**3. Photothermal deflection technique**

#### **3.1 Principle of the photothermal deflection technique PTD**

The sample is heated by modulated and uniform light pump beam. The optical absorption of the sample will generate a unidimentional thermal wave that will propagate into the sample and in the surrounding fluid near the surface of the sample, inducing a temperature gradient then a refractive index gradient in the fluid. The absorbed light is transformed into heat by a nonradiative de-excitation process. A laser beam skimming parallely the sample surface and passing through this refractive index gradient is deflected. This deflection is related to the thermal and optical properties of the sample.

The deflection of the probe laser beam <sup>ψ</sup> is complex number <sup>ψ</sup> ¼j <sup>ψ</sup><sup>j</sup> ej<sup>φ</sup> � � given by [11]:

$$\Psi(z,\ t) = \frac{L}{n\_0} \frac{dn}{dT\_f} \frac{\sqrt{2}}{\mu\_f} \quad |T\_0| \quad e^{-\frac{z\_0}{\mu\_f}} \ e^{j\left(\theta + \frac{z}{4} - \frac{z\_0}{\mu\_f}\right)} e^{jat} \tag{1}$$

T0 which is the periodic temperature rise at the sample surface is a complex number that written T0 <sup>¼</sup> <sup>∣</sup> T0 <sup>∣</sup> <sup>e</sup><sup>j</sup><sup>θ</sup>, Z0 is the distance between the probe laser beam axis and the sample surface, L is the sample length in the direction of the laser probe beam, n is the fluid refractive index. Where ð Þ <sup>μ</sup><sup>f</sup> <sup>¼</sup> Df*=*π<sup>f</sup> <sup>1</sup>*=*<sup>2</sup> is the thermal diffusion length of the fluid with Df the thermal diffusivity of the fluid and j<sup>2</sup> ¼ �1*:*<sup>∣</sup> <sup>ψ</sup><sup>∣</sup> and <sup>φ</sup> are respectively the amplitude and the argument of the laser pump beam deflection given by:

$$|\Psi(\mathbf{z})| = -\frac{L}{n\_0} \frac{dn}{dT\_f} \frac{\sqrt{2}}{\mu\_f} \begin{array}{c} \left| T\_0 \right| \text{ } \text{e}^{-\frac{x\_0}{\mu\_f}} \end{array} \tag{2}$$

$$
\rho = \frac{-z\_0}{\mu\_f} + \theta + \frac{\pi}{4} \tag{3}
$$

In order to determine the deflection of the probe laser beam, we have to calculate the periodic temperature T0 at the sample surface.
