Effects of thermal gradients on total internalreflection corner cubes

Scott D. Goodrow and Thomas W. Murphy, Jr.*Center for Astrophysics and Space Sciences, University of California, San Diego,

9500 Gilman Drive, MC-0424, La Jolla, California 92093-0424, USA

*Corresponding author: tmurphy@physics.ucsd.edu

Received 31 August 2012; revised 21 November 2012; accepted 23 November 2012;posted 26 November 2012 (Doc. ID 175367); published 20 December 2012

Uncoated corner cube retroreflectors (CCRs) operating via total internal reflection (TIR) are less suscep-tible to internal heating than their metal-coated analogs, lacking an absorber on the rear surface. Evenso, environments that induce differential heating within the CCR will result in thermal lensing of theincident wavefront, introducing aberrations that will generally reduce the central irradiance of thepolarization-sensitive far-field diffraction pattern (FFDP). In this paper, we characterize the sensitivityof TIR CCRs to axial and radial thermal gradients. We present simulated FFDPs for key input polariza-tions and incidence angles and provide a generalized analytic model that approximates the behavior ofthe central irradiance as temperature differences are introduced. © 2012 Optical Society of AmericaOCIS codes: 220.1010, 260.1960, 260.2710, 260.5430, 260.6970, 350.6830.

1. Introduction

Total internal reflection (TIR) corner cube retrore-flectors (CCRs) are often advantageous choices in thepresence of significant radiative flux—such as sun-light—due to their lack of absorptive coatings. Still,these optical devices can become subject to thermalgradients in the presence of dust deposition, surfaceabrasions, bulk absorption of light within the materi-al, and conduction from their mounting arrangement.Such differential heating will produce distortions ofthe emerging wavefront as a form of thermal lensingon their far-field diffraction pattern (FFDP). It isimportant then to understand the sensitivity of theFFDPs produced by TIR CCRs to perturbations fromthermal gradients. Our specific interest arises fromobservations of the degraded performance of the retro-reflector arrays placed on the Moon by the Apollo as-tronauts [1]. Although we concentrate on the impactof thermal lensing on TIR CCRs, many of our resultsapply to the metal-coated variety as well.

The literature contains a number of papers thatmention thermal sensitivity of TIR CCRs; however,none provide a full quantitative analysis. A contribu-tion to the final report on the Apollo 11 Laser RangingRetro-Reflector Experiment by Emslie and Strong(1971) [2] provides the closest quantitative handle,however lacks the computational power to give a com-plete view of how the FFDPs evolve as a function ofthermal gradient and angle of incidence. Faller (1972)[3] reports on the optimal dimensions for CCRs in ret-roreflector arrays, including thermal considerations,but does not detail their subsequent thermal lensingbehaviors. Zurasky (1976) [4] presents experimentalfindings for combined axial and thermal gradients inCCRs employing dihedral angles and evaluating fluxin an annulus well away from the central spot. Morerecently, in efforts to produce next-generation retrore-flector arrays, Currie et al. (2011) [5] mention the im-portance of minimizing thermal gradients to improvereturn signals. Dell’Agnello et al. (2011) [6] furtheremphasize the degradation of the return signal fromthermal lensing, but neither provide generalizedresults on the subject.

The goal of this paper is to provide a general char-acterization of the FFDPs produced by TIR CCRs

1559-128X/12/368793-07$15.00/0© 2012 Optical Society of America

20 December 2012 / Vol. 51, No. 36 / APPLIED OPTICS 8793

subjected to axial and radial thermal gradients, withapplication to our lunar ranging project [7]. Thesimulations herein build directly on our analysisframework for evaluating TIR CCRs (companionpaper [8]), and we make the computer code availableonline.

The quantitative results here target thermalconditions relevant to space environments, whileproviding a framework that can be applied to otherconditions. Additionally, we display thermally lensedFFDPs for a variety of key input polarizations andincidence angles. We detail the methodology andinclude analytic solutions to approximate the beha-vior of the central irradiance as thermal gradientsare introduced at normal incidence.

2. Method

While all paths through an isothermal CCR for agiven angle of incidence have the same geometricpath length, l, a solid glass CCR subject to thermalgradients will experience thermal expansion leadingto path-specific deviations, l0 � l�1� αδT�, where α isthe thermal expansion coefficient, δT is the depar-ture from some reference temperature, T0, and thebar represents a path average. Additionally, the aver-age wavelength within the medium, λ̄, can differ foreach path as the refractive index varies throughoutthe CCR according to the thermo-optic coefficient, β:

λ̄ � λ0n0 � βδT

; (1)

where λ0 is the vacuum wavelength and n0 is therefractive index at temperature T0.

Consequently, the emerging wavefront will havenonuniform phase shifts leading to thermal lensingof the FFDP. The absolute phase in radians can becalculated for an arbitrary path from its averagewavelength within the medium:

ϕ � 2πl0

λ̄� 2πl�1� αδT��n0 � βδT�

λ0: (2)

Expanding this and neglecting second-order termsin δT allows us to define a generalized thermalcoefficient η≡ αn0 � β and rewrite Eq. (2) as

ϕ � 2πlλ0

�n0 � ηδT�: (3)

As our analysis is aimed at the use of retroreflectorsin the lunar daylight environment, we are particu-larly interested in the sensitivity of fused silica CCRsto thermal gradients at temperatures in the range of300–350 K, which have a refractive index of n0 �1.46071 at our chosen wavelength of λ � 532 nm. Wetherefore use α ≈ 5 × 10−7 K−1 and β ≈ 10−5 K−1 in ouranalysis [2,9]. We see that thermal expansion plays aminor role in thermal lensing for Apollo CCRs and useη � 10−5 K in our analysis to provide easy scaling ofour results to other conditions.

The approximate cylindrical symmetry of the cir-cularly cut CCR suggests two primary modes ofthermal gradient distribution: axial and radial. Assuch, we modeled axial temperature differences,ΔTh, from the CCR’s front face to its vertex at theorigin and radial temperature differences,ΔTR, fromthe CCR’s center to its periphery as well as super-positions of the two. We use the geometry of theApollo CCRs, R � 19.05 mm and h � 29.80 mm, inour simulations and numerical results. As the radialdisplacement, r �

����������������x2 � y2

p, is independent of z, the

two gradients are orthogonal and can be computedseparately given the path-length-weighted averagesz̄ and r̄ of the ray path in question. The averagetemperature offset along the path is then

δT � ΔTh

hz̄�ΔTR

Rr̄: (4)

Both r̄ and z̄ have analytic expressions given thecoordinates of the end points of each path segment.We can determine the approximate peak-to-valleyphase difference of the wavefronts by calculating thedifference of the phase offsets for normally incidentpaths through the center (subscript c) and perimeter(subscript p) of the CCR under the assumption thatthe thermal lensing for both axial and radial thermalgradients will be dominated by the difference be-tween the central and peripheral paths. If the abso-lute difference in mean temperature between thesetwo paths is ΔδT � jδTc − δTpj, then the phase offsetdifference is

Δϕ � 2πlηΔδTλ0

: (5)

The central path has two path segments of equallength, from the front face to the vertex and back,making z̄c � h

2 ≈ 14.90 mm and r̄c � 0 mm. The peri-meter path is dependent on azimuth of entry, so wechoose the path that enters above an edge of theCCR. This path has three path segments of unequallength, leading to path-length-weighted averagesz̄p ≈ 8.83 mmand r̄p ≈ 12.59 mm. Thus, the perimeterray stays closer to the front surface and obviouslyspends time farther from the center than does thecentral ray. If we arbitrarily set ΔTh � ΔTR � 1 K,the differences of the average temperature offsets ofthe two paths are ΔδTaxial � 0.20 K and ΔδTradial �0.66 K. This yields peak-to-valley phase differencesofΔϕaxial � 1.43 radians andΔϕradial � 4.66 radians.In the radial gradient case, the wavefront perturba-tions vary with azimuth and our result correspondsto a minimum. The maximum, Δϕradial � 5.11 ra-dians, occurs for paths that enter or exit at an azimuth30° away from an edge.

We employed a ray tracing algorithm in Python totrack the phase offsets of each path through the CCR[8]. In the case of a metal-coated CCR—assumedto have perfect reflection at its rear surfaces—the

8794 APPLIED OPTICS / Vol. 51, No. 36 / 20 December 2012

phase of the input wave is shifted uniformly by π ra-dians at each interface. Consequently, the departureof the output wavefront from planarity deriving froma planar input wave can be used to characterize theeffects of the thermal gradient on the wavefront.Such is not the case for TIR CCRs, as TIR at the rearsurfaces induces phase shifts dependent on the angleof incidence. Here each of the six wedges within theaperture acquires a potentially different phase shiftin addition to the smooth perturbations from thethermal gradient. Three-dimensional representa-tions of these wavefronts are shown in Fig. 1 fornormal incidence.

3. Simulation Analysis

The orientation of our FFDPs represent directioncosines in the global frame, as specified in our com-panion work [8]. In this frame, the CCR is orientedto have a real edge extending toward the right.Horizontal and vertical polarization states are like-wise defined according to the same reference.

In order to visualize the evolution of the FFDPs atvarious angles of incidence, we present 4 × 5 grids inFigs. 2 and 3 that vary the incidence angle in steps of5° from left to right. Along the vertical axis, we showhow the FFDPs evolve with incrementally increasingtemperature differences. All irradiance values arenormalized to the central irradiance of the normalincidence isothermal frame in the top left.

Figure 2 shows FFDPs for right-handed circular,horizontal linear, and vertical linear input polariza-tions for the axial gradient case with the tempera-ture difference increasing in 1 K increments. Thepatterns for left-handed circular input polarizationare the same as that of right-handed except for a 180°rotation of each frame. Figure 3 depicts the FFDPsfor the radial gradient case with the temperaturedifference increasing in 0.5 K increments. We seeimmediately that the central irradiance is quicklydiminished when even modest thermal gradientsare present, exacerbated by off-normal viewing.

In Figs. 4 and 5, we present graphs of the centralirradiance as a function of temperature differencefor key incidence angles. Again, irradiance is normal-ized to the isothermal case. The solid curve in eachpanel corresponds to the response of a perfectlyreflecting (metal-coated) CCR for arbitrary inputpolarization. For TIR, both types of circular inputpolarization appear as the same dotted curve, anda gray envelope encompasses the range within whichall linear input polarizations fall. We find that, forisothermal TIR CCRs, the central irradiance is 26%that of the perfect reflector case at normal incidenceand that this proportion holds precisely as thermalgradients are introduced. This demonstrates thatthe temperature dependence of the central irradi-ance is independent of polarization state at normalincidence.

Fig. 1. Three-dimensional representations of wavefronts produced by a CCR subject to a thermal gradient at normal incidence. At left isthe perfect reflecting (metal-coated) case for an axial thermal gradient, resulting in a perfectly spherical wavefront. At center is the perfectreflector afflicted with a radial thermal gradient. At right is an example TIR case, depicting the combined effects of the TIR phase shiftsand the perturbations from an axial thermal gradient. If the wavefront travels upward, all panels correspond to the corner cube beingcooler on the front surface or on the periphery, although the results of this paper are insensitive to the sign of the thermal gradient.

Fig. 2. FFDPs produced by a TIR CCR with an axial thermal gradient. Right-handed circular input polarization is at left, followed byhorizontal and vertical linear input polarizations. In each grid, the incidence angle increases in 5° steps from left to right. Rows correspondto various front-to-vertex temperature differences, increasing in 1.0 K steps. Each frame is 50λ ∕7D radians across. Irradiance is normal-ized to the normal incidence isothermal frame in the top left. All results herein correspond to the behavior of fused silica at a wavelength of532 nm.

20 December 2012 / Vol. 51, No. 36 / APPLIED OPTICS 8795

Furthermore, at normal incidence, comparable ra-dial and axial thermal gradients of 0.079 Kmm−1

and 0.100 Kmm−1 reduce the central irradiance to15% that of the isothermal case, respectively. In theradial gradient case, this is the start of the slowlyvarying nearly linear feature that begins at atemperature difference of 1.5 K.

While differing materials and temperature re-gimes will have different thermal coefficients, thiswill only cause a linear scaling of our results due tothe linear dependence on η in Eq. (3). For example,we see in Fig. 4 that the central irradiance exhibitscomplete destructive interference when the axialtemperature difference is 4.4 K, corresponding to thespherical wavefront spanning exactly two Fresnelzones. If conditions double the thermal coefficientto η � 2 × 10−5 K−1, the extinction occurs insteadwhen the axial temperature difference is 2.2 K.

Likewise, the phase difference is inversely propor-tional to wavelength [Eq. (5)]. A nulling of thecentral irradiance at ΔTh ≈ 4.4 K at λ0 � 532 nmwould be pushed out to 5.2 K at a wavelength of633 nm.

A. Combined Axial and Radial Gradients

In Fig. 6 we present smooth curves of the centralirradiance in the presence of both axial and radialthermal gradients at normal incidence. As the axialtemperature difference increases, we hold the ratioΔTR ∕ΔTh constant. For example, the dotted–dashedline represents the case where the radial tempera-ture difference is half that of the axial temperaturedifference. Together, the two gradients work to re-duce the central irradiance at smaller temperaturedifferences. For reference, the solid line correspondsto a purely axial thermal gradient.

Fig. 3. FFDPs produced by a TIR CCR with a radial thermal gradient. Panels follow the conventions of Fig. 2 with the exception of thecenter-to-edge temperature difference increasing in steps of 0.5 K.

Fig. 4. Central irradiance of the FFDP as a function of axial temperature difference. The solid curve corresponds to a CCR employingperfectly reflecting (metalized) rear surfaces. For TIR, all rotations of linear input polarization fall within the gray envelope, and the dottedcurve corresponds to circular input polarization of either handedness. Incidence angle increases in 5° increments from left to right, andirradiance is normalized to the isothermal case in each panel. At normal incidence, the curves do not deviate from each other.

Fig. 5. Central irradiance of the FFDP as a function of radial temperature difference. Panels follow the conventions of Fig. 4. At normalincidence, the curves do not deviate from each other.

8796 APPLIED OPTICS / Vol. 51, No. 36 / 20 December 2012

B. Off-Center Irradiance

A prominent central lobe can be seen in the normalincidence cases of Figs. 2 and 3. The shape of this lobeclosely follows the Airy function in the isothermalcase, as explored in our companion work [8]. In Fig. 7,we show how the shape of the central feature evolvesas a result of thermal gradients by plotting envelopesencompassing the maximum variability of horizontaland vertical profiles of the central region. We haveselected temperature differences, ΔTh � 1.9 K andΔTR � 0.8 K, which individually drive the central ir-radiance to half that of the isothermal case. Includedis the perfect reflector (giving rise to the Airy func-tion in the isothermal case), right-handed circularinput polarization, and linear input polarization. Inthe linear case, the envelopes include variation dueto the angle of the input polarization as well. The pro-file of left-handed circular polarization is the sameas that of right-handed except for a left–right flip.

From this we learn that all normal incidence casesmimic the Airy function near the center of the FFDP,to varying degrees, and that thermal gradients donot cause immediate departure from this similarity.

4. Analytic Model of Central Irradiance

Having seen how similar the central region is to thatof the Airy function, we can confine our interest tothe central irradiance, knowing that we can extendthis result to points near the center of the diffractionpattern for modest temperature gradients. By doingso, we can analytically produce useful approxima-tions for the normal incidence case.

As outlined in our companion work [8], the FFDPcan be calculated from the Fourier transform ofthe complex amplitude and phase of the electric fieldexiting the CCR, with the square magnitude repre-senting the irradiance in the far field. A simpler ex-pression may be employed to investigate the centralvalue of the FFDP if one neglects off-axis field points:

I�0; 0� �����Z Z

apertureS�r; θ� exp�iϕ�r; θ��rdrdθ

����2: (6)

For TIR CCRs subject to thermal gradients, thephase function ϕ�r; θ� can be represented as thesuperposition of the set of TIR static phase shifts,fϕnjn∶1 → 6g, and the thermal lensing perturba-tions, ϕΔT�r; θ�, so that ϕ�r; θ� � ϕn � ϕΔT�r; θ�—asdepicted in the rightmost panel of Fig. 1. Addition-ally, as thermal gradients do not affect the amplitudeof the electric field, the amplitude function isdescribed entirely by the effects of TIR, makingS�r; θ� � Sn. Thus,

I�0; 0� �����Z

2π

0dθ

ZR

0rdrSn exp�iϕn� exp�iϕΔT�r; θ��

����2:

(7)

Since ϕn and Sn are constant within each wedge,we can pull them out of the integrals and sum overthe six wedges:

Fig. 6. Central irradiance of the FFDP of a TIR CCR subject toboth axial and radial thermal gradients, at normal incidence. Asthe axial temperature difference increases, the radial temperaturedifference follows proportionally. Irradiance is normalized to theisothermal case.

Fig. 7. Irradiance profiles of the central lobe of the FFDPs of TIR CCRs subject to thermal gradients at normal incidence. Temperaturedifferences correspond to those required to reduce the central irradiance to half that of the isothermal case. The perfect reflector (metal-coated CCR) is at left, followed by TIR cases given right-handed circular and linear input polarizations. Envelopes encompass maximumvariability in horizontal and vertical profiles, and for all rotations of the input polarization in the linear case. The isothermal case in the leftpanel corresponds to the Airy function.

20 December 2012 / Vol. 51, No. 36 / APPLIED OPTICS 8797

I�0;0�

�����X6n�1

�Sn exp�iϕn�

Z π3n

π3�n−1�

dθZ

R

0rdr exp�iϕΔT�r;θ��

�����2:

(8)

In Fig. 1, we saw that the thermal lensing pertur-bations are symmetric across each of the six wedges,allowing us to separate the integrals from the summa-tion and integrate only over the first wedge, leaving

I�0;0������X6n�1

fSn exp�iϕn�gZ π

3

0dθ

ZR

0rdrexp�iϕΔT�r;θ��

����2

:

(9)

In the absence of thermal gradients, ϕΔT�r; θ� � 0and the integral yields πR2 ∕6. For a perfectlyreflecting CCR, the summation contributes a factorof 6 as ϕn does not vary and Sn � 1; therefore,I�0; 0� � π2R4. The phases, ϕn, introduced by TIRand detailed in [8], result in I�0; 0� � 0.264π2R4 fora fused silica corner cube, making the central irradi-ance 26.4% that of the Airy function. Note that theintegrals and summation are now separable so thatthe normal incidence TIR response for nonzero ther-mal phase functions (displaying sixfold symmetry) is

ITIR�0; 0� � 9.505����Z

π ∕3

0dθ

ZR

0rdr exp�iϕΔT�r; θ��

����2:

(10)

For the axial gradient case, we approximate thespherical wavefront as a paraboloid centered at theorigin: ϕΔT�r� � kr2. We constrain this paraboloid tohave a phase offset of ϕR � kR2 at the perimeter—which we will later relate to the thermal gradient—and find that

ITIR�0; 0� � 0.528π2R4 1 − cos�ϕR�ϕ2R

: (11)

Similarly, we approximate the radial gradientcase’s pointed wavefront as a cone, ϕΔT � kr. Apply-ing a similar constraint, ϕR � kR, we get

ITIR�0; 0� � 1.056π2R4 ϕ2R − 2 cos�ϕR� − 2 sin�ϕR� � 2

ϕ4R

:

(12)

A comparison of these models with our simulationresults is shown in Fig. 8, with the axial thermalgradient model at left and the radial thermal gradi-ent model at right. The axial gradient result is instrong agreement with that reported by Emslie andStrong [2], which has the same functional form andextinction point when appropriately scaled for thedifference in wavelength.

Furthermore, we can combine Eqs. (4) and (5) todetermine the phase offset ϕR at the perimeter of theperturbed wavefront. The result simplifies greatlywhen we confine our interest to CCRs without a rimpad. In the axial gradient case, this phase offset cor-responds to the uniform perimeter of the paraboloidin Fig. 1:

ϕR ����2

p πηΔThRλ0

: �13�

The nonuniformity of the perimeter in the radialgradient case obscures an obvious scale for the cone.This variability is shown in Fig. 9 as a gray envelope.Here we explore two cases. In the first case, we usethe average of the extrema along the perimeter asthe edge phase offset (solid). For the second case, wechoose a scale that forms a cone with an equalamount of overestimation and underestimation ofthe actual area-weighted wavefront (dashed). Bothmodels are represented in Fig. 8. The phase offsetof the cone that extends to the average of the extremais then

ϕR � 3.737πηΔTRR

λ0: (14)

Fig. 8. Comparison of the central irradiance of simulated FFDPs (points) and the prescribed analytic models (curves), with axial thermalgradients at left and radial thermal gradients at right, for normal incidence. The axial wavefront is modeled as a paraboloid and the radialwavefront is modeled as a cone. Irradiance is normalized to the isothermal case.

8798 APPLIED OPTICS / Vol. 51, No. 36 / 20 December 2012

For CCRs with a rim pad, the scale of the perturbedwavefronts changes slightly. If we consider the rimpad size as a fraction of the radius of the CCR, in theaxial gradient case, the rim pad depreciates thevalue of ϕR by roughly 7% for every tenth of the ra-dius. In the radial gradient case, the rim pad appreci-ates the scale of the rim pad by about 12% for everytenth of the radius. For example, the Apollo CCRswe are modeling have a rim pad of 2.9 mm, or 15%of their radius. Here we see a depreciation of the axialwavefront perturbation by 10% and an appreciation ofthe radial wavefront perturbation by 17%. Applyingthese corrections brings the results of Eqs. (13) and(14) in line with the numbers reported in Section 2for the Apollo corner cubes.

5. Conclusions

The introduction of axial or radial thermal gradientswithin CCRs amounting to only a few degreesdiminishes the central irradiance of the FFDP by anorder of magnitude. The flux in the Airy-like centrallobe of the FFDP is well represented by the centralirradiance. This result is qualitatively independentof input polarization state, angle of incidence, or re-flector type. Together, minor thermal gradients alongboth principle axes drive the central irradiance to

near extinction regardless of input polarization andangle of incidence.

We provide generalized analytic models thatapproximate the behavior of the central irradiancesubject to axial or radial thermal gradients at normalincidence, enabling easy determination of the sensi-tivity of TIR CCRs in differing conditions. Thesemodels adequately capture the simulation results aswell as early models of the Apollo retroreflector ar-rays and provide quantitative guidance on tolerablethermal gradients in next-generation retroreflectors.The Python code used to generate these results maybe found at [10].

Part of this work was funded by the NASA LunarScience Institute as part of the LUNAR consortium(NNA09DB30A), and part was funded by theNational Science Foundation (grant PHY-0602507).

References1. T. W.Murphy, Jr., E. G. Adelberger, J. B. R. Battat, C. D. Hoyle,

R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs,and H. E. Swanson, “Long-term degradation of optical deviceson the Moon,” Icarus 208, 31–35 (2010).

2. A. G. Emslie and P. F. Strong, “Optical distortion caused by avertical temperature gradient in a retro-reflector,” ApolloLaser Ranging Retro-Reflector Experiment, SO78, final report(1971), pp. 272–286, http://hdl.handle.net/2060/19710011991.

3. J. E. Faller, “The Apollo retroreflector arrays and a newmulti-lensed receiver telescope,” in Space Research XII (Akademie-Verlag, 1972), pp. 235–246.

4. J. L. Zurasky, “Cube corner retroreflector test and analysis,”Appl. Opt. 15, 445–452 (1976).

5. D. G. Currie, S. Dell’Agnello, and G. O. Delle Monache, “A lunarlaser ranging retroreflector array for the 21st century,” ActaAstronaut. 68, 667–680 (2011).

6. S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori,C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N.Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G.Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O.Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W.Zagwodzki, “Creation of the new industry-standard space testof laser retroreflectors for the GNSS and LAGEOS,” Adv.Space Res. 47, 822–842 (2011).

7. T. W. Murphy, Jr., E. G. Adelberger, J. B. R. Battat, L. N. Carey,C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E.Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, andE. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detec-tions,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).

8. T.W.Murphy, Jr. and S. D. Goodrow, “Polarization and far-fielddiffraction patterns of total internal reflection corner cubes,”Appl. Opt., doc. id 175366 (to be published).

9. Haraeus Qurzglas, “Quartz glass for optics data and proper-ties,” (2011), http://optics.heraeus‑quarzglas.com.

10. http://physics.ucsd.edu/tmurphy/papers/ccr‑sim/ccr‑sim.html.

Fig. 9. Profiles of the radial thermal gradient phase perturba-tions and models. The gray envelope encompasses the variabilityof the wavefront perturbations. The solid line corresponds to thecone whose phase offset is the average of the extrema at the peri-meter, and the dashed line represents the cone which equally over-estimates and underestimates the area-weighted wavefront.

20 December 2012 / Vol. 51, No. 36 / APPLIED OPTICS 8799