t

1

(Sn) = (dnn/dT) lim pp Jro

Some typical numerical examples for platinum are as follows: when ro = 1μ, E1 = 20(J/cm2), and 7 = 30 ns, then the maximum values for (r) and (dn) occur at times 3μs and are respectively (r) 2μ and (dn) ~ 4 × 10-3. These values give an effective focal length of about f 14 cm. But from tables 6 and 7 the tensile stress at t=730 ns exceeds the theoretical strength of the host before the lens effect becomes most important. Consider now another example for platinum in which σe (max-tensile) is less than the theoretical breaking stress of the glass host. When ro= 1μ, E1 = 0.33 (J/cm2), 7 = 30 ns, and Th(ro, T) = 600 °C, then the maximum values of (r) and (on) occur at times about 3μs and are respectively (r) 2μ and (dn) ~ 1.2 × 10-4. These values give an effective focal length of about ƒ~ 1.6 × 104 cm. These focal lengths are very long.

~

cause

Hence the model predicts that the lens effect arising from heated inclusions probably does not damage. Those cases for which the maximum tensile strength is less than the theoretical tensile stress of the glass have minimum effective focal lengths which are much greater than any dimension of neodymium doped glass elements used in present glass laser systems. In those cases for which the tensile stress exceeds the theoretical strength, the tensile stress probably causes damage before the lens effect could cause damage by heating another inclusion or by initiating an intrinsic damage mechanism such as self-focussing.

Equations (34) and (35) and the results in tables 6 and 7 show that Sb and the fictitious Al2O3(a) are more likely to cause damage than Pt. However, this difference is marginal. Additional computations for Pt. spheres in fuzed silica, daeff+6.0 × 10−6 (1/°C); indicate that the maximum tensile stresses occur at

r=ro and t=7 and that they are comparable to those stresses reported in tables 6 and 7. However, the range of ro for which σ (max-tensile) exceeds the theoretical strength is much greater; namely, when

10-4 cm ro≥ 10-6 cm,

T=30 ns, and EL=20(J/cm2), then σe (max-tensile; Pt in silica) > 6 × 109(N/m2). Fused silica has a tensile strength comparable to glass.

The maximum tensile stress as a function of the thermal conductivity K and the thermal expansion coefficient an is studied also. These investigations are limited to Pt in Glass (B) and in Glass (U) for which the effective expansion coefficients are negative. It is found that increasing Kh in table 2 for Glass (B) from 0.008 to 0.04 (W/cm °C) decreases the maximum tensile stress from 9.0 × 109 (N/m2) to 4.0 × 109 (N/m2). Similarly, increasing K in table 2 for Glass (U) from 0.013 to 0.04 (W/cm °C) decreases the maximum tensile stress from 10.6 × 109 (N/m2) to 5.8 × 109 (N/m2). Again, all the other properties of Glass (B) and of Glass (U) are kept the same except for the thermal expansion coefficient and the resulting maximum tensile stress as a function of an for fixed r=5 × 10-5 cm, 7=30 ns, and E1 = 20 J/cm2 is reported in tables 8 and 9. Observe that the maximum tensile stress has a minimum at anan(min); namely, an (min, Glass (B)) ~ 10 × 10-6 (1/°C) and an (min, Glass (U))~8 × 10-6 (1/°C). Also, these tables indicate that the maximum tensile stress is a slowly varying function of the thermal expansion coefficient, an. Thus, for these cases, an influences greatly the behavior near the interface because it appears in the effective expansion coefficient and the thermal conductivity influences greatly the behavior near the region of maximum tensile stress.

Ꮮ

Intuitive arguments exist to explain why the temperature Th (ro, t) in tables 6 and 7 has a maximum value at some ror(max Th) for fixed E, and 7 and decreases for values of ro greater than r(max T) and less than r(max T). As (ro/r(max Th)) becomes larger than one, the volume increases much faster than the surface area of the sphere. The inclusion receives less energy per unit volume. Therefore, the maximum surface temperature at the end of the pulse decreases. As (ro/r(max Th)) becomes smaller than one, the equilibration time (r/a) approaches zero. Then the surface temperature at the end of the pulse cannot deviate much from the equilibrium temperature due to the extremely short equilibration time.

The above computations in table 10 suggest that examining incipient absorbing centers in laser glasses either by methods which employ the interference of two light rays (which experience different optical path length changes due to the local variation of the refrac tive index near the inclusion) or by methods which employ thermal stress birefringence are promising. However, because the propagation of light in the pres ence of a stress birefrigence which has spherical symmetry is complex, the interpretation of birefringence data will be more tedious than data from an interference method. The feasibility of combining the inter

ference method and holographic techniques has been demonstrated for 50 micron particles [2]. In addition the results of table 10 suggest that the use of laser pulses with pulse widths greater than a few microseconds may be more promising for the detection of small incipient absorbing centers than the use of nanosecond laser pulses. The longer pulses produce spatial changes in the refractive index which extend over greater distances in the host and thereby increase the probability of detecting small inclusions. Another approach is to employ pulse widths and observation times which are less than the relaxation times of the host. Relaxation times for laser glasses between 800 °C and 1100 °C are approximately between nanoseconds and microseconds. These are estimates of the time during which stress is proportional to strain in glasses which are not elastic for infinite time. Such short time observations will permit one to raise the inclusion surface temperature substantially above the present 600 °C reported here and still satisfy the assumptions of the present model.

The author thanks especially Alan D. Franklin for many helpful discussions and for his encouragement. While undertaking this study the author also had several discussions with many other researchers concerned with laser damage problems, some of whom are cited in the references. He thanks them and the others from whom he has learned about damage in laser materials. He also thanks M. J. Cooper and A. Kahn for their reading of the manuscript.

6. Appendix. Center-Host Interface

A laser beam with an energy flux per unit area I (J/cm2 s), with a pulse width 7(s) and with a wavelength λ(cm) impinges upon the inclusion. The inclusion exhibits an absorptance A (X, T), a reflectance R(λ, T), and a partial emissivity E(X, T), where A is the wavelength and T is the absolute temperature of the inclusion. The following relations among A, R, and E exist: (a) A+R=1; no energy is transmitted.

(b) E(X, T)/A (λ, T) = e(λ, T); Kirchoff's law. The function e(λ, T) is a universal function only of X and T and is independent of material and surface properties.

(c) e(λ, T) dλ= ST4: Stefan-Boltzmann law. The constant SB is SB=5.673 × 10-5 (erg/cm2 s K1). (d) AmaxT=0.2897 cm K; Wien's displacement law. The wavelength for which e(λ, T) has a maximum value for a fixed temperature T is Amax.

The values for the absorptance A(X, T) in table 4 are valid for ro ≥ λ ~ 1.06 μ. The results of Mie scattering theory [11] show that A(X, To) does not change appreciably for values of ro≤ and roy-1 where y is the absorption coefficient.

The spherical inclusion has a complex index of refraction mene-ine and the Poynting vector for the incident radiation in the absorbing center is proportional to the factor exp (-ye (ro-r)). The absorp tion coefficient Ye is given by (4πn'/λ). Because a sphere with index me is imbedded in the host with a real index m=n≈ 1.52, the values for me to use in the equations of ref. [11] are me(med)=(men). Similarly the wavelength in the host is λ(med) = (X1/m1⁄2) where λ= 1.06 μ. The sphere intercepts in accordance with Mie scattering theory Qextπr watts from the incident laser beam, independently of the polarization of the beam. There will be Qabsπrl watts absorbed by the sphere and Qscar watts scattered in all directions by the sphere. The conservation of energy gives Qext=Qabs+Qsca. The absorptance Qabs=A(X, T: ro small) replaces the absorptance A(X, T; ro= large) quoted in table 4, whenever ro becomes less than the wavelength (med). The variable X = (2πгo/λ(med)) is introduced and when X ≤ 0.6 Mie scattering theory relates X and Qext; namely,

[1] Bliss, E. S., in Damage in Laser Glass, A. J. Glass and A. H. Guenther, Eds. (ASTM Special Technical Publication 469. 1969), p. 9.

[2] Snitzer, E., private communication.
[3] Turnbull, D., private communication.
[4] Dietz, E. D., private communication.

[5] Mauer, R. D., private communication.

[6] Boley, B., and Weiner, J., Theory of Thermal Stresses (John) Wiley Sons, Inc., New York, 1960).

[7] Bennett, H. S., Heat diffusion near absorbing centers in laser materials, J. Res. Nat. Bur. Stand. (U.S.), 75 A (Physics and Chem.), No. 4, 261 (July-Aug. 1971).

[8] Gilman, J. J., in Physics and Chemistry of Ceramics, C. Klings berg, Ed. (Gordon and Breach, New York, 1963), p. 240. [9] Quelle, F., J. Applied Optics 5, 633 (1966).

[10] Uhlmann, D., private communication. [11] van de Hulst, H. C., Light Scattering by Small Particles (John Wiley and Sons, Inc., New York, 1957).

(Paper 75A4-666).

1. Introduction

In a previous paper [1], a model to describe the behavior of absorbing centers in laser materials has been developed. Computing the thermal stresses predicted by this model requires solutions to the heat diffusion equation for the temperature. The mathematical details for solving the heat diffusion equation are omitted in ref. 1 and are reported in the present paper. In particular, solutions to the heat diffusion equation for spherical absorbing centers in laser materials, series expansions of these solutions at the center-host interface when times are sufficiently small and large, and a proof that in the limit of small inclusions the temperature of the center-host interface is independent of the bulk thermal properties of the inclusion are presented. These solutions may be applied also to other diffusion problems such as oxygen diffusion in TiO2, Al2O3, ZnO, and other oxides [2].

When certain physical conditions exist, some solutions to diffusion equations contain error functions for complex arguments. For example, when the volume specific heat of a spherical absorbing center is greater than three fourths times the volume specific heat of the host, the analytic expressions for the temperature in the host contain error functions for complex arguments. This is the case for platinum inclusions in some neodymium doped laser glasses. The oxygen diffusion problem is a second example. When the volume diffusion coefficient is sufficiently greater than the surface exchange coefficient, the analytic expressions for the oxygen concentration profile also contain error functions for complex arguments. This may be the case for oxygen diffusion in zinc oxide [3]. Representations of the error function for complex arguments which yield efficient computer programs are not readily available in the literature and therefore are derived in the appendix.

2. Mathematical Description of the Model

The heat diffusion equation and the boundary conditions are summarized in this section.

A spherical absorbing center of radius ro(cm) is imbedded in a host of infinite extent. The absorbing center and host are initially at a uniform temperature To. An incoming spherical wave of light (laser beam) with an energy flux (TrEL/4πr2T) (W/cm2) falls at time t=0 upon the absorbing sphere. The quantity E is the energy density (J/cm2). The quantity r=|r| is the distance from the origin of the system, which is the center of the sphere. A square-wave light pulse of width 7(s) is assumed. The sphere absorbs an equivalent energy flux H(t) (W/cm2) uniformly over its entire surface. The energy flux for a square-wave laser pulse is

The Laplacian operator is denoted by V2 and the thermal diffusivity is a2= (K/pC) (cm2/s), where K is the thermal conductivity (W/cm °C), p is the density (g/cm3), and C is the heat capacity at constant volume (J/g °C). The subscript c refers to the absorbing center and the subscript h refers to the host. The diffusion equations (2) and (3) are valid only when local thermodynamic equilibrium exists. They become suspect for times comparable to the phonon-phonon collision time (~10-12s) and for distances comparable to atomic dimensions (~ 10-8 cm). The diffusion equations require a statement of the boundary conditions before solutions are uniquely defined. The temperature has the form,

{ (4π r3/3) pcCc (dTc/dt)/4πr}} = = M (dTc/dt).

Taking the Laplace transform of the conservation of heat flux, eq (26), which is valid when Kc> Kn, the author obtains the same expression for the Laplace transform Un as that given in eq (24). Hence, approximation (24) obtains either when ros/2/ac < 1 or when Kc> Kh. The latter inequality imposes no restriction on ro and s or equivalently on ro and t but the former inequality does impose such restrictions.

The volume specific heat ratio R= (4pcCc/3phCn) determines three regions of behavior. Namely, when R<1, the roots b are real and are not equal; when R=1, the roots are real and equal; and when R>1, the roots are complex and conjugate to one another. The latter is the region for which the arguments of the complementary error functions are complex. In the next Section, analytic expressions for the temperature Th (r, t) in each of these three regions are presented.

Th(r,t) = B, (r, t)u(t-7)B, (r, t-T). (29)

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