## 2 Experimental Setup
### Principle of Numerical Correction Method for Perfect Circularly Polarized Light  
The polarization state of a light beam can be represented using the Jones vector. According to Jones matrix theory, any polarization state of light can be represented as the superposition of components along two orthogonal directions:  
\[E=\left[ \begin{array}{c}A_{x}\\ A_{y}e^{i\delta}\end{array} \right] \tag{1}\]  
where \(\delta\) represents the phase difference of \(E_{x}\) with \(E_{y}\). The polarization state of linearly polarized light is expressed as follows:  
\[E=\left[ \begin{array}{c}\cos\theta\\ \sin\theta\end{array} \right] \tag{2}\]  
A 2 \(\times\) 2 complex matrix can be used to express the action of a quarter wave plate on a beam as follows:  
\[M_{QWP}=\left[ \begin{array}{cc}\cos^{2}\theta+i\sin^{2}\theta&\sin\theta \cos\theta(1-i)\\ \sin\theta\cos\theta(1-i)&\sin^{2}\theta+i\cos^{2}\theta\end{array} \right] \tag{3}\]  
Consequently, the optical system can be systematically traversed using Jones matrices. When the direction of the polarization of linearly polarized light makes an angle of 45\({}^{\circ}\) with the direction of the optical axis of the QWP, it can be given as  
\[E_{\theta=45^{\circ}}=\frac{1}{\sqrt{2}}\left[ \begin{array}{c}1\\ 1\end{array} \right] \tag{4}\]  
\[M_{QWP_{\theta=0^{\circ}}}=\left[ \begin{array}{cc}1&0\\ 0&-i\end{array} \right] \tag{5}\]  
Obtaining the Jones vector of a beam passing through a wave plate involves multiplying the incident beam by the Jones matrix of the wave plate. After passing through an optical element, the change in the polarization state of light is given by \(E_{out}=E_{in}\cdot E_{0}\). In the reference arm, a mirror reflects a beam passing through an achromatic lens without changing its polarization state. In the sample arm, before reaching the sample, the beam sequentially passes through a polarization controller, a collimator, a QWP, and chromatic aberration lens. The incident light can be transformed into circularly polarized light by adjusting the polarization axis of the incident linearly polarized beam to be oriented using the polarization controller. The incident light represented in the sample arm can be expressed as follows:  
\[E_{1}=M_{QWP_{\theta=0^{\circ}}}\cdot E_{\theta=45^{\circ}}=\frac{1}{\sqrt{2} }\left[ \begin{array}{c}1\\ -i\end{array} \right] \tag{6}\]  
This represents a circularly polarized beam. The expression for the returned light from the entire reference arm that carries the sample information is given as follows:  
\[E_{s}=M_{QWP_{\theta=0^{\circ}}}\cdot M_{s}\cdot M_{s}\cdot M_{QWP_{\theta=0^ {\circ}}}\cdot E_{\theta=45^{\circ}} \tag{7}\]  
Here, \(M_{s}(\delta,\theta)\) is the polarization state change induced by the sample. This explains how the system polarization properties influence the detected fringes. From the known polarization state of the incident light at the sample, one can extract information about the birefringence of the sample. The imaging method based on the Hilbert transform was used to obtain the reflectivity and birefringence images of the sample and apply the correction method mentioned above [29]. The interference light is divided into orthogonal components \((I_{1},I_{2})\) by a polarizing beam separator. \(I_{1}\) represents the horizontal polarization state, and \(I_{2}\) represents the vertical polarization state. The interference light is then collected by the detectors. Theinterference effect of two beams depends on the phase difference. Hence, the intensity of the light detected by a detector can be represented as follows:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos\delta \tag{8}\]  
The detected intensity recorded at the depth position \(z_{0}\), \(I_{k}(z)\), is expressed as follows:  
\[I_{k}(z)=I_{r,k}+I_{s,k}+2\sqrt{I_{r,k}\cdot I_{s,k}}\cdot\left|\gamma\left(z- z_{0}\right)\right|\cdot\cos(\delta_{k}) \tag{9}\]  
where \(I_{r,k}\) is the intensity of the reference arm, and \(I_{s,k}\) represents the intensity of the sample arm.