## 2 Materials and Methods  
### Problem of Integrating-Bucket Method for PSI  
With the integrating-bucket method, four raw interferograms, called buckets, are generally used to reconstruct one en face image at a given depth. Each bucket is captured after accumulating the interference signal during each quarter cycle of a single-phase modulation period [12], as  
\[E_{k}\ =\ D+\frac{4}{T}\int_{(k-1)T/4}^{kT/4}A\cos[\Phi+B\sin(\omega t+\theta)] dt,\ k\ =\ 1,\ 2,\ 3,\ 4, \tag{1}\]  
where \(A\) and \(\Phi\) are the amplitude and phase of the interferogram that we want to retrieve; \(B\) and \(\theta\) are the amplitude and initial phase of the modulation; \(T\) and \(\omega\) are the period and angular frequency of the phase modulation; \(D\) is the DC component of the bucket signal. An intensive mathematical manipulation using a Jacobi-Anger expansion [12] makes Equation (1) a linear sum of the cosine and sine functions of the phase \(\Phi\), as follows  
\[E_{k}\ =\ D+A(f_{k}\cos\Phi+g_{k}\sin\Phi). \tag{2}\]  
The two coefficients \(f_{k}\) and \(g_{k}\) of the \(k\)-th bucket signal \(E_{k}\) are given as functions of the modulation parameters \(B\) and \(\theta\), and can be represented as a series of Bessel functions of the first kind [12]. They have the following useful characteristics  
\[f_{1}\ =\ f_{3},\ \ f_{2}\ =\ f_{4},\ \ g_{1}\ =\ -g_{3},\ \ g_{2}\ =\ -g_{4}. \tag{3}\]  
During one complete period of the phase modulation, four integrations of Equation (1) are applied, which give the four bucket signals \(E_{1}\), \(E_{2}\), \(E_{3}\), and \(E_{4}\). From their linear combinations, using the relationships of Equation (3), we can obtain the following two intermediate signals  
\[I_{\cos}\equiv E_{1}-E_{2}+E_{3}-E_{4}\ =\ 2A(f_{1}-f_{2})\cos\Phi, \tag{4}\]  
\[I_{\sin}\equiv E_{1}-E_{2}-E_{3}+E_{4}\ =\ 2A(g_{1}-g_{2})\sin\Phi. \tag{5}\]  
By defining two constants as  
\[F\equiv 2(f_{1}-f_{2}), \tag{6}\]  
\[G\equiv 2(g_{1}-g_{2}), \tag{7}\]  
we can simplify the two intermediate signals of Equations (4) and (5) as follows  
\[I_{\cos}\ =\ AF\cos\Phi, \tag{8}\]  
\[I_{\sin}\ =\ AG\sin\Phi. \tag{9}\]  
As is well-known, the two constants, \(F\) and \(G\), are equal to each other when the modulation parameters, \(B\) and \(\theta\), have specific optimum values [12]. With these optimum parameters, the amplitude \(A\) of the interferogram can be easily retrieved by conducting simple trigonometric manipulations with the intermediate signals of Equations (8) and (9), as follows  
\[\sqrt{{I_{\cos}}^{2}+{I_{\sin}}^{2}}\ =\ AF\ =\ AG. \tag{10}\]  
It says that the amplitude \(A\) is obtained without being affected by the phase \(\Phi\). However, if the modulation parameters are not properly adjusted, \(F\) and \(G\) will differ to each other and the image reconstruction will be defective. In other words, there will be some residual interference fringes. It depends on the phase \(\Phi\), even after the en face image reconstruction  
\[\sqrt{{I_{\cos}}^{2}+{I_{\sin}}^{2}}\ =\ A\sqrt{{F}^{2}+(G^{2}-{F}^{2})\sin^{2} \Phi}. \tag{11}\]  
In a previous study, a filtering in the Fourier domain was applied to remove the interference fringes [14]. However, with this method, the separation of the frequency components caused by the artifacts is difficult to achieve, and data loss occurs. Thus, we propose a simple post-processing method that can remove or reduce the residual fringes without suffering data loss.