## 3 Experimental Results and Discussion  
A convex cone mirror was manufactured by the machining and high-quality polishing of a stainless-steel block, as shown in Figure 5. Micro-roughness measurement of the optical surface was 0.05 \(\upmu\)m RMS (Root Mean Square). An aluminium plate mounted in a rigid frame was used as a specimen. Deformations by dilatation in the plate were induced by heating it to 80 \({}^{\circ}\)C with an electric resistance set in a small cylinder of 6 mm in diameter and 4 mm in thickness in contact with the rear. This heating temperature was low enough to minimize out-of-plane displacements due to buckling deformations. In consequence, deformations by heat convection effects were neglected. Figure 4: (**a**) Total sensitivity of the system, (**b**) Total sensitivity in plane \(y=0\). A 552 nm laser was used as a divergent illumination source set to \(z_{s}=265\) mm in front of the specimen, as shown in Figure 1. This position was required with respect to the vertex of the conical mirror to avoid convergence and ensure full field analysis due to the geometrical design of the mirror. The retarder-polarized system \((fc-fl)\) generated circular and linear polarizing states for both the beam that crossed the polarizing ring and the beam that propagated freely. When the linear polarizer rotated, linear polarizing states were generated, which allowed for the interference of the two beams with different curvature radiuses, and this generated the observed patterns with radial symmetry. The generated interferograms were captured by a CCD camera. The images were a monochromatic \(640\times 480\) pixels resolution of 8 bits. To obtain the optical phase, \(\Delta\phi\), we used the polarizing [21] shifting phase technique [22]. For this, a deformation in the sample was generated by heating in such a way that temporal variations were longer than the capture time of the five interferograms. Each phase shift was produced by changing the angle of the linear polarized filter (_fl_) so that five interferograms with a relative shift of \(\frac{2\pi}{5}\) rad were obtained. The acquired images were filtered to normalize the contrast of the interference fringes with digital image processing. The result is shown in Figure 6. The optical unwrapping phase \(\Delta\phi\) was obtained by the algorithm of the unwrapping phase proposed by [PERSON] [23]. This algorithm had the advantage of no error propagation during the unwrapping phase by means of adaptation of the evaluation trajectory during the progression of the algorithm. Figure 5: The conic mirror manufactured in stainless steel. Figure 6: _Cont._  
Considering the system sensitivity shown in Figure 4 and using Equation (6), Figure 7 shows the radial deformation of the surface. As mentioned above, it was not possible to take measurements in the centre due to the convergence of the illumination at this point from the other directions. Figure 6: Interferograms obtained in distinct relative angles of the linear polarizer, (**a**) 0 rad, (**b**) \(\frac{\pi\alpha}{5}\) rad, (**c**) \(\frac{4\pi}{5}\) rad, (**d**) \(\frac{6\pi}{5}\) rad, (**e**) \(\frac{8\pi}{5}\) rad, (**f**) wrapped phase in module \(2\pi\) terms, (**g**) \(\Delta\phi\) optical phase in rad, (**h**) position of the heating source. Figure 7: Measurement of the deformation \(ur\) of the surface. The measurement achieved for the maximum radial deformation was 0.8 \(\upmu\)m when the plate was heated to 80 \({}^{\circ}\)C by the rear in the centre point. The deformation decreased according to the increasing radial distance. The contribution of the out-of-plane component of the sensitivity vector represented less than 3% towards the outer edges. In maximum deformations, its contribution was estimated at less than 1%, and that is why it was neglected. Numerical simulations were performed using ANSYS software for finite element analysis applied only to a three-dimensional (3D) geometrical model of the specimen plate.