Influence of Thermal Distortion in Mirrors on the Propagation of High Power Lasers LV Ke^{a, b} ZHENG Wei^{b} HUA Weihong^{a} ^{a }College of Optoelectric Science and Engineering, National University of Defense Technology, Changsha410073, China; ^{b} Unit 91404 of PLA, Qinhuangdao066326, China
ABSTRACT The rootmeansquare (RMS) phase distortion and the strehl ratio in the farfield influenced by the thermal distortion of mirrors in the propagation of high power lasers were calculated. Calculations with different CO_{2} laser beam intensity distribution indicate that compare with stable cavity whose output is Gaussian, unstable cavity’s beam quality influenced by the thermal deformation depends on the slope of the intensity distribution, that is, contents of H_{2}O, and the obscuration. Among the mirror materials such as SiO_{2}, BeO, CaF_{2} and Si, CaF_{2} provides the steadiest thermal character in the high power laser irradiance. When the SiO_{2} mirror is cooled, the RMS phase aberration brought by the thermal deformation would be less. For a high power mirror, cooled by the water is more efficacious than cooled by the wind.
Keywords: Thermal distortion, beam quality, intensity distribution, cooled mirror
1. INTRODUCTION When high power laser propagates through the transmitting optic, i.e., the optical train, thermal distortion exists in the high power mirrors. The influence of thermal distortion would be significant on the laser beam quality, and it has been a matter of the utmost concern in the design of high energy laser system.^{ }In order to evaluate the thermal distortion, one can use basic principles of thermal and structural static mechanics, calculate the thermal aberration with finiteelement method, derive 2D solution of the thermal conduction equation in the form of generalized Fourier series, and simulate thermal deformation as a planestress problem.^{ [1]}The result would be valuable to the design of adaptive optics in high power laser system.
An accurate assessment of the impact of the thermal distortion is a tedious and difficult calculation requiring wave optics and sophisticated analytical tools for modeling each mirror in the optical train. Obviously these are not very practical for use in system analysis application studies in which a large parameter space must be invested. Therefore, the development of simple but reasonably accurate models is required. In this work we will discuss how various features of the input beam and the materials of the mirror would influence the propagation of the high power lasers. We will propose a simple analytical expression for the thermal distortion and discuss its dependence on system parameters such as the beam intensity profile, the materials of the mirrors and their cooling status. 2. Analytical treatment Obviously, the thermal distortion of the mirror would influence the beam quality of high power lasers. Usually the optical quality of the input beam is specified by the ‘times diffraction limited’ factor, β. Unfortunately, this definition of beam quality does not allow one to easily incorporate the other beam quality degrading factors induced by the thermal distortion of the optical elements. Therefore, in this article, we use the rootmeansquare (RMS) phase distortion as the evaluation of the beam quality. Consider a laser beam containing small scale random phase variations whose RMS is σ, the peak intensity reduction is^{[2]} (1) The ideal thickness of the mirror is L. Consider a small unit of the mirror with S in area and m in mass. Compare with the perfect flat surface mirror, the thermal distortion ΔL leads to an extra phase aberration (2) According to the definition of the thermal expansion coefficient, (3) Where ΔT is the magnitude of the temperature fluctuations in the small unit. 2.1 Uncooled mirrors For an uncooled mirror, the energy absorption Q is the function of surface reflectivity R, the power P and the total pulse length t. (4) Take (2)(4) into account together, we get (5) Where ρ is the density of the mirror material. Equation (5) expresses the fact that with the same material and total pulse length, the extra phase aberration induced by the thermal distortion is proportional to the irradiance. So the same relationship appears between RMS phase aberration induced by the thermal distortion and RMS irradiance distribution. (6) 2.2 Cooled mirrors When the mirror is cooled by the water or wind flow, the energy of the mirror is consisted of two parts: absorption of some fraction of the incident laser beam, energy flow into the mirror from the front; convective between the mirror and the water or wind, energy flow out of the mirror from the back. According to Newton’s law of cooling, heat flux by the convection is (7) Where ΔT is the temperature difference between the mirror and the fluid, h is the heat transfer coefficient. Usually for forced convection gas, h=20~200 w/m^{2}K, and for forced convection liquid, h=500~15000 w/m^{2}K^{[3]}. Assuming that the initial temperature of the mirror is the same as the fluid, the magnitude of the temperature rise of the mirror is the same as the temperature difference between the mirror and the fluid. (8) Take (2) (3) and (8) into account together, we get (9) With the same mirror material and the total pulse length, the extra phase aberration induced by the thermal distortion is proportional to the irradiance. The same relationship appears between RMS phase aberration induced by the thermal distortion and RMS irradiance distribution. (10) 3. examples and results 3.1 Intensity distribution Now we analyze how intensity distribution of the same average intensity influences the beam quality by the thermal distortion of uncooled highpower mirror whose substrate is silicon. Some parameters of the silicon mirror are listed in table1. Table 1 Parameters of the Silicon Mirror^{ [4]} Property  Value  Unit  Density(ρ)  2329  Kg/m^{3}  Specific heat(c)  695  J/kg°C  Thermal expansion coefficient(α)  2.33·10^{6}  °C^{1}  Surface reflectivity (R)  99.8% 
 For the carbon dioxide laser, water is often added to the laser gas mix to suppress the dissociation of CO_{2} molecules into CO and oxygen.^{[5]} In order to get a higher gain, it is essential to add such a catalyst. The gain distribution around the nozzle is different as the amount of water vapor changes. When H_{2}O contents above a critical level, the energy should be picked up in a shorter distance, an unstable cavity is more convenient for the resonator design, the intensity distribution of the output beam is a plane or linearity from upper reaches to the lower reaches of the gas flow, slope k depends on H_{2}O contents. (11) On the contrary, if H_{2}O contents below a specific value, it is better to adopt a stable cavity, the intensity distribution is a Gaussian beam with spot size ω.^{[6]} (12) Assuming CO_{2} laser beam with a square profile D=0.1m, P=100kw, wavelength λ=10.6μm and average intensity I_{ave}= 10000kw/m^{2}. We take into account of some different intensity distribution of the same average intensity listed in table2. Table 2 Different Intensity Distribution Case  Style  Slope k  Obscuration ε 
 a  Linearity  5  0.25 
 b  Linearity  10  0.25 
 c  Gaussian     



 a)  b)  c)  Figure 1 Different intensity distribution of the initial laser beam: a) Linearity, k=5, ε=0.25; b) Linearity, k=10, ε=0.25; c) Gaussian beam According to Equation (1) and Equation (6), the RMS phase aberration in the wavefront after reflected from the silicon mirror and the strehl ratio in the farfield can be calculated as laser irradiating time increases.

 (a)  (b)  Figure 2 Beam quality decline versus laser irradiating time for different intensity distribution of the initial laser beam: a) RMS phase aberration; b) Strehl Ratio in the farfield Figure 2 shows different intensity distribution of the same average intensity vary in thermal distortion, lead to different decline of beam quality. RMS phase aberration is proportion to laser irradiating time. It can be concluded that when we use unstable cavity, the intensity distribution appear to be linearity, the greater slope, the greater RMS phase aberration it induces. Compare with stable cavity whose output is Gaussian, beam quality of the output of unstable cavity depends on the slope of the intensity distribution, that is, contents of H_{2}O and the obscuration. When the slope is high, the thermal distortion induced by the linearity intensity distribution is more critical. Another beam quality factor strehl ratio in the farfield degrades according to Equation (1). The more RMS phase aberration, the worse beam quality, the less strehl ratio in the farfield as time goes by. 3.2 Materials of the mirror In this section, we’ll discuss the thermal performance in the high power laser propagation of some mirror materials such as silicon (Si), beryllium oxide (BeO), calcium fluoride (CaF_{2}) and silica (SiO_{2}). Table 3 Parameters of the Materials^{ [1][7][8]} Property  Unit  Si  BeO  CaF_{2}  SiO_{2}  Density(ρ)  Kg/m^{3}  2329  3020  3180  2200  Specific heat(c)  J/kg°C  695  1000  9113  753  Thermal expansion coefficient(α)  °C^{1}  2.33·10^{6}  8·10^{6}  18.9·10^{6}  6.89·10^{6}  Thickness of the mirror body(L)  m  0.01  0.01  0.01  0.01  For convenient it is assumed that the transverse intensity distribution is cylindrically symmetric Gaussian beam shown in Figure 1c).After irradiating on the uncooled mirror for a few seconds, beam quality degrades because of the thermal distortion. The reflectivity of the mirror is related to the thin films instead of the substrate, so we suppose the surface reflectivity of all mirrors is a constant, R=99.8%. According to Equation (1) and Equation (6), the beam quality varies as the mirror material changes.

 a)  b)  Figure 3 Beam quality decline versus laser irradiating time for different uncooled mirror substrate materials: a) RMS phase aberration; b) Strehl Ratio in the farfield Beam quality decline as a function of laser’s irradiating time for different mirror substrate materials Si, BeO, CaF_{2} and SiO_{2}, respectively, is shown in Figure 3. Beam quality decrease slowest as the time goes by for CaF_{2} and quickest for SiO_{2}. It is indicated that the thermal deformation of CaF_{2} is more insensitive to the laser irradiating time than other materials because of its high specific heat. As for SiO_{2}, small density and specific heat lead to the largest thermal deformation, the strehl ration in the farfield declines to 0.88 within 5 seconds.
When the SiO_{2} mirror is cooled by the wind or water, things would be difference. Assuming the forced gas convection heat transfer coefficient h=180w/m^{2}K, the forced water convection heat transfer coefficient h=5000w/m^{2}K, comparison of the influence of thermal distortions on beam quality among uncooled, wind cooled and water cooled SiO_{2} mirror can be made according to Equation(1) and Equation(10).

 a)  b)  Figure 4 Beam quality decline versus laser irradiating time for cooling status of the mirror: a) RMS phase aberration; b) Strehl Ratio in the farfield Figure 4 shows that once the mirror is cooled, its thermal characteristic would be better; the RMS phase aberration brought by the thermal deformation would be less. For a high power mirror, cooled by water is more efficacious than cooled by the wind, because the forced water convection heat transfer coefficient is higher. But restricted by the manufacturing technology, the water cooled SiO_{2} mirror is difficult to produce. When choosing the mirror material in the high power laser experiments, the priority is the thermal characteristics of the material. When we take account of the cost and convenient factors, or dedicate to reduce the thermal aberration which influences the beam quality, cooling the mirror is a good choice. 4. Conclusions In this paper we have reported analytical expressions for the influence of the thermal distortion of the mirrors on the propagation of high power lasers. The decrease of the beam quality is closely related to the thermal expansion coefficient, the density, the specific heat and some other parameters of the material as well as the intensity distribution and wavelength of the initial laser beam. We can conclude from the calculation that compare with stable cavity whose output is Gaussian, the beam quality of the output of unstable cavity depends on the slope of the intensity distribution, that is, contents of H_{2}O and the obscuration. Among the different materials’ mirror, the mirror made of CaF_{2} provides the least thermal distortion after the same irradiance than the mirror made of silicon (Si), beryllium oxide (BeO) and silica (SiO_{2}).When the mirror is cooled, its thermal deformation would be less. The technique is a quick and unitary assessment to measure the impact of the mirror’s thermal deformation to the transmission of the laser, has a practical significance for the laser system design and beam quality control.
REFERENCES An Jianzhu, Li Youkuan, Du Xiangwan, “Influence of laser window’s thermal lensing effect on beam quality”, High Power Laser and Particle Beams, 16(4),429433(2004) Su Yi, Wan Min, “High energy laser system”, National Defense Industry Press, Beijing, 3959(2004) Wu Tianhua, Wang Xiaomo, Xu Guoliang, “Engineering heat transfer”, Huazhong University of Science and Technology Press, Wuhan, 34(2011) Peng Yufeng, Cheng Zuhai, Zhang Yaoning, “Temperature distributions and thermal deformations of mirror substrates in laser resonators”, Applied Optics,vol.40, 48244830(2001) Hagop Injeyan, Gregory D.Goodno, “Highpower laser handbook”, The McGrawHill Companies, New York,78(2011) Tan Hong, Zhu Zonghou, “Technology of aerodynamic lasers”, National Defense Industry Press, Beijing, 314319(1977) Peng Yufeng, Cheng Zuhai, “Finite element analyses of thermal distortions of mirror substrates for high power lasers”, High power laser and particle beams, vol.17, 58(2005) Marvin J. Weber, “Laser and optical science and technology series”, The CRC Press,Berkeley,2003 