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Tracking Down the Thermal Parameters of Sapphire

Tracking Down the Thermal Parameters of Sapphire

- Contributed by Ryan Lawrence

Around 2007, the LIGO interferometers will be subject to a major equipment upgrade geared towards increasing the broadband gravitational wave strain sensitivity of the instrument by more than a factor of 10. Improved suspensions and seismic isolation will better isolate the test masses from ground movement, a higher power laser will allow for better statistics in the light seen at the output of the instrument, and test masses will be constructed of materials with a lower mechanical loss (i.e., a higher Q value) in order to reduce broadband vibration of the optic's surfaces due to thermal excitation. The low internal loss material earmarked for the advanced LIGO test masses is sapphire (crystalline Al2O3), and research is underway throughout the LIGO Scientific Collaboration to gauge the suitability and attainability of LIGO-sized sapphire test masses (about 12" in diameter and 5" thick).

One major concern about using sapphire as a test mass material is its relatively large coefficient of thermal expansion (quoted anywhere from 4x10-6 to 8x10-6 per degree Kelvin), which is about 10 times larger than that of the present test mass material, fused silica (amorphous SiO2). Why do we care about thermal expansion? One reason is that absorption of laser light will induce a nonuniform temperature rise within the test masses, causing a "bump" to form on the surface, which then distorts the light circulating in the interferometer thus making it less efficient and more difficult to control. A more fundamental reason is that because of the coefficient of thermal expansion, very small random temperature fluctuations turn directly into very small random fluctuations of the optic's surface. Since LIGO is trying to detect gravity waves which induce length changes of about 10-18 meters, a coefficient of thermal expansion of 5x10-6 /K means that mean temperature fluctuations greater than 10-12 K will be seen instead of gravity waves! Statistical physics, along with the well-known quantities of heat capacity and density for sapphire, predicts temperature variations typically less than 10-14 degrees K in the measured volume of a LIGO test mass, so it is clear that this "thermoelastic noise" will not ruin everything, but it will most certainly enforce a limit on the sensitivity of the detector. Exactly where this limit lies depends on the value taken for thermal expansion, which, depending on who you ask, varies by a factor of 2. Here at MIT, we're trying to pin down a much more precise value for the thermal expansion coefficient of sapphire, in addition to other parameters such as thermal conductivity and the thermo-optic coefficient (dn/dT).

Figure 1. Figure 2.

To measure these parameters, we look at how a thermally isolated sapphire optic distorts when we abruptly begin to heat it with an external laser beam. We then compare the measured distortion with a computer simulation we've made of the optic's thermo-mechanical response, and adjust the material parameters in the model until the calculated behavior matches what we observe. Figure 1 at left above shows a snapshot of the numerical model, and Figure 2 at right is a photograph of the heating laser table (a 10 Watt carbon dioxide laser, typically used in industry for soldering). Then, Figures 3 and 4 below show some data taken as well as the model's fit to the data. We measure the distortion of the optic by reflecting a large beam of low power laser light off of the heated surface, then resolving the local change in slope of the returning wavefront with a Shack-Hartmann sensor.

Figure 3. Figure 4.

So far, we have gathered data on two samples: a 3" diameter by 1" thick high quality M-axis sample, and a 5" diameter by 4" thick low quality C-axis sample. Initial analysis of the M-axis data (data and fits are charted in Figures 3 and 4 above), fitting to a cylindrically symmetric model, yields effective values of:

k_th = 35.8 ( +/- 0.8 ) W/m/K
alpha_axial = 4.30 ( +/- 0.09 ) x10^-6 /K
alpha_radial = 6.34 ( +/- 0.12 ) x10^-6 /K

These values in general are quite close to other independent measurements of the materials, giving confidence in the method. Because the method relies on measures which relate directly to some of our application needs, we feel more confident that our models for the interferometer performance given material properties are correct.

Now the experimental apparatus will be applied to its original purpose, that of advancing the art of Active Thermal Compensation--the correction of lensing due to absorption in the substrate or coating of the main sensing laser beam by applying a complementary heating source. In the next phase of the experiment, that additional heating source will be a computer-directed CO2 laser beam, allowing compensation of high-order defects in the system, or for non-circularly-symmetric situations like the beamsplitter. Watch this space for future results!