Characterization of Absorption Losses and Transient Thermo-Optic E ﬀ ects in a High-Power Laser System

: (1) Background: The modeling, characterization, and mitigation of transient lasers, thermal stress, and thermo-optic e ﬀ ects (TOEs) occurring inside high energy lasers have become hot research topics in laser physics over the past few decades. The physical sources of TOEs are the un-avoidable residual absorption and scattering in the volume and on the surface of passive and active laser elements. Therefore, it is necessary to characterize and mitigate these e ﬀ ects in real laser systems under high-power operations. (2) Methods: The laboratory setup comprised a 10-kW continuous wave laser source with a changeable beam diameter, and dynamic registration of the transient temperature proﬁles was applied using an infrared camera. Modeling using COMSOL Multiphysics enabled matching of the surface and volume absorption coe ﬃ cients to the experimental data of the temperature proﬁles. The beam quality was estimated from the known optical path di ﬀ erences (OPDs) occurring within the examined sample. (3) Results: The absorption loss coe ﬃ cients of dielectric coatings were determined for the evaluation of several coating technologies. Additionally, OPDs for typical transmissive and reﬂective elements were determined. (4) Conclusions: The idea of dynamic self-compensation of transient TOEs using a tailored design of the considered transmissive and reﬂecting elements is proposed.


Introduction
High energy lasers (HELs) with an average power of 100 kW are required in the fields of science, technology, and the military [1][2][3][4][5][6][7][8]. In the case of laser weapon (LaW) applications, such systems do not operate as steady-state, continuous-wave (CW) systems, as the required interaction time with a target is only a few seconds. The transient laser, thermal stress, and thermo-optic effects (TOEs) occurring inside a HEL and laser beam-forming tracking have to be characterized [9,10] and compensated. The main physical sources of TOEs are the un-avoidable residual absorption and scattering in the volume and on the surface of dielectric coatings of passive and active laser elements, which have been theoretically and experimentally studied for several decades (see, e.g., [11][12][13]). Therefore, the characterization and mitigation of these effects are required in real laser systems operating under a high power [9]. Furthermore, developing a technology for optical elements with such applications remains a great

Experiments
In the examined laser system, we used two kinds of components-one part supplied by an external vendor and the other manufactured by our team (dichroic mirrors and lenses). We intended to validate both groups of optical elements for our specific purpose, so the information on temperature increases under a real high-power operation was of critical importance. Moreover, by applying the COMSOL model, we intended to estimate how these temperature profiles impact the beam quality of the overall optical train.
The state of the art in absorption loss measurements is a sensitivity below 1 ppm [12,13,16]. However, for the testing of high-power optics destined for LaW applications, the loss measurement sensitivity of 1-5 ppm seems to be satisfactory [15]. In our case, it was more important to validate the optics in the real laser system and characterize transient TOEs occurring during the typical exposition durations of 10-30 s. To achieve this goal, we used a method based on the comparison of dynamically registered 2D temperature maps, via an infrared camera (IRC), with a model of a sample implemented in COMSOL Multiphysics. The experimental setup is shown in Figure 1.
Following the approach in [14], we preliminary assessed the thermal response time τ TR of the examined samples according to the following formula: where l is the thickness of the laser element and κ is the thermal diffusivity (κ = 0.823 mm 2 /s for fused silica).

87
Following the approach in [14], we preliminary assessed the thermal response time τTR of the 88 examined samples according to the following formula: where l is the thickness of the laser element and κ is the thermal diffusivity (κ = 0.823 mm 2 /s for fused 90 silica).

91
A high-power laser beam of 9.5 kW and diameter 2wg = 19.6 mm were incident on an examined 92 sample, which was slightly tilted to reflect the laser beam to the beam dumper. The thermal image of 93 the examined surface under the incidence angle of ~5° was registered by means of an SC5600 IR 94 camera with a nominal temperature resolution of 0.025 K. However, we estimated that the realistic 95 temperature uncertainty was 0.2-0.3 K, which corresponds to our specific experimental conditions 96 (incident power of 10 kW with a beam diameter of less than 2 cm), with uncertainty in absorption 97 loss estimations of 2 ppm.

98
The exposition time for each sample (excluding DCH2*) was 60 s. Afterwards, the laser operation 99 was stopped. The exposition time was comparable to or shorter than the thermal response time for 100 each sample (Table 1); thus, we assumed that the examined sample operates in the so-called 'pre-101 diffusion phase'. In the experiments, the laser beam diameter was more than two times smaller than 102 the sample diameter, so truncation losses and heating of the mechanical mounts can be neglected. A high-power laser beam of 9.5 kW and diameter 2w g = 19.6 mm were incident on an examined sample, which was slightly tilted to reflect the laser beam to the beam dumper. The thermal image of the examined surface under the incidence angle of~5 • was registered by means of an SC5600 IR camera with a nominal temperature resolution of 0.025 K. However, we estimated that the realistic temperature uncertainty was 0.2-0.3 K, which corresponds to our specific experimental conditions (incident power of 10 kW with a beam diameter of less than 2 cm), with uncertainty in absorption loss estimations of 2 ppm.

104
The exposition time for each sample (excluding DCH 2* ) was 60 s. Afterwards, the laser operation was stopped. The exposition time was comparable to or shorter than the thermal response time for each sample (Table 1); thus, we assumed that the examined sample operates in the so-called 'pre-diffusion phase'. In the experiments, the laser beam diameter was more than two times smaller than the sample diameter, so truncation losses and heating of the mechanical mounts can be neglected. For each exposition, 2D thermal images were registered and the maximal temperature was found for the time interval of 0.5 s (Figures 2 and 3). To verify the measurement method, we used an additional focusing lens (focal length f = 11 m), and after thermal stabilization, processed the second exposition with a slightly decreased beam diameter of 2w g = 14 mm. The 2D temperature distributions were recorded and compared to the results of the model in COMSOL Multiphysics (red curves in Figure 2c

121
The red and blue curves represent the results of the model and experiment, respectively.    For each measurement point (sample corresponding to the column of results in Table 1), we compared two experimental temperature profiles to the model results for an assumed absorption coefficient. In the case of surface absorption losses, it was assumed that, on the given surface, absorption manifests as the "artificial" surface heat source (W/m 2 ) proportional to A surf and the power density. We do not enter here into the physics of absorption and scattering on a thin (a few microns) layer. The net effect in the form of surface absorption coefficient A surf evidences the quality of technology, i.e., preparing of the surface and coating process.
Assuming that the error in ∆T was 0.3 K, we obtained, for almost all cases (HR1, HR2, HR3, and DCH1), differences of less than 2 ppm between the cases of larger (19.5 mm) and smaller (14 mm) beam diameters. The change of 2 ppm caused changes in ∆T of about 0.5 K.
We are convinced that our approach is valid for a temperature increase of more than 1 K, which corresponds to about 10 ppm of total losses in our specific case for a 10-kW incident power. For the optical elements with losses of less than 10 ppm, this method is problematic. However, we claim that such an approach gives reasonable results for threshold losses of 10 ppm with an uncertainty of about 20% and enables the comparison and validation of several optical technologies for our specific applications.

Modeling of Transient Thermo-Optic Effects
As demonstrated in Section 2, the maximal temperature increased with the duration of exposition and the absorption loss coefficient. The impact of the transient TOEs on the laser wavefront distortion OPD is quite different for the case using a mirror compared to that using a transmissive element (window or lens).
Here, OPD mirr is the OPD of a mirror, OPD lens is the OPD of a transmissive element (window or lens), α CTE is the coefficient of thermal expansion, dn/dT is the thermal dispersion of the refractive index n, and ∆T is the temperature increase.
Because of the inhomogeneity of the laser beam profile (for our case, it was a fiber laser source with a close to Gaussian beam profile) and the requirement to model transient, un-stationary effects of the 3D temperature distributions and resulting thermal-elastic deformations of the facets, analytical solutions of the heat equation (see, e.g., [23][24][25][26]) cannot be used. Therefore, we decided to use COMSOL Multiphysics for the calculations [27] (Figures 4-7, Table 2). The mirror surface was convex due to the temperature profile, so the resulting wavefront diverged and paraxial thermal lensing occurred, with a negative sign proportional to the thermal expansion coefficient α CTE (dashed red curve in Figure 4). In the case of the transmissive element, due to the convex surfaces of both facets and positive dn/dT, the wavefront was concave and paraxial thermal lensing was positive (black curve in Figure 4). Therefore, both TOEs can balance the OPD in principle. In the case of fused silica, dn/dT = 9.6 × 10 −6 1/K and α CTE = 5.2 × 10 −7 1/K; thus, the thermally-induced OPD of the mirror is much weaker compared with that of the lens.
Photonics 2020, 7, x FOR PEER REVIEW 6 of 11 Here, OPDmirr is the OPD of a mirror, OPDlens is the OPD of a transmissive element (window or 147 lens), αCTE is the coefficient of thermal expansion, dn/dT is the thermal dispersion of the refractive 148 index n, and ΔT is the temperature increase.

149
Because of the inhomogeneity of the laser beam profile (for our case, it was a fiber laser source  Table 2). The mirror surface was 154 convex due to the temperature profile, so the resulting wavefront diverged and paraxial thermal 155 lensing occurred, with a negative sign proportional to the thermal expansion coefficient αCTE (dashed 156 red curve in Figure 4). In the case of the transmissive element, due to the convex surfaces of both 157 facets and positive dn/dT, the wavefront was concave and paraxial thermal lensing was positive (black 158 curve in Figure 4). Therefore, both TOEs can balance the OPD in principle. In the case of fused silica, 159 dn/dT = 9.6 10 −6 1/K and αCTE = 5.2 10 −7 1/K; thus, the thermally-induced OPD of the mirror is much

201
The sensitivity limits for the most important loss measurement techniques are presented in Table   202 3.

201
The sensitivity limits for the most important loss measurement techniques are presented in Table  Figure 7. Power in the bucket of 2 × (CL + 2M) vs. the radius for several exposition durations for the 9.5-kW beam, where 2w g = 19.7 mm. For the absorption coefficient data taken from Table 1, the effective OPD of the train with two components (two mirrors, followed by a compensating lens (CL)) is dynamically balanced with the maximal effective OPD of less than a 0.1 wavelength during 60-s exposition ( Figure 5).
To determine the influence of the OPD on the beam quality parameters (Strehl ratio and M 2 ), the following numerical approach was implemented. First, we calculated the intensity distribution I ff,OPD of the aberrated and truncated Gaussian beam in the far field as a Fourier-Bessel transform of the incident Gaussian beam with OPD aberration, as follows: and the corresponding intensity distribution I dl of the diffraction limited Gaussian beam truncated at the sample diameter 2r aper : where w g is the radius of the incident Gaussian beam, k is the wavelength, θ is the angle in the far field, and OPD is the determined experimental/numerical evaluation profile of the OPD (Figures 4 and 5).
Knowing these functions, we can determine the Strehl ratio SR as follows: To determine the beam quality parameter M 2 , we used the definition of power in bucket (PIB) of the beam radius, as follows: M 2 = r 86.5,OPD /r 86.5,dl .
Both radii r 86.5,OPD and r 86.5,dl were found by numerically solving the following equations of PIB distributions: PIB OPD (r 86.5,OPD ) = 0.865, PIB dl (r 86.5,dl ) = 0.865, where PIB OPD (r) = r 0 I f f (x; OPD)xdx Photonics 2020, 7, 94 8 of 10 The far field intensity plots ( Figure 6) and PIB curves ( Figure 7) were realized based on the OPD data, showing near-perfect compensation of the transient TOEs for the case of the train comprising two special compensating lenses (CLs) and four mirrors (2 × (CL + 2M)). Assuming a perfect Gaussian incident beam, the resulting M 2 parameter was less than 1.072 for the entire 60-s exposition duration (Table 2).

Discussion
The sensitivity limits for the most important loss measurement techniques are presented in Table 3.  [13] <1 <1 Cavity mode eigen frequency change [14,17,20,29] <1 <1 Thermal deflectometry [12] <1 <1 Thermal imaging + model (our approach) 2 5 The method of loss measurements proposed in this paper results in the inferior sensitivity compared to the best results obtained during the last few decades (see Table 3). However, as we have shown in part 2 and 3, this approach enables the estimation of residual thermo-optic aberrations and deterioration in the beam quality of the train of optical elements.
The results of the model show that, for a simple lens (Table 2 column 'L'), the transient TOEs are significant and result in a serious reduction in the beam quality (M 2 = 1.19). The simple combination of one lens and two mirrors ( Table 2, L + 2M) shows partial dynamic compensation of the transient TOEs (M 2 = 1.03). Further improvement by employing a special CL via tailored absorption in the coatings enables compensation of a system comprised of two CLs and four mirrors.
This effect was preliminarily confirmed in the experimental characterization of our laser system. First, we did not observe the paraxial thermo-optic-induced focus shift typical for industrial HELs. Therefore, regarding first-order paraxial optics, the system was dynamically compensated. Moreover, the residual higher-order transient TOEs were partly compensated. The observed deterioration in beam quality with power [17,18] was caused by the effects occurring inside the HEL, i.e., heating of gain fibers, transient mode instabilities, and uncompensated transient TOEs occurring in the fiber endcap.
Such an approach can be used in several highly demanding laser optic trains of HELs. Manipulation with substrates and coatings of transmissive element losses (windows, beam splitters, and lenses) and mirrors are attractive for the dynamic self-compensation of high-power laser systems, including the most demanding cases of LIGO.

Conclusions
Transient 2D temperature distributions in laser optical elements under a 10-kW laser beam exposition were measured and compared to those obtained via numerical modeling in COMSOL Multiphysics. By applying such an experimental/numerical approach, the effective absorption losses in the dielectric layers of typical mirrors and in the volume of the transmissive elements under a high laser power were determined. The layer absorption loss was determined as 20-50 ppm for highly reflective mirrors and less than 10 ppm for antireflective coatings.
The idea of dynamic self-compensation of transient TOEs by a tailored design of the transmissive and reflective elements was proposed and preliminarily verified in a model and experiment. We do not intend to propose any general solution leading to the full, dynamic compensation of thermo-optic effects. Instead of this, we have proposed a solution for a given power density profile and specific sample geometry and material, in order to find the control parameter (in our case, the value of surface absorption losses of AR coatings of transmissive elements), which gives the best possible compensation of thermally-induced OPDs occurring in a train of examined optical elements. This is only a proposal of basic idea, and how to change the effective absorption of coatings is a task or challenge for optical technology.
The numerical model of this concept for two dynamically compensated optical trains, with one comprising a single lens and two mirrors (L + 2M) and the second one comprising two special compensated lenses and four mirrors (2 × (CL + 2M)), has been presented.