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arXiv:quant-ph/0210143 · Physical Review A

Existence of superposition solutions for pulse propagation in nonlinear resonant media

Prasanta Kumar Panigrahi, G. S. Agarwal

Published Mar 28, 2003• 14 citations
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Abstract

Existence of self-similar, superposed pulse-train solutions of the nonlinear, coupled Maxwell-Schr\"odinger equations, with the frequencies controlled by the oscillator strengths of the transitions, is established. Some of these excitations are specific to the resonant media, with energy levels in the configurations of $\ensuremath{\Lambda}$ and N and arise because of the interference effects of cnoidal waves, as evidenced from some recently discovered identities involving the Jacobian elliptic functions. Interestingly, these excitations also admit a dual interpretation as single pulse trains, with widely different amplitudes, which can lead to substantially different field intensities and population densities in different atomic levels.

DOI
10.1103/physreva.67.033817

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Existence of self-similar, superposed pulse-train solutions of the nonlinear, coupled Maxwell-Schr"odinger equations, with the frequencies controlled by the oscillator strengths of the transitions, is established. Some of these excitations are specific to the resonant media, with energy levels in the configurations of \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ and N and arise because of the interference effects of cnoidal waves, as evidenced from some recently discovered identities involving the Jacobian elliptic functions. Interestingly, these excitations also admit a dual interpretation as single pulse trains, with widely different amplitudes, which can lead to substantially different field intensities and population densities in different atomic levels.
Investigating the behavior of an atom in response to the applied electric field when the latter frequency is in resonance with the natural frequency of the atom is an interesting subject. Near resonance, some of the most interesting optical phenomena such as dispersion, and absorption are more pronounced. Moreover, considering resonant and near resonant interaction of light with two-level atoms, uncovers fascinating physical phenomena such as area theorem and self-induced transparency describing stable pulse propagation regimes in which the pulses maintain their identity. Existence of these optical structures in on-resonance optical media has not been investigated in detail. In this thesis, the on-resonance interaction of optical field with atoms is considered and the formation of several novel self-similar and kink waves in linear and nonlinear resonant media is discovered and theoretically explored. First, self-similar pulse formation in homogeneous broadened linear amplifiers in a vicinity of an optical resonance is analyzed. It is demonstrated that the self-similar pulses serve as universal asymptotics of any near resonance short pulses propagating in coherent linear amplifiers. Second, ultrashort self-similar pulse propagation in coherent linear absorbers near optical resonance is investigated. Third, existence of self-similar optical waves with kink structure in resonant optical systems is discovered. Fourth, it is found that self-induced transparency quadratic solitons are realizable in the media with quadratic optical nonlinearities, doped with resonant impurities. Finally, stable spatial similaritons supported by homogeneous conservative optical media with quintic nonlinearities are explored. To experimentally realize the presented results, physical models are presented for all systems under consideration. The stability of the proposed near resonance optical systems is demonstrated through a series of numerical case studies.
The Maxwell–Schrödinger equations describing the ultrashort laser pulse propagation in multilevel molecular media are shown to be analytically solvable and to reduce to nonlinear multisine-Gordon equations for certain models of the field–molecule system. Special cases of level schemes are shown to lead to soliton propagation. A numerical study is presented in order to examine the effect of nonresonant (virtual) transitions on pulse propagation and amplification in a three-level molecular system as a function of intensity and detuning.
Nonlinear two-component electromagnetic pulse propagation through a resonant axially symmetric anisotropic medium having a permanent dipole moment is analyzed under conditions of strong coupling between the ordinary (short-wavelength) and extraordinary (long-wavelength) pulse components. It is shown that a pulse can propagate through the medium in regimes different from self-induced transparency if its ordinary component is detuned off resonance. In particular, a pulse propagating in the regime of self-induced super-transparency substantially changes quantum-level populations, but its group velocity remains almost equal to the linear velocity. If a pulse propagates in the extraordinary transparency regime and the carrier-frequency detuning from resonance is small, then its group velocity is substantially lower, while the level populations remain virtually invariant. Regimes of propagation through weakly excited media under quasi-resonance conditions are also identified.
A combined analytical and numerical analysis of the propagation of a step-function light pulse in a resonant medium is presented. It is found that the light pulse evolves into a steady-state amplitude-modulated wave train very closely approximated by the Jacobian elliptic function $\mathcal{E}=1.25 {\mathcal{E}}_{0}dn[(\frac{1}{\ensuremath{\tau}})(t\ensuremath{-}\frac{z}{V}); \ensuremath{\lambda}]$, where the modulus of the elliptic function $\ensuremath{\lambda}$ is equal to 45\frac{4}{5}54​ and E0{\mathcal{E}}_{0}E0​ is the amplitude of the input pulse. This result is consistent with an earlier analysis which showed that the elliptic dn\mathrm{dn}dn function represented one of two classes of functions which satisfy the equations of resonant-pulse propagation and correspond to the distortionless propagation of an amplitude-modulated wave train through an inhomogeneously broadened medium. Expressions for the period, velocity, and parameter $\ensuremath{\tau}$ of the elliptic function are found as functions of the intensity of the input pulse. The form of the steady-state wave train is found to be the same in either an attenuating or amplifying medium. It is found that the velocity of the pulse train becomes significantly less than the speed of light when the density of energy that can be stored in the resonant atoms becomes an appreciable fraction of the energy density of the electromagnetic field. The conditions necessary for observing the amplitude-modulated pulse train in gaseous SF6{\mathrm{F}}_{6}F6​ and Na vapor are discussed.
The authors develop a theoretical analysis which describes coherent nonlinear resonant effects where two optical pulses of different frequencies undergo two-photon self-induced transparency (two-photon SIT) in an inhomogeneously broadened three-level system. The field (Maxwell) and atomic (Schrodinger; density matrix representation) equations are coupled in a self-consistent manner by virtue of the resultant macroscopic polarizations at the two frequencies. This treatment is accomplished under circumstances where atomic coherence plays a predominant role. The resulting equations yield a new two-photon area theorem which gives first the intensity dependent, two-photon absorption law analogous to Beer's law for the one-photon transition. Secondly, a 2 pi Lorentzian pulse solution is derived which propagates without loss at the modified pulse velocity.
We show that self-similar optical waves with a kink structure exist in a wide class of resonant nonlinear media, adequately treated in the two-level approximation. The self-similar structure of the present kinks is reflected in the time evolution of the field profile, atomic dipole moment, and one-atom inversion. We develop an analytical theory of such kinks. We show that the discovered kinks are accelerating nonlinear waves, asymptotically attaining their shape and the speed of light. We also numerically explore the formation and eventual disintegration of our kinks due to energy relaxation processes. Thus, the present kinks can be viewed as intermediate asymptotics of the system.