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

Physics of Solid-State Lasers

∆ n by no more than n. With the development of the transition process and the approach to the stage of the stationary state, the behaviour of the photon concentration becomes less and less stochastic.

We now examine the non-linear period of the increase of the electron concentration in which it is necessary to take into account the saturation effect. The subsequent effect of noise can be ignored. In this situation, we can use equation (7.6) without the diffusion term

with the initial condition (7.17).

From equations (7.12), (7.70) and (7.19) at t > α –21 we obtain:

Multiple activation of pumping, being the source of lasing, formally corresponds to the ‘switching on’ of parameter α 2. The value determines the density of the tracks of the increase of lasing, corresponding to multiple ‘switching on’ of parameter α 2 and the concentration of the photons in the range from n to n + ∆ n at the moment of time t.

7.3 ANALYSIS OF THE RELATIONSHIP BETWEEN THE LASING CONDITIONS AND FLUCTUATIONS OF THE DURATION OF THE TRANSITION PROCESS

From the viewpoint of practical applications, of greatest interest are

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the fluctuations of the time of increase of the photon concentration to

is the equation of balance for the electron concentration, describing the non-linear stage of lasing.

Using the relationship (7.16), we obtain the expression for the distribution

~b g

function W t , corresponding to the arbitrary initial distributions of the electron concentration W(n0):

(7.24)

θ = nsη e−α 2t . n2

Using the representation of the modified Bessel function I0 (u) in the form of a series, the time dependence can be expressed in the explicit manner:

From equations (7.19)–(7.21) we obtain the following expression for

the function W t :

~b g

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The mean time ~t to the appearance of the given concentration of the photons n, determined by level η and dispersion σ 2t of this time, calculated using (7.27) are equal to respectively:

it is not possible to define more accurately the condition of rapid activation of the source of excitation taking into account the fluctuations of the time to establishment of the stationary regime. It is evident that

for this purpose it must be that the duration of radiation of the parameter

~ α −1 should be considerably less than t ~ 2 .

The relative fluctuations of the time to establishment of the given concentration of the photons, determined in accordance with (7.28), (7.29) by the relationship

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at the fixed values of n0 and η decrease logarithmically with increasing stationary concentration n2.

Thus, in the proposed physical model of the transition processes of laser lasing it is shown that to eliminate the fluctuations of the photon concentration, it is necessary to stabilise the number of seed particles to the excitation of lasing, and also in the initial stage of excitation with the duration of the order of –α –1.

7.4 THE STATISTICAL MODEL OF THE EXCITATION OF LASING IN THE ABSENCE OF INITIAL THERMODYNAMIC EQUILIBRIUM

We shall examine a statistical model of the excitation of a photoflux in the absence of the initial thermodynamic equilibrium. At the arbitrary initial distribution W(n0), the mean time of establishment of t and the dispersion σ 2t will be calculated using the expansion of (7.25). As a result, we obtain:

where ψ (k + 1) is the Euler psi function; ζ(2, k) is the Rieman ζ-function. It may be seen that the main dependence of t and σ 2t on the excess above the lasing threshold α 2 is the same for the functions can W(n0) is differing in the arbitrary manner, whereas at the given values of the parameters α 2 and n2, the quantities t and σ 2t depend on the coefficients Lk, i.e. determined by the form of the initial distribution W (n0).

The relationships (7.34) and (7.35) can be used for analysis of the statistical phenomena determined by the effect of the pumping pulses, separated in time, on the active medium. It is assumed that a right-angled initial excitation pulse is applied to the active medium in such a manner as to establish the asymptotics value of stationary lasing with the Gaussian distribution of the photons (7.9). After switching off this pulse, lasing attenuates and the distribution (7.9) changes smoothly over a relatively long period of time to the exponential distribution (7.8), chatacterising the thermodynamically equilibrium state of the system. The second pulse is applied after time T after the completion of the first excitation pulse, and the distribution (7.9), transformed to the moment t = T is a result

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of the relaxation system, will represent the seed distribution for the subsequent increase of lasing. Consequently, the quantities t and σ 2t, corresponding to this increase, depend in the given situation on the delay time T between the excitation pulses.

We shall explain the variation of the distribution (7.9) at the moment of time t = T, after rapid switching of the parameter of lasing to the position below the threshold of appearance of the lasing photons. We examine the case in which the role of noise is not large because the distribution of the photon concentration at the moment of collapse of lasing may be approximated by the delta function

This condition is realised at high values of the lasing parameters α 1 below the threshold.

The function n(t) can be easily determined from the balance equation describing the attenuation of lasing without taking the fluctuations into account

where α 02 is the lasing parameter corresponding to the first excitation pulse; N is the mean concentration of the photons in the stationary lasing conditions. Consequently, the dependence of the quantities t and σ 2 on the delay time T can be determined by means of the relationships (7.34) and (7.35) in which now

and n(T) is defined by equation (7.37) at t = T.

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