Error analysis for Mueller matrix measurement

The linear errors of Mueller matrix measurements are formulated for misalignment, depolarization, and incorrect retardation of the polarimetric components. The measured errors of a Mueller matrix depend not only on the imperfections of the measuring system but also on the Mueller matrix itself. The error matrices for different polarimetric systems are derived and also evaluated for the straight-through case. The error matrix for a polarizer-sample-analyzer system is much simpler than those for more complicated systems. The general error matrix is applied to null ellipsometry, and the obtained errors in ellipsometric parameters psi and delta are identical to the errors specifically derived for null ellipsometry with depolarization.     

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.     

Automated high-speed Mueller matrix scatterometer

A new scatterometer-polarimeter is described. It measures the angular distribution of intensity and of the complete Mueller matrix of light scattered by rough surfaces and particle suspensions. The measurement time is 1 s/scattering angle in the present configuration but can be reduced to a few milliseconds with modified electronics. The instrument uses polarization modulation and a Fourier analysis of four detected signals to obtain the 16 Mueller matrix elements. This method is particularly well suited to online, real time, industrial process control involving rough surfaces and large particle suspensions (an arithmetic roughness or particle diameter of >1 mum). Some results are given.     

Anisotropy coefficients of a Mueller matrix

Anisotropy coefficients α, β, and γ that describe the type and the relative amount of the three kinds of anisotropy generally present in a Mueller matrix are introduced. Their derivation, algebraic properties, and physical interpretation are discussed. In particular, they are shown to permit a geometrical representation for the anisotropy and polarizing characteristics of a Mueller matrix. Illustrative experimental examples are provided.     

Mueller matrix analysis of infrared ellipsometry

A Mueller analysis has been done of IR ellipsometry performed with imperfect optical components. Equations linking experimental and calculated Fourier coefficients have been derived and consistently solved. Correction routines for permittivity measurements are demonstrated and discussed with gold and SrTiO(3) as examples. It is shown that such effects as interferometer polarization, detector dichroism, transmission, and phase changes in polarizers can be calculated and effectively removed from the spectra. The problems of calibration and multiple reflections between IR polarizers are discussed, and error propagation in permittivity measurements is analyzed.     

Reduced form of a Mueller matrix

Abstract

Through a simple procedure based on the Lu–Chipman decomposition [S.-Y. Lu and R.C. Chipman, J. Opt. Soc. Am. A 13, 1106 (1996)], any depolarizing Mueller matrix can be transformed into a reduced form which accumulates the depolarization and polarizance properties into a set of six parameters. The simple structure of this reduced form provides straightforward ways for the general characterization of Mueller matrices as well as for the analysis of singular Mueller matrices.     

Dual-modulator broadband infrared Mueller matrix ellipsometry

A broadband mid-infrared Mueller matrix ellipsometer is described based on two photoelastic modulators and a step-scan interferometer. The data are analyzed using a combination hardware-software double Fourier transformation. Obtaining spectra of the Mueller matrix elements requires that the infrared wavelength-dependent retardation amplitude of the modulators be known through calibration and subsequently incorporated into the data processing. The spectroscopic capability of the instrument is demonstrated in transmission and reflection geometries by the measured Mueller matrices of air, an anisotropic quartz crystal, and the ZnSe-water interface, each from 2500-4000 cm^-1.     

Components of purity of a Mueller matrix

The degree of polarimetric purity of a Mueller matrix, also called "depolarization index" [Opt. Acta 33, 185 (1986)] is expressed as a quadratic average of two contributions of different nature. The contribution due to the polarizance and diattenuation properties is given by a unique parameter called "degree of polarizance," and the complementary contribution due to nonpolarizing properties is given by a parameter called "degree of spherical purity." These two intrinsic quantities are useful in order to analyze the sources of the polarimetric purity of a material sample whose Mueller matrix has been measured and provide criteria for the classification of Mueller matrices.     

Structure of a general pure Mueller matrix

Changes in the radiance and state of polarization of a beam of radiation can often be described by means of a pure Mueller matrix. Such a 4 × 4 matrix transforms Stokes parameters and can be expressed in terms of the elements of a 2 × 2 Jones matrix. Relations between the two types of matrix are discussed. Explicit expressions are given for changes of a pure Mueller matrix that are caused by certain elementary changes of its Jones matrix, such as when its transpose, complex conjugate, or Hermitian conjugate are taken. It is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements. All possible interrelations for the elements of a general pure Mueller matrix are derived from this simple structure.     

Mueller matrix error correction for a fringe-free interferometry system

We present an automated surface profiling system based on a shearing interferometer, in which precise measurement of the polarization states eliminates fringe ambiguity. A full error correction based on Mueller matrices allows comparatively inaccurate but rapidly switchable liquid-crystal wave plates to be used, enabling unambiguous profile information to be obtained in real time.     

Optimization of a dual-rotating-retarder Mueller matrix polarimeter

The dual-rotating-retarder configuration is one of the most common forms of the Mueller matrix polarimeter. I perform an optimization of this polarimeter configuration by minimizing the condition number of the system data reduction matrix. I find the optimum retardance for the rotating retarders to be 127 degrees. If exactly 16 intensity measurements are used for a Mueller matrix calculation, a complex relationship exists between the condition number and the sizes of the angular increments of the two retarders. If many intensity measurements are made, thus overspecifying the calculation, I find broad optimal ranges of angular increments of the two retarders that yield essentially equal performance. Experimental results are given.     

Infrared Mueller matrix acquisition and preprocessing system

An analog Mueller matrix acquisition and preprocessing system (AMMS) was developed for a photopolarimetric-based sensor with 9.1-12.0 microm optical bandwidth, which is the middle infrared wavelength-tunable region of sensor transmitter and "fingerprint" spectral band for chemical-biological (analyte) standoff detection. AMMS facilitates delivery of two alternate polarization-modulated CO(2) laser beams onto subject analyte that excite/relax molecular vibrational resonance in its analytic mass, primes the photoelastic-modulation engine of the sensor, establishes optimum throughput radiance per backscattering cross section, acquires Mueller elements modulo two laser beams in hexadecimal format, preprocesses (normalize, subtract, filter) these data, and formats the results into digitized identification metrics. Feed forwarding of formatted Mueller matrix metrics through an optimally trained and validated neural network provides pattern recognition and type classification of interrogated analyte.     

Transmission Mueller matrix ellipsometry of chirality switching phenomena

Mechanisms of molecular chirality induction are fundamental to many questions in chemistry. Interest in these mechanisms is shifting toward media of increasing complexity that simultaneously exhibit linear birefringence and dichroism and where the common assumption that optical activity is the only optical effect that affects light polarization is no longer valid. Light propagation through several of these anisotropic media can be appropriately studied with transmission Mueller matrix ellipsometry. The applications presented herein include the measurement of optical activity in stirred solutions of soft-matter nanophases and the determination of chiral domains in solid-state samples.     

On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

Abstract

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L ₂ KL ₁ where L ₁ and L ₂ are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.     

一种全偏振参数的显微测量系统

本文设计了一种用于Mueller矩阵测量的全偏振参数显微镜测量系统.该测量系统以液晶调制器键合于偏振分束棱镜表面的模块为核心器件,实现了以10次光强获取为周期的Mueller矩阵参数测量功能,能够方便地实现Mueller矩阵中全部参数图像的实时测量,比前人设计系统的测量效率有很大提高.基于该系统,本文进行了Mueiler矩阵快速调制测量方案的具体设计,并给出了测量结果.论文还对该系统进行了测量误差的分析,实验结果及误差分析表明,系统的相对测量误差能够控制在2%以内.     

Dielectric tensor measurement from a single Mueller matrix image

A technique for measuring dielectric tensors in anisotropic layered structures, such as thin films of biaxial materials, is demonstrated. The ellipsometric data are collected in a quasi-monochromatic Mueller matrix image acquired over a large range of incident and azimuthal angles by illuminating a very small area on the sample with a focused beam from a modulating polarization state generator. After the beam interacts with the sample, the reflected and/or transmitted light is collected using an imaging polarization state analyzer. An image of the exit pupil of a collection objective lens is formed across a CCD such that each pixel collects light from a different angle incident on the sample, thus acquiring ellipsometric data at numerous incident angles simultaneously. The large range of angles and orientations is necessary to accurately determine dielectric tensors. The small but significant polarization aberrations of the low-polarization objective lenses used to create and collect the focused beams provide a significant challenge to accurate measurement. Measurements are presented of a thin-film E-type polarizer and a stretched plastic biaxial film.     

Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase

The Mueller matrix (M) corresponding to the phase matrix in the backscattering region (scattering angles ranging from 175 degrees to 180 degrees) is investigated for light scattering at a 0.532-microm wavelength by hexagonal ice crystals, ice spheres, and water droplets. For hexagonal ice crystals we assume three aspect ratios (plates, compact columns, and columns). It is shown that the contour patterns of the backscattering Mueller matrix elements other than M11, M44, M14, and M41 depend on particle geometry; M22 and M33 are particularly sensitive to the aspect ratio of ice crystals. The Mueller matrix for spherical ice particles is different from those for nonspherical ice particles. In addition to discriminating between spherical and nonspherical particles, the Mueller matrix may offer some insight as to cloud thermodynamic phase. The contour patterns for large ice spheres with an effective size of 100 microm are substantially different from those associated with small water droplets with an effective size of 4 microm.     

Dynamic response of a fast near infra-red Mueller matrix ellipsometer

The dynamic response of a near infra-red ferroelectric liquid crystal-based Mueller matrix ellipsometer (NIR FLC-MME) is presented. A time-dependent simulation model, using the measured time response of the individual FLCs, is used to describe the measured temporal response. Furthermore, the impulse response of the detector and the pre-amplifier is characterized and included in the simulation model. The measured time-dependent intensity response of the MME is reproduced in simulations, and it is concluded that the switching time of the FLCs is the limiting factor for the Mueller matrix measurement time of the FLC-based MME. Based on measurements and simulations, our FLC-based NIR-MME system is estimated to operate at the maximum speed of approximately 16 ms per Mueller matrix measurement. The FLC-MME may be operated several times faster, since the switching time of the crystals depends on the individual crystal being switched, and to what state it is switched. As a demonstration, the measured temporal response of the Mueller matrix and the retardance of a thick liquid crystal variable retarder upon changing state is demonstrated.     

Time-resolved Mueller matrix analysis of a liquid crystal on silicon display

We present a full polarimetric characterization of a liquid crystal on silicon (LCoS) display, with time resolution measurements below the frame period of the device. This time-resolved analysis shows evidence of temporal fluctuations in the millisecond range in the state of polarization of the beam reflected by the display. We demonstrate that light reflected by the display is maintained fully polarized, but these temporal fluctuations result in an effective depolarization effect when detectors with long time integration intervals are used in the characterization of the display.