## 2 Theoretical Background
In applying the reflection geometry in the mirror, as observed in Figure 2, we can describe the relationship for the reflection of the illumination on the conic mirror on the plane \((r,z)\) as [19]:  
\[\overset{\rightarrow}{R}_{i}=\overset{\rightarrow}{I}_{i}-2\bigg{(}\overset{ \rightarrow}{I}_{i}\hat{N}_{i}\bigg{)}\hat{N}_{i},i=A,B \tag{2}\]  
where \(\overset{\rightarrow}{R}_{i}\) is the illumination vector reflected by the conic mirror to the specimen surface; \(\overset{\rightarrow}{I}_{i}\) is the illumination vector from the illumination source to the conic mirror; and \(\hat{N}_{i}\) is the normal unitary vector to the surface of the conic mirror, with the feature that it is a vector with a constant direction on the full surface, of which the component in cylindrical coordinates is \(\hat{N}(r,\theta,z)=[\cos(\delta),\theta,\sin(\delta)]\). Considering the coordinates of each element, the illumination vectors reflected, \(\overset{\rightarrow}{R}_{i}\), can be expressed as shown in the following equation:  
\[\overset{\rightarrow}{R}_{A}=\left( \begin{array}{c}r_{p}-r_{ma}\\ \theta+\pi\\ z_{p}-z_{ma}\end{array} \right)=\left( \begin{array}{c}r_{ma}-r_{s}\\ \theta+\pi\\ z_{ma}-z_{s}\end{array} \right)-2\left[\left( \begin{array}{c}r_{ma}-r_{s}\\ \theta+\pi\\ z_{ma}-z_{s}\end{array} \right)\cdot\left( \begin{array}{c}\cos(\delta_{a})\\ \theta+\pi\\ \sin(\delta_{a})\end{array} \right)\right]\left( \begin{array}{c}\cos(\delta_{ a})\\ \theta+\pi\\ \sin(\delta_{a})\end{array} \right)\right] \tag{3}\] \[\overset{\rightarrow}{R}_{B}=\left( \begin{array}{c}r_{p}-r_{ mb}\\ \theta\\ z_{p}-z_{mb}\end{array} \right)=\left( \begin{array}{c}r_{mb}-r_{s}\\ \theta\\ z_{mb}-z_{s}\end{array} \right)-2\left[\left( \begin{array}{c}r_{mb}-r_{s}\\ \theta\\ z_{mb}-z_{s}\end{array} \right)\cdot\left( \begin{array}{c}\cos(\delta_{b})\\ \theta\\ \sin(\delta_{b})\end{array} \right)\right]\left( \begin{array}{c}\cos(\delta_{ b})\\ \cos(\delta_{b})\\ \sin(\delta_{b})\end{array} \right)\]  
in such a way that the unitary vectors \(\hat{n}_{A}\) and \(\hat{n}_{B}\) can be calculated by means of the definition of the unitary vector. \[\hat{n}_{A}=\frac{\overset{\rightarrow}{R}_{A}}{|\overset{\rightarrow}{R} _{A}|},\hat{n}_{B}=\frac{\overset{\rightarrow}{R}_{B}}{|\overset{\rightarrow }{R}_{B}|} \tag{4}\]  
The optical phase \(\Delta\phi\) associated with the displacements \(\overset{\rightarrow}{d}=[u_{r},u_{\theta},u_{z}]\) from all of the points that confirm the specimen surface is related to the vector sensitivity \(\overset{\rightarrow}{e}\) (Equation (1)) by the expression [20]:  
\[\Delta\phi=\overset{\rightarrow}{e}\cdot\overset{\rightarrow}{d}. \tag{5}\]  
By analyzing the previous equations, we can simplify the optical phase into two components, since the sensitivity vector always coincides with the \(rz\) plane. The optical phase will be composed by in-plane and out-of-plane radial sensitivity depending on the displacement vector, \(\overset{\rightarrow}{e}(r,z)=e_{r}\hat{\rho}+e_{z}\hat{k}\), where \(\hat{\rho}\) and \(\hat{k}\) are the unitary vectors in the radial and \(z\) axis directions for the components \(e_{x}\) and \(e_{z}\)  
Figure 2: Reflection geometry for the calculation of the sensitivity vector with divergent illumination. respectively. Simplifying Equation (5) by disregarding the associated quantity to \(e_{z}\) and assuming that the contribution to sensitivity is small, we can rewrite it as:  
\[u_{r}(r,\theta)\approx\frac{\lambda\Delta\phi(r,\theta)}{2\pi(\hat{n}_{A}-\hat{n }_{B})}. \tag{6}\]  
A numerical analysis can be done by taking into account the dimensions shown in Table 1 for the optical arrangement of ESPI interferometry with in-plane radial sensitivity and divergent illumination. Considering the Equations (1), (3), and (4), we can calculate the in-plane and out-of-plane radial component of the sensitivity vector.