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

Authors: Heinz-S. Kitzerow; Heinrich Matthias; Stefan L. Schweizer; Henry M. van Driel; Ralf B. Wehrspohn
Journal: Advances in Optical Technologies
 (2008)

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## 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.


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## Body

## 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 [3–11], 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 [12–15]. 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 [13–15].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 [16–18]. 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: (1)χ=χ(1)+χ(2)E+χ(3)EE+⋯. 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] (2)(εav−ε1)(εav+2ε1)−1=f2(ε2−ε1)(ε2−2ε1)−1, 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 (3)neff=(Σfini2)1/2, 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 (Section2) 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 [20–24] 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 [25–28] and experimentally demonstrated for
colloidal crystals [29–34] before being applied to macroporous silicon [35–46]
and other PBG semiconductor structures, including tunable light sources
[47–50]. 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=ne−no between the extraordinary
refractive index ne and the ordinary refractive index no is of the order Δn≈0.2 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.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 [36–38].
Deuterium-nuclear magnetic resonance (2H-NMR) [36, 37] and
fluorescence confocal polarizing microscopy (FCPM) [39–42] were used in order to
analyze the director field of the liquid crystal inside the pores.For example,
Figure2 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=34°C. 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 λ≈12μm. 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 Δλ≈152 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).(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].
(a)(b)(c)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 indexno of 5CB. Increasing the
temperature above the clearing point causes an increase to the isotropic value
niso≈(1/3ne2+2/3no2)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(24°C)=no2 and εLC(40°C)=niso2]
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. Figure3 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 neff  of the
liquid crystal with increasing temperature and can be attributed to the
transition from an initially parallel aligned nematic phase (nLC,eff=no) to the isotropic state (nLC,eff=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.(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].
(a)(b)Comparison of
experimental2H-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.(a) Schematic relation
between parallel (P), planar polar (PP), or escaped radial (ER) director fields
and the respective2H-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].
(a)(b)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 I∝cos4α 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.(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].
(a)(b)(c)


## 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 [54–59] 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 (ℏωp) 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 susceptibilityχ(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 (4)ε=εSi−Ne2ε0ω2m*, 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.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.(a) The 2D Si photonic
crystal showing pump-probe beam geometry. The probe beam is incident along theΓ-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.
(a)(b)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.Figure8 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.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 coefficientn2 in
the silicon PhC from the relation [55] (5)ΔR=dRdλdλdnn21−RufI0, 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 Figure9 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 5.2×10−15 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.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(6)Δn=n2I, 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/∂T≈1×10−4 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.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 densityN being given by (7)N(z)=πβτp4ℏωpln2I2(z), 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 (8)I(z)=(1−Ru)I0f+βz(1−Ru)I0, 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(9)ΔR=dRdλdλdne22n0ωr2m*ε0πβτp4ℏωpln2I2(z), 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.


## 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 (4)ε=εSi−Ne2ε0ω2m*, 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.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.(a) The 2D Si photonic
crystal showing pump-probe beam geometry. The probe beam is incident along theΓ-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.
(a)(b)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.Figure8 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.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 coefficientn2 in
the silicon PhC from the relation [55] (5)ΔR=dRdλdλdnn21−RufI0, 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 Figure9 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 5.2×10−15 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.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(6)Δn=n2I, 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/∂T≈1×10−4 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.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 densityN being given by (7)N(z)=πβτp4ℏωpln2I2(z), 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 (8)I(z)=(1−Ru)I0f+βz(1−Ru)I0, 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(9)ΔR=dRdλdλdne22n0ωr2m*ε0πβτp4ℏωpln2I2(z), 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 (Table1). 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.Table 1
Spectral shifts and switching speeds of the different effects described in Sections2 and 3.
EffectInitial frequencyν (Hz)Frequency shiftδν or refractive index shift δn/control parameterRelative frequency shift|δν|/νTime constant (s)Temperature-induced shift
of the “silicon” band edge in a liquid crystal-infiltrated structure [36].2.32⋅1013δν/δT=−1.68⋅1011 Hz/16°C0.72%≈10−3Temperature-induced shift
of the “liquid crystal” band edge in a liquid crystal-infiltrated structure
[36].2.81⋅1013δν/δT=−3.74⋅1011 Hz/16°C1.33%Temperature-induced shift
of the microcavity resonance in a liquid crystal-infiltrated structure [38].4.05⋅1013δν/δT=−1.56⋅1011 Hz/1°C0.39%Temperature-induced shift
of the PL emission frequency in an Er-doped liquid crystal-infiltrated
structure [46].1.93⋅1014δν/δT=−8.70⋅1011 Hz/23°C0.45%Free charge carrier
injection induced by one-photon absorption [54].1.58⋅1014δν/δT=−2.53⋅1012 Hz/(2.1±0.4)  m J/cm21.60%≈10−6Tuning by Kerr-like
nonlinearities [55].δn≈10−3Instantaneous (10−13)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 [20–24]
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.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 largeχ(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.



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*Source: 780784-2008-06-22.xml*