Radial Polarisation Measurements

• Radial Polarisation Measurements are techniques that use polarisation-resolved extinction microscopy to determine the anisotropic optical response of nanoparticles and biological tissues.
• They integrate wide-field and confocal imaging with controlled polarisation states to extract extinction cross-sections and full Jones matrix reconstructions.
• Recent advances, including vectorial Fourier ptychography and multi-polarisation spectral fitting, enable sub-nanometre precision and high throughput in nanoparticle morphometry.

Quantitative polarisation-resolved optical extinction microscopy is a class of wide-field and confocal optical techniques that enable spatially resolved measurement of extinction cross-sections and full polarimetric properties of microscopic specimens, including individual plasmonic nanoparticles, biological tissue, and anisotropic materials. By systematically controlling and analyzing the polarisation state of both illumination and detection, these methods resolve the vectorial response of samples with high throughput and spatial resolution, attaining quantitative morphometric precision comparable to electron microscopy for nanoparticles and full Jones matrix imaging for biological or calibration samples. Recent advancements include integration with vectorial Fourier ptychography for large-area Jones matrix reconstructions (Dai et al., 2021), confocal schemes for >10⁸ cross-polarization extinction (Benelajla et al., 2020), and 3D nanoparticle aspect-ratio measurement with sub-nanometre precision using multi-polarisation spectral fitting (Payne et al., 21 Dec 2025).

1. Physical Principles of Polarisation-Resolved Extinction

Polarisation-resolved extinction microscopy relies on probing the interaction of polarized light with matter, capturing both the amplitude and polarisation-dependent attenuation (diattenuation) and phase retardance effects. At the single-particle nanoscopic level, the extinction cross-section $\sigma_{\rm ext}$ is defined by the dipole polarizability $\alpha(\omega)$ of the nanoparticle or the specimen features: $$\sigma_{\rm ext}(\omega) = \frac{k}{\varepsilon_0}\Im\{\alpha(\omega)\}$$ where $k = 2\pi n_m/\lambda$, $\varepsilon_0$ is the vacuum permittivity, and $n_m$ is the refractive index of the medium (Payne et al., 2013, Payne et al., 21 Dec 2025).

For anisotropic particles, the extinction depends on both the shape and the orientation relative to the incident polarisation. Ellipsoidal nanoparticles exhibit tensor polarizability: $$\alpha'_j(\omega) = V\varepsilon_0 \frac{\varepsilon(\omega) - \varepsilon_m}{\varepsilon_m + L_j[\varepsilon(\omega) - \varepsilon_m]},\quad j=1,2,3$$ where $V = 4\pi abc/3$ is the ellipsoid volume, $L_j$ are depolarisation factors, and $\varepsilon(\omega)$ is the complex permittivity including size-dependent surface damping (Payne et al., 21 Dec 2025).

At the macroscopic scale, biological specimens and optical elements are described by spatially varying 2×2 complex Jones matrices $J(x,y)$, encoding all local polarimetric responses, including diattenuation and birefringence (Dai et al., 2021).

2. Instrumentation and Experimental Implementation

Quantitative polarisation-resolved extinction microscopy utilizes wide-field or confocal transmission microscopes equipped with polarisation control elements and spectral filters:

• Illumination: Kohler illumination from a broadband lamp or LED array; high-NA oil-immersion condensers focus light onto the sample at variable angles (Payne et al., 21 Dec 2025, Dai et al., 2021).
• Polarisation Control: Motorised Glan–Thompson polarisers set in-plane polarisation angles ($\phi$, stepped in 45° increments for comprehensive coverage), and patterned liquid-crystal devices generate radial or complex polarisation states in the condenser back focal plane, enabling excitation along multiple spatial axes (Payne et al., 21 Dec 2025).
• Detection: Scientific CMOS or color cameras acquire sequential bright-field images at varied polarisation states; for confocal architectures, single-mode fibers and pinholes spatially filter higher-order modes (Benelajla et al., 2020).

The measurement protocol typically involves recording pairs of images (with/without specimen, shifted laterally) for each wavelength and polarisation state, followed by differential transmission analysis to extract $\sigma_{\rm ext}$ for hundreds of nanoparticles per field of view (Payne et al., 2013, Payne et al., 21 Dec 2025).

For vectorial Fourier ptychography, a microscope outfitted with an LED array and wire-grid polarizers at both illumination and detection achieves full synthetic aperture coverage and spatially resolved Jones matrix reconstruction (Dai et al., 2021).

3. Polarisation-Resolved Data Acquisition and Models

Data acquisition involves systematic polarisation state cycling and multi-wavelength imaging:

Measurement Workflow

Imaging Modality	Polarisation States	Data Quantity/Field
Wide-field extinction	Linear (0°, 45°, 90°, 135°), Radial	Up to 30 values/particle
Fourier ptychography	2–4 generator/analyzer pairs	900 frames, 29 mm² FOV
Confocal extinction	Cross-polarization, dark-field	Laser suppression $>10^8$

Wide-field: Acquisition of $\sigma_{\rm ext}$ for each particle via bright-field transmission leads to background-corrected integration and enables ensemble statistics over hundreds of particles (Payne et al., 2013).

Fourier Ptychography: vFP algorithm solves for $J(x,y)$ by fusing images from multiple LED angles and polariser/analyzer combinations, utilizing a sequential Gauss–Newton solver that alternately updates both object and system pupil aberrations (Dai et al., 2021).

Confocal Extinction: Introduction of a reflecting interface between polarizer and analyzer amplifies extinction via destructive interference and Imbert–Fedorov modal transformations, yielding cross-polarization ratios of $R>10^8$ (Benelajla et al., 2020).

4. Quantitative Reconstruction and Analysis

Analysis proceeds via direct extraction of extinction cross-sections, tensor polarizabilities, and Jones matrix elements:

• Extinction Cross-Section Extraction: For each NP, $\sigma_{\rm ext}$ is computed by integrating differential transmission $\Delta$ across the particle region, correcting for local background. Shot-noise floors reach $\sim10$–$590\,\mathrm{nm^2}$ depending on acquisition parameters (Payne et al., 2013, Payne et al., 21 Dec 2025).
• Polarisation-Dependent Modeling: For ellipsoidal NPs, extinction as a function of incident polarisation angle $\theta$ is modeled:

$$\sigma_{\rm ext}(\theta) = \sigma_0[1+\alpha \cos2(\theta-\theta_0)]$$

where $\alpha$ quantifies shape anisotropy, and $\theta_0$ gives principal axis orientation (Payne et al., 2013).

• Full Jones Matrix Imaging: vFP yields $J(x,y)$ at each pixel; eigenanalysis produces retardance $\delta(x,y)$, diattenuation $D(x,y)$, and fast-axis orientation $\omega(x,y)$:

$$D(x,y) = \left| \frac{|\xi_q|^2 - |\xi_r|^2}{|\xi_q|^2 + |\xi_r|^2} \right|$$

where $\xi_{q,r}$ are the Jones matrix eigenvalues (Dai et al., 2021).

• Spectral and Morphometric Fitting: Multi-wavelength polarisation-resolved datasets are fit using ellipsoid models, permittivity datasets, and surface damping parameters to recover all three NP semi-axes $(a, b, c)$ and 3D orientation with sub-nanometre precision (Payne et al., 21 Dec 2025).

5. Aberration Correction and Calibration

Polarisation-resolved extinction microscopy requires rigorous handling of optical aberrations and polarization leakage:

• Systematic Aberration Removal: vFP treats the vectorial pupil $P(k)$ as an unknown to be refined in the reconstruction, correcting for spatially varying and polarization-mixing aberrations that would otherwise contaminate quantitative maps (Dai et al., 2021).
• Polarizer Leakage and Optimization: In confocal schemes, polarizer-analyzer pairs are optimized with a beam-splitter or mirror; destructive interference suppresses leakage beyond Malus-law limits, with alignment and mirror phase critical for achieving $R\gtrsim10^8$ extinction (Benelajla et al., 2020).
• Calibration Protocols: Quantitative scaling and error analysis are performed by Monte Carlo simulations, shot-noise estimation, and analytical fitting over large ensemble datasets (Payne et al., 2013, Payne et al., 21 Dec 2025).

6. Applications, Limitations, and Extensions

Quantitative polarisation-resolved extinction microscopy is applied across several research domains:

• Nanoparticle Morphometry: Sub-nanometre determination of gold NP shape, aspect ratio, and orientation via polarisation-resolved spectral extinction, surpassing the typical accuracy of optical methods and approaching that of TEM (Payne et al., 21 Dec 2025).
• Ensemble Statistics: Rapid wide-field acquisition of hundreds of NPs per field enables statistical analysis of shape distributions and asphericity, verified by polarization anisotropy histograms (Payne et al., 2013).
• Biological and Calibration Specimens: Large-area Jones matrix imaging yields high-contrast retardance and diattenuation maps for tissue and birefringent samples, with built-in pupil aberration correction (Dai et al., 2021).
• Resonant Fluorescence Suppression: Confocal cross-polarization extinction techniques quantitatively suppress laser background and allow single-emitter spectroscopy with calibrated extinction floors (Benelajla et al., 2020).

Limitations include finite NA and field-of-view constraints, wavelength-dependent retardation corrections, and the need for precise polarisation and alignment control. Extensions to arbitrary particle shapes, advanced electrodynamic models, and broadband confocal architectures are anticipated (Payne et al., 21 Dec 2025, Dai et al., 2021, Benelajla et al., 2020).

7. Theoretical Developments and Future Directions

Recent advances introduce unified vectorial models, robust optimisation algorithms, and unprecedented precision in polarimetric quantification:

• Vectorial Fourier Ptychography (vFP): Enables full synthetic aperture polarimetric imaging for large specimens; the sequential Gauss–Newton loop jointly solves for Jones matrix and pupil aberrations (Dai et al., 2021).
• Imbert–Fedorov Modal Engineering: Confocal spatial filtering leverages topological beam transformations, converting subwavelength polarization shifts into high extinction ratios (Benelajla et al., 2020).
• 3D Nanoparticle Sizing: Integration of linear and radial polarisation extinction images with spectral fitting and retardation correction achieves <0.25 nm precision in all semi-axes, validated against electron microscopy (Payne et al., 21 Dec 2025).

A plausible implication is that further integration of computational electrodynamics, nonlocal material response, and automation of polarisation state cycling may drive the translation of these methods into routine nanomaterials quality control, structural biology, and advanced optical metrology.