Fig.5. Damage threshold versus tempera- lengths/33,34/ Fig.6. Frequency dependence of damage THE THEORY OF INCLUSION-INITIATED LASER DAMAGE IN OPTICAL M.F. Koldunov, A.A. Manenkov, and I.L. Pokotilo Academy of Sciences A model of laser induced damage based on the mechanism of thermal explosion of absorbing inclusions is analyzed. Conditions for occurrence of the thermal explosion are formulated. The influence of saturation of the absorption of laser radiation and the role of photoionization of a dielectric matrix by absorbing inclusion on thermal radiation in explosion development are considered. The kinetics of the thermal explosion are analyzed, and the pulse width dependence of the damage threshold is found. Numerical estimates of the damage thresholds for typical cases are presented, showing that the thermal explosion model considered is a rather realistic one for describing the laser-induced damage in optical materials containing absorbing inclusions. Key words: absorbing inclusions, damage threshold, laser induced damage, photoionization, pulse-duration dependence, thermal explosion, thermal radiation. The important result of the numerous experimental investigations which have been carried out until recently consists of the fact that absorbing inclusions strongly affect the laser-induced damage in dielectric materials /I/. Many laser damage features (low damage threshold of the dielectric surface as compared with its bulk, variation of the damage threshold in a sample and from one sample to another, damage statistics and so on) are explained from this standpoint. The pioneering theoretical investigations of laser damage due to absorbing inclusion were carried out in papers /2,3/ where laser heating of the absorbing inclusion was analyzed under the assumption that the material parameters, such as absorption and thermal conductivity coefficients, are independent of temperature. Expressions were found for a laser damage threshold in terms of different criteria of the critical temperature, T, corresponding to the melting point /2/ and the mechanical breakdown /3/. The models considered in papers /2,3/ qualitatively explain some important regularities of laser induced damage, such as the dependence of the damage threshold on pulse duration /2/, but they do not sufficiently correspond to physical processes of laser damage. First of all, the assumption that the material parameters are independent of temperature cannot be correct, since the temperature in an inclusion region can reach 10“ "K, when the intensity is equal to the damage threshold /I/. Besides, in a framework of the thermoelastic model /3/ the value and physical sense of the critical temperature remain indefinite because of difficulty in estimating critical stresses at which the mechanical breakdown occurs. Finally, from the viewpoint of the thermoelastic stress model a correlation between the damage threshold of the materials and their physicalmechanical parameters should take place, which does not agree with the experimental data. A substantially new approach to the laser damage problem has been proposed in paper /4/, where it was shown that an allowance for the temperature dependence of the material parameters yields a qualitative change in the character of laser-produced heating of the absorbing inclusions. In this case there is a threshold intensity ¶ (or an associated critical temperature T2) which if exceeded, leads to thermal explosion of the absorbing inclusion. The thermal explosion model explains the catastrophic character of laserinduced damage with a strictly defined damage threshold. If the energy absorbed by the inclusion is not sufficient to produce a macrodamage in the material (it can take place for inclusions of very small submicron size), the thermal explosion can serve as a source of absorption in the surrounding material through photoionization by thermal radiation. This mechanism was proposed in Ref. /5/ and developed in detail in Ref. /6/. Another mechanism of additional absorption is associated with thermal-ionization /7/ but it, obviously, may be effective only for the narrow bandgap materials. The aim of the present paper is to report a consistent treatment of the laser-induced damage model related to thermal explosion of the absorbing inclusions. In Section 2 there is an analysis of the conditions for the occurrence of thermal instability and the role of the absorption saturation in the inclusion and the conditions of thermal instability in the material associated with photoionization. In Section 3 we consider the kinetics of thermal explosion and the pulse-width dependence of damage threshold. In Section 4 major results obtained are discussed and numerical estimations of the damage threshold predicted by a thermal explosion mechanism are presented for a typical case. II. LASER HEATING OF ABSORBING INCLUSIONS The laser-induced damage in optical materials is related, as a rule, with the absorbing inclusion of a small size R10-5 cm (R is the inclusion radius). For this reason, it is rather difficult to control them in optical materials. Typically, they are the foreign material particles of metallic or oxide nature. The inclusions usually have very high absorption and in spite of their small size they can create high temperature in their vicinity. It is important that the absorption coefficient depends on temperature. For example, in the case of a semiconductor-type inclusion the temperature dependence of the absorption coefficient is described by where A is the activation energy, x and x1 are the constants. This example illustrates the principle laws of T-dependence of x(T) in the general case: at low temperatures x(T) is independent of T; x(T) increases rapidly in the region of T -A/2 and x(T) no longer increases at high temperatures. The specific feature of the behavior of x(T) defines a physical picture of the inclusion laser heating: increase of x(T) can give a thermal instability of the inclusion with rapid growth of its temperature (thermal explosion). The saturation of absorption leads to limitation of the temperature rise, temperature stabilization, and the suppression of thermal explosion. Below we shall analyze laser heating of the absorbing inclusions on the basis of solution of the thermal conductivity equation, taking into account the temperature dependence of absorption coefficient of the conclusion and surrounding dielectric. As was mentioned above, the increase of x(T) with increasing T results in thermal explosion of the inclusion and in sharp temperature increase in its vicinity. In such conditions qn is defined as the laser light intensity above which the thermal explosion occurs. For this reason, in the framework of the thermal explosion theory the mechanical properties of the material do not play any role in defining ¶n, and its value is determined completely by the nonlinear characteristics of absorption and some other material parameters related to thermal conductivity. Basic Equations Heating of inclusion with the radius R by a laser pulse with the maximum intensity q and temporal shape f(t) is described by the thermal conductivity equation: C is the heat capacity, Px is the density, x(T) is the thermal conductivity of the inclusion and matrix (subscript K I refers to the inclusion, and K 2 to the matrix), n(x) = I at x > 0, n(x) = 0 at x ≤ 0, and o(T) is the absorption cross-section of the inclusion at temperature T. Hereinafter the inclusion temperature is considered as a surface temperature. It is a good approximation for the inclusion with a strong thermal conductivity for which where AT is the temperature difference in the inclusion center and on its surface. For pulse duration and intensity q - ≤ ¶n (¶n is the damage threshold) the stationary temperature distribution around the inclusion is arranged, and the maximum heat flow potential is satisfied by the following equation: Heat flow potentials which are determined by Eqs. (4) and (5) depend on the inclusion temperature. Its value is defined by o(T) which depends on the temperature itself. Its self-coordination value is determined from the condition of equality of heat flow potentials at the inclusion surface (r R). It gives: Thermal Instability The laser heating character of the absorbing inclusion for laser pulse duration To >> 72 is determined by the (T) function, which, therefore, determines the solution behavior of Eq. (2). When Eq. (6) has a solution, it gives maximum temperature of the inclusion corresponding to the stationary temperature distribution around the inclusion. The function (T) has a maximum if the absorption cross-section grows rapidly with temperature. It can be found from the equation (T2) When Eq. (7) has the only solution T, the function (T) has single maximum 9n (see Fig.1, plot 1). Equation (6) has a solution only in the region q≤ ¶n' At q> ¶n Eq. (6) has no solution, i.e. the stationary temperature distribution around the inclusion cannot occur; then, a thermal explosion develops. So the temperature at which (T) is maximum is the inclusion critical temperature, which, if exceeded, leads to thermal instability. The maximum of (T) gives the threshold intensity an For this reason, function (T) will be called the threshold function. the It is of great interest to consider the influence of the saturation of absorption on the development of thermal instability. Let us investigate this influence for the semiconductor-type inclusion, the absorption cross-section of which is given by σ(T) - σ + σ2 exp(-4). (see (I)). Let x2 be independent of T; then Eq.(7) has the following form: In order to find the parameter region of this model where the explosion takes place, we shall follow the theory of catastrophes /9/. The region boundary is For the inclusion in a dielectric with parameters 98 and from region I (see O (9) Fig. 2) Eq. (9) has no real solution. The threshold function (T) increases monotonically with the temperature rise and, hence, there is no thermal instability. For the inclusion with the parameters from region II (see Fig.2), eq. (7) has two solutions. In this case the threshold function is shown in Fig. I (plot 2). At q> In the thermal instability occurs and the temperature grows up to T due to saturation of inclusion absorption. In the limiting case when - 0, Eq. (8) has an exact solution, substitution of which into Eq. (6) at A >> To yields the threshold intensity: |