Tailoring thermal emission via $Q$ matching of photonic crystal resonances

We develop a model for predicting the thermal emission spectrum of a two-dimensional metallic photonic crystal for arbitrary angles based on coupled-mode theory. Calculating the appropriate coupled-mode parameters over a range of geometrical parameters allows one to tailor the emissivity spectrum to a specific application. As an example, we design an emitter with a step-function cutoff suppressing long-wavelength emission, which is necessary for high-efficiency thermophotovoltaic systems. We also confirm the accuracy of the results of our model with finite-difference time-domain simulations.

Figure 1

Emissivity spectrum of flat, bulk tungsten (blue) and an ideal selective emitter (green) designed to match the 0.6 eV/$\hslash$ band gap of InGaAs.

Figure 2

A periodic array of cylindrical holes with period $a$, depth $d$, and radius $R$ etched into a slab of tungsten. The hole depth is less than the tungsten slab thickness.

Figure 3

(a) Resonant mode frequency in units of $2\pi c/a$ with $a=1$ $\mu$m, (b) $Q_{\mathrm{abs}}$, and (c) $Q_{\mathrm{rad}}$ of the lowest three modes. The absorptive and radiative $Q$ are plotted on the same color scale. $Q$ matching of the first resonance is indicated by the white line.

Figure 4

Absorption spectrum for the tungsten photonic crystal slab with $a=4.8\hspace{0.5em}\mu$m, $R=0.104a$, and $d=0.394a$. Diffraction appears as sharp peaks and resonant peaks due to photonic crystal modes appear at approximately 2.7 and 3.4 $2\pi c/a$.

Figure 5

Absorption spectra for photonic crystal slabs with various $a$ but $R$ and $d$ fixed at 0.5 and 1.89 $\mu$m, respectively.

Figure 6

A multimode resonator with resonant frequencies ${\omega }_i$ and absorptive and radiative lifetimes ${\tau }_{\mathrm{abs},i}$ and ${\tau }_{\mathrm{rad},i}$, respectively. Resonances radiatively couple to each other with rate $1/{\tau }_{\mathit{ij}}$.

Figure 7

Absorptivity spectrum for $a=1.2$ $\mu$m calculated via FDTD simulation (blue) and coupled-mode theory (red).

Figure 8

Absorptivity spectrum for the same structure as in Fig. 7, but with ${\mathbf{k}}_{\parallel }a/2\pi =(0.1776,0.1056)$. Peaks in the emissivity would correspond to polar angles of ${18}^{°}$ and ${15}^{°}$ and an azimuthal angle of ${31}^{°}$.