the great wedge: quantifying chromatic aberration in ...dent aberrations. our aim is to assess many...

The great wedge: Quantifying chromatic aberration in imaging spectroscopysystems and application to temperature measurements in thelaser-heated diamond anvil cell

Laura Robin Benedetti,1,a! Daniel L. Farber,1 and Abby Kavner21Lawrence Livermore National Laboratory, 7000 East Ave., L-396, Livermore, California 94550, USA2Department of Earth and Space Science and Institute of Geophysics and Planetary Physics,University of California, Los Angeles, California 90095, USA

!Received 7 April 2008; accepted 29 October 2008; published online 27 January 2009"

To aid in evaluating spatial and spectral imaging abilities of any imaging spectroradiometer system,we developed a spectral intensity gradient standard based on the behavior of a birefringent wedgeimaged between cross polarizers. By comparing calculated with observed images of the wedge, achromatic scrambling kernel was measured to generally estimate chromatic aberrations in anyspectral imaging optical system. This technique provides a quantitative method to compare spectralimaging quality of different optical systems and also provides a quick test for severe misalignmentsin the optical path. Applying this method to the spectroradiometric measurement of temperature andtemperature gradients in the laser-heated diamond cell, the observed scrambling kernel is used toinfer original hotspot information from measured behavior, to provide a quantitative evaluation ofthe ability to measure a temperature gradient in any spectral system, and to yield an objectivedetermination of precision of spectroradiometric temperature measurements. The birefringent wedgemethod and its application described in this paper are simple and inexpensive enough to be used onany spectroradiometric system. © 2009 American Institute of Physics. #DOI: 10.1063/1.3041632$

I. INTRODUCTION

Measurements of material properties at high pressuresand temperatures are required to understand the thermody-namic and elastic properties of materials and to interpret thebehavior of the deep interiors of planets. The laser-heateddiamond anvil cell !LHDAC" is uniquely well suited formeasuring material properties under static high-pressure,high-temperature conditions. This technique can achievehigh-pressure !20 GPa! P! %200 GPa" and high-temperature !1500 K!T! %5000 K" conditions for ex-tended periods of time, from seconds to days, by compress-ing material between two optical quality diamond anvils andintroducing an infrared laser into the sample chamber. Theinterpretation of measurements of material properties at theseextreme conditions depends on the ability of the optical sys-tem to measure the temperatures in the LHDAC by spectro-radiometric methods. In this paper we present a method toevaluate an optical system for suitability to measure tem-peratures in the LHDAC. Our method, based on the opticalproperties of a birefringent wedge, characterizes the com-bined spectral and spatial resolution properties of any spec-tral imaging system. From this measurement we further de-rive an experimental confidence band on temperaturemeasurements in the LHDAC.

For laser heating in the diamond anvil cell, the largevolume and high thermal conductivity of the diamond anvilsrequires a room temperature boundary condition just tens ofmicrons away from the peak temperatures of the hotspot,which generally are thousands of degrees. This introduces

steep temperature gradients of hundreds of K /"m. If thesetemperature gradients are not properly measured and inter-preted, they compromise measurement of physical propertiesin the LHDAC. In addition, the presence of temperature gra-dients combined with refracting optics !including the dia-mond anvil" diminishes the accuracy of the hotspot tempera-ture and temperature gradient measurement.1 Recently,systems designed to measure temperature gradients as wellas temperatures in the LHDAC have been developed.2,3

These systems and associated approaches, such as reducingthe numerical aperture of the optical system1,4 or collectingand separately focusing only a few wavelengths,3 must beevaluated by a reproducible, laboratory-invariant method.

As with any quantitative experimental technique, a mea-surement is only as good as the ability to quantify measuredvalues as well as their uncertainties. Unfortunately, tempera-ture gradients have proven difficult to measure. Chromaticaberration within spectral imaging systems has been blamedfor “scrambling” the spatial-spectral data.4 If, as a result ofchromatic scrambling, an optical spectroradiometer cannotresolve spatial variation in intensity at the approximately mi-cron scale for all wavelengths, then measurements of tem-perature will not be reliable if temperature varies over a simi-lar length scale. While it may seem self-evident thatmeasurements of temperature gradient or profile will not bereliable, even measurements of peak temperature will be in-accurate in the presence of chromatic aberration induced lossof spatial resolution.

For the purposes of this work, we define chromatic ab-erration as any wavelength-dependent behavior in the opticalpath. Admittedly, this definition is broader than the usualdefinition of wavelength-dependent index of refraction ef-a"Electronic mail: [email protected].

JOURNAL OF APPLIED PHYSICS 105, 023517 !2009"

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http://dx.doi.org/10.1063/1.3041632

http://dx.doi.org/10.1063/1.3041632

http://dx.doi.org/10.1063/1.3041632

http://dx.doi.org/10.1063/1.3041632

fects. However, in a real optical system, optical aberrationsarise from a variety of sources, including imperfections inthe lens shape and dispersion. Aberrations from these sourcescan be compounded by misalignments in the optical system.In addition, vignetting can also contribute spectrally depen-dent aberrations. Our aim is to assess many of these sourcesof chromatic bias in a single measurement, and this methodmeasures the total effect of all contributions to chromaticaberrations.

Spectroradiometers for laser heating are calibrated usinga lamp whose wavelength-dependent intensity is known !andusually traceable to NIST standards". As these lamps are of-ten based on the thermal emission of a tungsten filament at aknown temperature, this calibration method provides a directvalidation of temperature measurements that are subse-quently performed by measuring wavelength-dependent in-tensity of a sample at unknown temperature. The methodcould be extended to the calibration of temperature gradientmeasurements by using a thermally emitting standard lampthat not only has a known temperature but also a knowntemperature variation with position !a “calibrated tempera-ture gradient source”". In fact, the notion of a calibrated tem-perature gradient source has been a subject contemplated byresearchers for over a decade !e.g., GSECARS/COMPRESHigh-Pressure Workshop “Future Directions for the Laser-Heated Diamond Anvil Cell at the Advanced PhotonSource,” 2004". Our intention at the origin of this projectwas simply to design and build a light source that wouldproduce a temperature gradient by means of a current flow-ing through a tapered filament. However, the engineeringrequirement to provide a stable and well-calibrated tempera-ture gradient of thousands of kelvins over %10 "m lengthscales—in order to match the relevant scales of the extremethermal situation in the LHDAC—was difficult to achievebecause the thermal conductivity of most metals is too highto permit significant temperature gradients for any machin-able geometry.

The critical characteristic of a “temperature gradientstandard” is that it mimics the salient attributes of the un-known hotspots: large intensity gradients !approximately oneorder of magnitude" over a small spatial scale !%10 "m"and significant chromatic variation. Therefore, we developeda calibration source based on the spectral-spatial image char-acteristic of a birefringent wedge placed between twocrossed polarizers. This optical technique bypasses the ob-stacle of thermal conductivity by taking advantage of con-structive and destructive interference effects that producevariations in intensity over %10 nm in wavelength and%10 "m in distance when a highly birefringent wedge isilluminated between crossed polarizers.

In Secs. II and III, we describe the theory and applica-tion of our spectral calibration method, including the equa-tions governing its spatiospectral intensity behavior. Our ap-proach combines forward modeling of idealizedspectroradiometry systems with experimental measurementson a real system. We present calculated spectral images withand without chromatic aberrations. Then, in Sec. IV wepresent results showing experimental measurements of inten-sity transmitted through the wedge with different optical con-

figurations. Finally, in Sec. IV the difference between mea-surements and simulations is used to quantify the spatial-spectral aberrations of our temperature measurement system,and in Sec. V these results are used to calculate precisionerrors in temperature measurement in LHDAC experiments.We show that this optical technique is not limited in use tospectroradiometers designed for laser heating in the diamondanvil cell; it can be used to evaluate and calibrate sphericaland chromatic aberrations in any imaging spectroscopy ap-plication.

II. WEDGE METHOD OVERVIEW

Birefringence is an optical property in which the indexof refraction is not identical along all crystallographic axes.Consequently, light that is polarized parallel to one axis trav-els through the crystal at a different rate and is refracted to adifferent angle than light that is polarized parallel to anotheraxis. The best-known example of this is the doubling of im-ages traveling through calcite. The use of uniaxially birefrin-gent wedges in optical systems has been demonstrated to beuseful in system design !e.g., isolation5 and smoothing6" andalso component characterization !e.g., birefringence,7

polarization,8 and surface slope and strain9". Here, we showfurther utility of the birefringent wedge as we employ it toexamine spatiospectral resolution in a spectroradiometric op-tical system designed to generate and measure high tempera-tures and associated temperature gradients in the LHDAC.

The retardation, #, of the slow branch of light relative tothe fast is the product of the crystal thickness, t, and thebirefringence, $, the difference between the ordinary indexof refraction, n%, and the extraordinary index of refraction,n&: #= t$= t!n&!n%". When a uniaxially birefringent materialis placed between crossed polarizers, the two branches de-structively interfere when the retardation is !2n+1"' /2. Forany thickness of a birefringent crystal some wavelengths oflight are extinct. Therefore, a crystal with a smoothly varyingthickness #e.g., a wedge, see Fig. 1!a"$ oriented so that theprincipal axes of the indicatrix are perpendicular to an in-coming white light source exhibits interference fringing inboth the spatial and wavelength dimensions. When imageddirectly in white light, this effect is illustrated by the Michel–

FIG. 1. !Color online" Schematic of experimental setup. !a" The YVO4wedge is placed so that the front face is perpendicular to the optic axis. Thewedge is cleaved such that the major and minor axes of the indicatrix arealigned with the axis of increasing wedge thickness !“wedge axis”" Theoptics cluster: !b" light source, !c" diffuser, !d" polarizer, !e" YVO4 wedgewith wedge angle (, !f" diamond optic !in some experiments", and !g" ana-lyzer are used to project the interference pattern onto the spectrometer. They-axis is defined by !h", the optical path from the light source to the spec-trometer, while the z-axis is defined by the slit of the spectrometer and isintended to be identical to the wedge axis. Note that the optical path !h" alsorepresents the optical system to be characterized.

023517-2 Benedetti, Farber, and Kavner J. Appl. Phys. 105, 023517 "2009!


Lévy chart.10 When light is dispersed, as in a spectrometer,the sinusoidal quality of the interference at each wavelengthbecomes apparent. Neighboring fringes will have a peak-to-peak retardation difference equal to the wavelength of light,'=#1!#2=$!t1! t2". With a wedge angle (, the difference inthickness over an in-plane distance, d, is d tan (. So theequation governing the length scale between neighboringfringes is '=d$ tan (.

We designed an interference wedge consisting of ahighly birefringent material cut so that the major and minoraxes of the indicatrix lie in the plane of and parallel to theaxis of increasing wedge thickness. Two requirements gov-erned our choice of material: neighboring extinction distancenear the spatial length scale of a laser-heated hotspot!%10 "m" and a wedge angle small enough to prevent sig-nificant path deviation !Fig. 2". We chose an YVO4 wedgebecause its birefringence !+0.24" is higher than many othercommon birefringent materials. Furthermore, the wavelengthdependence of its birefringence is known.11,12 Our YVO4wedge has a wedge angle of 6° and is %100 "m thick at thethinnest point.

When this wedge is illuminated between crossed polar-izers, an interference pattern is produced and the ideal inten-sity as a function of wavelength and distance along thewedge can be calculated exactly. Differences between ob-served and predicted interference patterns indicate the spec-trally dependent spatial precision in optical measurements.From this information, the associated precision of tempera-ture and temperature gradient measurements can be deter-mined. In Sec. III we describe the calculations of idealwedge behavior.

III. MODEL CALCULATIONS

The spatially and spectrally varying light intensity, I!'",that passes through the birefringent wedge and polarizers canbe calculated using the following equation:13

I = I0#1 + cos 2) cos 2* + sin 2) sin 2* cos &$ , !1"

where I0 is the spectral intensity of the light source, ) and *are the angles between a reference position and the polarizerand analyzer, respectively, and & is the fractional phase dif-ference between the slow and fast paths. When the anglebetween the polarizer and the analyzer is constrained to be90°, this reduces to

I = I0!#1 ! cos &$ = I0!&1 ! cos2+$t

'' . !2"

As observed at an imaging spectrometer, this function variesboth in wavelength, ' !the horizontal axis of the spectralimage data" and position !the vertical axis of the spectralimage data". The wedge thickness, t, is related to the spec-trometer row variable, z, by t= t0+M tan ( cos ,z, where t0is the thickness of the wedge at z=0 !the edge of the spec-trometer"; M is the magnification of the optical system; ( isthe angle by which the thickness of the wedge increases; and, is the rotation angle of the wedge relative to the spectrom-eter slit !which is intended to be 0".

We use Eq. !2" to calculate spectral intensity profiles!Fig. 3" using parameters that mimic our experimental setup,including the wavelength-dependent birefringence ofYVO4,11,12 the angle and initial thickness of our wedge, thewavelength range of our spectrometer, and the image size ofour charge coupled device !CCD" array detector. Horizontaland vertical profiles show that the interference pattern is apure cosine in z, but a modulated cosine in wavelength. Theeffect of increasing the thickness of the wedge is to decreasethe horizontal !wavelength" distance between intensityminima !fringes" without affecting the vertical !constantwavelength" fringe distance.

Figure 3 depicts an idealized observation with no opticalaliasing !i.e., with no spatial or chromatic aberrations" and istherefore the baseline for a perfect optical system. The effectof chromatic aberration is to limit resolution in awavelength-dependent fashion since not all light is focusedequally well at the spectrometer entrance. To model the ef-fects of a chromatic aberration, we introduce a wavelength-dependent “scrambling kernel,” r!'", with dimensions oflength. It can be interpreted in two mathematically identicalways: as a length scale over which light emitted from a pointsource is smeared at the detector and as the radial distancefrom which light is observed at a single position !x ,z" on thedetector. In Appendix A, we show that the predicted effect ofa scrambling kernel is to reduce the contrast of thebirefringence-induced intensity oscillations. The direct trans-mission through the wedge and polarizers from Eq. !2" isaliased by a scrambling kernel to

I!r,',z" = I0!&1 ! cos(2+$

't)

-* 4++

v=0

1,1 ! v2#cos 2+crv$dv-' , !3"

where c= #$!'" /'$M tan ( cos ,, an observable parameter ofthe optical system and wedge having aggregate dimension!1/length". The integral in Eq. !3" is numerically solvable to

FIG. 2. Designing the birefringent wedge. The distance between interfer-ence oscillations depends on birefringence, wedge angle, and wavelength oflight as d=' /$ tan (. Here contours of fringe distance !in microns" depen-dence on birefringence and wedge angle are shown for green light !'=600 nm". Our wedge exhibits an interference distance of 25 "m at thiswavelength.



a simple function of the dimensionless quantity cr that de-creases monotonically from 1 at cr=0 to 0 at cr=0.6 !Fig.4". Thus the aggregate effect of chromatic aberrations is toreduce the amplitude of interference !i.e., contrast". Any ob-served loss of contrast in interference fringes can be relateddirectly to a wavelength-dependent distance over which thespatial intensity information is scrambled. Once c is mea-sured, then the observed amplitude of interference oscillationdetermines r directly.

We can use a simple optical system as a model to esti-mate the magnitude of loss of contrast that we might expectdue to chromatic aberrations. We calculated the chromaticaberration effect of a highly dispersive diamond window di-rectly behind the wedge #Fig. 1!f"$ for two simplified opticalsystems: one with a narrow numerical aperture !NA=0.03"and one with a wide aperture !NA=0.34", similar to the op-tical systems examined in Ref. 1. As shown in Fig. 5, thecalculated chromatic effect is severe, with a large depen-dence on system numerical aperture. In these models an ar-

bitrary focal point was chosen such that paraxial light at '=850 nm was focused: when red light is selectively focusedfringe contrast loss is significant for less-well-focused bluelight.

IV. EXPERIMENTAL PROCEDURE AND RESULTS

To compare the results from a real optical system withthe calculations shown in Sec. III, we place a birefringentwedge in the object plane of a microspectroradiometer sys-tem, illuminate it with divergent light, place it betweencrossed polarizers, and measure its spectral intensity using animaging spectroradiometer. The experimental configurationis shown in Fig. 1, and further details of the optical compo-nents are listed in Table I. The optical path used is a spec-troradiometric measurement system designed to measuretemperatures in the LHDAC consisting of a microscope ob-jective, two optical elements designed to reject the 1064 nmlaser light and to transmit visible light, and two pelliclebeamsplitters. A collimated light source incident on a dif-fuser was used to illuminate the polarizer and birefringentwedge in transmission. The diffuser was engineered to mimicthe Lambertian quality of blackbody radiation, although dueto geometric constraints, the diffuser was placed %1 cm be-hind the wedge, limiting the emitted angle range of thesource light to be slightly less than that of a hot sample. Weused Polarcor polarizers for their superior broadband perfor-mance, transmitting more at blue wavelengths than manyplastic polarizers. The birefringent wedge was mounted on astage allowing for translation in two dimensions perpendicu-lar to the optical axis. The analyzer was placed at the end ofthe optical path in front of the spectrometer slit, but all otheroptical elements in Fig. 1 are before the Mitutoyo objective.The spectral image data were collected by an Acton Research150 mm spectrograph equipped with a 300 g/mm grating,and imaged onto a PiXIS 100 CCD. Data were collected in

FIG. 3. !Color online" Calculated spectrograph images of birefringence-induced interference patterns as predicted for our experimental conditions using Eq.!2". Each image represents 30 !vertical" "m imaged by our CCD for two initial thicknesses: 110 and 250 "m. Projections of the horizontal and vertical slicesindicated by dashed lines are shown above and to the right, respectively, of the spectrograph images.

FIG. 4. Numerical solution to the integral factor.!4 /+"/v=0

1 ,1!v2#cos 2+crv$dv0 in Eqs. !3" and !A7" as a function of theparameters c and r. Chromatic aberrations act to reduce the amplitude ofinterference oscillations from the birefringent wedge by this factor that de-pends on r!'", the scrambling kernel, and c!'", the inverse of the period ofoscillation. c is entirely determined by the design characteristics of the op-tical system so interference contrast loss can be directly linked to r!'".



imaging mode to yield intensity as a function of both wave-length !horizontal axis on the CCD" and distance along thewedge axis !vertical on the CCD chip". In the imaging direc-tion, the magnification is %73x, with each pixel covering%0.27 "m in the spatial direction. The spectral resolution is0.380 nm/pixel. We collected several images as function ofwedge thickness, a technique that mitigated the difficulty inpositioning the wedge so that its thickness is preciselyknown. Finally, we adjusted the focus of each image to maxi-mize the fringe contrast on the CCD. This procedure ensuresthat the front surface of the wedge is maintained in the objectplane of the optical !microscope" system.

The data analysis has four components and is describedin detail in Appendix B: !1" normalization, !2" fitting, !3"evaluation and consistency check, and !4" determination of

the scrambling kernel, r!'". This normalization step is criti-cal since the scrambling kernel depends almost entirely onintensity. We then quantify the total chromatic bias of oursystem by comparing our measured output data with the pre-diction. The result of the data analysis procedure described inAppendix B is a wavelength-dependent scrambling kernel,r!'", with dimensions of length.

Plots of normalized spectral image data under two opti-cal configurations are shown in Figs. 6 and 7. Figure 6 im-ages the wedge in our optical system without the diamond#i.e., Fig. 1 with 1!f" missing$. Figure 7 has a diamond!%2 mm thick" inserted between the wedge and the analyzer#i.e., Fig. 1 with 1!f" present$. Our normalized data !Figs. 6and 7" appear qualitatively similar to the model predictions!Figs. 3 and 5" indicating that our simple model of birefrin-gent fringing based on Eqs. !2" and !3" captures the funda-mental physical behavior of the wedge technique. Both datasets—with and without a diamond anvil in the path—showwavelength-dependent intensity variations superimposedupon the interference oscillations. These variations were ob-served in our calculated data set only in the case when anoptical aberration was introduced. Therefore, we characterizethis intensity modulation signal as a hallmark of our opticalsystem’s chromatic aberrations and spectral biases, and weuse the intensity variation to determine the scrambling ker-nel. Our data exhibit small distortions near 580 and 700 nmand an additional periodic variation in intensity. We havedetermined that both of these apparent distortions are relatedto small errors in background subtraction and normalization.

FIG. 5. !Color online" Calculated interference pattern from the birefringent wedge under the influence of chromatic aberrations due to a 2 mm diamondwindow. Each image represents 30 !vertical" "m imaged by our CCD for an initial wedge thickness of 100 "m. !Top" objective NA=0.03; !bottom" NA: 0.34projections of the horizontal and vertical slices from the center of the image are shown above and to the right of each spectrograph image.

TABLE I. Principal components of the birefringent wedge experiment.

Description Manufacturer Model

Resolution kernel componentsDiffuser Thorlabs ED1-LBPolarizer Newport 05P109AR.16YVO4 wedge, 6° angle, ' /4 flatness Bank Photonics, Inc. Custom

Spectroradiometry componentsMicroscope objective Mitutoyo 378-803-2YAG Laser rejection optic Chroma Technology 880dcxrPellicle beamsplitter Thorlabs BP245B2Imaging spectrograph Princeton Instruments SP2150iCCD Princeton Instruments 7515-0001



While perfect normalization is desired, these distortionsyield quantitatively small errors in the determination of thewavelength-dependent scrambling kernel.

The scrambling kernel, determined by the methods de-scribed in Appendix B, is shown for the optical system withand without a diamond window in Fig. 8. Several points areapparent from this plot. First, the scrambling kernel is muchgreater than the ideal optical resolution of %1–2 "m andvaries from approximately 7 "m at the 550 nm to about13 "m at 850 nm. Second, the scrambling kernel is signifi-

cantly increased !worsened" by the addition of the diamond.Third, the spatial resolution is better at lower wavelengths inboth cases. This fact—that the scrambling kernel is mini-mized near 550 nm—suggests that the optical system wasbetter focused for light at these wavelengths. This is moreconsistent with the peak sensitivity of the human eye !555nm" being the driving force in our ability to focus the opticalpath rather than of the peak intensity of our calibration lamp!%800 nm" or the peak sensitivity of the optics and CCD!600–650 nm", a fact that is especially surprising since we

FIG. 6. !Color online" Observed birefringent interference patterns collected on our optical system. Projections of the horizontal and vertical slices indicatedby dashed lines are shown above and to the right of the spectrograph images. Wavelength-dependent offset !dark gray curve" and amplitude of the cosinefitting function !light gray shading" are shown with the horizontal projection.

FIG. 7. !Color online" Observed birefringent interference patterns collected on our optical system with an %2 mm thick diamond window inserted after thewedge. Wavelength-dependent offset !dark gray curve" and amplitude of the cosine fitting function !light gray shading" are shown with the horizontalprojection.



refined the focus to maximize contrast in our raw spectralsignal, which itself exhibited a maximum %600–650 nm.This is different than the model calculations, which werecompleted in advance of the data analysis and which wereperformed for an optical system that was arbitrarily focusedfor paraxial light at 850 nm. Nonetheless, the basic point thatlight cannot be adequately focused for all wavelengths !andincident angles" through an optical system that includes dis-persive optics remains true and supports the developmentand use of aberration-aware calibration methods.

This method can be used to estimate wavelength-dependent birefringence in an unknown material since thesevalues are independently determined as a by-product of theconsistency checks in the data analysis protocol. We repro-duced the expected wavelength-dependent birefringence ofour YVO4 material to within 10% !Fig. 9", and we proposethat the 10% overestimate of the birefringence can be con-sidered to be an estimate of the systematic error in this mea-surement.

We interpret the residual high frequency oscillations inFigs. 8 and 9 as estimates of experimental precision in thismeasurement because the amplitude and frequency of theseoscillations are related to the precision of the cosine fitting.Figure 8 shows that without the diamond, these fluctuationsin the spatial resolution average to be approximately 5%.With the diamond, the amplitude of these fluctuations ap-proximately doubles to about 10%. Similar fluctuations ap-pear in the measured birefringence.

Many laboratories have configured their spectroradi-ometric systems to measure a single thermal spectrum ratherthan a spectral profile by inserting a spatial aperture !pin-hole" into the optical system at a focal plane. It is possible!though perhaps tedious" to use the wedge method to mea-sure the wavelength-dependent scrambling kernel and toevaluate the suitability of an optical system for spectroradi-ometry even if the system is not normally configured forimaging. In this case, it is feasible to translate the wedge bya known amount !1 or a few microns" in the vertical !wedgeaxis" direction, collecting a single spectrum at each of sev-eral !%100" positions. From such a data set a two-dimensional image can be reconstructed and then analyzed,as described in Appendix B.

V. IMPLICATIONS FOR TEMPERATUREMEASUREMENT IN THE LASER HEATED DIAMONDANVIL CELL

Our primary goal is to use the measured scrambling ker-nel to calculate the effects of chromatic aberrations on tem-perature measurements of laser-heated hotspots. This can beaccomplished either by an inverse approach, i.e., using thekernel to “unscramble” the measured information to back outthe original hotspot from the measurement, or by a forwardapproach, i.e., propagating synthetic input hotspots throughthe scrambling kernel to calculate an “observed” hotspot.Since the inverse method cannot generate a unique solutionin the presence of experimental statistical noise, we use aforward approach to examine how the scrambling kernel in-fluences the peak temperature and hotspot width of variousinput temperature profiles.

These comparisons are contoured in Figs. 10!a" and10!b". Figure 10!a" contours the calculated output hotspot-center temperature measurements from a synthetic diamondcell laser heating experiment subject to the scrambling kernelshown in Fig. 8. In Fig. 10!b", the calculated output tempera-ture gradient full width at half maximum !FWHM" is shown.These values are shown as a function of the input peak tem-perature and input FWHM. These synthetic data sets showhow chromatic aberration systematically alters the tempera-ture measurement. In all cases, the output temperatures arehigher than the input temperatures, and the output peakwidths are broader than the input peak widths. These effectsare significant: For an input Gaussian hotspot with a peaktemperature of 3000 K and a FWHM of 30 "m, the outputhotspot has a peak temperature of %3600 K !20% differ-ence" and a FWHM of 33 "m !10% difference". The effecton observed peak width is exacerbated by narrower hotspotpeaks and lower temperatures. For example, at an input peak

FIG. 8. Observed scrambling kernel, r!'", for two experimental setups:wedge alone !black", indicating the chromatic aberration of our optical sys-tem, and wedge plus a 2 mm diamond window !gray", estimating the chro-matic aberration of our system for its intended use with a diamond anvilcell.

FIG. 9. Wavelength-dependent birefringence of YVO4 as determined in ourexperiment !solid line" and as given in literature !dashed line" !Refs. 11 and12". Experimentally determined wavelength-dependent birefringence of theYVO4 wedge agrees with literature values within 10%.



temperature of 2000 K and a FWHM of 20 "m, the outputpeak temperature is 2300 K !15% difference" and theFWHM is 28 "m !40% difference".

Although one might expect that a loss of resolutionwould generate a lower observed temperature than the actualtemperature, our results show that the output !observed" peaktemperature is higher than the input !actual" peak tempera-ture. This apparent temperature increase is due to the chro-matic dependence of the scrambling kernel. Since in our sys-tem the red end of the spectrum experiences morescrambling than the blue end, at the high-temperature centerof the hotspot, more of the emitted light from the blue por-tion of the spectrum arrives at the middle of the detectedportion of the hotspot. Interpreting this spectrally scrambledsignal with a blackbody radiation model generates a higherapparent temperature because of the scattering away of thered portion of the spectrum and consequent relative intensityweighting toward the blue portion of the spectrum.

Furthermore, at smaller hotspot widths, the output tem-perature profiles are no longer Gaussian. Interestingly, thescrambled !measured" hotspots have more of a “flat-top” ap-pearing profile, purely as a result of thermal scrambling atthe center of a hotspot. Therefore, spectroradiometric obser-vations of a flat-top temperature need to be interpreted with

care, considering the effects of chromatic aberrations on animaging spectroradiometric system. This problem can beminimized if a simultaneous two-dimensional intensity im-age of the hotspot is obtained.2,3

The scrambling kernel can be used to provide a quanti-tative estimate of the confidence band of the temperaturemeasurement as a function of the hotspot size and the wave-length range of the spectroradiometry measurement. By de-termining the range of possible temperature mismeasurement!Fig. 11", we show in Appendix C that the half-width of atemperature measurement confidence band, #T /T, is

#T

T=

$!1 ! $2"!'22 ! '1

2"!'2 ! '1"2 ! $2!'2 + '1"2 , !4"

where '1 and '2 are the two ends of the observation wave-length range and $ is a measure of the local temperaturegradient. We describe the determination of the local gradient,$, in detail in Appendix C. Assuming a Gaussian hotspotprofile, $ is mainly dependent on the hotspot size and thelocation within the hotspot: the gradient is most shallow atthe hotspot center, steepest at the standard deviation. Thus,the confidence band is also several times larger for tempera-tures measured at the edge of the hotspot than it is for centraltemperatures, and the confidence band increases !nonlin-early" as the hotspot size decreases. The absolute size of ahotspot is relevant only as it relates to the observed scram-bling kernel, r!'", so we define the hotspot size ratio, R=d / r, to be the ratio of the hotspot diameter to the averagescrambling kernel. The hotspot needs to be significantlylarger than the average scrambling kernel to be “immune” toaberration effects.

Figure 12 illustrates the relative effects of wavelengthrange, hotspot size ratio, and local temperature gradient onmeasurement confidence by plotting confidence band half-width !#T /T" as a function of the longest wavelength used intemperature fitting #'2 from Eq. !4"$ for different initialwavelengths '1. Using a wide wavelength window to deter-

a

b

FIG. 10. Contour plots of !a" peak temperatures and !b" FWHM that wouldbe observed from synthetic hotspot data that has been aliased by a 2 mmthick diamond window. Contours are shown as functions of the actualFWHM !vertical" and actual peak temperature !horizontal".

FIG. 11. Schematic of temperature determination by the !linear" Wien ap-proximation. Actual light emitted is converted to the Wien function !solidlines" and fit to a linear function where the slope indicates temperature andthe intercept !intersection of dash-dot lines" determines the emissivity. Ex-treme cases of optical scrambling described in Appendix C are shown asdashed lines.



mine temperature increases measurement precision !de-creases the confidence band width", though the effect ofmoderate wavelength range limitation !'2!'1%300 nm" isless severe than the effect of reducing the hotspot size ratioor making a measurement where the local temperature gra-dient is large.

Forward calculations indicate that R.10 is nearly suffi-cient to reproduce temperatures and temperature gradients,while a R.5 is often adequate for measuring peak tempera-ture, but is likely to cause significant error in profile. Ob-served ratios near 1 or 2 or 3 are certain to be poor repre-sentations of the actual thermal structure in the sample. Thisindicates that for our observed scrambling kernel !r=10.35 "m between 500 and 800 nm" the laser hotspotwould need to be %100 "m in size to achieve a 10% con-fidence band on central temperatures measured in the dia-mond cell and more than 200 "m to achieve even 20% con-fidence bands on temperature profile measurements.Alternatively, modifications in the LLNL optical system toreduce aberrations, perhaps by limiting the numerical aper-ture, and to increase the wavelength range of measurementsmay be sufficient to achieve reasonable precision in spectro-radiometric measurements for typical hotspot sizes.

These results highlight the importance of good design inLHDAC experiments to maximize the hotspot width, whichhas the dual effect of increasing the hotspot size ratio anddecreasing the local temperature gradient. More powerful la-

sers will play a role, but high-temperature mechanical stabil-ity of the high-pressure apparatus needs to be considered aswell. Since the width of the hotspot is also sensitive to thethermal conductivity and thickness of the insulationlayer,14–16 improvements in temperature measurement can bemade by improving the engineering of the sample to maxi-mize the insulation layer thickness, for example, by usingdiamond anvils designed to minimize the gasket compressionas the pressure is increased !e.g., Boehler–Almax typeanvils17". A broader hotspot also has the benefit of minimiz-ing systematic errors in temperature due to misalignmentwhen making in situ measurements !e.g., simultaneous x-raydiffraction and laser heating".

VI. HOW TO MAKE A BETTER MEASUREMENT:A CHECKLIST FOR TRANSPARENTLY MAKINGAND REPORTING SPECTRORADIOMETRICMEASUREMENTS

There are nearly as many temperature fitting and report-ing protocols as there are researchers who measure tempera-tures during laser heating by spectroradiometry. While thismay not be the origin of disagreement in controversial mea-surements, it is at least a factor that inhibits the direct com-parison of discrepant data. In order to close this “compara-bility gap,” we provide a description of the steps necessaryfor reporting both the characteristics of the optical systemand the certainty of the measurement.

1. Design a good optical system for spectroradiometry,keeping in mind that the qualities that make a good im-aging system are not necessarily the same as those thatpromote accurate spectroradiometry: Minimize aberra-tion effects by using low numerical aperture and reflect-ing optics. In addition to imaging spectroradiometry,consider focusing and collecting several wavelengths in-dependently as described in Ref. 3. Maintain awarenessthat illumination used for alignment and imaging !e.g.,transmitted light" is often nearly paraxial, and so doesnot experience any spherical aberration, while thermalemission, which is Lambertian in angular composition,is quite sensitive to spherical aberrations. If possible,center the optical focusing around the wavelength andangular distribution over which data will be collected.

2. Determine the wavelength-dependent scrambling kernel#r!'"$ for the optical system as described in this article.Report at least the average scrambling kernel over thewavelength range used in temperature fitting. Report theminimum, maximum, and average scrambling kernels ifpossible.

3. Calculate and report the experimental precision of tem-perature measurements using Eq. !4" and either Eq. !C8"!for central temperatures" or Eq. !C13" !for temperaturesmeasured off the center of the hotspot". This experimen-tal confidence band depends on the wavelength rangeused to calculate the temperature, the hotspot size ratioR, and the factor !Tmax!T0" /T. Procedures for estimat-ing the !Tmax!T0" /T factor are described in Appendix C.In the absence of more complete information, estimatesof !Tmax!T0" /T%0.9 when using Eq. !C8" and !Tmax

FIG. 12. Confidence band half-widths for temperatures measured by spec-troradiometry plotted as a function of the longest wavelength used in tem-perature fitting. All half-width curves are plotted for a Gaussian temperatureprofile with a central temperature of 3000 K. Confidence bands for central!r=0" temperatures with high hotspot size to scrambling kernel ratios, R=10, are plotted in solid black for three different initial wavelengths: 400nm !thin line", 600 nm !thick line", and 800 nm !thin line". In addition towavelength range, hotspot size ratio and location within the hotspot stronglyaffect the width of the confidence band. The dotted curve is the confidenceband for central temperatures !r=0" measured with an initial wavelength at600 nm but with a smaller hotspot size ratio, R=5. The gray curve shows theconfidence band for temperatures measured far from the center of thehotspot !r=/" with an initial wavelength at 600 nm and a large hotspot sizeratio, R=10. A vertical line is plotted at 1000 nm to indicate a long-wavelength limit that is often imposed by the presence of an infrared laserwith a wavelength near 1000 nm. !Variation in peak temperature from 3000K only changes the relative confidence band slightly."



!T0" /T%1.4 when using Eq. !C13" are reasonable. Be-cause hotspot size is strongly dependent on sample ge-ometry, the size of the hotspot should be measured atleast once for each distinct sample.

4. Determine and report the accuracy of each temperaturemeasurement by the standard deviation of the histogramof the two-color temperature, as described in Ref. 18.This component of uncertainty is a good reflection oflimitations on accuracy due to wavelength-dependentemissivity as well as being an indication of uncertaintydue to temperature gradients. Monitoring the residual ofthe Wien function during heating may also give an indi-cation of changes in the uncertainty due to changes inoptical properties or thermal gradients.19

Thus, most experimental uncertainties are accounted forby reporting both an experimental precision that is deter-mined fundamentally from the optical system and an accu-racy that is determined from each measurement by its degreeof non-Planck behavior due to sample specific propertiessuch as wavelength-dependent emissivity. It is our intentionthat the steps described here !if followed" will lead to clearcommunication of measurements among experimentalists inthe field as well as increased confidence in spectroradiomet-ric temperature measurements by nonspecialists, in much thesame way as the checkerboard resolution test in seismologyis an assessment of model performance that is accessible tononspecialists.

VII. CONCLUSION

The technique we have presented provides an experi-mental method to determine the chromatic scrambling ker-nel, r!'", of any optical system by using a birefringentwedge. When applied to the spectroradiometric measurementof temperatures, this technique allows optical systems to besystematically and objectively evaluated and optimized forefficacy at temperature measurement using the guideline thatthe ratio of laser-heated hotspot size to average scramblingkernel must be at least 5 to measure peak temperatures and atleast 10 to measure temperature profiles. Furthermore whenthe scrambling kernel and hotspot size are known, quantita-tive estimates of the experimental precision !confidenceband" of temperature measurements due to chromatic effectsare directly calculable. Consequently, the birefringent wedgetechnique !in concert with the associated calculation of anexperimental confidence band" makes possible direct com-parisons of spectroradiometric measurements made in differ-ent laboratories. Our vision was to provide a method forobjective, quantitative interlaboratory comparison of tem-perature measurements to the LHDAC community, and webelieve that spectroradiometric temperature measurementsthat report the average scrambling kernel and the optical con-fidence band !as described in this work" and also the accu-racy limit due to wavelength-dependent emissivity !as de-scribed in Ref. 18" satisfy this objective. In order to furtherpromote this goal of increased interlaboratory consistencyand comparability, we have provided a detailed descriptionof the steps necessary to document precision and accuracy ofa temperature measurement.

ACKNOWLEDGMENTS

M. Armentrout, C. Nugent, and D. Ruddle provided ex-perimental help. P. Bird and W. R. Panero furthered thiswork by stimulating discussions. A. Kavner acknowledgessupport from NSF Grant No. EAR0510914. This work wasperformed under the auspices of the U.S. Department of En-ergy by Lawrence Livermore National Laboratory underContract No. DE-AC52-07NA27344. !LLNL-JRNL-402546".

APPENDIX A: EFFECT OF CHROMATICABERRATIONS ON THE IMAGE OF ABIREFRINGENT WEDGE

We model chromatic aberration simplistically by way ofa wavelength-dependent scrambling kernel, r!'". That is, weposit that light observed !to be" at a given position on thebirefringent wedge actually originates from a circle of radiusr!'" around that position. Recall that the intensity transmit-ted through the birefringent wedge and polarizers is simply afunction of the wedge thickness and birefringence:

I = I0!&1 ! cos(2+$

'!t0 + M tan ( cos ,z")'

= I0!.1 ! cos#2+c!z ! z0"$0 , !A1"

where !z ,'" are the relevant observational variables: z is theposition on the vertical spectrometer slit and ' is the wave-length of light. Here we have collected the geometric vari-ables into a single parameter, c. Define x to be the horizontaldirection normal to the optic axis #see Fig. 1!a"$ at eachwavelength:

I!x,z" =I

I0!=

1+r2+

r!=0

r

I!x!,z!"dA

=1

+r2+r!=0

r

.1 ! cos#2+c!z! ! z0"$0dA . !A2"

The simplistic part of this analysis is the assumption thatlight arrives equally from the entire circle of radius r!'". Theeffect of real aberrations is likely to involve unequal weight-ing, and thus the scrambling kernel is a notion, and not ac-tually a parameter with real physical meaning.

The first term in the integral in Eq. !A2" is a constantand integrates directly to the area of the circle.

I!x,z" =+r2

+r2 !1

+r2+z!=z!r

z+r +x!=x!,r2!!z! ! z"2

x+,r2!!z! ! z"2

cos#2+c!z!

! z0"$dx!dz!. !A3"

Similarly, the x-integral is a constant because the transmittedintensity is only a function of the wedge thickness, which isproportional to z.



I!x,z" = 1 !1

+r2+z!=z!r

z+r

2,r2 ! !z! ! z"2cos#2+c!z! ! z0"$dz!.

!A4"

We define a variable of integration, v= !z!!z" /r !dv=dz! /r", and apply a trigonometric identity so that Eq. !A4"becomes

I!x,z" = 1 !1

+r2+v=!1

1

2r,1 ! v2cos#2+c!rv + z ! z0"$rdv ,

!A5"

and then

I!x,z" = 1 !2++

v=!1

1,1 ! v2#cos 2+crv cos 2+c!z ! z0"

! sin 2+crv sin 2+c!z ! z0"$dv . !A6"

In the previous equation, cos!2+crv" is symmetric about 0,while sin!2+crv" is antisymmetric, integrating to 0. Equation!A6" then reduces to

I!x,z" = 1 !4+

cos 2+c!z ! z0"+v=0

1,1 ! v2cos 2+crvdv .

!A7"

The integral in Eq. !A7" is not analytically solvable, but thenumerical solution is straightforward. As shown in Fig. 4, itis a function of the dimensionless quantity cr. r, the effectiveradius of point-source light, which we call the scramblingkernel, has units of length, while c is the inverse of theperiod of the fringes !c= #$!'" /'$M tan ( cos ,", and assuch has units 1/length.

APPENDIX B: DATA ANALYSIS PROTOCOL

Interference data are expected to satisfy the function I= I0!!1!0!cr"cos!2+$t /'"" while c= #$!'" /'$M tan ( cos ,,0!cr"=4 /+/v=0

1 ,1!v2#cos 2+crv$dv, t= t0+M tan ( cos ,z,and the objective is to determine r. Here we outline the de-tails of normalization and analysis of the spatiospectral in-tensity data.

1. Normalization

One of the key procedures was to find a consistent wayto normalize the data for inter- and cross-comparisons. Allmeasurements were performed as pairs. For each interferencepattern collected, Ifringe!' ,z", an additional pattern was col-lected under identical conditions, save one: The analyzer wasremoved from the optical path, yielding a baseline intensityIbase!' ,z", with no fringing. Each data set was normalized bycalculating the ratio Inorm= Ifringe / !Ibase-Tanalyzer", whereTanalyzer is the predetermined wavelength-dependent transmit-tance of the analyzer !in percent". This normalization schemeyields a data set that is between 0 and 1 for all !' ,z".

A complication is introduced when the sensitivity of thesystem is close to zero, in which case the normalized databecome dominated by noise. Thus the wavelength range ofour measurements is limited by the combined optical absorp-

tion and detector sensitivity to regions where the signal tonoise ratio was large enough to confidently fit the data to acosine curve.

2. Fitting

We fit each column !i.e., each wavelength" of the imagedata separately to a cosine functional form f =c1!c2 cos#2+!c3+c4z"$, where z is the pixel position and.c1 ,c2 ,c3 ,c40 are the four fitting parameters. We found thatfitting the columns to cos!z" was more efficient and repro-ducible than fitting the rows to cos!1 /'", even though it en-tailed approximately ten times as many curve fits. This pro-tocol yielded four functions, the ci!'", that are then evaluatedto find the scrambling kernel. The fit parameters map to thetheoretical fringe function as c1= Io!, c2= I0!0!cr", c3=$!'"t0 /', and c4=$!'"M cos ( cos , /', which is the c in0!cr".

3. Evaluation

Because of the periodic nature of cosine, the fit param-eter c3 is nonunique: c3+n= #$!'" /'$t0. To overcome thatchallenge, we constrained c3 to be between 0 and 1 duringthe curve fit. Then we added sequential integers to c3 in orderto make c3!'" a smooth and monotonic function. Next, weconsidered the ratio of the third and fourth fit parameters:

c3!'" + n

c4!'"=

$!'"t0

$!'"M tan ( cos ,=

t0

M tan ( cos ,.

That is, the ratio is not a function of wavelength andshould be a constant. We adjust n0, the initial integer in thesequence, until the ratio !c3+n" /c4 is as independent ofwavelength as possible. The degree to which this value is notwavelength independent may represent one source of error orestimate of uncertainty. We always were able to choose n0 tomake this quantity constant to within a few percent.

To determine t0, the thickness of the wedge at one end ofthe CCD, and the quantity M tan ( cos ,, which is a functionof the wedge and optic geometry, we need two images !a andb" separated by a known vertical distance, z. Then,

( c3 + n

c4)

a

( c3 + n

c4)

b

=

t0a

M tan ( cos ,

t0b

M tan ( cos ,

=t0a

t0b=

t0b + #t

t0b= 1 +

#t

t0b. !B1"

Since #t is known, #t=#z tan (, we can solve for t0b:

t0b =( c3 + n

c4)

b

( c3 + n

c4)

a! ( c3 + n

c4)

b

#t , !B2"

and then

M tan ( cos , = t0( c4

c3 + n) . !B3"

We checked the internal consistency of this determina-tion by comparing the quantity M tan ( cos , determined



from image a and image b. !M tan ( cos ,"a and!M tan ( cos ,"b should be very nearly equal, just as t0a andt0b should differ by #t. Furthermore, since M and ( areknown or measurable, the inferred value of , should besmall.

With this information, it is also possible to indepen-dently determine the wavelength dependence of birefrin-gence:

$!'" =c3 + n

t0=

c4

M tan ( cos ,.

In this experiment, birefringence determined in this man-ner reproduced that in Refs. 11 and 12 within 10% !see Fig.9", providing further support for the data reduction tech-nique. Thus this is a simple method for determining birefrin-gence approximately.

We have found that as long as the z-range of the spec-trometer is long enough to collect nearly one period of co-sine, then the fits are quite good. Our optical system imaged27 "m !in real space" along the 100 vertical pixels of theCCD chip. Variations in magnification or resolution may benecessary for different applications. It may then be necessaryto modify the birefringent wedge design parameters !wedgeangle, wedge material, and initial thickness" to ensure thatthe period of interference fringes along the wedge axis is lessthan or equal to the vertical size of the CCD and also that theinterference fringing is adequately sampled to determine itsintensity accurately. Consistency between different thick-nesses, as well as consistency in reproducing the expectedwavelength dependence of birefringence, is also good.

4. Determination of the scrambling kernel

Once internal consistency is confirmed, spatial reso-lution is determined from c1, the cosine offset, and c2, cosineamplitude. The ratio c2 /c1=0!cr", so the amplitude/offsetratio is mapped onto the numerical integral in Fig. 4 to de-termine cr, the phase fraction of the cosine that must beaveraged to decrease contrast to the observed value. Then cris divided by the fitting factor c4 !in 1 /"m" to give r, thescrambling kernel at each wavelength.

APPENDIX C: ESTIMATE OF CONFIDENCE BANDSFOR TEMPERATURE MEASUREMENTS

We find that it is possible to calculate a confidence bandfor temperature measurements by considering the extremesof spectrally scrambled temperature measurement. The re-sulting confidence band reflects errors in precision due to theeffect of optical aliasing, neglecting effects in accuracy dueto axial temperature gradients or wavelength-dependentemissivity !e.g., Ref. 18". Figure 11 shows a schematic ofhow temperatures are measured using the linear Wien ap-proximation to Planck’s law !e.g., Ref. 20". Observed inten-sity, I, may be linearized by the Wien function:

Wien =k

hcln(2+hc2

I'5 ) =1T

1'

!k

hcln & , !C1"

where h is Planck’s constant, c is the speed of light, & is theemissivity and is generally dependent on pressure, tempera-

ture, and wavelength, ' is the wavelength of light, k is Bolt-zmann’s constant, and T is the temperature. If emissivity isassumed to be wavelength independent, then the temperaturemay be determined from the slope of the Wien function plot-ted as a function of 1 /'. If a low-temperature region is ad-jacent to a high-temperature region, then optical aliasing mayact to scramble the light from the two regions. The worst-case scenario of scrambling would be if red light is measuredentirely from one region and blue light entirely from another.These two cases are shown as dashed lines in Fig. 11.

Therefore, we define an extreme-case confidence band asthe difference between the two extreme-optical-mixing sce-narios. If we use only two points to determine temperatureby the Wien approximation,

TWien =

1'1

!1'2

1T

1'1

!1T

1'2

. !C2"

The extreme false temperature measurements are then

Ta =

1'1

!1'2

1Th

1'1

!1Tl

1'2

and Tb =

1'1

!1'2

1Tl

1'1

!1Th

1'2

, !C3"

where Tl and Th are the actual temperatures of the low-temperature and high-temperature regions, respectively, andthe full width of the extreme-case confidence band is

Ta ! Tb = ( 1'1

!1'2)1 1

1Th

1'1

!1Tl

1'2

!1

1Tl

1'1

!1Th

1'22 . !C4"

Next we replace the high and low temperatures by a tempera-ture gradient, defined such that Th=T!1+$", while Tl=T!1!$", where T is then the average of the two temperatures and$ is the difference between the average and the high or low.Using this notation, the above equation simplifies to

Ta ! Tb = ( 1'1

!1'2)1 1

1T!1 + $"

1'1

!1

T!1 ! $"1'2

!1

1T!1 ! $"

1'1

!1

T!1 + $"1'22 , !C5"

which becomes

Ta ! Tb

T=

2$!1 ! $2"!'22 ! '1

2"!'2 ! '1"2 ! $2!'2 + '1"2 . !C6"

Thus the size of the confidence band, #T /T1 !Ta!Tb" /T, is not a function of absolute value of the tempera-ture but is a function only of the wavelength range used andthe temperature gradient between adjacent measurements. Inorder to interpret this further, we use the observed scram-bling parameter and hotspot size to provide informationabout the variation in temperature between adjacent mea-surements, letting Tl and Th be the temperatures of two



points separated by the average scrambling kernel length, r.Assuming a Gaussian temperature profile, T=T0+ !Tmax!T0"exp!!r2 /2/2", the minimum temperature gradient is atr=0 !where #T /#r=0" and the maximum temperature gradi-ent is at r=/ !where #2T /#r2=0". We address both casesbelow.

We introduce some notation: T0 is the ambient tempera-ture boundary condition, normally 300 K. For simplicity, wedefine T!=Tmax!T0. We have defined !in the main text" theratio of hotspot size to average scrambling kernel, R=d / r, tobe a relevant, observable parameter. d is the observed diam-eter !FWHM" of the hotspot and r is the average measuredscrambling distance. In a Gaussian distribution, FWHM=2,2 ln!2"/, so R=2,2 ln 2/ / r.

In the low gradient !r=0" case, our two temperaturescome from the hotspot center and one scrambling distanceaway from the center. Averaging and subtracting,

T = T0 +T!

2&1 + exp(!

r2

2/2)'= T0 +

T!

2&1 + exp(!

8 ln 2R2 )' , !C7"

and

$ =T!

2T&1 ! exp(!

r2

2/2)' =T!

T

12&1 ! exp(!

8 ln 2R2 )' .

!C8"

In the high gradient !r=/" case, our two temperatures comefrom r /2 to either side of r=/,

Th!l" = T(/ ! !+ "r

2) = T0 + T! exp3!

(/ ! !+ "r

2)2

2/2 4 .

!C9"

Simplifying slightly,

Th!l" = T0 + T! exp(! 12)exp(+ !! "r/

2/2 )exp(! r2

8/2) .

!C10"

Averaging,

T = T0 +T!

2exp(! 1

2)exp(! r2

8/2)*exp( r

2/)

+ exp(! r

2/)- . !C11"

In terms of the observable ratio, R,

T = T0 +T!

2exp(! 1

2)exp(! ln 2

R2 )*exp(,2 ln 2R

)+ exp(! ,2 ln 2

R)- . !C12"

Subtracting the two temperatures,

$ =T!

2Texp(! 1

2)exp(! r2

8/2)*exp( r

2/) ! exp(! r

2/)-

=T!

2Te!1/2e!ln 2/R2

#e,2 ln 2/R ! e!,2 ln 2/R$ . !C13"

The prefactor !T! /T" is dependent on peak temperature!Tmax" but very nearly independent of R !as long as R2%3". Furthermore, !T! /T" is well approximated by its valueat r=0: For the low gradient case, T! /T=T! /Tmax, whichincreases from %0.8 to 0.95 as Tmax increases from 1500 to5000 K, while for the high gradient case, T! /T=T! / #T0+T! exp!!0.5"$, which increases from %1.2 to 1.5 as Tmaxincreases from 1500 to 5000 K.

Thus, confidence bands on peak central temperatures canbe estimated by using the local temperature gradient, $, inEq. !C8" to calculate the full width of the confidence bandwith Eq. !C6", and confidence bands on temperature profilescan be estimated by using the gradient, $, in Eq. !C13" in-stead. The confidence band determined using Eq. !C13" isalways larger !generally several times larger" than that cal-culated using Eq. !C8". Figure 12 shows a contour plot of thehalf-width of the confidence band #0.5!!Ta!Tb" /T$ for tem-peratures measured at the center of the hotspot for an opticalsystem with a hotspot size to scrambling kernel ratio of 10.!T! /T=0.85, as for a peak temperature of 2000 K, was usedin the calculation." For such a configuration, a very satisfyingconfidence band of less than 10% can be achieved if a widewavelength range is used, but the use of either a small wave-length range or an optical system with a poor hotspot size toscrambling kernel ratio rapidly decreases the accuracy.

1L. R. Benedetti, N. Guignot, and D. L. Farber, J. Appl. Phys. 101, 013109!2007".

2A. Kavner and C. Nugent, Rev. Sci. Instrum. 79, 024902 !2008".3A. J. Campbell, Rev. Sci. Instrum. 79, 015108 !2008".4M. J. Walter and K. T. Koga, Phys. Earth Planet. Inter. 143–144, 541!2004".

5M. Shirashi and K. Asama, Appl. Opt. 21, 4296 !1982".6D. H. Munro, S. N. Dixit, A. B. Langdon, and J. R. Murray, Appl. Opt. 43,6639 !2004".

7T. Wakayama, H. Kowa, Y. Otani, N. Umeda, and T. Yoshizawa, Opt. Eng.!Bellingham" 45, 033601 !2006".

8K. Oka and T. Kaneko, Opt. Express 11, 1510 !2003".9S. Nakadate, Opt. Lasers Eng. 26, 331 !1997".

10W. D. Nesse, Introduction to Mineralogy, 3rd ed. !Oxford UniversityPress, Oxford, 1999".

11H. S. Shi, G. Zhang, and H. Y. Shen, J. Synth. Cryst. 30, 85 !2001".12L. G. DeShazer, Proc. SPIE 4481, 10 !2002".13C. Snell, Nature !London" 214, 78 !1967".14A. Kavner and W. R. Panero, Phys. Earth Planet. Inter. 143–144, 527

!2004".15W. R. Panero and R. Jeanloz, J. Geophys. Res., #Solid Earth$ 106, 6493

!2001".16W. R. Panero and R. Jeanloz, Rev. Sci. Instrum. 72, 1306 !2001".17R. Boehler and K. De Hantsetters, High Press. Res. 24, 391 !2004".18L. R. Benedetti and P. Loubeyre, High Press. Res. 24, 423 !2004".19L. R. Benedetti, D. Antonangeli, D. L. Farber, and M. Mezouar, Appl.

Phys. Lett. 92, 141903 !2008".20D. L. Heinz and R. Jeanloz, High-Pressure Research in Mineral Physics,

edited by M. H. Manghnani and Y. Syono !American Geophysical Union,Washington, DC, 1987", pp. 113–127.



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