Double Pulse Infrared Thermography

X. Maldague, A. Ziadi, M. Klein

Electrical and Computing Engineering Department

Université Laval, Quebec City (Quebec) G1K 7P4 Canada

• Corresponding author is: X. MaldagueElectrical and Computing Engineering DepartmentUniversité Laval, Quebec City (Quebec) G1K 7P4Canada

maldagx@gel.ulaval.ca, ph: ++ 1 418 656-2962, fax: ++ 1 418 656-3594

• Title for running head: Double Pulse Infrared Thermography

• Original date of receipt: _____________________________(as indicated by the Editor)

Keywords: thermal contrast, defect detection, heating pulse, thermal diffusivity

0. Abstract

In this paper we propose a simple enhancing protocol for pulsed infrared thermography. The

method is based on stimulating the specimen with two heating pulses separated by a short time

interval delta. This approach was tested on several specimens of low and high thermal diffusivity

and showed to provide higher thermal contrasts when an optimum interval delta is chosen. The

observed thermal contrast improvement is about a few percentage values. Lastly, a Matlab™ pro-

cedure is also proposed to link this interval to the thermal diffusivity of the specimen. In the

paper, the method is presented including theoretical basis, simulation and experimental results.

1. Introduction

Active infrared thermography is an established procedure for NDT. Basically, the specimen to

inspect is thermally stimulated and the subsequent temperature evolution is recorded to reveal

possible subsurface flaws. Details on active infrared thermography can be found into numerous

papers and conferences devoted on the topic, see for instance [1,2,3,4] among others. As for the

thermal stimulation schemes the common approaches in active infrared thermography are: pulsed,

modulated or vibrated with sometimes truly imaginative procedures described in the literature

(see for instance chap. 8 in [4]). In this paper, we concentrate on the pulsed approach based on

photo-thermal stimulation. In this case, the specimen is pulsed heated with lamps or flash lamps.

Figure 1 depicts this classical apparatus.

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Generally, a single pulse of variable length is applied to the specimen (to accommodate specimens

of low or high thermal diffusivity). In the present study, the hypothesis is that a double-pulse

approach could lead to improved thermal contrasts.

2. Theory

As it is well known, the time-frequency duality exists with mathematical tools such as the Four-

rier transform to go back and forth between these two domains [5]. In the case of pulsed infrared

thermography (PT), such duality was for instance exploited to link the pulsed and the modulated

approaches, giving birth to a technique called the pulsed phase thermography (PPT) [6].

When a specimen is pulsed heated, a burst of thermal waves are launched within the specimen.

These highly damped waves were first studied in the XIXth century [7] and have a propagation

speed which depends on their frequency (see for instance [4, section 9.1]). Of particular interest is

the thermal diffusion length µ expressed by:

(1)

with thermal conductivity k, density , specific heat C, modulation frequency in rad/s

( with f the frequency in Hertz) and thermal diffusivity . Equation (1) indicates that

thermal waves of low frequencies propagate deeper within the specimen. In fact for temperature

µ 2k ωρC⁄ 2α ω⁄= =

ρ ω

ω 2πf= α

2

images, inspection is limited to a depth under the surface of about one µ while phase images

probe 40 to 70 % deeper (in one study about graphite-epoxy specimens [8]).

Back to the time-frequency duality, Figure 2 shows this correspondence in the case of a square

pulse of amplitude 2A and duration ∆ (single-pulse case). Notice the sinc function in the fre-

quency domain. It is seen that the energy is not distributed uniformly across the frequency spec-

trum as it is in the case of an ideal Dirac pulse [5].

The basic idea of the double-pulse approach discussed here is thus to modify the frequency distri-

bution of the thermal waves launched into the specimen by changing the “shape” of the time-

domain thermal pulse. As result, a preferred frequency distribution of thermal waves is obtained

with enhanced results as discussed below.

Lets consider now two thermal pulses having duration ∆, amplitude A and separated by a specific

time interval δ. The new frequency distribution is shown on Figure 2 (double-pulse case). With

respects to the single-pulse case, it shows that the distribution is very similar at low frequencies

while high frequencies which do not contribute much to the detection due to their limited propa-

gation depths, eq. (1), are attenuated. This phenomena is very important since in fact it improves

the signal to noise ratio as shown below.

More formally, we can consider the Fourier transform F(ω) of a pulse f(t) in the time domain t.

Now, a summation of two of such thermal pulses with the second delayed by a δ time interval cor-

responds to:

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f (t) + f (t- δ) <-> F(ω) + F(ω)e-j ω δ = (1 + e-j ω δ) F(ω) (2)

where the exponential term corresponds physically to a phase shift (we recall that the imaginary

exponential decomposes itself as e-j ω δ = cos ωδ + j sin ωδ following the Euler’s relationship).

Next, simple mathematics indicates that eq. (2) implements in fact a filter whose transfer function

module is:

2 (1 + cos ω∆) (3)

with maximum response at ω = 2nπ/∆ (n is an integer) and first zero at ω = π/2∆. Clearly, in the

frequency span of interest:

[0, π/2∆] (4)

the behavior is similar to the one of a low pass-filter.

Hence, by modifying the spectral pulse content thanks to a delayed double stimulation, lower fre-

quencies are preserved and higher frequencies are reduced as it was shown in Fig. 2. Following

eq. (1), a better detection of deeper defects is thus expected with an improved signal to noise ratio.

For higher frequencies the filter effect does not affect much results since close to the surface

defects provides high response.

In a first analysis, one might think that the best would be to have a δ as large as possible since then

the span of frequencies reduces with enhanced thermal diffusion lengths, eq. (1, 4). However this

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is not the case since with well separated pulses the measurement simply repeats itself but with less

(half) energy. As it is well known, for a semi-infinite medium, after absorption of a Dirac pulse,

the temperature decay conforms in a first approximation to [10]:

(5)

where ∆T is the temperature increase of the surface, Q is the quantity of energy absorbed,

the thermal effusivity of the material and t the time. As this shows, with half injected

energy, one expects half surface temperature increase.

We thus have two phenomena competing against each other as δ enlarges and thus we might

expect δ to have an optimum value corresponding to the highest enhanced results. This will be

shown in the next sections.

3. Simulations

In this section, the analysis derived in the previous section is confirmed by a classical two-dimen-

sional thermal model developed in Matlab™ (details available in [9]). Various materials were

simulated. Figure 3 reports on the improvement I of the maximum thermal contrast ([4], eq. 5.33)

as function of δ interval in the case of a semi-infinite 4.5 mm thick aluminum plate with a flat-bot-

tom hole 1 cm diameter located 1 mm under the surface. On the figure, I is computed as the differ-

∆T Q

e πt------------=

e k ρC=

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ence between maximum thermal contrast of two-pulse and single-pulse cases. Considered cases

were a single 1000 kW - 20 ms duration thermal pulse and two 500 kW - 20 ms duration pulses

separated by a variable δ as indicated on the figure. Interestingly, the maximum thermal contrast

is a useful parameter for quantitative inversion of results (see chap. 10 in [4]).

As seen on Figure 3, the improvement steadily increases up to a maximum before offering worse

performance than the single-pulse case. This conforms to our discussion in previous section. This

behavior was also verified for all materials simulated. Table 1 lists the span of values for which

the two-pulse approach gives better thermal contrasts than the one pulse scheme (case of 1 mm

subsurface defects as in Figure 3).

4. Experimental results

Experiments were conducted to verify the previous findings. A classical active pulsed infrared

thermography set-up was used for this purpose (pictured on Figure 1). Basically, it comprises an

12 bit InSb liquid nitrogen-cooled 160 x 120 pixel 57 Hz infrared camera, two 6.4 kJ Balcar

Star*Flash 3 flashes all controlled and linked to a Windows PC computer so that experiments are

reproducible. The specimens (of slab types with flat-bottom holes) are mounted vertically 40 cm

away from the flashes. The light pulse was recorded with a fast NPN photo-transistor (Optek

OP804SL, 7 µs rise/fall times) connected to a memory scope (Tektronix TDX 320, 100 MHz).

Figure 4 shows a typical light pulse obtained with this experimental apparatus. It is about 20 ms

duration (at mid-amplitude). Either two of the flashes were fired simultaneously (single-pulse

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case) or they were fired one after the other with a δ time interval between the pulses (double-pulse

case). The variable δ was generated by a micro controller (MCS51) with pre-programmed δ val-

ues in ms.

Figure 5 shows the thermal contrast evolution obtained in the case of the 1010 steel slab for a 1

mm deep flat bottom hole subsurface defect with δ = 3, 10, 50 ms and also in the case of the sin-

gle-pulse case. Figure 6 shows how the improvement I of the maximum thermal contrast is

strongly related to δ. In fact, an optimum value (δoptimum) is observed as discussed previously.

Interestingly, the peek plotted on that figure corresponds to a thermal contrast increase of about 4

%.

The same phenomenon was observed on other materials as well. Table 2 summarizes the improve-

ment I of the maximum thermal contrast for both methods as function of δ, results are in the same

directions as those of Table 1. From Table 2, it is seen the experimental improvement I is related

both to the thermal diffusivity α of tested materials and also to δ. Interestingly, the "griddata"

function included in the curve fitting toolbox of Matlab™ (see Table 3) allows to “play” with val-

ues provided in Table 2 in order to optimize a given experiment.

Effect of the subsurface defect depth was also studied and showed not to affect δoptimum. However

the more the defect is deep, the less the energy reaches the defect and consequently the thermal

contrast difference gets smaller but still shows improvement in the double-pulse case (at δopti-

mum).

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5. Conclusion

In this paper we studied the effect of a double-pulse thermal stimulation and showed that such

procedure yields to an improved - by a few percentage values - thermal contrast when an optimum

δ separation is chosen (with respect to the classical single-pulse method). At equal injected

energy, this fact was explained by two phenomena competing against each other. First, separation

of the two thermal pulses reduces the importance of high frequency thermal waves launched into

the specimen with as result an enhanced signal to noise ratio. On the other hand, this separation

tends to reproduce the experiment but with less energy. An optimum δ separation thus exists for a

given experiment. Finally, a simple Matlab™ procedure linking δ to specimen thermal diffusivity

was also discussed.

6. Acknowledgement

Support of NSERC (Canada) is acknowledged.

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7. References

[1] QIRT conference proceedings (www.gel.ulaval.ca/qirt/)

[2] Thermosense 25th conference anniversary commemorative CD: I-XXV, vol. CDS91,

Bellingham: SPIE pub., 2003 (in press).

[3] Maldague X., “Introduction to NDT by Active Infrared Thermography,” Materials

Evaluation, 6[9]: 1060 -1073, 2002.

[4] Maldague X.P.V., Theory and Practice of Infrared Technology for Non Destructive Testing,

John-Wiley & Sons, 684 p., 2001.

[5] Gonzalez R.C., Wintz P., Digital Image Processing, Reading: Addison-Wesley, 431 p.,

1977.

[6] Maldague X., Marinetti S., “Pulsed Phase Infrared Thermography,” J. Appl. Phys, 79:

2694-2698, 1996.

[7] Ångstrom, MAJ, “New method of determining the thermal conductibility of bodies,” Phil.

Mag., 25: 130-142, 1863.

[8] Vavilov V. P., Marinetti S., “Pulsed Phase Thermography and Fourier analysis thermal

tomography,” Russian Journal of Nondestructive Testing, 35[2]: 134-145, 1999. [translated

from Defektoskopiya]

[9] Ziadi A., Étude et modification de l’impulsion de chauffage utilisée en évaluation non-

destructive des matériaux par thermographie infrarouge, M.Sc. thesis, Quebec city:

Université Laval, 107 p., 2003.

[10] Carlslaw H. S., Jaeger J. C., Conduction of Heat in Solids, Oxford University Press, 2nd

edition, 1959.

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Figure captions

Figure 1 - Classical experimental set-up for Pulsed Infrared Thermography experiments with

photograph of the rig used in this paper.

Figure 2- Time-frequency duality for two thermal pulses having duration ∆, amplitude 2A and

A and separated by a specific time interval δ.

Figure 3 - Simulated maximum thermal contrast difference as function of pulse separation δ in

the case of a semi-infinite 4.5 mm thick 1024 aluminum plate with a flat-bottom hole

1 cm diameter located 1 mm under the surface.

Figure 4 - Typical light pulse obtained with used experimental rig.

Figure 5 - Thermal contrast evolution obtained in the case of 1010 steel plate for a 1 mm deep

flat bottom hole subsurface defect with δ = 3, 10, 50 ms and also in the case of the sin-

gle-pulse case.

Figure 6 - Experimental maximum thermal contrast difference as function of pulse separation δ

in case of 1010 steel plate for a 1 mm flat bottom hole subsurface defect.

Table 1: Span of optimal ∆ values for simulated materials

Materialspan of optimal ∆

(ms)

Aluminum 1024 0 to 15

Steel 1010 0 to 50

Plexiglas 0 to 80

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Table 2: Evolution of the maximum thermal contrast difference (in 1/10 °C) as function of δ for some materials (defects at depth d from surface)

Material

δ (ms)

Aluminum 2024

(d = 3 mm)

Steel

(d = 3 mm)

Graphite-epoxy

(d = 1 mm)

Plexiglas

(d = 1 mm)

3 0.27 1.81 0.05 0.1

5 0.47 3.31 0.06 0.4

10 0.4 1.16 0.40 0.66

20 0.30 0 0.058 0.14

100 - -0.16 -0.22 -0.17

δoptim um (ms) 5 5 10 10

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Table 3: Simple Matlab™ procedure to optimize δ based on data of Table 2a

a. See Matlab™ help for more details (www.mathworks.com).

operation comments

ZI = griddata(xdelta, ydiffusivity, zcontrast, XI, YI, 'method');

where:• 'xdelta' is the δ vector called x-vector;• 'ydiffusivity ' is the thermal diffusivity vector (1D size) called y-vector;• 'zcontrast' is the 2D table (data of Table 2) called z - vector so that z = f(x,y);'method' is one of the available interpolation method in the Matlab™ curve fitting toolbox (lin-ear, cubic, nearest, v4).

XI and YI are 2 points (scalar value) for which an interpolated zcontrast is looked for. XI and YI can also be vectors, for instance:XI = [0 : 0.01 : 10] and YI = [2 : 0.3 : 7]. zcontrast is then a 2D table and represents f(XI,YI).

Note: in case of undefined - unavailable - values in zcontrast, it is necessary to provide Matlab™ with a NaN value (NaN = not a number). Ex: zcontrast = [4, 2, NaN, 5];

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