Chunnong Zhao, Li Ju, Jerome Degallaix, and David Blair
School of Physics, the University of Western Australia, Crawley, WA 6009, Australia
High optical power advanced laser interferometer gravitational wave detectors require thermal compensator plates to overcome the de-focussing due to thermal lensing in their test masses. Here we show that the use of negative thermo-optic coefficient materials provide short scale length compensation which cannot be achieved using positive coefficient materials. We show that the crystalline quartz is a promising candidate, and outline the physical design parameters required to achieve good compensation.
1.
Introduction
Several long baseline gravitational wave detectors, notably the LIGO detectors, are currently operating at close to design sensitivity. At this sensitivity they are sensitive to signals emanating from binary neutron star inspiral events to a distance ~ 10-20 Mpc. At this range the expected event rate for detectable signals is low, so that the probability of detection over several years observation is quite small. To increase the sensitivity to the level where frequent events are certain requires a substantial increase in optical power. For this reason Advanced LIGO is planned to use more than
100W of input power and to build this up to more than 1kW in the power recycling cavity, and to almost 1MW in the arm cavities. At this power level thermal lensing becomes a critical issue. The focal length of a typical thermal lens is comparable to the design focal length of the test mass mirrors. A proven solution to thermal lensing is to create a temperature gradient across the optics to counteract the thermal lens. This is normally applied by a CO2 laser which is strongly absorbed in the fused silica test masses.
One can consider the thermal lens in a test mass to be a positive thermal image of the beam profile, modified by thermal diffusivity and radiative heat loss. This thermal image has two components: a) a thermal image due to optical coating absorption and b) a thermal image due to bulk absorption.
The proven method of compensation consists of applying a heating pattern either directly to the test mass, or to a separate compensation plate. This heating pattern is designed to be a negative thermal image which, when added to the positive thermal image of the thermal lens, creates a null image across the incident beam wavefront. This has been demonstrated successfully in various situations
[1, 2, 3, 4]
. However, the method has intrinsic limitations. Consider the case of an ideal test mass which has small uniform absorption. One applies heat in a circumferential pattern near the outer rim to create a temperature profile with a minimum at the central beam axis. Heat is lost by radiation, while the laser provides a heat source which matches this loss so that the temperature profile over the central region can be moderately flat. Modelling shows the impressive compensation that can be achieved, as shown in Figure 1
[5]
. However compensation is impossible over the full beam profile since the thermal image of a Gaussian beam differs fundamentally from any negative thermal image that can be created by localised heating. More importantly, thermal compensation of localised absorption structures, such as surface contamination or point defects, absorb light strongly in one location. Such an absorber creates a cusp shaped positive thermal image. Thermal diffusivity makes it almost impossible to create a negative thermal image with a scale length that matches the positive thermal image of such a cusp. If a material was used that had a Negative Thermo-optic Coefficient
(NTC), such as crystalline quartz, then one could compensate for localised absorption by a positive thermal image. Intrinsically it is much easier to create matching positive thermal images than

positive-negative pairs. Even such compensation is not perfect, however, because the thermal diffusivity is likely to create different scale lengths for the images.
Fig.1: (a) Comparison between the optical path length difference with and without the thermal compensation; (b) the schematic of a thermal compensation scheme using heating ring in front of fused silica optics.
The idea of using NTC materials for compensating thermal lensing in high power solid state lasers was first proposed by Koch
[6]
and was demonstrated by Weber and colleagues
[7, 8,9]
. This has also been considered for high power Faraday isolators [10, 11] . It has been very successful in suppressing thermal lensing in the laser rod and the magneto-optical materials, but is limited to small beam diameters and uses rather high loss materials. For applications which require extremely low losses and large beam sizes, such as laser interferometer for gravitational wave detection, the range of suitable materials is limited.
For NTC thermal compensation it is necessary to consider both the magnitude of the NTC, and the coefficient of thermal expansion. To explain this we need to consider thermal lensing in further detail. When a Gaussian profile laser beam passes through an optical medium, the heating of the medium due to residual optical absorption gives rise to a thermal lens through three main mechanisms:
A.
Thermo-optic effects: the refractive index of the medium changes in proportional to the thermo-optic coefficient dn/dT and the local temperature variations. The laser beam profile combined with heat dissipation to the environment leads to a temperature gradient, which creates a consequent refractive index gradient in the medium. This leads to an Optical Path
Difference (OPD) between the centre of the beam and outer region which is analogous to a standard optical lens.
B.
Thermal expansion: the heating induced by a laser beam directly causes the optical component to expand in the heated region (by thermal expansion), normally creating convex deformation at the point where the laser beam enters and exits the component.
C.
Photo-elastic effects: Thermally induced mechanical stress can also directly cause refractive index change due to the photo-elastic effect. This phenomenon is often used for observation of stress in mechanical systems.
In the case of uniform heat deposition over a radius r p
which is smaller than the radius r of a cylindrical mirror, the OPD is proportional to (Tidwell)
  dn
 
C r ,
 n 3
0
 
( 1
 
)( n
0

1 ) (1) dT
Here dn/dT is the thermo-optic coefficient, n
0 is the index of refraction,
 is Poisson’s ratio,  is the coefficient of thermal expansion, and C r,
 is photo-elastic coefficient in the radial and tangential directions. Even though this is not the case of Gaussian beam heating (the usual case in a laser interferometer GW detector) we can still use (1) for an initial discussion of the effectiveness of
NTC materials for thermal compensation.

With NTC materials the negative thermo-optic coefficient creates a negative lens but the positive thermal expansion creates a positive lens. The lens created by the photo-elastic effect depends on the photo-elastic coefficients of the materials. For a linear polarized beam it is clear from equation 1 that the photo-elastic effect due to thermal stress is anisotropic. For an NTC material to be useful as a candidate thermal compensation material its coefficient

in equation 1 must be negative and its anisotropy must be either negligible or else it must be compensated for by an appropriate anisotopic heating pattern.
Hence we now restrict our attention to crystalline quartz. It has a NTC of 5.5/×10 -6 , but it has a rather large coefficient of thermal expansion 7.07/13.3×10
-6
. See Table 1 for a summary of its key parameters.
Below we present analysis of a crystalline quartz compensator. We show that such a compensator does indeed offer the capability of providing more versatile compensation without introducing astigmatism when used in an appropriate configuration.
Table 1 Material property of fused silica and crystalline quartz
Property
Expansion
Thermal conductivity
Unit
Refractive Index - dn/dT 10
-6
K
-1
Thermal 10
-6
K
-1
Wm -1 K -1
Heat capacity J kg
-1
K
-1
Elastic coefficient 10
9
Pa
Fused silica
1.45
8.7
0.51
1.38
746
Crystalline quartz Ref. n
0
=1.53, n e
=1.54
-5.5

11=

22=13.3

33=7.07
K11=K22=6.5
K33=11.7
733
C11=C22=86.6
C33=106.4
C12=6.7
C13=C23=12.4
C44=C55=58
C66=40
Youngs modulus 10
9
Pa
Poisson ratio -
72.6
0.164
76.5
0.18
Photo-elastic coefficient
- p11=p22=0.121; p12=p21=p13= p23=0.270; p11=p22=0.16 p12=p21=0.27 p13=p23=0.27 p44=-0.079
2.
Simulation of thermal lensing and thermal compensation
In Advanced LIGO, we expect ~1 kW inside the power recycling cavity and 830 kW inside the arm cavities. This will create substantial surface deformation that affects the TEM00 arm cavity modes, while substrate thermal lensing affects both the carrier power in the arm cavities, and the sideband power build-up in the power and signal recycling cavities. We will not discuss thermal compensation in the arm cavity as this can be corrected by direct heating of the test masses to correct their radius of curvature. We concentrate on thermal compensation in the recycling cavities due mainly to the ITM substrate thermal lensing. Figure 2 shows a schematic of the central part of

the interferometer with a compensation plate in one arm. The vertical arm is the same as the horizontal arm, and is omitted in the schematic.
Hello and Vinet
[12,13]
analytically modelled the thermal lensing due to the coating and bulk absorption but not including the photo-elastic effect. We will use Finite Element Modelling (FEM) to analyse the thermal lensing including all three effects mentioned above. The parameters used for modelling are listed in table 2.
Table 2 Parameters used in the simulation
ITM
Quartz plate
Material
Fused silica
Crystalline quartz
Diameter
34
34
Thickness
20
20
Figure 3 (a) shows the OPD of the ITM due to the coating and bulk absorption by setting the centre of the ITM as a reference (OPD=0). This is similar to a convex lens. Figure 3 (b) shows the OPD in a single crystalline quartz plate with 0.6 W power absorbed. The C-axis of the quartz is in the beam line and the beam polarization is horizontal. This is similar to a concave lens but with strong anisotropy. It is clear that simply using a single passive quartz plate can not compensate the ITM thermal lensing effectively because of the anisotropy. However, if we insert second identical quartz plate and a 90

quartz rotator in between, the beam will pass the first plate with horizontal polarization and then pass the second plate in vertical polarization. The beam will experience the same thermo-optic and thermal expansion effects. The photo-elastic effect will create an elliptical distortion with the long axis aligned with the vertical axis for the first plate, while it will be horizontal for the second plate. If we add thermal lensing effects in two plates together the anisotropy will disappear. Figure 3 (c) shows the sum OPDs of two quartz plates. Excluding the fluctuations due to the finite FEM meshing the OPD is isotropic. Figure 3 (d) shows the compensation results by adding together OPDs of the ITM and quartz plates. It is clear that the OPD is reduced about 2 orders of magnitude over all.
4
To vertical arm cavity
1 2 3 6
5
Horizontal
CO
2 arm cavity
Fig. 2. Thermal compensation scheme using crystalline quartz in the power recycling cavity; 1: the first quartz compensation plate; 2: half-wave plate; 3: the second quartz plate; 4: power recycling mirror; 5: beam splitter; 6: input test mass of horizontal arm; CO2: CO2 laser beam for extra heating.

(a) (b)
(c) (d)
Fig. 3, simulated thermal lensing results of ITM and compensation plate, and thermal compensation results; a) OPD in the fused silica ITM with 0.425W absorbed on the coating and 4mW absorbed in the substrate; b) OPD in the compensation plate with 0.6W absorbed and horizontally polarized interferometer beam pass through it once; c) sum of OPD in the compensation plates with 0.6W absorbed and horizontally polarized interferometer beam pass through the first one then the polarization rotated 90 degrees and pass through the second one; d) the sum of OPDs in the ITM and compensation plates.
3.
Conclusions
This report shows that crystalline quartz is a feasible negative dn/dT material for thermal compensation. A method has been proposed to avoid distortions due to the photo-elastic effect which yields good results in finite element modelling. In future we will demonstrate the technique experimentally.

4.
References
1 Ryan Lawrence, David Ottaway, Peter Fritschel and Mike Zucker, Op.
Lett.
, 29 2635, (2004)
2 H. Luck, A. Freise, S. Gosler, S. Hild, K. Kawabe and K, Danzmann,
Class. Quantum Grav. 21, S985 (2004).
3 D. Ottaway, K. Mason, S. Ballmer, M. Smith, P. Willems, C. Vorvick,
G. Moreno, D. Sigg, http://www.ligo.caltech.edu/docs/G/G050095-00/G050095-00.pdf
4 C. Zhao, et al, Phys. Rev. Lett . 96 , 231101 (2006)
5 J. Degallaix, C. Zhao, L. Ju, D. Blair, Appl. Phys. B 77 , 409 (2003)
6 R. Koch, Opt. Comm. 140 , 158 (1997)
7 R. Weber, Thomas Graf, and Heinz P. Weber , IEEE J. Quantum Electron., 36 , 757 (2000)
8 Th. Graf, E. Wyss, M. Roth, H. P. Weber, Opt. Commun . 190 , 327 (2001)
9 Eduard Wyss, Michelle Roth, Thomas Graf, and Heinz P. Weber, IEEE J. Quantum Electron., 38 , 1620 (2002)
10 G. Mueller, R. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and
D. B. Tanner, Methods for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers, Class. Quantum
Grav., Vol.19, 2002.
11 Efim Khazanov, et al, IEEE J. Quantum Electron . 40 , 1500 (2004)
12 P. Hello and J. Vinet, J. Phys. France 51 1267 (1990)
13 P. Hello and J. Vinet, J. Phys. France 51 2243 (1990)