Optimization-based hologram design for fine optical tweezer arrays and extension of super-resolution criteria

TL;DR

This work tackles the challenge of generating densely spaced, distortion-free light-spot arrays with high-NA holography. It introduces a optimization-based CGH design that uses a fidelity-of-intensity (FOI) cost and conjugate-gradient minimization, accounting for vector Debye focusing to operate under high-NA conditions. The authors demonstrate a $5×5$ spot array with spacing $0.952 μm$ using λ=820 nm and NA=0.75, and propose a VP-based extension of super-resolution that unifies Rayleigh, Sparrow, and Abbe criteria via spot spacing and separation. They show FOI outperforms traditional RSS in spot separability and uniformity, discuss stability against disturbances, and interpret FOI-designs as enabling controlled complex-amplitude modulation, with broad implications for optical tweezers, microscopy, and microfabrication.

Abstract

Aligning light spots into arbitrary shapes is a fundamental challenge in holography, leading to various applications across diverse fields in science and engineering. However, as the spot interval approaches the wavelength of light, interference effects among the spots become prominent, which complicates the generation of a distortion-free alignment. Herein, we introduce a hologram design method based on the optimisation of a nonlinear cost function using a holographic phase pattern as the optimisation parameter. We confirmed a spot interval of 0.952(1) $μ$m in a $5 \times 5$ multispot pattern on the focal plane of a high-numerical-aperture (0.75) objective by observing the near-infrared (wavelength: 820 nm) holographic output light from a spatial light modulator device, a result which overcomes the limitation of a few micrometres under similar conditions. Furthermore, the definition of the Rayleigh diffraction limit is refined by considering the separation of spots and the spot interval, thereby concluding the achievement of "super-resolution." The proposed method is expected to advance laser fabrication, scanning laser microscopy, and cold atom physics, among other fields.

Figures (7)

• Figure 1: (a) Schematic of CGH design procedure. On the SLM plane, user-defined phase $\theta_j$ is attached to incident light amplitude $a_j$ to make output light amplitude of SLM to be $\psi_j = a_j e^{i \theta_j}$. Light propagation, expressed symbolically by an illustration of a lens, acts as a unitary transformation on $\psi_j$ to yield light amplitude on the image plane, $\Psi_i$. Comparison of output pattern, $I_i = |\Psi_i|^2$, and target pattern, $T_i$, provides a guide to update $\theta_j$. (b) CGHs for the 5$\times$5 square lattice pattern optimized by CGM using FOI (left) and RSS (right) cost function.
• Figure 2: Schematic of experimental setup for hologram reproduction. LD: laser diode, $\text{L}_1$-$\text{L}_4$: achromatic lens, ND: neutral density filter, AP: aperture, BS: beam splitter, P: pinhole, M: mirror, $\text{OB}_1$-$\text{OB}_2$: objective lens, IS: CMOS image sensor.
• Figure 3: (a), (b): Numerically reproduced images of square lattice target patterns with different spot spacings. The upper row was calculated from FOI-designed CGHs, and the lower row from RSS-designed ones. The spot spacings are 1.6, 1.8, 2.1, and 2.5 pixels from left to right. (c), (d): Experimental results of images reproduced from FOI- and RSS-designed CGHs, respectively. The spot spacing was set to 1.8 pixels in a common target pattern, corresponding to an actual length of $0.952(1)$ µm on the image plane.
• Figure 4: (a) Focal pattern (top) and profile (bottom) of Airy disk [NA = 0.75(air)]. Red line in the bottom plot shows cross-sectional profile corresponding to the focal pattern, where $r_{\text{A}}$ refers to the Airy-disk radius. (b) Focal patterns and profiles of incoherent sums of two Airy disks aligned at distances of $d = r_{\text{A}}$ (left) and $2r_{\text{A}}$ (right). In the bottom plots, thick green lines display the profiles of the incoherent sums, whereas thin red lines indicate those of individual Airy disks. Definitions of peak (P) and valley (V) are illustrated as well. (c) The relationship between the VP ratio and spot spacing for incoherent sums of adjacent Airy disks is depicted (solid green line), where the traditional Sparrow, Abbe, and Rayleigh criteria are plotted as purple crosses. Open symbols represent VP ratios for numerically reproduced images from FOI- (red open circles) and RSS-designed (blue open triangles) CGHs, while closed symbols represent those for the corresponding observed patterns, i.e., Figs. \ref{['fig:rep_images']}(c) (red closed circle) and (d) (blue closed triangle), respectively. Error bars (SEs) are determined similarly to those in Table \ref{['tab:var_eff']} but are smaller than the size of the plot symbols, except for the experimental result of an RSS-designed CGH.
• Figure 5: Reproduced images of FOI-holograms with hexagonal, kagome, and triangular lattices as target patterns. The spot intervals are 1.9, 1.9, and 2.3 pixels, respectively.