Abstract

We established the angular conditions that maintain the quasi-phase matching conditions for enhanced second-harmonic generation. To do that, we investigated the equifrequency surfaces of the resonant Bloch modes of a two-dimensional periodic, hole-array photonic crystal etched into a GaN/sapphire epitaxial structure. The equifrequency surfaces exhibit remarkable shapes, in contrast to the simpler surfaces of a one-dimensional structure. The observed anisotropy agrees well with the surfaces calculated by a scattering matrix method. The equifrequency surfaces at fundamental and second-harmonic frequencies provide the values of polar and azimuthal angles that maintain quasi-phase matching conditions for enhanced second-harmonic generation over an extended tuning range. The predicted values for quasi phase-matching conditions show that frequency tuning for the two-dimensional case covers an about two times larger fractional bandwidth relative to the one-dimensional case.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  35. D. Coquillat, A. Ribayrol, R.M. De La Rue, M. Le Vassor d�??Yerville, D. Cassagne, J.P. Albert, �??Observations of band structure and reduced group velocity in epitaxial GaN-sapphire 2D photonic crystals,�?? Appl. Phys. B 73, 591-593 (2001).
    [CrossRef]
  36. M. Galli, M. Agio, L.C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, P. Bellutti, �??Spectroscopy of photonic bands in macroporous silicon photonic crystals,�?? Phys. Rev B 65, 113111-1 to 113111-4 (2002).
    [CrossRef]
  37. J.P. Mondia, H.M. Van Driel, W. Jiang, A.R. Cowan, J.F. Young, �??Enhanced second-harmonic generation from planar photonic crystals,�?? Opt Lett 28, 2500-2502 (2003).
    [CrossRef] [PubMed]

Appl. Phys. B (1)

D. Coquillat, A. Ribayrol, R.M. De La Rue, M. Le Vassor d�??Yerville, D. Cassagne, J.P. Albert, �??Observations of band structure and reduced group velocity in epitaxial GaN-sapphire 2D photonic crystals,�?? Appl. Phys. B 73, 591-593 (2001).
[CrossRef]

Appl. Phys. Lett. (6)

Baba, T. Matsumoto, �??Resolution of photonic crystal superprism,�?? Appl. Phys. Lett. 81, 2325-2327 (2002).
[CrossRef]

H.Y. Zhang, X.H. He, Y.H. Shih, M. Schurman, Z.C. Feng, R.A. Stall, �??Study of nonlinear optical effects in GaN:Mg epitaxial film,�?? Appl. Phys. Lett. 69, 2953-2955 (1996).
[CrossRef]

I.V. Kravetsky, I.M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, A. Eisenbach, H.L. Hartnagel, �??nonlinear optical response of GaN layers on sapphire: The impact of fundamental beam interference,�?? Appl. Phys. Lett. 76, 810-812 (2000).
[CrossRef]

A. Chowdhury, H.M. Ng, M. Bhardwaj, N.G. Weimann, �??Second-harmonic generation in periodically poled GaN,�?? Appl. Phys. Lett. 83, 1077-1079 (2003).
[CrossRef]

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

V.N. Astratov, R.M. Stevenson, I.S. Culshaw, D.M. Whittaker, M.S. Skolnick, T.F. Krauss, R.M. De La Rue, �??Heavy photon dispersions in photonic waveguides,�?? Appl. Phys. Lett., 77, 178-180 (2000).
[CrossRef]

J. Appl. Phys. (1)

D.N. Hahn, G.T. Kiehne, J.B. Ketterson, G.K.L. Wong, P. Kung, A. Saxler, M. Razeghi, �??Phase-matched optical second-harmonic generation in GaN and AlN slab waveguides,�?? J. Appl. Phys. 85, 2497-2501 (1999).
[CrossRef]

J. Opt Soc. Am. B (1)

For recent review see Focus Issue �??Nonlinear Optics and Photonic Crystals,�?? (NOPC), Edited by C.M. Bowden and A.M. Zheltikov, J. Opt Soc. Am. B 19, 2046-2296 (2002).
[CrossRef]

J. Opt. Soc. Am A (2)

B. Gralak, S. Enoch, G. Tayeb, �??Anomalous refractive properties of photonic crystals,�?? J. Opt. Soc. Am. A 17, 1012-1020 (2000).
[CrossRef]

J.M. Pottage, E. Silvestre, P. St. Russell, �??Vertical-cavity surface-emitting resonances in photonic crystal films,�?? J. Opt. Soc. Am. A 18, 442-447 (2001).
[CrossRef]

J. Opt. Soc. AM B (1)

J. Miragliotta, D.K. Wickenden, T.J. Kistenmacher, W.A. Bryden, �??Linear- and nonlinear-optical properties of GaN thin films,�?? J. Opt. Soc. Am. B 10, 1447-1456 (1993).
[CrossRef]

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

Opt. Lett. (1)

Opt. Quantum Electron. (1)

I. Shoji, T. Kondo, R. Ito, �??Second-order nonlinear susceptibilities of various dielectric and semiconductor material,�?? Opt. Quantum Electron. 34, 797-833 (2002).
[CrossRef]

Phys. Rev B (2)

A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, T.E. Tiwald, �??Free-carrier and phonon properties of n- and p-type hexagonal GaN films measured by infrared ellipsometry,�?? Phys. Rev B 62, 7365-7377 (2000).
[CrossRef]

M. Galli, M. Agio, L.C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, P. Bellutti, �??Spectroscopy of photonic bands in macroporous silicon photonic crystals,�?? Phys. Rev B 65, 113111-1 to 113111-4 (2002).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. B (11)

V. Pacradouni, W.J. Mandeville, A.R. Cowan, P. Paddon, J.F. Young, S.R. Johnson, �??Photonic band structure of dielectric membranes periodically textured in two dimensions,�?? Phys. Rev. B 62, 4204-4207 (2000).
[CrossRef]

R. Reinisch, M. Nevière, �??Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,�?? Phys. Rev. B 28, 1870-1885 (1983).
[CrossRef]

A.R. Cowan, J.F. Young, �??Mode matching for second-harmonic generation in photonic crystal waveguides,�?? Phys. Rev. B 65, 85106-1 to 85106-6 (2002
[CrossRef]

J. Miragliotta, D.K. Wickenden, �??Nonlinear electroreflectance from gallium nitride using optical second-harmonic generation,�?? Phys. Rev. B 53, 1388-1397 (1996).
[CrossRef]

D. M. Whittaker, I. S. Culshaw, �??Scattering-matrix treatment of patterned multiplayer photonic structures,�?? Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

A.M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L.C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, M. De Vittorio, �??Resonant second-harmonic generation in a GaAs photonic crystal waveguide,�?? Phys. Rev. B 68, 161306(R)-1 to 161306-4 (2003).
[CrossRef]

J. Torres, D. Coquillat, R. Legros, J.P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, M. Le Vassor d�??Yerville, E. Centeno, D. Cassagne, J.P. Albert, �??Giant second-harmonic generation in a one-dimensional GaN photonic crystal,�?? Phys. Rev. B 69, 85105-1 to 85105-8 (2004).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, �??Superprism phenomena in photonic crystals,�?? Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, S. Kawakami, �??Splitting of triply degenerate refractive indices by photonic crystals,�?? Phys. Rev. B 62, 1477-1480 (2000).
[CrossRef]

M. Notomi, T. Tamamura, Y. Ohtera, O. Hanaizumi, S. Kawakami, �??Direct visualisation of photonic band structure for three-dimensional photonic crystals,�?? Phys. Rev. B 61, 7165-7168 (2000).
[CrossRef]

M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Phys. Rev. E (3)

M. Centini, C. Sibilia, M. Scalora, G. D�??Aguanno, M. Bertollotti, M.J. Bloemer, C.M. Bowden, 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]

A.V. Balakin, V.A. Bushuev, B.I. Mantsyzov, I.A. Ozheredov, E.V. Petrov, A.P. Shkurinov, P. Masselin, G. Mouret, �??Enhancement of sum frequency generation near photonic band gap edge under the quasiphase matching conditions,�?? Phys. Rev. E 63, 46609-1 to 46609-11 (2001).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, A. Levenson, M.J. Bloemer, C.M. Bowden, J.W. Haus, M. Bertollotti, �??Photonic band edge effects in finite structures and applications to X(2) interactions,�?? Phys. Rev. E 64, 16609-1 to 16609-9 (2001)
[CrossRef]

Phys. Rev. Lett. (1)

M.C. Netti, A. Harris, J.J. Baumberg, D.M. Whittaker, M.B.D. Charlton, M.E. Zoorob, G.J. Parker, �??Optical trirefringence in photonic crystal waveguides,�?? Phys. Rev. lett. 86, 1526-1529 (2001).
[CrossRef] [PubMed]

Other (4)

J.D Jouannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton, NJ: Princeton University Press, (1995).

C.M. Soukoulis, Photonic crystals and light localization in the 21st Century (Kluwer Academic: Dordrecht, The Netherlands, (2001).

J.M. Lourtioz, H. Benisty, V. Berger, J.M. Gérard, D. Maystre, A. Tchelnokov, Les cristaux photoniques, ou la lumière en cage (Coll. Technique et scientifique des télécommunications, Lavoisier, 2003

S. Nakamura and G. Fasol, The Blue Green Diode (Springer, Berlin, 1997).

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Figures (8)

Fig. 1.
Fig. 1.

Schematic diagram illustrating the sample, coordinate system, and experimental geometry used for angular resolved reflection and transmission experiments to investigate the resonant Bloch modes dispersion.

Fig. 2.
Fig. 2.

Experimental transmission spectra for the two-dimensional GaN PhC for various angles of incidence, with s and p polarized light along the Γ-K and Γ-M lattice directions. The angle of incidence is varied from 0° to 39° with a step of 2° then 3°. (a) Γ-K, s-polarized incident light; (b) Γ-K, p-polarized incident light; (c) Γ-M, s-polarized incident light; (d) Γ-M, p-polarized incident light. The marked features in the higher angles spectra correspond to resonant Bloch modes. Along the symmetry directions the modes are excited by either s- or p-polarized light. For propagation directions away from the symmetry directions, these two polarizations are mixed and modes radiate an elliptically polarized field. For convenience in their labeling, the modes are referred to as s or p by continuity according to their polarization at φ=0°.

Fig. 3.
Fig. 3.

Reflection spectra at θ=25°, φ varies from 0° (Γ-K) to 30° (Γ-M). (a) s-s refers to incident-wave s-polarization and reflected-wave analyzer also s-polarization. (b) p-p refers to incident-wave p-polarization and reflected-wave also p-polarization.

Fig. 4.
Fig. 4.

Reflection spectra at θ=25° and φ=16° ; s-p refers to the incident and detected polarizations.

Fig. 5.
Fig. 5.

Photonic band structure of the two-dimensional GaN PhC for s-polarized light as determined from the experimental transmission spectra (solid dots) and from theoretical calculation (solid lines and open dots) along the high symmetry directions. The resonant Bloch modes 1p, 2s and 6s are respectively in red, in green, and in violet. The two large circles indicate the points involved in the QPM condition satisfied for fundamental field at 0.48 and SH field at 0.96. The solid lines show the edges of the cone of incident light.

Fig. 6.
Fig. 6.

EFS for different normalized frequencies as determined from the experimental transmission spectra (dots) and from calculation (solid), for the three resonant Bloch modes, and for s-polarized incident light; (a) 1p in red, (b) 2s in green, and (c) 6s in violet.

Fig. 7.
Fig. 7.

Experimental EFS used to predict angular conditions for enhanced SHG (red curve for fundamental frequency, blue curve for SH frequency). Black circles denote the achievement of QPM conditions. (a) EFS at ω from 1p at 0.51 and EFS at 2ω extracted from 10s at 1.02 plotted at half its in-plane wave-vectors. (b) Same but for fundamental frequency at 0.50 and SH frequency at 1.00. (c) Same but for fundamental frequency at 0.49 and SH frequency at 0.98. (d) The particular configuration where the coincidence point is along the Γ-M direction; fundamental frequency at 0.48 and SH frequency at 0.96.

Fig. 8.
Fig. 8.

Calculated EFS of the one-dimensional GaN PhC (a=500 nm) involved in enhanced SHG (red full squares for fundamental frequency, blue open circles for SH frequency). Black circles denote the achievement of QPM conditions. (a) EFS at ω from the mode 2p of the one-dimensional GaN PhC at frequency 0.629 (in units of ωa/2πc); the mode 2p is elliptically polarized for propagation along general directions. EFS at 2ω extracted from the mode4s at 1.258 (in units of ωa/2πc) plotted at half its in-plane wave-vectors. (b) Same but for fundamental frequency at 0.637 and SH frequency at 1.274. (c) Same but for fundamental frequency at 0.645 and SH frequency at 1.290.

Tables (1)

Tables Icon

Table 1. Second-order non linear coefficients dij for several dielectric materials.

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