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Physical Review A

Frequency-dependent cavity lifetime and apparent superluminality in Fabry-Pérot-like interferometers

Hsin-Yu Yao, N. C. Chen, Tsun‐Hsu Chang, Herbert G. Winful

Published Nov 28, 2012• 12 citations
Publisher

Abstract

Extraordinary group delays shorter than the transit time of light propagating at $c$ through an equal distance have been experimentally demonstrated in single-Fabry-P\'erot (FP) waveguide systems and cascaded-FP structures under off-resonant conditions. These ``superluminal'' phenomena are well explained by the multiple-reflection destructive interference that reduces the intracavity stored energy when operating off resonances. Excellent agreement between theory and experiment is obtained when the dispersive effects of reflective boundaries are considered. These results provide further insight into the nature of apparent superluminality in regions of allowed propagation.

DOI
10.1103/physreva.86.053832

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Extraordinary group delays shorter than the transit time of light propagating at ccc through an equal distance have been experimentally demonstrated in single-Fabry-P'erot (FP) waveguide systems and cascaded-FP structures under off-resonant conditions. These ``superluminal'' phenomena are well explained by the multiple-reflection destructive interference that reduces the intracavity stored energy when operating off resonances. Excellent agreement between theory and experiment is obtained when the dispersive effects of reflective boundaries are considered. These results provide further insight into the nature of apparent superluminality in regions of allowed propagation.
This work develops a time-domain approach to explain the origin of superluminality, i.e. , an apparent faster-than-light phenomenon in one-dimensional Fabry-Pérot cavities. The interferences between multiple-reflection echoes are tracked and analyzed. Under the quasi-static condition, the destructive interference is found to result in the advancement of the superposed pulse as compared to its parent signals. It is the key to the superluminal effect. On the contrary, constructive interference can lead to the retardation of the superposed pulse. For a cavity operated at the off-resonance state, the competition between roundtrip destructive interference (group advancement) and multiple-roundtrip constructive interference (group retardation) determines how short the group delay can be observed. Manipulating the competition between roundtrip interferences opens numerous opportunities in controlling the signal delays or shaping the pulse waveforms.
The exact expressions of finesse and full width at half maximum (FWHM) of the Airy distribution of a Fabry-P\'erot (FP) resonator are derived, which solves the breakdown problem at ultralow reflectivity in traditional formulas. We demonstrate that, when the cavity-mirror reflectivity $R\ensuremath{\rightarrow}0$, the FWHM approaches half of the free spectral range rather than infinity; moreover, the cavity finesse F\mathcal{F}F approaches 2 rather than zero. These expressions are useful for the development of bad-cavity lasers, such as the four-level active optical clock based on the strong atomic transition of cesium. Also, the exact expressions of the FWHM and finesse of the reflection distribution composed of reflected mode profiles are derived, which separately intersect with that of the Airy one at R=0R=0R=0. In addition, the characteristics of an antiresonant cavity, the frequency of which is exactly at the center of two adjacent resonant cavity modes, are analyzed. It is demonstrated that at the resonant and antiresonant frequencies the enhancement and inhibition of intracavity light intensity caused by the FP cavity are symmetrical in the logarithmic view. Furthermore, we provide a universal expression of the cavity-enhancement factor rather than the classical representation of ``$\text{2}\mathcal{F}/\phantom{\text{2}\mathcal{F}\ensuremath{\pi}}\phantom{\rule{0.0pt}{0ex}}\ensuremath{\pi}$,'' which is inadequate for the high-loss cavity because the cavity-enhancement factor should naturally reach 1 when $R\ensuremath{\rightarrow}0$. Finally, we extend the application of the antiresonant cavity to an inhibited laser, the frequency of which has a stronger suppression effect on the cavity-length thermal noise than the traditional resonant laser.
Pulse propagation through 1-dimensional stratified media is studied. The general features of pulse propagation in relation to the delay or advancement of the pulse are discussed. It is shown that in a lossless system with spatial symmetry the reflected and the transmitted pulses have the same delay or advancement. The phase times and the explicit output pulse profiles for an input Gaussian pulse are calculated. While recalling recent results for a variety of systems (resonant absorbers or heterogeneous medium in a Fabry-Perot cavity, a slab of negative index medium, a left-handed/right-handed periodic structure, etc.), new results involving coupled waveguides and a high finesse cavity containing an electromagnetically induced transparency (EIT) medium are presented. Calculations clearly show that for pulses with a carrier tuned at the resonance of the stratified media, one generally has subluminal propagation. Alternatively, in stop gaps or away from the resonances one ends up with superluminal transit. In the case of the EIT cavity it is shown that the large dispersion offered by the intracavity medium can lead to substantial superluminality (ng≃−106) with peak transmission as high as 20%. This is in sharp contrast to large signal attenuation for usual superluminal transit. Note also that EIT, to the best of the authors knowledge, has so far been used to achieve only slow light.
Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.
There have been many recent theoretical and experimental reports on the propagation of light pulses at speeds exceeding the speed of light in vacuum ccc within media with anomalous dispersion, either opaque or with gain. Superluminal propagation has also been reported within vacuum, in the case of inhomogeneous pulses. In this paper we show that the observations of superluminal and non-causal propagation of evanescent pulses under the conditions of frustrated internal reflection are only apparent, and that they can be simply explained employing an explicitly (sub)luminal causal theory. However, the usual one-dimensional approach to the analysis of pulse propagation has to be abandoned and the spatial extent of the incoming pulse along the directions normal to the propagation direction has to be accounted for to correctly interpret the propagation speed of these evanescent waves. We illustrate our theory with animations of the time development of a pulse built upon the Huygen's construction.