Direct calculation of thermal emission for three-dimensionally periodic photonic crystal slabs

We perform direct thermal emission calculations for three-dimensionally periodic photonic crystal slabs using stochastic electrodynamics following the Langevin approach, implemented via a finite-difference time-domain algorithm. We demonstrate that emissivity and absorptivity are equal, by showing that such photonic crystal systems emit as much radiation as they absorb, for every frequency, up to statistical fluctuations. We also study the effect of surface termination on absorption and emission spectra from these systems.

Figure 1

(Color) Band structure for a 3D periodic woodpile structure made of perfect metal rods with a square cross section of width $0.25a$. We show bands along $\Gamma \text{−}X$ and $\Gamma \text{−}Z$. The resolution is 30 grid points per $a$. We consider modes with all polarizations. We notice a large band gap in the system where light is forbidden from propagating, specifically from 0 to $0.42c∕a$.

Figure 2

(Color) Comparison between absorption and thermal emission (averaged over 40 runs) from a slab of 3D periodic woodpile made of metal rods, at normal incidence. We use a long computational cell with two unit cells of the woodpile structure in the $z$ direction. For the metal, we use the Drude model with parameters ${ϵ}_{\infty }=1$, $\gamma =0.3(2\pi c∕a)$, $4\pi \sigma =10(4{\pi }^2{c}^2∕a^2)$. The frequency resolution is $0.001c∕a$. We see good agreement between the emissivity (green and blue solid lines) and the absorptivity (black and red dashed lines). We note also that the emission of the woodpile structure exceeds that of a uniform slab at all frequencies. The greatest enhancement comes from the nongapped region above $0.4c∕a$, where the enhancement can be as high as a factor of 4. Translucent yellow shading indicates regions of pseudogap for such a woodpile slab structure made of imperfect metal rods, inferred from the absorption and/or emission spectrum.

Figure 3

(Color) Band structure for a 3D periodic metallodielectric structure made of perfect metal spheres of radius $0.177a$ in a background of Teflon $(ϵ=2.1)$. We show bands along $\Gamma \text{−}X$ and $\Gamma \text{−}L$. The resolution is 32 grid points per $a$. We consider modes with all polarizations. We note a complete band gap in the system where light is forbidden from propagating, specifically from $0.54c∕a$ to $0.63c∕a$.

Figure 4

(Color) Comparison between absorption and thermal emission (averaged over 40 runs) from a slab of 3D periodic metallodielectric structure made of metal spheres in a Teflon background, at normal incidence. We use a long computational cell with two unit cells of the metallodielectric structure in the $z$ direction. For the metal, we used the Drude model with parameters ${ϵ}_{\infty }=1$, $\gamma =0.3(2\pi c∕a)$, $4\pi \sigma =10(4{\pi }^2{c}^2∕a^2)$. Here, we use a lower frequency resolution of $0.01c∕a$ in order to decrease the duration of each run. We see good agreement between the emissivity (green and blue solid lines) and the absorptivity (black and red dashed lines). We notice also that the emission of the metallodielectric structure exceeds that of a uniform slab at all frequencies above $0.1c∕a$. The greatest enhancement comes from the nongapped region around $0.8c∕a$, where the enhancement can be as high as a factor of 6. Note that the emissivity in that region is close to unity. Note also that the decreased run time leads to lower frequency resolution, as evidenced by the smoother spectrum. However, the size of the fluctuations remains unchanged (compare with Fig. 2), since is determined by the number of runs used in ensemble averaging.

Figure 5

(Color) Absorbance and/or emittance spectrum for a woodpile PhC slab made of imperfect metal rods for five different surface terminations. Light polarized along $x$ is normally incident from the top of the cell. We use a long computational cell with two unit cells of the woodpile structure in the $z$ direction. For the metal, we used the Drude model with parameters ${ϵ}_{\infty }=1$, $\gamma =0.3(2\pi c∕a)$, $4\pi \sigma =10(4{\pi }^2{c}^2∕a^2)$. The inset is a schematic (lengths not to scale) indicating the surface terminations chosen. For all calculations, we keep the thickness of the slab to about two unit cells, so changing the surface termination amounts to shifting the structure within a two-unit-cell-thick slab “mask” which remains stationary as the structure is shifted, such that the total amount of material is kept constant. For instance, for ST0 the structure used is that between the two black lines, while for ST6, it is what lies between the two blue lines. ST7 appears to have the highest absorption and/or emission at all frequencies.