## 2 Methods
### The Center Spatial Frequency and Parameters Design Criterion  
Figure 3 is the sketch of the two reference arms with the pivot-offset galvo scanners. The color lines represent the light beams when the galvo scanners are at different scan angles. The pivot-offset distances are indicated as \(s_{1}\) and \(s_{2}\). The phase shift between successive axial scans (A-scan) induced by the galvo scanner can be expressed as [9]  
\[\delta\phi_{i}=\frac{8\omega T_{S_{1}}\pi}{\lambda_{0}},\qquad i=1,2 \tag{3}\]  
where \(\omega\) is the angular velocity of the galvo scanners, \(T\) is the exposure time of the camera in the spectrometer, and \(\lambda_{0}\) is the center wavelength of the light source. Considering a tomogram of \(N\) depth profiles covering a lateral range of \(\Delta x\), the two center spatial carrier frequencies can be calculated by  
\[\upsilon_{ci}=\frac{\delta\phi_{i}}{2\pi}\cdot\frac{N}{\Delta x}\qquad i=1,2. \tag{4}\]  
The lateral scanning range \(\Delta x\) is calculated by  
\[\Delta x=2N\omega Tf \tag{5}\]  
where \(f\) is the focal length of the sample objective. The positive spatial frequency range can be calculated by  
\[\upsilon_{x}=\frac{N}{2\Delta x}. \tag{6}\]  
Figure 3: Sketch of the two reference arms in the spatial frequency domain multiplexing SDOCT system. \(s_{1}\) and \(s_{2}\): the pivot-offset distances; L1 and L2: the focal lens in the reference arms; M1 and M2: the plane mirrors; f: focal length of the focal lens. For high-quality image reconstruction, another important parameter is the characteristic bandwidth (BW) of the obtained spatial frequency contents, which is determined by the inverse speckle size. The speckle size can be modeled by the transverse resolution, and BW can be expressed as [9]  
\[BW=\frac{\pi d}{4\lambda_{0}f} \tag{7}\]  
where \(d\) is the collimated beam diameter entering the sample objective. In theory, the proposed method could quadruple the effective imaging depth range compared with a conventional SDOCT system. However, practically some factors can affect the improvement of the range depth, which includes the crosstalk between the two spatial frequency contents of the two depths signal, the crosstalk between the spatial frequency content and its complex conjugate content, and the crosstalk between the spatial frequency contents and the interference term between the two reference arms' light fields. Crosstalk between the two spatial frequency contents corresponding to the two depths can occur when the tails of the spatial frequency contents superpose with each other as schematically represented in Figure 4a. In view of the Bedrosian theorem, spatial frequency content should be well separated from the introduced modulation frequency and the other content for successful reconstruction, as shown in Figure 4b. The crosstalk due to the reference arms' interference can be eliminated via background subtraction. According to Equations (3)-(7), the parameters including the angular velocity \(\omega\), the complementary metal oxide semiconductor (CMOS) exposure time \(T\), and the offset distance \(s_{i}\) are all responsible to the image reconstruction quality. Thus, to avoid crosstalk and fully utilize the proposed method for extended range imaging, the center spatial carrier frequencies and the system parameters must be chosen carefully. To quantify the crosstalk effect, we introduce the SCR defined as a function of the frequency-spacing/bandwidth ratio (\(\Delta\
u/BW\)) for two reference arms, where \(\Delta\
u=\
u_{c2}-\
u_{c1}\) as shown in Figure 4. According to the theoretical value, the SCR can be higher than 100 dB when the frequency-spacing/bandwidth ratio \(\tau\) satisfies the condition [12]  
\[\tau=\frac{\Delta\
u}{BW}\geq 3. \tag{8}\]  
Furthermore, to avoid aliasing terms, the bandwidth of the spatial frequency content should be narrow relative to the full spatial frequency range; in this case, the small phase fluctuations will not cause a reduction of the SCR.