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Physical review. B, Condensed matter

Atomic radiation rates in photonic crystals

Virginie Lousse, Jean‐Pol Vigneron, Xavier Bouju, J. M. Vigoureux

Published Nov 2, 2001• 42 citations
Publisher

Abstract

A photonic crystal, defined as a periodic dielectric structure, can influence the radiation rate of an embedded dipolar emitter. A theory of this effect is developed and the efficiency of a dipolar photonic source is calculated for a realistic three-dimensional crystal. Taking as a starting point the photonic band structure and its associated eigenfields, it is shown that the emission rate is strongly correlated with the density of modes, but also that the density of modes alone cannot explain all features found in the emission spectrum. For an infinite crystal, the computation of the field propagator confirms that the emission rate falls to zero in the frequency range defined by the photonic band gap. The emitter lifetime changes with the dipole location and orientation, leading to a radiation rate enhancement or inhibition according to the direction of the emission. These results may open routes to photonic sources with very high quantum efficiency.

DOI
10.1103/physrevb.64.201104

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•Oct 1, 2014•0 citations
A photonic crystal, defined as a periodic dielectric structure, can influence the radiation rate of an embedded dipolar emitter. A theory of this effect is developed and the efficiency of a dipolar photonic source is calculated for a realistic three-dimensional crystal. Taking as a starting point the photonic band structure and its associated eigenfields, it is shown that the emission rate is strongly correlated with the density of modes, but also that the density of modes alone cannot explain all features found in the emission spectrum. For an infinite crystal, the computation of the field propagator confirms that the emission rate falls to zero in the frequency range defined by the photonic band gap. The emitter lifetime changes with the dipole location and orientation, leading to a radiation rate enhancement or inhibition according to the direction of the emission. These results may open routes to photonic sources with very high quantum efficiency.
There has been a great deal of interest lately in periodic dielectric materials that exhibit photonic band structures. One of the more interesting features of such a substance is its ability to alter the emission rate of probe atoms embedded in the periodic lattice. Large enhancements and complete inhibition of emission rates can be obtained. It has been known for some time that the influence of a cavity on atomic emission is essentially a classical effect. Hence, we develop a general classical treatment of radiation rates in an inhomogeneous medium. Our results agree with those of a fully quantum calculation, and are applied to a simple scalar model of a dipole in a one-dimensional periodic lattice of the Kronig-Penney type.
It has been suggested that spontaneous emission can be inhibited if atomic transition frequencies fall inside photonic band gaps, that is, three-dimensional frequency stop bands of electromagnetic waves generated by three-dimensional periodic dielectric materials (photonic crystals). There has been a growing interest in how atomic emission spectra are changed quantitatively inside the photonic crystals. We develop a classical theory for the calculation of the emission power from the electric dipole located in three-dimensional photonic crystals by incorporating the plane-wave method, dyadic Green’s function, Poynting theorem, and tetrahedron k-space integration. With the method we perform numerical computations for the emission power of an electric dipole located in the photonic crystals of the fcc lattice structure with spherical atoms. The results show the total inhibition of emission in the photonic band gap as well as strong enhancement around the band edges. In addition, the data indicate the strong dependencies of the emission spectrum on the dipole position and the dipole moment in the photonic crystal.
We study the spontaneous emission of an atom embedded in a one-dimensional photonic crystal or superlattice using a classical electrodynamic theory of radiation. The rate of emission is a function of the frequency of the emitted photon, the dipole's position and orientation, as well as the geometric and material parameters of the superlattice. The emission spectrum shows an oscillatory behavior which follows the photonic band structure. For TE modes, there are frequency regions where radiative emission is completely prohibited due to the absence of modes with k//>omega/c; the radiation is then TM polarized. In addition to the radiative modes, there are always evanescent modes with k//>omega/c which are waveguided by the dielectric layers. The evanescent contribution to the spontaneous emission is dominant if a dielectric layer is in the near field region of the dipole. For TM modes, emission rates greatly vary for parallel and perpendicular dipole moments. In a photonic crystal with a high filling fraction of the dielectric and perpendicular dipoles located in the low-index layer, the decay rate can be as much as 76 times the free space value for a single atom and 50 times for a gas of atoms. We also find that the rate of emission presents a strong dependence on the atom's position.
Photonic crystals affect the photon emission of embedded emitters due to an altered local density of photon states (LDOS). We review the calculation of the LDOS from eigenmodes in photonic crystals and propose a rate equation model for fluorescent emitters to determine the changes in emission induced by the LDOS. We show how to calculate the modifications of three experimentally accessible characteristics: emission spectrum (spectral redistribution), emitter quantum yield, and fluorescence lifetime. As an example, we present numerical results for the emission of the dye Rhodamine B inside an opal photonic crystal. For such photonic crystals with small permittivity contrast, the LDOS is only weakly modified, resulting in rather small changes. We point out that in experiments, however, usually only part of the emitted light is detected, which can have a very different spectral distribution (e.g., due to a photonic band gap in the direction of detection). We demonstrate the calculation of this detected spectrum for a typical measurement setup. With this reasoning, we explain the previously not fully understood experimental observation that strong spectral modifications occurred, while at the same time only small changes in lifetime were found. With our approach, the mentioned effects can be quantitatively calculated for fluorescent emitters in any photonic crystal.
There has been a great deal of interest in periodic dielectric materials that exhibit photonic band structures. One of the more interesting features of such a substance is its ability to alter the emission rate of probe atoms embedded in the periodic lattice. Large enhancements and complete inhibition of emission rates can be obtained. It has been known for some time that the influence of a cavity on atomic emission is essentially a classical effect. Hence, we develop a general classical treatment of radiation rates in an inhomogenous medium. Our results agree with those of a fully quantum calculation, and are applied to a simple scalar model of a dipole in a one-dimensional periodic lattice of the Kronig-Penney type.
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. 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. 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.