Local density of optical states in the band gap of a finite photonic crystal

We study the local density of states (LDOS) in a finite photonic crystal, in particular in the frequency range of the band gap. We propose a new point of view on the band gap, which we consider to be the result of vacuum fluctuations in free space that tunnel in the forbidden range in the crystal. As a result, we arrive at a model for the LDOS that is in two major items modified compared to the well-known expression for infinite crystals. Firstly, we modify the Dirac delta functions to become Lorentzians with a width set by the crystal size. Secondly, building on characterization of the fields versus frequency and position we calculated the fields in the band gap. We start from the fields at the band edges, interpolated in space and position, and incorporating the exponential damping in the band gap. We compare our proposed model to exact calculations in one dimension using the transfer matrix method and find very good agreement. Notably, we find that in finite crystals, the LDOS depends on frequency, on position, and on crystal size, in stark contrast to the well-known results for infinite crystals.Comment: 22 pages, 8 figure

Finite size scaling of the density of states in photonic band gap crystals

Photonic crystals are tailored periodic dielectric media that allow for an unprecedented control in the manipulation of light-matter interactions. One of their outstanding features is the realization of a complete photonic band gap that drastically inhibits light propagation in all directions and for all polarizations. A band gap is associated with a complete vanishing of the density of optical states (DOS) in the crystal. As a necessary corollary, it implies a vanishing of the local DOS (LDOS) too, which leads to a complete inhibition of spontaneous emission everywhere inside such a crystal.In our paper, we will present our approach for the DOS in finite support crystals in both 2D and 3D spatial dimensions. We will show the surprising result that the DOS inside the bandgap decreases linearly with size irrespective of the crystal dimensionality as shown in Fig. 1(b) for the example of inverse woodpile crystals that are being pursued in our group [3]. Our work sets design rules for the sizes of photonic bandgap crystals for practical applications in cavity QED and quantum information processing (vacuum noise shielding). It also has the potential of establishing new methods for engineering finite crystals to enhance the suppression of DOS

Controlling emission and propagation of light with photonic band gap crystals

In certain three-dimensional crystals, a frequency range exist for all polarizations for which light is not allowed to propagate in any direction, called the 3D photonic band gap: a frequency range where the density of vacuum fluctuations vanishes in an ideal infinitely large and perfect system. The complete absence of vacuum fluctuations can be interpreted as zero density of states and vice versa. It is well known that the characteristics of spontaneous emission of light depend strongly on the environment of the light source. According to quantum electrodynamics, the emission rate of a two-level quantum emitter as described by Fermi’s “golden rule”, is factorized into one part describing the emitter’s properties and a second part describing the effect of the environment on the light field via the local density of optical state (LDOS). The radiative decay rate of the emitters is proportional to the LDOS. The absence of vacuum fluctuations or the zero LDOS leads to complete inhibition of the spontaneous emission.\ud While many great advances towards a band gap have been made, no complete inhibition has ever been observed to date. In this thesis, for the first time, we have studied the spontaneous emission of PbS quantum dots inside real 3D photonic band gap crystal as a function of frequency throughout the band gap. In our experiments we have observed an inhibited emission of 18x in the band gap. The time-resolved decay curves show evidence for intriguing finite-size effects whereby vacuum fluctuations tunnel into the band gap.\ud Unfortunately, most theories assume crystals of infinite extent, where the inhibition in the band gap is also infinite, as vacuum fluctuations don’t exist inside the crystal. Here we propose an original model to calculate the LDOS in the band gap of a finite photonic crystal based on tunneling of the vacuum fluctuations into the crystal. We validate our model for the one-dimensional situation with a direct comparison to the analytic calculations which exist for one-dimensional photonic crystals. For three dimensional crystals, the theory shows an excellent match to the experiment

Non-Rayleigh distribution of reflected intensity from photonic crystals with disorder

Structural disorder results in multiple scattering in real photonic crystals, which have been widely used for applications and studied for fundamental interests. The interaction of light with such complex photonic media is expected to show interplay between disorder and order. For a completely disordered medium, the intensity statistics is well-known to obey Rayleigh statistics with a negative exponential distribution function, corresponding to absence of correlations. Intensity statistics is unexplored however for complex media with both order and disorder. We study experimentally the intensity statistics of light reflected from photonic crystals with various degree of disorder. We observe deviations from the Rayleigh distribution and the deviations increase with increasing long-range order.Comment: 5 pages, 4 figure