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Physical Review Letters

Faraday Rotation and Complex-Valued Traversal Time for Classical Light Waves

V. Gasparian, M. Ortuño, J. Ruiz, Emilio Cuevas

Published Sep 18, 1995• 59 citations
Publisher

Abstract

We introduce a magnetic clock approach to measure the traversal and reflection times of an electromagnetic wave through a slab, when multiple reflections are taken into account. The traversal time is a complex quantity whose real component is proportional to the Faraday rotation and whose imaginary component is proportional to the degree of ellipticity. We conclude that the complex traversal time found for an electron by several different methods is not a consequence of the quantum nature of the particle, but due to the wavelike character of the entity involved.

DOI
10.1103/physrevlett.75.2312

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We introduce a magnetic clock approach to measure the traversal and reflection times of an electromagnetic wave through a slab, when multiple reflections are taken into account. The traversal time is a complex quantity whose real component is proportional to the Faraday rotation and whose imaginary component is proportional to the degree of ellipticity. We conclude that the complex traversal time found for an electron by several different methods is not a consequence of the quantum nature of the particle, but due to the wavelike character of the entity involved.
We show that the physical clock approach can be applied to the problem of optical pulse transmission in the Fabry–Perot cavity. Our theoretical analysis leads directly to a complex-valued traversal time for the pulse. Real and imaginary parts of the traversal time, referred to as the phase time and the loss time, are associated, respectively, with the rotation angle of polarization and the change in the polarization ellipticity of the outgoing pulse in the presence of a magnetic clock. The physical significance of the phase time and the loss time is discussed in relation to the superluminal group velocity and the spectral shift of the pulse.
We develop a Green's function formalism and with it we are able to obtain close expressions for the tunnelling and for the reflection times. For each problem, there appear two characteristic times which correspond to the real and imaginary parts of the integral of the Green's function at coinciding coordinates. The time related to the imaginary part represents the minimum uncertainty of the measurement. A strong analogy between the results of the different existing approaches is established, and we show that their main differences are due to finite-size effects.
Baz' and Rybachenko have proposed the use of the Larmor precession as a clock to measure the time it takes a particle to traverse a barrier. An applied magnetic field is confined to the barrier. The spin of the incident particles is polarized perpendicular to this field. The extent of the Larmor precession occurring during transmission is used as a measurement of the time spent traversing the barrier. However, the particles tunneling through an opaque barrier also acquire a spin component parallel to the field since particles with spin parallel to the field have a higher transmission probability than particles with spin antiparallel to the field. Similar effects are actually used to polarize electrons and neutrons. An interpretation of this experiment compares the results with an approach which determines the traversal time by studying transmission of particles through a time-modulated barrier. This leads to three characteristic times describing the interaction of particles with a barrier. A dwell time measures the average time interval during which a particle interacts with the barrier whether it is reflected or transmitted at the end of its stay, a traversal time measures the time interval during which a particle interacts with the barrier if it is finally transmitted, and a reflection time measures the interaction time of a reflected particle.
We calculate numerically the transmission coefficient and the traversal time for finite Gaussian wavepackets as a function of their widths. We consider electromagnetic waves crossing a slab and a periodic structure. We find that the periodic structure can be crossed at superluminal velocities for wavenumbers in the optical gap and sizes larger than the thickness of the system. The corresponding transmission coefficients are very small. The long wavepacket limit of the traversal time coincides with previous analytical results for the real component of the interaction time. The imaginary component of this time only affects the change in size of the wavepacket.
We study the use of the Faraday effect as a quantum clock for measuring traversal times of evanescent photons through magnetorefractive structures. The Faraday effect acts both as a phase shifter and as a filter for circular polarizations. Only measurements based on the Faraday phase-shift properties are relevant to the traversal time measurements. The Faraday polarization filtering may cause the loss of nonlocal (Einstein-Podolsky-Rosen) two-photon correlations, but this loss can be avoided without sacrificing the clock accuracy. We show that a mechanism of destructive interference between consecutive paths is responsible for superluminal traversal times measured by the clock.
The question of how much time a wave packet spends in a traversing region of space has been of both fundamental and practical interest associated with the old problem of quantum particle tunneling. The question was then renewed by the experiment of Steinberg et al. (1993), implying the superluminal tunneling of a photon and began extending to physical systems concerning optical and electromagnetic waves. In this study, the traversal time taken by an optical pulse to off-resonantly transmit in a Fabry-Perot cavity is theoretically investigated as one such extension.