Photonic crystals with anomalous dispersion: Unconventional propagating modes in the photonic band gap

We present an investigation of the optical properties of photonic crystals whose constituent materials exhibit anomalous dispersive behavior. In particular, the anomalous dispersion near resonances may lead to additional propagating modes in the gap of the undoped system for a localized region of wave-vector space. Such a system may be realized by infiltrating quantum dots in polymer suspensions into the pores of two-dimensional high-index photonic crystals. An evaluation of the absorption lengths associated with these unconventional modes and corresponding transmission calculations demonstrate that this effect can be observed in currently accessible structures.

Figure 1

(Color online) Model system of a 2D macroporous silicon PC infiltrated with quantum dots in a polymer suspension. This PC consists of a 2D macroporous silicon backbone (dielectric constant ${\epsilon }_{\mathrm{Si}}=12.0$) into which a square lattice of pores (radius $r∕a=0.45$) has been etched, which subsequently have been filled with a low-index polymer (dielectric constant ${\epsilon }_{\text{polymer}}=2.56$).

Figure 2

(Color online) Effective dielectric constant of a typical polymer doped with quantum dots for three different concentrations $\eta$. The quantum-dot parameters are given in the text. The resonance of this Maxwell-Garnett effective dielectric constant is shifted relative to the resonance frequency of isolated quantum dots (vertical dotted line).

Figure 3

(Color online) Photonic band structure for TM-polarized radiation propagating in the reference PC structure. The associated system parameters are listed in the caption of Fig. 1.

Figure 4

(Color online) A selection of maximally localized Wannier functions for the reference PC structure. The corresponding photonic band structure is displayed in Fig. 3 and the associated system parameters are listed in the caption of Fig. 1.

Figure 5

(Color online) Illustration of the partitioning of real space in order to facilitate an on-shell band structure calculation. For a fixed frequency $\omega$ and direction $\widehat{k}$ in reciprocal space, we partition the PC into identical rectangular supercells whose sides are given by ${\vec{s}}_{\parallel }=\alpha {\vec{a}}_1+\beta {\vec{a}}_2$ and ${\vec{s}}_{\perp }=\alpha {\vec{a}}_2-\beta {\vec{a}}_1$, where $\alpha$ and $\beta$ are integers such that $\vec{G}=\alpha {\vec{b}}_1+\beta {\vec{b}}_2$ is parallel to $\widehat{k}$. Periodic and Bloch-boundary conditions are, respectively, applied along ${\vec{s}}_{\perp }$ and ${\vec{s}}_{\parallel }$. For this illustration on a square lattice, we have chosen $\alpha =2$ and $\beta =1$, so that a supercell contains five unit cells of the PC.

Figure 6

(Color online) Geometric illustration of the iterative solution of the nonlinear eigenvalue problem associated with the band structure of a PC with dispersive constituent materials. The solid line depicts the frequency dependence of the real part of the model system’s dielectric constant for a quantum-dot filling ratio of $\eta =0.03$. The values of the other system parameters are given in the text. The dashed lines display the frequency line ${\omega }_1\left({\epsilon }_{\mathrm{fict}}\right)$ of the first band for three different but fixed wave vectors $\vec{k}$. The intersections of these frequency lines with the graph of the model system’s dielectric constant (circles) correspond to the iterative solutions of the nonlinear eigenvalue problem of Eq. (4).

Figure 7

Frequency dependence of the transmittance at normal incidence for finite-sized PC samples with different thicknesses ($N=1,11,21,31,41,51$ unit cells). The PC is oriented such that normal incidence corresponds to propagation along the $\Gamma \text{−}X$ direction. The inset demonstrates that once the sample thickness exceeds about 20 unit cells, the transmittance decays approximately exponentially with thickness. This behavior should be compared with the corresponding reflectance calculations displayed in Fig. 8. For actual calculations of the attenuation length, sample thicknesses in the range of $N=1,\dots ,100$ with a step size of $\Delta N=1$ have been analyzed. The PC parameters are given in the text.

Figure 8

Frequency dependence of the reflectance at normal incidence for finite-sized PC samples with different thicknesses ($N=11,21,31,41,51$ unit cells). The PC is oriented such that normal incidence corresponds to propagation along the $\Gamma \text{−}X$ direction. Once the sample thickness exceeds about 20 unit cells, the reflectance values (approximately) become independent of the sample thickness. This behavior should be compared with the corresponding transmittance calculations displayed in Fig. 7. The PC parameters are given in the text.

Figure 9

(Color online) Photonic band structure for the model system with quantum-dot concentration $\eta =0.03$ and when material absorption is ignored (solid lines with symbols). The anomalous dispersion of the pore dielectric constant near the band edge of the second band leads to the formation of propagating modes (lines with diamonds) inside the photonic band gap of the undoped system. For reference, the photonic band structure of the undoped PC is indicated by dashed lines. The horizontal dotted line again depicts the resonance frequency of isolated quantum dots.

Figure 10

(Color online) Band structure and attenuation length calculations for the model system with a quantum-dot concentration $\eta =0.03$. In the left panel, Wannier function (WF) calculations are compared to the extended band structure (extended BS) method. The Wannier function calculations yield both real and imaginary parts of the wave vector (middle panel). The latter can be converted into an attenuation length and can be compared with results from transmittance calculations through finite-sized PCs via the scattering-matrix method (right panel). The large values of the attenuation length for frequencies close to the bubble suggest that this effect can be observed experimentally.

Figure 11

(Color online) Right panel: evolution of the bubble (lines with diamonds) when the concentration $\eta$ of quantum dots in the model system is increased. The dashed lines depict the band structure of the undoped PC. Left panel: Wannier function (solid lines) and scattering-matrix (dashed lines) calculations of the corresponding attenuation lengths.