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Journal of physics. A, Proceedings of the Physical Society. General

The method of characteristics in the theory of resonant or nonresonant nonlinear optics

J. Chris Eilbeck, R. Bullough

Published Jun 1, 1972• 25 citations
Publisher

Abstract

The method of characteristics is used to analyse the system of five nonlinear partial differential equations describing ultrashort optical pulse propagation through a medium of two-level atoms. Contrary to previous suggestions it is shown rigorously that the system is causal: no physical speeds exceed c. The method is well adapted to numerical integration. It can take full account of both back scattering and the effect of a surface to the dielectric. A number of new physical results obtained this way are briefly reported.

DOI
10.1088/0305-4470/5/6/007

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The method of characteristics is used to analyse the system of five nonlinear partial differential equations describing ultrashort optical pulse propagation through a medium of two-level atoms. Contrary to previous suggestions it is shown rigorously that the system is causal: no physical speeds exceed c. The method is well adapted to numerical integration. It can take full account of both back scattering and the effect of a surface to the dielectric. A number of new physical results obtained this way are briefly reported.
The initial value problem for the propagation of a pulse through a resonant two-level optical medium is solved by the inverse scattering method. In general, an incident pulse decomposes not only into a special class of pulses to which the medium is transparent but also yields radiation which is absorbed by the medium. In this respect ``this problem'' has properties markedly different from other dispersive and reversible wave phenomena some of which are tractable by the inverse scattering method. Indeed, it is remarkable that in the present case the method still applies. In particular, we show that, while there are an infinite number of local conservation laws, the integrated densities, and in particular the energy, are only conserved for a very special class of initial conditions. The theoretical results obtained are in close agreement with all the qualitative features observed in the experiments on coherent pulse propagation. Finally, we also show that causality is preserved. Two new and novel features are introduced and briefly discussed. First, we show that if the homogeneous broadening effect is a function of position in the medium, the pulses may speed up and slow down accordingly, without losing their permanent identities. Second, we have found a new kind of solution mode corresponding to a proper eigenvalue of the scattering problem which is not a bound state.
We consider the propagation of ultrashort optical solitons in media described by a general Hamiltonian of multilevel atoms. Assuming that all transition frequencies of the medium are well below the typical wave frequency, i.e., only the contribution of infrared transitions is taken into account, we use a short-wave approximation and a rigorous application of the reductive perturbation formalism to derive a cumbersome coupled system of nonlinear partial differential equations describing ultrashort soliton evolution in such systems. The rather complicated set of coupled equations can be simplified to a generic double-sine-Gordon equation for a special case of identical three-level atoms, whereas for a special case of identical four-level atoms the system of coupled equations can be reduced to a generalized double-sine-Gordon equation. Numerical simulations showing the formation of robust breather-type solutions of both the standard double-sine-Gordon and of the generalized double-sine-Gordon equations from sinusoidal inputs with Gaussian envelopes are also presented.
Propagation and amplification of an ultrashort laser pulse in a two-level medium is numerically investigated with the use of the finite-difference time-domain (FDTD) procedure simultaneously solving Maxwell and Schrodinger equations with no assumptions characteristic of slowly varying envelope approximation. Comparison of the results of FDTD numerical simulations with the predictions of the pulse area theorem has demonstrated that the developed numerical procedure provides an adequate description of pulse evolution in a two-level medium. Amplification of ultrashort light pulses in a two-level medium is analyzed. Two methods of improving the gain of ultrashort pulses in a two-level medium by modulating the spatial distribution of dipole moments of resonant transitions and using initially chirped frequency-detuned pulses are explored.
We develop a theoretical approach for the full analytical description of dissipative optical solitons, which takes into account all relaxation processes in the medium. Most analytical solutions available today for ultrashort optical solitons are obtained upon neglecting a number of relaxation terms in the governing equations for the medium response. The suggested method is based on finding explicit analytical expressions for the extended soliton tails distant enough from the soliton central part, when taking into consideration all relaxation processes. These soliton tails can be described by the linearized equations for the medium response both at the soliton leading and trailing edges. Afterwards, the analytical expressions for the soliton tails and its central part can be matched to yield the full analytical description of ultrashort dissipative solitons with all relaxation processes in the resonant medium properly treated. The proposed approach is demonstrated for ultrashort dissipative solitons in a resonant medium consisting either of only amplifying resonant two-level centers or of a mixture of amplifying and absorbing two-level centers.