Thermal waves visualized by holographic interferometry

Guillermo H. Kaufmann and Charles M. Vest

The propagation of long thermal waves induced in thin aluminum plates by a slowly modulated heat sourcewas investigated by stroboscopic holographic interferometry. The effect of different modulation frequencieson the visualization of such waves is shown. Experimental results confirm analytical thermoelastic predic-tions. The application of long thermal waves to holographic nondestructive testing is demonstrated.

1. Introduction

When the surface of an absorbing medium is sub-jected to a periodic heat flux, the equation describingthe diffusion of heat into that medium has a wavelikesolution. The corresponding physical phenomenon isreferred to as a thermal wave. A periodic temperatureT exp(iwt) at sample depth x = 0 in a semi-infinitesolid results in a thermal wave, which in one dimensionis given by'

T(x,t) = To exp(-x/g) exp[i(wt - x)], (1)

where T(x,t) is the temperature rise above the equilib-rium ambient temperature and g is the thermal diffu-sion length

iu = (2a/w)112

(2)

with a = thermal diffusivity and w = 2 rf, being themodulation frequency. The thermal diffusivity isequal to

a = k/pc, (3)

where k = heat conductivity, p = density, and c =specific heat.

Equation (1) represents a thermal wave of wave-length given by

A = 27r/. (4)

As seen from Eq. (1), the thermal wave has a highlydamped amplitude. At a depth of one wavelength itsamplitude is reduced by a factor of exp(-27r) = 0.0019,

The authors are with University of Michigan, Department ofMechanical Engineering & Applied Mechanics, Ann Arbor, Michi-gan 48109-2116.

Received 26 January 1987.0003-6935/87-142799-05$02.00/0.© 1987 Optical Society of America.

which implies that the present semi-infinite solutioncan be used for a solid whose depth is one or twowavelengths. A few years ago, it was found that thethermal wave concept is necessary to understand pho-toacoustic signal generation.2 More recently it wasshown that these waves can be used to image micro-scopic subsurface inhomogeneities.3 ,4 By modulatinga laser beam to a frequency of -1 kHz, flaws weredetected several hundred micrometers beneath the ob-served surface of ceramic materials.

The purpose of this paper is to show that the propa-gation of long thermal waves through solid objects canbe investigated by means of holographic interferome-try, which displays the induced time-dependent ther-moelastic deformation of the object surface. By longwaves we refer to waves with lengths of the order ofseveral millimeters. The potential importance of theapplication of holography to thermal waves is that itproduces a whole-field display. Conceptually this isequivalent to parallel processing of the thermal wavesignature, as opposed to point-by-point scanning usedpreviously. Thermal waves induced in thin aluminumplates by a slowly modulated heat source were visual-ized by means of a stroboscopic technique, which in-volves two exposures of very short duration. Afterpresenting several interferograms, we show that theexperimental results confirm the analytical thermoe-lastic predictions. An application of long thermalwaves to flaw detection also is presented.

II. Experimental Technique

The specimens used for visualizing the thermalwaves were thin aluminum disks of thickness h = 0.5mm and 10 cm in diameter. They were clamped by a1-cm wide annular ring at their periphery. A smallspot, -0.5 cm in diameter, at the center of the backsurface of each disk was periodically heated at a verylow frequency. This surface was painted black to in-crease thermal absorbtivity. With this configuration,the thermal wave propagates radially away from this

15 July 1987 / Vol. 26, No. 14 / APPLIED OPTICS 2799

(a)

F(f)

Fo

J I- Laser2- Infrared Lamp

F.I

(b)

Fig. 1. Modulation of heat input and the laser beam used to recordstroboscopic holograms. (a) Schematic diagram of the chopper.The large opening is for the heat input, and the small openings arefor the laser. (b) The corresponding heat and laser modulations.

spot. The heat input was obtained by concentratinglight from a 250-W infrared lamp by means of a shortfocal length lens. The IR light was modulated bypassing it through a rotating chopper driven by a vari-able speed motor, as shown in Fig. (a). This chopperwas also used to shutter the laser utilized to record thehologram, as indicated in the same figure, In Fig. l(b)the corresponding square-wave heat input F(t) isshown, as is the pulse width of the laser light used torecord the hologram. The period of the heat input is f= 1/f, and its amplitude is F0. Two exposures, at timestj, and t2 , were recorded for each hologram, one at theend of the heating part of a period and the other at theend of the cooling part of the same period. Thus

t= 0.049a, t2 = tl + a12. (5)

For this square-wave heat input, the radial tempera-ture distribution T(r,t) is found by Fourier series solu-tion of the heat conduction equation to be

T(r,t) = k}; ierf[r/(2&g)

+2FoW; E~ 2n+ 1 exp[-rVl(2n + 1)w/(2a)J

-sin[(2n + 1)ct - r(2n + 1)col2a - r/41, (6)where r is the radial distance. For aluminum, a = 0.82X 10-4 m2/s and k = 2.07 X 102 W/mK.

The first term of the temperature distribution givenby Eq. (6) represents a monotonic temperature riseassociated with the average value of the square-waveheat input. The second one is the periodic part of thefield, which is the thermal wave.

Figure 2 is a schematic diagram of the setup used inthe experiments. A 50-mW He-Ne laser and Agfa

M irror

Ii rror

(Photographic Plate

- Pinhole

E MirrorFig. 2. Schematic diagram of the experimental setup.

Gavaert 10E75 plates were used to record the holo-grams.

The holographic system that was used providesfringes that are contours of the normal deformation wof the plate. These out-of-plane deflections were cal-culated from the fringe patterns by using5

co = N/(1 + cos), N = 0, 1, 2.... (7)

where N is the order of the bright fringes, X = 0.6328Atm, and 0 = 290 is the angle between the illuminatingand viewing directions.

Ill. Thermoelastic Deformation Analysis

To compute the thermoelastic response of the diskdiscussed above, a quasi-state solution in which theinfluence of the coupling of temperature and strainfields is neglected was used. This assumption is rea-sonable because of the low frequencies considered inthe present study. In the plane state of stress in a thinlinear elastic plate, the out-of-plane deflection w, inthe direction of the z axis, induced by thermal bendingis governed by the biharmomic equation

DV2 V2w = _V 2m.(8)

D is the flexural rigidity, which for a homogeneousisotropic plate is given by

D = Eh 3/[12(1 - 2)], (9)

where E is the Young's modulus of the plate material,and v is the Poisson's ratio.

In Eq. (8), m is given by

m 1 -j T(xy,z)zdz, (10)

where at is the coefficient of linear thermal expansion.For aluminum, E = 7.1 X 1010 Pa, v = 0.33, and at = 2.5X 10-5 K-1.

Assuming that the temperature distributionT(x,y,z) is constant in the direction of the z axis andthat it only depends on the radial coordinate r, thegoverning equation becomes

2800 APPLIED OPTICS / Vol. 26, No. 14 / 15 July 1987

~~~~~~~~~~~I

A

0.501

E

-o0.2

0 2 3 4

r(cm)

Fig. 3. Temperature distribution in an infinite thin aluminum diskdue to a square-wave heat input; f = 0.3 Hz, t = 60 s: , totaltemperature distribution [Eq. (4)]; - - -, monotonic rise [firstterm of Eq. (4)];--- - -, parabolic curve fitting of the solid curve.

Fig. 7. Two-exposure interferogram when the disk is heated steadi-ly.

(dr2 rdr [(dr2 rdr h

For the present case T(r) can be expressed as

T(r) = T(rto + t) - T(rto + t2),

(11)

(12)

where T(r,t), t, and t2 are given by Eqs. (6) and (5),respectively, to being the heating time before the firstholographic exposure was recorded.

Figure 3 is a plot derived from the temperaturedistribution given by Eq. (6) for an excitation frequen-cy f = 0.3 Hz and to = 60 s. It is seen that thetemperature gradient of the oscillating part of the field(the thermal wave) is significantly greater than that ofthe monotonically increasing part.

To simplify the integration of Eq. (11), the tempera-ture distribution given by Eq. (12) was fitted with aparabolic function by means of a least-squares pro-gram; hence

T(r) _ a + alr + a2r2.

Fig. 4. Stroboscopic interferogram: f = 0.3 Hz.

(13)

For comparison, a parabolic function which approxi-mately fits the temperature distribution T(r) shown inFig. 3 is also plotted in the same figure.

The boundary conditions of the clamped edge re-quire that the deflection and slope both be zero at r = b,where b = 4 cm. Thus

Wlrb = 0, dw 1r= = dr j~

(14)

Fig. 5. Stroboscopic interferogram: f = 0.6 Hz.

Substituting the temperature distribution T(r) givenby Eq. (13) into the biharmonic Eq. (11), carrying outthe integration, and using the boundary conditionsdefined in Eq. (14), we finally obtain the approximatethermoelastic deformation:

w(r) = 3a( + ) a2(r4-2b2 r2 + b4)h [16

+ C1 (2r3 - 3br + b

Fig. 6. Stroboscopic interferogram: f = 0.3 Hz. Laser pulse widthused to record interferogram 3 times that for Fig. 4.

(15)

IV. Experimental Results

Figures 4-7 are interferograms from a sequence ofexperiments in which the radial propagation of thethermal waves was visualized. The disk has a slot cutin it because we were interested in the interaction ofthe slot with the thermal wave. In all cases shown, the

15 July 1987 / Vol. 26, No. 14 / APPLIED OPTICS 2801

2

E

00 1 2 3 4

r(cm)

Fig. 8. Out-of-plane deflection as a function of radial distance:experimental results derived from an interferogram of a sam-

ple without a flaw; -- -, approximate thermoelastic analysis.

heating progressed for 60 s before the first holographicexposure was recorded.

Figure 4 shows the fringe pattern when the frequen-cy of the heat input was 0.3 Hz. This frequency corre-sponds to a wavelength of 5.86 cm and thermal diffu-sion length of 0.93 cm. The technique gives rise toseven interferometric fringes in this case. Figure 5 isequivalent to Fig. 4, except that the frequency was 0.7Hz. In this case, because the energy input is lower dueto the smaller period, only three fringes were obtained.

Figure 6 is an interferogram recorded in conditionsidentical to those of Fig. 4, except that the width of thelaser pulses used to record the hologram was 3 timesgreater than that leading to Figs. 4 and 5. Note thatthis affects the characteristic function of the fringes, sothat they are darker and more difficult to interpret.The extreme example of this effect is the use of time-average interferometry, which gives a very poor fringepattern in this case.

Figure 7 is a two-exposure holographic interfero-gram recorded when the plate is heated steadily ratherthan periodically. In this case there is no thermalwave, just a nonperiodic diffusion of heat. The sameheat source was used but without the chopper. Thelaser pulses used to record the hologram had exactlythe same timing as those used in Fig. 4. Even thoughtwice the amount of heat has been absorbed by theplate as in the case corresponding to Fig. 4, there areonly two fringes. This is clear empirical evidence thatthe fringes in Fig. 4 are indeed indicative of the ther-mal wave, i.e., the periodic part of the temperaturefield, which has steeper gradients than the nonperiodicfield. Furthermore, it suggests that only about onefringe in Fig. 4 is due to the nonperiodic component ofthe temperature field.

Figure 8 shows the out-of-plane deflection w plottedas a function of radial distance. The solid curve repre-sents experimental displacements determined by anal-

Fig. 9. Two-exposure interferogram when a flawed thick disk isheated steadily. One flaw extends radially upward at the top of thedisk, and a second flaw at the bottom of the disk is oriented in the

tangential direction.

Fig. 10. Stroboscopic interferogram of the same disk shown in Fig.9 when it is heated periodically with f = 0.1 Hz.

ysis of an interferogram of a disk with no slot for a heatinput frequency of 0.3 Hz. These displacements werecalculated by using Eq. (7). For comparison, the ap-proximate thermoelastic solution given by Eq. (15) isalso plotted in the same figure in the dashed line. Asseen, reasonable agreement is obtained between ex-perimental and approximate analytical results.

V. Application to Holographic Nondestructive Testing

The application of long thermal waves to holograph-ic nondestructive testing was demonstrated by using 5-mm thick aluminum disks as test objects. Flaws in theform of slots 1 mm deep with 4- X 12-mm cross sectionwere milled into the back surface of the disks. Eachdisk was clamped as in the previous experiments butuniformly heated from its back surface. Thus thethermal wave propagates normal to the observed ob-ject surface.

Figure 9 is a two-exposure holographic interfero-gram recorded while the plate was heated steadily.There is a small perturbation in the vicinity of theradially oriented slot at the top of this plate and also aclosed loop fringe near the slot at the bottom of the

2802 APPLIED OPTICS / Vol. 26, No. 14 / 15 July 1987

plate, which is oriented tangential to the radius. Thisfigure represents a typical result of holographic nonde-structive testing with thermal loading.

Figure 10 is the fringe pattern obtained with thesame test object as in Fig. 9 when a thermal wave isstimulated by a periodic heat input with 0.1-Hz fre-quency. It is seen that the perturbation of the fringesindicative of the radially oriented flaw is quite obvious.

VI. Conclusions

The propagation of long thermal waves throughmetal objects was investigated. These waves werecreated when the surfaces of thin aluminum plateswere subjected to a slowly modulated heat source. Inthis manner surface deformations large enough to beobserved by holographic interferometry were induced.A stroboscopic technique was used to visualize theradial propagation of the thermal wave. Predictedvariations of fringe properties, such as visibility anddensity, were verified experimentally. The measuredout-of-plane deflections agree quite well with the ana-lytical thermoelastic solution.

Preliminary results showed that flaw detection canbe improved when thermal waves, rather than unmod-ulated heating, are used in holographic nondestructivetesting. Increasing the modulation frequency is desir-able. However, as frequency is increased, either more

intense heating or increased interferometric sensitiv-ity is required. We are exploring the use of discretephase shift interferometry for the later purpose.

This paper was presented at the 1985 Annual Meet-ing of the Optical Society of America held in Washing-ton, DC, in October. The research was sponsored bythe U.S. National Bureau of Standards, Nondestruc-tive Evaluation Program. One of the authors, G. H.Kaufmann, is on leave from the Instituto de FisicaRosario, Republica Argentina, under a fellowship fromthe Consejo Nacional de Investigaciones Cientificas yTecnicas.

References1. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids

(Oxford U.P., London, 1959).2. A. Rosencwaig, Photoacoustics and Photacoustic Spectroscopy

(Wiley, New York, 1982).3. Y. H. Wong, R. L. Thomas, and J. J. Pouch, "Subsurface Struc-

tures of Solids by Scanning Photoacoustic Spectroscopy," Appl.Phys. Lett. 35, 368 (1979).

4. J. J. Pouch, R. L. Thomas, Y. H. Wong, J. Schuldies, and J.Srinivasan, "Scanning Photoacoustic Microscopy for Nonde-structive Evaluation," J. Opt. Soc. Am. 70, 562 (1980).

5. C. M. Vest, Holographic Interferometry (Wiley, New York,1979).

6. W. Nowacki, Thermoelasticity (Addison-Wesley, Reading, MA,1962).

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15 July 1987 / Vol. 26, No. 14 / APPLIED OPTICS 2803