Library of Congress Cataloging-in-Publication Data

Collett, Edward, 1934-

Field guide to polarized light / Edward Collett.

p. cm. – (SPIE field guides; v. FG05)

Includes bibliographical references and index.

ISBN 0-8194-5868-6

1. Polarization (Light) I. Title. II. Series.

QC441.C63 2005

535.5′2--dc22

2005006346

Published by

SPIE—The International Society for Optical Engineering

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## Introduction to the Series

Welcome to the * SPIE Field Guides!* This volume is one of the first in a new series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the

*is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these

*is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each*

**Field Guides***provide additional information such as related material outside the main scope of the volume, key mathematical relationships and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.*

**Field Guide**The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new

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Optical Sciences Center

The University of Arizona

## Field Guide to Polarized Light

The polarization of light is one of the most remarkable phenomena in nature and has led to numerous discoveries and applications. Today it continues to play a vital role in optics. Before the 1950s there was very little activity on the foundations of polarized light. For example, answers to questions such as the nature and mathematical formulation of unpolarized light and partially polarized light were not readily forthcoming. Today there is a very good understanding of polarized light. In particular, the mathematical difficulties that had hindered complex polarization calculations were finally overcome with the introduction of the Mueller-Stokes matrix calculus and the Jones matrix calculus. Research in polarized light continues with much vigor as witnessed by the continued appearance of numerous publications and conferences.

The primary objective of this Guide is to provide an introduction to the developments in polarized light that have taken place over the past half-century. In this Guide I have tried to present the most salient topics on the subject. Hopefully, this Field Guide will enable the reader to have a good grasp of the material and most of all to allow him or her to be comfortable and even delighted with the beauty and subject of polarized light.

Finally, this Field Guide is dedicated to my wife, Mary Ann, and my children Ron and Greg. Their encouragement and support greatly simplified the task of writing this Guide.

Edward Collett

Georgian Court University

Lakewood, New Jersey

## Table of Contents

**Glossary** x

**The Foundations of Polarized Light** 1

The Ray Theory of Light 1

The Polarization of Light 2

Malus’s Law 3

Brewster’s Law 4

**The Wave Theory of Light** 5

Fresnel’s Wave Theory 5

The Polarization Ellipse 7

Degenerate Polarization States 8

The Parameters of the Polarization Ellipse 9

The Poincaré Sphere 10

Degenerate States on the Poincaré Sphere 11

**The Observables of Polarized Light** 12

The Stokes Polarization Parameters 12

Stokes Parameter Relations 14

Classical Measurement of the Stokes Parameters 16

The Mueller Matrices for Polarizing Components 17

Polarizers 18

Wave Plates 20

Rotators 22

Mueller Matrices for Rotated Components 23

Mueller Matrix Applications-Malus’s Law 25

Mueller Matrix Applications-The Optical Shutter 26

Mueller Matrix Applications-Stokes Parameters 27

**The Observable Polarization Sphere** 28

The Observable Polarization Sphere 28

Plotting the Quarter-Wave Plate on the OPS 32

The Rotating Quarter-Wave Plate 34

The Babinet-Soleil Compensator 35

Linear and Circular Polarizers 36

The Generation of Elliptically Polarized Light 37

Measurement Methods of the Stokes Parameters 38

The Rotating Quarter-Wave Plate Measurement 39

Birefringent Crystals and Wave Plates 40

Multiple and Zero-Order Wave Plates 41

**Reflection and Transmission** 42

Mueller Matrices for Reflection and Transmission 42

Reflection and Transmission Stokes Parameters 43

Reflection and Transmission Mueller Matrices 47

Total Internal Reflection 48

The Fresnel Rhomb 49

Single and Multiple Dielectric Plates 50

Pile of Polarizing Dielectric Plates 52

Fresnel’s Reflection and Transmission Coefficients 55

**Other Polarization Matrix Calculi** 57

The Jones Matrix Calculus 57

Wolfs Coherency Matrix Calculus 62

**Optical Activity and Optical Rotation** 63

Optical Activity and Optical Rotation 63

Faraday Rotation 64

Optical Isolators 66

**Depolarizers** 72

Wave Plate Depolarizers 72

The Lyot Crystal Depolarizer 74

**Polarizing Materials** 75

Polarizers 75

Polarizing Prisms 76

Characterizing Polarizers 78

Wave Plate Materials 81

**Superposition and Decomposition of Polarized Beams** 82

Incoherent Superposition and Decomposition 82

Incoherent Decomposition-Ellipses 83

Coherent Superposition and Decomposition 84

**The Electro-Optical Effect** 85

The Electro-Optical Effect - Modulators 85

The Pockels Cell 87

**Refractive Index Measurements** 88

Incidence Refractive Index Measurement 88

**The Radiation Field** 91

Maxwell’s Equations 91

The Radiation Equation and the Stokes Parameters 92

The Linear Oscillating Bound Charge 93

The Randomly Oscillating Bound Charge 94

A Charge Moving in a Circle 95

A Charge Moving in a Magnetic Field 96

The Classical Zeeman Effect 98

Optical Scattering 101

**The Optics of Metals and Semiconductors** 105

The Optics of Metals and Semiconductors 105

Refractive Index and Absorption Coefficient 106

Incidence Angle Reflectivity 107

Complex Reflection Coefficients 109

The Principal Angle of Incidence Measurement 110

**Appendix** 114

Equation Summary 124

**Bibliography** 128

#### Glossary

**Frequently used variables and symbols:**

B

birefringence

**B**(**r**, t)

magnetic induction vector

**c**

speed of light in a vacuum

cp

circularly polarized

db

decibels

**D**(**r**, t)

electric displacement vector

* e*-

extraordinary ray

ɛ

permittivity constant

**E**_{0}_{x}

maximum amplitude in the * x* direction

**E**_{0}_{y}

maximum amplitude in the * y* direction

**E**_{x}(*r*, t)

* x* component of the optical field

**E**_{y}(*r*, t)

* y* component of the optical field

**E**

Jones vector

**E**(**r**, t)

electric field vector

**F**

force vector

**H**_{0}

Transmission of two parallel polarizers

**H**_{90}

Transmission of two crossed polarizers

**H**(**r**, t)

magnetic field vector

HWP

half wave plate

**i**

angle of incidence

**i _{B}**

Brewster angle

**i,j,k**

Cartesian unit vectors

**J**

Jones matrix

**j**(**r**, t)

electric current density vector,

**J**_{POL}

Jones matrix for a polarizer

**J**_{WP}

Jones matrix for a wave plate

**J**_{ROT}

Jones matrix for a rotator

**J**_{QWP}

Jones matrix for a quarter-wave plate

**J**_{HWP}

Jones matrix for a half-wave plate

**J**(θ)

Jones matrix for a rotated polarizing element

**k**

wave number

**k**

wave vector

**k**_{1}

major transmittance of a polarizer

**k**_{2}

minor transmittance of a polarizer

KDP

potassium dihydrogen phosphate

L−45P

linear −45 polarization

L+45P

linear +45 polarization

LCP

Left circular polarization

LHP

linear horizontal polarization

LVP

linear vertical polarization

μ

permeability constant

**M**

Mueller matrix

**M**_{HWP}

Mueller matrix of a half-wave plate

**M**_{LP}

Mueller matrix of a linear polarizer

**M**_{POL}

Mueller matrix of a polarizer

**M**_{QWP}

Mueller matrix of a quarter-wave plate

**M _{R}**

Mueller matrix for reflection

**M**_{ROT}

Mueller matrix of a rotator

**M**_{T}

Mueller matrix for transmission

**M**_{WP}

Mueller matrix of a wave plate

* M*(θ)

Mueller matrix of a rotated polarizing element

**n**

complex refractive index

**n _{e}**

refractive index of the extraordinary ray

**n _{o}**

refractive index of the ordinary ray

**n _{p}**

parallel refractive index

**n _{s}**

perpendicular refractive index

**n _{L}**

levo-rotary refractive index

**n _{R}**

dextro-rotary refractive index

**n**

refractive index

* o*-

ordinary ray

OPS

observable polarization sphere

* p*-

parallel polarization state

**p _{x}**

polarizer transmission coefficient (* x*)

**p _{y}**

polarizer transmission coefficient (* y*)

$\mathcal{P}$

degree of polarization

PBS

polarizing beam splitter

QWP

quarter wave plate

**r**

angle of refraction

**r**

radius vector

RCP

right circular polarization

* s*-

perpendicular polarization state

**S**

Stokes vector

**S**_{0}

first Stokes parameter

**S**_{1}

second Stokes parameter

**S**_{2}

third Stokes parameter

**S**_{3}

fourth Stokes parameter

**S _{R}**

Stokes vector for reflection

**S _{T}**

Stokes vector for transmission

TIR

total internal reflection

UNP

unpolarized

**v**(**r**, t)

velocity vector

**v _{x},v_{y},v_{z}**

principal velocities

**V**

Verdet’s constant

${V}_{\pi}$

half-wave voltage

**V _{m}**

maximum modulation voltage

WP

wave plate

**x,y,z**

Cartesian coordinate system

2α

coordinate on the observable polarization sphere

2_{Ψ}

coordinate angle on the Poincaré sphere

2_{X}

coordinate angle on the Poincaré sphere

α

auxiliary angle

ε

complex dielectric constant

ε*_{x}*, ε

*, ε*

_{y}

_{z}principal dielectric constants

δ

coordinate on the observable polarization sphere

δ

phase difference

δ_{x}

phase of the wave (* x*)

δ_{y}

phase of the wave (* y*)

θ

angle of rotation

κ

absorption coefficient

Ψ

orientation angle

X

ellipticity angle

ρ_{s,p}

Fresnel reflection coefficients

ρ(**r**,t)

electric charge density

σ

conductivity

τ_{s,p}

Fresnel transmission coefficients

$\text{\varphi}$

phase shift

${\text{\varphi}}_{x}$

phase shift (* x*)

${\text{\varphi}}_{y}$

phase shift (* y*)

ω

angular frequency

ω_{c}

cyclotron frequency

ω_{L}

Larmor’s frequency

ω_{m}

modulation frequency

$\nabla $

spatial vector operator