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.
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LIGO
LIGO-T03xxxx-00-D
1 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.
2 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 
wetm 
L

L

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1  L / Retm 
1  L / Ritm 1  1  L / Ritm 1  L / Retm 
4
1  L / Ritm 
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:
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2
2
witm
 wetm
TN 
witmwetm
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 1 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
ETM ROC m
2020
2040
2060
2080
2100
2120
2140
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.
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3 Model for the Recycling Cavities
We now ask whether the strategy of maintaining the ITM spot size at 6 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 6 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 6 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 4 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 4 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
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PRM. If the deformation of the beamsplitter 50/50 surface is ignorably small 1, then the contrast
defect will also be minimized.
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.
4 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
2 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.
1
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.
1
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