LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY LIGO Laboratory / LIGO Scientific Collaboration LIGO LIGO-T060214-01-D 9/25/2006 Test Mass Thermal Compensation Strategies Phil Willems Distribution of this document: LIGO Science Collaboration This is an internal working note of the LIGO Project California Institute of Technology LIGO Project – MS 18-34 1200 E California Blvd Pasadena, CA 91125 Phone (626) 395-2129 Fax (626) 304-9834 E-mail: info@ligo.caltech.edu Massachusetts Institute of Technology LIGO Project – NW17-161 175 Albany St Cambridge, MA 02139 Phone (617) 253-4824 Fax (617) 253-7014 E-mail: info@ligo.mit.edu LIGO Hanford Observatory P.O Box 1970 Mail Stop S9-02 Richland WA 99352 Phone 509-372-8106 Fax 509-372-8137 LIGO Livingston Observatory P.O Box 940 Livingston, LA 70754 Phone 225-686-3100 Fax 225-686-7189 http://www.ligo.caltech.edu/ LIGO LIGO-T03xxxx-00-D Introduction The simultaneous thermal compensation of the input test mass (ITM) and compensation plate (CP) in Advanced LIGO has been shown to be problematic If the test masses are not compensated directly, then the ROC of their HR surfaces is expected to change due to thermal expansion caused by absorption of arm cavity light power This change will nominally cause the spot size at both test masses to change from 6.0 cm at low laser input power to ~5.3 cm at full laser input power A fixed ring heater acting on the CP can only be optimized to compensate for one given spot size On the other hand, if the test masses are compensated to maintain the arm cavity mode spot size, then the resulting thermorefractive aberration in the ITM caused by its ring heater will cause the thermal lens to change from positive to negative In this case the ring heater can provide no compensation at all This technical note discusses what can be achieved if the arm cavity mode is compensated by acting only on the end test mass (ETM) 1.1 Version History 09/07/2006 -00 Initial version covering only maintenance of arm cavity modes 09/25/2006 -01 Updated version including maintenance of recycling cavity modes for carrier and sidebands Model for the Arm Cavity A simple two-mirror cavity with spherical mirrors suffices to model the system, as the thermoelastic deformations are very well approximated by pure radii of curvature in terms of their effect on the cavity modes The spot sizes at the mirrors of a Fabry-Perot cavity are given by: witm = (1 − L / Retm ) Lλ π (1 − L / Ritm ) (1 − (1 − L / Ritm )(1 − L / Retm ) ) wetm = (1 − L / Ritm ) Lλ π (1 − L / Retm ) (1 − (1 − L / Ritm )(1 − L / Retm ) ) For the nominal parameters of Advanced LIGO (L=4000m, λ=1064nm, Ritm= Retm =2076m), the spot size on both mirrors is 6.0cm Assuming that Brownian noise in the coatings dominates the thermal noise on the test masses, the thermal noise amplitude of a test mass is inversely proportional to the spot size Therefore, we can determine the relative thermal noise for the two test masses combined as being proportional to their root sum squared: LIGO LIGO-T03xxxx-00-D 2 witm + wetm TN ∝ witm wetm For 6cm spot sizes at both mirrors, TN=23.55 At full power, the uncompensated test masses will distort to an effective ROC of 2137m Since the coating contributes both the bulk of the absorbed power to the mass and the predominant part of the surface thermoelastic deformation, both masses have very nearly the same distorted ROC The spot sizes then become 5.3cm, and the thermal noise factor rises to 26.91, a 14% increase This may be tolerable, but is worth preventing if possible Figure shows the spots sizes at both mirrors when the ITM is allowed to thermally distort to a ROC of 2137 and the ROC of the ETM is tuned from 2137m to 2000m The spot size at the ITM is restored to its nominal 6.0cm value when the ETM ROC is 2033m The spot size at the ETM is 5.7cm for this ROC spot size at mirror m 0.062 2020 2040 2060 2080 2100 2120 2140 ETM ROC m 0.058 0.056 0.054 The thermal noise parameter for this configuration is 24.20, which is a 2.8% increase over the nominal thermal noise level The increase in the test mass temperature will also contribute to a rise in the thermal noise level The test mass ring heater design, applied equally to each test mass, would increase the temperature of each test mass by 5.4K, or 1.8% Since the noise amplitude density is proportional to the square root of the temperature, this would cause an overall 0.9% increase in the thermal noise from both test masses combined If all the compensation is applied to the ETM, then that mass will require approximately twice the applied heat, and so will rise 3.6% in temperature, but since the ITM temperature does not rise this again increases the thermal noise by 0.9%, to which we must add the 2.8% increase caused by spot size variation So, the net thermal noise increase for the ETM-only compensation strategy is 3.7%, compared to 0.9% for the balanced compensation strategy For comparison, the level of coating thermal noise itself depends upon the mechanical loss of the coating materials, which are typically known to ~10-20% This makes the thermal noise amplitude uncertain at the 5-10% level LIGO LIGO-T03xxxx-00-D Model for the Recycling Cavities We now ask whether the strategy of maintaining the ITM spot size at cm while allowing the ITM radius of curvature to vary from 2076 m to 2137 m is compatible with the needs for thermal compensation in the recycling cavities We start with a power-recycling Fabry-Perot interferometer for simplicity, and assume that at low power this interferometer is optimally aligned Since the ITM HR radius of curvature is 2076 m, then if the AR face is flat, the ITM radius of curvature as seen from the AR side is 2076/n m, where n is the ITM index of refraction The PRM and input beam to the interferometer are designed to deliver the beam with just this wavefront, and with cm spot size at the ITM For example, the wavefront at the HR face of the PRM will be (2076/n+8.3) m if a marginally stable cavity is used This is true for both the carrier and for the RF sidebands A marginally stable recycling cavity is not essential to this argument Now consider operation at high interferometer power We assume for simplicity that both thermal lensing and thermal compensation can be adequately characterized by a thermal focal length The arm cavity mode control strategy has been applied, so that the ITM spot size is still cm, but its HR radius of curvature is 2137 m In addition, thermal lensing produces a lens f ITM of order 10 km in the ITM and a lens f BS of order 100 km in the beamsplitter, and the thermal compensation on the compensation plate produces a lens f CP All these distortions are located within about m along the beam, which is very small compared to their focal lengths, so collectively they can be considered as a thin lens The wavefront of the arm cavity mode through these thermal lenses is ~1400 m, so along this m beam path the spot size is essentially unchanged Since the arm cavity mode spot size at the PRM and ITM are nearly independent of the thermal effects, the arm cavity will be optimally coupled to the input beam if the compensation plate, with the other thermal lenses, converts the radius of curvature at the PRM HR face to the 2137 m radius of curvature at the ITM HR face However, this is precisely the same requirement that must be satisfied if the RF sidebands are to be optimally coupled into the power recycling cavity, since the resonant mode of the power recycling cavity must also have the 2137 m radius of curvature of the ITM HR face Therefore, optimal compensation of the arm cavity coupling and the RF sideband power buildup and mode overlap can be simultaneously achieved- they are not inconsistent The model here can be equally well applied to the signal recycling cavity In this case, it is easy to show that the optimal compensation of the gravitational wave sideband coupling to the dark port is consistent with the optimal compensation for resonance of the RF sidebands in the signal recycling cavity Notice that the arguments above are completely independent of the precise amount of thermal lensing, so long as the thin lens assumption and constant spot size assumptions hold Therefore, in a Fabry-Perot Michelson interferometer with a recycling cavity, both arms can be simultaneously optimized Since both arms are matched to the PRM radius of curvature through the thermal lenses, they are matched to each other, there and upon recombining at the beamsplitter toward the PRM If the deformation of the beamsplitter 50/50 surface is ignorably small 1, then the contrast defect will also be minimized At the very least, it will likely be much smaller than the beamsplitter’s thermorefractive lens, since the beamsplitter absorption will be predominantly in the substrate LIGO LIGO-T03xxxx-00-D However, the conditions that optimize coupling in the power recycling cavity may not be the same as those that optimize coupling in the signal recycling cavity This is because the beamsplitter substrate contributes more optical path length to the signal recycling cavity than to the power recycling cavity, and so the power of the f BS lens will in general be different for the two cavities Implications for TCS This note shows that the spot size can be maintained at the ITM without direct actuation on the ITM itself However, the wavefront ROC of the cavity mode will still be given by the thermally distorted ITM ROC, which is 2137m, or 1.37x10 -5 diopters from the nominal 2076m ROC However, this aberration is small compared to the ITM thermal lens in the substrate, which is ~10 -4 diopters, so it requires only a small increase in the power needed from the TCS system Compared to the complete change in the required TCS heating profile required if the ITM is directly compensated, it is a much more practical option The simultaneous compensation of the arm cavity input coupling and RF sideband power buildup and overlap in the power recycling cavity can be naturally accomplished, as can the simultaneous compensation of the gravitational-wave sideband extraction efficiency and RF sideband power buildup and overlap in the signal recycling cavity Simultaneous optimization for both cavities is in principle not possible, due to the different effect of the beamsplitter in the two recycling cavities Since the beamsplitter contributes only ~10% of the thermal lens, this difference may be too small to cause any significant problems However, if we choose 0.5 ppm/cm-absorption glass rather than ppm/cm-absorption glass for the beamsplitter, it will contribute only ~2.5% of the thermal lens and be even less significant Choosing a TCS setpoint halfway between ideals for the two recycling cavities may be the best strategy Since the extraction efficiency for the gravitational wave sidebands sets the strictest threshold on the thermal lensing, optimization should be biased towards the signal recycling cavity ... the thermal noise on the test masses, the thermal noise amplitude of a test mass is inversely proportional to the spot size Therefore, we can determine the relative thermal noise for the two test. .. Introduction The simultaneous thermal compensation of the input test mass (ITM) and compensation plate (CP) in Advanced LIGO has been shown to be problematic If the test masses are not compensated... also contribute to a rise in the thermal noise level The test mass ring heater design, applied equally to each test mass, would increase the temperature of each test mass by 5.4K, or 1.8% Since the