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Possibility of a Light Pulse with Speed Greater than c

Xian-jian Zhou

Published Jan 1, 2000• 12 citations

Abstract

In two models it is shown that a light pulse propagates from a vacuum into certain media with velocity greater than that of a light in a vacuum (c). By numerical calculation the propagating properties of such a light are given. Recently L. J. Wang and his collaborators in their experiment [1] have found that the group velocity of a laser pulse in particularly prepared atomic caesium gas can much exceed that of light in a vacuum (c). Here we will explore the problem from a theoretical piont of view. Many years ago A. Sommerfeld rigorously proved [2] that the velocity of light pulse can not exceed c in absorption media. We call this conclusion as basic theorem afterwardz. In the following we will first examine why usually the basic theorem holds and in what condition it will be violated. Then two models are proposed, where with precise numerical calculation we will show that the basic theorem is indeed violated. Therefore the properties of light propagating in media of the models are given. We believe that these properties may appears in more realistic models. 1 Let a light pulse propagating along x axis toward its positive direction in a vacuum (x < 0), at time t = 0 arriving at x = 0, and then entering into a medium (x ≥ 0) afterwards. At x = 0, the amplitude of the light pulse changes with time t as ⎨ F(t) t ≥ 0 f(t) = ⎩ 0 t < 0. Usually f(t) can be rigorously expressed in Fourier integration as f(t) = Re A(n)e −int dn. (2) 0 For simplicity suppose there is no reflection of light at x = 0. The amplitude of the light pulse entering into the medium (x ≥ 0) is [2] f(t, x) = Re A(n)e −int+ikx dn, (3)

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In two models it is shown that a light pulse propagates from a vacuum into certain media with velocity greater than that of a light in a vacuum (c). By numerical calculation the propagating properties of such a light are given. Recently L. J. Wang and his collaborators in their experiment [1] have found that the group velocity of a laser pulse in particularly prepared atomic caesium gas can much exceed that of light in a vacuum (c). Here we will explore the problem from a theoretical piont of view. Many years ago A. Sommerfeld rigorously proved [2] that the velocity of light pulse can not exceed c in absorption media. We call this conclusion as basic theorem afterwardz. In the following we will first examine why usually the basic theorem holds and in what condition it will be violated. Then two models are proposed, where with precise numerical calculation we will show that the basic theorem is indeed violated. Therefore the properties of light propagating in media of the models are given. We believe that these properties may appears in more realistic models. 1 Let a light pulse propagating along x axis toward its positive direction in a vacuum (x < 0), at time t = 0 arriving at x = 0, and then entering into a medium (x ≥ 0) afterwards. At x = 0, the amplitude of the light pulse changes with time t as ⎨ F(t) t ≥ 0 f(t) = ⎩ 0 t < 0. Usually f(t) can be rigorously expressed in Fourier integration as f(t) = Re A(n)e −int dn. (2) 0 For simplicity suppose there is no reflection of light at x = 0. The amplitude of the light pulse entering into the medium (x ≥ 0) is [2] f(t, x) = Re A(n)e −int+ikx dn, (3)

Contemporary observational and theoretical studies on the temporal nature of microscopic measurements renewed the discussion about the fundamental constants, leading to the possibility of light speed variation and superluminal pulse propagation. Gain assisted experiments using anomalous dispersion near an absorption line in atomic gas, a "fast - light" medium, seem to lead to a wave group velocity vG exceeding c, the vacuum speed of light; moreover, definition of the information velocity vi sets the question of interpretation of the three speeds: one view is that vi = vG, but this violates Causality; another view is that vi = c in all situations, but this limits, a priori, the transport of information. Another view is that vi, vG and c are distinct. This contribution follows the last possibility. A draft discussion on space-time is given.

Manipulating the group velocity of light has resulted in many advances, including slowing light to a car’s pace, stopping light and even forcing pulses to propagate faster than the speed of light in a vacuum. While the group velocity is equal to or less than the speed of light in a vacuum (c) when propagating through a medium exhibiting normal dispersion, anomalous dispersion allows the group velocity to be greater than c, or even negative. Strangely, the peak of a pulse propagating with a negative group velocity through a medium appears to exit the medium before entering it. This is a result of interference from spectral components of a pulse propagating with different phase velocities in the medium.

The propagation of a Gaussian light pulse through a medium having a positive or negative absorption line is examined. Analytical approximations are obtained for the case where the spectral width of the pulse is much smaller than that of the line. It is shown that the pulse remains substantially Gaussian and unchanged in width for many exponential absorption depths, and that the locus of instants of maximum amplitude follows the classical expression for the group velocity, even if this is greater than the velocity of light, or negative. Numerical calculations have been used to examine what happens beyond the limit of usefulness of the analytical approximations.

Einstein's theory of relativity establishes the speed of light in vacuum, c, as a fundamental constant. However, in optically dense media, light can be made to travel at speeds lower than c, e.g. light in water propagates at 3/4c. More dramatically, the speed of light pulses can be altered by orders of magnitude in dispersive media such as cold atomic clouds [1], atomic vapours [2] and structurally engineered materials [3]. Under exotic conditions, the velocity of these pulses can be rendered greater or smaller than c, or even negative [4]. While significant control can be exerted over the speed of light in dispersive media, no experimental demonstration of altered light speed has hitherto been achieved in vacuum for propagating physical beams of finite energy. Here we show that pulsed beams of twisted light can exhibit sub-luminal velocities upon vacuum propagation. Our experiment reveals changes in the range of 0.1% in the velocity of propagating twisted pulsed beams relative to c. Though our work poses no challenge to relativity theory, it experimentally confirms a counterintuitive body of theoretical work on the group velocity of structured pulses in vacuum, and introduces a novel theory of pulse velocity for twisted light. Our results are particularly important given the recent surge of interest in the application of structured beams to telecommunication [5], quantum key distribution [6] and quantum information [7].

It is shown that the prediction of Garrett and McCumber that a light pulse's velocity may exceed the speed of light in a resonantly absorbing medium is due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light. It is also shown that in certain cases a pulse's maximum will propagate at the group velocity even when the pulse as a whole is distorted by dispersion.