Ξ21-F2cos Ξ21-F22.

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