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APPLIED PHYSICS LETTERS 87, 151111 2005

Nonlinear diffraction and second-harmonic generation enhancement in silicon-opal photonic crystals
A. A. Fedyanina and O. A. Aktsipetrov
Department of Physics, M. V. Lomonosov Moscow State University, 119992 Moscow, Russia

D. A. Kurdyukov and V. G. Golubev
Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia

M. Inoue
Toyohashi University of Technology, 441-8580 Toyohashi, Japan

Received 14 April 2005; accepted 9 August 2005; published online 4 October 2005 Nonlinear diffraction in three-dimensional silicon-filled photonic crystals of opals is studied. Efficient backward second-harmonic generation SHG is observed in the specular direction upon the fundamental radiation reflection from the 111 face of the face-centered-cubic opal lattice. Tuning the fundamental wavelength across the photonic band gap PBG results in the 20-times increase of the second-harmonic intensity. The SHG peak has the width of approximately 20 nm and is located at the long-wavelength edge of the PBG. © 2005 American Institute of Physics. DOI: 10.1063/1.2077836 One of prospective directions of the use of threedimensional 3D photonic crystals of opals1,2 is their impregnation by semiconductor,3­ 8 metallic,9,10 or magnetic11 materials with the final goal to fabricate photonic crystals with complete photonic band gap PBG or with PBG tunable under external impacts. The dielectric function periodicity imposed by initial opal matrix is utilized for the control of the optical field propagation inside such composite photonic crystals. Nonlinear-optical effects in opal photonic crystals are accompanied with linear and nonlinear light diffraction on the 3D face-centered-cubic fcc lattice of silica microspheres of submicron diameter. However, the evenorder nonlinear-optical processes, such as second-harmonic generation SHG , are forbidden in the bulk of amorphous silica microspheres due to their inversion symmetry. Effective nonlinear diffraction in SHG can be observed in opal photonic crystals after impregnation of opal voids by semiconductors. In this letter, nonlinear diffraction in SHG is observed in the opal photonic crystals impregnated by silicon. The second-harmonic SH intensity is increased if the fundamental wavelength is close to the Bragg reflection condition. Spectral dependence of the intensity of the SH light generated in reflection from the 111 faces of the fcc opal lattice demonstrates the 20-times enhancement when the fundamental wavelength is tuned across the PBG. Previous studies of the SHG Ref. 12 and thirdharmonic generation13 performed in colloidal 3D polystyrene photonic crystals were focused on the enhancement of the nonlinear-optical signal due to phase-matching fulfillment. Compensation of the phase mismatch between fundamental and second third -harmonic waves was achieved when the harmonic wavelength was tuned across the PBG. The fundamental wavelength was far redshifted in spectrum from the PBG and fundamental wave dispersion was similar to that of the homogeneous medium. In the present letter, efficient SHG is associated with nonlinear diffraction of the fundaa

mental radiation on the 3D lattice of opal voids possessing the spatial periodicity of both linear and second-order susceptibilities. Samples consist of microspheres of amorphous silicon dioxide with diameter a 245 nm. The microspheres are assembled in the close-packed fcc lattice. The voids between microspheres are impregnated by silicon with a filling factor larger than 0.9 by the thermal chemical vapor deposition technique based on the thermal decomposition of silane.14,15 X-ray diffraction and Raman scattering studies showed that silicon in opal voids is a mixture of amorphous silicon and silicon nanocrystals. The depth of homogeneous silicon impregnation is estimated to be at least 0.1 mm. The polished sample surface coinciding with the 111 plane with accuracy better than 5° is moistened by glycerol to decrease the diffuse light scattering. Figure 1 shows the field-emission scanning electron microscope FESEM images of the 111 face acquired with different resolution. A good periodicity of the close-packed microspheres and almost full filling of opal voids by silicon are seen. The FESEM images with the larger scale show the sample has the polydomain structure with the average domain size of approximately 40 m. Within the single domain, silica microspheres in the 111 layer possess

Electronic mail: fedyanin@shg.ru

FIG. 1. FESEM images of the 111 face of the silicon-opal photonic crystal. 87, 151111-1 © 2005 American Institute of Physics

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FIG. 2. Reflection spectrum of the silicon-opal photonic crystal. Diffused background is subtracted. Inset shows the configuration of linear and nonlinear diffraction from the 111 opal layers.

FIG. 3. Spectral dependence of the intensity of the SH wave reflected from the 111 face of the silicon-opal photonic crystal, filled circles, compared with fundamental radiation reflection spectrum, open circles, measured at 20°. Inset shows the schematic of the "elementary" nonlinear source-- the single silicon--filled opal void.

regular hexagonal arrangement, which is broken at the domain boundaries. However, periodicity of the 111 layers along the growing direction remains the same for different domains1 that allows consideration of the sample for light propagation along or close to the 111 direction as a single crystal opal. Figure 2 shows the reflection spectrum measured at the 5° using an optical microscope angle of incidence technique.15 The maximum of reflectance is achieved at the wavelength of PBG 870 nm and corresponds to the PBG for this light direction. The full width at half maximum FWHM of the reflection peak is approximately 90 nm. The PBG center wavelength PBG is monotonous blueshifted with the increase of the incident angle . The linear-polarized output of the a nanosecondparametric generator/amplifier with the idler wavelength tunable from 800 nm to 1000 nm is used as fundamental radiation. Pulse duration is approximately 2 ns and pulse energy is below 5 mJ. The fundamental radiation is directed on 20°. The SH light reflected in the the opal surface at specular direction is separated from the fundamental radiation and the residual light by narrow-band color filters and detected by the photon-counted photomultiplier tube PMT . Polarization of the SH light is controlled by the Glan prizm placed in front of the PMT. Figure 3 shows the spectral dependence of the SH intensity when the fundamental wavelength is tuned in the spectral region of the PBG. Both waves, fundamental and second-harmonic, are polarized perpendicular to the plane of incidence corresponding to the s-in, s-out polarization con880 nm that is figuration. The SH intensity peaks at redshifted at 20 nm from the PBG measured at the same angle of incidence. At fundamental wavelengths shorter than 850 nm and longer than 920 nm, the SH intensity value becomes negligible. The maximal SH intensity is approximately 20 times larger in comparison with that just outside the SHG peak and up to 50 times larger than the intensity 950 nm. The FWHM of measured outside the PBG at the SHG peak is approximately 20 nm, that is about five times narrower than the peak in the linear reflection spectrum. The magnitude and the spectral shape of the SHG peak are checked to be almost the same for other polarization combinations of the fundamental and SH waves. The azi-

muthal dependence of the SH intensity does not possess any rotational anisotropy within error bars. SHG is induced at the surfaces of the silicon-filled opal voids one of which is schematically shown at the inset in Fig. 3. The void surfaces possess the quadratic susceptibility ^ S2 as a result of the inversion symmetry breaking at surface. The quadratic polarization P22 at the void surface is given as P
2 2

= ^ S2 :E E ,

1

where E is the fundamental wave amplitude. The nonzero elements of the ^ S2 tensor are ^
2 S,

,

^

2 S,

,

^

2 S,

,

2

where and denote normal and tangential directions, respectively, taken at the particular point of the void surface. Macroscopically, opal void is centrosymmetric that should forbid SHG in the electric-dipole approximation. However, the SH wavelength inside the silicon-filled opal voids is comparable with the void size. In this case, the SH fields generated at the opposite Si-opal interfaces do not interfere destructively due to the wave retardation. This results in the nonzero SHG signal from the single opal void, which is similar to the coherent nonlinear scattering on dielectric spheres with surface quadratic susceptibility.12,16 The SH field in the Fraunhofer region at the distance R from the single opal void placed in the homogeneous medium can be estimated as follows:12 E
2

=F

k2 beik2 R 2 2 ^ E, R sin /2 S

3

where F is the scattering factor accounting for the void shape and the retardation of the fundamental and SH waves at the void size b, and is the angle between wave vector k of the incident fundamental radiation and wave vector k2 of the scattered SH wave. For the ordered array of opal voids, expression 3 should be modified taking into account interference of the SH fields from every silicon-filled void, that can be done, for example, using the Green function formalism17 or direct solving of the wave equation in the medium possessing 3D periodicity.18 In a medium with spatial periodicity of the second-order susceptibility, efficient


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Appl. Phys. Lett. 87, 151111 2005

SHG is achieved as the scattered SH wave vector k2 satisfies the condition of nonlinear diffraction,19 k2 = 2k + nGNL, with diffraction order n and one of the elementary reciprocal vectors GNL characterizing the spatial lattice of the second-order susceptibility. The condition of nonlinear diffraction of the SH wave from the series of the 111 opal layers has a form k
2

=2k +2G

NL 111

.

4

In silicon-opal photonic crystals, 3D lattices of dielectric function and effective quadratic susceptibility are equivalent and reciprocal vectors GNL = GL 111 111 G111. Here, G111 is the reciprocal vector of the fcc opal lattice directed along the direction of the photonic Brillouin zone with G111 =2 / d and d =2a 2 / 3 is the interlayer distance for the most closely packed 111 planes. Vectorial Eq. 4 expresses the condition of the efficient backward SHG when the wave vector -2k and the fundamental radiation approaches PBG, k2 i.e., when k -G111 / 2. The latter condition also responsible for the fact that nonlinear diffraction with only n =2 in Eq. 4 is achieved. Similarly, backward SHG can be observed when k
2

vg = / k, takes the minimum.23 The small value of vg is equivalent to the enhancement of the photonic mode density in this spectral region or changes in the photon lifetime,24 and the SH intensity, which depends on fundamental wave vg -2, is increased significantly at the group velocity as I2 17 PBG edge. The surprisingly narrow spectral width of the SHG peak is attributed to the fast variation of the photonic mode density in the spectrum at the PBG edge. In conclusion, nonlinear diffraction of the laser radiation at the 3D lattices of linear and second-order susceptibilities of silicon-opal photonic crystals leads to the efficient backward SHG if the fundamental radiation is tuned across the PBG edge. This work was supported by the Russian Foundation for Basic Research, the Presidential Grant for Leading Russian Scientific Schools, INTAS Grant No. 03-51-3784, ECfunded project PHOREMOST FP6/2003/IST/2-511616 and the Grant-in-Aid for Scientific Research S , No. 17106004, from Japan Society for the Promotion of Science.
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1

=2k ,

k =k +G

L 111

.

5

The second vectorial equation in Eq. 5 is the Bragg condition for the coherent backward scattering of the fundamental radiation. It corresponds to the minus-first-order of linear diffraction of the fundamental radiation utilizing the reciprocal vector GL of the 3D-dielectric opal lattice and is ful111 filled at the wavelength corresponding to PBG at the di2 1/2 , where eff is the rection, PBG =2d eff - sin effective dielectric constant of photonic crystal.20 In this case, the efficient SHG is a result of nonlinear interaction of the fundamental radiation with the wave vector k , which was first diffracted in the backward direction. The nonlinear Bragg condition 4 is also fulfilled as the fundamental wavelength PBG. Thus, vectorial Eqs. 4 and 5 are formally equivalent, but responsible for different effects. The SH intensity in the diffraction maximum is limited by the approximate fulfillment of the diffraction conditions 4 and 5 as a result of the SH light absorption in silicon nanocrystals. The escape depth of SHG in the silicon-opal photonic crystals can be estimated similar to the one-dimensional phodif k2 -2 k -1. The effective ditonic crystals21 as NL electric function of silicon-opal composite is taken in the form of eff 1- f SiO2 + f Si, with f is the volume fraction of silicon nanocrystals and SiO2 , Si are the dielectric functions of opal microspheres and silicon nanocrystals. Taking dispersion of silicon nanocrystals the same as for the bulk dif silicon, NL 500 nm for 900 nm and f 0.26, that is approximately five times smaller than penetration depth of dif the SH radiation. However, NL is one order of magnitude larger than the escape depth of the SHG in reflection from the homogeneous nonlinear medium, which can be estimated k2 +2 k -1.22 The increase of the thickness of as NL material constructively contributing to the SHG output results in the SH intensity enhancement. The spectral position of the maximal SHG achieves at the fundamental wavelength in the vicinity of the long-wavelength edge of the PBG, where group velocity of the fundamental radiation,