Emulating one-dimensional resonant $Q$-matching behavior in a two-dimensional system via Fano resonances

Through detailed numerical and analytical studies, we establish that the significant enhancement of thermal emission via $Q$ matching, which has been possible in one-dimensional (1D) systems only, can be extended to two-dimensional (2D) systems by means of Fano resonances in the 2D system. In particular, we show the existence of essentially 1D behavior in a 2D system—a case of reduced dimensionality. Moreover, we show how properties of these spectra can be controlled by changing the geometrical parameters of the 2D system.

Figure 1

(Color) Panel (a) shows some generic features associated with thermal radiation of common blackbodies or graybodies into air (the case of a perfect blackbody is denoted by solid lines, while an example of a graybody, a uniform Si slab of thickness $0.75\mu \mathrm{m}$, is denoted by dashed lines); plotted is the (thermal radiation intensity)/$d{k}_{x}d{k}_{y}d\lambda$ for an exemplary case: $k_x=0.838\mu {\mathrm{m}}^{-1}$ and $k_y=0$. First, blackbodies or graybodies have perfectly incoherent and ultrabroad bandwidth thermal-emission spectra. Next, as the temperature of such a body increases, the emission spectrum shifts to shorter wavelengths. Finally, bodies with lower absorption have weaker thermal emission; for example, a thin silicon slab is nearly transparent for infrared light even at fairly high temperatures, so its thermal-emission is very weak. Panel (b) shows FDTD calculations of the thermal emission spectra of the same Si slab as in panel (a), but this time patterned as a square-lattice 2D-periodic photonic crystal slab of holes with radius $r=0.3\mu \mathrm{m}$, and lattice constant $a=1.5\mu \mathrm{m}$ (structure schematic shown in the inset). Thermal radiation of such a body can display drastically different behavior than the one shown in panel (a). First, a photonic crystal can produce very coherent thermal radiation, as implied by the narrowness of the emission peaks. Next, as one increases the temperature, the peak emission can shift to longer (instead of shorter) wavelengths. Finally, despite the near-transparency of Si, emissivity can be comparable to that of a perfect blackbody for certain frequencies.

Figure 2

(Color) FDTD simulations of a photonic crystal slab (PhC) of thickness $0.5a$ with $\mathrm{Re}\left(ϵ\right)=12$ for incident electric field polarized in the $y$ direction. (a) Transmission and absorption spectra for a PhC slab, for $k_x=0.2(2\pi ∕a)$, with $\mathrm{Im}\left(ϵ\right)\approx 0.005$ (chosen to maximize absorption for the first peak), displaying four Fano resonance peaks. (b) Band diagram of the photonic crystal slab modes in the case of an infinitesimally small amount of absorption. Note that Fano resonances in panel (a) occur exactly at those frequencies for which there is a guided mode of the PhC that has $k_x=0.2(2\pi ∕a)$.

Figure 3

(Color) Comparison between theory and simulation, for the first two Fano peaks shown in Fig. 2a. The parameters of the theory (${\omega }_0$ and ${\gamma }_{\mathit{rad}}$) were calibrated by fitting the red line (theory with vanishingly small absorption) to the red circles (simulation with vanishingly small absorption). The black line gives the prediction of theory for the absorptive case; the black circles represent the results from simulation with absorption. (The fraction of energy in dielectric $\left(\xi \right)$ was 90% for the first peak and 95% for the second, done in a separate FDTD calculation.)

Figure 4

(Color) Variation of Fano absorption peak frequency as a function of angle of incidence (in degrees); the rotation angle $\theta$ is around the $y$ axis, and the electric field is polarized along $y$. (a) First peak. (b) Second peak. The change in frequency for the second peak is much smaller than that for the first. The angles are different for the top and bottom panels even though the $k_x$’s used are the same because $\theta$ is a function of both $k_x^{\prime }$ and $\omega$.