Fig. 1. Phase space models of Fresnel diffraction. (a) Although a PSF can be dense in space (x) and the frequency (ν) domain, it is sparse in phase space; it is color-coded using brightness for magnitude and hue for phase. (b) Side-by-side comparison of how propagating light will progressively shear over phase space. An example point is color-coded red in phase space. (a) Phase space of a PSF. (b) Fresnel diffraction shears phase space.

Fig. 2. Chirplet example with a quadratically changing phase.

Fig. 3. Examples of canonical dual windows for different values of ab, computed for a discrete signal with N=1024 samples and 64 modulations.

Fig. 4. Summary of the proposed Fresnel transform pipeline. Every row and column can be processed independently thanks to the separability of the Fresnel transform. Using the dual window, we obtain a structured collection of chirplets forming a Gabor frame. These are transformed to obtain the target Gabor coefficients, after which the resulting signal can be retrieved with an inverse Gabor transform. Reprinted from Fig. 20 in Ref. [33].

Fig. 5. Approximate Gabor coefficient mapping for fast Fresnel diffraction. The diagrams depict the discrete Gabor coefficients, showing translations along the x-axis and modulations along the ν-axis, color-coded using brightness for magnitude and hue for phase. The example hologram (a) is an idealized Gaussian-modulated random diffuser placed a few cm from the hologram plane, brought into focus in (b). The coefficient in (b) marked by a “+” sign is calculated using only the highlighted five rows of five coefficients in (a), chosen to best align with the parallelogram-shaped region. (a) Source Gabor space. (b) Destination Gabor space.

Fig. 6. Plots of the PSF of the discrete Fresnel diffraction, showing the real and imaginary parts and the magnitude. (a), (b) Spatial domain method, (c), (d) proposed Gabor domain method, and (e), (f) frequency domain method. (a) Spatial domain Fresnel transform (real/imaginary). (b) Spatial domain Fresnel transform (magnitude). (c) Gabor domain Fresnel transform (real/imaginary). (d) Gabor domain Fresnel transform (magnitude). (e) Frequency domain Fresnel transform (real/imaginary). (f) Frequency domain Fresnel transform (magnitude).

Fig. 7. Side-by-side comparison of an example hologram propagated with different algorithms, at d=5  mm. (a) The input hologram, followed by multiple reconstructions, using (b) the spatial domain method, (c) the proposed Gabor domain method, and (d) the frequency domain method. Reprinted from Fig. 21 in Ref. [33]. (a) Input hologram. (b) Output (spatial domain). (c) Output (Gabor domain). (d) Output (frequency domain).

Fig. 8. Example of scaled numerical diffraction with an offset and resolution change. This is done with (b) the spatial domain approach and (c) the proposed Gabor domain method. The figures depict the holograms with correct relative sizes. (a) Source hologram. (b) Destination hologram (spatial). (c) Destination hologram (Gabor).

Fig. 9. Log-log plot of calculation time, comparing the reference FFT-based Fresnel diffraction algorithm with the proposed algorithm as the hologram resolution increases.