Visualized Geometric Phase of Caustic Geometric Beams

TL;DR

The paper tackles visualizing Pancharatnam-Berry ($PB$) geometric phases in complex optical fields without interferometry or beam truncation. It introduces SU(2) coherent states constructed from Generalized Gaussian (GG) eigenmodes and mapped onto the SU(2) Poincaré sphere (MPS), enabling caustic-driven 3D wave-packet evolution to encode $PB$ and Gouy phases. A noninterferometric measurement framework is developed, leveraging the 3D wave-packet surface feature lines and a polynomial extrapolation to extract the $PB$ phase from experimental data, validated by an SU(2) mode evolution demonstration. The approach provides a direct, structure-based method to analyze geometric phases in structured light, with potential implications for beam shaping, optical metrology, and photonic quantum information processing. Throughout, the total $PB$ phase for an SU(2) mode emerges as a sum over constituent eigenmodes, linking geometric evolution on the SU(2) sphere to observable spatial patterns.

Abstract

Detecting Pancharatnam-Berry geometric phases of light typically requires interferometry or diffraction through a specially truncated aperture. Here, we introduce a simpler method that allows direct and fully visual detection of geometric phases in structured light without using interferometers or beam truncation. Our approach takes advantage of the geometric phase that naturally arises in SU(2) structured beams, where spatial wave packets follow caustic trajectories during propagation. By observing the evolution of these caustic-linked wave packets, we directly visualize both the geometric phase and the Gouy phase. This visual detection method provides new insight into geometric phases in complex optical fields and expands the possibilities for designing optical systems that exploit phase geometry.

Figures (8)

• Figure 1: Schematic of caustic light modes mapped on the SU(2) Poincaré sphere: (a) 3D wave packet of a caustic-coupled geometric mode. (b) This wave packet is spatially coupled to a Lissajous parametric surface. (c) The Lissajous surface is a ruled surface generated by a family of rays forming caustics; the coupled transverse Lissajous curve evolves upon propagation. (d) The generalized evolution of the Lissajous--trochoidal trajectories and their caustic-coupled modes, with the modes mapped onto the mode Poincaré sphere (MPS), illustrates the complex behavior and interactions of these light modes. A phase–intensity color disk serves as a legend indicating phase and intensity, where the hue represents the phase $\theta$ of the complex field and the brightness corresponds to the field intensity $|E|^2$.
• Figure 2: Transverse intensity profiles for different $M$ values. (a)--(d) correspond to $M = 6, 10, 20,$ and $30$, respectively. As $M$ increases, the mode profiles become sharper and the ray structures more distinct.
• Figure 3: Schematic illustration of the generation and transformation of an SU(2) wave packet through coherent superposition: The schematic depicts the creation of an SU(2) wave packet via the coherent superposition of its eigenstate wave packets. The summation symbol $\sum$ denotes coherent superposition, and the downward-pointing arrow signifies the transformation of upper modes into the specified lower modes. Starting from the top, the first row illustrates an SU(2) equatorial mode, termed the initial mode, generated by superposing HG modes. The mode indices in the two transverse directions, $(m_{{K}}, n_{{K}})$, are $(18,30)$, $(19,33)$, $(20,36)$, $(21,39)$, $(22,42)$, $(23,45)$, and $(24,48)$. These modes correspond to the equatorial points on the OAM PS with orders $N_{{K}} = 48$, $52$, $56$, $60$, $64$, $68$, and $72$, respectively. In the second row, an SU(2) polar mode is created from $M+1$ LG modes, sharing the same transverse indices and mode orders as the eigenstate modes of the SU(2) equatorial mode. The third row presents an SU(2) mode that emerges from the final HG modes, which also share the same transverse indices and mode orders as the previous rows. These HG modes acquire a PB phase through a complete cycle of mode transformation. Unlike the initial SU(2) mode in the first row, the SU(2) mode in the third row, termed the final mode, includes an additional PB phase. Importantly, any modes from the first and third rows that correspond to the same equatorial point on the OAM PS will also result in coherently superposed SU(2) modes corresponding to the same equatorial point on the SU(2) PS. A phase–intensity color disk serves as a legend indicating phase and intensity, where the hue represents the phase $\theta$ of the complex field and the brightness corresponds to the field intensity $|E|^2$.
• Figure 4: Illustration of the effect of the PB phase on the initial and final modes of HG and SU(2) under a complete cycle of mode transformation: This diagram represents the mode profiles and transformations. (a1) and (a2) depict the initial and final mode profiles of the HG mode, respectively, showing identical intensity distributions but differing phase distributions. This indicates that interferometric approaches are viable for determining the generated PB phase. (d1) and (d2) present the initial and final mode profiles of the Lissajous geometric mode, showcasing varying intensity and phase distributions, suggesting that non-interferometric methods are applicable to inferring the generated PB phase. (e1) and (e2) display the 3D parametric surfaces of the Lissajous geometric modes, while (f1) and (f2) illustrate the feature lines on these surfaces. The intersections of these feature lines serve as indicators for measuring the PB phase. (b) and (c) show the MPS with several modes along a closed path. Here, $\alpha$ denotes the azimuthal coordinate; $\alpha_{1}$, $\alpha_{2}$, and $\alpha_{3}$ are the azimuthal coordinates of the red, green, and blue segments, respectively; and $\gamma$ is the enclosed spherical angle of the closed path. A phase–intensity color disk serves as a legend indicating phase and intensity, where the hue represents the phase $\theta$ of the complex field and the brightness corresponds to the field intensity $|E|^2$. Collectively, these panels elucidate the different impacts of the PB phase on eigenstates versus SU(2) coherent state wave packets, highlighting the potential of non-interferometric measurement, whereby the PB phase is inferred from PB-induced changes in the intensity distribution and feature-line evolution.
• Figure 5: Comparison of SU(2) modes for $(s,t)=(1,3)$ before and after continuous evolution. (a) Initial mode. (b) Final mode after a full continuous evolution where the solid angle $\Omega = 140^\circ$. The fields are presented in the transverse plane $(x,y)$. A phase–intensity color disk serves as a legend indicating phase and intensity, where the hue represents the phase $\theta$ of the complex field and the brightness corresponds to the field intensity $|E|^2$. It can be seen that the PB phase alters both the SU(2) mode’s phase and intensity structure. This change is consistent with both the mode profiles obtained through continuous evolution and those predicted by Eq. \ref{['SU(2)_coordinate_with_PB']}.