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Frontiers in Optics

Simple Model of Spontaneous Emission in 1D Photonic Crystal

Adam Rudziński, Anna Tyszka-Zawadzka, Paweł Szczepański

Published Jan 1, 2005• 1 citation
Publisher

Abstract

We derive an analytic expression for spontaneous emission spectrum in a finite onedimensional photonic crystal. With this formula we obtain spontaneous emission spectrum characteristics of photonic crystal, which take the influence of defects into account.
DOI
10.1364/fio.2005.jwa69

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Acta Physica Sinica•Jan 1, 2016•0 citations
Abstract We derive an expression for the spontaneous-emission rate in finite one-dimensional photonic crystals with arbitrary defects, using the effective resonator model to describe electromagnetic field distributions in the structure. We obtain explicit formulae for contributions from different modes, i.e. radiation, substrate and guided modes. Formal calculations are illustrated by a few numerical examples, which demonstrate that application of the effective resonator model simplifies the interpretation of results.
The spontaneous emission properties of a Λ_type atom embedded in anisotropic photonic crystals are investigated. We find that the dynamical properties of the atom and the propagating characteristics of the radiation field are obviously influenced by the relative position of the upper level from the forbidden gap and the space between the two lower levels. With the help of numerical calculation and theoretical analysis, we plot the evolution of the total radiation field and its components with the time. Furthermore, we also study the influence of photonic crystals on radiative properties of the atom by means of spontaneous emission spectrum.
The spontaneous emission spectrum of a two-level atom in a one-dimensional (1D) anisotropic photonic crystal with symmetry of double helix is investigated using Laplace transform. Properties of spectrum of the radiation modes are discussed in detail under different values of detuning of atomic resonant frequency from a band-edge. Since spontaneous emission is proportional to the density of states (DOS), it is suppressed within the reflection band, corresponding to vanishing of the DOS. However, it is enhanced at the band edge. Therefore, spontaneous emission spectrum of the system is influenced at the edge of the stop band rather than at its centre.
Spontaneous emission in photonic crystals with anisotropic three-dimensional dispersion relation is studied. If the upper level is below a characteristic frequency omega(1), or above omega(2), or between omega(1) and omega(2), the radiation is a localized field with a frequency in the band gap, or a propagating field with a frquency in the band, or a diffusion field, respectively. An analytical expression for the Lamb shift is obtained. The Lamb shift for the current case is small compared to that in an ordinary vacuum or in one- or two-dimensional photonic crystals due to lower density of states.
The propagation and emission of light in a real photonic crystal is routinely affected by the intrinsic disorder and its finite-size. Thus, theoretical calculation based on the notion of a defect free crystal with an infinite thickness is irrational in the real world experiments. Here, we report an acute modification of light propagation and emission induced by the finite-size of the crystal. We have measured the emission intensity and rate from single domain of the crystal to study the finite size effect on the spontaneous emission suppression. The results show spatial fluctuation of spontaneous emission suppression across domains that have a linear scaling with crystal size. The results have implications in contriving devices pertaining to controlled spontaneous emission such as in cavity quantum electrodynamics.
The spontaneous emission field and spectrum of a two-level atom, located in an isotropic photonic crystal with dynamic band edges, are investigated by means of numeric calculation. The investigation is expected to help comprehend the characteristics of the atomic spontaneous emission in the dynamic photonic crystal, and provide a possible way to control dynamically the spontaneous emission in photonic crystal. The expression of the spontaneous radiation field is obtained without using the far-zone approximation and the Weisskopf-Wigner approximation, and expected to be applicable in other relevant researches. In the investigation, the spontaneous radiation field and spectrum are calculated when the band edge frequency is unmodulated, or modulated by a step function or triangle function. In the unmodulated situation, the radiation field intensity tends to a constant which is equal to the intensity of the localized field component. The radiation field pulse presents a wave packet behavior as propagation distance increases. The components of the radiation field correspond one-to-one to the peaks in the spontaneous radiation spectrum. When the band edge frequency is modulated by step function, the radiation field intensity tends to a steady-state value after the modulation has happened. And the steady-state intensity is affected by the time when the modulation happens. The components of the non-localized field and the frequency of the localized field after modulation depend on the atomic transition frequency and the band edge frequency, and are identical to those in the unmodulated situation with the same parameters. When the band edge frequency is modulated by a triangle function, the field intensity presents a decaying quasi-periodic oscillation after a long enough time. The modulation frequency determines the frequency of the oscillation, and influences the decay rate. The radiation energy becomes sharp peaks around a set of the discrete frequencies which are evenly spaced with the modulation frequency. The central frequency of these frequencies depends on the atomic transition frequency and the value range of the band edge frequency. The modulation initial phase affects the intensity of the radiation field emitted in an initial period of time.