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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Conservative form of the density of states of a photonic crystal with a pseudogap

Xiaodong Liu, Yiquan Wang, Bingying Cheng, Daozhong Zhang

Published Sep 18, 2003• 12 citations
Publisher

Abstract

We show that the total number of states in a photonic crystal in the entire allowed frequency regime will be conserved, and it is equal to that of its corresponding effective medium, i.e., if the density of states (DOS) has a valley(s) in some range(s) of frequencies, it must be compensated for by increases over some other range(s). This rule is of importance in developing a model pseudogap in order to describe the mean emission characteristics of the system when there is a collection of dependently emitting atoms or molecules with essentially random dipole orientations in a large volume and the spectrum of the active atoms is wide enough. This is because, with this rule, the states-conservative model always results in DOS-induced suppression, absolute enhancement, narrowing, spectrum splits, and redshift or blueshift of spontaneous-emission spectra.

DOI
10.1103/physreve.68.036610

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We show that the total number of states in a photonic crystal in the entire allowed frequency regime will be conserved, and it is equal to that of its corresponding effective medium, i.e., if the density of states (DOS) has a valley(s) in some range(s) of frequencies, it must be compensated for by increases over some other range(s). This rule is of importance in developing a model pseudogap in order to describe the mean emission characteristics of the system when there is a collection of dependently emitting atoms or molecules with essentially random dipole orientations in a large volume and the spectrum of the active atoms is wide enough. This is because, with this rule, the states-conservative model always results in DOS-induced suppression, absolute enhancement, narrowing, spectrum splits, and redshift or blueshift of spontaneous-emission spectra.
It is shown that there exist a conservation rule of the number of states in a photonic crystal with pseudogaps in the whole frequency regime, and a local conservation rule of the number of states in a photonic crystal with completely photonic band gaps in every band regime. These conservation rules state that, relative to the photonic density of states (DOS) in the effective medium of a photonic crystal, if a valley of the DOS in the photonic crystal appears in some range of frequencies, it must be compensated for by the increases over some other ranges. In this paper, using a model DOS with a state-conservative photonic pseudogap, we obtain naturally the enhancement and suppression of spontaneous emission of a two-level atom in a photonic crystal, and a DOS-induced suppression, enhancement, narrowing and redshift or blueshift of spontaneous emission spectra. Our theory will be especially appropriate in the problem of the spontaneous emission of a collection of dependently emitting atoms or molecules with essentially random dipole orientations in a photonic crystal with a photonic pseudogap.
We have investigated the analytical forms of the photonic density of states (DOS) in a photonic crystal in the interested frequency regime according to the rule of state conservation in a photonic crystal with pseudogaps, which states that, if a valley of the DOS appears in some range of frequencies, it must be compensated for by increasing over some other ranges. By using a model DOS with a state-conservative photonic pseudogap, we have also investigated the enhancement and suppression of spontaneous emission of two-level atoms with different frequency positions and widths of emission spectra in a state-conservative electromagnetic reservoir; and the DOS-induced suppression, enhancement, narrowing and redshifting or blueshifting of spontaneous emission spectra are naturally obtained.
The dynamical characteristics of the spontaneous decay of a single excited atom or molecule depend strongly on its position within a photonic crystal (PC) and the relative orientation of its dipole and the electric field there. Measuring its dynamical characteristics of the spontaneous decay in a PC can only obtain the local density of states (DOS) in the PC, but cannot reflect the total DOS in the PC. It is found, theoretically, that using different number density distributions of luminescent molecules within the PC, one can probe the partial information or even all of the total DOS in the PC. In this paper, we separate, for the first time to our knowledge, the total DOS into two parts, DOS in the scatterer and DOS in the matrix of the Wigner-Seitz unit cell, according to their origin, and demonstrate that a special number density distribution of fluorescent molecules can reflect the total scatterers DOS or total matrix DOS or even both of them, i.e. the total DOS in the PC. This is meaningful in explaining and designing the experiments of prohibiting and enhancing the spontaneous emission of excited molecules.
Photonic crystals (PCs) with relatively low dielectric contrast (i.e., with pseudogaps) have significant influence on the fluorescence decay of internal emitters. Fluorescence decays of ensembles of dye molecules measured at different positions in the PCs exhibit a nonexponential behavior, which is best fitted by a continuous distribution of decay rates. The most frequent decay rates of these distributions are smaller and their widths are narrower in a PC with a pseudogap acting in the emission range of the emitters than in a PC having the pseudogap out of this range. These experimental results have been well accounted for by calculations of the local density of states and rate constant for spontaneous emission.
A decomposition scheme for probing the total density of states (DOS) in an arbitrary photonic crystal (PC) having luminescent molecules is suggested. Some number distributions of the luminescent molecules can partially probe DOS information or even all the details of the total DOS by measuring the dynamic characteristics of the spontaneous decay of excited luminescent molecules in a PC. One can probe the total DOS directly when the ratio of the number densities of the homogeneously dispersed luminescent molecules in the scatterers and in the background material is equal to the ratio of the dielectric constants of the scatterer and the background material.
We present a formalism for the description of fluorescence from optically active materials embedded in a photonic crystal structure possessing a photonic band gap or pseudogap. An electromagnetic field expansion in terms of Bloch modes of the crystal is used to develop the equations for fluorescence in terms of the local density of photon modes available to the emitting atoms in either the high or low dielectric regions of the crystal. We then obtain expressions for fluorescence spectra and emission dynamics for luminescent materials in photonic crystals. The validity of our formalism is demonstrated through the calculation of relevant quantities for model photon densities of states. The connection of our calculations to the description of realistic systems is discussed. We also describe the consequences of these analyses on the accurate description of the interaction between radiative systems and the electromagnetic reservoir within photonic crystals.