Abstract

We report modal phase matched (MPM) second harmonic generation (SHG) in high-index contrast AlGaAs sub-micron ridge waveguides, by way of sub-mW continuous wave powers at telecommunication wavelengths. We achieve an experimental normalized conversion efficiency of ~14%/W/cm2, obtained through a careful sub-wavelength design supporting both the phase matching requirement and a significant overlap efficiency. Furthermore, the weak anomalous dispersion, robust fabrication technology and possible geometrical and thermal tuning of the device functionality enable a fully integrated multi-functional chip for several critical areas in telecommunications, including wavelength (time) division multiplexing and quantum entanglement.

© 2011 OSA

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    [CrossRef]
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    [CrossRef]
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  35. D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
    [CrossRef]
  36. A. Jaouad and V. Aimez, “Passivation of air-exposed AlGaAs using low frequency plasma-enhanced chemical vapor deposition of silicon nitride,” Appl. Phys. Lett. 89(9), 092125 (2006).
    [CrossRef]

2010 (1)

M. Volatier, D. Duchesne, R. Morandotti, R. Arès, and V. Aimez, “Extremely high aspect ratio GaAs and GaAs/AlGaAs nanowaveguides fabricated using chlorine ICP etching with N2-promoted passivation,” Nanotechnology 21(13), 134014 (2010).
[CrossRef] [PubMed]

2009 (2)

H. Ishikawa and T. Kondo, “Birefringent phase matching in thin rectangular high-index-contrast waveguides,” Appl. Phys. Express 2, 042202 (2009).
[CrossRef]

P. Abolghasem, J. Han, B. J. Bijlani, A. Arjmand, and A. S. Helmy, “Highly efficient second-harmonic generation in monolithic matching layer enhanced AlxGa1-xAs Bragg reflection waveguides,” IEEE Photon. Technol. Lett. 21(19), 1462–1464 (2009).
[CrossRef]

2007 (4)

A. M. Zheltikov, “Limiting efficiencies of second-harmonic generation and cascaded χ(2) processes in quadratically nonlinear photonic nanowires,” Opt. Commun. 270(2), 402–406 (2007).
[CrossRef]

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32(7), 826–828 (2007).
[CrossRef] [PubMed]

J. Meier, W. S. Mohammed, A. Jugessur, L. Qian, M. Mojahedi, and J. S. Aitchison, “Group velocity inversion in AlGaAs nanowires,” Opt. Express 15(20), 12755–12762 (2007).
[CrossRef] [PubMed]

2006 (5)

2005 (1)

2004 (2)

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[CrossRef]

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal and quasi-phase-matching techniques,” J. Opt. A, Pure Appl. Opt. 6(6), 569–584 (2004).
[CrossRef]

2002 (4)

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

A. Arie, K. Fradkin-Kashi, and Y. Shreberk, “Frequency conversion in novel materials and its application to high resolution gas sensing,” Opt. Lasers Eng. 37(2-3), 159–170 (2002).
[CrossRef]

R. F. Curl and F. K. Tittel, “Tunable infrared laser spectroscopy,” Annu. Rep. Prog. Chem. C 98, 217–270 (2002).
[CrossRef]

K. R. Parameswaran, R. K. Route, J. R. Kurz, R. V. Roussev, M. M. Fejer, and M. Fujimura, “Highly efficient second-harmonic generation in buried waveguides formed by annealed and reverse proton exchange in periodically poled lithium niobate,” Opt. Lett. 27(3), 179–181 (2002).
[CrossRef]

2001 (3)

S. Zollner, “Optical constants and critical-point parameters of GaAs from 0.73 to 6.60 eV,” J. Appl. Phys. 90(1), 515–517 (2001).
[CrossRef]

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

W. Petrich, “Mid-infrared and Raman spectroscopy for medical diagnostics,” Appl. Spectrosc. Rev. 36(2&3), 181–237 (2001).
[CrossRef]

2000 (1)

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

1998 (2)

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second-harmonic generation at λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

1996 (1)

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

1995 (2)

S. J. B. Yoo, R. Bhat, C. Caneau, and M. A. Koza, “Quasi-phase-matched second-harmonic generation in AlGaAs waveguides with periodic domain inversion achieved by wafer-bonding,” Appl. Phys. Lett. 66(25), 3410–3412 (1995).
[CrossRef]

T. C. Kowalczyk, K. D. Singer, and P. A. Cahill, “Anomalous-dispersion phase-matched second-harmonic generation in a polymer waveguide,” Opt. Lett. 20(22), 2273–2275 (1995).
[CrossRef] [PubMed]

1994 (1)

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47(5), 25–32 (1994).
[CrossRef]

1993 (1)

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

1991 (1)

1978 (1)

W. Sohler and H. Suche, “Second-harmonic generation in Ti-diffused LiNbO3 optical waveguides with 25% conversion efficiency,” Appl. Phys. Lett. 33(6), 518–520 (1978).
[CrossRef]

1971 (1)

D. B. Anderson and J. T. Boyd, “Wideband CO2 laser second harmonic generation phase matched in GaAs thin-film waveguides,” Appl. Phys. Lett. 19(8), 266–268 (1971).
[CrossRef]

Abolghasem, P.

P. Abolghasem, J. Han, B. J. Bijlani, A. Arjmand, and A. S. Helmy, “Highly efficient second-harmonic generation in monolithic matching layer enhanced AlxGa1-xAs Bragg reflection waveguides,” IEEE Photon. Technol. Lett. 21(19), 1462–1464 (2009).
[CrossRef]

Absil, P. P.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

Aimez, V.

M. Volatier, D. Duchesne, R. Morandotti, R. Arès, and V. Aimez, “Extremely high aspect ratio GaAs and GaAs/AlGaAs nanowaveguides fabricated using chlorine ICP etching with N2-promoted passivation,” Nanotechnology 21(13), 134014 (2010).
[CrossRef] [PubMed]

A. Jaouad and V. Aimez, “Passivation of air-exposed AlGaAs using low frequency plasma-enhanced chemical vapor deposition of silicon nitride,” Appl. Phys. Lett. 89(9), 092125 (2006).
[CrossRef]

Aitchison, J. S.

Anderson, D. B.

D. B. Anderson and J. T. Boyd, “Wideband CO2 laser second harmonic generation phase matched in GaAs thin-film waveguides,” Appl. Phys. Lett. 19(8), 266–268 (1971).
[CrossRef]

Arès, R.

M. Volatier, D. Duchesne, R. Morandotti, R. Arès, and V. Aimez, “Extremely high aspect ratio GaAs and GaAs/AlGaAs nanowaveguides fabricated using chlorine ICP etching with N2-promoted passivation,” Nanotechnology 21(13), 134014 (2010).
[CrossRef] [PubMed]

Arie, A.

A. Arie, K. Fradkin-Kashi, and Y. Shreberk, “Frequency conversion in novel materials and its application to high resolution gas sensing,” Opt. Lasers Eng. 37(2-3), 159–170 (2002).
[CrossRef]

Arjmand, A.

P. Abolghasem, J. Han, B. J. Bijlani, A. Arjmand, and A. S. Helmy, “Highly efficient second-harmonic generation in monolithic matching layer enhanced AlxGa1-xAs Bragg reflection waveguides,” IEEE Photon. Technol. Lett. 21(19), 1462–1464 (2009).
[CrossRef]

Baldi, P.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Berger, V.

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second-harmonic generation at λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Bhashi, M.

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

Bhat, R.

S. J. B. Yoo, R. Bhat, C. Caneau, and M. A. Koza, “Quasi-phase-matched second-harmonic generation in AlGaAs waveguides with periodic domain inversion achieved by wafer-bonding,” Appl. Phys. Lett. 66(25), 3410–3412 (1995).
[CrossRef]

Bijlani, B. J.

P. Abolghasem, J. Han, B. J. Bijlani, A. Arjmand, and A. S. Helmy, “Highly efficient second-harmonic generation in monolithic matching layer enhanced AlxGa1-xAs Bragg reflection waveguides,” IEEE Photon. Technol. Lett. 21(19), 1462–1464 (2009).
[CrossRef]

Boyd, J. T.

D. B. Anderson and J. T. Boyd, “Wideband CO2 laser second harmonic generation phase matched in GaAs thin-film waveguides,” Appl. Phys. Lett. 19(8), 266–268 (1971).
[CrossRef]

Bravetti, P.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 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 λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

Bristow, A. D.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Cahill, P. A.

Calligaro, M.

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[CrossRef]

Caneau, C.

S. J. B. Yoo, R. Bhat, C. Caneau, and M. A. Koza, “Quasi-phase-matched second-harmonic generation in AlGaAs waveguides with periodic domain inversion achieved by wafer-bonding,” Appl. Phys. Lett. 66(25), 3410–3412 (1995).
[CrossRef]

Chak, P.

Cheben, P.

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

Christodoulides, D.

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

Christodoulides, D. N.

Curl, R. F.

R. F. Curl and F. K. Tittel, “Tunable infrared laser spectroscopy,” Annu. Rep. Prog. Chem. C 98, 217–270 (2002).
[CrossRef]

De Angelis, C.

De Micheli, M.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

De Riedmatten, H.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

De Rossi, A.

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[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 λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

Dong, P.

Ducci, S.

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[CrossRef]

Duchesne, D.

M. Volatier, D. Duchesne, R. Morandotti, R. Arès, and V. Aimez, “Extremely high aspect ratio GaAs and GaAs/AlGaAs nanowaveguides fabricated using chlorine ICP etching with N2-promoted passivation,” Nanotechnology 21(13), 134014 (2010).
[CrossRef] [PubMed]

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

Ebrahimzadeh, M.

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal and quasi-phase-matching techniques,” J. Opt. A, Pure Appl. Opt. 6(6), 569–584 (2004).
[CrossRef]

El-Ganainy, R.

Fan, S.

Fejer, M. M.

Fiore, A.

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second-harmonic generation at λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Fradkin-Kashi, K.

A. Arie, K. Fradkin-Kashi, and Y. Shreberk, “Frequency conversion in novel materials and its application to high resolution gas sensing,” Opt. Lasers Eng. 37(2-3), 159–170 (2002).
[CrossRef]

Fujimura, M.

Fukatsu, S.

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

Gehrsitz, S.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

Gisin, N.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Goldhar, J.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

Gourgon, C.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

Grove, R.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

Han, J.

P. Abolghasem, J. Han, B. J. Bijlani, A. Arjmand, and A. S. Helmy, “Highly efficient second-harmonic generation in monolithic matching layer enhanced AlxGa1-xAs Bragg reflection waveguides,” IEEE Photon. Technol. Lett. 21(19), 1462–1464 (2009).
[CrossRef]

Harris, J. S.

Helmy, A. S.

P. Abolghasem, J. Han, B. J. Bijlani, A. Arjmand, and A. S. Helmy, “Highly efficient second-harmonic generation in monolithic matching layer enhanced AlxGa1-xAs Bragg reflection waveguides,” IEEE Photon. Technol. Lett. 21(19), 1462–1464 (2009).
[CrossRef]

Herres, N.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

Ho, P. T.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

Huo, Y.

Ibrahim, T. A.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

Ishigame, Y.

Ishikawa, H.

H. Ishikawa and T. Kondo, “Birefringent phase matching in thin rectangular high-index-contrast waveguides,” Appl. Phys. Express 2, 042202 (2009).
[CrossRef]

Ito, R.

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

Iwanow, R.

Iyer, R.

Janz, S.

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second-harmonic generation at λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

Jaouad, A.

A. Jaouad and V. Aimez, “Passivation of air-exposed AlGaAs using low frequency plasma-enhanced chemical vapor deposition of silicon nitride,” Appl. Phys. Lett. 89(9), 092125 (2006).
[CrossRef]

Johnson, F. G.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

Jugessur, A.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Kano, S. S.

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

Kirk, A. G.

Kondo, T.

H. Ishikawa and T. Kondo, “Birefringent phase matching in thin rectangular high-index-contrast waveguides,” Appl. Phys. Express 2, 042202 (2009).
[CrossRef]

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

Kowalczyk, T. C.

Koza, M. A.

S. J. B. Yoo, R. Bhat, C. Caneau, and M. A. Koza, “Quasi-phase-matched second-harmonic generation in AlGaAs waveguides with periodic domain inversion achieved by wafer-bonding,” Appl. Phys. Lett. 66(25), 3410–3412 (1995).
[CrossRef]

Kumar, S.

Kumata, K.

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

Kuo, P. S.

Kurz, J. R.

Lamontagne, B.

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

Lanco, L.

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[CrossRef]

Langrock, C.

Laurent, N.

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Locatelli, A.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

McGeehan, J. E.

Meier, J.

Modotto, D.

Mohammed, W. S.

Mojahedi, M.

Morandotti, R.

M. Volatier, D. Duchesne, R. Morandotti, R. Arès, and V. Aimez, “Extremely high aspect ratio GaAs and GaAs/AlGaAs nanowaveguides fabricated using chlorine ICP etching with N2-promoted passivation,” Nanotechnology 21(13), 134014 (2010).
[CrossRef] [PubMed]

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

G. A. Siviloglou, S. Suntsov, R. El-Ganainy, R. Iwanow, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, D. Modotto, A. Locatelli, C. De Angelis, F. Pozzi, C. R. Stanley, and M. Sorel, “Enhanced third-order nonlinear effects in optical AlGaAs nanowires,” Opt. Express 14(20), 9377–9384 (2006).
[CrossRef] [PubMed]

Moutzouris, K.

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal and quasi-phase-matching techniques,” J. Opt. A, Pure Appl. Opt. 6(6), 569–584 (2004).
[CrossRef]

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 λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Nishihara, H.

Ortiz, V.

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[CrossRef]

Ostrowsky, D. B.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Parameswaran, K. R.

Petrich, W.

W. Petrich, “Mid-infrared and Raman spectroscopy for medical diagnostics,” Appl. Spectrosc. Rev. 36(2&3), 181–237 (2001).
[CrossRef]

Pozzi, F.

Qian, L.

Rao, S. V.

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal and quasi-phase-matching techniques,” J. Opt. A, Pure Appl. Opt. 6(6), 569–584 (2004).
[CrossRef]

Reinhart, F. K.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

Ritter, K.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

Rosencher, E.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 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 λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Roussev, R. V.

Route, R. K.

Scaccabarozzi, L.

Shiraki, Y.

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

Shreberk, Y.

A. Arie, K. Fradkin-Kashi, and Y. Shreberk, “Frequency conversion in novel materials and its application to high resolution gas sensing,” Opt. Lasers Eng. 37(2-3), 159–170 (2002).
[CrossRef]

Sigga, H.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

Singer, K. D.

Sipe, J. E.

Siviloglou, G. A.

Smirl, A. L.

Sohler, W.

W. Sohler and H. Suche, “Second-harmonic generation in Ti-diffused LiNbO3 optical waveguides with 25% conversion efficiency,” Appl. Phys. Lett. 33(6), 518–520 (1978).
[CrossRef]

Sorel, M.

Stanley, C. R.

Stegeman, G. I.

Suche, H.

W. Sohler and H. Suche, “Second-harmonic generation in Ti-diffused LiNbO3 optical waveguides with 25% conversion efficiency,” Appl. Phys. Lett. 33(6), 518–520 (1978).
[CrossRef]

Suhara, T.

Suntsov, S.

Tanzilli, S.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Theilmann, S.

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Tittel, F. K.

R. F. Curl and F. K. Tittel, “Tunable infrared laser spectroscopy,” Annu. Rep. Prog. Chem. C 98, 217–270 (2002).
[CrossRef]

Tittel, W.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Upham, J.

Van, V.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

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 λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

van Driel, H. M.

Vodjdani, N.

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Volatier, M.

M. Volatier, D. Duchesne, R. Morandotti, R. Arès, and V. Aimez, “Extremely high aspect ratio GaAs and GaAs/AlGaAs nanowaveguides fabricated using chlorine ICP etching with N2-promoted passivation,” Nanotechnology 21(13), 134014 (2010).
[CrossRef] [PubMed]

Vonlanthen, A.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

Willner, A. E.

Xu, D.-X.

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

Yang, Z.

Yoo, S. J. B.

S. J. B. Yoo, R. Bhat, C. Caneau, and M. A. Koza, “Quasi-phase-matched second-harmonic generation in AlGaAs waveguides with periodic domain inversion achieved by wafer-bonding,” Appl. Phys. Lett. 66(25), 3410–3412 (1995).
[CrossRef]

Yu, X.

Zbinden, H.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Zheltikov, A. M.

A. M. Zheltikov, “Limiting efficiencies of second-harmonic generation and cascaded χ(2) processes in quadratically nonlinear photonic nanowires,” Opt. Commun. 270(2), 402–406 (2007).
[CrossRef]

Zollner, S.

S. Zollner, “Optical constants and critical-point parameters of GaAs from 0.73 to 6.60 eV,” J. Appl. Phys. 90(1), 515–517 (2001).
[CrossRef]

Annu. Rep. Prog. Chem. C (1)

R. F. Curl and F. K. Tittel, “Tunable infrared laser spectroscopy,” Annu. Rep. Prog. Chem. C 98, 217–270 (2002).
[CrossRef]

Appl. Phys. Express (1)

H. Ishikawa and T. Kondo, “Birefringent phase matching in thin rectangular high-index-contrast waveguides,” Appl. Phys. Express 2, 042202 (2009).
[CrossRef]

Appl. Phys. Lett. (7)

S. J. B. Yoo, R. Bhat, C. Caneau, and M. A. Koza, “Quasi-phase-matched second-harmonic generation in AlGaAs waveguides with periodic domain inversion achieved by wafer-bonding,” Appl. Phys. Lett. 66(25), 3410–3412 (1995).
[CrossRef]

A. Jaouad and V. Aimez, “Passivation of air-exposed AlGaAs using low frequency plasma-enhanced chemical vapor deposition of silicon nitride,” Appl. Phys. Lett. 89(9), 092125 (2006).
[CrossRef]

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, E. Rosencher, and J. Nagle, “Second-harmonic generation at λ = 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72(23), 2942–2944 (1998).
[CrossRef]

W. Sohler and H. Suche, “Second-harmonic generation in Ti-diffused LiNbO3 optical waveguides with 25% conversion efficiency,” Appl. Phys. Lett. 33(6), 518–520 (1978).
[CrossRef]

D. B. Anderson and J. T. Boyd, “Wideband CO2 laser second harmonic generation phase matched in GaAs thin-film waveguides,” Appl. Phys. Lett. 19(8), 266–268 (1971).
[CrossRef]

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84(16), 2974–2976 (2004).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, N. Laurent, S. Theilmann, N. Vodjdani, and J. Nagle, “Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides,” Appl. Phys. Lett. 68(10), 1320–1322 (1996).
[CrossRef]

Appl. Spectrosc. Rev. (1)

W. Petrich, “Mid-infrared and Raman spectroscopy for medical diagnostics,” Appl. Spectrosc. Rev. 36(2&3), 181–237 (2001).
[CrossRef]

Electron. Lett. (1)

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

P. Abolghasem, J. Han, B. J. Bijlani, A. Arjmand, and A. S. Helmy, “Highly efficient second-harmonic generation in monolithic matching layer enhanced AlxGa1-xAs Bragg reflection waveguides,” IEEE Photon. Technol. Lett. 21(19), 1462–1464 (2009).
[CrossRef]

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grove, J. Goldhar, and P. T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14(1), 74–76 (2002).
[CrossRef]

J. Appl. Phys. (3)

M. Bhashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficient of AlxGa1-xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74(1), 596–601 (1993).
[CrossRef]

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigga, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87(11), 7825–7837 (2000).
[CrossRef]

S. Zollner, “Optical constants and critical-point parameters of GaAs from 0.73 to 6.60 eV,” J. Appl. Phys. 90(1), 515–517 (2001).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A, Pure Appl. Opt. (1)

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal and quasi-phase-matching techniques,” J. Opt. A, Pure Appl. Opt. 6(6), 569–584 (2004).
[CrossRef]

Nanotechnology (1)

M. Volatier, D. Duchesne, R. Morandotti, R. Arès, and V. Aimez, “Extremely high aspect ratio GaAs and GaAs/AlGaAs nanowaveguides fabricated using chlorine ICP etching with N2-promoted passivation,” Nanotechnology 21(13), 134014 (2010).
[CrossRef] [PubMed]

Nature (1)

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

Opt. Commun. (1)

A. M. Zheltikov, “Limiting efficiencies of second-harmonic generation and cascaded χ(2) processes in quadratically nonlinear photonic nanowires,” Opt. Commun. 270(2), 402–406 (2007).
[CrossRef]

Opt. Eng. (1)

D. Duchesne, R. Morandotti, P. Cheben, B. Lamontagne, D.-X. Xu, S. Janz, and D. Christodoulides, “Group-index birefringence and loss measurements in silicon-on-insulator photonic wire waveguides,” Opt. Eng. 46(10), 104602 (2007).
[CrossRef]

Opt. Express (4)

Opt. Lasers Eng. (1)

A. Arie, K. Fradkin-Kashi, and Y. Shreberk, “Frequency conversion in novel materials and its application to high resolution gas sensing,” Opt. Lasers Eng. 37(2-3), 159–170 (2002).
[CrossRef]

Opt. Lett. (5)

Phys. Today (1)

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47(5), 25–32 (1994).
[CrossRef]

Other (2)

R. W. Boyd, Nonlinear Optics Third Edition, (Academic Press, New York 2008).

D. Duchesne, R. Morandotti, G. Siviloglou, R. El-Ganainy, G. Stegeman, D. Christodoulides, D. Modotto, A. Locatelli, C. De Angelis, F. Pozzi, and M. Sorel, “Nonlinear photonics in AlGaAs photonics nanowires: self phase and cross phase modulation,” in International Symposium on Signals, Systems and Electronics, 475–478 (2007).

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Figures (4)

Fig. 1
Fig. 1

(a) Scanning electron microscopy image of the fabricated waveguide device (width 650nm). (b) Corresponding heterostructure design, with widths W varying from 300nm to 1000nm. (c) Major electric field component of the horizontally (Ex ) polarized fundamental mode at λ = 1582.32nm. (d) Major electric field component of the vertically (Ey ) polarized fundamental mode at λ = 1582.32nm. (e) Major electric field component of the horizontally polarized (Ex ) second harmonic mode at λ = 791.15nm.

Fig. 2
Fig. 2

Second harmonic signal power obtained in the 650nm-wide waveguide while scanning the CW input fundamental wavelength (experiment dashed red curve), showing a phase matching wavelength at 1582.6nm (791.3nm for the SH), and an allowable wavelength mismatch of ~1nm. The coupled input power was estimated to be 155μW. The high frequency fringes are a result of the fundamental wavelength Fabry-Perot effect. The blue solid curve is the theoretical fitting used to estimate the SH losses. Inset: Second harmonic signal power in the 600nm-wide waveguide when excited with a pulsed 100fs input laser source. An enhancement in the SH power is clearly observed at the phase matching wavelength of ~1529nm.

Fig. 3
Fig. 3

Quadratic dependence of the generated second harmonic signal power on the fundamental frequency input CW power. An experimental normalized conversion efficiency of 13.8%/W/cm2 is obtained.

Fig. 4
Fig. 4

Modal dispersion of the first order modes at the fundamental frequency and the third order mode at the second harmonic (plotted at twice the wavelength here) for a 650nm-wide photonic wire showing a phase matching wavelength at 1582.32nm for a type II interaction. (b) Theoretical phase matching wavelength vs waveguide width (for the same type II interaction).

Equations (7)

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z E F F 1 = i η exp ( i Δ β z ) E S H E F F 2 * α F F 1 E F F 1 / 2 z E S H = i 2 η exp ( i Δ β z ) E F F 1 E F F 2 α S H E S H / 2 z E F F 2 = i η exp ( i Δ β z ) E F F 1 * E S H α F F 2 E F F 2 / 2
P S H = η 2 L 2 P F F 2 sin c 2 ( Δ β L / 2 )
Γ P S H L 2 P F F 2
χ x y x ( 2 ) = χ x x y ( 2 ) = χ y x x ( 2 ) = χ y z z ( 2 ) = χ z z y ( 2 ) = χ z y z ( 2 ) χ ( 2 )
η = χ ( 2 ) 2 ω 2 ε 0 c 3 n F F 1 n F F 2 n S H W G G d x d y | F F F 1 | 2 d x d y | F F F 2 | 2 d x d y | F S H | 2 d x d y G = F F F 1 x F F F 2 x F S H y + F F F 1 x F F F 2 y F S H x + F F F 1 y F F F 2 x F S H x F F F 1 y F F F 2 z * F S H z F F F 1 z * F F F 2 z * F S H y F F F 1 z * F F F 2 y F S H z
η χ ( 2 ) 2 ω 2 ε 0 c 3 n F F 1 n F F 2 n S H 1 Δ x Δ y
P S H = η 2 L 2 P F F 2 g / u F F 1 u F F 2 g exp ( - α SH L ) [ ( 1-exp ( - α L / 2 ) ) 2 + 4exp ( - α L / 2 ) sin 2 ( Δ β L / 2 ) ] ( Δ β L ) 2 + ( α L / 2 ) 2 u i ( 1 R i exp ( α i L ) ) 2 + 4 R i exp ( α i L ) sin 2 ( β i L )

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