Flying spot thermography is an efficient thermographic non-destructive technique that has been widely applied to detect surface breaking cracks on large parts in a fast way. It consists in heating the sample surface with a laser spot that moves at a constant speed and recording the surface temperature with an infrared camera. The presence of a crack hinders heat propagation and produces a discontinuity in the temperature at the surface, that is, the signature of the crack. In this tutorial, we address two quantitative applications of flying spot thermography: the measurement of thermal diffusivity and the determination of the width of cracks. We present derivations of the analytical expressions of the surface temperature, from which specific methods for the quantitative assessment of thermal diffusivity and crack width are introduced. We show that the methodology is also valid for the complementary configuration, with a static laser spot exciting a moving specimen, as in industrial production chains. The methodology is illustrated with experimental data on sound and artificially cracked samples. We conclude the tutorial by discussing the limitations and future perspectives of this technique.

## I. INTRODUCTION

The so-called photothermal effect consists in heat generation by light absorption. In the case of opaque samples, heat is generated just at the sample surface and then propagates, mainly by conduction, into the bulk and into the surroundings. Consequently, several physical effects take part: temperature rise, thermal expansion, generation of acoustic waves, change of the refractive index, etc. By measuring these effects, the thermal properties of the sample can be retrieved. In particular, the temperature rise of the sample surface can be recorded in a noncontact way by means of infrared video cameras, the so-called infrared (IR) thermography.

Optically excited IR thermography is an efficient tool for measuring the thermal diffusivity of solids as well as detecting and sizing hidden defects in industrial parts.^{1} By using a flat and homogeneous illumination of the whole sample surface, heat propagates in the in-depth direction, and, therefore, the surface temperature recorded by an IR camera is sensitive to the in-depth thermal diffusivity of the sample. Moreover, the presence of subsurface defects (voids, inclusions, corrosion, delaminations, etc.) modifies the heat flux in such a way that the surface temperature above the defect is modified with respect to the healthy zones, betraying the presence of these defects. On the other hand, a focused laser beam produces a lateral heat flux and, therefore, a radial gradient of the surface temperature, which is sensitive to the in-plane thermal diffusivity of the sample. Besides, this lateral heat flux is modified by the presence of a vertical crack, a hard to detect flaw, producing a discontinuity in the surface temperature that reveals the fissure position. For this application, a spherical lens, which produces a circular laser-spot at the sample surface, is usually employed. However, a cylindrical lens, producing a laser-line at the surface, has been proposed to sense long cracks more efficiently.

In the last few decades, taking advantage of improvements in IR cameras and lasers, several methods to measure thermal diffusivity and size buried defects have been developed. In most applications, both the laser and the sample remain at rest. However, in the nineties of the last century, with the aim of testing large surfaces in short times, and following the seminal work by Kubiak,^{2} the so-called “flying spot” thermography was introduced. It consists in heating the sample surface with a moving laser spot and detecting the surface temperature field with an infrared video camera.^{3} Since then, flying spot thermography has been widely developed and has been proven as a suitable technique for detecting and sizing vertical cracks^{4–19} and buried delaminations^{20–22} in a very fast and efficient way. This technique has also been used to measure the local thermal diffusivity of big parts,^{23–25} obtaining the thermal diffusivity map of the sample as a final product.^{26}

Recently, a complementary setup has been proposed, where the laser spot is at rest and the sample is moving at a constant speed. This configuration is addressed to in-line production or in-line quality control processes in factories, where defects must be detected in real time without stopping the production chain. This configuration has been used to detect and size the width of vertical cracks^{27,28} and to measure the thermal diffusivity^{29,30} in order to assess the homogeneity of the production.

In the first part of this tutorial (Secs. II–IV), we present a comprehensive overview of the theoretical framework to calculate the surface temperature of a sample when it is illuminated by a laser spot moving at a constant speed. We show that the temperature profiles along the laser direction of motion and in the direction perpendicular to it behave linearly as a function of the distance to the laser spot, from whose slope the thermal diffusivity of the sample can be measured. This linear relation allows us to extend the concept of thermal diffusion length (the distance with respect to the laser spot at which the temperature is still significant), which is typically defined for both sample and laser at rest, to the case of laser in motion. We also obtain an analytical expression of the surface temperature for a sample containing an infinite vertical crack when the laser spot is moving in the direction perpendicular to the crack. Finally, we demonstrate that the surface temperature in the complementary configuration (the moving sample and the laser at rest) can be obtained by performing a mere change of coordinates.

In the second part of this tutorial (Secs. V–VII), we explain in detail the experimental setups for flying spot thermography and for moving samples. Then, we show several examples of thermal diffusivity measurements and crack width sizing. Finally, we present the future trends in flying spot thermography taking advantage of the latest technological advances: IR cameras with enhanced spatial resolution and frame rate, IR macro-lenses with high optical quality, laptops with improved processor speed, image treatment software, etc.

## II. BACKGROUND: SAMPLE AND LASER AT REST

Let us consider a homogeneous, isotropic, semi-infinite, and opaque sample that is adiabatically isolated from the surroundings. The sample surface is illuminated by a laser beam of the Gaussian profile and focused to a radius *a* (at 1/e^{2}), as depicted in Fig. 1. We look for the surface temperature corresponding to two time profiles of the illumination corresponding to the most successful configurations in laser spot IR thermography: a harmonically modulated laser beam and a brief laser pulse. Then, we calculate the surface temperature of a sample containing an infinite vertical crack.

### A. A harmonically modulated laser beam

*P*, harmonically modulated at a frequency

_{o}*f*(

*ω*= 2π

*f*), heats the sample. As a result, the sample temperature oscillates at the same frequency as the excitation: $T(r,z,t)=T(r,z) e i \omega t$, where $r= x 2 + y 2$. By introducing this expression in the heat diffusion equation, the spatial dependence of the temperature satisfies a modified Helmholtz equation,

*q*

^{2}=

*iω*/

*D*, with

*D*being the thermal diffusivity of the sample. Due to the cylindrical symmetry of the problem, it is useful to work in the Hankel space,

^{31}where the Helmholtz equation, a partial differential equation, reduces to

*β*

^{2}=

*δ*

^{2}+

*iω*/

*D*,

*δ*is the conjugate variable of the radial coordinate,

*r*, in the Hankel space, and $ T ~( \delta , z)$ is the Hankel transform of the temperature $T( r , z)$. The general solution of Eq. (2) is

*A*and

*B*are obtained from the boundary conditions. As the second term on the right hand side of Eq. (3) increases exponentially when penetrating the sample, coefficient

*B*(

*δ*) has to be null in the current case of a semi-infinite specimen. Factor

*A*is obtained from a boundary condition with a prescribed heat flux,

*η*is the power fraction absorbed by the sample, and

*K*is the thermal conductivity. Proceeding in this way, the Hankel transform of the temperature writes

*a*= 0), Eq. (6) reduces to

*e*with respect to the position of the laser spot. Therefore, this parameter is an estimation of the distance at which the temperature oscillation vanishes. Note that

*μ*increases with the thermal diffusivity of the sample and decreases with the modulation frequency.

As can be observed in Eq. (7), both $ ln( | T | R)$ and Ψ behave linearly as a function of the distance to the excitation point, with the same slope $m=1/\mu = \pi f / D$. These linearities provide a simple method, based on a linear fit, to measure the in-plane thermal diffusivity of solids. By performing calculations using Eq. (6), it can be seen that a finite laser spot size does not modify the above mentioned slopes. This result is shown in Fig. 2. Calculations have been performed for the surface temperature of AISI-304 stainless steel (*D* = 4 mm^{2}/s, K = 15 W m^{−1} K^{−1}) with *f* = 1 Hz and *a* = 0.5 mm.

### B. A brief laser pulse

*Q*impinges the sample surface. We look for the time evolution of the temperature rise above the ambient after the pulse. A common method to solve transient thermal problems is to work in the Laplace space. The Laplace transform of the heat diffusion equation is

_{o}^{33}

*p*

^{2}=

*s*/

*D*,

*s*is the conjugate variable of time in the Laplace space, and $ T \xaf( r , z , s)$ is the Laplace transform of the temperature $T( r , z , t)$. By comparing Eqs. (1) and (8), we conclude that $ T \xaf$ can be obtained from the modulated solution by changing the power

*P*and the factor

_{o}*iω*by the pulse energy

*Q*and by

_{o}*s*. Accordingly, starting from Eq. (7), we obtain the Laplace transform of the temperature in the case of a brief laser pulse of negligible size,

^{34}

*a*= 0) reduces to

We define the thermal diffusion length as the distance from the laser spot at which the temperature is reduced by a factor *e* with respect to the temperature at the center of the laser spot at the same instant. According to Eq. (13), it is $\mu = 4 D t$. Moreover, Eq. (12) allows measuring the in-plane thermal diffusivity of a sample by fitting the temperature profile at a given instant to a Gaussian curve^{35,36} or by fitting Ln(*T*) to a parabolic curve.^{23} This behavior is shown in Fig. 3 for AISI-304 with *a* = 0.5 mm, *Q _{o}* = 1 J, and

*η*= 1.

### C. The sample contains an infinite vertical crack

*x*= 0, as plotted in Fig. 4. The sample is illuminated by a modulated laser beam of power

*P*and the Gaussian profile of radius

_{o}*a*(at 1/e

^{2}of the intensity). The center of the laser spot is located at a distance

*d*from the crack. Both sample and laser are at rest. Adiabatic boundary conditions at the sample surface are assumed. The crack hinders heat propagation and acts as a thermal resistance. The temperature at any point of the sample is given in Eq. (9) in Ref. 37, where a detailed step by step demonstration is presented,

*x*and

_{o}*y*represents the contribution of any surface point illuminated by the laser spot weighted by its intensity, $ R o= ( x \u2212 x o ) 2 + ( y \u2212 y o ) 2 + z 2$ and $ r o= ( x \u2212 x o ) 2 + z 2$.

_{o}*R*is the thermal contact resistance of the crack, which is related to the crack width,

_{th}*w*, through the equation

*R*=

_{th}*w*/

*K*(see page 20 in Ref. 33).

_{air}*x*axis does,

^{38}

*R*always appears multiplied by

_{th}*K*, the thermal conductivity of the sample. This means that it is easier to detect narrow cracks in good thermal conductors than in thermal insulators.

As can be observed that the temperature of the cracked sample given by Eq. (15) has two terms: the first one corresponds to the homogeneous sample while the second one accounts for the effect of the crack. As an illustration of the effect of the crack, Fig. 5 shows calculations, using Eq. (15), of the temperature profile along the *x* axis corresponding to an AISI-304 sample with a vertical crack at plane *x* = 0. The energy of the laser pulse is *Q _{o}* = 1 J (

*η*= 1), the radius is

*a*= 0.4 mm, and the distance to the crack is

*d*= 1 mm. Temperature profiles are shown for two times after the laser pulse: 75 and 100 ms. For each time, three crack widths are analyzed:

*w*= 0.25, 2.5, and 25

*μ*m. Note that even submicrometer wide cracks can be detected.

## III. THEORY OF FLYING SPOT THERMOGRAPHY: LASER IN MOTION AT CONSTANT SPEED

In this section, we calculate the temperature field of a sample when a CW laser spot scans the sample surface along a straight line at constant speed. Figure 6 shows a scheme of the problem. The laser spot was switched on at *t _{o}* and is moving at constant speed

*v*along the

*x*axis, crossing the origin of coordinates at

*t*= 0. The goal is to calculate the temperature at any time

*t*.

*dt*by a CW laser of power

*P*centered at point (

_{o}*b*,0,0). The resulting small temperature rise with respect to the ambient associated with an energy

*dQ*=

_{o}*P*is obtained from Eq. (11),

_{o}dt*v*along the

*x*axis, the temperature at a point (

*x*,

*y*,

*z*) at time

*t*due to the energy delivered by the laser at time

*τ*(

*τ*<

*t*) is given by

Note that for *τ* = 0, the laser is at the origin of coordinates.

*x*,

*y*,

*z*) at time

*t*is obtained by integrating Eq. (17) since the laser was switched on at

*t*until

_{o}*t*,

Equations (18) and (19) give the temperature evolution at any time after the laser was switched on. This evolution consists in a transient period, when the temperature is increasing, followed by a steady state characterized by a constant temperature.

*a*= 0), when the steady state has been reached (

*t*= −∞) and when the laser spot is at the origin of coordinates (

*t*= 0), Eq. (18) has an analytical solution (see page 267 in Ref. 33),

*T*), for an AISI-304 stainless steel sample, which is illuminated by a CW laser moving along the

*x*axis at

*v*= 2 cm/s using Eq. (20). On the left, we show the surface temperature (

*x*,

*y*,0), and on the right, the cross-section temperature at plane

*x*= 0. As can be observed, the laser movement induces a strong asymmetry in the surface temperature, which increases with the laser speed but decreases with the sample thermal diffusivity. However, the cross section remains symmetric.

*x*. In front of the laser spot (

*x*> 0), the slope of $ Ln ( T|x|)$ verifies

*x*< 0), $ Ln ( T|x|)$ is flat without any information on the sample diffusivity or laser speed. Anyway, this flat behavior is useful for the fine-tuning of the experimental setup. This behavior is shown in Fig. 8(a) for an AISI-304 sample and three values of the laser speed.

*y*axis and the in-depth profile write, respectively,

This behavior is shown in Figs. 8(b) and 8(c) for the same material and laser speeds as in Fig. 8(a). Note that for a better comparison between the three figures, the same spatial scale has been used. In particular, by applying Eq. (25) to the symmetric branches of the transverse profile, the thermal diffusivity of the sample can be obtained easily. In fact, as this slope is half of the value corresponding to the longitudinal profile and taking into account the symmetry at both sides of the origin of coordinates, more experimental data can be used before reaching the noise level. Accordingly, in isotropic samples, using the transverse profile is more reliable than the longitudinal one. In the case of a finite laser spot radius (*a* ≠ 0), there is some rounding of the temperature profiles around the laser spot, but the slopes remain unchanged.

*T*I

*x*I,

*T*I

*y*I, and

*T*I

*z*I are reduced by a factor

*e*with respect to the value at the center of the laser spot,

By comparing these expressions to the one corresponding to a modulated laser beam remaining at rest, it can be concluded that the laser speed plays a similar role as the frequency: the higher the speed (frequency) the shorter the thermal diffusion length. In this way, following the usual criterion, a sample behaves as thermally thick when its thickness *L* > 2*μ _{z}*, while it behaves as thermally thin if

*L*< 0.5

*μ*. Accordingly, a slab of a given material behaves as thick or thin depending on the laser speed.

_{z}As mentioned above, the linearities of the profiles of $ Ln ( T|x|)$, $ Ln ( T|y|)$, and $ Ln ( T|z|)$ are only valid when the steady state has been reached. As an illustration in Fig. 9(a), we show the longitudinal profile of $ Ln ( T|x|)$ for AISI-304, using a tightly focused laser spot (*a* = 0) moving at *v* = 1 cm/s. The profile is calculated when the laser spot is at the origin of coordinates [*t* = 0 in Eq. (19)]. Four switching on times are considered: *t _{o}* = −0.1 s, −0.2 s, −0.5 s, and −∞. As can be observed, for times shorter than 1 s, the linearity is not yet established. From Eq. (19), we can determine precisely the minimum time to reach the steady state,

*t*

_{o}_{min}, as a function of the laser velocity and sample diffusivity. Figure 9(b) shows the values of

*t*

_{o}_{min}vs the laser speed for four thermal diffusivities covering a wide range from thermal insulators to good thermal conductors. As can be seen, the absolute value of this minimum time decreases with the laser velocity but increases with the sample diffusivity. Anyway, as the flying spot method is developed to study large surfaces in short times, velocities

*v*≥ 1 cm/s are used. For these speeds, the steady state is reached in a few seconds, even for the most conducting samples.

*v*along the

*x*axis to the left while the laser spot remains at rest centered at the origin of coordinates (see Fig. 10), is obtained directly from Eq. (18) by performing a mere change of the frame from the sample to the laser spot,

It is worth mentioning that Gupta and co-workers found a similar solution for a moving fluid with an alternative approach by solving the heat diffusion equation with a transport term.^{39} Note that the integral starts at *t *= 0, indicating that the laser was switched on at that time. As a direct consequence, all the results and conclusions of this section are of direct application to this configuration (laser at rest and sample moving to the left, see Fig. 10). In particular, Figs. (7) and (8) as well as Eqs. (22), (25), and (26) are also valid for this configuration. This means that the linear relationships involved in Eqs. (22) and (25) can be applied to measure the thermal diffusivity of a moving sample.

In this tutorial, for the sake of clarity, we have dealt with the simplest case of an isotropic, opaque, and semi-infinite sample. Expressions for the surface temperature in the most general case, where the sample is either isotropic or anisotropic, opaque or transparent, thick or thin, can be found in Refs. 25 and 29. Moreover, in these works, the influence of heat losses by convection and radiation is also taken into account. As a practical summary, the linear relations of the temperature profiles shown in Fig. 8 are valid for all kind of samples. Accordingly, Eqs. (22) and (25) can be applied to measure the thermal diffusivity of a sample regardless its optical, geometrical, and thermal properties, and they are unaffected by heat losses.

## IV. THEORY OF FLYING SPOT THERMOGRAPHY ON SAMPLES CONTAINING VERTICAL CRACKS

*x*= 0. A CW laser of power

*P*and the Gaussian profile of radius

_{o}*a*(at 1/e

^{2}) is moving along the

*x*axis to the right at constant velocity

*v*. The temperature profile along the

*x*axis at time

*t*is given by the convolution integral of Eq. (15),

Note that in the convolution integral, *t _{o}* indicates the time when the laser was switched on, and that at

*t*= 0, the laser is at the origin of coordinates, i.e., on top of the crack. Moreover, by performing a simple Galilean transformation in Eq. (28), we obtain the temperature profile when the laser remains at rest while the material is moving at constant velocity along the

*x*axis to the left [see Fig. 11(b)]. This is obtained by replacing exponents (

*x*−

*vτ*)

^{2}and (

*x*−

_{o}*vτ*)

^{2}by (

*x*+

*vt*−

*vτ*)

^{2}and (

*x*+

_{o}*vt*−

*vτ*)

^{2}, respectively.

*x*axis for an AISI-304 sample with an infinite vertical crack located at

*x*= 0. The laser spot is moving from left to right at constant speed

*v*= 10 mm/s. The power of the laser is

*P*= 1 W and its radius is

_{o}*a*= 0.4 mm. Four laser positions are shown:

*x*= −0.8, −0.4, 0.4, and 0.8 mm. For each position, three crack widths are considered:

*w*= 25, 2.5, and 0.25

*μ*m. As in the case of a static laser spot, the signature of the crack is an abrupt temperature discontinuity, which increases with the crack width. Besides, it can be seen that the temperature jump depends on the distance between the laser spot and the crack. To quantify the optimum position of the laser with respect to the crack to obtain the highest temperature jump, we define the normalized temperature contrast at the crack, Δ, as the temperature difference at both sides of the crack divided by the temperature at the position of the laser spot,

*P*.

_{o}Figure 13 shows the dependence of the temperature contrast, Δ, as a function of the distance of the laser spot with respect to the crack, *d*. Calculations are performed for the same material and parameters as in Fig. 12, i.e., AISI-304, *v* = 10 mm/s and *a* = 0.4 mm. As can be observed that there are two extrema, one maximum when the laser spot is reaching the crack and a minimum when the laser spot has already crossed the crack. The minimum is produced when the distance between the laser spot and the crack verifies $d\u2248a$, whereas the maximum appears for $|d|\u2248a/2$. This last expression means that the laser spot overlaps the crack. However, from an experimental point of view, it is better to avoid the laser spot overlapping the fissure to prevent the light from entering inside the crack and disturbing the temperature profile. Accordingly, the best option is using $|d|\u2248a$, i.e., when the laser spot is just touching the crack without overlapping it. On the other hand, note that the temperature contrast Δ is very sensitive to the crack width, *w*. This result suggests that the width of the crack could be obtained from the direct measuring of Δ, provided *v*, *a*, and *d* are known. However, as it will be shown in Sec. VII, the sharpness of the temperature jump is softened due to optical aberrations of the IR lens and by diffraction.^{40} Consequently, it is wiser to fit the whole temperature profile along the direction of the laser motion to Eq. (24) to retrieve the crack width. More precisely, the temperature profile when the laser spot is just reaching the crack (*d* ≈ −*a*) or when the laser spot has just overpassed the crack (*d* ≈ *a*).

## V. EXPERIMENTAL SETUPS

Figure 14 shows the schemes of two complementary IR thermography setups: laser or sample in motion. The common part of both configurations consists in a CW laser of the Gaussian profile, a lens to focus the laser beam on top of the sample surface, an IR video camera, and a Ge window (or a dichroic mirror) placed in front of the camera lens. This element is used to direct the laser beam perpendicularly to the sample surface while not blocking the infrared radiation emitted by the sample. The use of a microscope lens for the IR camera is highly recommended in order to enhance the spatial resolution. In our laboratory, we use a 320 × 256 pixel sensor together with a microscope lens with a magnification ratio of 1:1, leading to a spatial resolution of 30 *μ*m/pixel. It is also advisable to use an IR camera with a high acquisition speed (above 1 kHz frame rate). Regarding the samples, it is highly advisable to cover the surface of metallic specimens by a thin graphite layer to enhance both the absorption to the laser and the IR emissivity. Moreover, this graphite layer serves to homogenize the sample surface (avoiding the disturbing effect of stains, color spots, scratches, etc.) and to reduce the Narcissus effect.^{41}

Figure 14(a) shows the configuration where the sample is at rest while the laser beam is scanning the sample surface. A dual axis galvanometer optical scanner is used to control the spatial displacement of the laser beam. The laser beam is focused by means of an F-Theta lens, which is specially designed for laser scanning systems, minimizing distortion and vignetting. Its main feature is that the laser spot displacement at the focal plane is proportional to the laser beam angle. Figure 14(b) shows the configuration where the laser beam is kept fixed while the sample is mounted on a cart, which slides along a track, mimicking the in-line production chain at laboratory scale. An electric engine moves the cart at constant speed.

As an example, we show in Fig. 15, the amplitude thermograms corresponding to a laser spot lock-in thermography experiment performed under static conditions, i.e., where both laser and sample remain at rest. The radius of the laser spot is *a* ≈ 200 *μ*m, the frequency is *f* = 1 Hz, and the laser power varies in the range *P _{o}* = 50 mW–0.5 W depending on the thermal conductivity of the sample. Figure 15(a) corresponds to an AISI-304 stainless steel sample, which is an intermediate thermal conductor (

*D*= 4.0 mm

^{2}/s and

*K*= 15 W m

^{−1}K

^{−1}). As it can be seen, the isotherms are concentric circumferences indicating the thermal isotropy of this material. Figure 15(b) refers to a composite where the polymeric matrix is reinforced by unidirectional carbon fibers up to a volume fraction of 61%. Fiber orientation introduces an anisotropy in the thermal transport properties: they are enhanced in the fibers direction but reduced in the perpendicular direction. Accordingly, the thermogram shows elongated isotherms, indicating that the fibers are aligned horizontally. Finally, Fig. 15(c) corresponds to two AISI-304 parallelepiped blocks that are put in contact, one on top of the other one, under some pressure. The two surfaces in contact are mirror-like polished. In this way, a very narrow vertical crack (≈0.5

*μ*m wide), which acts as a thermal barrier, is simulated. In Fig. 15(c), the laser spot is focused on the lower block, close to the interface. As can be observed, the isotherms show a clear discontinuity, allowing detection of even such a narrow crack.

Figure 16(a) shows a thermogram corresponding to an AISI-304 sample moving to the left at 4.0 cm/s while the laser spot remains at rest. The radius of the laser spot is *a* ≈ 200 *μ*m and the laser power is *P _{o}* = 1.0 W. This thermogram has been recorded under steady state conditions, which, according to Fig. 9(a), are reached in less than 1 s after the laser was switched on. Even though the material is isotropic, the isotherms are elongated in the direction of the sample movement. However, unlike the case of an anisotropic sample under static conditions, these elongated isotherms are not concentric with respect to the laser spot, showing a comet-like feature. Note that this thermogram is quite noisy. As the system is in the steady state, the signal to noise ratio can be improved by averaging a number of thermograms. Figure 16(b) shows the result of averaging several hundreds of consecutive thermograms. This resulting thermogram shows a remarkable noise reduction that allows retrieving the sample thermal diffusivity with high accuracy using the linear relations corresponding to the longitudinal and transverse profiles whose slopes are given by Eqs. (22) and (25). Note that the price to be paid for this noise reduction is that it becomes impossible to perform local thermal diffusivity measurements. An alternative noise reduction procedure, which allows local diffusivity measurements, is using post-processing methods on a single thermogram, such as singular value decomposition (SVD) or others.

^{42}As a final comment on this figure, exactly the same thermogram would be obtained if the sample were at rest and the laser spot were moving to the right at 4.0 cm/s.

Figure 17 shows a sequence of thermograms corresponding to an AISI-304 sample containing an infinite vertical crack of 2.5 *μ*m wide, when the sample is moving at 4.0 cm/s to the left while the laser is at rest. The infinite vertical crack of calibrated width was produced with two parallelepiped blocks of AISI-304 with two metallic tapes of 2.5 *μ*m thick in between. This configuration guarantees that the air gap between the AISI-304 blocks is of the same width as the tapes thickness. We have selected eight thermograms showing the crack crossing the laser spot. As in the case of static conditions, the crack produces a temperature discontinuity, which is a clear fingerprint of the flaw.

It is worth noting a more subtle effect related to the sample motion. In Fig. 12, the temperature profile along the direction of the laser motion (the *x* axis) is calculated using Eq. (28). At all times, the temperature at the discontinuity is higher in the laser side. However, as it can be seen in Figs. 17(d)–17(g), once the crack has crossed the laser spot, the non-central longitudinal thermal profiles away from the laser spot show an inverted discontinuity with respect to the central one. This means that for those non-central profiles the temperature at the discontinuity is higher at the other side of the laser spot, as it is clearly visible in the outer part of the thermogram.

## VI. THERMAL DIFFUSIVITY MEASUREMENTS

In this section, we show several examples of the measurement of the thermal diffusivity using the linear behaviors studied in Sec. III. Starting from averaged thermograms like the one in Fig. 16(b), we have analyzed two orthogonal temperature profiles crossing the center of the laser spot: one in the direction of the sample motion and the other one in the transverse direction. In Fig. 18, we plot the temperature profiles for three metallic samples covering a wide range of thermal diffusivity values. Measurements have been performed at the following speeds: AISI-304 stainless steel (*v* = 1.65 cm/s), Zn (*v* = 7.89 cm/s), and Al 2024-T6 alloy (*v* = 9.43 cm/s). Dots are the experimental data and the continuous lines are the corresponding linear fits. Figure 18(a) shows the longitudinal profiles, where the left branches are almost horizontal straight lines, with no information on the thermal diffusivity, as predicted by the theory. Anyway, it is of practical interest to assess the tuning of the experimental setup. The right branch, instead, shows a linear behavior whose slope gives the thermal diffusivity using Eq. (22). Figure 18(b) shows the corresponding transverse profiles which are symmetric with respect to the position of the laser spot (*y* = 0). Moreover, beyond the region affected by the laser spot, the profiles are linear and using Eq. (25), the thermal diffusivity is obtained. All the results, which agree very well with the literature values, are summarized in Table I.

Sample, speed . | D (transverse profile)
. | D (longitudinal profile)
. | D (literature)^{43,44}
. |
---|---|---|---|

AISI-304, 1.65 cm/s | 4.0 ± 0.2 | 4.1 ± 0.3 | 3.95 |

Zn, 7.89 cm/s | 43 ± 2 | 40 ± 4 | 41.8 |

Al 2024-T6, 9.43 cm/s | 73 ± 5 | 70 ± 7 | 73.0 |

Sample, speed . | D (transverse profile)
. | D (longitudinal profile)
. | D (literature)^{43,44}
. |
---|---|---|---|

AISI-304, 1.65 cm/s | 4.0 ± 0.2 | 4.1 ± 0.3 | 3.95 |

Zn, 7.89 cm/s | 43 ± 2 | 40 ± 4 | 41.8 |

Al 2024-T6, 9.43 cm/s | 73 ± 5 | 70 ± 7 | 73.0 |

It is worth noting that the uncertainty of the diffusivity values obtained from the transverse profiles (the *y* axis) is smaller than those obtained from the longitudinal ones (the *x* axis). This is because in the transverse profile there are more data before reaching the noise since its slope is half of the longitudinal one. Finally, let us remark that this method is not only valid for opaque and bulk samples, but it is of general validity, i.e., opaque or semitransparent, bulk or thin slabs, isotropic or anisotropic sample (see Ref. 29).

In Fig. 19, we show the temperature profiles for AISI-304 at two different speeds to check the robustness of the method. Dots are the experimental data and the continuous lines are the linear fits. Using Eqs. (22) and (25), we obtained the same thermal diffusivity value, *D* = 4.0 ± 0.3 mm^{2}/s, within the experimental uncertainty. Anyway, note that as the sample speed increases the number of data in the linear region decreases, and, therefore, the reliability of the diffusivity value is reduced. In fact, the lack of spatial resolution is the main drawback of this method. Accordingly, the use of IR cameras with a high resolution (640 × 512 or even 1280 × 1024 pixels) provided with microscope lenses allows spatial resolutions near the diffraction limit (5 *μ*m).

## VII. CRACK WIDTH SIZING

In Fig. 17, we plotted a series of thermograms obtained for an AISI-304 sample containing a calibrated 2.5 *μ*m wide vertical crack. A clear discontinuity in the temperature reveals the presence of the fissure. In this section, we are going a step further, demonstrating the ability of the flying spot thermography to size the width of vertical cracks by fitting the theoretical temperature profile perpendicular to the crack and crossing the center of the laser spot, given by Eq. (28), to the experimental profile. Figure 20(a) shows two temperature profiles corresponding to two positions of the crack, before and after reaching the laser. Data were recorded for the sample moving to the left at *v* = 7.5 mm/s. Dots are the experimental data and the continuous lines are the fits in Eq. (28) using a nonlinear least square fitting. Temperature profiles described by Eq. (28) depend on six parameters: *P _{o}η*/

*ε*,

*KR*=

_{th}*Kw*/

*K*,

_{air}, D*v*,

*a*, and

*t*. The latter governs the distance between the laser spot and the crack. The thermal properties of the sample (

*D*and

*K*) and the thermal conductivity of air,

*K*are taken from the literature. Parameters

_{air},*v*and

*t*are measured very precisely from the IR video recorded by the camera due to the high acquisition frame rate. The radius of the laser spot

*a*= 350 ± 15

*μ*m is measured optically. Accordingly, only two parameters are involved in the fittings of the temperature profiles:

*P*/

_{o}η*ε*and

*w*. The values in Fig. 20(a) correspond to the retrieved crack width. The uncertainty is 10%. Note that the two values are consistent and in good agreement with the nominal one, within the experimental uncertainty, indicating the reliability of the method. It is worth mentioning that the fitting of each profile is performed in less than 1 min in a laptop.

Similarly, in Fig. 20(b), we plot two temperature profiles in the case of a nominal crack width, *w* = 10 *μ*m. Dots are the experimental data and the continuous lines are the fits to Eq. (28). The retrieved widths are given in the figure with an uncertainty of 8%. As before, the agreement with the nominal value is very good.

It is worth noting that this method to size the width of cracks is quite restrictive since it is only valid for infinite vertical cracks when the sample (or the laser spot) is moving in the direction perpendicular to the crack. For other configurations that are more realistic, no analytical solution has been found, and, therefore, we have to proceed numerically. This issue will be addressed and discussed in detail in Sec. VIII.

Some authors have suggested using the temperature jump at the crack position to size the width of the crack avoiding the fitting of the whole temperature profile. However, this method must be disregarded since the theoretically abrupt temperature jump at the crack position is systematically rounded due to the combined effect of lens aberrations and diffraction. This effect can be observed in Fig. 21, which is a close up near the crack position of the complete temperature profile shown in Fig. 20(b). The temperature contrast is reduced with respect to the predicted value: smaller temperature at the hot zone and higher temperature at the cold region. To reduce this effect as much as possible, IR lenses of excellent optical quality must be used, reaching the diffraction limit. Note that this effect is inherent to any image forming system and cannot be overcome by just increasing the spatial resolution of the thermogram.

## VIII. OUTLOOK

Throughout this tutorial, the capabilities of flying spot thermography as an efficient and fast tool for quantitative assessment have been presented. The exposition has also brought to light some of the limitations and challenges of this technology. Regarding thermal diffusivity measurements, as mentioned in Sec. III, the methodology is quite general as it allows dealing with thick and thin samples, opaque and semitransparent materials, and is insensitive to heat losses. However, an eventual lack of spatial resolution is the main limitation of this technique, especially when working at high speed, which is highly desirable and the main strength of this methodology. This drawback is especially harmful when dealing with low diffusivity materials. The availability of high spatial resolution (≥1 Mpx) cameras combined with microscope lenses can push the limits beyond the speeds handled in this tutorial.

As for cracks, the state-of-the-art casuistry presented in this tutorial is very restrictive in terms of material properties, thermal interactions with the surroundings, and crack geometry: the analysis was limited to infinite vertical cracks in thick, opaque materials under adiabatic conditions, for which the heat diffusion equation has an analytical solution, as presented in Sec. IV. Even within this framework, the exposed methodology requires an additional constraint: the direction of motion of the laser needs to be perpendicular to the crack.

In order to overcome the above-mentioned restrictions, a different approach is required, and therefore, there is plenty of room for future progress. The simple presence of heat losses in the ideal system analyzed in this tutorial impedes writing the surface temperature in an analytical form. Likewise, in order to model the effect of an infinite tilted crack, a real vertical crack of finite dimensions, or non-flat cracks, numerical approaches are required. Finite elements modeling (FEM) has been used to characterize real cracks typically wider than 30 *μ*m.^{13,17,45} The characterization of early stage, narrow cracks (below 20 *μ*m) by means of conventional FEM is challenging, because in this approach the heat diffusion problem is modeled as a transmission problem in a medium with two domains, the bulk and the air filling the crack, that require defining conforming meshes. Therefore, for very thin cracks, extremely fine meshes are needed, which entail meshing and computation difficulties and a very high computational cost.

To overcome this problem, the implementation of discontinuous Galerkin finite elements has been proposed,^{46,47} which allows temperature calculations for very thin cracks of any size and shape. In this approach, thin cracks are considered as an interface characterized by a thermal resistance. They enable discontinuous temperature distributions over the crack and are efficient in terms of computation. These methods open a wide field of research for the characterization of real cracks.

In the case of finite cracks, the temperature discontinuity at the crack position is expected to be smaller than that of infinite cracks. As a consequence, the absolute temperature at the other side of the crack with respect to the position of the laser spot would be higher than that of an infinite crack of same width. Accordingly, a sensitivity analysis should be performed in order to verify whether the quantities characterizing the crack (length, depth, and width) remain uncorrelated.

Another interesting issue for future work is to analyze the ability of flying spot thermography to detect and characterize hidden cracks (not surface breaking cracks), which cannot be detected by optical or dye penetrant methods. In this regard, the thermal diffusion length *μ _{z}* defined in Eq. (26) is expected to represent a limit for detecting hidden cracks.

Finally, it is worth mentioning that, as illustrated in Sec. VII, whichever the required theoretical approach, there is an experimental difficulty related to the smoothing effect of the point-spread-function of the imaging system on the temperature discontinuity that characterizes the crack. This issue is particularly significant when dealing with wide (and deep) cracks that produce large temperature contrasts, especially in good thermal conductors. The characterization of the point-spread-function of the imaging system and subsequent deconvolution of the thermographic images might help sharpening the temperature contrast produced at the crack and enable a more accurate assessment of the crack width.

As a summary, we can conclude that the combination of flying spot thermography and discontinuous Galerkin finite elements promises a wide growing potential for methodologies to address the challenge of fast quantitative characterization of narrow cracks in an early development stage.

## ACKNOWLEDGMENTS

This work has been supported by the Ministerio de Economía y Competitividad (No. DPI2016-77719-R, AEI/FEDER, UE), Gobierno Vasco (No. PIBA 2018-15), and Universidad del País Vasco UPV/EHU (No. GIU16/33).

## REFERENCES

*Nondestructive Evaluation of Materials by Infrared Thermography*

*Photoacoustic and Photothermal Phenomena II*

*The Fourier Transform and Its Applications*

*Conduction of Heat in Solids*

*Theory and Practice of Infrared Technology for Nondestructive Testing*

*Photothermal Science and Techniques*