Lu et al. Vol. 24, No. 3/March 2007/J. Opt. Soc. Am. A 659 Wavefront distortion of the reflected and diffracted beams produced by the thermoelastic deformation of a diffraction grating heated by a Gaussian laser beam Patrick P. Lu, Amber L. Bullington, Peter Beyersdorf,* Stefan Traeger,† and Justin Mansell‡ Ginzton Laboratory, Stanford University, Stanford, California 94305-4085, USA Ray Beausoleil HP Laboratories, 13837 175th Place NE, Redmond, Washington 98052-2180, USA Eric K. Gustafson,** Robert L. Byer, and Martin M. Fejer Ginzton Laboratory, Stanford University, Stanford, California 94305-4085, USA Received April 20, 2006; revised September 13, 2006; accepted September 18, 2006; posted September 26, 2006 (Doc. ID 70078); published February 14, 2007 It may be advantageous in advanced gravitational-wave detectors to replace conventional beam splitters and Fabry–Perot input mirrors with diffractive elements. In each of these applications, the wavefront distortions produced by the absorption and subsequent heating of the grating can limit the maximum useful optical power. We present data on the wavefront distortions induced in a laser probe beam for both the reflected and dif- fracted beams from a grating that is heated by a Gaussian laser beam and compare these results to a simple theory of the wavefront distortions induced by thermoelastic deformations. © 2007 Optical Society of America OCIS codes: 000.2780, 050.1950, 120.3180, 120.6810. 1. INTRODUCTION that are not optically transparent, thus increasing the list of possible materials available for this application. Many optical systems contain transmissive components In this paper, we present a model for the thermoelastic such as lenses and beam splitters. Laser power incident wavefront distortions caused by surface heating of reflec- on these is partially absorbed by their substrates. At high tive optics and compare it with experimental results for powers, substrate absorption produces wavefront distor- both the reflected and diffracted beams from a thermally tions through thermal-lensing and thermoelastic distor- loaded grating. We also make calculations comparing tions. This is especially relevant for advanced thermoelastic deformations of a surface-heated grating gravitational-wave interferometers,1–4 which will require with thermal lensing due to substrate absorption in a several kilowatts of circulating power at the beam splitter transmissive beam splitter. to obtain the required phase sensitivity. Both the genera- tion and the handling of such high optical power will be challenging. Kilowatt-class laser designs for gravitational-wave interferometry based on Nd:YAG 2. HELLO AND VINET’S ANALYSIS OF have been proposed,5 and it is likely that lasers for this THERMOELASTIC DISTORTIONS application will be a reality within the current decade. Thermoelastic distortions arise from nonuniform heating However, high laser power will produce wavefront distor- caused by light absorbed in the reflective and antireflec- tions in both the beam splitters and Fabry–Perot input tive coatings on the surface of a mirror and in the mirror mirrors; these distortions are more likely to limit the cir- substrate. These problems were examined by Winkler et culating power in advanced gravitational-wave detectors al. in 199113,14 using a simple thermal model of this effect than the available laser power. While advanced based on dimensional analysis in gravitational-wave in- gravitational-wave interferometers will probably use terferometers and produced estimates of the size of the ef- some form of active thermal compensation to reduce the fect agreeing to a factor of ϳ2 with the exact results pre- effects of laser beam heating-induced distortion,6 it could viously obtained by Hello and Vinet15,16 and discussed be advantageous to eliminate the degradation caused by below. substrate absorption by using reflective rather than Hello and Vinet15,16 developed an exact result for the transmissive components.7–12 The use of reflective optics distortions induced in the surface of a right circular cyl- not only eliminates substrate absorption and thermal inder of radius a and thickness h, under thermal loading lensing, it also allows for the use of substrate materials by surface absorption of a Gaussian light beam of 1/e2 ra- 1084-7529/07/030659-10/$15.00 © 2007 Optical Society of America 660 J. Opt. Soc. Am. A/Vol. 24, No. 3/March 2007 Lu et al. dius w, including both conductive and radiative cooling. In the steady state, the difference of the temperature field inside the mirror and the surrounding temperature field, T=Tint−Text, obeys the Laplace equation, 2T͑r,z͒ =0, ͑1ٌ͒ subject to the boundary conditions on the barrel ͑r=a͒,at the front surface (the plane z=−h/2), and at the back sur- face (the plane z=h/2): T͑r,− h/2͒ץ T͑a,z͒ץ − ␬ =4␴ЈT3 T͑a,z͒,−␬ zץ z extץ 2P = ⑀ exp͑−2r2/w2͒ −4␴ЈT3 T͑r,− h/2͒, ␲w2 ext T͑r,h/2͒ץ − ␬ =4␴ЈT3 T͑r,h/2͒. ͑2͒ z ext Fig. 2. Fractional deviation between the simplified theory andץ the exact model of Hello and Vinet versus optic size. The x axis indicates the scaling of the optic relative to a fixed beam size of These dictate the balance between the power absorbed w=3.5 cm. The aspect ratio of the optic was held fixed, with h from a Gaussian beam of power P and spot size w with an =a. Fractional deviations were calculated at r=2w by taking the intensity profile, difference between the simplified and exact theories and dividing by the exact theory. An emissivity of 0.5 was used. 2P I͑r͒ = exp͑−2r2/w2͒, ͑3͒ ␲w2 3. SIMPLIFIED ANALYSIS Here, we develop a simplified analysis, which reproduces and the power radiated according to the Stefan– the results of Hello and Vinet for most cases of interest Boltzmann relation linearized in T, where ⑀ is the fraction and is convenient for design calculations. The approxi- of the incident power absorbed, ␴Ј is the Stefan– mate computation is based on the observation that cylin- Boltzmann constant corrected for the emissivity of the drical mirrors of radius aϾ2w and thickness hϾ2w are material, and ␬ is the thermal conductivity of the mirror expected in all currently discussed advanced substrate. The coordinate system, geometry of the mirror, gravitational-wave interferometers due to considerations and the orientation of the Gaussian heating beam are of optical losses and mirror thermal noise. Under these shown in Fig. 1. In the analysis of Hello and Vinet, an ex- conditions, we assume that the temperature and elastic act solution for the temperature field is expressed as a distortions in a finite mirror can be approximated by the Dini series, and then this temperature field is used to solutions for a half-infinite mirror out to radii where the compute the thermoelastic surface distortions. intensity of the light beam becomes negligible, ϳrՆ2w. We have compared Hello and Vinet for gravitational-wave mirrors and reflective beam splitters of silicon, sapphire, and fused silica to our formalism. Over the range rϽ2w, the two theories differ by less than 11% in the worst case. Figure 2 compares the two theories for different materials and optic sizes. For the case described in our experiment, where a=12.7 mm, w=1.1 mm, and the substrate is BK-7, the two theories differ by ϳ0.3%. For this paper, we have used our simplified theory and believe that for studying gravitational-wave detector design, when somewhat more accuracy than is obtainable with the approach of Winkler et al.13,14 is required, the simpler expressions shown here may be more convenient than the exact expressions of Hello and Vinet. Our model uses the temperature profile in the half- infinite optic to estimate the distortions in a finite-sized mirror. Normalizing the temperatures to a characteristic value, 2⑀P T = , ͑4͒ Fig. 1. Coordinate system and geometry used to calculate the c ␲w␬ temperature field and the thermoelastic distortions. Note that we have selected the same coordinate system as Hello and Vinet ¯ (Ref. 16) in which the heated surface is in the plane z=−h/2. The according to T=T/Tc, and expressing the spatial dimen- heating beam is normally incident on the grating. sions in terms of the laser beam waist w as ¯r=r/w and Lu et al. Vol. 24, No. 3/March 2007/J. Opt. Soc. Am. A 661 ¯z=z/w, the boundary condition for the front face given in ings capable of handling high power have been demon- Eq. (2) can be written strated, for example, at Lawrence Livermore National Laboratory17 for stretching and compressing high-energy ¯ T 1 laser pulses. Several effects must be addressed before aץ − + T¯ = exp͑−2¯r2͒, ͑5͒ grating can be considered for use as a diffractive beam z ¯l¯ץ th splitter in a gravitational-wave interferometer.10 The ¯ ␬ ͑ ␴ 3 ͒ wavelength dependence of the diffraction angle couples where lth=wlth= / 4 ЈText characterizes the relative im- portance of radiation versus conduction for removing heat laser frequency noise to beam pointing fluctuations; the from the illuminated region and is the only parameter in spatial profile of the diffracted beam is in general ellipti- the solution for the thermal field. Appendix A shows that cal; and thermoelastic deformation of the substrate af- for a half-infinite mirror with a radiative boundary condi- fects the diffracted beam differently than the reflected tion on the reflective surface, which is also absorbing a beam. As discussed in Ref. 10, using the grating near the fraction of the laser light, the dimensionless temperature Littrow configuration reduces the ellipticity of the dif- can be written fracted beam, and the dispersion in the grating can be compensated for by using an additional grating or by ϱ ͓ 2 ͑ ¯ ͔͒ ͑ ͒ 1 exp − k /8 − k ¯z + h/2 J0 kr¯ double passing the grating to null the dispersion.