Abstract

We theoretically analyze a new class of aperiodic phase mismatch. The phase-matching function that is chosen depends on the calculated second-harmonic amplitude generated in the device during the propagation of the fundamental beam at given input intensity and wavelength. We show that, in such a configuration, the fields evolve toward the eigenmodes of a χ(2) two-wave mixing process. Hence a constant pump and an enhanced nonlinear phase shift that grows linearly with propagation length are obtained. We also discuss the feasibility of this scheme that provides an alternative approach for the realization of optical switching devices or Kerr-effect compensators.

© 1999 Optical Society of America

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  1. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, and E. W. Van Stryland, “Self-focusing and self defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
    [CrossRef] [PubMed]
  2. J. Khurgin, A. Obeidat, S. J. Lee, and Y. J. Ding, “Cascaded optical nonlinearities: microscopic understanding as a collective effect,” J. Opt. Soc. Am. B 14, 1977–1983 (1997).
    [CrossRef]
  3. P. St. J. Russell, “All optical high gain transistor action using second order nonlinearities,” Electron. Lett. 29, 1228–1229 (1993).
    [CrossRef]
  4. D. J. Hagan, Z. Wang, G. Stegeman, E. W. Van Stryland, M. Sheik-Bahae, and G. Assanto, “Phase-controlled transistor action by cascading of second-order nonlinearities in KTP,” Opt. Lett. 19, 1305–1307 (1994).
    [CrossRef] [PubMed]
  5. Y. Baek, R. Schiek, G. Stegeman, and G. Assanto, “All optical mode mixer spatial switch based on cascading in lithium niobate,” Appl. Phys. Lett. 72, 3405–3407 (1998).
    [CrossRef]
  6. Y. Baek, R. Schiek, G. Stegeman, G. Krinjnen, I. Baumann, and W. Sohler, “All-optical integrated Mac-Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055–2057 (1996).
    [CrossRef]
  7. G. Assanto, G. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “All-optical switching devices based on large nonlinear phase shifts from second harmonic generation,” Appl. Phys. Lett. 62, 1323–1325 (1993).
    [CrossRef]
  8. C. Paré, A. Villeneuve, and S. LaRochelle, “Modulational instability in a communication link exploiting a negative nonlinearity for compensation of self phase modulation,” in Nonlinear Guided Waves and Their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 86–88, paper NWE9–3.
  9. C. Paré, A. Villeneuve, P.-A. Bélanger, and N. J. Doran, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. 21, 459–461 (1996).
    [CrossRef] [PubMed]
  10. G. 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]
  11. G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “Coherent interactions for all-optical signal processing via quadratic nonlinearities,” IEEE J. Quantum Electron. 31, 673–681 (1995).
    [CrossRef]
  12. A. E. Kaplan, “Eigenmodes of χ(2) wave mixings: cross-induced second-order nonlinear refraction,” Opt. Lett. 18, 1223–1225 (1993).
    [CrossRef] [PubMed]
  13. S. Trillo, S. Wabnitz, R. Chisari, and G. Cappellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
    [CrossRef] [PubMed]
  14. M. Cha, “Cascaded phase shift and intensity modulation in aperiodic quasi-phase-matched gratings,” Opt. Lett. 23, 250–252 (1998).
    [CrossRef]
  15. G. D. Landry and T. A. Maldonado, “Efficient nonlinear phase shifts due to cascaded second-order processes in a counterpropagating quasi-phase-matched configuration,” Opt. Lett. 22, 1400–1402 (1997).
    [CrossRef]
  16. Y. J. Ding and J. B. Khurgin, “Second-harmonic generation based on a quasi-phase matching: a novel configuration,” Opt. Lett. 21, 1445–1447 (1996).
    [CrossRef] [PubMed]
  17. K. Koynov and S. Saltiel, “Nonlinear phase shift via multistep χ(2) cascading,” Opt. Commun. 152, 96–100 (1998).
    [CrossRef]
  18. C. G. Trevinio-Palacios, D. Ortega, G. Stegeman, and J. S. Aitchison, “Spatial chirping of wavevector mismatch in lithium niobate segmented waveguide for engineering of specific second-harmonic generation detuning curves for cascading applications,” in Conference on Lasers and Electro-Optics, Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 105–106, paper CtuJ2.
  19. Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), pp. 87–93.
  20. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Peshan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  21. H. Kestelman, Modern Theories of Integration, 2nd ed. (Dover, New York, 1959), pp. 30–66.
  22. L. Perko, Differential Equations and Dynamical Systems, 2nd ed. (Springer-Verlag, Berlin, 1996), pp. 70–79.
  23. L. E. Myers, R. C. Eckart, M. M. Fejer, and R. L Byer, “Quasi-phasematched optical parametric oscillators using bulk periodically poled lithium niobate,” in Solid State Lasers and Nonlinear Crystals, L. K. Cheng, L. Esterowitz, and G. J. Quarkes, eds., Proc. SPIE 2379, 154–163 (1995).
    [CrossRef]
  24. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase matched waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435–436 (1993).
    [CrossRef]
  25. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Buer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  26. M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi phase-matched second harmonic generation in LiNbO3 waveguide,” Electron. Lett. 30, 34–35 (1994).
    [CrossRef]
  27. Crystal Technology, Lithium Niobate. Optical Waveguide Substrate (Crystal Technology, Palo Alto, Calif., 1995).

1998 (3)

Y. Baek, R. Schiek, G. Stegeman, and G. Assanto, “All optical mode mixer spatial switch based on cascading in lithium niobate,” Appl. Phys. Lett. 72, 3405–3407 (1998).
[CrossRef]

M. Cha, “Cascaded phase shift and intensity modulation in aperiodic quasi-phase-matched gratings,” Opt. Lett. 23, 250–252 (1998).
[CrossRef]

K. Koynov and S. Saltiel, “Nonlinear phase shift via multistep χ(2) cascading,” Opt. Commun. 152, 96–100 (1998).
[CrossRef]

1997 (2)

1996 (3)

1995 (2)

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “Coherent interactions for all-optical signal processing via quadratic nonlinearities,” IEEE J. Quantum Electron. 31, 673–681 (1995).
[CrossRef]

L. E. Myers, R. C. Eckart, M. M. Fejer, and R. L Byer, “Quasi-phasematched optical parametric oscillators using bulk periodically poled lithium niobate,” in Solid State Lasers and Nonlinear Crystals, L. K. Cheng, L. Esterowitz, and G. J. Quarkes, eds., Proc. SPIE 2379, 154–163 (1995).
[CrossRef]

1994 (2)

D. J. Hagan, Z. Wang, G. Stegeman, E. W. Van Stryland, M. Sheik-Bahae, and G. Assanto, “Phase-controlled transistor action by cascading of second-order nonlinearities in KTP,” Opt. Lett. 19, 1305–1307 (1994).
[CrossRef] [PubMed]

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi phase-matched second harmonic generation in LiNbO3 waveguide,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

1993 (5)

P. St. J. Russell, “All optical high gain transistor action using second order nonlinearities,” Electron. Lett. 29, 1228–1229 (1993).
[CrossRef]

G. 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]

G. Assanto, G. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “All-optical switching devices based on large nonlinear phase shifts from second harmonic generation,” Appl. Phys. Lett. 62, 1323–1325 (1993).
[CrossRef]

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase matched waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435–436 (1993).
[CrossRef]

A. E. Kaplan, “Eigenmodes of χ(2) wave mixings: cross-induced second-order nonlinear refraction,” Opt. Lett. 18, 1223–1225 (1993).
[CrossRef] [PubMed]

1992 (3)

1962 (1)

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

Armstrong, J. A.

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

Assanto, G.

Y. Baek, R. Schiek, G. Stegeman, and G. Assanto, “All optical mode mixer spatial switch based on cascading in lithium niobate,” Appl. Phys. Lett. 72, 3405–3407 (1998).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “Coherent interactions for all-optical signal processing via quadratic nonlinearities,” IEEE J. Quantum Electron. 31, 673–681 (1995).
[CrossRef]

D. J. Hagan, Z. Wang, G. Stegeman, E. W. Van Stryland, M. Sheik-Bahae, and G. Assanto, “Phase-controlled transistor action by cascading of second-order nonlinearities in KTP,” Opt. Lett. 19, 1305–1307 (1994).
[CrossRef] [PubMed]

G. Assanto, G. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “All-optical switching devices based on large nonlinear phase shifts from second harmonic generation,” Appl. Phys. Lett. 62, 1323–1325 (1993).
[CrossRef]

G. 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]

Baek, Y.

Y. Baek, R. Schiek, G. Stegeman, and G. Assanto, “All optical mode mixer spatial switch based on cascading in lithium niobate,” Appl. Phys. Lett. 72, 3405–3407 (1998).
[CrossRef]

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

Baumann, I.

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

Bélanger, P.-A.

Bloembergen, N.

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

Bortz, M. L.

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi phase-matched second harmonic generation in LiNbO3 waveguide,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

Buer, R. L.

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

Byer, R. L

L. E. Myers, R. C. Eckart, M. M. Fejer, and R. L Byer, “Quasi-phasematched optical parametric oscillators using bulk periodically poled lithium niobate,” in Solid State Lasers and Nonlinear Crystals, L. K. Cheng, L. Esterowitz, and G. J. Quarkes, eds., Proc. SPIE 2379, 154–163 (1995).
[CrossRef]

Cappellini, G.

Cha, M.

Chisari, R.

DeSalvo, R.

Ding, Y. J.

Doran, N. J.

Ducuing, J.

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

Eckart, R. C.

L. E. Myers, R. C. Eckart, M. M. Fejer, and R. L Byer, “Quasi-phasematched optical parametric oscillators using bulk periodically poled lithium niobate,” in Solid State Lasers and Nonlinear Crystals, L. K. Cheng, L. Esterowitz, and G. J. Quarkes, eds., Proc. SPIE 2379, 154–163 (1995).
[CrossRef]

Fejer, M.

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

Fejer, M. M.

L. E. Myers, R. C. Eckart, M. M. Fejer, and R. L Byer, “Quasi-phasematched optical parametric oscillators using bulk periodically poled lithium niobate,” in Solid State Lasers and Nonlinear Crystals, L. K. Cheng, L. Esterowitz, and G. J. Quarkes, eds., Proc. SPIE 2379, 154–163 (1995).
[CrossRef]

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi phase-matched second harmonic generation in LiNbO3 waveguide,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

Fujimura, M.

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi phase-matched second harmonic generation in LiNbO3 waveguide,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

Hagan, D. J.

Jundt, D. H.

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

Kaplan, A. E.

Khurgin, J.

Khurgin, J. B.

Koynov, K.

K. Koynov and S. Saltiel, “Nonlinear phase shift via multistep χ(2) cascading,” Opt. Commun. 152, 96–100 (1998).
[CrossRef]

Krinjnen, G.

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

Landry, G. D.

Lee, S. J.

Magel, G. A.

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

Maldonado, T. A.

Myers, L. E.

L. E. Myers, R. C. Eckart, M. M. Fejer, and R. L Byer, “Quasi-phasematched optical parametric oscillators using bulk periodically poled lithium niobate,” in Solid State Lasers and Nonlinear Crystals, L. K. Cheng, L. Esterowitz, and G. J. Quarkes, eds., Proc. SPIE 2379, 154–163 (1995).
[CrossRef]

Nada, N.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase matched waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435–436 (1993).
[CrossRef]

Obeidat, A.

Paré, C.

Peshan, P. S.

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

Russell, P. St. J.

P. St. J. Russell, “All optical high gain transistor action using second order nonlinearities,” Electron. Lett. 29, 1228–1229 (1993).
[CrossRef]

Saitoh, M.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase matched waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435–436 (1993).
[CrossRef]

Saltiel, S.

K. Koynov and S. Saltiel, “Nonlinear phase shift via multistep χ(2) cascading,” Opt. Commun. 152, 96–100 (1998).
[CrossRef]

Schiek, R.

Y. Baek, R. Schiek, G. Stegeman, and G. Assanto, “All optical mode mixer spatial switch based on cascading in lithium niobate,” Appl. Phys. Lett. 72, 3405–3407 (1998).
[CrossRef]

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

Sheik-Bahae, M.

Sohler, W.

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

Stegeman, G.

Y. Baek, R. Schiek, G. Stegeman, and G. Assanto, “All optical mode mixer spatial switch based on cascading in lithium niobate,” Appl. Phys. Lett. 72, 3405–3407 (1998).
[CrossRef]

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

D. J. Hagan, Z. Wang, G. Stegeman, E. W. Van Stryland, M. Sheik-Bahae, and G. Assanto, “Phase-controlled transistor action by cascading of second-order nonlinearities in KTP,” Opt. Lett. 19, 1305–1307 (1994).
[CrossRef] [PubMed]

G. Assanto, G. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “All-optical switching devices based on large nonlinear phase shifts from second harmonic generation,” Appl. Phys. Lett. 62, 1323–1325 (1993).
[CrossRef]

G. 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. Stegeman, and E. W. Van Stryland, “Self-focusing and self defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

Stegeman, G. I.

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “Coherent interactions for all-optical signal processing via quadratic nonlinearities,” IEEE J. Quantum Electron. 31, 673–681 (1995).
[CrossRef]

Trillo, S.

Van Stryland, E.

Van Stryland, E. W.

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “Coherent interactions for all-optical signal processing via quadratic nonlinearities,” IEEE J. Quantum Electron. 31, 673–681 (1995).
[CrossRef]

D. J. Hagan, Z. Wang, G. Stegeman, E. W. Van Stryland, M. Sheik-Bahae, and G. Assanto, “Phase-controlled transistor action by cascading of second-order nonlinearities in KTP,” Opt. Lett. 19, 1305–1307 (1994).
[CrossRef] [PubMed]

G. Assanto, G. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “All-optical switching devices based on large nonlinear phase shifts from second harmonic generation,” Appl. Phys. Lett. 62, 1323–1325 (1993).
[CrossRef]

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

Villeneuve, A.

Wabnitz, S.

Wang, Z.

Watanabe, K.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase matched waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435–436 (1993).
[CrossRef]

Yamada, M.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase matched waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435–436 (1993).
[CrossRef]

Appl. Phys. Lett. (4)

Y. Baek, R. Schiek, G. Stegeman, and G. Assanto, “All optical mode mixer spatial switch based on cascading in lithium niobate,” Appl. Phys. Lett. 72, 3405–3407 (1998).
[CrossRef]

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

G. Assanto, G. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “All-optical switching devices based on large nonlinear phase shifts from second harmonic generation,” Appl. Phys. Lett. 62, 1323–1325 (1993).
[CrossRef]

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase matched waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435–436 (1993).
[CrossRef]

Electron. Lett. (2)

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi phase-matched second harmonic generation in LiNbO3 waveguide,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

P. St. J. Russell, “All optical high gain transistor action using second order nonlinearities,” Electron. Lett. 29, 1228–1229 (1993).
[CrossRef]

IEEE J. Quantum Electron. (2)

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. W. Van Stryland, “Coherent interactions for all-optical signal processing via quadratic nonlinearities,” IEEE J. Quantum Electron. 31, 673–681 (1995).
[CrossRef]

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

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

Opt. Commun. (1)

K. Koynov and S. Saltiel, “Nonlinear phase shift via multistep χ(2) cascading,” Opt. Commun. 152, 96–100 (1998).
[CrossRef]

Opt. Lett. (9)

C. Paré, A. Villeneuve, P.-A. Bélanger, and N. J. Doran, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. 21, 459–461 (1996).
[CrossRef] [PubMed]

G. 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]

A. E. Kaplan, “Eigenmodes of χ(2) wave mixings: cross-induced second-order nonlinear refraction,” Opt. Lett. 18, 1223–1225 (1993).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, R. Chisari, and G. Cappellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
[CrossRef] [PubMed]

M. Cha, “Cascaded phase shift and intensity modulation in aperiodic quasi-phase-matched gratings,” Opt. Lett. 23, 250–252 (1998).
[CrossRef]

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

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

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

D. J. Hagan, Z. Wang, G. Stegeman, E. W. Van Stryland, M. Sheik-Bahae, and G. Assanto, “Phase-controlled transistor action by cascading of second-order nonlinearities in KTP,” Opt. Lett. 19, 1305–1307 (1994).
[CrossRef] [PubMed]

Phys. Rev. (1)

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

Proc. SPIE (1)

L. E. Myers, R. C. Eckart, M. M. Fejer, and R. L Byer, “Quasi-phasematched optical parametric oscillators using bulk periodically poled lithium niobate,” in Solid State Lasers and Nonlinear Crystals, L. K. Cheng, L. Esterowitz, and G. J. Quarkes, eds., Proc. SPIE 2379, 154–163 (1995).
[CrossRef]

Other (6)

Crystal Technology, Lithium Niobate. Optical Waveguide Substrate (Crystal Technology, Palo Alto, Calif., 1995).

H. Kestelman, Modern Theories of Integration, 2nd ed. (Dover, New York, 1959), pp. 30–66.

L. Perko, Differential Equations and Dynamical Systems, 2nd ed. (Springer-Verlag, Berlin, 1996), pp. 70–79.

C. Paré, A. Villeneuve, and S. LaRochelle, “Modulational instability in a communication link exploiting a negative nonlinearity for compensation of self phase modulation,” in Nonlinear Guided Waves and Their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 86–88, paper NWE9–3.

C. G. Trevinio-Palacios, D. Ortega, G. Stegeman, and J. S. Aitchison, “Spatial chirping of wavevector mismatch in lithium niobate segmented waveguide for engineering of specific second-harmonic generation detuning curves for cascading applications,” in Conference on Lasers and Electro-Optics, Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 105–106, paper CtuJ2.

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Figures (6)

Fig. 1
Fig. 1

(a) Evolution of the fundamental beam amplitude for constant PM and for two other kinds of aperiodic PM distribution. The distributions that are chosen depend on the amplitude of the generated second harmonic. For aperiodic PM, the fundamental beam amplitude reaches a steady state after a given propagation length. (b) Evolution of the asymptotic amplitude (uas) of the fundamental beam as a function of Γ and n.

Fig. 2
Fig. 2

(a) NLPS’s on the fundamental beam that result from several aperiodic phase matches and from a constant PM distribution. The dependence of the sign of the NLPS on the sign of Δk is still valid. (b) Evolution of the derivative of the asymptotic nonlinear phase shift (dφas/dz) with Γ2 and n, where Γ2 is proportional to the input intensity. As expected, the plots are very close to square-root functions.

Fig. 3
Fig. 3

Period distribution of an inversion grating in lithium niobate necessary for a phase-mismatch distribution given by Δk=(π/L)v2. The refractive indices at 1.55 and 0.775 µm have been evaluated at room temperature.25

Fig. 4
Fig. 4

(a) Evolution of the amplitude of the fundamental beam in an equivalent scheme close to the asymptotic behavior of Δk=(π/L)v2. The design criterion of the scheme was to implement initially a 13-bit Barker code overmodulation of the inversion grating to increase the bandwidth of the device. The area between the two crystals has not been taken into account. The residual oscillation is less than 3.5% (b) Evolution of the NLPS. NLPS growth is clearly linear in the asymptotic equivalent part of the beam.

Fig. 5
Fig. 5

(a) Evolution of amplitude as a function of wavelength of the input beam for the equivalent scheme close to the asymptotic behavior of Δk=(π/L)v2 as defined Fig. 4. After 1551.6 nm, the inversion grating of the second crystal of the grating begins to match the wavelength of the input beam. The strong oscillations in the beam amplitude are characteristic of such a process. (b) Evolution of the nonlinear phase shift as a function of the input beam. The sign inversion at 1551.9 nm can be related to the same process that we mentioned for (a). Taken together, (a) and (b) allow us to define a bandwidth of 3.6 nm.

Fig. 6
Fig. 6

(a) Evolution of amplitude as a function of the input intensity for the equivalent scheme close to the asymptotic behavior of Δk=(π/L)v2 as defined in Fig. 4. The operation point corresponds to the initial wavelength and input intensity configuration. From the operation point to the half input intensity point the variation of the normalized output amplitude of the pump beam is less then 10%. (b) Evolution of the nonlinear phase shift as a function of the input intensity. Between the operation point and the half input power point the NLPS displays a good linear fit with respect to input intensity and consequently can emulate a Kerr coefficient.

Tables (1)

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Table 1 Propagation Lengths (in mm) after Which the Pump Oscillation is Less Than 0.001

Equations (33)

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dudξ=-uv sin(θ),
dvdξ=u2 sin(θ),
dθdξ=h(ξ)+sin(θ)cos(θ)ddξln(u2v),
h(ξ)=1ΓΔkξΓ+ξ ddξΔkξΓ,
Γ=12Wω2[χ(2)(-2ω, ω, ω)]20c3ηω2η2ω1/2.
u2(ξ)v(ξ)cos(θ)=Ξ(ξ),
Ξ(ξ)=120ξh(ξ) dv2(ξ)dξdξ.
dv2dξ=2([v(1-v2)2-Ξ2(ξ)])1/2=2([v(1-v2)+Ξ(ξ)]([v(1-v2)-Ξ(ξ)])1/2)1/2,
[v(1-v2)]2-1/4M2v4g(v, ξ)[v(1-v2)]2-1/4m2v4,
p1(v, ξ)=-Δk0(n+2)Γvn+2-AΔk02Γv2-nΔk0Γ×0ξ ξdvdξ2vndξ+(1-v2)v,
p2(v, ξ)=Δk0(n+2)Γvn+2+AΔk02Γv2+nΔk0Γ×0ξ ξdvdξ2vndξ+(1-v2)v,
0ξξvndvdξ2dξ
-Δk0(n+2)Γvn+2-AΔk02Γv2-nΔk0Γ0ξ ξdvdξ2vndξ
p1(v)1-nΔk0Γ0ξ ξdvdξ2vndξ.
p(v)=-Δk0(n+2)Γvn+2-AΔk02Γv2+(1-v2)v-qΔk0Γqξvqdvdξ2ξ,
dp1dv=-Δk0Γvn+1-Δk0AvΓ+1-3v2-nΔk0Γξvn dvdξ.
-Δk0Γvn+1-Δk0AvΓ+1-3v2=0.
d2Easdξ2+iΓhas(ξ) dEasdξ-(1-2|Eas|2)Eas=0,
Eas=uas exp[-iφas(ξ)].
dφasdξ+=-12Δk(uas)Γ+12Δk(uas)Γ2-4(1-2uas2)1/2,
dφasdξ-=-12Δk(uas)Γ-12Δk(uas)Γ2-4(1-2uas2)1/2.
χ(2)(-2ω, ω, ω, z)2πχ(2)(-2ω, ω, ω)exp-2πΛz,
Δk=4πη2ωλ-4πηωλ-2πΛ.
Λn=2πλ4π(η2ω-η)-λΔk0fvi=0n-1 Λi.
h(ξ)+(-1)puas2vas-2vas=0,
Δk=Γ(-1)p+11-3vas2vas,
u(ξ)=uas,
v(ξ)=vas,
2φω(ξ)-φ2ω(ξ)=π
u2(ξ)v(ξ)cos(θ)+12ΔkΓv2(ξ)=0,
cos(ΔkL)=(-1)p+12ΔkΓvasuas2.
uas=sech(ΓL),
D=14(2m+1)λ2η2ω(medium)-ηω(medium),

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