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U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and

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THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consist- ent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of the Office of Measurement Services, and the following center and divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center for Radiation Research — Laboratory Astrophysics 2 — Cryogenics 2 — Electromagnetics 2 — Time and Frequency 2

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Self-Study Manual on

Optical Radiation Measurements:

Part I —Concepts, Chapter 6

Fred E. Nicodemus, Editor

ATIOWAL BU2EAB

J UN 3 1977

/Id* 3/0-3

77

C/.-2—

Optical Physics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

Chapter 6. Distribution of Optical Radiation with

Respect to Polarization, John B. Shumaker

4?" 0P cot

0ffeA u o*

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued June 1977
National Bureau of Standards Technical Note 910-3

Nat. Bur. Stand. (U.S.), Tech. Note 910-3, 62 pages (June 1977) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $2.10 Stock No. 003-003-01785-0
PREFACE

This is the third in a series of Technical Notes (910- ) entitled "Self-Study Manual

on Optical Radiation Measurements." It contains the sixth chapter of that Manual. Inadvertently, this chapter has been completed before chapters 4 and 5, which are nearly ready

for publication in NBS TN 910-2. Chapter 4, "More on the Distribution of Optical Radiation

with Respect to Position and Direction," deals with all of the radiometric quantities of

Table Al-3 in Appendix A of NBS TN 910-1 that were not covered in detail in the first three

chapters. Chapter 5, "The Measurement Equation," introduces the central topic that ties

together our entire treatment of optical radiation measurements. The measurement-equation

concept is introduced by deriving the appropriate measurement equations for three different

illustrative problems. All of these chapters are still in Part I—Concepts, but our plans

call for us next to turn to one or two chapters in Part III—Applications. We feel that we now have established enough of the fundamentals to be able to usefully deal with some specific applications, the topic in which we realize many readers will be most interested.

For background information on the whole project and on the plans for the "Self-Study

Manual" (SSM) , we reproduce here (immediately following) the Preface to the first Technical

Note 910-1, of March 1976. Although the exact details of the outline and plans for the

individual chapters are still developing, our principal aims and overall plans are still

as set forth in that Preface.

Here, in chapter 6 of Part I, we are primarily concerned with establishing the concepts needed to deal with polarization in relation to radiometry (the measurement of

optical radiation) . There already exist excellent treatments of polarization that are concerned primarily with polarization phenomena, and we have not tried to compete with them.

This is not a treatise on polarization phenomena, as such, but only on as much about them

as is needed to deal adequately with the measurement of power propagated in the form of

optical radiation — with radiometry. To our knowledge, this is a topic that has not

previously been treated in the literature.

We are grateful to many for valuable comments and criticisms received, both informally

as well as in formal reviews, concerning all phases of the SSM. In connection with this

polarization chapter, we are particularly indebted to Drs. Jean M. and Harold E. Bennett

of the Michelson Laboratory, Naval Weapons Center, China Lake, California, and

Dr. Elio Passaglia of NBS for helpful discussions and critical reviews. Mrs. Betty Castle

has again done an outstanding job of typing a difficult text.

Fred E. Nicodemus, Editor

Henry J. Kostkowski, Chief,

Optical Radiation Section

PREFACE to NBS TN 910-1

This is the initial publication of a new series of Technical Notes (910) entitled

"Self-Study Manual on Optical Radiation Measurements." It qontains the first three chapters of this Manual. Additional chapters will be published, similarly, as they are completed. The Manual is being written by the Optical Radiation Section of NBS. In addition

to writing some of the chapters, themselves, Fred E. Nicodemus is the Editor of the Manual

and Henry J. Kostkowski, Chief of the Section, heads the overall project.

In recent years, the economic and social impact of radiometric measurements (including

photometric measurements) has increased significantly. Such measurements are required in

the manufacture of cameras, color TV's, copying machines, and solid-state lamps (LED's).

Ultraviolet radiation is being used extensively for the polymerization of industrial coatings, and regulatory agencies are concerned with its effects on the eyes and skin of

workers. On the other hand, phototherapy is usually the preferred method for the treatment

of jaundice in the newborn. Considerable attention is being given to the widespread utilization of solar energy. These are just a few examples of present day applications of optical radiation. Most of these applications would benefit from simple measurements of one to

iii
a few per cent uncertainty and, in some cases, such accuracies are almost essential. But

this is rarely possible. Measurements by different instruments or techniques commonly disagree by 10% to 50%, and resolving these discrepancies is time-consuming and costly.

There are two major reasons for the large discrepancies that occur. One is that optical radiation is one of the most difficult physical quantities to measure accurately.

Radiant power varies with the radiation parameters of position, direction, wavelength, time,

and polarization. The responsivity of most radiometers also varies with these same radiation parameters and with a number of environmental and instrumental parameters, as well.

Thus, the accurate measurement of optical radiation is a difficult multi-dimensional problem. The second reason is that, in addition to this inherent difficulty, there are few

measurement experts available. Most of the people wanting to make optical radiation

measurements have not been trained to do so. Few schools have had programs in this area

and tutorial and reference material that can be used for self-study is only partially available, is scattered throughout the literature, and is generally inadequate. Our purpose in

preparing this Self-Study Manual is to make that information readily accessible in one place

and in systematic, understandable form.

The idea of producing such a manual at NBS was developed by one of us (HJK) in the

latter part of 1973. Detailed planning got under way in the summer of 1974 when a fulltime editor (FEN) was appointed. The two of us worked together for about one year developing an approach and format while writing and rewriting several drafts of the first few

chapters. These are particularly important because they will serve as a model for the rest

of the Manual. During this period, a draft text for the first four chapters was distributed, along with a questionnaire, for comment and criticism to some 200 individuals representing virtually every technical area interested in the Manual. About 50 replies were

received, varying widely in the reactions and suggestions expressed. Detailed discussions were also held with key individuals, including most of the Section staff, particularly

those that will be writing some of the later chapters. In spite of the very wide range of

opinions encountered, all of this feedback has provided valuable guidance for the final

decisions about objectives, content, style, level of presentation, etc.

In particular, we have been able to arrive at a clear solution to difficult questions

about the level of presentation. Both of us started out with the firm conviction that,

with enough time and effort, we should be able to present the subject so that readers with

the equivalent of just elementary college mathematics and science could easily follow it.

That conviction was based on our experience of success in explaining the subtleties of

radiometric measurements to technicians at that level. What we failed to consider, however, was that, in making such explanations to individuals we always were able to relate

what we said to the particular background and immediate problem of the individual. That's

just not possible in a text intended for broad use by workers in astronomy, mechanical

heat-transfer engineering, illumination engineering, photometry, meteorology, photo-biology

and photo-chemistry, optical pyrometry, remote sensing, military infrared applications, etc. To deal directly and explicitly with each individual's problems in a cook-book approach

would require an impossibly large and unwieldy text. So we must fall back on general

principles which immediately and unavoidably require more knowledge and familiarity with

science and mathematics, at the level of a bachelor's degree in some branch of science or engineering, or the equivalent in other training and experience.

In its present form, the Manual is a definitive tutorial treatment of the subject that

is complete enough for self instruction. This is what is meant by the phrase "self-study"

in the title. The Manual does not contain explicitly programmed learning steps as that

phrase sometimes denotes. In addition, through detailed, yet concise, chapter summaries,

the Manual is designed to serve also as a convenient and authoritative reference source. Those already familiar with a topic should turn immediately to the summary at the end of

the appropriate chapter. They can determine from that summary what, if any, of the body

of the chapter they want to read for more details.

The basic approach and focal point of the treatment in this Manual is the measurement

equation. We believe that every measurement problem should be addressed with an equation

relating the quantity desired to the data obtained through a detailed characterization of

the instruments used and the radiation field observed, in terms of all of the relevant
parameters. The latter always include the radiation parameters, as well as environmental

and instrumental parameters, as previously pointed out. The objective of the Manual is to develop the basic concepts and characteristics required so that the reader will be able

to use this measurement-equation approach. It is our belief that this is the only way that

uncertainties in the measurement of optical radiation can generally be limited to one, or

at most a few, per cent.

Currently, the Manual deals only with the classical radiometry of incoherent radiation.

The basic quantitative relations for the propagation of energy by coherent radiation (e.g.,

laser beams) are just being worked out [19-22] . * Without that basic theory, a completely

satisfactory general treatment of the measurement of coherent (including partially coherent)

optical radiation is not possible. Accordingly, in spite of the urgent need for improved

measurements of laser radiation, we won't attempt to deal with it now. Possibly this situation will be changed before the current effort has been completed and a supplement on laser measurements can be added.

As stated above, we first hoped to prepare this Manual on a more elementary level but

found that it was impossible to avoid making use of both differential and integral calculus

of more than one variable. However, to help those that might be a bit "rusty" with such

mathematics, we go back to first principles each time a mathematical concept or procedure

beyond those of simple algebra or trigonometry is introduced. This should also throw additional light on the physical and geometrical relationships involved. Where it seems inappropriate to do this in the text, we cover such mathematical considerations in appendices.

It is also assumed that the reader has had an introductory college course in physics, or the equivalent.

The Manual is being organized into three Parts, as follows:

Part I. Concepts

Step by step build up of the measurement equation in terms of the radiation parameters, the properties and characteristics of sources, optical paths, and receivers, and the

environmental and instrumental parameters. Useful quantities are defined and discussed

and their relevance to various applications in many different fields (photometry, heattransfer engineering, astronomy, photo-biology, etc.) is indicated. However, discussions

of actual devices and measurement situations in this Part are mainly for purposes of

illustrating concepts and basic principles.

Part II. Instrumentation

Descriptions, properties, and other pertinent data concerning typical instruments,

devices, and components involved in common measurement situations. Included is material

dealing with sources, detectors, filters, atmospheric paths, choppers (and other types of

optical modulators), prisms, gratings, polarizers, radiometers, photometers, spectroradiometers, spectrophotometers, etc.

Part III. Applications

Measurement techniques for achieving a desired level of, or improving, the accuracy

of a measurement. Included will be a very wide variety of examples of environmental and

instrumental parameters with discussion of their effects and how to deal with them. This

is where we deal with real measurements in the real world. The examples will also be

drawn from the widest possible variety of areas of application in illumination engineering,

radiative heat transfer, military infrared devices, remote sensing, meteorology, astronomy,

photo-chemistry and photo-biology, etc.

figures in brackets indicate literature references listed at the end of this Technical

Note.
Individual chapter headings have been assigned only to the first five chapters:

Chapter 1. Introduction

Chapter 2. Distribution of Optical Radiation with respect to Position and

Direction — Radiance

Chapter 3. Spectral Distribution of Optical Radiation

Chapter 4. Optical Radiation Measurements — a Measurement Equation

Chapter 5. More on the Distribution of Optical Radiation with respect to Position and Direction

Other subjects definitely planned for Part I are thermal radiation, photometry, distribution with respect to time, polarization, diffraction, and detector concepts. It is not our intention, however, to try to complete all of Part I before going on to Parts II and III.

In fact, because we realize that a great many readers are probably most interested in the

material on applications to appear in Part III, we will try to complete and publish some chapters in Parts II and III just as soon as adequate preparation has been made in the

earlier chapters of Part I. However, because our approach to radiometry differs so much

from the traditional treatment, we feel that unnecessary confusion and misunderstanding

can be avoided if at least the first nine chapters of Part I are published first and so are available to readers of later chapters.

Finally, we invite the reader to submit comments, criticisms, and suggestions for

improving future chapters in this Manual. In particular, we welcome illustrative examples

and problems from as widely different areas of application as possible.

As previously stated, we are indebted to a great many individuals for invaluable

"feedback" that has helped us to put this text together more effectively. Notable are the

inputs and encouragement from the Council on Optical Radiation Measurements (CORM)

,

especially the CORM Coordinators, Richard J. Becherer, John Eby, Franc Grum, Alton R. Karoli, Edward S. Steeb, and Robert B. Watson, and the Editor of Electro-Optical Systems

Design, Robert D. Compton. In addition, for editorial assistance, we are grateful to Donald A. McSparron, Joseph C. Richmond, and John B. Shumaker, and particularly to Albert T. Hattenburg.

We are especially grateful to Mrs. Betty Castle for the skillful and conscientious

effort that produced the excellent typing of this difficult text. We also want to thank

Henry J. Zoranski for his capable help with the figures.

Fred E. Nicodemus, Editor

Henry J. Kostkowski, Chief,

Optical Radiation Section

March 1976

vi
Contents

Page

Part I. Concepts 1

Chapter 6. Distribution of Optical Radiation with Respect to Polarization 1

In this CHAPTER 1

THE STOKES COMPONENTS of POLARIZATION 2

STOKES VECTORS and MUELLER MATRICES 9

THE SIMPLE SPECTRAL FILTER 12

THE IDEAL POLARIZER 12

THE IDEAL LINEAR RETARDER 16

DETERMINING STOKES COMPONENTS with IDEAL INSTRUMENTS .... 17

THE IDEAL CIRCULAR POLARIZER 20

RADIOMETER CALIBRATION 22

THE ROTATION MATRICES 24

REAL OPTICAL COMPONENTS 28

POLARIZER-ATTENUATORS 35

DEPOLARIZERS 36

A SIMPLIFIED CHARACTERIZATION of POLARIZATION 37

EXPERIMENTAL CONCERNS 39

SUMMARY of CHAPTER 6 40

References 45

Appendix 6. Matrix Multiplication 47

Appendix 7. The Measurement of Mueller Transmittance Matrices 49

List of Figures

6.1 Electric-field-vector behavior for monochromatic polarized radiation 4

6.2 The components of L in the L , L and L ' , L' coordinate systems .... 26

vii

SELF-STUDY MANUAL on OPTICAL RADIATION MEASUREMENTS

Part I. Concepts

Chapter 6. Distribution of Optical Radiation with Respect to Polarization

by John B. Shumaker

This chapter develops and illustrates the concepts necessary

to include polarization rigorously in classical radiometry. The

treatment is based upon the Stokes polarization vector of spectral

radiance and Mueller transmittance matrices. The Mueller matrices

of many common polarizing optical components are discussed. Considerable attention is paid to the measurement of the Stokes

spectral-radiance-vector components and to the measurement of the

Mueller-transmittance-matrix elements. The concepts and techniques

are illustrated by discussions of such subjects as radiometer calibrations, three-polarizer attenuators, depolarizers, and the

characterization of Polaroid-type polarizers. Also included are an appendix on matrix multiplication and an appendix showing one way by which, in principle, any Mueller transmittance matrix may

be measured.

Key Words: Mueller matrix; polarization; polarizer; Stokes

parameters.

In this CHAPTER . We discuss polarization, another of the radiation parameters introduced

in Chapter 1 [1]. The other radiation parameters — position, direction, wavelength, and

time — are continuous variables and we have used functional notations like L(x,y,8 , <j>, A)

to show the dependence of radiometric quantities upon these parameters. Polarization, however, enters somewhat differently. We include polarization by introducing four polarization components of spectral radiance, L n

(x,y,8,<j>, A) , L , (x,y,6,4>, A) , L „ (x,y,8 ,(j>, A) , A , U ^ , _1_ A , Z and L (x,y,8, <)>, A) . The first of these, L n , is identical with the spectral radiance

A , j A , U we have been using in earlier chapters. The other three, L , L , and L Q , proA,l " > Z " , J vide the polarization information. The expressions that result from this four-fold

extension of the quantities appearing in the measurement equation are adequate to handle

any kind of polarizing optical element and any polarization state within the approximations

of classical (geometrical optics) radiometry of incoherent radiation.

We will not in this chapter discuss in detail specific polarizing or depolarizing

instruments. These will be described in Part II and are also covered in several of the

references [2,3,4,5]. We concentrate here on developing and illustrating the use of the

concepts and equations necessary to include polarization rigorously in classical radiometry. We begin the chapter with a review of the relationship between the oscillating

electric field and the polarization state of radiation. We then introduce the four com- ponents of spectral radiance and study how they are transformed by ideal polarizers and

retarders. Next we discuss the response of a radiometer to polarized light and how it may

be calibrated. After this we examine real, non-ideal, optical components and finally con- clude the chapter with a brief look at some experimental precautions in the use of polarizing optics.
The object of this chapter is to extend the equations of radiometry which have been

developed in preceding chapters so that they can include polarization rigorously. We make

little attempt to provide an intuitive understanding of the polarization phenomenon. Our

approach is directed toward the radiometrist who may regard polarization as an unfortunate

complication which, like the common cold, has been inflicted upon him by the whimsical

gods. We therefore have tried to present a systematic measurement approach to the polarization problem which does not require the reader to become an expert on polarization optics.

As a consequence, we illustrate our equations and the kinds of measurement information

involved with conceptual measurements which employ simple, easily available instruments.

Although some very precise relative polarization measurements can be made using high

quality optical components and specialized instruments such as ellipsometers, such precision is usually not required in radiometry. These special devices and techniques are described in some of the references [2,5,6] and will also be considered in Parts II and III. This chapter also does not discuss the many remarkable uses which man and nature have made

of polarized light. These are briefly discussed in reference [2] which includes extensive

further literature references.

THE STOKES COMPONENTS of POLARIZATION

Polarization can be thought of in terms of the orientation of an electromagnetic field

vector oscillating about the line of sight between an observer and a light source. All of

the consequences of polarization can be derived from this starting point and the names "linear" polarization and "circular" polarization originate in this description. However,

within the approximations of geometrical optics a more direct phenomenological approach is

possible. This simply requires the reinterpretation of spectral radiance as being fully

characterized by four independent polarization components, called Stokes components. All

four components are measurable, as we will see, and form a phenomenological basis for the

quantitative inclusion of polarization in radiometry and photometry. There are other ways

of describing polarization [2,4,5] but for our purposes the use of Stokes components is the

simplest because for light with a negligible degree of polarization — frequently the most

important situation in radiometry — the treatment reduces directly to the equations we have been considering in the preceding chapters. The Stokes components provide no particular insight into why or how polarizing optical components work but we will not be interested in that aspect of polarization in this chapter. We are interested in how the polarization state of a beam affects the response of a radiometer and in how to describe the

flux of a polarized beam so that the beam is uniquely defined from a radiometric point of

view — so that it could be used to calibrate a radiometer, for example. The four Stokes

components of spectral radiance at every point and direction in a radiation field provide

that kind of description. Since most people prefer to think in the familiar terms of

electric field vectors and linear and circular polarization, we will indicate how the

Stokes components are related to these common concepts, but we emphasize that this is not

essential to radiometry.
We begin with the familiar microscopic description of polarization. We consider a single steadily radiating oscillator at a distant point on the negative Z-axis of a rectangular coordinate system and imagine that we can continuously record the instantaneous

electric field 1 present at the origin due to this oscillator. Except at distances from the

light source of a few wavelengths or less, the Z component of the electric field — the

component along the direction of propagation — will be completely negligible and the field

will lie in the X-Y plane. The X and Y field components then will be of the form

E = V 'cosUir'vt + 6 ) X X X [V-nT 1

] (6.1)

E = V -cos(27T-vt + 6 ) y y y

where V and V are the amplitudes [V'm-1

], v is the frequency [Hz], 6 and 6 x y x y are the phases t rad] of the electromagnetic wave train, and t is the time [s]. In figure 6.1 are shown some examples to illustrate how the electric-field vector changes with

time for some special combinations of amplitudes and phases. In each case the sketch on

the left shows the pattern traced out in the X-Y plane by the tip of the electric-field

vector and the graphs on the right show the time behavior of the X and Y components. In diagrams (a) we see light which is described as linearly polarized 2 in a vertical direction;

the horizontal component, E , of the electric field is always zero. Diagrams (b) show a more general example of linear polarization. E and E are exactly in phase with one another, 6 = 6 , but their amplitudes V and V differ. Diagrams (c) show right

y x y x —°

circularly polarized light ; the electric vector appears to trace a circle in the clockwise

The electric and magnetic field vectors are perpendicular to each other and to the direction of propagation of the radiation. Our development could equally well have been based

upon the magnetic field, as it is in much of the older literature.

Linear" and "plane" polarization are synonymous terms. "Linear" seems more consistent

with the terms "circular" and "elliptical" and so is used exclusively here.

3 The terms "horizontal" and "vertical" linear polarization and "right" and "left" circular

polarization are somewhat arbitrary and are used differently by different authors. Light

which contains no vertical component of the electric field may be described as horizontally

polarized; however, if the magnetic field vector is emphasized instead of the electric

vector then such light would be called vertically polarized. Right and left handedness are likewise ambiguous descriptions of circular polarization. Fortunately, as long as one is

consistent within his own calculations it rarely matters what conventions one adopts for

these definitions. To insure consistency one must usually take all his formulas and equations from the same source. Our conventions used in this chapter agree with those of

Shurcliff [2] where, for instance, light reflected from horizontal surfaces such as from a desk top or the ocean is defined as possessing enhanced horizontal polarization. Polarizing

sunglasses, consequently, are vertical polarizers as customarily worn.
Figure 6.1

Electric-field-vector behavior for

monochromatic polarized radiation

(a) Linearly polarized in the Y-Z plane

(b) Linearly polarized at +30'

-^v

(c) Right circularly polarized

(d) Right elliptically polarized
direction (as we look back toward the source) . E and E have equal amplitudes but are

y x exactly a quarter cycle out of phase with one another: 6=6+ it/2. If 6 = 6 - tt/2 y x y x the electric vector will rotate in a counterclockwise direction and the polarization is

called left circular . The final example shows elliptically polarized light which is the

general case of eq. (6.1). In this case no special relationships exist between V and

V and between 6 and 6 . E and E have unequal amplitudes and are out of phase

by an angle y = 6 - 6 . If y is positive the polarization is called right elliptical

y x and if y is negative it is called left elliptical. Elliptical polarization represents

the most general expression of a perfectly monochromatic ray which is possible. Linear and

circular polarizations are special cases of elliptical polarization.

As an aid to visualizing polarization we might think of a classical source mechanism

in terms of the oscillations of charged particles. At radio wavelengths a vertically

polarized field can be produced by a vertical transmitting antenna with a cluster of electrons sloshing up and down at the radio frequency. At optical wavelengths we can similarly

imagine a vertical vibration of electrons at the source as one way of generating vertically

polarized light. Circular polarization then can be thought of as being generated by

charges moving in circles at the source. If the charges are moving in circular orbits as seen by an observer in one direction then observers in other directions will see the

charges moving in ellipses or even vibrating in straight lines thus observing elliptical or linear polarization. Such directional dependence of the polarization form is actually

observed in the Zeeman effect, for example, where some spectral lines emitted by a gas in a magnetic field show circular polarization if viewed along the direction of the magnetic

field and change continuously through elliptical to linear polarization as the viewing

direction changes toward the perpendicular to the magnetic field.

Three parameters are required to describe the general polarization ellipse. They can be chosen in several ways. For example the magnitude of the parameter y = 6 - 6 y x specifies the ellipticity of the polarization and its sign specifies its handedness — left

or right. The parameter V /V specifies the orientation of the ellipse in the X-Y plane.

And the parameter V

2 + V2 specifies the size of the ellipse. A set of parameters which

x y

has proven to be somewhat more useful, however, is the set of Stokes parameters 2

:

Y is further required to be adjusted by adding or subtracting integral multiples of 2tt

if necessary to give |y| < ^

In eq. (6.2) that follows, the proportionality constant k is given by k = n/(2yc)

where n is the refractive index, u the magnetic permeability and c is the vacuum

velocity of light. If V and V are expressed in volts/meter and E * in watts/m2

then </n = 1.327 x 10~ 3 [W*V-2

] for non-magnetic materials. Equations (6.2) contain less

information than eqs. (6.1). The absolute phase information has been lost. Ultimately

this will result in the exclusion of interference effects from our treatment and in the

additivity of radiometric quantities such as spectral radiance. These, of course, are

just the limitations (and advantages) of geometrical optics.

5
E n = k(V 2 + V

2

) [W-m" 2

] n,0 x y

E . = k(V 2 - V

2

) n,l x y

E . = k(2V -V -cosy) n,2 x y

E _ = k(2V -V -siny). n,3 x y

(6.2)

There are four of these parameters, the first of which is redundant in the present context 1

but, as we shall see later, forms the basis for including unpolarized light. The second

Stokes parameter E -. is a measure of the excess of horizontal linear polarization over n,l

vertical linear polarization. It is called the horizontal preference parameter . The third

parameter E . is a measure of the excess of linear polarization at 45° (from the horin, i. zontal) to that at 135°. It is called the +45° preference parameter . The last Stokes

parameter is a measure of the excess of right-circular component to left-circular component.

It is called the right-circular preference parameter . These parameters all have the unitdimensions of irradiance and constitute the four Stokes polarization components of normal

irradiance from a single steady oscillator.

A real, non-monochromatic, non-point radiation source will also generate a varying

electric field vector at any point. But the motion of this electric field vector will

generally not be easy to describe. For one thing the motion need not be confined to a plane. For another, it may not be periodic. In the limit as the solid angle and wavelength

interval approach zero the motion of the electric field vector will become planar and

elliptical and so the Stokes parameters are appropriate for describing the polarization

state of the radiometric quantity spectral radiance. Accordingly, the average values of the

Stokes spectral radiance components that a real radiometer would measure are given by

2

A,0 AA-At

AA

x + V

At

Aoi'AA dx-dy-dt

J X,1

J X,2

AA-At

AA-At

x AAJ At Aw-AA dx-dy-dt

AA At

2V -V -cosy x y Aw-AA dx-dy-dt

(6.3)

A, 3 AA-At

AAJ At

2V -V -siny

x y ' Aoo-AA dx-dy-dt

where At is the time required for a measurement, AA is the area of the detector surface

perpendicular to the ray direction, Aw is the solid angle from which rays can illuminate

because E

2 =E 2 +E 2 +E 2 n,0 n,l n,2 n,3
the detector and AA is the wavelength band reaching the detector. V , V and y are

x y the field amplitudes and phase difference at the detector surface.

In the mathematical limit as AA, At, Aw and AA approach zero

. L

A

2,0 " L

A

2,1 + L

A

2,2 + h2 ,3- (6 ' 4)

This is the condition for complete polarization. With sufficient radiometer resolution in

these four parameters, A, t, oj and A, all measurement results would show complete

polarization. However, as was discussed in Chapter 2, [1] radiometric measurements cannot

be made if any of these resolution intervals goes to zero because zero power would then be

received. The practical limits on the smallness of AA, At, Aid and AA result in

values of Stokes spectral-radiance components which obey

L

A

2,0 * L

A

2,1 + L

A

2,2 + L

A

2,3'

Thus, we speak of a polarized component of spectral radiance

and an unpolarized component

L A,P = V

L

A

2,1 + L ^2 + L

A

2,3

(6 ' 5)

L = L - L.

A ,u A ,0 A,p

and define the degree of polarization as

P " L

A, P /L

A,0- (6 - 6)

Although at first glance it would seem that a definition of unpolarized light which

obviously depends upon instrument resolution would be highly unsatisfactory, it turns out

that this is rarely a problem. Usually unpolarized light arises because of inadequate

By considering the coherence properties of radiation one can deduce that the state of

polarization in the detector plane remains essentially constant over areas of the order of

6A 'v A

2 * (6oj) -1

. Similarly, the state of polarization is essentially constant in time over

a time interval of the order of 6t ^ (6v) _1

= A

2 *(c*6A) _1

. Therefore, the resolution

intervals need be no smaller than these values. A perfect radiometer with these resolution

intervals would require a source with spectral radiance

h*c / , 9 A L, n * 10 4 h-v/(6u)-6A-6A-5t) = lO 4 •—.—•( A 2 -—) <\. 2-10 19 [W-m-3 -sr" 1

] A ,1) Aye/

to achieve a 1% measurement accuracy at A = 500 [nm] . This is a factor of 10 6 times the

solar spectral radiance. For any less intense source the radiometer resolution intervals

must be larger if useful accuracy is desired.
resolution in the time dimension. For typical spectrometers and for steady incandescent

sources (so-called natural light), measurement times less than 10-11 seconds would be

required to observe complete polarization. That is, V , V , and y will not change

x y

appreciably over periods of 10-11 seconds and non-vanishing values of L , ,, L „ or

A , 1 A , z

L, „ can be expected. If At is very much longer, however, the fluctuations in V , V

A,

J

x y and y may cause cancellations in computing the mean values of these three Stokes components. L, „, being the sum of positive quantities V

2 and V

2

, will not be reduced by A,0 x y averaging. Thus, L will appear to include a contribution which has disappeared from

A , U the L , L „ and L components and is consequently called unpolarized. Since the A,i A,z A,

j

time resolution required to resolve the unpolarized component is so far beyond the capabilities of customary radiometry there is, for all practical purposes, no ambiguity in the

definition of unpolarized light. Occasionally a (usually periodic) rapid variation in

polarization over t or over one of the other parameters A, oj or A is deliberately

introduced to simulate the "natural" unpolarized light we have just described. These

variations are usually sufficiently coarse that they could conceivably be resolved by some radiometers. For this reason such light is frequently called depolarized instead of

unpolarized.

The first Stokes component, L n , is the spectral radiance that we have been conA , u sidering in earlier chapters. It is a measure of the total radiant flux associated with

a ray and is therefore usually the quantity of greatest interest. It is the value of

spectral radiance which a polarization-indifferent radiometer would measure. L n can

A , U never be negative. The other three Stokes components L , , , L „ and L may be

A, I A , z A , j

either negative or positive. If the radiation contains more vertical linear polarization

than horizontal linear polarization then the horizontal preference component L will be

A,

I

negative. If the radiation is completely vertically polarized then L .. = -L . If the

A , I A , u radiation contains equal horizontally and vertically linearly polarized contributions as in

unpolarized light or in 45° polarized light then L = 0. If the horizontal content is

A,l

in excess then L is positive; its maximum value is L when the radiation is comA , 1 A , U pletely horizontally linearly polarized. The +45°-preference Stokes component of spectral

radiance, L „, describes the +45° and +135° linearly polarized content of the radiaA ,z tion in an exactly analogous manner. And finally, the right-circular preference component,

L , similarly takes on values from -L _ for completely lef t-circularly polarized

A , 3 A , U light through zero for light with no net circular polarization to +L _ for completely

A , U right-circularly polarized light. Radiation linearly polarized at other angles (other than

0° and 45° and the complementary angles 90° and 135°) will have non-zero values for

both L, . and L. „. Likewise if either or both of L, .. and L are non-zero and in

A,

I

A,Z A , J. A,Z

addition L is non-zero then the radiation contains attributes of both linear and cirA, 3 cular polarization and is, of course, elliptically polarized.

Although for spectral radiance the four Stokes components provide a complete and, for

all practical purposes, unique description of the polarization state of radiation, the same cannot be said for the integrals of spectral radiance such as radiance, spectral intensity,

8
spectral irradiance, etc. For such cases, as we have indicated, the trajectory of the tip

of the electric vector can no longer be described, in general, by three parameters. The

only Stokes component of these integrated radiometric quantities which is always unambiguous is the first one — the integrals of L „. As in earlier chapters where polarization

A , U was neglected, an integral of spectral radiance does not uniquely characterize a complete

radiation field; one must, in general, trace the spectral radiance by determining the

propagance, ray by ray, from source to receiver and then, at the receiver, perform the

desired integration of spectral radiance. Fortunately, at the ultimate detector the usual

quantity of interest is the total power, and to calculate this requires only an appropriate

integral of L . Therefore, we shall discuss polarization only in terms of rays and

A , U spectral radiance.

STOKES VECTORS and MUELLER MATRICES

The complete description of polarization in classical radiometry requires the specification of four parameters such as the Stokes components of spectral radiance associated with

each ray. These Stokes components are measurable with, for example, a linear polarizer and

a quarter-wave plate. We will return to this later after we have generalized the concept

of propagance or transmittance to include polarization.

In Chapter 3, eq. (3.14) [1] we related 2 the spectral radiance L' of a ray emerging

from an optical medium to the incident spectral radiance L by means of a transmittance or propagance T:

^For simplicity we'll usually speak only of transmittance and won't distinguish between

propagance and transmittance. However, our results apply equally well when reflectance is

involved provided some care is exercised in defining the coordinate system used for the

incident and reflected rays. The simplest coordinate system to use is one with the origin

at the point of reflection of the ray. The incident ray is referred to such a coordinate

system with its Z-axis along the ray in the direction of propagation, its Y-axis in the

plane of incidence — that is, the plane defined by the incident and reflected rays — and

its X-axis perpendicular to the plane of incidence and, therefore, lying in the reflecting

surface or, more generally, tangent to it. The reflected ray is referred to a coordinate

system with its Z-axis in the direction of propagation of the reflected ray, with* its X-axis

coincident with the X-axis of the incident ray and with its Y-axis in the plane of incidence

but with its sense chosen to preserve the handedness of the coordinate system. If the

incident-ray coordinate system is right-handed then that of the reflected ray is also.

2 In the interests of typographical simplicity several of the symbols used in earlier chapters will be modified in this chapter. In particular, since we will be dealing only with

spectral radiance we will have no need to distinguish between L and L , so we will

A usually omit the subscript A on this quantity. The spectral-radiance input to a device,

formerly denoted L . , will usually be written L and the output, L , will usually be

written L' . The transmittance, t, or propagance, t* , will usually be written T.

9
L'(x,y,e,<M) = T(x,y,0,(f>,A)-L(x,y,e,(t>,A).

L and L' we now see consist of four components each so we must expect four equations of

this type — one for each component. Moreover, since the propagance may differ for different forms of polarization and since every component of L may in general contribute to each component of L' the propagance T must now contain sixteen elements:

L

= T00' L + T0r L

l + T02* L2 + T03' L

3

L

l = T10* L + T

ll' L

l + T12" L2 + T13' L

3

L

2 " T20- L + T21' L1 + T22* L2 + T 23' L

3

L

3 T 30* L + T31* L

1 + T32* L2 + T33* L3"

(6.7)

This is the generalization of eq. (3.14) which is required to describe the changes in the

Stokes spectral-radiance components of a ray upon passing through an optical medium. Just

as the spectral radiance in a beam may vary from ray to ray and from wavelength to wavelength, so may the propagance elements Tnn , Tm , . . . vary from ray to ray, etc. Thus

the values of these elements depend upon the ray we are examining and we should write them

as Tnn

(x,y, 6, <j>, A) , etc., where the x,y,6, and <j> are the same reference-surface

quantities which identify the ray of spectral radiance L n (x,y,6 ,cj>, A) . However, throughA , U out most of this chapter we deal with the behavior of a single ray so we will usually not

explicitly display this dependence.

Because the four Stokes polarization components are independent [within the restricL

Q

> (L^ + L^ + L^]

Stokes spectral-radiance vector

tion L > (L 2 + L

2 + L2

) 2

] it is common to describe the set of four as a vector, the

L =

~ L

o L

l L

2

_ L

3

(6.8)

By expressing spectral radiance this way, as a column vector, eqs. (6.7) can be compactly

written as

L' - T • L . (6.9)

where is the 4x4 matrix

10
T =

T T T T 00 01 02 03

T T T T 10 11 12 13

T T T T 20 21 22 23

T T T T L 30 31 32 33

(6.10)

and eq. (6.9) means exactly the same thing as eqs. (6.7) — no more and no less. We shall

generally use this vector and matrix notation 1 throughout the rest of this chapter, not

because of any useful four-dimensional mental image that might be envisioned, but simply for

economy of notation.

There is one precaution about matrix equations which must be observed. That is that

the product f

• |_ is not always the same as • ] and the product of two matrices

Ti "To i- s not generally the same as To*T-i • Tne rule for writing the equations correctly

in accordance with the conventional matrix multiplication formulas is to write the incident

radiation vector and the transmittance matrices in the order, from right to left, in which

the ray encounters the optical elements they represent. So, if a ray with spectralradiance vector |_ passes through one optical device with transmittance matrix J., and

then through a second with transmittance matrix J„ , the emerging spectral radiance will

be

L'(x,y,9,(j),A) = T2 (x,y,6,<»,A) • ^(x.y.e.^.A) • |_(x,y,e,<f>,A) (6.11)

The effect of a succession of optical devices can be combined into a single effective transmittance matrix by simply multiplying the individual matrices together in the proper order.

Thus, eq. (6.11) can be written

L' = T-L . (6.12)

where the product matrix f = To'Tt can t> e obtained by following the rules for matrix

multiplication. Of course the same result would be obtained by writing out eqs. (6.7) for

the radiance components passing device 1 and then again using eqs. (6.7) inserting these

expressions for the radiance input to device 2 to obtain expressions for the final emerging

radiance components.

The matrices which describe the transmittance or propagance through optical elements

which influence polarization are known in the literature as Mueller matrices . In very

simple cases the proper form of the Mueller matrix is obvious by inspection. For optical

elements for which equations can be written relating the amplitudes and phases of the

headers who are not familiar with matrix manipulations, or who tend to forget which subscript means what, will find everything needed to convert these matrix equations to workable

formulas in Appendix 6.

11
emerging radiation field to the amplitudes and phases of the entering field, the matrix can

be derived by a straightforward procedure [7,8]. This includes all perfect polarizing

optical elements such as ideal polarizers, quarter-wave plates, mirrors, etc. For a real,

non-ideal device, however, the Mueller matrix can only be obtained ultimately by experiment.

Mueller matrices for ideal optical components have been tabulated by many authors [2,7].

THE SIMPLE SPECTRAL FILTER

Let us now study some examples of the use of Mueller matrices and Stokes components.

Perhaps the simplest useful optical element is just a glass absorbing spectral filter with

transmittance t(A). The Mueller transmittance matrix for this should have the effect of

simply multiplying each component of |_ by t with no mixing by or interaction with other

components. That is, we want

L

o' = x(A)-L

L

l

- t(A)-L

L

2 = x(A)-L

L

3 = t(A)-L

(6.13)

Comparing these equations with eq. (6.7) we see that T00x(A), Tn

= t(A),

T„_ = t(A) and all other T = 0. Thus, for a simple filter we have

T

22

= x(A),

T =

t(A)

t(A)

t(A)

t(A)

= t(A)

1

1

1

1

(6.14)

THE IDEAL POLARIZER

A more interesting optical element is the ideal linear polarizer . An ideal polarizer

possesses a polarization axis (perpendicular to the ray direction) with the property that

the transmittance to light linearly polarized in this direction is 1 and the transmittance

to light linearly polarized at right angles to this direction is 0. Aside from surfacereflection effects, crystal-prism polarizers such as the Nicol-prism polarizer come close

to this ideal. If the angle between the polarization axis of the polarizer and the horizontal reference axis or x-axis of the optical system is <j> the Mueller transmittance

matrix of an ideal linear polarizer is

POD) =

1 cos2tf> sin2<f>

cos2tf) cos 22<j) sin2<f>*cos2<j)

sin2<|> sin2(f>' cos 2(f) sin 2

2cj> (6.15)

12
Let us use eq. (6.9) to see the effect of such a polarizer on unpolarized light. Since

unpolarized light is characterized by L^ = L2 = L3 = 0, we have

L' = P(*)-L = h

1 cos2<J> sin2<J> L cos2$ cos 2<j> sin2<f>'cos2tj>

sin2<fi sin2(j)*cos2(}) sin 2

2<f> = h l.

1

cos2c|

sin2<| (6.16)

The vector equation (6.16) is simply a shorthand for the set of four equations:

L

x

' = JiL -cos2<j)

L

2

' = %L -sin2<j>

L

3 "

°-

By using the trigonometric identity cos 22<j) + sin2

2cf) = 1 we see that

L' = (l' 2 + L' 2 + L' 2

)1 , This shows [eq. (6.4)] that the output radiance is completely

polarized. Since L ' = there is no circular component so the polarization is strictly

linear. If we turn the polarizer so that its axis is parallel to the x axis we would

expect horizontally polarized light. Putting <j> = 0° in eq. (6.16) we obtain

L' = P(o°)-L = k\

Thus, light with Stokes components like

is pure horizontally polarized light. If the polarizer is turned to light polarized in that orientation. From eq. (6.16) we get

= 45° we expect

L' = hL

Q

Thus, light whose Stokes components are a multiple of

13
is linearly polarized at 45°. Similarly inserting if) = 90° and 135° in eq. (6.16)

shows that vertically polarized light is a multiple of

1 -1

and light polarized at 135° is characterized by Stokes components proportional to

1

-1

Notice that the spectral radiance L' is always positive, as its name suggests it must be,

while the other components may have either sign.

Suppose instead of starting with unpolarized light we had put arbitrarily polarized

light through the ideal linear polarizer. We have

L' = P(*)-L = h

1 c s L

o

c c

2 s-c L

l

s s-c s

2 L

2

_ L L 3-

= hCL

Q + i^-.c + l

2 «s) (6.17)

where we have used the abbreviations cos2<f> = C and sin2<f> = S. As with initially

unpolarized radiation, the transmitted radiation is completely linearly polarized at the

orientation <j) of the polarizer axis. Equation (6.17) takes on an especially simple form

if the initial ray is completely linearly polarized. In this case, as we have seen, |_ has the form

L-Lr

1

cos2<|>

sin2<J;

where 4> is the polarization direction. Substituting from this expression L = L 'cos2^

and L = L «sin2^ in eq. (6.17) we find

14
|_' = ^L "[l + cos2(<HC)] L

Q «cos 2

((t.-^)

1

cos2<j>

sin2<J> (6.18)

When <$> = ty the polarizer has no effect on the ray but when <J> = \p ± 90° the ray is com- pletely extinguished. Equation (6.18) is known as Malus ' Law and expresses the familiar

behavior of a pair of ideal polarizers.

Before going on to consider other polarizing components we should remark that the

discussion above can be turned around and used to derive the Mueller transmittance matrix

of an ideal polarizer [eq. (6.15)]. The argument runs as follows: We define an ideal

linear polarizer as an object (1) which cannot be used to distinguish circularly polarized

light from unpolarized light, (2) whose output is always completely linearly polarized in

a direction parallel to its polarization axis, and (3) which attenuates completely linearly

polarized light in accordance with Malus' Law [eq. (6.18)]. If we now examine eqs . (6.7) we see that, because the first property says that the L ' components on the left do not

depend upon the value of L , the last column of the matrix must consist entirely of zeros. The second property says that the output of the polarizer is proportional to

1

cos2<j

sin2<j

where <Ji is the orientation of the polarizer axis. Combining this with the third property

we see that if the input ray has a radiance vector

L=Lr

1 cos2a

sin2a

the output ray will be described by

L

1 = L -cos 2

(<j)-a)

1

cos2<{>

sin2<J>

Therefore the Mueller matrix P(<f>) for an ideal linear polarizer must satisfy

15
L' = *

Q

1 + cos2<f>*cos2a + sin2<j>'sin2a

cos2<j) + cos 2 2<j>«cos2a + sin2<j>*cos2<f)'sin2a

sin2cf) + sin2<}>*cos2<J>*cos2a + sin2 2<j>*sin2a = P(+)'Lr

1 cos2a

sin2a

where we have substituted cos 2

(cf>-a) = %(1 + cos2())-cos2a + sin2<}>'sin2a) . Comparing this

equation with eqs. (6.7) and remembering that the . last column of P(4>) must be zero we immediately obtain eq. (6.15). This derivation illustrates the possibility of deducing

the form of a Mueller matrix from the observed experimental behavior of the object it is

meant to describe. We will later use this approach again with a Mueller matrix form which

more accurately describes real (non-ideal) polarizers.

THE IDEAL LINEAR RETARDER

A linear retarder is an optical device frequently made of a thin plate of an anisotropic crystal such as calcite or quartz with its faces cut parallel to the crystal optic

axis. Such plates have the (birefringence) property that light linearly polarized in one direction, called the fast axis of the retarder, travels with a higher velocity than light

linearly polarized in a direction at right angles to this, called the slow axis. A mixture

of both polarization forms entering a retarder then will emerge with the components phase

shifted with respect to one another and consequently with its polarization form changed.

The amount of phase shift is called the retardance , 6. The retardance is related to the

thickness, a, of the plate and the indices of refraction n^ and n for the fast and

f s slow polarizations, respectively, according to

2-Tra (n

f - n

s

) [rad] (6.19)

The Mueller transmittance matrix for an ideal retarder is:

R(6,<f>) = 10C

2 + S

2 -cos6 S-C-(l - cos6) -S-sin6

S-C- (1 - cos6)

S«sin6

S

2 + C

2 -cos6

-Osin6

Osind

cos6

(6.20)

where C = cos2<J>, S = sin2<f> and c|> is the angle between the fast axis of the retarder

and the horizontal.

Retarders in which the retardance is one-quarter cycle (6 = tt/2) or one-half cycle

(5 = it) are called quarter- and half-wave plates. Quarter-wave plates are useful for

converting linearly polarized light to circularly polarized light, and vice-versa; half- wave plates , for rotating the direction of polarization of linearly polarized light. If we set 6 = tt/2 in eq. (6.20) we obtain the Mueller matrix for an ideal quarter-wave plate:

16
1

c

2 s-c _c

s-c s

2 c

s -c

Q($) = R(tt/2,4>) = „ ... o2 „ (6.21)

DETERMINING STOKES COMPONENTS with IDEAL INSTRUMENTS

With an ideal linear polarizer and an ideal quarter-wave plate we are now in a position to measure, in principle, the four Stokes components of an arbitrary light ray. We

pass the ray through the quarter-wave plate and then through the polarizer and measure the

output with a polarization-indifferent detector. The Mueller transmittance matrix for the

combination of quarter-wave plate at angle <f> = a and polarizer at angle <j> = 3 is

A

= P(3)-Q(a)

1 cos23

cos23 cos 2 23

sin23 sin23 - cos2f

sin23

sin23-cos23

sin2 23

10cos 2a sin2occos2a -sin2a

sin2a*cos2a sin 2a cos2a

sin2a -cos2a

1 cos2a # cos2(3-a) sin2a*cos2(3-a) sin2(3-a)

cos23 cos23*cos2a'Cos2(3-a) cos23 # sin2a*cos2 (3-a) cos23 # sin2 (3-a)

sin23 sin23*cos2a'cos2(3-a) sin23*sin2a-cos2(3-a) sin23*sin2(3-a)

(6.22) 1

If a ray with spectral-radiance vector

L

L 3J

passes through this combination, the spectral radiance vector [_' of the output ray will

be given by

1 cos23

L' sin26 = A*L = ^[L + L -cos2a-cos2(3-a) + L

2 'sin2a-cos2(3-a) + L

3 "sin2 (3-a)] (6.23)

Obviously, unless one can find simplifying trigonometric identities a long train of

polarizing optics will quickly lead to an almost unmanageable problem. The operator form

of Mueller matrix manipulation described by Priebe [9] permits many simplifications to be

performed before the matrix elements become too unwieldy. His technique is especially

useful for weakly (linearly) polarizing components, small retardances , and small angles

of rotation.

17
Since we are only interested in the response of a polarization-indifferent detector to this

ray we actually need only the L ' component of [_' which is

L

Q

' = %[L

Q + L

1 -cos2cccos2(6-a) + L • sin2a-cos2(B-a) + L -sin2(3-a)] . (6.24)

Had we been interested in passing the output into or through any further polarizationsensitive components we would, in general, have needed all four components of |_' . From

eq. (6.24) we see that if we make four measurements of L ' at four combinations of a and 3, we can solve the four resulting equations for the Stokes components L_, L , L„, and L of the original ray. A particularly simple solution is obtained when a and

B are set at 0°, 45°, and 90° as follows:

LJ (a = 0, 6 = 0) = lg(L + L

1

) L' (a = 0, 6 = 90°) = H(L - L.

)

(6.25)

L

Q

' (a = 45°, B = 45°) = %(L

Q + L2

) Lj (a = 0, B = 45°) = %(L

Q + 1^).

From these four measured values of L ' the polarization components of the original ray

can be obtained at once.

In general the result of passing a ray through an ideal retarder followed by an ideal

linear polarizer is a ray whose spectral radiance is a sum of the Stokes components of the

original ray each multiplied by sines and cosines of the orientation angles and the

retardance. 1 Such a relationship readily lends itself to Fourier analysis, and many automatic techniques for measuring some or all of the Stokes components are based on this idea

[10,11].

It is possible to deduce relative values of L , L , and L by measuring the pair

of angular settings a and B of the quarter-wave plate and polarizer which simultaneously minimize L ' . This is a common technique in ellipsometry, for example, because sl

a determination requires only monotonicity of detector response — not linearity or, indeed,

not even significe

adjusting a and

ously minimize L ' . This is a common technique in ellipsometry, for example, because such

re not even significant stability. The minimum value of L ' which can be achieved by

-J - ^0 - <h

2 + S

2 + h^\

This occurs when

tan2a = L„/L

and (6.26)

tan2(B-a) = 1^/(1^ + L^) 2

,

1 The complete expression is given in eq. (6.41)

18
as can be obtained by differentiation of eq. (6.24). From a measurement of the angles a and B-a at this minimum 1 then we can calculate the relative values of the polarization

components from eq. (6.26):

L, /L = cos2a-cos2 (6-ot)

1 P L„/L = sin2a'cos2(3-a)

2 P L_/L = sin2(g-a),

3 P

(6.27)

where L = (L,2 + L

2 + L2

) is the polarized component of L. . Unfortunately the

p 1 2 3 determination of L.Opand L cannot be carried out by measurements of angles only but

requires flux measurements with the ubiquitous problems of temporal, positional, directional, and wavelength responsivity and of detector linearity which characterize and

bedevil radiometric measurements.

An alternative way to measure relative values of L , L , and L is by using a variable retarder — usually called a compensator . This is a retarder composed essentially

of two wedge-shaped retarders arranged so that sliding them across one another changes the

total thickness of the composite block and hence the retardance of the device. By simultaneously adjusting the retardance and angular orientation of a perfect compensator,

arbitrarily polarized light can be converted to any other arbitrary polarization state

which possesses the same degree of polarization. Thus, we may pass radiation, whose Stokes

components are to be determined, through a compensator and then through a fixed perfect

polarizer and adjust the compensator so that the polarizer is able to extinguish the polarized component of the light. From the retardance and orientation settings of the compensator when this adjustment has been achieved the Stokes component values can be computed.

Taking for simplicity a polarizer with its polarization axis horizontal we have from

eqs. (6.15) and (6.20)

P(0°).R(6,4>).|_ = h

1 1 o o" 110

0_

10C

2 + S2 -cos6 S«0(l-cos6) -S-sin^

S-0(l-cos5) S

2 + C

2 *cos6 C-sin<!

S«sin6 -Osin6 cos6

J

.S-.

The output radiance is

L

» = % [l + L

1 -(C 2 + S

2 -cos5) + L

2 -S-C-(l-cos6) - I^-S-sinfi],

*In the literature of ellipsometry the angles a and x = &-<* defined by eq. (6.26) are

known respectively as the azimuth and ellipticity of the polarized component. See reference [6] for a review of ellipsometry.

19
By differentiation we find that L ' has a minimum value of

L

Q

' -h [Lq-CL,2 + 1^ +L/) 1 *] =>2 (L -L

p

)

at values of the orientation angle <f> and retardance 6 which satisfy:

L n /L = -(cos 2

2<j> + sin2 2(j)'cos6) 1 P

L_/L = -sin2d> > cos2<j>- (1 -cos6)

2 P

L_/L = sin2d)'sin6.

3 P

(6.28)

These ellipsometric techniques can be extremely accurate and are used extensively in

studying surfaces and surface films by the analysis of the relative polarization form of

reflected light [5,6]. Note that these techniques require an accurate means of sensing a minimum flux level and that L and the more important component L„ remain undetermined,

THE IDEAL CIRCULAR POLARIZER

In eqs. (6.25) we saw how a quarter-wave plate followed by an ideal linear polarizer

with its polarization axis at 45° to the fast axis of the retarder could be used to gather

information about the Stokes component L , which is the circular polarization component.

In this case the Mueller transmittance matrix [eq. (6.22)] reduces to:

Acir

= P(3)'Q(a = 6+45°) =

1 ±1

C ±c

S ±s (6.29)

where C = cos 23, S = sin 23, 3 describes the orientation of the linear polarizer and

the + or - signs are chosen according to the sign of 6 — a = +45°. Arbitrarily

polarized light passing through this combination will emerge with Stokes vector

L' = Acir

~v 1 L

l L

2 " hiL +

- V

' cos2

sin2

_ L3_

(6.30)

Suppose, for example, that the incoming light is completely circularly polarized so that

L_ = ±L„ and that its handedness is opposite to that of A . . We will then observe

3 cir

[_' = 0, i.e. complete extinction. Notice that although a measurement of the spectral

radiance of the emergent ray yields information primarily about the circular polarization

component L„ the emergent ray itself is linearly polarized. In order to generate circularly polarized light we must reverse the retarder-polarizer combination: the light must

first pass through the linear polarizer and then through the quarter-wave plate. Taken in

20
this order the combination has the Mueller matrix

Pcir

= Q(3 ±45°).P(3) = h

1 C S

±1 +C ±S

(6.31)

where C = cos2g, S = sin23 and 6 is the orientation of the linear polarizer. The

quarter-wave plate is at the angle a = 8 ±45°. Now if arbitrarily polarized light passes

through this combination the result is

L' -P.cir

L

L

3

kCL

Q + L cos28 + L sin2g)

1

±1

(6.32)

which represents pure circularly polarized light. In addition to illustrating the mathematical quirk that the product of two matrices may depend upon the order in which they are

taken, the difference between A . and P . [(or eqs. (6.29) and (6.31)1 is important cir cir ltv/ t- experimentally since commercial circular polarizers are usually made by cementing a quarter-wave plate to a linear polarizer with their axes 45° apart. Such a device will

act as a circular polarizer only if light enters from the linear polarizer side and will

act as an analyzer of circularly polarized light only if the light enters from the quarter- wave plate side.

An ideal circular polarizer without these idiosyncracies can be made by adding a second quarter-wave plate to make a sandwich with the linear polarizer in the middle.

Each element must be turned 45° from the previous one to give

cir

= Qfcj, ±90°)-P(^ ±45°)-QO) (6.33)

where Q(<j>) is the Mueller matrix [eq. (6.21)] for a quarter-wave plate and P(t))) is that

for a linear polarizer [eq. (6.15)]. If we multiply out the matrices we find

P1 . = h

cir

1 ±1

±10 1

(6.34)

Plus signs in eq. (6.33) lead to plus signs in eq. (6.34) and a right-circular polarizer.

Minus signs throughout give a left-circular polarizer. This circular polarizer now has

properties analogous to those we associate with the ideal linear polarizer: it can be

used as polarizer or analyzer from either side; it will pass circularly polarized light

of the proper handedness but will extinguish any component of the opposite handedness; and

21
it cannot distinguish between linearly polarized light and unpolarized light.

RADIOMETER CALIBRATION

Equation (6.32) is a special case of

[_' = Q(a)-P(0)-L

1 cos2g sin23

cos2a*cos2 (a-g) cos2a-cos2g«cos2(a-g) cos2a*sin2g'cos2(a-g)

sin2a*cos2(a-g) sin2a«cos2g-cos2(a-g) sin2a-sin2g«cos2(a-g)

sin2(a-g) cos2g sin2(a-g) sin2g-sin2(a-g)

(6.35)

%(L + L -cos2g + L «sin2g)

cos2occos2 (a-g)

sin2a*cos2 (a-g)

sin2(a-g)

which can be obtained by multiplying the matrices of eqs. (6.21) and (6.15). This

describes the behavior of a polarizer at orientation angle g followed by a quarter-wave

plate at orientation angle a. By the proper orientation of the quarter-wave plate and

polarizer any arbitrary state of polarization of the emerging beam can be achieved. If

g = a, for example, the output is linearly polarized at the orientation a, and if

g = a ±45° the output is right (+) or left (-) circularly polarized. Such a beam can be

used to measure Mueller transmittance matrix elements and to calibrate a radiometer.

The complete Mueller transmittance matrix for an optical component must be known if

the polarization state of a ray after passing through this component is wanted. However,

in many cases all that is needed is a prediction of the response of a complete instrument

to the input radiation. This is a simpler problem because the instrument response depends

only upon the spectral-radiance component L ' which triggers the final electronic,

thermal, or chemical response. 1 Here, only the top row of the Mueller matrix of the

instrument is required. That is, only the first of eqs. (6.7) is needed. We can write

dS = (R

0() .L + R

01

. Ll + R

Q2

.L

2 + R03

.L

3

) -dG-dA

where dS is the element of radiometer signal output associated with the ray whose Stokes

spectral-radiance components are L , L , L , and L and whose elements of throughput

^Some detectors may respond to properties of the electromagnetic radiation other than

power — angular momentum, for example. It appears that such detectors can still be

treated in a purely formal way as we describe here, although the resulting responsivities,

R , may not be proportional to the Mueller matrix elements of any physically realizable

optical component.

22
and wavelength are dO = dwcosO'dA and dA, respectively. The coefficients R

nn > Rni ' etc., are the first-row Mueller transmittance-matrix elements of the optical part of the

radiometer, including the optical polarization characteristics of the detector- or transducer-element responsivity, multiplied by the polarization-indifferent responsivity

of that element in combination with any signal-processing and display components of the

instrument. The total response of the instrument is the integral of this expression over

all rays which can reach the photo-sensitive surface.

S = AA AA

(RQ0

-L + R

Q1

'L + Rq

2 'L

2 + R

03

'L )-dwcose«dA"dA (6.36)

Acj

where AA, AA, and Aw are the acceptance intervals of the radiometer. If these acceptance intervals are small enough that the Stokes radiance vector can be considered uniform

over all of them then 2

S = *00' L +

*0l'

L

l + *02- L

2 + *03' L3

(6 - 36a)

where Rn . = Oi Rn

_. 'du'cosO *dA'dA. These four responsivity factors RnA can be

AA; AA Aw J determined by observing the response of the instrument to four sufficiently different,

known polarization states of a test beam with fixed geometry and spectral distribution.

We thus obtain four equations of the form of eq. (6.36a) to be solved for the four responsivity factors Rn-' The L. are the Stokes components of the input-beam radiance produced, for example, as described above, following eq. (6.35), and measured, if necessary,

as described in connection with eq. (6.24) using a polarization-indifferent detector and

a second quarter-wave plate and linear polarizer.

If the acceptance intervals of the instrument are too large to assume uniform radiance,

then it may be that the coefficients R in eq. (6.36) can be considered constant, independent of wavelength and of ray position and direction. In this case

Must as the electrical engineer replaces an actual generator by an equivalent circuit

consisting of either an ideal voltage generator and series impedance or an ideal current

generator and shunt impedance, it is convenient, analytically, to replace the actual

detector or transducer element by an equivalent combination of (1) a pure transmittance

(whose Mueller transmittance matrix can be combined with that of the other optical elements by matrix multiplication) followed by (2) a polarization-indifferent detector element that, in combination with the following signal-processing and display components,

determines the magnitude of the overall responsivity.

As in earlier chapters we assume linearity throughout this discussion. That is, R_

.

may depend upon ray position, direction and wavelength but not upon Stokes spectralradiance components.

23
where $

.

1

R •$ + R •$ + R •$ + R •$ (f, 16M

00 01 1 02 2 03 3 ko.JOD;

L . -dwcose *dA-dA. Again the four responsivity coefficients can be

' L\' AAJ Aw determined from observations of the output response S to four beams with known, different,

Stokes spectral-radiance-vector distributions. A common way of achieving approximately

isotropic (directionally uniform) responsivity, at least over the acceptance solid angle

Aw, is, as we have seen in Chapter 5 [12], to introduce a diffuser as the first component

of the radiometer. In this case, since such a diffuser is also a depolarizer, we have the

further simplification: R .. = R = R = 0. The quantity <3> is the total radiant flux

in the wavelength pass band of the radiometer.

If neither the R„ . nor the L. are uniform, the dependence of the R„ . upon ray

Oi l Oi

direction, position and wavelength can still be determined by probing the instrument with

well-defined, narrow pencils of rays with uniform Stokes spectral-radiance vector. The

general usefulness of such a radiometer, however, is questionable since its response is a sum of weighted integrals of the spectral radiance distribution falling upon it. For com- parison of sources with identical relative spectral radiance distribution over direction,

position and wavelength, of course, such a radiometer might be satisfactory and its calibration in this case again calls for only four measurements.

THE ROTATION MATRICES

If we look back at eqs. (6.15) and (6.21) we notice a similarity in the way the angle

<j) enters into the matrices for a polarizer and for a quarter-wave plate. This is not

accidental. It arises in the following way: suppose we somehow were given numerical

Mueller propagance-matrix-element values for an optical component at a particular orientation, say <j) = 0. How can we generalize this to other orientations, 4> 4- 0? We do it in

a kind of backward way by first finding the Stokes components of an entering ray in a coordinate frame which has been rotated through the angle cj> so as to match the orientation for which we know the Mueller matrix of the (rotated) optical component. Then we can multiply this Stokes vector by the Mueller matrix and finally rotate the coordinate frame

back to the original "horizontal" orientation.

Consider an arbitrarily polarized ray described by

I

J L

l L

2 L

L

3

If we rotate our coordinate frame, only the linear polarization components, L and L , will be affected because these are the only components whose definitions require specification of a reference, horizontal direction. So, thinking of L and L for the moment as

24
orthogonal components of a two-component vector (see fig. 6.2) we can easily find the new components of this vector in a rotated frame of reference. In real space we know that if we rotate the "horizontal" direction by 45° we will have rotated the direction of L into

that of ±L . In our two-dimensional vector space, however, the angle between L and L

is 90°; therefore a rotation through the angle <j> in real space corresponds to a rotation

through the angle 2<J> in our (L ,L ) vector space. We see from figure 6.2 that such a rotation of our coordinate axes will leave us with new components

L' = L •cos2<() + L 'Sin2<}>

and

L' = -L 'Sin2cj) + L •cos2<j>.

This result, combined with

LJ=L

and

L

3 =L

3

is very suggestive of eq. (6.7): that is, by defining a matrix M(<j>) we can write

L" = M(*)-L10

cos2<(> sin2<J)

-sin2<}> cos2cf>

1

(6.37)

to describe this rotation. The procedure for handling an optical component turned at an angle (f> is now fairly clear. Rotating the coordinates of through the angle <j> so that "horizontal" agrees with the optical-component orientation, applying its Mueller

propagance matrix, and rotating again through -<j> to get back to the real horizontal is

equivalent to the following

L" = 10

cos(-2<|>) sin(-2(j))

-sin(-2tj)) cos(-2<fi)

1

10

cos2<j> sin2<j>

-sin2cj> cos2<(>

1

(6.38)

in which ; is the Mueller matrix. The matrix M(<}>) defined in eq. (6.37) is known as

a rotation matrix . If the multiplication of the three 4x4 matrices in eq. (6.38) is

carried out we obtain the equivalent Mueller matrix evaluated at the angle <j>. That is,

25
L ' = OD = OA + AD = OA + BC

= L 'cos2(j) + L «sin2q

L

2

' = DP = CP - DC = CP - AB

= L 'COs2<f> - L «sin2c

Figure 6.2. The components of £ in the 1^, L

2 and L * L

' coordinate systems.

26
T«>> = M(-*)-T(o)-M<*)

oo

T10

C T 20

S

T10* S + T 20- C

"30

T r

C " T02' S T

0l"

S + T02* C

Tn

-A

T21 + B

12

T22+ A

T

3l'

C "

T 32* S T

3l'

S + T 32* C

"03

T13* C "' T23* S

T13

.S + T23

.C

33

(6.39)

where A = (T - T^-S2 + (T

2 + T^-S-C

B = (j - T

22 )-S-C - (T + T 21 )-S 2

S = sin2cj>

C = cos2<f>

and Tnn » Tm ' T

n?' etc «> are taken to be the matrix elements [eq. (6.10)] at <}> = 0. Thus, in their dependence upon the orientation angle $, all Mueller matrices must

resemble the matrix product of eq. (6.39). This multiplication has already been performed

for most of the matrices we discuss in this chapter and accounts for the angular orientation dependence built into them. Many Mueller matrices in the literature, however, are written only for a special angle, usually <j> = 0, and require the transformation of

eq. (6.39) before they can be used at other orientations.

In addition to being useful in providing the dependence of the transmittance of

optical components upon orientation, the rotation matrix M(<j)) of eq. (6.37) is the

Mueller transmittance matrix for an ideal circular retarder . Circular retarders are made

of materials which exhibit the property called optical activity or circular birefringence . Crystal quartz and solutions of dextrose are examples of such materials. Linearly polarized light entering a circular retarder will emerge with its direction of polarization

rotated by the angle $ appearing in ^\(<$>) . This angle is the circular retardance and is

related to the wavelength X and the thickness a of the material by

<p = -~- (nr

- n

£

) [rad]

[compare with eq. (6.19)] where n is the refractive index for right-handed circularly

polarized light at wavelength A and n is for left-handed. For solutions of an optically active material n - n. will be proportional to the concentration of solute so that

measurements of the angular rotation of the direction of polarization for a fixed path

length provide a convenient means of measuring solute concentrations. This technique has

been developed to a high degree of accuracy in instruments called saccharimeters or polarimeters.

27
A different kind of rotation problem arises if we have the Mueller matrix elements of an optical component or train of components and ask what happens if the radiation passes

in the opposite direction through the system. How does the Mueller matrix change? The answer is that for an original matrix of elements T.. given by eq. (6.10) the matrix for

the reversed optical system is

(r)

00

01

02

03

'10 -T

20 '30

l ll-T

21 '31 -T

12 22 -T

"13 -T

23

32

33

(6.40)

the rows and columns have been interchanged 1 and the signs of the matrix elements are changed once for every subscript 2 which appears. The interchange of rows and columns is

understandable because the individual columns operate on the individual components of the

Stokes vector of the input radiation while the rows generate the components of the Stokes

vector of the output radiation. So if we interchange the roles of input and output it

isn't surprising that the roles of the rows and columns must be interchanged too. The

change of sign associated with the subscript 2 is caused by the transformation of the +45°

polarization component into its complementary +135° component when an object is reversed. 2 Imagine a plate containing milled slots in horizontal, vertical, and 45° directions and

also a hole tapped for a right-handed screw. Now if the plate is viewed from the back the

horizontal and vertical slots and the tapped hole will appear unchanged but the 45° slot

will look like a 135° slot. Thus, the +45° and +135° directions change roles upon reversal, and this requires the sign changes shown in eq. (6.40).

REAL OPTICAL COMPONENTS

So far in this chapter we have seen that, if we have a source of unpolarized light, a polarization-indifferent detector, a perfect linear polarizer, and a perfect quarter-wave

plate, we can determine the state of polarization of any arbitrary ray and by varying the

polarization of such a ray we can use it to characterize or calibrate the response of an instrument to polarized light. Unfortunately real polarizers and quarter-wave plates

exhibit Mueller transmittance matrices which differ somewhat from those given in

*If the rows and columns of a matrix are interchanged the result is known as the transpose

of the original matrix.

2 It is not obvious that the relationship between the matrix for the combination of a polarizer followed by a quarter-wave plate in eq. (6.35) and its reverse in eq. (6.22) satisfies

eq. (6.40) This is because the angles used in eq. (6.35) are the supplements of those in

eq. (6.22); when an optical component at orientation <f> is reversed it will find itself at

orientation (180° - <j>) . If the matrices in eq. (6.35) and (6.22) are both expressed in

terms of the same angles it will be found that eq. (6.40) is obeyed.

28
eqs. (6.15) and (6.21). High quality crystal polarizers and retarders will probably

approach the matrices of eqs. (6.15) and (6.20) closely except for the effects of surface

reflections. So, for many radiometric applications, where polarization complications are

no more than small corrections to measurements often made with complete neglect of polarization, such optical elements can be used and the Mueller matrices of eqs. (6.15) and (6.20)

will be adequate to account for their behavior (to within 10% or so)

.

Regardless of whether a polarizer and retarder are perfect, if their Mueller matrices

are known the Stokes components of a ray can still generally be measured essentially as described in connection with eq. (6.24). Four or more combinations of angular settings of

the polarizer and retarder are needed as before. The difference is that the Mueller matrix

of the combination of imperfect elements will differ from eq. (6.22) so that eq. (6.24)

will not be valid. However, the analogue to eq . (6.24) will still be linear in the com- ponents L„, L.. , L„, and L and the four measurements at suitable angular settings

will permit these components to be evaluated. In the simplest case the retarder will not

be exactly a quarter-wave plate due, for example, to its use at a wavelength other than the one for which it was designed. For this case eq. (6.24) becomes [using eq. (6.20) in (6.22)

instead of (6.21)]:

{ L + L • [cos 2a*cos 2(B-a) - sin 2a*sin 2 (6-a) # cos6]

+ L -[sin 2a*cos 2(g-a) + cos 2a*sin 2(g-a)'cos6] (6.41)

+ L • [sin 2(3-a)-sin6]|

Obviously if 6 is known, from eq. (6.19) for example, this equation is no more difficult

to work with than eq. (6.24).

The real problem arises when one doesn't know the Mueller matrices of his polarizers

or retarders and has no sources of polarized light with known Stokes components which can be used to calibrate them. One can, with considerable confidence, produce unpolarized

light and a polarization- indifferent detector (using diffusers, such as integrating

spheres [13], for instance). With such a source and detector, one can determine only

element T of the Mueller matrix of an optical component or combination of components.

However, as we have seen in eq. (6.41), for example, when polarizing components are com- bined the transmittance to unpolarized radiation of the combination depends upon additional

matrix elements of the individual components. Thus, we might expect that measurements of

the transmittances of all combinations of several polarizing components as a function of

their orientation angles can yield considerable information about the matrix elements of

the individual components. Using this idea we show in Appendix 7 how almost any Mueller

matrix can be measured using nothing more than an unpolarized light source and a polarization-indifferent detector. In the following paragraphs we show how the same technique can

be applied to measure a simple matrix which approximately describes the common Polaroid

type of polarizer.

29
Our procedure simultaneously measures three linear-polarizing filters. 1 We assume

that each is mounted in such a way that it can be reproducibly inserted and removed from

the unpolarized measurement radiation beam and that it can be rotated about the beam axis,

with its angular orientation read from a scale fixed to the filter mount. Real polarizers,

especially sheet polarizers, can be expected to exhibit a small amount of retardance.

Likewise, real retarders may be slightly linearly polarizing. Therefore, for the Mueller

transmittance matrix of such real components we assume a form which is general enough to include both functions:

F(*) =

s d«C d-S

d-C s-C 2 + p-S 2 (s-p)-S-C -q-S

d-S (s-p)-S-C s-S 2 + p-C 2 q-C

q-S -q«C p

(6.42)

In this expression

the filter mount. C = cos2(tj)-<t> ) and S = sin2(cf>-<f) ), where cf>

is the orientation of the filter as read from the scale attached to

is the orientation of

the polarization axis of the polarizing filter. The angle <J>~ and the parameters s, d,

p, and q are to be determined by experiment. The form of FC^) > °f course, includes the

<j> -dependence of eq. (6.39).

The parameters s, d, p, and q represent a model for dichroic polarizers of the

Polaroid type with allowance for some retarder behavior. 2 In this model

s = h(K + kj

Vv k *cosd

where the retardance of the filter is 6 principle transmittances of the filter. 3

s(k, - kj

V

k *k "sin6,

and the quantities k and

(6.43)

k are known as the

The maximum transmittance to linearly polarized

: and the minimum transmit tanc

90°) is k . The relationships (6.43) imply that

light is k and the minimum transmittance (when the direction of polarization is rotated

q

2 = s2 - d

2 - p2

. (6.44)

We use the word "filter" here and in the rest of this chapter in a general sense to refer

to any optical element which may modify the distribution of radiation with respect to any

parameter, including polarization.

2 Equation (6.42) can also be used to describe the polarizing effects of reflecting surfaces

[4,14]. If the coordinate system described in the footnote on page 9 is used then

, o

3 Nominal values of k.. and k„ have been published [2] for the common Polaroid polarizers,

30
Thus we really have only three parameters to measure — say s, d, and p — in order to define the Mueller matrix p completely. Note that the matrix includes, as special cases, both perfect polarizers (k = 1, k = 0) and perfect retarders (k =1, k = l) . It also provides for the most important imperfections encountered in real linear polarizers

and retarders.

We introduce the notation F(<j>) to indicate the result of a measurement with our polarization-indifferent detector of the transmittance to unpolarized radiation of filter

F when it is turned to an orientation angle d> on its mount. Likewise GF(d> ,*_) is the

g f measured transmittance of the pair of filters F and G at angular settings of <j> and

<() , respectively. Our notation also indicates the direction of radiation travel — from

right to left through the symbols; i.e. through filter F first, then through filter G

to the detector.

We first measure the transmittances F(d>_), G (<i> ), and H(<i, ) of the three individual

r g h

filters. From eq. (6.42) we see that these are given by

F(+

f

) s

f

, G ) = s , and H (<j>, ) = s,

; s h h

where the subscripts on s identify the filter to which it pertains. This result should

be independent of i> , i> , and d>, , so, in principle, a measurement at any orientation

f g h

should suffice. However, a real filter will probably have small inhomogeneities which may

move into and out of the beam as the filter rotates. In this case the best that can be

done is to obtain an average value of s by sampling many orientations <£. These averages

of F((|> ) , etc., we will simply write as F, G, and H. Thus,

s_ = F, s = G, and s, = H

f g h (6.45)

We now measure the transmittance FH(<J> f

,<j> ) of a pair of the filters. From eq. (6.42) we have

F(* f)*H<*h

> =

[ s

f*

s h + Wc fh]

LVVcf + VVVc fh + VVVs

f h ] [Ws

f + s f-VVc fh ~ Pf-VV s

f h ] [Ws fh]

(6.46)

where we have shown only the first column of the product transmittance matrix because it

turns out that this is all that enters into the final expressions. We have also used the

shorthand notation

C fh

= cos2[(cD

f - d, f°) - Uh

- c)> h

)] and

S,, = sin2[U f - 4>°) - ( fh f

Y

f

K hT h

°>]

31
The transmittance FH(d),.,<f), ) is just the (0,0) element of this matrix or:

t n

ra<*f.*h> = S f'

S h + WC fh

By expanding C,., we can rewrite this as

1 h

FH< > fl <j> h

) = FH

X + FH2 -cos2(<t>f

|> h

) + FH

3 .sin2(4>

f - <j> h

) (6.47)

where FH

1 " S f'

Sh

FH2 = d

f -d

h 'cos2(*

f° - 4» h

°)

and FH

3 = d

f .d

h .sin2(*

f° -*

h

°)

If we measure FH ($ ,<j> ) at two or more sets of angles <J> and <f> we will be able to solve the resulting set of equations (6.47) — either by the usual methods for simultaneous

equations or, better, by least squares fitting — for the coefficients FH and FH . FH of course, is already known from eq. (6.45). 1 In this way the orientation-independent coefficients FH and FH are measured. From these we obtain

tan2(<

^h > = FH

3 /FH

2

and V V

d h

=: V

FH

2

2 +FH

3

2

(6.48)

(6.49)

By repeating this process for the other two possible ways of pairing the three filters we obtain the other polarization-axis angle differences and the other d*d products like

eq. (6.49). The three equations of the form (6.49) can be further solved for the individual values of d:

-i u

d

f =

d =

dh

=

2 2\ / 2 — 2 GH

2 + GH

3 j-(^FH2 + FH3 — 2 — 2 HG

2 +.HG

3

2 — 2 HG

2 + HG

3

GF

2

2 + GF

3

2

• 2 — 2 GF

2 + GF

3 FH

2

2 + FH

3

2

FH

2

2 + FH

3

2J-(hG22 + HG

3

2

(6.50)

If FH is determined from eqs. (6.47) and does not agree with F»H [eq. (6.45)] then

the model of eq. (6.42) may be inadequate. Most likely some of the (0,3) or elements don't vanish as assumed in eq. (6.42).

32

(3,0)
So far our measurements have given us the orientation of the three filter polarization

axis directions <J> with respect to one another but have not revealed the absolute orientation of any one of them. This is obviously because there is no unique transverse reference

direction associated with our unpolarized measurement beam. The absolute polarization axis

orientation can be determined, if desired, by passing a beam known to be partially horizontally linearly polarized through one of the filters and rotating the filter for maximum

transmission. At this orientation <}> = <J> which then serves to determine the <\> values

of the other two filters from eqs. (6.48). A suitable source of partially horizontally

polarized light is light reflected from a glass surface oriented so that the plane of

incidence (the plane defined by the incident and reflected rays) is vertical.

The final measurements are of the three-filter transmittances HGF(<j> ,<J> ,$> f ). From

eqs. (6.46) and (6.42) we find, after some simplification, that the (0,0) element of the

three-filter product matrix is:

HGF(<j>, ,(f> ,((>,,) = s -s >s c + s -d -d -C c + d-d -s -Q

h g f hgf hgfgf hgfhg

+ d «s -d «C C

'C, - d «p «d «S C'S, . h g f gf hg h

r g f gf hg

(6.51)

As before C = cos2 [(<J> - cf> °) - (<j>. - <j> °)] = cos2 [(<f> - $ ) - (<J> ° - <j> °)] , etc. Everything in this equation is now known [from eqs. (6.45), (6.48), and (6.49)] except p , so

p can be calculated directly:

P„ =

+ G

H-G-F - HGF(<h ,<(> ,«j) ) + H-(GF„

2 + GF2

)

2 -C + F-(HG

2 + HG.

2 )^-C,

h g i I 3 g_f 2 J hg

2 — 2 k

(FH„ + FH_ ) -S -S

2 3 gf hg

gf hg

S f' S U ' gf hg

(6.52)

Again this result should be averaged over many settings of 4> , (f> , and (j> in order to minimize errors due to filter inhomogeneities. Similar three-filter measurements with

filters F and H in the middle will permit p

f and p to be determined. Finally the

values of q can be calculated 1 from eq. (6.44) and with that the three matrices corresponding to eq. (6.42) are completely evaluated.

The algebraic sign of q is not determined by the procedure we have outlined here. In many applications this will be of no consequence. The ambiguity is, in part, due to the

arbitrariness in the definition of circular handedness. The sign of q can be determined

if a beam containing circularly polarized light of specified handedness is passed through

the filter and the output is analyzed with a linear polarizer. See also the discussion of

these sign ambiguities in Appendix 7.

33
The procedure we have outlined applies only to filters which are at least partial

linear polarizers. This is because the right-hand side of eq. (6.48) can become indeterminate (0/0) for a pure retarder or a simple attenuating spectral filter. Any such

component can, however, be measured by putting it in the middle of a three-filter train

between two polarizing filters which have already been characterized. Equation (6.51) is

still valid and the symbol G now refers to the unknown filter. The quantity s can be

obtained by the single filter transmittance measurement as before. This leaves d , p , o o and <j> (appearing in C

f

, etc.) to be determined. The angle $ can be established

g

o

g

o 8 by setting <j> = <j>, + <j> f - <j> (i-e., the polarization axes of F and H are made

parallel to each other) and rotating the unknown filter G until a maximum transmittance

is obtained. With the polarization axes of F and H parallel to each other the transmittance is

HGF(<f>. ,4> ,<fr. + <J> ° - <f>,°) Y = s, -s -s^ + d, -p -d, + (s -d. + d, •s_) »d -C. h' Y g

T h

T fT h h g f h

r g f h f h f g hg

+ d, 'd.-(s - p )-C 2 h f g g hg

(6.53)

This expression has a maximum at d> = <f>, + d> - 6, . 1 This serves to determine the angle

g h g h

(j) , which once again should be averaged for many settings of d>, . Once d> is known the

g h g other parameters of filter G can easily be obtained by making many measurements and using

linear-least-squares fitting in eq. (6.51) or by making measurements at two suitable angles,

4> , and solving the resulting two equations given by eq . (6.53) for the unknowns d and

p . q then follows as before from eq. (6.44). If the transmittance, eq. (6.53),

exhibits no significant dependence on <t>, - <t> after averaging over d>, then d =0,

h g h g and p = s . Such a filter is just an attenuating spectral filter.

Needless to say, with real measurements on real filters some uncertainties in

determining s, d, p, and q can be expected. For example, if 1% measurements are being

made of a good linear polarizer the uncertainties in s, d, p, and q will all be

similar and of the order of s/100. But since p ^ q ^ the relative uncertainty in these

two parameters may appear alarmingly large. Fortunately, the relative contribution to any

final instrument response from a single matrix element T . . of an optical component cannot

be larger 2 than the fraction |t../T„_|. Thus, in the example of the 1% measurements of a ° ' ij 00

linear polarizer the uncertainties in p and q can never contribute more than about a

1% uncertainty to any measurement made using this polarizer.

If d is small but not zero there may be a lesser maximum 90° removed from this angle.

2 This is true if the ultimate instrument detector responds to a property of the electromagnetic radiation such as power, linear momentum, or photon number, but need not be true

if it responds to a property characteristic of a particular polarization state such as angular momentum.

34
Once the Mueller matrix elements have been determined for two real linear polarizers

and two approximate quarter-wave plates the polarizer and wave-plate pairs can be used to generate and analyze a wide range of polarization states of light. And this in turn makes

possible the characterization of more complicated Mueller matrices — in particular those

of circular polarizers, which are not included in our general imperfect filter expression

p in eq. (6.42) because we have set F = F . =0. We thus see how we can start with

real, inexpensive, polarizing filters and build up a complete polarization measurement

capability.

POLARIZER-ATTENUATORS

A three-polarizer train is sometimes used as an absolute or predictable variable optical attenuator [15,16]. The first and last polarizers of this train are aligned with their

polarization axes parallel to each other and the middle element is rotated to achieve the

desired degree of attenuation. Equation (6.53) describes just this arrangement. The trans- mittance of the train relative to its maximum transmittance is therefore:

HGF(cJ>°,<j> <t>°) t = —-—S = [ S . s .- + d, -d.-p + (s,-d£ + d -s,).d -C, o,o, o x HGF((|)

L hgf h f *g hf hfghj

h

,(J. g

,<j) f

)

+ d, -d.-(s - p )'C, 2

] /[s, -s -s^ + s -d -d^ + d -s -d^ hf v g Vhghgf hgf hgf (6.54)

+ d-d -sJ

h g f

his expression for the relative transmittance, of course, is valid only for unpolarized

light and a polarization- indifferent detector. If the polarizers are all perfect linear

polarizers we have s^ P, = s = s, = 1, d,. = d = d, = 1, and p,. f g h f g h

v

l = d, =0. Then ' h

1 + 2C + C = cos (c ') •

If the outer polarizers are perfect and the middle component is a perfect half-wave plate

instead of a polarizer the only change is that d =0 and p = -1. This results in

2C

-£- = cos 2

2(<J) - <j> °)

In both these cases the light emerging from the attenuator is always completely linearly

polarized in the direction determined by the final polarizer. The relative transmittance

depends only on the easily measured angle <J> . g Unfortunately, with real optical elements the values of s, d, and p must be known

if the transmittance is to be calculated accurately as a function of <j> from eq. (6.54).

Moreover, the state of polarization of radiation emerging from the attenuator will depend

35
weakly upon the angle $ and upon the state of polarization of the entering radiation.

Thus, the Mueller matrix of the rest of the optical system must be known and additional

matrix elements of the attenuator must be evaluated if the effective relative transmittance

is to be calculated exactly.

DEPOLARIZERS

We briefly mentioned depolarized radiation in the early part of this chapter and

would now like to say a few words about the means of achieving this. The Mueller transmittance matrix for a depolarizer is obviously proportional to 10

(6>5M

Unfortunately no single optical element exhibits this property. However, if we consider a complete measurement situation with a source radiance vector |_, an optical component with

Mueller matrix [), and a radiometer with responsivity matrix ^P , with the elements

(responsivity coefficients) R^j Rni , etc., we have for the radiometer output signal:

S = At AX AA Aw /^•D-L ,aw ' cose ' dA ' dX ' dt (6.56)

where Aio, AA, AA, and At are the resolution intervals of the radiometer in these

parameters. (As in our discussion of the calibration of a radiometer, we actually use only

the upper row of the responsivity matrix /^ to obtain the output signal S.) If the

responsivity of the radiometer and the Stokes spectral radiance vectors of the source are sufficiently uniform over the radiometer resolution intervals, they can be removed from

this integral to give

S - £• At AA AA

D'du'cosfi 'dA'dA-dt

AcoAfAA'AA'AO

= /R- <D ># L' At ' AA ' AA ' A0

•L'Af AA-AA'AG

(6.57)

where

<D> =

<D00

> ^Ol* <D02

> <D

03

>

<D

1Q

> <DU

> <D

12

> <D

13

>

<D

2Q

> <D n> <D

22

> <D

23

>

<D

3() > <D31

> <D

32

> <D

33

>

36
and <D . . > = ————

lj At-AA'AA'AG

At AA J AA .'.w

D. •dorcos6'dA«dA'dt. Now we see that if all <D .

.

except <Dnn

> can be made to vanish, the Mueller matrix <Q> will become 00

<D>

<D > 00

(6.58)

which is the form for a depolarizer. Most depolarizers consist of two retarders [eq. (6.20)]

placed one behind the other. The (0,0) element of the resulting product matrix is 1 and

all the other elements are zero or are trigonometric functions of the <£ ' s and 6's of the

two retarders. By carefully choosing the <j> values (or the 6 values) and averaging over

a range of 6's (or <j>'s) these trigonometric matrix elements can be made negligibly

small. The 6's (or <j>'s), over which the device averages, are arranged to vary in time,

wavelength, position or direction. Depolarizers which depolarize by averaging over wavelength (^100 nm) [17] or over time (^1 sec) [17] or over the cross-sectional area of the

beam (^1 cm2

) [3] have been used and will be described more completely in Part II. Depolarizers are frequently known as pseudo-depolarizers or as polarization scramblers .

A SIMPLIFIED CHARACTERIZATION of POLARIZATION

Many radiometric measurements involving polarized light are carried out using just two measurements — one with a linear polarizer set in a horizontal orientation in front of the

detector and the second with the polarizer turned 90°. Let us see what the limitations

of such measurements are and under what conditions they will be sufficient to characterize

a light ray. In order not to complicate matters unduly, let us assume that the polarizer

is perfect so that its Mueller transmittance matrix in the horizontal orientation is:

110 110

P(0°) =

and in the vertical orientation:

P(90°) = 1-10

-110

We will also assume that the detector is polarization indifferent or has been compensated

or calibrated for horizontally and vertically polarized radiation. When an arbitrarily

polarized ray with Stokes vector

37
passes through the linear polarizer in the two orientations, we have

L*(o°) = P(o°).[_ =

L

o + L

i L

o + Li

and

L'(90°) = P(90°)-L =

Lo" L

i "L

o + Li

The detector will record signals proportional to L '(0°) = L_ + L and L ' (90°) = L~ - L, , respectively. Thus the original components L and L can be calculated from

%[L'(0°) + L ' (90°)] and %[L ' (0°) - L ' (90°)]. If we can assume that L then

the light beam has been completely characterized. Usually L , the spectral radiance, and

the degree of polarization [eq. (6.6)]

P = L '(0°) - L '(90°)

L

Q L '(0°) + L '(90°)

are the quantities reported. Thus we see that the two-measurement description of polarized

light is limited to those situations in which (as we could have guessed) there is no 45°

(or 135°) linearly polarized component and no circularly polarized component. The elimination of the +45°-preference component can always be arranged by suitable choice of the

reference "horizontal" direction. Note, however, that unless one has some a priori knowledge

of this direction this entails the equivalent of at least one additional measurement to find

this direction and there is thus no real saving in attempting to eliminate this component.

In many situations, such as scattering studies of isotropic samples irradiated, first, by

radiation polarized in the plane of incidence and then by radiation polarized perpendicular

to the plane of incidence, a reference direction parallel to or perpendicular to the plane

of incidence can be taken and the +45°-preference component may then be safely neglected

(but only if the sample is isotropic) . The neglect of the circularly polarized component

is probably usually justified in most radiometric measurements where uncertainties of a

38
percent or so are tolerable. For the highest accuracies circular polarization must be

included, however, because it can creep in whenever partially polarized light at non-normal

incidence is reflected from mirrors or transmitted through films. Thus, any but the

simplest instruments will be likely to exhibit some circular-polarization selectivity.

When circular polarization can be neglected most of the discussions of this chapter

can be greatly simplified. For example, no measurements involving retarders such as quarter-wave plates need be considered; sources, optical components and radiometers can be

completely characterized using only linear polarizers. Also the procedure outlined in

eqs. (6.42) to (6.53) for measuring the Mueller matrix of a polarizer can be simplified by

omitting the three-filter measurements because we can now set q = and p

2 = s2 - d2

for each filter.

EXPERIMENTAL CONCERNS

We have tried in this chapter to present a complete, exact treatment of polarization

and an indication of how this treatment can be applied in the laboratory. We have not

mentioned the experimental problems which make high accuracy polarization measurements

difficult — these properly belong in later chapters where we deal with specific radiometric

instruments. However, some brief mention of some of these problems seems appropriate here.

The proper experimental inclusion of polarization in measuring instrumentation, as we have

seen, requires the insertion into the optical path of one, two, or three polarizing filters

of one kind or another. These must be rotatable and it must be possible to measure the

angles of rotation, or at least to set each filter reproducibly at several preselected

angular positions. Unless the filters are absolutely uniform there will be a dependence of

filter transmittance on angular orientation which may be unrelated to the polarization

property being investigated. For anything more than qualitative observations it is therefore important that polarization measurements be carried out and suitably averaged over many orientations of the polarizing components. Another experimental difficulty common to all measurements using multiple filters in tandem but which is exacerbated with polarizing

filters is that of filter-filter interref lections . Such interref lections have the effect

of increasing the apparent transmittance of the filter train and thereby introducing errors.

In the case of polarizing filters this problem can be eliminated by the usual technique of

tilting the filters, but the axis of rotation of the filters must remain parallel to the

beam axis. Thus, the filter must wobble as it is rotated. If a tilted polarizing filter

were rotated about an axis perpendicular to its face, then entering radiation would be

partially linearly polarized at a fixed orientation by the tilted first surface, resulting

in observations of partial polarization even of unpolarized light. If the tilt rotates

with the filter, however, then this polarizing effect becomes simply an integral property

of the filter and will be accounted for in the measured Mueller transmittance matrix elements. Of course, the filter must always be used at the same tilt orientation. Another

complication which may require experimental attention is that the lateral displacement of

the beam as it passes through the tilted filter now rotates with the filter.

39
SUMMARY of CHAPTER 6 . Polarization can be included rigorously in classical radiometry (geometrical optics)

by treating spectral radiance as a four-component mathematical column vector called a Stokes vector

Lx -

X,0

Vi

L A,2

L

l

a,3

(6.8a)

The four components of |_ are known as Stokes components . The Stokes vector and its

components, in general, depend upon ray position and direction and upon wavelength, i.e.,

upon x,y,0,(j>, and A. The first Stokes component L n is the spectral radiance which a

A , (J polarization- indifferent detector would measure. It is the measure of the spectral radiant

flux associated with the ray and is identical to the spectral radiance discussed in earlier

chapters. Each of the other three components of |_ represents the excess of one form of

A polarization over its complementary form. L ., describes the excess of horizontal over

A,i

vertical linear polarization, L „ describes the excess of +45° over +135° linear polarA,Z

ization and L _ describes the excess of right over left circular polarization. The

A, J spectral radiance L n , is the sum of two parts, a polarized part, L , given by

A , U A ,p

,p 1 A,. + L

A,

2

+ L

A,

3

(6.5)

and an unpolarized part

The fraction

A ,u A ,0 A ,p

A,p A,0

(6.6)

is known as the degree of polarization.

In this description of polarized radiation the transmittance or propagance of an optical path is represented by a 4 x 4 matrix known as a Mueller matrix. For example

(compare eq. (3.14) [1])

l» - t-l, (6.9a)

relates the spectral radiance |_ at one point on a ray and the propagance ) along the

A path between that point and a second point to the spectral radiance |_ ' at the second

A point. The Mueller propagance matrix in this equation is given by

40
T-

"00

30

T T T 01 02 03

T T T T

10 11 12 13

T T T T 20 21 22 23

T T T 31 32 33

(6.10)

so that eq. (6.9a) is equivalent to

LJ

T00' L + T 01- L

l + T02' L2 + T03' L

3

L

l " T 10- L + T11' L

1 + T 12- L2 + T 13' L

3

L

2

= T 20' L + T

2l' L

l + T 22* L2 + T23* L

3

(6.7)

L

3 T 30' L + T

3l' Ll + T 32' L2 + T 33* L

3

where we have dropped the subscript A for simplicity.

When polarization is included, the spectral radiance of a ray of radiation transmitted

by a train of optical components depends upon the order in which the propagating flux in

the ray encounters the components. This is reflected in matrix equations by the order of

writing the Mueller matrices and the Stokes vectors. The entering Stokes vector is written

on the right and the Mueller matrices are written from right to left in the order in which

the optical components they represent are encountered. The resulting matrix equations can then be evaluated by the usual matrix multiplication rules (Appendix 6)

.

The Mueller transmittance matrix of an ideal linear polarizer is given by

P(4>) = h

1 c s

c c

2 s-c

s s-c s

2 (6.15a)

where C = cos2<J>, S = sin2<{>, and <j> is the angle between the polarization axis of the

polarizer and the horizontal polarization reference axis of the ray. When a ray of arbitrary spectral radiance is passed through such a polarizer the polarization state of the

transmitted ray can be predicted by performing the matrix multiplication indicated in

eq. (6.9a):

|_i = P(<|>).|_ = P(<

_ 3_

k(.L

Q + L

1 -C + L

2 «S) (6.17)

41
The transmitted ray has a spectral radiance of ^(L. + L -cos2<}> + L «sin2<J>). It has no net

circularly polarized component (L ' = 0) but is completely linearly polarized at the angle

<j>, [P = 1, from eq. (6.6)] .

The Mueller transmittance matrix of an ideal linear retarder is

Res.*) =

1

C

2 + S2 -cos6 S-C- (1 - cos6) -S-sin6

S-C- (1 - cos6) S

2 + C2 -cos6 C-sin6

S-sin6 -C-sin6 cos6

(6.20)

where C = cos2cj>, S = sin2cf>, and <f> is the angle between the fast axis of the retarder

and the horizontal reference axis of the ray. 6 is the retardance of the device and is

proportional to the thickness of the retarder plate. If 6 = it/2 [rad] the retarder is

called a quarter-wave plate , which has a Mueller transmittance matrix

QUO =

1

c

2 S-C -s S-C s

2 c

s -c

(6.21)

The combination of an ideal quarter-wave plate and an ideal linear polarizer can be

used either to determine the state of polarization of an arbitrarily polarized ray or to produce any given state of polarization. If the radiation passes first through a rotatable

quarter wave plate at angle a and then through a fixed polarizer at angle 3 the transmitted ray is given by

L' = P(3)-Q(a;

J o

L

L 3J

= h[L + L • cos2a«cos2(8-a) + L •sin2a«cos2($-a) + L •sin2($-a)]

(6.23)

1 cos23

sin2g

If the spectral radiance of the transmitted ray is measured at four suitable orientations

a of the quarter-wave plate the four simultaneous linear equations of the form of

eq. (6.23) can be solved for the Stokes components L , L.. , L„ , and L„ of the original

ray. Since the transmitted beam is always completely linearly polarized at the fixed

orientation 3, the measurement can, in principle, be carried out by a detector of any

arbitrary polarization selectivity. For absolute measurements, however, a calibration of

the detector will be required using (usually) a differently polarized source. In this case

it will be simpler if the detector is polarization-indifferent so that the spectral-radiance,

42
Ln

' , is the quantity measured,

If the radiation passes first through the polarizer and then through the quarter-wave

plate, the transmitted ray will be described by

|_' = Q(a) -P(3) h(L

Q + L

1 -cos23 + L2 -sin2g)

cos2a*cos2(a-$)

sin2cccos2(a-3)

sin2(a-3)

(6.35)

By suitable choice of the orientation angles of the quarter -wave plate and polarizer any

pure polarization state from circular (|a-g| = 45°) to linear (a-3 = 0°) can be realized,

The final electrical, thermal, or chemical response, S, of a radiometer depends upon

all four Stokes components of the entering radiance:

S = AA J AA Aw

(R

Q0

-L + Rq-^I^ + Rq

2 -L

2 + R03

-L

3 )-dco-cos6-dA-dA, (6.36)

The quantities AA, AA, and Aoj are the acceptance intervals of the radiometer. If these

acceptance intervals are sufficiently small that the Stokes spectral-radiance components

can be assumed constant and removed from the integrals in eq. (6.36) then we can write

S= *00' L +/?01- L

l +i?02- L

2 +/?03- L 3- (6.36a)

The values of the responsivity factors R can be determined experimentally from observations of the radiometer output for a beam in four different, known states of polarization

such as can be obtained from the polarizer and quarter-wave plate combination described by

eq. (6.35). If the first optical component of the radiometer is a perfect diffuser,

eq. (6.36) reduces to

S = R

where

00

AA AAJ Ago

(6.36c)

L 'dwcose *dA*dA is the total radiant flux falling on the diffuser

within the wavelength band pass of the radiometer. If neither of these simplifications is

appropriate the dependence of the coefficients R upon ray position, direction and wavelength can, in principle, be determined by probing the radiometer with rays of known polarization state. With these functions known, then, a radiometer measurement of an unknown

beam provides one relationship among integrals of the four L.(x,y,0,<j>,A) functions. If

the use of the radiometer is restricted to comparisons of beams with identical relative

spectral-radiance distributions then such detailed knowledge of its responsivity is unnecessary and four measurements of different known polarization states will characterize it

for this purpose.

43
The ideal Mueller transmlttance matrices of eqs . (6.15) and (6.20) are only approximations to the transmlttance matrices of real optical components. Many real linear

polarizers and retarders can, however, be adequately represented by Mueller matrices of the

form

F(+) = s-C 2 + p-S 2

s d-C

d-C

d-S (s-p)-S-C

q-S

d-S

(s-p)-S-C

s-S 2 + p-C 2

-q-C

-q-S

q-C

P

(6.42)

where

and

C = cos2(<})-cf> ) , S = sin2(<j>-c|> )

q

2 = s

2 - d2 - p2

. (6.44)

The values of s, d, and p as well as the filter polarization axis direction $ can be

determined experimentally by appropriate transmittance measurements. If no sources of

known variable polarization state are available it is possible to measure these parameters

with sufficient accuracy for most radiometric applications by making transmittance measurements in unpolarized light on three or more such objects as a function of their orientation

angles. Once the Mueller transmittance matrices of a polarizer and an approximate quarter- wave plate have been established they can be used for analyzing or generating a wide range

of arbitrary polarization states by generalizing eqs. (6.23) and (6.35).

44
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Billings, Bruce H. , "A Monochromatic Depolarizer", J. Opt. Soc. Am., 41 , No. 12, 966-975 (Dec. 1951).

Margenau, H. and Murphy, G. M. , "The Mathematics of Physics and Chemistry",

Vol. 1, 2nd ed. (D. Van Nostrand Co., Inc., New York, 1956).

Walther, A., "Radiometry and Coherence", J. Opt. Soc. Am. _58, No. 9, 1256-1259

(Sept. 1968).

45
[20] Walther, A., "Radiometry and Coherence", J. Opt. Soc . Am. 63_, No. 12, 1622-1623

(Dec. 1973).

[21] Marchand, E. W. and Wolf, E. , "Radiometry with Sources of any State of Coherence",

J. Opt. Soc. Am. 64, No. 9, 1219-1226 (Sept. 1974).

[22] Marchand, E. W. , and Wolf, E. , "Walther ' s Definitions of Generalized Radiance",

J. Opt. Soc. Am. 64, No. 9, 1273-1274 (Sept. 1974); and Walther, Adriaan, "Reply

to Marchand and Wolf", ibid., p. 1275.

46
Appendix 6. Matrix Multiplication

by John B. Shumaker

In this appendix we review the rules for matrix multiplication. We will confine our attention to the 4-component Stokes polarization vectors and the 4*4 Mueller matrices

encountered in the study of polarization in Chapter 6. For a more general treatment the

reader should consult mathematics or mathematical physics texts [7,18].

A Mueller matrix is a two-dimensional array of 16 numbers, the symbol for any one of

which may be labeled by two subscripts. The first subscript denotes the row and the second

the column of the element in question. We can write the matrix f\ in any of the following

equivalent ways:

A - [Arc

] =

A

00

A

01

A

02

A

03

A

10

A

ll

A

12

A

13

A A A A 20 21 22 23

L

A

30

AAA

31 32 33

(A6.1)

The product of two matrices /\ and $ is a new matrix

C = A-B

each of whose elements is computed by the formula:

(A6.2)

C rc

= 5>rk-Bkc , k=0

r ,c = 0, 1, 2, 3. (A6.3)

Unfortunately our page isn't wide enough to display the complete product matrix Q in one

block; however, if we may be permitted to break it between the second and third columns we

can show it all.

C

= [c ] = re

A00' B00 + A01' B 10 + A02- B 20 + A03- B 30

A10- B00+A11- B 10

+A12- B 20

+A13' B 30

A20' B00 + A

2l'

B10 + A

22

#B 20 + A

23

#B 30

L

A30- B

00. + A31' B10 + A32' B 20 + A33' B30

A00' B01 + A01' B 11 + A02- B 21 + A03' B 31

A10' B 01 + Air Bll + A12' B 21 + A13' B 31

A20' B 01 + A

2l'

B ll + A22- B 21 + A23' B 31

A30* B01 + A

3l' Bll + A32- B 21 + A33* B 31

(A6.4)

A00- B02- + A01' B 12 + A02- B 22 + A03' B 32

A10- B 02

+All' B12

+A12' B 22 +A13- B 32

A20' B02+A21- B 12

+A22- B

22 +A23- B32

A30* B

02 + A

3l' B12 + A32" B 22 + A33* B32

A00* B 03 + A01* B13 + A02* B 23 + A

03

#B

33

A10' B03 + A

ll'

B13 + A12- B 23 + A13* B 33

A20' B03 + A21' B13 + A22' B

23 + A23

' B33

A30' B 03 + A

3l'

B13 + A32* B 23 + A33* B 33

47
We may think of a Stokes vector

L - [Lr

] -

L

L 3J

as a matrix in which all columns after the first column are zero. Viewed this way

product of a Mueller matrix by a Stokes vector

(A6.5)

the

L' = A'L (A6.6)

follows the formula given above. Alternatively we may write a special formula for this

product as

-' = Ea

k=0

rk Lk

r = 0, 1, 2, 3

L : A00' L + A

0l' L

l + A02' L

2 + A03* L

3 L

l A10' L + A

ll' L

l + A12* L

2 + A13* L

3 L

2 A20* L + A2l' L

l + A22' L

2 + A23* L

3 L

3 " A30* L + A31' L1 + A32' L

2 + A

3 3* L

3

(A6.7)

The meaning of the product of a matrix or vector by a scalar [which is any non-matrix,

non-vector quantity such as \, t(A), or ^(L„ + L *cos23 + L 'Sin2g)] is that each

element of the matrix or vector must be multiplied by the scalar. For example,

•An = teA ] rc

00

JsA10

1 hk

02 ^ A

03 ^ A

ll

4l

hA

±2

hAu

^ A

20 '

'

hA

3Q

-2 A

22

%A

23

%A

31

35A32

%A „

(A6.8)

Similarly,

t(X)-|_ =

x(X)-L

tW-Lj

t(X)-L,

t(X)-L

3 _

(A6.9)

48
Appendix 7. The Measurement of Mueller Transmittance Matrices

by John B. Shumaker

In this appendix we show how Mueller filter transmittance matrices consisting of sixteen arbitrary matrix elements may be determined experimentally using an unpolarized source

of radiation and a polarization-indifferent detector. The technique consists essentially

of making unpolarized transmittance measurements of polarization filters singly, in pairs,

and in triples, as a function of the angular orientations of the filters. In practice, in

the presence of experimental measurement noise, difficulties may arise in the numerical

solution of the equations which follow, but we ignore such problems here. The purpose of

this appendix is merely to demonstrate in principle the measurability of an arbitrary

Mueller matrix using only an unpolarized source and a polarization-indifferent radiometer

with no previously characterized polarizing optical components.

The procedure simultaneously measures three polarization filters. We assume that each

is mounted in such a way that it can be reproducibly inserted into and removed from the

measurement radiation beam and that it can be rotated about the beam axis with its angular

orientation read from a scale fixed to the filter mount. If the filters are labeled F, G,

and H we take their Mueller transmittance matrices to be perfectly general:

F(o°) =

6(o°) =

00 01 02 03

F F F F 10 11 12 13

P "C1 "C1 "C1 20 21 22 23

F F F F

30 31 32 33

G

00

G01G02 G

03

G

10

Gll

G

12

G13

G

20 G21

G

22 G23

G

30

G

31 G32 G33

(A7.1)

and similarly for H. Their dependence upon the orientation angles <f> , <J> , and <}>, is

given by eq. (6.39). These angles are just the orientation angles read from the scales

attached to the filter mounts. We make no assumption about the relationship between the

<j) = 0° directions (at which the matrices written above apply) and the polarization axes of

the filters, if any.

The simplest measurements to make are the transmittances of the individual filters as measured with our beam of unpolarized light and our polarization-indifferent detector. We

shall denote these measured transmittances by simply writing a bar over the filter label:

F, G, and H. These transmittances give us directly

F " F

oo' G = G

oo> and H = H00' (A7.2)

49
The next simplest set of measurements we can make is of the filters taken in pairs. Let us consider the pair F and H and pass the measurement beam through in the direction from F to H. The measured pair transmittance, which we will call HF, is just the (0,0) element

of the matrix product

H(*h

)

• F(* f

).

That is

TM (Hoo- Foo + WV + (Hoi'

Fio + ^-V-Sf + (Hoi'

F20 - "oz'V^hf (A7 ' 3)

where C, c = cos2(d), -d) £ ) , S, £ = sin2(d> -<fc.) , and the angular dependence comes from eq.

nr h r nr h r (6.39). If this measurement is made at a number of angular differences, cfv -^* the results

h f can be solved as simultaneous equations or least-squares fitted to obtain values for the

coefficients (appearing in parentheses) of the constant and cosine and sine factors in

eq. (A7.3). Let us call these quantities HF HF and HF :

ffil = K00* F00 + H03' F30

(A7 ' 4a)

^2 " Hor

F10 + H02' F20

(A7 ' 4b)

^3 = H01' F

20 " H02* F

10- (A7 ' 4C)

We can also reverse either filter and repeat the measurements. Using eq. (6.40) we see that we will get two more sets of equations:

^1 = H00' F00 + H03' F03

(A7 ' 5a)

52 = H

0l' F01 " H02' F02

(A7 ' 5b)

53 = -H0r F02- H02' F01

(A7 ' 5C)

and

^1 " H00' F00 + H30' F

30

(A7 ' 6a)

^2 = H10- F10 " H20- F

20

(A7 ' 6b)

^3 " H10' F

20 + H20- F10

(A7 ' 6C)

where we have used a prime (') to indicate a reversed filter. Using the other two pairs of

filters F,G and G,H we can obtain similar equations in products of F and G matrix

elements and products of G and H matrix elements. Since the (0,0) elements of p,

G, and H are known [eq. (A7.2)], eq. (A7.5a) and its analogs in FG and GH can be

solved for F , G ., and H _. Likewise from eq. (A7.6a) and its analogs F , G

Q

,

50
and H

n can be obtained. The result for filter G is

3

(FG ' - F-G)' (GH ' -G-H)

HF' - H-F

(A7.7a)

3

(F'G - F-G)- (G'H - G-H)

HTF

1 - H-F

(A7.7b)

The other solutions can be obtained by cyclic permutations of the symbols F, G, and H. Notice that when G (or G ) is calculated we may take either the positive or negative

sign for the square root. This ambiguity arises because we have not defined in our experiment what we mean by right-circular polarization. We are free to choose either direction

to be associated with a positive value for the L„ Stokes component. If consistency with

an independent standard of handedness is necessary then a filter or beam of known handedness is required. Once one sign has been chosen then the signs of all the other (0,3) and

(3,0) matrix elements are uniquely defined by equations (A7.4a), (A7.5a), (A7.6a) and their

analogs.

Equations (A7.5b) and (A7.5c) and the similar equations written for the other two filter pairs constitute six equations in six unknowns. The solution of this system for

filter G, for example, is

Yl+ Y; 1 HF

2

' 2 + HF ' 2 (A7.8a)

, 2 =

"0 2

Y1+ Y

2 HF„' 2 + HF ' 2 (A7.8b)

where

and

= [(GH' 2 + GH ' 2 )-(HF' 2 + HF ' 2 )'(FG„'

2 + FG _' 2 )]^

Y

2 = HF

3'-(GH

2

' -FG

3

' + GH^ -FG

2')+ HF2

' • (GH^ -FG^ - GH^-FGp

The solutions for F and H can be obtained by permuting the symbols. The solution of

the six equations like eqs. (A7.6b) and (A7.6c) for the (1,0) and (2,0) matrix elements

follows immediately by symmetry. Again we see that the sign of G , for example, can be

chosen arbitrarily after taking the square root of eq. (A7.8a). As with G we are free

to take either sign but not if we want to be consistent with other polarization optics. If,

for example, G is primarily a linear polarizer which is more transparent to horizontally

polarized light when 4> = than to vertically polarized light then G should be chosen

51
positive. A suitable source of partially horizontally polarized light is light specularly

reflected from a glass surface oriented so as to produce a vertical plane of incidence.

Similarly, if at <j> =0 it is more transparent to +45° polarized light than to +135°

polarized light then G should be taken positive. Once these two signs are established

all the others are determined by eqs. (A7.4), (A7.5), (A7.6), etc.

Finally we measure the transmittance of all three filters together. Let us assume

that the beam passes through them in the order F,G,H so that the measured transmittance,

HGF is given by the (0,0) element of the matrix product

HUh

)-G(*

g)-F(*f

) r

If we perform the matrix multiplication we find that this (0,0) element is given by

HGF = HGF1 + HGF -C , + HGF -S , + HGF '(I + HGF -S, 1 2 gf 3 gf 4 hg 5 hg

+ HGF.-C -C, + HGF -C . •

S

u + HGF-S -C. + HGF -S -S, 6 gf hg 7 gf hg 8 gf hg 9 gf h£

(A7.9)

where

HGF

1 = H00- G00' F00 + H00' G03- F 30 + H03' G30* F00 + H03- G33' F 30

^2 " H00* G01- F

10 + H00- G02' F20 + H03' G 31- F10 + H03' G32- F20

^3 " H00' G0l' F

20 " H00' G02- F10 + "oS^Sl'^O " H03' G32' F

10

HGF

4 = H01

-G10

-F

0() + H01

-G13-F 30 + H^-G^-F^ + H^-G^-F^

^5 =

-H02- G 10- F00 " H02- G13' F 30 + ^/So^OO + H r

G 23' F

30

(A7 ' 10)

HGF

6 = H01

-Gu

-F10 + H

Q1

-G12-F 20 + H^-G^-F^ + H^-G^-F^

HGF

7 = "H02* G

ll" F10 " H02' G12' F 20 + H01* G2l'

F10 + H

0l" G22' F 20

HGF

8 = H0l' Gll'

F 20 " H

0l'

G12' F10 + H02* G

2l' F

20 " H02* G22' F10

HGF

9 = -H

02

-Gn-F 20 + H

Q2

-G12-F10 + H^-G^-F^ - H

Q1

-G 22-F 10

and C = cos2((() -§ ), S = sin2(<J) -$ ) , etc. Again we assume that by least squares

fitting or by solution of simultaneous equations we have obtained numerical values for the

coefficients HGF , HGF„, etc., of all the trigonometric factors in eq. (A7.9); that is, we assume that we know the left-hand sides of eqs. (A7.10). In the first of eqs. (A7.10) we now know everything except G so it can be solved for directly. The second and third

of eqs. (A7.10) are a pair of equations in which the only remaining unknowns are G and

G„_. Likewise the next two equations contain only two unknowns, G and G . The last

four equations of the set (A7.10) contain the four unknowns G , G „, G , and G . Thus,

52
all the rest of the matrix elements of the Mueller matrix for the filter G can be determined from these nine coefficient values. Measurements of the three-filter train in the

orders G, H, F and H, F, G will permit the remaining matrix elements of the other two filters to be determined. We won't attempt to write out explicit solutions for the unknowns

appearing in the set of equations (A7.10). The procedure to obtain the solutions is

obvious and straightforward, but the entire data reduction process for this general case is

clearly best left to a computer, if possible.

We have assumed that the matrix elements in the first row and first column are non-zero.

If zeros appear here some of the matrix elements may become indeterminate. For example, in

eq. (A7.7a) if either F or H vanishes the denominator and one factor of the numerator will both vanish and G cannot be evaluated. The physical reason for this is clear.

The experimental procedure, in effect, requires generating partially circularly polarized

light with one filter and then analyzing it with a second filter. If the first filter fails

to produce circularly polarized light, no information can be obtained about the circular

polarization behavior of the second filter. Similar arguments apply to the other first

column and first row matrix elements. The extreme case occurs when all the first column

and first row elements of one filter, say G, vanish. Then G is totally opaque and

obviously contributes nothing to the measurement of filters F and H. With only two filters, F _ and H „ can be measured as before, but equations (A7.4), (A7.5), and A7.6)

provide only 9 equations in 12 unknowns and contain no information about the 18 other matrix

elements which do not lie in the first row or column. When the equations become insoluble

because of the occurrence of zero denominators one must either abandon the attempt to measure the inaccessible matrix elements or find other filters which don't have such limitations.

When the matrix elements for an optical component have been determined as outlined

above they can be inserted in eq. (6.39) to obtain the orientation dependence of the Mueller

matrix. In general, even for ideal optical components the result will not agree with, for

example eq. (6.15), but this is only because the angle between the polarization axis of the

device and the <j> = 0° direction was assumed arbitrary. The direction of the polarization

axis can be found, if desired, by finding the angle cf>° at which the experimental Mueller

matrix most nearly resembles the matrix for the ideal component. For example, for an ideal

linear polarizer at <J> = 0° the (0,2) element vanishes so for a real polarizer we must

find the angle <j>° at which the (0,2) element [from eq. (6.39)] of the experimental

matrix vanishes. That is, G •sin2<)) + G -cos2cf> =

or

<f>° = -Jsarctan(G

02

/G01

).

This is the direction of the polarization axis and if the orientation angle for an ideal

component is measured from this direction instead of from <f> = 0° its experimentally

determined Mueller matrix should agree with that given by eq. (6.15).

53
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN 910-3

2. Gov't Accession

No.

4. TITLE AND SUBTITLE

SELF-STUDY MANUAL ON OPTICAL RADIATION MEASUREMENTS

Part I. Concepts

Chapter 6. Distribution of Optical Radiation with Respect

to Polarization

7. AUTHOR(S)

Fred E. Nicodemus and John B. Shumaker

3. Recipient's Accession No.

5. Publication Date June 1977

6. Petforming Organization Code

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DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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Same as No. 9

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16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) This is Chapter 6 of a SELF-STUDY MANUAL ON OPTICAL RADIATION MEASUREMENTS.

It develops and illustrates the concepts necessary to include polarization rigorously in classical radiometry. The treatment is based upon the Stokes polarization

vector of spectral radiance and Mueller transmit tance matrices. The Mueller matrices

of many common polarizing optical components are discussed. Considerable attention

is paid to the measurement of the Stokes spectral radiance vector components and to the measurement of the Mueller transmittance matrix elements. The concepts and

techniques are illustrated by discussions of such subjects as radiometer calibrations, three-polarizer attenuators, depolarizers, and the characterization of

Polaroid-type polarizers. Also included are an appendix on matrix multiplication

and an appendix showing one way by which, in principle, any Mueller transmittance

matrix may be measured.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Mueller matrix; polarization; polarizer; retarder; Stokes parameters.

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