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.
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 \ensuremathℏ\ensuremathω/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.
Over the past 15 years, several groups have engineered media that are both strongly dispersive and roughly transparent for some finite bandwidth. Relationships and intuitive models that are satisfactory when it is reasonable to neglect dispersion may then fail. We analyze three such cases of failure. First, a simple generalization of the Abraham and Minkowski momenta to dispersive media entails multiplying each per-photon momentum by n/ng, where n is the refractive index and ng is the group index. The resulting forms are experimentally relevant for the case of the Abraham momentum, but not for the Minkowski momentum. We show how dispersion modulates the displacement of a sphere embedded in a dispersive medium by a pulse. Second, pulse transformation in a nonstationary medium is modulated by the presence of dispersion. Using an explicit description of the kinetics of dispersive nonstationary inhomogeneous media, we show how the group velocity can modulate pulse response to a change in the refractive index and how Doppler shifts may become large in a dispersive medium as the velocity of the Doppler shifting surface approaches the group velocity. We explain a simple way to use existing technology to either compress or decompress a given pulse, changing its bandwidth and spatial extent by several orders of magnitude while otherwise preserving its envelope shape. Finally, we note that the nature of a single optical cavity quasimode depends on intracavity dispersion. We show that the quantum field noise associated with a single cavity mode may be modulated by dispersion. For a well-chosen mode in a high-Q cavity, this can amount to either an increase or a decrease in total vacuum field energy by several orders of magnitude. We focus on the "white light cavity," showing that the quantum noise of an ideal white light cavity diverges as the cavity finesse improves.
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.
We analyze the effect of a highly dispersive element placed inside a modulated optical cavity on the frequency and amplitude of the modulation to determine the conditions for cavity self-stabilization and enhanced gyroscopic sensitivity. Hence, we model cavity rotation or instability by an arbitrary AM/FM modulation, and the dispersive element as a phase and amplitude filter. We find that anomalous dispersion may be used to self-stabilize a laser cavity, provided the magnitude of the group index of refraction is smaller than the phase index of refraction in the cavity. The optimal stabilization is found to occur when the group index is zero. Group indices with magnitudes larger than the phase index (both normal and anomalous dispersion) are found to enhance the sensitivity of a laser gyroscope to rotation. Furthermore, our results indicate that atomic media, even coherent superpositions in multilevel atoms, are not useful for these applications, because the amplitude and phase filters work against one another, i.e., decreasing the modulation frequency increases its amplitude and vice versa, with one exception: negative group indices whose magnitudes are larger than the phase index result in negative, but enhanced, beat frequencies. On the other hand, for optical resonators the dispersion reversal associated with critical coupling enables the amplitude and phase filters to work together under a greater variety of circumstances than for atomic media. We find that for single over-coupled resonators, or in the case of under-coupled coupled-resonator-induced absorption, the absorption and normal dispersion on-resonance increase the contrast and frequency of the beat-note, respectively, resulting in a substantial enhancement of the gyroscopic response. Moreover, for cavity self-stabilization, we propose the use of a variety of coupled-resonator induced transparency that is accompanied by anomalous dispersion.
We analyze the effect of a highly dispersive element placed inside a modulated optical cavity on the frequency and amplitude of the output modulation to determine the conditions for enhanced gyroscopic sensitivities. The element is treated as both a phase and amplitude filter, and the time dependence of the cavity field is considered. Both atomic gases (two level and multilevel) and optical resonators (single and coupled) are considered and compared as dispersive elements. We find that it is possible to simultaneously enhance the gyro scale factor sensitivity and suppress the dead band by using an element with anomalous dispersion that has greater loss at the carrier frequency than at the sideband frequencies, i.e., an element that simultaneously pushes and intensifies the perturbed cavity modes, e.g. a two-level absorber or an undercoupled optical resonator. The sensitivity enhancement is inversely proportional to the effective group index, becoming infinite at a group index of zero. However, the number of round trips required to reach a steady state also becomes infinite when the group index is zero (or two). For even larger dispersions a steady state cannot be achieved, and nonlinear dynamic effects such as bistability and periodic oscillations are predicted in the gyro response.
The theory of the laser linewidth is formulated to account for arbitrarily large output couplings and spatial hole burning. We show explicitly that the linewidth can be interpreted in terms of either spontaneous-emission noise or the amplification of vacuum field modes leaking into the cavity, depending on the ordering of operators in the correlation function determining the laser spectrum. This allows us to derive the Petermann K factor associated with ``excess spontaneous-emission noise'' in a physically transparent and mathematically simple way, without the need to introduce adjoint modes of the resonator. It also allows us to straightforwardly include spatial-hole-burning effects, which are found to increase the K factor and the linewidth in high-gain systems appreciably.
The measurement sensitivity of a cavity could be enhanced for sensing applications in an anomalous dispersion condition. In this paper, we find that when injecting a squeezed vacuum light into a dispersive cavity, the sensitivity could be improved further, even beating the standard quantum limit at the detuning frequencies via controlling the dispersion condition. It is amazing that normal dispersion is superior to anomalous dispersion in improving the measurement sensitivity in the presence of a squeezed vacuum field. It implies this paper could be used to realize high-precision sensing applications.