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

Physics of Solid-State Lasers

pulses with the intensity lower than the threshold of single-pulse failure. In the presence of absorbing inclusions in the medium, this may be caused by the build-up of irreversible changes in the medium. In the absence of inclusions, the build-up effect is also observed but the reasons for this have not as yet been determined [49].

All the previously mentioned mechanisms also operate on the surface of the active medium. However, because of a considerably higher concentration of defects (in comparison with the concentration in the volume of the medium) in the subsurface layer there is a large scatter of the values of the breakdown threshold from specimen to specimen; in most cases, the thresholds of laser failure of the surfaces are lower (by a factor of 2 or more) than the threshold of volume failure [49].

It has been confirmed reliably that when self-focusing of radiation in the medium is avoided and the energy of the light quanta is smaller than the half width of the forbidden band, and there are no spatialtime fluctuations of laser radiation (single-mode and single-frequency radiation), the threshold of natural breakdown is a constant of the medium that does not depend on the radiation parameters and for K8 glass its value is 1013 W/cm2 [49].

5.8 NEW OPTICAL CIRCUITS OF SOLID-STATE LASERS

Recently, new circuits and design of solid-state lasers have been developed ensuring high power and energy of radiation to be obtained with a high degree of spatial and time coherence. In solid-state lasers, a large part of the pumping energy is not converted to lasing radiation and is transferred to thermal energy, including heating of the active element. Thermal energy is removed from the active element from its surface by liquid flows. Consequently, thermal gradients and thermo-optical strains form in the active medium and lead to the distortion of the radiation wavefront, passing through the active medium.

The application of a circuit with the mechanical removal of the heat from the excitation channel makes it possible to eliminate the thermo-optical strains of the active medium [50] and obtain a high mean lasing power (> 1 kW) at the diffraction divergence of radiation (< 1 mrad). The application of thin flat sheets instead of round bars [51] under the condition of the homogeneity of pumping has greatly increased the lasing energy parameters with a significant improvement of the spatial characteristics of radiation.

In Ref. 52, the authors proposed a laser circuit with a plate-shaped active element with the zigzag passage of radiation through the plates. Consequently, it was possible to compensate the thermal optical distortions, induced in the active rod, by pumping radiation. The pumping radiation is induced in a plate-shaped active elements by bifocal lens with

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birefringence [49]. In circuits with the zigzag passage, these distortions are compensated and added up together during the passage of the beam from one plane to another with the opposite sign. Consequently, compensation results in high spatial and angular characteristics of radiation. The output radiation power in optimised, compact circuits is higher than 0.5 kW, with a high beam coherence [53].

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Physics of Solid-State Lasers

Chapter 6

Principles of lasing of solid-state lasers

6.1 QUANTUM KINETIC EQUATION FOR THE DENSITY MATRIX

In a semiclassic approximation in which the quantum fluctuations of the radiation field are ignored, the resonance interaction of the atom (or molecule) with the electromagnetic wave E(r, t) can be described by the Schrödinger equation

for the wave function Ψ (r, ξ , t), where r is radius-vector of the centre of inertia of the particles; ξ is the population of its internal co-ordinates. The particle energy is determined by the eigenvalues of the Hamiltonian

which is represented by the sum of the operator of the energy of the non-perturbed electron shell and the operator of interaction with external

fields D V (r, ξ , t).

In the statistical examination of the effect of the environment on the examined system, it is efficient to transfer from the wave function to the density matrix

which makes it possible to describe by a simple procedure the completely and partially defined quantum mechanics state.

The Neuman equation for the density matrix

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obtained from (6.1) is, as is well known, the most general form of the quantum mechanics description of the evolution of different systems [1, 2]. In accordance with the principles of quantum theory, the calculations of the mean quantum mechanics values of the physical quantities are carried out using the equation

where is the operator corresponding to quantity A.

A

When describing the interaction by internal variables and after averaging with respect to the degrees of freedom of the environment, excluding the field with a resonance interaction with the emission particle E(r, t), equation (6.4) is greatly simplified and assumes the following form

where j, l are the indices of the energy levels. The interaction of the electromagnetic wave E (r, t) with the examined particle is described in this case by the operator Vjl(r, t), whereas the operator Rjl takes into account the averaged-out effect of the environment on the particle which is usually assumed to be a stationary random process.

In the model of the relaxation constants, the quantum kinetics equations (6.6) have the following form

Here Γ jl are the elements of the relaxation matrix in the case in which their correlation time is short in comparison with the characteristic time of variation of ρ jl(t). The diagonal elements of the density matrix determine the population of the energy levels of j; non-diagonal elements ρ jl characterise the correlation of the states j and l; the quantities qj take into account the excitation of levels j.

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Physics of Solid-State Lasers

For the simplest two-level model of the active particles, the system of equations for the elements of the density matrix, corresponding to the m–n transition, has the following form

here Γ m, Γ n are the total widths of the working levels m, n; Γ is the constant of polarisation relaxation, induced by the radiation resonant in respect of m–n; Amn is the Einstein probability of spontaneous decay. Equations of this type were obtained for the first time by Bloch for describing the precession of the nucleus spin in the magnetic field. From the physics of nuclear magnetic resonance, investigators transferred to the quantum electronics of the concept of transverse and longitudinal relaxation T2 = Γ –1, T1 = Γ –m1 (and Γ m = Γ n), respectively.

In the electric dipole approximation, the operator of interaction with

where dmn is the matrix element of the operator of the electric dipole moment d; ω mn is the Bohr frequency of the m–n transition. The polarisation of the medium P is the total dipole moment of the unit volume, and can be calculated from the following equation:

where the spur is calculated from the variables of the entire ensemble of the particles in the unit volume. In particular, the following relationship for polarisation corresponds to the two-level approximation:

The formal solution of the last equation of the system (4.8) has the following

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Principles of lasing of solid-state lasers

form

N = ρ mm −ρ nn

is the difference of the populations of the working levels. Consequently

After double differentiation of these equations with respect to time, we obtain the following equation for the polarisation vector:

This equation, together with the equation for the difference of the populations N(t) and the Maxwell equations for the electromagnetic field E, can be used for describing the laser generation process in a self-consistent manner.

6.2 EQUATIONS FOR THE ELECTROMAGNETIC FIELD

The electromagnetic field in a laser resonator can usually be represented in the form of two running waves:

where Es(t) are the slowly changing functions of time; the index s indicates the direction of propagation of the running wave with the frequency ω and the wave vector k = ω /c. In a laser with a ring resonator, the lasing conditions form when Es ≠ E–s. For a laser with a Fabry– Perot resonator we have: Es = E–s.

The field E(z, t) is governed by the Maxwell equations which can be written in the following convenient form:

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Physics of Solid-State Lasers

Here εαβ is the tensor of dielectric permittivity; for the medium with no optical activity εαβ = δ αβ . All losses in the resonator are taken into account by introducing the tensor of effective conductivity σ αβ . Consequently, it is the necessary to solve the boundary problem because the description of the losses by ohmic conductivity gives the same results.

The vector of polarisation of the active medium P is expressed by the density matrix using equation (6.7), which makes it possible to represent it in the form identical to (6.15):

Substituting (6.15) and (6.70) into (6.16), and carrying out averaging with respect to high-frequency oscillations, we obtain an equation for slow amplitudes of the field:

Subsequently, we examine the case of linearly polarised radiation and σ αβ = σδ αβ . Consequently, the equations can be greatly simplified.

6.3 MODELLING OF LASER SYSTEMS

The effect of an illuminating filter or some other non-linear medium under the effect of coherent radiation is modelled quite easily in the two-level approximation. The main equations, describing the behaviour of the twolevel system in the field of the running wave, taking into account equation (6.18), have the following form

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