Observation of spatially structured Montgomery effect in free space

TL;DR

The paper addresses lensless self-imaging of aperiodic structures by introducing the Montgomery carpet, a generalization of the Talbot effect in which transverse spectra are discretized as $k_r(n)=k_L\sqrt{n}$. Using a phase-only SLM to implement a dynamic holographic spectrum, the authors realize self-imaging over 30–100 mm with independent control of the self-imaging distance $z_s$ (via $k_L$) and the beam width/DOF (via the number of rings $N$). They demonstrate: (i) direct observation of Montgomery carpets, (ii) 3D Talbot-like repetition in cylindrical coordinates, and (iii) self-imaging of diverse structured light—HG, LG with OAM, Ince-Gaussian, and Airy beams—while preserving topological features through an overlay function $\widetilde{A}(k_r,\chi)$. The work offers a programmable, lensless platform for multi-plane microscopy, optical trapping, and structured-quantum systems, with potential extensions to non-paraxial and vector fields via metasurfaces and polarization control.

Abstract

We report the first direct observation of the spatially structured Montgomery effect, a lensless self-imaging phenomenon that generalizes the Talbot effect to aperiodic structures, unfolding repeated tightly focused spots (~10 $μ$m) in free space. Using a dynamic optical hologram to discretize radial spatial frequencies, we demonstrate self-imaging at distances ranging from 30 to 100 mm. Our method independently controls the focal spot size and self-imaging period, enabling dynamic three-dimensional light patterns. We also show the arbitrary tunability of the transverse profile by demonstrating revivals of Laguerre-Gaussian, Hermite-Gaussian, Ince-Gaussian modes, and Airy beams. These findings open opportunities for multi-plane microscopy, optical atom traps, and quantum atomic systems.

Figures (5)

• Figure 1: The concept of the Talbot and Montgomery effects. Column I displays the Fourier space in which the spatial spectra of the corresponding field lay on the intersections of horizontal iso-$k_z$ planes and the paraboloid, which represents the dispersion relation $k_{o}\!-\!k_{z}\!\!=\!\!(\!k_x^2\!+\!k_y^2)/2k_{o}$; the colormap of the surface and the lines represent the spectral phase $\phi(k_x,k_y)$. Column II depicts the intensity distributions $I(x,y,z)$ of the corresponding field in 3D physical space, and Column III cross-sections at $z\!=\!z_{\mathrm{s}}$ and $x\!=\!0$. (a) The Talbot effect produced via a periodic sampling of radial frequencies $k_{r}$. The Montgomery effect generated from a aperiodic sampling of radial frequencies $k_{r}$ with (b) a constant spectral phase and (c) spiral phase $\widetilde{\psi}(k_{r},\chi)\!=\!e^{i\chi}$. Here $k_{r}\!=\!\sqrt{k_x^2\!+\!k_y^2}$ and $\chi\!=\!\arctan{(k_y/k_x)}$.
• Figure 2: Experimental realization of Talbot and Montgomery effects. (a) Schematic of the synthesis and characterization setup consisting of a (i) phase‑only SLM to imprint the hologram, (ii) a 4-f system with two spherical lenses ($f_{1}\!=\!f_{2}\!=\!250$ mm) with an (iii) iris at the Fourier plane for spatial filtering and to image the SLM plane, where (iv) a CCD camera axially scans the evolution of the intensity profile $I(x,y,z)$. (b) Observation of the Talbot effect, Montgomery effect with (c) a flat phase profile and (d) a spiral phase profile $e^{i\ell\chi}$ ($\ell\!=\!1$). In (b-d) left panels depict the transverse intensity profiles $I(x,y,z_{\mathrm{s}})$ at the Talbot planes (depicted by a dashed white line on the right panel), the right panel corresponds to the axial intensity profiles $I(y,z)$ at $x\!=\!0$. The inset on the left panel of (d) depicts the measured phase profile $\phi(x,y)$ at $z=25$ mm. We use a log-scale colormap for the intensity plots throughout the figure for clearer visualization.
• Figure 3: Measured axial intensity distributions of Montgomery carpets. Columns correspond to the radius of the Montgomery ring $k_{\mathrm{L}}\!=\!47$, 57, and 66 rad/mm; rows correspond to the number of rings in the spatial spectrum $N\!=\!3$, 5, and 7. White line plots at the bottom of each panel represent normalized axial intensity profiles at the beam center $I(0,0,z)$. The beam size $w$ and depth of focus (DOF) $\Delta z$ are defined as the full width at half maximum of the intensity in the transverse and axial directions, respectively, as shown in (i). We use a log-scale colormap throughout the figure for clearer visualization.
• Figure 4: Measured characteristic parameters of the Montgomery effect. (a) self-imaging distance $z_{\mathrm{s}}$, (b) beam size $w$, and (c) the depth of focus (DOF) $\Delta z$ as a function of $k_{\mathrm{L}}$. Points correspond to the measured data, and a line with the corresponding color indicates the theoretical plot. In (b,c) different colors correspond to different numbers of rings $N$ in the Montgomery effect.
• Figure 5: Demonstration of the spatial structuring of the transverse profile of the Montgomery carpet. Montgomery carpet with the transverse profile in the form of Hermite-Gaussian modes (a) HG$_{01}$ mode and (b) HG$_{24}$ mode; (c) Ince-Gaussian mode IG$_{66}$ mode with ellipticity $\epsilon=0.5$; (d) 1D Airy beam with cubic phase structure along the $y$-axis. Column I represents the calculated complex overlay function $\widetilde{A}(k_x,k_y)$, where the colormap depicts the phase. Column II corresponds to the transverse intensity profile $I(x,y)$ at the first self-imaging plane $z\approx25$ mm. Columns III and IV show the measured 2D intensity plots $I(x,z)$ at $y=0$ (horizontal dashed line in (a) Column II) and $I(y,z)$ at $x=0$ (vertical dotted line in (a) Column II), respectively.