# Polarized Light and the Mueller Matrix Approach

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## Book Description

*An Up-to-Date Compendium on the Physics and Mathematics of Polarization Phenomena*

**Polarized Light and the Mueller Matrix Approach** thoroughly and cohesively integrates basic concepts of polarization phenomena from the dual viewpoints of the states of polarization of electromagnetic waves and the transformations of these states by the action of material media. Through selected examples, it also illustrates actual and potential applications in materials science, biology, and optics technology.

The book begins with the basic concepts related to two- and three-dimensional polarization states. It next describes the nondepolarizing linear transformations of the states of polarization through the Jones and Mueller–Jones approaches. The authors then discuss the forms and properties of the Jones and Mueller matrices associated with different types of nondepolarizing media, address the foundations of the Mueller matrix, and delve more deeply into the analysis of the physical parameters associated with Mueller matrices.

The authors proceed to interpret arbitrary decomposition and other interesting parallel decompositions as well as compare the powerful serial decompositions of depolarizing Mueller matrix M. They also analyze the general formalism and specific algebraic quantities and notions related to the concept of differential Mueller matrix. The book concludes with useful approaches that provide a geometric point of view on the polarization effects exhibited by different types of media.

Suitable for novices and more seasoned professionals, this book covers the main aspects of polarized radiation and polarization effects of material media. It expertly combines physical and mathematical concepts with important approaches for representing media through equivalent systems composed of simple components.

## Table of Contents

**Polarized electromagnetic waves **Introduction: Nature of polarized electromagnetic waves

Polarization ellipse

Analytic signal representation and the Jones vector

Coherency matrix and Stokes vector

2D space–time and space–frequency representations of coherence and polarization

Poincaré sphere

Polarimetric interpretation of the Pauli matrices

Intrinsic coherency matrix

Polarimetric purity

Composition and decomposition of 2D states of polarization

Classification of 2D states of polarization

Invariant quantities of a 2D polarization state

Quantum description of 2D states of polarization

**Three-dimensional states of polarization**

Introduction

3D Jones vector

3D Coherency matrix

3D Stokes parameters

Composition and decomposition of 3D states of polarization

3D space–time and space–frequency representations of coherence and polarization

Intrinsic 3D Coherency matrix

Intrinsic 3D Stokes parameters

3D polarimetric purity

Interpretation of the coherency matrix for 3D polarization states

Invariant quantities of a 3D polarization state

Quantum formulation for 3D polarization states

**Nondepolarizing media**

Introduction

Basic polarimetric interaction: Jones calculus

Pure Mueller matrices

Singular states of Polarization

Normality and degeneracy of Jones and Mueller matrices

**Nondepolarizing media: Retarders, diattenuators, and serial decompositions**

Introduction

Retarders

Diattenuators

Other mathematical representations of the polarimetric properties of nondepolarizing systems

Polar decomposition of a nondepolarizing system

General serial decomposition of a nondepolarizing system

Dual linear retarder transformation of a nondepolarizing system

Constitutive vectors of a nondepolarizing Mueller matrix

Invariant polarimetric quantities of a nondepolarizing Mueller matrix

Particular forms of nondepolarizing Mueller matrices

**The concept of Mueller matrix**

Introduction

The concept of Mueller matrix

Covariance and coherency matrices associated with a Mueller matrix

Changes of reference frame and rotated Mueller matrices

Characterization of Mueller matrices: Covariance criterion

Normal form of a Mueller matrix

Reciprocity properties of Mueller matrices

Passivity constraints for Mueller matrices

Vectorial partitioned expression of a Mueller matrix

Spectral and characteristic decompositions of a Mueller matrix

Polarimetric purity of a Mueller matrix

**Physical quantities in a Mueller matrix**

Introduction

Components of purity of a Mueller matrix

Indices of polarimetric purity

Invariant quantities of a Mueller matrix

Purity space

Anisotropy coefficients of a Mueller matrix

From a nondepolarizing to a depolarizing Mueller matrix

**Parallel decompositions of Mueller matrices **

Introduction

Additive composition of Mueller matrices

Arbitrary decomposition of a Mueller matrix

On the rank of the covariance matrix of a parallel composition

Characteristic decomposition of a Mueller matrix

Polarimetric subtraction of Mueller matrices

Passivity constraints

Optimum filtering of measured Mueller matrices

**Serial decompositions of depolarizing Mueller matrices **

Introduction

Generalized polar decomposition

Symmetric decomposition

Passivity constraints in serial decompositions of depolarizing Mueller matrices

Invariant-equivalent Mueller matrices

Arrow decomposition of a Mueller matrix

Singular Mueller matrices

Serial-parallel decompositions

**Differential Jones and Mueller matrices**

Introduction

Differential Jones matrices and elementary polarization properties

Differential Mueller matrices

Differential decomposition of Mueller matrices

Differential Mueller matrix of a homogeneous depolarizing medium

**Geometric representation of Mueller matrices**

Introduction *P*-image and *I*-image of a Mueller matrix

Representative ellipsoids of a Mueller matrix

Ellipsoids associated with some special Mueller matrices

Characteristic ellipsoids of a depolarizing Mueller matrix

Intrinsic ellipsoids of a Mueller matrix

Topological properties of the characteristic ellipsoids

Five-vector representation

Geometric view of depolarization, diattenuation and polarizance

Geometric representation of nondepolarizing Mueller matrices

Experimental examples

References

Index

A summary appears at the end of each chapter.

## Author(s)

### Biography

**José Jorge Gil** is a professor at the University of Zaragoza, where he leads R&D projects in physics and e-learning technologies and methodologies. He has developed an original dual-rotating-retarder absolute Mueller polarimeter, introduced new concepts such as the depolarization index, and developed wireless systems for interactive meetings, which earned the Tecnova award from the Spanish Industry Ministry. Dr. Gil has also been a recipient of the G.G. Stokes Award from the International Society for Optics and Photonics (SPIE) in recognition of his "groundbreaking collection of rigorous mathematical descriptions of polarization that are used widely to interpret experimental data." He received his PhD in physics from the University of Zaragoza.

**Razvigor Ossikovski** is an associate professor at the Ecole Polytechnique, where he is the leader of the fundamental polarimetry and Raman spectroscopy activities in the Laboratory of Physics of Interfaces and Thin Films. His current research interests are the theory of polarimetry (Mueller matrix algebra) and experimental tip-enhanced Raman spectroscopy. He received his PhD in physics (materials science) from the Ecole Polytechnique.

## Reviews

"The best-ever treatise on the main concepts of both polarization states of light and Mueller matrices of media, illustrated with numerous figures, tables, and experimental examples. The comprehensiveness, clarity, and rigor make it essential for anyone interested in this field."

—Tiberiu Tudor, Professor, Faculty of Physics, University of Bucharest"Gil and Ossikovski have gathered together in one source a wealth of information on the intricacies of the interpretation of representations of polarized light. Current research on the mathematics of polarized light is thoughtfully presented. This is an essential reference for the serious student or researcher in the field."

—Dr. Dennis Goldstein, Polaris Sensor Technologies, Inc.