Radiative local density of states in three-dimensional photonic band-gap crystals to interpret time-resolved emission

TL;DR

This work addresses how spontaneous emission in 3D photonic band-gap crystals can be interpreted through the radiative local density of states (RLDOS). It compares two common calculation methods—plane-wave expansion (PWE) for infinite crystals and finite-difference time-domain (FDTD) for finite samples—and demonstrates that they agree on frequency trends within about 12%, with discrepancies arising from finite-size boundary effects. By computing RLDOS at many emitter positions using PWE, the authors derive distributions of decay rates and construct the corresponding time-resolved decay curves for ensembles, linking theoretical RLDOS to TCSPC measurements. The findings establish practical upper bounds on decay-rate enhancements, quantify ensemble-averaged emission behavior in 3D photonic crystals, and provide a framework for designing optical devices that exploit RLDOS for controlled emission.

Abstract

We investigate the spontaneous emission of light in three-dimensional (3D) photonic crystals through theoretical calculations and simulations. It is well known that spontaneous emission depends on the radiative local density of states (RLDOS). Photonic band-gap crystals radically modulate the RLDOS, thereby controlling spontaneous emission. We compare two different methods to calculate the RLDOS: the plane-wave expansion (PWE) method and the finite-difference time-domain (FDTD) method. The PWE method directly calculates the RLDOS of an infinite photonic crystal, whereas the FDTD method simulates the RLDOS through the power emitted by a dipole in a finite photonic crystal. We demonstrate that the methods yield similar frequency-dependent trends in the RLDOS, with relative differences of less than 12% that originate from the different boundary conditions. We employ the plane-wave expansion method to compute distributions of emission rates that are relevant to many optical experiments where quantum emitters are distributed within a crystal. Such distributions of emission rates enable us to compute and directly interpret the time-resolved decay as observed in experiments. We expect that our results promote the RLDOS to the realm of optical design and products.

Figures (5)

• Figure 1: Illustrations of non-primitive unit cells. (Left) The spheres of the fcc colloidal opal with pitch $a$. (Right) The pores of the inverse-woodpile structure with pitches $a$ and $c \equiv a/\sqrt{2}$. The RLDOS locations, due to symmetry equivalent to those in the text, are denoted by $L$.
• Figure 2: Radiative local density of states (RLDOS) calculated at $(\frac{a}{4},\frac{a}{4},0)$ inside the colloidal photonic crystal using the (red) PWE and (blue) FDTD method. (Black solid line) Calculations by Nikolaev et al. using the PWE method Nikolaev2009JOSAB. (Black dashed lines) RLDOS of homogeneous media, namely TiO$_2$ (dashed, $\varepsilon = 7.35$) and water (dash-dotted, $\varepsilon = 1.77$).
• Figure 3: Radiative local density of states (RLDOS) calculated at (0,$\frac{c}{4}$,$\frac{a}{8}$) inside the inverse-woodpile photonic crystal using the (red) PWE and (blue) FDTD method. The pores of the silicon ($\varepsilon = 12.1$) crystal are filled with toluene ($n = 1.48$). (Black lines) RLDOS of the homogeneous media (dash-dotted) toluene, and (dashed) silicon.
• Figure 4: (Red) Radiative local density of states (RLDOS) averaged over all positions in the pores of the inverse-woodpile photonic crystal using the PWE method. (Blue shaded area) Standard deviation of the RLDOS around the average value. (Black dashed lines) LDOS of homogeneous media, namely toluene (the pores' material, $n = 1.48$) and silicon ($\varepsilon = 12.1$). (Blue bars) Regions in which we determine $\zeta(\Gamma)$ in Fig. \ref{['fig:ldos_zeta_ft_wnr6500']}.
• Figure 5: (a) The $\zeta(\Gamma)$ distribution calculated at $\tilde{\nu} =$ 6500±100 (red solid) and $\tilde{\nu} =$ 9200±100 (blue dashed) inside the pores of the photonic crystal ('Ph. C.') with an inverse woodpile structure. The input for the calculations (red/blue) is the reference $\delta$ distribution ('Ref.', black) with amplitude 1 and decay rate 1µs. (b) The corresponding decay curves calculated from Eq. \ref{['eq:ldos_ft_to_zetaGamma']} on a semi-logarithmic scale. The integrals over the total curves are the same.