Mueller Matrix Formalism Research Articles

Use of Dirac matrices in polarization optics

Several matrix methods have been developed for studying polarization properties of light. Jones was the first to apply the matrix method to the study of polarization optics. In Jones matrix formalism the polarized wave field is represented by 2-element column matrix known as Jones Vector and the polarization device encountered by light is represented by a 2×2 matrix, known as the characteristic Jones matrix of the device. Mueller introduced a new matrix method where the wave field is represented by a 4-dimensional vector. The elements of the vector are the Stokes parameters of the beam. In Mueller matrix formalism the optical device is represented by a 4×4 real matrix known as ‘Mueller Matrix’ of the device. The use of coherency matrix also proves to the useful in the study of partially polarized light. Pauli spin matrices have been used to unify the different matrix treatments of polarization optical phenomena. The present article is an attempt to unify the analysis of polarization phenomena using Dirac matrices used by Dirac in quantum mechanics. We have however redefined the set of Dirac matrices in terms of the Kronecher product of Pauli spin matrices.

Measurement of optical rotation and phase retardance of optical samples with depolarization effects using linearly and circularly polarized probe lights

This paper presents an experimental technique for measuring the optical rotation, depolarization, and phase retardance of optical samples. In the proposed approach, the optical properties of the sample are derived using a Stokes–Mueller matrix formalism in conjunction with linearly and circularly polarized probe lights. For a compound sample comprising a half-wave plate positioned in front of glucose solutions with concentrations ranging from 0 to 1.2 g/dl, the average normalized error in the measured rotation angle is determined to be 3.11% when using a linearly polarized light. The average surviving linear and circular polarization fractions of the glucose solutions are determined to be 1.0252 and 0.9945. The average normalized error in the measured retardance of the half-wave plate is 3.45%. When measuring a compound sample comprising a half-wave plate positioned in front of the scattered glucose solutions with turbidities from 0% to 50% by the addition of milk, experimental results show that the induced rotation angle increased as the turbidity increased, whereas both surviving linear and circular polarization fractions decreased as the turbidity increased. The effect of the turbidity on rotation angle is more significant than that on both surviving linear and circular polarization fractions. The average normalized error in the measured retardance of the half-wave plate is 1.43%. Consequently, a simplified geometry of the polarimeter is proposed to independently estimate the rotation angle, surviving linear and circular polarization fractions, and retardance from the derived viable algorithm.

Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size

In this paper, we present an approach based on Mueller polarimetry for dimensional characterization of periodic structures with sample sizes smaller than the illuminating beam spot. The relevant theoretical background based on Mueller matrix formalism is presented. The sample, a photoresist grating box surrounded by silicon substrate, was measured in conical configuration by means of a spectroscopic Mueller polarimeter whose beam was illuminating both the grating and the substrate. By translating the boundary between these two regions through the beam we could follow the depolarization effect related to the incoherent superposition of the light reflected by the grating and the substrate. The grating’s optical response was modeled by rigorous coupled-wave analysis. The optimized geometrical parameters of the grating (the line width, the grating depth, and the side-wall angle) show very good consistency for all sample positions with 15% or more of the spot area covered by the grating. Another source of depolarization, the finite spectral resolution of the polarimeter was also taken into account in the analysis, leading to a good quantitative agreement with the sharp “depolarization lines” observed in the measured Mueller spectra.

A Mueller-Matrix Formalism for Modeling Polarization Azimuth and Ellipticity Angle in Semiconductor Optical Amplifiers in a Pump–Probe Scheme

This paper presents a Mueller-matrix approach to simulate the azimuth and ellipticity trajectory of a probe light in a tensile-strained bulk semiconductor optical amplifier (SOA) in a conventional pump-probe scheme. The physical mechanisms for the variations of polarization azimuth and ellipticity angle of the probe originate from the significant nonuniform distributions of carrier density across the active region in the presence of an intense pump light. Due to this carrier-density nonuniformity, the effective refractive indexes experienced by transverse-electric (TE) and transverse-magnetic (TM) modes of the probe are different. This results in a phase shift between TE and TM modes of the probe upon leaving the SOA. Simulations of the carrier distributions along the cavity length at different pump-light levels are demonstrated using multisection rate equations, which take into account the longitudinal nonuniform carrier density. The optical gain is considered via the parabolic band approximation. The influences of the spontaneous recombination and carrier-dependent material loss on the amplifier performance are included. The Mueller-matrix formalism is utilized to predict the variations of azimuth and ellipticity angle, which greatly reduces the complexity of the simulations in comparison with Jones-matrix formalism. The suggested approach is beneficial to experimental investigations due to the fact that during the optical-tuning process, Stokes parameters are virtually measurable on the Poincareacute sphere, and the Stokes vector of the incoming probe can be adjusted by a polarization controller and monitored by a polarization analyzer. Based on these carrier-induced nonlinearities in SOAs, an optical and gate with extinction ratio larger than 14 dB and Q-factor larger than 25 is presented at a bit rate of 2.5 Gb/s

Systematic effects induced by a flat isotropic dielectric slab

The instrumental polarization induced by a flat isotropic dielectric slab in microwave frequencies is discussed. We find that, in spite of its isotropic nature, such a dielectric can produce spurious polarization either by transmitting incoming anisotropic diffuse radiation or emitting when it is thermally inhomogeneous. We present evaluations of instrumental polarization generated by materials usually adopted in radio astronomy, by using the Mueller matrix formalism. As an application, results for different slabs in front of a 32 GHz receiver are discussed. Such results are based on measurements of their complex dielectric constants. We evaluate that a 0.33 cm thick Teflon slab introduces negligible spurious polarization (<2.6 x 10(-5) in transmission and <6 x 10(-7) in emission), even minimizing the leakage (<10(-8) from Q to U Stokes parameters, and vice versa) and the depolarization (approximately 1.3 x 10(-3)).

Improved signal analysis for the calibration of the motional Stark effect diagnostic

A Stokes vector and Mueller matrix formalism are adopted to model the optical train of the motional stark effect (MSE) system. The method to find all four elements of Stokes vectors and optical parameters of MSE system is presented in this article. A new fitting model for MSE calibration is discussed. A method for the measurement of offset angle of MSE is also presented. The nonideal effects of the background unpolarized and polarized light of non-Stark effect are analyzed.

Improved signal analysis for motional Stark effect data

Nonideal effects in the optical train of the motional Stark effect diagnostic have been modeled using the Mueller matrix formalism. The effects examined are birefringence in the vacuum windows, an imperfect reflective mirror, and signal pollution due to the presence of a circularly polarized light component. Relations for the measured intensity ratio are developed for each case. These relations suggest fitting functions to more accurately model the calibration data. One particular function, termed the tangent offset model, is found to fit the data for all channels better than the currently used tangent slope function. Careful analysis of the calibration data with the fitting functions reveals that a nonideal effect is present in the edge array and is attributed to nonideal performance of a mirror in that system. The result of applying the fitting function to the analysis of our data has been to improve the equilibrium reconstruction.

Design aspects of a MSE diagnostic for ITER

The motional Stark effect (MSE) diagnostic is unique in its ability to measure the current profile and will be essential in the International Thermonuclear Experimental Reactor (ITER) for detailed analysis of Advanced Tokamak and other types of discharges. However, design of a MSE diagnostic for ITER presents many unique challenges. Among these is optical analysis for the convoluted optical path, required for effective neutron shielding, that employs several reflective optics arranged to form a labyrinth. The geometry of the diagnostic has been laid out and the expected Doppler shifts and channel resolution calculated. A model of the optical train has also been developed based on the Mueller matrix formalism. Unfolding the pitch angle for this complicated geometry is not straightforward and possible methods are evaluated. The CORSICA code is used to model a variety of ITER discharges including start-up, Ip-ramp, and reverse shear. The code also incorporates a synthetic MSE diagnostic that can be used to evaluate different viewing locations and optimize channel locations for the above discharges. Simulation of the optical emission spectrum is also under way.

First mirror contamination studies for polarimetry motional Stark effect measurements for ITER

The motional Stark effect (MSE) diagnostic on the International Thermonuclear Experimental Reactor will need to guide the light through a labyrinth of mirrors to provide neutron shielding. Knowledge of how the mirrors change the polarization is essential for accurate determination of the q profile. The optical properties of the plasma facing mirror are also expected to change with time due to deposition/erosion. For the purpose of examining this experimentally a detector system, identical to the JET MSE system, using twin photoelastic modulators was constructed. Measurements have been performed on freshly prepared mirrors, on mirrors after exposure to plasmas in Tore Supra, and labyrinth designs. The result shows a significant effect on the optical properties and demonstrate the need for in situ monitoring. The measured properties of the labyrinth closely follow the Mueller matrix formalism. With a correct choice of material the angle change introduced by the four mirrors furthest away from the plasma will be below 1°.

Mueller matrix spectroscopic ellipsometry: formulation and application

Mueller matrix spectroscopic ellipsometry (MMSE) can be a powerful technique for the study and accurate determination of the dielectric function, optical properties and geometric characteristics of anisotropic materials and complex systems. In this work we present the formalism we developed in order to be able to calculate the sample Mueller matrix using phase-modulated ellipsometry. In order to test the formalism and check the information we can derive spectroscopically by a Mueller matrix, in comparison with standard spectroscopic ellipsometry, we proceeded to the study of an isotropic c-Si wafer and tested the ability of the formulation to calculate correctly its dielectric function using the M 33 and M 43 components in the energy region 1.5–6.5 eV. Also, we have used this formalism to obtain the MMSE spectra of a biaxially anisotropic SnSe bulk sample. Finally, we applied the Mueller matrix formalism for the study of biaxially stretched poly(ethylene terephthalate) (PET) film, as a representative complex anisotropic material. MMSE spectra were obtained for various angles of film's orientation, determined by the angle between the axis of preferential orientation of PET's macromolecules and the plane of incidence where the optical anisotropy of PET was verified. The interpretation of the MMSE spectra is found to provide more insights to the study of anisotropic materials.

The need of polarization modeling for ozone profile retrieval from backscattered sunlight

In the ultraviolet and visible part of the spectrum, measurements of space‐borne grating spectrometers are in general sensitive to the state of polarization of the observed light. The correction for this polarization sensitivity is based on broadband polarization measurements. In parts of the spectrum where the state of polarization is varying rapidly with wavelength this correction is not sufficient and severely limits the accuracy of the atmospheric parameters retrieved from the polarization corrected measurements. In this paper we demonstrate that the problems due to instrument polarization sensitivity can be solved in a natural way by the use of polarization modeling. For the forward model of a retrieval algorithm we propose the combination of a vector radiative transfer model to simulate the transport of radiation in the probed atmosphere and a straightforward simulation of the instrument polarization sensitivity by use of the Mueller matrix formalism. The use of a vector radiative transfer model also overcomes another common bias in retrieval algorithms, caused by the widely used scalar approximation of atmospheric radiative transfer. The capability and need of the proposed approach are demonstrated for ozone profile retrieval from measurements of the Global Ozone Monitoring Experiment (GOME). A comparison of retrieved profiles with 123 ozonesonde profiles shows that the use of a polarization forward model yields a significant improvement in root‐mean square difference of about a factor 1.5 in the stratosphere as well as in the troposphere. Also, a solar zenith angle dependence in the differences is reduced significantly.

A Mueller matrix formalism for modeling polarization effects in erbium-doped fiber

The effects of erbium anisotropy in erbium-doped fiber lasers, sources, and amplifiers are examined. Starting from basic ion properties, inversion and gain equations are derived analytically to describe polarization dependencies. A novel matrix form of the Er/sup 3+/ rate equations is presented to propagate powers and polarization states. These equations are then numerically integrated and compared to experimentally observed polarization hole burning and polarization dependent gain. The theoretical predictions agree strongly with experiment in all cases.

Comments on "A Mueller matrix formalism for modeling polarization effects in erbium-doped fiber"

This comment presents a means of extending to the paper cited above, the effects of circular polarization anisotropy for the evolution of optical signals in erbium-doped fiber amplifiers (EDFA's) and lasers.

Polarization model for a helically wound optical fiber

A polarization model for a helically wound optical fiber is presented. The model is based on a Mueller-matrix formalism that includes the effects of the winding configuration of the optical fiber on the polarization properties. A comparison between experimental polarimetric data for short lengths of polarization-maintaining optical fiber wound in a helical configuration upon the surface of a shell and the polarization model is presented. The effects of static radial displacements of a cylindrical shell are considered. Finally, the model is used to predict the effects of a range of displacements on the polarization properties of the transmitted light, hence demonstrating its use as a design tool for polarimetric sensors utilizing nontrivial geometric configurations.

Analysis of depolarizing optical targets by Mueller matrix formalism

The polarizing and depolarizing properties of an optical system can be derived from the experimental determination of its Mueller matrix. Constructing an algorithm to treat an experimental Mueller matrix based on the knowledge of the measurement errors, and connected to the study of the depolarization effects in the Poincare space, allows us to characterize and classify any optical system. Different samples of materials and surfaces are analysed by this method while polarimetric image coding provides the spatial distribution of the depolarization for each target.

Mueller matrix analysis of light depolarization by a linear optical medium

An attempt is made to investigate how incident light propagating through a linear non-image-forming optical device, or scattered by a linear medium is depolarized using the Mueller matrix formalism. By considering incident light in the form of pure states of polarization, we are able to show that a particular Mueller matrix decomposition is advantageous for analysis of the physical mechanisms of depolarization of light propagating through linear media. We illustrate our analysis by examining several physically realizable Mueller matrices reported in the optics literature.

Small-angle light scattering from an anisotropic sphere in the Rayleigh-Gans-Debye approximation: the Mueller matrix formalism

An explicit form of the Mueller matrix is given for an anisotropic sphere fulfilling the requirements of the Rayleigh-Gans-Debye (RGD) approximation. The angular dependences of single matrix elements are compared with Lorenz-Mie predictions for an isotropic sphere. Good agreement between the Lorenz-Mie model and that of RGD is observed where the latter is applicable. The magnitude and sign of the matrix elements depend on the size parameter kR (k = 2pi/lambda;R is the radius of the sphere) as well as on the magnitude and sign of the (ñ - 1)/Deltan parameter, where ñ is the relative refractive index and Deltan is the relative anisotropy of the sphere.

Target decomposition theorems in radar scattering

By converting the coherent scattering matrix into a target vector, the concept of a coherency matrix may be introduced for characterising statistical scattering problems involving polarised waves. This technique is compared with the more conventional Mueller matrix formalism and a unique decomposition theorem developed for modelling dynamic scattering problems.

Diffraction ellipsometry at microwave wavelengths

A Mueller matrix formalism is used for finding the relations between the reflection coefficient of a diffracting half-plane and the polarization of the diffracted wave. Experiments were made on diffracting consisting of graphite-pine rosin or graphite-paraffin mixture in the microwave region of the electromagnetic spectrum. Measurements on the polarization of diffracted waves were made; the results were used for calculation of the reflection coefficient and then the complex permittivity of the diffracting object.

The Mueller matrix formalism in diffraction phenomena

Electromagnetic diffraction at the edge of a plane screen is described by the use of the Mueller matrix. The transformation law is found for the Stokes parameters of the incident and diffracted wave by applying the Mueller method to several cases of diffraction.