3D Stokes polarimetric imaging at nanoscales

Abstract

Optical fields polarized along three dimensions are frequent in optical microscopy and nanophotonics, and yet retrieving their polarization distribution is challenging. We present the experimental implementation of three-dimensional (3D) Stokes polarimetric imaging of nonparaxial optical fields with nanoscale spatial resolution. This approach extends classical Stokes polarimetry (traditionally limited to paraxial fields) into the nonparaxial regime. We use an array of gold nanospheres, each acting as a localized electric dipolar scatterer, to probe 3D polarization states over a field of view of tens of micrometers. The scattered signal is collected by a high numerical aperture objective lens and separated into its circular polarization components, providing a very simple imaging system. We introduce a computational algorithm to efficiently extract the physical parameters from the generated dipole spread functions with a high throughput across the whole field of view. Finally, we show that this method can also be applied to single-molecule localization and orientation fluorescence microscopy.

Figures (5)

• Figure 1: 3D Stokes polarimetric imaging principle. a Illustration of a fully polarized elliptical dipole ($P=1$) (red arrow: spin vector), whose major axis has a polar angle $\eta=45^{\circ}$ and an in-plane angle $\xi=45^{\circ}$, and whose minor axis is along $x$ with aspect ratio $r_{ba}=0.5$. Decomposition of the corresponding Circ-DSF on the SGM basis $\{\mathcal{I}_{j}\}$. b Matrix containing the inner product of the basis elements weighted by the inverse of $\mathcal{I}_0$, calculated for the fully unpolarized state ($S_{j \neq 0}=0$). The sub-matrix enclosed in blue is a representative case of the FIM. c Schematic of the set-up. BS: unpolarized (50:50) beam splitter; SF1,SF2: spatial filters; QWP: quarter-wave plate; WP: Wollaston prism. d Illustration of three different polarization states and their corresponding Circ-DSFs. From left to right: a fully polarized ($P=1$) rectilinear dipole with angles $\eta=45^{\circ}$, $\xi=45^\circ$; a fully ($P=1$) and a partially ($P=0.6$) polarized dipoles whose main eigenvector's major axis is oriented at $\eta=45^{\circ}$, $\xi=45^{\circ}$, its minor axis is parallel to the $x$-$y$ plane, and with aspect ratio of $r_{ba}=0.5$. The basis, FIM and DSFs were computed assuming in-focus dipoles in a continuous medium with refractive index $n_0=1.518$, an objective aperture of $\mathrm{NA}=1.49$, a wavelength $\lambda=532$ nm and a pixel size of 52.72 nm.
• Figure 2: SGM parameter estimation illustration and precision limits. a Vector form of the combined Circ-DSFs. b Construction of the matrix $\mathcal{I}$, including a constant column accounting for a uniform background. c Illustration of the method for SGM parameter retrieval. d Top: synthetic ground-truth Circ-DSF pair; Middle: synthetic Circ-DSF pair containing 7500 signal photons and a uniform background noise of 10 photons per pixel; Bottom: best fit using the retrieval procedure described in the text and illustrated in c. e and f Estimation precision for the normalized SGM parameters in terms of the number of signal photons, for fully and partially polarized states, respectively. The dotted lines show the CRLB, while the solid lines show the mean precision obtained with Monte Carlo simulations and the shaded areas indicate the corresponding standard deviation. Due to the system's axial symmetry, the graphs for each pair $(s_{21}, s_{22})$ and $(s_{31}, s_{32})$ overlap. The Circ-DSFs and the basis were computed assuming in-focus dipoles in a continuous medium with refractive index $n=1.518$, an effective aperture at the detection path of $\mathrm{NA}=1.49$, $\lambda=532$ nm and a pixel size of 52.72 nm.
• Figure 3: C3Pol on 3D linearly polarized states.a C3Pol images of the nanosphere array. Scale bar: 1.5 $\mu$m. b and f Illustration of the generation of uniform p- and s-polarized fields (inset: BFP scheme). c and g, representative examples of measured Circ-DSFs and their best fit for incident p- and s-polarized fields, respectively. Scale bar: 365 nm. d and h Top view of 2D localized elliptical dipoles generated by p- and s-polarized fields; the ellipse's color indicates the value of the major axis' in-plane angle $\xi$. Scale bar: 1.5 $\mu$m. e and i Lateral view of the measured 3D oriented polarization ellipses within an area of $5.8 \mu$m$\times6.3\mu$m located at the central part of the images shown in d and h, respectively. j and l, image composed of the detected ellipses generated with eight incident angles (see insets), for p- and s-polarizations, respectively. In j the in-plane angle $\xi$ is color-coded from $0$ to $360^{\circ}$, to differentiate tilted dipoles whose in-plane angle differs by $180^{\circ}$. k and m, histograms (over 632 and 717 nanospheres, respectively) of the recovered parameters for all the ellipses presented in Figs. j and l.
• Figure 4: C3Pol on elliptically polarized states.a and e: Illumination schemes using LHC and RHC polarized light (inset: BFP scheme). The resulting 3D polarizations at the test plane ideally have the same orientation and ellipticity, but opposite spin vectors. b and f Superposed recovered polarization ellipses and spin vectors for 11 and 16 nanospheres, respectively, seen from two points of view. c and g Histograms of the measured aspect axis ratio $r_{ba}$ and degree of polarization $P$. d and h Examples of measured Circ-DSFs and their best fit. Scale bar: 365 nm. i Generation of a field with subwavelength polarization variations by focusing two beams at opposite extremes at the BFP of the objective (see BFP scheme). j Measured spatially-varying polarization distributions over the vertical plane. The color indicates the sign of $s_{32}$, namely the spin vector's $y$ component. Scale bar 26 nm. k Examples of measured Circ-DSFs and their corresponding best fit. Scale bar 365 nm.
• Figure 5: Measurement of 3D oriented single fluorophores deposited on a coverslip.a and f illustration of the s- and p-polarized illuminations. For each of these two illuminations: b and g reconstructed 3D oriented molecules represented as sticks colored according to the measured polar angle $\eta$. The region of interest is $5.7\times5.7\mu\mathrm{m}^2$. c and h 2D localized molecules across the complete FOV of $\sim 36\mu$m of diameter. The molecules are represented as sticks colored according to the measured angles $\eta$ and $\xi$ (modulo $180^\circ$), and degree of polarization $P$, respectively. The total amount of detected molecules is 10257 and 8138, respectively. Scale bar 9 $\mu$m. d and i Polar histograms for the recovered angles $\eta$, $\xi$ and histogram for the degree of polarization $P$. e and j Examples of measured Circ-DSFs and their best fit. Scale bar 365 nm.