## 2 Phase-Resolved Doppler Model in SD-OCT  
### The Relation Between the Doppler Phase Shift and Axial Sample Velocity  
The starting point of the phase-resolved Doppler OCT (PR-DOCT) in SD-OCT is the single complex depth scan \(\Gamma_{j}(z)\) resulting from the Fourier transform of the detected spectrally shaped interference signal \(\mathrm{I}_{j}(k_{z}z)\). This A-scan with index \(j\) comprises the amplitude \(A(z)\) and the random phase \(\varphi(z)\) information of the backscattered light from different depth positions \(z\) of the volume scattering sample, as shown by Equation (1):  
\[FT\{\mathrm{I}_{j}(k,z)\}=\Gamma_{j}(z)=A_{j}(z)\mathrm{e}^{i\varphi_{j}(z)} \tag{1}\]Here, the amplitude enables the structural imaging, while the phase of a scattering sample relates to a random fluctuation term. Since an axial sample movement between consecutive A-scans separated by the period \(T_{\text{A-scan}}\) causes a Doppler frequency shift, and with this a phase shift \(\Delta\varphi(z)\) of the interference spectrum, the phase information can be used for the measurement of the axial sample velocity \(v_{z}(z)\), in consideration of the center wavelength \(\lambda_{0}\), the refractive index n of the sample, and the time \(T_{\text{A-scan}}\) needed for one A-scan relating to the sum of the integration time \(T_{\text{Int}}\) of the line detector of the spectrometer and the detector dead time \(\Delta T\) due to a possible operating shutter control [21]. \[v_{z}(z)=\frac{\Delta z}{T_{\text{A-scan}}}=\frac{\Delta\varphi(z)\lambda_{0}} {4\pi nT_{\text{A-scan}}}\text{ with }\Delta z=v_{z}\cdot T_{\text{A-scan}}\text{ and }T_{\text{A-scan}}=T_{\text{Int}}+\Delta T \tag{2}\]  
The bias-free Doppler phase shift \(\Delta\varphi(z)\) is calculated by the multiplication of the complex Fourier coefficient of one A-scan \(\Gamma_{j+1}(z)\) and the previous conjugate complex measurement \(\Gamma_{j}^{*}(z)\)[20; 22], where the resulting complex data contains the Doppler phase shift \(\Delta\varphi(z)\) as the argument:  
\[\Gamma_{\text{res}}(z)=\Gamma_{j+1}(z)\cdot\Gamma_{j}^{*}(z)=A_{j+1}(z)A_{j}( z)\text{e}^{i[\varphi_{j+1}(z)-\varphi_{j}(z)]} \tag{3}\]  
Since the noise of the Doppler phase shift increases with the amount of transverse displacement in the case of an oblique sample motion [19; 20], low signal-to-noise ratio (SNR) [16], and detector dead time [21], the amplitude weighted averaging by Equation (4) is recommended. \[\overline{\Delta\varphi(z)}=\arg\Bigg{\{}\frac{1}{K}\sum_{j=1}^{K}\left[ \Gamma_{j+1}(z)\cdot\Delta\Gamma_{j}^{*}(z)\right]\Bigg{\}} \tag{4}\]  
The discrepancy of the relationship of \(\Delta\varphi(z)\) and \(v_{z}(z)\) for any oblique sample motion detected with spectrometer-based OCT systems has been theoretically described in detail in previous studies [19; 20; 21]. To summarize the general description of the Doppler phase shift \(\Delta\varphi\) independent of the OCT system parameters, dimensionless coordinates are used as follows:  
\[\delta z=\frac{2n\Delta z}{\lambda_{0}},\ \delta x=\frac{\Delta x}{w_{0}},\ t^{ \prime}=\frac{t}{T_{\text{A-scan}}} \tag{5}\]  
where \(w_{0}\) corresponds to the width of the Gaussian sample beam (FWHM) of the OCT system. Since the dependence of \(\Delta\varphi\) on \(\delta z\) and \(\delta x\) cannot be calculated in an analytical way [19; 20; 21], the numerical result is given by a universal contour plot, where \(\Delta\varphi\) is drawn in steps of \(\pi/6\) for \(\delta z\) and \(\delta x\) ranging from zero to four. Additionally, \(\Delta\varphi\) is limited to its principle value between 0 and \(2\pi\), for which reason phase shifts exceeding \(2\pi\) are simply wrapped. For the case of no detector dead time \(\Delta T/T_{\text{A-scan}}=0\), the numerically determined \(\Delta\varphi\) is shown by the contour plot in Figure 1a [20]. By considering a partial duty cycle [21] with \(\Delta T/T_{\text{A-scan}}=0.05\), as for the system used in this study, \(\Delta\varphi\) as a function of \(\delta x\) and \(\delta z\) is changed as presented in Figure 1b.