Abstract

We derive an analytical expression for the conversion efficiency of second-harmonic generation (SHG) in a photonic crystal. The results obtained for the undepleted-pump limit allow us to describe the role played by the feedback and dispersion introduced by the photonic crystal and hence to optimize the SHG process.

© 2001 Optical Society of America

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  1. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
    [Crossref]
  2. C. L. Tang and P. P. Bey, “Phase matching in second harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. QE-9, 9–17 (1973).
    [Crossref]
  3. P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propa-gation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. B 67, 423–438 (1977).
    [Crossref]
  4. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. B 67, 438–448 (1977).
    [Crossref]
  5. 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, 2631–2654 (1992).
    [Crossref]
  6. R. Schiek, L. Friederich, H. Fang, G. I. Stegeman, K. R. Parameswaran, M. H. Chou, and M. M. Feier, “Nonlinear directional coupler in periodically poled lithium niobate,” Opt. Lett. 24, 1617–1619 (1999).
    [Crossref]
  7. A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. 23, 506–508 (1998).
    [Crossref]
  8. G. Schreiber, D. Hofmann, W. Grundkotter, R. Ricken, and W. Sohler, “Near-infrared cascaded difference frequency generation in periodically poled Ti:LiNbO3 waveguides,” presented at the OSA Meeting on Nonlinear Guided Waves and Their Applications, Dijon, France, September 1–3, 1999.
  9. B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. 24, 1410–1412 (1999).
    [Crossref]
  10. C. Conti, S. Trillo, and G. Assanto, “Bloch function approach for parametric gap solitons,” Opt. Lett. 22, 445–447 (1997).
    [Crossref] [PubMed]
  11. T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
    [Crossref]
  12. A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Masselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
    [Crossref]
  13. G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
    [Crossref]
  14. 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, 3166–3172 (1997).
    [Crossref]
  15. M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
    [Crossref]
  16. J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
    [Crossref]
  17. M. A. Afromowitz, “Refractive index of AlxGa1-xAs,” Solid State Commun. 15, 59–63 (1974).
    [Crossref]
  18. S. Adachi, “GaAs, AlAs and AlxGa1-xAs: material parameters for use in research and device applications,” J. Appl. Phys. 58, R1–R29 (1985).
    [Crossref]
  19. 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, 3410–3412 (1995).
    [Crossref]
  20. A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
    [Crossref]

2000 (1)

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
[Crossref]

1999 (5)

1998 (3)

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. 23, 506–508 (1998).
[Crossref]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[Crossref]

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

1997 (2)

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, 3166–3172 (1997).
[Crossref]

C. Conti, S. Trillo, and G. Assanto, “Bloch function approach for parametric gap solitons,” Opt. Lett. 22, 445–447 (1997).
[Crossref] [PubMed]

1995 (1)

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, 3410–3412 (1995).
[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, 2631–2654 (1992).
[Crossref]

1985 (1)

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

1977 (2)

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propa-gation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. B 67, 423–438 (1977).
[Crossref]

A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. B 67, 438–448 (1977).
[Crossref]

1974 (1)

M. A. Afromowitz, “Refractive index of AlxGa1-xAs,” Solid State Commun. 15, 59–63 (1974).
[Crossref]

1973 (1)

C. L. Tang and P. P. Bey, “Phase matching in second harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. QE-9, 9–17 (1973).
[Crossref]

1970 (1)

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[Crossref]

Adachi, S.

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

Afromowitz, M. A.

M. A. Afromowitz, “Refractive index of AlxGa1-xAs,” Solid State Commun. 15, 59–63 (1974).
[Crossref]

Assanto, G.

Balakin, A. V.

Bang, O.

Barthelemy, A.

Berger, V.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

Bertolotti, M.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[Crossref]

Bey, P. P.

C. L. Tang and P. P. Bey, “Phase matching in second harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. QE-9, 9–17 (1973).
[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, 3410–3412 (1995).
[Crossref]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[Crossref]

Bloemer, M. J.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[Crossref]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[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, 3166–3172 (1997).
[Crossref]

Boucher, D.

Bourliaguet, B.

Bowden, C. M.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[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, 3166–3172 (1997).
[Crossref]

Bravetti, P.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

Bushuev, V. A.

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, 2631–2654 (1992).
[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, 3410–3412 (1995).
[Crossref]

Centini, M.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[Crossref]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[Crossref]

Chou, M. H.

Cole, J. D.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[Crossref]

Conti, C.

Couderc, V.

D’Aguanno, G.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[Crossref]

De Angelis, C.

de Sterke, C. M.

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
[Crossref]

Delobel, L.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (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, 3166–3172 (1997).
[Crossref]

Fang, H.

Feier, M. M.

Fejer, M. M.

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, 2631–2654 (1992).
[Crossref]

Fiore, A.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

Friederich, L.

Grundkotter, W.

G. Schreiber, D. Hofmann, W. Grundkotter, R. Ricken, and W. Sohler, “Near-infrared cascaded difference frequency generation in periodically poled Ti:LiNbO3 waveguides,” presented at the OSA Meeting on Nonlinear Guided Waves and Their Applications, Dijon, France, September 1–3, 1999.

Hanna, D. C.

Haus, J. W.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[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, 3166–3172 (1997).
[Crossref]

Hofmann, D.

G. Schreiber, D. Hofmann, W. Grundkotter, R. Ricken, and W. Sohler, “Near-infrared cascaded difference frequency generation in periodically poled Ti:LiNbO3 waveguides,” presented at the OSA Meeting on Nonlinear Guided Waves and Their Applications, Dijon, France, September 1–3, 1999.

Hong, C. S.

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propa-gation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. B 67, 423–438 (1977).
[Crossref]

Iizuka, T.

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491–4499 (2000).
[Crossref]

Janz, S.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

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, 2631–2654 (1992).
[Crossref]

Kalocsai, A. G.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[Crossref]

Kivshar, Y. S.

Kobyakov, A.

Koroteev, N. I.

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, 3410–3412 (1995).
[Crossref]

Lederer, F.

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, 2631–2654 (1992).
[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, 3166–3172 (1997).
[Crossref]

Mantsyzov, B. I.

Masselin, P.

Nagle, J.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

Nefedov, I.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[Crossref]

Ozheredov, I. A.

Parameswaran, K. R.

Ricken, R.

G. Schreiber, D. Hofmann, W. Grundkotter, R. Ricken, and W. Sohler, “Near-infrared cascaded difference frequency generation in periodically poled Ti:LiNbO3 waveguides,” presented at the OSA Meeting on Nonlinear Guided Waves and Their Applications, Dijon, France, September 1–3, 1999.

Rosencher, E.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

Ross, G. W.

Scalora, M.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[Crossref]

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[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, 3166–3172 (1997).
[Crossref]

Schiek, R.

Schreiber, G.

G. Schreiber, D. Hofmann, W. Grundkotter, R. Ricken, and W. Sohler, “Near-infrared cascaded difference frequency generation in periodically poled Ti:LiNbO3 waveguides,” presented at the OSA Meeting on Nonlinear Guided Waves and Their Applications, Dijon, France, September 1–3, 1999.

Shkurinov, A. P.

Sibilia, C.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[Crossref]

Sievers, A. J.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[Crossref]

Smith, P. G. R.

Sohler, W.

G. Schreiber, D. Hofmann, W. Grundkotter, R. Ricken, and W. Sohler, “Near-infrared cascaded difference frequency generation in periodically poled Ti:LiNbO3 waveguides,” presented at the OSA Meeting on Nonlinear Guided Waves and Their Applications, Dijon, France, September 1–3, 1999.

Stegeman, G. I.

Tang, C. L.

C. L. Tang and P. P. Bey, “Phase matching in second harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. QE-9, 9–17 (1973).
[Crossref]

Theimer, J.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[Crossref]

Trillo, S.

van der Mer, P.

A. Fiore, S. Janz, L. Delobel, P. van der Mer, 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, 2942–2944 (1998).
[Crossref]

Viswanathan, R.

J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[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, 3166–3172 (1997).
[Crossref]

Yariv, A.

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propa-gation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. B 67, 423–438 (1977).
[Crossref]

A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. B 67, 438–448 (1977).
[Crossref]

Yeh, P.

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Figures (3)

Fig. 1
Fig. 1

Refractive indices of the semiconductor layer at the fundamental frequency (solid curve) and at the SH component (dashed–dotted curve) that are necessary to fulfill the resonance conditions Φ1L=π and Φ2L=2π in the photonic crystal.

Fig. 2
Fig. 2

Ratio of the effective nonlinear coefficient of the photonic crystal to the intrinsic nonlinear coefficient of the semiconductor layers as a function of the device length [η=(deff/dAlGaAs)].

Fig. 3
Fig. 3

Conversion efficiency in the photonic crystal: (a) Amplitude of the forward SH component at the device output (z=L). Solid curve, approximate analytical formula; dashed curve, exact numerical results; dashed–dotted curve, SH field generated in an homogeneous crystal of AlGaAs, provided that the phase-matching condition is satisfied. (b) Amplitude of the backward SH component at the device input (z=0). Solid curve, approximate analytical formula; dashed curve, exact numerical results.

Equations (22)

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n2(z, ω)=n02(ω)+2δ1 cos(Pz)+2δ2 cos(2Pz),
n102=n02(ω0),n202=n02(2ω0),
χ(z)=(n20λ/2π)[d0+2d1 cos(Pz)],
E1(z)=E1f(z)exp(-jk1z)+E1b(z)exp(jk1z),
k1=(2π/λ)n10.
jE1fz=C1E1b exp(jΔ1z),
-jE1bz=C1E1f exp(-jΔ1z),
E1f(z)=[A exp(-jΦ1z)+B exp(jΦ1z)]exp[j(Δ1/2)z],
E1b(z)=[C exp(-jΦ1z)+D exp(jΦ1z)]exp[-j(Δ1/2)z],
Φ1=[(Δ1/2)2-C12]1/2.
A=(Δ1/2+Φ1)E0 exp(jΦ1L)2[Φ1 cos(Φ1L)+j(Δ1/2)sin(Φ1L)].
E1f exp[-j(Δ1/2)z]=cos(Φ1z)-j Δ1L2π sin(Φ1z)×E0=αE0,
E1b exp[j(Δ1/2)z]=j C1Lπ sin(Φ1z)E0=βE0.
jFz-Δ22F=C2B+E02(d0α2+2d1αβ),
-jBz-Δ22B=C2F+E02(d0β2+2d1αβ).
F(L)=E02L8π d0 Δ1Δ2L22π+2d01+Δ1L2π2π+d1 C1L2π(2C2-Δ2-2Δ1),
B(0)=E02L8π 2C2Ld0 Δ1L2π-d1 C1Lπ+C1Lπ(d1Δ2L-2d1Δ1L+2d0C1L).
1Δ1L2πC1Lπ=e1 sign(Δ1),
1Δ2L2πC2Lπ=e2 sign(Δ2),
F(L)=E02Ldeff,
deff=d04e1(e1+e2)±e1 d14[e2(±1-1)-2e1].
deff=e1d04(e1+e2)-d12e1.

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