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Abstract
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.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With ever-increasing power and pulse duration demands for industrial and scientific lasers, more lasers are being designed around frequency conversion and parametric amplification because of their scalability and wavelength flexibility [1,2]. One downside to their operation is that the performance of many nonlinear components involved in these processes is more sensitive to temperature fluctuations than conventional laser amplifiers, which becomes an increasing concern at higher powers [3,4]. Heating can come from any number of sources from nearby hardware to absorption, and establishing a detailed understanding of these sources and appropriate techniques for measurement is essential in developing the next generation of high-power lasers.
One of the more-relevant and well-documented sources of heating in nonlinear amplifier systems is bulk absorption [5–7]. Efforts can be made to work with materials with low absorption, but with harmonics, signal, and idler beams, crystals are exposed to a broad range of wavelengths; often, one of these results in heating that is significant enough to affect performance. Another complicating factor is that absorption is often characterized at low powers by photothermal techniques [8], but the heating measured by thermal imaging and compared with models indicates there can be greater absorption at higher powers [9]. Research in this area has led to ingenious advancements in thermal management involving directly bonding sapphire to crystal surfaces [10], serving to efficiently carry heat away, which has led to record power levels in ultrafast UV sources [11]. The resulting temperature distribution in these cases of bulk heating is often a rounded thermal bump that extends through the crystal along the beam propagation direction, the specific shape and magnitude of which are determined by the beam, the crystal dimensions, and boundary conditions.
One form of heating that has received less study is the highly localized type that can result from absorption in crystal defects, surface contamination, or damage and is sometimes not as easily quantified through transmission-based spectroscopic techniques when an optical surface is compromised, such as at a damage site. This heating can be characterized by sharp thermal features, which also have the potential to damage optics and disrupt phase matching. In this work, we present measurements of the effects of heating from localized surface absorption in lithium triborate (LBO) crystals. Heating induces both index changes and thermal expansion, and the resulting changes in optical-path lengths are measured interferometrically [12]. The absorbers in this case are single laser-induced damage spots identified through microscopy to be at or near the center of the crystal faces. Results from these measurements are then compared to numerical and analytical models to determine the associated temperature distribution. Details of the analytical model will be shown to be helpful in determining the total power absorbed at a given site, serving as a sensitive indirect method of measuring localized absorption.
The technique utilized here has the benefit of being capable of measuring very small (one part in 105) levels of absorption and changes in temperature at high resolution. It has advantages over thermal imaging methods because it is sensitive to changes in temperature throughout the bulk of the material. For this reason, although not demonstrated in this work, it is also capable of detecting localized absorption within the material, not just on the surface. The specific interferometer geometry used here is not essential to the measurement, and the measurement can be easily adapted to utilize other wavefront measuring tools, such as lateral shearing interferometry [13] or a Shack–Hartmann sensor [14]. It should be noted that the sensitivity of this technique depends greatly on not only the properties of the material of study, but also on the dimensions of the material and cut. Both index change and thermal expansion play a role, and if the related coefficients are too small or the probed axis is not long enough, sensitivity can be limited.
2. Experiment
The crystals chosen for this experiment were 1-cm cubes of LBO, principal cut such that each crystallographic axis was perpendicular to a face. They were obtained from multiple vendors and were characterized by photothermal common path interferometry (PCI) to have absorption at the parts per million (ppm) level at both 1064 and 532 nm. Additional reference samples with much higher absorption around 50 ppm were obtained as well. While this is not the most realistic preparation for a material intended for use in a laser, it is an ideal model system for facilitating the measurement. The highest levels of sensitivity require samples with less potential for unintended heating. Coatings have been shown to be a source of absorption capable of heating to levels that interfere with nonlinear performance [3]; as such, they were not applied to samples here. Since the signal amplitude is dependent on both probe propagation direction and polarization, a geometry that allows propagation along one single axis and polarization along another presents an opportunity for multiple unambiguous measurements that can differentiate between the effects of thermal expansion and the temperature dependence of the index of refraction in a heated sample.
Since changes in optical-path length depend on both index change and thermal expansion, the efficacy of such a measurement can depend greatly on the properties of the material, its dimensions, and which axes are involved in the measurement. Table 1 shows expected changes in optical-path length in a 1-cm path in LBO per °C change in temperature for all possible propagation directions and polarizations for a 632-nm beam, calculated using published material properties [15,16]. In the case of LBO, the dominating material property is thermal expansion, which means probe propagation direction determines not only the magnitude, but also the sign of the change in optical-path length. The contribution from dn/dt is significant, but a secondary effect. It is clear from this table that a temperature fluctuation of a given magnitude can be much more easily detected with a probe beam propagating along the x or y axes than z.
Table 1. Expected Changes in Optical-Path Length for a 632-nm Beam in a 1-cm-thick Sample of LBO per °C (Radians)
The interferometer used to perform these measurements is a Mach–Zehnder type, shown schematically in Fig. 1. The sample lies in one arm, oriented with its faces perpendicular to the probe beam, which has been expanded to overfill the crystal face. The reference beam is pointed to create ~50 fringes across the studied region of the crystal. Upon exiting the interferometer, the beam is imaged onto a 1388 × 1038 camera with a filter to block any scattered pump light. The surface defect is heated with a 1064-nm fiber laser run at a power of 1040 W. The laser’s output is first collimated, then passed through a 1.25-m focusing lens and directed through the sample at an angle of ~4° with respect to the probe. The beam comes to a focus roughly 50 cm before the crystal and expands to 2 mm by the point at which it reaches the crystal.
Fig. 1 Experimental setup. An LBO sample lies in one arm of a Mach–Zehnder interferometer, with the outgoing probe beam imaged onto a camera. The pump beam—a 1040-W, 1064-nm fiber laser with a 2-mm diameter—passes through the crystal at an angle of 4° with respect to the probe.
Since this interferometer is sensitive and is measuring small amounts of absorbed power relative to the pump, a number of efforts have been taken to minimize the effects of unintended heating to the crystal, surrounding air, and interferometer optics that can occur when working with high-power beams. Working with a slowly diverging pump to achieve a 2-mm beam, for example, was preferable to down-collimating in order to avoid distortions from localized heating [17]. Similar concerns exclude a co-propagating pump–probe geometry facilitated by dichroic mirrors. The interferometer itself is housed in an acrylic box containing no high-power optics, and the water-cooled beam dump used to terminate the kW beam is located a few feet away from the box with shielding in place to block scattered light and any convective heat from reaching it. Because the LBO is hygroscopic [18] and uncoated, the interferometer box is actively purged with nitrogen between measurements to protect from humidity.
The camera, which records every 2-D interferogram, is set for an exposure time of 0.2 ms—a time scale that is very small compared to the ~5 min over which the crystal thermalizes. Every recorded 2-D interferogram contains information about the difference in wavefronts between the probe and reference beams of the interferometer. This information is extracted through a standard method of apodizing the interferogram’s Fourier transform at the positive spatial frequency of the interference fringes, unwrapping the phase of the inverse transform, and subtracting the linear term resulting from the interference fringes [19]. The greatest source of noise at this level of analysis is variation in probe-beam intensity. These often manifest themselves in the data as oscillations with an rms variation of ~0.07 rad as a result of the inherent Fourier filtering that occurs in the computation of phase, and is the ultimate determining factor in this measurement’s sensitivity. Because this signal is affected by every optic within the interferometer, it can have an amplitude on the scale of a wavelength and measurements of smaller heat-induced effects must be performed differentially. One recording is made before the pump beam is turned on, and another is taken at a given delay of interest. The difference between these two recordings represents the effect of laser-induced heating. All phase distributions shown in this study were taken 10 min after the laser had been turned on to ensure complete thermalization.
An initial test of the interferometer’s accuracy was performed on a 5-mm-thick poly(methyl methacrylate) (PMMA) window. With absorption at 1064 nm of a few percent per centimeter, a significant signal can be generated with a 1-W beam without much possibility of inadvertent heating from scattered beams interfering with the measurement. The 2-D change in optical-path length measured for this window when exposed to 1 W of pump light is shown in Fig. 2. It consists of a smooth Gaussian dip roughly the size of the pump beam. The associated change in temperature between the center and a point 4 cm out can be calculated with material parameters [20] to be 2.6°C. A thermal image of the same plate showed a difference in temperature over the same distance to be 2.5°C. The discrepancy between the two is within the resolution of the thermal camera, and a smaller number is not a surprise since the interferometer measures temperature averaged longitudinally through the material, which one would expect to be slightly higher than the surface as a result of convective cooling.
Fig. 2 Steady-state changes in optical-path length (radians) in a PMMA test window while exposed to 1-W, 1064-nm pump beam. Smoothly varying heat distribution results from bulk absorption throughout the material. The corresponding thermal gradient was calculated from material properties to be 2.6°C, which matched thermal imaging to within 0.1°C.
The LBO samples that are the subject of this study each have single laser-induced damage spots near the center of their faces, which serve as localized surface absorbers. The damage spots were unintentionally created with exposure to cw intensities of the order of 3 × 104 W/cm2. Since this is well below the expected damage threshold of LBO, it is possible that this damage was caused from subsurface contamination, either a consequence of the polishing process or mishandling. The absorption at this site is then likely a combination of that from the original contamination and heat-induced chemical changes in the LBO itself. The magnitude of the optical-path length changes induced by the pump laser in these crystals in general varied between samples but was typically of the order of a few tenths of radians. Figure 3 shows a representative 2-D phase distribution for a sample probed along the x axis with polarization along the z axis. It is markedly sharper than the bulk-absorption signal in the PMMA sample and demonstrates more heating than would be expected from PCI-measured absorptivity. Another noticeable feature of these temperature distributions is their asymmetry in the vertical dimension, which is most easily seen from lineouts shown in Fig. 4. This is the result of the crystals losing heat through their mount. Delrin was chosen as a mount material for its thermal insulating properties, but heat flow through it is still more efficient than to the surrounding air. A slight large-scale fringe pattern is also somewhat visible in this image. It is the result of multiple reflections of the probe beam off the uncoated surfaces of the crystal and is more significant in samples with poor parallelism. Although weak, these reflections interfere with the probe beam and produce faint variations in intensity, which the analyzing algorithm can interpret as phase, presenting the biggest hurdle in performing accurate measurements. Fringe position is random with each measurement, so the effect can be reduced with averaging. For this reason each LBO phase measurement described in this work was the result of averaging five separate measurements.
Fig. 3 (a) Raw data showing interference fringes produced in a typical measurement (b) Measured change in optical-path length (radians) in optically pumped LBO with a surface defect.
Fig. 4 (a) Horizontal and (b) vertical lineouts of optical path-length changes in laser-pumped LBO. Blue curves show changes for a crystal with a surface defect, while red curves show changes for a sample with higher bulk absorption, but no surface defect.
Although these heat distributions vary both qualitatively and quantitatively from those expected from bulk absorption, a number of additional observations can be made to support the fact that they are the result of localized absorption from defects within the crystal. Single damage spots had been identified through microscopy (see Fig. 5) on only one surface of a particular axis of study. While heated, these surfaces showed hot spots when viewed with a thermal imager, while the opposite undamaged side remained cool. Multiple measurements were also performed with a crystal slightly translated horizontally within the beam with all other parameters fixed. Each scan showed the peak of the distribution moving correspondingly, indicating that the heat was flowing from a specific point in the crystal, not the center of the pump beam.
Fig. 5 Images of the (a) first and (b) second surface defects studied. Laser-induced changes in optical-path length measured from the first are shown in Figs. 3 and 4.
Finally, interferometric measurements were performed on a pristine reference sample that had been characterized to have ~50-ppm bulk absorption at 1064 nm. The crystals were pumped with 500 W, which corresponds to roughly 20 × the bulk absorbed power in the samples with surface defects. Lineouts from these scans are shown in Fig. 4 in contrast to the defect-heated samples. They show no measurable change in temperature distribution, indicating that bulk absorption in the samples with surface defects does not contribute to their signal.
It is here that the principal cut of these samples becomes useful in qualifying these measurements since a change in probe polarization will change the magnitude of a thermally induced signal by an amount that can be calculated with material properties. To test this, measurements were taken with identical parameters, but with probe polarization rotated 90°. In each case, the amplitudes of these signals were different, and their ratios could be matched to calculated ratios to within 10%. This served as confirmation that the measured signals are the result of material heating and no other optical, mechanical, or thermal effects elsewhere in the device related to laser operation.
3. Modeling and discussion
While it is clear from the data and morphology of the damage sites that the source of heat is well localized, a more detailed understanding of its dimensions, the total power absorbed, and heat flow from the site requires calculation. Within resolution limitations placed by computational resources, a full solution to the heat equation carried out through finite element analysis (FEA) can serve this purpose because it can take all anisotropic material parameters and boundary conditions into account. To this end, a model was established using the MATLAB Partial Differential Equation ToolboxTM. In this calculation, it was assumed that the heat source was hemispherical in shape and was located in the middle of the front surface of the crystal and that the power absorbed was uniform throughout.
To treat boundary conditions correctly, it was necessary to establish heat transfer coefficients for the interfaces between the LBO and both air and the delrin mount. These were measured using a combination of thermal imaging and the model itself. The crystal was first placed on a small hot plate set to an elevated temperature and allowed to reach equilibrium, at which point the temperature at the top of the crystal was recorded. The model was then used to simulate this arrangement while adjusting the heat transfer coefficient for the air interface so the temperature at the top matched the measured value. This resulted in a heat transfer coefficient of 13 W/m2K for the interface. The heat transfer coefficient for the delrin interface was established in a similar manner with the crystal on a hot plate and a delrin block on the top; the steady-state temperature distribution in the crystal was used to determine the heat transfer coefficient for the delrin interface to be 100 W/m2K.
Figure 6 shows the results of the calculation with a spot size of 500 μm and a total absorbed power of 50 mW. While the size of the absorbing spot is undoubtedly smaller than 500 μm, memory limitations did not allow for accurate calculations with smaller spot sizes without more-sophisticated meshing. The model shows a highly localized hot spot on the surface of the crystal with a peak temperature change of ~8°C. This 3-D temperature distribution was interpolated onto a uniform grid and was used to calculate an expected change in optical-path length associated with it. Lineouts of these modeled data are shown in Fig. 6. This calculation is able to reproduce the basic features of the measured changes in optical-path length, with some notable difference. First, there is a more-rounded peak, which is a consequence of the chosen spot size; second, there is a smaller vertical asymmetry than in the measurement. This could potentially result from more conduction into the mount than is assumed in the model.
Fig. 6 (a) Schematic showing the geometry of the numerical model with an LBO crystal in thermal contact with a platform mount. (b) Results of a 3-D finite element analysis calculation of heat flow in LBO heated at the surface in a 500-μm spot, with a total absorbed power of 50 mW. This model reproduces the vertical asymmetry in temperature measured interferometrically, but it lacks spatial resolution to properly model smaller heat sources. (c) Horizontal lineout of calculated phase. (d) Vertical lineout of calculated phase.
The greatest shortcoming of the numerical model as employed is its inability to simulate the effects of smaller absorption sites. Luckily, the geometry is simple enough that an analytical model can provide a good deal of insight. Consider the heat flow from a localized spherical source within the bulk of an isotropic infinite material: From symmetry, one can safely assume that the heat-flux density will be radial and uniform at a given distance from the source and is related to the temperature through Fourier’s law, which takes the form Q = –(dT/dr), where Q is the heat-flux density, κ is the thermal conductivity of the material, r is the radial distance from the heat source, and T is temperature. Conservation of energy dictates that the inner product of this flux with an enclosing surface is equal to the enclosed power dissipated by the source. If a spherical surface is chosen, the temperature can then be calculated by multiplying each side of the equation by the surface area, solving for dT/dr and integrating, resulting in
where P is the total absorbed power. This solution is valid with any additive constant, which is chosen to be zero, since this study is more concerned with distribution than absolute temperature. Assuming that the heat source is spherical and flow from it is uniform, this same approach can be applied to calculate the heat distribution in regions within the absorbing site, resulting inwhere R is the radius of the source and the integration constant is chosen for continuity with the solution outside the source.The thermal distribution described above is that which is expected for a localized heat source within the bulk of an isotropic material but can be extended to anisotropic materials with a scaling of axes. Because the size of the spot in our sample is so small compared to the dimensions of the crystal and heat flow within the crystal is much greater than through the exposed surface, this is a reasonable approximation and can be used to model data for arbitrarily small spot sizes. Finally, because the absorber of focus in this study is on the surface, the heat flows over only half of this model volume, so the distribution is multiplied by a factor of 2 for our analysis.
Just as with the numerical model, this 3-D analytical thermal distribution can be used to calculate expected changes in longitudinal optical-path length. These calculated changes share a common shape with the data. Figure 7 shows a number of these curves calculated for a power of 15 mW, but with varying spot size. It illustrates a universal feature of the heat distribution generated from a localized source that is clear from the equations: outside the source, the curves are solely a function of total absorbed power and are independent of the spot size. Also worthy of note is that the amplitude of the peak does not vary considerably with significant changes in spot size, but it is proportional to absorbed power. This implies that even without detailed information about the absorbing spot’s geometry, a decent estimate of power absorbed at the site can be made from the amplitude of the signal.
Fig. 7 Simulated changes in optical-path length using a simple analytical model with a power of 15 mW and a number of surface defect sizes. Although the signal magnitude is directly proportional to absorbed power, it changes only by a factor of ~2 with a factor-of-50 change in spot size.
In the case of the damage spot shown in Fig. 5(a), lineouts from the signal place a ~100-μm upper bound on the size of the absorbing region. With this assumed spot size, the data match most closely with a model curve with a total absorbed power of 15.5 mW, shown in Fig. 8. The measured changes in optical-path lengths corresponding to the second damage spot tested [Fig. 5(b)] were significantly smaller, of the order of 0.1 radians, near the threshold of detection. This amplitude corresponds to an absorbed power of around 3 mW. Microscopy revealed that this defect was 1.5 mm from the center of the crystal and peak intensity of the pump. At this position, the pump intensity is 32% of its maximum, which implies that the absorption at this site is actually comparable to the first. While the overlap between the model and measured signal is quite sensitive to absorbed power, it is important to remember these estimated values are not direct measurements of absorbed power, as assumptions in the model affect its predictive accuracy in real-world systems. Loss of heat at the front surface, although smaller than flow through the crystal, likely decreases the overall effect of absorption on changes in optical path length, making these values a lower bound. More detailed modeling and analysis taking surface loss into account can be used to increase accuracy.
Fig. 8 Comparison of measured optical-path length change in LBO with an analytical model for 15.5 mW.
These small amounts of absorbed power represent nearly one part in 105 of the total laser power, which is difficult to measure directly through changes in transmission. This is not to say that the absorptivity of the material around the defect site is so low. A more spatially resolved transmission measurement would also provide accurate information on local absorption, but it is not feasible in this situation since the surface in this region has significant structure and collecting all transmitted and scattered light would prove difficult. Based on the estimated total absorbed power and the intensity distribution of the pump beam, a 22-μm lower bound on the spot size can be made by assuming absorption is complete within its cross section. Similarly, with a 100-μm upper bound on spot size, we can say that absorption averaged across its face cannot be less than ~5%. The reality is likely between these two extremes with absorptivity that varies with position.
According to the simple model, the peak change in surface temperatures associated with the level of heating in the first defect described above is ~15°C. Measurements of surface temperature with a thermal imager showed no gradients higher than 6°C, but this should not be a surprise since the effective resolution of the thermal image was 375 μm, and surface losses are not taken into account in the model. In either case, this rise in temperature is not large enough to cause damage, but it can significantly affect phase matching. A quick calculation can show that this temperature distribution in a 1-cm-long LBO crystal cut for type-1 second-harmonic generation with an input intensity of 1 × 109 W/cm2 will not have a significant effect on harmonic production. Interestingly, the effect on efficiency is slightly stronger for localized heat sources within the bulk and on the back surface of the crystal because the point in which the harmonic-generation process is interrupted is different. Even in these cases, the absorbed power must be in excess of 45 mW for there to be a 1% change in harmonic output. The small effect on harmonic production has more to do with the spatial extent of the thermal bump than its magnitude, which drops to half its maximum value at a depth of less than 1% into the crystal.
While the heating measured in this experiment was small, the peak temperatures reached in any given localized heat source will depend greatly on the absorption levels and defect size. This along with material properties is an important factor to take into account when considering whether an optic will damage or cease to function properly. It is also important to point out that the thermal distributions measured here are steady state from a cw source, which can describe only average power effects. Most applications involving nonlinear optics use pulsed sources, in which case peak powers are higher, and the potential for damage propagation is much higher.
4. Conclusion
When working with nonlinear optics, especially at high average powers, a detailed understanding of heat sources and how heat flows through nonlinear crystals is essential. We have demonstrated interferometric measurements of thermal distributions resulting from localized surface defects in LBO in the presence of high average optical powers. This technique offers both high sensitivity and spatial resolution and has provided valuable information on the distribution of heat that arises from such absorbers. Measurements of low (~10−5) levels of absorption from spots <100 μm have been performed and compared with both FEA and analytical models. While numerical modeling can properly handle the effects of boundary losses, it requires significant computational resources to accurately calculate the finer features that define these heat distributions. The simple analytical model, on the other hand, does not consider boundaries but provides good insight into how these measurements depend on properties of the absorbing site, such as size and total absorbed power. With the aid of this model, one can use measurements of the bulk response of the material to indirectly determine total power absorbed and estimate resulting changes in temperature.
Funding
Department of Energy National Nuclear Security Administration (NNSA) (DE-NA0003856), University of Rochester; New York State Energy Research and Development Authority.
Acknowledgements
This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.
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