## 1 What is the type of hologram?
### Optimizing Space-Frequency Bandwidth Product  
The part of the signal in phase space, that is actually conveyed by a pixel-matrix depends exclusively on its pixel pitch \(p\) and number of pixels \(N_{\xi}\times N_{\eta}\). Due to physical limitations on the pixel pitch and addressable number of pixels, the phase space of visible light holograms is never captured in full. In case of in-line or on-axis holography, these limitations of any row/column of said matrix defines a rectangular, axis-aligned bounding box around the optical axis \((\xi,\eta)=0\) in the phase space. If we assume effectively a complex-valued modulation/detection capabilities per pixel, the bounding box is given by the pixel matrix--bandwidth \(1/p\) times the spatial extent of the hologram given by \(Np\) in absolute units of frequency [Hz] and space [m]. This holds for digital and optical setups [36] alike, if aberrations and intermediate limiting numerical apertures of optical setups are neglected. The space-bandwidth product (SBP) of a spatial light modulator (SLM) is given as \(\text{SPB}_{\text{SLM}}:=N_{\xi}N_{\eta}\) and cannot be enlarged with additional optics. It characterizes a passive optical system through its entendu (active area times angular spread) [36]. Figure 8: **(a,b)** Front and rear back-propagation of a hologram containing two objects. Only by joint consideration the entire scene is visible. (c) The S-method evaluated on a 1D cross-section of the hologram, shows a clear separation of (all) objects in phase space. Objects are indistinguishable in either spatial or frequency domain alone. (d) For reference, the point-cloud model of the scene is given. Parts of this figure are reused from [30]. We illustrate the bounding box using normalized coordinates in Figure 9c. For this we selected a hologram depicting a single model of a diffusive scattering \"Earth\" (from the INTERFERE-II database [46], see Figure 9a) with a resolution of \(8192\times 8192\), 1 \(\upmu\)m pixel pitch, and a wavelength of 633 nm. The model is located at a depth of 1.18-1.64 cm and therefore comparatively shallow. The ramifications of this bounding box interpretation are substantial. It allows to predict and subsequently tune systems towards more effective use of the SBP by maximizing the ratio  
\[\frac{\text{SPB}_{\text{H}}}{\text{SPB}_{\text{SLM}}}\,, \tag{16}\]  
where \(\text{SPB}_{\text{H}}\) is the SPB of a signal \(H\)--see [13], p. 33f. Conceptually \(\text{SPB}_{\text{H}}\) equals the area occupied by the signal within the bounding box and we will discuss two ways of obtaining a numerical estimator in Appendix A. For example, a holographic capture of only object 2 of Figure 8 requires about half the frequency bandwidth (i.e., only a hologram of double the pixel-pitch and half the number of pixels) than a recording of object 1. All signal components that exceed the maximal supported frequency of a DH \(1/(2p)\) (also: Nyquist frequency) cause aliasing. Aliased frequency components appear as being wrapped around said frequency range. During the reconstruction of weakly aliased holograms, parts of the scene appear mirrored along the boundaries. See for example Figure 9b where we reconstructed the hologram (without any aperture and increased contrast) after resampling the hologram with a pixel pitch of \(1/0.65\)\(\upmu\)m. As the hologram is now undersampled, aliasing artifacts arise along the borders. For more more severe undersampling, the aliased images and the original will overlap and greatly reduce visual quality. Figure 9: **(a)** Standard back-propagation of a hologram of a diffuse Earth recorded in the Fresnel limit using a numerical aperture of \(1/4\) of its size. (b) Back-propagation after undersampling the hologram by a factor of \(1/0.65\) without using a numerical aperture. Phase space footprints of 1D cross-sections are shown after refocusing to different distances in (c-e). In (c) the space-frequency bandwidth boundary box is marked. (f) Shows a hologram of the same object in a different diffraction regime.