Abstract

The parametric quadratic nonlinear interaction is considered within three-dimensional photonic crystals. A theoretical model that includes the full three-dimensional aspect of such nonlinear interaction is developed. Results from the study prove that second-order processes are possible in centrosymmetric three-dimensional photonic crystals and that the contribution to this nonlinear interaction is localized at the interfaces separating the two materials of the photonic lattice. In fact, such structures provide an independent solution to some of the most basic requirements for an efficient second-order nonlinear interaction: a nonvanishing interaction in the dipole approximation, a phase-matching mechanism, and a high nonlinear susceptibility not linked to the specific properties of the crystalline structure. Numerical results show that efficient parametric processes are achievable by use of short three-dimensional photonic crystals when realistic parameters for such nonlinear structures are used.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
    [CrossRef] [PubMed]
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  8. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
    [CrossRef] [PubMed]
  9. J. Trull, R. Vilaseca, J. Martorell, and R. Corbalán, “Second-harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
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  16. J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
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  17. J. Martorell, R. Vilaseca, J. Trull, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Opt. Photon. News 8, 34 (1997).
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    [CrossRef]
  24. Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
    [CrossRef]
  25. M. de Sterke, S. M. Saltiel, and Y. Kivshar, “Efficient collinear fourth-harmonic generation by two-channel multistep cascading in a single two-dimensional nonlinear photonic crystal,” Opt. Lett. 26, 539–541 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  28. As is well known, bound states in 3-D lattices are obtained only when the strength of the defect exceeds a critical threshold value, see, for instance, P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic, San Diego, Calif., 1995).
  29. See, for instance, E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).
  30. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [CrossRef]
  31. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  32. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Bayer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  33. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483 (1970).
    [CrossRef]
  34. Amnon Yariv and Pochi 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]
  35. J. P. Van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
    [CrossRef]
  36. J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
    [CrossRef]
  37. N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87(10), 103902 (1–4) (2001).
    [CrossRef] [PubMed]

2001 (4)

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

M. de Sterke, S. M. Saltiel, and Y. Kivshar, “Efficient collinear fourth-harmonic generation by two-channel multistep cascading in a single two-dimensional nonlinear photonic crystal,” Opt. Lett. 26, 539–541 (2001).
[CrossRef]

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87(10), 103902 (1–4) (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (3)

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999), http://www.osa.org/opticsexpress.
[CrossRef] [PubMed]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

1998 (2)

1997 (4)

J. Martorell, R. Vilaseca, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. 70, 702–704 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, J. Trull, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Opt. Photon. News 8, 34 (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 (3)

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in two-dimensional photonic lattices,” Phys. Rev. B 54, 5722–5749 (1996).
[CrossRef]

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear response of layered composite materials,” Phys. Rev. A 53, 2792–2798 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (2)

J. Martorell and R. Corbalán, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (2)

C. J. Herbert, W. S. Capinski, and M. S. Malcuit, “Optical power limiting with nonlinear periodic structures,” Opt. Lett. 17, 1037–1039 (1992).
[CrossRef] [PubMed]

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

1990 (1)

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1977 (1)

Amnon Yariv and Pochi 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]

1976 (1)

J. P. Van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

1970 (1)

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

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light 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]

Angerer, W. E.

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87(10), 103902 (1–4) (2001).
[CrossRef] [PubMed]

Armstrong, J. A.

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

Balakin, A. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Bayer, R. L.

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

Bloembergen, N.

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

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

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

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]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Boyd, R. W.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear response of layered composite materials,” Phys. Rev. A 53, 2792–2798 (1996).
[CrossRef] [PubMed]

Bushuev, V. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Campillo, A. J.

Capinski, W. S.

Centini, M.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Corbalán, R.

J. Martorell, R. Vilaseca, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. 70, 702–704 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, J. Trull, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Opt. Photon. News 8, 34 (1997).
[CrossRef]

J. Trull, R. Vilaseca, J. Martorell, and R. Corbalán, “Second-harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
[CrossRef] [PubMed]

J. Martorell and R. Corbalán, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
[CrossRef]

D’Aguanno, G.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Dadap, J. I.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

de Sterke, M.

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]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Ducuing, J.

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

Dumeige, Y.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Eisenthal, K. B.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Fedotov, A. B.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Fejer, M. M.

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

Fischer, G. L.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear response of layered composite materials,” Phys. Rev. A 53, 2792–2798 (1996).
[CrossRef] [PubMed]

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]

Gehr, R. J.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear response of layered composite materials,” Phys. Rev. A 53, 2792–2798 (1996).
[CrossRef] [PubMed]

Golovan, L. A.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

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]

Heinz, T. F.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Herbert, C. J.

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]

Ilegems, M.

J. P. Van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

John, S.

S. John, “Strong localization in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Jundt, D. H.

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

Kashkarov, P. K.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Kivshar, Y.

Koroteev, N. I.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Lawandy, N. M.

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

Lee, Reginald K.

Levenson, J. A.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Lin, H.-B.

Lisachenki, M. G.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Magel, G. A.

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

Malcuit, M. S.

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]

Mantsyzov, B. I.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Martorell, J.

J. Trull, J. Martorell, and R. Vilaseca, “Angular dependence of phase-matched second-harmonic generation in a photonic crystal,” J. Opt. Soc. Am. B 15, 2581–2585 (1998).
[CrossRef]

J. Martorell, R. Vilaseca, J. Trull, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Opt. Photon. News 8, 34 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. 70, 702–704 (1997).
[CrossRef]

J. Trull, R. Vilaseca, J. Martorell, and R. Corbalán, “Second-harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
[CrossRef] [PubMed]

J. Martorell and R. Corbalán, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
[CrossRef]

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

Masselin, P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Mouret, G.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Naumov, A. N.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Ohtaka, K.

K. Sakoda and K. Ohtaka, “Sum-frequency generation in two-dimensional photonic lattices,” Phys. Rev. B 54, 5722–5749 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

Ozheredov, I. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light 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]

Petrov, E. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Sagnes, I.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Sakoda, K.

K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999), http://www.osa.org/opticsexpress.
[CrossRef] [PubMed]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in two-dimensional photonic lattices,” Phys. Rev. B 54, 5722–5749 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

Saltiel, S. M.

Sauvage, S.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Scalora, M.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[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]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Shan, J.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Shkurinov, A. P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Sibilia, C.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Sidorov-Biryukov, D. A.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Sievers, A. J.

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

Sipe, J. E.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear response of layered composite materials,” Phys. Rev. A 53, 2792–2798 (1996).
[CrossRef] [PubMed]

Timoshenko, V. Yu

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Tonucci, R. J.

Trull, J.

Van der Ziel, J. P.

J. P. Van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

Vidakovic, P.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

Vilaseca, R.

J. Trull, J. Martorell, and R. Vilaseca, “Angular dependence of phase-matched second-harmonic generation in a photonic crystal,” J. Opt. Soc. Am. B 15, 2581–2585 (1998).
[CrossRef]

J. Martorell, R. Vilaseca, J. Trull, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Opt. Photon. News 8, 34 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. 70, 702–704 (1997).
[CrossRef]

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

J. Trull, R. Vilaseca, J. Martorell, and R. Corbalán, “Second-harmonic generation in local modes of a truncated periodic structure,” Opt. Lett. 20, 1746–1748 (1995).
[CrossRef] [PubMed]

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]

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]

Xu, Y.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yang, N.

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87(10), 103902 (1–4) (2001).
[CrossRef] [PubMed]

Yariv, A.

Yariv, Amnon

Amnon Yariv and Pochi 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, Pochi

Amnon Yariv and Pochi 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]

Yodh, A. G.

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87(10), 103902 (1–4) (2001).
[CrossRef] [PubMed]

Zheltikov, A. M.

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Appl. Phys. Lett. (4)

J. Martorell, R. Vilaseca, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. 70, 702–704 (1997).
[CrossRef]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[CrossRef]

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

J. P. Van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

J. Opt. Soc. Am. B (3)

JETP Lett. (1)

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenki, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of second optical harmonic in porous silicon-based structures with a photonic band gap,” JETP Lett. 69, 300–305 (1999).
[CrossRef]

Opt. Commun. (1)

J. Martorell and R. Corbalán, “Enhancement of second harmonic generation in a periodic structure with a defect,” Opt. Commun. 108, 319–323 (1994).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Opt. Photon. News (1)

J. Martorell, R. Vilaseca, J. Trull, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Opt. Photon. News 8, 34 (1997).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (3)

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear response of layered composite materials,” Phys. Rev. A 53, 2792–2798 (1996).
[CrossRef] [PubMed]

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]

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

Phys. Rev. B (2)

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in two-dimensional photonic lattices,” Phys. Rev. B 54, 5722–5749 (1996).
[CrossRef]

Phys. Rev. E (1)

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (1–11) (2001).
[CrossRef]

Phys. Rev. Lett. (7)

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

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87(10), 103902 (1–4) (2001).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

Other (6)

For a recent review see, for instance, C. M. Soukoulis, ed., Photonic Crystals and Light Localization in the 21st Century (Kluwer Academic, Dordrecht, The Netherlands, 2001).

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second harmonic light from small spherical particles,” in Quantum Electronics and Laser Science Conference (Optical Society of America, Washington, D.C., 1995) Vol. 16, p. 32.

J. Martorell, “Second-harmonic scattering from sites of a crystalline lattice,” in Photonic Band Gap Materials, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 529–534.

As is well known, bound states in 3-D lattices are obtained only when the strength of the defect exceeds a critical threshold value, see, for instance, P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic, San Diego, Calif., 1995).

See, for instance, E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).

J. Martorell, “Quadratic nonlinear interactions in three-dimensional photonic crystals” in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 589–599.

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Figures (9)

Fig. 1
Fig. 1

Spheres distributed on a plane triangular lattice corresponding to one of the (111) planes of a fcc lattice. The separation between spheres on that plane is a, and the diameter of the spheres is D. a1=ax and a2=a[-(1/2)x+(3/2)1/2y] are the primitive lattice vectors on the xy plane. x and y are unit vectors.

Fig. 2
Fig. 2

Numerically determined transmission of light at 532 nm through a 1-mm-thick photonic crystal made of 115-µm (with a 4.7% diameter dispersion) diameter polystyrene spheres (thick solid curve) when the parameters of the actual crystal from Fig. 2 of Ref. 26 are used. The index of polystyrene and water at 532 nm and 1064 nm is 1.59 and 1.333, respectively. The numerical calculation performs an average over the different contributions that are due to the 4.7% dispersion, as well as accounting for the effective absorption that is due to scattering and was measured in Ref. 26 to be of the order of 10 cm-1. Reflection for the same crystal with only 100 planes with no dispersion in the sphere diameter (thin solid curve) and with a 4.7% dispersion included (dashed curve) are also shown. Both reflection curves were reduced by a factor of 0.1. Fresnel reflection losses from the four interfaces of the glass cuvette containing the crystal were included in only the transmission curve.

Fig. 3
Fig. 3

Photon-dispersion curve in the neighborhood of the frequency of the SH when the sphere diameter is 115 nm and there is an infinite number of planes (short-dashed curve) and when the number of planes is 50 (thin solid curve). The photon-dispersion curve is also shown for a sphere diameter of 170 nm and 100 planes (dotted–dashed curve) and a sphere diameter of 190 nm and 50 planes (thick solid curve). The forbidden band for light propagating in the direction normal to the (111) planes is shown for a crystal of 115-nm-diameter spheres.

Fig. 4
Fig. 4

Effective index of refraction at 532 nm plotted as a function of the angle of incidence in the vicinity of a stop band centered near 532 nm for 225 crystalline planes when the sphere diameter is 115 nm (thick solid curve), 170 nm (dashed curve), and 190 nm (thin solid curve). The effective index at 1064 nm is also shown (dotted–dashed curve) and is seen to be unaffected by the stop band and essentially independent of the sphere diameter.

Fig. 5
Fig. 5

Dielectric sphere of a diameter comparable with the wavelength of the generated light. The surface is covered by a layer of nonlinear molecules that on average point in the radial direction.

Fig. 6
Fig. 6

Reflected amplitude (a) from one single plane and (b) the transmitted amplitude through 1000 crystalline planes of the SH field normalized to the input field for three sphere sizes plotted as functions of the angle of incidence. The sphere diameters are 115 nm (dotted curve), 170 nm (short-dashed curve), and 190 nm (solid curve). The spacings between planes considered here are different than the fixed spacing used for Fig. 4; as a consequence, the angles of phase matching between both figures bare no relation.

Fig. 7
Fig. 7

Transmitted SH field amplitude as a function of the number of planes when the sphere diameter is 240 nm with CV=0% (solid curve), 2% (dashed curve). The separation between planes was set to 1.16 times half of the wavelength of the generated light in water.

Fig. 8
Fig. 8

Effective index of refraction plotted as a function of the angle of incidence in the vicinity of a stop band centered at 1064 nm for 50 crystalline planes when the sphere size is 170 nm (thin solid curve), 280 nm (dotted curve), and 300 nm (thick solid curve). The effective index at 532 nm is also shown (dotted–dashed curve), which is essentially the same for any given sphere diameter.

Fig. 9
Fig. 9

Down-converted field amplitude for light generated at 1064 nm from the difference of a wave at 500 nm and a wave at 943.26 nm plotted as a function of the angle of incidence when the Bragg resonance is tuned at a wavelength close to 1000 nm and the sphere diameter is 280 nm. The numerical calculation was performed with 225 planes.

Equations (15)

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

(·E)-2E=ω20c2(r)E,
(r)-1=GG(z)exp(iG·ρ),
1Acj exp(-iGρi)AcdA[r(R, z)-1]exp(-iGR),
E(r)=d3kE(k)exp(ikr),
d2dz2E(k)+ω2c2-kρ2E(k)+ω20c2G0GE×(k-G)exp(ikρρ)exp(ikz)dkρdkz=0,
d2dz2E(z, kρ)+ω2c2-kρ2E(z, kρ)=-ω2c2GGG(z)E(z, kρ-G).
E(z, ks)=-G(z, z)R(z)dz,
G(z, z)=-i2kzexp[-ikz(z-z)].
E(z, kxxˆ-G)=Ein(kxxˆ-G)exp(ikzz),
E(z, kx)=i exp(-ikzz)8π33ωc2(D/2)3kza2×(r-1)GF1(|g|D/2)Ein(kxxˆ-G),
F1(|g|D/2)=3(|g|D/2)3[sin(|g|D/2)-(|g|D/2)cos(|g|D/2)]
Pω1(r)=0χ(2)(r) : Eω3(r)Eω2*(r),
Pω1(r)=0Gdk exp(ikρρ)exp(ikzz)χG(2)×exp(iGρ)χG(2)(z) : (Eω3Eω2)k.
d2dz2Eω1(z, kρ)+ω12c2-kρ2Eω1(z, kρ)=-ω12c2G[GEω1(z, kρ-G)+χG(2)(z) : [Eω3Eω2*(z)],kρ-G].
Eω1(z, k1x)=exp-ik1zz2π3ω1c2δDk1zk1sa2χrrr(2Eω3Eω2*×[(2 sin θi cos θr-sin θr cos θi)cos θiF1FH(k1sD/2)+2 sin2 θi sin θr F2SH(k1sD/2)],

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