In an experiment to simulate the conditions in high optical power advanced
gravitational wave detectors such as Advanced LIGO, we show that strong thermal
lenses form in accordance with predictions and that they can be compensated
using an intra-cavity compensation plate heated on its cylindrical surface. We
show that high finesse ~1400 can be achieved in cavities with internal
compensation plates, and that the cavity mode structure can be maintained by
thermal compensation. It is also shown that the measurements allow a direct
measurement of substrate optical absorption in the test mass and the
compensation plate.Comment: 8 page

A. Brooks

A. E. Siegmann

B. J. J. Slagmolen

C. Cutler

C. M. Mow Lowry

C. Zhao

D. E. McClelland

D. G. Blair

D. Hosken

D. J. Hosken

D. Mudge

G. Billingsley

J. Degallaix

J. Munch

L. Ju

M. A. Barton

M. B. Gray

P. Hello

P. J. Veitch

Y. Fan

English

arXiv

Crossref

Compensation of Strong Thermal Lensing in High-Optical-Power Cavities

In an experiment to simulate the conditions in high optical power advanced gravitational wave detectors, we show for the first time that the time evolution of strong thermal lenses follows the predicted infinite sum of exponentials (approximated by a double exponential), and that such lenses can be compensated using an intracavity compensation plate heated on its cylindrical surface. We show that high finesse ~1400 can be achieved in cavities with internal compensation plates, and that mode matching can be maintained. The experiment achieves a wave front distortion similar to that expected for the input test mass substrate in the Advanced Laser Interferometer Gravitational Wave Observatory, and shows that thermal compensation schemes are viable. It is also shown that the measurements allow a direct measurement of substrate optical absorption in the test mass and the compensation plate.C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, D. E. McClelland, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J. Veitch, M. A. Barton and G. Billingsle

Zhao, C.

Degallaix, J.

Ju, L.

Fan, Y.

Blair, D.

Slagmolen, B.

Gray, M.

Mow-Lowry, C.

McClelland, D.

Hosken, D.

Mudge, D.

Brooks, A.

Munch, J.

Veitch, P.

Barton, M.

Billingsley, G.

Adelaide Research & Scholarship

PUBLISHED VERSION  Zhao, C.;...; Hosken, David John; Mudge, Damien Troy; Brooks, Aidan Francis; Munch, Jesper; Veitch, Peter John; ... et al.  Compensation of strong thermal lensing in high-optical-power cavities Physical Review Letters, 2006; 96(23):231101  ©2006 American Physical Society  http://link.aps.org/doi/10.1103/PhysRevLett.96.231101           http://link.aps.org/doi/10.1103/PhysRevD.62.093023                        http://hdl.handle.net/2440/23569   PERMISSIONS http://publish.aps.org/authors/transfer-of-copyright-agreement   “The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright Act, 17 U.S.C. §101, the employer named [below], shall have the following rights (the “Author Rights”): [...] 3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.”    9th May 2013  PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006Compensation of Strong Thermal Lensing in High-Optical-Power CavitiesC. Zhao, J. Degallaix, L. Ju, Y. Fan, and D. G. BlairSchool of Physics, The University of Western Australia, 35 Stirling Highway, Nedlands, Western Australia 6009, AustraliaB. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, and D. E. McClellandCentre for Gravitational Physics, The Australian National University, Canberra, 0200, AustraliaD. J. Hosken, D. Mudge, A. Brooks, J. Munch, and P. J. VeitchDepartment of Physics, The University of Adelaide, Adelaide, South Australia, 5005 AustraliaM. A. Barton and G. BillingsleyLIGO, California Institute of Technology, Pasadena, California 91125, USA(Received 28 February 2006; published 12 June 2006)0031-9007=In an experiment to simulate the conditions in high optical power advanced gravitational wavedetectors, we show for the first time that the time evolution of strong thermal lenses follows the predictedinfinite sum of exponentials (approximated by a double exponential), and that such lenses can becompensated using an intracavity compensation plate heated on its cylindrical surface. We show thathigh finesse 1400 can be achieved in cavities with internal compensation plates, and that mode matchingcan be maintained. The experiment achieves a wave front distortion similar to that expected for the inputtest mass substrate in the Advanced Laser Interferometer Gravitational Wave Observatory, and shows thatthermal compensation schemes are viable. It is also shown that the measurements allow a directmeasurement of substrate optical absorption in the test mass and the compensation plate.DOI: 10.1103/PhysRevLett.96.231101 PACS numbers: 04.80.Nn, 95.55.YmITM ETM PM SM PD BS Laser FIG. 1. Simplified schematic diagram of the optical layout ofan advanced interferometer: ETM, ITM, power recycling mirror(PM), signal recycling mirror (SM), BS, photodiode (PD).Shading illustrates the thermal lensing in the various compo-nents; the beamwidth signifies light intensity.The first generation laser interferometer gravitationalwave detectors are taking data [1–5]. The Laser Inter-ferometer Gravitational Wave Observatory (LIGO) detec-tors have achieved their target design sensitivity, and yetthis sensitivity is estimated to be only sufficient to detectlarge rare gravitational wave signals. To achieve frequentdetection of signals from firmly predicted sources, a ten-fold improvement in sensitivity is required. This will in-crease the sensitive range from 14 Mpc today to about200 Mpc, leading to event rates of many per year [6].To achieve this improvement the Advanced LIGO [7]and other second generation detectors require a 100-foldincrease in circulating laser power, in addition to improve-ment in the interferometer configuration, test mass thermalnoise, and vibration isolation. This high power level isproposed to be achieved as follows: 125 W of stabilizedlaser power is injected into the interferometer. This is builtup to 2 KW inside the power recycling cavity and afurther enhanced to more than 800 kW in the 4 km longFabry-Perot arm cavities. Such power levels create twosignificant challenges. The first is to control parametricinstabilities in the form of radiation pressure mediatedopto-mechanical oscillation [8]. The second is the controlof strong thermal lensing in the beam splitter (BS) andinboard test mass (ITM), which is the topic of this Letter.Figure 1 [7] shows the core optical components of anadvanced interferometer, with shading showing the tem-perature build up in the core optical components. Thermallensing arises whenever absorbed light in an optical sub-strate or coating creates a temperature gradient, which, via06=96(23)=231101(4) 23110thermal expansion and the thermo-optic coefficient, leadsto wave front distortion of the optical modes of the inter-ferometer [9–15].In the Advanced LIGO the absorption in the ITM sub-strate and coatings is predicted to create a strong thermallens with focal length 4:9 km [16], which is comparableto the ITM mirror focal length of 1.038 km. Such strongthermal lensing will normally cause an interferometer toeither severely lose sensitivity or become inoperable [10].Already, compensation of thermal lensing due to excessabsorption in an ITM has been required in the initial LIGO1-1 © 2006 The American Physical SocietyFIG. 2. A schematic diagram of the experiment setup. Thebeam from 10 W laser is mode matched into the Fabry-Perotcavity formed by the back-surface ITM and the ETM. The CP isplaced near the ITM and heated by 6 turns of nichrome wireinsulated by Teflon tape. The Fourier transform of the beamtransmitted through the ETM is recorded by a CCD cameraplaced near the back focal plane of the 50 mm diameter, 500 mmfocal length lens.PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006interferometer [17,18]. In that case, a CO2 laser beam wasused to heat the back surface of the ITM to cancel thepositive lens produced by absorption. Similar low levelcompensation has been applied to the GEO 600 gravita-tional wave detector [19].In this Letter we report on the observation and charac-terization of strong thermal lensing in a sapphire ITM, inwhich the magnitude of the distortion is similar to thatexpected in the Advanced LIGO. We demonstrate com-pensation using direct heating of a fused silica compensa-tion plate inside the cavity. We show for the first time thatthe thermal lens evolves according to the predictions ofHello and Vinet [13], and show that the results may be usedto determine the average optical absorption of both the testmass and the compensation plate.The experiment described here was performed at theGingin High Optical Power Test Facility (HOPTF) [20]in Western Australia, which has been developed with thespecific goal of investigating the effects of high opticalpower as part of the LIGO collaboration [21]. It has amaximum arm length of 80 m and currently has a single-frequency 10 W Nd:YAG laser [22]. The optical cavity wasdesigned to produce a maximum distortion that is as closeas possible to the Advanced LIGO conditions.To achieve this, we use a Fabry-Perot cavity with areversed input mirror so that the high reflective coating ison the outside. This allows high power 1 kW to traversethe substrate. The layout of the experiment is shown inFig. 2 and the parameters of the cavity are listed in Table I.An intracavity compensation plate (CP) placed within theTABLE I. ParametITMRadius of curvature (m) R1 Materials sapphDiameter (mm) 100Thickness (mm) 50high-reflection coated transmission (ppm) 1840antireflection coated reflectivity (ppm) 29Cavity internal power (kW) 1Cavity length (m) 7723110cavity is conductively heated on its cylindrical surface andis located near the ITM. While the CP in our experiment isdirectly mounted on a breadboard, in an advanced detectorit would be suspended and radiatively heated to avoid noisecoupling. The measured cavity lifetime was about 118 ms,giving a finesse of about 1400. The transmitted beamobserved with a CCD camera was used to characterizethe thermal lens development.The thermal lensing arises from radial temperature gra-dients in the test mass due to the Gaussian profile laserbeam and radiative heat dissipation through the sides of themirror. This produces a net radial heat flow. The thermalvariation of the material refractive index and the elasticthermal expansion at the mirror surfaces are the two domi-nant thermal aberrations. Hello and Vinet [13,14] predictedboth the steady state and transient temperature distributionthroughout a cylindrical mirror for the case where thechange in temperature due to absorption of optical powerremains small compared to the external ambient tempera-ture. The time evolution of the thermal aberration per unittotal absorbed power for a concentric laser beam passingthrough a cylindrical test mass of radius a and height h, isgiven by an infinite sum of exponentials as follows: t; r  4a4KhXp;mdmm1 epmt1 cosuph=aup  sinuph=a2m  u2pJ0mr=acpup; (1)where   4"T30a=K. Here m is mth solution of equa-tion, xJ1x  J0x, and up is the pth solution of equa-tion, u   cotuh=2a. The terms dm are the coefficientsof a Dini series expansion of the azimuthally symmetricincident intensity distribution, while pm  KCa2 	2m  u2p and cp  1 auph sinuph=a. The constants, K, ", and  are the thermo-optical coefficient, thethermal conductivity, the emissivity, and the density ofthe test mass material, respectively, and  is the Stefan-Boltzmann constant. We show below that Eq. (1) can beapproximated as a sum of just two exponentials.To enable modeling of arbitrary geometry we used finiteelement modeling (FEM) to calculate the time evolution ofa thermal lens. Figure 3 compares the time dependenters of the cavity.ETM CP1 R2  720 Flatire sapphire fused silica150 16080 17100 2020 12 12 <100.01-2PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006optical path difference between the beam center and thebeam waist position calculated using Eq. (1) and a FEMmodel. We assumed an absorbed power of 0.5 W, corre-sponding to a substrate absorption coefficient of50 ppm=cm and 1 kW circulating power.The analytical results summed over the first 60 terms(dashed line) and the FEM modeling results (asterisks)show excellent agreement. The radius of curvature due tolaser beam heating of the ITM substrate is about 2.5 km,even stronger than that expected (4.9 km) for the AdvancedLIGO using fused silica as the test mass material [16]. Thesolid line shown in Fig. 3 is the sum of two exponentialsfitted to the FEM model result, with time constants 1 and2. This result is clearly a good approximation to both theanalytical and FEM results. We interpret 1 as due to thetime constant for heating the Gaussian beam profile withinthe test mass, while 2 characterizes the much longer timeit takes the test mass to radiate heat through its sides andcome into steady state equilibrium with the heated beamvolume. For our sapphire ITM, we find that 1  0:76 sand 2  5:91 s. It is interesting to note that the value ofthe time constants 1 and 2 depend slightly on the mirrorgeometry and so are only valid for this particular ITM. Thethermal lensing of the fused silica CP can be similarlymodeled, yielding the much longer time constants of 1 7:40 s and 2  50:1 s.The thermal lensing in the ITM causes the beam size onthe end test mass (ETM) to change correspondingly. Theradius of the beam (!) on the ETM can be expressed as thefunction of the laser beam wavelength, the cavity length(L) and the radii of curvature of the ITM (R1) and ETM(R2) [23],! R2s LR2  L1=41 R2R11=4: (2)0 10 20 30 40 50 600246810121416Time (seconds)Optical Path Difference (nm)Analytical resultsFEM resultsFitted with sum of two exponentialsFIG. 3 (color online). Simulation results of ITM thermal lens-ing time dependence when 0.5 W was absorbed; the dots aresimulated results and the solid line is the fitted the curve usingthe sum of two exponentials with time constants of 0.76 and5.91 s.23110In our case, R1  R2, and Eq. (2) can be approximated as! R2s LR2  L1=41 R24R1: (3)Equation (3) implies that the beam radius ! reduceslinearly with the optical path difference because the latteris directly proportional to 1=R1.Thus we expect the time dependence of ! to follow thesum of exponentials predicted above, with the same timeconstants. Using the configuration illustrated in Fig. 2, thebeam radius at the ETM can be calculated from the mea-sured beam radius on the CCD using Gaussian beampropagation law. The experiment consisted of measuringthe beam radius as a function of time since cavity locking.The time dependence of the beam radius at the ETM, andthe transmitted power are plotted in Fig. 4. The imagedspots could be accurately described by a fundamentaltransverse mode Gaussian intensity profile and no obviousasymmetric distortion was observed. The cold-cavity beamradius at the CCD was 1.0 mm, corresponding to the beamradius of 9.3 mm at the ETM. As expected, thermal lensingat the ITM decreases the beam size at the ETM.Since the time constants for the thermal lensing areknown, only 2 free parameters, the powers absorbed bythe ITM and CP, are needed to fit the expected timedependence to the measurements. Thus the time evolutionof the thermal lens provides a means of measuring theoptical absorption of the ITM. The solid line in Fig. 4shows the fitted time dependence, which yields absorbedpowers of 0.51 W and 5.5 mW, respectively. Because thecoating absorption is much less than the substrate bulkabsorption in the ITM the above absorbed power corre-sponds to an average bulk absorption of about 51 ppm=cmin the ITM. The power absorbed in the CP is about 5.5 ppm.0 10 20 30 40 50 60 70 800246810Transmitted Power (a.u.)88.28.48.68.899.29.49.69.810Time (Seconds)Beam Radius (mm)Transmitted power   Measured Beam Radius Simulated Data FIG. 4 (color online). A plot of the measured time dependenceof the beam radius at the ETM (dots) and the result of asimulation in which only the power absorbed by the ETM andCP were optimized (solid line). The cavity circulating power isshown on the top of the graph.1-30 1000 2000 3000 4000 5000 6000 700088.28.48.68.899.29.49.69.810Time (seconds)Beam Radius (mm)02468101214Transmitted Power (a.u.)Transmitted PowerMeasured Beam Radius Simulated Data FIG. 5 (color online). Time dependence of the measured (dots)and simulated (squares) ETM beam radius during thermal com-pensation. Optical power storage commenced at t  0; CPheating power was turned on at 450 s and turned off at time4026 s. The power transmitted through the ETM, which isproportional to the stored cavity power, is also shown.PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006Thermal compensation was simulated using a modelsimilar to that used for the simulation of the thermallensing with the addition of heating from the heatingring. The results, shown as squares in Fig. 5, agree reason-ably well with the measurements.In conclusion, we have observed significant wave frontdistortion due to optical absorption in the substrate of amirror, using conditions that are traceable to those ex-pected in the Advanced LIGO. The observed distortion isconsistent with that expected from modeling, and couldlead to a significant reduction in sensitivity and perhapsinstrument failure. The time evolution of thermal lensingshows a double exponential dependence in agreement withthe predictions of our finite element model [12] and theanalytical model of Vinet and Hello [13]. Measurement ofthe thermal time constants allow an accurate estimate ofthe test mass optical absorption. We have also shown thatthe distortion can be compensated using a conductivelyheated, fused silica compensation plate. Furthermore, itappears feasible that the compensation could be main-tained using a system in which the transmitted beam sizeis used to define an error signal for feedback control.An off-axis Hartmann wave front sensor will shortly beinstalled at the HOPTF, which will allow spatially re-solved, sensitive measurements of the wave front distortionin the ITM and CP [24]. An autoalignment system [20] willbe implemented to prevent the thermal drift effects ob-served in Fig. 5.We would like to thank the International AdvisoryCommittee of the ACIGA/LIGO High Power TestFacility for their encouragement and advice, and HelenaArmandula for technical support. This research was sup-ported by the Australian Research Council and theDepartment of Education, Science and Training and bythe U.S. National Science Foundation. It is a project of23110the Australian Consortium for Interferometric Gravita-tional Astronomy in collaboration with LIGO. We thankespecially Barry Barish, Stan Whitcomb, and David Reitzewhose support made this project possible. J. Degallaix issupported in part by Whitfeld.1-4[1] B. Abbott et al. (LIGO Scientific Collaboration), Phys.Rev. Lett. 94, 181103 (2005).[2] B. Abbott et al. (LIGO Scientific Collaboration), Phys.Rev. Lett. 95, 221101 (2005).[3] B. Abbott et al. (LIGO Scientific Collaboration), Phys.Rev. D 73, 062001 (2006).[4] B. Abbott et al. (LIGO Scientific Collaboration), Phys.Rev. D 72, 122004 (2005).[5] B. Abbott et al. (LIGO Scientific Collaboration), Phys.Rev. D 69, 082004 (2004).[6] C. Cutler and K. S. Thorne, in Proceedings of the 16thInternational Conference on General Relativity &Gravitation, edited by N. T. Bishop and S. D. Maharaj(World Scientific, Singapore, 2002), p. 72.[7] P. Fritschel, http://www.ligo.caltech.edu/docs/T/T010075-00.pdf.[8] C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. Blair, Phys.Rev. Lett. 94, 121102 (2005).[9] W. Winkler, K. Danzmann, A. Ru¨diger, and R. Schilling,Phys. Rev. A 44, 7022 (1991).[10] R. C. Lawrence, http://www.ligo.caltech.edu/docs/P/P030001-00.pdf.[11] R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, andD. Shoemaker, Classical Quantum Gravity 19, 1803(2002).[12] J. Degallaix, C. Zhao, L. Ju, and D. Blair, Appl. Phys. B77, 409 (2003).[13] P. Hello and J. Vinet, J. Phys. (France) 51, 1267 (1990).[14] P. Hello and J. Vinet, J. Phys. (France) 51, 2243 (1990).[15] Ryan Lawrence, David Ottaway, Peter Fritschel, and MikeZucker, Opt. Lett. 29, 2635 (2004).[16] J. Degallaix, Ph.D. thesis, The University of WesternAustralia, 2006 (to be published).[17] D. Ottaway, K. Mason, S. Ballmer, M. Smith, P. Willems,C. Vorvick, G. Moreno, and D. Sigg, http://www.ligo.caltech.edu/docs/G/G050095-00/G050095-00.pdf.[18] D. J. Ottaway, J. Betzwieser, S Ballmer, S. Waldman, andW. Kells, Opt. Lett. 31, 450 (2006).[19] H. Luck, A. Freise, S. Gosler, S. Hild, K. Kawabe, andK Danzmann, Classical Quantum Gravity 21, S985(2004).[20] B. J. Slagmolen, M. Barton, C. Mow-Lowry, G. de Vine,D. S. Rabeling, J. H. Chow, A. Romann, C. Zhao, M. B.Gray, and D. E. McClelland, Gen. Relativ. Gravit. 37,1601 (2005).[21] L. Ju et al., Classical Quantum Gravity 21, S887 (2004).[22] D. Hosken, D. Mudge, C. Hollitt, K. Takeno, P. Veitch,M. Hamilton, and J. Munch, Prog. Theor. Phys. Suppl.151, 216 (2003).[23] A. E. Siegmann, Lasers (University Science Books,Sausalito, California, 1986).[24] A. Brooks, P. Veitch, J. Munch, and T.-L. Kelly, Gen.Relativ. Gravit. 37, 1575 (2005).

Physical Review Letters

Compensation of strong thermal lensing in high-optical-power cavities

In an experiment to simulate the conditions in high optical power advanced gravitational wave detectors, we show for the first time that the time evolution of strong thermal lenses follows the predicted infinite sum of exponentials (approximated by a double exponential), and that such lenses can be compensated using an intracavity compensation plate heated on its cylindrical surface. We show that high finesse ~1400 can be achieved in cavities with internal compensation plates, and that mode matching can be maintained. The experiment achieves a wave front distortion similar to that expected for the input test mass substrate in the Advanced Laser Interferometer Gravitational Wave Observatory, and shows that thermal compensation schemes are viable. It is also shown that the measurements allow a direct measurement of substrate optical absorption in the test mass and the compensation plate

Blair, D. G.

Slagmolen, B. J. J.

Gray, M. B.

Mow Lowry, C. M.

McClelland, D. E.

Hosken, D. J.

Veitch, P. J.

Barton, M. A.

Caltech Authors - Main

PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006Compensation of Strong Thermal Lensing in High-Optical-Power Cavities
C. Zhao, J. Degallaix, L. Ju, Y. Fan, and D. G. Blair
School of Physics, The University of Western Australia, 35 Stirling Highway, Nedlands, Western Australia 6009, Australia
B. J. J. Slagmolen, M. B. Gray, C. M. Mow Lowry, and D. E. McClelland
Centre for Gravitational Physics, The Australian National University, Canberra, 0200, Australia
D. J. Hosken, D. Mudge, A. Brooks, J. Munch, and P. J. Veitch
Department of Physics, The University of Adelaide, Adelaide, South Australia, 5005 Australia
M. A. Barton and G. Billingsley
LIGO, California Institute of Technology, Pasadena, California 91125, USA
(Received 28 February 2006; published 12 June 2006)0031-9007=In an experiment to simulate the conditions in high optical power advanced gravitational wave
detectors, we show for the first time that the time evolution of strong thermal lenses follows the predicted
infinite sum of exponentials (approximated by a double exponential), and that such lenses can be
compensated using an intracavity compensation plate heated on its cylindrical surface. We show that
high finesse 1400 can be achieved in cavities with internal compensation plates, and that mode matching
can be maintained. The experiment achieves a wave front distortion similar to that expected for the input
test mass substrate in the Advanced Laser Interferometer Gravitational Wave Observatory, and shows that
thermal compensation schemes are viable. It is also shown that the measurements allow a direct
measurement of substrate optical absorption in the test mass and the compensation plate.
DOI: 10.1103/PhysRevLett.96.231101 PACS numbers: 04.80.Nn, 95.55.YmITM 
ETM 
PM 
SM 
PD 
BS 
Laser 
FIG. 1. Simplified schematic diagram of the optical layout of
an advanced interferometer: ETM, ITM, power recycling mirror
(PM), signal recycling mirror (SM), BS, photodiode (PD).
Shading illustrates the thermal lensing in the various compo-
nents; the beamwidth signifies light intensity.The first generation laser interferometer gravitational
wave detectors are taking data [1–5]. The Laser Inter-
ferometer Gravitational Wave Observatory (LIGO) detec-
tors have achieved their target design sensitivity, and yet
this sensitivity is estimated to be only sufficient to detect
large rare gravitational wave signals. To achieve frequent
detection of signals from firmly predicted sources, a ten-
fold improvement in sensitivity is required. This will in-
crease the sensitive range from 14 Mpc today to about
200 Mpc, leading to event rates of many per year [6].
To achieve this improvement the Advanced LIGO [7]
and other second generation detectors require a 100-fold
increase in circulating laser power, in addition to improve-
ment in the interferometer configuration, test mass thermal
noise, and vibration isolation. This high power level is
proposed to be achieved as follows: 125 W of stabilized
laser power is injected into the interferometer. This is built
up to 2 KW inside the power recycling cavity and a
further enhanced to more than 800 kW in the 4 km long
Fabry-Perot arm cavities. Such power levels create two
significant challenges. The first is to control parametric
instabilities in the form of radiation pressure mediated
opto-mechanical oscillation [8]. The second is the control
of strong thermal lensing in the beam splitter (BS) and
inboard test mass (ITM), which is the topic of this Letter.
Figure 1 [7] shows the core optical components of an
advanced interferometer, with shading showing the tem-
perature build up in the core optical components. Thermal
lensing arises whenever absorbed light in an optical sub-
strate or coating creates a temperature gradient, which, via06=96(23)=231101(4) 23110thermal expansion and the thermo-optic coefficient, leads
to wave front distortion of the optical modes of the inter-
ferometer [9–15].
In the Advanced LIGO the absorption in the ITM sub-
strate and coatings is predicted to create a strong thermal
lens with focal length 4:9 km [16], which is comparable
to the ITM mirror focal length of 1.038 km. Such strong
thermal lensing will normally cause an interferometer to
either severely lose sensitivity or become inoperable [10].
Already, compensation of thermal lensing due to excess
absorption in an ITM has been required in the initial LIGO1-1 © 2006 The American Physical Society
FIG. 2. A schematic diagram of the experiment setup. The
beam from 10 W laser is mode matched into the Fabry-Perot
cavity formed by the back-surface ITM and the ETM. The CP is
placed near the ITM and heated by 6 turns of nichrome wire
insulated by Teflon tape. The Fourier transform of the beam
transmitted through the ETM is recorded by a CCD camera
placed near the back focal plane of the 50 mm diameter, 500 mm
focal length lens.
PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006interferometer [17,18]. In that case, a CO2 laser beam was
used to heat the back surface of the ITM to cancel the
positive lens produced by absorption. Similar low level
compensation has been applied to the GEO 600 gravita-
tional wave detector [19].
In this Letter we report on the observation and charac-
terization of strong thermal lensing in a sapphire ITM, in
which the magnitude of the distortion is similar to that
expected in the Advanced LIGO. We demonstrate com-
pensation using direct heating of a fused silica compensa-
tion plate inside the cavity. We show for the first time that
the thermal lens evolves according to the predictions of
Hello and Vinet [13], and show that the results may be used
to determine the average optical absorption of both the test
mass and the compensation plate.
The experiment described here was performed at the
Gingin High Optical Power Test Facility (HOPTF) [20]
in Western Australia, which has been developed with the
specific goal of investigating the effects of high optical
power as part of the LIGO collaboration [21]. It has a
maximum arm length of 80 m and currently has a single-
frequency 10 W Nd:YAG laser [22]. The optical cavity was
designed to produce a maximum distortion that is as close
as possible to the Advanced LIGO conditions.
To achieve this, we use a Fabry-Perot cavity with a
reversed input mirror so that the high reflective coating is
on the outside. This allows high power 1 kW to traverse
the substrate. The layout of the experiment is shown in
Fig. 2 and the parameters of the cavity are listed in Table I.
An intracavity compensation plate (CP) placed within theTABLE I. Paramet
ITM
Radius of curvature (m) R1 
Materials sapph
Diameter (mm) 100
Thickness (mm) 50
high-reflection coated transmission (ppm) 1840

antireflection coated reflectivity (ppm) 29

Cavity internal power (kW) 1
Cavity length (m) 77
23110cavity is conductively heated on its cylindrical surface and
is located near the ITM. While the CP in our experiment is
directly mounted on a breadboard, in an advanced detector
it would be suspended and radiatively heated to avoid noise
coupling. The measured cavity lifetime was about 118 ms,
giving a finesse of about 1400. The transmitted beam
observed with a CCD camera was used to characterize
the thermal lens development.
The thermal lensing arises from radial temperature gra-
dients in the test mass due to the Gaussian profile laser
beam and radiative heat dissipation through the sides of the
mirror. This produces a net radial heat flow. The thermal
variation of the material refractive index and the elastic
thermal expansion at the mirror surfaces are the two domi-
nant thermal aberrations. Hello and Vinet [13,14] predicted
both the steady state and transient temperature distribution
throughout a cylindrical mirror for the case where the
change in temperature due to absorption of optical power
remains small compared to the external ambient tempera-
ture. The time evolution of the thermal aberration per unit
total absorbed power for a concentric laser beam passing
through a cylindrical test mass of radius a and height h, is
given by an infinite sum of exponentials as follows:
 t; r  4a
4
Kh
X
p;m
dm
m
1 epmt

1 cosuph=a
up
  sinuph=a
2m  u2p

J0mr=a
cpup
; (1)
where   4"T30a=K. Here m is mth solution of equa-
tion, xJ1x  J0x, and up is the pth solution of equa-
tion, u   cotuh=2a. The terms dm are the coefficients
of a Dini series expansion of the azimuthally symmetric
incident intensity distribution, while pm  KCa2 	
2m  u2p and cp  1 auph sinuph=a. The constants
, K, ", and  are the thermo-optical coefficient, the
thermal conductivity, the emissivity, and the density of
the test mass material, respectively, and  is the Stefan-
Boltzmann constant. We show below that Eq. (1) can be
approximated as a sum of just two exponentials.
To enable modeling of arbitrary geometry we used finite
element modeling (FEM) to calculate the time evolution of
a thermal lens. Figure 3 compares the time dependenters of the cavity.
ETM CP
1 R2  720 Flat
ire sapphire fused silica
150 160
80 17
100 20
20 12
 12 <100
.0
1-2
PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006
optical path difference between the beam center and the
beam waist position calculated using Eq. (1) and a FEM
model. We assumed an absorbed power of 0.5 W, corre-
sponding to a substrate absorption coefficient of
50 ppm=cm and 1 kW circulating power.
The analytical results summed over the first 60 terms
(dashed line) and the FEM modeling results (asterisks)
show excellent agreement. The radius of curvature due to
laser beam heating of the ITM substrate is about 2.5 km,
even stronger than that expected (4.9 km) for the Advanced
LIGO using fused silica as the test mass material [16]. The
solid line shown in Fig. 3 is the sum of two exponentials
fitted to the FEM model result, with time constants 1 and
2. This result is clearly a good approximation to both the
analytical and FEM results. We interpret 1 as due to the
time constant for heating the Gaussian beam profile within
the test mass, while 2 characterizes the much longer time
it takes the test mass to radiate heat through its sides and
come into steady state equilibrium with the heated beam
volume. For our sapphire ITM, we find that 1  0:76 s
and 2  5:91 s. It is interesting to note that the value of
the time constants 1 and 2 depend slightly on the mirror
geometry and so are only valid for this particular ITM. The
thermal lensing of the fused silica CP can be similarly
modeled, yielding the much longer time constants of 1 
7:40 s and 2  50:1 s.
The thermal lensing in the ITM causes the beam size on
the end test mass (ETM) to change correspondingly. The
radius of the beam (!) on the ETM can be expressed as the
function of the laser beam wavelength, the cavity length
(L) and the radii of curvature of the ITM (R1) and ETM
(R2) [23],
! 

R2

s 
L
R2  L

1=4

1 R2
R1
1=4
: (2)0 10 20 30 40 50 60
0
2
4
6
8
10
12
14
16
Time (seconds)
O
pt
ica
l P
at
h 
Di
ffe
re
nc
e 
(nm
)
Analytical results
FEM results
Fitted with sum of two exponentials
FIG. 3 (color online). Simulation results of ITM thermal lens-
ing time dependence when 0.5 W was absorbed; the dots are
simulated results and the solid line is the fitted the curve using
the sum of two exponentials with time constants of 0.76 and
5.91 s.
23110In our case, R1  R2, and Eq. (2) can be approximated as
! 

R2

s 
L
R2  L

1=4

1 R2
4R1

: (3)
Equation (3) implies that the beam radius ! reduces
linearly with the optical path difference because the latter
is directly proportional to 1=R1.
Thus we expect the time dependence of ! to follow the
sum of exponentials predicted above, with the same time
constants. Using the configuration illustrated in Fig. 2, the
beam radius at the ETM can be calculated from the mea-
sured beam radius on the CCD using Gaussian beam
propagation law. The experiment consisted of measuring
the beam radius as a function of time since cavity locking.
The time dependence of the beam radius at the ETM, and
the transmitted power are plotted in Fig. 4. The imaged
spots could be accurately described by a fundamental
transverse mode Gaussian intensity profile and no obvious
asymmetric distortion was observed. The cold-cavity beam
radius at the CCD was 1.0 mm, corresponding to the beam
radius of 9.3 mm at the ETM. As expected, thermal lensing
at the ITM decreases the beam size at the ETM.
Since the time constants for the thermal lensing are
known, only 2 free parameters, the powers absorbed by
the ITM and CP, are needed to fit the expected time
dependence to the measurements. Thus the time evolution
of the thermal lens provides a means of measuring the
optical absorption of the ITM. The solid line in Fig. 4
shows the fitted time dependence, which yields absorbed
powers of 0.51 W and 5.5 mW, respectively. Because the
coating absorption is much less than the substrate bulk
absorption in the ITM the above absorbed power corre-
sponds to an average bulk absorption of about 51 ppm=cm
in the ITM. The power absorbed in the CP is about 5.5 ppm.0 10 20 30 40 50 60 70 80
0
2
4
6
8
10
Tr
an
sm
itt
ed
 P
ow
er
 (a
.u.
)
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10
Time (Seconds)
Be
am
 R
ad
iu
s 
(m
m)
Transmitted power
   Measured 
Beam Radius 
Simulated Data 
FIG. 4 (color online). A plot of the measured time dependence
of the beam radius at the ETM (dots) and the result of a
simulation in which only the power absorbed by the ETM and
CP were optimized (solid line). The cavity circulating power is
shown on the top of the graph.
1-3
0 1000 2000 3000 4000 5000 6000 7000
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10
Time (seconds)
Be
am
 R
ad
iu
s 
(m
m)
0
2
4
6
8
10
12
14
Tr
an
sm
itt
ed
 P
ow
er
 (a
.u.
)
Transmitted Power
Measured Beam Radius 
Simulated Data 
FIG. 5 (color online). Time dependence of the measured (dots)
and simulated (squares) ETM beam radius during thermal com-
pensation. Optical power storage commenced at t  0; CP
heating power was turned on at 450 s and turned off at time
4026 s. The power transmitted through the ETM, which is
proportional to the stored cavity power, is also shown.
PRL 96, 231101 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending16 JUNE 2006Thermal compensation was simulated using a model
similar to that used for the simulation of the thermal
lensing with the addition of heating from the heating
ring. The results, shown as squares in Fig. 5, agree reason-
ably well with the measurements.
In conclusion, we have observed significant wave front
distortion due to optical absorption in the substrate of a
mirror, using conditions that are traceable to those ex-
pected in the Advanced LIGO. The observed distortion is
consistent with that expected from modeling, and could
lead to a significant reduction in sensitivity and perhaps
instrument failure. The time evolution of thermal lensing
shows a double exponential dependence in agreement with
the predictions of our finite element model [12] and the
analytical model of Vinet and Hello [13]. Measurement of
the thermal time constants allow an accurate estimate of
the test mass optical absorption. We have also shown that
the distortion can be compensated using a conductively
heated, fused silica compensation plate. Furthermore, it
appears feasible that the compensation could be main-
tained using a system in which the transmitted beam size
is used to define an error signal for feedback control.
An off-axis Hartmann wave front sensor will shortly be
installed at the HOPTF, which will allow spatially re-
solved, sensitive measurements of the wave front distortion
in the ITM and CP [24]. An autoalignment system [20] will
be implemented to prevent the thermal drift effects ob-
served in Fig. 5.
We would like to thank the International Advisory
Committee of the ACIGA/LIGO High Power Test
Facility for their encouragement and advice, and Helena
Armandula for technical support. This research was sup-
ported by the Australian Research Council and the
Department of Education, Science and Training and by
the U.S. National Science Foundation. It is a project of23110the Australian Consortium for Interferometric Gravita-
tional Astronomy in collaboration with LIGO. We thank
especially Barry Barish, Stan Whitcomb, and David Reitze
whose support made this project possible. J. Degallaix is
supported in part by Whitfeld.1-4[1] B. Abbott et al. (LIGO Scientific Collaboration), Phys.
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Compensation of Strong Thermal Lensing in High Optical Power Cavities

Compensation of Strong Thermal Lensing in High Optical Power Cavities

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