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arXiv:2009.13625 · Physical Review A

Negative superluminal velocity and violation of Kramers-Kronig relations in causal optical systems

Mehmet Emre Taşgın

Published Jan 5, 2021• 0 citations
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Abstract

We investigate nonanalyticities (e.g., zeros and poles) of refractive index $n(\ensuremath{\omega})$ and group index ${n}_{g}(\ensuremath{\omega})$ in different optical media. We first demonstrate that while a Lorentzian dielectric has no nonanalyticity in the upper half of the complex frequency plane (CFP), its group index---which governs the pulse-center propagation---possesses zeros in the same region. Thus we classify the nonanalyticities as in the (a) first order (refractive index or reflection) and (b) second order (group index or group delay). The latter contains the derivative of the former. Then, we study a possible connection between the negative superluminal velocities and the presence of nonanalyticities in the upper half of the CFP. We show that the presence of nonanalyticities in the upper half of the CFP for (a) the first-order and (b) the second-order responses are accompanied by the appearance of negative (a) phase and (b) group velocities, respectively. We also distinguish between two kinds of superluminosity, $v>c$ and $v<0$, where we show that the second one ($v<0$) appears with the violation of KKRs.

DOI
10.1103/physreva.103.013504

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We investigate nonanalyticities (e.g., zeros and poles) of refractive index n(\ensuremathω)n(\ensuremath{\omega})n(\ensuremathω) and group index ng(\ensuremathω){n}_{g}(\ensuremath{\omega})ng​(\ensuremathω) in different optical media. We first demonstrate that while a Lorentzian dielectric has no nonanalyticity in the upper half of the complex frequency plane (CFP), its group index---which governs the pulse-center propagation---possesses zeros in the same region. Thus we classify the nonanalyticities as in the (a) first order (refractive index or reflection) and (b) second order (group index or group delay). The latter contains the derivative of the former. Then, we study a possible connection between the negative superluminal velocities and the presence of nonanalyticities in the upper half of the CFP. We show that the presence of nonanalyticities in the upper half of the CFP for (a) the first-order and (b) the second-order responses are accompanied by the appearance of negative (a) phase and (b) group velocities, respectively. We also distinguish between two kinds of superluminosity, v>cv>cv>c and v<0v<0v<0, where we show that the second one (v<0v<0v<0) appears with the violation of KKRs.
Abstract This paper considers the question of superluminal (or negative) group velocities of signals in dispersive media, as well as negative group delay times of signals passing through linear filters. It is shown that information can be transmitted only with the help of jumps in the time dependence of a function or its derivatives and, therefore, the information propagation velocity in any (including dispersive, absorbing, or amplifying) medium exactly coincides with the speed of light in vacuum. As for the group velocity of a wave packet, it is only the propagation velocity of its infinitely differentiable envelope, by means of which the transmission of information, strictly speaking, is impossible. Therefore, in regions with anomalous dispersion, the group velocity can be superluminal or negative without violating the “light limitation” of the theory of relativity or the principle of causality. Examples of situations in which superluminal or negative group velocity occurs are given.
The existence of anomalous, i.e. superluminal or negative, group velocities [1] provides one of the most amazing phenomenon of wave propagation which is compatible with, and even a consequence of, causality [2]. Anomalous wave propagation generally relies on the resonant interaction of light with an atomic medium, i.e. it exploits the dispersive properties of absorptive [3] or inverted [4,5] atomic media near resonances. Evanescent wave propagation across photonic barriers [1] provides the other wide framework where anomalous group velocities have been observed, however in such cases the achievement of a negative group velocity seems unlikely. By exploiting the dispersive properties of a gain doublet, a narrow spectral region of transparent anomalous dispersion around the dip center of two Lorentzian gain lines, leading to negative transit times and no pulse attenuation, can be created [4]. A recent demonstration thereof has been indeed reported in an experiment by Wang and coworkers using Cs vapor [5].
This study investigates the first experimental observation of faster-than-c group velocities in a passive system with neither absorption nor reflection. The investigation shows that superluminal and even negative group velocities can exist because of interference, instead of absorption or reflection. Results suggest that causal superluminal propagation can potentially be observed in any system in which input energy can escape into other modes. These results also demonstrate a particularly unusual application of the amplitude-phase Kramers-Kronig (K-K) relations; for the system being described, an infinitesimal adjustment of parameters radically affects the validity of the K-K transformation between amplitude and phase.
The propagation of narrow packets of electromagnetic waves(EMWs) in frequency dispersive medium with the consideration of the complex refractive index is studied. It is shown that counting in the dispersion of the complex refractive index within the context of the conventional expression of the group velocity of narrow wave packets of EMWs propagating in a dispersive medium results in the appearance of additional constraints on the group velocity, which dictates that the physically acceptable group velocity can only be realized in the case of a negligible imaginary part of the group index. In this paper, the conditions that allow one to realize the physically acceptable group velocity are formulated and analyzed numerically for the relevant model of the refractive index of a system of two-level atoms in the optical frequency range. It is shown that in the frequency band where superluminal light propagation is expected, there is a strong dispersion of the refractive index that is accompanied with strong absorption, resulting in a strongly attenuated superluminal light.
The propagation of narrow packets of electromagnetic waves (EMWs) in frequency dispersive medium with the consideration of the complex refractive index is studied. It is shown that counting in the dispersion of the complex refractive index within the context of the conventional expression of the group velocity of narrow wave packets of EMWs propagating in a dispersive medium results in the appearance of additional constraints on the group velocity, which dictates that the physically acceptable group velocity can only be realized in the case of a negligible imaginary part of the group index. In this paper, the conditions that allow one to realize the physically acceptable group velocity are formulated and analyzed numerically for the relevant model of the refractive index of a system of two-level atoms in the optical frequency range. It is shown that in the frequency band where superluminal light propagation is expected, there is a strong dispersion of the refractive index that is accompanied with strong absorption, resulting in a strongly attenuated superluminal light.
Subluminal and superluminal propagation of a light pulse in Kerr materials has been investigated. Group velocities as slow as much less than 1 mm per second to as fast as negative several thousands meters per second can easily be obtained in the Kerr medium, which possesses a large nonlinear refractive index and long relaxation time, such as Cr3+-doped alexandrite, ruby, and GdAlO3. The physical mechanism is the strong highly dispersive coupling between different frequency components of the pulse.