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Journal of Modern Optics

Spontaneous emission from a planar microcavity below threshold and the wolf effect

Hiroyuki Yoshimura, Toshiaki Iwai

Published Apr 1, 1997• 2 citations
Publisher

Abstract

Equivalence between a planar microcavity below threshold and a partially coherent primary source is pointed out for the first time. Spatial coherence properties of spontaneous emission from the planar microcavity are controllable by varying two cavity parameters of the cavity length and the mirror reflectivity. The spontaneous emission gives rise to the Wolf effect which is spectral changes induced by the spatial coherence. It is clearly shown from our theoretical analyses that the spectrum in the far field shifts toward higher frequencies than that on the emitting plane of the planar microcavity. In particular, the blue shift of the far-field spectrum takes a maximum value at the centre of the far-field plane and decreases gradually far away from the centre. Moreover, the blue shift increases with decreasing values of the two cavity parameters.

DOI
10.1080/09500349708230688

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Equivalence between a planar microcavity below threshold and a partially coherent primary source is pointed out for the first time. Spatial coherence properties of spontaneous emission from the planar microcavity are controllable by varying two cavity parameters of the cavity length and the mirror reflectivity. The spontaneous emission gives rise to the Wolf effect which is spectral changes induced by the spatial coherence. It is clearly shown from our theoretical analyses that the spectrum in the far field shifts toward higher frequencies than that on the emitting plane of the planar microcavity. In particular, the blue shift of the far-field spectrum takes a maximum value at the centre of the far-field plane and decreases gradually far away from the centre. Moreover, the blue shift increases with decreasing values of the two cavity parameters.
We study the transient regime of spontaneous emission by an atom excited in a planar high-Q microcavity. The variation of the spatial distribution of the radiation outside the cavity is determined as a function of the increasing number of internal reflections for transition dipole moments parallel and perpendicular to the mirrors and for mirror separations equal to $\ensuremath{\lambda}/2$ and $\ensuremath{\lambda}/4.$ Particular attention is given to the parallel dipole in the $\ensuremath{\lambda}/2$ cavity, where the number of reflections is closely proportional to the elapsed time. It is shown that the transient regime extends over times of the order of the cavity decay time, in contrast to the effectively one-dimensional cavities previously studied, where the steady state is achieved after the cavity round-trip time. The field distribution inside the cavity tends to a steady-state form whose dimension parallel to the mirrors is of the order of the transverse coherence length. Despite this transverse localization of the field excitation, it is shown that the conditions for the achievement of strong atom-field coupling and the observation of Rabi oscillations cannot be met in the planar microcavity.
A three-dimensional quantum nonperturbative theory of spontaneous emission in a planar microcavity is developed on the basis of a complete set of orthogonal standing-wave mode functions. Earlier results on the spontaneous decay rate and far-field emission pattern which have been obtained in a perturbative fashion are shown to be contained in our theory. Time evolution of the initially excited atomic state is considered and comparison with perturbative results is given. It is found that though the Fabry-Perot cavity is of an open type, vacuum-field Rabi oscillations are still possible. Conditions that are favorable for this strong coupling regime are discussed. The near-field spontaneous-emission pattern and the transition from the near field to the far field are investigated. For moderate mirror reflectivities, the spread in the near-field emission pattern is found to be in good agreement with the effective mode radius concept. When the mirror reflectivity increases, however, one has a strong coupling regime and together with it, multiple reabsorptions and reemissions of the photon may occur, leading to a better localization of the photon around the atomic position. These considerations may be useful in designing microlasers of the planar type.
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In this chapter, the author calculates the modification of the spontaneous emission pattems and the corresponding rate change using classical physics. To derive the spontaneous emission pattems or rates the readers must average the emission rate at each frequency over this distribution function. It is important to understand that this assumption is only valid if the emission line is homogeneously broadened (albeit not necessarily at its natural linewidth limit due to the radiative decay). One more thing needs to be pointed out before the readers move on to a more exact calculation of the spontaneous emission pattern and the emission rate of a dielectric cavity. In the ideal mirror the reflection takes place at a well-defined plane. Integration of the emission over all solid angles and all wavelengths gives us information about the total spontaneous emission rate, or, equivalently, about the spontaneous emission lifetime of an emitter in a dielectric cavity.
The transverse distributions of intensity inside and outside an optical cavity produced by spontaneous emission from atoms inside the cavity are reviewed. The mode properties inside a closed passive one-dimensional cavity are summarised and the spatial distributions of emitted light for a passive cavity with partially transmitting mirrors are discussed. A qualitative theory describes the effects of an amplifying medium inside the cavity, covering the regimes both below and above the lasing threshold. Expressions are derived for the angular spread of the output light in the presence of gain. A theory based on rigorous general relations between gain and noise provides more complete results for the spontaneous emission noise from an amplifying slab within an optical cavity.
A complete set of cavity modes in planar dielectric microcavities is presented which naturally includes guided modes. We show that most of these orthonormal fields can be derived from a coherent superposition of plane waves incoming on the stack from the air and from the substrate. Spontaneous emission of a dipole located inside the microcavity is analyzed, in terms of cavity modes. Derivation of the radiation pattern in the air and in the substrate is presented. The power emitted into the guided modes is also determined. Finally, a numerical analysis of the radiative properties of an erbium atom located in a Fabry-P\'erot multilayer dielectric microcavity is investigated. We show that a large amount of light is emitted into the guided modes of the structure, in spite of the Fabry-P\'erot resonance, which increases the spontaneous emission rate in a normal direction. \textcopyright{} 1996 The American Physical Society.