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U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

SELF-STUDY MANUAL

ON

1 RADIATION MEASUREMENTS

Part l--Concepts

Chapters 1 to 3
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 technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent 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, the Office of Radiation Measurement and the following Center and divisions:

Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center

for Radiation Research: Nuclear Sciences; Applied Radiation — Laboratory Astrophysics 2 — Cryogenics " — Electromagnetics ~ — Time and Frequency \

THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized materials needed by industry, commerce, educational institutions, and Government; provides

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THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

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1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

2 Located at Boulder, Colorado 80302.
PV «TASflDAJU>a

APR 5 1976 Self-Study Manual on

Optical Radiation Measurements:

Part l-Concepts, Chapters 1 to 3

Fred E. Nicodemus, Editor

Optical Physics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

Chapter 1. Introduction, F. E. Nicodemus, H. J. Kostkowski, and A. T. Hattenburg

Chapter 2. Distribution of Optical Radiation with Respect to Position and Direction --Radiance, F. E. Nicodemus, and

H.J. Kostkowski Chapter 3. Spectral Distribution of Optical Radiation, F. E. Nicodemus and H. J. Kostkowski

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U.S. DEPARTMENT OF COMMERCE, Elliot L Richardson, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued March 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $2.10
National Bureau of Standards Technical Note 910-1

Nat. Bur. Stand. (U.S.), Tech. Note 910-1, 93 pages (Mar. 1976)

v CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

(Order by SD Catalog No C13. 46:910-1). Price S (Add 25 percent additional for other than U.S. mailing).
PREFACE

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

"Self-Study Manual on Optical Radiation Measurements." It contains 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 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 characterizations 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 [1,2,3,4]. 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

1 Figures in brackets indicate literature references listed at the end of the Technical Note.

iv
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.

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.

Contents

Page

Part I. Concepts 1

Chapter 1. Introduction 1

In this CHAPTER 1

OPTICAL RADIATION 1

The MEASUREMENT EQUATION ... 4

UNITS and NOMENCLATURE 7

SUMMARY of CHAPTER 1 8

Chapter 2. Distribution of Optical Radiation with respect

to Position and Direction — Radiance 10

In this CHAPTER 10

The OPTICAL-RAY APPROACH . 10

RADIANCE (through two apertures) 12

RAY RADIANCE (at a point in a direction) 19

The ELEMENT of FLUX and the GEOMETRICAL INVARIANCE

of RADIANCE 23

APPLICATIONS of RADIANCE INVARIANCE 26

An EXAMPLE of RADIANCE INVARIANCE ........ 29

GEOMETRICAL INVARIANCE of THROUGHPUT 30

OPTICAL PROPAGATION — INTERACTIONS with MATTER 37

TRANSFORMATION from KNOWN RADIANCE DISTRIBUTION

to FLUX at ANOTHER LOCATION 40

SUMMARY of CHAPTER 2 . 42

Chapter 3. Spectral Distribution of Optical Radiation 45

In this CHAPTER 45

SPECTRAL RADIANCE . 45

GEOMETRICAL INVARIANCE of SPECTRAL RADIANCE 51

TOTAL FLUX in a RADIATION BEAM 52

TRANSFORMATION from KNOWN SPECTRAL-RADIANCE

DISTRIBUTION to FLUX at ANOTHER LOCATION 53

REMAINING RADIATION PARAMETERS — TIME and

POLARIZATION' 54

SUMMARY of CHAPTER 3 54

vii
Page

Appendix 1. Units and Unit Symbols 56

Appendix 2. Spherical Coordinates and Geometrical Relationships 62

SPHERICAL COORDINATES (solid polar coordinates) 62

A SOLID ANGLE 62

A SOLID ANGLE in SPHERICAL COORDINATES 67

A PROJECTED SOLID ANGLE 70

OTHER SOLID-ANGLE MEASURES and/or APPROXIMATIONS 77

References 81

List of Figures

1.1 Spectrum of electromagnetic waves 2

1.2 A sophisticated spectroradiometer configuration for

precision laboratory measurements of spectral radiance 5

1.3 Lunar radiation measurement configuration 6

2.1 Rays from lamp source to detector-receiver 11

2.2 Experiment to develop the concept of radiance 13

2.3 Tilted apertures 14

2.4 Notation for subdivided apertures 17

2.5 Computation of flux A$ 1

i+ 18

2.6 The solid angle Ao) 12 subtended at AAj by AA2 20

2.7 Geometry of ray-surface intersection (for the definition

of radiance ) 22

2.8(a) A single ray 24

2.8(b) An elementary beam of radiation 24

2.9 Invariance of basic radiance along a refracted ray 27

2.10 Sources A, B, or C, all of the same (uniform, isotropic)

radiance L, are indistinguishable when "viewed" from P (only one present at a time) 31

2.11 Slide-projector throughput 35

2.12 Burning-glass throughput 36

2.13 Transmittance of ray path 39

3.1 Large variations of radiometric quantities with respect to wavelength 46

viii
Page

3.2 Ideal filter set with spectral transmittances all

centered about the same A 48

c 3.3 Spectral distribution of emission for three typical

light sources 50

A2-1 Spherical and rectangular coordinates of P(p,6,<j>) = P(x,y,z) 63

A2-2 The same plane angle 9 is subtended at by:

1. the circular arc CD

2. the straight line EF

3. the curve E F

4. the plane figure E F 64

A2-3 A solid angle 66

A2-4 An element of solid angle dw = dA /p 2 = sin9*d9-d(J> [sr] 68

A2-5 A right circular cone about the polar axis with

half-vertex angle 9 71 h

A2-6 An arbitrarily oriented surface element dA2 subtends a solid-angle element dw^ at ^Al at a distance D 72

A2-7 Sketch of Cassegrain-type optics 73

A2-8 Geometrical construction for measures of an element of

solid angle do) and an element of projected solid angle dQ 75

A2-9 Wiener's construction for measures of the solid angle w and the projected solid angle Q, subtended at dA by an irregular object 76

A2-10 Measures of an element of solid angle dw and an element

of projected solid angle dfl (unit-diameter tangent sphere

construction) 78
List of Tables

Page

Al-1 SI Base Units 56

Al-2 SI Prefixes .......... ... 57

Al-3 SI Derived Units for Radiometry 58

Al-4 SI Derived Units for Photometry 59

Al-5 Some Additional non-SI Photometric Units 60

Al-6 Units for Photon-Flux Radiometry .... 61

A2-1 Values of projected solid angle ft, solid angle w, common approximations TT-tan 2 6h and ff*9h2

> an^ related measures of "light-gathering power" (f-number or relative

aperture f/#, and numerical aperture N.A.) for right

circular cones of half-vertex (polar) angle 8^ about the

normal to a reference surface (aperture stop, field stop,

etc.), i.e., about the optic axis of a cylindrically

symmetric optical system . 80
SELF-STUDY MANUAL on OPTICAL RADIATION MEASUREMENTS

This is the initial publication of a new series of Technical Notes

(910) entitled "Self -Study Manual on Optical Radiation Measurements."

It contains the first three chapters of this Manual. Additional chapters

will be published, similarly, as they are completed. The Manual is a definitive tutorial treatment of the measurement of incoherent optical

radiation that is complete enough for self instruction. Detailed chapter summaries make it also a convenient authoritative reference source.

The first chapter is an introduction that includes a description

of optical radiation and the ray approach to its treatment in this Manual (based on geometrical optics) , a discussion of relevant parameters

and their use in a measurement equation as a systematic technique for

analyzing measurement problems, and a presentation of the system of

units and nomenclature used.

The second chapter, on the distribution of optical radiation with

respect to position and direction, introduces the basic radiometric

quantity, radiance, and its important invariance properties. It is

shown how to determine the total power in a beam from the radiance

distribution and to determine the distribution of radiance at any

surface, through which the beam passes, in terms of the distribution

at any other surface that also intersects the entire beam.

The third chapter, on the spectral distribution of optical radiation, develops the concept of spectral radiance. Its invariance

properties and the evaluation of flux in a beam from a known distribution of spectral radiance are developed in a treatment parallelling

that for radiance in Chapter 2.

These are the first chapters of Part I, in which are developed

the basic concepts, essential for the subsequent discussions of

instrumentation in Part II, and of applications in Part III.

Key Words: Optical radiation measurement; photometry; radiometry;

spec troradiome try.

Part I. Concepts

Chapter 1. Introduction

by Fred E. Nicodemus, Henry J. Kostkowski, and

Albert T. Hattenburg

In this CHAPTER . We describe optical radiation and the ray approach to its treatment in

this Manual (based on geometrical optics) . We discuss relevant parameters and their use

in a measurement equation as a systematic technique for analyzing measurement problems. We also present the system of units and nomenclature used herein.

OPTICAL RADIATION . Energy propagated in the form of electromagnetic waves or particles

(photons), which can be reflected, imaged, or dispersed by optical elements, such as mirrors, lenses, or prisms, is referred to here as optical radiation. In the spectrum of

electromagnetic waves, shown in figure 1.1, 'optical radiation lies between x-rays and

microwaves, i.e., in the interval from about one nanometer to about one millimeter. 1

^Commonly used physical units and symbols are listed in Appendix 1.

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The most familiar portion is visible light, which lies between about 400 and 700 nanometers, the approximate limits of the human-eye response in full daylight. None of these

limits is very sharp; only approximate boundaries exist between the wavelength regions of

figure 1.1.

Physical theories and models of physical phenomena are abstractions or idealizations

that approximate, more or less, what actually takes place in the "real" world. For example, Newton's laws of motion are quite adequate for dealing with most ordinary mechanical

devices of everyday experience. But in cases involving atomic particles or astronomical

bodies moving at very high velocities, the more sophisticated relations of Einstein's theory

of relativity are needed. Quite possibly that theory, in turn, will be found to be only

approximately true and will need to be supplemented, for certain applications, by still more

sophisticated treatments. Similarly, there are three approaches or models for dealing with

optical phenomena, each with its region of useful validity. They are: geometrical optics

(ray optics) , physical optics (wave optics) , and quantum optics (particle optics)

The simplest approach to optics, and the one used in this Manual almost exclusively, is that of geometrical optics or ray optics. It accounts very well for the way in which optical radiation is propagated from most common sources, such as incandescent lamps (light

bulbs), fluorescent lamps, arcs, discharges, light-emitting diodes (LED's), and laboratory

blackbodies. However, geometrical or ray optics can't account for the patterns (diffraction or interference) which are produced at the edges of certain shadows, in focal regions

where rays sharply converge, or by devices called interferometers. Then we require the treatment of what is usually called physical optics or wave optics. Finally, when we deal

with interactions with matter in microscopic detail, it is necessary to recognize that

energy exchanges take place in discrete amounts. Then we find it useful to consider optical radiation as being propagated in discrete "packets" or photons, whose distribution in

large numbers produces average energy distributions in time and space corresponding to the waves of physical optics. For our purposes, it is adequate to treat "classical" optical

radiation measurements or radiometry in terms of geometrical optics, with occasional rec- ognition of wave-optics or quantum-optics phenomena as perturbations of the ray-optics

relations. Both geometrical and physical optics are based on the wave theory of light,

although they can also be reconciled with corpuscular theories. The distinction between

them involves mainly the phenomenon of coherence.

Electromagnetic waves consist of periodic variations in interrelated electric and magnetic fields, variations that are periodic in space along the direction of propagation and

in time at any single point along the path. Emission of electromagnetic radiation can be

considered as involving oscillations of individual charged particles in the atoms or molecules of the material of the radiation source. If these are random oscillations, such as those produced by thermal excitation in heated matter, the resulting waves will be similarly random in phase. Then they will combine and propagate as incoherent radiation that

obeys the laws of geometrical optics, where waves passing through the same point in different directions seem to be completely independent of each other and do not interfere.
However, if the particles are somehow made to oscillate together, "in step" with each

other, the resulting ^waves will be coherent. They will have consistent phase relations

so that they reinforce or cancel each other over many periods of oscillation, producing the

interference patterns' that deviate from the laws of geometrical optics. The relations

based on geometrical optics do not apply, in many instances, to strongly coherent radiation

such as that produced' by lasers. However, as previously stated, we can use geometrical

optics almost entirely in connection with measurements of common sources, such as those

listed, treating the wave- and quantum-optics phenomena as occasional perturbations.

This Manual will deal principally with incoherent optical radiation from about 200 nan- ometers to about 20 micrometers. We will exclude, at least initially, all laser radiation

(see Preface) and the" vacuum ultraviolet and far infrared spectral regions. For brevity,

we'll refer to this reduced region of incoherent radiation as just "optical radiation."

The MEASUREMENT EQUATION . Every measurement of optical radiation involves a beam of radiation originating at a source, propagating along an optical path, and impinging upon a radiometric instrument. The source may emit radiation or it may be an irradiated object

that reflects or scatters radiation incident upon it from another source. The propagation

path may traverse a vacuum, or it may pass through a number of different media and involve

a variety of interactions with matter, such as reflection, refraction, scattering, absorption, and even emission (by fluorescence). Finally, the radiometric instrument canitake

many forms, e.g., a bare photocell or a sophisticated spectroradiometer . Two measurement

configurations illustrating different combinations of some of these possibilities ate shown

in figures 1.2 and 1.3. They are not intended for detailed comprehension at this point but

only to emphasize the wide range of complexity that can be encountered. We can cope with

such complexity only through an orderly, systematic approach, and our approach is based on

a measurement equation. $

The measurement equation is the mathematical expression that quantitatively relates the

output of a measuring instrument to the radiometric quantity that is being measured, taking

into account all of the pertinent factors contributing to the measurement result. The main

part of that measurement equation relates the radiation input at the receiving aperture of

the instrument to the; resulting output in terms of the instrument responsivity (output

"signal" per unit incident radiation input). The complete equation also accounts, as needed, for the effects of interactions between matter and radiation at the source and

along the optical path of the radiation beam as well as at the instrument. Setting all of

this down systematically in a quantitative equation helps to insure that all pertinent

factors will be appropriately considered and will not be inadvertently overlooked. It also facilitates the evaluation or estimation of the effects of individual parameters by

the investigator in order to achieve needed simplifications. The complete equation is

usually unmanageably complex until simplifying data or assumptions are introduced to make

it tractable. Fortunately, the effects of a number of parameters or variables will often

be negligible; the effects of others will be small and easily evaluated. The basic problem is to identify and accurately assess the effects of all significant factors, making
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The moon M is imaged on a photocell detector

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A and the optics of T. Proportions greatly

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simplifications wherever possible in order to obtain tractable expressions. The measurement equation provides a mechanism for approaching this systematically, thereby minimizing

the chance that any significant factor will be inadvertently overlooked.

The quantities used in the measurement equation are expressed in terms of the radiation

parameters and pertinent instrumental and environmental parameters. The distribution of

radiant power (radiant energy flow) or flux in a beam of incoherent optical radiation is

completely described or specified in terms of five radiation parameters: position, direction, wavelength, time, and polarization. The measurement of optical radiation is a multidimensional problem that always involves these five variables and, possibly, others as well. The strength of radiation can be different at different points in space, in different directions from any one point, for different wavelengths, and for different polarizations, and it can vary greatly with time. In addition, the interactions between radiation

and matter — absorption, emission, scattering or reflection, and refraction — may also

depend upon these same five radiation parameters. Instrumental and environmental parameters may also affect a measurement (e.g., temperature, humidity, magnetic fields). Although they can't be so conveniently and exhaustively listed, we'll try to provide systematic approaches for identifying and dealing with all those of significance in a wide

variety of situations, particularly when we reach Part III, Applications (see Preface).

UNITS and NOMENCLATURE . A mechanical engineer, concerned with dissipating frictional

heat, wants to know how much energy flows away from an exposed hot surface as optical

(heat) radiation. He is concerned with the rate of energy flow — the power — in the

radiation beam, usually expressed in watts. An atomic physicist, concerned with the light

emitted by individual particle interactions, wants to know the number of photons flowing

in a beam, the number of quanta p-er second. An illumination engineer, trying to provide

adequate lighting on a desk for comfortable and efficient reading and writing, measures

light in terms of its effect upon the average human eye, using lumens 1 for units. Thus,

there are different ways of stating the amount of optical radiation determined by a measurement; we can use different units of flux , the general term for the quantity of radiation per unit time flowing in a beam.

This Manual will be mainly concerned with radiant power measured in watts. Lumens and

related units are used when discussing photometry, and photon flux is utilized when we are concerned with the quantum aspects of interactions between radiation and matter. It cannot be too strongly emphasized, however, that everything said here about the fundamentals

of radiometry, even though stated in terms of watts, applies equally to all forms of optical radiation measurements. For example, the measurement of illumination for application

to vision needs involves all of the fundamentals, not just those discussed in the chapter

on photometry. And, conversely, there is much that is pertinent to, say, military applications of infrared radiation in that chapter on photometry. After all, photometry is

just the measurement of optical radiation with detectors having a specified spectral

^The lumen is defined and discussed in a later chapter on "Photometry."

7
responsivity that is related to that of the human eye. Accordingly, the problems of photometry are, many of them, equally pertinent to measurements with other spectrally-selective

sensors, many of which are used extensively in the infrared. Also, many military infrared

devices, e.g., night-vision devices, have visual displays where the photometric considerations, as such, are directly applicable. It's all optical radiation in spite of differences

in terminology and units.

As a matter of fact, the great diversity of nomenclature that has grown from the use of

optical radiation measurements in so many different fields of application is a perennial

problem. Different terms are used for the same concept and, conversely, the same term is

often used for different concepts. We will employ, as much as possible, the nomenclature

of the CIE International Lighting Vocabulary [5] 1 as the most comprehensive and least con- troversial authority available. Exceptions will be clearly noted when we do depart from or add to the CIE nomenclature at times when we find it inadequate. We will also mention

alternative terms and practices that are widely used in the literature where, at least for

a long time to come, complete standardization in so many different areas of application

just isn't going to take place. For example, it will be a long time, if ever, before

astronomers stop using star magnitudes. 2 Accordingly, others who need to use published

star radiation data must learn to convert them to the equivalent values of point brilliance

or of illuminance or irradiance.

One of the most useful techniques for coping with this unavoidable diversity of nomenclature is the regular use of unit-dimensions and routine unit-dimension-consistency checks

for all radiometric quantities and their mathematical relationships [6] . This Manual will

follow the practice of associating the proper units and their dimensions with each physical

quantity that enters into an important equation. This will facilitate verification of the

consistency of the unit-dimensions of the final result. All unit symbols are enclosed in

square brackets to emphasize their dimensionality in this connection. Standard SI units

and symbols are used wherever possible and exceptions are noted.

SUMMARY of CHAPTER 1 . Optical radiation is defined as energy propagated as electromagnetic waves or photons which can be manipulated and studied by optical elements (e.g.,

mirrors and prisms) . This Manual treats incoherent optical radiation in the wavelength

region between approximately 200 nanometers and 20 micrometers. The method of treatment employs geometrical optics or ray optics.

Incoherent optical radiation is completely specified in terms of the five radiation

parameters: position, direction, wavelength, time, and polarization. Its interaction with

matter, as it traverses an optical path and is directed and measured by an instrument, is

also governed by these radiation parameters, as well as by environmental and instrumental

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

2 A star magnitude is a logarithmic unit of incident flux per unit area.

3 See Appendix 1.

8
parameters that cannot be exhaustively listed. The quantitative interrelationship between

the incident radiation and the measured result in a measurement equation, in terms of all

relevant parameters, is the basic approach in this Manual. The complete measurement equation, usually too complex for a complete general solution, facilitates the evaluation of

the effects of simplifying data and assumptions used to obtain more tractable expressions

for particular applications. It provides a systematic approach that helps to minimize the

chance that any significant factor may be inadvertently overlooked.

Radiometric relations are usually given in this Manual in terms of flux in watts, with

photometric and photon-flux quantities also employed, when appropriate. Nomenclature

follows the CIE system, with exceptions and additions noted and with alternate terms also

supplied, if they are widely used. Unit-dimension checks are recommended for dealing with

nomenclature confusion and diversity. SI units and symbols are used wherever possible and

exceptions are noted.
Chapter 2. Distribution of Optical Radiation with respect to Position

and Direction — Radiance.

by Fred E. Nicodemus and Henry J. Kostkowski

In this CHAPTER . We introduce the basic radiometric quantity radiance . 1 This quantity

allows us to explicitly characterize the distribution of radiant power from point to point

and direction to direction throughout a beam of optical radiation. This is particularly

important when both the power distribution, and the sensitivity (responsivity) of a radiometer used to measure this power, are non-uniform or non-isotropic. We also show how the

total power in the beam can be obtained from the radiance distribution. Finally, we show

how to determine the distribution of radiance at any surface, through which the beam passes,

in terms of the distribution at any other surface that also intersects the entire beam.

The OPTICAL-RAY APPROACH . In figure 2 . 1 we see a side view of a lamp with a flat, glowing,

tungsten-ribbon filament, and a detector-receiver that is irradiated by the lamp. A ray

drawn from a point on the lamp filament to a point on the receiver surface represents the

optical radiation originating at a small area element around this point, propagating along

the line drawn, and impinging on the receiver area element. Thus a ray is a line or direction along which optical radiation flows. In terms of waves, it is the direction in which

that particular part of the light wave is traveling.

From experience, we know that different parts of the glowing tungsten surface will not appear equally bright to the eye. The ends of the ribbon filament are cooler than the rest

so point 1 in figure 2.1 will typically be brighter than point 2. Moreover, though to a lesser degree, the brightness of any point, such as point 2, will also change with direction. It will be different when viewed from directions a, b, and c. Thus, the positional

and directional distribution of optical radiation can be associated with rays if we can

find a way of associating a definite quantity or concentration of radiant power or flux with

each ray. This can be done through the quantity called radiance , a concept which we now

develop.

*The purely geometrical aspects of radiance are identical to those of the photometric quantity luminance and are approximately the same as those of the familiar psychophysical quantity brightness . Radiance pertains to flux (amount of radiation flowing) measured in watts

or other power units, and luminance to flux measured in lumens, units that are related to standardized eye response (defined and discussed in the chapter on Photometry) . Brightness

is related to luminance in that incandescent sources of equal luminance usually appear

equally bright. However, a common "optical illusion" shows that the apparent brightness of an area can be strongly affected by a background or "surround" of a different brightness.

Also, the familiar photographic "gray scales," with steps of apparently equal brightness

difference, actually have luminance or radiance steps of approximately equal ratio, i.e.,

a logarithmic scale of luminance or radiance. Introduction of color makes things even more

complicated. A completely satisfactory theory still eludes the experts in color imetry and

vision research.

10
TUNGSTEN RIBBON

LAMP

Figure 2.1. Rays from lamp source to detector-recelver,

11
RADIANCE (through two apertures) . Consider an experiment (figure 2.2) with a large-area

visible-wavelength source that appears uniformly bright from all directions, two black

screens each having a small aperture, and a photocell next to the second screen. The photocell responds to all the radiant power reaching it from the source through both apertures.

The beam from source to photocell consists of all rays in the shaded region between the extreme rays through both apertures, as shown in the figure. It is assumed that the medium

(air) is perfectly transparent, with a negligible loss of radiant power from the beam by

scattering or absorption. If the power A$ in the beam is measured for different aperture

areas AAj and AA2 (perpendicular to the plane of figure 2.2), and for different distances D between the apertures, it is found, over a wide range of values, to be proportional to the quantity

AAi *AA2 (2.1) D

z

The measured power or flux A$ also changes when the brightness of the source changes. We denote the radiometric quantity that corresponds to that brightness by the letter L. Then

we can write

AAi 'AA2 A$ = L

. — (2.2) D

2

Also, if the apertures aren't kept perpendicular to the central ray through both of them, we find that the measured flux also varies with the cosines of the angles of tilt, shown in

figure 2.3. This makes our final expression

AA} 'COs6i • AA2 'cos6 2

A<f> = L • 5 [W]. 1 (2.3)

Next we note that, holding the apertures fixed, the measured flux doesn't change when the source is moved farther away from, or closer to, the apertures or tilted, just as long as all rays in the beam through both apertures come from the uniformly bright emitting surface.

Only if we change the source so that it is no longer uniform and isotropic does the measured flux vary with the position and orientation of the source. We can again make the

measured flux less sensitive to source position and orientation, at least to small shifts,

by making the apertures AAj and AA2 small compared to their separation distance D and

small compared to the distances on the source surface between points of significantly different brightness.

The experiments show that the quantity L is not only related to the source brightness

but also to the small bundle of rays leaving the source in a region of uniform brightness.

In fact, it appears to have the same value anywhere along such a bundle of rays. Its value

can be obtained for our sample situation by solving eq. (2.3) for L, thus:

1 [W] denotes unit-dimensions of watts (see Appendix 1)

12
Source

HJ

Photocell

Screen Screen

Figure 2.2. Experiment to develop the concept of radiance.

13
Figure 2.3. Tilted apertures

14
L = ^ . (2. A) AAj 'cos 6

j •AA2 *cos9 2

Actually, careful measurements, and measurements with sources of more uneven brightness,

show that these equations yield only average values of L. On the other hand, if we try to evaluate L for a smaller beam, representing just a portion of the larger beam, by making

the apertures smaller and the distance D larger, we reach a point where there is no longer enough power in the beam reaching the photocell to make a measurement at all. Even before that happens, we may encounter diffraction effects where our geometrical-optics model

of propagation along rays no longer adequately describes the situation. Nevertheless, if we

are careful to observe the limitations of geometrical optics, not applying our equations,

without corrections, to situations involving significant diffraction or interference effects, we can obtain relations having a very wide useful range of application by the mathematicalanalysis methods of calculus. To do this, we assume an underlying continuous distribution

of flux among the rays of the radiation beam, even when the apertures are made arbitrarily

small [7].

We define the quantity L, then, as the limit of the quotient of flux A4>, passing

through both AAj and AA2 , by the geometrical quantity (AAj 'cosej «AA2 •cos9 2 )/D2 as

AAi and AA2 are made smaller and smaller. This is written

Lim

L = AAJ-* ^ g (2.5) AA2

-*- (AA 1 'cos9 1 -AA2 -cos9 2 )/D2

But this is just the defining equation for a second derivative, so that

d

2 $ L = . (2.6)

cos9;i -cos9 2 dAi-dAo- - D

2

In the limit, as the apertures become vanishingly small, there remains only a single ray through both of them, so radiance, so defined, is associated with an elementary beam collapsed to just a single ray. This doesn't mean however, that it's ever possible to actually measure the radiance of just a single ray. Real measurements always involve apertures

and beams of finite dimensions, as in eq. (2.4), which then yield only average values of

radiance. Such an average value equals the actual radiance only when the beam is completely

uniform and isotropic with all rays having exactly the same value of radiance.

There is now no limit to how large the apertures may be. By using integral calculus, we can relate the ray-radiance distribution to the total flux or power flowing in a beam of

any size. We begin by imagining that each large aperture is divided into many small areas, each small enough so that, through any pair of these small areas (one in each aperture),

all rays have the same radiance. The flux through each such pair of small areas can be

15
calculated by eq. (2.3) and the results added up, for all such pairs to account for the full area of each aperture, to obtain the total flux in the entire beam. A simple example of

just three small areas in one aperture and four in the other is illustrated in figure 2.4. Then if, for example, the flux in the portion of the beam between area 2 of the first aperture and area 3 of the second aperture is designated as A$ 23 , we can write the expression

for the total flux in the beam through both apertures as

$ = h$i\ + A$ 12 + A$ 13 + A$m +

A$2i + A$22 + A$23 + A<J> 2 i* +

A<J> 31 + A$ 3 2 + A$33 + A$ 3

tt

. (2.7)

The flux in the beam through each pair, in turn, is evaluated by eq. (2.3). This is illustrated for the pair AAj in the first aperture and AA^ in the second aperture, in fig- ure 2.5 and the following expression:

AAi 'cose^-AA^-cose^!

A$ 14 = L lV (2.8) D

More generally, if each aperture is divided into an arbitrarily large number of small areas,

the first into i areas and the second into j areas, the total flux in the beam can then

be written as

AA. -cosG .

. -AA. -cost

= r**ii -ZZv' 1J 2

3—— w (2 - 9)

ij J i i D--

Again using calculus, we let the areas AA. and AA. become arbitrarily small so the numbers i and j become, at the same time, arbitrarily large. This is written as

Lim

$ =. AA.-*- I A$ [W], (2.10)

1 4 1 xl

j AA.-*- 1J

i -> oo

j

-»- 00

where I A$ can be expanded as in eq. (2.9). This is simply the defining equation for

ij ^

the double integral over the two apertures:

cosOi •c,os02 * = f / L • -j • dAx-dAa [W]. (2.11) A2 Aj D

In practice, if we know both the radiance L and the slant distance D along the ray

between the area elements dAj and dA2 as functions of the position coordinates of those

elements over the full areas of both apertures, the integral for the flux $ can be evaluated, at least on a computer. Usually, however, we try to make measurements under conditions where the radiance L is the same, or approximately so, for every ray through both

apertures. Then, as a constant, it can come outside the integrals, leaving

16
I2

: 3'

2

3

4

1

•A*23

Figure 2.4. Notation for subdivided apertures.

17
A*14

Figure 2.5. Computation of flux A^j^.

18
cos9i *cos6 2 L ' /A JAi • dAj-dA2 [W]. (2.12) 'A 2

J A!

Evaluation of these double integrals can become quite complicated. However, integrals of

this type have been evaluated and tabulated for many different geometrical configurations,

with different sizes, shapes, orientations, and separations of A± and A2

, as we'll see

in more detail in Chapter 4 and the accompanying Appendix 3.

RAY RADIANCE (at a point in a direction) . The defining equation for radiance, eq. (2.5), can be written in a different way by recognizing that (AA2 *cos6 2 ) /D 2 = Aw^ i- s the solid

angle 1 in steradians [sr] subtended at AAj by AA2

, as shown in figure 2.6. Accordingly, radiance can also be defined as

Lim

L = AA -y A$

Aid -> AA'cosG'Ato

[W-m"2 -sr

-1

]. (2.13)

.dA*cos6 »d(D

In this form, it is easier to recognize that the units of radiance are watts per square meter and steradian, as shown. This form is also more general, since it defines L at a point in the direction of a ray through that point, rather than between two points. With

this approach we need not assume a perfectly transparent medium with no attenuation between

the two points. However, we have presented both approaches because this concept, involving

simultaneous variation and distribution of flux in both position and direction, is a difficult one. Many find the first approach easier to understand while others prefer the second

approach which, in any event, has definite advantages for many applications.

Radiance is a ray-associated field quantity. What this means is that its value depends

on the point in space where it is evaluated and on the ray direction through that point.

It is the concentration of propagated optical flux or power, with respect to both position

and direction, as a function of both position and direction. We'll define it more explicitly in mathematical terms and use the definition to relate the radiance along an emitted

ray at the surface of a source to its value at subsequent points along that ray, particularly at the point of incidence on the surface of a receiver. At the same time, we develop

the concept of the element of propagated flux associated with each ray as the product of

(1) the radiance and (2) the associated element of throughput, which is also defined. The

distribution of these quantities, as functions of position and direction, incident on the

receiving aperture of a radiometer or radiometric device can then be combined with the flux

responsivity of the instrument, which may also be a function of position and direction of the incident flux element, to obtain the instrument output in terms of the incident radiation input and the spatial parameters. However, we won't take that step until Chapter 4.

The precise, explicit definition of radiance [8] is given first in words and then

See Appendix 2 for definition and discussion of the concept of solid angle.

19
Figure 2.6. The solid angle Aw 12

subtended at AAj by AA2

P^Ni is normal to AA^

P 2 N2 is normal to AA2

20
mathematically (refer to figure 2.7). The radiance at a point on a surface in the direction of a ray through that point is defined as the radiant flux or power per unit projectedarea-perpendicular-to-the-ray-at-the-point and unit solid-angle-in-the-direction-of-theray-at-the-point

where

L(x,y,6,<j>) [W'm-2, sr

-1

] is the radiance at the point x,y in the direction 6,4>;

x and y [m] are the position coordinates, on the surface, of the point of

intersection with the ray (usually, but not necessarily, it is convenient

to have a plane reference surface, in which case x and y are cartesian

coordinates)

9 and <(> [rad] are spherical coordinates; 1 9 is the polar angle between the

ray and the normal (perpendicular) to the surface at the point x,y and

<j) is the azimuth angle about the point x,y in the plane tangent to the

surface at the point x,y;

d

2 $(x,y,9,<}>) [W] is the element of radiant flux through the surface element

dA = dx'dy [m2

] about the point x,y and within the element of solid

angle 1 dw = sin9'd0'd<j> [sr] in the direction 9,<J>; and

dA # cos9 [m2

] is the element of projected area perpendicular to the ray direction 9, (J).

As we saw in eq. (2.13), radiance as defined here is the limit, as AA and Aw both

approach zero, of the quotient in the first line of that equation, where A$ is the

radiant flux or power flowing through the area AA within the solid angle Aw. In order

for this quotient to converge (approach a definite limiting value) at the point x,y in

the direction 9,<f>, the flux A<t> must also become vanishingly small as AA and Aw both approach zero. Thus, as we've already pointed out,- we can never exactly measure this

quotient at a point and in a given direction; all we can ever measure in reality is its

average value over small intervals of area and solid angle through which enough flux can

pass to produce a measurable output "signal" in a radiometer. (Of course, if the radiance

has the same value throughout a beam, the average value will equal the value along any

single ray, so this is the way we try to make the most accurate measurements.) It is very

important to understand this limitation clearly; but it certainly doesn't destroy the usefulness of the concept of radiance.

Actually, there are many such "point functions" that can't be exactly measured. Some are so familiar as part of our everyday experience that we never stop to think about this

•^Spherical coordinates, solid angles, etc., are discussed in Appendix 2 for those who may

with to refresh their memories on these topics.

21
Figure 2.7. Geometry of ray-surface intersection

(for the definition of radiance )

22
limitation. A good example would be the concept of density, i.e., of mass per unit volume.

It's obviously impossible to measure density exactly at a point in a region of changing

density, as in a flowing gas or liquid where the density varies rapidly from point to point.

Nevertheless, most of us have no difficulty with the idea that there is a definite value

associated with each point along a path through such a region of (continuously) changing

density. But any measurement we make can give us only the average value over a finite volume, not the exact value at a point. In the same way, a radiometer gives an average

response to inputs spread over intervals of area and solid angle or intervals of position

and direction. But it's still very useful to think in terms of the radiance at a single

point and in a single direction through that point as the basis for analysis. Furthermore,

as already suggested, we are usually able to detect or indirectly control the uniformity

of such a quantity with much greater precision than we can measure its absolute value.

Accordingly, by arranging to have the beam for a measurement as uniform and isotropic

(constant radiance) as possible, we can make the measured average correspond very closely

to the actual value of that constant radiance for all points and directions within the beam.

The ELEMENT of FLUX and the GEOMETRICAL INVARIANCE of RADIANCE . Now let's see how we can use our definition of radiance to express the element of radiant flux associated with a single ray. In figure 2.8 we show first (a) a single ray between two points P^ and P 2 and

second (b) the elementary beam, made up of all the rays between two area elements dAj and

dA2 about the points Pj and P 2 , respectively. As we've seen, an element of area dA

is just a small area AA that can be made arbitrarily small in the process of approaching

a limit, the limiting value of a quotient for a derivative or the limiting value of a summation for an integral. It may be helpful, at first, to think of dA 1 as being part of the

emitting surface of a source, such as a tungsten ribbon filament, and, similarly, of dA2 as being part of the surface of a receiver on which the ray P1P2 is incident. However,

they can just as well be apertures in real or imaginary screens through which the elementary

beam passes. Everything we say now, in the following discussion, is equally applicable to any pair of two (imaginary) surface elements intersecting any two points P^ and P 2 along

the path of a single ray. These points may be arbitrarily chosen anywhere along the ray.

In figure 2.8(b), we have also drawn thenormals PiNj and P 2 N2 , perpendicular to the surface elements dAj and dA2

, respectively. The ray and normal that intersect at

Pj form an angle 9^; 9

2 is the angle between the ray and normal intersecting at P 2

. In many cases, when the surfaces of interest are parallel to each other, 6j = 9 2

. However, we don't want formulas that are too restricted in their application, so we've chosen the more

general case where these angles of tilt may (or may not) be unequal. On the other hand, we

do want to restrict ourselves, at first, to a medium in which the index of refraction is

everywhere the same, that is, where radiation flows at the same velocity everywhere and in

all directions (uniform and isotropic), so that all rays are straight lines. Later, we'll

see how to deal with the still more general case where the refractive index is found to vary.

The elementary beam of radiation between dA^ and dA2 , the beam defined by those

two surface elements, consists of all of the rays along which radiation flows or is

23
(a)

(Source) (Receiver)

Figure 2.8 (a) A single ray

Figure 2.8 (b) An elementary beam of radiation

PlNi is normal to dAi

P 2 N2 is normal to dA2

(b)

24
propagated between the two surface elements. In other words, all of the rays joining

points on dAj to points on dA2

, taken as a whole, constitute the beam. If we were to try to draw them all, we'd just have a solid black, band that wouldn't show the detail in

which we're interested. Instead, we've just suggested what's involved by drawing in these

rays for only three points on each surface element, one at Pj or P 2 , respectively, near

the center, and the other two at the edge on opposite sides in each case. Even this ends

up as a fairly "busy" figure, with nine rays altogether, three from each point on one sur- face element to each of the three points on the other surface element.

Rays that all intersect at a common point are said to form a pencil of rays. If we consider the radiation in figure 2.8(b) to be flowing from dAj to dA2

, the diverging

rays from any one point on dAj to all points on dA2 form an exitent pencil , while

those converging at any one point on dA2 from all points on dAj form an incident pencil . The extreme rays between a point on either area element and the entire edge of the other area element form a cone bounding the solid angle subtended at the point by that area element. 2 The element of solid angle subtended by dA2 at Pj is given by

dw 12 = cos9 2 *dA2 /D 2 [sr], where D [m] is the distance between P} and P 2

• Similarly,

the element of solid angle subtended at P 2 by dAj is given by da) 2 i = cos9i*dAi/D 2 [sr]

When the area elements are small enough, the solid angle subtended by either one of them at

a point on the other is the same for all such points.

With these geometrical relations established, we can turn our attention to the flow of

radiant energy in the beam of figure 2.8(b). When the area elements are small enough, there

will be no significant differences in radiance between the rays through different points

across a surface element or in different directions within the pencil of rays to the other

surface element through any single point of the first element. Accordingly, we assume that

all of the rays leaving dAj toward dA2 are of radiance Lj and that those same rays all arrive at dA2 with radiance L2

. Of course, we've already seen that experiments show

that L 1 = L 2. But we ignore that, for the moment, so that we can also show, by analysis,

that L 1 = L2 follows just from our definition of radiance. We next write expressions for

the element of radiant flux or power in the elementary beam through each area element.

If all rays leaving dA 1 are of radiance Lj watts per square meter of projected

area and steradian of solid angle, and they emerge through a projected area (perpendicular

to the ray P1P2) °f cos6 1 «dA 1 square meters and within a solid angle of dtu 12 steradians, the flux or power in the exitent elementary beam is the product of these quantities,

or

d$i = Lj^ 'cosSi *dAj •du)i 2

= L 1 'cos-dA1 'cose 2 'dA2 /D 2 [W]

(2.15)

^'Exitent" was coined as an antonym of "incident" by Richmond [9].

2 Spherical coordinates, solid angles, etc., are discussed in Appendix 2 for those who may

wish to refresh their memories on these topics.

25
which looks very much like eq. (2.3) or eq. (2.8). Similarly, the element of flux or power

reaching dA£ in this same elementary beam is

d$2 = L2 , cos62'dA2'da)2i

= L 2 'cose 2 'dA2 -cose 1 'dA 1 /D2 [W]

(2.16)

If there is no loss of radiant power or flux in the intervening medium, so that all of the

flux leaving dAj in the elementary beam toward dA2 arrives at dA2

, we can set

d$i = d$2. Then

Li 'cosG! -dAj •cose2'dA2 /D2 = L2 *cos9 2 "dA2 *cos9 j -dAj /D2

which reduces to

Lj = L2

. (2.17)

Since no restriction was placed on the choice of the points Pj and P2 along the

ray, eq. (2.17) must apply to any pair of points, i.e. to all pairs of points, along that

ray. This means that the radiance in the direction of a ray is the same at every point

along that ray in the absence of any energy losses or new sources of energy. Radiance is

geometrically invariant along a ray in a passive, lossless, uniform, isotropic medium.

Accordingly, if we know the value of exitent radiance at the surface of a source for a particular ray, this also means that we know its value at any subsequent point of that

ray, including the point where it is finally incident on a receiver, providing there are no losses of energy (or new sources of energy) along the intervening path. Moreover, if

such losses exist, they can be accounted for by an appropriate factor, the propagance of

the path, which we'll define and discuss in detail later. 1

APPLICATIONS of RADIANCE INVARIANCE . Although we won't get into a thorough discussion

of applications until Part III, we want to look at some of the useful applications of the

invariance property of radiance now because it will help to clarify the significance of

this important quantity. First, however, we need to examine what happens when a ray traverses different media with different refractive indices. Even though the atmosphere is

often reasonably uniform and isotropic, especially for measurements in the laboratory, rays

frequently pass also through lenses, prisms, or other optical elements with quite different

refractive indices. We need to know how this affects the value of radiance along a ray.

Most of the optical elements with which we are concerned have relatively smooth sur- faces so we'll analyze the situation for regular (specular) transmission, with refraction,

at a smooth boundary surface between two media of different refractive indices, as depicted in figure 2.9. We define a "smooth" surface as any surface where it is possible

to construct a tangent plane, i.e. where every surface element dA can be treated as common to the surface and to a plane tangent to the surface at that point. This figure

l See eq. (2.37) (on p. 38.)

26
Figure 2.9. Invariance of basic radiance

along a refracted ray.

27
shows a vertical plane (plane of the paper) containing the normal (perpendicular) to a surface element dA of the smooth surface and an incident ray, inclined at an angle B± to the normal, within an element of solid angle dcuj , in the medium of refractive index n^ above the surface. The incident radiance is Li [W«m-2 -sr" 1

]

, and the element of radiant

flux or power in the elementary beam incident on dA through dwj is d<J>j [W] .

Below the surface, in the second medium of refractive index n2, the refracted ray

is inclined at an angle 82 and the refracted elementary beam fills an element of solid

angle da)2 • This solid-angle element dw2 differs slightly from du)j because the rays

bounding duj are refracted by slightly different amounts. The radiance here is L 2 and

the element of refracted radiant flux or power is d$2- As before, we can write the elements of flux in terms of the radiance and the geometrical quantities as

d$j = Lj 'cosSj -dA'du^ = Lj 'dA-cosOj 'sinOj •d8 1 *dc|> [W] , and

(2.18)

Also, as before, we are interested only in the effects connected with ray and beam geometry

so, again, we assume there are no losses in either medium. However, we know that, even with so-called anti-reflection coatings, there will always be some of the incident flux

reflected at a smooth surface so that only part of it will be transmitted and refracted.

If the total incident element of flux is d<f>i' with a radiance Lj ' and the reflected

portion is p*d$i' with radiance p'Li', then the remainder that is transmitted and

refracted without loss is d^ = (l-p)'d$i' with Lj = (l-p)*!,!'. Accordingly, setting

d$j = d$2> since it is transmitted and refracted without loss, we have, from eqs. (2.18),

d$j Lj 'dA'cosBj 'sinGj 'dBj 'dif)

d<J>2 L,2 , dA'C0s62 , sin62 , d62'd<f>

Lj •sin9 1 'cos

± 'dBj L2 *sin62 •cos62'd9 2 (2.19)

a relation involving only radiance and beam geometry and ignoring the portion of the incident flux that is reflected.

The angle of incidence Q\ for every incident ray is related to the angle of refraction 9 2 for the corresponding refracted ray by Snell's law of refraction, which can be

stated mathematically as

ni*sin6i = n2'sin62- (2.20)

By differentiating with respect to angle, we also have

ni^cosg^dSj = n2"cos6 2 'd82. (2.21)

Rearranging eqs. (2.20) and (2.21), we can write

sin9i cos9i 'dSi n? i— = i — = —±-. (2.22) sin92 cosS2*d9 2 n^

28
Finally, combining eqs. (2.19) and (2.22), we have

L 1

, n2

2 Lj

= 1; or —— = —=-. (2.23) L2 *ni 2 n^ 2 n2

2

Accordingly, in going from one medium to another, the invariant quantity is not radiance L. Instead, it is the basic radiance L/n2 (where n is the index of refraction of the

medium) that has the same value in the direction of a ray at all points along that ray.

In fact, more sophisticated proofs [10] show that the invariance of basic radiance is a completely general geometric property, even along a ray traversing a non-uniform, nonisotropic medium in which the index of refraction varies continuously from point to point.

It must be reemphasized that this is a purely geometric property and that the actual radiance or basic radiance is usually further modified by interactions with matter, being

attenuated (reduced) by absorption, reflection, or scattering out of the beam, and also

possibly augmented (increased) by emission or scattering into the beam. The simplest case

of attenuation will be treated briefly at the end of this chapter. More details on such

interactions will come later.

For the moment, the important point is that, from eq. (2.23), we can see that radiance

L is geometrically invariant along all parts of the same ray that lie in the same medium

(same refractive index), regardless of intervening passage through an optical element (with

smooth surfaces) of a different material (different index) . The radiance within the

material of the optical element changes, keeping the basic radiance constant, but it returns to the original value upon reemerging into the same medium (usually air) again.

The only effect, then, is possible attenuation by the optical element, which will be discussed briefly at the end of this chapter and later in more detail.

This is such an important point that it may be helpful to restate it explicitly in

mathematical terms. Given a ray of radiance Lj in a medium of index tij that passes

through an optical element with smooth surfaces of a material with index n2 and out into

a third medium of index n3, the radiance L2 internal to the optical element and the

final radiance L3 in the third medium satisfy the following (based on eq. (2.23)):

Li/nj 2 = L2 /n2

2 = L 3 /n 3 2 [W-m~2

• sr" 1

]

(2.23a)

Furthermore, if the ray emerges, without attenuation losses, again into the same medium,

e.g., into air, so that n 3 = n^, this means that L3 = Lj and we can ignore the fact

that the ray traversed a different medium. As will be shown later, if there are attenuation losses, this will reduce the final value of propagated radiance by the fractional

amount of loss.

An EXAMPLE of RADIANCE INVARIANCE . Many photographic exposure meters for measuring

reflected light from a scene or object (not the incident light on the object or scene)

consist of a photocell mounted behind a baffle, grid, and/or lenses that define the

receiving aperture area and solid angle through which rays from the scene or object can

29
reach the photocell, just as the apertures in the two screens of figure 2.2 define the beam

from the source to that photocell. The solid angle of acceptance or field angle of such an exposure meter is typically about 30° to each side or a cone with a total vertex angle of

about 60°, roughly the cone subtended by a circular object at a distance equal to its

diameter. You can use such a meter to verify the part of the earlier experiment that establishes the invariance of radiance along any ray. Point the meter at a uniformly bright wall

from a distance that is clearly less than the height or width of the wall, whichever is the

smaller. Note that, over fairly wide limits, the "reading" of the meter doesn't change as you move it in or out or tilt it to "view" different parts of the wall. Only when you get

so close that you shade part of the wall in the field of view, or so far away or tilted so far that some of the radiation from the surroundings beyond the uniform wall reaches the

photocell, do you see any change. Accordingly, since the rearrangement or substitution of

rays within the field of view makes no difference, they must all be of the same radiance

as they reach the instrument, regardless of distance or angle, as long as they originate

from the uniformly bright wall surface.

The invariance of radiance along a ray enables us to immediately write down a very

general rule that is often obtained through a fairly involved mathematical derivation. The

rule is that the flux per unit area reaching a point (e.g., on the surface of a receiver)

from a distant extended (source) surface of uniform, isotropic (constant) radiance depends

only on the value of that radiance and on the solid angle subtended by the (source) surface

at that point, the solid angle enclosed by the rays from the extremities of the surface as "seen" from the point. It is otherwise completely independent of the geometrical configuration. For example, in figure 2.10 the heavy lines at A, B, and C, represent three

possible configurations for such an extended source with, in each case, the same uniform,

isotropic (constant) radiance. It is clear from the figure that, since radiance is

invariant along every ray, the configuration of incident ray radiance converging at P will

be exactly the same, and so will produce the same flux per unit area, regardless of which

of the three sources, A, B, or C, is present, as long as that surface has the same radiance L [W-m~2 'sr _1

]

GEOMETRICAL INVARIANCE of THROUGHPUT . Although radiance is the complete distribution

function describing or specifying the spatial distribution of optical radiation in both

position and direction, the physical quantity that flows in a beam of optical radiation is

energy, and the flux is the energy-per-unit-time or power flowing, e.g. , through some reference surface that intersects the beam. It is to energy or power that radiation de- tectors usually respond. We said earlier that we need to be able to associate an amount

or concentration of propagated flux or power with a ray. Actually, we've already done

this in the expressions for the elements of flux or power d$ [W] in eqs. (2.15), (2.16),

and (2.18). However, we want to go back for another look to get a clearer idea of this

quantity and its significance.

From our defining equation for radiance, eq. (2.14), we can directly write the

expression for the element of flux d$(x,y,6,cj>) along a ray of radiance L(x,y,6,4>)

30
'LiL

#* 0) 3 a H •H O

4-1 O) 4J

•H

H cfl c ,-Q 3 n) 4-> ^^ ,-C ca

03

<u

H

u

r 3 G

cfl

M <D

COG

w

•H <u

01 P '-< .<" en p.

4-1 •HT3 0)

'4-1 (3

g o i I o H <U >. u r—

cfl 03 co O i-4 Ph M CU o a e a o

cUu « 'H 4-4 PQ T3

cd *

f-l -d

ff\ 0) <tj /-s. 3 o ^

•H •rl CO P< >

cu o z o u U 4-1 ti d o cu o w ,g

C/3 -rl & o

rH

CM

a)n3

60

•H

tn

31
through an element of surface dA = dx'dy at its point of intersection x,y with the ray

and within an element of solid angle dto = sin9'd9*dtj> about the ray in the direction 9,cf>, as 1

d$(x,y,9,<{>) = L(x,y,9,<f>)'cos9'du) dA [W] . (2.24)

Furthermore, if we know the distribution of radiance L(x,y,6,(j>) across any reference surface over an area A that includs all points of intersection x,y between rays of a given

beam and that reference surface, and over a solid angle oj at each point x,y, that contains all directions 6,<J> for rays of the beam that pass through that point, then by sum- ming up all elements of flux d$ we can obtain the total flux $ in the beam. This can

be done by using integral calculus where

$ =

/ / L(x,y,9,<t>)-cos6-du)-dA [W]. 2 (2.25)

A oj

Note that this is just eq. (2.11) in slightly different form.

In a uniform, isotropic beam, where the radiance has the same constant value L for

all rays of the beam, as with eq. (2.12), this simplifies to

$ = L • / / cos9-dwdA = L-0 [W] , (2.26) A oj

where

H / / cose-dwdA [m2 »sr] (2.27) A oj

is the throughput of the beam [11].

It is clear from eq. (2.27) that the element of throughput is

d0 = cosB-dwdA [m2 -sr], (2.28)

Consequently, the element of flux, given in eq. (2.24), can be written simply as

d$ = L-dG [W]. (2.29)

It is the product of the radiance of a ray and the associated element of throughput.

In a lossless, uniform, isotropic medium, flux or power is conserved and d$ remains

^ome would prefer that the left side of eq. (2.24) be shown as a second-order differential

d <i>, but this raises mathematical questions that we don't want to get involved with. In any

event, the order of the differential is somewhat arbitrary, since d<J> = L-d0 = L-dA-dfi = L* dx'dycosS'dw = L'dx-dycos9'sin9'd9'd<)>

2 This is expanded as $ = / / / / L(x,y,9 ,<j>) -cos9 -sin9 «d9 'd^-dx-dy [W] . (see also Appeny x <$> 9 dix 2).

32
unchanged along an elementary beam (associated with a ray). We equated d<J>i of eq. (2.15)

and d$2 of eq. (2.16) when the points Pj and P2 of figure 2.8(b) were arbitrarily

chosen as any two points along the same ray in such a medium. We then found, in eq. (2.17),

that radiance L is invariant along such a ray. Consequently, if both d<J> and L are invariant along a ray and they satisfy eq. (2.29), then the element of throughput do must

also be similarly invariant along the elementary beam associated with that ray. Furthermore, since ray geometry is not altered in any way by attenuation of the flux propagated

along those rays, the fact that dO is invariant along a ray in a lossless medium with no attenuation means that it must always be so, even in the presence of attenuation. 1

If dG is thus invariant along every elementary beam or ray that makes up a given beam

of radiation, then its integral for the throughput of the entire beam in eq. (2.27) must be

similarly invariant. What this means is that, at any reference surface that intersects the

entire beam, the integral of eq. (2.27) will have the same value as long as no rays have

been added to or taken away from the beam—as long as it is still made up of exactly the same rays. Thus throughput, a purely geometrical quantity, is the geometrical invariant

characterizing any given beam of optical radiation. The larger the throughput the larger

the flux propagated through an optical system. In fact, when the radiance is everywhere

the same throughout the beam, eq. (2.27) shows that the propagated flux is directly proportional to the throughput. Accordingly, the throughput has been found useful as a figure

of merit in making comparisons between different optical systems on the basis of their ability to propagate or transmit radiant flux or power [12].

Another aspect of the geometrical invariance of throughput is seen more readily if we

further simplify eq. (2.27) by assuming that the solid angle u filled by rays of the

beam is exactly the same at every point x,y, where rays of the beam intersect the reference surface, over the entire area A. In other words, there is no vignetting. Then the two integrals comprising the double integral on the right-hand side of the equation are independent of each other, making them separable, so that we can write

= (j dA^-(/ cose-doo) = A'Q [m2 'sr], (2.30)

where

n = / cose-dco [sr] (2.31)

(0

is called a projected solid angle (or, sometimes, a weighted solid angle; for more discussion, see Appendix 2). Here the throughput is exactly equal to the product of the area A

*As a purely geometrical property, it is possible to establish the invariance of throughput, independently of the invariance of radiance, by purely geometrical reasoning. However, the logic used here is quite correct and a great deal simpler.

2 See any standard text on geometrical optics [14,15] for definitions and discussions of

apertures, beams, stops, vignetting, etc. 33
of intersection between the beam and the reference surface and the projected solid angle fi corresponding to (and, for small solid angles, approximately equal to) the solid angle 10 filled by the rays of the beam at its intersection with the reference surface. And even when there is vignetting and the solid angle w varies with position at different intersection points x,y across the area A, it is still true that each element of throughput

dQ can be written as the product of the area element dA and the element of projected

solid angle dfi = cos6*da):

d6 = dA-cos6'do) E dA-dfi [m2 «sr]. (2.32)

In fact, Jones [13] has proposed the term "area-solid-angle product" for the quantity we call throughput.

The foregoing is not just an exercise in terminology and notation. What eqs. (2.30)

and (2.32) tell us is that, since throughput is invariant for a given beam and is also

roughly equal to the product of cross-sectional area and solid angle filled by the beam at its intersection with a reference surface, if the area is reduced at the intersection with

another reference surface, the solid angle must be correspondingly increased to keep the

throughput the same, and vice versa. For example, as illustrated in figure 2.11, the solid

angle o)_, subtended at the slide in a slide projector by the projection optics is

much larger than the solid angle u subtended by those same optics at the distant projection screen. If the area of the slide is A , and that of the projected image is A , the approximate relationships are given by

% A

s -«s % AT 'Q [m2 -sr], (2.33)

where the projected solid angle Q = / cos9-do) % w [sr], Rays leave each point of the

small-area slide surface through a fairly large solid angle and arrive at each point of the

large-area projected image through a correspondingly small solid angle, so that the throughput or "area-solid-angle product" at each reference surface is the same.

A good illustration, with still a different reference surface, other than that at the source or its image, is provided by a simple lens used as a "burning glass" to focus the sun's rays into a very small, hot image of the sun (see figure 2.12). The area A^ of the

lens is very much larger than the area As of the sun's image, but the solid angle or projected solid angle (they are practically the same for small angles where cos9 % 1) w % tt subtended by the sun at the lens near the earth's surface is correspondingly

very much smaller than the projected solid angle Q corresponding to the solid angle of sL converging rays subtended at the sun's image by the lens. For the throughputs to be the

same, the k'Q products at both reference surfaces, i.e., at L and s, must be approximately equal:

A-fiTC % A -ft, [m2 -sr]. (2.34)

Ti Lb s sL i

The exact relationship, which is not so easily evaluated, is

34
Figure 2.11. Slide-projector throughput.

S = slide

= projection optics

1 = image on projection screen

Throughput fy Aq* fl c ^ AT* fi T

[ m2 * sr l> where

projected solid angle 0, = j cos9»du) % to [sr],

35
Figure 2.12. Burning-'g lass throughput.

36
/ / cose -dw -dA = / / cos6 -dw «dA [m2 'sr], (2.35)

wt c A a) T LS s sL

where the quantities on the left-hand side are evaluated over a reference surface at the

lens L and those on the right-hand side are evaluated over a reference surface at the sun's image s, and where the subscript L refers to the lens, S to the sun, and s to the sun's image.

As with radiance, we've considered throughput first for uniform, isotropic media where

the velocity of propagation (hence, the refractive index) is the same everywhere and in all

directions so that rays are all straight lines. In the more general case, where the index

of refraction changes from point to point along a ray, it is the element of basic throughput n

2 *dQ, rather than the element of throughput dG, that is invariant. This follows 1

immediately from the invariance of flux d<J> = L-dO [= (L/n2

)

• (n 2 -dG) ] and the invariance

of basic radiance L/n 2

, when only ray geometry is considered and interactions with matter

that may produce attenuation or augmentation are ignored. We won't make much use of basic

throughput, however, It is mentioned here primarily for completeness. It is basic radi- ance that is usually more useful. Its geometrical invariance along each ray, together with

functions taking into account any interactions with matter along the ray, provide the basis

for transforming from the distribution of radiance across the intersection of a beam with

one reference surface to the distribution at a second intersecting reference surface. Once

the second distribution is known, the total flux in the beam at the second reference surface is correctly given by eq. (2.25), in terms of radiance (not basic radiance), even when

the index of refraction n also varies from point to point over that reference surface.

OPTICAL PROPAGATION — INTERACTIONS with MATTER . The propagation of optical radiation,

especially the interactions with matter along the propagation path that can either attenuate (reduce) or augment (increase) the flux in a radiation beam, or do both simultaneously,

could take up a separate chapter, or more. In fact, the complete, highly sophisticated

treatment of radiative energy transfer, or just radiative transfer as it is usually termed,

is beyond the scope of this Manual, but we will give it some attention later. Right now, we'll limit our treatment to the attenuation of radiance by absorption and/or by scattering

or reflection into other directions. This will be adequate for a large majority of common measurement situations which do not involve optical paths through emitting or strongly

scattering material nor the observation of weak sources that are close, at least in direction, to very much stronger ones. For example, amateur photographers are always cautioned, at least as beginners, to take pictures out of doors on clear days and not to point

the camera near the sun. These restrictions still leave them with plenty of opportunities

As pointed out in an earlier footnote concerning throughput, it is similarly possible to establish the invariance of basic throughput along a ray by purely geometrical reasoning,

independently of the invariance of basic radiance. However, again, the logic used here is

quite correct and much simpler.

37
for taking satisfactory pictures. Incidentally, photography is a rough form of optical

radiation measurement. An ordinary photograph can be, and sometimes is, used as a measurement of the directional distribution of the radiance of the rays converging on the camera

lens (the receiving aperture) from different parts of the angular field (the scene). However, reproducibility and calibration accuracy are not as good as with most other techniques. Nevertheless, it is frequently helpful to think of radiometric problems in terms

of parallel situations in photography, with which many people have at least some familiarity.

To further simplify things in considering attenuation, we'll also confine our attention

to optical paths that begin and end in the same medium. Then we can treat radiance L as the geometrical invariant. We need not concern ourselves with the refractive index and

basic radiance as long as we are only interested in the initial and final values and not in

the radiance within intermediate optical elements of different index.

Since an element of throughput d0 is always invariant and is a purely geometrical

quantity, unaffected by attenuation, any attenuation of the element of flux d<J> = L*d0, as

a ray propagates along an optical path, requires corresponding attenuation of the radiance

L. For example, consider an element of flux d$j reduced to d$ 2 = T*d$ 1 in traversing

the path shown from surface 1 to surface 2 in figure 2.13. At each location, the flux

element is the product of the radiance and the element of throughput: d^ = Iq-dOj and

d$2 = L2 -dQ2 . But dQ x = dQ 2 = d0, so

L2 = d$2 /d0 = T-dfi/dG = fLj (2.36)

In this instance, x = d$ 2 /d<f>i = L 2 /Li, the ratio of final radiance to initial radiance or

the fraction of the initial radiance that is- (successfully) transmitted, is the transmittance of the particular ray path. More generally, a complicated ray path may also include

one or more points of regular (specular) reflectance where attenuation also takes place.

A similar quantitative relation describes that situation, with reflectance p = d$ /d$. =

L /L., the ratio of reflected to incident radiance or the fraction of incident radiance

r i that is (successfully) reflected, in place of transmittance. The corresponding ratio of

final to initial radiances, as a measure of the attenuation due to all transmittances and

reflectances over an extended path, the fraction of incident radiance that is (successfully)

propagated over the entire path, is the propagance

x* E d$pi/d$. = L /L., (2.37) pi

where d$ and L are the propagated quantities reaching the end of the ray path and

d$. and L. are the initial quantities at the beginning of the ray path.

If the propagance is the same over all ray paths that make up a larger beam, it will

have that same value for the entire beam. If not, the overall propagance for the entire

beam, as the fraction of the total incident flux that is propagated to reach the end of

the path, will be an average of the individual ray or elementary-beam propagances. It will

38
Figure 2.13. Transmit tance of ray path

from dAq to dA2 is

t = d<J> 2 /d$i = L2

L l

where d$. = L.dA. «cos0 . *du). 1 IX 11

= L.'dO; i = 1 or 2

39
be a simple average if the incident beam is uniform and isotropic, with all rays starting

out with the same initial value of radiance L.. If they do not, it will be a weighted

average, with the spatial (position and direction) distribution of radiance L. (x. ,y. ,9 .,<(>.

as the weighting function.

Again, since we can't measure the radiance of a single ray or elementary beam, but

only the average value for a beam of rays, we can't measure the ray or elementary-beam

propagance, only the average value for a beam. However, the individual-ray or elementarybeam concept is regarded as the more basic for analysis. It is only in terms of individual

rays that we can completely analyze and account for the behavior of non-uniform distributions interacting with non-uniform media and, finally, with radiometric instruments whose

responsivities may also vary with the position and direction of each incident elementary

beam at their receiving apertures.

TRANSFORMATION from KNOWN RADIANCE DISTRIBUTION to FLUX at ANOTHER LOCATION . A frequently

encountered problem in making optical radiation measurements is to transform from a known

distribution of radiance across the intersection of a beam with one reference surface to the

corresponding distribution and its integral, the flux, across the intersection of the same beam with another reference surface. For example, given the distribution of radiance from

a known source, e.g., a standard for which a certificate has been issued by NBS, what is

the distribution of incident radiance at the surface of a receiver after the beam has

passed through an atmosphere and optical elements for which the overall path propagance is

known for each ray of the beam? Or, conversely, given the measured values of radiance as

a function of position and direction over a reference surface, measured by a scanning or imaging radiometer, and the measured values of path propagance to a reference surface at the source, what is the corresponding radiance distribution and its integral, the emitted

flux, at the source?

The first step, in either case, is to establish corresponding coordinates for ray

position (point of intersection) and direction, between reference surfaces intersecting the

beam at the two locations. We need to know, for all rays of the beam, the value of the

coordinates of the point X2,y2 and direction 92,<J>2 where a ray intersects the second

reference surface when given the coordinates of the point xj ,y\ and direction 9 j

,

<(>

for the same ray where it intersects the first reference surface, or vice versa. While

not trivial, a table or formula for providing these corresponding coordinates for the same ray is often not a very difficult problem. In any event, it is a ray-tracing problem for

which adequate treatments should be readily available in texts and references on geometrical optics, so we won't go into it any further at this point. Particular situations will

be covered in Part III, when we go more thoroughly into applications. What concerns us now is that either set of coordinates, x\ ,y\ ,&i ,<$>i or X2 ,y2 >®2 >^Z > unambiguously identifies the same ray. Hence we may express any property of that ray in terms of either set

of coordinates, or, even another set, as convenient. For example, we can express the

radiance of a given ray at the second location as a function of the coordinates of that ray

at the first location, L2 (xi ,yi ,9j ,$1 ) , and vice versa, Li (X2 ,y2

e 2 Al) • Accordingly,

40
we'll drop the subscripts from the coordinates and show them just as x,y,9,cj>. They can be

the coordinates at any convenient reference surface that intersects the beam. Of course,

the same set must be used consistently for all quantities relating to the same rays in a given expression in order to establish meaningful relationships.

On this basis, we can write the propagance for a given ray, as a function of its

coordinates, from eq. (2.37), as

T*(x,y,0,<|>) = L (x,y,e,<J>)/L

i (x,y,6,<j)). (2.38)

Then, if we know the exitent propagated radiance L (x,y,9,<t>) and the ray-path propagance

P

t*(x,y,6,<}>) , the incident initial radiance is

L

i (x,y,6,<J)) = L (x,y,6,<}>)/T*(x,y,e,<i)) [W-m-2 -sr" 1

] . (2.39)

Conversely, when the incident radiance L . (x,y,6 ,<(>) and propagance x*(x,y,0,<t>) are known,

the exitent propagated radiance is found as

L (x,y,6,<i>) = T*(x,y,9,<f>)-L.(x,y,9,<f>) [W-m-2-sr" 1 ]. (2.40)

P !

Often the quantity that is finally desired is not the detailed distribution of radiance but the integrated total flux in the beam at the desired location. The corresponding

expressions are the integrals of the quantities in eqs. (2.39) and (2.40), respectively.

The integration is carried out over the area A containing all points of intersection x,y

on the selected reference surface and over the solid angle w at each point x,y that

includes the directions 9, (J) for all rays of the beam that pass through that point of

intersection.

*. = / / [L (x,y,e,<|>)/T*(x,y,e,«|))]-cose«du)«dA [W] . (2.41)

1 A to P

$ / / L.(x,y,9,<t>)-T*(x,y,9,<}))-cos0'dwdA [W] . (2.42)

P A to 1

It should be reemphasized that these equations are based on the assumption that each pair of

values of the radiances L. and L for the same ray (same value of x,y,8,<t>) exist at

1 P points of that ray where the refractive index is the same (in the same medium) . If this is not the case, the transformations must be modified to take into account the different

refractive indices at the two locations. 1

If n. is the refractive index at the beginning of the propagation path, n the index

1 P at the end of the path, and n the index at the location of the intersecting reference

surface for the coordinates x,y,8,<j>, these relations become:

41
T*(x,y,e,<|>) = d* (x,y,8,«|»)/d*

1

(x,y,e,«(.) = (L »d0 )/(L

± 'de

±

)

(L /n 2 )-(n 2 -d0 ) "

P

, P

7 h

2— = (Ln/M'W), (2.37a)

(L./nJ 2 )-(n. 2 -d0j P x x P

x i 11

since, by the invariance of basic throughput, n

2, d0 = n

2 -d0.. Then

p p i 1

$. = / / [L (x,y,e,(f>)/T*(x,y,e,<|>)]-(n2 /n 2

) -cos9 -d(WA [W] , and (2.41a)

1 A a) P P

$ = / / L.(x,y,e,<j>)-T*(x,y,6,ct>)-(n2 /n. 2 )-cose-dwdA [W] . (2.42a)

P A w x 1

SUMMARY of CHAPTER 2 . In order to obtain an expression for the amount of radiant (luminous

or photon) flux propagated along a ray, we first introduce radiance (luminance) as the com- plete distribution of optical radiation with respect to the spatial parameters of position

and direction. It is defined at a point on a reference surface in the direction of a ray

through that point as (see figure 2.7)

?.7.e.» 'teta' [»-»-2 --]. (2.U)

This quantity, radiance, is geometrically invariant along any ray, in the direction of the

ray, in a uniform, isotropic, passive, lossless medium. Across smooth boundaries between

different media, or in media with varying refractive index, it is the basic radiance L/n

(where n is the index of refraction at the point where the radiance is L) that is similarly invariant along any ray. However, in the same medium (same refractive index), even after propagation through another, e.g., through an optical element of different index,

just the radiance L is invariant (neglecting attenuation), with the same value in the

direction of a ray at all points of the ray in that medium (e.g., air).

The element of flux associated with a ray is given by

d$(x,y,e,(f1 ) = L(x,y,e,<j>)'cos6'd(WA [W] (2.24)

= L-d0 [W], (2.29)

where d0 = cosB-dwdA [m2 -sr] (2.28)

is the element of throughput associated with the ray through the point x,y in the direction 9,<(>. The throughput element d0 is also geometrically invariant along the ray at all points in the same medium (same index n)

In order to evaluate the total flux $ in a beam of radiation where it intersects

some convenient reference surface, it is necessary to know the distribution of radiance

over both the full area A, and the solid angle u at each point of that area, that

42
include all of the rays that make up the beam. The total flux in the intersecting beam is,

then,

$ =

/ / L(x,y,9,<j>)-cose-dwdA [W] . (2.25) A oo

If that radiance distribution is known only at another reference surface, it can be transformed to the desired values through the invariance of (basic) radiance along each ray of

the beam, taking into account also interactions with matter over the intervening propagation paths that may attenuate or augment the radiance of each ray. Usually, the points of

interest will both lie in the same medium (same refractive index; e.g., in air) where radiance is invariant, with no need to resort to basic radiance and refractive indices. Also, most radiation measurement situations involve direct paths through passive media where the

only interactions are those that produce attenuation. For our purposes, the most convenient measure of the result of that attenuation is the propagance

T*(x,y,e,<|>) = L (x,y,6,<}.)/L

i

(x,y,9,(t>) (2.38)

over the ray path from the point x.,y., where an incident ray, of radiance L. (x. ,y . ,9 . ,6 . )

11 11111

intersects a reference surface in the direction 9 . ,<)> . , to the point x ,y , where the same 11 p p ray intersects a second reference surface in the direction 9 ,<j> with propagated radiance

P P L (x ,y ,9 ,<t> ). Either set of coordinates x. ,y . ,9 . ,d> . or x ,y ,9 ,A> , or even those at

p p

J p p

Y p i' J i i i p"p p p a third reference surface (also in the same medium—same index) that intersects the entire

beam, may be used in eq. (2.36), so the subscripts have been dropped there. It is only

necessary to use, consistently, for all quantities in the same equation, such coordinates

that uniquely identify each ray by its point of intersection and its direction with respect

to the same reference surface.

Given (1) the incident radiance distribution L . (x,y,9 ,<j>) at a reference surface, e.g.,

the surface of a source, and (2) the propagance T*(x,y,8,<t>) of each ray over the path from

its intersection with the first reference surface to its intersection with a second reference surface, e.g., the surface of a receiver, the transformation to the distribution of

propagated radiance L (x,y,9,<t>) across the second reference surface is

P

L (x,y,9,<f>) = L.(x,y,9,<J>)-T*(x,y,9,cj>) [W-nT2 -sr" 1

] . (2.40)

P i

Conversely, if the propagated radiance distribution has been measured and the trans- formation back to the incident radiance distribution at the first reference surface is

desired, it is

L.(x,y,9,4>) = L

p (x,y,9,4>)/T*(x,y,9,<i>) [w-nT2 sr

-1

]. (2.39)

Finally, the corresponding transformations to obtain the integrated total flux in the

beam at the second location, in each case, are, respectively,

43
$ = / / L.(x,y,e,<t>)-T*(x,y,9,ct>)-cose-dwdA [W] (2.42)

p A to i

and

$. = / / [L (x,y,e,<t>j7T*(x,y,e,<j>)]'cose-d<WA [W]

(2.41)

1 Am P

It is assumed that both reference surfaces are in the same medium (same refractive index) as well as any third reference surface that might be used for the ray coordinates in an unusual

situation. If this is not the case, see eqs. (2.41a) and (2.42a) in the footnote at the end

of the last paragraph preceding this Summary.

44
Chapter 3. Spectral Distribution of Optical Radiation

by Fred E. Nicodemus and Henry J. Kostkowski

In this CHAPTER . We develop the concept of spectral radiance . This is the basic quantity

for specifying the distribution of radiation relative to wavelength and, at the same time,

relative to position and direction. We examine the interrelationships between spectral

radiance and radiance, particularly in relation to geometrical invariance along a ray. Finally, we look at the way in which the total flux in a beam is evaluated from the distribution of spectral radiance. In the main, this is almost a duplication of the treatment in

Chapter 2 except for the addition of the new variable. However, we've spelled it all out

in much the same detail again for those who want it. Those who prefer to do so can skip

directly to the summary at the end of the Chapter and use it to decide what, if any, details

they need to review more fully in the body of the Chapter.

SPECTRAL RADIANCE . In general, the optical radiation emitted by most sources, the propagation over many paths, and the responsivity of many detectors, all can vary greatly with the

spectral parameter, wavelength. The combined result of all such effects of wavelength

(spectral) variations involved in a measurement is usually substantially greater than the

effects of geometrical variations (variations in ray position and direction) . An example

of the degree of variation that can occur is seen in figure 3.1 which shows the spectral

distribution of the radiation emitted by the central, uniform portion of a 2750-K tungsten

strip lamp and the spectral responsivity of a frequently used photomultiplier detector.

The radiance of the lamp increases by a factor of 7.5 from 450 to 800 [nm] while the

photomultiplier responsivity decreases by a factor of 30 over that same interval.

In order to extend the concept of radiance so that it also covers distribution with

respect to wavelength, consider again the experiment illustrated in figures 2.2 and 2.3,

] The spectral parameter is commonly given in three different ways. They are (1) frequency

v (=c/X ) [THz], (2) wavelength X [nm] , and (3) wave number a (=1/X Q

) [cm-1

] (where

c % 3xl0 8 [m's-1

] is the vacuum speed of electro-magnetic radiation, X Q (=n*A) [nm] is

the wavelength in vacuum, and n is the index of refraction). The units shown are those

typically used in each case, but they are not consistent. For v = c/A Q to be in [THz]

with A Q in [nm] , we must use c ffc 3*10 5 [km«s-1

], and for o = 1/AQ to be in [cm-1

] it is obvious that A must be given in [cm]. Incidentally, these wave-number units,

widely used by spectroscopists, are called "reciprocal centimeters." Although frequency v

is the basic spectral parameter, that remains unchanged as a ray passes through different

media, we will follow common practice by expressing most spectral quantities in radiometry

in terms of wavelength. Note, also, that the term "spectral," itself, can be ambiguous;

there are also space-frequency spectra, referring to repeating patterns in the spatial distribution of radiance, and modulation- or scintillation-frequency spectra, referring to frequencies f <<v [Hz] of variation in the 'average radiant flux or power in a radiation

beam. We will consistently use "spectrum" or "spectra" and "spectral" alone, without a modifier, only to refer to the radiation parameters v, X, and/or a. 45
SSION FROM

IGSTEN LAMP

LlJ

>

<

-J

UJ

RESPONSE

PHOTOMULTIPLSER

Qnm]

Figure 3.1. Large variations of

radiometric quantities

with respect to wavelength.

46
where we found that the flux in the beam defined by two small apertures is given approximately by

AAi -cosei -AAo •cos9o

A$ % L • —— — [W] (2.3)

D

2

(this time we use the "approximately equals" sign to clearly recognize that this is an approximation). Now, in this same experiment, we insert into the beam, one at a time,

filters that are transparent (transmittance t = 1) only to certain desired wavelengths

and are opaque (transmittance T = 0) to all other wavelengths, completely blocking them, as shown in figure 3.2. Of course, real filters can only approximate those ideal characteristics, but it's useful to make the assumptions for analysis to clarify the related concepts

With such a set of filters, of successively decreasing spectral intervals (bandwidths) AX about the same central wavelength X c , we find that the measured flux A$ in the beam is now approximately proportional to the spectral-wavelength bandwidth AA, as well as to the

spatial factors, as before. Also, as with the spatial parameters, as the wavelength interval decreases, the proportionality becomes more exact. In fact, we find that, even though

we are stopped again by the minimum amount of flux A$ required for any measurement to be

made, it is once more analytically useful to assume a continuous underlying distribution to which the limiting process of calculus is applicable. Accordingly, we assume that, when

the spectral interval is made arbitrarily small, the proportionality is exact and the proportionality constant is called spectral radiance (more explicitly, spectral -wavelength

radiance) and is denoted and defined as

Lim A<i> L^ = AX -+ AA1+ (AX-AAj •cose 1 'AA2 'cos9 2 )/D2 AA2

->-

d

3 $ = — (3-D

(dX'dA 1 'cos9 1 'dA2'cos9 2 )/D2

These relations correspond to those in eqs. (2.5) and (2.6) and, like them, are important

aids for many who find this approach to the concepts easier to understand. Later, however,

for useful applications, we'll go back to the approach of eq. (2.12) and employ the equivalent expressions in terms of position and direction at a single location.

From eqs. (2.6) and (3.1), it is clear that spectral radiance is the spectral distribution 1 of radiance, the spectral concentration per unit wavelength interval as a function of

^The CIE-IEC International Lighting Vocabulary [5] does not use the term "spectral radiance."

Instead, it speaks of the value at a particular wavelength as a "spectral concentration of

radiance" and of the spectral concentration as a function of wavelength as the "spectral

distribution of radiance." However, we follow the widespread practice in this country in our use of the term "spectral radiance" for both of these quantities.

~47
I- u

z

<

CO

z

<

<

or

Io

bJ

CL

(/)

WAVELENGTH X

Figure 3.2. Ideal filter set with

spectral transmittances

all centered about the same X .

48
wavelength

dL MA') =

A

x

' dX (3.2)

A*

Accordingly, it provides a means for specifying the distribution of optical radiation with

respect to wavelength and, at the same time, with respect to position and direction. Furthermore, by integrating both sides of eq. (3.2), the value of radiance in a wavelength

band from Xj to X 2 is obtained as

X 2 L„._^ = / VdX. (3.3)

'(Ai-A 2 ) \

For example, the wavelength distributions of emitted radiation for three frequently used

light sources are plotted on relative scales in figure 3.3. If the ordinate scales were

adjusted to represent values of spectral radiance, the areas under each curve between

limiting wavelengths X

1 and X 2 , i.e., the integrals of eq . (3.3) between those limits,

would represent the emitted radiance in that spectral-wavelength band for each lamp.

As with radiance, we want to define and work with spectral radiance as the property of

a ray at its intersection with a reference surface (figure 2.7). Again, recognizing that

(AA2 *cos6 2

) /D 2 = A(jj 12 is the solid angle 1 in steradians [sr] subtended at AAj by AA2 , as shown in figure 2.6, we can rewrite eq. (3.1) as

Lim A$

L, = AA >

X . „ AA«cos6'9wAX

Aw *

AX -+

= ^a' d A A^ [W-m-^sr-i-nnT1 ]. (3. A) dA*cos9 'dw'dX

The SI units for spectral radiance are often given as [W-m-3 'sr-1

]

, using the same unit of

length (meter [m]) for wavelength as for other distances. However, it then appears, misleadingly, to be a volume concentration, which it certainly is not. (In fact, we'll see later that there is a radiometric quantity called "sterisent" that is correctly given just

See Appendix 2 for definition and discussion of the concept of solid angle.

Similarly, if the wave-number unit is given as the reciprocal meter [m-1

], the unit of

spectral (wave-number) radiance L 5 dL/da would be [w«m_1

• sr

-1

] , rather than

[W-m~ -sr" -cin], as we prefer it because it correctly suggests a simultaneous distribution

with respect to area, solid angle, and wave number. An earlier footnote suggests that

frequency v [THz] be regarded as the basic spectral parameter. From that standpoint,

wavelength and wave number can be considered as indirect measures of frequency, rather than

as lengths or reciprocal lengths, per se, as further justification for treating them as having different dimensionality from other lengths.

49
r

i c. O

O

-rro

u

43 o M •u •HrH cuo

rH d

cd cfl o •H

CJ •H ^ o CX>+-i • >, >H CO «» 4-1 C M ** t>o cu

cu •H ,fl cu cn -u u o

XI o

4J d cu •< >-l 4=

cn -u o •H

4-1 O X e

CU 4-1 U

o h- o

•H

cu cu o o en

en w 4-J <J) z •H CO rH UJ

cu . CU

cn u _I o

4-J •H CU UJ d

c >

d m

o d > •H cu o

> o

o

<

5

3

•HH

•U

elati f

any

1^ cn •H

>-i o

X) d 4J •H X! rH toO CO cn -rl u cu cu u x

o J-i cu d cu a O XI

C/J cn -m o

o

ID

O

ro

CO

CO

cuu

d

•H

Pn

N0ISSIIAI3

~IVHlD3dS 3AllVH3a

50
those units and is quite correctly treated as a volume concentration.) Accordingly, we prefer to use a different length unit (also of more appropriate magnitude) , the nanometer

[nm] , for wavelengths and thus make it easier to recognize and keep track of the different

dimensionality of the spectral parameter in checks of unit-dimension consistency.

The fully explicit defining equation for spectral radiance, then, is

d

3 <J>(x,y,9,<J>,A) „ , . L

A (x,y,e,4>,A) e dA. cos9 . du) . dX [W.m-z.ar-i.nm-1], (3.5)

where

x,y,9,<(> are defined, relative to the intersecting ray and surface element dA, in

connection with eq. (2.13) (and figure 2.7);

L. (x,y,6 ,<j> , A) [W*m-2

'sr -1 -nm

-1

] is the spectral radiance at the point x,y in the

direction 9,<j> and at the wavelength A; and

d

3 $(x,y,9 , <j>, A) [W] is the radiant flux through the surface element dA = dx*dy [m2

] about the point x,y within the element of solid angle dw = sin6 # d6*d<j> [sr]

in the direction 9,<|> and within the elementary wavelength interval dA [nm]

about the wavelength A.

GEOMETRICAL INVARIANCE of SPECTRAL RADIANCE . In the last chapter, considering only the

spatial parameters of position and direction, we found that the basic radiance L/n is invariant along any ray, in the direction of that ray, in lossless passive media where the

index of refraction may vary and where its value is n at the point where the radiance is

L. The same is true for the basic spectral radiance L /n2 or L /n2

, in terms of frev a quency v or wave number a, respectively. However, wavelengths depend on the refractive

index of the medium, so that A(n) = A Q /n where A(n) is the wavelength in a medium of

refractive index n, and A is the wavelength of the same ray in a vacuum. To see how

this affects the invar iance of the spectral radiance L = dL/dA, let's go back to the

A situation depicted in figure 2.9. We now assume that the incident ray contains only wavelengths in an elementary spectral interval dAj = dA /nj so that its radiance, in terms of

the incident spectral radiance L, , is given by

A l

dLi = L. -dAj = L, -dA / ni . (3.6)

A l A l

Similarly, the refracted radiance in the second medium, of index n2

, is now given, in terms of the refracted spectral radiance L, , by

*2

dL2 = L

A -dA 2 = L

A -dA /n2

. (3.7)

According to eq. (2.22), these values of radiance are related by

dLi dL?

(3.8)

^ee Footnote 2 on p. 49.

2 2 n l n2

51
in which we can substitute from eqs. (3.6) and (3.7) and cancel the common value of dA

to obtain

A l A 2 n l 3 n 2

3 (3.9)

The invariant or basic spectral-wavelength radiance, in terms of local wavelength A(n) in

a medium of varying refractive index n, is, thus, L/n3 (not L/n2 ). Some workers,

A A however, prefer to deal with this problem by expressing all spectral quantities in terms of vacuum wavelength X = n*A(n) and then using the basic spectral radiance in the form

L /n2

. Also, in a dispersive medium, where the refractive index varies significantly with

° wavelength as n(A) or n(A ), the value of n that would make the basic radiance L/n2 invariant depends on the spectral content of the radiance L = JL *dA, which may also vary

A along a ray. If the medium is highly dispersive, the variation of n with A can also

significantly affect L/n 3

, although the discrepancy is insignificant in air at room temA peratures. However, the basic spectral radiance in terms of A Q , v, or a is L /n2

,

i 2 instancesL /n, or L /n 2

, respectively, and is geometrically invariant along each ray in all

TOTAL FLUX in a RADIATION BEAM . From eq. (3.5), we can write the expression for the element of flux d$(x,y,9,cf>, A) along a ray of spectral radiance L (x,y ,9 ,<{> , A) through an

A element of surface dA = dx'dy at its point of intersection x,y with the ray and within

an element of solid angle dw = sin9*d9'd<j> about the ray in the direction 9,<j> and also

within a spectral-wavelength element dA about the wavelength A. It is

d$(x,y,9,cj>, A) = L, (x,y,6,<J>, A) 'sin6 'dwdA-dA

= L

A -d0-dA [W], (3.10)

the counterpart of eqs. (2.23) and (2.28), now also taking account of spectral-wavelength

variations and functional dependence. Then, if we know the distribution of spectral radiance L (x,y,9,<j>, A) over an area A that includes all points of intersection x,y between

A rays of a given beam and some reference surface, over a solid angle to at each point x,y

that contains all directions 9,<f> for rays of the beam that pass through that point, and over a wavelength interval AA that includes all wavelengths A for which there is a significant amount of spectral radiance, 1 the total flux $ in the beam at its intersection

with the reference surface is given by

$ = / / / L (x,y,9,,4>,A)-dA-cos9'dwdA f w l

• (3.11)

A w AA

Strictly, the integration should cover all wavelengths A for which L / 0. Also, use

A of the symbol A A in no way implies that the interval is small; it may be of any size.

52
This is the counterpart of eq. (2.24), to which the spectral-wavelength dependence has now

been added. For a more completely explicit expansion of this triple integral, see the

footnote below eq. (2.24) or Appendix 2.

At the beginning of this Chapter, we called attention to the wide spectral variations

that commonly occur. It is quite unusual for the spectral radiance L (A) of a source to

A be a constant over a wide band of wavelengths. However, there are many situations where

the radiance L = / L (A)«dA is a constant, to a useful degree of approximation (practically

A

A uniform and isotropic) throughout a beam of radiation. Hence, even when there is a significant spectral variation, the relationships given in eqs. (2.25) through (2.33) may be applicable and are often useful.

TRANSFORMATION from KNOWN SPECTRAL-RADIANCE DISTRIBUTION to FLUX at ANOTHER LOCATION . We've

already seen that detector response can be highly variable as a function of the spectral

parameter wavelength. The same is true of other interactions with matter that produce

attenuation along a propagation path. Accordingly, eq. (2.36) must now be rewritten to define the spectral-directional propagance for a ray through the point x,y in the direction

0,<f> and of wavelength A (at that point and in that direction) as

T*(x,y,6,<|>,X) = L. (x,y,9,<(.,X)/L. . (x,y ,9 ,<j) , A) . (3.12)

A,p A,i

Similarly, eqs. (2.37) and (2.38) become, respectively,

L, .(x,y,6,<|),A) = L, (x,y,e,<)),A)/T*(x,y,9,<}>,A) [W-m-2 -sr" 1 -nra" 1] (3.13) A,l A,p

and

L, (x,y,6,<f,,A) = L. .(x,y,6,«(.,X)-T*(x,y,e,«|>,X) [W-m"2 -sr" 1 -nm" 1] (3.14)

A,p A,i

for the transformation from a known spectral radiance at one location to that at another

location along the same ray when both points are in the same medium (same refractive index)

Likewise, for the integrated total flux in the beam at the desired location, eqs. (2.39)

and (2.40) become, respectively, * =

/ / / [L, (x,y,e,<j,,A)/T*(x,y,e,((),A)].dA-cos6-dwdA [W] (3.15)

1 A to AA

A ' P

and

$ = / / / L, .(x,y,9,<£,A)-T*(x,y,e,<|>,A)-dA-cose-da>-dA [W] . (3.16)

P A a) AA *»*

We reemphasize that these relations hold only when the point of incidence, where L . A , 1 exists at the beginning of each ray path, and the point of exitence, where L exists

A ,P at the end of the ray path, are in the same medium (same refractive index). Also, because

of the geometrical invariance of both the radiance and the associated element of throughput

53
along each ray, the reference surface for the coordinates x,y,6,<}>,A may be at either

location, or even at a third location, as long as the same coordinates and the area A,

the solid angle u>, and the wavelength interval A A (all containing only, and all, rays

of the beam) are all consistently referred to the same reference surface at its intersection with the beam. Thus it is possible to choose the reference surface where these

quantities are easiest to evaluate.

In the more general case, where the refractive indices at the points of incidence and

exitence are not the same, the transformation must be based on the geometrical invariance

of basic spectral radiance, taking into account those refractive indices, rather than on

the invariance of spectral radiance as above. Otherwise, everything proceeds in the same

way. It's all quite straightforward, but the expressions are longer and more complicated

and they are seldom needed.

REMAINING RADIATION PARAMETERS — TIME and POLARIZATION . The spatial and spectral parameters, which have now been covered, have traditionally received the most attention since

their effects are almost always significant. On the other hand, we may often safely ignore

the remaining parameters, time or frequency of fluctuation or scintillation, and polarization, especially when an uncertainty of a few per cent or more is adequate. Accordingly, we will put off our discussion of the remaining parameters for a while and turn, next, to the measurement equation, to more about spatial distributions, to thermal radiation, and to photometry. But we strongly caution the reader that, in doing so, we definitely are not recommending this as the ultimate or correct approach for accurate measurements. It is our conviction that many of the problems and inconsistencies that arise every day in connection with optical radiation measurements can be traced to these remaining parameters, particularly to polarization effects, that are too often ignored. We present things in this

order because we feel that it will be easier for the reader to grasp the new concepts when

they are presented in this way. But we strongly emphasize that, until all of the parameters are included, the treatment of the measurement equation and related topics must be

regarded as only preliminary and incomplete. It will have a substantial area of usefulness,

but the limitations on its usefulness must not be forgotten.

SUMMARY of CHAPTER 3 . The distribution of optical radiation with respect to position,

direction, and wavelength—the spectral-geometrical or spectral-spatial distribution—is

spectral radiance

L, (A') = dL/dAl , [W-m-^sr-^-nm- 1 ] (3.2)

A A

or , more completely

The spectral parameter of wavelength A [nm] may be replaced in these defining equations

by the more basic spectral parameter of frequency v [THz] , or by the other spectral

parameters of vacuum wavelength A Q

[nm] and wave number a [cm-1

], both of which also

54
remain constant during passage through different media (different refractive indices).

Like radiance L, spectral radiance L. is geometrically invariant along any ray, in

the direction of the ray, in an isotropic, passive, lossless medium. In media of varying

refractive index n, the invariant basic spectral radiance depends on the spectral parameter. For frequency v, for wave number a, and for those who transform always to the vacuum wavelength X , it is L /n2

, L /n2, and L /n2

, respectively. However, for

v o Aq

the spectral parameter of "local" wavelength X(n), where the refractive index is n, the

basic spectral-wavelength radiance is L /n . A The element of flux associated with a ray of spectral radiance L is given by

A

d$(x,y,e,<f>,A) = L, (x',y,6,<j>,A)«cose«dco'dA«dA [W]

A

= L -dO-dA [W], (3.10)

A

The total flux in a beam, given the distribution of spectral radiance at its intersection

with a convenient reference surface, is

$ =

/ / / L (x,y,6,<j>,X)-dX-cose-dwdA [W] . (3.11)

A w AX

Except for the fact that the additional radiation parameters, time or frequency of modulation or fluctuation and polarization, have been ignored, eqs. (3.10) and (3.11) are general

relations of wide validity. In particular, they are not in any way dependent on the index

of refraction.

For the simple transformations from a known spectral radiance or spectral-radiance distribution for the rays of a beam at one location to the distribution or the integrated

value of flux in the same beam at another location, based on the geometrical invariance of

spectral radiance, it is necessary that both locations (both intersecting reference surfaces)

be in the same medium (same refractive index). The transformation relations are:

L

x>i (x,y,e,<M) = L^

p (x,y,6,c|>,X)/T*(x,y,e,<KX) [W-m-^-sr" 1 -nm" 1

] , (3.13)

L (x,y,6,<j>,X) = L^ .(x,y,e,(|>,X)-T*(x,y,e,4>,X) [W-nT2 -sr" 1 -nnT 1

] , (3.14) A,P A,l

$. = / / / [L. (x,y,e,<j>,A)/T*(x,y,e,<j),X)]-dA-cose-daj-dA [W] , (3.15) 1 . . , A , p A u> AX "' P

and

* = / / / L .(x,y,6,(|),X)-T*(x,y,e,<f),X)-dA-cose-dwdA [W]

(3.16)

p . ., A.i A a) AX '

When the refractive indices at the two locations are not the same, similar but somewhat more

complicated expressions must be used, based on the geometrical invariance of basic spectral

radiance, taking into account those refractive indices.

^ee footnote on first page of Chapter 3.

55
Appendix 1. Units and Unit Symbols

The base units of the International System (SI) are given in Table Al-1, which follows Table 1 of [16], except that we have added the square brackets enclosing each unit

symbol. That practice was adopted to emphasize the dimensionality of the units and the

usefulness of that dimensionality in routine unit-dimension-consistency checks and analyses to cope with the great diversity of nomenclature in the literature on optical radiation measurements.

Table Al-1

SI Base Units

Quantity Name Symbol

length meter [m]

mass kilogram [kg]

time second [s]

electric current ampere [A]

thermodynamic temperature . . kelvin [K]

amount of substance ..... mole [mol]

luminous intensity candela [cd]

Of the SI base units, the one of particular interest in the branch of optical radiation

measurements known as photometry is the candela, defined officially as follows: "The

candela is the luminous intensity, in the perpendicular direction, of a surface of

1/600 000 square meter of a blackbody at the temperature of freezing platinum under a pressure of 101 325 newtons per square meter {13th CGPM (1967). Resolution 5}." [16].

56
10

1015

10 12

10 9

10 6

10 3

10 2

10 1

exa

peta

tera

giga

mega

kilo

hecto

deka

Table Al-2

SI Prefixes

Factor Prefix Symbol

E

P

T

G

M

k

h

da

Factor Prefix

10" 1

lO"2

icr 3

icr 6

10- 9

io- 12

lO" 15

10-18

deci

centi

milli

micro

nano

pico

f emto

atto

Symbol

The names and symbols listed above are used, in combination with the names and symbols, respectively, of the SI units, as prefixes to form decimal multiples and submultiples of those units. (This table is based on Table 7 of [16] to which we have added

the two recently adopted 1 prefixes "exa" and "peta" for 10 18 and 10 15

, respectively.)

l See, for example, NBS Dimensions, Vol. 59, No. 10, October 1975, p. 229.

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59
Table Al-5

Some Additional non-SI Photometric Units

Quantity

luminance

ergolumic

intensity*

illuminance

luminance

luminance

luminous flux

illuminance

luminance

The CIE [5] seems not to have any term for this quantity, nor does there seem to be any

in general use other than the term for the units, although "beam-candlepower-second" is

also used at times. This term for the quantity is taken from Jones' "phluometry" proposal [17]

Unit Symbol SI Equivalent

apostilb [asb] = ft -1 [cd*m-2

] candela-second [cd-s] = l[cd-s]

footcandle [fc] = [lm-ft -2

] = 10.764[lm-m_2

] footlambert [fL] = Tr-^cd'ft -2

] = 3.426[cd-m-2] lambert [L] = tt -1 [cd-cm-2

] = lO^-fr-ifcd-m-2

] light-watt** [£W] = |K_1

1 |-[lm] m ' % 680_1 [lm]

phot [ph] = lflm-cm-2

] = lO^flm-m-2] stilb [sb] lfcd'cm-2

] = lO'+tcd-m-2]

**The light-watt [£W] is related to the unit of radiant flux, the watt [W] , by

760

•

" = / V(X)-$ (A)-dA [JIW]-,

x, 3g0

e,A

where

V(A) [dimensionless] is the photopic spectral luminous efficiency [5,18],

A [nm] is the wavelength,

(A) [W«nm_1

] is a distribution of spectral radiant flux as a function of

e,A wavelength, and

$ [&W] is the luminous flux in light-watts of the radiation described by

the spectral distribution $ \(^ '

The luminous flux, in lumens, in this same beam of radiation is given by

v where

K •« % 680-1> [lm], m x. Jo

t K % 680 [lm«W_1

] at A % 555 [nm] is the maximum spectral luminous m

efficacy (of radiation) [5,18].

Note that both the lumen and the light-watt are units of luminous flux. They have the same dimensionality and differ only by the numerical scale factor |k | ty 680. There

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61
Appendix 2. Spherical Coordinates and Geometrical Relationships

SPHERICAL COORDINATES (solid polar coordinates ) . In almost all optical radiation measurement situations, we are concerned with the flux through a reference surface that intersects

a beam of radiation. We need coordinates for specifying, at each point of that reference

surface, the direction of every intersecting ray incident from (or exitent into) the entire

hemisphere above the plane tangent to the reference surface at that point. That is the

plane containing the element of surface dA at the given point. The most useful coordinate

system for this purpose is illustrated in figure A2-1. The given point is taken as the

origin. The normal to dA at is chosen as the polar axis for spherical coordinates

or the Z-axis for rectangular coordinates. The tangent plane, containing the element dA,

is then the X-Y plane. A straight line extending from in some convenient direction

in that plane is selected as the azimuth reference or X-axis. The spherical coordinates of

any point P in space, as shown in the figure, are then p,9,<f>, where p is the length

of the line OP, 6 is the polar angle between the line OP and the polar axis, and <j) is the azimuth angle in the tangent plane between OP', the projection of OP on that

plane, and the azimuth reference (the X-axis). The corresponding rectangular coordinates

of P are x,y,z, where

x = p«sin9'cos<f>,

J y = p'sin6'sin<j>, and 1 (A2-1)

z = p«cos9, I

so that

p = (x2 + y

2 + z

2

) 2

,

I

6 = tan^Kx2 + y

2

)

Js /z], and . (A2-2)

<j> = tan_1 (y/x). '

Usually, we are concerned only with the direction of a ray rather than the coordinates of

a particular point on the ray. The direction of the ray through and P is specified

by just the two angles 6,<j>. Also, when they are often confined to the hemisphere above

the tangent plane, < 6 < tt/2 [rad] and < <j> < 2tt [rad].

A SOLID ANGLE . One of the best ways to grasp the concept of a solid angle is by analogy

with the corresponding features of the more familiar plane angle. Accordingly, we'll start with a brief look at plane angles.

A plane angle, formed by two straight lines that meet at a point, the vertex, is

defined as the locus of all directions that may be occupied by either line as it is rotated

about the vertex to bring it into directional coincidence with the other line. For example,

in figure A2-2, the lines OA and OB form the angle 6 at the vertex 0. That is the

angle filled by all of the intermediate positions that could be occupied by either line as

it is rotated about to bring it into coincidence with the direction of the other line.

Note that the acute angle 9, as shown, is covered by the most direct sense for such

rotation. However, these lines also define an exterior obtuse angle of (360 - 9) [deg]

62
r = (x +y ) = p*s\r\9 = z* tan 9

Figure A2-1. Spherical and rectangular

coordinates of P(p,e,cf>) = P(x,y,z)

63
"I •.

F B

Figure A2-2. The same plane angle is subtended at by:

1. the circular arc CD,

2. the straight line EF,

3

the curve E ' F

'

4. the plane figure E F .

64
or (2tt - 6) [rad] if the rotation is in the opposite sense [the units are defined in the next paragraph]

The length of each line is immaterial, just so it's long enough to be a straight-line

segment. In fact, an angle is also formed when two curved lines meet if it's possible to draw the tangent to each curve at the point of intersection, the vertex. The angle between

the curves is then the angle between the tangents. Hence, it is also the angle between the

infinitesimal elements of the curved lines at that point. In figure A2-2, the lower

straight line OB intersects a transverse straight line at F, a circular arc at D, an irregular curve at F , and the extreme point of a plane figure at F". The upper straight

line OA is too short to meet any of these same lines or the plane figure, but its extension intersects the lines at E, C, and e', respectively, and is tangent to the plane

figure at its other extremity at E". Then we can say that the angle 6 intercepts the

line segments EF, CD, and E'F' and just encloses or contains the plane figure e"f".

Conversely, the line segments and the plane figure all subtend the same angle 6 at 0. In particular, the circular arc CD has its center at 0, so its length is a measure of

the angle 9. In radians [rad], the size of the angle is given by the ratio of the length

of the arc CD to its radius 0C or 0D:

6 = CD/OC [rad]. (A2-3)

The magnitude of the same angle in (circular) degrees [deg] is given by

9 = (180/tt)- (CD/OC) [deg]. (A2-4)

A solid angle is similarly formed at a point, also called the vertex, by a conical surface or cone . The cone, in turn, is the surface that contains all possible straight lines

(i.e., is the locus of those lines) that extend from the vertex point to a point on some closed, simply-connected curve in space that does not pass through the vertex. Such a "curve," using the term in its broadest sense, may include straight-line segments and discontinuous changes of direction or angles. It is only required that, starting from any

point on the "curve" and traveling along it far enough in either direction, you return to the starting point after passing once, and only once, through every other point on the "curve." In particular, when the "curve" is made up entirely of straight-line segments that

form a polygon, the "cone" is a pyramid. On the other hand, when a pencil of rays converges

on the axis of a cylindrical optical system, with circular optical components, the solid

angle formed at the focus is bounded by a right circular cone and a solid angle is very

often so depicted. In fact, speaking loosely, we often say that the solid angle is a right

circular cone. However, there is really no such limitation on the concept of a solid angle

which, as we have just seen, can be formed at the vertex of a pyramid, or of the conical

surface formed by the straight lines joining the vertex to a closed curve of almost any

shape (see figure A2-3)

Like a plane angle, a solid angle can be defined as a locus of directions; it is the

65
Figure A2-3. A solid angle.

66
locus of all directions lying within the defining cone. Also, just as with a plane angle,

there are always two possibilities: the acute interior solid angle and the obtuse exterior

solid angle. However, unless the contrary is stated, it is usually assumed to be the acute

interior solid angle.

Any plane through the vertex of a solid angle that intersects the cone enclosing the

solid angle will do so along two straight lines that meet at the vertex to form a plane

angle. Thus, we can consider figure A2-2 as depicting such an intersection with a solid

angle having its vertex at and with the points A and B lying on the closed curve in

space that defines the bounding cone and, accordingly, does not pass through 0. Then the

straight-line segment EF represents a plane area, the arc CD a spherical-surface area,

the line segment E'F 1 an irregular surface area, and the plane figure E"F" a section

through an irregular solid object, all of which subtend the same solid angle to at 0.

The measure of a solid angle is sometimes confused with the solid angle itself. The

solid angle is that which exists at the vertex, regardless of the extent of the bounding

conical surface, just as a plane angle can be formed at a vertex by very short straight-line

segments, or even infinitesimal line elements. The measure of the solid angle, on the other

hand, is provided by the area intercepted by the bounding cone, or its extension beyond the

defining curve if necessary, on the surface of a sphere centered at the vertex, just as the

plane angle is measured by the intercepted arc of a circle centered at its vertex. The

magnitude of a solid angle to in steradians [sr] is just the ratio of this intercepted

spherical-surface area A to the square of the radius p of the sphere:

s

to = A /p 2 [sr]. (A2-5)

s

A SOLID ANGLE in SPHERICAL COORDINATES . To express a solid angle to in spherical coordinates, we will start, first, with just an element of solid angle dco. In figure A2-4, we show the angle 6 increased by an infinitesimal element d6 and the angle cj> similarly

increased by d<j>. The point P, always at the same distance p from the origin 0, moves

over the surface of a sphere of radius p. The element of angle d6 intercepts an arc of

length p*d8 on the spherical surface. The element d<j>, however, represents rotation

about the polar or Z-axis. Hence, the point P does not move on a great circle. Instead,

it follows an element of arc of radius p

# sin9 (the projection of OP, of length p, onto

the X-Y plane, or onto a plane parallel to the X-Y plane through P) . The length of that

circular arc on the spherical surface is p*sin6*dc)>, and it subtends an element of angle

sin6«d<)) [rad] at the origin 0. Two pairs of these elements of arc enclose the element of

spherical-surface area dA = p 2, sin6 -d9 - d<j), as shown in the figure. The element of solid

angle dto, subtended at the origin by that area element dA is, by eq. (A2-5),

s

dto = dA /p 2 =' sin6'd8-dc|> [sr]. (A2-6)

s

It is also clear, since each of the angle elements d6 and sin9*d<}> is in units of

radians [rad], that their product is in units of square radians [rad 2 ], just as the

67
/> = sin 6 -6<f>

p-dd

Y

\^A /o*sin 9 *d<f>

p'

p sin 8

Figure A2-4. An element of solid angle

do) = dA /p 2 = sin9*d8*d<J>

68
corresponding arc lengths p«d9 and p'sin9«d(j>, each in meters [m] , have a product

dA = p

2 'sin8*d9*d4> that is an area in square meters [m2

]. Thus, a steradian [sr] is a square radian [rad2

].

Although the relationship between radians and degrees, as plane-angle units, is familiar enough so that most people feel they understand it, there is confusion about their

dimensions. In the case of a solid angle, similar confusion exists concerning dimensions

and, furthermore, concerning the relationship between steradians and square degrees. Moon

and Spencer [19] have made a helpful suggestion, that we recognize the orthogonality of

direction between radial lengths and transverse lengths by designating their dimensions as

[L ] and [L ], respectively. Then a plane angle has the dimensions [L # L

_1

] rather

than being a dimensionless ratio as in the usual treatment. The distinction may not seem

too important when the same length units are used for both the arc and the radius to eval- uate the size of an angle in radians as the quotient of these two quantities. However, if we measure the arc in smaller units, equal to ir/180 % 1.745 329 x 10~2 times the length

unit for the radius, the quotient will be the size of the same angle in degrees. Carrying

this one step further, we can now relate the unit of solid angle, the steradian or square

radian, to the square degree quite simply through this same relationship, since we have

established the equivalence of a steradian and a square radian.

l[deg] = (n/180)[rad] ^ 1.745 329 * 10_2 [rad] (A2-7)

l[deg2

] = (Tr/180) 2 [rad2] = (ir/180) 2 [sr]

% 3.046 174 x lO-^fsr]. (A2-8)

Incidentally, the usefulness and validity of the treatment based on the suggestion by

Moon and Spencer is well brought out by the way in which it clarifies the confusion about

the dimensions of work or energy and of torque or moment , something that bothers almost

every physics student when first introduced to dimensional analysis. In the notation used

above, work or energy has the dimensions of force times colinear length—either [F -L ] or

[F «L ] —while torque or moment has the dimensions of force times orthogonal length

usually [F «L ]. But work or energy is generated when a torque or moment acts through a plane angle. Dimensionally, that situation is described by [f *L ]

• [L 'L l

] = [F *L ], t r t i which is gratifyingly self-consistent.

So much for the digression into units and dimensions. We obtained the expression for

the element of solid angle in spherical coordinates in eq. (A2-6) . The integral of that

quantity over the appropriate limits then provides us with a general expression for any

solid angle in spherical coordinates:

a) = / / sinB-d9-d<J> [sr]. (A2-9) fS

For example, it is often useful to have the expression for the solid angle at the vertex of

a right circular cone of half-vertex angle 9 . If we choose the polar axis along the axis

69
of the cone and the origin at the vertex, as in figure A2-5, it is easy to see that

r w = }

27r

}Asin9-d0-d<j> = 2tt(1 - cos9.) = 4n«sin2 (6/2) [sr]. (A2-10)

o o n n

For a hemisphere, 6, = tt/2 [rad] , cos9, = 0, and sin(6,/2) = 1//T,

n, h h

so that

"hemisphere = 2 *

[sr] *

2 ' 062 648 * 10" [de^] (A2"n>

and

w

s here

= 4 * [sr1 ^ 4 * 125 296 X 1Qh tde8

2j ' (A2-12)

Finally, we want a general expression for the element of solid angle du> 12 [sr] subtended at a surface element dAj by an arbitrarily oriented surface element dA2 [m2

] at

a distance of D [m] . This situation is illustrated in figure A2-6, where dA2 is shown

with its normal making an angle 2 with the line joining the two area elements. It is

evident in the figure that the solid-angle element do)i 2 subtended at dAj by dA2 intercepts an area element dA2 -cos9 2 on the surface of a sphere of radius D about dA^

Then, from eq. (A2-5) , we can write immediately

dw 12 = (dA2 'cos9 2 )/D2 [sr]. (A2-13)

Up to this point, we have considered only the simplest form of solid angle where the cone enclosing the angle is determined by a simply-connected curve. However, two or more

such simple solid angles can be combined in various ways to produce much more complicated

angles. For example, as illustrated in figure A2-7 , the solid angle filled by converging

rays at the focus of a Cassegrain-type optical system is a "hollow" cone. There are con- verging rays only within the solid angle between two coaxial right circular cones; there are none within the smaller inner cone.

A PROJECTED SOLID ANGLE . In Chapter 2, we defined radiance in eq. (2.14) in terms of the

element of flux through an element of projected area cos9'dA and an element of solid

angle dto. There, the obliquity factor cos9 is clearly associated with the element of area dA. However, when we obtain from this definition the expression for the flux in an entire beam, eq. (2.25), particularly in the case of uniform isotropic radiation, with constant L throughout the beam, and no vignetting so that the integrals are separable, as in

eq. (2.30) » it is convenient to associate the obliquity factor cos 9, instead, with the

element of solid angle dw as cos9-da> = cos9 •sin9 , d9«d(f>. We have followed Jones [16] in

calling

fi = / dfi

' = / cos9-do) [sr] (2.31a)

to u

the projected solid angle associated with the solid angle u>. It is a weighted solid angle

(and is sometimes so-called), with the obliquity factor cos9 as the weighting function.

70
X

Figure A2-5 A right circular cone

about the polar axis,

with half-vertex angle

71
V Plane tangent to

o u x j- ^

x . sphere Sphere of radius D

about dA,

Normal to dA 2

dAos cos0«

Figure A2-6. An arbitrarily oriented surface element dA2 subtends

a solid-angle element dco 12 at dAj at a distance D,

dco 12 = (dA2*cos62)/D2 [sr].

72
<u fa wu

<u a) 4J > .fl rd c <VJ o cn o > >

-QW QJ 0) J-l C

>> XI

O

cO CD

M CJ

M 44C

a •H

M

i-H

OM 0/ 0) iH S-i C H ,£> ai C > •H

CO <U c

>-(

U iH

03 CO o CO CX •H

cn 0)

X

•u ^X CO C CO 4-1 a

cu

U CJ T3 M-l

•rl 4-1 o QJ c .C C QJ cu 4-> i—

X)H

•rl 00 c

cj c rl • c ccj cn H ca o -a >^: rl CU H CO

4-1 E H

u

CX o o o cn cn oc

0) cu cu ex to x -G

>•> 3 4J 4J 4J cfl •rl 1 CJ S c 0) « •H pq W n

CO QJ

t-i

C

M • S-i o

<U fa o o

en

u

en

u : CO cn •H "s u a S o o -J

H-l o >mH O -M S-i CO .G

4= 0)

a : O -C c

4J

U o 03 CU u

^i 4-J QJ Cfl CyQ CO M • H i CM<

(UM3

00

fa

73
The name is suggested by the analogy with an element of projected area cos8-dA. However,

while the projection of an area onto another surface can be constructed so that there exists

an area that is the projected area, don't try to picture or visualize anything similar for

the projected solid angle. It doesn't exist in that sense, although there are at least two geometrical constructions for the measure of a projected solid angle that can be very helpful.

The first of these has been called Nusselt's construction [20] by Jakob [21], who also

cites Seibert [22] who published in the same year as Nusselt [20]. However, Gershun [23]

cites a much earlier publication by Wiener [24]. This, incidentally, is a good illustration of how badly scattered and disjointed things can be in the literature on optical radiation measurements. Wiener's construction is presented in figures A2-8 and A2-9. In fig- ure A2-8, a hemisphere of unit radius above a surface element dA is shown both in vertical elevation and in horizontal plan view. An element of spherical-surface area on this

hemisphere is labelled dw because its area is equal to the element of solid angle dw [sr]

that it subtends at dA at the center of the hemisphere. The rectilinear projection of

that spherical-surface area element onto the circular base of the hemisphere in the tangent

plane containing dA is, in turn, labelled dft = cosB'dw [sr], its area being equal to the projected solid angle in steradians. The sides of that element of projected area on

the base are also shown to be cos8«d8 [rad] radially and sin8-d<f> [rad] in the orthogonal

direction.

Figure A2-9 shows, similarly, how the conical projection of an irregular object onto

the surface of the unit-radius hemisphere is equal in total area to the solid angle w, in

steradians, subtended by that object at dA at the center of the hemisphere. Also, since

the relationship shown in figure A2-8 holds for every element dco of the entire sphericalsurface area to, the rectilinear projection of that entire area onto the base of the hemisphere is equal in area to the total projected solid angle 0. =

J cos9*dw in steradians

subtended at dA by this same object. In fact, the relationship holds for the complete

hemisphere of 2ir [sr] solid angle (the area of the unit-radius hemispherical surface).

The corresponding projected solid angle in steradians is equal to the area of the projection of the hemisphere onto the plane, of dA — the full unit-radius circular base — which

is -n [sr].

We can verify that last result analytically. First, we need the general expression for

the projected solid angle at the vertex of a right circular cone of half -vertex angle 8 with its axis normal to dA. Again, as in figure A2-5, we choose the polar axis along the

axis of the cone and the origin at the vertex to write for the projected solid angle

a r J J

h

cos8-sin8'd9-d(f) = (tt/2) • (1 - cos29

h

>

= TT-sin 2

8, [sr]. (A2-14)

n

For the full hemisphere, 8, = it/2 [rad], cos28 = -1, and sin9 = 1,

74
2 2 • «

/> -dw= ^> . sm 0-d0-d<£

= element of area on surface

of sphere of radius p

= dw [sr] for p=\

= P

2 «sin0 -cos0-d0-d<£

= projection of area element on base of hemisphere

= d& [sr] for /> = l

p-cos 9-d9

Figure A2-8. Geometrical construction for measures of an element of solid angle

dto and an element of projected solid angle dQ.

du = sin6«d6'd<}>; dfi cos9'do) = cos9 'sinS'de -dif

75
OBJECT

A Measure of the

SOLID ANGLE SUBTENDED AT

JJ

sin0-d0 '6<f> [sr]

= CONICAL PROJECTION OF OBJECT

ON SURFACE OF SPHERE

OF UNIT RADIUS (p = \)

A Measure of the

\ PROJECTED SOLID ANGLE

cos0*do

= Jj cos0 -sin© -d0 -d<£ [sr]

= RECTILINEAR PROJECTION

OF a* ON BASE OF HEMISPHERE (plane of dA)

c

Figure A2-9. Wiener's construction for measures of the solid angle id and the

projected solid angle fl subtended at dA by an irregular object.

P is an arbitrarily chosen point on the object; P' is its projection (conical) on the unit-radius hemisphere above dA; and P"

is the projection (rectilinear) of P' on the base of the hemisphere

(plane of dA)

76
so that

J2. . . = ir [sr] % 1.031 324 x 104 [deg2 ], (A2-15) hemisphere u ° '

in agreement with the conclusion from Wiener's construction.

Wiener's construction provides valuable insight into the relationships between solid

angles and the corresponding projected solid angles and facilitates analytical computations.

It has also been the basis for analog computers to evaluate projected solid angles or the

related configuration factors (angle factors, view factors, etc.) in illumination and heattransfer engineering [25]. (The relationships between projected solid angles, throughputs,

and configuration factors, etc., are discussed in Appendix 3 ..) A second construction,

while not as useful for computations, may seem somewhat more satisfying and mathematically

elegant, since it involves an area intercepted by the cone of the solid angle itself rather

than a second projection, as in Weiner's construction.

The unit-diameter tangent sphere construction is illustrated in figure A2-10. A unitdiameter sphere is constructed tangent to the surface element dA at the origin 0. Its center C, accordingly, lies on the normal to dA (the polar axis) midway between dA and

the point where the unit-diameter sphere is also tangent to the unit-radius hemisphere about

0. The elementary cone, subtended at by a remote surface element dS at P, intercepts an element of area dw [p 2, dto, where p = 1] on the surface of the unit-radius

hemisphere at P , as before. It also intercepts an area da = dwcosS = dfl at P" on the surface of the unit-diameter sphere [da = 4r2 'cos6*dw, where r = 1/2], an area that

is numerically equal to the subtended projected solid angle. This relationship holds true

for every element of solid angle in the entire hemisphere above dA, so it is also true

for the integrated total projected solid angle subtended at dA by a more extended body or surface. Again, we can verify its validity for the projected solid angle of the entire

hemisphere. This would be numerically equal, in steradians, to the total surface area of

the unit-diameter sphere:

Q , . , = 4irr 2 for r = 1/2 hemisphere

= tt [sr]. (A2-15a)

The unit-diameter tangent sphere construction is based on some relations that appear

to have been first worked out by Sumpner [26] and have been summarized, more recently and

accessibly, by Nicodemus [27]. This construction is used or referred to in a number of texts and papers on illumination and heat-transfer engineering as something that is rather

widely known and we have not succeeded in finding any citations or references that help to establish who first devised and published it.

OTHER SOLID-ANGLE MEASURES and/or APPROXIMATIONS . There are two commonly used measures of

a solid angle or projected solid angle, when it represents the "light-gathering power" of

Appendix 3 will appear in a subsequent Technical Note along with Chapter 4.

77
unit- diameter sphere

(P *da>)

(da = 4r 2 -cos0-dw

= 4r 2 -dil)

Figure A2-10. Measures of an element of solid angle dto and an element of projected

solid angle dft (unit-diameter tangent sphere construction)

dw = sin6*d6«d(J) = cose -da/ (2r -cose) 2 = da/ (4r 2 -cosG) .

If r = 1/2 and p = 2r =1, then

da = cos6'dw = dfi.

78
an optical system or, more exactly, the solid angle or projected solid angle subtended at

the focus by the converging pencil of rays. That pencil is contained within the cone subtended by the edges of the exit pupil, usually a right circular cone. For such a cone of

half-vertex angle 9, , the two quantities are: h

1. the numerical aperture : N.A. = n«sin8 [dimensionless]

(A2-16)

where n is the refractive index of the medium, and

2. the relative aperture or f-number:

f/# = (2'sine,} -1 [dimensionless]. (A2-17)

h

The popular definition of f-number as the (effective) focal length of a lens divided by its

diameter would lead to the approximation f/# % (2«tan8, ) _1

. However, in well-corrected

h

optical systems (that satisfy Abbe's sine condition [14,28]), the height h of an incident

axial ray above the axis is given by h = f'sinS, where f is the effective focal length

and 9 the angle the ray makes with the axis at the focus. Then for the extreme ray, f/# = f/2h = (2*sin9 ) -1 for the maximum value of h and the corresponding 8=9. These

h h two measures are related to the projected solid angle as follows:

= ir-sin 2 8^ = TT-[4(f/#) 2

] -1 = tt- (N.A. ) 2 -n-2 [sr] . (A2-18)

h

A common approximation to a solid angle u or a projected solid angle ft, with ref- erence to the angular field of an optical system, is the area of the field stop or exit

window in the focal plane divided by the square of the focal distance, which reduces to

ft ^J iftan2

8 1 [sr] in terms of the half-field angle 6, [rad] . Another approximation is to h h

express the solid angle, or projected solid angle, as tt G 2 [sr] . When 8, is given in

h h

degrees, rather than radians, this last approximation also yields a value in square degrees.

Table A2-1 lists the values of each of the foregoing measures or approximations along

with the corresponding values of projected solid angle ft [sr] for the same values of halfvertex angle 8, over the range < 8, < ir/2 [rad] or < 8, < 90 [deg] . The percentage

n h n of each approximation with respect to the projected solid angle ft is also given.

The range of validity of a particular approximation can be estimated from the table, or

it can be evaluated directly. For example, to establish the range for using the approximation ft % tt • 9 2 without exceeding a given error of, say one per cent, set h

ft - tt-6, 2 = -0.01'ft or tt'8 2 = (l.Ol)-ft = (1.01)'TT-sin2 8^, h h h

hence 8,_ = (1. 01)

2 -sin6^ = (1.004 987 562)-sin8, , h h n

so that 6, = 0.1727 [rad] = 9.895 [deg]. h

Accordingly, rounding off conservatively, the approximation ft % ^"^h

2 wiH De accurate within one per cent for 8, < 0.172 [rad] or 8, < 9.89 [deg].

n n

79
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References

[1] Walther, A., "Radiometry and Coherence", J. Opt. Soc. Am. 58.. No. 9, 1256-1259

(Sept. 1968).

[2] Walther, A., "Radiometry and Coherence", J. Opt. Soc. Am. 63^, No. 12, 1622-1623

(Dec. 1973).

[3] 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).

[4] 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.

[5] "International Lighting Vocabulary", 3rd Edition, common to the CIE and IEC (International Electrotechnical Commission) , International Commission on Illumination

(CIE), Bureau Centrale de la CIE, 4 Av. du Recteur Poincare, 75-Paris 16e, France,

Publication CIE No. 17 (E-l.l.) (1970).

[6] Nicodemus, Fred E. , "Optical Resource Letter on Radiometry", J. Opt. Soc. Am. 59 , No. 3, 243-248 (March 1969); also Am. J. Phys. 38, No. 1, 43-50 (Jan. 1970); also

reprinted in [29].

[7] Jones, R. Clark, "Appendix [on Radiometry]", pp. 19-36 of "Atmospheric Transmission:

Concepts, Symbols, Units and Nomenclature", by Spiro, Jones, and Wark, Infrared

Physics _5, No. 1, 11-36 (March 1965); also reprinted in [29], where a number of

unfortunate typographical errors have been corrected.

[8] Nicodemus, Fred E., "Radiance", Am. J. Phys. 31, No. 5, 368-377 (May 1963).

[9] DeWitt, D. P., and Richmond, J. C, "Theory and Measurement of the Thermal

Radiative Properties of Metals", a chapter in "Techniques of Metals Research",

series ed. R. R. Bunshah, Vol. IV, Part 1, "Measurement of Physical Properties.

Part 1 Some Special Properties", ed. E. Passaglia, (Wiley-Interscience, New York,

N. Y., 1972).

[10] Liebes, Sydney, Jr., "Brightness — On the Ray Invariance of B/n2 ", Am. J. Phys. 37 , No. 9, 932-934 (Sept. 1969).

[11] Steel, W. H., "Luminosity, Throughput, or Etendue?", Appl. Opt. 13, No. 4, 704-705

(April 1974).

[12] Zachor, A., "Throughput: A Figure of Merit for Optical Systems", J. Opt. Soc. Am. 54 , No. 1, p. iv (Adv. of Block Engineering, Inc.) (Jan. 1964).

[13] Jones, R. Clark, "Immersed Radiation Detectors", Appl. Opt. 1, No. 5, 607-613

(Sept. 1962).

[14] Jenkins, Francis A., and White, Harvey E. , "Fundamentals of Optics", 3rd ed. (McGraw-Hill, New York, N. Y., 1957).

[15] Ditchburn, R. W. , "Light", Vol. 1, 2nd ed. (John Wiley-Interscience, New York, N. Y.

1963).

[16] Page, Chester H. , and Vigoureux, Paul, Eds., "The International System of Units (SI)",

Nat. Bur. Stand. (U.S.), Spec. Publ. 330, 43 pages, (July 1974).

[17] Jones, R. Clark, "Terminology in Photometry and Radiometry", J. Opt. Soc. Am. 5_3, No. 11, 1314-1315 (Nov. 1963); also reprinted in [29].

81
[18] "American National Standard — Letter Symbols for Units Used in Science and Technology", ANSI Y10.19—1969 (Revision of USAS Y10.19---1967) (IEEE No. 260), The

American Society of Mechanical Engineers, 345 East 47th Street, New York, N. Y. 10017.

[19] Moon, Parry and Spencer, Domina Eberle, "A Study of Photometric Nomenclature",

J. Opt. Soc. Am. 36, 666-675 (1946).

[20] Nusselt, W., Z. Ver. deut. Ing. 72_, 673 (1928).

[21] Jakob, Max, "Heat Transfer — Vol. II" (Wiley, 1957) (See Fig. 31-35 on p. 66).

[22] Seibert, 0., Arch. Warmewirt. % 180 (1928).

[23] Gershun, A., "The Light Field", Moscow (1936); English translation by Parry Moon

and Gregory Timoshenko, J. Math. Phys. 18, 51-151 (May 1939) (MIT, Elect. Eng.

Dept. Contrib. No. 164, 1939).

[24] Wiener, Christian, "Lehrbuch der darstellenden Geometrie", Leipzig, 1884 (Vol. I, Chapter on "Bleuchtungslehre mit ihrer Anwendung auf evenflachige KSrper".)

[25] Several such devices are described on pp. 67-70 of [21].

[26] Sumpner, W. E. , Proc. Phys. Soc. (London) 12^, 10 (1892); also Phil. Mag. 15, 81

(1893).

[27] Nicodemus, Fred E. , "Emissivity of Isothermal Spherical Cavity with Gray Lambertian

Walls", Appl. Opt. ]_, No. 7, 1359-1362 (July 1968); see Appendix, pp. 1361-1362.

[28] Sears, Francis W. , "Optics", 3rd ed. (Addison-Wesley , Reading, Mass., 1949).

[29] Nicodemus, Fred E. , "Radiometry — Selected Reprints", published for the American

Association of Physics Teachers (AAPT) by the American Institute of Physics (AIP)

New York, N. Y., (July 1971). Included are reprints of [6,7,17].

82
fjlBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN 910-1

2. Gov't Accession

No.

3. Recipient's' Accession No.

4. TITLE AND SUBTITLE - SELF-STUDY MANUAL ON OPTICAL RADIATION

MEASUREMENTS — Part I — Concepts

Chapter 1. Introduction Chapter 2. Distribution of

Optical Radiation with respect to Position and Direction

Chapter 3. Spectral Distribution of Optical Radiation

5. Publication Date March 1976

6. Performing Organization Code

7. AUTHOR(S)

Fred E. Nicodemus, Henry J. Kostkowski, and Albert T. Hattenburg

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as No.

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15. SUPPLEMENTARY NOTES

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

"A Self-Study Manual on Optical Radiation Measurements." It contains the first three

chapters of this Manual. Additional chapters will be published, similarly, as they are

completed. The Manual is a definitive tutorial treatment of the measurement of incoherent optical radiation that is complete enough for self instruction. Detailed chapter

summaries make it also a convenient authoritative reference source.

The first chapter is an introduction that includes a description of optical radiation and the ray approach to its treatment in this Manual (based on geometrical optics)

a discussion of relevant parameters and their use in a measurement equation as a systematic technique for analyzing measurement problems, and a presentation of the system of

units and nomenclature used.

The second chapter, on the distribution of optical radiation with respect to position and direction, introduces the basic radiometric quantity, radiance, and its

important invariance properties. It is shown how to determine the total power in a beam from the radiance distribution and to determine the distribution of radiance at any surface, through which the beam passes, in terms of the distribution at any other

surface that also intersects the entire beam.

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)

Optical radiation measurement; photometry; radiometry; spectroradiometry.

18. AVAILABILITY QF Unlimited

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The third chapter, on the spectral distribution of optical radiation, develops

the concept of spectral radiance. Its invariance properties and the evaluation of

flux in a beam from a known distribution of spectral radiance are developed in a treatment parallelling that for radiance in Chapter 2.

These are the first chapters of Part I, in which are developed the basic concepts, essential for the subsequent discussions of instrumentation in Part II, and

of applications in Part III.

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