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Journal of the Optical Society of America B

Gain effects on microsphere resonant emission structures

Vitaly V. Datsyuk

Published Jan 1, 2002• 10 citations
Publisher

Abstract

The effects of light absorption and amplification on the transition rates of the electric-dipole emission of atoms or molecules embedded in micrometer-sized dielectric spheres are investigated in the framework of classical electrodynamics. The expression for the form factor of a resonant emission line is derived. Its dependence on the imaginary part of the light’s refractive index is investigated. It is discovered that the active sphere does not act as an optical resonator if light amplification is strong.

DOI
10.1364/josab.19.000142

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The effects of light absorption and amplification on the transition rates of the electric-dipole emission of atoms or molecules embedded in micrometer-sized dielectric spheres are investigated in the framework of classical electrodynamics. The expression for the form factor of a resonant emission line is derived. Its dependence on the imaginary part of the light’s refractive index is investigated. It is discovered that the active sphere does not act as an optical resonator if light amplification is strong.
We have calculated the effect of material absorption on the rate of spontaneous emission by an embedded atom using a macroscopic Green-function approach and a microscopic Hamiltonian method that includes reservoir damping. When local field effects are neglected, the free-space emission rate is modified by the real part of the refractive index at the transition frequency of the embedded atom. Local field effects are introduced via a local field correction factor. This modification of the spontaneous emission rate generalizes a well-known result for transparent media and is of crucial importance in assessing the degree of inhibition of spontaneous emission for frequencies close to the resonances of the dielectric.
The transition frequency shift and the spontaneous decay rate (the radiative linewidth) variation of an atom placed in a transparent dielectric microsphere are presented. It is demonstrated that with the refractive index of the microsphere material (e.g. diamond) being high enough, the size of the microsphere can be selected such that the spontaneous decay rate decreases (or increases) several times, depending on the position of the atom. At the same time, the atomic transition frequency shift proves to depend more strongly on the position of the atom, reaching high values near the surface of the microsphere.
The rate of spontaneous electromagnetic emission in a cavity is the normalized output radiation rate in the classical electrodynamics and the Einstein AAA coefficient in a quantum theory. We reveal the difference and similarity between the quantum and classical rates by calculations of the relative output power of the resonant spontaneous electric-dipole emission as a function of light absorption in a high-QQQ cavity. We show that the relative resonant output power is equal to the relative width of a morphology-dependent resonance in the spectral dependence of the Einstein AAA coefficient. The validity of the theory is demonstrated by numerical verification of an experimental result on a microsphere.
The study of optical effects in active media is rich with unexpected consequences. For instance, one may imagine that absorption and amplification in a dielectric medium will exhibit some sort of symmetry because both are related to the same imaginary part of the dielectric constant, except for a sign change. It turns out that such a symmetry does not exist [1]. This issue has also been investigated in detail for random or disordered gain media, with varying viewpoints [2]. Traditionally, much of the research on amplifying media has considered the interaction of light within the entire volume of such a medium. Recent interest in metamaterials and other esoteric structures in which plasmons are used to manipulate optical signals has brought attention to the role of surface waves in active dielectrics [3, 4]. From a fundamental perspective, the main difference between active and passive media is that spontaneous emission cannot be avoided in gain media. As a result, a rigorous analysis of gain media demands a quantum-mechanical treatment. Spontaneous emission in a gain medium depends not only on the material properties of that medium but also on the optical modes supported by the structure containing that material [5]. By a clever design of this structure (e.g., photonic crystals or microdisk resonators with a metallic cladding), it is possible to control the local density of optical modes and the spontaneous emission process itself. In Chapter 4, we discuss how to model a gain medium under such conditions by deploying optical Bloch equations.
Spontaneous decay of an excited atom near a dispersing and absorbing microsphere of given complex permittivity that satisfies the Kramers-Kronig relations is studied, with special emphasis on a Drude-Lorentz permittivity. Both the whispering gallery field resonances below the band gap (for a dielectric sphere) and the surface-guided field resonances inside the gap (for a dielectric or a metallic sphere) are considered. Since the decay rate mimics the spectral density of the sphere-assisted ground-state fluctuation of the radiation field, the strengths and widths of the field resonances essentially determine the feasible enhancement of spontaneous decay. In particular, strong enhancement can be observed for transition frequencies within the interval in which the surface-guided field resonances strongly overlap. When material absorption becomes significant, then the highly structured emission pattern that can be observed when radiative losses dominate, reduces to that of a strongly absorbing mirror. Accordingly, nonradiative decay becomes dominant. In particular, if the distance between the atom and the surface of the microsphere is small enough, the decay becomes purely nonradiative.