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

Principles of lasing of solid-state lasers

Here ω r denotes the eigen frequency of the resonator which can differ both from the frequency of resonance transition ω mn and from the frequency of the electromagnetic field ω ; d is the reduced matrix element of the dipole moment.

The spatial heterogeneity of the inversion of populations might become evident in the field of coherent counter waves. At not too high intensities it is accompanied by the oscillator spatial modulation of the inversion of the populations of the type

where N1(z, t) is the component of the inversion of the populations, slowly changing along the z axis; N2 (z, t) is the amplitude of the rapidly changing part of the inversion of the populations, reflecting the presence of its spatial modulation. The system of self-consistent equations in this case has the form

The systems of equations (6.90)–(6.21) and (6.23)–(6.26) are analysed taking into account the boundary conditions, determined by laser geometry, and the initial conditions.

We examine a ring-shaped running wave laser on the condition that the rate of polarisation relaxation is high:

In this case, it may be assumed that the polarisation of the medium tracks the changes of the field:

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In this case, the ring-shaped running wave laser can be described by the balance equations:

where Us is the density of radiation equal to the amount of energy of the electromagnetic field in the unit volume of matter;

is the spectral Einstein coefficient characterising the probability of forced transition; BUs is the quantity proportional to the probability of forced transition; BN describes the resonance amplification (N > 0) or absorption (N < 0) in the medium.

The density of radiation Us in J/cm3 is linked with the intensity of radiation Is (W/cm2) and with the number of photons in the unit volume ns in cm–3 by the following equation:

In a laser with counter waves, if the periodic population grating does not form, it may be assumed that N2 = 0, N = N1, and the lasing behaviour at ω = ω r = ω mn in the balance approximation can be described by the equation:

b∂ t+ γ gN= N0γ − BNU, U= U+1+ U−1 (6.33) and by two equations of type (6.30) at s = ±1. For the processes that

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Principles of lasing of solid-state lasers

are slow in comparison with the duration of double passage of radiation along the resonator T, these equations transform into a single equation for the total density of radiation U:

Here σ 0 is the coefficient of laser radiation losses; µ is the degree of filling of the resonator by the active medium. The system of equations (6.33), (6.34) corresponds to the simplest model of the solid-state laser and makes it possible to examine important relationships governing generation.

The systems of the energy levels of the active media enable the inversion of populations to be developed between two levels in the generation channel using external energy sources. For controlling lasing, in addition to the active medium, it is necessary to introduce laser media with the non-linear dependence of the difference of the populations of two levels of the intensity of laser radiation. Figure 6.1 shows the schema of the energy levels in cases of laser media that are of greatest interest for practice.

For the displayed two-, threeand four-level systems with the working m–n transition on the condition that polarisation tracks the resonance field of radiation, the behaviour of laser lasing is described by the balance equations (6.33), (6.34). However, in these schemes, the relaxation constant of the inversion of populations γ , the initial value of the inversion N0 and coefficient B show different dependences of the probability of spontaneous decay Ajl, the probabilities of non-optical transition ν jl and the probability of forced transitions.

1. For the two-level system (Fig. 6.1a), we have

where the coefficient is described by equation (6.31).

2. A ruby laser operates on the basis of the three-level schema (6.1b). Taking into account the probability of forced absorption BpUp in the pumping channel n → l, the parameters of equation (6.33) have the following form:

Quantity η denotes the fraction of particles falling from the level l on

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Fig. 6.1 Diagrams of energy levels and probability of transition between them in two-level (a), three-level (b) and four-level (c) laser systems.

the upper working level m.

3. The 4-level schema (6.1c), is described by the relationships:

In the initial stage of initiation of lasing when the radiation density U is not high and the level n can be regarded as not populated, the constants γ and B are expressed by the equations:

In the lasing regime, radiation density increases and together with it

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Principles of lasing of solid-state lasers

the populations n, and in this case

The four-level schema is used in the majority of lasers in which the media are activated by the ions of rare-earth elements (in particular, aluminium–yttrium garnet and glasses activated by Nd ions).

6.4 FREE LASING

In the previous section, it was shown that the simplest model of the solidstate laser is defined by the system of equations (6.33), (6.34) for the inversion of the populations N and radiation density U. The system permits the following stationary solutions:

The trivial solution of (6.40) corresponds to the case of the maximum attainable excitation of the medium in the absence of lasing. The solution of (6.41) describes the usual stationary lasing in which the gain in the system is equal to the losses.

Analysis of the stability of the stationary state of relatively small perturbations ∆ U, ∆ N is carried out by means of linearisation with respect to these perturbations. In this case, the solution of the linearised system of equations is found in the form

The roots of the quadratic characteristic equation in the case of the trivial solution

β N0 > σ 0, condition (6.40) is unstable and becomes a saddle-type singular point.

The stationary state (6.41) with non-zero intensity exists if N0 > Nc. The characteristic equation for this case

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