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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Spectral Properties of Vacuum Energy and Spontaneous Emission in Fabry-Perot Micro-/Nano-Resonators

Daniel Repp, Markus Pollnau

Published Jun 1, 2019• 0 citations
Publisher

Abstract

Summary form only given. It is often assumed that spontaneous emission by an emitter placed inside a Fabry-Perot resonator into one of the longitudinal resonator modes occurs with the spectral shape of that mode, because it is triggered by a vacuum photon that has the same spectral shape as the resonator mode. More generally, broadband luminescence by many such emitters would then have to occur with the spectral shape of the Airy distribution of the Fabry-Perot resonator [1]. The assumption is incorrect in several respects. (i) The uncertainty principle for energy and time predicts that a vacuum photon at 1 μm wavelength survives approx. τ = 1 fs, i.e., it travels a hundred nanometers, shorter than the length of most optical resonators, except for nano-resonators. Consequently, it does not experience interference after one round trip, hence vacuum energy should occur as white light. (ii) Spontaneous emission typically occurs over a time that is much longer than one resonator-round-trip time [2]. If the emission time were infinite, then white-light vacuum energy would induce spontaneous emission that establishes the Airy distribution, because the emission interferes with itself over an infinite number of round trips. (iii) If vacuum energy resembled the spectral properties of the mode, the spectral distribution would be narrower. (iv) When also considering the finite emission lifetime, we find surprising relations between different resonator parameters. We investigate a single emitter (e.g. rare-earth ion, quantum dot) with a lifetime r inside a resonator of length t and mirror reflectivities R1 = R2 = R, triggered by white-light vacuum energy. The electric field is modelled as a superposition of plane waves emitted at integer multiples of the round-trip time. We calculate the amplitude of the newly emitted wave, taking into account the loss of energy of the emitter, and calculate the normalized spectral-intensity profile. For a sufficient number of round trips, we find convergence and calculate the FWHM linewidth A v. The results for different ratios of Liz and different R are shown in Fig. 1 The position on the x-axis for different combinations of emitters and resonator lengths is outlined in Table 1.

DOI
10.1109/cleoe-eqec.2019.8871672

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Summary form only given. It is often assumed that spontaneous emission by an emitter placed inside a Fabry-Perot resonator into one of the longitudinal resonator modes occurs with the spectral shape of that mode, because it is triggered by a vacuum photon that has the same spectral shape as the resonator mode. More generally, broadband luminescence by many such emitters would then have to occur with the spectral shape of the Airy distribution of the Fabry-Perot resonator [1]. The assumption is incorrect in several respects. (i) The uncertainty principle for energy and time predicts that a vacuum photon at 1 μm wavelength survives approx. τ = 1 fs, i.e., it travels a hundred nanometers, shorter than the length of most optical resonators, except for nano-resonators. Consequently, it does not experience interference after one round trip, hence vacuum energy should occur as white light. (ii) Spontaneous emission typically occurs over a time that is much longer than one resonator-round-trip time [2]. If the emission time were infinite, then white-light vacuum energy would induce spontaneous emission that establishes the Airy distribution, because the emission interferes with itself over an infinite number of round trips. (iii) If vacuum energy resembled the spectral properties of the mode, the spectral distribution would be narrower. (iv) When also considering the finite emission lifetime, we find surprising relations between different resonator parameters. We investigate a single emitter (e.g. rare-earth ion, quantum dot) with a lifetime r inside a resonator of length t and mirror reflectivities R1 = R2 = R, triggered by white-light vacuum energy. The electric field is modelled as a superposition of plane waves emitted at integer multiples of the round-trip time. We calculate the amplitude of the newly emitted wave, taking into account the loss of energy of the emitter, and calculate the normalized spectral-intensity profile. For a sufficient number of round trips, we find convergence and calculate the FWHM linewidth A v. The results for different ratios of Liz and different R are shown in Fig. 1 The position on the x-axis for different combinations of emitters and resonator lengths is outlined in Table 1.
The processes of absorption and stimulated emission are investigated via (i) energy conservation in a Fabry-Perot resonator, (ii) the Lorentz-oscillator model, and (iii) the Kramers-Kronig relations. Vacuum fluctuations are added and the relation between stimulated and spontaneous emission is clarified. It is shown that previous models start from incorrect assumptions about the physical foundations of spontaneous emission. The non-orthogonality of longitudinal optical modes in a linear resonator is investigated and its consequences for the laser linewidth are discussed. Based on these results, a semi-classical approach to the laser linewidth below, around, and above the laser threshold which is consistent with resonator theory is presented. These results also shed new light on the usually considered linewidth-narrowing and -broadening factors.
Summary form only given. Over the last three decades of the 20 th century many experiments have demonstrated that the rate of spontaneous emission for an emitter is determined by its electromagnetic surrounding. This phenomenon has been usually studied for atoms or molecules placed in the vicinity of mirrors or in optical resonators. Furthermore, it is also well-known that the fluorescence lifetime of an emitter changes after being embedded in dielectric media where aside from a change in the refractive index, it experiences the local fields in the dielectric. For a spherical medium larger than the order of a wavelength it is known that the coupling to Mie modes of this dielectric resonator has a large influence on the emission rate of an emitter inside. However, when the medium becomes comparable or smaller than a wavelength one can no longer speak of resonator modes. Here we report on an experiment where we have studied the variation of the spontaneous emission rate as the dielectric medium surrounding an emitter is made smaller and smaller down to a few nanometers.
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.
Optical mimcavities are resonators that have at least one dimension on the order of a single optical wavelength. These structures enable one to control the optical emission properties of materials placed inside them. They can for example, modify the spatial distribution of radiation power, change the spectral width of the emimd light, and enhance or suppress the spontaneous emission rate [l]. In addition to being attractve for studying the fundamental physics of the interaction between luminescent atoms and vacuum field, optical mimavities hold technological promise for constructing nove1 kinds of light emitting devices.
We discuss the frequency dependence of the spontaneous-emission rate (power) in a Fabry-Perot microcavity containing a Lorentz dielectric, with emphasis on modifications of the spectrum under the circumstances of the strong medium-cavity coupling. With the standard-cavity tuning, the total spontaneous-emission power spectrum consists of a multiwave version of the familiar low-frequency vacuum Rabi split peak and a broad structure on the high-frequency side coming from the enhanced spontaneous emission into the continuum of the obliquely propagating waves. This form of the spectrum derives from the properties of the (single-wave) radiation-intensity spectrum of the molecule. Depending on the observation direction, the radiation-intensity spectrum exhibits either a double-peak structure characteristic of the vacuum Rabi splitting, as observed in light transmisssion and reflection experiments, or a single-peak structure describing the enhanced spontaneous emission at the corresponding cavity resonant frequency.
Anomalously dispersive cavities, particularly white-light cavities, may have larger bandwidth to finesse ratios than their normally dispersive counterparts. Partly for this reason, they have been proposed for use in laser interferometer gravitational-wave observatory (LIGO)-like gravity-wave detectors and in ring-laser gyroscopes. In this paper we analyze the quantum noise associated with anomalously dispersive cavity modes. The vacuum field energy associated with a particular cavity mode is proportional to the cavity-averaged group velocity of that mode. For anomalously dispersive cavities with group index values between 1 and 0, this means that the total vacuum field energy associated with a particular cavity mode must exceed $\ensuremath{\hbar}\ensuremath{\omega}/2$. For white-light cavities in particular, the group index approaches zero and the vacuum field energy of a particular spatial mode may be significantly enhanced. We predict enhanced spontaneous emission rates into anomalously dispersive cavity modes and broadened laser linewidths when the linewidth of intracavity emitters is broader than the cavity linewidth.