Download Technical Note LIGO- T030087-00-R x/xx/99

LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY
-LIGOCALIFORNIA INSTITUTE OF TECHNOLOGY
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Technical Note
LIGO- T030087-00-R
x/xx/99
Revised Report to the April 15, 2004
Core Optics Downselect Committee
Meeting
Phil Willems
This is an internal working note
of the LIGO Project.
California Institute of Technology
Massachusetts Institute of Technology
LIGO Project – MS 51-33
LIGO Project – MS 20B-145
Pasadena CA 91125
Cambridge, MA 01239
Phone (626) 395-2129
Phone (617) 253-4824
Fax (626) 304-9834
Fax (617) 253-7014
E-mail: [email protected]
E-mail: [email protected]
WWW: http://www.ligo.caltech.edu
File /home/blacke/documents/T9000xx.ps – printed November xx, 1999
ABSTRACT
We discuss the current understanding of thermal lensing issues for an Advanced LIGO with
fused silica test masses. The general conclusion is that thermal compensation should be
relatively simple so long as unexpected inhomogeneities are not too large. The case when
inhomogeneities are present is still under study.
POWER ABSORPTION AND TEMPERATURE RISE
The requirement for a fused silica ITM is not clearly specified in the COC Design Requirements
Document, so I make predictions based upon two alternative assumptions: case 1, ultralow
absorption glass (.5ppm/cm) and case 2, low absorption glass (2ppm/cm). The coating
absorption in both cases is assumed to be .5ppm. For maximum power operation, there will be
850kW circulating in each arm cavity, and 2.1kW in each ITM substrate. The ITM is 20cm
thick. The absorbed powers are:
Case 1: .446W
Case 2: .509W
Clearly, ultralow absorption glass in a fused silica ITM will not substantially reduce the total
absorbed power. In finite element models, the maximum temperature rise in the optic will be
1.8K at the center of the HR face, and the temperature at the circumference will be less than .2K.
The most significant dependency here is on coating absorption. Double this, and the
temperatures quoted here nearly double as well. The main risk of failure due to temperature rise
is at the silicate bond, but silica/silica bonds are routinely heated by 35K without harm.
For sapphire, we assume the same coating absorption and laser beam parameters, but the optic is
13cm thick, and we assume a bulk absorption of 40ppm/cm, which is typical of the large samples
at hand. The absorbed power then is 1.517W, and leads to a temperature rise of about 1.0K
which is quite uniform over the whole sapphire, the variation being about 10%. Again, it is hard
to see any risk due to temperature rise. Sapphire/silica bonds have good strength in standard
laboratory conditions, in which the temperature fluctuates at this level daily.
MAXIMUM COMPENSATION ACHIEVABLE
Ryan Lawrence modeled the optimum compensation of 1W of power absorbed in a fused silica
ITM using a compensation plate and a shielded optimized ring heater. His figure of merit was
the decrease in scattered power from a 5cm waist TEM00 mode due to compensated thermal
aberration, relative to the scatter without compensation. The best compensation he was able to
achieve in his model decreased the scattered power by a factor of 3.2x10 -4. He deemed a
correction of only 10-2 necessary to correct the worst-case absorption in a fused silica Advanced
LIGO. I do not expect these numbers to change substantially for a 6cm beam. Therefore, a ring
heater on a compensator plate should be adequate for Advanced LIGO, without additional laser
compensation. The ring heater temperature would rise about 17K in operation, but would be
suspended by wire loops. Excess noise as the compensator plate expands in its sling could be an
issue.
How much absorbed power can be tolerated? Ryan Lawrence’s worst case assumed .27W
absorbed power in a fused silica ITM (my estimate assumes higher interferometer power). Since
the scatter scales roughly as the square of the absorbed power, a factor of 10 -2 correction of .27W
thermal aberration would scatter as much power as a factor of 3.2x10-4 correction of 1.5W
thermal aberration. Thus, I estimate 1.5W absorbed power as the maximum tolerable in a fused
silica Advanced LIGO. Note that correction at the 10-2 level requires the ring heater power to be
set with comparable accuracy. For this reason, an additional external laser may prove difficult to
adjust properly.
Note that all of the above assumes homogeneous absorption in the fused silica substrate and
coating. The case of inhomogeneous absorption is discussed at the end of this document.
SURFACE DISTORTION
Equation 2.3 in Ryan Lawrence’s thesis estimates the thermally induced radius of curvature
(ROC) of the ITM HR face, but does so only for bulk absorption, whereas in fused silica the
coating is dominant. I built a FEMLAB model of the fused silica ITM in Advanced LIGO using
the heating estimate above, and compared to the Hello-Vinet analytical model for axisymmetric
heating. The resulting radius of curvature was estimated by fitting a 8th order polynomial to the
deformation of a diameter of the HR face and taking the 2nd order coefficient as the inverse of
twice the radius (see Figure 1). The resulting value is 72.2km. This estimate of ROC is
conservative in that the surface distortion is gentler than a simple parabola starting about 4-5cm
from the optic center. Because the nominal radius of curvature is 2076m, if the TMs (yes, both,
since coatings dominate the absorption) are to have this ROC during hot operation, then when
cold, their ROC will be 2018m, reducing the g-factor from -.926 to -.982, and increasing the spot
size from 6cm to about 8.5cm. This is a large change in spot size.
A similar calculation for sapphire optics shows that the thermal radius of curvature is 85.1km for
the ITM, and 167km for the ETM. In this case, a point design that works at high power would
require cold radii of curvature of 2026m and 2050m for the ITM and ETM, respectively. The gfactors for the cold cavity would be -.974 and -.95, with spot sizes of 7.0cm and 7.1cm. This is
much less of a change than would be seen in fused silica ITMs, and could come down still
further if the sapphire absorption can be reduced.
In order to compensate these deformations, it would be necessary to actuate directly onto the
ITMs and ETMs, likely onto the HR surfaces. Silica is more likely to require this compensation
than sapphire, and ETMs of sapphire may not require compensation at all. The following table
shows the waist sizes for various compensation schemes for silica and sapphire. Note that any
HR surface compensation is assumed to be ideal. The following table shows how the spot sizes
at the optics will vary in several advanced LIGO configurations. Note that the waist location
will also move in configurations without compensation of both optics, though this is not shown
in the table.
Silica
ITM spot size ETM spot size
Hot, or both compensated 6.0cm
6.0cm
Cold
Cold, ITM compensated
Sapphire Hot, or both compensated
Cold
Cold, ITM compensated
8.5cm
6.8cm
6.0cm
7.0cm
6.3cm
8.5cm
6.6cm
6.0cm
7.1cm
6.2cm
Figure 1: surface deformation of fused silica ITM at high power
NONUNIFORM SURFACE DISTORTION
This analysis is more open-ended and incomplete, but preliminary results suggest that a small
number of nonuniformities in coating/surface absorption can be tolerated without direct
compensation onto the HR surface.
I begin by assuming the most likely form of coating absorption nonuniformity to be point
absorbers. Until I can learn how to improve my code, the best approximation I can make to a
point in my FEM is a gaussian of 4mm waist. This at least is much smaller than the IFO spot
size. Points can be distributed arbitrarily over the surface of a real mirror, but to focus my
analysis I model only one point absorber at a time.
HR surface compensation is required only to correct figure errors seen by the arm cavity.
Aberration seen from the recycling cavity is assumed to be fixed with a compensation plate.
The figure of merit in this analysis is the overlap of a 6cm waist arm cavity mode with itself
upon reflection from the deformed mirror, in which all other sources of phase variation are
absent (e.g. the cavity is ideally made, the ROC has been perfectly compensated). The
difference between this quantity, squared, and unity is the loss to non-TEM00 modes. For
analysis of off-axis spots, the overlap of the TEM00 with the TEM01 and TEM10 modes is also
calculated and subtracted from the loss, since ASC is expected to correct for this. Changes in the
ROC should also be considered but are not yet.
The following table summarizes the loss from the cavity when 100mW of heat are dissipated by
the ‘point’ absorber, for various places on the HR face.
Spot location
centered spot
off 2cm in x
off 4cm in x
off 6cm in x
off –2cm in x
off –4cm in x
off –6cm in x
off 2cm in y
Loss from cavity, single bounce
.29%
.2%
.1%
.04%
.22%
.1%
.04%
.22%
Clearly, the closer the spot is to the center of the optic, the more optical loss it causes.
Furthermore, the closer the spot is to the center of the optic, the more highly illuminated it will
be, so a smaller increase in absorption has more effect. The following table scales the results of
the previous table to determine how much excess absorption a spot must have at each location to
induce 1ppm loss from the cavity at maximum power. To scale these values to higher losses,
remember that the cavity loss scales as the square of the absorbed power.
Spot location
Centered spot
off 2cm in x
off 4cm in x
off 6cm in x
off –2cm in x
off –4cm in x
off –6cm in x
off 2cm in y
Absorption causing 1ppm loss
.25ppm
.37ppm
1.0ppm
4.9ppm
.35ppm
1.0ppm
4.9ppm
.35ppm
I expect that the results in the tables above are roughly invariant if the size of the ‘point’ absorber
is reduced while its peak absorption is increased such that the net absorbed power is unchanged.
Comparisons with the analytical Hello-Vinet model for the cylindrically symmetric case of a
central absorber show very little variation in the deformation for absorbers of 4mm, 2mm, and
1mm waist if the power is held constant. Thus, smaller spots can have higher peak absorptions.
I recommend that this analysis be used to set limits on the acceptable level and number of point
defects on the coated optics. This is because correction of them would require CO2 laser
compensation on all four test masses, with stringent noise requirements on the laser, and the
difficult task of identifying the individual spots on each optic to correct. These limits seem to be
very stringent.
The following tables repeat the calculation for sapphire.
Spot location
centered spot
off 2cm in x
off 4cm in x
off 6cm in x
Loss from cavity, single bounce
.054%
.041%
.02%
.008%
Spot location
Centered spot
off 2cm in x
off 4cm in x
off 6cm in x
Absorption causing 1ppm loss
.57ppm
.81ppm
2.3ppm
11ppm
We see that the requirements on coating absorption inhomogeneity are less stringent for
sapphire, but only by a factor of about 2.
RECYCLING CAVITY ABERRATIONS DUE TO INHOMOGENEOUS SURFACE
ABSORPTION
We assume in what follows that, since the absorption in the bulk of the fused silica adds only a
small amount of heat to that produced by the coating, any inhomogeneities in the bulk of the
fused silica add only a small amount of nonaxisymmetric aberration to that produced by coating
inhomogeneities. This assumption seems justified by comparison of fused silica absorption
maps from Lyon to recent data from Stanford on coating absorption. In practice, so long as
direct compensation of the HR surface of the ITM is not used, the remedy- CO2 laser
compensation on the compensation plate- will be used regardless of the location of the
inhomogeneity.
The aberration in the recycling cavity is much worse than in the arm cavity, because the thermooptical coefficient is so much larger than the thermal expansion coefficient. If a 4mm spot at the
center of the coating absorbs at twice the nominal level (that is, at 2ppm rather than 1ppm), the
excess power will be about 2.4mW. Modeling of the thermorefractive aberration due to this
heating yields an overlap integral of 99.88%, or .12% loss in the recycling cavity, assuming all
homogeneous thermal lensing has been ideally corrected. CO2 laser compensation of
inhomogeneity causing this level of loss was modeled and demonstrated experimentally by Ryan
Lawrence, where he was able to reduce this loss by a factor of ten. By analogy to the surface
distortion results, I expect that off-axis spots are less critical by similar factors. The thermal
aberration for the central spot is shown in Figure 2.
Aberration (m)
x 10
1.4
-8
0.15
1.2
0.1
1
0.05
0.8
0
0.6
-0.05
0.4
-0.1
0.2
-0.15
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0
Figure
2: thermal aberration in OPL for a central absorbing spot dissipating 2.4mW
The actual requirement on inhomogeneous thermal aberration is not yet well defined. One
reasonable approach is to require that inhomogeneous thermal aberrations not exceed the residual
static refractive index inhomogeneity after the compensating polish. This is specified in the
COC DRD as less than 10nm rms for adequate coupling of carrier light into the arm cavity, along
with the requirement that the RF sideband power buildup not be significantly reduced. Precisely
what ‘significantly’ means in this context is not yet defined. Nevertheless, the overall amplitude
of thermal aberration in the above case is peaked at about 10nm in a small region around the
center of the optic, so the rms is presumably far less. It would appear that, so long as point
absorbers in the ITM do not dissipate more than about a few mW and are rare, CO2 laser
compensation of inhomogeneity seems a viable option.
ACTUATOR NOISE
If all the actuation is to applied to a compensation plate in the recycling cavity, there is only one
serious actuator noise issue. This is the noise induced by a scanning laser system. Ryan
Lawrence analyzed this effect in chapter 5.1.5 of his thesis and concluded this would not be a
problem, even though the actual optical path variations could be very large (60nm peak-peak),
because the variations would all be below the Advanced LIGO bandwidth. However, he
implicitly assumed the optical path variations would be an unmodulated sinusoid, which would
have no higher harmonics. In fact, the scanned laser compensation he prototyped would induce a
modulated sawtooth variation in the optical path, with enormous amounts of upconverted noise
throughout the Advanced LIGO band. It may be possible to redesign the scanned laser system to
avoid this noise coupling, but we have not found one yet that would not be very delicate to
design. Therefore, we instead endorse a ‘staring’ laser system, in which the CO2 laser intensity
pattern is fixed (perhaps by apertures outside the vacuum as in initial LIGO), and only the
overall intensity varies, on slow thermal time scales.
CONCLUSIONS
The amount of thermal compensation required for a fused silica Advanced LIGO would be very
large, and the requirements on coating uniformity strict. Unless we directly actuate upon the HR
surfaces of the both the ITMs and ETMs, the spot sizes will vary by ~10% between high and low
power operation; if we compensate none of the HR surfaces, the spot sizes will vary by ~40%.
This is in addition to the large (99%) thermal compensation of the thermal aberrations inside
the substrate as seen by the recycling cavity, and CO2 laser compensation of any hot spots due to
the coating absorption inhomogeneities at the HR surface. Finally, the requirements on number
and size of absorption inhomogneities could be quite strict to maintain high arm cavity finesse at
high power.
The amount of thermal compensation required of sapphire could be far less.