Three-dimensional polarization algebra

Three-dimensional polarization effects in optical tunneling

We consider the three-dimensional (3D) polarimetric properties of an evanescent optical field excited in the gap of a double-prism system by a random plane wave. The analysis covers the case of frustrated total internal reflection (FTIR), i.e., optical tunneling, and relies on the characteristic decomposition of the 3×3 polarization matrix. We find in particular that, for any incident partially polarized plane wave, the evanescent field inside the gap is necessarily in a nonregular, genuine 3D polarization state. We also show that the 3D polarimetric properties of the field at the second boundary are sensitive to the changes of the gap width and that the relevant effects occur for the smaller widths when the angle of incidence of the plane wave becomes larger. The results of this work uncover new aspects of the polarimetric structure of genuine 3D evanescent fields and may find applications in near-field optics and surface nanophotonics.

Spatial tomography of light resolved in time, spectrum, and polarisation

Measuring polarisation, spectrum, temporal dynamics, and spatial complex amplitude of optical beams is essential to studying phenomena in laser dynamics, telecommunications and nonlinear optics. Current characterisation techniques apply in limited contexts. Non-interferometric methods struggle to distinguish spatial phase, while phase-sensitive approaches necessitate either an auxiliary reference source or a self-reference, neither of which is universally available. Deciphering complex wavefronts of multiple co-propagating incoherent fields remains particularly challenging. We harness principles of spatial state tomography to circumvent these limitations and measure a complete description of an unknown beam as a set of spectrally, temporally, and polarisation resolved spatial state density matrices. Each density matrix slice resolves the spatial complex amplitude of multiple mutually incoherent fields, which over several slices reveals the spectral or temporal evolution of these fields even when fields spectrally or temporally overlap. We demonstrate these features by characterising the spatiotemporal and spatiospectral output of a vertical-cavity surface-emitting laser.

The work harnesses principles of spatial state tomography to fully characterise an optical beam in space, time, spectrum, and polarisation. Analysis of the output of a vertical-cavity surface-emitting laser illustrates the technique’s capabilities.

Optimal nonlinear Stokes-Mueller polarimetry for multi-photon processes

In this Letter, we present an optimization model for nonlinear Stokes-Mueller polarimetry (SMP) to improve the precision in estimating the nonlinear Mueller matrix (MM) for two- and three-photon processes. Although nonlinear polarimeters can measure the polarization properties of multi-photon processes or materials, existing methods are suboptimal, leading to low measurement precision. Based on the model and its solution, we have designed a new measurement strategy to substantially reduce the estimation variance of nonlinear MM coefficients by approximately 58.2% for second-harmonic generation polarimetry and 78.7% for third-harmonic generation polarimetry. The model and measurement method can be directly applied to multi-photon processes to improve the precision of SMP.

Purity of 3D polarization

Measures of purity for 3D partially polarized fields, and in particular, the separation into circularly and linearly polarized contributions, are reexamined, and a new degree of total linear polarization introduced. Explicit expressions for the characteristic decomposition in terms of coherency matrix elements are presented, including the special case of an intrinsic coherency matrix. Parameterization of the coherency matrix in terms of ellipticity, and the directions of the ellipse normal and major axis are investigated. Phase consistency is discussed. A comprehensive collection of results regarding intrinsic polarization properties is presented.

Circular Intensity Differential Scattering for Label-Free Chromatin Characterization: A Review for Optical Microscopy

Circular Intensity Differential Scattering (CIDS) provides a differential measurement of the circular right and left polarized light and has been proven to be a gold standard label-free technique to study the molecular conformation of complex biopolymers, such as chromatin. In early works, it has been shown that the scattering component of the CIDS signal gives information from the long-range chiral organization on a scale down to 1/10th-1/20th of the excitation wavelength, leading to information related to the structure and orientation of biopolymers in situ at the nanoscale. In this paper, we review the typical methods and technologies employed for measuring this signal coming from complex macro-molecules ordering. Additionally, we include a general description of the experimental architectures employed for spectroscopic CIDS measurements, angular or spectral, and of the most recent advances in the field of optical imaging microscopy, allowing a visualization of the chromatin organization in situ.

The field of values of Jones matrices: classification and special cases

The concept of field of values (FoV), also known as the numerical range, is applied to the 2 × 2 Jones matrices used in polarization optics. We discover the relevant interplay between the geometric properties of the FoV, the algebraic properties of the Jones matrices and the representation of polarization states on the Poincaré sphere. The properties of the FoV reveal hidden symmetries in the relationships between the eigenvectors and eigenvalues of the Jones matrices. We determine the main mathematical properties of the FoV, discuss the special cases that are relevant to polarization optics, and describe its application to calculate the Pancharatnam-Berry phase introduced by an optical system to the input state.

Eigenvectors of polarization coherency matrices

Calculation of the eigenvectors of two- and three-dimensional coherency matrices, and the four-dimensional coherency matrix associated with a Mueller matrix, is considered, especially for algebraic cases, in the light of recently published algorithms. The preferred approach is based on a combination of an evaluation of the characteristic polynomial and an adjugate matrix. The diagonal terms of the coherency matrix are given in terms of the characteristic polynomial of reduced matrices as functions of the eigenvalues of the coherency matrix. The analogous polynomial form for the off-diagonal elements of the coherency matrix is also presented. Simple expressions are given for the pure component in the characteristic decomposition.

Polarization in reflectance imaging

The Sinclair and Kennaugh matrices are widely used in the remote sensing discipline for signals detected in the backward direction. The connections between the Jones matrix and the Sinclair matrix, and between the Mueller matrix and the Kennaugh matrix, are explored. Different operations on the Jones matrix and their corresponding effects on the Mueller matrix, coherency matrix, and coherence vector are derived. As an example, the Sinclair matrix leads to a Mueller-Sinclair matrix, and a transformed coherence vector. The Kennaugh matrix is not, however, a Mueller matrix, but can be determined from the Mueller or Mueller-Sinclair matrices. We consider backscattering through a medium on a perfect mirror. We propose that backscattering from a uniform medium can be modeled as an effective uniform medium situated on a perfectly reflective substrate, and the elementary polarization properties derived. In this way, the concept of a uniform polarizing medium can be extended to the reflectance geometry. An experimental Mueller matrix from the literature is considered as an example.

Polarimetric nonregularity of evanescent waves

Three-dimensional polarization states of random light can be classified into regular and nonregular according to the structure of the related 3×3 polarization matrix. Here we show that any purely evanescent wave excited in total internal reflection of a partially polarized plane-wave field is always in a nonregular polarization state. The degree of nonregularity of such evanescent waves is also studied in terms of a recently advanced measure. Nonregular evanescent waves uncover new aspects of the polarimetric structure and dimensional character of electromagnetic near fields, with potential applications in nanoscale surface optics.

Spinning spin density vectors along the propagation direction

It has been known that light possesses both spin and orbital angular momenta (AM) arising from the twisting behaviors of the electric (and magnetic) field vector and the wavefront of the field, respectively. The spin (AM) density is also a vector in the field of three dimensions (3D), since its orientation can be in any direction. In this Letter, we show that through focusing a Gaussian beam with both on-axis and off-axis vortices in a high-numerical-aperture system, the spin (AM) density vector in the focal region exhibits nontrivial behaviors: rotating around the central axis along the propagation direction. We demonstrate that this helical behavior of the spin (AM) density vector is mainly caused by the different Gouy phases of the three field components. By changing the position of the off-axis vortex and the semiaperture angle, the helical shape and the helical length can be adjusted. This is a new type of optical twist, to the best of our knowledge, and it may supply another rotational degree of freedom in optical tweezers.

Soleillet's formalism of coherence and partial polarization in 2D and 3D: application to fluorescence polarimetry

We review here the pioneering research conducted by Paul Soleillet on the statistical properties of light in his doctoral thesis from 1929 [Ann. Phys.10, 23 (1929)]. Soleillet's wide-reaching work on polarization, coherence, fluorescence scattering, and three-dimensional fields has remained largely unrecognized; yet, his original contributions rival the modern rediscoveries in both generality and form. Only now, 89 years after Soleillet's original publication and stimulated by our current research on fluorescence polarimetry, have we been able to fully understand and recognize the significance of his results.

Three-dimensional polarization algebra for all polarization sensitive optical systems

Using three-dimensional (3D) coherency vector (9 × 1), we develop a new 3D polarization algebra to calculate the polarization properties of all polarization sensitive optical systems, especially when the incident optical field is partially polarized or un-polarized. The polarization properties of a high numerical aperture (NA) microscope objective (NA = 1.25 immersed in oil) are analyzed based on the proposed 3D polarization algebra. Correspondingly, the polarization simulation of this high NA optical system is performed by the commercial software VirtualLAB Fusion. By comparing the theoretical calculations with polarization simulations, a perfect matching relation is obtained, which demonstrates that this 3D polarization algebra is valid to quantify the 3D polarization properties for all polarization sensitive optical systems.