Abstract

We summarize and discuss the results of our second-harmonic generation experiments in titanium-indiffused lithium niobate optical channel waveguides. The wave-vector mismatch in the nonlinear wave interaction was varied with temperature tuning around the second-harmonic resonances. Fundamental depletion and second-harmonic tuning curves show a strong power dependence, which is an indication of an intensity-dependent wave-vector modification of the interacting modes. This nonlinear refractive effect (which is called cascaded nonlinearity) is characterized with interferometric measurements of the resulting nonlinear phase shift of the fundamental. Large nonlinear phase shifts (>2π) appear in regions of low fundamental depletion (<10%) because of a nonuniform wave-vector mismatch along the waveguide. At resonance a maximum fundamental depletion of more than 90% was observed. All the measured results are explained well theoretically with a coupled-mode model that has proved to be a reliable design tool for fabricating waveguide devices for applications of the cascaded nonlinearity.

© 1998 Optical Society of America

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    [CrossRef]

1997 (3)

1996 (5)

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities—a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

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]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

R. Schiek, Y. Baek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, “All-optical switching in lithium niobate directional couplers with cascaded nonlinearity,” Opt. Lett. 21, 940–942 (1996).
[CrossRef] [PubMed]

1995 (2)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Y. Baek, R. Schiek, and G. I. Stegeman, “All-optical switching in a hybrid Mach–Zehnder interferometer as a result of cascaded second-order nonlinearity,” Opt. Lett. 20, 2168–2170 (1995).
[CrossRef]

1994 (1)

1993 (3)

1992 (2)

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

1989 (1)

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

1988 (1)

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with a scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[CrossRef]

1987 (1)

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, “Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators,” IEEE J. Quantum Electron. 23, 42–51 (1987).
[CrossRef]

1984 (1)

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[CrossRef]

1979 (1)

W. K. Burns, P. H. Klein, and E. J. West, “Ti diffusion in Ti:LiNbO3 planar and channel optical waveguides,” J. Appl. Phys. 50, 6175–6182 (1979).
[CrossRef]

1962 (1)

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

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Armstrong, J. A.

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

Asobe, M.

Assanto, G.

Baek, Y.

Baumann, I.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

R. Schiek, Y. Baek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, “All-optical switching in lithium niobate directional couplers with cascaded nonlinearity,” Opt. Lett. 21, 940–942 (1996).
[CrossRef] [PubMed]

Bava, G. P.

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with a scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[CrossRef]

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, “Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators,” IEEE J. Quantum Electron. 23, 42–51 (1987).
[CrossRef]

Belashenkov, N. R.

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

Beranek, M. W.

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

Bierlein, J. D.

Bloembergen, N.

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

Bosshard, C.

Burns, W. K.

W. K. Burns, P. H. Klein, and E. J. West, “Ti diffusion in Ti:LiNbO3 planar and channel optical waveguides,” J. Appl. Phys. 50, 6175–6182 (1979).
[CrossRef]

DeSalvo, R.

Ducuing, J.

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

Edwards, G. J.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[CrossRef]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Gagarskii, S. V.

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

Hagan, D. J.

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

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Inochkin, M. V.

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

Itoh, H.

Kaino, T.

Kim, D. Y.

Klein, P. H.

W. K. Burns, P. H. Klein, and E. J. West, “Ti diffusion in Ti:LiNbO3 planar and channel optical waveguides,” J. Appl. Phys. 50, 6175–6182 (1979).
[CrossRef]

Kobyakov, A.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities—a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Krijnen, G.

R. Schiek, Y. Baek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, “All-optical switching in lithium niobate directional couplers with cascaded nonlinearity,” Opt. Lett. 21, 940–942 (1996).
[CrossRef] [PubMed]

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

Krug, W.

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

Lawrence, M.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[CrossRef]

Lederer, F.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities—a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Levenson, J. A.

Lovering, D. J.

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Miao, E.

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

Montrosset, I.

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with a scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[CrossRef]

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, “Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators,” IEEE J. Quantum Electron. 23, 42–51 (1987).
[CrossRef]

Pershan, P. S.

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

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Rochford, K. B.

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

Russell, P. St. J.

Schiek, R.

Seibert, H.

Sheik-Bahae, M.

Sohler, W.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

R. Schiek, Y. Baek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, “All-optical switching in lithium niobate directional couplers with cascaded nonlinearity,” Opt. Lett. 21, 940–942 (1996).
[CrossRef] [PubMed]

R. Schiek, M. L. Sundheimer, D. Y. Kim, Y. Baek, G. I. Stegeman, H. Seibert, and W. Sohler, “Direct measurement of cascaded nonlinearity in lithium niobate channel waveguides,” Opt. Lett. 19, 1949–1951 (1994).
[CrossRef] [PubMed]

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, “Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators,” IEEE J. Quantum Electron. 23, 42–51 (1987).
[CrossRef]

Stegeman, G. I.

R. Schiek, Y. Baek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, “All-optical switching in lithium niobate directional couplers with cascaded nonlinearity,” Opt. Lett. 21, 940–942 (1996).
[CrossRef] [PubMed]

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

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]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Y. Baek, R. Schiek, and G. I. Stegeman, “All-optical switching in a hybrid Mach–Zehnder interferometer as a result of cascaded second-order nonlinearity,” Opt. Lett. 20, 2168–2170 (1995).
[CrossRef]

R. Schiek, M. L. Sundheimer, D. Y. Kim, Y. Baek, G. I. Stegeman, H. Seibert, and W. Sohler, “Direct measurement of cascaded nonlinearity in lithium niobate channel waveguides,” Opt. Lett. 19, 1949–1951 (1994).
[CrossRef] [PubMed]

M. I. Sundheimer, C. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides due to cascaded second-order processes,” Opt. Lett. 18, 1397–1399 (1993).
[CrossRef] [PubMed]

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

Strake, E.

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with a scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[CrossRef]

Suche, H.

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, “Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators,” IEEE J. Quantum Electron. 23, 42–51 (1987).
[CrossRef]

Sundheimer, M. I.

Sundheimer, M. L.

Torner, L.

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

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Van Stryland, E.

Van Stryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

M. I. Sundheimer, C. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides due to cascaded second-order processes,” Opt. Lett. 18, 1397–1399 (1993).
[CrossRef] [PubMed]

Vanherzeele, H.

Vidakovic, P.

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Webjörn, J.

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

West, E. J.

W. K. Burns, P. H. Klein, and E. J. West, “Ti diffusion in Ti:LiNbO3 planar and channel optical waveguides,” J. Appl. Phys. 50, 6175–6182 (1979).
[CrossRef]

Yokohama, I.

Zanoni, R.

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

AEÜ Int. J. Electron. Commun. (1)

R. Schiek, “Soliton-like pulse propagation and second harmonic generation in waveguides with second-order optical nonlinearities,” AEÜ Int. J. Electron. Commun. 51, 77–86 (1997).

Appl. Phys. Lett. (1)

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
[CrossRef]

IEEE J. Quantum Electron. (2)

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, “Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators,” IEEE J. Quantum Electron. 23, 42–51 (1987).
[CrossRef]

K. B. Rochford, R. Zanoni, G. I. Stegeman, W. Krug, E. Miao, and M. W. Beranek, “Pulse-modulated interferometer for measuring intensity-induced phase shifts,” IEEE J. Quantum Electron. 28, 2044–2050 (1992).
[CrossRef]

J. Appl. Phys. (1)

W. K. Burns, P. H. Klein, and E. J. West, “Ti diffusion in Ti:LiNbO3 planar and channel optical waveguides,” J. Appl. Phys. 50, 6175–6182 (1979).
[CrossRef]

J. Lightwave Technol. (1)

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with a scalar finite-element method,” J. Lightwave Technol. 6, 1126–1135 (1988).
[CrossRef]

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

Opt. Lett. (8)

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

M. I. Sundheimer, C. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides due to cascaded second-order processes,” Opt. Lett. 18, 1397–1399 (1993).
[CrossRef] [PubMed]

R. Schiek, M. L. Sundheimer, D. Y. Kim, Y. Baek, G. I. Stegeman, H. Seibert, and W. Sohler, “Direct measurement of cascaded nonlinearity in lithium niobate channel waveguides,” Opt. Lett. 19, 1949–1951 (1994).
[CrossRef] [PubMed]

Y. Baek, R. Schiek, and G. I. Stegeman, “All-optical switching in a hybrid Mach–Zehnder interferometer as a result of cascaded second-order nonlinearity,” Opt. Lett. 20, 2168–2170 (1995).
[CrossRef]

P. Vidaković, D. J. Lovering, J. A. Levenson, J. Webjörn, 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]

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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Phys. Rev. (1)

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[CrossRef]

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[CrossRef] [PubMed]

Phys. Rev. E (1)

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[CrossRef]

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[CrossRef]

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R. Schiek, H. Fang, and G. I. Stegeman, “Measurement of the non-uniformity of the wave-vector mismatch in waveguides for second-harmonic generation,” in Nonlinear Guided Waves and Their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 256–258.

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Figures (24)

Fig. 1
Fig. 1

Experimental setup for SHG, fundamental depletion, and nonlinear phase-shift measurements: pol’s, polarizers.

Fig. 2
Fig. 2

Modes of a 15-μm-wide waveguide: (a) fundamental TM00, (b) SH TE00, (c) SH TE10, (d) SH TE01, (e) SH TE02, (f) SH TE06. All figures have the same scale. We measured the mode intensity profiles by focusing the output surface of the waveguide with a 10× microscope objective into a vidicon camera.

Fig. 3
Fig. 3

TM00 mode intensity in 8-, 11-, and 20-μm-wide waveguides at λ=1.32 μm: EXP, experiment; FD, finite-difference theory.

Fig. 4
Fig. 4

TE00, TE10, TE01, and TE02 mode intensity in a 15-μm-wide waveguide at λ=0.66 μm: EXP, experiment; EI, effective-index theory.

Fig. 5
Fig. 5

Normalized SHG tuning curve of a 15-μm-wide waveguide with a temperature scan speed of +36 K/h and cw fundamental input PF=1 mW. The temperature is the controller set-point temperature.

Fig. 6
Fig. 6

Interference between SH modes in a 15-μm-wide waveguide: (a) TE10 and TE06 at ϑ=350.15 °C; (b) TE10, TE20, and TE06 at ϑ=350.6 °C.

Fig. 7
Fig. 7

Calculated effective indices of modes in a 15-μm-wide waveguide.

Fig. 8
Fig. 8

Normalized SHG tuning curve of the TM00-to-TE00 resonance from a 15-μm waveguide with negligible fundamental depletion for a scan speed of +36 K/h at a cw fundamental input of PF=5 mW.

Fig. 9
Fig. 9

Effective temperature profile in 10- and 15-μm-wide waveguides for several scan speeds and a controller set-point temperature of ϑSET=340 °C.

Fig. 10
Fig. 10

SH cw power development along a 15-μm-wide waveguide for temperatures at the minima and maxima in the tuning curve. The power is normalized to the output SH power at phase-matching, low-depletion-case, +36 K/h temperature profile.

Fig. 11
Fig. 11

Normalized fundamental depletion in a 10-μm-wide waveguide for three scan speeds: (a) experiment, (b) theory. All three scans were taken with a fundamental input peak power of 60 W in 90-ps long ML pulses.

Fig. 12
Fig. 12

SHG tuning curve of the TM00-to-TE00 resonance from a 10-μm waveguide with negligible fundamental depletion for three scan speeds at a cw fundamental input of PF=0.5 mW. The SH power is normalized to the maximum SH power of a tuning curve with a speed of +36 K/h.

Fig. 13
Fig. 13

SHG in a 15-μm-wide waveguide (WG) at the temperature ϑM1=336.64 °C for maximum SH output.

Fig. 14
Fig. 14

Fundamental depletion curves of a 15-μm-wide waveguide for three input powers. The fundamental power is normalized to the maximum fundamental output power; cw theory.

Fig. 15
Fig. 15

Power-dependent mode propagation along a 15-μm-wide waveguide at 336.6 °C; cw theory: (a) nonlinear wave-vector modification, (b) the resultant phase-matching conditions, (c) SH power normalized to the fundamental input power, (d) accumulation of the nonlinear phase shift.

Fig. 16
Fig. 16

Power dependence of the tuning curve and the nonlinear phase shift of a 15-μm-wide waveguide for cw theory: (a) nonlinear phase shift, (b) SH tuning curves. Each SH curve is normalized to 100%.

Fig. 17
Fig. 17

SHG tuning curve and fundamental depletion in a 15-μm-wide waveguide with 30-W peak input power in a pulsed input with 90-ps-long ML pulses. The scan speed was 36 K/h, and the curves are normalized to the fundamental input power.

Fig. 18
Fig. 18

Fundamental depletion and normalized SHG tuning curves in a 15-μm-wide waveguide for three input peak power with 90-ps-long ML pulses and a scan speed of 36 K/h: (a) depletion curves normalized to the maximum output fundamental, (b) SH tuning curves normalized to 100%, (c) theoretical SH tuning curves.

Fig. 19
Fig. 19

Fundamental depletion in a 15-μm-wide waveguide for several input peak powers with 90-ps-long MLQS pulses at a scan speed of 36 K/h. The power is normalized to the maximum output fundamental.

Fig. 20
Fig. 20

Fixed temperature fringes of the fundamental at a temperature of ϑ=336.48 °C from a 15-μm waveguide: (a) 4-W high peak power, ϕNL=(0.1±0.014)π; (b) 20-W high peak power, ϕNL=(0.24±0.03)π; (c) 40-W high peak power, ϕNL=(0.34±0.054)π.

Fig. 21
Fig. 21

Interference fringes and nonlinear phase shift from a 10-μm-wide waveguide taken at a scan speed of +36 K/h: (a) high (60-W peak power) and low (1-W peak power) power fringes and (b) nonlinear phase shift versus temperature, which is equal to the difference between the instantaneous phases of the two fringe sets.

Fig. 22
Fig. 22

Nonlinear phase shift in a 15-μm-wide waveguide for three input peak powers with 90-ps-long MLQS pulses at a scan speed of 36 K/h.

Fig. 23
Fig. 23

Power dependence of the nonlinear phase shift at three temperatures.

Fig. 24
Fig. 24

Cw depletion curve and nonlinear phase shift in a uniform waveguide with 200-W fundamental input.

Tables (3)

Tables Icon

Table 1 Effective Mode Indices at ϑ0=365 °C

Tables Icon

Table 2 Calculated (calc) and Experimental (exp) Phase-Matching Temperature Differences ΔT=ϑPM Mode-ϑPM TE00 and Relative Maximum Power P in the Phase-Matched Second-Harmonic Modes in a 15-μm-Wide Waveguide for Low Fundamental Depletion a

Tables Icon

Table 3 Parameters for the Effective Temperature Profiles in Fig. 9

Equations (14)

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Et(x, y, z)=a1(z)e1t+a2(z)e2t,
Ht(x, y, z)=a1(z)h1t+a2(z)h2t.
da2dz+jβ2a2=-jω24p0 χ(2)Ka1a1-α22 a2.
da1dz+jβ1a1=-jω14p0 2χ(2)Ka2a1*-α12 a1.
c(x, y)=c0f(u)g(s),u=yDy,s=2xW,
f(u)=exp(-u2),
g(s)=0.5erfW2Dx (1+s)+erfW2Dx (1-s).
c0=6.412×1022cm-3 τDy.
n1(ϑ)=nTM(ϑ, 1.32μm)=nTM(ϑ0)+9.8×10-6K-1(ϑ-ϑ0)+6.66×10-9K-2(ϑ-ϑ0)2,
n2(ϑ)=nTE(ϑ, 0.66μm)=nTE(ϑ0)+100×10-6K-1(ϑ-ϑ0)+75.55×10-9K-2(ϑ-ϑ0)2,
ϑ0=365 °C.
ϑ(z)=ϑSET-i=1iendΔTi(z),
ΔTi(z)=TNizui-zzNipizzui0zui<zzoiTNiz-zoizNipizoi<z,
zui=zc-zFi,zoi=zc+zFi,zc=23.5mm.

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