## 2 Point-Spread Functions in Phase Space  
Now, we introduce the fundamental concepts of DH required for the subsequent phase space analysis. The diffraction pattern, recorded on a given manifold, for example a (detector-)plane, generated by a single point-source is called point-spread function (PSF) \(P\). A vast amount (\(>10^{5}\)) of super-positioned PSFs, each from a different illuminated point-source in the scene, is what we refer to as a hologram \(H(\xi,\eta,0)\). Within scalar diffraction theory PSFs are complex-valued and described in every point of the recording manifold by one amplitude and one phase. Given an amplitude-phase distribution \(h(x,y,z)\in\mathbb{C}\) of point-sources located between the \((x,y,z=const)\) planes with \(z\in[z_{1},z_{2}]\), \(H\) can be calculated via the first Rayleigh-Sommerfeld integral solution as  
\[H(\xi,\eta,0)=\overline{E_{R}}(\xi,\eta,0)\underbrace{\int_{z_{1}}^{z_{2}}\int _{-\infty}^{\infty}\int_{-\infty}^{\infty}h(x,y,z)P(\xi,\eta,0;x,y,z)dxdydz}_{O (\xi,\eta,0)}, \tag{1}\]  
whereas \(\overline{E_{R}}\) is a conjugated reference wave used for recording. \(O\) is called object wave and this point-wise multiplication of two complex-valued wavefields is called interference. Interference is vital in practical setups. For the sake of clarity, we will split the discussion of the phase space of the object and reference waves. The main part of this paper will be concerned only with the object wavefield \(O\). Reference waves and the interplay of both waves will be covered in Section 4. For now, we shall assume a plane wave with phase 0, i.e., \(E_{R}=1\). Given the complex-valued, continuous object wavefield on a surface, a digital hologram can be obtained through pixelation [36]--a sampling with a given pixel pitch \(p\) and discretization by some bit-depth. A complex-valued matrix of size \(N_{\eta}\times N_{\xi}\) is obtained where \(\xi\) enumerates the columns.  
For a PSF, the spatial frequency distribution, i.e., the way spatial frequencies vary with space, encodes the position of the corresponding point-source in scene space and its relative phase offset. We visualized the dependence on lateral and longitudinal positioning in Figure 2 for two different expressions of the PSFs.  
The two most common PSF expressions are the rigorous form, which corresponds to working with spherical wavefronts, and the Fresnel approximation, corresponding to quadratic wavefronts. Rarely, a more crude PSF approximation finds use as well which is called the Fraunhofer approximation. These approximations are used to simplify analysis and substantially speedup calculations as through Equation (1) a different diffraction regime, i.e., a way of (approximate) light propagation, is associated to each of the specific PSF shapes.  
Figure 2: 1D hologram genesis starting from 3 specific points, and phase space footprint in the rigorous form at \(z_{Ex}\) and within the Fresnel approximation at \(z_{Fr}\).  
One way to classify diffraction regimes is by the Fresnel number \(N_{Fr}\), see [13], defined as  
\[N_{Fr}(r):=\frac{(0.5\ \varnothing(\text{aperture}))^{2}}{\lambda\left|z\right|}= \frac{0.5\max(\left|\xi-x\right|)\cdot 0.5\max(\left|\eta-y\right|)}{\lambda\left|z \right|}\,. \tag{2}\]  
If \(N_{Fr}>1\) the PSF takes the rigorous form and its Fourier transform, the phase accommodation kernel of the angular spectrum method, is typically used for propagation of holograms in this regime. If \(N_{Fr}\lesssim 1\) the Fresnel approximation is sufficiently accurate for propagation and if \(N_{Fr}\ll 1\) propagation can be approximated even further by the Fraunhofer approximation. The reminder of this section will be devoted to analyzing the phase space behavior of these PSF forms.