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Journal of the Optical Society of America B

Generic propagation of sharp boundaries electromagnetic signals in any linear dispersive medium

Er’el Granot

Published Feb 9, 2016• 1 citation
Publisher

Abstract

A generic formalism for the propagation of a pulse with sharp boundaries in any linear medium is derived. It is shown that such a pulse experiences generic deformations. For any given linear medium, the pulse deformation is expressed as a generic differential operator, which characterizes the medium and which operates on the pulse at the singular points (the sharp boundaries). The theory is then applied to a Fabry–Perot etalon and to dispersive media with second order dispersion, third order dispersion, and a combination of both. Simple approximate expressions are also derived for a relatively short, i.e., low dispersive, medium and compared with exact numerical solutions.

DOI
10.1364/josab.33.000334

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A generic formalism for the propagation of a pulse with sharp boundaries in any linear medium is derived. It is shown that such a pulse experiences generic deformations. For any given linear medium, the pulse deformation is expressed as a generic differential operator, which characterizes the medium and which operates on the pulse at the singular points (the sharp boundaries). The theory is then applied to a Fabry–Perot etalon and to dispersive media with second order dispersion, third order dispersion, and a combination of both. Simple approximate expressions are also derived for a relatively short, i.e., low dispersive, medium and compared with exact numerical solutions.
To investigate the physical meaning of pulse peak in fast and slow light media, we investigated propagation of differently shaped pulses experimentally, controlling the sharpness of the pulse peak. Symmetric behavior with respect to fast and slow light was observed in traditional Gaussian pulses; that is, propagated pulses were advanced or delayed, respectively, whereas the pulse shape remained unchanged. This symmetry broke down when the pulse peak was sharpened; in the fast light medium, the sharp pulse peak propagated with luminal velocity, and the transmitted pulse deformed into a characteristic asymmetric profile. In contrast, in the slow light medium, a time-delayed smooth peak appeared with a bending point at t=0t=0t=0. This symmetry breaking with respect to fast and slow light is a universal characteristic of pulse propagation in causal dispersive systems. The sharp pulse peak can be recognized as a bending nonanalytical point and may be capable of transferring information.
The asymptotic description of the propagation of a step-modulated signal of fixed angular frequency omegacinto the half-space z > 0 that is occupied by a single-resonance active Lorentzian medium, which is a classical model of an inverted two-level atom, has been presented. The steepest descent method as an asymptotic technique provides the detailed time evolution of the pulse propagation in a linear, temporally dispersive active Lorentzian medium. The analysis and the numerical results show the relation of the location of the saddle points and the topography of the phase function in the complex omega-plane with the transient portion of the pulse that can be used to control the transient response of the medium
Effects of propagation of an extremely short (of one or several oscillation periods) electromagnetic pulse in a medium whose resonance transition is characterized by diagonal as well as nondiagonal matrix elements of the dipole moment operator have been studied numerically. The Maxwell-Bloch system of equations is employed without using the approximation of slowly varying envelopes. An analog of the McCall and Hahn area theorem is discussed as applied to the division of the initial extremely short pulse into subpulses. The solution is obtained in the form of a solitary stable bipolar signal with a nonzero pulse area (nonzero breather).
Pulse propagation through 1-dimensional stratified media is studied. The general features of pulse propagation in relation to the delay or advancement of the pulse are discussed. It is shown that in a lossless system with spatial symmetry the reflected and the transmitted pulses have the same delay or advancement. The phase times and the explicit output pulse profiles for an input Gaussian pulse are calculated. While recalling recent results for a variety of systems (resonant absorbers or heterogeneous medium in a Fabry-Perot cavity, a slab of negative index medium, a left-handed/right-handed periodic structure, etc.), new results involving coupled waveguides and a high finesse cavity containing an electromagnetically induced transparency (EIT) medium are presented. Calculations clearly show that for pulses with a carrier tuned at the resonance of the stratified media, one generally has subluminal propagation. Alternatively, in stop gaps or away from the resonances one ends up with superluminal transit. In the case of the EIT cavity it is shown that the large dispersion offered by the intracavity medium can lead to substantial superluminality (ng≃−106) with peak transmission as high as 20%. This is in sharp contrast to large signal attenuation for usual superluminal transit. Note also that EIT, to the best of the authors knowledge, has so far been used to achieve only slow light.
The propagation of electromagnetic pulses in a linear, dispersive Lorentz medium with two isolated resonance frequencies is presented within the context of the classical theory due to Sommerfeld and Brillouin. Particular attention is given to the modification of the dynamical field evolution when one proceeds from a single-resonance to a double-resonance medium. The appearance of a new field structure in the precursor field evolution is obtained, and the conditions for its evolution are described. The signal velocity for a pulsed field is obtained and compared with the energy-transport velocity in a double-resonance medium both when the conditions for the evolution of the modified precursor field structure are satisfied as well as when they are not satisfied and it is absent from the total propagated field structure.
In recent years longitudinal coherence of electron and neutron wave packets has been investigated,1,2 and it was shown that their coherence length remains unchanged even though the length of the wave packet increases on propagation.3,4 As pointed out in Ref. 3, one may expect that a similar result will hold for light that propagates in a dispersive medium. We present some new general theorems relating to an ensemble of identical 1-D optical pulses which propagate in a linear homogeneous medium of any given dispersive properties. In particular we derive a general expression for the mutual coherence function and for the cross-spectral density function of the pulse field in the medium in terms of the Fourier spectrum of a single pulse shape. The analysis brings out clearly the distinction (already noted in Ref. 4) between the coherence length and the coherence time of fields in dispersive media. We also show that the pulse field is spatially completely coherent at each frequency, even though the shape of each pulse changes on propagation. The results have potential application to interferometry with optical pulses and with de Broglie wave packets of elementary particles.