## 2 Point-Spread Functions in Phase Space
### Rigorous Form  
Let \(x,y,z\) be a right-handed Cartesian coordinate system in scene space and let \(\xi,\eta\) be spatial coordinates of the hologram plane located in \(z=0\), such that \(\xi,\eta\) are parallel to \(x,y\), respectively and the \(z\) axis is aligned with the optical axis of our system. Without loss of generality let the scene be located at \(z>\lambda\gg 0\) where \(\lambda\) is the wavelength of the illuminating light. Then the PSF \(P(r)=P(\xi,\eta,z;x,y,0)\) of a spherical wave diverging from \((x,y,z)\) towards \(z=0\) is given as  
\[P(r):=\frac{ze^{i\varphi(r)}}{i\lambda r^{2}}\quad\text{with}\quad\varphi(r): =-\frac{2\pi}{\lambda}r\quad\text{and}\quad r:=\sqrt{(\xi-x)^{2}+(\eta-y)^{2} +(0-z)^{2}}\,. \tag{3}\]  
\(\varphi\) is its (instantaneous) phase and can be used for the definition of its (instantaneous) frequencies \(f_{\xi}\) and \(f_{\eta}\).  
\[f_{\xi}(r):=\frac{1}{2\pi}\frac{\partial\varphi}{\partial\xi}=-\frac{(\xi-x)} {\lambda r}\quad\text{and}\quad f_{\eta}(r):=\frac{1}{2\pi}\frac{\partial \varphi}{\partial\eta}=-\frac{(\eta-y)}{\lambda r}\,, \tag{4}\]  
with \(r\) given by Equation (3). This definition is based on the principle of stationary phase, which states that the (instantaneous) phase \(\varphi\), Equation (3) is approximately sinusoidal while varying \(\xi,\eta\) over several \(\lambda\). It is worth mentioning, that other definitions for instantaneous frequency, which are not based on the principle of stationary phase, do exist. One example is the definition based on marginals evaluated for the WV PSR [14] or higher order derivatives of the phase functions. A detailed list is provided in [37].  
Irrespective of the chosen definition, the spatial frequencies of a monochromatic wave may not exceed \(\pm 1/\lambda\). For a DH, it furthermore may not exceed \((2p)^{-1}\). Equation (4) describes the rigorous mapping from space \(\xi,\eta\) to frequency \(f_{\xi},f_{\eta}\) for an ideal PSF without any aberrations. Its phase space is shown in Figure 3c for a 1D PSF, i.e. with \(\Delta x:=(\xi-x)\) and \(r:=\sqrt{(\Delta x)^{2}+(0-z)^{2}}\). The real part of the same PSF, which is shown in spatial domain in Figure 1a, is presented in Fourier space in Figure 3a. Its instantaneous phase function in spatial domain is depicted in Figure 3b.  
Spatial frequencies are linked to diffraction angles by the grating equation. For example a discretized \(f_{\xi}\) determines the angle \(\theta\) in the \(\xi-z\) plane  
\[\sin(\theta)=\lambda f_{n}\quad\text{with}\quad f_{n}=\frac{n}{Np}\quad\text {and}\quad n\in[-N/2+1,N/2] \tag{5}\]  
where \(N\) is the number of pixels of the hologram along \(\xi\) and \(p\) is its pixel pitch.  
Let us now study the effects of translations along \(x,z\) on the instantaneous frequency \(f_{\xi}\). Translations along \(x\) correspond directly to translations of \(f_{\xi}\) along \(\xi\) as \(f_{\xi}=0\Leftrightarrow\xi=x\). Translations along the optical axis \(z\) change the steepness of the frequency transition. As \(z\) appears only in the denominator for \(z\to\infty\) the change of frequencies will happen imperceptibly slow, i.e., \(\frac{\partial f_{\xi}}{\partial\xi}\big{|}_{\xi=x}\approx 0\). For \(z\approx 0\) the step function is obtained. If \(z<0\) then \(\xi\) and \(x\) trade places and thereby induce a sign change of \(f_{\xi}\). If as shown, a down-chirp (decreasing frequency) is present, the recorded wave is diverging from a point-source located at \(z>0\) towards \(z=0\). A spherical wave diverging from \(z<0\) towards \(z=0\) is approximated by up-chirps. In Figure 2 we color-coded the footprints of the PSFs belonging to 3 different point-sources.