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Advances in Optical Technologies
Volume 2008 (2008), Article ID 780784, 12 pages
http://dx.doi.org/10.1155/2008/780784
Review Article

Tuning of the Optical Properties in Photonic Crystals Made of Macroporous Silicon

1Department of Chemistry, Faculty of Science, University of Paderborn, 33095 Paderborn, Germany
2Institute of Physics, Martin-Luther-University Halle-Wittenberg, 06099 Halle, Germany
3Department of Physics and Institute for Optical Sciences, University of Toronto, Toronto, ON, Canada M5S 1A7

Received 9 January 2008; Accepted 11 April 2008

Academic Editor: D. Lockwood

Copyright © 2008 Heinz-S. Kitzerow et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is well known that robust and reliable photonic crystal structures can be manufactured with very high precision by electrochemical etching of silicon wafers, which results in two- and three-dimensional photonic crystals made of macroporous silicon. However, tuning of the photonic properties is necessary in order to apply these promising structures in integrated optical devices. For this purpose, different effects have been studied, such as the infiltration with addressable dielectric liquids (liquid crystals), the utilization of Kerr-like nonlinearities of the silicon, or free-charge carrier injection by means of linear (one-photon) and nonlinear (two-photon) absorptions. The present article provides a review, critical discussion, and perspectives about state-of-the-art tuning capabilities.

1. Introduction

Artificial structures exhibiting a spatially periodic structure with lattice constants comparable to the wavelength of light [1, 2] are extremely promising materials for integrated optical devices. These structures, referred to as photonic crystals [311], are characterized by an unusual dispersion relation which might show a photonic band gap (PBG), that is, a frequency range in which the propagation of light is not permitted. Properly designed defects within these structures may serve, for example, as optical waveguides, frequency filters, optical switches, or resonant microcavities with high quality factor, which can be used for low-threshold lasing. Many excellent works have demonstrated the potential capabilities of these structures, but their fabrication is still elaborate.

Electrochemical etching of silicon turned out to be a very reliable and versatile technique to fabricate two-dimensional periodic arrays of macropores [1215]. Similarly to the method of electrochemical polishing, silicon wafers are dipped into hydrofluoric acid and a DC voltage between the wafer and a counter-electrode is applied. The generation of free charge carriers in the doped silicon is assisted by exposure to infrared radiation. However, in contrast to the common polishing process, the parameters are chosen in a narrow range where the etching process does not lead to flattening of the surface, but instabilities cause the self-organized growth of pores. Spontaneously, the pores tend to arrange in a two-dimensional hexagonal lattice with spacings between 0.5𝜇m and 10𝜇m. Additional pretreatment of the surface by photolithography can be used to alter the symmetry of the lattice, to create extremely high correlation lengths, and to design the arrangement of defects with very high precision. The depth-to-width ratio of the pores can be as large as 500. In addition, it is possible to modulate the width of the pores along the pore axes by controlling the current through the sample during the growth process, thereby creating even three-dimensional structures [1315].

In addition to utilizing the benefits of the unusual dispersion relation, it is highly desirable to change the dielectric properties and to control them by external parameters, thereby achieving even a tunable dispersion relation [1618]. This might be necessary in order to compensate for fabrication tolerances and to fine-tune the properties of the produced photonic crystal, or be motivated by targeting active switching devices. An obvious way to achieve tunable properties is altering the dielectric susceptibility of the silicon. As usual, the dielectric susceptibility can be expanded in a power series as follows: 𝜒=𝜒(1)+𝜒(2)𝐸+𝜒(3)𝐸𝐸+.(1) Thus, linear [𝜒(1)] or nonlinear effects [𝜒(2),𝜒(3),] may be considered. In addition, the macroporous silicon structures can be filled with a dielectric compound. If the silicon (𝜀=𝜀1) is infiltrated with a different compound (𝜀=𝜀2), the most fundamental change is a shift of the average dielectric constant 𝜀av, which is approximately given by the Maxwell-Garnett relation [19] 𝜀av𝜀1𝜀av+2𝜀11=𝑓2𝜀2𝜀1𝜀22𝜀11,(2) where f2 is the volume fraction of component 2. If the denominators in (2) are not too different, the effective refractive index of the composed material is approximately given by 𝑛e=Σ𝑓𝑖𝑛2𝑖1/2,(3) where fi is the volume fraction of component i of the heterogeneous system. Thus, changing the refractive index of one of the two components has an effect on the properties of the entire structure. However, the influence on the dispersion relation and on the linear and nonlinear photonic properties is much more subtle than a simple change of the average refractive index. Due to their sensitivity to external parameters, liquid crystals proved to be very efficient as a dielectric liquid yielding addressable photonic properties.

In this paper, we would like to review the methods of tuning by means of liquid crystal infiltration (Section 2) as well as all-optical effects that are due to the properties of the silicon (Section 3). The latter effects can be based on charge carrier injection due to one- or two-photon absorption or on Kerr-like nonlinearities (the latter being described by the real part of 𝜒(3)).

2. Tuning by Means of Liquid Crystal Infiltration

Liquid crystals [2024] exhibit very sensitive electro- and thermo-optical properties. Filled into the pores of a photonic crystal, they provide the opportunity of adjusting the effective refractive index by external parameters. This method was proposed by Busch and John [2528] and experimentally demonstrated for colloidal crystals [2934] before being applied to macroporous silicon [3546] and other PBG semiconductor structures, including tunable light sources [4750]. The sensitivity of liquid crystals is due to a preferred uniform alignment of their typically rod-like molecules, which in turn leads to birefringence. The local structure of the least complicated liquid crystalline mesophase, the nematic phase (Figure 1), can be described by the director n (a pseudovector) and a scalar order parameter S, which indicate the local molecular alignment (i.e., the optical axis) and the degree of orientational order, respectively. External fields can rotate the optical axis, while an increasing temperature results in a decreasing order parameter and thus a decreasing birefringence Δ𝑛. Typically, the difference Δ𝑛=𝑛𝑒𝑛𝑜 between the extraordinary refractive index ne and the ordinary refractive index 𝑛𝑜 is of the order Δ𝑛0.2 if the temperature is several Ks below the nematic-isotropic phase transition (clearing temperature). No birefringence appears above the clearing temperature. Thus, relatively large thermally induced changes of the effective refractive index are observed at this phase transition.

780784.fig.001
Figure 1: The chemical structure of 4-cyano-4-pentyl-biphenyl, arrangement of rod-like molecules in the (nonchiral) nematic phase (N), and temperature dependence of the ordinary refractive index no (effective for light that is linearly polarized with its electric field perpendicular to the director n), the extraordinary refractive index ne (effective for light that is linearly polarized with its electric field parallel to the director n), and the isotropic refractive index 𝑛iso.

Leonard et al. [35] were the first to infiltrate a two-dimensional structure made of macroporous silicon with a liquid crystal and observed changes of the photonic band edge for light propagating in the plane of the silicon wafer. This effect might be very useful for integrated optical waveguides in silicon. Subsequent experiments were focused on three-dimensional (3D) structures consisting of macroporous silicon that are filled with a liquid crystal. The latter structures show also a stop band for light propagating perpendicular to the plane of the wafer. Two-dimensional hexagonal or rectangular arrays of pores with an extremely high aspect ratio (diameter ≤1𝜇m, depth ≥100𝜇m) were fabricated by a light-assisted electrochemical etching process using HF [12, 13], and a periodic variation of pore diameter was induced by variation of the electric current during the etching procedure, thereby yielding in a three-dimensional photonic crystal (PhC) [14, 15]. The macroporous structure was evacuated and filled with a liquid crystal. The photonic properties for light propagation along the pore axis were studied by Fourier transform infrared (FTIR) spectroscopy [3638]. Deuterium-nuclear magnetic resonance (2H-NMR) [36, 37] and fluorescence confocal polarizing microscopy (FCPM) [3942] were used in order to analyze the director field of the liquid crystal inside the pores.

For example, Figure 2 shows the infrared transmission of samples that show a two-dimensional hexagonal array of pores with a lattice constant 𝑎=1.5𝜇m. Along the pore axis, the diameter of each pore varies periodically between 𝐷min=(0.76±0.10)𝜇m and 𝐷max=(1.26±0.10)𝜇m with a lattice constant 𝑏=2.6𝜇m. The pores were filled with the nematic liquid crystal 4-cyano-4-pentyl-biphenyl (5CB, Figure 1) which shows a clearing temperature of 𝑇NI=34C. For light propagation along the pore axes, the FTIR transmission spectrum of the silicon-air structure shows a stop band centered at 𝜆=(10.5±0.5)𝜇m. Filling the pores with 5CB decreases the dielectric contrast to silicon and results in a shift of the stop band to 𝜆12𝜇m. The band edge was found to be sensitive to the state of polarization of the incident light. For linearly polarized light, rotation of the sample with respect to the plane of polarization was found to cause a shift of the liquid crystal band edge by Δ𝜆152 nm (1.61 meV). This effect can quantitatively be explained by the square shape of the pore cross-section, which brakes the threefold symmetry of the hexagonal lattice. Due to the presence of the liquid crystal, the band edge at lower wavelengths (“liquid crystal band” edge) can be tuned by more than 140 nm (1.23 meV) by heating the liquid crystal from 24°C (nematic phase) to 40°C (isotropic liquid phase).

fig2
Figure 2: (a) SEM top-view and side-view of a photonic crystal made of macroporous silicon containing a two-dimensional hexagonal array of pores with periodically modulated diameter. (b) Transmission spectra of the same photonic crystal for light propagation along the pore axes if the sample is filled with (—) air, (- - -) 5CB in its nematic phase, and () 5CB in its isotropic liquid state, respectively. (c) Comparison of the calculated dispersion relation using the plane wave approximation and the experimental spectra. For details, see [36].

The shift of the photonic band edge towards larger wavelengths indicates an increase of the effective refractive index with increasing temperature. This effect can be explained by a predominantly parallel alignment of the optical axis (director) of the nematic liquid crystal along the pore axis. For a uniform parallel alignment, the effective refractive index of the nematic component corresponds to the ordinary refractive index no of 5CB. Increasing the temperature above the clearing point causes an increase to the isotropic value 𝑛iso(1/3𝑛2𝑒+2/3𝑛2𝑜)1/2, where ne is the extraordinary refractive index of the liquid crystal (𝑛𝑒>𝑛𝑜). For a very crude approximation, the average dielectric constant 𝜀av of the heterogeneous structure can be calculated from the respective dielectric constants [𝜀LC(24C)=𝑛2𝑜 and 𝜀LC(40C)=𝑛2iso] using the Maxwell-Garnett relation (2) [19]. The relative shift of the stop band edge towards larger wavelengths corresponds approximately to the relative increase of the average refractive index by 0.65%. However, more precise analysis of the data shown in Figure 2 indicates that the shift of the two band edges is not the same (the shift of the left band edge is larger). The reason for the difference is that the overlap of the electric field with the pores is different at the two band edges (it is larger at the short wavelength edge), therefore changes of the refractive index in the pores translate into different shifts of the band edge. This is correctly predicted by the calculated dispersion relation shown in the right part of Figure 2, but cannot be explained by the Maxwell-Garnett relation [36].

Planar microcavities inside a 3D photonic crystal appear when the pore diameter is periodically modulated along the pore axis, stays constant within a defect layer, and is continued to vary periodically. Figure 3 shows a structure where a defect layer is embedded between five periodic modulations of the pore diameter. The pores are arranged in a 2D square lattice with a lattice constant of 𝑎=2𝜇m. The pore width varies along the pore axis between 𝐷min=0.92𝜇m and 𝐷max=1.55𝜇m. The length of a modulation is 𝑏=2.58𝜇m. The defect has a length of 𝑙=2.65𝜇m and pore diameters 𝐷def=0.82𝜇m. Within the defect layer, the filling fraction of the liquid crystal is 𝜉def=0.17. For infrared radiation propagating along the pore axes, a fundamental stop gap at around 13𝜇m and a second stop gap at around 7𝜇m are expected from calculations using the plane wave approximation [51]. The experiment shows a transmission peak at 𝜆=7.184𝜇m in the center of the second stop band, which can be attributed to a localized defect mode. Filling the structure with the liquid crystal 4-cyano-4-pentyl-biphenyl (5CB, Merck) at 24°C causes a spectral red-shift of the stop band. Together with the stop band, the wavelength of the defect state is shifted by 191 nm to 𝜆=7.375𝜇m. An additional shift of Δ𝜆=20 nm to 𝜆=7.395𝜇m is observed when the liquid crystal is heated from 24°C (nematic phase) to 40°C (isotropic liquid phase). Again, the shift towards larger wavelengths indicates an increase of the effective refractive index 𝑛e of the liquid crystal with increasing temperature and can be attributed to the transition from an initially parallel aligned nematic phase (𝑛LC,e=𝑛𝑜) to the isotropic state (𝑛LC,e=𝑛iso). During continuous variation of the temperature, a distinct step by 20 nm is observed at the phase transition from the nematic to the isotropic phase. The quality factor Q of the investigated structure, 𝑄=𝜆/𝛿𝜆=52, is rather small and thus the shift by 20 nm appears to be small compared to the spectral width of the defect mode. However, the same order of magnitude of the temperature-induced wavelength shift can be expected for structures with a much higher quality factor and might be quite large compared to the band width of the defect mode.

fig3
Figure 3: (a) SEM image (bird’s eye view) of a silicon structure with modulated pores including a planar defect layer without modulation. (b) Transmission spectra of this structure if the sample is filled with (—) air, (- - -) 5CB in its nematic phase, and () 5CB in its isotropic liquid state, respectively. For details, see [38].

Comparison of experimental 2H-NMR results and calculated spectra (Figure 4) confirms a parallel (P) alignment of the director along the pore axis for substrates that were treated like the samples described above. However, also an anchoring of the director perpendicular to the silicon surfaces (“homeotropic” anchoring) can be achieved if the silicon wafer is cleaned with an ultrasonic bath and a plasma-cleaner and subsequently pretreated with N,N-dimethyl-n-octadecyl-3-aminopropyl-trimethoxysilyl chloride (DMOAP). NMR data indicate the appearance of an escaped radial (ER) director field in the latter case.

fig4
Figure 4: (a) Schematic relation between parallel (P), planar polar (PP), or escaped radial (ER) director fields and the respective 2H-NMR-lineshapes. 2H-NMR spectrum of α-deuterated 5CB in cylindrical pores with perpendicular anchoring [thin line: spectrum expected for an escaped radial (ER) structure with weak anchoring]. For details, see [36, 37].

For the first time, optical microscopic studies of the director field in pores with a spatially periodic diameter variation could be achieved by means of a nematic liquid crystal polymer that shows a glass-like nematic state at room temperature [39, 40]. For fluorescence polarizing microscopy, the polymer was doped with N,N-bis(2,5-di-tert-butylphenyl)-3,4,9,10-perylene-carboximide (BTBP). After filling the photonic crystal in vacuum, the sample was annealed in the nematic phase at 120°C for 24 hours and subsequently cooled to room temperature, thereby freezing the director in the glassy state. The silicon wafer was dissolved in concentrated aqueous KOH solution and the remaining isolated polymer rods were washed and investigated by fluorescence confocal polarizing microscopy (FCPM). The transition dipole moment of the dichroic dye BTBP is oriented along the local director of the liquid crystal host. The incident laser beam (488 nm, Ar+) and the emitted light pass a polarizer, which implies that the intensity of the detected light scales as 𝐈cos4𝛼 for an angle α between the local director and the electric field vector of the polarized light. Thus, the local fluorescence intensity indicates the local orientation of the liquid crystal director with very high sensitivity. For a template with homeotropic anchoring and a sine-like variation of the pore diameter between 2.2𝜇m and 3.3𝜇m at a modulation period of 11𝜇m, the FCPM images of the nematic glass needles (Figure 5) indicate an escaped radial director field. Comparison with numerical calculations based on a tensor algorithm [52, 53] reveals some characteristic features that differ from nonmodulated pores. In the cylindrical cavities studied previously, point-like hedgehog and hyperbolic defects appear at random positions and tend to disappear after annealing, due to the attractive forces between defects of opposite topological charges. In contrast, the modulated pores stabilize a periodic array of disclinations. Moreover, disclination loops appear instead of point-like disclinations.

fig5
Figure 5: (a) Nematic escaped radial (ER) director field, calculated using the algorithm described in [52, 53]. (b) Theoretical and experimental fluorescent confocal polarizing microscopy (FCPM) images for polarized light with its electric field parallel to the tube axis. (c) Theoretical and experimental FCPM images for polarized light with its electric field perpendicular to the tube axis. For details, see [39].

3. All-Optical Tuning

Altering the optical properties by optical irradiation has been the subject of intense research efforts related to the potential development of active photonic crystal components. Here, the impact of an optical pump beam on a photonic crystal consisting of a two-dimensional array of macropores in silicon [5459] is reviewed. Figure 7(a) shows a sketch of such a crystal with photo-electrochemically etched straight pores with an aspect ratio of 100 [8, 12, 14]. If the photon energy (𝜔𝑝) of the pump beam is larger than the electronic band gap of silicon, absorption causes a free charge carrier generation in the semiconductor which in turn changes the dielectric constant due to the Drude relation [54] (Section 3.1). These free carriers, generated by photon absorption, can be injected either by a single photon absorption or, in the presence of very high pump intensities, by two-photon absorption. In contrast to the changes achieved by liquid crystal reorientation, this direct optical addressing of the silicon is very fast. Whereas the former occurs on time scales ranging from milliseconds to seconds, the optical tuning takes place in the subpicosecond regime.

Because of the centrosymmetric space group of silicon, bulk-contributions corresponding to the second-order nonlinear susceptibility 𝜒(2) are ruled out, but surface effects and third-order, that is, Kerr-like nonlinearities, corresponding to 𝜒(3) can be found. In addition, two-photon absorption of photons with low energy can cause a charge carrier injection like the one-photon absorption of photons with high energy. Silicon has an indirect band gap of 1.1 eV (𝜆=1.1𝜇m) at 295 K and a direct band gap of 3.5 eV (𝜆=355 nm). Thus, a relatively weak phonon-assisted linear absorption or a two-photon one occurs across the visible and near infrared. If the pump intensity is sufficiently high, nonlinear optical effects can cause changes of the dielectric constant even for photon energies that are smaller than the electronic band gap of silicon [55].

3.1. Charge Carrier Injection by One-Photon Absorption

To explore the influence of an electron-hole-plasma, a high density of carriers in a native macroporous silicon sample had been optically injected using optical techniques and by this monitoring the shift of a stop-gap edge [54]. As the (indirect band gap) absorption edge of silicon is at 1.1𝜇m for room temperature, electron-hole pairs can be efficiently produced at shorter wavelengths. The presence of the high density (N) carriers is expected to alter the real part of the dielectric constant 𝜀 at probe frequency ω through the expression 𝜀=𝜀Si𝑁𝑒2𝜀0𝜔2𝑚,(4) where 𝜀Si (11.9) is the quiescent dielectric constant of silicon, 𝜀0 is the permittivity of free space, and 𝑚 is the optical effective mass of the electrons and holes. The carrier relaxation rate (~6 THz) has been neglected in comparison with the probing frequencies of interest. In the infrared region of the spectrum for wavelengths between 1𝜇m and 5𝜇m, this is a reasonable approximation. Similarly, the influence of the imaginary part of the dielectric constant that would contribute to loss was neglected. From the frequency dependence in (1) one observes that for probing wavelengths in the near infrared region, changes in the dielectric constant of the order of 10% can be obtained for our peak carrier densities. These substantial changes are expected to modify the location of the band gaps. For the experiments, the lowest stop band was investigated, which occurs in the photonic crystal described above between 1.9𝜇m and 2.3𝜇m for E-polarized light propagating along the Γ-M direction. In particular, the shift of the shorter wavelength band edge was measured. Similar shifts are expected for the longer wavelength band edge. The experimental results were obtained with a parametric generator pumped by a 250 kHz repetition rate Ti-sapphire oscillator/regenerative amplifier which produces 130 fs pulses at 800 nm at an average power of 1.1 W. The signal pulse from the parametric generator is tunable from 1.2𝜇m to 1.6𝜇m and the idler pulse is tunable from 2.1𝜇m to 1.6𝜇m. Reflection measurements were made using 150 fs pulses with center wavelength of 1.9𝜇m and of sufficient bandwidth to probe the dynamical behavior of this edge. The pulses were focused onto the silicon photonic crystal with fluence per pulse up to ~2 m J cm−2. Simple estimates based on anticipated absorption properties of the photonic crystal at this wavelength indicate that the peak density of electron hole pairs is >1018 cm−3. The focal spot diameter of the probe beam was 30𝜇m, reasonably small compared to the ~100𝜇m spot diameter of the H-polarized pump beam. Figure 6 shows how the probe reflection characteristics change with the fluence of the pump beam.

780784.fig.006
Figure 6: Variation of the air band edge of the silicon photon crystal following optical pumping by a 300 fs, 800 nm pulse of variable fluence.
fig7
Figure 7: (a) The 2D Si photonic crystal showing pump-probe beam geometry. The probe beam is incident along the Γ-M direction. (b) Band structure of the photonic crystal for the Γ-M direction. The solid and dashed lines correspond to E and H polarized lights, respectively.

The major effect of the optical pumping is to shift the band edge to shorter wavelengths as expected since the Drude contribution decreases the dielectric constant of the silicon. Detailed calculations based on absorption characteristics of the photonic crystal at the pump wavelength and the variation of the photonic crystal dispersion curves with injected carrier density are in agreement with the maximum shift of about 30 nm (at the 3 dB point) that is observed here. Indeed, the shift of the edge scales linearly with the pump fluence, or injected carrier density, as expected theoretically. It should be noted however that the shift of the edge is not rigid. The shift is less for higher values of the reflectivity. This is presumably related to the fact that the 800 nm pump radiation is inhomogeneously absorbed, with an absorption depth of a few microns. In the spectral range near the peak value of the reflectivity associated with a stop gap, the reflectivity originates from lattice planes over a considerable depth within the crystal. In contrast, Fresnel reflectivity of the surface region dictates the reflection characteristics in the spectral range with higher transmission. To overcome this problem, other pumping schemes have to be used.

It was also possible to time resolve the reflectivity behavior by monitoring the probe reflectivity of the band edge as a function of delay between the probe and pump beams. Not surprisingly, the probe reflectivity change is virtually complete within the duration of the pump beam as charge carriers accumulate in the silicon. However, the recovery of the induced change occurs on a much longer time scale (at least nanoseconds) in our photonic crystal reflecting the electron-hole carrier recombination characteristics.

3.2. Tuning by Kerr-Like Optical Nonlinearities

The Kerr effect was used to tune the short wavelength edge of a photonic band gap. In these experiments, a 2D photonic crystal was used to demonstrate the all optical tuning. Both the short wavelength edge (1.3𝜇m) and the long wavelength edge (1.6𝜇m) could be redshifted by the Kerr effect (𝜒(3)). But for high pump intensities, the two-photon absorption was significantly generating free carriers, leading to a blueshift of the photonic band edge via the Drude contribution to 𝜒(1).

The 2D silicon PhC sample has a triangular lattice arrangement of 560 nm diameter, 96𝜇m deep air holes with a pitch, a, equal to 700 nm. Figure 7(a) shows a real space view of the sample while Figure 7(b) illustrates the photonic band structure for the Γ-M direction, which is normal to a face of the PhC. Of particular interest is the third stop gap for E-polarized (E-field parallel to the pore axis) light. Lying between 1.3𝜇m and 1.6𝜇m, this gap falls between two dielectric bands that are sensitive to changes in the silicon refractive index. The purpose was to optically induce changes to the two edges with idler pulses from the parametric generator and probe these changes via time-resolved reflectivity of the signal pulses. Note that, because of the link between the signal and idler wavelengths, different pump wavelengths (2.0𝜇m for a 1.3𝜇m probe; 1.76𝜇m for a 1.6𝜇m probe) must be used when the probe wavelength is changed. However, as will be shown in what follows, small changes in the pump wavelength can lead to significant changes in the induced optical processes.

Figure 8 shows the time-dependent change in reflectivity at 1.3𝜇m for a 2.0𝜇m pump pulse and the cross-correlation trace of both pulses. The pump and probe intensities are 17.6 and 0.5 GW/cm2, respectively. The decrease in reflectivity is consistent with a redshift of the band edge due to a positive nondegenerate Kerr index. The FWHM of the reflectivity trace is 365 ± 10 fs which is 1.83 times larger than the pump-probe cross-correlation width as measured by sum frequency generation in a beta-barium borate (BBO) crystal. This difference can be explained in terms of pump and probe beam transit time effects in the PhC as discussed above. Indeed, from the pump group velocity and probe spot size, one can deduce that the reflected probe pulse is delayed by 110 fs within the PhC sample. After these effects are taken into account, the intrinsic interaction times are essentially pulse width limited, consistent with the Kerr effect.

780784.fig.008
Figure 8: Temporal response of reflectivity change at the 1.3𝜇m band edge when the PhC is pumped with a 2.0𝜇m pulse at 17.6 GW/cm2. Also shown is the cross-correlation trace of the pump and probe pulses.

One can estimate a value for the nondegenerate Kerr coefficient n2 in the silicon PhC from the relation [55] Δ𝑅=𝑑𝑅𝑑𝜆𝑑𝜆𝑛𝑑𝑛21𝑅𝑢𝑓𝐼0,(5) where I0 is the incident intensity, f is the filling fraction, and Ru is the reflectivity of the sample. The experimental values of the steepness of the band edge reflectivity, 𝑑𝑅/𝑑𝜆=0.04 nm−1, and the differential change in band edge wavelength with refractive index, 𝑑𝜆/𝑑𝑛=174 nm, are relatively large. Thus, induced reflectivity changes in the vicinity of the 1.3𝜇m band edge are found to be 70 times more sensitive than that in bulk materials for the same refractive index change, a degree of leverage also noted by others [54, 60]. Indeed, when the PhC is replaced by bulk crystalline silicon, no change in reflectivity is observed for the range of pump intensity.

The inset to Figure 9 shows there is good correlation between the change in probe reflectivity and the steepness of the band edge reflectivity (measured separately) at different wavelengths and for a range of pump intensities. Figure 9 shows the change in reflectivity with pump intensity at zero time delay. The linear dependence is consistent with the Kerr effect and the nondegenerate Kerr index is estimated to be 5.2×1015 cm2/W. This is within an order of magnitude of the degenerate Kerr index reported [61, 62] at 1.27𝜇m and 1.54𝜇m and represents reasonable agreement considering uncertainty in the lateral position (x) of the pump pulse and its intensity at the probe location. It should also be noted that linear scattering losses as the pump pulse propagates through the PhC along the pore axis have not been taken into account.

780784.fig.009
Figure 9: Dependence of probe reflectivity change at 1.316𝜇m on pump intensity at zero delay. The inset shows the spectral characteristic of reflectivity change at the 1.3𝜇m band edge at zero delay for different pump intensities at 2.0𝜇m. Also shown in the inset is the 𝑑𝑅/𝑑𝜆 curve, which measures the steepness of the band edge reflectivity.
3.3. Tuning by Kerr-Like Nonlinearities and Two-Photon Absorption

In general, overall pulse-width limited response can only be achieved using nonresonant, nonlinear induced changes to material optical properties such as the optical Kerr effect (a third-order nonlinearity). In this case, the change in refractive index for a probe beam is given by Δ𝑛=𝑛2𝐼,(6) where I is the intensity of the pump beam and n2 is the Kerr coefficient associated with the pump and probe frequencies. If the probe light intensity is limited to values like in the experiment of Leonard et al. [54], the imaginary terms in the dielectric function arising from free-carrier absorption and intervalence-band absorption are very small.

Results from experiments used to probe the 1.6𝜇m band edge when the sample is pumped with 1.76𝜇m pulses are illustrated in Figure 10, which shows the temporal response of the change in probe reflectivity at different pump intensities for a probe intensity of 0.13 GW/cm2. There is an initial increase and decrease in probe reflectivity on a subpicosecond time scale followed by a response that decays on a time scale of 900 picoseconds and partially masks the Kerr effect near zero delay. At this band edge, the subpicosecond behavior is consistent with a Kerr effect similar to the previous experiments. The long time response could possibly be due to thermal or Drude contributions to the dielectric constant due to the generation of free carriers. Using a peak pump intensity of 120 GW/cm2 and a 0.8 cm/GW two-photon absorption coefficient [61, 62] for 1.55𝜇m as an upper limit, one can estimate the surface peak carrier density to be <1019 cm−3 and the maximum change in temperature to be <0.15 K. From the thermo-optic coefficient 𝜕𝑛/𝜕𝑇1×104 K−1 at the probe wavelength [63], the change in silicon refractive index is on the order of 10−5 and the (positive) induced change in reflectivity is expected to be about the same. From free carrier (Drude) contributions to the refractive index at the probe wavelength, changes to the imaginary part of the dielectric constant are about 2 orders of magnitude smaller than that of the real part [64], which is about −10−3. Hence, free carrier absorption of the probe pulse as well as thermally induced changes can be neglected in what follows and the change in reflectivity is ascribed to changes in the real part of the dielectric constant due to Drude effects.

780784.fig.010
Figure 10: Temporal response of the reflectivity change at the 1.6𝜇m band edge for different pump intensities at 1.76𝜇m. The inset shows the dependence of the carrier-induced reflectivity change on pump intensity for low pump powers.

At low pump powers, the change in probe reflectivity scales quadratically with pump intensity. This can be explained by free carrier generation due to two-photon absorption, with the charge carrier density N being given by 𝑁(𝑧)=𝜋𝛽𝜏𝑝4𝜔𝑝𝐼ln22(𝑧),(7) where β is the two-photon absorption coefficient, 𝜏𝑝 is the temporal FWHM pulse width of the pump pulses, and 𝜔𝑝 is the pump frequency. However, at higher pump intensities, there is an apparent deviation from this quadratic dependence (see inset in Figure 10) due to pump saturation effects, since the pump intensity I varies along the z-direction as 𝐼(𝑧)=1𝑅𝑢𝐼0𝑓+𝛽𝑧1𝑅𝑢𝐼0,(8) according to attenuation by two-photon absorption. With increasing intensity in a two-photon absorption process, an increasing fraction of the carriers are created closer to the surface where the pump pulse enters and the probe region develops a reduced and increasingly nonuniform carrier density. It can be estimated that at a depth of 60𝜇m, the expected saturation pump intensity is about an order of magnitude larger than the maximum pump intensity used in this setup. The carrier lifetime of 900 picoseconds is most likely associated with surface recombination within the PhC sample with its large internal surface area.

The reflectivity change due to the Drude effect is given by Δ𝑅=𝑑𝑅𝑑𝜆𝑑𝜆𝑒𝑑𝑛22𝑛0𝜔2𝑟𝑚𝜀0𝜋𝛽𝜏𝑝4𝜔𝑝𝐼ln22(𝑧),(9) where 𝜔𝑟 is the probe frequency, 𝑚 is the effective optical mass of the electrons and holes (=0.16𝑚0), and 𝜀0 is the permittivity of free space. Thus, from the low intensity behavior in the inset to Figure 10, the two-photon absorption coefficient, β, can be estimated to be 0.02 cm/GW, which is within an order of magnitude but smaller than that reported [61, 65] for wavelengths near 1.55𝜇m. For the 2𝜇m pump wavelength, the upper limit for β is estimated to be 2×10−3 cm/GW from the signal to noise and the fact that no measurable long-lived response at the highest pump intensity used is observed. This value is an order of magnitude smaller than what is determined at 1.76𝜇m and it is not a surprise since β is expected to decrease rapidly with increasing wavelength as the indirect gap edge is approached.

4. Summary

In conclusion, either the infiltration of macroporous silicon with liquid crystals and subsequent control of the thermodynamic variables or the use of light absorption of Kerr-like optical nonlinearities can be used to achieve tunable properties in photonic crystals made of macroporous silicon. For both methods, the relative frequency shift of photonic bands, band edges, or resonance frequencies of microcavities is roughly of the order of 1% of the absolute frequency (Table 1). The effect is limited, but can nevertheless be much larger than the linewidth of modes to be tuned, since microresonators with very large Q factors can be fabricated. The methods summarized in Sections 2 and 3 may find different applications. The use of liquid crystals has the advantage that the control parameters temperature and electric fields are easily available. However, the greatest disadvantage is probably the limited speed of director reorientation, which corresponds to time constants in the millisecond range. In contrast, absorption and nonlinear effects lead to very fast changes of the photonic properties (with time constants below 1 picosecond) and can be used for all-optical switching. However, very large intensities are required for the nonlinear optical effects.

tab1
Table 1: Spectral shifts and switching speeds of the different effects described in Sections 2 and 3.

Besides further technical developments that make use of the effects studied, so far, a couple of novel, fundamentally interesting systems deserve to be explored in more detail:

(1) Chiral liquid crystals: cholesteric phases and blue phases [2024] show a helical superstructure of the local alignment, thereby leading to a spatially periodic director field 𝐧(𝐫). This intrinsic periodicity can be combined with two-dimensional arrays of pores, thereby leading to novel three-dimensional heterogeneous structures [41]. As an example, Figure 11 shows the FCPM image of a sample which shows an inherent periodicity within the pores. Such structures may show enhanced nonlinear optical effects or may be used for switching between a three-dimensional and a two-dimensional periodicity of the optical density. Additional work on these systems is in progress.

780784.fig.011
Figure 11: FCPM image of a cholesteric cylinder with perpendicular anchoring and a helix pitch smaller than the cylinder diameter. The regular fringes are perpendicular to the pitch axis and their distance corresponds to one half of the pitch. For further details, see [41].

(2) Liquid crystals can exhibit both second- and third-order optical nonlinearity. Thus, infiltration of photonic crystals with liquid crystals that exhibit large 𝜒(2)- or 𝜒(3)-values may be used for frequency conversion or all-optical switching, respectively. A considerable enhancement of second harmonic generation (SHG) intensity is known to appear in spatially periodic structures, where both the fundamental frequency and the second harmonic are close to photonic stop bands [66, 67]. In addition, suitable liquid crystals are known for their giant optical nonlinearity (GON) [68], that is, a huge Kerr effect which is due to collective reorientation of the liquid crystal molecules induced by the optical electric field strength.

Fundamental studies of these effects in the environment of a silicon photonic crystal appear to be challenging. In conclusion, the development of tunable photonic crystals based on silicon is still in progress.

Acknowledgments

This work was supported by the company E. Merck (Darmstadt) with liquid crystals. Also, the funding by the German Research Foundation (KI 411/4 and SPP 1113) and the European Science Foundation (EUROCORES/05-SONS-FP-014) is gratefully acknowledged.

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