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arXiv:1107.2254

Scattering of TE- and TM-polarized wave packets on a dielectric layer: true and false group-delay times

Nikolay Leonidovich Chuprikov

Published Jul 12, 2011• 1 citation
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Abstract

We consider an oblique incidence on a uniform dielectric layer of the plane monochromatic TE- and TM-waves, as well as TE- and TM-polarized wave packets consisting of waves with the same angle of incidence. For each polarization the stationary model is presented, which allows one to uniquely restore the dynamics of the transmitted and reflected components of the incident wave packet at all stages of scattering. On this basis we introduce the concepts of ``true group-delay time"\/ and ``false group-delay time"\/ (the corresponding quantities for transmission and reflection are the same). The former is defined as the difference of two instants in the evolution of the same wave packet, one of which cannot be measured directly. Conversely, the latter is defined as the difference of two instants in the evolution of different wave packets, both can be measured directly. The false group-delay time does not in itself have a physical sense, but plays a key role for an indirect measurement of the true group-delay times of both polarizations.

DOI
10.48550/arxiv.1107.2254

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•Jun 1, 2013•0 citations
We consider an oblique incidence on a uniform dielectric layer of the plane monochromatic TE- and TM-waves, as well as TE- and TM-polarized wave packets consisting of waves with the same angle of incidence. For each polarization the stationary model is presented, which allows one to uniquely restore the dynamics of the transmitted and reflected components of the incident wave packet at all stages of scattering. On this basis we introduce the concepts of true group-delay time"\/ and false group-delay time"/ (the corresponding quantities for transmission and reflection are the same). The former is defined as the difference of two instants in the evolution of the same wave packet, one of which cannot be measured directly. Conversely, the latter is defined as the difference of two instants in the evolution of different wave packets, both can be measured directly. The false group-delay time does not in itself have a physical sense, but plays a key role for an indirect measurement of the true group-delay times of both polarizations.
The time delay of an almost monochromatic wave packet can be defined unambiguously via the energy derivative of the phase of the amplitude defining the superposition of stationary states making up the wave packet. We illustrate how this concept, originally formulated by Eisenbud and Wigner in the context of scattering theory, can be applied to quantum reflection by the nonclassical region of a long-ranged attractive potential tail. We pay special attention to the case of a wave packet incident from the near side of the nonclassical region and interpret the results with the help of regularized step potentials.
A complete one-dimensional scattering of a spinless particle on a time-independent potential barrier is considered. To describe separately transmitted and reflected particles in the corresponding subsets of identical experiments, we introduce the notions of scattering channels for transmission and reflection. We find for both channels the (unitary) scattering matrices and reconstruct, by known out asymptotes (i.e., by the transmitted and reflected wave packets), the corresponding in asymptotes. Unlike the out asymptotes for transmission and reflection, their in asymptotes represent nonorthogonal functions. As is shown, the position distributions of to-be-transmitted and to-be-reflected particles, except their average positions, are unpredictable. At the same time, the momentum distributions of these particles are physically meaningful and can be observed to the full. We show that both the subensembles of particles must start (on the average) from the same spatial point, and the momentum distributions of to-be-scattered and scattered particles must be the same for each scattering channel. Taking into account these properties, we define the (individual) delay times for transmission and reflection for wave packets of an arbitrary width. Besides, to estimate the duration of the scattering event, we derive the expression for a (total) scattering time.
The time history of subsequent tunneled wave packets can represent a more meaningful way to describe signal evolutions and to determine the group delay than the detection of a demodulated monopulse envelope. In the present experimental research, the delay of transmitted tunneled packets was found to be shorter than that measurable in vacuum, in good agreement with the phase time and Esposito's equation and independent of the barrier length. The wave packets did not show narrowing and "reshaping," so these phenomena did not appear to be the reason of the short delay experienced. The latter was not found to be proportional to the amplitude of the incident signal.
We develop a new variant of the wave-packet analysis and solve the tunneling time problem for one particle. Our approach suggests an individual asymptotic description of the quantum subensembles of transmitted and reflected particles both at the final and initial stage of tunneling. We find the initial states of both subensembles, which are non-orthogonal. The latter reflects ultimately the fact that at the initial stage of tunneling it is impossible to predict whether a particle will be transmitted through or reflected off the barrier. At the same time, in this case, one can say about the to-be-transmitted and to-be-reflected subensembles of particles. We show that before the interaction of the incident packet with the barrier both the number of particles and the expectation value of the particle's momentum are constant for each subensemble. Besides, these asymptotic quantities coincide with those of the corresponding scattered subensemble. On the basis of this formalism we define individual delay times for both scattering channels. Taking into account the spreading of wave packets, we define the low bound of the scattering time to describe the whole quantum ensemble of particles. All three characteristic times are derived in terms of the expectation values of the position and momentum operators. The condition which must be fulfilled for completed scattering events is derived. We propose also the scheme of a {\it gedanken} experiment that allows one to verify our approach.
A new variant of the packet analysis to solve the tunneling time problem for the so-called completed elastic scattering (a potential barrier of an arbitrary form is located in the finite spatial interval) is presented. In particular, we propose an explanation of the "paradoxes" of wave packet tunneling, and elaborate a rule enabling one to extract the tunneling times for a particle from the analysis of tunneling the corresponding wave packet. It is shown that the quantum ensembles of transmitted and reflected particles are statistically independent not only after the scattering but also before this event. We find the corresponding counterparts for the transmitted and reflected packets, which describe separately both scattering channels at early times, and define the expectation values of delay times for each channel. Besides, to describe jointly all particles at the stage when they cannot be divided into two independent subensembles, we find the scattering time taking into account broadening the packets. We derive the condition to be satisfied for the completed scattering. The times above are defined in terms of the well-known quantum-mechanical averages for the position and wave-number operators.
A new approach to the problem of scattering the plane electromagnetic TE wave on a homogeneous dielectric layer is presented, which does not predict, unlike the standard model, the Hartman effect for the tunneling time in the case of scattering the TE wave in the regime of a frustrated total internal refection (FTIR). The basic idea of this approach is that a correct definition of the tunneling velocity and time is possible if only the dynamics of both its subprocesses — transmission and reflection — is known at all stages of the scattering process investigated. It is shown that the Wigner (group) tunneling time was introduced without taking into account of this requirement, and, as a consequence, both this characteristic itself and the associated with it Hartman effect have no relation to transferring the light energy through the layer. The dwell transmission time, which is directly related to it, does not lead to the Hartman effect.