Abstract

We deal with phase matching of three-wave mixing by total internal reflection in isotropic semiconductors. This technique makes use of the large relative phase lag between the three interacting waves at total internal reflection, as described by Augustin Fresnel. This is why we denote this technique as Fresnel phase matching. The theory of Fresnel phase matching is developed with a propagation matrix method: It allows us to describe the conditions (sample thickness, polarization, tuning angles, etc.) for phase matching, the influence of surface roughness, and the walk-off effects due to Goos–Hänchen shifts. Moreover, we show that nonresonant phase matching strongly alleviates the phase-matching tolerance while keeping good conversion yields. The potential of this technique is demonstrated by largely tunable mid-infrared generation (between 7 and 13 µm with a single sample) by use of difference-frequency mixing of two near-infrared sources. Excellent agreement between the presented theory and experiments is demonstrated both in GaAs and ZnSe samples.

© 2004 Optical Society of America

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2003 (5)

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “Quasi-phase-matched difference frequency generation (8–13 μm) in isotropic semiconductors using total internal reflection,” Appl. Phys. Lett. 82, 1167–1169 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, and E. Rosencher, “Non-resonant quasi-phase matching in GaAs plates by Fresnel birefringence,” Appl. Phys. Lett. 83, 1506–1508 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “New mid-infrared optical sources based on isotropic semiconductors (ZnSe and GaAs) using total internal reflection quasi-phase matching,” in International Conference on Solid State Crystals 2002, J. Rutkowski and A. Rogalski, eds., Proc. SPIE 5136, 335–343 (2003).

R. Haïdar, N. Forget, and E. Rosencher, “Optical parametric oscillations in microcavities based on isotropic semiconductors: a theoretical study,” IEEE J. Quantum Electron. 39, 569–576 (2003).
[CrossRef]

M. M. J. W. Van Herpen, S. E. Bisson, and F. J. M. Harren, “Continuous-wave operation of a single-frequency optical parametric oscillator at 4–5 μm based on periodically poled LiNbO3,” Opt. Lett. 28, 2497–2499 (2003).
[CrossRef] [PubMed]

2002 (2)

O. Levi, T. J. Pinguet, T. Skauli, L. A. Eyres, K. R. Parameswaran, J. S. Harris, Jr., M. M. Fejer, T. J. Kulp, S. E. Bisson, B. Gerard, E. Lallier, and L. Becouarn, “Difference frequency generation of 8-μm radiation in orientation-patterned GaAs,” Opt. Lett. 27, 2091–2093 (2002).
[CrossRef]

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

2001 (1)

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

2000 (2)

D. Yang, J. B. Khurgin, and Y. J. Ding, “Cascaded waveguide phase-matching arrangement,” Opt. Lett. 25, 496–498 (2000).
[CrossRef]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

1999 (1)

E. Rzepka, J. P. Roger, Ph. Lemasson, and R. Triboulet, “Optical transmission of ZnSe crystals grown by solid phase recrystallisation,” J. Cryst. Growth 197, 480–484 (1999).
[CrossRef]

1998 (5)

H. Komine, W. H. Long, J. W. Tully, Jr., and E. A. Stappaerts, “Quasi-phase-matched second-harmonic generation by use of a total-internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23, 661–663 (1998).
[CrossRef]

E. Lallier, L. Becouarn, M. Brévignon, and J. Lehoux, “Efficient second-harmonic generation of a CO2 laser with a quasi-phase-matched GaAs crystal,” Opt. Lett. 23, 1511–1153 (1998).
[CrossRef]

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear otpical materials,” Nature 391, 463–466 (1998).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

1996 (2)

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

A. N. Baranov, V. V. Sherstnev, C. Alibert, and A. Krier, “New III–V semiconductor lasers emitting near 2.6 μm,” J. Appl. Phys. 79, 3354–3356 (1996).
[CrossRef]

1995 (1)

1994 (1)

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

1992 (1)

M. M. Fejer, M. 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]

1988 (1)

J. B. Khurgin, “Second-order non-linear effects in asymetric quantum-well structures,” Phys. Rev. B 38, 4056–4066 (1988).
[CrossRef]

1986 (1)

1985 (1)

S. Adashi, “GaAs, AlAs and AlxGa1−xAs: material parameters for use in a research and device applications,” J. Appl. Phys. 58, R1–R29 (1985).
[CrossRef]

1984 (1)

H. H. Li, “Refractive index of ZnS, ZnSe and ZnTe and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 13, 103–150 (1984).
[CrossRef]

1976 (1)

D. E. Thomson, J. D. McMullen, and D. B. Anderson, “Second-harmonic generation in GaAs stack of plates using high power CO2 laser radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[CrossRef]

1975 (1)

J. P. van der Ziel, “Phase-matched harmonic generation in a laminar structure with wave propagation in the plane of the layers,” Appl. Phys. Lett. 26, 60–62 (1975).
[CrossRef]

1971 (3)

D. Chemla, Ph. Kupecek, Ch. Schwartz, C. Scwhab, and A. Goltzene, “Nonlinear properties of cuprous halides,” IEEE J. Quantum Electron. QE-7, 126–132 (1971).
[CrossRef]

H. Shih and N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
[CrossRef]

P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971).
[CrossRef] [PubMed]

1966 (1)

G. D. Boyd and C. K. N. Patel, “Enhancement of optical second harmonic generation (SHG) by reflection phase matching in ZnS and GaAs,” Appl. Phys. Lett. 8, 313–315 (1966).
[CrossRef]

1962 (2)

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

J. A. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Lett. 8, 19–20 (1962).
[CrossRef]

Adashi, S.

S. Adashi, “GaAs, AlAs and AlxGa1−xAs: material parameters for use in a research and device applications,” J. Appl. Phys. 58, R1–R29 (1985).
[CrossRef]

Aigle, M.

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

Alibert, C.

A. N. Baranov, V. V. Sherstnev, C. Alibert, and A. Krier, “New III–V semiconductor lasers emitting near 2.6 μm,” J. Appl. Phys. 79, 3354–3356 (1996).
[CrossRef]

Anderson, D. B.

D. E. Thomson, J. D. McMullen, and D. B. Anderson, “Second-harmonic generation in GaAs stack of plates using high power CO2 laser radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[CrossRef]

Angell, M. J.

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

Armstrong, J. A.

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

Baillargeon, J. N.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Baranov, A. N.

A. N. Baranov, V. V. Sherstnev, C. Alibert, and A. Krier, “New III–V semiconductor lasers emitting near 2.6 μm,” J. Appl. Phys. 79, 3354–3356 (1996).
[CrossRef]

Barbieri, S.

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

Bauer, G.

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

Beck, M.

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

Becouarn, L.

Berger, V.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear otpical materials,” Nature 391, 463–466 (1998).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Birman, J. L.

Bisson, S. E.

Bloembergen, N.

H. Shih and N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
[CrossRef]

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

Bortz, M. L.

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

Bosenberg, W. R.

Boyd, G. D.

G. D. Boyd and C. K. N. Patel, “Enhancement of optical second harmonic generation (SHG) by reflection phase matching in ZnS and GaAs,” Appl. Phys. Lett. 8, 313–315 (1966).
[CrossRef]

Bravetti, P.

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear otpical materials,” Nature 391, 463–466 (1998).
[CrossRef]

Brévignon, M.

Byer, R. L.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
[CrossRef]

M. M. Fejer, M. 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]

Capasso, F.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Chemla, D.

D. Chemla, Ph. Kupecek, Ch. Schwartz, C. Scwhab, and A. Goltzene, “Nonlinear properties of cuprous halides,” IEEE J. Quantum Electron. QE-7, 126–132 (1971).
[CrossRef]

Cho, A. Y.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Chu, S.-N. G.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Collot, Ph.

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

Delobel, L.

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Ding, Y. J.

Ducuing, J.

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

Eckardt, R. C.

Emerson, M. L.

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

Eyres, L. A.

O. Levi, T. J. Pinguet, T. Skauli, L. A. Eyres, K. R. Parameswaran, J. S. Harris, Jr., M. M. Fejer, T. J. Kulp, S. E. Bisson, B. Gerard, E. Lallier, and L. Becouarn, “Difference frequency generation of 8-μm radiation in orientation-patterned GaAs,” Opt. Lett. 27, 2091–2093 (2002).
[CrossRef]

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

Faist, J.

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Fejer, M. M.

O. Levi, T. J. Pinguet, T. Skauli, L. A. Eyres, K. R. Parameswaran, J. S. Harris, Jr., M. M. Fejer, T. J. Kulp, S. E. Bisson, B. Gerard, E. Lallier, and L. Becouarn, “Difference frequency generation of 8-μm radiation in orientation-patterned GaAs,” Opt. Lett. 27, 2091–2093 (2002).
[CrossRef]

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
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M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

M. M. Fejer, M. 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]

Fiore, A.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear otpical materials,” Nature 391, 463–466 (1998).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Forget, N.

R. Haïdar, N. Forget, and E. Rosencher, “Optical parametric oscillations in microcavities based on isotropic semiconductors: a theoretical study,” IEEE J. Quantum Electron. 39, 569–576 (2003).
[CrossRef]

Gerard, B.

Gibbons, J. F.

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
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J. A. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Lett. 8, 19–20 (1962).
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D. Chemla, Ph. Kupecek, Ch. Schwartz, C. Scwhab, and A. Goltzene, “Nonlinear properties of cuprous halides,” IEEE J. Quantum Electron. QE-7, 126–132 (1971).
[CrossRef]

Haïdar, R.

R. Haïdar, N. Forget, and E. Rosencher, “Optical parametric oscillations in microcavities based on isotropic semiconductors: a theoretical study,” IEEE J. Quantum Electron. 39, 569–576 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “New mid-infrared optical sources based on isotropic semiconductors (ZnSe and GaAs) using total internal reflection quasi-phase matching,” in International Conference on Solid State Crystals 2002, J. Rutkowski and A. Rogalski, eds., Proc. SPIE 5136, 335–343 (2003).

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “Quasi-phase-matched difference frequency generation (8–13 μm) in isotropic semiconductors using total internal reflection,” Appl. Phys. Lett. 82, 1167–1169 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, and E. Rosencher, “Non-resonant quasi-phase matching in GaAs plates by Fresnel birefringence,” Appl. Phys. Lett. 83, 1506–1508 (2003).
[CrossRef]

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

Harren, F. J. M.

Harris Jr., J. S.

Heiss, W.

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

Hoyt, J. L.

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

Hutchinson, A. L.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Janz, S.

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

Jundt, D. H.

M. M. Fejer, M. 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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D. Yang, J. B. Khurgin, and Y. J. Ding, “Cascaded waveguide phase-matching arrangement,” Opt. Lett. 25, 496–498 (2000).
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A. N. Baranov, V. V. Sherstnev, C. Alibert, and A. Krier, “New III–V semiconductor lasers emitting near 2.6 μm,” J. Appl. Phys. 79, 3354–3356 (1996).
[CrossRef]

Kruck, P.

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

Kulp, T. J.

Kupecek, Ph.

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “New mid-infrared optical sources based on isotropic semiconductors (ZnSe and GaAs) using total internal reflection quasi-phase matching,” in International Conference on Solid State Crystals 2002, J. Rutkowski and A. Rogalski, eds., Proc. SPIE 5136, 335–343 (2003).

R. Haïdar, Ph. Kupecek, and E. Rosencher, “Non-resonant quasi-phase matching in GaAs plates by Fresnel birefringence,” Appl. Phys. Lett. 83, 1506–1508 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “Quasi-phase-matched difference frequency generation (8–13 μm) in isotropic semiconductors using total internal reflection,” Appl. Phys. Lett. 82, 1167–1169 (2003).
[CrossRef]

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

D. Chemla, Ph. Kupecek, Ch. Schwartz, C. Scwhab, and A. Goltzene, “Nonlinear properties of cuprous halides,” IEEE J. Quantum Electron. QE-7, 126–132 (1971).
[CrossRef]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Lallier, E.

Lehoux, J.

Lemasson, Ph.

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “New mid-infrared optical sources based on isotropic semiconductors (ZnSe and GaAs) using total internal reflection quasi-phase matching,” in International Conference on Solid State Crystals 2002, J. Rutkowski and A. Rogalski, eds., Proc. SPIE 5136, 335–343 (2003).

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “Quasi-phase-matched difference frequency generation (8–13 μm) in isotropic semiconductors using total internal reflection,” Appl. Phys. Lett. 82, 1167–1169 (2003).
[CrossRef]

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

E. Rzepka, J. P. Roger, Ph. Lemasson, and R. Triboulet, “Optical transmission of ZnSe crystals grown by solid phase recrystallisation,” J. Cryst. Growth 197, 480–484 (1999).
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Li, H. H.

H. H. Li, “Refractive index of ZnS, ZnSe and ZnTe and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 13, 103–150 (1984).
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Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Magel, M. A.

M. M. Fejer, M. 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]

McMullen, J. D.

D. E. Thomson, J. D. McMullen, and D. B. Anderson, “Second-harmonic generation in GaAs stack of plates using high power CO2 laser radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[CrossRef]

Mennerat, G.

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

Mustelier, A.

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

Myers, L. E.

Nagle, J.

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear otpical materials,” Nature 391, 463–466 (1998).
[CrossRef]

Oesterle, U.

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

Parameswaran, K. R.

Pascher, H.

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

Patel, C. K. N.

G. D. Boyd and C. K. N. Patel, “Enhancement of optical second harmonic generation (SHG) by reflection phase matching in ZnS and GaAs,” Appl. Phys. Lett. 8, 313–315 (1966).
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J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pierce, J. W.

Pinguet, T. J.

Puri, A.

Roger, J. P.

E. Rzepka, J. P. Roger, Ph. Lemasson, and R. Triboulet, “Optical transmission of ZnSe crystals grown by solid phase recrystallisation,” J. Cryst. Growth 197, 480–484 (1999).
[CrossRef]

Rosencher, E.

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “New mid-infrared optical sources based on isotropic semiconductors (ZnSe and GaAs) using total internal reflection quasi-phase matching,” in International Conference on Solid State Crystals 2002, J. Rutkowski and A. Rogalski, eds., Proc. SPIE 5136, 335–343 (2003).

R. Haïdar, N. Forget, and E. Rosencher, “Optical parametric oscillations in microcavities based on isotropic semiconductors: a theoretical study,” IEEE J. Quantum Electron. 39, 569–576 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “Quasi-phase-matched difference frequency generation (8–13 μm) in isotropic semiconductors using total internal reflection,” Appl. Phys. Lett. 82, 1167–1169 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, and E. Rosencher, “Non-resonant quasi-phase matching in GaAs plates by Fresnel birefringence,” Appl. Phys. Lett. 83, 1506–1508 (2003).
[CrossRef]

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear otpical materials,” Nature 391, 463–466 (1998).
[CrossRef]

Rzepka, E.

E. Rzepka, J. P. Roger, Ph. Lemasson, and R. Triboulet, “Optical transmission of ZnSe crystals grown by solid phase recrystallisation,” J. Cryst. Growth 197, 480–484 (1999).
[CrossRef]

Schwartz, Ch.

D. Chemla, Ph. Kupecek, Ch. Schwartz, C. Scwhab, and A. Goltzene, “Nonlinear properties of cuprous halides,” IEEE J. Quantum Electron. QE-7, 126–132 (1971).
[CrossRef]

Schwarzl, T.

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

Scwhab, C.

D. Chemla, Ph. Kupecek, Ch. Schwartz, C. Scwhab, and A. Goltzene, “Nonlinear properties of cuprous halides,” IEEE J. Quantum Electron. QE-7, 126–132 (1971).
[CrossRef]

Sherstnev, V. V.

A. N. Baranov, V. V. Sherstnev, C. Alibert, and A. Krier, “New III–V semiconductor lasers emitting near 2.6 μm,” J. Appl. Phys. 79, 3354–3356 (1996).
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H. Shih and N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
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C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Sivco, D. L.

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

Skauli, T.

Springholz, G.

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

Stappaerts, E. A.

Thomson, D. E.

D. E. Thomson, J. D. McMullen, and D. B. Anderson, “Second-harmonic generation in GaAs stack of plates using high power CO2 laser radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[CrossRef]

Tien, P. K.

Triboulet, R.

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “New mid-infrared optical sources based on isotropic semiconductors (ZnSe and GaAs) using total internal reflection quasi-phase matching,” in International Conference on Solid State Crystals 2002, J. Rutkowski and A. Rogalski, eds., Proc. SPIE 5136, 335–343 (2003).

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “Quasi-phase-matched difference frequency generation (8–13 μm) in isotropic semiconductors using total internal reflection,” Appl. Phys. Lett. 82, 1167–1169 (2003).
[CrossRef]

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
[CrossRef]

E. Rzepka, J. P. Roger, Ph. Lemasson, and R. Triboulet, “Optical transmission of ZnSe crystals grown by solid phase recrystallisation,” J. Cryst. Growth 197, 480–484 (1999).
[CrossRef]

Tully Jr., J. W.

Van der Meer, P.

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
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J. P. van der Ziel, “Phase-matched harmonic generation in a laminar structure with wave propagation in the plane of the layers,” Appl. Phys. Lett. 26, 60–62 (1975).
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Vavra, I.

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos–Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[CrossRef]

Yang, D.

Appl. Opt. (1)

Appl. Phys. Lett. (10)

G. Springholz, T. Schwarzl, W. Heiss, G. Bauer, M. Aigle, H. Pascher, and I. Vavra, “Midinfrared surface-emitting PbSe/PbEuTe quantum-dot lasers,” Appl. Phys. Lett. 79, 1225–1227 (2001).
[CrossRef]

J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5 μm) quantum cascade lasers operating above room temperature,” Appl. Phys. Lett. 68, 3680–3682 (1996).
[CrossRef]

C. Sirtori, P. Kruck, S. Barbieri, Ph. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, “GaAs/AlxGa1−x As quantum cascade lasers,” Appl. Phys. Lett. 73, 3486–3488 (1998).
[CrossRef]

J. P. van der Ziel, “Phase-matched harmonic generation in a laminar structure with wave propagation in the plane of the layers,” Appl. Phys. Lett. 26, 60–62 (1975).
[CrossRef]

D. E. Thomson, J. D. McMullen, and D. B. Anderson, “Second-harmonic generation in GaAs stack of plates using high power CO2 laser radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[CrossRef]

M. J. Angell, M. L. Emerson, J. L. Hoyt, J. F. Gibbons, L. A. Eyres, M. L. Bortz, and M. M. Fejer, “Growth of alternating 〈100〉〈111〉-oriented II–VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates,” Appl. Phys. Lett. 64, 3107–3109 (1994).
[CrossRef]

G. D. Boyd and C. K. N. Patel, “Enhancement of optical second harmonic generation (SHG) by reflection phase matching in ZnS and GaAs,” Appl. Phys. Lett. 8, 313–315 (1966).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. Van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second harmonic generation at 1.6 μm in AlGaAs/AlOx waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2944 (1998).
[CrossRef]

R. Haïdar, Ph. Kupecek, E. Rosencher, Ph. Lemasson, and R. Triboulet, “Quasi-phase-matched difference frequency generation (8–13 μm) in isotropic semiconductors using total internal reflection,” Appl. Phys. Lett. 82, 1167–1169 (2003).
[CrossRef]

R. Haïdar, Ph. Kupecek, and E. Rosencher, “Non-resonant quasi-phase matching in GaAs plates by Fresnel birefringence,” Appl. Phys. Lett. 83, 1506–1508 (2003).
[CrossRef]

IEEE J. Quantum Electron. (3)

M. M. Fejer, M. 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]

D. Chemla, Ph. Kupecek, Ch. Schwartz, C. Scwhab, and A. Goltzene, “Nonlinear properties of cuprous halides,” IEEE J. Quantum Electron. QE-7, 126–132 (1971).
[CrossRef]

R. Haïdar, N. Forget, and E. Rosencher, “Optical parametric oscillations in microcavities based on isotropic semiconductors: a theoretical study,” IEEE J. Quantum Electron. 39, 569–576 (2003).
[CrossRef]

J. Appl. Phys. (3)

R. Haïdar, A. Mustelier, Ph. Kupecek, E. Rosencher, R. Triboulet, Ph. Lemasson, and G. Mennerat, “Largely tunable mid-infrared (8–12 μm) difference frequency generation in isotropic semiconductors,” J. Appl. Phys. 91, 2550–2552 (2002).
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[CrossRef]

J. Cryst. Growth (1)

E. Rzepka, J. P. Roger, Ph. Lemasson, and R. Triboulet, “Optical transmission of ZnSe crystals grown by solid phase recrystallisation,” J. Cryst. Growth 197, 480–484 (1999).
[CrossRef]

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

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

J. Phys. Chem. Ref. Data (1)

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Figures (17)

Fig. 1
Fig. 1

Scheme of the Fresnel phase-matching configuration.

Fig. 2
Fig. 2

Geometry of the Fresnel phase-matched plate: (a) side view for geometrical description and (b) view from above for angle definition.

Fig. 3
Fig. 3

Resonant Fresnel phase matching: The distance L between two successive bounces is optimized [L=(2p+1)ΛC, where p is an integer].

Fig. 4
Fig. 4

Fresnel phase shift ΔϕF computed for various polarization configurations in GaAs and ZnSe plates.

Fig. 5
Fig. 5

Fine tuning of the Fresnel phase-matching angle.

Fig. 6
Fig. 6

Fresnel phase-matching experimental scheme. OPO, optical parametric oscillator.

Fig. 7
Fig. 7

DFG in ZnSe. Two polarization configurations are explored: (a) spp and (b) pss.

Fig. 8
Fig. 8

There is a similarity between the Fresnel phase matching and a cavity surrounding a gain medium.

Fig. 9
Fig. 9

Nonresonant Fresnel phase matching: The distance L between two successive bounces is not optimized.

Fig. 10
Fig. 10

Fresnel phase shift ΔϕF combined with the δϕ can virtually reach any value between 0 and 2π(mod 2π), thus greatly alleviating the quasi-phase-matching conditions.

Fig. 11
Fig. 11

Two nonresonant quasi-phase-matching configurations are presented here for generation of λ3=11.6 µm in a GaAs plate.

Fig. 12
Fig. 12

Single GaAs plate is enough to cover the whole 9–13-µm spectral range. The only resonant cases are highlighted (circles) to demonstrate that the hyperwide tunability is obtained thanks to the nonresonant technique.

Fig. 13
Fig. 13

This scheme illustrates some nonresonant cases that prove more efficient than the resonant one.

Fig. 14
Fig. 14

(a) Goos–Hänchen effect and (b) effect of the Goos–Hänchen shift on the three interacting waves (walk-off effect).

Fig. 15
Fig. 15

This scheme illustrates the roughness parameters, as well as the Strehl ratio.

Fig. 16
Fig. 16

(a) Conversion losses due to surface roughness and (b) evolution of the efficient number of bounces Neff as a function of the Strehl ratio R.

Fig. 17
Fig. 17

Illustrating scheme of the Fresnel phase-matching computation formula.

Tables (4)

Tables Icon

Table 1 Theoretically Determined Resonant Fresnel Phase-Matching Conditions in (110) GaAs and ZnSe Platesa

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Table 2 Some Typical Cases of Nonresonant Quasi-Phase-Matching Situations in a GaAs Plate and Comparison with a Resonant Case at θ=45° a

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Table 3 Roughness Standard Deviation Measurements of the ZnSe and GaAs Plates Used for Parametric Conversion in the Resonant Quasi-Phase-Matching Situationsa

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Table 4 Conversion Efficiency Measurements of ZnSe and GaAs Plates in the Resonant Quasi-Phase-Matching Configurationsa

Equations (44)

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Δϕ=ΔkL+ΔϕF+δϕ.
Δϕ=ΔkL+ΔϕF+δϕ=2π(mod 2π).
η1sin(ΔkL/2)Δk/22.
η2sin(NΔϕ/2)N sin(Δϕ/2)2,
N=EP1+l-ttan θt tan θ,
ηsin(ΔkL/2)Δk/22sin(NΔϕ/2)N sin(Δϕ/2)2.
I3out=Z0ω322c2 (Ndeff)2n1n2n3 sin(ΔkL/2)Δk/22 × sin(NΔϕ/2)N sin(Δϕ/2)2I1inI2in,
I3out=2Z0ω32π2c2 (Ndeff)2n1n2n3ΛC2I1inI2in.
Δk(ω1,2,3)L(t, θ)=π,
ΔϕF(θ, ω1,2,3, P1,2,3)+δϕ(θ, φ, P1,2,3)=π,
Δk(ω3)L(t, θ)=π,
ΔϕF(θ, ω3, P1,2,3)+δϕ(θ, φ, P1,2,3)=π.
ΔkL(t, θ)=π,
ΔϕF(θ, P1,2,3)+δϕ(θ, φ, P1,2,3)=π.
dspp=d[12 cos2 θ(1+3 cos 2φ)+sin2 θ]sin φ,
dpss=14d(cos φ+3 cos 3φ)cos θ.
(1.9 µm, 2.3 µm)DFG11-µm
(1.949 µm, 2.343 µm)DFG11.6-µm
R=exp-σ 4πnλ cos θ21-σ 4πnλ cos θ2.
I3out=Z0ω322c2 (deff)2n1n2n3 sin(ΔkL/2)Δk/22 × r2N 1-2rN cos(NΔϕ)+r2N1-2r cos Δϕ+r2I1inI2in.
sin(NeffΔϕ/2)sin(Δϕ/2)2=r2N 1-2rN cos(NΔϕ)+r2N1-2r cos Δϕ+r2.
Em(z, t)=Re{m(z)exp[i(ωmt-kmz)]},
d2dz2m(z)kmddzm(z).
z[νL-L, νL],ddzA2,ν(z)=-jγνA3,ν*(z)exp{-j[Δkz+Φ1(0)+(ν-1)ϕF1]},
z[νL-L, νL],ddzA3,ν(z)=-jγνA2,ν*(z)exp{-j[Δkz+Φ1(0)+(ν-1)ϕF1]}.
γν=deffc ω1ω2ω3n1n2n31/2r1ν-1|A1(0)|exp[-j(ν-1)δϕ]=γr1ν-1 exp[-j(ν-1)δϕ]
A2,ν-=A2,ν+-j 2γr1ν-1 exp[-j(ν-1)(ΔkL+ϕF1+δϕ)]Δk × A3,ν+*exp[-jΦ1(0)]sinΔk2Lexp-j Δk2L
Zm,ν(z)=Am,ν(z)expjΔk2(z-νL+L)+Φ1(0)2.
Zm,ν+=Am,ν+ expj ν-12(ΔkL+ϕF1+δϕ),
Zm,ν-=Am,ν- expj ν2(ΔkL+ϕF1+δϕ),
Z2-Z3-*ν=exp[j 12(ΔkL+ϕF1+δϕ)]γr1ν-1ΔkCγr1ν-1ΔkC*exp[-j 12(ΔkL+ϕF1+δϕ)]Z2+Z3+*ν=Pν*(Z2+Z3+*)ν,
Z2+Z3+*ν+1=r2 exp(-jϕF2)00r3 exp(+jϕF3)
×Z2-Z3-*ν=B*Z2-Z3-*ν.
Z2+Z3+*ν+1=B*Pν*Z2+Z3+*ν=Tν*Z2+Z3+*ν,
Tν=r2 exp[+j 12(ΔkL+ϕF1+δϕ-2ϕF2)]r2 γr1ν-1ΔkC exp(-jϕF2)r3 γr1ν-1ΔkC* exp(+jϕF3)r3 exp[-j 12(ΔkL+ϕF1+δϕ-2ϕF3)].
Z2+Z3+*N+1=TN*TN-1**T1*Z2+Z3+*1.
Tν=rexp[+j12(ΔkL+ϕF1+δϕ-2ϕF2)]00exp[-j12(ΔkL+ϕF1+δϕ-2ϕF3)]+γrνΔk 0C exp(-jϕF2)C* exp(+jϕF3)0,
Tν=rD+γrνΔkA,
TN*TN-1**T1=rNDN+γΔk m=0N-1rmDmADN-m-1.
Z3,N+1+=rN γΔkC* exp(jϕF3) × expj N-12(ΔkL+ϕF1+δϕ+-2ϕF2)Z2,1+m=0N-1rm exp(-jmΔϕ),
Δϕ=ΔkL+(ϕF1-ϕF2-ϕF3)+δϕ.
I3out=Z0ω322c2 (deff)2n1n2n3 sin(ΔkL/2)Δk/22 × r2N 1-2rN cos(NΔϕ)+r2N1-2r cos Δϕ+r2I1inI2in,
I3out=Z0ω322c2 (Ndeff)2n1n2n3 sin(ΔkL/2)Δk/22 × sin(NΔϕ/2)N sin(Δϕ/2)2I1inI2in.
I3out=Z0ω322c2 ||deff|1+|deff|2r exp(jΔϕ)|2n1n2n3 sin(ΔkL/2)Δk/22 × rN 1-2rN/2 cos(NΔϕ/2)+rN1-2r cos Δϕ+r2I1inI2in.

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