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arXiv:cond-mat/0505469 · Physical Review B

Electromagnetic wave propagation in an active medium and the equivalent Schrödinger equation with an energy-dependent complex potential

H. Bahlouli, Abdulaziz D Alhaidari, A. Al Zahrani, Eleftherios N. Economou

Published Sep 28, 2005• 19 citations
arXivHTMLPDF

Abstract

We study the massless limit of the Klein-Gordon (K-G) equation in $1+1$ dimensions with static complex potentials in order to provide an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in an active medium. In the case of a dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy dependent. We study also the nonrelativistic Schr\"odinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of the solutions of this Schr\"odinger equation is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. In particular, both equations exhibit a discrepancy between the time-dependent and stationary results; our study attributes this discrepancy to the appearance of time-growing bound eigenstates corresponding to poles in the transmission and reflection amplitudes located in the upper half of the wave-number plane. The omission of these bound states in the expansion in stationary states leads to the observed discrepancy. Furthermore, it was demonstrated that there is a frequency- (energy) -and-size-dependent gain threshold above which this discrepancy appears. This threshold corresponds to the value of the gain at which the first pole crosses the real axis.

DOI
10.1103/physrevb.72.094304

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We study the massless limit of the Klein-Gordon (K-G) equation in 1+11+11+1 dimensions with static complex potentials in order to provide an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in an active medium. In the case of a dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy dependent. We study also the nonrelativistic Schr"odinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of the solutions of this Schr"odinger equation is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. In particular, both equations exhibit a discrepancy between the time-dependent and stationary results; our study attributes this discrepancy to the appearance of time-growing bound eigenstates corresponding to poles in the transmission and reflection amplitudes located in the upper half of the wave-number plane. The omission of these bound states in the expansion in stationary states leads to the observed discrepancy. Furthermore, it was demonstrated that there is a frequency- (energy) -and-size-dependent gain threshold above which this discrepancy appears. This threshold corresponds to the value of the gain at which the first pole crosses the real axis.
Adding a constant energy offset leaves classical dynamics unchanged. In quantum mechanics it changes the phase velocity of the wavefunction. The inclusion of the constant rest energy in the Klein-Gordon formulation leads to significantly higher phase velocities compared with the Schrödinger equation. The Schrödinger equation predicts an attenuation of the wavefunction along one of the paths in a Sagnac interferometer when a beamsplitter's trajectory along that path includes a segment where its speed exceeds the phase velocity of a free particle. Such an attenuation does not occur for electromagnetic waves nor for eigenstates of momentum in the Klein-Gordon equation since the speed of the beamsplitter cannot then exceed the phase velocity of the wave. This attenuation reduces the amplitude without introducing a phase shift, preserving the overall structure of the transmitted wave group. While a Klein-Gordon wave group undergoes three traversals of the beamsplitter that moved, it experiences attenuation equivalent to only a single pass, whereas the Schrödinger equation predicts the expected attenuation for three passes. This discrepancy highlights a fundamental incompatibility between the Klein-Gordon equation and the Schrödinger equation in a regime where their predictions should converge.
We study the classification of electromagnetic fields using the equivalence relation on the set of all 4-potentials of the Schrodinger equation. In the general case we find the relations among the equivalent fields, currents, and charge densities. Particularly, we study the fields equivalent to the null field. We show that the non-stationary state function for a particle in arbitrary uniform time-dependent magnetic field is equivalent to a plane wave. We present that the known coherent states of a free particle are equivalent to the stationary states of an isotropic oscillator. We reveal that the only constant magnetic field is not equivalent to the null field (contrary to a constant electrical field) and we find other fields that are equivalent to the constant magnetic field. We establish that one particular transformation of the free Schrodinger equation puts a plane wave and Green's function in a equivalence relation. c
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.
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Expressions are derived for the electromagnetic fields of guided waves which are analogous to the quantum-mechanical equations representing barrier tunneling. This analogy is achieved by comparing the propagation constant of the Schrödinger time-independent wave equation with that of the electromagnetic wave equation in waveguide and by comparing the de Broglie wavelength of a particle with the wavelength of the waves that propagate the energy. This results in an expression relating the form of an arbitrary one-dimensional energy barrier to the physical dimensions of a section of waveguide. The analogy is tested by the propagation of energy in the TE10 mode at both the 3- and 6-cm bands for the cases of rectangular and hyperbolic barriers. Although evanescent modes are present at the discontinuous regions, the analog for the rectangular barrier, which is considered to be the worse case, is verified when an effective barrier length leff of about 1.2l is used. This experimental verification demonstrates the possibility of waveguide simulation of quantum-mechanical energy barriers and the practicality of utilizing an electromagnetic analog for demonstrating the tunneling phenomenon and provides a method for measurement of the transmission coefficient through an arbitrarily shaped barrier.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.