Analysis of spontaneous emission in a 1D photonic crystal with effective resonator model

Abstract

Abstract-This letter discusses the application of effective resonator model as a tool for analysis of spontaneous emission in one-dimensional photonic crystals built of a finite number of layers and with arbitrary defects of layer's width or refractive index. It is pointed out that quantities defined in the model allow analyzing the properties of the structure without calculating field distributions.Photonic crystals are periodically arranged dielectric materials which possess photonic band gaps -frequency ranges for which propagation of light inside the structure is forbidden. They have been actively investigated since the works of Yablonovitch [1], who suggested using them to suppress spontaneous emission, and John [2], who pointed out possible localization of light if a proper defect was introduced. One-dimensional photonic crystals (1DPC) are the simplest structures of this kind. They are a particularly good subject of studies, because they are of practical importance (they can be used to build LEDs with increased efficiency [3], DFB or VCSEL lasers [4,5], filters [6,7] and other devices) and are characterized by the same features as more complex photonic crystal structures, but unlike them, they can be modelled analytically. Periodical structures can be described with the help of the Floquet-Bloch theorem, as in e.g. [8]. However, real structures are never perfectly periodical, first of all because their dimensions are finite, but also because they contain intentionally introduced defects. Therefore, in general, they should be considered a specific case of a planar multilayer waveguide (see Fig. 1, where j -index of the layer, n (j) -refractive index). These structures can be modelled with a generalized CarnigliaMandel model [9,10], in which the modes of electromagnetic field are obtained as combinations of plane waves, coming from a plane wave incident on the structure from the outside (incoming modes) or propagating away from it (outgoing modes) [11], which determines amplitudes of all the other plane waves in layers of the structure (they can be calculated e.g. with * E-mail: arudzins@poczta.onet.pl the translation matrix method [12,13]) -see Fig. 2 (a), (b). This approach is correct and widely used, but it does not naturally fit the description of spontaneous emission from within the structure, because modes fixed by waves from the outside of the structure do not explicitly show the properties of the region surrounding the emitter -to investigate them it is necessary to manipulate and analyze the obtained field distributions. A much more relevant approach is provided by the effective resonator model [14][15][16][17], in which the construction of modes starts with plane waves inside the same layer the emitter is situated in (Fig. 2c). In this model, a parameter, so-called mode spectrum is defined, which clearly and explicitly shows the properties of the layer interesting for a designer of the structure. Analysis of spontaneous emission in a 1D photonic crystal with effective resonator model 80The tran...

Local density of optical states calculated by the mode spectrum in stratified media

“…The ERM can convert the 1D multi-layer waveguide to an effective Fabry-Pérot (FP) resonator, [23][24][25][26] as shown in Fig. 1(a).…”