Infiltrating the air pores of a two-dimensional macroporous silicon photonic crystal with the nematic liquid-crystal mixture E7 converts the photonic band gap into a thermally tunable one (Leonard et al., 2000), but the observed temperature-dependent band-edge shifts could only be reproduced qualitatively, because the surface anchoring of E7 on the silicon pore walls was never measured and the full quantitative modeling chain was never completed. The open problem is: experimentally determine the anchoring type and quantitative anchoring strength of E7 on macroporous silicon (the actual director-field realization, axial versus escaped-radial, including its defect structure), and incorporate the measured anchoring into an anisotropic band-structure and mode calculation with the temperature-dependent refractive-index dispersion of E7 that quantitatively reproduces the measured band-edge shifts of the macroporous-silicon photonic crystal. The third issue raised by the source paper, fast electrically driven tuning blocked by free-carrier screening of the doped silicon, has since been resolved by later work using electrode configurations that avoid field screening.

Background

A photonic crystal is a periodic dielectric structure whose photonic band gap, a frequency interval in which propagation of light is forbidden, can be engineered by the lattice geometry and the refractive indices of its constituents. Macroporous silicon, obtained by electrochemical etching of n-type silicon under backside illumination, yields high-aspect-ratio cylindrical pores arranged in a triangular lattice of micrometer pitch, and supports robust two-dimensional photonic band gaps in the mid-infrared; two independent polarization channels are distinguished, denoted H and E in the source literature (transverse-electric and transverse-magnetic, respectively, with respect to the pore axis). Because silicon's refractive index is fixed, tuning the gap requires changing the index of the material filling the pores. A nematic liquid crystal such as E7, a eutectic mixture of cyanobiphenyls, is a uniaxial fluid whose molecules align on average along a unit vector, the director, giving ordinary and extraordinary refractive indices non_o and nen_e with a birefringence that decreases with temperature and vanishes at the nematic-isotropic transition (about 59 degrees C for E7); the resulting average index seen by light can then be tuned thermally. Busch and John (1999) predicted theoretically that such infiltration turns a photonic crystal into a tunable electromagnetic vacuum, that is, a photonic band-gap material whose gap and effective modes shift when the liquid crystal is reoriented, and Leonard et al. (2000) demonstrated it experimentally: after infiltration with E7, the H-polarized gap of a macroporous silicon crystal shifted from 3.3-5.7 micrometers to 4.4-6.0 micrometers, the E-polarized gap collapsed, and heating through the nematic-isotropic transition moved the short-wavelength H-gap edge by up to about 70 nanometers. In a cylindrical pore, the director configuration is set by the liquid-crystal anchoring at the silicon wall, that is, by the preferred orientation (planar, parallel to the wall; or homeotropic, perpendicular) imposed on the director at the surface and by the anchoring strength, commonly quantified by a coefficient WθW_\theta that measures the energy penalty per unit area of deviating from the preferred angle. Two candidate configurations dominate in such geometry: an axial state, in which the director is uniformly parallel to the pore axis, and an escaped-radial state, in which the director lies roughly radial near the wall but escapes into the axial direction at the pore center, possibly with a disclination (director singularity) line along the axis. The source paper's plane-wave band-structure calculation, a plane-wave expansion in which the electromagnetic fields and the periodic dielectric function are expanded in Fourier components on the lattice, assumed axial alignment and overestimated the air-band-edge shift, the shift of the short-wavelength gap edge, which borders the band whose states concentrate in the low-index (air-filled) pores, while an escaped-radial model with a representative weak-anchoring parameter reconciled theory and experiment only qualitatively. Classical local probes such as deuterium nuclear magnetic resonance, NMR on deuterium-labeled liquid-crystal molecules whose spectral line shapes report the orientation distribution of the labeled bonds and hence the director field, have determined director fields and anchoring of nematics confined in silica aerogels and cylindrical polymer cavities (Kralj et al., 1993; Polak et al., 1994), but the anchoring type and strength of E7 on the electrochemically etched, oxidized silicon pore walls of macroporous silicon were not experimentally determined for these samples. A further practical obstacle noted in the source paper is that the macroporous silicon is highly doped for the etching process, so its free carriers, the mobile charges of the doped crystal, screen applied low-frequency electric fields and prevent electrically driven reorientation of the director; whether alternative doping levels, insulating substrates, or fabrication routes could permit fast electrically driven tuning in comparable two-dimensional photonic-crystal geometries was left open.

Problem Statement

For the liquid-crystal-infiltrated macroporous silicon two-dimensional photonic crystal of Leonard et al. (triangular pore lattice of 1.58 micrometer pitch, E7 filler, H- and E-polarized mid-infrared gaps), establish the surface anchoring of E7 on the silicon pore walls experimentally, and use it to close the quantitative theory gap: (1) determine the anchoring type (planar, homeotropic, or degenerate) and the quantitative anchoring strength WθW_\theta of E7 on the electrochemically etched, oxidized pore walls of macroporous silicon, together with the actual director-field realization in the pores, distinguishing an axial state from an escaped-radial state, including any disclination or spatially periodic director structure; (2) build the full quantitative modeling chain that couples the measured anchoring parameters and pore-surface chemistry, the exact temperature-dependent ordinary and extraordinary refractive indices of E7 at the wavelengths of the band gap, and the corresponding director field, into anisotropic three-dimensional (finite-depth) electromagnetic band-structure and mode-profile calculations, and show that this chain quantitatively reproduces the measured band-edge shifts of the infiltrated crystal, including the roughly 70 nm thermal shift of the short-wavelength H-gap edge, resolving the discrepancy that idealized axial-alignment calculations overestimate the air-band-edge shift. The two deliverables share one objective, a quantitative, experimentally anchored description of how surface anchoring determines the effective anisotropic refractive index seen by light in the pores and hence the tunability of the band gap; the companion practical question of the source paper, whether electrically driven fast tuning is possible in comparable two-dimensional photonic-crystal geometries despite free-carrier screening of the doped silicon, has been answered affirmatively by later electrode-configuration designs and is not part of this task.

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