Suppressing electromagnetic local density of states via slow light in lossy quasi-one-dimensional gratings

We propose a spectral-averaging procedure that enables the computation of bandwidth-integrated local density of states (LDOS) from a single scattering calculation, and exploit it to investigate the minimum extinction achievable from dipolar sources over nonzero bandwidths in structured media. Structure-agnostic extinction bounds are derived, providing analytical insights into scaling laws and fundamental design tradeoffs with implications to bandwidth and material selection. We find that perfect LDOS suppression over a nonzero bandwidth $\mathrm{\Delta }\omega$ is impossible. Inspired by limits which predict nontrivial $\sqrt{\mathrm{\Delta }\omega }$ scaling in systems with material dissipation, we show that the pseudogap edge states of quasi-one-dimensional bullseye gratings can—by simultaneously minimizing material absorption and radiation—yield arbitrarily close to perfect LDOS suppression in the limit of vanishing bandwidth.

Figure 1

Spectral window function $W\left(\omega \right)$ used in calculating average LDOS around a “center” frequency ${\omega }_0$, Eq. (6), for various values of the bandwidth parameter $\mathrm{\Delta }\omega =\{1,\frac{1}{5},\frac{1}{10},\frac{1}{20},\frac{1}{50}\}{\omega }_0$ (top to bottom). For ease of visualization, $W\left(\omega \right)$ is normalized by $W\left({\omega }_0\right).$

Figure 2

(a) and (b) Lower bounds on bandwidth-integrated LDOS. All curves and markers refer to TM line sources located at the center of $L\times L$ design domains with (a) $\chi =4$ for $L/{\lambda }_0=\{1,1.59,10\}$ (blue, orange, green) and (b) $\chi =4+0.1i$ for $L/{\lambda }_0=\{5,10,80\}$ (blue, orange, green) where ${\lambda }_0=2\pi c/{\omega }_0$. Solid lines are lower bounds for finite $L\times L$ design domains, while dotted lines pertain to infinite systems, $L/{\lambda }_0\to \infty$, computed via Eq. (11). All results are normalized by the corresponding bandwidth-integrated LDOS in vacuum. Markers correspond to structures discovered via topology optimization, where circles ($\circ$) are over rotationally symmetric structures, stars ($★$) are for truncated 2Dd square-lattice photonic crystals of period ${\lambda }_0$, and crosses ($\times$) allow for arbitrary structuring. Insets of (a) and (b): Representative inverse designs corresponding to $\mathrm{\Delta }\omega /{\omega }_0={10}^{-7}$ for a $10{\lambda }_0\times 10{\lambda }_0$ design region. (c) and (e) Representative spectra for $L/{\lambda }_0\gg 1$ [orange for $10{\lambda }_0\times 10{\lambda }_0$ crystal, blue for $80{\lambda }_0\times 80{\lambda }_0$ bullseye grating (not shown)]; optimizing over cylindrical geometries, the lossless and dispersionless case converges to a bullseye geometry of radial periodicity $a=\frac{{\lambda }_0}{4}\frac{1+\sqrt{1+\chi }}{\sqrt{1+\chi }}$ and ring thickness $h=\frac{{\lambda }_0}{4\sqrt{1+\chi }}$ (a radial quarter-wave stack) supporting a pseudogap centered around ${\omega }_0$ [see (c)]; the lossy case converges to a radial chirped grating of initial spacing $a={\lambda }_0/2$ and tapered thickness, causing the net absorbed power to come arbitrarily close to zero at ${\omega }_0$ as $L\to \infty$, in the vicinity of which the system exhibits square-root dispersion [see (e)]. (d) and (f) The bandwidth-averaged LDOS of optimized bullseye structures within a design region of diameter $L$. The lossless (lossy) case demonstrates exponential (inverse square root) approach as a function of $L$ to a positive saturation value due to a nonzero bandwidth $\mathrm{\Delta }\omega /{\omega }_0={10}^{-7}$.

Figure 3

LDOS spectrum, Eq. (13), of a 1D PhC cavity with period $a={\lambda }_0/2$, material layers of thickness $h=a/50$, and a central air gap of thickness $d=\{a/3,a-h\}$ (solid blue, dashed orange, respectively) with loss $\text{Im}\left[\chi \right(\omega \left)\right]=0.1$. The dotted green curve shows a square-root fit around ${\omega }_0.$ The inset is a representative intensity profile $|E(z;{\omega }_0){|}^2$ at the band edge, exhibiting minima in the material layers (vertical black bars), as well as $|E(z;{\omega }_m){|}^2$ at the pseudomidgap, exhibiting minima adjacent to the material layers for $d=a/3$.