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

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100

.U5753

No.910-^4

1979
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's technical work is performed by the National Measurement Laboratory, the National Engineering Laboratory,

and the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

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The National Bureau of Standards was reorganized, effective April 9, 1978.
National Bureau of Standard

JUN 4 I97g

Self-Study Manual on

Optical Radiation Measurements:

Part I —Concepts, Chapters 7, 8, and 9

Fred E. Nicodemus, Editor

Radiometric Physics Division

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, D.C. 20234

Chapter 7. The Relative Spectral Responsivity and Slit-Scattering Function of a Spectroradiometer, Henry J. Kostkowski

Chapter 8. Deconvolution, John B. Shumaker

Chapter 9. Physically Defining Measurement-Beam Geometry by Using Opaque Barriers,

Fred E. Nicodemus

*< °>

*"'«u of

*'

' mm jih i /

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

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1979
National Bureau of Standards Technical Note 910-4

Nat. Bur. Stand. (U.S.), Tech. Note 910-4, 134 pages (June 1979) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1979

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

Stock No. 003-003-02067-2 Price $4.25

(Add 25 percent additional for other than U.S. mailing)
PREFACE

Preparation of the NBS Self-Study Manual on Optical Radiation Measurements is a major

part of our effort to meet the needs of the National Measurement System with respect to the measurement of optical electromagnetic radiation. Needs exist for measurements with

uncertainties of about one per cent, but the uncertainties actually achieved are often 5 to 10 per cent, or even more. These relatively poor accuracies, as compared to those in

many other types of physical measurements, result to a great extent from the multidimensional character of optical radiation; radiant power is distributed and may vary with

position, direction, wavelength, time, and polarization. In addition, many of those who

make measurements of optical radiation have little or no training in optics and are limited

in the amount of time they can spend to make up this deficiency. Moreover, there are few

schools that offer courses in radiometry, and there are almost no adequate texts or references dealing with this entire subject.

The idea of producing a self-study manual at NBS to try to fill some of this void 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 full-time editor (FEN) was appointed. NBS Tech. Note 910-1

with the first three chapters appeared in March 1976; TN 910-3 with chapter 6 came out in

June 1977; and TN 910-2 with chapters 4 and 5 in February 1978. This Tech Note 910-4

includes chapters 7, 8, and 9. Three more chapters, 10, 11, and 12, are planned for

publication by September 30, 1979. New budgetary limitations mean that no funds for con- tinuing the Manual will be available after that date although, unfortunately, substantially more is needed to cover just the basic fundamentals of radiometry in Part I —Concepts.

(The detailed outline, with chapter headings and topics to be covered in future chapters

under a plan that had been developed when we were expecting to continue publishing about

three chapters a year at least until September 1981, are given later in this preface.)

Various possibilities for continuing to work on the Manual with outside support are being

explored.

Our aim has been to provide a comprehensive tutorial treatment that is complete enough

for self instruction. That is the meaning of the phrase "self-study"; the Manual does not

contain explicitly programmed learning steps as that phrase often connotes. In addition,

through detailed, yet concise, chapter summaries, the Manual is designed to serve also as a convenient 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 material in the Manual is presented at the level of a college graduate in science

or engineering but, even for those with facility in college mathematics and a first course

in physics, it's not at all easy reading in spite of our best efforts at clarity and simplicity. This is an unavoidable result of the primary aim ("to make one-per-cent measurements commonplace") coupled with the fact that it must serve the needs of so many different

fields, including astronomy and astrophysics, mechanical heat-transfer enginineering,

illumination engineering, photometry, meteorology, photo-biology and photo-chemistry,

optical pyrometry, remote sensing, military infrared applications, etc.

Apparently, it is very difficult for those who have not been directly involved to realize the full implications of that situation. Each of us tends to think of radiometry

and radiometric measurements in terms of his own immediate experience and requirements.

What he wants is a set of simple carefully designed procedures, with appropriate cautions

concerning likely sources of error, for making his particular measurements. However, the

next reader wants the same thing, but for his, entirely different measurements. The

desired radiometric quantities to be measured are different, the instrumentation is different, the ambient conditions are different — the possible ways in which significant

differences may exist, in terms of the radiation parameters (position, direction, spectrum,

time or frequency of modulation or fluctuation, and polarization) as well as instrumental

and environmental parameters, are so numerous that any attempt to cover them all with a "cookbook" treatment of specific measurement procedures would be impossibly unwieldy and

could never be completed within any conceivable budget limitations short of Utopia. The

only way in which we can hope to effectively assist every reader who needs to make one-percent measurements is to provide him with material which, with sufficient effort on his
part, will give him enough of an understanding and grasp of basic general principles and

techniques to solve his own particular problems. That's why we have concentrated on the

basic material of Part I—Concepts and why it is not easy reading. The problems of radiometry in general just aren't simple ones, even when limited to classical radiometry in the

domain of geometrical optics (ray optics) . Although we have kept within that limitation so far, we recognize the increasingly urgent need for similar treatment of the problems of

coherent radiation with strong interference and diffraction effects that do not follow the

laws of ray optics. The basic theoretical work underlying the radiometry of coherent

radiation is just now being developed at a much more sophisticated level and we had planned,

as part of the proposed three-year program, to study that material and, if possible, use it to work out and present a chapter on the effects of coherence in radiometric measurements

to go along with the rest of this Manual. But it is not at all clear that sufficient

progress has been made in the basic work -to make this feasible yet. If production of the

Manual continues for a few more years, we will try to prepare such a chapter.

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

equation, first introduced in detail in Chapter 5. 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. Those parameters always include

the radiation parameters (listed above) , as well as environmental and instrumental parameters peculiar to each measurement configuration. The objective of the Manual is to develop

the basic concepts 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.

The overall plan for the Manual organized it into three Parts:

Part I--Concepts:

Step-by-step buildup 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, heat-transfer

engineering, astronomy, photo-biology, etc.) is indicated. However, discussions of actual

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

concepts and basic principles.

Part II—Instrumentation:

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

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

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

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

Part III—Applications:

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

a measurement. Included will be a very wide variety of examples of environmental and instrumental parameters with discussion of their effects and how to deal with them. This is

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

drawn from the widest possible applications in illumination engineering, radiative heat

transfer, military infrared devices, remote sensing, meteorology, astronomy, photo-chemistry

and photo-biology, etc. — That was our rather ambitious plan when we started out; limitations of support and

available resources, particularly available authors, have determined how much, or how

little, we could accomplish. So far, as indicated earlier, we have concentrated on Part I

Concepts. The chapters already published are:

1. Introduction,

2. Distribution of Optical Radiation with Respect to Position and

Direction—Radiance,
3. Spectral Distribution of Optical Radiation,

4. More on the Distribution of Optical Radiation with Respect to Position and Direction,

5. An Introduction to the Measurement Equation,

6. Distribution of Optical Radiation with Respect to Polarization.

Chapters 1, 2, and 3 were in TN 910-1; 4 and 5 in TN 910-2; and 6 in TN 910-3. In the

present Tech. Note 910-4, are three more chapters:

7. The Relative Spectral Responsivity and Slit-Scattering Function

of a Spectroradiometer

,

8. Deconvolution,

9. Physically Defining Measurement-Beam Geometry by Using Opaque

Barriers.

Currently scheduled for completion in the fiscal year ending September 30, 1979, are three more chapters on the following topics:

Linearity Calibrations,

Physical Photometry,

Blackbody Radiation and Temperature Scales.

This is a departure from the topics listed for FY 79 in our earlier plan for activities up

to September 1981. Even though that plan has been abandoned, we present it here to show

what coverage we had considered as providing a reasonably complete and useful treatment of

basics in Part I—Concepts and as a beginning of Part II —Instrumentation and Part III

Applications: in FY-79:

Part III chapter on Linearity Calibrations,

Part I chapter on Physical Photometry,

Part III chapter on Spectroradiometry of Continuous UV Spectra

(including an appendix on Uncertainty),

study and decision on feasibility of a basic Part I chapter

on Coherence in Radiometry vs. a more limited treatment of

Diffraction Corrections to Throughput Calculations,

preparatory study and completion of detailed outline for a Part I chapter on Distribution of Optical Radiation with

Respect to Time;

in FY-80:

Part I chapter on Distribution of Optical Radiation with Respect

to Time,

Part I chapter on Spectrophotometry,

Part I chapter on Coherence in Radiometry or on Diffraction

Corrections to Throughput Calculations,

Part III chapter on Spectroradiometry of Spectral Lines;

in FY-81:

Part I or II chapter on Detectors of Optical Radiation,

Part I chapter on Blackbody Radiation and Temperature Scales,

Part II chapter on Radiometric and Photometric Source Standards.

The plan then called for a reevaluation to decide whether to continue with more chapters,

particularly those for Parts II and III.
Incidentally, in preparing material for the Manual, we have rediscovered the fact that

the best way to learn something is to try to teach it to someone else. The exercise of

preparing tutorial material for such wide general application has required us to analyze

our own measurement activities in a different way that has broadened our understanding,

resulting in improved methods and more accurate results. Also, note that all references to measurement accuracy or uncertainty in this preface are concerned not only with precision

(relating to the repeatability of measurement results) but also with accuracy (relating to agreement with the "truth" which, while unknowable in the last analysis, is approximated by

analysis and estimates based on the widest possible experience including agreement with

measurements of the same quantities by others, particularly when using different instrumentation and measurement methods)

.

We are indebted to a great many individuals for invaluable "feedback" that has helped

us to put these chapters together more effectively. We renew our invitation to all readers

to submit comments, criticisms, and suggestions. In particular, we would welcome illustrative examples and problems from as widely different areas of application as possible.

We particularly acknowledge the inputs from members of CORM (the Council for Optical

Radiation Measurements), especially the CORM Coordinators. We are also grateful to Donald A. McSparron, Joseph C. Richmond, John B. Shumaker, Albert T. Hattenburg, and

Diana Nyyssonen, all of NBS, for helpful discussions and editorial assistance.

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

effort that produced the excellent typing and layout of this difficult text. We also want

to thank Paul M. Beachley for his capable help with the figures.

Fred E. Nicodemus, Editor

Henry J. Kogtkowski, Program Director

April 1979
Contents

Page

Part I. Concepts 1

Chapter 7. The Relative Spectral Responsivity and Silt-Scattering

Function of a Spectroradiometer 3

In this CHAPTER 3

NEED for the RELATIVE SPECTRAL RESPONSIVITY in

SPECTRAL MEASUREMENTS 3

TYPICAL and IDEALIZED RELATIVE RESPONSIVITY FUNCTIONS .... 5

DIRECT DETERMINATION of RELATIVE RESPONSIVITY FUNCTION

using SPECTRALLY TUNABLE MONOCHROMATIC BEAMS 12

INDIRECT DETERMINATION of RELATIVE RESPONSIVITY

FUNCTIONS using FIXED WAVELENGTH MONOCHROMATIC BEAMS —

SLIT-SCATTERING FUNCTION and RESPONSIVITY FACTOR 14

EXAMPLES of an EXPERIMENTALLY DETERMINED RESPONSIVITY

FACTOR and of SLIT-SCATTERING FUNCTIONS .'...' 21

RESPONSIVITY-FUNCTION EFFECTS 22

When either the RADIOMETRIC QUANTITY or the

RESPONSIVITY FUNCTION or both VARY with

POSITION and DIRECTION 26

MEASUREMENT of SPECTRAL-LINE RADIATION 27

SUMMARY of CHAPTER 7 32

References 34

Chapter 8. Deconvolution 35

In this CHAPTER 35

INTRODUCTION 35

A SIMPLE ITERATIVE SOLUTION 38

DECONVOLUTION in MATRIX NOTATION 40

A NUMERICAL EXAMPLE 44

END-EFFECTS 47

ACCURACY 48

The INVERSE SLIT-SCATTERING FUNCTION 50

UNIQUENESS and NOISE in DECONVOLUTION 55

A GENERALIZED LEAST-SQUARES DECONVOLUTION 57

An ITERATIVE SOLUTION for LARGE NUMBERS of POINTS 63

Vii
Page

CONFIDENCE BANDS and the RESIDUAL SPECTRAL

RESPONSIVITY FUNCTION 69

SUMMARY of CHAPTER 8 75

References 80

Appendix 8A. Deconvolution Programs 82

Chapter 9. Physically Defining Measurement-Beam Geometry

by Using Opaque Barriers 91

In this CHAPTER 91

BEAM GEOMETRY in RADIOMETRY 92

SIMPLE BEAM-DEFINING APERTURES 93

VIGNETTING 95

A BEAM-DEFINING APERTURE PAIR 95

A THIN LENS between TWO APERTURES:

BEAM-DEFINING APERTURE/APERTURE-IMAGE PAIRS 98

OBJECT SPACE and IMAGE SPACE 100

APERTURE STOPS, FIELD STOPS, and THEIR IMAGES:

PUPILS and WINDOWS 101

DESIGNATIONS of STOPS, PUPILS, and WINDOWS

in RADIOMETRY 103

OPTICAL SYSTEM FOCUSED on INFINITY 104

FLEXIBILITY in CHOICE of REFERENCE SURFACE

for SPATIAL PARAMETERS 105

ENTRANCE PUPILS as RECEIVING APERTURES —

SOME PRACTICAL PROBLEMS 105

DIFFUSERS 107

BAFFLES 109

MORE on VIGNETTING 109

RESPONSIVITIES and BEAM GEOMETRY 112

SUMMARY of CHAPTER 9 112

Appendix 9A. Numerical Evaluation Confirming

Invar iance of Throughput for Beam through Thin Lens

between Two Beam-Defining Apertures 117

References 119

ERRATA and ADDENDA for NBS Tech Notes 910-1 and 910-2 120
List of Figures

Page

7.1 Schematic diagram of a typical spectroradiometer used for

spectral-irradiance measurements 2

7.2 Relative responsivity function of a spectroradiometer

containing a monochromator with the image of the entrance slit

having the same width as the exit slit 5

7.3 Relative positions of exit slit and image of entrance slit,

showing why the responsivity function has a spectral shape

approximating a triangle 7

7.4 Illustration of radiation scattering giving rise to the wings

of the relative responsivity function 8

7.5 Triangular relative responsivity function with a spectral slit width of AA 9 w

7.6 Relative responsivity function of a spectroradiometer containing

a monochromator with the image of the entrance slit having a different width than the exit slit 9

7.7 Trapezoidal relative responsivity function with entrance-slit

image that is narrow compared to the exit slit 10

7.8 -Trapezoidal relative responsivity function with entrance-slit

image that is wide compared to the exit slit 11

7.9 Illustration of why the output signal of a spectroradiometer

viewing a narrow spectral line is proportional to the relative

responsivity function r(A , A'), where A is the wavelength

setting of the instrument and A' is the wavelength of the line 13

7.10 Relative positions of exit slit and image of entrance slit when a spectroradiometer spectrally scans a monochromatic beam of wavelength

A for determining the slit-scattering function at this wavelength .... 16

7.11 Spectral plots of a narrow spectral line, a typical slitscattering function, and a responsivity factor, indicating

why a spectral scan of such a line with the spectroradiometer

results in the mirror image of the slit-scattering function

and not the slit-scattering function itself 18

7.12 The responsivity factor and central portions of slit-scattering

functions for a spectroradiometer consisting of a double

monochromator with a quartz prism and grating, a photomultiplier

detector with a tri-alkali cathode, and an averaging-sphere

receiving element coated on the inside with BaSO, 21

7.13 Long-wavelength side of experimentally determined

slit-scattering functions for three different spectroradiometers, obtained by using the mercury line at 254 [nm] and scanning between 254 and 200 [nm] 23

7.14 Examples of how the responsivity function distorts or modifies the spectral distribution being measured . 25

IX
Page

7.15 Representative functions when determining the irradiance

produced by a narrow spectral-line source 28

7.16 Representative functions when determining the irradiance

produced by a source" emitting a wide spectral line or two narrow spectral lines separated by a wavelength interval

comparable to the spectral slit width 30

8.1 A three-dimensional representation of the spectral

responsivity function 36

8.2 Spectral irradiance distribution (a), and spectral

responsivity function (b) , assumed for numerical examples

in tables 8-1, 8-la, 8-2, and 8-3 45

8.3 Responsivity function, spectral irradiance, and

their product r for X = 6 49

o 8.4 Residual responsivity functions for example of table 8-la 74

9.1 Evaluation of the throughput between two apertures 94

9.2 Vignetting regions in a beam between two circular apertures 96

9.3 An example of vignetting in a beam between two circular apertures 97

9.4 Two apertures with a thin lens between them 99

9.5 Diagramatic horizontal section of measurement configuration

for spectral-radiance comparison measurements 102

9.6 A simple optical system focused on infinity

(for field use with distant targets or sources) 104

9.7 Dif fuser-detector configurations for measuring irradiance 108

9.8 Two cases of vignetting 110

9.9 Vignetting by an aperture in a thick wall Ill

List of Tables

8-1 Simple iterative deconvolution 46

8-la Simple iterative deconvolution — a significant portion

of the computations of table 8-1 repeated with more data

and smaller wavelength intervals 51

8-2 Iterative calculation of inverse slit-scattering function 54

8-3 Generalized least-squares deconvolution 61
SELF-STUDY MANUAL on OPTICAL RADIATION MEASUREMENTS

Part I. Concepts

This is the fourth in a series of Technical Notes (910- ) entitled

"Self-Study Manual on Optical Radiation Measurements". It contains

Chapters 7, 8, and 9 of this Manual. Additional chapters will continue

to be published, similarly, as they are completed. The Manual is a com- prehensive tutorial treatment of the measurement of incoherent radiation

that is complete enough for self instruction. Detailed chapter summaries

make it also a convenient authoritative reference source.

The manner in which the spectral responsivity of a spectroradiometer

containing a monochromator varies with wavelength is treated in Chapter 7

.

The nature and characteristics of this function, hew it is determined, and

the effects it has on spectroradiometric measurements are discussed in

detail. The traditional, indirect determination, involving the slit-scattering function, is shown to be highly inaccurate except for wavelengths

in the central portion of the function. An introduction to the measurement of spectral line radiation is also presented.

Deconvolution, discussed in Chapter 8, is the numerical process of

recovering an improved spectral distribution from spectroradiometric

measurements inevitably smeared spectrally by the spectral-responsivity

function of the radiometer. A simple, iterative technique, which is

extensively used, will be completely satisfactory in almost all radiometric situations. A more sophisticated technique, which is founded on

a sounder theoretical basis, should succeed in many of the few remaining

instances where the simpler technique fails. Examples of computer programs for the two techniques are given in an appendix. Both are applied

to a simple illustrative numerical example with brief mention of some of

the sources of difficulty and the limitations of deconvolution. Finally, we touch on the question of errors and describe a couple of useful measures of accuracy.

The physical definition of measurement beams, including the integration limits in the measurement equation, is treated in Chapter 9 in

terms of geometrical-optics quantities and concepts. The usual definitions of geometrical optics are extended to situations where defining

the measurement beam may not involve the imaging of any source. The

importance of aperture pairs or equivalent aperture/aperture-image

pairs, in defining measurement beams without vignetting, is developed.

Also discussed are entrance pupils as receiving apertures in radiometry (their positions and dimensions), diffusers, vignetting, and the

effects of beam geometry on overall instrument radiance and irradiance

responsivities. The treatment is in terms of simple, ideal, geometrical

optics, with sharp image and shadow boundaries, and only passing mention

is made of perturbations caused by imperfect imaging, aberrations,

scattering, and diffraction.

Key Words: Aperture/aperture- image pair; aperture pair; calibration;

convolution; deconvolution; geometrical-optics radiometry; inversion;

measurement-beam geometry; relative spectral responsivity; slitscattering function; slit-scattering function corrections; spectral

line radiometry; spectroradiometer characterizations; spectroradiometry

;

vignetting.
ENTRANCE

SLIT

WAVELENGTHS

OTHER THAN Xo

Figure 7.1. Schematic diagram of a typical spectroradiometer used for spectral-irradiance

measurements. Monochromator is shown set for a wavelength X . Note that,

for simplicity, dispersion is shown as taking place at the second focusing

(curved) mirror rather than at the ruled surface of the diffraction grating.

E

A (x,y) Spectral irradiance at the point x,y in the receiving aperture

- Averaging sphere

- Focusing optics

- Detector

- Electronics

Readout
Chapter 7. The Relative Spectral Responsivity and Slit-Scattering Function of a Spectroradiometer

by Henry J. Kostkowski

In this CHAPTER . We are concerned with how the spectral responsivity of a spectroradiometer containing a monochromator varies with wavelength. The need for this information and some typical relative spectral responsivities are presented. The relationship between the

spectral responsivity and the slit-scattering function, involving the responsivity factor,

is developed, and a method for experimentally determining these quantities is outlined.

Examples are given of the types of spectral distortions produced by the responsivity

function, but details on how to correct for these distortions are postponed until Chapter 8. Finally the radiometric measurement of spectral-line radiation, which also requires spectralresponsivity information, is introduced.

NEED for the RELATIVE SPECTRAL RESPONSIVITY in SPECTRAL MEASUREMENTS . When one is interested in making spectral radiometric measurements, particularly at many different wavelengths, a spectroradiometer containing a monochromator is usually used.

If the quantity to be measured is spectral irradiance, a suitable instrument is the one used

in problem 2 of Chapter 5 [7.1] 1 and shown schematically in figure 7.1. The measurement

equation for spectral irradiance when using such an instrument was shown, in Chapter 5, to be [eq. (5.23)]

S(A,AA,A ) = / / E (x,y,A)-R (x,y,A ,A)-dA-dA [S], (7.1)

AA A

where S(A,AA,A ) and R (x,y,A , A) are the output signal and spectral-flux responsivity

function2

, respectively, of the instrument for a wavelength setting of A , A is the area

of the receiving aperture, and AA is the wavelength interval for which the value of R

is not zero. For simplicity in the present discussion, we assume that the spectral irradiance E and the responsivity R are each uniform throughout A, and we focus attention

on just the spectral considerations for any given A by writing

S(AA,A ) = A-/ E,(A)-R.(A ,A)-dA [S]

,

(7.1a)

O . . A $0 AA

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

2 The word "function" is added to "spectral-flux responsivity" or "spectral- irradiance

responsivity" when we wish to emphasize the spectral shape of these quantities. In general

R(x,y,A ,A) may have a different spectral shape; i.e., be a different function of wavelength A, for each wavelength setting A of the instrument and for each point x,y. For

brevity we will often refer to a term such as "spectral-flux responsivity function" as simply the responsivity function.
or

S(AA,A ) = / E,(A)-R (A ,A)-dA [S] (7.1b)

o . , A ho AA

-1 2 where R^CA ,A) = A*R (A , A) [S'W 'in ] is the spectral-irradiance responsivity first

fcj o *p o introduced in Chapter 5 {eq. (5.49) [7.1]}.

In problem 2 of Chapter 5, we assumed that AA was sufficiently small so that E..A changed negligibly over this wavelength range. In this case

S(AA,A ) = E,(A )•/ RH ,A)-dA [S]. (7.2)

o A o AX~E o

We also assumed that a similar equation would be valid when calibrating the instrument with

a standard of spectral irradiance resulting in

S S (AA,A

q

) = E^(A

o )-/ RE

(A

Q

,A).dA [S] (7.3) AA

where the superscript s refers to the standard. Combining eqs . (7.2) and (7.3) we obtain

S(AA,A ) E,(A ) = ^— • E,(A ) [W-m -ran X

]

.

AO (7.4)

r. S / A -, 1 \ AO S (AA,A )

This is the simplest and most frequently used relation for spectral measurements when

utilizing a spectroradiometer containing a monochromator . However, when AA cannot be

made sufficiently small for eq. (7.4) to be valid, we must utilize eq. (7.1b) directly in

determining E . A There are two ways of obtaining a solution for E,(A) from eq. (7.1b). When the

A relative spectral distributions of E (A) and R„(A ,A) are known, the solution can be

A to

obtained in a manner similar to that used in problem 3 of Chapter 5. Then, instead of eq. (7.4), the equation for E. is

A

S(AA,A) / <(A).r(A

o ,A).dA

E (A ) = — • -^ • e (A ) [W-m 2 nm

1

] (7.5)

° S S (AA,A ) J e,(A)-r(A ,A)-dA °

° AT A O AA

where r(A , A) = R_(A , A) /K [dimensionless] , e.(A) = E, (A)/Kn [dimensionless] , and

O t o A A I -12' -2

the arbitrary factors K[S*W e m ] and K1 [W*m ] are constants with respect to wavelength. The quantity r(A ,A) is called the relative responsivity function .

If the relative spectral irradiance e.(A) is not known, it is still possible to

A obtain E.(A ). The technique for doing this utilizes eq. (7.1b) and is referred to as a

4
deconvolution . Now however, measurements are required at a number of wavelengths rather

than at just the wavelength X at which E. is desired. Also, the relative responsivity

function must be known at each of these wavelengths . Details relative to the deconvolution

technique will be presented in Chapter 8.

Even if eq. (7.4) is thought to be sufficiently accurate for obtaining E in a particular application, it is highly desirable to confirm this with one of the two above

described methods. Thus for reliable measurements of spectral irradiance or any other

spectral quantity, the corresponding relative spectral responsivity is required.

TYPICAL and IDEALIZED RELATIVE RESPONSIVITY FUNCTIONS . The relative responsivity function

r(X ,X) of a spectroradiometer may be obtained by irradiating the instrument, while it is set on a wavelength X , with a succession of monochromatic beams having the same irradi- ance 1 but different wavelengths, which are distributed over a range large enough to cover

the entire responsivity function. A plot of the resulting output signal against wavelength

is the relative responsivity function r(X , X) . Figure 7.2 is an example of such a plot

for the spectroradiometer in figure 7.1 when the width of the image of the entrance slit is

equal to the width of the exit slit. In general, this function will be different for different wavelength settings X .

r(X ,X)

Figure 7.2. Relative responsivity function of a spectroradiometer containing

a monochromator with the image of the entrance slit having the

same width as the exit slit. The wavelength setting is X . Wavelengths shown correspond to those in figure 7.3.

x We assume that the responsivity does not vary with the magnitude of the irradiance; that

i is, the instrument is "linear". Otherwise, the relative spectral responsivity determined

in this manner would be valid only for the irradiance level used in the determination.

5
In order to understand why the typical responsivity function has the spectral shape

shown in figure 7.2, let us examine closely the relative positions of the exit slit and the

images of the entrance slit for a few of the monochromatic beams with wavelengths close to

X . The images we are talking about are located at the intersections, in the plane of the

exit slit, of the optical rays shown as dashed lines in figure 7.1. Another view of these

images, looking in a direction perpendicular to the plane of the exit slit, is shown in

figure 7.3. Here the exit slit is represented by solid vertical lines and the monochromatic images of the entrance slit, each of different wavelength, by dashed lines. The cross hatching represents the radiant flux in the monochromatic image.

As can be seen in figure 7.3, when the wavelength of the monochromatic beam is X , the wavelength setting of the instrument, flux fills the exit slit resulting in a signal

from the instrument corresponding to the peak value shown in figure 7.2. When the monochromatic beam has a wavelength different than but close to X , the flux in this image

will only partially fill the exit slit. t In figure 7.3, flux associated with wavelengths A and X_ fills half the exit slit and gives rise to about half the output signal or relative

responsivity. The spectral propagance of the spectroradiometer between the entrance aperture and the exit slit, and the spectral responsivity of the detector, usually change only

slightly over a small wavelength range such as X -X„, and therefore, the output signal is

approximately proportional to the portion of the exit slit filled with flux. In figure 7.3

it appears that the output signals for the monochromatic beams of wavelengths X and A, should be zero. However, due to imperfect imaging, diffraction, and scattering, a small

amount of flux of these wavelengths does fall on the exit slit and produces the output

signal indicated in figure 7.2. Even when the image of the entrance slit is positioned far

from the exit slit, as illustrated in figure 7.4b, some flux associated with this wavelength

is scattered and diffracted onto the exit slit resulting in a small but significant output

signal. That is, the relative responsivity function is not zero even for wavelengths such

as X in figure 7.4a.

In the limit of perfect imagery and no scattering and when the instrument dispersion,

detector responsivity, and propagance do not vary with wavelength over the wavelength range

indicated in figure 7.3, the plot of the output signal against wavelength of the monochromatic beams would be triangular in shape. That is, the relative responsivity function would

have the triangular shape shown in figure 7.5.

When the exit slit and the image of the entrance slit do not have the same width, the

relative responsivity function has a spectral shape similar to that in figure 7.& rather

than that in figure 7.2. The sloping top is due to small variations with respect to wavelength in the detector responsivity and instrument propagance. Under the ideal conditions

referred to in the previous paragraph, the spectral shape of this relative responsivity

function would be trapezoidal. This can be confirmed by studying figures 7.7 and 7.8.

The width of the relative responsivity function is often of interest. It is called the

spectral slit width with the symbol AX , as shown in figures 7.5, 7.7 and 7.8, and is

6
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Figure 7.4. Illustration of radiation scattering giving rise to the wings

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r(X ,X ) 7

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Figure 7.5. Triangular relative responsivity function with a spectral slit

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r(X ,X)

Figure 7.6. Relative responsivity function of a spectroradiometer containing

a monochromator with the image of the entrance slit having a different width than the exit slit. Sloping top is due to change in

detector responsivity and/or instrument propagance over this wave- length interval. Wavelength setting of spectroradiometer is A .
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defined as either (1) the width of the function at half its peak height or (2) as the

ratio of the integral of the function with respect to wavelength to the peak height of the

function; i.e., J r(X ,X)dX/r(X ,X ). For the idealized functions, triangular or trapezoidal, the two definitions result in the same value. For the triangular shape

1 [nm] (7.6) a / dx

where Ax is the width of the exit slit, and dx/dX is the linear dispersion of the

instrument. 'For the trapezoidal responsivity function

AX ./<Ax £/of [mn] (7.7)

where Ax is either the width of the entrance slit image or the exit slit, whichever is

larger. The width of the base AX of the trapezoidal responsivity function is

B

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ent. slit

[nm] (7.8)

and the width of the flat top AX of this function is

AX„ Ax image of ent. slit

Axexit

slit

dx

dX [nm] (7.9)

These various equations may be confirmed by studying figures 7.3, 7.5, 7.7 and 7.8. From

eqs. (7.8) and (7.9), one sees that as one of the slits becomes very narrow relative to the

other, the relative responsivity plot approaches a rectangle.

DIRECT DETERMINATION of RELATIVE RESPONSIVITY FUNCTION using SPECTRALLY TUNABLE

MONOCHROMATIC BEAMS . The relative responsivity function may be obtained experimentally by

the technique described in the preceding section; that is, irradiating the spectroradiometer

with a monochromatic beam whose wavelength can be varied over the wavelength range for which

the magnitude of the responsivity function is not zero and whose irradiance at the receiving

aperture of the instrument can be kept constant or can be measured as the wavelength is

changed. The validity of this technique is easily confirmed starting with the spectral

measurement equation,

S(AX,X

q

) = / E

x (X)-RE

(X

Q

,X)-dX [S] AX (7.1b)

or

S(AX,X ) = K/ E (X)-r(X ,X)-dX [S] (7.10)

AX

12
AX

Figure 7.9. Illustration of why the output signal of a spectroradiometer

viewing a narrow spectral line is proportional to the relative

responsivity function r(X ,A'), where A is the wavelength

setting of the instrument and A' is the wavelength of the

line. Varying A' throughout the wavelength range A A while

the irradiance of the heam is unchanged, results in a determination of the relative responsivity function for the instrument wavelength setting A .

where r(A , A) is the relative responsivity function and K is a constant with respect to wavelength, both introduced in connection with eq. (7.5). Figure 7.9 shows a typical

relative responsivity function and the spectral irradiance of a monochromatic beam, represented by a very narrow spectral line at A'. The spectral line should be sufficiently narrow so that r(A ,A) may be treated as constant, reducing eq. (7.10) to

S(AA' . ,A ) line o K«r(A ,A')«f E,(A)-dA [S] o ., . A AX' line

(7.11)

where the integral in eq. (7.11) is the irradiance of the monochromatic beam at the re- ceiving aperture of the radiometer. Thus the output signal is proportional to the relative

responsivity function for the wavelength setting of the spectroradiometer A and evaluated

at the wavelength of the monochromatic beam A'. Varying A' throughout the wavelength

13
range 1 AA while the irradiance of the beam is unchanged, results in a determination of the

relative responsivity function for the instrument wavelength setting A .

This direct method of determining the relative responsivity function has been used

infrequently because an easy-to-use, spectrally tunable, monochromatic beam has not been

available. The only tunable monochromatic incoherent beam currently available is that

emerging from the exit slit of a monochromator . The entrance slit of the monochromator is

irradiated with the image of a high radiance tungsten ribbon lamp or carbon arc and the

exiting beam is monitored with a detector calibrated for relative spectral flux responsivity. This is certainly not a convenient, easy-to-use source. Alignment of the components is critical, the source is not readily portable, and it is not as spectrally narrow

or as high in irradiance as is often desired for relative responsivity measurements. However, as a result of the recent availability of small, high-throughput double monochromators

and of silicon detector standards of spectral responsivity with 1.5-5% uncertainty, this

direct method, using a tunable incoherent beam, is more appealing now than it was a few

years ago.

A coherent source that can be used in the direct method is the tunable dye laser. This

monochromatic beam can be obtained with a much narrower spectral band width and with higher

irradiance than that from a monochromator. However, it is less portable, not available at all wavelengths, and significantly more expensive. In addition, in using it, one must take

steps [7.2] to insure that the resulting relative responsivity function is not affected by

the coherence of the beam. Another possible problem is that some lasers seem to include

radiation at wavelengths other than the primary wavelength with irradiances of about 10

that of the primary spectral line. However, this interfering radiation can be measured and

removed by a filter or dispersive device when it presents a problem. In spite of these

limitations, the tunable dye laser appears to be an attractive device for this application.

INDIRECT DETERMINATION of RELATIVE RESPONSIVITY FUNCTIONS using FIXED WAVELENGTH

MONOCHROMATIC BEAMS — SLIT-SCATTERING FUNCTION and RESPONSIVITY FACTOR . Most measurements of the relative responsivity functions of spectroradiometers con- taining monochromators have been made using an indirect method where a fixed-wavelength,

monochromatic beam and a standard source of continuous radiation are spectrally scanned with

the monochromator. This indirect method is simpler to carry out and requires less expensive

apparatus than the direct method, but certain conditions must exist for the resulting relative responsivity function to be accurate. In this section these conditions and two new

quantities, the slit-scattering function and the responsivity factor , that are required in

the indirect method, are discussed in considerable detail.

An insight into why and when the simpler indirect method works can be obtained from

examining, again, the relative positions of the exit slit and the image of the entrance slit

*Note that AA is the wavelength interval in which r(A ,A) 4 0; AA' is the much narrower

interval in which E ¥ 0; AA ' << AA.

A line

14
for this method and for the direct method presented in the last section. Figures 7.3 and

7.4 give these positions for the direct method where the spectroradiometer is set on the

wavelength A and output signals are obtained for an incident monochromatic beam whose

wavelength is changed throughout AA, the wavelength range for which the responsivity

function has a significant value. We showed that when the irradiance of these beams was

constant the output signals were proportional to the relative responsivity function. Figure

7.10 gives the slit positions for the indirect method where the wavelength of the monochromatic beam is fixed at A and the wavelength setting of the spectroradiometer is changed

o (scanned) throughout AA. The relative positions of the exit slits and the images of the

entrance slits in figures 7.3 and 7.10 are the same. Also, even though the wavelengths

associated with the corresponding "look-alike" figures are different, except for case c, the

wavelength differences are, respectively, the same. This is the case because it was assumed, in preparing figure 7.10, that the dispersion of the instrument was the same for all

the wavelengths indicated. This, then, is one of the required conditions for the indirect

method to apply. Another condition, which is not apparent from the two figures, is that the

optical aberrations, scattering and diffraction must be the same for the look-alike diagrams

in figures 7.3 and 7.10. Even if these conditions hold, however, the relative output signals corresponding to each of the two figures may be different. This is because the spectral propagance of the monochromator and of the fore optics and the spectral responsivity of

the detector will not, in general, be the same for all the wavelengths indicated. However,

when the above conditions hold, these factors depend only on the wavelength of the flux and

not on the wavelength setting of the monochromator.

All the above conditions that are required for the indirect method to apply can be

incorporated into one mathematical statement. This is that the relative responsivity function can be written as a product of two functions, one depending only on the difference

between the wavelength setting of the monochromator and the wavelength of the flux and the

other function depending only on the wavelength of the flux; that is,

r(A ,A) = z(A-A)-r f (A) [dimensionless]

,

(7.12)

where we call z the slit-scattering function 1

» 2 and r the responsivity factor . The

^he slit-scattering function in eq. (7.12) has often been referred to as simply the slit

function. However, we are not continuing this practice because the CIE committee TC-1.2 on Photometry and Radiometry has proposed that the name "slit function" be used for the "relative spectral transmittance of a monochromator as a function of wavelength for a given

setting of wavelength and slits". In addition, our new name for z(A -A) is more descriptive because, although the central portion of this function is determined primarily by the

slit widths and dispersion of the monochromator, the wings of the function are determined

primarily by internal radiation scattering.

(See page 17 for Footnote 2

) 15
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16
2 Sometimes one sees the argument of the slit-scattering function in eq. (7.12) written

\-X rather than A -X. Physically, this difference is unimportant. We have adopted A -X because this is the form in which convolution integrals are normally expressed in an extensive literature on deconvolution which is cited in Chapter 8. A minor disadvantage "of this

form is that it establishes the rather unappealing sign convention by which the argument of

the slit function is negative for the long wavelength portion of the function and vice

versa. However, we felt that the extensive use of this form in various mathematical theo- rems and in the deconvolution literature outweighed this slight disadvantage.

essence of the indirect method and eq. (7.12) is that the relative responsivity function for

a particular wavelength setting of a monochromator X can be obtained by varying X rather than by varying X as in the conventional direct method. Since the value of the

si it -scattering function z, when the X -X dependence in eq. (7.12) is valid, depends

only on the difference between X and X, it is immaterial in determining this function

°

f whether one varies X or X. The advantage of varying X is that then r (X) is

constant, and therefore a spectral scan of a single, fixed-wavelength, monochromatic beam

(constant X) can be used to obtain z. The explicit relationship between the slit-scattering function z(X -X) and the output signal of a spectroradiometer when scanning 1 a narrow, isolated spectral line can be derived by starting with the spectral portion of the

measurement equation

S(AX,X

q

) = /E

x (X).R

E

(X o ,X)-dX [S]. (7.1b)

Assuming that IL,(X ,X) = K>r(X ,X) = K-z(X -X)-r

f (X) [S«W «m ], we obtain

t o o o

S(AX,X

o

) m K-J E

A (X).z(X

o-X)-rf (X)-dX [S]. (7.13)

AX

Letting E. (X) be the spectral irradiance of a narrow spectral line of wavelength X . and designating the wavelength setting of the spectroradiometer as X', as indicated in

figure 7.11,

S(AX. . ,X») = K-J E,(X)-z(X'-X)-rf (A)-dX [S] (7.14) line . , A AX, . line

where AX.. . is the wavelength range for which the spectral irradiance of the line is

significant. If both z(X'-X) and r (X) change negligibly over AX. . ,

^he terms "scanning" and "spectral scan" are consistently used here for an instrumental

scan, where the wavelength setting of the instrument is varied or "scanned".

17
^ line = ^o ^ -Ao + tAAw

Figure 7.11. Spectral plots of a narrow spectral line, a typical slit-scattering

function, and a responsivity factor, indicating why a spectral scan

of such a line with the spectroradiometer results in the mirror

image of the slit-scattering function and not the slit-scattering

function itself. Note that the values shown for X, . and X' line

are those for the situation depicted in figure 7.10(a).

S(AX line ,X') K. Z (X'-Xline).r

f (X line)./ E

A (XWX [S] AX. . line

(7.15)

and

z(X'-X 1

. ) = C-S(AX 1

. ,X') [dimensionless] line line (7.16)

where C is a constant for a given X . Since one usually is interested only in relative values of z, the constant C is frequently set equal to the reciprocal of the peak

value of S, making the peak value of z unity. Varying the wavelength setting of the

instrument X' over a sufficiently wide range provides, from eq. (7.16), a determination of

the slit-scattering function z(X'-X. ). Implicit in this determination is that the slitscattering function does not change when the wavelength setting of the instrument, in this case X', changes.

18
Note that when A' > A, . , as indicated in figure 7.11, the monochromator wavelength

line

setting is in the long wavelength wing of the spectral line. However the output signal for

this setting is proportional to the value of the slit-scattering function evaluated at

A, . which is in the short wavelength wing (where (A'-A) is positive) of the slit-scat- lme

tering function. This spectral mirror-image relationship between the scan of a spectral

line (monochromatic beam) and the slit-scattering function is caused by the fact that in the

indirect method the wavelength setting of the spectroradiometer is varied rather than the

wavelength of the monochromatic beam as in the direct method. It might help to clarify this

situation further if we compare figures 7.3a and 7.10a. Figure 7.3a depicts the direct

determination of the relative responsivity function at a wavelength which is AA less than

w the peak wavelength A . For the equivalent of figure 7.3a in the indirect method, the

wavelength setting must still be greater than the wavelength of the flux by AA . However,

since the wavelength of the flux in the indirect method is fixed and in this case, as shown

in figure 7.10a, equal to A , the wavelength setting in figure 7.10a must be A +AA . Moreover, in terms of the spectral scan of a spectral line representing the required monochromatic beam, the spectral position represented by figure 7.10a is in the long wavelength

portion of the scan at a wavelength greater than the peak wavelength. Thus the long wavelength side or wing of the spectral scan of the monochromatic beam corresponds to the short

wavelength side or wing of the slit-scattering function and vice versa.

In order to determine the relative responsivity function from eq. (7.12) one must

determine the responsivity factor r as well as the slit-scattering function. It is easy

to demonstrate how the responsivity factor can be obtained from a spectral scan of a spectral standard of continuous radiation. Starting again with the spectral portion of the

measurement equation

S(AA,A ) = / E fA)-R„(A ,A)-dA [S]. (7.1b)

o . , A to AA

f s We assume that R„(A , A) = K*z(A -A) -r (A) and E A (A) = E (A), the spectral irradiance of

hi o O A A a standard of continuous radiation. Thus eq. (7.1b) becomes

S

S (AA,A ) = K'f Ef(A)-z(A -A)-r f (A)-dA [S], (7.17)

° a i A O AA

s and if S (AA,A ) is determined for a sufficiently large range and number of monochromator

° ' s f wavelength settings A , eq. (7.17) can be solved for E, *r by deconvolution (see Chapter

O A s f 8) . Since E is known, this will also be a. solution for the responsivity factor r .

Sometimes a sufficiently accurate solution for r can be obtained without a deconvos f lution. If E «r - is approximately linear over the wavelength range for which z(A -A)

is significant and z is approximately symmetrical with respect to A , eq. (7.17) can be

written as

19
and

S

S (AX,A ) - K-E^(X )-r

f (X )•/ z(X -X) -dX [S] (7.18)

o A o ° AX °

S

S (AX,X ) r (X ) = [dimensionless] (7.19)

o K«Ef(X )•/ z(X -X)-dX

AX

where all the wavelength-dependent quantities on the right-hand side of eq. (7.19) are known or measurable.

Obtaining the relative responsivity function by the indirect method, that is, via the

slit-scattering function and responsivity factor, is significantly easier than by the

direct method. Only a single wavelength monochromatic beam and a spectral standard of

continuous radiation are required. The single-wavelength monochromatic beam can be obtained

from an incoherent source having a strong, isolated, narrow spectral line such as the 253.7

[nm] line of mercury or from a single laser spectral line, provided coherence effects are adequately accounted for [7.2]. The major problem or limitation of the indirect method is

the validity of the X -X dependence in eq. (7.12). In the relatively small wavelength

range corresponding to the central portion of the relative responsivity function (approximately X ± AX ) , eq. (7.12) is generally quite accurate. In addition, in this central

o w

region, the spectral line can usually have a very different wavelength than the X for

which the relative responsivity function is desired. The major requirement is that the

monochromator dispersion at the two wavelengths not be significantly different. The situation is quite different, however, for the distant wings of the responsivity function. We

have obtained spectral scans of monochromatic beams with wavelengths 90 [nm] apart where

the spectral values differed by 100% at 50 [nm] from the peak. Of course, if eq. (7.12)

had been valid over this wavelength range, the spectral shape of these scans would have

been the same. The differences are mainly due to a difference in internal scattering. In addition, spectral scans of monochromatic beams sometimes exhibit spectral bumps and weak

spectral lines 25 [nm] and more from the peak of the scan. We believe these are due,

respectively, to multiple reflections from the sides and various parts of the instrument

and to grating ghosts. The wavelengths of the grating ghosts have a clear-cut regularity

and can be predicted from measurements of a few monochromatic beams with different wavelengths, but this does not appear to be the case with their amplitudes. Thus, in general,

the indirect method is not very accurate for determining the distant wings of the relative

responsivity function.

It appears that spectral scans of monochromatic beams using monochromators possessing

high-quality holographic (photo-resist) gratings are much smoother than those with ruled

gratings. With monochromators having these gratings, it may be possible to define a more

general slit-scattering function and apply the indirect method even when the X -X dependence of eq. (7.12) does not exist. Research directed at this possibility is being

pursued by the author.

20
EXAMPLES of an EXPERIMENTALLY DETERMINED RESPONSIVITY FACTOR and of SLIT-SCATTERING

FUNCTIONS. Since the responsivity factor of a spectroradiometer depends on the spectral

propagance of the instrument and its detector responsivity, the spectral shape of this

factor will vary greatly with instruments having different types of optical elements and

detectors. However, a fairly common spectral shape for the responsivity factor of a spectroradiometer is shown in figure 7.12. This is a plot of the responsivity factor for the

NBS spectroradiometer used to realize and maintain the NBS scale of spectral irradiance [7.3]

100 c

O.i

Z(700-X)

--3.7 n m -

300 500

X(nm)

700

Figure 7.12. The responsivity factor and central portions of slit-scattering functions

for a spectroradiometer consisting of a double monochromator with a quartz

prism and grating, a photomultiplier detector with a tri-alkali cathode,

and an averaging-sphere receiving element coated on the inside with BaSO,

.

The slit widths are set at one millimeter.

21
The responsivity factor was determined with eq. (7.19) using data from a spectral scan of a deuterium lamp standard and a tungsten quartz halogen lamp standard and the manufacturer's

published data on the dispersion of the instrument. This particular instrument is similar

to the one shown in figure 7.1 except that the foreoptics utilizes mirrors and the dispersing portion of the instrument is a double monochromator containing a quartz prism and a grating. The inside surface of the averaging sphere in the NBS instrument is coated with

barium sulphate and the detector is a photomultiplier with a tri-alkali cathode and quartz

window. The rapid decrease in the responsivity factor below 300 [nm] , that is shown in

figure 7.12, is primarily due to the averaging sphere. The decrease at long wavelengths is

largely due to the spectral responsivity of the detector. The relative spectral efficiency

of the grating also contributes to the short- and long-wavelength falloffs.

Also shown in figure 7.12 for the NBS spectroradiometer are the central portions of

the slit-scattering functions determined at 300 [nm] and 700 [nm] for a slit width of one millimeter. The dispersion of the double monochromator of this instrument, and therefore

the spectral slit width of the instrument, varies by about 50% over this wavelength range,

as indicated. An all-grating instrument would have a much smaller change in dispersion

over this wavelength range and an all-prism instrument a much larger change.

Figure 7.13 shows the long wavelength portion of the slit-scattering functions at 254

[nm] for two single grating monochromators with a ruled and holographic grating, respectively, and for a double monochromator with two holographic gratings. These were obtained

by spectrally scanning a low pressure mercury source between 200 [nm] and 254 [nm] . In the case of the double monochromator, a short wavelength cut-off filter was also used because

radiation from the mercury source for other than the 254 [nm] line was also significant.

There was continuum and line radiation below 250 [nm] that produced a signal ranging from

10 to 500 times that produced by the slit-scattering function; i.e., scattering of the 254

[nm] line. This situation was even worse in the long wavelength wing of the 254 [nm]

mercury line; that is, in the short wavelength wing of the slit function which is not shown

in the figure. The values of the 254 [nm] slit-scattering functions in the wings at about

— 5 —8

50 [nm] from their peak are 4 x 10 and 10 for the single and double holographic

grating instruments, and are representative values in the UV for good-quality instruments.

RESPONSIVITY-FUNCTION EFFECTS . The effects of the responsivity function on spectroradiometric measurements can be illustrated by performing calculations that simulate the measurement of a few characteristic spectral irradiance distributions. We use the simplified

measurement equation

S(AX,X ) = K-/ E (X)-r(X ,X)«dA [S] (7.10)

° AX °

and the corresponding equation for a calibration measurement

S

S (AX,X ) = K-f Ef(X)-r(X ,X)-dX [S] (7.10a)

o

J .. A o AX

22
I. Or

SINGLE RULE0-6RATIN6

MONOCHROMATOR

SINGLE HOLOGRAPHIC-GRATING

MONOCHROMATOR

DOUBLE HOLOGRAPHIC -GRATING

MONOCHROMATOR

250 270 290 310 nm

WAVELENGTH

Figure 7.13. Long-wavelength side of experimentally determined slit-scattering

functions for three different spectroradiometers, obtained by using

the mercury line at 254 [nm] and scanning between 254 and 200 [nm]

.

23
for these calculations. In order that the calculations be independent of the spectral

shape of the standard, we assume that E^(A) is constant over AX. Then

S

S (AX,X ) = K-Ef(A )•/ r(X ,X)-dX [S]

.

(7.20)

o A o ' , , o AX

s Dividing eq. (7.10) by eq. (7.20) and multiplying by E. we obtain

J E

x (X)-r(XQ

,X)-dX — • E^ = — [W-m^-nm"1

]. (7.21)

S

S

f r(X ,X)dX

AX °

But the left-hand side of eq. (7.21) is just the irradiance we would obtain [eq. (7.4)] from

g measurements of S and S without taking into account the convolution of E (A) and

A r(X ,A). Designating this "observed" irradiance as E,(A ), we have

O A O

/ E

x (X). r (X

o ,X)-dX

E°(X ) = — [W-m^-nm-1

]. (7.22)

/ r(X ,A)-dA

AX °

Thus by normalizing the measurement equation with the integral of the responsivity function,

the result is the irradiance we would obtain if we utilized the simple equation

E (X ) = — • Ef(X ) [W.m~2 -nm_1

]. (7.4)

X o s X o

Figure 7.14 illustrates the result of calculating E from eq. (7.22) for the relative

A responsivity functions and spectral irradiance distributions shown in the figure. Figure

7.14a is the typical spectral scattering situation. A substantial value is obtained for

E^ for wavelengths less than X , even though the actual spectral irradiance is zero or very small at these wavelengths. The integrand in the numerator of eq. (7.22) is zero for

wavelengths up to X.. (see figure 7.14a) because E,(X) is zero in this range. But from

X. to longer wavelengths E (X) becomes so large that, even though r(X , X) is, for the

J. A O most part, very small in most of this region, the cumulative contribution of their product

to the integral over all wavelengths can be very large. Physically, E^(X ) represents an

A O accumulation over this large wavelength range, of the scattering, illustrated earlier in

figure 7.4b for one wavelength, X^,. An extreme example of spectral scattering of this

type occurs when making terrestrial solar irradiance measurements below 320 [nm] . Absorption of solar radiation by atmospheric ozone produces an E distribution similar to that

A shown in figure 7.14a. Spectral irradiance measurements at 300 [nm] when utilizing a single monochromator and at 290 [nm] with a double monochromator are typically too large by

about a factor of two when the responsivity function is not taken into account; that is,

when eq. (7.22) or (7.4) is used.

24
(a)

r(X ,X)

X fl X o A i

(b)

^(Xq.X)

Ev a e

SYMMETRICAL

RELATIVE TO X

(O

r(X 2l X)

Figure 7.14. Examples of how the responsivity function distorts or modifies the spectral distribution being measured.

25
If the spectral distribution being measured is linear over the wavelength range AA covered by the relative responsivity function r(A ,A) and this function is symmetrical

with respect to A , as pictured in figure 7.14b, no distortion results. The observed

spectral irradiance is identical to that being measured. This is so because E. , as seen

A in eq. (7.22), is simply the weighted mean of E. over AA, weighted by the relative

responsivity function r(A ,A). For the linearity and symmetry in figure 7.14b, the result

at A is the same as having an E that is constant over AA. Put another way, the

smaller contribution on the "low" E^ side (where A < A ) is exactly compensated by the

larger contribution on the "high" E.. side (where A > A ) . A O For the spectral distribution e a(^) shown in figure 7.14c, the observed irradiance

obtained from eq. (7.22) will be larger at A„ and smaller at A, than the actual spectral

irradiances at these wavelengths. There is no compensation as in figure 7.14b. The weighted mean of E, at A„ is higher and at A, , lower than the actual values at these wavelengths.

At A_, E.(A) is linear and the situation is similar to that in figure 7.14b so that

E (A ) is equal to E, (A ) . Note that the curvature at any point on the observed spectral

A J A o distribution in figure 7.14c is either the same as or less than that of the actual or true

spectral distribution.

Estimates for the magnitude of the distortion produced in any particular measurement

can be calculated using eq. (7.22) if the relative responsivity function and an estimated

spectral distribution for the quantity being measured are available. In fact such calculations form the basis of one deconvolution technique described in Chapter 8.

When either the RADIOMETRIC QUANTITY or the RESPONSIVITY FUNCTION or both VARY with

POSITION and DIRECTION . The treatment of the responsivity and slit-scattering function in

this chapter has assumed that both the radiometric quantity being measured and the responsivity function are uniform and isotropic throughout the beam of radiation incident on the

monochromator. Ordinarily this is easily achieved when measuring spectral irradiance by incorporating an averaging sphere for the first optical element of the spectroradiometer . In this section, we examine the consequences when such an element is not used.

Let us address this situation by starting with the general measurement equation 1 {eq.

(5.30) [7.1]}

S(AA,A,w,A ) =""/

/ / L

x (x,y,e,<j>,A)-R (x,y,6,<|>,A , A) -cosS -dwdA-dA [S] . (7.23)

AA A oj

When the responsivity is uniform and isotropic throughout A and w, but the spectral

radiance is not, then R (x,y,9,<j>,A ,A) = ^(A ,A) and eq. (7.23) may be written as

S(AA,A,co,A ) = / L,(A)«Rr (A ,A)-dA [S] (7.24)

O ' , A L O AA

*We are still assuming that polarization does not have an effect; that is, either the incident beam is unpolarized or the spectroradiometer is not polarization sensitive.

26
where

/ J

L. (x,y,6,<J>,X) •cosO'dwdA

L (X) = -A-2 tW-m"

2 .sr"

1 -nm"1

] (7.25)

/ / cosG'dwdA

A to

is the weighted average of the spectral radiance in A and to, weighted with respect to cos6, and R.(X , X) = R,(X ,X)«/ / cos6'du)«dA. Since eq. (7.24) is equivalent to eq. A to (7.1b), the responsivity and slit-scattering function material developed in this chapter for

spectral irradiance also applies to the weighted average spectral radiance of eq. (7.25), as long as the responsivity is uniform and isotropic. When the responsivity is not uniform and

Isotropic but the spectral radiance is; that is, L. (x,y,0,<f>, A) = L^(X) , eq. (7.23) becomes

S(AX,A,u>,X ) = / L

A (X)-RL

(A,to,X o ,X)-dX [S] (7.26)

AX

where

RL

(A,to,X

Q

,X) = / / R

$ (x,y,9,(f),Xo ,X)-cose«d(0'dA [S«W

_1

'cm

2 «sr] (7.27)

A to

is the radiance responsivity. Equation (7.26) is also equivalent to eq. (7.1b), and therefore the material of this chapter also applies to the case of a non-uniform, non-isotropic

flux responsivity provided the spectral radiance is uniform and isotropic throughout the

beam being measured.

When neither L. nor R is uniform and isotropic throughout the beam of interest,

the beam to be measured must be reduced in size (A and to) until at least one of the

above quantities is sufficiently uniform and isotropic for the accuracy required. This

amounts to determining the spectral radiance (and responsivity) as a function of direction

and position and requires many more measurements and more difficult measurements than when

uniformity and isotropy exist.

MEASUREMENT of SPECTRAL-LINE RADIATION . A chapter on the relative responsivity or slitscattering function of a spectroradiometer would not be complete without at least an introduction to the measurement of spectral-line radiation. In radiometry it is usually not

necessary to obtain the spectral distribution of a spectral line but only the radiance of

the line radiation or the irradiance produced by the line radiation. ' These are defined 1 as

L(X ) = / L

A (X)-dX [W-m"2 -sr_1

] (7.28)

AA line

and

*In eqs. (7.28) and (7.29), it is assumed that there is no continuum radiation present,

27
E(A. . ) lxne

-2. = / E (A)«dA [Win ] (7.29)

AA line

In order to illustrate the manner in which the measurement equation is solved when

determining the irradiance produced hy narrow spectral-line radiation, we assume that the

spectroradiometer is set on the wavelength A , corresponding to the peak of the line, and

that eq. (7.13) repeated below is valid,

S(AA,A ) = K«J E (A)«z(A -A) «r (A) «dA [S] (7.13)

AA

A plot of the various functions used in determining E(A. . ) is shown in figure 7.15.

Figure 7.15. Representative functions when determining the irradiance

produced by a narrow spectral-line source.

28
Using these functions, eq. (7.13) reduces to

S(AX,X ) « K-z(X -X )-r

f (A )•/ E ( A) -dX [S] (7.30)

O O O O • , A AX, . line

where r (X) and z(X -X ) {= z(0)} have been removed from within the integral because

o o line"

sivity factor evaluated at A and the constant K can be eliminated by making a measurement of a standard of spectral ir radiance at A . In such a measurement

o

S S (AA,A ) = K>f Ef(A)-z(A -X) -r

f (X) -dX [S] (7.13a)

° AA

X °

and with the usual approximations

S

S (AA,A ) c K-r

f (A ) «E?(A )

• J z(A -A) -dA [S]. (7.31)

o o X o

J

, , o AX

Combining eqs. (7.30) and (7.31),

S(X ) E(LJ = j E.(X)-dX - - • E^(X )-AX [Wm L

\ , (7.32) line ' ,, A „s,.., Xo err

AX n . S (X ) line o

where

/ z(X -X)«dX

AX eff

=

AA

2 J.X) [nm]. (7.33)

o o

All the quantities on the right-hand side of eqs. (7.32) and (7.33) are known or can be

measured. If eq. (7.31) is not sufficiently accurate, S may be obtained for a large

number of wavelengths throughout AX in order to determine Kt from a deconvolution of

eq. (7.13a). The E(X ) may then be determined from eqs. (7.29) and (7.30).

If the spectral line is too wide to be treated in the above manner or if there are two

or more spectral lines within the central portion of the slit-scattering function, as shown

in figures 7.16a or 7.16b, one can obtain an approximation for the irradiance produced by

the line or lines by spectrally scanning and integrating the resulting output signal over the wavelength region for which the signal is significant. This is demonstrated mathematically in the following manner. Integrating eq. (7.13) over the wavelength range AX for which the signal S(AX,X ) is significant,

J S(AX,X )-dX = K'J / E (X)-z(X -X) -r

f (X) -dX -dX [S-nm]. (7.34)

AX ° ° AX AX

A ° °

29
E

S

X

(X)

(o) Z(X„-X)

(b)

Z (X n-X)

Figure 7.16. Representative functions when determining the irradiance

produced by a source emitting a wide spectral line or two narrow spectral lines separated by a wavelength interval

comparable to the spectral slit width.

30
Assuming that the responsivity factor does not change significantly over either integration

interval, the responsivity factor can be removed from within both integrals and evaluated at the center of the AA scan at wavelength A , o mean

/ S(AA,A )'dX « K-r

f U )•/ / E '(X)-z(A -A) -dA-dA [S-nm]. (7.35)

» -. o o mean J

A , .-, A o o AA AA AA

o o

The only factor in the integrand of eq. (7.35) which depends upon A is z(A -A). The

integral of this slit-scattering function over the entire range AA of A values for

which there is any instrument response, yields a fixed number independent of A, which can

be removed from the integral over A. Thus we obtain

f S(AA,A )-dA « K«r

f (A )•/ z(A-A)-dA f E 1 (A)-dA [S-nm]. (7.36)

. > o o mean J

. , o o

;

A , A AA AA AA

o o

The responsivity factor at A and the constant K in eq. (7.36) may be obtained by a r J mean

measurement of a spectral r irradiance standard at A . This measurement is represented by mean

a slight modification of eq. (7.31) or

S

S (AA,A ) K-rf (A )'E?(X )•/ z( A -A) -dA ' [S]

.

(7.31a) mean mean A mean J

, , o AA

Note that the integration limits of the slit-scattering function integrals in eq. (7.31a)

and (7.36) are different. However, since the magnitude of AA was chosen sufficiently

large so as to include any significant portion of the relative responsivity function

{or z(A -A)}, integration of z(A -A) over the larger wavelength range AA is negligibly different from that over AA. That is, from eq. (7.33),

/ z(A -A)«dA a / z(A -A)»dA = z(0)-AA ££ [nm] . (7.37)

a-v ° °

J „ o eff ' AA AA

o

Combining eqs. (7.36), (7.31a) and (7.37) results in

/ S(AA,A ) -dA AA

/ E.(A)-dA = • Ef(A ) [W-ni ]. (7.38)

AA

X S

S (AA,A ) A mean

mean

Refinements and additional details on the determination of the irradiance produced by sources containing individual or multiple spectral lines and a mixture of lines and continua

are planned for a future chapter of the Self -Study Manual.

31
SUMMARY of CHAPTER 7 . For reliable spectral irradiance measurements, it is necessary to know the relative spectral responsivity function r(A , A) in the spectral portion of the

measurement equation {eq. (7.1b)}

S(AA,A ) = / E (A)'K-r(A ,A)«dA [S], (7.10)

AA

where K«r(A ,A) = Rp(A , A) and K is a constant with respect to wavelength.

The direct method for obtaining r(A ,A) is by irradiating the spectroradiometer,

with it set on wavelength A . with a succession of monochromatic beams, each having the same irradiance but a different wavelength. These wavelengths should be distributed over a range large enough to cover the entire relative responsivity function. A plot of the

resulting output signal of the spectroradiometer against wavelength is the relative responsivity function r(A , A) of the instrument at A . For a spectroradiometer containing a monochromator, when the width of the exit slit is equal to the width of the image of the

entrance slit (in the neighborhood of the exit slit), the relative responsivity function,

for a particular instrument setting A , is a bell-shaped curve similar to that shown in

figure 7.2. The width of the relative responsivity function is called the spectral slit

width which is defined as either (1) the width at half-height or (2) the ratio

/oo r(A ,A)dA/r(A , A ). The width of the central portion of the responsivity function is

-oo o o o determined primarily by the linear dispersion and the width of the exit slit and the

entrance-slit image. The value of the relative responsivity function far from the central

region is primarily due to internal radiation scattering. Diffraction and optical aberrations also contribute to the shape of the responsivity function. If scattering, diffraction and aberration effects are negligible compared to the effects of the slits; and if the

instrument dispersion, detector spectral responsivity and spectral propagance change

negligibly over the central (wavelength) portion of the relative responsivity function,

this function has approximately a triangular or trapezoidal shape depending on whether or not the slit widths mentioned above are equal. The width of the base AA of such an

B ideal relative responsivity function is

AA, Ax. ,. image of +

Ax .. exit

ent. slit slit [nm] (7.8)

where dx/dA is the linear dispersion of the instrument and the Ax's are the slit widths

referred to above. The width of

widths) responsivity function is

referred to above. The width of the top portion AA of the trapezoidal (unequal slit

AA, Ax.image of ent . slit

Axexit

slit

[nm] (7.9)

32
Since suitable monochromatic beams for which the wavelength is tunable have not been

readily available, most measurements of the relative spectral responsivity function have

been made by using an indirect approximate method rather than the above described direct

method. In this indirect method, a fixed wavelength monochromatic beam is spectrally

scanned by the spectroradiometer. The spectral plot of this output signal is the mirror

image of the plot of a major factor of the relative responsivity function, a factor that we

call the slit-scattering function . The remaining factor, which we call the responsivity

factor , can be obtained by spectrally scanning a continuous source standard of spectral

irradiance.

The indirect method for determining r(A , A) is accurate only when the various factors

affecting the shape of the slit-scattering function: dispersion, scattering, diffraction

and optical aberrations, change negligibly over the entire wavelength range of the function.

This condition is equivalent to the slit-scattering function being a function only of the

difference between the wavelength setting of the monochromator A and the wavelength of

the flux. Then

r(A ,A) = z(A-A)«rf (A) [dimensionless]

,

• (7.12)

where z(A -A) is the slit-scattering function and r (A) is the responsivity factor

referred to earlier. The A -A dependence in eq. (7.12) is highly accurate only for small

values of |A -A|, approximately equal to the spectral slit width or less. For much larger

values of |A -A|, the relative responsivity function measured by the indirect method may

be in error by 100% or more.

The slit-scattering function in the neighborhood of A has a spectral shape similar

to that of the responsivity function because, there, the responsivity factor generally

changes much more slowly with wavelength than the slit-scattering function. The distant

-4

wings of the slit-scattering function typically have values relative to their peak of 10

—8

and 10 for a single and double monochromator, respectively. Examples of such slit

functions as well as a typical example of a responsivity factor are shown in figures 7.13

and 7.12.

The distortion produced by the relative responsivity function in spectroradiometric

measurements is illustrated (figure 7.14) for a few characteristic spectral distributions.

These include the familiar "out-of-band" spectral scattering so common in UV spectroradiometry and the reduction of spectral curvature of convex- and concave-shaped spectral

distributions.

The responsivity and slit-scattering function material developed in this chapter for

spectral irradiance measurements can also be applied to spectral radiance measurements

provided either the spectral radiance or the responsivity function is uniform and isotropic

throughout the measurement beam selected by the spectroradiometer. This can always be

realized by making this beam sufficiently small, but then many more measurements are required in order to cover the original sized beam.

33
Since information about the relative responsivity function is required for radiometric

measurements of spectral lines, an introduction to this subject is included. It is shown

that the irradiance produced by an isolated spectral line of width small compared to the

spectral slit width and without continuum radiation is given by

E S(A )

i" z (VA ) dX

(X., ) E / E,(X)-dX - — .E*(XJ- — [Wm~2

] (7.32a) line AX* S

S (X ) X ° z(X -X ) lxne o o o

where S(X ) and S (X ) are the output signals, respectively, of the spectroradiometer

when measuring a spectral line source and a spectral standard continuum source at a waves length setting X corresponding to the wavelength of the spectral line. E (X ) is the

o A o spectral irradiance at X of the standard, z(X -X) is the slit-scattering function of

the instrument at a wavelength setting X , and z(X -X ) is the peak value of the slitscattering function.

When the spectral width of the line is not small compared to the spectral slit width

or when there are two or more lines separated by a wavelength interval roughly equal to the

spectral slit width, the irradiance of the line (or lines) is given approximately by

/ S(AX,X ) dX

AX _„ E( , „*. ) « £- • E^(X ) [Wm Z

] (7.38a)

of all lxnes

S (AX X ) mean

' mean

where the output signal S(AX,X ) must now be integrated over a sufficently large wave- length range AX so as to include all significant contributions to the signal from the

spectral lines. The wavelength X at which the signal and spectral irradiance from ° mean

the standard is obtained is at the center of the wavelength interval AX and it is again

assumed in eq. (7.38a) that no continuum radiation is present.

References

[7.1] Henry J. Kostkowski and Fred E. Nicodemus, "An Introduction to the Measurement

Equation", Chapter 5 of "Self-Study Manual on Optical Radiation Measurements:

Part I—Concepts", Nat. Bur. Stand. (U.S.), Tech Note 910-2, 118 pages

(Feb. 1978) 58-92.

[7.2] Louis Sica, "Use of Laser Radiation in the Calibration of Spectrometer Scanning

Functions", J. Opt. Soc. Am. 57, No. 11 (Nov. 1967) 1366-1372.

[7.3] R. D. Saunders and J. B. Shumaker, "Optical Radiation Measurements: The 1973 NBS

Scale of Spectral Irradiance", Nat. Bur. Stand. (U.S.), Tech Note 594-13,

36 pages (Apr. 1977).

34
Chapter 8. Deconvolution

by John B. Shumaker

In this CHAPTER . We discuss deconvolution, the numerical process of recovering an improved

spectral distribution from spectroradiometric measurements which inevitably are smeared

spectrally by the spectral-responsivity function of the radiometer. We first present a simple iterative technique which is extensively used and will be completely satisfactory in

almost all radiometric situations. We also present a more sophisticated technique which

should succeed in many of the instances where the simpler technique fails and which is

founded on a sounder theoretical basis. Examples of computer programs for the two techniques are given in an appendix. We apply both techniques to a simple numerical example and

briefly mention some of the sources of difficulty and the limitations of deconvolution.

Finally, we touch on the question of errors and describe a couple of useful measures of

accuracy.

INTRODUCTION . Deconvolution 1 refers to the problem of extracting spectral radiometric

quantities such as spectral irradiance from measurements in which the spectral shape of the

measured quantity has been distorted by the effects of a responsivity function. This is an example of a general problem which arises in any measurement situation where the resolution

of the measuring instrument is less than perfect. Other examples in radiometry include

measurements of the angular, temporal, or spatial (area) dependence of radiance using any

real radiometer possessed of non- infinitesimal resolvabilities in these dimensions. In the case of spectral resolution we have {see eq. (5.32) [8.1]

}

2

S(A

q

) = /R

E

(A

o ,A).E

A (A)-dA [S], (8.1)

-2 -1

where E (A) [W«m *nm ] is the spectral irradiance whose value is desired. (Similar

A relationships, of course, exist for the other spectral radiometric quantities.)

-1 2 R^(A , A) [S*W *m ] is the spectral irradiance responsivity {eq. (5.49) [8.1]} of the

radiometer as a function of the wavelength A of the radiation entering the instrument and

the (mechanical) setting A of the instrument itself. In principle, as we have noted in

Chapters 5 [8.1] and 7, Rp,(^ >^) can be measured directly with monochromatic beams.

Strictly speaking, the terms "convolution" and "deconvolution" apply to integrals in which

the kernel — our responsivity function — depends only upon the difference between its two

arguments. However, there seems to be no good reason to use a different name to cover the

usual radiometric case in which Rp(A , A) , in addition to depending mainly upon the difference A -A, may also depend on where on the wavelength scale this difference occurs,

i.e., on A or A , and can be handled by a trivial generalization of the more restricted

case.

2 Figures in square brackets indicate literature references listed at the end of this chapter.

35
co

•Huoc3

"4-1

>>

4-1

toCo

a.

CO

a)

COM

4-1 a

0)a

0)oco

•Hc

0)caa

<u§

•H

ca13 IIcu

aiu

01M3

60 •H

36
Figure 8.1 is an attempt at a 3-dimensional sketch of the surface Rp(^ >a) as a function

of X and X. S(X ) is the output signal in volts or other appropriate units when the

instrument is set at X and views the source of spectral irradiance E^(X). The integration is carried out over all X [nm] for which neither R„(X ,X) nor E,(X) is zero.

£. O A Thus, S(X ) and R„(X , X) are the quantities which can be directly measured and the

o tt o problem is to calculate the desired quantity E i(^) from the integral equation, eq. (8.1).

If the radiometer had perfect spectral resolution we could write 1

R

e

(X

q

,X) = R^(X).6(X

Q

-X) [S-W

_1

.m

2

] (8.2)

and eq. (8.1) would become

S(X

q

) = jR^(X)-6(X

o-X).E

A (XWX = R^(^ )'E

x (X

o

) [S]. (8.3)

In this case E.(X) can be computed directly from the measurements by division by a simple

f spectral irradiance responsivity factor RI(X) . The assumption of eq. (8.2) leading to eq. (8.3) is equivalent to ignoring the effects of the responsivity function. With less-thanperfect spectral resolution, R-(X , X) can no longer be represented by a delta function and

t o both the measurement of the responsivity function Rtj(^ ,X) and the calculation of E (X)

become much more difficult.

Before going on, we should stop for a moment to note the distinction between this

situation and the similar one of a measurement with a broad-band (broad-spectral-band)

radiometer covered in the discussion following eq. (5.32) in Chapter 5 [8.1]. Equation

(5.32) is quite similar to eq. (8.1) [or eq. (5.13)], the important difference being that

in eq. (8.1) the responsivity is not only a function of wavelength X, but also of a monochromator wavelength setting X . For any one value of X , i.e., for just one measurement at a single monochromator setting, the solution to eq. (8.1) or (5.13) is just as indeterminate as that to eq. (5.32) and everything said in the discussion of eq. (5.32) is

equally applicable to the monochromator situation. However, with the monochromator, the

incident radiation at each value of X affects the output at a number of different values

of X . In other words, the outputs at several values of X all contain information about

o

> r the input at a given value of X (along with information about inputs at other values of

X, as well) . It is this additional overlapping information which enables us to say some- thing about the dependence of E^(X) upon X.

6(x) is the Dirac delta function with the properties:

6(x) = 0, for x i 0, and

(8.4)

J 6(x)«dx = 1, for any a > 0.

37
A SIMPLE ITERATIVE SOLUTION . Although many techniques 1 have been devised for the solution

of the integral equation, eq. (8.1), the only one which has been extensively applied in

optical spectrometry is based upon an iterative method for the solution of simultaneous

linear equations [8.2 - 8.10]. If we assume that measurements S(X ) are made at n

o equally spaced wavelength settings A and if we approximate the integral in eq. (8.1) by a summation we can rewrite eq. (8.1) as

n

S

k

= AX- £ \± '*

± [S]» k=l,2,...,n. (8.5)

i=l

In this equation AX [run] is the wavelength spacing, S [S] is the spectroradiometer

output S(X ) when set at the k wavelength, E. [W*m 'nm ] is the desired spectral

t-Vi irradiance E (X.) (the subscript X is omitted for simplicity) at the i wavelength

A i

• -1 2 and EL [S*W *m ] is the instrument spectral irradiance responsivity R (X , X.) (the

spectral output per unit incident spectral irradiance) to monochromatic radiation at the

i wavelength when the instrument is set to the k wavelength. In figure 8.1 the value

of any R, . is just the height of the surface, R-p(A ,X), at the point of intersection of

the line drawn on the surface at constant X = X, with the line drawn on the surface at

o k

constant X = X .

.

l Equations (8.5) consist of n equations in n unknowns where n in typical applications may be as small as 50 or as large as 10 . Because of the size of n and because, as we shall see later, a mathematically exact solution of eqs. (8.5) is not needed, an iterative approximation technique is used for solving this set of equations. In this procedure 2

an assumed solution E. for the E. together with the R^ . yields a computed set of

(0) 1 1

S, from eqs. (8.5). Based upon a comparison of these computed values with the measured

values of S, the E. are corrected and the process is repeated. The detailed steps are

1 (0) as follows: First we assume trial spectral irradiance values E. which we obtain from

E^ 0) = S./(AX-Tr..) [W-m"2 -nm_1

], i=l,2,...,n. (8.6)

i i * il

J

The value of E. is just the value which would have been obtained for E. if deconvolution were not attempted [i.e., by solving eq. (7.2)]. Next, using eqs. (8.5), we

calculate what effect the responsivity function would have on this spectrum:

hundreds of papers have been written on deconvolution and related subjects. The references

cited in this chapter give only a sampling of this literature. Further references are given

in [8.11,8.12,8.13]. Some interesting techniques which are not even hinted at in this

chapter are described in [8.13-8.16].

2An illustrative computer program for this procedure is given in Appendix 8A.

38
Th

s[0) = b\'l \± -z[0) [S], k=l,2,...,n. (8.7)

i

ese values of S^ will in general not agree with the measured S, . Therefore, we

correct the E. using the difference between S, and S, so as to improve our estimated values for E.. Thus, we compute 1

e£

X) = e£0) + VtSk~

Sk

0)] fW*111"

2,1"11"1 ^ k=l,2,...,n. (8.8)

where A, is a convergence factor whose value is not critical. Frequently a good value for

A, [as will be derived in eq. (8.81)] is

A, = R, k

/(AX-J R?.) [W'm~2 -nm"

1 -S~1

], k = 1,2, ... . ,n. (8.9)

We shall return to a further brief discussion of A, below. Now we continue in the same

(1) (2)

way using values of E. as a trial solution to obtain better estimates E. , etc. In general 2

e£v) = e£V_1) + A^-tS^S^"1 ^ [W-m"

2 -nm_1

], k = l,2,...,n, (8.10)

with S k

V_1)

= AA '? R

ki'

E

iV_1)

[S] ' k=l,2,...,n, (8.11)

i

where the summation over i runs from i = 1 to i = n.

In many applications two or three iterations (i.e. v = 2 or 3) will suffice to

reduce the correctable errors due to responsivity-function effects to less than the measurement noise. In fact, a criterion based upon consideration of the measurement noise

should be used to terminate the iterations: when the computed S , given by eq. (8.11),

agree with the measured S, to within the experimental noise, for example, when [8.10]

*It would have been nice to be able invariably to associate the subscript k with the

instrumental setting X and the subscript i with the wavelength A, but the varying

requirements for these dummy indices in the different equations make any such consistent

usage impossible to maintain.

^Iterative multiplicative corrections instead of additive corrections have been used [8.13,

8.17]:

^ " Ek

V" 1) ' S k

/Sk

V~ 1)

' (8 - 12)

Although rarely used it appears that this technique might be advantageous where S(X ) spans several orders of magnitude of signal level with similar relative accuracy.

39
k=i

(S

(v) .

s,)' < n [dimensionless]

,

(8.13)

with a [S] being the estimated random experimental uncertainty in S , then the iterations should be stopped. Further iterations will begin to have a deleterious effect —

introducing spurious spectral distortions into the E. which increase with v and which

are due entirely to noise in the S and to computational round-off error.

The convergence factors A, are sometimes increased (or decreased) during the iteration process in an effort to achieve faster convergence. At convergence the factor

[S, - S, ] in eq. (8.10) will have decreased to zero so that the value of E, reached

no longer depends upon A, . Consequently, the only role played by A, is in determining

the number of iterations required to achieve convergence by setting the magnitude of the

correction to be applied in each iteration. Normally, all A, 's would be given the same value based upon a typical evaluation of eq. (8.9) and then, if after 3 or 4 iterations

convergence is not close, this value would be modified upward (or downward) by trial-anderror to hasten convergence (or stem divergence) . Somewhat similarly the ultimate values of

E. achieved at convergence should not depend upon the starting values E. . The initial

values given by eq. (8.6) are an obvious starting point but starting, for example, with all

E. =0 will often require only one or two more iterations. If a better set of initial

values is available it should, of course, be used.

DECONVOLUTION in MATRIX NOTATION. Before turning to a simple numerical example to illustrate the application of eq. (8.10), let us digress for a moment to rewrite the equations

as matrix equations. As we pointed out in Chapter 6 [8.18], no numerical advantage is

gained by this; however, matrix equations are extremely useful in organizing calculations,

particularly since some computer languages contain explicit matrix operations which can greatly reduce the chore of programming. We write the n values of the observed signal S, as an n-dimensional vector

S = [sk

] = [S] (8.14)

and similarly, the n desired values E of spectral irradiance

40
E = [E.] = tit "2 - 1 ! (8.15)

The responsivities R, . form an n by n matrix

R = [R^l -

R

ll

R

12 R

21 R22

In

2n

R . R „ . . . R

nl n2 nn

[S'W

-1 ^

2

] (8.16)

where the matrix elements, R, . , are just the heights of the intersection points on the

Rp(A ,A) surface as sketched in figure 8.1. Now, using matrix "shorthand", the n simultaneous equations represented by eq. (8.5) are exactly what is meant by the one equation

S = AA-R-E [S] (8.17)

or

= AA

R

ll R

12 R

21 R

22

hi ha

R , R

nl n2

li

2i

hi '

ni

RmE

l R

2n

E

2

* E

i

R

kn •

RnnEn _ _

[S]. (8.18)

A typical member of this set of equations is 1

S k "

AX * (RkrE

l + R

k2*

E

2 + ••• + \i*E

i + ••• +Rkn'

En

) (8.18a)

1 This n-dimensional matrix multiplication is an obvious generalization of the 4-dimensional

formulas given in Appendix 6 [8.19].

41
Thus the signal obtained at the k wavelength setting of the radiometer is associated

with the k row of the spectral responsivity matrix R. If the radiometer is set at the

k wavelength and an isolated spectral line source of known irradiance at wavelength A. is observed (i.e., all E = except E.) then the observed signal will be just

S„ = AA'R, . *E . . If this line source is tunable so that it can be moved from A. to X a complete

K kirow of

1

IL . values can be determined. Moving the radiometer wavelength

In

setting to other wavelengths and repeating these measurements will provide other rows of the spectral

responsivity matrix. On the other hand,- if the spectral line is fixed at the i wavelength while the radiometer setting is moved from A. to A a complete column R, . , R_ . , 1 n li' 2i

ki' R . of R values can be determined, ni

Going on now to eqs. (8.10) we see that in matrix notation they become

E

(v) „ E^+A-ts-S^30] (8.19)

with S

(V_1)

= AA.R.E

(V_1) (8.20)

and

A =

A

l A

2 A fT7 -2 -1 _-l, LW*m *nm *S J (8.21)

In the slit-scattering function approximation we approximate the responsivity function

by a product [eq. (7.12); K is defined following eq. (7.5)]

R(A ,A) = K«z(A -A)-r f (A) [S-W

1 «m2

] o o (8.22)

In terms of matrices we can say that the matrix R is proportional to the product of a slit-scattering function matrix

z

=

'-1

J -l

z z z 1-n 2-n 3-n

z i z o 1-n 2-n

n-1

n-2

'-1

'n-1

n-3 n-2

[dimensionless] (8.23)

42
and a responsivity-factor matrix

f r

l °.

. .

r

2

. .

'..

. . . r

[dimensionless] (8.24)

So eq. (8.5) becomes

S

= AA.K-Z-R

f 'E [S]. (8.25)

Usually it is convenient to perform the final multiplication and define a quantity

E' = R

f -E =

r r

E

i

r

f E

2 #t2

r

f .E n n

-2 -1

[W«m 'nm J (8.26)

so that, in terms of this quantity eq. (8.5) becomes

S = AA-K-Z'E' [S], (8.27)

which is identical in form to eq. (8.17) and can be solved by the iterative technique we are discussing. If the responsivity factors r, are known, this then permits the E. to be

f 1 1 calculated from the E'. . If the r. are not known it is necessary to repeat the measurements and the solution of eq. (8.27) for a standard of known spectral irr'adiance. Then the

r can be calculated or they may be eliminated by using the ratios E'. /E'. (s) . That is,

E

I -2 -1

E >( s )

* E

i

( s ) t w ' m " nm !• (8.28)

[The argument (s) , here is used to denote values associated with the standard of known

spectral irradiance.] Forming the ratios S,/S, (s) before deconvolution in "order to avoid

deconvolution of the standard source data will not give the same results unless the relative

spectral distributions of both sources are identical. Returning for a moment to the slitscattering function matrix of eq. (8.23), notice that a column of this matrix (associated

with a radiometer wavelength scan of a source of a fixed wavelength spectral line) contains

the same elements as a row (associated with viewing a spectral line of changing wavelength

at a fixed radiometer setting) but in reverse order. This, of course, is simply another way

of looking at the apparent slit-scattering function reversal with fixed-wavelength spectral

43
lines which has already been mentioned in Chapter 7. The great advantage of the slitscattering function approximation when it is appropriate is that at most only 2n + 1 values must be stored in the computer to specify the ~]_ matrix; any element of the array is

identical to one of these. For large n or small computers this is a vital consideration.

A NUMERICAL EXAMPLE . In this section we wish to illustrate the iterative deconvolution

described by eqs. (8.10) with a simple numerical example. For our example we assume a "true" spectral- irradiance distribution and an instrument-responsivity function as shown in

figures 8.2a and 8.2b. The responsivity function depends only upon the difference A - A. The product of the simple, piecewise, algebraic expressions which define R(A ,A) and

E,(A) can be integrated to give exact values of the "observed" signal {eq. (8.1)}:

S(A ) = / R(A ,A)«E (A)-dA [S] (8.29)

at any value of A . Such values are shown as the second column of table 8-1 for wavelength

intervals of 2 (AA = 2) . Assuming that measurements with our radiometer had yielded

these values of S, we wish to obtain values of E. using the technique of eqs. (8.10).

If all goes well these should approximate values of E^(A) given in figure 8.2a. First we

A need a table of responsivity values at wavelength intervals of 2. These can be seen by

inspection of figure 8.2b to be non-zero only for A = A and A = A ±AA. Thus we have

If we arrange kk

= 4, and R, , , = 2 for any k and all other R, . = 0, Vk-l "

2 ' the R, . in a matrix it will look like

R - iv -

4 2

2 4 2

2 4 2

2 4

2

[S«W

_:L -m

2

] (8.30)

We choose A, = 1/12. This is the value given by eq . (8.9) for all k except k = and k = n of the R array of eq. (8.30). Since the values of A, only affect the rate of

convergence of the iterations and not the final results, it is usual to choose A, to be

constant, independent of k as we have done. The zeroth approximation to the values of E. according to eq. (8.6) is simply the corresponding value of S. multiplied by 1/16. Again

we ignore possible errors in this factor at the ends of the wavelength range. These values

of E. are given in column 3 of table 8-1. They are exactly the values of spectral

irradiance which would be obtained if the responsivity-function correction were ignored.

Now we proceed to the calculation of the first iterative correction according to eqs. (8.10), At each instrumental setting, k, we first calculate the sum £ R «E. and then the

44
8-

(a)

f\. E

X

=8

E^=-Z8+6\-\2A

40 50 60

(b)

2-

R(X ,X> = 4 + (X -X) /\ R(X ,X) =4- (X -X)

R(X ,X)=0 \ / \ / R(X ,X) =0

„ ,

, i ^y i \<

-10 10

Figure 8.2. Spectral irradiance distribution (a), and spectral responsivity function (b)

,

assumed for numerical examples in tables 8-1, 8-la, 8-2, and 8-3.

45
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46
correction A, *[S, - AA-£ R^ -E. ]. For k = 3, for example, (at A = 4) we have

i

7 R, .-E^

0) = . . . + 0.0 + 2*0.02 + 4*0.33 + 2-1.60 + 0.0 . . . = 4.56 and S, = 5.33.

h ki i k

1 (1) So the correction is (5.33 - 2-4.56)/12 = -0.32. Thus, we obtain E^ = 0.33 - 0.32 = 0.01

in agreement with the third entry in the fourth column of table 8-1. In a similar fashion

the other entries in the E column are calculated. Then new corrections are calculated

(2) and the second approximation E. obtained. That is, still for . k = 3,

I

IL .E^ 1) = 2-(-0.08) + 4-0.01 + 2-1.23 «= 2.34, (5.33 - 2-2.34)/12 = 0.05, (8.31)

i

E. = 0.01+0.05 = 0.06. Table 8-1 shows successive approximations E through

v = 9. The next-to-last column is the "true" spectral irradiance taken from the expression

in figure 8.2a and should be compared to the various approximations, E . The final

(9) column shows how well the "observed" signal is reconstructed from the final iteration E and the responsivity array R^ . using eqs. (8.11). The discrepancies between S, and

(9)

S are quite small in the upper part of the table and since the correction to be applied

to E is proportional to the difference S - S [eq. (8.10)] this means that

convergence is essentially complete i:i this part of the table.

END-EFFECTS . In the lower part of table 8-1 it is clear that there is a problem which is

becoming worse as the number of iterations increases. This is caused by our having ar- bitrarily terminated the table of S values at A = 60 before S, had returned to zero,

and by the fact that the computational algorithm, in effect, proceeds as though all E for

A . > 60 (and for A . < 0) are zero. If the data had, in fact, exhibited a step from

S = 128 to S = at A = 60 the solution being generated would be correct. In table 8-1

the iterative solution is clearly valid up to within 2 or 3 spectral slit widths of the end

of the data, although with significantly more iterations this will no longer be true. To

understand why the iteration algorithm functions as though all values of E outside the

measured range are zero consider the last equation implied by eq. (8.17) [see eq. (8.18a)]:

S = AA-( . . . + R -E ). (8.32) n nn n

If we had made one more measurement we would be writing this equation

S = AA-( . . . + R -E + R -E ._). (8.33) n nn n n,n+l n+1

[There would, of course, now be one more equation:

S = AA-( . . . +R -E + R ,. *E .,) (8.34) n+1 n+l,n n n+1, n+1 n+1

but we aren't interested in it at the moment.] Since S and the R, . and E, are all

n Tci i fixed numbers, obviously both eqs. (8.32) and (8.33) can't be correct unless E = 0. n+1

47
Thus, if we use eq. (8.30) for the matrix R we imply that all E. = for i < 1 and for

i > n. The best way to avoid this problem is by extending the wavelength range of the

measurements or by the use of spectral filters to assure that the instrument output will be

negligible outside the measurement range. If this is impractical then it may be possible to extend the table of S values by measurement or estimation far enough that the end-effect

after deconvolution will not affect the values of E . in the wavelength region of interest.

In the example of table 8-1 the wavelength span in which end-effects can be seen is given

approximately by

AX '/v, (8.35) w

where AA is the spectral slit width (eq. 7.34). So, for v iterations, measurements or w

estimates of S, should be extended this far beyond the spectral region of interest on either end if the signal is not negligible there. In the present example we might assume

that the wavelength region of interest is from A = to A = 46 and that the S values from A = 48 to A = 60 were included only to avoid the end-effect. Since the

data run smoothly to zero at A = there is no end-effect problem there.

The^ last row of table 8-1 gives the rms error in reconstructing S from the corresponding column of E. values, i.e., the quantity

jl ECSi-S.^) 2 (8.36)

i-1

for each iteration. We have omitted from the sum the values in the lower half of the table

(A > 30) in order to exclude the end-effects region. If the measurements S are assumed

(3) to be accurate to within 0.1% of their maximum then three iterations (E ) are evidently

sufficient to achieve convergence to within the accuracy of the data and' the iterations

should be terminated there. Of course, the rms value of the error E. - E. ' would be a more valuable statistic but this is never available in a real measurement situation since

E (A) would be unknown. In the next paragraph and near the end of this chapter we briefly

discuss errors in the deconvoluted results, E

ACCURACY . In' spite of the fact that convergence is essentially complete after a few

iterations in the upper part of table 8-1 there still remains a significant discrepancy

P C9)

J k ~"~ T """" *V

attributable primarily to the error in approximating the exact integral of eq. (8-1) by the

summation of eq. (8.5). Equation (8.5) is exact if the integrand, R(A ,A) 'E(A) , is linear

between data points. In figure 8.3 we show this product as a function of A for A =6.

Clearly, in any one segment, say from A = 4 to A = 6, the integrand is far from linear;

however, the errors are positive in some parts of the range of integration and negative in

others so that they tend to cancel. For this reason more elaborate integration formulas

improve the accuracy very little. The certain way to improve the accuracy in the

48
Responsivity function, spectral irradiance, and their

product for A =6. From example of figure 8.2.

49
neighborhood of rapid changes in S(X ) is to take data with a smaller AX, i.e., to take more data. Of course, even with a perfect mathematical solution there may still be errors

if there exist measurement errors or if eq. (8.1) is not obeyed due, for example, to a nonlinear detector. The result of taking more data in our example is illustrated in table 8-la

in which AX was taken as %. The accuracy of deconvolution after a few iterations is dramatically improved over the results given in table 8-1. The rate of convergence is essentially the same. Table 8-la shows only the portion of the calculations corresponding to the

upper portion of table 8-1. In the end-effects region we would see that the amplitude of

the oscillations is much greater with the smaller sampling interval but the range of wavelengths over which the calculations are perturbed by the end-effect is the same.

The INVERSE SLIT- SCATTERING FUNCTION . When the slit-scattering function approximation, eq. (8.22), is appropriate a technique is available which saves considerable computational

effort if two or more spectra are to be deconvoluted using the same slit-scattering function

[8.20]. Let us write eqs. (8.5) in terms .of the slit-scattering function

S k

= AX-K-E z k

_.-E]_ [S], k=l,2,...,n. (8.37)

i

As in eq. (8.23) we have used z, . = z(X,-X ) [dimensionless] where X [nm] is the

instrumental setting and, as in eq. (8.27), we have combined E. with the responsivity

factor r. = r (X.) [dimensionless] to give E' = r . *E . [W«m *nm ]. The summation is over all values of E', and we take the slit-scattering function z, to be zero any time

k-i takes on a value outside the range for which the slit-scattering function has been

tabulated. Let us also assume that n is an odd number and redefine i and k so that

numbering begins at the mid-points of E and S. Thus if n„ = (n-l)/2 we rewrite eq. (8.37)

n

2

S k

= AX-K- J2 z k_i'E

l IS]. k = -n

2 ,-n2+l,... ,-1,0,1,...,

n

2

. (8.38)

i=-n

2

Now we multiply both sides of this equation by a quantity v., (yet to be determined) and sum over k

n

2 n

2 / n2 \

I Vk'S k = AX * K \£ E

l ( £ Zk-i*V

£-k • (8.39) k=-n

2 i=-n2 \k=-n2 /

We see that if the values of v , could be chosen in such a way that 1

1 5., in eq. (8.40) is the Kronecker delta function. It has the value 1 if i = k and

if i f k. The Kronecker delta function has the same effect in summations as the Dirac

delta function [eq. (8.4)] has in integrals.

50
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51
n

2 K'^J z v_n' v o_v

= &

o-i [dimensionless]

,

i = -n9

, . . . ,n

? (8.40)

k=-n„

i_ , • • • ,"2

'2 £ = -n

2

, . . . ,n

2

then eq. (8.39) would become

n

2 n

2 -2. 5: v

.-k*

s k

= AX ;£ E i-«ii = AX * E

1 tw ' m" ] (8 - 41)

or

k=-n„ i=-n

2

n

2 1 V- „ r„ -2 -1,

V

l2

E * " IX\L v

£_ k

.S k [Wm-.nm-], ft = -n,, V (8.42)

k=-n.

which is a simple, direct solution of eq. (8.38) for E' . If the z are known the

-1 -2 -2 -1

v. [S •W , m ] are uniquely determined by eq. (8.40) and the quantities El [W # m *nm ] i *- can be evaluated for any number of spectra, S, using eq. (8.42) without recomputing the

v. as long as the slit function remains unchanged. Comparison of eqs. (8.38) and (8.42)

1 -1 -2

shows that the vector of v. [S •Wtti ] values constitutes a kind of reciprocal responsivity function and it also possesses the appropriate unit dimensions for this interpretation. To obtain the v. from eq. (8.40) we first multiply both sides by AX and change

the signs of i, k, and £:

-n

2 AX-6 M = AX-K-£ z ±_ k -\. l [TM>] < 8 - 4 3)

k=n

(by its definition, of course 6 = { = 6 ). Then letting £ = and observing that' as long as we sum over all values of k in the range from -n. to n the limits on the

summation can be interchanged, we obtain

n

2 AA-6

io

= AX-K •£ z ±_k

- v

k W. i = -n2 ,-n

2+l,..., -1,0,1,...

n

2

. (8.44)

k=-n

2

But eq. (8.44) has exactly the same form as eq. (8.38) so that any technique for solving

eqs. (8.38) [or eqs. (8.5)] can be used to obtain the quantities v. simply by starting

with a "signal" which is AX at the center and zero everywhere else.

So, in summary, to use this technique, we apply eqs. (8.10) to solve iteratively for

values of E. when S, is given numerically by

S k

= AX-S

kn , k= 1,2 n(odd) (8.45)

52
where n„ = (n-l)/2 and IL . = K*z .. Now denoting this solution after v iterations by

(.. Z. KX K.~ X v. , i.e., numerically

(v) _ (v) (8.46) V . - SL . 1 X

we can use these v. to compute a v order deconvolution of any set of real measurements from

; (V) = ik't^Zl'K [W-m^-nm-1

], £=l,2,...,n. (8.47)

£ AX ,*-' £-k k

k=l

For want of a better name we will call the set of values v, the v order inverse

th slit-scattering function . The idea of the v order inverse slitrscattering function

could presumably be generalized to apply to any arbitrary responsivity matrix by iteratively

computing each column V of a complete v order inverse responsivity function matrix

using the appropriate S. = AA-6.. for j = 1,2,.. . ,n. However, as far as we know, this has never been done. In the slit-scattering function approximation a full matrix of the V,kJ values can, of course, be assembled looking exactly like the ~[_ matrix of eq. (8.23) with

z everywhere replaced by v : in this approximation the inverse slit-scattering function

matrix V nas the same kind of symmetry 1 as that possessed by the slit-scattering

function matrix. This matrix y would satisfy the matrix equation

E

.(v) = _1 . v

(v).

g [w . m

-2. nm

-l K (8>48)

In table 8-2 we show the results of inverse slit-scattering function calculations using

the same slit-scattering function as in the example of table 8-1. In columns 3 to 12 we show successive approximations v for the delta-function-like S given in column 2. These are computed using eqs. (8.10) exactly as the corresponding columns of table 8-1 were

computed. In the last column we show the results of computing

31

Ek

9) = ~t'% V

k-i ,S i [W-TM"

2 -TM"1

]. k - 1,2,... ,31, (8.49)

'The symmetry we are talking about in this chapter is not the symmetry that a mathematician

has in mind when he speaks of a symmetric matrix. A symmetric matrix in that sense means

that the matrix and its transpose are identical. Thus, a symmetric matrix is symmetric

about its principle diagonal, which is the diagonal running from upper left to lower right.

The special slit-scattering function matrix symmetry, or row-by-row invariance with which we

are concerned is a kind of symmetry in which all elements of any diagonal parallel to the

principle diagonal have the same value. Such a matrix will also exhibit the customary

symmetry if the slit-scattering function is symmetric but we make no use of that matrix

property in this chapter.

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54
(9) where v. = if j < or j > 32 and where S is now the "observed" signal given in

J (9) column 2 of table 8-1. A comparison of this last column of table 8-2 with the E column of table 8-1 shows no differences aside from the end-effects at the bottom of the

table. The advantage of the calculation of the inverse slit-scattering function v of

table 8-2 is that any set of data taken using this slit-scattering function may now be

deconvoluted using eq. (8.42), whereas without this calculation the entire iterative process

must be repeated for each new set of data. There are two minor disadvantages, however:

(1) to see convergence one must convolute the inverse slit-scattering functions with data; one usually cannot detect convergence in the sequence of inverse slit-scattering functions

of successive orders. (2) The inverse slit-scattering function approach, as we have outlined it, is valid only when the responsivity function can be approximated by the product of

a slit-scattering function and a responsivity factor.

UNIQUENESS and NOISE in DECONVOLUTION . If R(X , X) is accurately known and S(X ) is

measured it would seem, in principle, that enough information should be available to calculate E(X) with about the same degree of accuracy as S(X ) possesses. Unfortunately,

we cannot extract this much information from eq. (8.1). First of all, the solution of eq. (8.1) is not mathematically unique. Consider, for example, the simple rectangularly shaped

responsivity function

R^X ,X) = 1 for |X -X| < 1/2

(8.50)

= otherwise.

Then eq. (8.1) becomes

X +1/2

S(X ) = J ° E.(X)-dX [S]. (8.51)

° X -1/2 X o

Now suppose we have a solution E (X) which satisfies eq. (8.51) for some set of observations S(X ) . Clearly

o

oo E,(X) + £ (A -cos2TmX + B •sin2imX) (8.52)

a , n n n=l

will also satisfy eq. (8.51) for any values of the A and B because

n n

X +1/2 X +1/2

J ° cos 2imX-dX = J

° sin 2-rmX'dX = 0. (8.53)

X -1/2 X -1/2

o o

Thus there are an infinity of solutions to eq. (8.51) represented by eq. (8.52) and the one

"true" solution cannot be determined from eq. (8.51) alone. Appeal must be made to other

desired properties of the solution such as smoothness or non-negativity to find an acceptable solution. In the iterative deconvolution technique, for example, we obtain a solution

55
which has the property that it satisfies eqs. (8.5) within the measurement uncertainty and

exhibits, in some sense, minimal departures from the spectral distribution of the measured

signal. Later we will discuss a deconvolution technique which picks out the solution

which, within limits, optimizes a measure of the smoothness of E.(A). Such solutions may

still not precisely describe the true spectral irradiance but they will usually be acceptable solutions in terms of one's intuitive notions about the nature of the spectral irradiance distribution.

In principle there is another similar cause of non-uniqueness in a real deconvolution

due to the wavelength interval of sampling, AA. When S(A ) is known only at a number of

discrete points, the curve of S(A ) may have any conceivable shape in the intervals AA between measured points and each of these possible S(A ) curves will give rise to a different solution E.(A); even the values of E.(A.) at the measured points, A., will

A A i i depend upon how S(A ) is interpolated between the measured points. Fortunately, in

practice one intuitively takes the wavelength interval AA small enough so that usually no

significant excursions of S(A ) remain unsampled. In any case the choice of the wavelength interval is controlled by the experimenter so that non-uniqueness from this source

can be minimized easily whereas that discussed above, which is due to the breadth and shape

of the responsivity function, can be reduced only by modifying the responsivity function —

by narrowing the slits, for example. Presumably, however, if the responsivity function

could have been narrowed, the experimenter would already have done this, balancing the

improved spectral resolution against the increased noise accompanying the lower signal

levels to arrive at over-all optimum experimental conditions for the total measurement time

available.

Another related aspect of the problem is that when the spectroradiometer physically

converts E.. (A) into S(A ) it performs a smoothing operation, flattening out the hills

A O and filling in the valleys of E i( A ) (see, e.g., figure 7.14c). In fact, the sharper the

detail in E.(A) the more it is smoothed out by the instrument. Therefore any mathematical

scheme which reconstructs E.(A) from S(A ) must amplify any small bumps and dips of

A o S(A ) to rebuild the hills and valleys of E. (A) . And the finer the detail, the greater

O A must be the amplification. The result of this is that random experimental noise in the

S(A ) values will receive great amplification so that in the presence of noise any exact

solution of eq. (8.1) is likely to produce a distorted spectral irradiance distribution

containing a wealth of spurious spectral detail. In many respects deconvolution may be

thought of as equivalent to carrying out the same experiment in the same length of time but

with narrowed spectroradiometer slits and a smaller wavelength interval: the total number

of photons collected during the experiment is the same but it is now distributed among more

samples resulting in a large increase in the relative noise in the high resolution samples

compared to that in the original, actually measured samples. Any attempt to achieve perfect

spectral resolution must result in infinite relative noise. A further consequence of the

physical smoothing performed by the spectroradiometer is that any spectral detail in E , (A)

which is smoothed to the point where it is lost in the measurement noise [as the sinusoidal

56
terms in the example of eq. (8.52) would be] will be irretrievably lost: it is not recoverable by any conceivable mathematical manipulation.

We can easily see how the iterative procedure selectively amplifies sharp spectral

features. Suppose the measurements S, contain a sharp spectral feature at X . Then, of

(0) course, E^ will contain this same structure [see eq. (8.6)]. However, the calculated

values of S, in the neighborhood of X will contain a much broader version of this

feature because the calculation S = AA*£ R «E simply performs mathematically the

( ) i same smoothing operation on the E as the radiometer does on the physical spectrum.

Therefore the correction term S, - S will still contain the feature at X in a slightly attenuated version. When such corrections are added again and again in the course

of generating E the magnitude of the excursion at X will have increased by a factor

of nearly A *v. A very broad spectral feature, on the other hand, will be reproduced quite

faithfully in computing S so that the correction term S - S will quickly

vanish as v increases.

A GENERALIZED LEAST-SQUARES DEC0NV0LUTI0N . We have described the iterative deconvolution

method in detail because of its intuitive appeal, its frequent appearance in the literature,

and because it will work in almost all radiometric applications. The number of iterations

required to reach a particular degree of precision depends mainly upon the amount of distortion that must be removed (and, of course, upon the degree of precision required).

Assuming that the ultimate spectral resolution sought is comparable to the wavelength

interval AX, if AX is wider than or comparable to the spectral slit width then a few

iterations will suffice. However, if AX is much smaller than the spectral slit width

then dozens or even hundreds of iterations may be required to achieve a satisfactory

deconvolution. Although successful iterative deconvolutions can be effected in these cases

[8.2,8.20] — usually by making use of techniques for accelerating convergence and for

smoothing to minimize the amplification of experimental noise which accompanies the iterations — this approach to deconvolution is not generally recommended [8.11,8.21] if more

than 5 or 10 iterations prove to be required. Occasionally the method fails by divergence

of the iterations before the condition of eq. (8.13) is satisfied. The method is most

likely to fail [8.21] with markedly asymmetric responsivity functions, with noisy data, and

when spectral detail narrower in width than the spectral slit width is sought. In the next

sections we present a deconvolution technique which is somewhat less likely to fail. This

is not to say that the solution will always be satisfactory: attempts to extract fine

spectral details from noisy data taken with coarse slits will always lead to solutions with

a substantial credibility gap. The technique to be described is a generalized least-squares

or constrained least-squares deconvolution method and has been described in various forms by

many authors [8.13,8.21,8.22,8.23]. We will not attempt a derivation of the method here but

will simply outline the approach.

57
Consider the following expression evaluated for any arbitrary estimate of the spectral

irradiance function E ^(^)

2 > 2

/ <S(X

q

) - /R(X

o ,X)-E

x (X)-dXj -dX

o + y-/ ^— * dX tS*nm]. (8.54)

I ^ I

The integrand of the first integral is the square of the discrepancy between the measurement

S(X ) at X and the instrument output calculated 'from this E.(X). The integral over

X is then a measure of the total error throughout the spectrum. The last integral is a measure of the roughness of the function E (X): if this E (X) has sharp excursions then

the second derivative will take on large values and this integral will be large. The com- plete expression is thus simultaneously a measure of the magnitude of the error and the

2-2 4 4 roughness or irregularity of the function E^(X). The parameter y [S «W «m «nm ] determines how much relative weight is to be given to these two attributes of E (X) (it

also absorbs some dimensional constants since the dimensions of the two components of ex- pression (8.54) are otherwise obviously different). The minimization of expression (8.54)

with respect to the parameters which describe E\(X) then yields a generalized leastsquares solution which simultaneously minimizes errors and roughness with a relative weighting determined by y. In the case of discrete measurements, expression (8.54) takes the form

I (s k

- ax.^.e.) 2 + r-I ( E

±_r2Ei + E i+1 ) 2 /(^)

2 rs

2

] < 8 - 55 ) k i i

and the minimization leads to the solution

E

= aI

* (R

T -R + vH)

_1

-R

T

-S [W-nf2 -TM-1

]. (8.56)

This is a matrix equation in which the quantities £ an^ S» the spectral irradiance and

instrument output vectors, and R the responsivity matrix, have already been introduced.

The superscript T on a matrix symbol means the matrix transpose of that matrix — obtained

T by interchanging rows and columns: i.e., R = R . The superscript -1 on a matrix

r , c c , r symbol means the matrix inverse of that matrix. The product of a matrix by its inverse

results in a matrix known as the identity or unit matrix 1:

*The notation for the matrix inverse and for the unit matrix brings out the analogy between

these quantities and the corresponding algebraic quantities, reciprocal and unity. Unfortunately the matrix inverse is difficult to calculate especially for large matrices and the

analogy is further marred by some of the other idiosyncrasies of matrix expressions such as

the importance of the order of the factors in multiplication. Since we are not attempting to derive the equations in this section, familiarity with matrix algebra should not be necessary. For interested readers, brief introductions to matrix algebra are given in [8.13] and

[8.24]. Complete treatments can be found in textbooks on matrix algebra or linear algebra.

58
A

_1

-A - I -

100

10

1 1

1

[dimensionless] (8.57)

The matrix H is given by

H =

1 -2 1

2 5 -4 1

1 -4 6 -4 1

1 -4 6 -4 1

1 -4 6 -4 1

r ~ 4 [nm i

] (8.58)

The matrix is symmetrical about both diagonals and except for the first two rows and the

last two rows, all rows are similar.

The matrix \\ given in eq. (8.58) tends to force E^(^) in the direction of linearity. Other choices for the matrix \\ are possible corresponding to different choices for

the smoothing term in expression (8.54). For example, the matrix

H =

1 -3 3 -1

-3 10 -12 6 -1

3 -12 19 -15 6 -1

-1 6 -15 20 -15 6 -1

-1 6 -15 20 -15 6 -1

[nm ] (8.59)

corresponding to a term y

J

d VA)

i dA3 •dA [S *nm] in expression (8.54) tends to force

E,(A) toward a quadratic. Usually there is little difference in the final results. Other

examples are given in [8.21,8.22,8.25], and especially [8.13].

When y = in eq. (8.56) the solution reduces to

E

= AT (R

T 'R)"*-R

T-S [W-nf^nm- 1

], (8.60)

59
which is the least-squares solution for a set of equations like eq. (8.17). When Y = and the number of equations is exactly equal to the number of unknowns, as it is in our

case, eq. (8.60) further reduces to

E

= aX*R

_1

*S [W-m-2 -nm_1

], (8.61)

which is just the formal, exact solution of eq. (8.17). As we have noted this solution is

likely to be unsatisfactory in deconvolution because of the problem of noise amplification.

As Y increases eq. (8.56) gives solutions of increasing smoothness. Eventually, for large

Y, the calculation of the matrix inverse (R *R + Y'H) will generally fail. This

appears to be a numerical problem associated with the fact that the determinant of \\ is zero.

The practical difference between eq. (8.56) and eq. (8.61) as solutions of the deconvolution problem is that because of the smoothing represented by the term Y'H [S #W «m ] eq. (8.56) may be solved by exact methods. If explicit matrix operations are available in

the computer language employed and if n is not too large the matrix operations can be

carried out directly. In languages without explicit matrix operations the calculation is more trouble, but still straightforward: the matrix inversion is the most difficult step

and algorithms for this are available in most computer centers.

So, in summary, in the generalized least-squares technique we choose a value of 2-244 -2-1

Y [S *W *m *nm ] anc* compute the values of E . (Y) [W*m "'ran ] using eq. (8.56). One

2 might start, for example, with Y * .01R* where R* is the largest element in the respon-

—1 2 sivity matrix R [S*W ' *m ]. We then evaluate some measure of the agreement between the

reconstructed signal and the measured signal, such as the standard deviation:

o(Y) =

} ^I'l [S k

- AA-^.-E.Cy)] 2 " [Si. (8.62)

Now, depending upon whether o(y) is larger or smaller than the estimated random measurement uncertainties, a smaller or larger value of y is chosen — say by a factor of 10 —

and the calculation is repeated. The goal is to introduce as much smoothing as possible —

as large a value of Y as possible — consistent with the measured signal, allowing for its

uncertainty. The solution is not very sensitive to Y and a less-than-optimum choice by an order-of-magnitude will' not seriously affect the results.

In table 8-3 we show the results of solving the deconvolution problem of table 8-1

using this generalized least-squares technique. The last four columns of the table are solutions obtained using different values of Y- Notice that we still have an end-effect at the bottom of the table. This is because the technique attempts to recreate the sudden jump

in S at X = 60 and, as we have said, the results would be perfectly correct if the measured S, values actually dropped abruptly to zero at this point. The similarity between

this family of solutions and the successive iterates of table 8-1 is apparent. The last row

60
Table 8-3. Generalized least-squares deconvolutlon

S y=.Q1 y=.1 y=1 y=10

.00 . 00 .00 . 04 -. 10

2 . 33 -. 00 . 00 -. 06 -. 16

4 5. 33 . 08 . 07 . 06 . 20

6 25. 67 1. 17 1. 19 1. 29 1. 59

8 64. 00 4. 00 4. 00 4. 00 4. 00

10 102. 33 6. 33 6. 31 6. 71 6. 41

12 122. 67 7. 91 7. 93 7. 95 7. 31

14 127. 67 8. 01 3. 00 8. 04 8. 16

16 128. 00 7. 99 8. 00 7. 93 3. 07

18 123. 00 8. 01 8. 00 3. 00 7. 99

20 128. 00 7. 99 8. 00 3. 00 7. 93

22 128. 00 8. 00 8. 00 8. 00 8. 00

24 123. 00 3. 00 3. 00 3. 00 8. 00

26 123. 00 7. 99 8. 00 8. 00 8. 00

28 123. 00 8. 00 3. 00 8. 00 8. 00

30 123. 00 3. 00 8. 00 8, 00 8. 00

32 128. 00 7. 9S 3. 00 3. 00 8. 00

34 128. 00 3. 06 8. 00 8. 00 8. 00

36 123. 00 7. S9 8. 00 8. 00 8. 00

38 128. 00 3. 13 7. 99 8. 00 3. 00

40 123. 00 7. 73 3. 02 8. 00 8. 00

42 123. 00 8. 37 7. 97 8. 00 3. 00

44 123. 00 7. 54 8. 01 7. 99 3. 00

46 128. 00 3. 51 8. 04 8. 00 3. 00

48 123. 00 7. 53 7. 84 8. 02 8. 01

50 123. 00 3. 26 8. 34 7. 95 8. 03

52 123. 00 3. 19 7. 52 7. 99 8. 01

54 123. 00 7. 02 8. 39 8. 29 7. 83

56 123. 00 10. 17 8. 30 7. 47 7. 70

58 123. 00 4. 20 6. 05 7. 62 8. 49

60 128. 00 13. 30 12. 66 11. 43 10. 34

.02 . 05 .34 1. 66

61
of table 8-3 is again the rms deviation of the computed AX*^R , 'E. from the measured S, for the first sixteen values of the four columns of E values. For measurement uncertainties in S, of 0.1% something between y = .1 and y = 1 would apparently be appropriate. The matrices R and \\ used in this solution have been given in eqs. (8.30) and

(8.58). As an aid to readers unaccustomed to working with matrices we will briefly indicate

the appearance of the rest of the matrices involved in this example. Since R is symmetric

about the diagonal running from upper left to lower right R and R are the same. The

T product R *R is

R

T -R

20 16 4 16 24 16 4

4 16 24 16 4

4 16 24 16 4

4 16 24 16 4

2-2 4

[S «W V]. (8.63)

For Y

= 1 the matrix R *R + Y*H becomes

R-R + H

21 14 5 14 29 12 5

5 12 30 12 5

5 12 30 12 5

5 12 30 12 5

[S

2 -W"

2 «m

4

] (8.64)

A portion of the middle row (or column) of the inverse of the matrix of eq. (8.64) is

10

_6

(, -73 395 -425 -1125 4375 -3125 -15625 46875-15625 -3125 . . .) (8.65)

Neighboring rows (or columns) are similar but displaced just as in eq. (8.64); however, the

departures from this typical row at the edges of the array are more pronounced than in eq. (8.64). Finally, the product of (R *R + H) by R for y = 1 gives a matrix whose

middle row (or column) is

(. . ,0004 .0006 -.0032 .0034 .009 -.035 .025 .125 .025 -.035 .). (8.66)

This matrix is functionally the equivalent of the inverse slit-scattering function matrix y described earlier.

62
There is no requirement in this procedure that the number of E

.

values be equal to

the number of S, values or that the wavelength spacings be uniform. In our numerical

example, for instance, if an additional measurement were made at A = 5, it would only be

necessary to insert an appropriate additional row into R

R =

4 2

2 4 2

2 4 2

1 3 3 1

2 4 2

(8.67)

expressing how S(A = 5) is to be calculated from the E(A) (A = 0,2,4,6,8,...). The

matrix R now will have 32 rows and 31 columns. Matrix multiplication requires that the

number of columns of the factor on the left be equal to the number of rows of the factor on the right and then the product matrix will contain as many rows as the first matrix and as many columns as the second. The following diagram illustrates this rule schematically

*******

*******

*******

*****

*****

*****

*****

*****

*****

*****

*****

*****

*****

(8.68)

If the matrix R has m (> n) rows and n columns then R will have n rows and m

T columns and matrix R *R will be an n by n square matrix. The matrix H then must be

chosen as an n by n matrix and we must invert an n by n matrix to solve the deconvolution problem. Note that the size of the matrix to be inverted is given by the number

of equally spaced E. values desired and AA is the spacing of these values.

An ITERATIVE SOLUTION for LARGE NUMBERS of POINTS . If the number of measurement points,

n, is large — greater than something like 100 — limitations of computer memory capacity

and increasing inaccuracies of matrix inversion routines make the direct evaluation of £

by eq. (8.56) extremely difficult if not impossible. In this case an iterative solution

similar to that of eqs. (8.10) and (8.11) can be carried out. Let us write

T R

T

-R + y-H [s

2 -w"

2 .m4

] (8.69)

63
and

Q = R

T 'S [S

2

--W-X

-m

2l; (8.70)

then eq. (8.56) can be written

E =

"^x'T" *Q [W'm •nm ] (8.71)

or Q = AXrJ'E which is the same form as eq. (8.17). Thus the iterative solution can be

written [by analogy with eq. (8.19)]

E

(v) = E(v-1) + a.|q .. AX . T

. E

(v-D]

LW-rn *nm J

.

(8.72)

E is computed from S by eqs. (8.6) exactly as in the simple iterative deconvolution we discussed earlier. The execution of the iteration scheme of eq. (8.72) normally can be

carried further toward convergence than can the simple iterative deconvolution scheme of eq. (8.10) because the term Y'H prevents noise build-up. Notice that if the matrix A were

taken as the exact inverse of AX •J, that is, if /\ = tt'T we would have

r(v) r(V-l) . 1 T-l n T r(v-l) 1 T-l n rT7 -2 -1- E - E + TT'T *Q ~ I*E = TT'T *Q [W*m -nm ], AX AX (8.73)

which is just eq. (8.71) or (8.56). In this case E \E) ^ s independent of the trial

spectral irradiance vector E an<^ consequently convergence is immediate with no iterations required. Therefore, if f\ can be chosen as a good approximation to

"rr*J the iterations of eq. (8.72) can be expected to converge very quickly. The folAXlowing procedure may

We are looking for a matrix A which satisfies

be used to compute a good approximation to TT'T [S 'W'm 'nm ]

.

A-T - aH [nm"

1

], (8.74)

where \ is the unit matrix. In obtaining A we make use of the fact that the elements of

the matrix ! are everywhere nearly zero except for a narrow band along the principle

diagonal

T -

9 -7 4 [S -W -m ] (8.75)

64
and that each row is identical or nearly identical to its neighbor except for a one-step

horizontal displacement. Then we determine the elements of a matrix /\ [S *W *m *nm ],

with a similar narrow diagonal band of identical non-zero rows which approximately satisfies

eq. (8.74). Thus we have a matrix equation which looks something like this

_1

AX

1

1 .

1

[nm ] ,

1

(8.76)

A

where the diagonal bands, because of their assumed row-by-row invariance look like

-2 -1

A

-2 A

-l

-2 -1

-2 -1

A

i v • '

A A

l A

2

(8.76a)

and

T-2 T-l

T

o T

l V • •

• • T-2 T-l

T

o T

l V • •

. . . T-2 T-l

T

o T

l V • •

. . . T

-2 T-l

T

o T

l T

2

(8.76b)

65
Consider now the equations which generate a single column near the middle of the unit

matrix:

VAi = VA

i + Tl «A2 A = VA

i + Tl »Ai A

o + VA-i

=

-1 A

l + VA

o" + Tr

A

-i

= 1/AX

T «A -2 A

l + T-r

A

o + VA

-i

=

T-2* A + T-i*A

-i

T-2* A

-1

=

=

(8.77)

In eqs. (8.77) we have assumed that only the | elements from T „ to T„ are significantly different from zero and that we desire a matrix /\ with only 3 non-zero diagonals.

In general, with the number nT of such non-zero diagonals in the matrix and the number

n. of /\ elements desired we will have n„ + n. - 1 equations in the n. unknown /\ matrix elements. These equations can be solved in a least-squares sense to give a set of j\ elements which will define a good band-matrix approximation to tt'T • The least-squares

solution is given by [compare eq. (8.60)]

/ |T ,v . ,T X = (T -T)"

1 -! 1/AX r -2

T72 -4 -1,

[S «W *m *nm J

,

(8.78)

where the elements of vector X» from top-to-bottom are the values A ,

i = . . . 2,1,0,-1,-2, and | is the abbreviated n + n -1 by n. matrix

A

66
(n columns)

-1

[ -2 -1

C -2 -1

[ -2

(nT + n - 1 rows) (8.79)

iT , The matrix I *J in eq. (8.78) which now must be inverted is only an n by n matrix.

Thus, by using this approximation and accepting the complication of iteration we are able to reduce the dimensions of the matrix which must be inverted from n by n (n is the total

number of measured points) to n by n where n is a small number usually taken equal

to the number of significantly non-zero elements in a row of the responsivity matrix R. Dr 3 iT r should pose no problem. Of course, n may

Thus, n will normally be equal to 2 or 3 times the nominal spectral slit width in units

I ,T , Vl of AA, so that the calculation of \J »J I always be chosen smaller, if necessary, at the expense of additional iterations

If n = 1 eqs. (8.77) reduce to

T

2

' A

o =

T

l

' A

o =

T

o

• A

o = 1/AA

T •A

1 A

o =

T_ 2* A =

(8.80)

with the least-squares solution

A

Q = T /[AA.( 2 2 2 2 2 T + T + T + T + T

2 10 -1 -2 -2 2 -4 -1

. )] [S

Z -WZ -m 'nm ^] (8.81))

67
This obviously does not solve eqs. (8.80) very well but it represents the best available

approximation to

A

=

AA

in a least-squares sense which consists of a single diagonal.

A

o A

o A

o A * (l/AA)-]"1

[S

2 -W

2 -m4 -nm

1

] (8.82)

This solution, for n = 1, when applied to the responsivity matrix of the simple iterative

deconvolution discussed earlier leads to eq. (8.9). More elaborate j\ matrices corresponding to n > 1 can be used in the simple iterative deconvolution just as in the iterations of the generalized least-squares method and will usually result in more rapid con- vergence. However, one of the most appealing features of the iterative deconvolution technique is its simplicity, both computational and conceptual, and, in our opinion, the effort

required to provide an improved /\ matrix would be better devoted to applying the generalized least-squares technique or a Fourier technique [8.6,8.7,8.26,8.27,8.28] to one's data.

As in the simple iterative deconvolution the value of the matrix /\ should only affect

the rate of convergence, not the final values of spectral irradiance obtained at convergence.

If the rows of matrix are not invariant [as they are in eq. (8.79)], which means that R

cannot be approximated by a slit-scattering function, then /\ will depend upon which part ticular group of n columns of ' are chosen for the submatrix [eq. (8.79)]. Thus

i "

any particular may lead to a matrix /\ which is inappropriate for other spectral

regions of the calculation in the sense that convergence may be slow in those regions. In extreme cases it may be necessary to calculate different matrices /\ for use in different

spectral regions; in other words, to break the problem into smaller problems in each of which

1 -1 a single /\ ^ "TT'T provides adequate convergence.

For brevity we have presented the generalized least-squares method, for the most part,

in matrix notation. We wish to stress that the implementation of the technique on a computer

need not use matrix operations except for the single matrix inversion, which can usually be

handled by a call to a suitable subroutine available in most computer-center program libraries. Moreover, to the extent that the matrices are band matrices with invariant rows, the

matrix multiplications can be replaced by convolutions. Convolution calculations reduce

computer-memory requirements by storing only the non-zero elements of one row of such a matrix and obtaining other matrix elements by juggling the indices of this row. For example,

instead of

68
\ = (T-E) r = E Trk -Ek

k=l

r = 1,2, ...,n

and (8.83)

Prc

= (R

T -R)rc " ^

R

rk'

R

kc

= £ R

kr'

R

kc k=l k=l

r,c = 1,2,. . .,n

we can write

W(i) = £ T(k)-E(i-k)

k=-n„

i = -n

2 ,..., -1,0,1 n

2

and n

2 _ n2 P(i) = Y, R (k)-R(i-k) = £ R(-k)-R(i-k) i = -n -1,0,1, ... ,n.

k=-n. k—n.

(8.84)

In these expressions (T ), (R ) and (P ) are n by n matrices with n assumed to be an odd number and n„ = ^(n - 1). The quantities T(k)

,

R(k) and P(k) are the entries

along the invariant rows of these matrices as in eqs. (8.76) and (8.76a). That is, for any

r

:

T(k) = Tr,r+k> R(k) = R

r,r+k' P < k> = Pr,r+k

= (R

T *R)r, r+k

and (8.85)

E(k) = En2+k+l> W(k) = W

n2+k+r

Examples of computer programs for both the simple iterative deconvolution technique and the

iterative generalized least-squares technique are given in Appendix 8A. Both these programs

use convolution instead of matrix multiplication and hence, are applicable only when the

slit-scattering function approximation is adequate. The only matrix operations used are

those associated with evaluating eq. (8.78).

CONFIDENCE BANDS and the RESIDUAL SPECTRAL RESPONSIVITY FUNCTION . If we can obtain an estimate of the matrix which converts the measured signal vector S into the deconvoluted

spectral irradiance vector £ then we can calculate how the random uncertainties in the

values of E. are related to the uncertainties in the measured S. [8.21]. Calling this

matrix |_ we need to know in the equation

E

=

|_"S [W*m *nm ] (8.86)

where £> here, is the calculated set of spectral irradiance values which result from deconvolution. Luckily, we already have expressions for |_. For the simple iterative deconvolution we have [from eq. (8.48)]

69
|_ = "TT*V IS •W*m *nm ]. (8.87)

For the generalized least-squares solution [from eqs. (8.70) and (8.71)]

L =

•ki'T"

1 -Rt [s^'Wi-2 TM-1

] (8.88)

or, if the iterative routine is used

L * A*R

T tS'-'-'W'm^'nm"1

] (8.89)

Now if £ and $ are related by eq. (8.86) then the uncertainties in E are related to the

uncertainties in 3 by

MM = L* cov (S)*L [w *m mm ], (8.90)

where cov($) means the convariance matrix of $, etc. For our purposes we shall make the

usual assumption that the random errors in $ are uncorrelated, that is, that the magnitude

and direction of the error in S

f

, say, doesn't depend upon the error that was made in

measuring any other value, S, . In such a case cov($] is just the diagonal matrix

•(s)-

a"

o

2

ot [S

2

], (8.91)

where a,, as in eq. (8.13), is the standard deviation of the measurement of S, . The

diagonal elements of the matrix cov(£) computed from eq. (8.90) are similarly the squares

of the standard deviations, £ , of the values of E . The off-diagonal elements of covlM

describe the correlation between errors at different wavelengths but we will not discuss

these quantities here. In the simplest case where all S. are subject to the same random

error a, covf$J reduces to a "I [S ] where \ is the unit matrix. Then eq. (8.90)

reduces to

'(E)-

2 , ,T rtT2 -A -2,

[W *m Tim J (8.92)

whence

2 Vt t T =

°

-iL

ik ,Lki

k

2 Vt 2ik

ttt2 ~4 -2 ^ [W «m »nm J (8.93)

70
Applying this formula to the example of table 8-2 using eq. (8.87) we find that, since all rows of the matrix V are identical we need only sum the squares of the column corresponding

to the appropriate number of iterations. For v = 3 iterations this leads to

g =a . — .^ |v^ 'J [W «m »nm ] (8.94)

or 5. = 0.18 a for any i. Assuming an uncertainty in the S measurements of 0.1% of the

maximum or 0.13 we obtain £. = 0.023 or about 0.3% of the maximum of E. Once the standard

deviations, E,

.

, of the E are known, confidence bands can be drawn around the calculated

E. values. For example, the probability is 0.997 that the "true" values (deconvoluted

random-error-free data) are within three standard deviations of the calculated values —

between E. - 3E,. and E. + 3£ . (The corresponding probability for two standard deviations

is 0.954 and for one standard deviation, 0.683.)

The uncertainties E, . are due entirely to the random error in S and do not include

the difference between the deconvoluted spectrum and the true spectral irradiance caused, for

example, by premature termination of the iteration or smoothing. These additional errors are described by a function we shall call the residual spectral responsivity function . This is

the effective responsivity function of the overall process including measurement and deconvolution. If Q is tne true spectral irradiance vector of a source and R is the responsivity matrix of our spectroradiometer , we will observe a signal $ which, even assuming

linearity, no error from replacing integration by summation, etc., is not given exactly by

S = AA-R-E [S] (8.95)

because of the introduction of random experimental noise both in the beam reaching the

radiometer and in the measuring process. In following this measurement by a deconvolution we

endeavor to reverse the instrumental smearing described by R. As we have noted, even where

it is mathematically and numerically possible to solve eq. (8.95) exactly for the vector £,

it is rarely desirable to do so because the $ we observe is not the quantity which satisfies eq. (8.95) and the discrepancies — the noise — will be enormously amplified. Consequently, in deconvolution we actually calculate a quantity 1 E> which is some approximation to £» °y a Process

E =. L'S [W-m"

2 -nm_1

] (8.96)

which is some approximation to the exact inverse

E - aX -R^'S [W-m^-nm-1

]. (8.97)

*In this section we distinguish between a true spectral irradiance £ an^ a calculated

approximation to it which we call E« We have not had to make this distinction previously

because it is usually obvious from the context whether E or E i- s intended.

71
We may think of the matrix |_ in eq. (8.96) as an explicit deconvolution matrix as given in eqs. (8.87), (8.88), or (8.89) or we may think of it merely as signifying the process of

deconvolution of $ by whatever means — i.e., as an operator. The degree to which the

1 —1 A process |_ approximates the exact VT'R and consequently the degree to which p approximates p is dependent indirectly upon the estimated noise level in the data $ because the

noise level limits the number of iterations v or governs the amount of smoothing y used

in the deconvolution process \_. We define the residual spectral responsivity function p

as the function which describes the relationship between p and p. That is

P

= AA'F'P [W-m~2 'nm"

1

] (8.98)

By substituting eq. (8.95) in eq. (8.96) and comparing with eq. (8.98) we see that the matrix

p is given by

-1,

F = L'R tnm"

1

], (8.99)

Although F and R in this equation are matrices they can be treated column-by-column as vectors. In other words eq. (8.99) can be written as n matrix equations, one for each

column of R and corresponding column of p:

li

2i

ni

li R

2i

R . ni

[nm ], i=l,2,...,n (8.100)

Since these equations are exactly the form of eq. (8.96) — the deconvolution process — we

see that the columns of the residual responsivity function can be obtained by deconvoluting

the corresponding columns of the spectral responsivity matrix as though the columns of the

responsivity matrix were observed signals, $• The deconvolution process, |_, in eq. (8.100) must be exactly the same matrix or represent the same sequence of calculations which

was used in deconvoluting the measurements $ to give the computed spectral irradiance distribution p; it must use the same values of any parameters such as y or A, and the same number of iterations, if any. The final result, the complete function p, depending upon

both A and A , is the responsivity function of an imaginary spectroradiometer which would

yield directly, without deconvolution, a spectral irradiance output identical to the computed

results of the real system consisting of radiometer plus deconvolution.

72
The significance of the residual spectral responsivity function is that it describes the

the spectral averaging which remains in the deconvoluted results and which is normally not

removable by further deconvolution: it describes the ultimate 1 resolution which can be

extracted from the data because of the limits imposed by noise and imprecision in the data

itself. Consequently, if it is desired to compare deconvoluted experimental spectral irradiance measurements with, for example, a theory, one should first convolute the theoretical

spectrum with the residual responsivity function — in other words, calculate, using eq. (8.98), what the theoretical spectrum would look like if measured with the imaginary radiometer exhibiting a responsivity function given by the residual responsivity function — and

then compare these results with an appropriate confidence band of the deconvoluted experimental results.

Normally the residual responsivity function is used merely to indicate the improvement

in spectral resolution achieved by the deconvolution process. For this purpose usually the

calculation of a single representative column of p is adequate. In the limit of a perfect

deconvolution where |_ - TT*R tne matrix p becomes

F = L-R = atR^-R = aT 1 tnm

"

1] (8 - 101)

where \ is the identity matrix. Each column of p is then the vector equivalent of a delta function 6(A -A). At the opposite extreme, where no deconvolution at all is carried

o

/\ out, we may take p proportional to $> which is equivalent to setting 2 [ a

J [(eq.

(8.96)] and we obtain

F = L-R - I-R = R. (8.102)

In this case each column of p is simply proportional to a column of the responsivity matrix

R which, in turn, is proportional to the observed response of the spectroradiometer when

scanned over a sharp, fixed-wavelength spectral line. Any real deconvolution, then, will

lead to a function F(A , A) whose dependence upon A at any fixed A. lies somewhere

between a delta function and the raw instrument distortion of a spectral line. Examples of

the calculation of this function are shown in figure 8.4 for the example of table 8-la.

Since R(A ,A) normally depends strongly upon A -A and only weakly (not at all in our example) upon A or A separately, it is usual to assume a similar behavior for F(A ,A).

Then the abscissa can be labeled A -A, even though in our example it was calculated using

the specific column of the responsivity matrix for A = 30. The steady improvement in the

overall spectral resolvability of the process as the number of iterations increases is

*we assume here that the iterations were terminated or the degree of smoothing was established by reference to the noise content of the data. If not, then, presumably, some further

resolution enhancement is still possible.

The symbol <*, here, is the symbol of proportionality, and is read, "is proportional to".

73
F .3-

Xn-X

Figure 8.4. Residual responsivity functions for example of table 8-la.

Simple iterative deconvolution using 0, 3, and 9 iterations.

74
clearly shown in the figure by the sharpening of the central peak with increasing v. Usually the most important feature of these curves from the point of view of characterizing

the resolution improvement due to deconvolution is the full width at half maximum of the

curve. In figure 8.4 this is seen to change from 4 at v = (no deconvolution) to 1.1

at v = 9 which is roughly equivalent to carrying out the measurements with an instrument

whose slit width was nearly a quarter that of the original instrument. Just as the measurement S(A ) of a source is of limited usefulness unless information about the instrument

o spectral bandpass is available, so the deconvoluted spectral irradiance E(A) is not really

complete unless information about the overall spectral resolution is available. Thus, any

careful work involving deconvolution should include a computation of the residual spectral

responsivity function.

SUMMARY of CHAPTER 8 . By the process of deconvolution we mean the recovery of a spectral

distribution such as spectral irradiance, E^(X), from spectroradiometric measurements

S(A ) when the two functions are related by

S(X

o

) = jR

E

(A

o ,A).EA (AWA [S]. (8.1)

In this equation R_(A ,A) is the spectral irradiance responsivity function of the measuring

b o spectroradlometer and the variable A is the wavelength setting of the radiometer. The

integral is understood to extend over all wavelengths, A, for which neither Rp(^ >^) nor

E.(A) vanish. The solution of this integral equation for E.(A) is complicated by non- uniqueness and, in the case of real measurements, by the amplification of measurement noise.

In any application of eq. (8.1) to n real discrete measurements it is necessary to interpret the integral as a summation, so the starting point of a practical treatment of

deconvolution is the set of equations

n

Sk

= £ R

krVAA [Si. k=l,2,...,n, (8.5)

i=l

where S, = S(A = A, ) [S] k o k

E. = E,(A - A.) [W-m^-nm-1

] i A i

and AA = A. -A. [nm] is the spacing between wavelengths, which we shall assume is 1 1 -1 2 constant. For the quantities R. . [S«W «m ] we use values of R(A ,A) at A = A, and ki o o k

A = A., thus passing from eq. (8.1) to eq. (8.5) by the trapezoidal integration approximation.

More elaborate integration formulas can still be expressed as eq. (8.5) provided the interpretation of the R, . is slightly modified. As written, eq. (8.5) is a set of n equations

in n unknowns and can be solved exactly. Because of the non-uniqueness and noise amplification problems of deconvolution, however, this solution will often prove to be erratic and

wholly unacceptable physically. What is needed is an approximate solution of eqs. (8.5)

which also meets additional criteria of acceptability such as smoothness.

75
Probably the most commonly used deconvolution technique in spectroscopy and spectroradiometry is based on requiring the spectral distribution E.(A) to resemble the distribution S(A ). This is a simple iterative technique which starts by assuming a trial set of

E. values

1

E

i0) = S i /(AA '£ R

i- ) [W-m"2 -nm"

1

], i = 1,2,... ,n, (8.6)

which is just the set which would be accepted if no corrections for instrumental spectral

broadening were to be made. This set is substituted into eqs. (8.5) and the computed values

of S, are compared with the measured ones to calculate a correction to the trial E. values. In general, starting with the v-1 set of E. values an improved v set is

computed from

Ek

V> = Ek

V_1) + VK~S kV_1)

} [W.m"2 .nm_1

], k= 1,2,. .

. ,n, (8.10)

where

n

S (v-D , A ,,V^ „ ..(v-D

k

= AA-£ Rki -E^" i; [S], ' k= 1,2,. ...n. (8.11)

1=1

The A, , here, are convergence factors whose values only affect the rate of convergence

toward the final solution. Frequently these factors are modified by trial-and-error during

the course of the iterations. A reasonable starting value is

\

= Rkk/(AX '£ RkV [W-m~

2 -nm"

1 -S~

1

] k=l,2,...,n. (8.9)

J

The iterations should be stopped when the reconstructed values, S, agree with the

measured S values within the estimated experimental error. Further iterations tend to make the solution more nearly mathematically exact and less physically acceptable. This

simple iterative deconvolution is usually not advocated if more than five or ten iterations

are required because of the problem of noise amplification; however, digital smoothing can be added to each iteration stage to permit large numbers of iterations if necessary [8.3,

8.4,8.20].

A useful technique, when the spectral responsivity function can be represented by the

slit-scattering function approximation

R(A ,A) = K-z(A -A)-r

f (A) [S-W

-1 ^

2

]. (8.22)

o o

is to solve eqs. (8.5) for numerical values S, = A ^ ,|S k

n> k = l,2,...,n where I is n/2

or (n-l)/2, whichever is an integer. The numerical solution which results after v iterations

76
v<

v)

E E<

v) (8.46)

is effectively an inverse slit-scattering function with the property that, for any set of

real measurements S, , a v -order solution can be calculated from

k

e

; (V> = TTEvS-S, [W-m^-nm-1

] £=l,2,...,n (8.47)

SL AA ,-H, £-k k

k=l

and

E

(v) _ E ,(v), f

where v. . [S 'W-m ] is taken to be zero whenever l-k is outside the range from 1 to

n. There is thus no further need for iterations as long as the slit-scattering function

doesn't change.

A different deconvolution technique in which the additional criterion of acceptability

is more explicitly related to the smoothness of the result is known by such names as generalized least-squares or constrained least-squares deconvolution. In this technique the

E are chosen in such a manner as to minimize a sum of two terms, one of which is a measure

of the error in reconstructing the measured S, , and the other of which is some measure of

the non-smoothness of the E. For example, in the expression

I (S

fc - AA-.jR^.E > 2 + Y-I(Ei_ 1-2Ei+E. +1 ) 2 /(AA)

2 [S2

] (8.55)

k i i

the first term is the sum of the squares of the errors in reconstructing S while the

second term is the sum of squares of an approximate measure of the departure from linearity 2-2 4 4 of E in the neighborhood of X . The parameter y [S 'M *m «nm ] simply sets the relative importance to be attached to these two measures contained in expression (8.55). The

minimization of such an expression with respect to the values E. leads to a solution for

the E. which is most compactly expressed by the matrix equation

E

= aT(RT *R + ^H)"

1 -^^ [W-m^.nnf 1

], (8.56)

T —1 ? / T \ —1 in which R [S«W «m ] denotes the matrix transpose of the matrix R and (R *R + Y'H)

/ T \ -4

denotes the matrix inverse of the matrix (R *R + Y'H)- The matrix \\ [nm ] in eq. (8.56) depends upon the form chosen for the term describing non-smoothness in expression

(8.55). The commonly used term shown above leads to

77
H =

1 -2 1

2 5 -4 1

1 -4 6 -4 1

1 -4 6 -4 1 [nm ] , (8.58)

but other choices for H are often used [8.13, 8.21, 8.22, 8.25]. Usually there is little

difference in the final solution for E.

The complete numerical procedure for generalized least-squares deconvolution, then, is

o to evaluate q from eq. (8.56) for a rang.e of values of y — say for y« 0.001R* , 2 2 0.01R* , and 0.1R* where R* is the largest element of the spectral responsivity matrix

R. For each solution the standard deviation of the error in reconstructing the measured

signal

TM = i^i'l [sk

- AA -Ki-V^ ] T

2 [s] (8.62)

should be calculated. The proper value of y to use is then the largest one (the most

smoothing) for which o(y) is not larger than the estimated measurement uncertainty.

Generally, additional values of y outside the range indicated above will have to be tried;

however, the solution is not critically dependent upon y and values within an order-ofmagnitude of the optimum one will yield essentially the same set of spectral irradiance

values, E .

If the number of data points is so large that the matrix operations of eq. (8.56) are not feasible, an iterative technique

T = R

t -R + y-H [s2 .w2

. m

4

], (8.69)

Q = R

T

-S is

2 *-1

.*2

], (8.70)

E

( V) = E( V_1) +A.|q. AX . T

. E

(v-1)| [w.,-2.^-1^ (8.72)

can be applied in which A t s «W •m -nm ] is only an approximate inverse of AX«J. This procedure can be expressed in much the same way as eq. (8.10) and requires little more

computer memory than the simple iterative deconvolution.

The dependence of errors in the computed values of E. upon measurement errors in S. can be estimated provided the deconvolution calculations which produce q from $ permit

the explicit evaluation of the matrix elements L., in |_ where |_ is defined by

78
E

= L

• S [W-m 2 -mn

X

] (8.86)

The matrix |_ is an approximation to an inverse responsivity matrix (1/AA) *R . Its elements usually can be easily evaluated during deconvolution {see eqs. (8.87), (8.88), and

(8.89)} and then

'(E) = L'cov(s)-L

T

[W

2 .nf4 -nnf2

] (8.90)

In this equation covf$) is the covariance matrix of the measured S. — usually just the

diagonal matrix

cov (s)-

a

l

2 °2

a

[s

2

], (8.91)

where a' is the measured or estimated standard deviation of S.. The diagonal elements

covfEJ of the covariance matrix cov(E) are likewise the squares of the standard

deviations of the computed E . The off-diagonal elements of covfE) describe how errors

at one wavelength are correlated with errors at another wavelength and are generally ignored.

Even if there were no measurement errors, there would still remain, after deconvolution, some spectral smearing of the spectral irradiance distribution. The residual spectral

responsivity function is a measure of this residual spectral distortion. This is the

computed response, to a monochromatic input beam, of the overall system including measurement and deconvolution. It is obtained by deconvoluting a "signal" numerically equal to the

responsivity function, using exactly the same computational steps as the real deconvolution

used, including the same values, if any, of A, and y, and the same number of iterations.

The computed spectral distribution which results is the effective spectral responsivity of

the total measurement-and-deconvolution process and normally cannot be improved by further

computations; the residual responsivity function represents the best spectral definition

which can be achieved in view of the noise content of the measurement data.

79
References

[8.1] Henry J. Kostkowski and Fred E. Nicodemus, "An Introduction to the Measurement

Equation", Chapter 5 of "Self-Study Manual on Optical Radiation Measurements:

Part I—Concepts", Nat. Bur. Stand. (U.S.), Tech Note 910-2, 118 pages

(Feb. 1978) 58-92.

[8.2] A. L. Khidir and J. C. Decius, "Numerical Methods for the Correction of

Apparent Band Shapes due to Finite Slit Width", Spectrochim. Acta 18 , (1962) 1629-1639.

[8.3] P. A. Jansson, "Method for Determining the Response Function of a HighResolution Infrared Spectrometer", J. Opt. Soc. Am. 60, No. 2 (Feb. 1970)

184-191.

[8.4] Peter A. Jansson, Robert H. Hunt, and Earle K. Plyler, "Resolution Enhancement

of Spectra", J. Opt. Soc. Am. 60, No. 5 (May 1970) 596-599.

[8.5] S. H. Sheen and J. G. Skofronick, "Deconvolution Methods in Elastic Total

Cross Section Measurements", J. Chem. Phys. 61 , No. 4 (15 Aug. 1974)

1430-1434.

[8.6] J. S. Rollet and L. A. Higgs, "Correction of Spectroscopic Line Profiles

for Instrumental Broadening by a Fourier Analysis Method", Proc. Phys. Soc.

79, (1962) 87-93.

[8.7] G. K. Wertheim, "Deconvolution and Smoothing: Applications in ESCA",

J. Electron Spectrosc. Relat. Phenom. 6_, (1975) 239-251.

[8.8] Sabri Ergun, "Direct Method for Unfolding Convolution Products — Its Application to X-ray Scattering Intensities", J. Appl. Cryst . 1_, (1968)

19-23.

[8.9] Fukuo Zenitani and Shigeo Minami, "An Analysis of the Iterative Method for

Deconvolving Spectroscopic Data Containing a Random Noise", Jap. J. Appl. Phys. 12, No. 3 (Mar. 1973) 379-387.

[8.10] L. D. Skarsgard, H. E. Johns, and L. E. S. Green, "Iterative Response

Correction for a Scintillation Spectrometer", Radiation Research 14 , (1961) 261-280.

[8.11] A. F. Jones and D. L. Misell, "The Problem of Error in Deconvolution",

J. Phys. A3, (1970) 462-472.

[8.12] P. S. Schoenfeld and J. R. DeVoe, "Statistical and Mathematical Methods

in Analytical Chemistry", Anal. Chem. 48, No. 5 (Apr. 1976) 403R-411R.

[8.13] S. Twomey, "Introduction to the Mathematics of Inversion in Remote Sensing

and Indirect Measurements" (Elsevier Scientific Pub. Co., New York, 1977).

[8.14] G. Backus and F'. Gilbert, "Uniqueness in the Inversion of Inaccurate Gross

Earth Data", Phil. Trans. Roy. Soc. Lond., A 2_66 , (Mar. 1970) 123-192.

[8.15] Arend den Harder and Leo de Galan, "Evaluation of a Method for Real-Time

Deconvolution", Anal. Chem. 46, No. 11 (Sept. 1974) 1464-1470.

[8.16] B. V. Bronk and W. B. Whitten, "Partial Deconvolution as a Means of Studying

the Limitations on Resolution of Spectra Imposed by Noise", Nucl. Instr. and Meth. 106, (1973) 319-331.

80
[8.17] R. N. Jones, R. Venkataraghavan, and J. W. Hopkins, "The Control of Errors

in Infrared Spectrophotometry — I The Reduction of Finite Spectral Slit

Distortion by the Method of 'Pseudo-Deconvolution' ", Spectrochim. Acta 23A,

(1967) 925-939.

[8.18] John B. Shumaker, "Distribution of Optical Radiation with Respect to Polarization", Chapter 6 of "Self-Study Manual on Optical Radiation

Measurements: Part I—Concepts", Nat. Bur. Stand. (U.S.) Tech Note 910-3,

62 pages (June 1977) 1-46.

[8.19] John B. Shumaker, "Matrix Multiplication", Appendix 6 of "Self-Study Manual

on Optical Radiation Measurements: Part I—Concepts", Nat. Bur. Stand.

(U.S.) Tech Note 910-3, 62 pages (June 1977) 47-48.

[8.20] G. Halsey and W. E. Blass, "Deconvolution of IR Spectra in Real Time",

Appl. Optics 16, No. 2 (Feb. 1977) 286-288.

[8.21] P. Paatero, S. Manninen, and T. Paakkari, "Deconvolution in Compton

Profile Measurements", Philos. Mag. 30, (1974) 1281-1294.

[8.22] S. Twomey, "The Application of Numerical Filtering to the Solution of

Integral Equations Encountered in Indirect Sensing Measurements",

J. Franklin Inst. 279 , No. 2 (Feb. 1965) 95-109; "On the Numerical

Solution of Fredholm Integral Equations of the First Kind by the

Inversion of the Linear System Produced by Quadrature", J. Assoc.

Comp. Mach. 10, (1963) 97-101.

[8.23] D. Louer, D. Weigel, and P. Louboutin, "Methode Directe de Correction

des Profils de Raies de Diffraction des Rayons X", Acta Cryst. A25 , (1969) 335-350.

[8.24] H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry",

Vol. 1, 2nd ed. (D. Van Nostrand Co., Inc., New York, 1956).

[8.25] Joel N. Franklin, "Well-Posed Stochastic Extensions of Ill-Posed Linear

Problems", J. Math. Anal, and Appl. 31_> (1970) 682-716.

[8.26] L. Moore, "Deconvolution of Physical Data", Brit. J. Appl. Phys.

(J. Phys. D) (1968) Ser. 2 1_, 237-245.

[8.27] H. Schrijver, "Unfolding of Spectra by Fourier Transformation and the

Application of a Numerical Filter", Physica 49_, (1970) 135-140.

[8.28] Gary Horlick, "Resolution Enhancement of Line Emission Spectra by

Deconvolution", Appl. Spectrosc. 26, No. 3 (1972) 395-399.

81
Appendix 8A. Deconvolution Programs

by John B . Shumaker

In this appendix we present two computer programs for performing deconvolution. The

first is the simple iterative deconvolution method of eq. (8.10). The second is the iterative generalized least-squares method of eq. (8.72) with the non-zero elements of a row of

the band matrix A computed by eq. (8.78). They are written in the BASIC language and as far as possible the variable names agre'e with the notation of Chapter 8. The programs are intended primarily to provide an alternative description of the deconvolution algorithms and

a working starting point for programming them. Each program is followed by a cross-reference table of variable names and line numbers. The inefficient use of memory by not writing

over program variables when they are no longer needed and especially the data input and

output are illustrative rather than practical.

Both programs employ the slit-scatte'ring function approximation and denote by R and

E the quantities z and E' of Chapter 8. Thus, if the responsivity function depends

upon the instrumental setting A and wavelength A only through their difference, A -A,

then the program variable R may be taken as the spectral responsivity and the program

variable E as the spectral irradiance. The programs also use D, G, and N for the

quantities AA (wavelength interval), y (smoothing parameter), and v (number of

iterations) of Chapter 8.

The number of points in R, E, S and the other vectors is always taken as an odd

integer. The slit-scattering function data are assumed to be in the order from top-tobottom of a column of the responsivity matrix — as if obtained by observing a fixed wavelength monochromatic source while tuning the radiometer through the spectrum in the same direction as the measured signal data were taken.

All matrix multiplications are performed by the same convolution routine based on eqs. (8.83), (8.84), and (8.85): the convolution of X with Y to form Z assumes that the

lengths of all three vectors are odd integers and the indexing is arranged so that the

summation which produces the central entry in Z includes the product of the central

entries of X and Y. Algebraically, if the lengths of vectors X, Y, and Z are

2x +1, 2y +1, and 2z +1, we have

h

Z. = £ X^-Y. . j = -z

2 ,-z5 +l,...,-l,0,l,...,z2 (8A.1)

L l

J i M

A

i M-i J "2' "2 •-*"•* *»"»* "2 i=i.

where i = j-y for j > y

2 -x

2 otherwise i- = -x.

and ±

2 = j + y2 for j < x

2 -y

? otherwise i

2 = x?

.

82
In BASIC this translates to

DIM X<2*X2+1), Y(2*Y2+1), Z(2*Z2+1>

FOR J=-Z2 TO Z2

LET Z(J+Z2+1)=0

LET Il=-X2

LET I2=X2

IF J>Y2-X2 LET I1=J-Y2

IF J>X2-Y2 LET I2=J+Y2

FOR 1=11 TO 12

LET Z(J+Z2+1)=Z<J+Z2+1)+X<I+X2+1)*Y(J+Y2+1-I)

NEXT I NEXT J

In iterative calculations, using convolution instead of matrix multiplication, there is

always a question of how to avoid increasing the vector lengths with every iteration. The

increases come about because the number of significant elements of Z which can be calculated by the convolution of X with Y is approximately equal to the sum of the numbers

of significant elements in X and Y. In these programs the following scheme is used: in

calculating the quantity 1 W = S - D*R*E (or W = Q - D*T*E in the second program) the

lengths of W and S (or Q) are made large enough to accommodate all significant elements in the convolution of R (or T) with E. In computing the improved E by con- volution of A with W, only the desired elements are retained.

The programs print out the complete set of E values after each iteration followed by

the rms value of the error in reconstructing the S values from this set of E values. If end-effects (see Chapter 8) are important the rms-error calculation should be restricted to the central wavelength region which is free of this problem. In the simple iterative deconvolution the iterations should be terminated when the rms error reaches a minimum or becomes comparable with the experimental error. In the generalized least-squares program

the iterations should be continued well beyond the point where the improvement per iteration

is comparable with experimental accuracy and then the rms error can be used to judge which

value of the smoothing constant G is most suitable.

^he asterisk stands for the convolution operation. That is, R*E = Yr. . # E., or the

x equivalent integral. This notation is commonly used in the literature, probably because it suggests matrix multiplication, which is what convolution becomes when the slit-scattering

function approximation is inappropriate and the full matrices must be used.

83
:=: I D :3r. y &. y~7 '5>

00 1 0020

0030

0040

0050

0060

0070

00SO

0090

1 00

OLIO

0120

1 30

0140

1 50

0160

0170

01.30

1 90

0200

02 1 0;.'20 0230

0240

0250

0260

0270

0280

0290

300

0310

0320

0330

340

0350

0360

0370

0380

0390

0400

0410

0420

0430

0440

0450

0460

0470

0430

0490

0500

05 1 1520 0530

0540

0550

0560

0570

0580

0590

0600

REM A PROGRAM FOR SIMPLE ITERATIVE DECONVOLUTIGN

PRINT "WAVELENGTH SPACING".:

INPUT D

PRINT

REM READ SLIT FUNCTION DATA

REM FIRST ENTRY IN DATA BLOCK IS # OF DATA POINTS TO FOLLOW.

REM ORDER OF SLIT-FUNCTION DATA IS AS IF A SHARP LINE

REM WERE SCANNED EXACTLY AS NORMAL MEASURED SIGNAL IS SCANNED.

OATA 3

DATA 2, 4, 2

PRINT

READ R9

LET R2= ( R9- 1 ) /2

DIM R(R9)

FOR 1=1 TO R9

READ R ( I

)

NEXT I REM READ MEASURED SIGNAL DATA

DATA

DATA

DATA

DATA

DATA

READ

:i O, 333333, 5. 33333, 25. 6667, 64, 102. 333, 122. 667

127. 667, 128, 128, 123, 123, 128, 123, 128

128,

128,

128, 123, 128, 123, 128, 123

123, 128

128, 123, 128,

128, 128, 128,

E9

LET S9=E9+R9~

I

DIM S(S9), E(E9), W(S9)

FOR 1=1 TO S9

LET S ( I

) =0

NEXT I FOR 1=1 TO E9

READ S( I+R2)

NEXT I LET R0=0

LET R1=0

FOR 1=1 TO R9

LET RO=RO+R( I)

LET R1=R1+R( I)~2

NEXT I LET A=R(R2+1)/R1/D

LET S2=(S9-l)/2

LET E2=(E9-l)/2

REM CALL ZEROTH ESTIMATE OF E

FOR J=-E2 TO E2

LET E < .J+E2+ 1 ) =S ( J+S2+ 1 ) /RO/D

NEXT J

REM CONVOLUTE R WITH E AND CALC W=S-D#R*E

LET X=0

FOR J=-S2 TO S2

LET W( J+S2+1 )=S( J+S2+1)

LET Il=-R2

LET N=0

LET I2=R2

IF ...D-E2-R2 LET I1=.J-E2

IF .J<R2-E2 LET I2=.J+E2

FOR 1=11 TO 12

LET W(.J+S2+1 )=W( J+S2+1 )-R< I+R2+1 ) *E<J+E2-I+1 )*D

NEXT I LET X=X+W(.J+S2+1)-S 2

NEXT J

REM CALC E=E+A#W

84
0610 FOR J=-E2 TO E2

0620 LET E<J+E2+l)=E(.J+E2+l)+A*W(J+52+i ) 0630 PRINT E(J+E2+1),

0640 NEXT J

0650 PRINT

0660 PRINT SQR<X/E9)

0670 LET N=N+1

06:30 IF N<11 GOTO 0460

0690 END

85
o

10

IT' in t

rtl noOro>6

CM

<i

in

CO

ft\ in

fO cm en

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in 0 M IT' in in m in o Q £• q Q ,-. o

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o 1* O o o i^i o 1* o >*i o m t ~H QTl ft\ i> cm in ft\ o CM

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86
I iZ-i L_ :=: O ef. S /=.. S"7" '5>

00 1 0020

0030

0040

0050

0060

0070

0080

0090

0100

0110

0120

1 30

0140

0150

1 60

0170

1 30

1 90

0200

02 1 0220

0230

0240

0250

0260

0270

0230

290

0300

03 1 0320

0330

0340

0350

0360

0370

0380

0390

0400

04 1 0420

0430

0440

0450

0460

0470

0480

0490

0500

05 1 0520

0530

0540

0550

0560

0570

0580

0590

0600

REM A

PRINT

INPUT

PRINT

PRINT

INPUT

PRINT

REM FOR

PROGRAM FOR

"WAVELENGTH

D

"GAMMA";

G

ITERATIVE

INTERVAL"

GENERALIZED LEAST-SQUARES DECONVGLUTION

INCREASE Al

BLOCK IS #

MORE ACCURATE A,

A 1=0

READ SLIT FUNCTION

FIRST ENTRY IN DATA

ORDER OF SLIT-FUNCTION DATA IS

WERE SCANNED EXACTLY AS NORMAL

3

2, 4, 2

OF

AS

DATA POINTS TO FOLLOW

IF A SHARP LINE

MEASURED SIGNAL IS SCANNED.

LET

REM

REM

REM

REM

DATA

DATA

READ R9

LET T2=R9-1

LET R2=(R9-l)/2

DIM R<R9), T<2*T2+1)

LET R0=0

FOR 1=1 TO R9

READ R ( I

)

LET RO=RO+R < I

)

NEXT I REM CONVOLUTE R WITH R-TRANSPOSE TO GIVE T

FOR J=l -R9 TO R9-1

LET T(.J+R9)=0

LET Il=-R2

LET I2=R2

IF J>0 LET I1=J-R2

IF J<0 LET I2=.J+R2

FOR 1=11 TO 12

LET T ( J+R9 ) =T ( J+R9 ) +R ( R2- I + 1 ) *R ( J+R2- I + 1

)

NEXT I NEXT J

REM ADD IN SMOOTHING TERM G*H

T ( R9 ) =T ( R9 ) +6*G

T ( R9+ 1

) =T < R9+ 1

) -4*G

T<R9-l)=T(R9-i)-4*G

T ( R9+2 ) =T ( R9+2 ) +G

T(R9-2)=T(R9-2)+G

COMPUTE APPROX INVERSE BY LEAST-SQUARES

A9=R9+2*INT(A1

)

B ( A9, A9 ) , A ( A9, 1 ) , Z ( A9, 1 ) A2=(A9-1 )/2

CONVOLUTE T-TRANSPOSE WITH T AND FILL OUT MATRIX

B=ZER

J=-A9 TO A9

ABS < .J ) >A2 GOTO 0520

Z(.J+A2+1, 1 )=0

LET

LET

LET

LET

LET

REM

LET

DIM

LET

REM

MAT

FOR

IF

LET

IF J+R9>0 IF .J+R9C=2*T2+1 LET Z ( J+A2+1, 1 )=T( J+R9)/D

LET B0=0

LET Il=-T2

LET I2=T2

IF .J>0 LET I1=.J-T2

IF J<0 LET I2=J+T2

FOR 1=11 TO 12

LET BO=BO+T ( T2- 1+ 1 ) *T ( J+T2- 1+ 1

)

NEXT I FOR L=-A2 TO A2

87
0610 LET E1=J+L+A2+1

0620 IF B1>0 IF BK=A9 LET B(L+A2+1 , Bl ) =B0

0630 NEXT L

0640 NEXT J

0650 MAT B=INV(B)

0660 MAT A=B*Z

C670 REM READ SIGNAL DATA

0630 DATA 31

0690 DATA 0, . 333333, 5. 33333, 25. 6667, 64, 102. 333, 122. 667

0700 DATA 127. 667, 123, 123, 123, 123, 123, 123, 128

0710 DATA 123, 123, 128, 123, 128, 128, 123, 123, 123, 128

0720 DATA 123, 123, 123, 128, 123, 123

C730 READ 39

0740 DIM 3(39), E( 39)

0750 FOR 1=1 TO 39

0760 READ 3(1)

0770 NEXT I 0780 LET Q9=2*R9+S9-2

0790 REM CONVOLUTE R-TRANSPOSE WITH 3 TO GIVE Q

0300 DIM Q(Q9), W(Q9)

0310 LET Q2=(Q9-l)/2

0820 LET 32=<S9-l)/2

0330 FOR J=-Q2 TO 02

0840 LET Q(J+Q2+1)=0

0350 LET Il=-R2

0860 LET I2=R2

0370 IF J>S2-R2 LET I1=J-S2

0880 IF JCR2-S2 LET I2=J+S2

0890 FOR 1=11 TO 12

0900 LET Q ( J+G2+ 1 ) =Q ( J+Q2+ 1 ) +R ( R2- 1 + 1 ) *3 ( .J+S2- 1 + 1 ) 0910 NEXT I 0920 NEXT J

0930 REM CALC ZEROTH ESTIMATE OF E

0940 FOR 1=1 TO 39

0950 LET E ( I

) =S ( I ) /RO/D

0960 NEXT I 0970 LET N=0

0930 REM CONVOLUTE T WITH E AND CALC W=Q-D*T*E

0990 FOR J=-Q2 TO Q2

1000 LET K=J+Q2+1

1010 LET W(K)=Q<K)

1020 LET Il=-T2

1030 LET I2=T2

1040 IF J>S2-T2 LET I1=J-S2

1050 IF ..KT2-S2 LET I2=J+S2

1060 FOR 1=11 TO 12

1070 LET W(K)=W(K)-T( I+T2+1 )*E< J+S2-I+1 )*D

1080 NEXT I 1090 NEXT J

1100 REM CONVOLUTE A WITH W AND CALC E=E+A*W

1110 FOR J=-S2 TO 32

1120 LET Il=-A2

1130 LET I2=A2

1140 IF ._I>Q2-A2 LET I1=J-Q2

1150 IF J<A2-Q2 LET I2=J+Q2

1160 FOR 1=11 TO 12

1 1 70 LET E ( J+32+ 1

) =E ( J+S2+ 1 ) +A < A2- 1 + 1 , 1 ) *W ( J+Q2- 1 + 1

)

1180 NEXT I U90 PRINT E ( .J+S2+ 1 )

,

1200 NEXT .J 1210 PRINT

1220 REM CONVOLUTE R WITH E AND CALC ERRORS: S-D*R*E

88
1230 LET X=0

1240 FOR J=-S2 TO S2

1250 LET XQ=S(J+S2+1)

1260 LET Il=-R2

1270 LET I2=R2

1280 IF J>S2-R2 LET Il=J-32

1290 IF JCR2-S2 LET I2=J+S2

1300 FOR 1=11 TO 12

1310 LET XO=XO-R( I+R2+1 )«E< J+S2-I+1 )*D

1320 NEXT I 1330 LET X=X+X0'N 2

1340 NEXT J

1350 PRINT SQR(X/S9)

1360 LET N=N+1

1370 IF N<10 GOTO 0980

1330 END

89
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90
Chapter 9. Physically Defining Measurement-Beam Geometry by Using Opaque Barriers

by Fred E. Nicodemus

In this CHAPTER. In previous discussions of the distribution of optical radiation with

respect to position and direction (Chapter 2 [9.1] and Chapter 4 [9.2]), we merely assumed

that opaque barriers, such as the opaque screens in figure 2.2 [9.1], could be used to define

beams by blocking all rays that are not a part of the desired beam. Now we want to examine more closely the way in which such opaque barriers are used to physically define or limit the

extent of radiation beams in making radiometric measurements. We'll do this mostly in terms

of throughput, the integrated product of projected area and solid angle (or area and projected solid angle), and the limits of the throughput integral(s), because they are the

quantitative measures of beam geometry. Even though exact numerical evaluation of these

quantities may not be needed very often, it is only in these terms that an adequate understanding of their significance can be achieved, including the ability to confidently decide,

in all cases, whether or not the numerical evaluation is required.

The geometrical-optics or ray-optics limits provided by opaque barriers are the starting

point for assigning the limits of integration for the spatial parameters of area and solid

angle (position and direction) in the measurement equation {e.g., eq. (5.5) [9.3]}. When

accuracy requirements are not too stringent, these limits may be adequate without refinement.

However, in most instances, the effects of scattering, diffraction, or optical-system aberrations (imperfect imaging) may extend the responsivity of a radiometer substantially beyond

the nominal limits of the ray-optics beam boundaries. Hence, for optimum accuracy, the integrals of the measurement equation must likewise be extended beyond those nominal limits, to the extent determined by experimental checks of instrument responsivity as a function of

position and direction. 2 The opaque-obstacle-shadow boundaries that we discuss in this

chapter, then, while they are very important for design purposes and as a starting point for

analysis, are no more than that — a starting point or first approximation — and must never

be considered as absolute limits that cannot be exceeded. With that caveat, however, we'll

devote the rest of this chapter, except for an occasional reminder about the real world, to the simpler domain of ideal geometrical optics with sharp ray-defined shadows and perfect

images in clear non-scattering media.

The chapter starts with a discussion of the simplest physical definition of a radiation

beam — by two apertures in opaque screens in a homogeneous (iso-refractive-index) medium

where all rays are straight lines. The concept of vignetting is also illustrated for this

simple case.

^Figures in square brackets indicate literature references listed at the end of this chapter.

n^Another approach, and in some ways a sounder approach, is to always extend the integration

limits as far as possible — e.g., < 8 < tt/2 [rad] and < <j> < 2tt [rad] — even though

the integrand may vanish over much of each interval.

91
There follows a discussion of the beam defined by two apertures with a thin lens between

them. This brings out the way in which aperture and/or aperture-image pairs can be equivalent to the earlier simple case of two apertures defining a beam of straight-line rays in a homogeneous medium. The concepts of object space and image space are developed, bringing out that both members of a beam-defining aperture/aperture- image pair exist in the same "space".

The geometrical-optics concepts of aperture stop and field stop, and their images in

object space, the entrance pupil and entrance window, respectively, are developed. This is

followed by a discussion of the somewhat "loose" way in which these terms are used in radiometry. The exit pupil and exit window, the images of the respective stops in image space

are also briefly defined. In addition, the special case of an optical system focused on infinity (entrance window at infinity) is treated.

Some practical problems relating to the role of entrance pupils as receiving apertures

in radiometric measurements are discussed, followed by a discussion of diffusers and their use in connection with the measurement of irradiance. Attention is given to the problem of

the exact position of the entrance pupil (receiving aperture) and the effects of diffusers

empirically shaped to achieve a "cosine response" characteristic.

Finally, there are brief discussions of the role of baffles, of some more aspects of

vignetting, and of the effects of beam geometry on overall instrument radiance and irradiance

responsivities.

BEAM GEOMETRY in RADIOMETRY . In Chapter 5 of this Manual, we found [9.3 (p. 90)] that the

measurement equation has basically the same form for all radiometric measurements {eq.

(5.30b)}

S(A,o>,AA) = / J / R (x,y,e,<(>,A)-L , (x,y, 6 ,<f>, A) -cos9 «dwdA«dA [S] . (9.1)

AA A w

Here S(A,w,AA) [S] is the output signal of an instrument with spectral-ray flux responsivity R (x,y,9 , c|>, A) [S«W ] for the incident flux element

d$(x,y,e,c)>,A) = L (x,y,6,(j),A) •cosG'dtWA'duj [W]

-2 -1 -1

associated with the ray, of spectral radiance L (x,y, 9,<j>, A) [W # m 'sr *nm ], incident on

2 the receiving-aperture-area element dA [m ] at the point x,y, within the solid-angle

element dw [sr] from the direction 6,<j>> and within the spectral-wavelength element

dA [nm] at the wavelength A. The effects of the radiation parameters of time or frequency-f (of fluctuation or modulation; f << v) , and of polarization (see Chapter 6 [9.4])

have been ignored here and, for simplicity, we will continue to ignore them in this chapter

while we examine the way in which measurement-beam geometry is physically defined. We'll

look at the physical considerations that determine the values of such quantities as the

spatial limits of integration, the area A and the solid angle cj [actually, w(x,y)], in

eq. (9.1) that define the beam and its throughput (0 =

J Jcos9*du>*dA, see next section).

92
We'll discuss the choice of reference surface to facilitate the beam (throughput) definition and will emphasize the purely ray-optics or geometrical-optics considerations.

Even when the physical limits that define the throughput are clearly and unambiguously

identified, accurate measurements to establish the values of the area(s) and solid angle(s)

involved can be quite difficult. However, we won't go into such precision-measurement

problems in this chapter but will concern ourselves primarily with the ways in which the

physical limits are established.

Just because these measurements are so difficult, we try to avoid the need for accurate

evaluation of the throughput wherever possible in making radiometric measurements, particularly when the best accuracy is required. That's why, in all three illustrative problems

in Chapter 5 [9.3], calibration measurements against a standard source are made under

conditions that are enough like those of the actual measurement so that all of the geometrical factors cancel and do not appear in the final expression for the measured quantity.

However, a thorough understanding of the physical limits is needed: (1) to be sure that

the beam geometry is, in fact, not significantly changed between the calibration measurement of a standard and the corresponding measurement of an unknown radiometric quantity,

(2) to insure adequate specification of the measurement beam in any complete report of a radiometric measurement, (3) to evaluate the adequacy of throughput estimates used in

planning a measurement, particularly in establishing the required detector capability, and

(4) as an essential factor in an error analysis to estimate the uncertainty of the results

of a radiometric measurement.

SIMPLE BEAM-DEFINING APERTURES . In the "thought experiment" to develop the concept of

radiance in Chapter 2 (figure 2.2 [9.1]) we introduced one of the simplest beam-defining

aperture configurations—a beam of all rays through a pair of apertures, one in each of two opaque screens. With no other obstructions from source to receiver, the selected beam is

clearly and unambiguously defined at either aperture as consisting of all rays through

every point of that aperture within the solid angle subtended at each point by the other

aperture. If we need to evaluate the throughput of the aperture pair filled by the

beam {see eq. (2.27) [9.1]}, we can choose either aperture (plane) as the reference surface

(see figure 9.1) and evaluate

= / / cos6'doj'dA [m

2 -sr], (9.2)

A w(x,y)

where

2 A [m ] is the reference-surface aperture area (includes the intersection of every

ray of the beam with the reference surface)

,

oj(x,y) [sr] is the solid angle subtended by the other aperture A' at the point

x,y within the aperture area A (includes every ray of the beam

that intersects that point)

,

2 dA [m ] is an element of reference-surface area at the point x,y [dA = dx*dy],

93
6 [rad] is the polar angle from the normal to dA at x,y that, together with

the azimuth angle <j> [rad] from the X-axis in the reference-surface

plane, constitute spherical co-ordinates about x,y, and

do) [sr] is an element of solid angle in the direction 0,<f> from x,y (within to); [dw = sin0«d0«d<j>]

.

Figure 9.1. Evaluation of the throughput between two apertures.

The beam consists of all rays (straight lines) through the apertures A

(chosen as the reference surface) and A' . Its throughput 8 is evaluated by the double integral of eq. (9.2), integrating or summing over all solid-angle elements dw within oj, the solid angle subtended by

A

1 at each point x,y of A, and over all area elements dA within

A. Each solid-angle element dm makes an angle 6 with the normal to the area element dA at the point x,y.

94
VIGNETTING . Away from the two aperture screens of figure 9.1, across any other plane that

intersects the entire beam, there is vignetting so that the solid angle w(x,y) at some points x,y in that plane is not defined by just one of the beam-defining apertures. This

situation, and the meaning of the term "vignetting", is best explained by reference to figures 9.2 and 9.3. Figure 9.2 shows a plane section through the centers of two parallel

coaxial circular apertures, A.. and A.. The extreme rays a - a through corresponding

points, together with extreme rays b - b through opposite points, bound different regions

of the defined beam (made up of all rays that pass through both apertures) . When a point

x,y of an intersecting plane, such as the plane c - c, lies within one of the central

regions, (2), (4), or (6), there is no vignetting because the solid angle w(x,y) filled

by all rays of the beam that intersect at x,y is a solid angle subtended at that point by

just one of the apertures. In regions (2) [as shown] and (6) this is aperture A~ ; in

region (4) it is A.. . However, except in the planes of the apertures, every such

intersecting plane c - c also crosses regions (1) , (3) , or (5) , where things are different. Consider the point x,y on the intersecting plane c - c in figure 9.3, where it

is shown in region (5) with the extreme rays bounding the solid angle u(x,y): the

uppermost ray through the apertures is limited by aperture A„ while the lowest one is

limited by aperture A.. . Thus the rays of the beam that intersect at this particular point

x,y do not completely fill either of the apertures, a condition called "vignetting". It

is easily seen that the same holds true for all points in the cross-hatched regions, (1),

(3), and (5). When vignetting occurs, as described here, it becomes extremely difficult to specify the limits of the solid angle w(x,y) in order to evaluate the throughput by eq. (9.2). A good example of the complexities in a case involving just a small amount of

relatively simple vignetting is found in [9.5]. Thus, there is a clear advantage, when

defining or analyzing any beam and its parameters, to choosing one of the two aperturescreen planes, where there is no vignetting, as the reference surface.

A BEAM-DEFINING APERTURE PAIR . Similar vignetting complications arise if any of the rays

from source to receiver, that would otherwise pass through both apertures, are blocked by

any obstruction or obstacle at a third location, whether between the apertures or external

to them. Accordingly, in the design of instruments and experimental configurations,

especially when the highest accuracy is needed, and particularly when it is necessary to compute the value of the throughput, it is desirable to arrange to have a beam defined by

just two apertures with no other obstructions to cause vignetting. We'll designate such an unvignetted pair of apertures as a beam-defining aperture pair or, most of the time, as just an aperture pair .

So far, we've looked only at the situation where both apertures and all of the space

between them are in the same uniform, isotropic medium so that all rays between them are

straight lines. We intend that those conditions be implied by the term aperture pair , except when explicitly stated otherwise. However, there are many optical systems where

^ee "More on Vignetting" later in this chapter, especially figure 9,

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there are focusing optical elements that deflect the rays between two beam-defining apertures. In the next section, we analyze such a case. We find that it is always possible,

for purposes of analysis and throughput computation, to replace any such pair of beamdefining apertures that has intervening focusing optical elements by an equivalent aperture/

aperture- image pair that has no focusing elements so that all rays between this equivalent

pair are straight lines.

A THIN LENS between TWO APERTURES: BEAM-DEFINING APERTURE/APERTURE-IMAGE PAIRS . Figure 9.4 depicts two vertical screens with coaxial circular apertures A., and A„

aligned along the horizontal optical axis of a thin circular lens L located between them.

In (a) , the extreme rays between the margins of the two apertures are again shown by solid

lines a - a and b - b, as in figure 9.2, but now they are deflected by refraction as they pass through the lens. On the right-hand side of the lens, the rays from the lefthand side have also been extended through the lens as undeviated straight dashed lines that

intersect at points on the margin of a virtual image 1 A' (also shown by dashed lines)

corresponding to similar points of intersection of the solid lines at the margin of aperture A„

.

It is clear that the rays of the beam that cross at aperture A., are exactly the same there at A.. as they would be if the lens were removed and the aperture A„ were replaced

by its virtual image A'. That is, the rays between A., and A„ that intersect at A., to form the subtended solid angles a), (x ,y ) at points x.. ,y 1 within the aperture A., in the expression for the throughput of the aperture-lens combination

G =

J / cos9 -dw -dA [m

2 -sr], (9.8)

A

l wl^ Xl' yl^

'•A real image is formed when rays from each object point converge, after deviation by a focusing mirror or lens, to intersect at a corresponding image point. If strong enough

visible light is used, such a real image can be seen on a ground glass or other scattering

or diffusing material placed at the location of the image. A virtual image is formed when

the rays diverge, after deviation by the lens or mirror, and intersect only when extended

back, as straight lines, again past the deflecting mirror or lens. Since the intersecting

lines at each virtual image point are only extensions of the rays, which do not actually

pass through that point, they cannot be seen there on a ground glass like those of a real

image. Instead, for example, the virtual image A' of the aperture A„ would be seen, and appear to be located at the position of A', only by looking back through the lens

from the left-hand side at the rays coming through it from the real object, the aperture

A„. In the vicinity of A,, looking back through the lens, those rays appear to be coming

from A', rather than from A„ , because of the refraction by the lens. The most familiar

virtual images are, of course, those that we see of ourselves and our surroundings in a common plane mirror or looking glass.

98
(a) Throughput between A and A^ (virtual image of A

2

)

.

(b) Throughput between A' (virtual image of A^ and k^

,

Figure 9.4. Two apertures with a thin lens between them.

(Vertical scale exaggerated by a factor of 2.)

99
are exactly the same as those that would come from corresponding points of an aperture at the virtual image A' if the lens and A were removed. Consequently, we can evaluate

the throughput most readily on that basis — between A.. and A' in the absence of the

lens L and aperture A»

.

In figure 9.4(b), in exactly the same way, is shown the virtual image A' of A.. . And it combines, similarly, with A_ , in the absence of A1 and L, to establish solid

angles w_(x„,y„), subtended by intersecting straight-line rays from A' at points x„,y„ across A , for evaluating the throughput

= / / cose 9 -dw

9 -dA9 [m -sr]. (9.9)

A co (x ,y ) 2 2 2»2 ^

2 ^2

,y 2

)

In Chapter 2 [9.1], in the discussion following eqs. (2.28) and (2.29), we establish

the geometrical invariance of throughput for any beam as long as no rays are added or removed. Consequently, the throughput at the two locations, given by eqs. (9.8) and (9.9),

must be exactly the same. For those who find a numerical example more satisfying than this

analytical conclusion, computations of the two throughputs are carried out, in Appendix 9A,

for both aperture/aperture- image pairs, using eq. (A3-15) [9.6], for an assumed configuration of aperture and lens dimensions, positions, and focal length.

Generally, when there are focusing optics between two apertures that, together, define

a radiation beam, an equivalent beam-defining aperture pair (joined by undeviated straightline rays) is formed by either aperture and the image of the other aperture produced by all

of the intermediate optical elements.

OBJECT SPACE and IMAGE SPACE . Note that, in figures 9.4(a) and (b) , the straight-lineequivalent rays between each of the equivalent beam-defining aperture/aperture- image pairs

coincide with the actual rays (solid lines) on just one side of the lens. They are extended as straight dashed lines past the lens and the second stop as though those optical

elements were not there, to intersect at the virtual image of that stop. .This situation is

briefly described in geometrical optics by stating that both the aperture and the aperture

image of a pair, and the straight-line rays between them, exist in the same space: the

other pair exists in a second space. One "space" is associated with the rays on the left

side of the lens, where both actual and equivalent rays coincide; the other is associated

with the actual (refracted) rays on the right side of the lens where, again, the actual and

equivalent rays coincide.

The conventions and terminology of geometrical (ray) optics are, typically, related to situations involving imaging. In radiometry, however, we often use focusing optics in situations where we are not concerned with forming an image of an object or source of radiation but, rather, ( with defining a beam of radiation from that source, when focusing optics

are used for imaging, the space associated with the rays between the object or source and

the focusing optics, and the straight-line extensions of those rays past any optical elements as though they were not present, is referred to as object space . Similarly, the

100
space associated with the deviated (refracted) rays emerging from the last focusing element

and their straight-line extensions back past all optical elements as though they were not

present, is called image space , since it is these rays that may intersect to form a real

image (or their extensions may intersect to form a virtual image) of the source or object.

Both spaces overlap completely and coexist on both sides of the optics. The determining

factor is the side on which the rays themselves (not their straight-line extensions) actually

exist.

Customarily, a source or object is illustrated as being on the left of a focusing lens

or lenses, with the radiation propagating through from left to right to form a real image,

if any, of the source or object on the right-hand side of the optics (or a virtual image on the left side by straight-line extensions of the right-hand-side rays). Accordingly,

unless explicitly stated otherwise, we will usually refer to the space associated with the

actual rays on the left as object space and that associated with actual rays on the right

as image space. For example, the aperture pair A.. -A' [figure 9.4(a)] would be said to be

in object space, while the pair A'-A„ [figure 9.4(b)] is in image space. Note that this

is consistent with the fact that, in visible light, the rays between A1 and kl appear

perfectly straight when viewed from the object side of the lens, while A' can't even be seen from the image (right) side, i.e., it doesn't "exist" in image space, etc.

Similar relationships exist with respect to a focusing or imaging mirror element, but

then we can't speak of tie rays on different sides of the element because the reflection

returns all rays to the same side. Instead, we must then distinguish between the incident

rays in object space (those coming from the object or source toward the element) and the

exitent rays in image space (those leaving the element), and their respective straight-line

extensions past the element to form virtual images beyond (behind) it.

APERTURE STOPS, FIELD STOPS, and THEIR IMAGES; PUPILS and WINDOWS . In Chapter 5 [9.3] of

this Manual, figure 5.1 (repeated here as figure 9.5) shows the physical configuration used

in a laboratory measurement of the radiance of the glowing filament of a tungsten-ribbon

lamp. In simple cases like this, where a relatively small source on the optic axis (the

target area W) is imaged on a receiver (the entrance slit S.. ) , the aperture that limits

the pencil or solid angle of rays from the source which is accepted by the instrument is

called the aperture stop , wherever it may be located in the instrument. In this case, the

aperture A in the lens mounting limits the size of the accepted beam to which the instrument responds and, hence, the solid angle that it subtends at the target W, so it is the

aperture stop of the system. With more complicated multi-element optics, where the aperture

stop may be located between some of the elements, we refer to the image of the aperture

stop, formed by all elements that precede it (those between it and the source), as the

entrance pupil of the system. A second aperture that, along with the aperture stop, determines the beam that can be accepted by the instrument, i.e., its throughput, is called the

field stop . It limits the solid angle w(x,y) filled by the intersecting rays of the beam

at each point x,y across the aperture stop. In this case (figure 9.5), the field stop is

the entrance slit S, . The image of the field stop, formed by all focusing elements that

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precede it, is the entrance window , in this case the target area W. Accordingly, in this

simple case (small axial source imaged on receiver) the accepted beam or throughput is

defined by the entrance window W in combination with the entrance pupil A, both in

object space.

More complicated, multi-element optical systems may also involve repeated reimaging.

However, if the instrument is sharply focused to accept radiation only from a well-defined

portion of an extended source, this means that some stop or limiting aperture is imaged

there, at the source. It is that stop that is designated as the field stop and its image

there on the source (in object space) is the entrance window. Similarly, at least in most

well-designed optical systems for radiometric instruments, another stop or aperture re- stricts the accepted solid angle or pencil of rays from any point of the entrance window

that can pass through the optical system without obstruction. It determines the transverse

dimensions of the beam entering the instrument. It is called the aperture stop and its

image in object space is the entrance pupil. Together, the entrance window and entrance

pupil form a beam-defining aperture(-image) pair in object space. They establish the

maximum accepted beam, the one containing all possible rays that can pass through the

optical system without obstruction. The measure of this beam is the throughput 9 of the

optical system.

DESIGNATIONS of STOPS, PUPILS, and WINDOWS in RADIOMETRY . The defined throughput or maximum accepted beam of the optical system, just described, exists whether or not there is

actually an extended source at the entrance window. In radiometry, it is not uncommon to find situations where focusing or imaging of a source does not take place but where, nevertheless, an optical system serves to define the accepted beam in the way just described,

i.e., by two stops whose images in object space form a beam-defining aperture(-image) pair.

The member of this pair farthest 2 from the instrument is loosely called the entrance window

and the associated stop, the field stop ; the other image and stop, similarly, being termed

the entrance pupil and the aperture stop , respectively. We say "loosely" because the

conventions of geometrical optics apply, strictly, only to situations involving an object

and its image. In fact, the rules for assigning the designations of aperture stop and

field stop are based on the object-point location and the designations may be different for

different locations of an object with respect to the same optical system [9.7,9.8,9.9].

However, for radiometry it isn't really important which member of the beam-defining aperture(-image) pair is the entrance pupil and which is the entrance window. Accordingly, we won't attempt to go into the geometrical-optics rules here. Those who need them will find

them in the cited texts [9.7,9.8]. Meanwhile, we will continue, in most instances, to follow the "loose" but convenient practice just described in designating the aperture and

field stops of most optical systems. Incidentally, since they are also used occasionally

^hus, S.. determines . the target area W or "field" from which the instrument accepts

radiation for measurement; hence the designation "field stop".

2 It may even be "at infinity" — see next section.

103
in radiometry, we should mention that the image of the aperture stop formed by all optical

elements that follow it (conventionally to the right) is called the exit pupil and, similarly, the image of the field stop formed by all following optical elements is called the

exit window . The exit pupil and exit window, thus, form a beam-defining aperture(-image)

pair in image space.

OPTICAL SYSTEM FOCUSED on INFINITY . There remains one important special case that is not

adequately covered by the foregoing treatment. When an instrument, usually one intended

for use in the field, is focused on infinity, the field-stop image that is the entrance

window is at infinity. Consequently, it's no longer accessible so we can't evaluate the

throughput as determined by the entrance window and entrance pupil as a beam-defining

aperture(-image) pair.

Actually, things are made simpler by the fact that, when the system is focused on infinity, the angular spread of the incident rays of the beam that intersect at each point

of the entrance pupil is the same for all such points. The solid angle co(x,y) , containing

all rays of the beam intersecting at the point x,y of the entrance-pupil area A is then

\ /

Figure 9.6. A simple optical system focused on infinity

(for field use with distant targets or sources)

2 A — entrance-pupil area [m ]

S — aperture stop

5 — field stop

to — field solid angle [sr]

6 — half-field angle [rad]

104
a constant for all such points: to(x,y) = to, a constant. This is evident in figure 9.6,

illustrating how a thin lens focuses incident parallel beams of dashed rays at opposite

points on the margin of the field stop in its focal plane. These extreme rays enclose the same solid angle to wherever they intersect at the entrance pupil which, in this simple

case, is the aperture stop formed by the margins of the lens, itself. Also, in this case, evaluation of the solid angle oj is quite simple; it is just the solid angle subtended at the center of the lens by the undeviated rays from the margins of the field stop. More

generally, with a thick lens or with multiple primary optical elements, this would be the

solid angle subtended at the rear nodal point 1 of the primary optics (in most cases, with

all elements in the same medium, air, this is the same as the rear principal point 1 )

,

rather than at the center of the lens 0. Once to is established, the throughput is quite

simply evaluated as the product of the entrance-pupil area A and' the projected solid

angle Q =

J cos6*dto {eq. (2.30) [9.1]}. If, with the usual circular field stop, to is the solid angle at the apex of a right circular cone of half-apex angle 9, (see figure

A2-5 [9.10]), we found in Appendix 2, eq. (A2-14) [9.10] that the corresponding projected

solid angle is

? n = it* sin 6, [sr] . (9.10)

n

Then, by eq. (2.30) [9.1], the throughput is

= A'fi = A-TT-sin

2 ©^ [m

2 -sr]. (9.11)

n

FLEXIBILITY in CHOICE of REFERENCE SURFACE for SPATIAL PARAMETERS . Although we consistently use the receiving aperture or entrance pupil of a radiometer as the logical reference

surface for the incident radiation to which an instrument responds, it was pointed out in

Chapter 2 [9.1], particularly in connection with the transformation equations, eqs. (2.41)

and (2.42) or (2.41a) and (2.42a), that any convenient reference surface that intersects

the entire beam may be used for the designation of the spatial parameters of position x,y

and direction 6, <f>. This follows from the invariance of throughput which, accordingly,

may be evaluated at any surface that intersects the entire beam. From what we have seen here about the effects of vignetting, the most convenient choices will surely be a stop or beam-defining aperture, or an image of such a stop, but it may be any one, depending on where the pertinent lengths and angles are most easily evaluated.

ENTRANCE PUPILS as RECEIVING APERTURES — SOME PRACTICAL PROBLEMS . We have seen that, in an imaging or focusing optical system, it is the entrance pupil that constitutes the re- ceiving aperture from the standpoint of limiting the transverse dimensions of the incident

beam of rays (in object space) that is accepted by the instrument. When the entrance pupil

is a virtual image of an aperture stop, inaccesibly located behind a lens, it may be difficult to measure its dimensions and position when these data are needed. Fortunately,

^ee any standard text on geometrical optics [9.7,9.8]

105
they are not required for solving the measurement equation in the illustrative problems of

Chapter 5 [9.3]. For radiance measurements, such as those of Problem 1, it is only necessary to be sure that the dimensions and position of the entrance pupil or receiving aperture remain exactly the same for all of the measurements. They then cancel out, so their

values don't enter into the final result, as in eq. (5.11) or eq . (5.20) [9.3]. For irradiance measurements, as in Problems 2 and 3, uniform, isotropic irradiation of the monochromator entrance pupil and constant . internal-beam geometry are insured by the use of a diffuser ahead of the focusing optics. An aperture in front of the diffuser is then the

receiving aperture for the overall system. In the next section, we discuss some considerations relating to such diffusers and the apertures in front of them. However, when field

measurements are made, with an instrument focused on infinity, of the irradiance from

distant targets or sources, it is often not feasible to use a diffuser and then it may be

quite important to know the exact position of the entrance pupil or receiving aperture.

There are two compelling reasons for not using a diffuser for these field measurements.

First, a diffuser reduces the detectivity of the instrument substantially, often beyond the

point where it is adequate to respond to the low irradiance from very distant sources.

Second, isolation of the radiation of the source or target from that of other unwanted

background sources in the uncontrolled field environment, along with an adequately large

receiving aperture, can be achieved only with focusing optical elements.

For field measurements of distant sources that subtend very small angles at the radiometer, it may no longer be essential to have isotropic responsivity, even across a relatively narrow field of view of the instrument. However, if the responsivity is not isotropic, it is important to know the value of the responsivity as a function of direction

together with the direction of the source within the instrument field at the time of the

measurement. This poses some difficult practical problems for moving sources, such as aircraft and missiles, in addition to the problem of dealing with background radiation,

e.g., that from the sky, and its contribution to the output of the measuring instrument.

Returning to the problem of the dimensions and position of the entrance pupil or receiving aperture, its area A cancels out and does not appear in the final solution of

the measurement equation for irradiance. For example, it appears in both eqs. (5.27) and

(5.28), so that it is eliminated when the quotient between them is taken in eq. (5.29).

However, the exact position of the entrance pupil may still be very important. An irradig ance-standard lamp is usually calibrated and certified for the values of irradiance E , or spectral irradiance E,(A), at a given distance, so we must be able to verify that the

reference surface established by the entrance pupil is, in fact, at that specified distance

from the standard source (lamp). 1 As already pointed out, this may be difficult if the

entrance pupil is a virtual image of an aperture stop and is inaccessibly located behind a lens or lenses. The simplest solution for precision work is to install an accessible

^his problem may not arise with a collimated-beam standard of spectral irradiance if the

unknown is also collimated or very far away.

106
aperture stop in front of the entire optical system, making sure that it becomes the

limiting aperture by making it just small enough so that no rays of the measurement beam

can quite reach the edges of the internal stop, which is then no longer the aperture stop

for that measurement. This will somewhat reduce the flux in the beam but it will probably

be a small price to pay for the improved accuracy in establishing the dimensions and location of the aperture stop (serving also as entrance pupil and receiving aperture) . It is

important to make sure, also, that it does form a beam-defining aperture pair with the

field stop or its image, i.e., that from every point within the aperture stop rays can

pass, without obstruction at some other point, through every point within the field stop.

The field solid angle to, or, with circular optical elements and stops, the halffield angle 0, , is also relatively easy to measure directly. A distant small source, at least optically distant (e.g., by merns of a collimator), is observed at different angles

from the optical axis (with the vertex of the angle at the center of the entrance pupil

or receiving aperture). The edge of the field solid angle or the angle 6, is that angle

where there is no longer any instrument response. With a steady source, this can also

be made a measurement of the directional responsivity characteristic, the relative responsivity as a function of direction r(0,<|>). With good symmetry, the responsivity may be

independent of <j> and a function only of the angle from the axis r(0). In any case, this

information will help to insure that the irradiance measurement is actually made with the source well within a central region of isotropic directional responsivity to the extent

that this is required (see Chapter 5, problems 2 & 3 [9.3]).

DIFFUSERS . In problems 2 and 3 of Chapter 5 [9.3], we have already seen that instruments

used for measuring irradiance (including spectral irradiance) often employ a diffuser, such

as an averaging sphere, as the first optical element in order to achieve the required isotropic directional flux responsivity (or, as an approximate equivalent, a lambertian response

that varies as the cosine of the angle of incidence when irradiated by a well-collimated,

iso- radiance, incident beam that is wide enough to completely fill the receiving aperture

at all times). When used as the first optical element, a diffuser terminates the incident

beam and becomes a new source for the exitent beam; there is no ray-to-ray continuity

between the two beams. For this purpose, the best diffuser for measurements of optimum

accuracy is probably a good integrating or averaging sphere with its entrance port (receiving aperture) defined by a sharp-edged aperture of well-defined position and dimensions

(see "More about Vignetting" below, especially figure 9.9). Even then, internal baffles or screens may be needed to insure that all rays of the exitent beam have not been reflected

directly from points exposed to the direct incident beam.

Sometimes, when the use of a sphere is not convenient, other forms of diffuser may be

employed, such as a flat-plate diffuse reflector or a transmitting ground-glass, opalglass, or translucent-plastic element. The result may be greater variation in directional

responsivity or narrowing of the solid angle within which the response to incident radiation

is acceptably isotropic. The problem of adequate baffling to insure that the exitent beam

for measurement includes radiation only from a desired portion of the diffusing element may

107
also be aggravated. On the other hand, the directional responsivity of a transmitting

diffuser is sometimes "improved" by adjusting its shape to achieve a more nearly lambertian

or "cosine" response, empirically compensating for the fact that it is not a perfect diffuser.

A typical empirically "shaped" diffuser is something like that depicted in figure

9.7(b). The width of a uniform (iso-radiance) collimated beam of parallel rays striking

the flat diffuser of figure 9.7(a) is reduced by the factor cos9 as the angle of incidence 9 is increased. However, most diffusing materials are not perfectly diffusing.

Also, reflection loss at the first surface tends to increase with increasing G. In general, such diffusers become less efficient diffuse transmitters at larger angles of

incidence so that the flux reaching the detector beneath the diffuser is reduced by more

than the factor of cos9. By raising the center of the element and adjusting its shape by

trial and error, it is possible to intercept a slightly wider beam at the larger angles of

incidence, as shown in figure 9.7(b), to compensate for the losses. In this way, the flux

reaching the detector, and hence its response, can be made to follow quite closely a desired

"cosine response" as the angle of incidence of a uniform collimated (parallel) beam is

changed. However, the beam from a relatively close lamp of finite dimensions can't be

expressed as the sum of such conveniently uniform beams of perfectly parallel rays. At present, we don't know how to establish a receiving-area or reference-surface distance or location for such shaped diffusers that will give consistent results for measurements of

beams with quite different spatial distributions of flux or beam geometries.

Figure 9.7. Dif fuser-detector configurations for measuring irradiance.

Note that, for isotropic flux responsivity over the widest

possible angle, it is usually desirable to make the detector

wide enough to collect as much as possible of the diffusely

transmitted radiation coming from the back (underside) of the

diffusing element, as shown.

108
When it is known that the beam geometry and the spatial distribution (both position

and direction) of flux within the beam are exactly the same for both the unknown source and

the calibration source, there is, of course, no problem. Then any sample of the same portion of each beam can serve as a measurement of the entire beam or any portion of it in

each case, so it's only necessary to use the same reference surface consistently for both

measurements. The problem of identifying the exact receiving-area distance for the irradi- ance measurement assumes importance only to the extent that the two beam geometries (unknown-source and calibration-standard spatial distributions of flux) differ. Unfortunately,

we have not been able to devise any general method for establishing the receiving area or reference-surface distance in such situations.

In summary, most real diffusers fall considerably short of the ideal, producing truly

isotropic flux responsivity for the instrument only over a limited solid angle or range of

directions rather than the desired full hemisphere. This makes it particularly important

to measure the directional responsivity and to make sure that all significant sources (both

unknown and standard) are clearly within the isotropic-responsivity region. On rare occasions, if the beam geometry of both unknown and standard sources is identical, measurements

of samples of the beam taken in the same way can establish an adequate comparison regardless of variations in responsivity as a function of direction as long as it is the same for

both measurements. However, in almost all instances, with a standard source that has a geometrical configuration quite different from that of the unknown source, accurate irradi- ance measurements cannot be made without an instrument of adequately isotropic flux responsivity.

BAFFLES . When the defined beam completely fills both of just two beam-defining apertures

or stops, with no other obstruction or vignetting, this means that all other apertures

through which that beam passes are then large enough so that they do not block any rays

that are passed by both beam-defining apertures. The other apertures are then either

baffles , introduced to intercept and reduce scattered radiation, or they are supports for

the optical elements. Baffles are used to block strong rays that, while they cannot pass

directly through all of the stops or beam-limiting apertures, might reach the detector or transducer by reflection or scattering from the internal surfaces of the instrument chamber.

MORE on VIGNETTING . Vignetting takes place, not only at the edges of a beam beyond or between the stops as illustrated in figures 9.2 and 9.3, but also at the nominal stops when

some of the rays between points within the nominal stops are blocked by obstructions elsewhere, either between or beyond the nominal stops, as in figures 9.8(a) or (b) . For those

blocked rays, the nominal stop is no longer the actual stop; instead, the obstacle becomes

the stop (beam-limiting-aperture margin) for the blocked rays, so that there is no longer

just a beam-defining aperture pair or pair of stops.

Even when we do have just a beam-defining aperture pair or pair of stops, it can be

difficult enough to evaluate the integrals in

109
Figure 9.8. Two cases of vignetting.

A and B are the nominal stops (beam-defining aperture pair)

.

Introduction of aperture C causes vignetting where unobstructed

rays of the beam through each point of A can no longer reach all

points of B, and vice versa. There are now extreme rays a - c on one side and a - b on the other.

110
/ / cos6«du)*dA [m *sr]

A u>(x,y)

(9.2)

because the solid angle co(x,y), filled by rays of the beam that intersect each area

element dA (at each point x,y over the area A) is often, as indicated, a significant

function of the position coordinates x,y of that element. That's why we're always glad

to take advantage of the published tables of configuration factors, as described in Appendix 3 [9.6], to evaluate such throughputs as that between two parallel coaxial circular

apertures in eq. (A3-15) [9.6]. When the limits of w involve the margins of more than

one aperture or barrier at different distances (i.e., when there is vignetting), evaluation

of the throughput becomes even more complicated and difficult [9.5].

Note also that, unless the stops have thin sharp edges, there can be vignetting of

oblique rays at the edges of a stop (see figure 9.9). On the near side, they are limited

by the front edge of the stop; on the far side by the back edge. If the edge is rounded,

the plane of the limiting margin changes continuously around the stop for an oblique, wellcollimated beam. A well-designed stop ordinarily has sharp edges lying in a plane perpendicular to the optical axis.

Figure 9.9. Vignetting by an aperture in a thick wall.

The front (upper) edges and the rear (lower) edges form circular stops

in different planes. Oblique parallel rays, as illustrated, are limited

by the front (upper) edge on the near (left) side and by the rear (lower)

edge on the far (right) side.. Of course, if these were just slightly

diverging rays from a point source, or one smaller than the aperture,

the rear (lower) edge would become the limiting aperture if the source were accurately aligned on the axis, perpendicularly above the aperture.

However, in general, radiation from a more extended source will include

parallel bundles of rays, incident at angles away from the normal, with

the effect shown.

Ill
RESPONSIVITIES and BEAM GEOMETRY . In an instrument with uniform, isotropic, linear flux

responsivity [R (x,y,0,c|>) = R , a constant, for all rays that reach the detector], the

radiance responsivity R.^ is directly related to the flux responsivity R by the throughput 0:

R

$

= S/* = S/(L-0) [s-w

-1

],

R^ = S/L = 0-R [S-W

-1

-m

2 -sr]

.

(9.12)

Note that, although we don't have a different symbol, these are now overall instrument

responsivities. They would be average responsivities if the individual-ray responsivity

R (x,y,6,4>) = dS/d$ were not the same for all rays. Here $ [ = //l-cos6 -dim-dA] is the

total flux in the beam to which the instrument responds, S is the resulting output signal,

L is the radiance of an iso-radiance incident beam that completely fills the throughput

(or the average radiance of such a beam if the radiance is not everywhere the same throughout the beam)

.

Equation (9.12) is true, regardless of vignetting [as long as the effect of the vignetting is correctly accounted for in the evaluation of the throughput 0, e.g., by choosing

the correct limits for co(x,y) as it varies from point to point in eq. (9.1)]. However,

the corresponding relation between flux responsivity R and irradiance responsivity R^

in eq. (5.49) [9.3]

R,, = S/E = A-RA [S-W

_1

-m

2 ], (9.13)

is valid, for the full aperture or entrance-pupil area A, only when there is no vignetting. This is apparent if we stop to think in terms of a uniform incident beam that is

well collimated (parallel rays from nearly the same direction) . If there is vignetting and

such an incident beam swings far enough away from the normal or axial direction, so that

rays through part of the nominal receiving aperture or entrance pupil are blocked by the

vignetting obstacle elsewhere in the optical system, the result is a reduction of the

actual aperture area A and a corresponding reduction in the irradiance responsivity R^

given by eq. (9.13). Accordingly, such instruments should be used, if possible, only for

incident beams or sources that lie well within the angular field or entrance window (away

from directions that involve the vignetting) where the irradiance responsivity R is

constant. We won't even attempt to deal here with all of the problems that can arise if

vignetting can't be avoided.

SUMMARY of CHAPTER 9 . In radiometry, a radiation beam is usually defined physically by

two apertures. The beam then consists of all rays that pass through both apertures. They are called beam-defining apertures if rays from each point of the plane across one aperture

reach all points within the other aperture and if all such rays through both apertures

extend for the full length of the beam, from source to receiver, without encountering any

112
other obstacle. If, in addition, the medium between the apertures is homogeneous (isorefract ive- index) , so that all rays between them are straight lines, we call them a beamdefining aperture pair , or just an aperture pair . When focusing optics are used and there

are one or more focusing elements between the beam-defining apertures, the rays between

them will be refracted (deviated) by the focusing element ( s) . Then the beam (throughput)

at either aperture (chosen as the reference aperture) is identically the same as that

defined by an equivalent aperture pair consisting of that (reference) aperture paired with

the image of the other aperture formed by the intervening optical element (s) {see figure

9.4(a) and (b) } . The straight-line rays joining this equivalent pair coincide with the

actual rays of the beam in the region about the reference aperture and extend, undeviated,

past the intervening optical elements as though those elements were not present.

Beam-defining apertures in a focusing optical system are called stops . The stop that,

regardless of its location in the system, determines the width of the beam where it enters

or is incident on the optical system is designated as the aperture stop . The image of the

aperture stop, formed by any optical elements that precede it (elements traversed by the

incident beam before reaching it) , is called the entrance pupil . If there are no such

elements, the stop itself is the pupil (see figure 9.5). The entrance pupil serves as the

receiving aperture of the instrument (with respect to incident rays and their undeviated

straight-line extensions past any focusing optical elements preceding the entrance pupil)

.

The entrance pupil is the image of the aperture stop in object space, the space associated

with the source or object, usually shown to the left of the optical system in illustrations

with incident rays coming from the left. Object space also includes the undeviated straightline extensions of those incident rays past any focusing optical elements into the region

to the right of them {as though they were not present, as in figure 9.4(a)}.

Similarly, the stop that, regardless of its location in the optical system, determines

the solid angle filled by intersecting incident rays of the beam at each point across the

entrance pupil is called the field stop . The image of the field stop formed by all optical

elements that precede it, i.e., the image of the field stop in object space, is called the

entrance window . The entrance window and entrance pupil, both in object space, form a beam-defining aperture(-image) pair. When the magnitude of the beam throughput is needed,

it is often convenient to evaluate it as the throughput of this aperture(-image) pair

(entrance window and entrance pupil) . A simple illustration of such numerical evaluations

is presented in Appendix 9A. In addition, the solid angle subtended by the entrance window

at the center of the entrance pupil in object space is called the angular field , or sometimes just the field , of the optical system. For systems with coaxial circular elements

and stops, the half-vertex angle of the right circular cone formed by the entrance window

as its base and the center of the entrance pupil as its vertex is called the half-field

angle of the optical system.

Although not often used in radiometry, our list of related geometrical-optics quantities and concepts also includes the exit pupil , the image of the aperture stop in image

space, i.e., the image formed by all optical elements that follow it (aperture stop), and

113
the exit window , the image of the field stop in image space. They are included here primarily for completeness.

In the special case of an instrument focused on infinity (entrance window at infinity)

,

it is not feasible to evaluate the throughput as that of the aperture (-image) pair formed

by the entrance window and entrance pupil in object space. However, in this case the field

solid angle has the same value at all points of the entrance pupil (see figure 9.6) so the

corresponding projected solid angle is also a constant as a function of position across the

entrance pupil. Accordingly, the throughput which, in general, is the integral of the product of projected area and solid angle (or area and projected solid angle) over the entrancepupil reference surface (see figure 9.1) then becomes simply the product of the entrance

pupil area and the constant projected solid angle {6 = J/dA-cosG «dto = A*fi [m «sr]}.

Vignetting takes place when a beam does not completely fill both of two nominal stops

or beam-defining apertures, i.e., when it is limited in one transverse direction by one aperture margin and in other transverse directions by one or more additional apertures or obstacles (see figures 9.3 and 9.8). When vignetting occurs, it becomes very difficult to evaluate the beam throughput or related measures of beam geometry, or even to make sure

that they are the same (physically unchanged) for two different radiometric measurements.

In most of this chapter, the geometrical-optics quantities and concepts are treated in

terms of throughput and the limits of the throughput integral (s) , because they are the

quantitative measures of beam geometry. It is only in these terms that an adequate understanding of their significance can be achieved, including the ability to confidently

decide, in all cases, whether or not a numerical evaluation of the throughput may occasionally be required.

In an instrument with uniform, isotropic, linear flux responsivity, where

R (x,y,G,<})) = R is constant for all rays that reach the detector (are not blocked by the

optical system), the radiance responsivity R^ is directly related to the flux responsivity R by the throughput G:

R A

= S/$ = S/(L«0) [S-W

_1

]

R^ = S/L = G-R

$

[S«W

_1

-m

2 -sr]. (9.12)

This is true regardless of vignetting, as long as the effect of the vignetting is correctly

accounted for in the evaluation of the throughput 0. However, the corresponding relation

between flux responsivity R and irradiance responsivity R^,

R^ = S/E = A-RA [S-W

_1

-m

2 ], (9.13)

is valid, for the full aperture or entrance-pupil area A, only when there is no vignetting.

114
Customarily, when the defined beam completely fills both of just two beam-defining

apertures or stops, with no other obstruction or vignetting, all other apertures through

which that beam passes are either baffles or supports for the optical elements. Baffles are employed to reduce or eliminate scattered radiation by blocking strong rays that, while

they cannot pass directly through all stops or beam-limiting apertures, might reach the

detector or transducer by reflection or scattering from the internal surfaces of the instrument chamber.

The receiving aperture or entrance pupil of a radiometer is consistently used, here,

as the logical reference surface for incident radiation to which the instrument responds.

However, any unvignetted stop or beam-defining apeiture (of a pair), or an image of same,

i.e., any pupil or window, may be the reference surface for the spatial parameters of position and direction, depending on where the pertinent lengths and angles are most easily

evaluated.

Diffusers are often used to achieve the isotropic flux responsivity needed for irradi- ance measurements (see problems 2 and 3 of Chapter 5 [9.3]). They are usually located just

behind a well-defined receiving aperture where they terminate the incident beam and become

a new source for the internal beam through the remaining optical components. Ideally, the

radiance of every ray in that internal beam is directly proportional to the total incident

flux and is not affected by any changes in the spatial distribution of that flux in the

externally incident beam. The use of diffusers involves a number of special problems that are discussed in some detail in this chapter, including the uncertainties in the receivingaperture location when empirically "shaped" diffusers are used in attempts to compensate

for imperfectly diffusing materials.

The treatment of geometrical-optics quantities and concepts in this chapter is all in

terms of ideal optical elements that produce perfectly sharp images and shadow boundaries.

In addition, we've ignored attenuation losses, since we've already seen how, at least in

principle, to introduce the propagance (see Chapter 2 [9.1]) in order to account for them.

However, real instruments also suffer from aberrations that produce imperfect images and

from scattering and diffraction that produce attenuation of some rays and augmentation of

others, resulting in a spatial redistribution of flux. We hope to treat the resulting

problems in future chapters of this manual. Accordingly, we must re-emphasize the statement at the beginning of this chapter that the material presented here, while it is very

important for design purposes and as a starting point for analysis, is no more than that —

a starting point or first approximation — and must never be considered as establishing

exact values or absolute limits (e.g., integration limits in the measurement equation) that

cannot be exceeded.

Finally, although the geometrical-optics concepts of pupils and windows are conventionally defined in terms of an object (source) and its image, they are defined here, for

radiometry, with respect to a radiation beam which may or may not involve imaging of the

radiation source. These designations are rather widely used in the literature on radiometry

115
and such an extension of the customary geometrical-optics definitions is needed to provide

a consistent basis for distinguishing between aperture stops and pupils and field stops and

windows when there is no source imaging involved. However, the really important consideration in radiometry is to clearly recognize and understand the significance of beam-defining

apertures and equivalent aperture/aperture-image pairs, regardless of which is designated

as the aperture stop or pupil and which the field stop or window.

116
Appendix 9A. Numerical Evaluation Confirming Invariance of Throughput for Beam through

Thin Lens between Two Beam-Defining Apertures

by Fred E. Nicodemus

In Chapter 9, we have treated the case of a thin lens between two beam-defining apertures. We find two aperture/aperture- image pairs, one in object space and one in image

space, that define the beam of all rays through both apertures. The corresponding throughput values are given, respectively, by eqs. (9.8) and (9.9). The argument for the invari- ance of throughput in a given beam, when no rays are added or removed, which follows eqs. (2.28) and (2.29) in Chapter 2 [9.1], is of general validity and requires no further proof

as the basis for asserting that the throughput given by eqs. (9.8) and (9.9) must be exactly

the same. Nevertheless, some readers may find it satisfying to corroborate this analytical

conclusion numerically. Also, carrying out the numerical throughput calculations for both

aperture/aperture- image pairs, using eq. (A3-15) , may help to bring out more clearly the

way in which apertures and aperture images enter into the evaluation of the throughput as well as to physically define a beam.

Both figures, 9.4(a) and (b) , are drawn approximately to scale for the dimensions

given below except that, for clearer illustration of the relationships, the vertical

dimensions are exaggerated by a factor of two. We choose the lens focal length as

f = 7.00 [cm] and its radius as r « 1 [cm] (large enough to pass all necessary rays

without obstruction). A.. is a circular aperture of radius r.. = 0.500 [cm] in a vertical

plane at a distance x = 3.00 [cm] to the left of the center of the lens, which we designate as the origin 0. A» , similarly, has a radius r„ = 0.600 [cm] at a distance

x_ = 2.000 [cm] to the right of 0. In order to determine the size and position of each

virtual image, we make use of the thin-lens formula 1

-£- + -£- = -j- [ cm-1 ]> ( 9A - X ) o i

where x is the object distance (from 0), x. is the image distance (also from 0),

and f is the focal length of the lens. We also need the fact that image size is in

direct proportion to distance from 0, since each image point must lie on the undeviated

ray (drawn with long dashes) through and the corresponding object point. Thus A' has a radius r' = 0.840 [cm] and is located at a distance x' = 2.800 [cm] to the right

of 0; A' has a radius r' = 0.875 [cm] and is located x' = 5.250 [cm] to the left of

0.

A1 and A' then constitute a circular aperture pair of radius r.. = 0.500 [cm]

and r' = 0.840 [cm], respectively, separated by a distance D = x, + x' = 5.800 [cm].

Using eq. (A3-15) [9.6], the throughput for this aperture pair is evaluated as

2

n = 0.0503 [cm *sr]. In the same way, A' and A constitute a circular aperture

'See any standard text on geometrical optics [9.7,9.8], or almost any general-physics text,

117
pair of radius r' = 0.875 [cm] and r = 0.600 [cm], respectively, separated by

D = x,' + x = 7.250 [cm]. The throughput for this second aperture pair is evaluated

2 similarly as 9 = 0.0507 [cm «sr].

Since the thin-lens formula is an approximation, derived for paraxial rays (rays deviating from the axial direction by angles small enough so that we can use the approximations

sin6 *** 8, for 6 measured in [rad] , and cosG "=> 1), this agreement to less than one per

cent for the two throughput values is gratifying confirmation of the validity of this

approach. .

118
References

[9.1] Fred E. Nicodemus and Henry J. Kostkowski, "Distribution of Optical Radiation

with Respect to Position and Direction — Radiance", Chapter 2 of "Self-Study

Manual on Optical Radiation Measurements: Part I—Concepts", Nat. Bur. Stand.

(U.S.), Tech. Note 910-1, 93 pages (March 1976) pp. 10-44.

[9.2] Fred E. Nicodemus, "More on the Distribution of Optical Radiation with Respect

to Position and Direction", Chapter 4 of "Self-Study Manual on Optical Radiation Measurements: Part I—Concepts", Nat. Bur. Stand. (U.S.), Tech. Note 910-2,

118 pages (Feb. 1978) pp. 1-57.

[9.3] Henry J. Kostkowski and Fred E. Nicodemus, "An Introduction to the Measurement

Equation", Chapter 5 of "Self-Study Manual on Optical Radiation Measurements:

Part I—Concepts", Nat. Bur. Stand. (U.S.), Tech. Note 910-2, 118 pages

(Feb. 1978) pp. 58-92.

[9.4] John B. Shumaker, "Distribution of Optical Radiation with Respect to Polarization", Chapter 6 of "Self-Study Manual on Optical Radiation Measurements:

Part I—Concepts", Nat. Bur. Stand. (U.S.), Tech Note 910-3, 62 pages

(June 1977).

[9.5] "Optical Radiation Measurements: The 1973 Scale of Spectral Irradiance",

Nat. Bur. Stand. (U.S.), Tech. Note 594-13, 36 pages (Apr.. 1977),

Appendix, pp. 20-28.

[9.6] Fred E. Nicodemus, "Projected Solid Angles, Throughputs, and Configuration

Factors", Appendix 3 of "Self-Study Manual on Optical Radiation Measurements: Part I—Concepts", Nat. Bur. Stand. (U.S.), Tech. Note 910-2,

118 pages (Feb. 1978) pp. 93-100.

[9.7] Francis A. Jenkins and Harvey E. White, "Fundamentals of Optics", 3rd ed.

(McGraw-Hill, New York, 1957).

[9.8] Miles V. Klein, "Optics" (Wiley, New York, 1970).

[9.9] R. H. Anderson, "Closeup Imaging of Documents and Displays with Lens

Arrays," Appl. Opt. 18, No. 2 (15 Feb 1979) 477-484; there is a good

example of the dependence of the entrance pupil on object position

described on p. 480.

[9.10] Fred E. Nicodemus, "Spherical Coordinates and Geometrical Relationships",

Appendix 2 of "Self-Study Manual on Optical Radiation Measurements:

Part I—Concepts", Nat. Bur. Stand. (U.S.), Tech. Note 910-1, 93 pages

(March 1976) pp. 62-80.

119
ERRATA and ADDENDA for NBS Tech Notes 910-1 and 910-2:

1. Table Al-3 (p. 58 of TN 910-1): Add note: A more complete version of this

table appears as table 4-1 on p. 3 of TN 910-2.

2. Table Al-4 (p. 59 of TN 910-1): Add note: This table also appears as table 4-2 on p. 5 of TN 910-2.

3. Table Al-6 (p. 61 of TN 910-1): Add note: This table also appears as table 4-3 on p. 6 of TN 910-2.

4. Table Al-6 (p. 61 of TN 910-1) and table 4-3 (p. 6 of TN 910-2):

The modifier for the first three entries should be changed from "photon"

to "photon-flux", just as in the fifth and succeeding entries. Note

that this change is particularly important in the case of the first

entry, photon-flux energy Q (= /$ *dt) , because of the very wide

and well established use of the term "photon energy" for hv, the

energy per photon, i.e. the energy of a single photon or quantum.

5. Table Al-4 (p. 59 of TN 910-1) and Table 4-2 (p. 5 of TN 910-2):

Add alternative term for the first quantity listed so that it reads,

"luminous energy; quantity of light".

6. Table Al-6 (p. 61 of TN 910-1) and Table 4-3 (p. 6 of TN 910-2):

Add alternative term for the first quantity listed so that it reads,

"photon-flux energy; number of photons".

ADDENDUM to tables Al-6 (p. 61 of TN 910-1) and 4-3 (p. 6 of TN 910-2),

Units for Photon-Flux Radiometry

The relationship of the photon-flux quantities of Tables Al-6 and 4-3 to the corresponding radiometric quantities of Tables Al-3 and 4-1 may be stated explicitly as follows;

= / \ ,(A)-(h-c/A) 1 -d\ [q-s

1

],

A l

where 4> [q # s ] is the photon flux in the spectral interval X. to X [nm] in a beam

p

_ 1 L characterized by the spectral radiant flux $ (A) [W*mn ], and where Planck's constant

-34 -, e ' h = (6.626 176 ± 0.000 036) x 10 [J«s], the vacuum speed of electromagnetic radiation

c = (2.997 924 580 ± 0.000 000 012) x 10

17 [nm-s"

1

] * 3 x 10

8 [m-s

-1

] , and. X [nm] is

the wavelength, so that the energy per quantum (per photon) is (h-c/A) [J-q ].* Similar

*Strictly, the energy per quantum is (h-c/A ) = {h-c/(n*A)} [J«q ], where A = n-X [nm]

is the vacuum waveTength and n [dimensionless] is the index of refraction of the medium

in which the wavelength is A [nm] . Also, v = c/A [THz] for c % 3 x 10 [km-s ] and

A in [nm] ; and a = 1/A [cm ] only when A is given in [cm].

120
relations exist for all of the other corresponding quantities and their spectral distribu-

-1 -2 tions, e.g., for photon-flux (surface) density E [q*s *m ], where the spectral irradi-

-2 -1 P ance is E , (A) [W*m 'nm ], e, A

^9 -1 -1 ? E = / E ,(A)-(h-c/A) -dA [q-s -m ]

.

p

J e,A

A l

If the spectral distribution is with respect to frequency v [THz] , this becomes

v

? -1 -1 -? E = / E (v)'(h'v) -dv [q-s -m ], P ^ e,v

where the energy per quantum is now given as h*v [J*q ]. Similar relations can also be

given for spectral distributions in terms of wave number a [cm ]

.

121
ERRATA: Several errors have been noted in the labelling in Figure A2-4 on page 68 of

NBS TN 910-1. It should be corrected to appear as follows:

P •sin 9 • d <£ P'69

<JA S

= p

z »s\r\9*d8*d<f>

Y

I I

« 1

S ^ ^ -j

6<p f \^A p»s\r\Q»d<p

/''

L—^>«sin 9

Figure A2-4. An element of solid angle

2 2

din = dA /p = sin6'd6«d<|> [rad ]

122
NBS-1UA (REV. 9-76)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. TN 910-4

&<S»y'fc' AebBSalon No. S. Resident's Accession No,

4. TITLE AND SUBTITLE _ SELF-STUDY MANUAL ON OPTICAL RADIATION

MEASUREMENTS: Part I—Concepts, CH. 7, The Relative Spectral

Responsivity and Slit-Scattering Function of a Spectroradiometer, CH. 8, Deconvolution, CH. 9, Physically Defining Measurement-Beam Geometry by Using Opaque Barriers

*. Performing Organization Code

5. Publication Date June 19 79

7. AUTHOR(S) chapter 7, Henry J. Kostkowski;

Chapter 8, John B. Shumaker; Chapter 9, Fred E. Nicodemus

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

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Same as item 9

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15. SUPPLEMENTARY NOTES

I | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

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 fourth in a series of Technical Notes (910- ) entitled "Self-Study Manual

on Optical Radiation Measurements". It contains Chapters 7, 8, and 9 of this Manual.

Additional chapters will continue to be published, similarly, as they are completed.

The Manual is a comprehensive tutorial treatment of the measurement of incoherent

radiation that is complete enough for self instruction. Detailed chapter summaries

make it also a convenient authoritative reference source. The manner in which the spectral responsivity of a spectroradiometer containing a monochromator varies with wavelength is treated in Chapter 7. The nature and characteristics of this function, how it is determined, and the effects it has on spectroradiometric measurements are discussed in detail. The traditional, indirect

determination, involving the slit-scattering function, is shown to be highly inaccurate except for wavelengths in the central portion of the function. An introduction to the measurement of spectral line radiation is also presented.

Deconvolution, discussed in Chapter 8, is the numerical process of recovering an improved spectral distribution from spectroradiometric measurements inevitably

smeared spectrally by the spectral-responsivity function of the radiometer. A

simple, iterative technique, which is extensively used, will be completely

_(abstract con' t)

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated fay 3 em ,co/onSj Aperture ^ aperture_ image pair . aperture pair ; calibration; convolution;

deconvolution; geometrical-optics radiometry; inversion; measurement-beam geometry;

relative spectral responsivity; slit-scattering function; slit-scattering function

v r J (key words con'tj

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16. ABSTRACT

satisfactory in almost all radiometric situations. A more sophisticated technique,

which is founded on a sounder theoretical basis, should succeed in many of the few

remaining instances where the simpler technique fails. Examples of computer programs for the two techniques are given in an appendix. Both are applied to a simple

illustrative numerical example with brief mention of some of the sources of difficulty and the limitations of deconvolution. Finally, we touch on the question of errors and describe a couple of useful measures of accuracy.

The physical definition of measurement beams, including the integration limits in

the measurement equation, is treated in Chapter 9 in terms of geometrical-optics

quantities and concepts. The usual definitions of geometrical optics are extended

to situations where defining the measurement beam may not involve the imaging of

any source. The importance of aperture pairs or equivalent aperture/aperture-image

pairs, in defining measurement beams without vignetting, is developed. Also discussed are entrance pupils as receiving apertures in radiometry (their positions

and dimensions), diffusers, vignetting, and the effects of beam geometry on overall

instrument radiance and irradiance responsivities. The treatment is in terms of

simple, ideal, geometrical optics, with sharp image and shadow boundaries, and only

passing mention is made of perturbations caused by imperfect imaging, aberrations,

scattering, and diffraction.

17. KEY WORDS

corrections; spectral line radiometry; spectroradiometer characterizations;

spectroradiometry ; vignetting.

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