Ghost projection via focal-field diffraction catastrophes

TL;DR

Ghost projection aims to synthesize an arbitrary target intensity $S(\mathbf{r})$ by nonnegative weights $w_k$ applied to a basis of illuminating patterns $I_k(\mathbf{r})$; this work replaces a mask-based speckle basis with a focal-field diffraction-catastrophe basis generated from phase modulations built from a truncated Zernike set and focused to produce $I_k = |\mathcal{F}[A(\rho) e^{i \phi_k}]|^2$. The authors demonstrate, via simulation, that NNLS can select a sparse subset of catastrophes to approximate $S$ (up to a pedestal $\mathcal{P}$) and show how modifying the target toward the average pattern $\langle I(\mathbf{r})\rangle$ improves reconstruction fidelity, especially under Poisson noise. They also analyze the influence of Poisson noise and discuss strategies for pedestal flattening, practical implementation across optical and matter-wave regimes, and potential applications in dynamic on-demand beam shaping, aberration correction, lithography, and tomographic manufacturing.

Abstract

Ghost projection is the reversed process of computational classical ghost imaging that allows any desired image to be synthesized using a linear combination of illuminating patterns. Typically, physical attenuating masks are used to produce these illuminating patterns. A mask-free alternative form of ghost projection is explored here, where the illuminations are a set of caustic-laden diffraction patterns known as diffraction catastrophes. These are generated by focusing a coherent beam with spatially modulated phase having random Zernike-polynomial aberrations. We demonstrate, via simulation, that a suitable linear combination of such random focal-field intensity patterns can be used as a basis to synthesize arbitrary images. In our proof-of-concept ghost-projection synthesis, the positive weighting coefficients in the decomposition are proportional to exposure times for each focal-field diffraction catastrophe. Potential applications include dynamic on-demand beam shaping of focused fields, aberration correction and lithography.

Figures (11)

• Figure 1: Schematic of diffraction-catastrophe ghost projection. (a) Parallel lines on the left represent a coherent plane wave traveling to the right, with curved wavefronts indicating the wavefield phase has been modulated (aberrated). The transverse intensity pattern at the focal plane is the "diffraction catastrophe" kravtsov2012caustics. (b) Superposing many different focal-plane diffraction catastrophes, for a variety of modulations, is the essence of our technique for "ghost projection" synthesis of an arbitrary target image paganin2019writingceddia2022ghostceddia2022ghostIIceddia2023universalkingston2025neutron. Note that the "pedestal" is a smooth background that is well approximated by the ensemble average of the utilized focal-field catastrophes.
• Figure 2: Illustration of the ghost-projection concept, for the case of spatially-random speckle maps, in the form given by Eq. (\ref{['eq:VanillaGhostProjection2']}). Here, the smoothed target pattern (the smiling face) is $S(\mathbf{r})\otimes PSF(\mathbf{r})$, the pedestal (or offset) is $\langle B_k\rangle \overline{I}$, and each speckle pattern $I_k(\mathbf{r})$ is weighted by the positive coefficient $B_k$.
• Figure 3: (a) A phase function $\phi_k$ generated by a random linear combination of the 2nd to 21st Zernike polynomials, (b) the corresponding diffraction-catastrophe illuminating pattern $I_k$, (c) a zoomed region of $I_k$.
• Figure 4: (a) A $100 \times 100$ pixel image of the letter "R" used as a target image, (b) a simulated ghost-projection reconstruction of the target image using 40 000 patterns, weighting the catastrophes by their dot product with the target image, (c) a reconstruction using the non-negative least squares algorithm to generate weights, using 40 000 illuminating patterns. The reconstructions have been normalized, so that the minimum value is rendered black and the maximum value is rendered white. (d) There is low correlation between the two methods for generating the weights, $w_k$, with the Pearson correlation coefficient $r$ being 0.05.
• Figure 5: The average diffraction catastrophe, plotted as (a) a 1D profile versus radial distance and (b) a 2D grayscale image. The 2D image above has not been normalized. As may be expected from the central limit theorem, the average illuminating pattern is well approximated by a Gaussian distribution.