Файл: Antsiferov V.V., Smirnov G.I. Physics of solid-state lasers (ISBN 1898326177) (CISP, 2005)(179s).pdf

Stochastic and transition processes in solid-state lasers

Chapter 7

Stochastic and transition processes in solid-state lasers

7.1 STATISTICAL MODELLING OF LASING

The lasing properties of solid-state lasers are determined to a large degree by stochastic processes which have been the subject of special attention in recent years [1, 2]. In this chapter, we examine the statistical model of the transition processes in lasing. The initial and non-linear stages of lasing are analysed in detail. A dependence is found between the lasing conditions and the fluctuations of the time of realisation of the transition process. In the approximation of weak saturation, the stochastic behaviour of lasing can be stimulated using the Langevin equation for the total number of lasing photons in the unit volume n, having the form [3,4]:

dt

Here G is the mean value of the lasing rate; f(t) is the part of the rate of lasing fluctuating in a random manner. In the stochastic non-linear differential equation (7.1), the function f(t) describes the δ -correlated random process, determining the properties of the latter in a Langevin source of fluctuations [3–5]:

The origin of these fluctuations is determined by the fact that the photons can be excited by non-equilibrium radiation, and also in the absorption of the quanta of vibrations of the lattice (phonons) and of the photons of thermal radiation which are in thermal equilibrium with the active medium. The region of applicability of the balance equation (7.1) for the concentration of the photons is restricted by the limits of taking into

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account only the linear and quadratic [with respect to n] terms, corresponding to taking into account the process of amplification of radiation and the saturation effect. Term α n describes the lasing of photons as a result of optical pumping. α denotes the effective coefficient of amplification of radiation. The dependence of coefficient α on radiation frequency ω reflects the specific features of the heterogeneous medium, including solid-state nanostructures with quantum wells. The term –γ n2 determines the first correction non-linear with respect to the amplitude of the light field in the polarisation of the active medium taking saturation into account. The fluctuations of the number of seed photons and different random processes determine, during the multiplication period, the stochastic behaviour of the dynamics of increase of the photon concentration. This behaviour is reflected in the fact that in multiple and rapid lasings (α > 0), in comparison with the time

the increase of the photon concentration to some level below the asymptotic value

takes place in different periods of time.

We shall note several special features of the increase of lasing intensity resulting directly from (7.1). The point n = 0.5ns, being the inflection point of the curve n = n(t), and the first derivative of the function n(t) at this point is equal to

In multiple realisation of the transition process this quantity does not change. Consequently, the slope of the curves of increase of the photons concentration n(t) in the vicinity of the value 0.5 ns should be the same; the fluctuations of the number of seed photons result in the random parallel displacement of the curves n(t) in the vicinity of the level n = 0.5 ns. For the statistical analysis of the transition processes it is convenient [6,7] to transfer from the stochastic differential equation (7.1) to the corresponding Focker–Planck equation for the photon distribu-

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Stochastic and transition processes in solid-state lasers

tion function W(n, t):

This transition and the criteria of applicability of the non-linear FockerPlanck equation have been examined in detail in [3–5]. Equation (7.6) is valid for not too high concentrations n because it was derived using only the first and second moments of this random quantity. It should be noted that equations of the same type were used previously when describing the gas-discharge plasma [8] and quantum fluctuations of lasing [9, 10].

The probability of finding the value of the photon concentration in the range from n to n + dn at the moment of time t is W(n, t)dt. Equation (7.6) describes the evolution of the initial distribution of the ‘seed’ number

Function W(n, 0) can be easily found by assuming that the stationary state is established below the threshold. The solution of equation (7.6) in the form

eter G below the lasing threshold. Since the number of photons prior to excitation is small, the role of saturation below the lasing threshold is also negligible. From (7.7) we have the exponential dependence of the normalised distribution function on n:

After supplying a right-angled excitation pulse, the asymptotic state of stationary lasing with the Gaussian distribution of the photons

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is established. This distribution was obtained from (7.6) at ∂ W/∂ t = 0. Here W0 is the normalisation multiplier; N = α 2 /γ is the mean number of the photons in the stationary lasing regime; the parameter characterises the dispersion of photon distribution. The switching off of the pumping is accompanied by the attenuation of lasing and a smooth transition of the distribution (7.9) to the exponential distribution (7.8) at relatively long times.

7.2 INITIAL AND NON-LINEAR STAGES OF THE LASING PROCESS

We shall examine the process of increase of the photon concentration at α > 0. The exact solution of equation (7.6), describing this process, is not possible. Therefore, we shall use a number of simplifications which, as shown later, are fully acceptable in the examined case. In the initial stage of lasing, where the photon concentration is low, it is possible to ignore the saturation effect, assuming that

To find the solution, in accordance with Feller [11], we use the Laplace transform:

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Stochastic and transition processes in solid-state lasers

the equation (7.11) in the Laplace variables is transformed to the form:

in this case, the function ϕ (t) = 0 because of the general properties of W(n, t), and to determine the Green function, we use the solution

where I0(u) is the Bessel function of the apparent argument. Consequently, the solution of equation (7.11) is the initial function

(7.8) at t0 = 0 can be written in the following form

n0 = n1 − n2 .

This equation is derived from equation (7.6) at γ = 0 without the diffusion term, if we re-determine the initial distribution (7.8), replacing n1 by n0 . At t > α 2−1 , the term with G2 in equation (7.6) can be ignored, and the effect of the noise at t > 0 is reduced to the effective increase of the initial seed by the value n2.

Equation (7.18) describes the process of transition from the stationary state below the lasing threshold, characterised by the photon distribution function (7.8), to the state of the initial stage of excitation of lasing with the linear increase of the photon concentration, accompanied by a decrease of the fluctuations of this concentration. As indicated by (7.18), this corresponds to decrease at t → ∞ of the probability W(n, t)∆ n of obtaining, as a result of excitation, a photon concentration differing from

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