Russian Physics Journal, Vol. 37, No. 10, 1994

SOLITON FORMATION

P R O C E S S E S IN O P T I C A L L Y D E N S E

MEDIA

Yu. V. Kistenev

UDC 535.345

Numerical methods are used to investigate the characteristics of soliton formation when optical pulses of arbitrary shape and duration propagate in resonantly absorbing media of large optical thickness. It is shown that a regime of soliton propagation in a resonant medium, characterized by a nonlinear Schr6dinger equation model, changes to a self-induced transparency regime. Incoherent steady-state interaction conditions are then replaced by coherent interaction. This effect can be utilized to compress laser pulses.

INTRODUCTION Two classes of soliton are known in optics, depending on the characteristic features of their interaction with the medium. In the absence of resonant absorption a soliton regime results from competition between the diffusive spreading of a pulse and its nonlinear self-compression. Such solitons are described by a nonlinear Schr6dinger equation model [1]. Under resonant absorption conditions an optical soliton involves coherent excitation of a medium in the leading part of the pulse and the return of the medium to the initial state on its trailing edge due to stimulated re-emission. This regime is referred to as self-induced transparency and arises when the pulse duration is much shorter than the relaxation times in the medium for pulse shapes in the form of a hyperbolic secant and when the pulse amplitude exceeds a certain threshold value [2]. As a rule, real laser pulses possess characteristics which differ substantially from those of solitons. In view of this the question arises of the possibility of solitons being formed when such pulses propagate in a medium. Individual aspects of this problem, when supplementary factors which hinder soliton formation may be considered as perturbations, were discussed in

[3-5]. In the present paper we consider the characteristics of soliton formation in a spectral region near resonance in two-level media of large optical thickness. The analysis is based on the Maxwell-Bloch system of equations (for example see [2])

0r

=

L 2

~ p (a') g (a - a') aa',

dP ~ . - = - - "~P + i w ~ , O'q O~ w -- w e t~ - - - I m [~*P] ,

0n

Q

(1. l) (1.2) (1.3)

where e is the normalized complex amplitude of the optical pulse; P is the complex polarization amplitude of the two-level particle; g is the distribution function of these particles over the resonant transition frequencies; w is the population difference of the levels of the resonant transition; we is its equilibrium value; r is the optical thickness of the medium; ,5 is the detuning from resonance;/z = Tz/rp, where T2 is the phase memory time of the medium and rp is the pulse duration; Q = T1/T2, where T1 is the longitudinal relaxation time; ~ = (t - Z/c)/rp; 7 = 1 - iAT2.

V. D. Kuznetsov Siberian Physicotechnical Institute at the State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 95-99, October, 1994. Original article submitted January 27, 1994. 1064-8887/94/3710-0997512.50 ©1995 Plenum Publishing Corporation

997

I a

so/

ll g < -

II J

Fig. 1. Dependences of the shape of a pulse transmitted through a medium on the optical thickness r of the medium. Conditions of calculation: q = 4, ~t = 1, S = 127r, A = 0 (a); A'T2 = 1 (b). Here 1(7) and S are respectively the intensity and initial area of the pulse.

It should be noted that the soliton propagation regimes result from corresponding limiting transitions: self-induced transparency solitons are formed when t~ --" oo ; nonlinear Schrodinger equation solitons are formed in the adiabatic following limit [6], in the weak nonlinearity approximation and for a sufficiently large detuning from resonance.

M E T H O D OF SOLUTION The system of equations (1) was solved numerically. The Bloch equations (1.2) and (1.3) can be represented in the form (for example see [7]):

OX = A (~q) X (-q) + R,

P 0~q

(2)

where X T = (P, P*, w); A is the matrix of the coefficients for X; R is the relaxationmatrix. For /~ < 1 this system is singularly Perturbed. The presence of a boundary layer for 0 _< ~l
OX

= aj X

+ R,

(3)

where Aj = A(ej) and ej is an approximation of the pulse shape in thej-th node of the grid ~7 = j.A~, A~/is the grid pitch. For example, for a step-function approximation, ej = e(jAr/). System (3) was solved using a Laplace transformation. The inverse Laplace transform was found from the theory of residues. The poles of the inverse matrix Aj-1 were found numerically or analytically if the equation detA = 0 degenerated into a quadratic equation. The latter can occur for T1 = T2 or for A = 0. The discrete value ~ was found as follows: ~ = X(~ = jan). It can be shown that the algorithm given above is equivalent, when transforming from the system to one differential equation, to the uniformly converging method of an exponential fit [9]. A numerical solution of Eq. (1.1) was found by a second order prediction-correction method. The initial pulse shape was taken as the function e(0, t)=eo[sin(at/Xp)]%

t)=0, t

t~[0,'%],

[0,

As the parameter q was varied the pulse shape changed from quasi-Gaussian to quasi-rectangular. 998

-r(U I

\

\

a /'

\

,oo/

/

Fig. 2. Dependences of the shape of a pulse transmitted through a medium on the optical thickness 7 of the medium. Conditions of calculation: q = 4, ~ -- 0.5, S = 307r, A ' T 2 = 1 (a); q = 4, /x = 0.1, S = 2507r, A = 0 (b).

RESULTS OF C A L C U L A T I O N S The calculations showed that in the case of pulses having steep leading and trailing edges the soliton formation process is considerably distorted in the time domain of the coherent transient processes in the medium which appear when the interaction is nonadiabatically switched on and off. The subsequent analysis is therefore performed for pulses of a smooth shape. The soliton formation process for/~ > 1 and A = 0 differs little from the conclusions of analytical calculations [3]. The influence of relaxation is manifested only in the number of solitons. For tx = 1, relaxation is no longer simply a perturbing factor. As can be seen from Fig. la, under these conditions one first observes an increase in the steepness of the pulse leading edge and a modulation of its shape. Then, as a result of energy being transferred to the leading part of the pulse and of absorption in its trailing part, pulses of area 2~r are formed having a shape close to that of a hyperbolic secant. Since their duration is much shorter than the phase memory time of the medium they will subsequently propagate as self-induced transparency solitons, their number being determined by competition between the above-mentioned processes of energy transfer and absorption. It should be noted that the amount of detuning has little influence on the soliton formation process. Thus, relaxation leads to a reduction in the number of solitons formed and an increase in the optical thickness r c required for this process relative to the classical results [2] when ~ ~ co. When the pulse duration is increased (Fig. 2) the characteristics of the soliton formation process are all manifested still more strikingly. In particular, it should be mentioned that the magnitude of ~'c increases practically linearly as the pulse duration zp increases. The calculating conditions for Fig. 2a are similar to those of the nonlinear Schr6dinger equation interaction model (except for taking account of the imaginary part of the medium polarization which describes the absorption of the pulse). As can be seen, the initial phase of the propagation process is indeed reminiscent of the collapse of a pulse on the nonlinear Schr6dinger equation model [4], although the absorption then causes a transition from the nonlinear Schr6dinger equation soliton regime to the self-induced transparency soliton regime. Figure 2b is of interest because in this case the initial conditions correspond to steady-state incoherent interaction. As is well known (for example see [2]) the nonlinearity can then cause only an equalization of the level population of the resonant transition (the saturation effect). The figure shows that in the case of large optical thicknesses this regime is replaced by a coherent interaction regime which, in particular, is characteristic of the self-induced transparency phenomenon. In contrast to the steady-state interaction conditions, it is possible for the level population to be inverted in this regime. Consequently the limits of applicability of the nonlinear propagation equations are bounded in terms of the saturation effect for media having a large optical thickness. The transition from incoherent to coherent interaction for tx < < 1 is accompanied by a considerable reduction (by factors of ten) in the pulse duration. This effect can be utilized in order to compress laser pulses. The duration of the output pulses is 7p < < T2, and so it can be controlled by changing the phase memory time of the resonant medium. For example, T2 - 109-1010 sec for the gaseous components of the atmosphere [10], and so in the case of atmospheric propagation the pulse duration is reduced to tens of picoseconds. 999

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