# High Bandwidth-Utilization Digital Holographic Reconstruction Using an Untrained Neural Network
## 2 Principle  
In DH wavefront recording, a Mach-Zehnder interference light path is designed, as shown in Figure 1a. The sample is illuminated by a laser at 532 nm, and the complex object optical field (\(O(x,y)=A_{O}(x,y)\exp(i\varphi_{O}(x,y))\)) passing through the objective lens is encoded into intensity modulations by interfering with the reference wave (\(R(x,y)=A_{R}(x,y)\exp(i\varphi_{R}(x,y))\)). The intensity distribution of the recorded hologram can be expressed as:  
\[f(O,R)=|O(x,y)+R(x,y)|^{2}=|O|^{2}+|R|^{2}+O^{*}R+R^{*}O \tag{1}\]  
where \(A_{O}(x,y)\) and \(A_{R}(x,y)\) denote object wave amplitude and reference wave amplitude, respectively. Moreover, \(\varphi_{O}(x,y)\) and \(\varphi_{R}(x,y)\) are the phase of the object wave and reference wave, and \"*\" denotes the complex conjugation. The hologram is captured by a CCD detector (The Imaging Source DMK 23U274, 1600 \(\times\) 1200, 4.4 \(\upmu\)m), as shown in Figure 1b. In this work, we turn the DH system into a slightly off-axis state while causing the zero-order term to partially overlap with the \(\pm\)1-order terms, as shown in Figure 1c. The +1-order term (\(R^{*}O\)) can be extracted by the spectral filtering window. The recovered wrapped phase can then be recovered by inverse Fourier transform. However, the phase aberration is inevitably caused by the objectives in the lens system. This problem can be addressed by employing the phase aberration compensation technology based on principal component analysis (PCA) [35; 36]. Then, the phase unwrapping method based on reliability [37] analysis is used to obtain the final phase result.  
Of note, in the DH system, the hologram can demonstrate various spatial carrier frequencies due to different modulation methods, which will influence the SBU of the sensor. According to the relationship between object bandwidth and sensor's bandwidth, the SBU can be defined by the ratio of the area of cross-correlation terms to camera bandwidth (\(1/\Delta\times 1/\Delta\), where \(\Delta\) is the pixel size) in the SFD. Figure 1d-g  
Figure 1: **(a)** Slightly off-axis interference optical path. **(b)** The captured low-carrier-frequency fringe under the slightly off-axis system. **(c)** The spectrum of the hologram. **(d–g)** The space bandwidth utilization (SBU) calculation results under the four different spectrum states.  
four different spectrum states. Figure 1d shows that under the quasi-off-axis situation with a horizontally modulated reference beam, the bandwidth of the object is \(B_{o}=0.25B_{c}\) (where \(B_{c}\) is the the bandwidth of camera). Assuming the spatial resolution of the DH system is limited by the spatial sampling of the CCD, and the area of the distribution of the spectrum can be calculated as \(\pi\left(B_{o}/2\right)^{2}\) in the diffraction-limited microscopy system. By definition, the SBU of the DH system can be written as:  
\[SBU=2\pi\Delta^{2}\left(B_{o}/2\right)^{2} \tag{2}\]  
Therefore, if the SFD of the hologram has \(M\times N\) pixels, the area of the cross-correlation terms is \(2\pi\left(M/8\right)\left(N/8\right)\). The SBU is \(2\pi\left(MN/64\right)/\left(MN\right)=9.82\%\). Additionally, under the situations shown in Figure 1e-g, the bandwidths of the object are \(B_{o}=0.5B_{c}\), \(B_{o}=2B_{c}/\left(2+3\sqrt{2}\right)\), and \(B_{o}=2B_{c}/\left(2+\sqrt{2}\right)\), respectively. Moreover, the corresponding SBU can be calculated as 39.27%, 16.12%, and 53.90%, respectively. It can obviously demonstrate that the slightly off-axis scheme has a higher SBU compared to the quasi-off-axis scheme.