Abstract

Playing a central role in microelectronics and optoelectronics, semiconductors almost stand apart from applications involving second-order nonlinear effects such as frequency converters, tunable sources, parametric amplifiers, and switches. The reasons are twofold: their strong chromatic dispersion, which prevents the interacting waves from propagating with the same phase velocity (phase mismatch), and the shortness of the semiconductor devices, which adds more difficulty to achieving reasonable nonlinear efficiencies. By exploiting the unique properties of photonic crystals, we demonstrate simultaneous phase matching and enhancement of the fields under nonlinear interaction. We demonstrate a second-harmonic efficiency growth faster than the fifth power of the structure length.

© 2002 Optical Society of America

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2001 (5)

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

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 bandgap edge under the 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]

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]

D. Pezzeta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

2000 (3)

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[CrossRef] [PubMed]

W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, “Soliton effects in photonic crystal fibres at 850 nm,” Electron. Lett. 36, 53–55 (2000).
[CrossRef]

J. K. Ranka, R. Windeler, and A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

1999 (3)

L. A. Golovan, A. M. Zheltikov, P. K. Kashkarov, N. I. Koroteev, M. G. Lisachenko, A. N. Naumov, D. A. Sidorov-Biryukov, V. Yu Timoshenko, and A. B. Fedotov, “Generation of the second optical harmonic in porous-silicon-based structures with a photonic bandgap,” JETP Lett. 69, 300–305 (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. 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 bandgap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

1998 (3)

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463–465 (1998).
[CrossRef]

Y. S. Wu, R. S. Feigelson, R. K. Route, D. Zheng, L. A. Gordon, M. M. Fejer, and R. L. Byer, “Improved GaAs bonding process for quasi-phase-matched second-harmonic generation,” J. Electrochem. Soc. 145, 366–371 (1998).
[CrossRef]

1997 (5)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

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

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (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]

C. Simonneau, J. P. Debray, J. C. Harmand, P. Vidakovic, D. J. Lovering, and J. A. Levenson, “Second-harmonic generation in a doubly resonant semiconductor microcavity,” Opt. Lett. 22, 1775–1777 (1997).
[CrossRef]

1996 (3)

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1995 (2)

1994 (2)

J. Martorell and R. Corbalan, “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 bandgap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Hundt, and R. L. Byer, “Quasi-phase-matched second-harmonic generation: tuning and tolerances,” IEEE J. Quant. 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 (3)

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

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–1489 (1987).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, “Gap solitons and the nonlinear response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

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, 483–485 (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]

Aitchison, J. S.

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]

Arriaga, J.

W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, “Soliton effects in photonic crystal fibres at 850 nm,” Electron. Lett. 36, 53–55 (2000).
[CrossRef]

Assanto, G.

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[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 bandgap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609, 1–11 (2001).
[CrossRef]

Becouarn, L.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79, 904–906 (2001).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Berger, V.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463–465 (1998).
[CrossRef]

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[CrossRef]

Bertolotti, M.

Bloembergen, N.

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

D. Pezzeta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[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 bandgap 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, 3166–3174 (1997).
[CrossRef]

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

Bowden, C. M.

D. Pezzeta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[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. 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 bandgap 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, 3166–3174 (1997).
[CrossRef]

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

Bravetti, P.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463–465 (1998).
[CrossRef]

Broderick, N. G. R.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[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 bandgap edge under the quasi-phase-matching conditions,” Phys. Rev. E 63, 046609, 1–11 (2001).
[CrossRef]

Byer, R. L.

Y. S. Wu, R. S. Feigelson, R. K. Route, D. Zheng, L. A. Gordon, M. M. Fejer, and R. L. Byer, “Improved GaAs bonding process for quasi-phase-matched second-harmonic generation,” J. Electrochem. Soc. 145, 366–371 (1998).
[CrossRef]

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

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]

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 bandgap 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]

Chatenoud, F.

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Conti, C.

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

Corbalan, R.

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

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

J. Martorell and R. Corbalan, “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]

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]

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N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
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Y. S. Wu, R. S. Feigelson, R. K. Route, D. Zheng, L. A. Gordon, M. M. Fejer, and R. L. Byer, “Improved GaAs bonding process for quasi-phase-matched second-harmonic generation,” J. Electrochem. Soc. 145, 366–371 (1998).
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W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, “Soliton effects in photonic crystal fibres at 850 nm,” Electron. Lett. 36, 53–55 (2000).
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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).
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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).
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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]

D. Pezzeta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[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 bandgap 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, 3166–3174 (1997).
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[CrossRef]

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D. Pezzeta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
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[CrossRef]

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Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
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B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
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Figures (9)

Fig. 1
Fig. 1

Geometry of the PC studied. The incident wave vector and the incident angle are represented with respect to the two characteristic axes (x, z). The incident wave vector ki is expanded into two orthogonal projections (ki,x, ki,z) related to the symmetry of the structure.

Fig. 2
Fig. 2

Photograph of the whole sample with several PCs (bars and pillars) and close-up of one PC clearly showing the layered sequence. Dark (bright) regions correspond to the oxidized (nonlinear) layers.

Fig. 3
Fig. 3

(a) Reduced-dispersion relation ω(kzeff) showing the gaps. (b) Another representation of the dispersion relation showing clearly the corresponding effective refractive indices ω(kzeffc/ω). (c) The DOM of the PC studied. The arrows indicate the FF and the SHF operating positions.

Fig. 4
Fig. 4

Experimental setup: The two lasers can be alternately focused on the sample by use of microscope objectives. The SHF and the reflected FF are collected by means of a multimode fiber. The spectrum analysis is made with an optical spectrum analyzer (OSA).

Fig. 5
Fig. 5

Spectra of the SHF and the reflected FF plotted as functions of half of the FF wavelength: (a) with the FF wavelength detuned from perfect phase matching, and (b) with a FF wavelength resonant.

Fig. 6
Fig. 6

Second-harmonic intensity plotted as a function of its wavelength (solid curve) for different FF central wavelengths (in steps of 5 nm). The arrows highlight the maxima of the transmission spectrum at the second-harmonic wavelength. The linear transmission spectrum around the SHF is represented by the dotted curve. The thick (dashed) curve corresponds to the measured (calculated) result at phase matching.

Fig. 7
Fig. 7

Power dependence of the generated SHF with the polarization of the FF beam. The circles represent measurements, and the solid curve represents the fit made by use of the Green function analysis. The angles 0° and 180° correspond to TM polarization, whereas 90° represents TE polarization of the FF.

Fig. 8
Fig. 8

The power of the second harmonic as a function of the power of the incident fundamental in a loglog scale. The circles represent the measured points, and the solid line is a quadratic fit.

Fig. 9
Fig. 9

SHF intensity, normalized to the value of the 10-period sample, plotted as a function of the number of PC unit cells N. The circles and the dashed curve correspond to the experimental results and the best fit, respectively. The solid curve is the theoretical prediction, whereas the usual quadratic law is represented by the dotted–dashed curve for comparison. The dotted curve is the calculated spectral width of the FF transmission resonance.

Equations (4)

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T-1=1+sin(Nβ)sin(β)2(Tu-1-1).
DOM(ω)=keffω=1vg.
kzeff(ω)=1Lsϕt(ω)=1iLslogt|t|,
ESH(i)(z)=-+G(z, z)·PNL(i)(z)dz.

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