Topological Expansion of Boehm's Brushes via Structured Light

TL;DR

The study demonstrates a novel entoptic phenomenon where spin–orbit structured light expands Boehm's brushes from two lobes to multiple lobes on the retina. The lobe count follows the topological relation $N = |\ell - 2|$, and the pattern location shifts depending on whether $\ell$ is greater or less than 2, revealing polarization‑dependent scattering in peripheral retinal structures. Psychophysical measurements show an exponential decrease in contrast thresholds with retinal eccentricity, reaching robustness around $r_{50} = 1.03^\circ$, consistent with scattering by isotropic, non‑foveal retinal components. These findings establish a new class of topology‑driven entoptic phenomena and suggest avenues for noninvasive retinal diagnostics and studies of light–tissue interactions using structured light.

Abstract

We report a novel entoptic phenomenon in which the classical two-lobed Boehm's brushes are transformed into a multi-lobed structure by projecting spin-orbit coupled light onto the human retina. These structured beams, composed of non-separable superpositions of circular polarization and orbital angular momentum (OAM), produce azimuthally modulated entoptic patterns through polarization-dependent scattering in the retina. Unlike Haidinger's brushes, which arise from dichroic absorption in the macula, the observed effect is driven by angular variations in scattering strength relative to the local polarization direction. In regions where scattering centers exhibit polarization orientations that converge toward a common point, their combined contributions reinforce one another, producing brighter and more sharply defined entoptic lobes whose number and orientation vary systematically with the topology of the spin-orbit stimulus. Psychophysical measurements across retinal eccentricities from 0.5$^\circ$ to 4$^\circ$ in eleven participants revealed that contrast detection thresholds decreased exponentially with eccentricity, consistent with polarization-sensitive scattering by isotropic structures in the non-foveal retinal regions. From the psychophysical fits, the mean eccentricity at which the entoptic pattern reached a 50 % threshold was $r_{50} = 1.03^\circ$ with a 95 % confidence interval of [0.72, 1.34]$^\circ$, indicating that the spin-orbit-induced entoptic structure becomes perceptually robust at approximately 1$^\circ$ retinal eccentricity. Together, these findings demonstrate that spin-orbit light modulates scattering-based visual phenomena in previously unrecognized ways, enabling new approaches for probing retinal structure and visual processing using topological features of light.

Figures (4)

• Figure 1: a) Conceptual illustration of how a structured entoptic pattern emerges from spin–orbit light carrying orbital angular momentum (OAM) $\ell = -2$. For illustrative clarity, the stimulus is shown here as a discrete ring of points, each with a polarization orientation determined by the local structure of the spin–orbit state, see Eq. \ref{['Eqn:PsiInGeneral']}. Each point elicits Boehm’s brushes through polarization-sensitive scattering, and the perceived global pattern arises from the incoherent sum of these local responses. b) Simulated entoptic profiles corresponding to structured beams with OAM ranging from $\ell = -4$ to $\ell = +6$, viewed from an annular (ring-shaped) stimulus. The number of visible lobes follows $N = |\ell - 2|$. For $\ell < 2$, the polarization modulation is pronounced outside the aperture while for $\ell > 2$ the modulation shifts inward, and the structured pattern appears within the dark central region. The annular aperture profile allows access to the inner polarization-dependent scattering response that is otherwise masked by the high central intensity of a disk stimulus.
• Figure 2: Experimental setup for projecting structured stimuli onto the retina. Two broadband white light sources were combined at a beamsplitter (BS); one arm included a rotating polarizer and shutter, while the other served as a DC offset. This allowed modulation of polarization contrast at fixed intensity. The combined beam passed through an annular aperture and was imaged onto a Q-plate to generate a spin–orbit beam with OAM $\ell = +4$, which was flipped to $\ell = -4$ by a half-wave plate (HWP). The Q-plate output was imaged onto the retina using a f=$125$ mm lens placed in front of the eye. An imaging insert before the final lens enabled alignment with fundus photos to calibrate aperture radius to retinal eccentricity. A 530 nm filter set the stimulus wavelength.
• Figure 3: (a) Group-averaged contrast thresholds for entoptic pattern detection across retinal eccentricity. Black points indicate the mean threshold across participants (11 total), where each point represents the average of data within eccentricity bins centered at 0.5$^\circ$, 1.0$^\circ$, 1.5$^\circ$, 2.0$^\circ$, 2.5$^\circ$, 3.0$^\circ$, 3.5$^\circ$, and 4$^\circ$ with a bin width of ±0.25$^\circ$. The solid curve shows the best-fit exponential function, $T(r) = 0.13 + 0.87\, e^{-0.85\, r}$, which captures the group-averaged decay in contrast threshold with increasing eccentricity. The inset shows the individual participant data and fits, illustrating the consistency of the exponential decay across subjects. The red points denote instances where the staircase reached the ceiling (maximum contrast) in two or more of the last six reversals. Note that these represent underestimates of the true threshold. The stimulus contrast range in the setup spanned from 80% to 2.5%. (b) Summary of fitted model parameters across participants. The upper panel shows the distribution of decay constants ($b$) extracted from the individual fits, with a group mean of $b = 1.42~\mathrm{deg}^{-1}$ and a 95% confidence interval of [0.80, 2.04]$~\mathrm{deg}^{-1}$. The lower panel shows the eccentricity at which the fitted models reached a contrast threshold of 50%, with a mean value of $r_{50} = 1.03^\circ$ and a 95% confidence interval of [0.72, 1.34]$^\circ$. These results indicate that the entoptic polarization pattern becomes perceptually robust at approximately 1$^\circ$ retinal eccentricity. Across all participants, thresholds were highest near the fovea and decreased (improved) with increasing retinal eccentricity, consistent with a polarization-sensitive scattering mechanism in the retina.
• Figure 4: Flowchart of the fully automated experimental procedure. A Python program controlled all aspects of the task, including loading calibration data, setting aperture order, managing the rotating spin–orbit stimulus, implementing the interleaved staircases, and logging responses. Participants initiated each trial and provided feedback via a three‑button keypad (right button = clockwise, left button = counterclockwise), while all other steps—including contrast adjustment, stimulus presentation, and progression through apertures—were executed automatically by the system.