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SIT Solitons

Self-Induced Transparency (SIT) is a marvellous manifestation of coherent nonlinear interaction between radiation and matter. The essence of the phenomenon can be described as follows: when the pulse power exceeds some critical value, a high intensity ultrashort optical pulse propagating through media whose decoherence lifetimes greatly exceed the pulse duration, travels at a reduced speed and unchanged shape with anomalously low energy loss. The physical explanation of the SIT effect is based on the representation of the absorber by an ensemble of two-level atoms, whose time evolution is governed by stimulated processes only due to the condition of much shorter pulse duration compared to the characteristic relaxation times of the medium. In the ideal SIT picture the atoms are driven to the excited state by absorbing ultrashort pulse energy and by reradiating this energy to the field they return to the ground state. Thus the optical energy is carried through the medium not by the electromagnetic field but by a polariton wave. The polariton is a quasiparticle which combines both light excitation and the matter response via absorption and emission thereby inducing medium polarisation. In other words, polariton is a real photon in a medium together with the polarisation cloud accompanying the propagation of electromagnetic field in the medium. The pulse travels with anomalously low energy loss and unchanged shape due to the re-emission of the absorbed radiation in phase with the driving strong resonant optical field. The solitary pulse propagation is accompanied by temporal oscillations of the carrier density in the material known as optical Rabi oscillations. 

We investigate the optical nonlinearities in planar waveguides in the high-intensity excitation regime when the electric field itself rather than the electric field envelope drives the light-matter interaction. For linearly polarised monochromatic one-photon optical excitation, the active medium is modelled by an ensemble of saturable absorbers with degenerate three-level discrete energy-level structure. Thus the time evolution of the quantum system in the external (not necessarily small) perturbation is described by an eight-dimensional real pseudospin vector. The resulting Maxwell-pseudospin equations are solved in the time domain using the Finite-Difference Time-Domain (FDTD) method on a specially constructed modified 2D Yee-grid. Using the model we have numerically demonstrated formation of 2D-SIT solitons in planar nonlinear optical waveguides. We have generalised the Pulse Area Theorem which establishes a general criterion for soliton (lossless) regime of propagation to more than one spatial dimension and three levels. Numerical evidence of multidimensional SIT-solitons localised both in space and time is given (Fig.1). 

Fig.1. Snapshot of a propagating plane-polarised TEM mode soliton in a nonlinear parallel mirror waveguide formed through the SIT effect

In Fig. 2 a cross-section of the ultrashort pulse from Fig. 1 is shown along with the population inversion, driven through a full Rabi flop by the high-intensity pulse and the medium polarisations. The leading edge of the pulse excites the 3-level system and the trailing edge de-excites it bringing the population back to the ground state).

Fig. 2.  Cross-section of a SIT-soliton showing the full Rabi flop of the carrier density S7=r22  -r11  driven by the optical field. S1 and S4 are medium polarisations

Fig. 3(a) demonstrates the pulse dispersion and energy loss during the propagation when the pulse intensity does not correspond to a 2p-area and Fig. 3(b) shows the stable 2p-pulse representing a stable soliton solution, according to the Pulse Area Theorem. 
                                                                                                            (a)                                                                                            (b)
Fig. 3. (a) Snapshots of the cros-section (along z-axis) of a propagating ultrashort pulse through an absorbing medium along z-direction of a planar waveguide at four time moments. Pulse spread out (dispersion) and pulse shape distortion occurs when the conditions for self-induced transparency are not satisfied; (b) Snapshots of a propagating SIT-soliton sampled at the same times as in (a). The pulse shape is preserved during the propagation in the optically absorbing medium. The pulse travels without losses as if the medium were transparent.

Fig 4 and Fig.5 (a,b) show the time evolution of the TM1 guided mode SIT-soliton and the corresponding time-evolution of the population inversion in a degenerate three-level system. The nearly lossless ultrashort pulse propagation and the preservation of the pulse shape in the SIT regime is extremely attractive for a wide range of applications, such as optical storage and processing of information, optical communication and pulse generation and compression techniques.

            (a)                                                                                                           (b)

Fig. 4.  Time evolution of a 2D SIT-soliton (a) TM1 mode at the simulation times t=90, 125, 155 fs showing the conservation of the 2D-pulse shape during the propagation in resonantly absorbing medium; (b) Corresponding population inversion showing 2D Rabi-flopping along y and z directions.
(a) (click on the picture to see the movie) 

(b) (click on the picture to see the movie) 

Fig. 5. Time evolution of the TM1 guided mode of a parallel mirror waveguide (2p--pulse with hyperbolic secant envelope) (a), and the population inversion in a degenerate 3-level system (b)

The 2D model has been validated against the predictions of the Pulse Area Theorem in 1D. In particular, it has been shown that an arbitrary shaped (e.g. Gaussian) pulse in absorbing medium evolves into a SIT 2p-soliton with hyperbolic secant envelope (Fig.6). 

Fig. 6. 
Pulse reshaping during propagation in resonantly absorbing medium: longitudinal (along z-axis) cross-section of a 2D-Gaussian pulse with initial pulse area 1.6p. at consecutive time moments vs z-coordinate of propagation.

Due to the resonant coherent interaction with the absorbers the optical pulse group velocity decreases with respect to the free space one and the pulse slows down, demonstrated in (Fig. 7) by a TEM plane-polarised pulse and the driven population inversion (click on the plots below to see animation)


Fig.7.  (a) pulse Eelectric field component reshaping ; (b) Population profile reshaping during the pulse  propagation in absorbing medium.