Imaging polarimetry through metasurface polarization gratings.

Metasurfaces-subwavelength arrays of phase-shifting elements-present new possibilities for polarization optics and polarimetry. In particular, a periodic, polarization-sensitive metasurface diffraction grating can enable full-Stokes imaging polarimetry with a single polarization-sensitive component. In this work, we show that a suitably-designed metasurface grating can serve as a polarimetric "attachment" to an existing intensity-only imaging system, converting it into one capable of full-Stokes imaging polarimetry. Design rules and tradeoffs governing this adaptation are described and demonstrated using a machine vision imaging system as an example.

Simultaneous balancing of geometric transformation and linear polarizations using six-fold-mirror geometry over the visible region.

The presence of optical rotation due to the Berry phase and coating-induced linear polarizations across a set of rays deteriorate the form of a point spread function. In this Letter, we model an optical system with six aluminum coated fold mirrors, and we show how to balance geometric transformation, linear retardance, and linear diattenuation simultaneously. We minimize the linear polarizations by arranging the six fold mirrors in three pairs of crossed-fold-mirror geometry. In the same optical model, we also show that geometric transformation induced by the first set of three mirrors is canceled out by the second set of three mirrors by rotating the eigen-polarizations in opposite direction with the same amount. We perform experimental verification using a spectroscopic Mueller matrix polarimeter over the visible spectral range.

Broadband extended source imaging Mueller-matrix polarimeter.

An imaging Mueller matrix polarimeter, named the red-green-blue (RGB)950, takes images of medium-sized (tens of centimeters) objects by using a very bright source, large polarization state generator, and high-quality camera. Its broadband extended light source switches between red, green, blue, and near-infrared light to allow taking polarimetric images for comparison with RGB camera images. The large diffuse source makes shadow transitions gradual and spreads out the specular reflected spot into a larger less conspicuous area.

Calibration and validation of Airborne Multiangle SpectroPolarimetric Imager (AirMSPI) polarization measurements.

The Airborne Multiangle SpectroPolarimetric Imager (AirMSPI), a precursor to the future Multi-Angle Imager for Aerosols satellite instrument, is a remote-sensing instrument for the characterization of atmospheric aerosols and clouds. To help discriminate between different aerosol particle types, which is crucial to improve our understanding of their impact on climate and air quality, AirMSPI acquires imagery over multiple view angles in the ultraviolet, visible, and near-infrared, and it employs dual photoelastic modulators (PEMs) to target an uncertainty requirement of ±0.005 in the degree of linear polarization (DoLP) at selected wavelengths. Laboratory polarimetric calibrations using a second-generation Polarization State Generator-2 (PSG-2) and validation measurements at 0

Binary classification of Mueller matrix images from an optimization of Poincaré coordinates.

A new binary classification method for Mueller matrix images is presented which optimizes the polarization state analyzer (PSA) and the polarization state generator (PSG) using a statistical divergence between pixel values in two regions of an image. This optimization generalizes to multiple PSA/PSG pairs so that the classification performance as a function of number of polarimetric measurements can be considered. Optimizing PSA/PSG pairs gives insight into which polarimetric measurements are most useful for the binary classification. For example, in scenes with strong diattenuation, retardance, or depolarization certain PSA/PSG pairs would make two regions in an image look very similar and other pairs would make the regions look very different. The method presented in this paper provides a quantitative method for ensuring the images acquired can be classified optimally.

Balancing polarization aberrations in crossed fold mirrors.

The polarization aberrations of a fold mirror can be compensated by orienting a second fold mirror's p-polarization with the first mirror's s-polarization. This crossed-mirror configuration compensates the polarization for a single angle to zero and leaves a linear variation of diattenuation and retardance for a spherical wavefront. Two sets of crossed fold mirrors when properly oriented compensate the remaining linear variation and leave a much reduced quadratic variation in a large compensated field of view.

Compound dichroic polarizers with wavelength-dependent transmission axes.

Compound dichroic polarizers using liquid crystal polymers and dichroic dyes have been studied for their variation of polarization with wavelength. Compound dichroic polarizers are composed of multilayer polarizers, where each layer has a different orientation and absorption spectrum. Several wavelength-dependent transmission axes configurations are proposed and investigated. Polarization properties, such as linear and circular diattenuation of the compound dichroic polarizers, for different wavelengths are measured and compared with simulation. Depending on the configuration, the compound dichroic polarizers can convert polarization signatures into color coding in unique ways.

Spectral density response functions for modulated polarimeters.

Conventional imaging devices are often compared using their optical transfer functions (OTFs) in space and their impulse responses in time. Modulated polarimeters cannot be directly compared this way, since they are frequency multiplexed. Here we define a spectral density response function that describes how the spectral density matrix of the Stokes parameters for an object transfers through a modulated polarimeter. This response function facilitates the objective comparison of polarimeters in a way that is analogous to the OTF for conventional imaging systems. The spectral density response is used to calculate a Wiener filter for a rotating analyzer polarimeter as an example of filter optimization for modulated polarimetry.

Liquid crystal polymer full-stokes division of focal plane polarimeter.

A division-of-focal-plane polarimeter based on a dichroic dye and liquid crystal polymer guest-host system is presented. Two Stokes polarimeters are demonstrated: a linear Stokes and the first ever Full-Stokes division-of-focal-plane polarimeter. The fabrication, packaging, and characterization of the systems are presented. Finally, optimized polarimeter designs are discussed for future works.

Role of the null space of the DRM in the performance of modulated polarimeters.

Imaging polarimeters infer the spatial distribution of the polarization state of the optical field as a function of time and/or wavelength. A polarimeter indirectly determines the polarization state by first modulating the intensity of the light field and then demodulating the measured data to infer the polarization parameters. This Letter considers passive Stokes parameter polarimeters and their inversion methods. The most widely used method is the data reduction matrix (DRM), which builds up a matrix equation that can be inverted to find the polarization state from a set of intensity measurements. An alternate strategy uses linear system formulations that allow band limited reconstruction through a filtering perspective. Here we compare these two strategies for overdetermined polarimeters and find that design of the null space of the inversion operator provides degrees of freedom to optimize the trade off between accuracy and signal-to-noise ratio. We further describe adaptive filtering techniques that could optimize the reconstruction for a particular experimental configuration. This Letter considers time-varying Stokes parameters, but the methods apply equally to polarimeters that are modulated in space or in wavelength.

Mueller matrix roots depolarization parameters.

The Mueller matrix roots decomposition recently proposed by Chipman in [1] and its three associated families of depolarization (amplitude depolarization, phase depolarization, and diagonal depolarization) are explored. Degree of polarization maps are used to differentiate among the three families and demonstrate the unity between phase and diagonal depolarization, while amplitude depolarization remains a distinct class. Three families of depolarization are generated via the averaging of different forms of two nondepolarizing Mueller matrices. The orientation of the resulting depolarization follows the cyclic permutations of the Pauli spin matrices. The depolarization forms of Mueller matrices from two scattering measurements are analyzed with the matrix roots decomposition-a sample of ground glass and a graphite and wood pencil tip.

Band limited data reconstruction in modulated polarimeters.

Data processing for sequential in time polarimeters based on the Data Reduction Matrix technique yield polarization artifacts in the presence of time varying signals. To overcome these artifacts, polarimeters are designed to operate at higher and higher speeds. In this paper we describe a band limited reconstruction algorithm that allows the measurement and processing of temporally varying Stokes parameters without artifacts. An example polarimeter consisting of a rotating retarder and polarizer is considered, and conventional processing methods are compared to a band limited reconstruction algorithm for the example polarimeter. We demonstrate that a significant reduction in error is possible using these methods.

Skew aberration: a form of polarization aberration.

We define a new class of aberration, skew aberration, which is a component of polarization aberration. Skew aberration is an intrinsic rotation of polarization states due to the geometric transformation of local coordinates, independent of coatings and interface polarization. Skew aberration in a radially symmetric system has the form of a circular retardance tilt plus coma aberration. Skew aberration causes undesired polarization distribution in the exit pupil. We demonstrate statistics on skew aberration of 2383 optical systems described in Code V's U.S. patent library [Code V Version 10.3 (Synopsys, 2011), pp. 22-24]; the mean skew aberration is 0.89° and the standard deviation is 1.37°. The maximum skew aberration found is 17.45° and the minimum is -11.33°. U.S. patent 2,896,506, which has ±7.01° of skew aberration, is analyzed in detail. Skew aberration should be of concern in microlithography optics and other high NA and large field of view optical systems.

Formulation of rigorous coupled-wave theory for gratings in bianisotropic media.

A formulation of rigorous coupled-wave theory for diffraction gratings in bianisotropic media that exhibit linear birefringence and/or optical activity is presented. The symmetric constitutive relations for bianisotropic materials are adopted. All of the incident, exiting, and grating materials can be isotropic, uniaxial, or biaxial, with or without optical activity. The principal values of the electric permittivity tensor, the magnetic permeability tensor, and the gyrotropic tensor of the media can take arbitrary values, and the principal axes may be arbitrarily and independently oriented. Procedures for Fourier expansion of Maxwell's equations are described. Distinctive polarization coupling effects due to optical activity are observed in sample calculations.

Three-dimensional polarization ray-tracing calculus I: definition and diattenuation.

A three-by-three polarization ray-tracing matrix method for polarization ray tracing in optical systems is presented for calculating the polarization transformations associated with ray paths through optical systems. The method is a three-dimensional generalization of the Jones calculus. Reflection and refraction algorithms are provided. Diattenuation of the optical system is calculated via singular value decomposition. Two numerical examples, a three fold-mirror system and a hollow corner cube, demonstrate the method.

Three-dimensional polarization ray-tracing calculus II: retardance.

The concept of retardance is critically analyzed for ray paths through optical systems described by a three-by-three polarization ray-tracing matrix. Algorithms are presented to separate the effects of retardance from geometric transformations. The geometric transformation described by a "parallel transport matrix" characterizes nonpolarizing propagation through an optical system, and also provides a proper relationship between sets of local coordinates along the ray path. The proper retardance is calculated by removing this geometric transformation from the three-by-three polarization ray-tracing matrix. Two rays with different ray paths through an optical system can have the same polarization ray-tracing matrix but different retardances. The retardance and diattenuation of an aluminum-coated three fold-mirror system are analyzed as an example.

Achromatic athermalized retarder fabrication.

A method for fabricating an achromatic, athermalized quarter-wave retarder is presented that involves monitoring retardance during polishing. A design specified by thicknesses alone is unlikely to meet specification due to uncertainties in birefringence. This method facilitates successful fabrication to a retardance specification despite these uncertainties. A retarder made from sapphire, MgF(2), and quartz was designed, fabricated, and its performance validated for the 0.470 to 0.865 µm wavelength region. Its specifications are as follows: at wavebands centered at 0.470, 0.660, and 0.865 µm, the band-averaged retardance should be 90°±10° for all fields and retardance should change less than 0.1° for a 1° change in temperature. Retarder fabrication accommodated birefringence and thickness uncertainties via the following steps. The first plate was polished to a target thickness. The retardance spectrum of the first plate was then measured and used to determine a retardance target for the second plate. The retardance spectrum of the combined first and second plates was then used to specify a retardance target for the third plate. The retardance spectrum of the three plates in combination was then used to determine when the final thickness of the third plate was reached.

Polarization state generator: a polarimeter calibration standard.

A polarization state generator (PSG) was built to generate light having a degree of linear polarization (DoLP) varying from 0.0005 to 0.4 with 0.0005 uncertainty. The PSG operates by tilting a plane parallel SF11 glass plate in a nearly unpolarized beam. The DoLP of collimated, unpolarized light passing through a plane parallel plate over a defined range of field angles can be calculated from theory, and the PSG was intended to act as a calibration standard based on this calculation. Several effects make the DoLP distribution as a function of field and tilt plate difficult to model to the desired 0.0005 uncertainty: source DoLP and intensity nonuniformity, lens surface diattenuation, and errors in optical alignment. Because of these effects, modeled DoLP was not used as a standard. Instead, DoLP was characterized with repeatability of 0.0005.

Analysis of static and time-varying polarization errors in the multiangle spectropolarimetric imager.

Multiangle Spectropolarimetric Imager (MSPI) sensitivity to static and time-varying polarization errors is examined. For a system without noise, static polarization errors are accurately represented by the calibration coefficients, and therefore do not impede correct mapping of measured to input Stokes vectors. But noise is invariably introduced during the detection process, and static polarization errors reduce the system's signal-to-noise ratio (SNR) by increasing noise sensitivity. Noise sensitivity is minimized by minimizing the condition number of the system data reduction matrix [Appl. Opt.41, 619 (2002)]. The sensitivity of condition numbers to static polarization errors is presented. The condition number of the nominal MSPI data reduction matrix is approximately 1.1 or less for all fields. The increase in the condition number above 1 results primarily from a quarter wave plate and mirror coating retardance magnitude errors. Sensitivity of the degree of linear polarization (DoLP) error with respect to time-varying diattenuation and retardance error was used to set a time-varying diattenuation magnitude tolerance of 0.005 and a time-varying retardance magnitude tolerance of ±0.2°. A Monte Carlo simulation of the calibration and measurements using anticipated static and time-varying errors indicates that MSPI has a probability of 0.9 of meeting its 0.005 DoLP uncertainty requirement.

Polarization conversion cube-corner retroreflector.

Polarization conversion cube-corner retroreflectors rotate the major axis of the incident electric field by 90°. We show that polarization conversion cube-corner retroreflectors cannot be created with isotropic reflecting surfaces, but can be created by incorporating nonisotropic reflecting surfaces. Two types are considered-cube corners with surfaces having elliptical eigenstates and cube corners having subwavelength gratings. Implementations that use subwavelength surface relief phase gratings are investigated, and three examples are shown.