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JETP Letters, Vol. 75, No. 1, 2002, pp. 15­19. Translated from Pis'ma v Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 75, No. 1, 2002, pp. 17­21. Original Russian Text Copyright © 2002 by Dolgova, Maoedykovski, Martemyanov, Fedyanin, Aktsipetrov.

Giant Third-Harmonic in Porous Silicon Photonic Crystals and Microcavities
T. V. Dolgova, A. I. Maoedykovski, M. G. Martemyanov, A. A. Fedyanin, and O. A. Aktsipetrov*
Moscow State University, Vorob'evy gory, Moscow, 119899 Russia * aktsip@shg.ru
Received December 4, 2001

A giant enhancement (no less than by 103) of the optical third-harmonic generation in one-dimensional porous silicon microcavities and photonic crystals was observed experimentally. The enhancement is due to the resonant enhancement of the fundamental field in the cavity mode and the fulfillment of the phase matching condition at the photonic band gap edges of the photonic crystal and in the vicinity of the microcavity mode. © 2002 MAIK "Nauka / Interperiodica". PACS numbers: 42.65.Ky; 42.70.Qs

In recent years, the nonlinear optics of photonic crystals (PCs) and PC-based microcavities (MCs) has been progressing intensively [1]. Giant nonlinear-optical phenomena caused by giant effective dispersion at the edges of the photonic band gap and in the vicinity of the cavity mode, and also by the enhancement of optical fields inside PCs and MCs under optimal frequency­angular conditions, can be observed in such microstructures. In particular, multiple reflection interference in PCs can compensate the phase mismatch for fundamental and second-harmonic waves when one of the waves falls on the edge of the photonic band gap in the space of frequencies or wave vectors [2]. Such an effective fulfillment of the phase matching conditions leads to a resonant enhancement of second-harmonic generation in PCs composed, for example, of dielectric polystyrene spheres [3], alternate GaAs­AlGaAs [4] or ZnS­SrF2 [5] layers, or alternate layers of porous silicon differing in porosity [6, 7]. At the same time, the spatial distribution of the electromagnetic field inside PCs and MCs can be adequately controlled. For example, the interference of waves with opposite projections of the wave vector onto the periodicity direction of the PC mirrors at a resonance of the external field with the MC mode leads to the formation of a standing wave inside the MC with an amplitude that resonantly increases in the vicinity of the cavity layer. The degree of field localization (enhancement), which is the measure of the quality factor of the MC, is determined by the parameters of the mirrors (photonic crystals) surrounding the cavity layer. An increase in the laser fluence inside the MC in the spectral (frequency or angular) vicinity of the MC mode leads to a resonant enhancement of the nonlinear-optical MC response (for example, to the giant second-harmonic generation in porous silicon [7] or zinc selenide [8] MCs) and to an

enhancement of Raman scattering in GaAs­AlAs [9] or porous silicon [10] MCs. However, the enhancement of second-harmonic generation is limited by the destructive interference of second-harmonic waves generated by the cavity layer of a half-wavelength (for fundamental radiation) thickness; in this case, the PC layers nearest to the cavity layer make the main contribution to the second-harmonic signal. The nonlinear-optical effects that depend on a higher degree of the fundamental amplitude (for example, the optical third-harmonic generation (THG)) are free from this limitation. This work presents the results of an experimental study on the giant THG in PC-based microcavities made of porous silicon. The resonant THG enhancement was found in angular TH intensity spectra in the vicinity of the cavity mode and at the edges of the photonic band gap. It is shown that the THG resonance in the mode is due to the combined action of the spatial localization of fundamental radiation in the vicinity of the cavity layer and the fulfillment of the phase matching condition. The enhancement of the standing fundamental wave amplitude in the resonant increase in the TH intensity in the MC mode is directly confirmed by near-field optical microscopy. The THG resonances at the edges of the photonic band gap are caused by the spatially uniform enhancement of the fundamental wave in PC mirrors of the MC and by the compensation of the phase mismatch between the fundamental and TH waves due to multiple reflection interference in the PC. The samples of microcavities were prepared according to the conventional electrochemical procedure [11] by etching a wafer of heavily doped p-type silicon with the (100) crystallographic orientation and a resistivity of 0.01 cm. A solution containing 15% of hydrofluoric acid, 27% of water, and 58% of ethyl alcohol was

0021-3640/02/7501-0015$22.00 © 2002 MAIK "Nauka / Interperiodica"


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hence, the etching rate and the porosity (the volume fraction of air in the porous silicon) at the leading etch front also vary. As a result, the prescribed current density­time profile is transferred to the porosity­depth profile. A topographic image of a sample cleavage obtained using a scanning microscope with a piezoelectric quasi-friction force detector confirms the periodicity of the sample structures and the existence of welldefined boundaries between layers [7]. The microcavity samples under study with the microcavity mode at normal incidence MC 1300 nm or MC 620 nm represent a half-wavelength cavity layer confined between two PC mirrors composed of five pairs of quarter-wavelength porous silicon layers. Each pair consists of layers obtained by etching at current densities of 25 mA/cm2 and 87 mA/cm2. The corresponding porosities fhigh 0.60 and flow 0.70 are determined by porosity­current density calibration curves obtained in advance. The refractive indices of layers calculated with these porosities within the framework of the effective medium model at the fundamental wavelength comprise 2.0 and 1.65, respectively. The porosity of the cavity layer fres = flow. Uniform porous silicon films about 1.5 µm in thickness are prepared as reference samples. A pulsed YAG:Nd3+ laser generating pulses with a duration of about 10 ns, the wavelength = 1064 nm, and a pulse energy of about 6 mJ was used. The polarized radiation of the laser passes through an infrared filter extracting the pumping and is directed at a sample clamped on a goniometer at an incidence angle . The goniometer provides a consistent revolution of the sample and the detection system in the range of incidence angles 0° < < 90° with a minimum step of 0.5°. The radiation reflected from the sample passes through a system of ultraviolet filters 11 mm in total thickness, which extracts the radiation at the TH wavelength, and through a Glan prism, which controls the TH polarization state. Next, the signal is detected by a photomultiplier tube (PMT) and an electronic gated recording system connected with a computer. The possibility of mounting a monochromator with slits 0.05 mm in width after all the elements of the optical system is provided for checking the frequency spectrum of the reflected radiation. The angular spectrum of the linear reflection coefficient is measured in the identical alignment. For this purpose, the frequency of the fundamental radiation reflected from the sample is doubled by a quartz plate, because the PMT is not very sensitive in the IR region. To obtain the absolute normalization of the reflection coefficient, the fundamental radiation is directed to a PMT through all the elements of the optical system, excluding the sample. The distributions of local fields inside a MC were characterized by near-field scanning optical microscopy. The sample cleavage under study is placed on a ceramic piezoelectric three-dimensional tube of the scanning microscope. A probing apertureless tip with a
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Fig. 1. (a) Intensity of the s-polarized TH radiation I3 measured in a porous silicon MC with MC 1300 nm as a function of the angle of incidence of the s-polarized fundamental radiation ( = 1064 nm). Dashed lines indicate the angular shift of the THG resonance from the position of the cavity mode. Inset: the spectrum of an optical signal in the vicinity of the TH wavelength measured at the THG resonance ( = 55°). (b) Angular spectrum of the reflection coefficient Rs of the s-polarized fundamental radiation from MC.

used as an electrolyte. The silicon wafer previously etched on both sides to remove the natural oxide was placed onto a flat copper cathode, and a platinum coil placed in the electrolyte above the silicon wafer surface in an electrochemical cell serves as the anode. The direct electric current was controlled by a galvanostat. To obtain a multilayer structure, the density of the current through the silicon wafer varies periodically, and,


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radius of curvature of about 50 nm and made of an optical single-mode fiber is placed directly above the sample cleavage. The surface topography is traced by a feedback system with a tuning-fork resonant quasi-friction force detector. The fundamental radiation is supplied by a multimode optical fiber perpendicular to the sample surface as close to the MC cleavage as possible. The radiation collected by the probe is directed to the PMT cathode through a monochromator. The THG experiments were performed for the geometry of fundamental and reflected TH waves polarized in the sample plane (ss-geometry). The choice of the geometry of an experiment is determined by the most pronounced manifestation of photonic properties for s-polarized waves. The corresponding component of the bulk cubic susceptibility of porous (3) silicon yyyy is not equal to zero, as distinct from yyy 0 where the y axis is perpendicular to the plane of incidence. Thus, the dipolar volume polarization P(3)(3) is the main source of third-harmonic generation in microcavities. The intensity I3 of the third-harmonic reflected from a microcavity expressed in arbitrary units is shown in Fig. 1a as a function of the angle of incidence of the fundamental radiation on the sample. An enhancement at the edges of the photonic band gap ( = 17° and = 75°) and a narrow peak in the region of the cavity mode ( = 55°) are observed in the angular spectrum I3(). The enhancement of the TH intensity in the MC mode comprises no less than 103 as compared with I3 in the photonic band gap. The inset in Fig. 1 shows the frequency constitution of the radiation reflected from the MC and passed through a set of filters of the recording system. The composition was obtained with the use of a monochromator. A narrow (with a half-width of about 0.1 nm) spectral peak is observed at a wavelength of 354.7 nm. This peak corresponds to the third-harmonic for pumping with a wavelength of 1064 nm. Measurements of the reflected radiation spectrum in a wide range showed the reliability of the selected set of filters for extracting the third-harmonic: radiation at wavelengths differing from the TH was not detected. The linear reflection coefficient Rs of the infrared s-polarized fundamental radiation is shown in Fig. 1b as a function of the angle of incidence on a microcavity sample. The region of the photonic band gap in angular variables starts at approximately = 25° and corresponds to vanishingly small values of the third-harmonic intensity. A dip is observed in the angular reflection spectrum at an incidence angle of 58°. Analogous measurements of the angular spectra I3() and Rs() were carried out for a uniform silicon wafer with a thickness of 1.5 µm, which is comparable with the total thickness of the microcavity sample (Fig. 2). Effects associated with multiple reflection interference, namely, a broad peak in the angular third-harmonic
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(2)

Fig. 2. Angular spectrum of the s-polarized TH radiation intensity I3 measured for a uniform porous silicon wafer 1.5-µm thick as a function of the angle of incidence of the s-polarized fundamental radiation. The TH intensity units are the same as in Fig. 1a. (b) Angular spectrum of the reflection coefficient Rs of the s-polarized fundamental radiation from a uniform porous silicon wafer.

spectrum and oscillations in the reflection coefficient are also observed in the curves obtained. The features of the angular spectrum of the thirdharmonic generated by a microcavity are determined by the phase matching conditions, which signify the constructive interference of third-harmonic waves from each MC layer or, in other words, the equality of the effective refractive indices at the fundamental and third-harmonic frequencies [12]. These conditions are fulfilled in the regions of high effective dispersion, which correspond to fast variations in the angular spectrum of the linear reflection coefficient, and determine the enhancement of third-harmonic generation at the photonic band gap edges at = 17° and = 75°. The


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MC Si(001)

cavity and a uniform wafer, admit a similar consideration in terms introduced for a cavity. The interfaces of a uniform wafer with air and with the substrate serve as mirrors in this case. The reflection coefficients of such mirrors are considerably smaller than those for multilayer structures. In fact, when characterizing microcavities by the quality factor determined through the halfwidths of THG resonance peaks at their half-maximum, one can see that this value in angular variables equals 10° for a uniform wafer and is 5 times smaller for a microcavity. The third-harmonic intensity reached in the MC mode is 15 times higher than in the maximum of the I3() curve for a porous silicon film. The localization of the field in the vicinity of the cavity layer was directly demonstrated by the measurement of the spatial two-dimensional distribution of the fundamental field intensity using a near-field microscope (Fig. 3). The image was obtained for a MC cleavage with MC 620 nm at resonance excitation by the radiation of a He­Ne laser with = 633 nm at a normal to the MC surface. The bright strip in the vicinity of the cavity layer corresponds to the resonant increase in light intensity detected by the microscope probe. The dark strips in the image correspond to intensity minima and fall on the centers of the PC mirrors. The contrast between the minimum and maximum intensities, which characterizes the degree of field localization in the cavity, reaches 10. The bright spot in the near-surface region of the sample (z < 200 nm) is apparently a consequence of the scattering of radiation by the sample surface. The cross-section of the image in the direction perpendicular to the microcavity surface correlates well with the envelop of the model spatial distribution of the standing fundamental wave intensity calculated within the propagation matrix formalism (Fig. 3, at the bottom). Thus, the localization of laser radiation in the vicinity of the cavity layer of microcavities with photonic crystal mirrors (at a resonance with the MC mode) and also inside photonic crystals (when the radiation wavelength and wave vector fall on the edge of the photonic band gap) leads to a giant enhancement of their nonlinear-optical response, in particular, to the generation of a giant third-harmonic. The matching of the phase velocities of the fundamental and third-harmonic radiations due to multiple reflection interference in the photonic crystals of the mirrors is another factor of the enhancement of the third-harmonic generation in microcavities. The authors are grateful to N. Ota for help in preparing the microcavity samples and to V.A. Yakovlev for fruitful discussions. This work was supported by the Russian Foundation for Basic Research, project nos. 00-15-96555, 01-0216746, 01-02-17524, and 01-02-04018.
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120 100 80 60 40 20 0 400 800 1200 1600 z(nm) I

Fig. 3. At the top: spatial distribution of the optical field intensity at an MC cleavage with MC 620 nm measured by a near-field optical microscope at resonance excitation by the radiation of a He­Ne laser at a normal to the MC. At the bottom: cross-section of the intensity image along the direction of PC periodicity (dots) and a model spatial distribution of the standing wave intensity in the cavity mode (solid lines). Dashed lines indicate the boundaries of the cavity layer.

weak manifestation of the photonic band gap edge in the angular dependence Rs at large angles is apparently associated with the rapid growth of the Fresnel coefficients in this region. The THG peaks at the photonic band gap edges are due to interference in the PC structure and are not associated directly with the existence or nonexistence of the cavity layer. The giant enhancement of third-harmonic generation for a microcavity at = 55° is provided by a combination of the fulfillment of the phase matching condition in the region of the cavity mode and the localization of the fundamental field in the cavity and nearest layers. The position of the dip in the linear reflection coefficient (Fig. 1b) is determined by the angular position of the microcavity mode at which the localization of the field in the cavity layer is most strongly pronounced. The peak in the TH intensity spectrum is shifted with respect to this position by a value determined by the fulfillment of the phase matching condition. Both samples, namely, a photonic crystal micro-


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REFERENCES
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7. T. V. Dolgova, A. I. Maoedykovskioe, M. G. Martem'yanov, et al., Pis'ma Zh. èksp. Teor. Fiz. 73, 8 (2001) [JETP Lett. 73, 6 (2001)]. 8. V. Pellegrini, R. Colombelli, I. Carusotto, et al., Appl. Phys. Lett. 74, 1945 (1999). 9. A. Fainstein, B. Jusserand, and V. Thierry-Mieg, Phys. Rev. Lett. 75, 3764 (1995). 10. L. A. Kuzik, V. A. Yakovlev, and G. Mattei, Appl. Phys. Lett. 75, 1830 (1999). 11. O. Bisi, S. Ossicini, and L. Pavesi, Surf. Sci. Rep. 38, 1 (2000). 12. M. Centini, C. Sibilia, M. Scalora, et al., Phys. Rev. E 60, 4891 (1999).

Translated by A. Bagatur 'yants

JETP LETTERS

Vol. 75

No. 1

2002