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Annalen der Physik

Front propagation in evanescent media

Μ. Büttiker, H. Thomas

Published Dec 1, 1998• 21 citations
Publisher

Abstract

We investigate the time evolution of waves in evanescent media generated by a source within this medium and observed at some distance away from the location of the source. The aim is to find a velocity which describes a causal process and is thus, for a medium with relativistic dispersion, limited by the velocity of light. For a source with a sharp onset in time, the wave function consists of a forerunner generated by the onset of the source, and of a monochromatic front. The forerunner is dominated by a frequency which decreases with time, and the monochromatic front carries the oscillation frequency of the source into the evanescent medium. For a medium with Schrödinger-like dispersion the velocity of the front is infinite and the monochromatic front propagates with a velocity which is in agreement with the traversal time for tunneling. In the relativistic case the forerunners travel with the velocity of light and the velocity the monochromatic front is smaller than the velocity of light and only for special energies equal to the velocity of light. For sources with a sharp onset, the forerunners are not attenuated and in magnitude far exceed the monochromatic front. This renders the detection of the monochromatic front difficult. To avoid the different behavior of forerunners and monochromatic fronts, sources which are frequency-band limited can be considered or a short-time Fourier transform of the field at the observation point can be taken. Both discussions suggest that the traversal time can be determined only up to a factor of 𝒪(1).

DOI
10.1002/(sici)1521-3889(199812)7:7/8<602::aid-andp602>3.0.co;2-g

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Front propagation in evanescent media

Μ. Büttiker, H. Thomas

Abstract We investigate the time evolution of waves in evanescent media generated by a source within this medium and observed at some distance away from the location of the source. The aim is to find a velocity which describes a causal process and is thus, for a medium with relativistic dispersion, limited by the velocity of light. For a source with a sharp onset in time, the wave function consists of a forerunner generated by the onset of the source, and of a monochromatic front. The forerunner is dominated by a frequency which decreases with time, and the monochromatic front carries the oscillation frequency of the source into the evanescent medium. For a medium with Schrödinger‐like dispersion the velocity of the front is infinite and the monochromatic front propagates with a velocity which is in agreement with the traversal time for tunneling. In the relativistic case the forerunners travel with the velocity of light and the velocity the monochromatic front is smaller than the velocity of light and only for special energies equal to the velocity of light. For sources with a sharp onset, the forerunners are not attenuated and in magnitude far exceed the monochromatic front. This renders the detection of the monochromatic front difficult. To avoid the different behavior of forerunners and monochromatic fronts, sources which are frequency‐band limited can be considered or a short‐time Fourier transform of the field at the observation point can be taken. Both discussions suggest that the traversal time can be determined only up to a factor of 𝒪(1).

Annalen der Physik•Dec 1, 1998•1 citation

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Forerunners and shapes of the signal in the propagation of radiation in structured nonlinear dispersive dielectrics

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Propagation of radiation in nonlinear dispersive media has been analyzed numerically without standard assumptions (like the envelope approximation or the rotating wave approximation) in the near-resonant regime in one spatial dimension. The medium has been modeled as an assemblage of two-, three-, or four-level systems. An attempt to include inhomogeneous broadening has been made with the help of random spatial modulation of the frequencies of transitions in those systems. Particular attention has been directed to the propagation in spatially structured media. Non-trivial dynamics of forerunners have been obtained in the cases of "stopped" light and "superluminal" propagation. The shapes of the signal, and, in particular, the propagation of the wings of the signal have been investigated. The time- and parameter-dependence of the group velocity have been studied in detail, and the vagueness of the concept of the group velocity has been stressed.

International Conference on Transparent Optical Networks•Sep 12, 2005•0 citations
We investigate the time evolution of waves in evanescent media generated by a source within this medium and observed at some distance away from the location of the source. The aim is to find a velocity which describes a causal process and is thus, for a medium with relativistic dispersion, limited by the velocity of light. For a source with a sharp onset in time, the wave function consists of a forerunner generated by the onset of the source, and of a monochromatic front. The forerunner is dominated by a frequency which decreases with time, and the monochromatic front carries the oscillation frequency of the source into the evanescent medium. For a medium with Schrödinger-like dispersion the velocity of the front is infinite and the monochromatic front propagates with a velocity which is in agreement with the traversal time for tunneling. In the relativistic case the forerunners travel with the velocity of light and the velocity the monochromatic front is smaller than the velocity of light and only for special energies equal to the velocity of light. For sources with a sharp onset, the forerunners are not attenuated and in magnitude far exceed the monochromatic front. This renders the detection of the monochromatic front difficult. To avoid the different behavior of forerunners and monochromatic fronts, sources which are frequency-band limited can be considered or a short-time Fourier transform of the field at the observation point can be taken. Both discussions suggest that the traversal time can be determined only up to a factor of 𝒪(1).
Abstract We investigate the time evolution of waves in evanescent media generated by a source within this medium and observed at some distance away from the location of the source. The aim is to find a velocity which describes a causal process and is thus, for a medium with relativistic dispersion, limited by the velocity of light. For a source with a sharp onset in time, the wave function consists of a forerunner generated by the onset of the source, and of a monochromatic front. The forerunner is dominated by a frequency which decreases with time, and the monochromatic front carries the oscillation frequency of the source into the evanescent medium. For a medium with Schrödinger‐like dispersion the velocity of the front is infinite and the monochromatic front propagates with a velocity which is in agreement with the traversal time for tunneling. In the relativistic case the forerunners travel with the velocity of light and the velocity the monochromatic front is smaller than the velocity of light and only for special energies equal to the velocity of light. For sources with a sharp onset, the forerunners are not attenuated and in magnitude far exceed the monochromatic front. This renders the detection of the monochromatic front difficult. To avoid the different behavior of forerunners and monochromatic fronts, sources which are frequency‐band limited can be considered or a short‐time Fourier transform of the field at the observation point can be taken. Both discussions suggest that the traversal time can be determined only up to a factor of 𝒪(1).
The time dependence of quantum evanescent waves generated by a point source with an infinite or a limited frequency band is analyzed. The evanescent wave is characterized by a forerunner (transient) related to the precise way the source is switched on. It is followed by an asymptotic, monochromatic wave which at long times reveals the oscillation frequency of the source. For a source with a sharp onset the forerunner is exponentially larger than the monochromatic solution and a transition from the transient regime to the asymptotic regime occurs only at asymptotically large times. In this case, the traversal time for tunneling already plays a role only in the transient regime. To enhance the monochromatic solution compared to the forerunner we investigate (a) frequency-band-limited sources and (b) the short-time Fourier analysis (the spectrogram) corresponding to a detector which is frequency band limited. Neither of these two methods leads to a precise determination of the traversal time. However, if they are limited to determine the traversal time only with a precision of the traversal time itself both methods are successful: In this case the transient behavior of the evanescent waves is at a time of the order of the traversal time followed by a monochromatic wave which reveals the frequency of the source.
We obtain the solution of a relativistic wave equation and compare it with the solution of the Schr\"odinger equation for a source with a sharp onset and excitation frequencies below cutoff. A scaling of position and time reduces to a single case all the (below cutoff) nonrelativistic solutions, but no such simplification holds for the relativistic equation, so that qualitatively different ``shallow'' and ``deep'' tunneling regimes may be identified relativistically. The nonrelativistic forerunner at a position beyond the penetration length of the asymptotic stationary wave does not tunnel; nevertheless, it arrives at the traversal (semiclassical or B\"uttiker-Landauer) time $\ensuremath{\tau}.$ The corresponding relativistic forerunner is more complex: it oscillates due to the interference between two saddle-point contributions and may be characterized by two times for the arrival of the maxima of lower and upper envelopes. There is in addition an earlier relativistic forerunner, right after the causal front, which does tunnel. Within the penetration length, tunneling is more robust for the precursors of the relativistic equation.
In a causally dispersive medium the signal arrival appears in the dynamical field evolution as an increase in the field amplitude from that of the precursor fields to that of the steady-state signal. The interrelated effects of phase dispersion and frequency dependent attenuation and/or amplification alter the pulse in such a fundamental way that results in the appearance of precursor fields. Although superluminal group velocities have been found in various dispersive media, the pulse "front" and associated precursors will never travel faster than c , and hence these are the vehicles through which relativistic causality is preserved. While many rigorous studies of wave propagation and associated abnormal group velocities in passive Lorentzian media have been performed, the corresponding problem in active media has remained theoretically unexplored. This problem is addressed in the present paper, by employing the steepest descent method for the determination of the response of an active Lorentzian medium to a step modulated pulse. The steepest descent method provides a detailed description of the propagation of the pulse inside the dispersive medium in the time domain. Moreover, the evolution of the saddle points illuminates the relation between the medium parameters and the temporal evolution of the propagating pulse within the medium. Hence, useful physical insights are obtained and the interesting differences between the passive and active case are deduced.
We show that the temporal peak of a quantum wave may arrive at different locations simultaneously in an absorbing medium. The arrival occurs at the lifetime of the particle in the medium from the instant when a point source with a sharp onset is turned on. We also identify other characteristic times. In particular, the ``traversal'' or ``B\"uttiker-Landauer'' time (which grows linearly with the distance to the source) for the Hermitian, non-absorbing case is substituted by several characteristic quantities in the absorbing case. The simultaneous arrival due to absorption, unlike the Hartman effect, occurs for carrier frequencies under or above the cutoff, and for arbitrarily large distances. It holds also in a relativistic generalization but limited by causality. A possible physical realization is proposed by illuminating a two-level atom with a detuned laser.
Propagation of radiation in nonlinear dispersive media has been analyzed numerically without standard assumptions (like the envelope approximation or the rotating wave approximation) in the near-resonant regime in one spatial dimension. The medium has been modeled as an assemblage of two-, three-, or four-level systems. An attempt to include inhomogeneous broadening has been made with the help of random spatial modulation of the frequencies of transitions in those systems. Particular attention has been directed to the propagation in spatially structured media. Non-trivial dynamics of forerunners have been obtained in the cases of "stopped" light and "superluminal" propagation. The shapes of the signal, and, in particular, the propagation of the wings of the signal have been investigated. The time- and parameter-dependence of the group velocity have been studied in detail, and the vagueness of the concept of the group velocity has been stressed.