Measurement of the thermal diffusivity of nonlinearanisotropic crystals using optical interferometry
Robert A. Morgan, K. 1. Kang, C. C. Hsu, Chris L. Koliopoulos, and Nasser Peyghambarian
We describe a simple common-path self-referencing interferometric method requiring no auxiliary opticalelements which accurately and directly measures the thermal diffusivity of anisotropic crystals in nonsteady-state conditions. We determine the thermal diffusivity of a lithium niobate crystal as a function oftemperature ranging from room temperature up to 500'C.
1. Introduction
The thermal diffusivity D is an important consider-ation when second harmonic generation (SHG) isneeded from a high energy laser source.' Optical ab-sorption causes self-heating and temperature gradi-ents to occur in the second harmonic generator. Theresulting change in temperature of the crystal is detri-mental to the phase-matching condition. A nonuni-form temperature distribution can also have devastat-ing effects on the beam profile and the phase fronts.Lithium niobate is a negative uniaxial crystal with anextraordinary index that varies much more rapidlywith temperature than does its ordinary index.2 Thiscreates potential problems for maintaining a uniformphase match when the optical absorption is sufficient-ly high and the thermal diffusivity is sufficiently low.Thermal diffusivity measurements are also of concernin other areas of nonlinear optics since thermal varia-tions in the refractive indices can impose limits onnonlinear optical devices. Good data on the thermaldiffusivity for nonlinear materials are not readilyavailable nor are most optics labs usually equipped tomake such determinations. In this paper we demon-strate that an all-optical interferometer can be used tomake the required measurements.
The interferometric configuration (a variation of aFizeau interferometer) used, which we call the rear-shear interferometer (RSI), interferes the front andrear reflections off the test crystal. The interference
The authors are with UniversityCenter, Tucson, Arizona 85721.
takes place directly at the test crystal, which acts as theinterferometer; thus, no auxiliary optics is needed.The necessary tilt needed to analyze the fringe data isbuilt-in due to the inherent wedge angle of the testsample. The real-time common-path capabilities ofthe RSI greatly enhance our ability to make measure-ments that require temperature variation; system sim-plicity increases the ease and accuracy of the experi-ment.
It has recently been pointed out that a shearinginterferometer can be used to measure the thermaldiffusivity of solids.3 However, this technique re-quires a prior knowledge of index and thermal expan-sion coefficients. One also needs to perform criticalmeasurements of the power of an electrically heatedmetal strip and refraction angles. Furthermore, thematerial requirements of size and optically flat polish-ing make this method prohibitively unattractive (if notimpossible) for most nonlinear optical materials.However, our all-optical technique requires one tomeasure optical path differences (OPD) only, and thematerial requirements are minimal (only small thincrystals are needed). This technique measures thethermal diffusivity directly, superseding the require-ment to measure thermal conductivities, specificheats, and densities separately. One can determinethe thermal diffusivity along two nondegenerate crys-tal axes if required. This method is quite general andwill work for a large variety of linear and nonlinearmaterials, which need not be crystalline nor aniso-tropic.
The method we used to determine the thermal diffu-sivity of lithium niobate was to insert the crystal intothe RSI and heat the crystal via its absorption with aninput pulse from a focused carbon dioxide laser operat-ing at 10.6 m (Fig. 1). As the heat diffuses outward inthe crystal, the refractive index increases and the crys-tal expands. The resulting optical path change can bemonitored interferometrically in real time, and from
Aperture lens LiNbO3 I Beam splitter Mirror(f=tO )
Fig. 1. Schematic of the experimental configuration used to mea-sure the thermal diffusivity of lithium niobate. Note that twoslightly different configurations were used. First, to collect datawith the I-D CCD array and OMA a collimating lens Li is necessary.Also it aids in the accuracy and alignment to have a slit which isparallel to the appropriate crystal axis and CCD array. Second, tocollect data with a TV camera, Li is not necessary if the crystal is
properly imaged (and magnified) into the camera.
the crystal manifesting itself as a change in OPD.This is given to first order by
OPD' = OPD0 + 2d an 5T + 2n d 6T,DT T
(3)
where T is the change in the temperature from itsunperturbed value, a quantity we allow to be a generalfunction of space and time. Noting that the thermalexpansion coefficient a (1)1(d) (d)/(T), we canrewrite Eq. (3) as
(4)OPD' = OPD + c6T(x,y,t),
where the constant c is given by
c 2d( + na) (5)
From this it is clear that a large perturbation withtemperature requires large thermal expansion and re-fractive-index coefficients (also of the same sign).
From Eq. (4) it is clear that the change in the OPDdue to a perturbing change in temperature is simplyproportional to the change in temperature,
AOPD = cbT(x,y,t).
fringe data the thermal diffusivity can be determined.The static variations due to optical imperfections andpoor polishng are substracted out in the data analysis,determining only the temporally varying optical pathchange due to the thermal diffusion. This not onlyincreases the accuracy but also places less stringentrequirements on the crystal under test.
Diffusion is a phenomenon common to many areas ofphysics and engineering, the effects of which are im-portant to measure and visualize. A unique advantageof an interrferometric mapping of the thermal distri-bution is that one can physically see how the heatdiffuses outward in the crystal.
In the RSI of Fig. 1 destructive interference fringesoccur when
OPD = 2nd + 2ny tan. = mX, (1)
where n is the refractive index of the crystal in thedirection of the incoming laser polarization, d is thethickness of the crystal, m is an integer, X is the laserwavelength, and 0 is the relative wedge angle betweenthe two parallel surfaces, which open in the y direction.The net result is straight parallel reference fringescommonly used to determine the wedge angle of aparallel plate. In reality, imperfect polishing andnonuniformity of the refractive index results in a varia-tion of these perfect straight parallel reference fringesby some arbitrary function of the transverse coordi-nates, f(x,y), which results in the following OPD:
OPD0 = 2nd + 2ny tano + f(x,y). (2)
This general form of the unperturbed OPD, as we willsee, also allows for a wedge angle that opens in a direc-tion different from a crystal principal axis.
Equation (2) expresses the OPD at a given steady-state temperature. A change in the temperature re-sults in a change in refractive index and thickness of
(6)
The temperature distribution in an anisotropic me-dium follows the diffusion equation5 given generally intwo dimensions (in the diagonalized coordinate sys-tem) by
abT(x,y,t) -Da2 02________= (Dx -+ DY -T(xy,t) + S(x,y,t) -R(x,y,t),at xa 2 ~ay2
(7)
where bT(x,y,t) is the change in the temperature distri-bution from the steady-state value To, Dx and Dy arethe respective thermal diffusivities in the x and y prin-cipal directions, S(x,y,t) is a general heat source, andR(x,y,t) refers to a nondiffusive loss of heat, e.g., ther-mal radiation. To form a retractable solution of Eq.(7) consistent with experiment we let the source termgo to zero and assume radiative cooling to be negligible.The diffusion equation can thus be written simply as
a5T(xyt) _ a2 + Da
at Dx ay+ D ) 6T(x,y,t). (8)
The solution of Eq. (8) to a heat impulse (assuminginfinite boundaries) is given by a 2-D Gaussian:
where C is a constant related to the heat capacity andto the total heat energy absorbed, (xo,yo) are the trans-verse coordinates of the impulse, and 47r(DxDY)1 /2t is anormalization constant reflecting the conservation ofenergy.
When the lithium niobate crystal is irradiated by theCO2 laser it will absorb essentially all the 10.6-Amradiation. Equation (9) is valid under the assumptionthat the effects of surface conduction and radiativecooling are negligible and that the transverse dimen-sions of the crystal are much larger than its depth (so a
2-D model is legitimate). The measurements are per-formed in conditions for which the above assumptionshold. This Gaussian model was also verified empiri-cally.
The effect of the heat pulse on the OPD is easilydetermined by substituting Eq. (9) into Eq. (6):
where we have incorporated all the constants into theproportionality constant C'.
The refractive indices of lithium niobate increasewith increasing temperature.2 In SHG the importantparameter for phase matching is the relative changewith temperature of the index at the laser frequencywith respect to the index at the second harmonic (SH)frequency. This is known as the temperature tuningcoefficients fi defined as
a (ns,e - nf,0 )
I3 T (11)
where the ordinary index at the laser frequency isdenoted nf,o and the extraordinary index at the SHfrequency is denoted nse. Thus, the index differencein the crystal changes with changing temperature as
An = Ano + f36T(xy,t), (12)
where Ano = nse - nfj, at the steady-state temperatureTo. In lithium niobate as in many nonlinear crystals 03is nonzero.2 6,7 The effect of self-heating on the phase-matching condition can be catastrophic to efficientSHG for these materials. The thermal diffusivity thusbecomes an important consideration for doubling highenergy lasers.
II. Experimental Setup
The experimental setup is depicted in Fig. 1. A 10-mW He-Ne laser beam is expanded and directed to-ward the test crystal. The polarization of the light wasadjusted parallel to the extraordinary axis, maximiz-ing the temperature response. 2 A cw CO2 laser isfocused onto the crystal with a ZnSe lens [f = 25 cm (10in.)]. Precise focusing is unimportant. An adjustableaperture was placed in front of the CO2 laser to aid inits attenuation. A number of different power levelswere used in various phases of the experiment rangingfrom 0.7 to 4 W. A shutter was used to control the CO2pulse length.
As illustrated in Fig. 1, the probe beam is collimatedvia Li and directed by a beam splitter toward thecrystal. The reflected fringe pattern is subsequentlydirected to a 1-D CCD array. To increase accuracyand aid in locating the approximate center of the CO2pulse (for maximum deviation) we passed the fringepattern through a slit (width _ 0.5 mm) oriented paral-lel to the proper crystal axis and to the detector array.The CCD array contained 1024 elements in 2.54 cm (1in.) and was controlled and analyzed using an EG&Gmodel 1460 optical multichannel analyzer (OMA).The detector scan time of 16.633 ms permitted us totake many sets of fringe data before and after the heat
pulse. The fringe center locations were typically de-termined to 1/30th of a fringe with the sample used.Alternatively, the crystal could be imaged via a lensafter the beam splitter into a TV camera. Collimationof the probe beam with Ll is thus unnecessary. Thefringes were subsequently monitored in real time withthe option to either record on video tape with the VCRor to store in the computer for analysis. A precisionruler in the object plane was imaged to determine themagnification.
The lithium niobate sample [2.54 X 2.54 X 0.32 cm (1X 1 X 0.125 in.)I was grown by Crystal Technology,Palo Alto, CA, from a congruent melt and was of opti-cal grade. It was cut for noncritically (900) phase-matched SHG, i.e, with the crystallographic b axisperpendicular to its aperture. It was enclosed be-tween two Watlow half-cylindrical fiber heaters, whosetemperature was controlled via a variable transformerby a Watlow 808 series temperature controller. Oneither side of the oven windows were placed, whichwere properly oriented so as not to produce confusinginterference fringe patterns. Closest to the CO2 laserwe used a NaCl window, which is transmissive at 10.6Am and throughout the visible and is relatively inex-pensive. The remaining window was quartz.
Since the NaCl window is water soluble, some spe-cial precautions were used to avoid any moisture con-densation. An infrared heat lamp was used overnightand while heating the oven to extend the lifetime of theNaCl window. Gradually increasing the voltage of thevariable transformer allowed for a slow rise in oventemperature, which minimized condensation. Thevariable transformer also allowed for better tempera-ture stability and control.
Various shutter speeds (from 0.125 to 1 s) were used,all resulting in temperature distributions that fit ex-tremely well to the proposed Gaussian model. Forpractical purposes a shutter speed of 0.125 s was usedmost often. We found that temporary surface damageoccurred at room temperature due to excessively tightfocusing or excess irradiation by the CO2 pulse (for CO2powers > 1.7 W forf = 13 cm (5 in.) and 5.5 W forf = 25cm at a shutter speed of 0.125 s). This damage fadedquickly on heating or after a few minutes at ambientand presented no serious problem.
111. Data Analysis and Results
Two methods of data analysis were attempted. Thefirst method is a simple one that was attempted usingthe fringe data taken from the 1-D CCD array andOMA. Representative fringe data at times before andafter the pulse are shown in Fig. 2. By measuring thedeviation of the fringes from their steady-state valuesthe thermal diffusivity was determined. Note fromEq. (2) that the location of the unperturbed fringeminima is given when
OPD0 = 2nd + 2n tanOy + f(x,y) = moX. (13)
Likewise from Eq. (10) the location of the perturbedfringe minima at a given time t1 is given when
Fig. 2. Illustrations of interference fringes from this experiment.(a) Reference fringes as seen in the crystal plane before heating(OPDO). Note that perfectly straight parallel reference fringes werenot required. (b) Interference fringes after the crystal has beenirradiated with the CO2 heat pulse deviating the fringes from theirunperturbed positions. (c) The 1-D CCD array scan of the referencefringes at to (solid line) and superimposed are the perturbed fringesat a later time t (dotted line). The intensity in arbitrary units is
plotted vs the distance along the y direction.
OPD(tl) 1 = OPD0 + - exp[ ( 4 jo)2 ]
ex-(v - yo)21 1 .(14)
X ex[- 4D t]
Clearly the change in OPD can be determined by sub-tracting Eq. (13) from Eq. (14). At t for fixed xposition, say x = x0, AOPD is given by
OPD(t1 ) = 9t exp[- 4D t] = (ml - o)X. (15)
Similarly, at a later time t2 , AOPD becomes
AOPD(t2 ) = - exp[- 4Dt = (M2-MO (16)
We note that the y axis here is representative of acrystal axis and is generally not in the same direction asthe wedge angle, although for maximum sensitivity it ispreferred. The difference in fringe order numberAm, 2 = m 1,2 - mo can be found by linear interpolationbetween the fringes. Therefore, if we divide Eqs. (15)and (16) by X and linearize them by taking the naturallogarithm, the diffusion coefficient can be determinedvia a least-squares regression fit:
(C, -1 -1n(AM1,2) = ln \tl 2 J 4DYt1,2
250 650y (channel no.)
Fig. 3. Data (denoted by circles) and the resulting quadratic fit [seeEq. (17)] computed from a least-squares regression. The curvaturedetermines the parameter 4Dyt for a given time t after the pulse.
After collecting and rearranging constants this be-comes
ln(Aml, 2 ) = A1 ,2 + Bl,2 + C1,2y , (17)
where 4Dytl,2 -1/C 1 ,2 - If the center (y0,x0 ) of theCO2 laser can be adequately determined visually andcross-checked from fitted data, one can use a simplerlinear regression (yo = 0) to determine D, simplifyingthe data analysis. The inverse of the slopes [ln(Am) vsy2] gives directly the coefficients 4Dytl,2 In eithercase, since At = t 2 - t is known either from the TVframe rate of 1/30 s or from the OMA scan rate, DY caneasily be determined from
1 1
C1y = C1 2 (18)4At
The diffusion coefficient along the x axis Dx may beobtained similarly.
Typically, -3 W of CO2 laser power was pulsed for0.125 s. Data were subsequently taken about 11/2 slater for t followed by a second set t2 -1 s later (i.e., At- 1 s). At least eight fringes were perturbed suffi-ciently to be used for analysis. A number of fringedata sets at various temperatures were analyzed in thismanner (about ten for each temperature) with a repre-sentative graph being shown in Fig. 3. We note thatfor our sample the fringes were approximately spacedalong the crystallographic c axis (extraordinary axis).The thermal diffusivity value corresponding to D inthe above equations is actually associated with thecrystallographic c axis (D, is measured). The resultswere consistent with the only available thermal diffu-sivity data at room temperature for lithium niobatethat were obtained using thermal conductivity mea-surements.8 -"1 To the best of our knowledge, this isthe first direct measurement of thermal diffusivity ofLiNbO3 at temperatures higher than room tempera-ture. (We note that anisotropy is reported to be <5%for lithium niobate.11) The results are shown in Fig. 4for temperatures ranging from 20'C up to 500'C.Precision values better than 10% were typical.
A second method attempted to analyze the data andto still keep costs reasonable is to make use of a TV
Fig. 4. Thermal diffusivity of lithium niobate as a function oftemperature as determined using the RSI. (The dashes connecting
the data points are for visual guidance.)
camera and to take advantage of commercially avail-able software. Modifications to an existing, commer-cially available fringe analysis software package'2 al-lowed both an automatic means of acquiring the dataand a means to determine the fringe center positions.
A video digitizer was used to digitize two video im-ages separated in time by a user-specified amount (inunits of 1/60th s). The digitization thus captured animage of the interference pattern produced shortlyafter the heating pulse has ended and another interfer-ence pattern some At later. Each fringe pattern isdigitized to 8 bits of intensity.
An IBM PC/AT microcomputer is then used to auto-matically determine the coordinates of the dark fringecenter positions and assign to them a fringe ordernumber. The fringe center positions are found tosubpixel resolution by using an adaptive fitting func-tion to the local intensity distribution near the darkfringe regions. Repeatability of this approach aftersmoothing varying fringe patterns is better than X/50peak-to-valley, with an accuracy in the determinationof the fringe center coordinates better than X/10 peak-to-valley.
Once an interferogram has been digitized, a set ofZernike polynomials4 "3 is used in a least-squares fit tothe x,y, and order number of the fringe center posi-tions. The Zernike polynomials are used in opticaltesting due to their orthogonality over the unit circle.The polynomial coefficients are subsequently used togenerate a uniform 63 X 63 grid of OPD values. Fromthis map, the overall tilt of the OPD map due to thefixed wedge of the crystal may be removed. Further-more, before thermally perturbing the crystal a refer-ence map of the OPD values of the crystal at equilibri-um is taken (OPDO). These values are then subtractedfrom each of the two thermally perturbed measure-ments [OPD'(t, 2)]. The resulting OPD map (AOPD)is due only to the heat distribution in the crystal. Thisis simply a 2-D Gaussian [Eq. (10)]. We subsequentlyfit the resulting AOPD(x,y,tl, 2) maps using standardleast-square regression techniques to a general 2-DGaussian. The resulting variances should thus deter-
mine both D. and DY as in Eq. (18); however, certainaspects associated with the analysis prevented theiraccurate determination, as discussed below. The av-erage results of this method at 201C and 3750C werefound to agree well with the first method of analysisshown in Fig. 4.
IV. Discussion
A number of drawbacks should be noted in the abovetwo methods of data analysis. First the logarithmused as part of the regression procedure accentuateserrors for small deviations (Am). This resulted inpoorer precision when end fringes were included in thedata set, limiting the number of useful fringes to eightor nine near room temperature. At elevated tempera-tures due to the increased temperature response2 morefringe data could be incorporated. Also to minimizethe effects of errors in our determination of D it wasgenerally found advantageous to take a large At.However, this method is still very accurate, giving aprecision of better than 10%.
A couple of potential problems were encountered inthe second data analysis procedure. The first is aconsequence of the fact that determinations of theOPD are ambiguous, since absolute phase informationis lost for intensity measurements. Since it is knownfrom Eqs. (4) and (5) that the OPD increases withincreasing temperature, the experimenter can unam-biguously identify the narrow part of the crystal wedgeas being in the direction in which the reference fringesbend on heating. One can therefore modify the soft-ware sufficiently to guarantee an upright Gaussianphase map. However, due to noise the absolute baseor zero level of the Gaussian had to be determined inthe fitting process (see Fig. 5). This was done via avisual check of the data and the fit.
The second problem in this second analysis wasassociated with using Zernike polynomials in that theyautomatically fit to a zero mean (setting piston to 0).This created problems not only in finding the absolutebase of the Gaussian phase distribution but also indetermining its anisotropy, since the base level of the xdirection differs from that of the y direction if anisot-ropy exists in the phase distribution. This is a definitedrawback of applying currently available commercialfringe data analysis packages and ultimately limitstheir effectiveness if used as is for this particular appli-cation. The logarithm created similar problems forthe regression as noted above for small phase values.We believe that alterations of this second method ofanalysis could result in a highly accurate determina-tion of the thermal diffusivity in two principal direc-tions of an anisotropic crystal with a single measure-ment, independent of fringe orientation.
V. Conclusions
We have demonstrated a relatively simple and all-optical nonsteady-state method of determining thethermal diffusivity of a material directly. The methodis quite general and should work for a variety of materi-als given the proper laser probe and (heat) pump wave-
Fig. 5. (a) Perturbed phase distribution (OPD) at a given timefollowing the CO 2 laser heat pulse. This is also simply proportionalto the temperature distribution. The data were determined byfitting fringe data to Zernike polynomials after irradiating the crys-tal with a heat impulse (OPD1) and subsequently subtracting out thereference phase (OPDO). (b) The x and y profiles of OPD1 (denotedby dots) and their corresponding fits to a 2-D Gaussian (solid line).
lengths. By measuring only the OPD we supersedethe need to measure temperature distributions, pow-ers, critical crystal parameters, dimensions, and an-gles. The method requires equipment that is not un-common for most optics labs and may be carried outwith relative ease. Possible analysis techniques toextend the accuracy and to determine the thermaldiffusivity in two nondegenerate crystal axes in a sin-gle measurement were pointed out. The thermal dif-fusivity of lithium niobate was measured as a functionof temperature for elevated temperatures and found todecrease by over a factor of 2 at 5000C compared toroom temperature.
This work is supported by Kirtland Air Force Baseunder contract F29601-84-0065. The authors wouldlike to acknowledge Frederic A. Hopf, Chih-Li Chu-ang, and David L. Kaplan of the Optical Sciences Cen-ter for their efforts and ideas in preliminary versions ofthis experiment; Daniel Johnson of AFWL for stimu-lating discussions; and Seung-Han Park for his assis-tance in using and taking data with the OMA.
References
1. D. Hon, "High Average Power Efficient Second Harmonic Gen-eration," in Laser Handbook, M. L. Stitch, Ed. (North-Holland,New York, 1979), Vol. 3, pp. 421-484.
2. R. A. Morgan and F. A. Hopf, "Measurement of the Tempera-ture Tuning Coefficient of Lithium Niobate Using NonlinearOptical Interferometry," Appl. Opt. 25, 3011 (1986).
3. S. E. Gustafsson, A. J. Hamdani, and E. Karawacki, "New Meth-od for Measuring Thermal Properties of Transparent Solids," J.Phys. 12, 387 (1979).
4. D. Malacara, Optical Shop Testing (Wiley, New York, 1978).5. C. Kittel and H. Kromer, Thermal Physics (Freeman, San Fran-
cisco, 1980).6. F. A. Hopf and G. Stegeman, Applied Classical Electrodynam-
ics: Nonlinear Optics, Vol. 2 (Wiley, New York, 1986).7. F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wi-
ley, New York, 1973).8. As quoted in A. Sigrist and R. Balzer, "Untersuchungen zur
Bidung von Tracks in Kristallen," Helv. Phys. Acta 50, 49(1977).
9. V. V. Chechkin, "Method of Measuring Thermal Conductivity,"Zavod. Lab. (Indust. Lab.) 46, 146 (Feb. 1980).
10. Yu. N. Venevtsev, S. A. Fedulov, Z. I. Shapiro, and V. P. Klyuev,Barium Titanate (Nauka, Moscow, 1973), p. 118, in Russian.
11. V. V. Zhdanova, V. P. Klyuev, V. V. Lemanov, I. A. Smirnov, andV. V. Tikhonov, "Thermal Properties of Lithium Niobate Crys-tals," Sov. Phys. Solid State 10, 1360 (1968).
12. The program used is called FAST! Video. Video fringe analysissoftware tools are available from Phase Shift Technology, Inc.,2601 N. Campbell Ave., Tucson, AZ 85719.
13. M. Born and E. Wolf, Principles of Optics (Pergamon, NewYork, 1980).
