Role of a Quarter-Wave Plate in Confocal Microscopy: Signature of Spin-Orbit Interactions

Abstract

Spin-orbit interactions of light couple polarization and spatial degrees of freedom, underpinning phenomena such as the spin Hall effect of light. Although widely explored at interfaces and in tightly focused beams, their impact in nominally paraxial confocal systems remains largely unexamined. Here we show that a single quarter-wave plate embedded in a simple confocal geometry between polarizers can strongly reshape the transverse structure of a Gaussian beam. We observe an enhancement of the polarization extinction ratio by more than two orders of magnitude, accompanied by a transformation of the Gaussian intensity profile into a first-order Hermite-Gaussian-like two-lobe mode. The orientation of this pattern is continuously tunable via rotation of the wave plate, evidencing polarization-controlled reorientation of the transverse field. To explain these observations, we introduce a minimal extension of Jones matrix formalism incorporating complex parameters that quantitatively reproduces the measurements. Our results uncover a previously overlooked form of spin-orbit-mediated mode control in standard confocal optics and establish a simple route to on-demand spatial mode engineering for applications in resonant spectroscopy, optical imaging and quantum optics.

Figures (10)

• Figure 1: Experimental configuration and polarization-dependent confocal response.a Schematic of the confocal setup. A Gaussian beam is prepared by a polarizer ($P$) at angle $\beta$ and transmitted through a quarter-wave plate (QWP) with fast-axis angle $\gamma$, followed by an analyzer ($A$) at angle $\alpha$. The transmitted field is coupled into a single-mode fiber (SMF2) and detected by a photodiode, while the fiber position is scanned in the focal plane. Unless otherwise stated, all elements operate at normal incidence. b Extinction ratio curves are shown for both cases, with (orange) and without (blue) a QWP. The polarizer, QWP and analyzer are initially set to $\beta = 0^\circ$, $\gamma = 0^\circ$ and $\alpha=90^\circ$, respectively. After optimization, the extinction ratio is enhanced to $\sim 10^7$, compared to $\sim 10^5$ without a QWP. In addition, the extinction maximum is shifted by $\sim 0.209^\circ$ relative to the case without a QWP. Co- and cross-polarized confocal mappings without (c) and with (d) a QWP are shown. The extinction ratio maps in c and d are obtained by dividing, pixel by pixel, the co-polarized data by the cross-polarized data for both cases. In cross-polarization, mode splitting into two lobes along the $x$ direction is observed when a QWP is introduced. The cyan dotted circle indicates the non-convoluted focal spot 1/e$^2$ waist diameter at the collecting fiber end.
• Figure 2: Confocal mapping of linearly polarized ($\beta = 0^\circ$) Gaussian laser beams with a QWP for different analyzer angles under cross-polarization. Measured and fitted evolution of the modal confocal mapping for different analyzer angles $\delta\alpha$. The extinction ratio curve is also presented to guide the extinction levels. At maximum extinction ratio, the mode splits into two lobes at $\delta\alpha=0$. With comparable extinction between $10^5$ and $10^6$ at angles $\delta\alpha=-0.08^\circ$ and $0.08^\circ$, the mode shifts opposite along $x$ axis. When the extinction ratio close to $10^5$ at angles $\delta\alpha=-0.16^\circ$ and $0.16^\circ$, the beam shifts back to center position indicated by two dashed white lines.
• Figure 3: Control of first-order HG-like mode orientation. Measured and fitted cross-polarized confocal modal maps for different combinations of the incident polarization and the QWP fast-axis orientation. Panels with solid frames correspond to incident linear polarization aligned parallel to the QWP fast axis, while panels with dotted frames correspond to incident linear polarization perpendicular to the QWP fast axis. The center positions of the co-polarization references are indicated by two dashed white lines. The orange arrow and blue dotted line indicate the polarization direction and QWP fast axis, respectively.
• Figure S0: Schematic of a finite-size Gaussian beam incident on a quarter-wave plate (QWP). The central wavevector is denoted by $k_0$, while $u$ and $v$ represent the angular coordinate of the constituent plane-wave components. The fast axis of the QWP is oriented at an angle $\theta$ with respect to the $z$ axis.
• Figure S1: Optimized extinction ratio as a function of the QWP retardance $\Phi_0$ for different angular tuning windows of the polarizer and analyzer: (a) $\pm0.1^\circ$, (b) $\pm0.2^\circ$, and (c) $\pm0.3^\circ$. Within each window, the optimization is performed on a discrete angular grid, where the angular step size $\Delta\alpha=\Delta\beta$ is indicated in the legend. The finite tuning window and discrete angular sampling impose a practical upper bound on the achievable extinction ratio, which is on the order of $10^7$ for the experimentally relevant window of $\pm0.2^\circ$.