# Incoherent Digital Holography: A Review
## 2 Principle of OSH
### Mathematical Model of OSH  
On the basis of the setup depicted in Figure 1, the signal detected by PD1 is proportional to the total power at the active area of PD1, and the general expression of the signal is written as follows [7; 13]:  
\[i(x^{\prime},y^{\prime},t)=\iiint_{\Sigma}\big{|}[E_{1}(x,y,t;z)+E_{2}(x,y,t;z )]T(x+x^{\prime},y+y^{\prime};z)\big{|}^{2}dxdydz \tag{1}\]  
where \((x,y,z)\) are the spatial coordinates, and \((x^{\prime},y^{\prime})\) denotes the transverse shift of the object with respect to the complex amplitude of the scanning beam. This shift is introduced by the \(X\)-\(Y\) scanner in Figure 1. \(\Sigma\) denotes the active area of the detector; \(T(x,y;z)\) is the 3D amplitude transmittance of the object, where we have assumed the 3D object is a collection of sectional images along \(z\). \(E_{1}(x,y,t;z)=E_{1}(x,y;z)\exp[j(\omega+\Omega)t]\) and \(E_{2}(x,y,t;z)=E_{2}(x,y;z)\exp(j\omega\,t)\) are the optical fields that make up the scanning beams. The signal expressed in Equation (1) is first band-pass filtered (see  
Figure 1: Schematic setup of optical scanning holography (OSH). EOM: electro-magnetic modulator; PBS: polarizing beamsplitter; M’s: mirrors; BEs: beam expanders; HWP: half-wave plate; L’s: lenses; \(f_{2}\) and \(f_{3}\) are the focal lengths of lenses L2 and L3, respectively; PDs: photodetectors; BPF@\(\Omega\): band-pass filter tuned at frequency \(\Omega\). the bandpass filters, BPF@\(\Omega\), in Figure 1) to reject the direct current (DC) term and demodulated by the lock-in amplifier to obtain both the amplitude and phase of the time oscillation term at the heterodyne frequency \(\Omega\). Finally, a complex hologram can be obtained. Depending on the recording geometry, there are two modes of OSH, the coherent mode and the incoherent mode [52; 53; 54; 55]. For the coherent mode of OSH, the active area of PD1 must be infinitesimal, or \(\Sigma=\delta(x,y)\). The resulting coherent complex hologram is expressed as  
\[H_{c}(x,y)=\int E_{1}(x,y;z)E_{2}^{*}(x,y;z)\otimes T(x,y;z)dz \tag{2}\]  
where \(\otimes\) denotes the correlation involving the \(x\) and \(y\) coordinates. In the case for which a spherical wave and a plane wave are involved (Figure 1), the correlation kernel under the paraxial approximation can be expressed as follows [56]:  
\[E_{1}(x,y;z)E_{2}^{*}(x,y;z)\approx\text{circ}\Big{(}\frac{r}{a}\Big{)}\exp \bigg{(}\frac{-jkr^{2}}{2z}\bigg{)} \tag{3}\]  
where \(k\) is the wave number of light, \(r=\big{(}x^{2}+y^{2}\big{)}^{1/2}\), and \(a=z\times NA\) is the radius of the interference fringe pattern, with \(NA\) being the numerical aperture of the spherical wave. It should be noted that this kernel is identical to that in conventional DH. Therefore, the properties and the reconstruction method of coherent OSH are the same as those of conventional DH [7]. For the incoherent mode of OSH, the active area of PD1 must be infinite, or \(\Sigma=1\). The resulting incoherent complex hologram is expressed as  
\[H_{ic}(x,y)=\int E_{1}(x,y;z)E_{2}^{*}(x,y;z)\otimes|T(x,y;z)|^{2}dz \tag{4}\]  
Now the correlation kernel is also the same as that of coherent OSH. However, the coded object function is the intensity transmittance, that is, \(|T(x,y;z)|^{2}\). Therefore, the phase of the object cannot be reconstructed in this mode of OSH. This is also the reason that this mode is called the \"incoherent\" mode. Figure 2 shows the real part and imaginary part of a complex hologram of two coins recorded under the incoherent mode of OSH [57]. The hologram size is \(550\times 550\) pixels with a pixel pitch of \(30\,\mathrm{\SIUnitSymbolMicro m}\). The object distances of the two coins are \(90\) and \(115\,\mathrm{mm}\). The scanned hologram can be digitally reconstructed to an arbitrary plane of distance \(z_{r}\) by  
\[E_{r}(x,y;z_{r})=H_{ic}(x,y)\odot E_{1}^{*}(x,y;z_{r})E_{2}(x,y;z_{r}) \tag{5}\]  
where \(\odot\) denotes the convolution involving the \(x\) and \(y\) coordinates.