A priori information and optimisation in polarimetry

Analysis of Polarization Detector Performance Parameters on Polarization 3D Imaging Accuracy

Three-dimensional (3D) reconstruction of objects using the polarization properties of diffuse light on the object surface has become a crucial technique. Due to the unique mapping relation between the degree of polarization of diffuse light and the zenith angle of the surface normal vector, polarization 3D reconstruction based on diffuse reflection theoretically has high accuracy. However, in practice, the accuracy of polarization 3D reconstruction is limited by the performance parameters of the polarization detector. Improper selection of performance parameters can result in large errors in the normal vector. In this paper, the mathematical models that relate the polarization 3D reconstruction errors to the detector performance parameters including polarizer extinction ratio, polarizer installation error, full well capacity and analog-to-digital (A2D) bit depth are established. At the same time, polarization detector parameters suitable for polarization 3D reconstruction are provided by the simulation. The performance parameters we recommend include an extinction ratio ≥ 200, an installation error ∈ [-1°, 1°], a full-well capacity ≥ 100 Ke-, and an A2D bit depth ≥ 12 bits. The models provided in this paper are of great significance for improving the accuracy of polarization 3D reconstruction.

Robust polarimetry via convex optimization

We present mathematical methods, based on convex optimization, for correcting non-physical coherency matrices measured in polarimetry. We also develop the method for recovering the coherency matrices corresponding to the smallest and largest values of the degree of polarization given the experimental data and a specified tolerance. We use experimental non-physical results obtained with the standard polarimetry scheme and a commercial polarimeter to illustrate these methods. Our techniques are applied in post-processing, which complements other experimental methods for robust polarimetry.

Stokes polarimetry-based second harmonic generation microscopy for collagen and skeletal muscle fiber characterization

The complete polarization state of second harmonic (SH) light was measured and characterized by collagen type I and skeletal muscle fiber using a Stokes vector-based SHG microscope. The polarization states of the SH signal are analyzed in a pixel-by-pixel manner and displayed through two dimensional (2D) Stokes vector images. Various polarization parameters are reconstructed using Stokes values to quantify the polarization properties of SH light. Also, the measurements are extended for different input polarization states to investigate the molecular structure of second harmonic generation (SHG) active molecules such as collagen type I and myosin.

Polarization resolved second harmonic microscopy

Second harmonic (SH) microscopy has proven to be a powerful imaging modality over the past years due to its intrinsic advantages as a multiphoton process with endogenous contrast specificity, which allows pinhole-less optical sectioning, non-invasive observation, deep tissue penetration, and the possibility of easier signal detection at visible wavelengths. Depending on the relative orientation between the polarization of the incoming light and the second-order susceptibility of non-centrosymmetric structures, SH microscopy provides the unique capacity to probe the absolute molecular structure of a broad variety of biological tissues without the necessity for additional labeling. In addition, SH microscopy, when working with polarimetry, provides clear and in-depth insights on the details of molecular orientation and structural symmetry. In this review, the working principles of the polarization resolving techniques and the corresponding implements of SH microscopy are elucidated, with focus on Stokes vector based polarimetry. An overview of the advancements on SH anisotropy measurements are also presented. Specifically, the recent progresses on the following three topics in polarization resolved SH microscopy will be elucidated, which include Stokes vector resolving for imaging molecular structure and orientation, 3-D structural chirality by SH circular dichroism, and correlation with fluorescence lifetime imaging (FLIM) for in vivo wound healing diagnosis. The potentials and challenges for future researches in exploring complex biological tissues are also discussed.

Intrinsic coincident full-Stokes polarimeter using stacked organic photovoltaics

An intrinsic coincident full-Stokes polarimeter is demonstrated by using strain-aligned polymer-based organic photovoltaics (OPVs) that can preferentially absorb certain polarized states of incident light. The photovoltaic-based polarimeter is capable of measuring four Stokes parameters by cascading four semitransparent OPVs in series along the same optical axis. This in-line polarimeter concept potentially ensures high temporal and spatial resolution with higher radiometric efficiency as compared to the existing polarimeter architecture. Two wave plates were incorporated into the system to modulate the S₃ Stokes parameter so as to reduce the condition number of the measurement matrix and maximize the measured signal-to-noise ratio. Radiometric calibration was carried out to determine the measurement matrix. The polarimeter presented in this paper demonstrated an average RMS error of 0.84% for reconstructed Stokes vectors after normalized to S₀. A theoretical analysis of the minimum condition number of the four-cell OPV design showed that for individually optimized OPV cells, a condition number of 2.4 is possible.

Equalized estimation of Stokes parameters in the presence of Poisson noise for any number of polarization analysis states

Estimation of the Stokes vector is based on projecting the input light on a number N of polarization analysis states. We address the optimization of the distribution of these analysis states on the Poincaré sphere in the presence of signal-dependent Poisson shot noise for an arbitrary value of N. We show that if this distribution forms a spherical 3 design, the Stokes vector is estimated with minimal equally weighted variance and with estimation variances of the last three Stokes parameters equal and independent of the input Stokes vector. We also demonstrate that in the presence of Poisson shot noise, the estimation signal to noise ratio is independent of N, whereas in the presence of signal independent additive noise, it is proportional to 1/N, which means that there is a precision loss in increasing the number of measurements.

Integrated plasmonic metasurfaces for spectropolarimetry

Plasmonic metasurfaces enable simultaneous control of the phase, momentum, amplitude and polarization of light and hence promise great utility in realization of compact photonic devices. In this paper, we demonstrate a novel chip-scale device suitable for simultaneous polarization and spectral measurements through use of six integrated plasmonic metasurfaces (IPMs), which diffract light with a given polarization state and spectral component into well-defined spatial domains. Full calibration and characterization of our device is presented, whereby good spectral resolution and polarization accuracy over a wavelength range of 500-700 nm is shown. Functionality of our device in a Müller matrix modality is demonstrated through determination of the polarization properties of a commercially available variable waveplate. Our proposed IPM is robust, compact and can be fabricated with a single photolithography step, promising many applications in polarization imaging, quantum communication and quantitative sensing.

Optimal Frames for Polarization State Reconstruction

Complete determination of the polarization state of light requires at least four distinct projective measurements of the associated Stokes vector. Stability of state reconstruction, however, hinges on the condition number κ of the corresponding instrument matrix. Optimization of redundant measurement frames with an arbitrary number of analysis states, m, is considered in this Letter in the sense of minimization of κ. The minimum achievable κ is analytically found and shown to be independent of m, except for m=5 where this minimum is unachievable. Distribution of the optimal analysis states over the Poincaré sphere is found to be described by spherical 2 designs, including the Platonic solids as special cases. Higher order polarization properties also play a key role in nonlinear, stochastic, and quantum processes. Optimal measurement schemes for nonlinear measurands of degree t are hence also considered and found to correspond to spherical 2t designs, thereby constituting a generalization of the concept of mutually unbiased bases.

On the performance of the physicality-constrained maximum-likelihood estimation of Stokes vector

We address the estimation of the Stokes vectors taking into account the physical realizability constraint. We propose a fast method for computing the constrained maximum-likelihood (CML) estimator for any measurement matrix, and we compare its performance with the classical empirical physicality-constrained estimator. We show that when the measurement matrix is based on four polarization states spanning a regular tetrahedron on the Poincaré sphere, the two estimators are very similar, but the CML provides a better estimation of the intensity. For an arbitrary measurement matrix, the CML estimator does not always yield better estimation performance than the empirical one: their comparative performances depend on the measurement matrix, the actual Stokes vector and the signal-to-noise ratio.

Stokes vector based polarization resolved second harmonic microscopy of starch granules

We report on the measurement and analysis of the polarization state of second harmonic signals generated by starch granules, using a four-channel photon counting based Stokes-polarimeter. Various polarization parameters, such as the degree of polarization (DOP), the degree of linear polarization (DOLP), the degree of circular polarization (DOCP), and anisotropy are extracted from the 2D second harmonic Stokes images of starch granules. The concentric shell structure of a starch granule forms a natural photonic crystal structure. By integration over all the solid angle, it will allow very similar SHG quantum efficiency regardless of the angle or the states of incident polarization. Given type I phase matching and the concentric shell structure of a starch granule, one can easily infer the polarization states of the input beam from the resulting SH micrograph.

Polarization-resolved second harmonic generation microscopy with a four-channel Stokes-polarimeter

We developed a four-channel photon counting based Stokes-polarimeter for spatial characterization of polarization effects in second harmonic generation (SHG). We have implemented a calibration technique allowing quantitative measurement of polarization parameters, such as the degree of polarization (DOP), degree of linear polarization (DOLP), degree of circular polarization (DOCP), as well as anisotropy from the acquired Stokes parameters. The technique is used as contrast mechanism to characterize the polarization properties from two potassium dihydrogen phosphate (KDP) micro-crystals and collagen type-I in SHG microscopy.

Signal, noise, and bias for a broadband, division-of-amplitude Stokes polarimeter

We analyze estimation error as a function of spectral bandwidth for division-of-amplitude (DoAm) Stokes polarimeters. Our approach allows quantitative assessment of the competing effects of noise and deterministic error, or bias, as bandwidth is varied.We use the signal-to-rms error (SRR) as a metric. Rather than calculating the SRR of the estimated Stokes parameters themselves, we use the singular-value decomposition to calculate the SRRs of the coefficients of the measured data vector projected onto the measurement matrix left singular vectors.We argue that calculating the SRRs for left singular vector coefficients will allow development of reconstruction filters to minimize Stokes estimation error. For the example case of a source with constant polarization over a relatively wide band, we show that as the spectral filter bandwidth is increased to include wavelengths significantly different than the design wavelength, the SRRs of the estimated left singular vector coefficients will a.) increase monotonically if relatively few photo-detection events (PDEs) are recorded, b.) after a sharp peak close to the design wavelength, decrease monotonically if relatively many PDEs are recorded, and c.) have well-defined maxima for nominal PDE counts. Given some idea of the source brightness relative to detector noise, one can specify a spectral filter bandwidth minimizing the variance and bias effects and optimizing Stokes parameter estimation. Our approach also allows one to specify the bandwidth over which the response of "achromatic" optics must be reasonably invariant with wavelength for rms Stokes estimation error to remain below some desired maximum. Finally, we point out that our method can be generalized not only to other types of polarimeters, but also to any sensing scheme that can be represented by a linear system for limiting values of a certain parameter.

Stokes polarimeter optimization in the presence of shot and Gaussian noise

An error minimization method is presented for Stokes polarimeters applicable when the detected signals are affected by a combination of shot and Gaussian noise. The expectation of the Stokes vector variance is used as a performance measure. This measure is compared with the condition number of a polarization state analyzer matrix that is commonly used as a figure of merit. We show that a polarimeter with the minimum condition number is not necessarily optimal. The approach is used to optimize existing prism based polarimeters giving improvements in the performance when shot-noise cannot be neglected.

Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise

We address the optimization of Stokes polarimeters in the presence of signal-dependent shot noise, which is the dominant type of noise in certain imaging systems. We show that in some precise sense, the polarimeters optimal for additive noise are also optimal for such noise and propose polarimeter architectures in which noise variances are equalized and independent of the input polarization state.