Abstract

Beginning with a recently proposed bidirectional beam propagation method based on scattering operators, we develop an accurate and efficient method for the analysis of periodic microstructured waveguides in a nonlinear regime. This novel numerical tool allows us to describe the role played by losses that are due to diffraction and the effects of the finite size of the beams in nonlinear optical devices with strong lateral confinement of the optical intensity. We demonstrate the effectiveness and the efficiency of our method with numerical examples.

© 2003 Optical Society of America

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References

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  1. A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
    [CrossRef]
  2. A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
    [CrossRef]
  3. A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
    [CrossRef]
  4. H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830–832 (1999).
    [CrossRef]
  5. H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood, Jr., “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
    [CrossRef]
  6. H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389–391 (2000).
    [CrossRef]
  7. P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316–1318 (2001).
    [CrossRef]
  8. P. L. Ho and Y. Y. Lu, “A bidirectional beam propagation method for periodic waveguides,” IEEE Photon. Technol. Lett. 14, 325–327 (2002).
    [CrossRef]
  9. H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
    [CrossRef]
  10. F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101, 760–766 (1997).
    [CrossRef]
  11. Y. Y. Lu, “A complex coefficient rational approximation of 1+x,” Appl. Numer. Math. 27, 141–154 (1998).
    [CrossRef]
  12. D. Yevick and D. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784–2790 (2000).
    [CrossRef]
  13. G. Agrawal, Nonlinear Fiber Optics: Principles and Applications (Academic, New York, 1989).
  14. M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  15. H. Rao and R. M. Osgood, Jr., “Bidirectional beam propagation method for computation in nonlinear optics,” in Integrated Photonics Research, Vol. 58 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001).
  16. A. D. Capobianco, D. Brillo, C. De Angelis, and G. F. Nalesso, “Fast beam propagation method for the analysis of second-order nonlinear phenomena,” IEEE Photon. Technol. Lett. 10, 543–545 (1998).
    [CrossRef]
  17. C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18, 348–351 (2001).
    [CrossRef]
  18. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
    [CrossRef]
  19. B. Shi, Z. M. Jiang, and X. Wang, “Defective photonic crystals with greatly enhanced second-harmonic generation,” Opt. Lett. 26, 1194–1196 (2001).
    [CrossRef]
  20. C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
    [CrossRef]

2003 (1)

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

2002 (3)

P. L. Ho and Y. Y. Lu, “A bidirectional beam propagation method for periodic waveguides,” IEEE Photon. Technol. Lett. 14, 325–327 (2002).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
[CrossRef]

2001 (3)

2000 (4)

H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood, Jr., “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389–391 (2000).
[CrossRef]

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

D. Yevick and D. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784–2790 (2000).
[CrossRef]

1999 (1)

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830–832 (1999).
[CrossRef]

1998 (2)

A. D. Capobianco, D. Brillo, C. De Angelis, and G. F. Nalesso, “Fast beam propagation method for the analysis of second-order nonlinear phenomena,” IEEE Photon. Technol. Lett. 10, 543–545 (1998).
[CrossRef]

Y. Y. Lu, “A complex coefficient rational approximation of 1+x,” Appl. Numer. Math. 27, 141–154 (1998).
[CrossRef]

1997 (2)

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101, 760–766 (1997).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

1996 (1)

C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

1978 (1)

Aitchison, J. S.

Baronio, F.

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

Betty, I.

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389–391 (2000).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Brillo, D.

A. D. Capobianco, D. Brillo, C. De Angelis, and G. F. Nalesso, “Fast beam propagation method for the analysis of second-order nonlinear phenomena,” IEEE Photon. Technol. Lett. 10, 543–545 (1998).
[CrossRef]

Brooke, G. H.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101, 760–766 (1997).
[CrossRef]

Capobianco, A. D.

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
[CrossRef]

A. D. Capobianco, D. Brillo, C. De Angelis, and G. F. Nalesso, “Fast beam propagation method for the analysis of second-order nonlinear phenomena,” IEEE Photon. Technol. Lett. 10, 543–545 (1998).
[CrossRef]

Collino, F.

C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

De Angelis, C.

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
[CrossRef]

C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18, 348–351 (2001).
[CrossRef]

A. D. Capobianco, D. Brillo, C. De Angelis, and G. F. Nalesso, “Fast beam propagation method for the analysis of second-order nonlinear phenomena,” IEEE Photon. Technol. Lett. 10, 543–545 (1998).
[CrossRef]

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

El-Refaei, H.

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389–391 (2000).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Gringoli, F.

Haus, J. W.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Ho, P. L.

P. L. Ho and Y. Y. Lu, “A bidirectional beam propagation method for periodic waveguides,” IEEE Photon. Technol. Lett. 14, 325–327 (2002).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316–1318 (2001).
[CrossRef]

Jiang, Z. M.

Locatelli, A.

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
[CrossRef]

Lu, Y. Y.

P. L. Ho and Y. Y. Lu, “A bidirectional beam propagation method for periodic waveguides,” IEEE Photon. Technol. Lett. 14, 325–327 (2002).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316–1318 (2001).
[CrossRef]

Y. Y. Lu, “A complex coefficient rational approximation of 1+x,” Appl. Numer. Math. 27, 141–154 (1998).
[CrossRef]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Midrio, M.

Milinazzo, F. A.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101, 760–766 (1997).
[CrossRef]

Modotto, D.

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
[CrossRef]

C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18, 348–351 (2001).
[CrossRef]

Nalesso, G. F.

C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18, 348–351 (2001).
[CrossRef]

A. D. Capobianco, D. Brillo, C. De Angelis, and G. F. Nalesso, “Fast beam propagation method for the analysis of second-order nonlinear phenomena,” IEEE Photon. Technol. Lett. 10, 543–545 (1998).
[CrossRef]

Osgood Jr., R. M.

H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood, Jr., “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830–832 (1999).
[CrossRef]

Pigozzo, F. M.

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
[CrossRef]

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
[CrossRef]

Rao, H.

H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood, Jr., “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830–832 (1999).
[CrossRef]

Scalora, M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Scarmozzino, R.

H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood, Jr., “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830–832 (1999).
[CrossRef]

Shi, B.

Steel, M. J.

Thomson, D.

D. Yevick and D. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784–2790 (2000).
[CrossRef]

Vassallo, C.

C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

Viswanathan, R.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Wang, X.

Yevick, D.

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389–391 (2000).
[CrossRef]

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

D. Yevick and D. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784–2790 (2000).
[CrossRef]

Zala, C. A.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101, 760–766 (1997).
[CrossRef]

Appl. Numer. Math. (1)

Y. Y. Lu, “A complex coefficient rational approximation of 1+x,” Appl. Numer. Math. 27, 141–154 (1998).
[CrossRef]

Appl. Opt. (1)

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

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 8, 440–447 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (7)

A. Locatelli, F. M. Pigozzo, D. Modotto, A. D. Capobianco, and C. De Angelis, “Novel bidirectional propagation method for quadratic nonlinear multilayers,” IEEE Photon. Technol. Lett. 14, 1536–1538 (2002).
[CrossRef]

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389–391 (2000).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316–1318 (2001).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A bidirectional beam propagation method for periodic waveguides,” IEEE Photon. Technol. Lett. 14, 325–327 (2002).
[CrossRef]

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approximants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12, 158–160 (2000).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830–832 (1999).
[CrossRef]

A. D. Capobianco, D. Brillo, C. De Angelis, and G. F. Nalesso, “Fast beam propagation method for the analysis of second-order nonlinear phenomena,” IEEE Photon. Technol. Lett. 10, 543–545 (1998).
[CrossRef]

J. Acoust. Soc. Am. (2)

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101, 760–766 (1997).
[CrossRef]

D. Yevick and D. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784–2790 (2000).
[CrossRef]

J. Lightwave Technol. (2)

C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood, Jr., “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18, 1155–1160 (2000).
[CrossRef]

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

Opt. Lett. (1)

Opt. Quantum Electron. (1)

A. Locatelli, F. M. Pigozzo, F. Baronio, D. Modotto, A. D. Capobianco, and C. De Angelis, “Bidirectional beam propagation method for second-harmonic generation in engineered multilayered waveguides,” Opt. Quantum Electron. 35, 429–452 (2003).
[CrossRef]

Phys. Rev. A (1)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Other (2)

H. Rao and R. M. Osgood, Jr., “Bidirectional beam propagation method for computation in nonlinear optics,” in Integrated Photonics Research, Vol. 58 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001).

G. Agrawal, Nonlinear Fiber Optics: Principles and Applications (Academic, New York, 1989).

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Figures (7)

Fig. 1
Fig. 1

Two-dimensional model of the periodic structure in the Bi BPM based on scattering operators.

Fig. 2
Fig. 2

Schematic view of the devices designed for SHG: bottom, defective and top, not defective.

Fig. 3
Fig. 3

Total field at the FF in a seven-period device.

Fig. 4
Fig. 4

Total field at the SH in a seven-period device.

Fig. 5
Fig. 5

Total field at the FF in a defective device.

Fig. 6
Fig. 6

Total field at the SH in a defective device.

Fig. 7
Fig. 7

Transmission coefficient T versus wavelength in a linear regime (left) and dependence of T from the maximum intensity of the field for a fixed wavelength λ=1543.9 nm in a nonlinear regime (right) in a 14-period device.

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

Lj=[xx+k02nj2(x)]1/2,
R(z)u+(x, z)=u-(x, z),
T(z)u+(x, z)=u+(x, a+),
R(zj-1+)=PjR(zj-)Pj,
T(zj-1+)=T(zj-)Pj,
C=Lj-1Lj+1[I-R(zj+)][I+R(zj+)]-1,
R(zj-)=(I+C)-1(I-C),
T(zj-)=T(zj+)[I+R(zj+)]-1[I+R(zj-)],
-2ik¯zu+(x, z)+zzu+(x, z)+xxu+(x, z)
+[k02nj2(x)-k¯2]u+(x, z)+34ω02c2 χj(3)(x)
×[|u+(x, z)|2+2|u-(x, z)|2]u+(x, z)=0
2ik¯zu-(x, z)+zzu-(x, z)+xxu-(x, z)
+[k02nj2(x)-k¯2]u-(x, z)+34ω02c2 χj(3)(x)
×[|u-(x, z)|2+2|u+(x, z)|2]u-(x, z)=0
zzu±(x, z)2ik¯zu±(x, z)+xxu±(x, z)
+[nj2(x)k02-k¯2]u±(x, z)=0.
zu+(x, z)=-iγj(x)[|u+(x, z)|2+2|u-(x, z)|2]u+(x, z)
zu-(x, z)=iγj(x)[|u-(x, z)|2+2|u+(x, z)|2]u-(x, z)
γj(x)=2πn2j(x)nj(x)/(λn¯);
n2j(x)=3χj(3)(x)/[8nj(x)]
u+(x, z+Δz)=exp{-iγj(x)[|u+(x, z)|2+2|u-(x, z)|2]Δz}u+(x, z),
u-(x, z-Δz)=exp{-iγj(x)[|u-(x, z)|2+2|u+(x, z)|2]Δz}u-(x, z).
pj,mNL+=exp{-iγj(x)[|u+(x, z)|2+2|u-(x, z)|2]Δz}I
pj,mNL-=exp{-iγj(x)[|u-(x, z)|2+2|u+(x, z)|2]Δz}I
Pj±=m=1Mpj,mNL±pjLIN.
nj±(x)=nj(x)+n2j(x)[|u±(x, zj-)|2+2|u(x, zj-)|2],
nj+1±(x)=nj+1(x)+n2j+1(x)[|u±(x, zj+)|2+2|u(x, zj+)|2],
u+(x, zj-)=Pj+u+(x, zj-1+),
u-(x, zj-1+)=Pj-u-(x, zj-),
R(zj-1+)=Pj-R(zj-)Pj+,
T(zj-1+)=T(zj-)Pj+.
C1=(Lj+)-1[Lj+1+-Lj+1-R(zj+)][I+R(zj+)],
C2=(Lj-)-1[Lj+1+-Lj+1-R(zj+)][I+R(zj+)],
R(zj-)=(I+C2)-1(I-C1),
T(zj-)=T(zj+)[I+R(zj+)]-1[I+R(zj-)].
zzuω±(x, z)2ik¯zuω±(x, z)+xxuω±(x, z)
+[nj2(x, ω)k02-k¯2]uω±(x, z)
=-k02χj(2)(x)[uω±(x, z)]*u2ω±(x, z),
uω±(x, z±Δz)=ulinear,ω±(x, z±Δz)-iΔz k02χj(2)(x)4k¯ {[uω±(x, z±Δz)]*×u2ω±(x, z)+[uω±(x, z)]*u2ω±(x, z±Δz)}.
zzu2ω±(x, z)4ik¯zu2ω±(x, z)+xxu2ω±(x, z)
+4[nj2(x, 2ω)k02-k¯2]u2ω±(x, z)
=-2k02χj(2)(x)[uω±(x, z)]2.
u2ω±(x, z±Δz)=ulinear,2ω±(x, z±Δz)-iΔz k02χj(2)(x)2k¯ uω±(x, z)×uω±(x, z±Δz).
Et+(x, zj-)=Pj,tEt+(x, zj-1+)+Xj,t+,
Et-(x, zj-1+)=Pj,tEt-(x, zj-)+Xj,t-,
Xj,t±=m=1Mpj,t(M-m)(xj,t±)m.
Rt(z)ut+(x, z)+Wt(x, z)=ut-(x, z),
Tt(z)ut+(x, z)+Kt(x, z)=ut+(x, a+),
Kω(zj-1+)=Kω(zj-)+Tω(zj-)Xj,ω+,
Wω(zj-1+)=Pj,ωRω(zj-)Xj,ω++Pj,ωWω(zj-)+Xj,ω-.
Wω(zj-)
=[I+R(zj+)][-I+R(zj+)]-1-I[I+R(zj+)][-I+R(zj+)]-1Lj+1-1Lj-I Wω(zj+),
Kω(zj-)
=Tω(zj+)[I+R(zj+)]-1[Wω(zj-)-Wω(zj+)]+Kω(zj+).

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