## 2 Principles of Doppler OCT for Otology  
The measured interfered spectrum (\(I(k)\)) is simply expressed by subtracting the DC term:  
\[I(k)=2\!\int_{-\infty}^{\infty}\sqrt{I_{R}I_{S}}\cos(2 kz+\varphi)\mathrm{d}z, \tag{1}\]  
where \(k=2\pi/\lambda\), \(z\) is the in-depth coordinate direction, \(\varphi\) is the random phase noise, and \(I_{R}\) and \(I_{S}\) are the obtained back-scattered signals from the reference arm and sample arm, respectively. To obtain an A-scan of OCT, which is a depth-resolved signal, a Fourier transform is applied to \(I(k)\) over \(k\) into the time-domain interfered signal (\(i(z)\)):  
\[i(z)=FT[I(k)]=A(z)\exp[j\Phi(z)] \tag{2}\]where \(z\) is the depth and can be expressed as \(z=2 ct\) (\(c\) is the speed of light and the refractive index of the sample is assumed as 1). In addition, \(A(z)\) and \(\Phi(z)\) are the amplitude and the phase of signals located at depth \(z\). The optical phase angle is expressed as:  
\[\Phi(z,t)=\frac{4\pi}{\lambda}\delta z\sin(2\pi f_{0}t)+\varphi, \tag{3}\]  
where \(\delta z\) and \(f_{0}\) are the amplitude and vibration frequency, respectively, and \(\lambda\) is the center wavelength of the source. To extract the value of Doppler phase shift, Doppler OCT uses the following equation:  
\[\Delta\Phi(z,t)=\frac{d\Phi(z,t)}{dt}=\frac{\theta_{i+1,z}-\theta_{i,z}}{T}, \tag{4}\]  
where \(\theta_{i,z}\) and \(\theta_{i+1,z}\) are phases of the obtained signal at the z pixel in-depth direction of \(i\) and \((i+1)\) and \(T\) is the time interval between adjacent A-lines. Moreover, the equation of the Doppler frequency shift is expressed after differentiation of Equation (3) as:  
\[\Delta\Phi(z,t)=\frac{8\pi^{2}}{\lambda}\delta zf_{0}\cos(2\pi f_{0}t)\Delta t +\xi(t), \tag{5}\]  
where \(\xi(t)\) is the intrinsic phase noise, which is comparably larger than \(\Delta\Phi(z,t)\), and \(\Delta t\) is reciprocal of the sampling frequency. According to the autocorrelation algorithm, Equation (4) becomes:  
\[\Delta\Phi(z,t)=\frac{1}{T}\tan^{-1}\left[\frac{\text{Im}(i{i^{\prime}}_{i+1} )}{\text{Re}(i{i^{\prime}}_{i+1})}\right]=\frac{1}{T}\tan^{-1}\left[\frac{ \text{Re}(i_{i,z})\text{Im}(i_{i+1,z})-\text{Re}(i_{i+1,z})\text{Im}(i_{i,z}) }{\text{Re}(i_{i+1,z})\text{Re}(i_{i,z})+\text{Im}(i_{i+1,z})\text{Im}(i_{i,z })}\right], \tag{6}\]  
where \(\text{Re}(i_{i})\) and \(\text{Im}(i_{i})\) are the real and imaginary parts of the obtained complex signal, respectively. As shown in Equation (6), the phase-resolved Doppler OCT method utilized the calculated phase information to extract a Doppler frequency shift. To enhance the signal-to-noise ratio (SNR) of the Doppler system, the averaging mask, which consists of multiple sequential A-lines in the transverse direction with pixels in the axial direction, was applied to Equation (6). Averaging in the lateral (temporal) direction for N lines and in the depth direction for M lines can be expressed as:  
\[ \begin{array}{l}\Delta\Phi(z,t)=\frac{1}{T}\tan^{-1}\left[\frac{\text{Im}( \sum\limits_{i=1}^{N}\frac{M}{\sum\limits_{i=1}^{N}i^{i}{i^{\prime}}_{i+1})}}{ \text{Re}(\sum\limits_{i=1}^{N}\frac{M}{\sum\limits_{i=1}^{N}i^{i}{i^{\prime}} _{i+1})}}\right]\\ \\ =\frac{1}{T}\tan^{-1}\left\{\begin{array}{l}\sum\limits_{i=1}^{N}\frac{M}{ \sum\limits_{i=1}^{N}\left[\text{Re}(i_{i})\text{Im}(i_{i+1,z})-\text{Re}(i_{ i+1,z})\text{Im}(i_{i,z})\right]}\\ \sum\limits_{i=1}^{N}\frac{M}{\sum\limits_{i=1}^{N}\left[\text{Re}(i_{i+1,z}) \text{Re}(i_{i,z})+\text{Im}(i_{i+1,z})\text{Im}(i_{i,z})\right]}\\ \end{array}\right\}\end{array}  \tag{7}\]  
By averaging for both directions (lateral and depth), Equation (7) reduces the background subtle noise, which leads to the image quality enhancement. Though a larger size of the phase mask (\(N\) and \(M\)) increases SNR, it requires a large processing time and reduces resolution. Therefore, the appropriate size of the phase mask is needed to choose according to applications.