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Physical Review Letters

Counterintuitive Dispersion Violating Kramers-Kronig Relations in Gain Slabs

Li‐Gang Wang, Lin Wang, M. Al-Amri, Shi‐Yao Zhu, Muhammad Suhail Zubairy

Published Jun 11, 2014• 20 citations
Publisher

Abstract

We demonstrate the counterintuitive dispersion effect that the peaks (dips) in the gain spectrum correspond to abnormal (normal) dispersion, contrary to the usual Kramers-Kronig point of view. This effect may also lead to two unique features: a broadband abnormal dispersion region and an observable Hartman effect. These results are explained in terms of interference and boundary effects. Finally, two experiments are proposed for the potential experimental verification.

DOI
10.1103/physrevlett.112.233601

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We demonstrate the counterintuitive dispersion effect that the peaks (dips) in the gain spectrum correspond to abnormal (normal) dispersion, contrary to the usual Kramers-Kronig point of view. This effect may also lead to two unique features: a broadband abnormal dispersion region and an observable Hartman effect. These results are explained in terms of interference and boundary effects. Finally, two experiments are proposed for the potential experimental verification.
In this paper, we study theoretically a pump–probe model for the Kramers–Kronig (KK) relations during laser operation. A solution of laser equations reveals that the dispersion profile of a saturated laser gain medium is non-analytical at the boundary between the lasing and the non-lasing spectral region. Such a non-analyticity cannot be explained in terms of the KK relations. In order to interpret this situation, it is important to consider carefully the physical basis of the KK relations. We conclude that the KK relation is expected to apply only to an independent probe applied to the medium, which is under excitation by the pump producing the gain as well as the lasing mode. The absorption/gain and dispersion profiles are then analytical, and satisfy the KK relations. Specifically, these are variants of the so-called Mollow–Ezekiel spectra of probe absorption/gain and dispersion in the presence of a pump, with the exception that in this case the medium is inverted.
Anomalous dispersion cannot occur in a transparent passive medium where electromagnetic radiation is being absorbed at all frequencies, as pointed out by Landau and Lifshitz. Here we show, both theoretically and experimentally, that transparent linear anomalous dispersion can occur when a gain doublet is present. Therefore, a superluminal light-pulse propagation can be observed even at a negative 7group velocity through a transparent medium with almost no pulse distortion. Consequently, a negative transit time is experimentally observed resulting in the peak of the incident light pulse to exit the medium even before entering it. This counterintuitive effect is a direct result of the rephasing process owing to the wave nature of light and is not at odds with either causality or Einstein's theory of special relativity.
Two theorems on the group velocity are presented in this paper. First a simple proof is given that for any dispersive dielectric, there must be a frequency at which the group velocity of an electromagnetic pulse becomes abnormal, i.e., greater than the vacuum speed of light, infinite, or negative. Second, at the frequency at which the attenuation (or gain) is a maximum, the group velocity must be abnormal (or normal). This second theorem is more widely applicable, e.g., to propagation in waveguides or through multilayer dielectrics. To illustrate these theorems we discuss dispersion in a medium with two resonance lines, one absorption and the other gain. We find that the group velocity is abnormal within the absorption line and in a transparent region outside the gain line.