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arXiv:quant-ph/9906017 · Journal of Lightwave Technology

Dipole emission in finite photonic bandgap structures: an exactly solvable one-dimensional model

J.P. Dowling

Published Nov 1, 1999• 29 citations
arXivHTMLPDF

Abstract

This paper considers an exact model of atomic spontaneous dipole emission and classical dipole radiation in a finite photonic bandgap (PBG) structure. The full three-dimensional (3-D) or two-dimensional (2-D) problem is reduced to a finite one-dimensional (1-D) model, which then is solved for analytically using algebraic matrix transfer techniques. The results give insight to the electromagnetic emission process in periodic dielectrics, quantitative predictions for emission in 1-D dielectric stacks, and qualitative formulas for the 2-D and 3-D problem.

DOI
10.1109/50.803005

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This paper considers an exact model of atomic spontaneous dipole emission and classical dipole radiation in a finite photonic bandgap (PBG) structure. The full three-dimensional (3-D) or two-dimensional (2-D) problem is reduced to a finite one-dimensional (1-D) model, which then is solved for analytically using algebraic matrix transfer techniques. The results give insight to the electromagnetic emission process in periodic dielectrics, quantitative predictions for emission in 1-D dielectric stacks, and qualitative formulas for the 2-D and 3-D problem.
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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.
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.
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.
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.