Applied Optics Letters to the Editor

Letters to the Editors should be addressed to the Editor, APPLIED OPTICS, AFCRL, Bedford, Mass. 01730, and should be accompanied by a signed Copyright Transfer

Agreement. If authors will state in their covering communications whether they expect their institutions to pay the publication charge, publication time should be shortened

(for those who do).

Contribution of multiple reflections to thermal runaway in germanium Walter Wild and Kalman Wilner

IIT Research Institute, Chicago, Illinois 60616. Received 23 July 1979. 0003-6935/79/233880-03$00.50/0. © 1979 Optical Society of America. During the past few decades, laser technology has evolved

from a laboratory system into field systems with a wide range of industrial and military applications. The attainable power levels, in particular with IR lasers, have led to substantial ef­forts by the scientific community to better understand in­teractions between lasers and materials. From a theoretical viewpoint, at high laser power levels, one can no longer assume that the physical properties (thermal, optical, etc.) of the materials remain constant during the interaction. The time-dependent temperature variable must be incorporated into any credible analysis.

The analysis of laser-material interactions is further com­plicated if the material is transparent to the laser, since it is then necessary to take into account the effect of the internal multiple reflections. The intent of this Letter is to demon­strate the effects of temperature-dependent absorption and the multiple internal reflections on the temperature rise of a transparent material via the solution of the heat-conduction equation. We have chosen germanium as an example to il­lustrate the influence of the above factors on the thermal runaway time for a given laser power density, and we compare the results with those for which only one laser path is con­sidered.

The general form of the heat-conduction equation in one dimension with source term A, specific heat C(T), density ρ, and thermal conductivity K(T) is

For a transparent material with surfaces of reflectivity R (where R is constant if the index of refraction is assumed constant over the temperature range of interest) we must determine the explicit form of the source term so that it takes into account the temperature-dependent aspect of the ab­sorption coefficient of the material as.well as the internal multiple reflections. This is best achieved by writing the contribution to A(x,t;T) from several reflections and then summing up the resulting series into a closed-form expres­sion.

For the initial path into the material following the entrance of the beam, the laser power flux at any given point in the material (defined to be situated in the coordinate range 0 ≤ x ≤ d, where d is the thickness of the sample) is given by the solution of the well-known expression

which is

where I is the power flux in watts per square centimeter, α(T) is the absorption coefficient as a function of temperature, and the (1 - R) term indicates that portion of the flux not reflected at the surface. The temperature is an implicit function of depth in the material, though under certain circumstances it may be possible to approximate the above integral as α(T)x, for instance, when the material is thin. I0 is the power flux incident upon the material.

The source term for the initial path is simply

where we have taken into account the reflection of the incident beam from the surface at x = 0. The reflected component of the beam effectively goes to infinity and does not contribute to the heating of the material.

Upon reflecting at the back surface at x = d, the next term in the series, A2, will be attenuated further, since there will be an added factor for the reflectance in Eq. (4). Further, the integral in Eq. (4) will remain, although the upper limit will be d rather than the depth coordinate. The second term in the series will therefore be of the form

The last factor is from Eq. (3), although here we are consid­ering the beam as it goes back up into the material after re­flecting from the bottom surface.

This process can be continued indefinitely. Fortunately, the contributions from the many terms can easily be summed to give the following expression for the source term:

Equations (1) and (6) form a partial integrodifferential equation for absorption and conduction of heat through a transparent material. As it stands, this equation is of ex­traordinary complexity, and to consider analytical solutions one must of necessity resort to approximation methods. It is for this reason that previous workers have only considered one pass of the beam through the material, and even then approximations were required.1

We wish to investigate the solutions of Eqs. (1) and (6) where the sample is a slab of germanium. Specifically, we have asked ourselves the question: How significant is the

3880 APPLIED OPTICS / Vol. 18, No. 23 / 1 December 1979

Fig. 1. Absorption coefficient as a function of temperature for germanium.

Fig. 2. Laser heating of a germanium sample. Note the thermal runaway effect at the surface of the material.

Fig. 3. Comparison of thermal runaway times for a model assuming internal reflections with those for a model neglecting the effect of internal reflections. These results are for the surface of the material

at x = 0.

inclusion of multiple internal reflections in a model of thermal runaway? Intuitively one can envision an effect, since the internal reflections bring more radiation to the x = 0 surface than would exist for a single pass. One would also expect that thermal runaway would occur at the top of the material before it would occur deeper, due to the greater heating at the sur­face.

The reason that thermal runaway occurs in semiconductor materials such as germanium is because of the highly tem­perature-dependent nature of the absorption coefficient α(T). The absorption coefficient for intrinsic germanium can be written in two terms:

where αlat refers to the lattice contribution, and αfc(T) refers to the free-carrier contribution. The latter term is temper­ature-dependent and can be expressed in the following form:

where k is Boltzmann's constant, and Eg = 0.803 eV is the band gap. The constant term was chosen to give the best approximation to the experimental data in Young's paper.2

Further, we set αlat 0.091. Figure 1 illustrates the tem­perature dependence of α(T) in the range of interest. It should be noted that long before the germanium window reaches high temperatures (~500 K), the material will most likely shatter due to thermal stresses.

Before the solutions are discussed, we will make the fol­lowing assumptions:

(1) We will assume that the temperature distribution within the sample is constant, so that the integral will reduce to α(T)x. This assumption can be justified by saying that the material is thin or else that the thermal conductivity is large. Because we are more interested in the thermal runaway time predicted by this model, the deviations in temperature just preceding thermal runaway (at the surface) at the various depths in the material, as well as those that occur afterward, should not significantly influence the result.

(2) The incident laser beam is assumed to be uniform. In order to consider spatial variations in the beam, it becomes necessary to resort to 3-D models.

As an initial condition we have set T = 300 K. For the boundaries we have the standard blackbody boundary con­ditions, which are3

where σ is the Stefan-Boltzmann constant, and E(Λ0) is the emissivity at the IR frequency λ0 = 10.6 μm. We will assume for simplicity that the emissivity is unity. Here we set T0 = 300 K, the temperature of the ambient medium. We further assume that convection losses are negligible.

Figure 2 illustrates the solution of Eqs. (1) and (6) for a sample 1 cm thick subjected to a power flux of 104 W/cm2. For germanium R = 0.36.4 The method of solution was an explicit finite-difference technique implemented on a com­puter. The temperature rise for various depths is fairly uni­form, though deviations occur about the thermal runaway time (defined when the surface exhibits the effect). Here ρ = 5.325 g/cm3, C = 0.31 J • g • K, and K = 0.59 W • cm • K are assumed constant.

Figure 3 shows the disparity in results when multiple re­flections are taken into account and when they are ignored.

1 December 1979 / Vol. 18, No. 23 / APPLIED OPTICS 3881

Notice that there is a deviation of about 25% in the predicted thermal runaway times. It is anticipated that when other materials for which R is larger are modeled in this manner, the runaway times (or, if this does not occur, heating above a critical limit) will occur much faster with internal reflec­tions.

We believe that the form of the heat-conduction equation we developed is useful inasmuch as it gives insight into laser heating of transparent media. Future work that remains may focus on different materials, spatial and temporal variations in the beam (such as for a pulsed Gaussian beam profile), variable thermal conductivity, reflectivity, and heat capacity. Perhaps the most important generalization to the relatively simple model presented here will be a 3-D model where edge effects and convection can be built into the boundary condi­tions. This will allow an even more accurate reflection of phenomena that occur in the real world and may give some basic physical insight into the mechanism of laser-material interactions.

The authors would like to express their gratitude to Tim­othy Rynne for a most useful conversation concerning the content of this paper. They would also like to thank Dimitri Gidaspow for his helpful suggestions.

References 1. R. L. Johnson and J. D. O'Keefe, Appl. Opt. 11, 2926 (1972); see

also E. M. Epstein, Sov. Phys. Tech. Phys. 23, 983 (1978). 2. P. A. Young, Appl. Opt: 10, 638 (1971). 3. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids

(Oxford U.P., New York, 1959). 4. For a discussion of the change in reflectivity, see R. M. Herman,

C. L. Chin, and E. Young, Appl. Opt. 17, 520 (1978).

3882 APPLIED OPTICS / Vol. 18, No. 23 / 1 December 1979