Adaptive beam shaping by controlled thermal lensing inoptical elements
Muzammil A. Arain, Volker Quetschke, Joseph Gleason, Luke F. Williams, Malik Rakhmanov,Jinho Lee, Rachel J. Cruz, Guido Mueller, D. B. Tanner, and David. H. Reitze
When an optical element is subjected to nonuniformspatial heat distribution (as even a weakly absorbingelement is when a high-power Gaussian laser beampasses through it), its physical surface profile andrefractive index change. The nonuniform heat distri-bution can arise from heating with an optical beamincident on the optical element or from some externalheat source. The absorbed power creates a nonuni-form temperature distribution in the element, caus-ing both shape deformations and spatially dependentrefractive indices to appear. The dominant effects aresurface deformation, governed by the coefficient oflinear expansion ��T�, and changes in refractive in-dex, depending upon the thermo-optic coefficient�dn�dT�. For most optical materials, other effects, thethermoelastic coefficient, and the photoelastic effectare smaller and therefore are neglected.1 However,for materials with large thermal expansion coeffi-
cient, this effect should be included. The dominanteffects may create substantial thermal aberrations,i.e., position-based optical path-length change, in theoptical beam. This behavior is typically modeled byintroducing imaginary lenses at the surface and inthe substrate. Hence, this effect is termed “thermallensing.” The focal length [or the radius of curvature(ROC)] of this lens varies according to the geometryand physical properties of the material, the temper-ature profile, and the incident optical beam.
The thermal lens can be used to realize adaptiveoptical components provided that the heating of theelement is precisely controlled.2,3 The ROCs of theseelements can be changed by heating them with apump beam or heating beam in conjunction with theprobe or original beam. (In this paper, we use theterm “probe beam” to refer to the beam that suffersthe thermal lens on account of its own absorption. Weuse interchangeably “pump beam” or “heating beam”to refer to the beam used to compensate the thermallensing and to control the profile of the probe beam.)By controlling the intensity and the beam diameter ofthe heating beam, the optical element can be trans-formed into a powerful modal shaping system, cor-recting in situ errors due to undesired changes in theoptical systems as well as serving as a fine-focusingsystem. Recently such an adaptive system was pro-posed, and initial results were presented.4 Here we
The authors are with the Department of Physics, University ofFlorida, Gainesville, Florida 32611, USA. M. A. Arain’s e-mailaddress is [email protected].
Received 23 August 2006; accepted 24 November 2006; posted 4December 2006 (Doc. ID 74353); published 3 April 2007.
present a comprehensive investigation of thermallyinduced adaptive beam shaping, including detailedtheory and enhanced experimental results. A numberof new techniques is proposed to implement the pos-itive as well as negative lensing element. Issues suchas the correct ROC estimation, higher-order losses,response time, and beam deviations are studied bothexperimentally and theoretically.
2. Theoretical Foundations
We begin by calculating the three-dimensional (3D)temperature distributions in optical elements. Whilea number of techniques are available to model thisbehavior, the basis of all these methods is the ther-mal diffusion equation. This equation can be solvedby both analytical and numerical finite-element-analysis methods.5–8 Some simpler methods basedupon geometrical considerations also exist but can bequite inaccurate for high-power applications.9,10
Hello and Vinet5,6 provided an analytical solutionfor the thermal aberrations in a finite cylinder due toabsorption from an optical beam. The model is gen-eral and can be applied to a large number of situa-tions; it is particularly suitable for thermal lensingcalculations in large-scale gravitational wave inter-ferometers. One specific limitation is in the case ofnonuniform absorption or point absorption, wherefinite-element-analysis methods become necessary.
We use a MATLAB-based implementation of the Helloand Vinet model11 as applicable to the geometry oflaser gravitational wave interferometers, shown inFig. 1. Light is incident on a cylindrical optical elementof thickness h and radius a with an optical coating onthe surfaces. The optical element is assumed to be invacuum where the only heat escape mechanism isthrough thermal radiation. The front surface has anantireflection coating at the probe beam wavelengthwhile the back surface has a high reflective coating.The optical beam has power P and wavelength �0. Theelement has coating (or surface) absorption ac and
substrate (or bulk) absorption coefficient �s (in m�1),refractive index n, thermo-optic coefficient dn�dT �K�1�,thermal conductivity �T �W m�1 K�1�, total emissivity�, and Poisson ratio �. The temperature distributionTs due to substrate heating throughout the materialand temperature distribution Tc due to surface heat-ing is given by11
Ts�r, z� � P�sa
2
kt�k
pk
�k2 � �1 � 2Ak cosh��k
za��J0��k
ra�,
(1)
Tc�r, z� � Pacakt
�k
pk � �Ak cosh��k
za�� Bk sinh��k
za��
� J0��k
ra�, (2)
where 4�Text3�a��t� with � as the Stefan–
Boltzmann constant. The coefficients Ak and Bk aregiven by
Ak �1
2�k sinh��k� � cosh��k��,
Bk �1
2�k cosh��k� � sinh��k��, (3)
where �k �k�h�2a�. The terms in these series arecharacterized by the roots of the equation
�J1��� � J0��� � 0. (4)
To increase the accuracy of our calculations, numer-ical techniques are used to find the exact roots of theequation instead of using the approximation �k � �k� 1�4� for k � 0, 1, 2, . . .� as done in Ref. 11. Theconstants pk are the coefficients of a Dini series5 ex-pansion of the incident intensity distribution I�r�.These coefficients can be calculated numerically
pk �2�k
2
��k2 � 2�J0
2��k�
1
a2 �0
a
rJ0��k
ra�I�r�dr. (5)
For example, a normalized TEM00 Gaussian beam, a“top-hat” beam, and their Dini series expansions aregiven in Fig. 2; good agreement is seen.
Once the temperature distribution is obtained, thesubstrate thermal aberrations can be calculated by
��r� 2
�0
dndT �
�h�2
h�2
T�r, z�dz. (6)
By using the temperature distributions calculatedearlier, the contributions of coating and substrate
Fig. 1. (Color online) Non-uniform-intensity beam incident on anoptical element. The power absorbed in the coating or the surfacecreates thermal aberrations on the surface due to the thermalexpansion coefficient �T and due to the dn�dT in the substrate.
Another effect of this nonuniform distribution is thechange in surface profile of the optical element.Here only the axial direction surface change istaken into account. The surface deformation uc dueto coating or surface absorption and us due to sub-strate absorption is11
uc�r, �h�2� � �acP�T�1 � v�a2
kT�k
pk
�k2 �
sinh��k��k
�� sinh��k��k � sinh��k�cosh��k�
� Ak�� �J0��k
ra�� 1�,
us�r, �h�2� � ��shP�T�1 � v�a2
kT�k
pk
�k2
� Ak cosh��k� � Bk sinh��k��
��J0��k
ra�� 1��
32 �shP
��T�1 � v�a2
kT�k
r2pk
a2�k2 Bk
� �sinh��k��k
� cosh��k��J1��k�. (8)
The surface deformation given by Eq. (8) combinedwith the original ROC gives the modified ROC of bothsides of the optical element. The solution presentedhere is a steady-state solution and therefore once aspecified amount of heat is deposited, the resultantthermal lens will retain its modal properties as longas the deposited heat distribution and amount re-mains same.
One notable limitation of this treatment is the useof the linearized radiative boundary condition. Theradiative heat loss of an element at temperature T isgiven by
F � �<T 4 � Text4=, (9)
where Text4 is the ambient temperature and �� is the
Stefan–Boltzmann constant corrected for emissivity. Ifthe temperature rise �T of the element is not exces-
Fig. 2. (Color online) Dini series representation of a top-hat beam and a Gaussian beam using the first 40 terms of the expansion.
Table 1. Nominal Values for Advanced LIGO Cavity Mirrors(Test Masses)
Property Unit Value
Diameter cm 34Thickness cm 30Material — FSHot ROC m 2076Beam size cm 6.0Power incident on HR side kW 850Power through substrate kW 2.1Refractive index at 1 �m — 1.45Bulk absorption at 1 �m parts per 106 cm�1 2Thermo-optic coefficient 10�6 K�1 8.7Thermal conductivity W m�1 K�1 1.37Heat capacity J kg�1 K�1 739Thermal expansion 10�6 K�1 0.55Density Kg m�3 22,010Poisson ratio — 0.17
sive, usually the case, Eq. (9) can be approximated as
F � 4�Text3�T. (10)
An ideal spherical lens has a quadratic variation ofphysical path length with radius; an ideal graded-index lens has a refractive index, which varies qua-dratically with radius. In contrast to this, thethermal aberrations may not be exactly quadratic.To pick a concrete example, we first model the ex-pected thermal lensing in the proposed advancedLIGO interferometer, specifically the arm-cavity in-put test mass.12 Table 1 gives the mirror and cavityparameters while Fig. 3 shows the substrate ther-mal lens for a typical coating absorbed power of0.5 W. It is evident that the induced substrate aber-rations, plotted as a solid curve, are not quadratic.Therefore, an approximation is required to includethe thermal lens in the cavity design. For homoge-neous absorption, the surface deformation or thethermal aberrations inside the substrate can be rep-resented by a higher-order polynomial:
s�x� � �m�0
M
Amxm. (11)
Instead of applying geometrical approximations orusing the coefficient of the quadratic term in Eq. (11)as a representative thermal lens,10 a general methodthat can be applied to estimate the ROC in thermal
lensing due to both surface deformation and heatingin the substrate is presented.
Consider an optical beam with a fundamentalGaussian TEM00 profile incident on an optical surfaceof ROC R1 with a minimum beam waist of w0 atz � 0, beam waist w�z� at a distance z from the origin,and phase front radius of curvature R�z� as shown inFig. 4. The incident electric field in the fundamentalGaussian beam is given by
E1�x, z� � �2 �1�4 1
�w�z�exp � x2� 1
w2�z�� i
�R�z��.
(12)
The actual reflected and transmitted beam from thesurface is given by
E2�x, z� � �2 �1�4 1
�w�z�exp � x2� 1
w2�z�� i
�R�z�
� i2
�R1� i
�
4s�x�x2 �. (13)
Here � ��1�4s�x��x2� is the optical phase differenceintroduced by the actual thermal lens. In case of re-flection, this term represents the surface deforma-tions while in the case of transmission, this termrepresents the sum of surface and substrate aberra-tions. Now, the electric field E3�x, z� reflected andtransmitted by the imaginary perfect thermal lens
Fig. 3. (Color online) Thermal aberration in the substrate mirror due to 0.5 ppm absorption in the coating for the case of Table 1 dataplotted as a solid curve. The approximation using the 12th degree polynomial is plotted as circles, and the optimal solution is plotted asa dotted–dashed curves.
where Aopt is the optimized value of the coefficientused to represent the thermal lens and is equal toAopt � 1�2Ropt and Ropt is the optimized ROC of thethermal lens. To get the optimal value of Aopt, theoverlap integral I between E2 and E3 is evaluated as
I�Aopt� ����
�
E2�x, z� � E3*�x, z�dx
� �2 �1�2 1
w�z� ���
�
exp � x2� 2
w2�z�� i
�
4s�x�x2
� i
�4Aopt�dx. (15)
The overlap integral of Eq. (15) represents the 1Damplitude coupling from the thermally aberrated re-flected beam in the case of Fig. 4(a) and transmittedbeam in the case of Fig. 4(b) into the fundamentalGaussian mode because E3 has the form of a purefundamental Gaussian mode. This perfect mode oc-curs because the thermal lens has been representedby a perfect spherical lens. The optimal value of Aopt
is obtained by maximizing the overlap integral I; Aopt
will yield the optimal spherical approximation to thethermal lensing effect. The above integral can bewritten as
I�A� � �2 �1�2 1
w�z� ���
�
exp � x2� 2
w2�z��� exp i� 4
�1s�x� �
�4Aoptx
2�dx. (16)
Next, using Euler’s identity, ei� � cos � � i sin � andnoting that the integral I is an even function of x, wewrite Eq. (16) as
Fig. 4. Geometry of the incident, reflected, and transmitted electric fields from an optical material with thermal lensing. In case ofreflection, the thermal aberrations consist of surface deformation as shown in (a) while in transmission, thermal deformations representthe sum of surface and substrate aberrations as depicted in (b).
This integral can be evaluated numerically and themaximum of the integral is used to determine theoptimal value of the ROC associated with the thermallensing. This is applicable to both the thermal lensingin substrate and on the surface.
Continuing with the example of the advanced LIGOinput test mass mirror, with the thermal aberrationsshown in Fig. 3, the overlap integral and correspond-ing mode mismatch are shown in Fig. 5. The maximumof this overlap integral occurs at a thermal ROC of 6.4km. The optimal quadratic solution is plotted in Fig. 3as a dotted–dashed curve. The optimal value of theintegral also gives the maximum value of the 1D am-plitude coupling coefficient; subtracting that amountfrom unity gives the 1D amplitude losses suffered bymode matching into an advanced LIGO optical cavity.The absolute losses increase as the optimal thermallensing increases with increase in coating or bulk ab-sorption, since as the thermal lensing increases, thedifference between a quadratic profile and the actuallens profile created by the heating beam becomeslarger and larger. This difference produces higher-order losses when the beam is either reflected or trans-mitted through such optics.
It is inevitable that when a high-power beam isreflected or transmitted by a mirror that is part ofsome Fabry–Perot cavity, a portion of the energy isdeposited as heat in the coatings, giving rise to aphysical deformation at the surface that changes theROC of that element and thus changing the funda-mental mode of that cavity.2,3,13 If the input beam isproperly mode matched to the cavity at low powers, it
becomes progressively more poorly matched at higherpowers. To keep the coupling into this cavity maximalone can introduce an adaptive mode-matching tele-scope to control the beam size and waist location. Forproper operation, a tunable optical element is re-quired that can provide both positive and negativelensing elements. Such a system can be developed byusing the technique proposed and demonstratedhere, i.e., by controlled heating of elements with spe-cific optical and physical properties.
3. Application to Adaptive Optical Elements
The phenomenon of thermal lensing can be used ad-vantageously for creating adaptive optical elements.This can be done by either direct or indirect compen-sation. In direct compensation, an auxiliary heatingbeam is applied to the same optical element that pro-duces the thermal distortion. In the case of indirectcompensation, the heating beam is applied to a com-pensation element inserted near the thermally aber-rated element to correct the thermal distortions.Control of the thermally induced optical profile can berealized by manipulating any of the following: (i) theintensity of the heating beam, (ii) the beam shape ofthe heating beam, (iii) the wavelength of the heatingbeam, or (iv) the properties of the compensation optics.
Typically, the intensity and the beam shape of theheating beam are the most easily adjustable adaptivecontrol knobs. Nonetheless, the heating beam wave-length and the properties of the optical element canalso greatly aid in designing an effective compensa-tion scheme; of course, their adaptive control may belimited for practical purposes. Proper selection ofthese two factors can greatly reduce the require-ments on the intensity and beam shape of the ele-ments. As the optical properties (absorption) of thematerial depend upon the wavelength, changing thewavelength provides another means of control for agiven material. For example, changing the heating
Fig. 5. The amplitude 1D coupling coefficient [Eq. (17)] for the case of substrate thermal lensing in LIGO test mass. The maximum occursat a thermal ROC of 6.4 km indicating the optimal ROC associated with the thermal aberrations.
beam wavelength to a more absorptive wavelengthrequires less heating or smaller thickness of theadaptive optical element for a given performance re-quirement.
A truly adaptive system will require both tunablepositive and negative lenses. A positive lens has aquadratically decreasing radial optical path lengthwhereas a negative lens has a quadratically increasingradial optical path length. As discussed above, heatingof an optical element produces primarily two effects,namely, surface deformation and substrate refractiveindex variation. Based upon the beam shape, the ma-terial characteristics of that element, and the opticalelement orientation, the combination of these factorscan produce either a positive or a negative lensingelement. A figure of merit (FOM), based upon the ma-terial properties of the element is
FOMM � �dndT � �T�1 � �� � �n � 1� � �T
n3
4�1 � ���1 � ��
� �p11 � p12��. (18)
Here p11 and p12 are the thermoelastic coefficients ofthe optical material specific for the [001] direction.14
FOMM depends upon the signs and values of dn�dT,�T, and thermoelastic coefficients that may vary ac-cording to the geometry and orientation of the opticalelement. Here the spatial dependence of the thermaldistortion is neglected in comparing the relativestrengths of the thermal expansion and thermo-optic
effects. If the value of FOMM is positive (negative),the optical element will act as a positive (negative)lens when heated by a traditional Gaussian pumpbeam with a radially symmetric, monotonically de-creasing intensity pattern. The situation can be re-versed by applying an “inverse” Gaussian intensitypattern or in fact, in general, by heating with anypump beam whose intensity increases radially. How-ever, this may require applying excessive heat ascompared with the Gaussian beam case for the sameamount of thermal lensing on account of the inwardflow of heat in the element.2,13 Therefore, using amaterial with negative FOMM is a better alternativeto the radial heating.
There are a few potential disadvantages with such asystem. The first are the higher-order losses owing tothe deviation of the thermal lens profile from the op-timal parabolic shape. Several solutions are available.A simple solution is to use a heating beam with adiameter larger than the probe beam. Then, aligningthe probe beam perfectly around the optical axis of theheating beam ensures that the probe beam only seesthe central portion of the thermal aberrations; theseare very close to a quadratic profile. This method wassuccessfully demonstrated in Ref. 4. However, increas-ing the pump beam diameter requires more power tocreate the same thermal lens. Another solution is touse a non-Gaussian shaped heating beam. For exam-ple, using a flat-top or “top-hat” beam as a heatingbeam will create a nearly ideal quadratic profile. Fig-ure 6 shows the computed thermal lens created by
Fig. 6. (Color online) Thermal lens created through top-hat and Gaussian beam plotted using dashed curve and solid curve, respectively.The dotted line is an ideal thermal lens of 6.4 km focal length.
Gaussian and a top-hat heating beams. The thermallens created by the top-hat beam mimics an ideal lensfor a much larger probe beam diameter; thus, thehigher-order losses for top-hat beam are lower than fora Gaussian beam with the same probe beam diameter.
A second potential problem is the thermal depolar-ization associated with the thermoelastic effect. Thecomplete theory of thermal depolarization is presentedin Ref. 14. For materials such as fused silica (FS), witha very low thermal expansion coefficient, we calculatethat the thermal depolarization is of the order of a fewtens of parts per 106. This effect becomes important forthe negative thermo-optic coefficient materials, whichusually have large thermoelastic coefficients. How-ever, this effect can be reduced significantly by select-ing appropriate orientations.
The third potential problem comes about from thespecific choice of implementation geometry. The reg-istry of the heating and probe beams on the compen-sation element can be performed either collinearlyusing dichroic beam combining optics or via irradia-tion by the heating and probe beams on the oppositesides of a reflective element. However, there may besituations where nonnormal incidence of the probebeam is unavoidable. In this case, the beam pointingwill change as a function of pump beam power due tothe induced index gradient. A first-order estimate ofthe beam deviation can be made by treating the ther-mal lens as a thick lens. A general ABCD matrix of athick lens is given by
�A BC D�� � 1 � �h�n��P1� h�n
��P1 � P1 � P1P2h�n� 1 � �h�n��P1��,(19)
where P1,2 � �n � 1��R1,2 with R1�2� being the ROC ofthe front (back) surface of the lens. The displacementand angular deviation of the transmitted and re-flected beam from the thermal lens can then be cal-culated by using ABCD propagation laws. If x1 �x2�
and �1 ��2� are the incident (final) beam displacementfrom the optical axis and the angle of incidence (re-fraction), respectively, then these quantities arerelated by
� x2
tan �2�� �A D
B C�� x1
tan �1�, (20)
where the ABCD matrix is given in Eq. (19). Notethat this model is only a first-order estimate of thebeam walking. More precise models can be developedby using beam propagation equations in inhomoge-neous anisotropic medium.
Another important issue to consider is the time re-sponse of the system. The Hello and Vinet theory sug-gests a characteristic response time of c � C�a2��,where c is the characteristic time and � is the densityof the medium.5,6 This formula is applicable to thesystems where the heating beam is the probe beamitself. In reality, the temperature evolution of theoptical element varies with the radial and axial loca-tion. For example, the temperature of the center ofthe optical element increases faster than the off-axislocations. In our case, the heating is due to an aux-iliary heating beam and the probe beam only samplesa very small portion of the heated volume. Thereforea reduced diameter �re� should be used to predict theresponse time. One way to determine this reducedradius is to measure the response time and infer thethermal coefficients from the results. However, intu-itively, the location of the probe beam should bewithin rc.
Furthermore, as the amount of probe beam dis-placement increases with increased heating-beampower, the response time should also increase. Here aparallel can be drawn with bolometer theory, wherethe response time also depends upon the externalheat dissipation.15 Therefore, the response time PBD
can be defined as
Fig. 7. Experimental arrangement of the adaptive optical system. SMF, single-mode fiber; L, lens; M, mirror; PD, photodetector; QPD,quad photodetector; GL, graded index lens; BS, beam splitter; and RC, scanning Fabry–Perot analyzer cavity.
Here � is a parameter that represents the effect ofauxiliary beam power absorption in units of K�1 m�1.For small values of P, Eq. (22) can be approxi-mated as
PBD �C�re
2
��
C�rc2
�2 P �C�re
2
�� �rP. (22)
Here re and �r can be determined by curve fitting aswe show in Section 4.
4. Experimental Demonstration
The experimental architecture is shown in Fig. 7. A 5cm diameter 0.95 cm thick UV-grade FS mirror isused as the adaptive compensation element. The FSmirror is mounted in a holder such that it rests onthree Teflon contacts. Hence negligible heat flowsthrough conductance. Although FS is in air but in thelimit of low temperature rise, the dominant heat es-cape mechanism is radiation. Relevant material pa-rameters for FS are shown in Table 1. For the probebeam, a 1.064 �m single longitudinal mode Nd:YAGlaser is used. The transverse mode of the probe beamis “cleaned” using a single-mode fiber. A pair of lensesis used to expand the beam. The beam diameter of theprobe beam at the front face of the FS mirror is1.8 mm. The test mirror is oriented such that thehigh-reflectance surface of the mirror is at the rearside with regard to the incident probe beam, allowing
us to take advantage of the integrated thermal opti-cal path deformation in the substrate in a double-pass geometry. The probe beam hits the test mirror atan angle of � � 3.0° with the normal. After reflectingfrom the mirror, the beam is focused by a 75 cm lensonto a pyroelectric beam scanner.16 The Gaussianparameters are measured at various positions in theoptical system to track the beam waist and locationthroughout the system.
A 10 W CO2 gas laser with a wavelength of 10.6 �mand a Gaussian profile is used as a heating beam. TheFS mirror substrate is essentially transparent for the1.064 �m wavelength probe beam, while the penetra-tion length at 10.6 �m is approximately 56 �m. Thus,all the energy in the heating beam is absorbed verynear to the surface. The heating beam is expanded to2.0 cm with a telephoto lens system, providing a heat-ing beam to probe beam diameter ratio of more than10, minimizing induced higher-order losses in theprobe beam.
The test plate was heated with the CO2 laser run-ning at various power levels, and a beam scan wasperformed on the probe beam after it had been re-flected from the mirror and passed through a focusinglens. Two sets of data were taken; one at 0, 1, 2, 3� Wand a second at 0, 0.5, 1.5, 2.5� W heating-beampower. The two different data sets correspond to twoslightly different beam scanning positions. The probebeam power is approximately 100 mW and has a neg-ligible effect on the thermal lensing in the test mirror.The beam divergence data are shown in Fig. 8 for the
Fig. 8. (Color online) Beam profile data and the corresponding Gaussian fit at 0, 1, 2, 3� W absorbed power.
first data set. The data points at various power levelsare represented by markers while a Gaussian beamfit is shown as solid curve. The back focal length, i.e.,the distance from the focusing lens to the minimumbeam waist position, is determined by the Gaussianfit parameters. Using this back focal length data andABCD matrices, the effective focal length of the testplate is calculated. Figure 9 shows the experimen-tally determined back focal length plotted on the leftaxis and the resultant focal length of the test plateplotted on the right y axis as a function of pump beampower.
The theoretical model presented earlier is used tocompute the focal length of the test plate at variouspower levels. It is important to note that there are noadjustable parameters; only the known optical andphysical parameters are used. The theoretical valueof the lens power (in diopter or m�1) as a function ofheating beam power is shown in Fig. 10 along withthe experimentally measured data. The two valuesare in excellent agreement. The slope of the curvegives a value of 0.082 diopter�W of pump beam.
The loss of power in higher-order modes can becalculated theoretically by evaluating the overlap in-tegral defined in Eq. (17) and subtracting it fromunity. The theoretically predicted power loss is shownin Fig. 11 as a function of absorbed power (left axis).The modeled ROC is also shown (right axis). Notethat the calculated decrease in power at 5 W is only0.015%, close to the measurement noise. Hence, itcan be assumed that the system has no appreciablehigher-order mode content.
Experimentally, this was verified by using a Fabry–Perot cavity to analyze both the tilt and higher-ordermode content.17 Thermally induced beam pointing isinferred from the loss in the fundamental mode andthe power transferred to higher-order TEM10,01 modesof an initially mode-matched cavity. We measuredthis using a combination of a photodiode and beamprofiling of the transmitted or reflected mode. Thealignment was optimized such that only the TEM00mode is resonant in the test cavity at zero heating-beam power. As the test mirror was heated by thepump beam, higher-order tilt modes are induced dueto the changes in the propagation and angle and dis-placement of incoming probe beam with respect to thecavity axis while higher-order Laguerre Gauss16
(mainly bull’s-eye) modes is introduced due to changein waist size and location of the probe beam inside thecavity. The coupling to the cavity is then reoptimizedby changing the tilt and the location of the cavity fordifferent pump beam powers. At all heating-beampower levels, we find that virtually the same amountof probe-beam power is stored in the cavity. The neg-ligible decrease in the resonant TEM00 mode afterincreasing the pump-beam power and realigning thecavity, indicates that no appreciable higher-ordermodes (beyond tilt modes induced by beam steeringand bull’s-eye mode caused by wavefront curvaturechanges) are produced by the adaptive lens over therange of experimental parameters explored.
We also attempted to measure the beam deflectionangle with a quadrant photodetector located 1 m fromthe compensation mirror. At even the highest heat-
Fig. 9. (Color online) Calculated and measured back focal distance as a function of absorbed power (left axis) and the corresponding focallength of the thermal lens (right axis) as a function of the absorbed power. The measured data are shown as symbols.
ing power, the beam deflection was experimentallyunresolved ��0.15 mrad�W�, so the angular beamdeviations were estimated by approximating theadaptive lens as a thick lens using Eq. (20). Numer-ically, an angular shift of 0.028 mrad�W was deter-mined. To enhance the amount of induced beamsteering, an incident angle of 18.5° was used, wherea substantial amount of tilt is expected. The probebeam is first centered on the quad photodetector(QPD) and the reading on the attached micrometeris noted. Next the auxiliary heating beam is turnedon, heating the test mirror. As a result, the probebeam suffers a displacement and the reading on theQPD is changed. The micrometer is then used torecenter the probe beam on the QPD. The change inmicrometer reading gives the direct measurementof probe-beam displacement. To confirm this mea-
surement, the heating beam is turned off and theprobe beam was observed to move in the oppositedirection as the test plate cooled to room tempera-ture. Again, the probe beam moved from the centerposition on the QPD and the micrometer was usedto recenter the QPD. An average of these two read-ings was used as the beam displacement caused bythermal lensing of the auxiliary heating beam.These data are shown in Fig. 12, and agree wellwith the predicted results using the thick lensapproximation. Curve fitting to Fig. 12 provides abeam deviation of 0.16 mrad�W. The cavity analyzerwas also used to measure the amount of power inhigher-order modes. At each power level, the cavitywas reoptimized by changing the x-axis and y-axis tiltof the beam through mirrors to minimize the inducedthermal steering and shifting the cavity along the
Fig. 10. (Color online) Measured values of lens power as a function of absorbed power is shown as symbols, with error bars. A linear fitto the data gives a slope of 0.082 diopter�W.
Fig. 11. (Color online) Prediction of higher-order losses in the experimental setup of Fig. 7 (left axis) and the corresponding focal length(right axis) as a function of absorbed power.
axial direction and�or changing the position of thefocusing lenses to remode-match the beam. Morethan 99% of the power can be maintained in the
resonant TEM00 mode by tilt and position control,showing that there is no appreciable higher-ordermode content beyond tilt.
Fig. 12. Deviation of the probe beam from its base position at room temperature as a function of absorbed power. The solid line is thetheoretical prediction using thick lens approximation. The slope gives 0.16 mrad�W deviation at 18.5° incidence angle of the probe beam.
Fig. 13. (Color online) Response time of the demonstrated system as a function of absorbed power. The slope is 2.715 s�W over anessential characteristic time delay of 52.916 s.
The inset in Fig. 12 shows data as a time series andthus gives valuable insight into the thermal con-stants or the response time of the system. The mea-sured response time m defined as the time requiredto reach 95% of the final value is plotted in Fig. 13. Alinear curve fit provides a value of m � 2.715P� 52.916 where P is the power of the auxiliary beam.Using the material properties in Table 1 and compar-ing the curve fit equation with Eq. (22) gives a re-duced radius of 2.1 mm, slightly larger than the offsetof the probe beam from the center of the optical ele-ment. This reduced radius is somewhat intuitivelyexpected. Thus the response time has a base valueof 53 s and �2.7 s additional delay per Watt of theheating beam power absorbed in this experiment. Ofcourse, these response times are specific to our par-ticular choices of experimental parameters and willchange for differing values for the probe radius andthe size of the adaptive compensation plate.
5. Conclusion
In conclusion, an adaptive system for control of first-order laser modal properties is presented that can beused to change dynamically the wavefront radius ofcurvature of an optical beam. Heating of the opticalelement can be used to create a positive or negativelensing element based upon the shape and the mate-rial properties of the adaptive optical element. Adetailed model for the thermal lensing is presented.A new technique based upon the overlap integralis presented for correct ROC estimation. A simplemodel for beam displacement is presented. The ther-mal response time has also been measured and foundto be close to a minute. Experimental results agreewell with theory. Applications include adaptive beamcoupling into high-power laser systems and thermallensing compensation.
The authors gratefully acknowledge the support ofthe National Science Foundation through grantsPHY-0555453 and PHY-0354999. We also thank theLIGO Science Collaboration internal review for acareful review of our manuscript.
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