## 2 Theoretical Background  
Figure 1 shows a diagram of the optical arrangement used for ESPI interferometry with radial-in-plane sensitivity. A laser illumination beam strikes an optical fibre at one end, emerging from the other end in such a way that it can be considered as a point source (spherical wavefront), which generates divergent illumination, _SiL_. Figure 1: _Cont_. There is an extensive literature explaining the polarization phase shifting method; e.g., [PERSON] et al. [17] show in detail the mathematical formulation, theory, and implementation of this technique. To implement our polarization phase shifting system, the illumination beam passes through a quarter-wave retarding plate (_fc_). With its fast axis tilted in a 0 rad angle, the linearly polarized filter (_fl_) is set in a \(\pi/4\) rad angle in front the first polarizer filter. A ring-shaped film with linear polarization is used to polarize only the illumination beam _bc_ reflected in the upper half of the conic mirror, while the illumination beam _ac_ remains with the illumination passed through the one-quarter wave retarder film. An opaque body, _Of_, blocks the illumination that would otherwise strike on the observation surface. A flat mirror, _Mf_, placed in front the opaque body allows for observation of the specimen surface through a charge-coupled device (CCD) camera. The two illumination beams, _ac_ and _bc_, converge at point \(c\) (Figure 1b), where a Speckle pattern can be observed. The conic mirror is made from polished stainless steel and provides a way of magnifying the image of an object on an opaque screen so that it can be viewed from directly in front of the screen, with a short projecting distance necessary to produce the magnification. The central part of the specimen surface has been excluded due to the convergence of the illumination reflected by the conic mirror. The unitary vector of illumination, \(\hat{n}_{A}\), keeps a circular polarization, while the unitary vector, \(\hat{n}_{B}\), is linearly polarized. The difference between the illumination unitary vectors is related to the sensitivity vector, \(\overrightarrow{e}\), by the following equation [18]:  
\[\overrightarrow{e}=\frac{2\pi}{\lambda}(\hat{n}_{A}-\hat{n}_{B}) \tag{1}\]  
where \(\lambda\) is the illumination beam's wavelength. The illumination unitary vectors \(\hat{n}_{A}\) and \(\hat{n}_{B}\) are obtained by considering: the polar coordinates from the illumination source, \(SiL(r_{s},0,z_{s})\); the cutting surface of the conical mirror at points \(a\) and \(b\), where the coordinates are \(Mc(r_{mn},\theta,z_{mn})\) and \(Mc(r_{mb},\theta+\pi,z_{mb})\), respectively; and, finally, a point in the plane of the specimen surface \(c\) with the coordinate \(Spc(r_{p},\theta,z_{p})\). The quantities \(r_{s}\), \(z_{s}\), \(r_{p}\), \(z_{p}\), \(r_{ma}\), \(z_{mn}\), \(r_{mb}\), and \(z_{mb}\) represent radial coordinates and the _z_-direction of the source, the specimen, and the conic mirror at points \(a\) and \(b\), respectively, while \(\theta\) is the angular  
Figure 1: **(a)** The proposed optical arrangement for ESPI interferometry with radial-in-plane sensitivity and divergent illumination. **(b)** The scheme of optical arrangement, where \(SiL\) is the source of the illumination laser; _fc_, one-quarter wave retarder film; _fl_, linear polarizer; _Of_, opaque filter; _Mf_, flat mirror; _Mc_, conical mirror; _Spc_, surface specimen; \(\overrightarrow{e}\) is the sensitivity vector; and \(\hat{n}_{A}\) and \(\hat{n}_{B}\) are unitary vectors related to the illumination beams. coordinate.