Figure 6. Occurrence of photomultiplier signal (top row) versus time corresponding to laser pulse shape in time; (a) corresponds to coating damage, (b) corresponds to substrate damage.

Figure 7. Laser induced damage thresholds of CdTe samples versus laser pulse width

Figure 8.

Plot showing temperature use of sample relative to single shot heating versus number of laser pulses for different pulse repetition frequencies.

Figure 9. LIPT of AR-coated CdTe versus number of laser pulses, theoretical curves show four different PRF's starting at a mean LIDT of 37J/cm2. The different symbols denote different beam sizes and PRF's. All work was done at 1Hz except those noted at 5Hz.

Manuscript Received

1-12-89

Microcomputer Finite Difference Modeling of Laser Heating and Melting*

J. O. Porteus

Physics Division, Research Department

Naval Weapons Center, China Lake, CA 93555-6001

An existing method of microcomputer finite difference modeling has been
adapted to laser-heating and melt-threshold computations. Versatility compared
to exact-solution methods permits various applications of moderate complexity
with arbitrary pulse waveform and an optional axisymmetric or one-
dimensional spatial profile. Thermal diffusion is modeled in two dimensions
based on optical and thermal transport properties that can
vary with
temperature, time, and space. Simplicity and freedom from mainframe
requirements make
make the method accessible, flexible, and easily learned.
Examples presented here were performed on XT- and AT-type PC-compatible
computers equipped with floating-point math coprocessor running Turbo Pascal
4.0. Comparisons are given with previously reported surface melt thresholds of
diamond-turned Cu and Au measured and calculated by earlier methods. Other
examples are pulsed-laser bulk heating of semiconductors, including a GaAs
coating on a quartz substrate and a comparison of flashlamp heating of laser
glass in rod and slab configurations.

Key words: diffusion; GaAs; heating; laser damage; melting; metal mirror;
thermal model.

1. Introduction

Calculation of temperatures produced by laser heating of optical materials is a common requirement in damage testing and other areas of laser technology. Simple analytic thermal models such as those provided in Carslaw and Jaeger [1] are capable of representing only the most elementary experimental situations. On the other hand, more powerful numerical models such as SINDA [2] are often too formidable for the casual user, or may not be readily available. This paper presents examples of a very capable microcomputer finite-difference modeling (MFDM) method [3] that offers a compromise between these two extremes. The new method is applied to melt thresholds of Cu and Au surfaces with intrinsic properties. Comparisons are made with previously computed results and laboratory measurements on essentially defect-free samples. Other examples include laser damage of a highly absorbing semiconductor, bulk heating of a semiconductor film, and flashlamp heating of laser glass. Computations were performed on XT- and AT-type PC-compatible computers equipped with floating-point math coprocessors. The typical computation time for heating a Cu surface to melt by single-pulse absorption, using 12x12 nodes, is about 5 minutes.

The mathematical basis for the method and the general finite-difference code in BASIC is available in a monograph and on disk [3]. The computations performed for this paper were

*This work was supported in part by Navy Independent Research funds. 1Numbers in brackets indicate the literature references at the end of the paper.

executed in a Turbo Pascal 4.0 version of the code that was adapted specifically to heating by optical absorption and was extended to include temperature-dependent material properties. The temperature dependence of absorption is especially important in laser heating of metal surfaces. Surface absorption of laser flux is treated in terms of a surface transfer coefficient (W cm-2 °C-1) with an arbitrary surface temperature drop of 105 °C, which is large compared to any calculated temperature excursions. Bulk absorption is treated as a decrement of transmitted laser flux at each internal node (calculation point representing a space cell in the beam path). An internal heat source at each node is equivalent to the absorbed energy in the corresponding cell. The model accepts any waveform and any axisymmetric or linear spatial profile, expressed either in tabular form or as mathematical formulas. Thermal diffusion can be treated in either one or two

dimensions of space.

2. Metal Surface Melting

Comparisons were made with melt thresholds of diamond-turned Cu and Au surfaces previously measured at the Naval Weapons Center and calculated by other methods [4-6]. The surfaces were sufficiently free of defects that intrinsic laser-induced melt thresholds could be measured [5]. Results of absorptance measurements on these metals at ambient and elevated temperatures are reproduced from reference 6 in table 1. A linear correction to the ambient absorptance A, represented as a fraction of A in the table by A-1 dA/dT, was applied at time intervals of 1 to 10 ns, depending on the laser waveform.

Thermal properties used here correspond closely to inputs used in the previously reported calculations. Constant values of conductivity K, density p, and specific heat C are equivalent to inputs used in reference 4, while dK/dT, the linear approximation to the temperature dependence of conductivity, is based on tabulated values of K [7]. Dependence of p and C on temperature are neglected. Thermal property inputs are summarized in table

2.

Initial comparisons involved the energy density for melting Cu by a 100-ns rectangular pulse of 2.7 μm (HF) wavelength neglecting dK/dT. The MFDM approach produced a value of 53.2, the same as an earlier SINDA result [5], and in excellent agreement with a value of 52.2 computed by Sparks and Loh, who used an analytic model [4]. Further comparisons between MFDM and later SINDA computations [6] use actual waveforms and include dK/dT. The waveforms were approximated by a sequence of linear segments as illustrated for the HF pulse in figure 1. The original waveform data entered into SINDA were unavailable, requiring use of figures reproduced from reference 6. Computed and measured energy densities (J cm-2) required to melt Cu and Au with the HF pulse are given in table 3.

aSpot-size correction from Dobrovol'skii and Uglov [8] applied to one-dimensional results.