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Journal of Physics C Solid State Physics

The coherent non-linear light propagation in crystal in the exciton range of the spectrum

S. N. Belkin, P. I. Khadzhi, S. A. Moskalenko, Anatol Rotaru

Published Oct 10, 1981• 11 citations
Publisher

Abstract

The phenomenon of self-induced transparency (SIT) in the exciton range of the spectrum consists of the propagation through the crystal of solitary wave packets of coherent excitons and photons with anomalous small losses. In the running frame of reference at the site occupied by the soliton a single aperiodic transformation of photons into excitons and a reverse motion of the energy into the field takes place. The solutions of the SIT equations have been received in the slowly-varying amplitude approximation. Taking into account the phase modulation, the authors have shown the appearance of the branch of the carrier wave frequency with anomalous dispersion not only at the given pulse width but at the given pulse amplitude. They have also found the exact solution of the above mentioned equations taking into account the effect of saturation of the dipole momentum transition from the crystal ground state to the exciton state. It leads to the increasing of the soliton time width, to the decreasing of its amplitude and worsens the SIT observation.

DOI
10.1088/0022-3719/14/28/012

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Self-induced transparency of excitons and the dispersion law of steady-state exciton-photon pulses

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We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort (picosecond) laser pulses which propagate near the band edge of the nonlinear photonic band gap (PBG) material doped with rare-earth atoms such as erbium. Solitary wave formation involves the combined effects of group velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We derive the general Maxwell-Bloch equations for a nonlinear PBG system and then demonstrate the existence of elementary solitary wave solutions for frequencies far outside the gap where GVD effects are negligible and for frequencies near the photonic band edge where GVD effects are crucial. We find two distinct new types of propagating SIT solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the nonlinear Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ However, when the host nonresonant Kerr coefficient is nonzero, we obtain the first new type of soliton. In this case, the optical soliton envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation (chirping). We derive the dependence of the solitary wave structure on the Kerr coefficient ${\ensuremath{\chi}}^{(3)},$ the resonance impurity atom density, and the detuning of the average laser frequency from the atomic transition. When the laser frequency and the atomic transition frequencies are near the photonic band edge we obtain the second type of soliton. To illustrate the second type of soliton we consider two special cases. In the first case, GVD facilitates the propagation of an unchirped SIT-gap soliton moving at a velocity fixed by the material's parameters. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. In the second illustrative case we set the Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ In this case, the solution is a chirped pulse which arises from the balance between GVD and the resonance interaction with the dopant atoms. Finally, we show that under certain circumstances, these solitary wave solutions may persist even in the presence of (subpicosecond) dipolar dephasing of the dopant atoms and absorption losses of the host PBG material, provided that the system is incoherently pumped. These results may be relevant to the application of PBG materials as optical devices in fiber-optic networks.

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Steady propagation of a coherent light pulse in a dielectric medium. II. The effect of spatial dispersion

Kensuke S. Ikeda, O. Akimoto

For pt.I see ibid., vol.10, p.425 (1977). The effect of spatial dispersion or exciton formation on the steady propagation of a coherent light pulse in a dielectric medium is studied. The optical Bloch equation for a lattice of atomic dipoles coupled to one another by an exchange-type interaction is solved simultaneously with the Maxwell equation by the method of power-series expansion developed in the preceding paper. Two kinds of pulse solutions are obtained in general: one propagating by the medium of the radiation field and the other by excitation transfer between atomic dipoles. The former solution is SIT-like in the short pulse limit, but a polariton-soliton or a standing wave of non-linear polariton if the pulse width is long. The latter is an exciton-soliton containing little photon component. When the pulse width is extremely long, there appears a polariton-soliton of a new type in which the two propagation mechanisms are mixed. It is found that the spatial dispersion prevents the polariton-soliton solutions from existing in a certain frequency range. This effect is explained on the basis of the functional behavior of the dispersion relation of the non-linear polariton.

Journal of Physics A Mathematical and General•Jul 1, 1979•16 citations
The phenomenon of self-induced transparency (SIT) in the exciton range of the spectrum consists of the propagation through the crystal of solitary wave packets of coherent excitons and photons with anomalous small losses. In the running frame of reference at the site occupied by the soliton a single aperiodic transformation of photons into excitons and a reverse motion of the energy into the field takes place. The solutions of the SIT equations have been received in the slowly-varying amplitude approximation. Taking into account the phase modulation, the authors have shown the appearance of the branch of the carrier wave frequency with anomalous dispersion not only at the given pulse width but at the given pulse amplitude. They have also found the exact solution of the above mentioned equations taking into account the effect of saturation of the dipole momentum transition from the crystal ground state to the exciton state. It leads to the increasing of the soliton time width, to the decreasing of its amplitude and worsens the SIT observation.
We determine the dispersion law of the carrier wave of an exciton-photon pulse which propagates as a solitary wave. For short pulses, four branches $\ensuremath{\Omega}=\ensuremath{\Omega}$ ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}$) appear: a branch which goes through the intersection point $\ensuremath{\hbar}cK={E}_{o}+\frac{{\ensuremath{\hbar}}^{2}{K}^{2}}{2M}$ of the uncoupled modes of the free exciton and the free light field, two polaritonlike branches and an excitonlike branch. The polaritonlike branches have a normal dispersion, whereas the branch which goes through the intersection point of the free modes has an anomalous dispersion for sufficiently short pulse width. This branch contains the solution of McCall and Hahn as a special case. For very short pulses the branch with anomalous dispersion continues into the two polaritonlike branches. For increasing pulse width a forbidden energy gap appears at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}=0$. For long pulses, only the two polaritonlike solutions exist.
The results of theoretical research of the propagation effect of an ultrashort optical pulse under the conditions of coherent interaction with two resonant transitions are presented. One of the transitions is tuned to exact resonance with the pulse carrier frequency; the other is distanced by much more than the pulse spectral width, thus the adiabatic following approximation may be applied to it. The interaction between the pulse and the far-from-resonance transition is described by the wave equation with the nonlinear index of refraction in the form of the Lorentz function of the field amplitude. This wave equation is solved together with the Bloch equations for the exact resonant transition. The solution in the form of the stationary phase-modulated pulse is obtained, and (pi) -pulse of self-induced transparency is the special case of this solution. The obtained solution has the area under the pulse envelope less than (pi) , and with increasing the absorption coefficient of the far-from-resonance transition the pulse area decreases. The effect can be used for reducing the transparency threshold of the resonant medium below (pi) .
Theoretical research of propagation effect of ultrashort optical pulse under conditions of coherent interaction with two resonant transitions simultaneously is presented. One of the transitions is tuned in exact resonance with pulse carrier frequency; the other is distanced much more than spectral pulse width, thus the adiabatic following approximation may be applied. Interaction between the pulse and far-resonant transition is described by wave equation with nonlinear index of refraction in the form of Lorentz function for field amplitude. This wave equation is solved together with Bloch equations for the exact resonant transition. The solution in the form of stationary phase-modulated pulse is obtained, and 2π-pulse of self-induced - transparency is the special case of this solution. The obtained solution has a square under pulse envelope less than 2π, and with increasing of far-resonant transition absorption coefficient the pulse square decreases. The effect can be used for reducing of the transparency threshold of the resonant medium below pi.
The form invariant coherent pulse propagation in semiconductors excited at 1s-exciton resonance is studied analytically using the reduced semiconductor Maxwell-Bloch equations. The sech-shaped pulse solution for excitonic self-induced transparency (SIT) is presented, showing significant difference in comparison with the well known SIT solution for non-interacting two-level systems. In contrast to 2π pulses in atomic systems, the phenomenon of SIT of interacting excitons in semiconductors occurs for the pulses of area 1.07π. Possible applications of the SIT solitons in semiconductor all-optical switching devices are discussed.
We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort (picosecond) laser pulses which propagate near the band edge of the nonlinear photonic band gap (PBG) material doped with rare-earth atoms such as erbium. Solitary wave formation involves the combined effects of group velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We derive the general Maxwell-Bloch equations for a nonlinear PBG system and then demonstrate the existence of elementary solitary wave solutions for frequencies far outside the gap where GVD effects are negligible and for frequencies near the photonic band edge where GVD effects are crucial. We find two distinct new types of propagating SIT solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the nonlinear Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ However, when the host nonresonant Kerr coefficient is nonzero, we obtain the first new type of soliton. In this case, the optical soliton envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation (chirping). We derive the dependence of the solitary wave structure on the Kerr coefficient ${\ensuremath{\chi}}^{(3)},$ the resonance impurity atom density, and the detuning of the average laser frequency from the atomic transition. When the laser frequency and the atomic transition frequencies are near the photonic band edge we obtain the second type of soliton. To illustrate the second type of soliton we consider two special cases. In the first case, GVD facilitates the propagation of an unchirped SIT-gap soliton moving at a velocity fixed by the material's parameters. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. In the second illustrative case we set the Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ In this case, the solution is a chirped pulse which arises from the balance between GVD and the resonance interaction with the dopant atoms. Finally, we show that under certain circumstances, these solitary wave solutions may persist even in the presence of (subpicosecond) dipolar dephasing of the dopant atoms and absorption losses of the host PBG material, provided that the system is incoherently pumped. These results may be relevant to the application of PBG materials as optical devices in fiber-optic networks.
For pt.I see ibid., vol.10, p.425 (1977). The effect of spatial dispersion or exciton formation on the steady propagation of a coherent light pulse in a dielectric medium is studied. The optical Bloch equation for a lattice of atomic dipoles coupled to one another by an exchange-type interaction is solved simultaneously with the Maxwell equation by the method of power-series expansion developed in the preceding paper. Two kinds of pulse solutions are obtained in general: one propagating by the medium of the radiation field and the other by excitation transfer between atomic dipoles. The former solution is SIT-like in the short pulse limit, but a polariton-soliton or a standing wave of non-linear polariton if the pulse width is long. The latter is an exciton-soliton containing little photon component. When the pulse width is extremely long, there appears a polariton-soliton of a new type in which the two propagation mechanisms are mixed. It is found that the spatial dispersion prevents the polariton-soliton solutions from existing in a certain frequency range. This effect is explained on the basis of the functional behavior of the dispersion relation of the non-linear polariton.