## 2 Basic Concepts of Optical Thermography NDT
### Pulsed Thermography (PT)  
The concept of a PT system for defect detection consists of applying a very short time and powerful energy pulse to an object and then recording the temperature rise, decay, or both curves in transient mode According to the thermal diffusion effect, pulsed thermal  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Thermal Source** & **IRT Terminology** & **Acronym** & \multicolumn{2}{c}{**Major Advantages**} & \multicolumn{2}{c}{**Major Limitations**} \\ \hline Photographic & Pulsed & & - & Very fast inspection time & - & Non-uniform heating \\ Flashes & Thermography & & & (few ms) & - & Relatively expensive \\  & & & - & Wide inspection area & - & Limited to some field \\  & & & - & High power energy & applications \\  & & & - & Beneficial for flat defects & \\ \hline Halogen Lamp & Lock-in & & - & Wide inspection area & - & Considering the optimal \\  & & & - & Beneficial for flat defects & \\  & & & - & Less non-uniform heating & \\  & & & - & and environmental factors & \\  & & & & Detecting defects located at & - & Long heating time \\  & & & & various depths & - & Limited to some field \\  & & & & & applications \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of thermal heating devices most commonly used for active thermography for NDT application.  
energy incident on the surface propagates under the surface. If a defect exists below the surface, it interferes with the diffusion of the pulse energy [6].  
Consequently, each defect will appear differently from the other defects located at different depths. Dirac pulse is defined as an intense unit-area pulse of such a short duration that no measuring equipment can differentiate it from a shorter pulse. However, in practice, producing an ideal Dirac pulse is not possible [22].  
Pulsed thermography (PT) is among the most prevalent thermographic techniques in TNDT. Figure 2 depicts the principle of PT where a high-power pulse heating is employed to the sample under inspection, and the response of the sample is recorded with an infrared camera. The continuance of the pulse depends on the sample's thermal conductivity under inspection and runs between 2 to 15 ms, making PT well known for its quickness in testing. As soon as the sample becomes excited, the surface absorbs the light energy, and temperature increases instantly. Due to the propagation of thermal waves inside the sample, the surface temperature starts to decay. Imperfections could be seen if there is variance in the thermal decay rate across the sample surface. The transient temperature field T(z,t) of PT can be expressed by an in-homogeneous one-dimensional (1D) Equation (1).  
\[\frac{\partial\mathrm{T(z,t)}}{\partial z^{2}}-\frac{1}{\alpha}\frac{\partial \mathrm{T(z,t)}}{\partial t}=-\frac{\mathrm{g(z,t)}}{\mathrm{k}},\ t>0 \tag{1}\]  
\[\alpha=\frac{\mathrm{k}}{\rho.\mathrm{C_{p}}} \tag{2}\]  
where \(\alpha\) is the material's thermal diffusivity with k being the thermal conductivity, \(\rho\) is the density, and \(\mathrm{C_{p}}\) is the specific heat capacity. The one-dimensional solution of the Fourier's law for the propagation of a Dirac pulse in a semi-infinite isotropic solid by conduction can be expressed by Equation (3) [6; 22].  
\[\mathrm{T(z,t)}=\mathrm{T_{0}}+\frac{\mathrm{Q}}{\mathrm{e}\sqrt{\pi\mathrm{t }}}\exp\left(-\frac{\mathrm{z}^{2}}{4\alpha\mathrm{t}}\right) \tag{3}\]  
\[\mathrm{e}=\sqrt{\mathrm{k}\rho\mathrm{c_{p}}} \tag{4}\]  
where T is the temperature rise at the time after the flash heating, T\({}_{0}\) is the initial temperature, Q is the energy density deposited on the surface, and e is the material's thermal effusivity to exchange heat with its surrounding.  
Figure 2: Configuration of a pulsed thermography inspection system.  
At the surface (z = 0 mm), Equation (3) can be rewritten as Equation (5).  
\[\text{T}(\text{t})=\frac{\text{Q}}{\text{e}\sqrt{\pi\text{t}}} \tag{5}\]