Measurements of Heat Flow from Surface Defects in Lithium

We present what is, to our knowledge, the first measurement of temperature distributions in a nonlinear optic resulting from absorption in a localized surface defect. These measurements were performed on principal cut samples of lithium triborate with damage spots centered on their front surfaces, pumped by a kW-scale continuous-wave laser. The changes in optical-path length associated with this heating were measured with a Mach-Zehnder interferometer, from which the temperature distribution could be inferred. These distributions have sharper features with larger magnitudes than would be expected with bulk-absorption heating. Comparison with both numerical and analytical models is used to qualify the measurements and to estimate the total power absorbed at a given site using this bulk material response. While sensitivity is dependent on the properties of the material of study, we demonstrate measurements of absorption levels of one part in 105.

The materials used in this study were all uncoated low-bulk-absorption lithium triborate (LBO) crystals, principally cut into 1-cm cubes and polished on four sides.This allowed for pump and probe beams to sample single crystallographic axes.This was used to qualify the measurements, in that a particular heat distribution could be probed with two separate polarizations, yielding two similar signals, but with different amplitudes.The ratio of these amplitudes could be calculated from material parameters and were shown to agree with measured values to within +10%.The localized absorbers-the focus of our study-were single unintentionally produced damage spots on the crystal surfaces.
The 2-D changes in optical path length that were measured from these crystals showed a distinctive shape.It consisted of a sharp central peak, no larger than 100 nm, with rapidly decaying amplitude moving outward from the source.This stands in stark contrast to the more Gaussian-like distributions seen when heating is the result of absorption in the bulk or in a coating.Thermal imaging measurements showed that the heating occurred on only one face of the crystal, indicating that it was solely the result of localized absorption.Also observable was an asymmetry in the vertical direction, but not the horizontal, indicating that the crystal was warmer at the top surface than at the bottom, which was the result of heat loss into the mount.
With access to finite element analysis tools, calculations of heat flow can be made, but the higher resolutions required to accurately model such a small source require significant memory.We instead employed an analytical model that can be easily derived by relating the total absorbed power to heat flow through a spherical surface at distance r from the absorber through Fourier's law.This results in a temperature distribution of the following form for all points outside the absorber:

Measurements of Heat Flow from Surface Defects in Lithium Triborate
D. Broege and J. Bromage Laboratory for Laser Energetics, University of Rochester T r r and can be shown to be for all points inside the absorber, where r is the observation radius, l is the thermal conductivity, R is the radius of the absorber, and P is the total absorbed power.For this model to be valid, the source must be small compared to the length of the crystal, and the heat flow through the crystal must be large compared to convective cooling at the surface.
What is immediately evident from the functional form of this distribution is that outside the absorber, its amplitude is proportional to the absorbed power, but completely insensitive to the size of the absorber.Calculations show that a 20-fold decrease in the size of the heat source results in only a factor-of-2 increase in the resulting peak temperature.This implies that without detailed knowledge of the morphology of the absorber, a decent estimate can be made of the total power absorbed.Figure 1 shows an overlay of measured changes in optical path length and a simulated data set using the simple model, with an absorbed power of 15.5 mW.The matching between the two is sensitive to the mW level, allowing for a very sensitive estimation of absorbed power at a localized defect.With this knowledge, one can use microscopy to determine the maximum spatial extent of the defect and determine a range of possible absorbances and sizes.In the case of our test spot, with a maximum possible size of 100 nm, assuming absorption is uniform across its face results in an average absorbance of 5%.If, on the other hand, absorbance is 100%, the size of the spot can be no smaller than 22 nm.
This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority.

Figure 1
Figure 1Comparison of measured optical path length change in LBO with an analytical model for 15.5 mW.This model was used along with interferometric measurements of temperature to estimate power absorbed at a localized spot with a sensitivity of one part in 10 5 .