## 2 Motivation and Methods
### Angular Spectrum Method for Digital Focusing of OCTA  
To obtain a volumetric blood microvasculature of a mouse brain, multiple B-scan OCT images were acquired at the same position, and then repeatedly in the C-scan direction. From each B-scan image, the complex OCT signal, denoted by \(C(x,z)\), was calculated with a well-known conventional method [23; 24]. The spectral-domain OCT provides not only a depth resolved OCT image but also the phase information at each depth. A cross-sectional blood flow image was acquired by extracting dynamic signals from four repeated B-scan OCT images using an OCTA technique given by [22]. \[I_{Flow(x,z)}=\ \frac{1}{N-1}\sum_{i=1}^{N-1}\left|C_{i+1}(x,z)-C_{i}(x,z)\right|, \tag{1}\]  
where \(C_{i}(x,z)\) is the complex OCT signal in the \(i^{th}\) repeated B-scan, and \(N\) is the number of repeated B-scans; in our case, \(N=4\). Since the SD-OCT scheme provides phase information of a subject at all 3D positions, we can use the angular spectrum method (ASM) for simulating wave propagation [25; 26]. For example, if a lateral plane at certain depth of a 3D subject is out-of-focus upon imaging, we can refocus it using the ASM. Of course, the lateral planes at various depths can have independent refocusing parameters that can be adjusted at every depth of the lateral plane. The digital refocusing based on ASM does not affect the numerical resolution of the system [27]. In the ASM, a complex wave field at a certain depth is decomposed into a sum of a number of plane waves by using a Fourier transform. The decomposed plane waves propagate separately along their directions and recombine at a lateral plane. When the propagation distance of the wave along the z direction is \(d\) (optical distance in air), the complex wave field at another depth is given as [25; 26]:  
\[C_{i}(x,z=d)=\mathfrak{I}^{-1}\left\{\mathfrak{I}[C_{0}(x,z=0)]\exp[id\sqrt{k^{ 2}-k_{x}^{2}}]\right\}, \tag{2}\]  
where \(C_{i}(x,z;d)\) is the line complex field at the depth of z after propagating as \(d\) along the z direction, \(\mathfrak{I}[C_{0}(x,z=0)]\) is the angular spectrum at the original depth of the 3D OCT data, \(k\) denotes the wave number in the medium, and \(k_{x}\) is the x directional wave number. It should be noted that \(C_{i}(x)\) corresponds to the line field distribution along the B-scans, in the fast scanning direction. In the  
Figure 1: Illustration showing a decrease in lateral resolution for the region outside DOF, on mouse brain imaging using optical coherence tomography (OCT) and OCT angiography (OCTA). Scale bar is 1 mm. calculations, the field distribution along the C-scan (thus, \(y\) direction) was excluded as the phase along the C-scan was unstable owing to long measurement time (13 s in our case). To acquire a well-focused field distribution by using Equation (2), it is necessary to determine the propagation distance \(d\) which comprises of multiplication of the physical length and the refractive index of the medium. For this, the point spread function (PSF) of the OCT system and the refractive index of the biological tissues must be considered, which are difficult to obtain accurately. Such requirements complicate the implementation of the digital refocusing algorithm. In order to circumvent these requirements, an algorithm that is capable of automatically determining the appropriate propagation distance \(d\) was included. While iteratively evaluating the contrast of the digitally refocused OCTA image, the propagation distance \(d\) is determined as the value that provides the OCTA image with the highest contrast. In other words, in the OCTA image with the highest contrast, the average of the square of the differences between adjacent A-scans (gradient in the x direction) is maximized. As the contrast evaluation function, the \"squared gradient\" method was utilized as in [28]:  
\[F_{sq.grad}(d)=\sum_{x}\sum_{z}(I_{Flow}(x+1,z;d)-I_{Flow}(x,z;d))^{2}, \tag{3}\]  
where \(I_{Flow}\) is the OCTA signal.