The Art of Radiometry - J.M. Palmer B. G. Grant [PDF]

Zitiervorschau

~

The

Art Radiometry James M. Palmer Barbara G. Grant

SPIE

PRESS

Bellingham, Washington USA

Library of Congress Cataloging-in-Publication Data Palmer, James M. Art of radiometry / James M. Palmer and Barbara G. Grant. p. cm. -- (Press monograph; 184) Includes bibliographical references and index. ISBN 978-0-8194-7245-8 1. Radiation--Measurement. 1. Grant, Barbara G. (Barbara Geri), 1957- II. Title. QDl17.R3P352009 539.7'7--dc22 2009038491

Pub lished by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: + 1 360.676.3290 Fax: + 1 360.647.1445 Email: [email protected] Web: http://spie.org Copyright © 2010 Society of Photo-Optical Instrumentation Engineers (SPIE) All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the work and thought of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America.

On the cover: A Crooke radiometer and the equation of radiative transfer.

SPIE

Contents Foreword ....... .... ......... ...... .... ... .. ... ... ........ .. .... .. .................. .. ... ... ..... .. .. ....... xi Preface ....... ................ ........ ..... ... ....... ..... ..... ................ ... ......................... xiii

Chapter 1 Introduction to Radiometry 11 1.1 Definitions ...................... .... ......................................... ......................... 1 1.2 Why Measure Light? ........................................................................... 2 1.3 Historical Background ............ ..................................... ......................... 4 1.4 Radiometric Measurement Process ............. .. .... ......... .. ...... .. ...... ........ 5 1.5 Radiometry Applications ......... ... .. .... .. ... .. .. .. .. .............. ...... ............. ..... . 7 References .. .... .......... .. ..... ... .... ...... .. ... ... ..... ....... ..... ........ .. ..... ........... ....... .. 9

Chapter 2 Propagation of Optical Radiation 111 2.1 Basic Definitions ..................................................... ........................... 11 2.1 .1 Rays and angles ..... ............................................................... 11 2.1 .2 System parameters ................................ ... .... ... ... ..... ............. 19 2.1 .3 Optical definitions .. .. .............. .. ..................... .. ....................... 23 2.2 Fundamental Radiometric Quantities .... .................... .... ......... ........... 24 2.2.1 Radiance .. .. ..... ... ... ............... ..... .. .. ............. .... ....... ...... .... .. .... 24 2.2.2 Radiant exitance .. ... .... .. .......................... ... .... ... ... .. ................ 26 Irradiance ............ .......................... ............... .. ....................... 28 2.2.3 2.2.4 Radiant intensity .. ... ............................................................... 29 2.3 Radiometric Approximations .............................................................. 30 2.3.1 Inverse square law ................................................................ 30 2.3.2 Cosine 3 law .......... ................................................................. 31 2.3.3 Lambertian approximation ..................................................... 32 2.3.4 Cosine4 law ..... ..... .. ................................... .... ..... ................... 33 2.4 Equation of Radiative Transfer ..................... .. ... .......... ....... .... ........... 36 2.5 Configuration Factors .. ...... ...... .. ............ .... .... .. ..... ........ ....... .... .. .... .. .. 38 2.6 Effect of Lenses on Power Transfer ...... ..... ............ ...... .............. ... ... . 40 2.7 Common Radiative Transfer Configurations ............ .. .......... ......... .... 42 2.7.1 On-axis radiation from a circular Lambertian disc .... .. ........... 42 2.7.2 On-axis radiation from a non-Lambertian disc ...................... 43 2.7.3 On-axis radiation from a spherical Lambertian source .......... 44 2.8 Integrating Sphere .................................................... ....... .................. 46 2.9 Radiometric Calculation Examples .................................................... 48 Intensities of a distant star and the sun .. ..... .... .................... .. 48 2.9.1 v

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Contents

2.9.2 Lunar constant. .......... .. ................... .... ................ .. ................. 50 2.9.2.1 Calculation .......... ................... .................. .. .................... 50 2.9.2.2 Moon-sun comparisons ........... .................. ................ ... . 51 2.9.3 "Solar furnace" ...... ....... ..... ....... .. .... .. ............. .. .... .. ............... .. 52 2.9.4 Image irradiance for finite conjugates .. .. ...................... .. .. ..... 53 2.9.5 Irradiance of the overcast sky ...... .. ................ .... .. .. ............ .. . 55 2.9.6 Near extended source .................. .... ............... .. .................... 55 2.9.7 Projection system ...... ... .. ... .......... ..... .. ................ .. ................. 56 2.10 Generalized Expressions for Image-Plane Irradiance .. .. .. .... ... ........ 57 2.10.1 Extended source .......... ......................................................... 57 2.10.2 Point source ............ ... ................ .. ..... .......... ......... ................. 58 2.11 Summary of Some Key Concepts .......... ................ .. .. .. ................... 58 For Further Reading ...... ............ .... ..... ......... .. .... ... .. ............. ..... .............. . 59 References ..... ...... ............... .. ...... ..... ............ .... .............. ........................ . 59

Chapter 3 Radiometric Properties of Materials I 61 3.1 Introduction and Terminology .. .......................................................... 61 3.2 Transmission .. ... ... ..... ......... .. .. .. ................ ................ .... .................. ... 62 3.3 Reflection ........ ... ... ... ............ ..... .... ..... ...... .. .................. ................ ... .. 63 3.4 Absorption ......... ... ................. ... ................ .................. .... ................... 69 3.5 Relationship Between Reflectance, Transmittance, and Absorptance ...... .. .................... .......................................................... 69 3.6 Directional Characteristics ... .. ....... ....... ... ..... ..... ..... ...... ... ...... ..... ..... ... 69 3.6.1 Specular transmittance and reflectance .............. .... ............. . 69 3.6.2 Diffuse transmittance and reflectance .................................. . 73 3.7 Emission ................................... ........................................ ................. 76 3.8 Spectral Characteristics .................................................................... 77 3.9 Optical Materials Checklist .. .... ..... ... ..... ... .. .... ... ... ...... .. .... .... .... .......... 79 For Further Reading .. ... ................ ...... .............................................. ....... 80 References ................. .. ............. ..... ................ .. .................. .. ................ ... 80

Chapter 4 Generation of Optical Radiation I 83 4.1 Introduction ............. ... .... ........ ..... ..... ... .. .... ..... ..... ....... ...... ...... ......... ... 83 4.2 Radiation Laws .................................................................................. 84 4.2.1 Planck's law ............. ... ...................................... .. ................... 84 4.2.2 Wien displacement law ............... .. .. .................... .. .............. .. . 86 4.2.3 Stefan-Boltzmann law ..... .. ........... .... ... ............. ... .. .............. .. 89 4.2.4 Laws in photons ...... .......... .......... .. ................... .............. .. .... . 89 4.2.5 Rayleigh-Jeans law ..................... .. ........................................ 92 4.2.6 Wien approximation .... ... ................. ................. .. .................... 93 4.2.7 More on the Planck equation ...... ...... ................. .. .. ... ............. 93 4.2.8 Kirchhoff's law ...... .... ... .......... ..... ... .... ..... .... ... ..... ..... ... ......... .. 97 4.3 Emitter Types and Properties .. ....... .......... .. ... ............... ........... ... ..... 102 4.3.1 Metals ...................... ............................................. ............... 102

.. Contents

vii

4.3.2 Dielectrics ............................................................................ 102 4.3.3 Gases .................................................................................. 103 4.4 Practical Sources of Radiant Energy ............................................... 104 4.4.1 Two major categories .......................................................... 104 4.4.2 Thermal sources ............... ...................................... ............. 105 4.4.2.1 Tungsten and tungsten-halogen lamps .................. ...... 105 4.4.2.2 Other metallic sources .................................................. 108 4.4.2.3 Dielectric thermal sources ............................................ 108 4.4.2.4 Optical elements ........................................................... 109 4.4.2.5 Miscellaneous thermal sources ...................... ....... ....... 109 4.4.3 Luminescent sources ..... ......................... ............................ 110 4.4.3.1 General principles ........................................................ 110 4.4.3.2 Fluorescent lamps ........................................................ 115 Electroluminescent sources ......................................... 117 4.4.3.3 LED sources ...... ... ......................... ........... ..... ............... 117 4.4.3.4 4.4.3.5 Lasers .......................................................................... 118 Natural sources ................................................................... 119 4.4.4 Sunlight ........................................................................ 119 4.4.4.1 Skylight, planetary, and astronomical sources ............. 120 4.4.4.2 4.4.4.3 Application: energy balance of the earth ........... ........... 121 4.5 Radiation Source Selection Criteria ................................................. 121 4.6 Source Safety Considerations ......................................................... 123 4.7 Summary of Some Key Concepts .................. ................................. 123 For Further Reading .............................................................................. 123 References ................... ....... ........... ...... .................. ................. ... ...... ..... 124

Chapter 5 Detectors of Optical Radiation 1127 5.1 Introduction ....................................................... ............................... 127 5.2 Definitions ... .. ......... ... .. .. .. ........... .... .. .............. ..... .... .. ....................... 128 5.3 Figures of Merit .................................................... ...................... ..... 131 5.4 #N$O%&I*S@E-A ........................................................................... 133 5.4.1 Introduction to noise concepts ............................................. 133 5.4.2 Effective noise bandwidth .................................................... 136 Catalog of most unpleasant noises ... .. ... ..................... .. ...... 137 5.4.3 5.4.3.1 Johnson noise ......... ..................................................... 137 5.4.3.2 Shot noise .................................................................... 139 5.4.3.3 1If noise ........................................................................ 139 5.4.3.4 Generation-recombination noise .................................. 140 5.4.3.5 Temperature fluctuation noise ...... .. ............... .. .. ........... 141 5.4.3.6 Photon noise ................................................................ 141 5.4.3.7 Microphonic noise ........................................................ 142 5.4.3.8 Triboelectric noise ........................................................ 142 CCD noises ..................................................... ............. 142 5.4.3.9 5.4.3.10 Amplifier noise ......... .. ................. .. ......... .... .... ... .......... .. 143 5.4.3.11 Quantization noise ........................................................ 143

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Contents

5.4.4 Noise factor, noise figure, and noise temperature ............... 143 5.4.5 Some noise examples ........................ .. ............................... 144 5.4.6 Computer simulation of Gaussian noise .............................. 147 5.5 Thermal Detectors ........................................................................... 147 Thermal circuit ..................................................................... 147 5.5.1 Thermoelectric detectors ............................. ........................ 150 5.5.2 5.5.2.1 Basic principles ............................................................ 150 5.5.2.2 Combinations and configurations .. .. ..................... ........ 153 5.5.3 Thermoresistive detector: bolometer .... .. ............................. 155 Pyroelectric detectors ....... ..... .... ........... ... ............................ 157 5.5.4 5.5.4.1 Basic principles ................................ ............................ 157 Pyroelectric materials ................................................... 160 5.5.4.2 5.5.4.3 Operational characteristics of pyroelectric detectors ... 162 Applications of pyroelectric detectors ........................... 162 5.5.4.4 Other thermal detectors ....................................................... 163 5.5.5 5.6 Photon Detectors ..... .... .................................................................... 164 5.6.1 Detector materials .................. ........................ .. .. .. .. .... .... .... . 164 Photoconductive detectors .... .. .... .. .......... .. .. ... .. .. ........ ......... 169 5.6.2 5.6.2.1 Basic principles ...... ....... .. ... ......... ... .................... .... .... .. 169 5.6.2.2 Noises in photoconductive detectors ........................... 173 5.6.2.3 Characteristics of photoconductive detectors .............. 174 Applications of photoconductive detectors ................... 175 5.6.2.4 Photoemissive detectors ................... .... .............................. 175 5.6.3 5.6.3.1 Basic principles ............................................................ 175 Classes of emitters ....................................................... 176 5.6.3.2 5.6.3.3 Dark current ..................................... .. .......................... 181 5.6.3.4 Noises in photoemissive detectors .. ... ......... .. ............... 182 Photoemissive detector types ...... ... .......... ................... 183 5.6.3.5 Photovoltaic detectors ................................ .. ....................... 185 5.6.4 5.6.4.1 Basic principles ............................................................ 185 5.6.4.2 Responsivity and quantum efficiency ........................... 195 5.6.4.3 Noises in photovoltaic detectors .. .. .............................. 196 Photovoltaic detector materials and configurations ...... 198 5.6.4.4 5.7 Imaging Arrays ......... ... ...................................... .............................. 199 Introduction .... ...................................................................... 199 5.7.1 Photographic film ....................................... .. .. .... .. ................ 199 5.7.2 History ...... ...................... ... ..... .. ... ................ .. ........ .. .. ... 199 5.7.2.1 Physical characteristics .... ... ... .... ... ... ............ ........ ... .... . 201 5.7.2.2 5.7.2.3 Spectral sensitivity .... ................. ..... ............................. 201 5.7.2.4 Radiometric calibration ................................................. 201 5.7.2.5 Spatial resolution .......................................................... 202 5.7.2.6 Summary ......................................... ............................. 202 5.7.3 Electronic detector arrays ....................... ............................. 203 5.7.3.1 History ........................................... .. ............................. 203 5.7.3.2 Device architecture description and tradeoffs .............. 203

Contents

ix

5.7.3.3 Readout mechanisms ......................... ......................... 204 5.7.3.4 Comparison ..... .. ..................................... ...................... 207 5.7.4 Three-color CCOs ... ............................................................ 207 5.7.5 Ultraviolet photon-detector arrays ....................................... 208 5.7.6 Infrared photodetector arrays ..................... ... .... ....... ... ........ 209 5.7.7 Uncooled thermal imagers ........ .................. ... ..................... 210 5.7.8 Summary .... ....... ..... ... ........ ... ............. .. .... ........ ..... .. ....... .. .... 211 For Further Reading ... ..... ..... .... ........... ................ .. ....... ..... ... ................. 211 References ......................... ............................................................. ...... 213

Chapter 6 Radiometric Instrumentation I 215 6.1 Introduction ........................ .................................... ..... ..................... 215 6.2 Instrumentation Requirements ...................................... .. ................ 215 Ideal radiometer .. ......................................... ....................... 215 6.2.1 6.2.2 Specification sheet ................................ ... ... ........................ 215 Spectral considerations ..... .... .. .... .. .... ...... ... .... .. ... ........... .. ... 216 6.2.3 Spatial considerations .. ........ .. ........ ... ......... .... ... .. ............. .. . 217 6.2.4 Temporal considerations ............... ............ ... ... ... ..... ...... ...... 217 6.2.5 6.2.6 Make or buy? .. ... .... ....................................... ...................... 218 6.3 Radiometer Optics ............ ..... .................................. ... ..................... 218 6.3.1 Introduction .......................................................................... 218 6.3.2 Review of stops and pupils .................................................. 218 6.3.3 The simplest radiometer: bare detector. .... ... ..... .................. 219 6.3.4 Added aperture ........ ........... ....................... ..... ..................... 219 6.3.5 Basic radiometer .. ... ... ... ..................................... ..... .. .......... 221 6.3.6 Improved radiometer ........ .. ....... ......... ... ...... ... ... ... ......... ...... 223 Other methods for defining the field of view .... ...... .......... .... 224 6.3.7 6.3.8 Viewing methods .... ................................. ... ..... .................... 224 6.3.9 Reference sources ... .. .................................. ... .................... 226 6.3.10 Choppers .. ........ .. .......................................... .... ................... 226 6.3.11 Stray light ............ ... ............................................................. 227 6.3.12 Summing up ........................................................................ 228 6.4 Spectral Instruments .. .................................................. ................... 228 Introduction .... .. .... ...................................... .......................... 228 6.4.1 6.4.2 Prisms and gratings .................................... .. .... .... .. ............. 230 6.4.3 Monochromator configurations .. .. ..... ... .. ........... .... ... ............ 231 6.4.4 Spectrometers ......... ... ........ .. ............ ... ...... ...... .................... 234 6.4.5 Additive versus subtractive dispersion ..... .. ..... .. ..... ......... .... 235 6.4.6 Arrays .............................................................. ... ................. 236 Multiple slit systems ... ................................... ............. ......... 236 6.4.7 Filters .................... .. ............................................................. 236 6.4.8 6.4.9 Interferometers .... .. .............................................................. 237 6.4.10 Fourier transform infrared .................................................... 237 Fabry-Perot ..... ... .................................... .. .... ....................... 238 6.4.11

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Contents

For Further Reading ............................................................................. . 240 References ................................ ................... ......................................... 240

Chapter 7 Radiometric Measurement and Calibration 1241 7.1 Introduction .... ... ................. ......... ..................................................... 241 7.2 Measurement Types ........................................................................ 241 7.3 Errors in Measurements, Effects of Noise, and Signal-to-Noise Ratio in Measurements .................. ................. .......................................... 241 7.4 Measurement and Range Equations ... ....................... ..................... 250 7.5 Introduction to the Philosophy of Calibration ................. .... .... .......... 253 7.6 Radiometric Calibration Configurations ........................................... 257 7.6.1 Introduction .......................................................................... 257 7.6.2 Distant small source ............................................................ 258 7.6.3 Distant extended source ............. ......................................... 260 7.6.4 Near extended source ............. ........................ ....... ............. 261 7.6.5 Near small source .. ............................................................. 262 7.6.6 Direct method ...................................................................... 262 7.6.7 Conclusion ........................................................................... 263 7.7 Example Calculations: Satellite Electro-optical System .................. 263 7.8 Final Thoughts .................... .. ................ ....... ................. .... ............ ... 267 For Further Reading .............................................................................. 268 References ............ ................................................................................ 268

Table of Appendices 1 269 Appendix A: Systeme Internationale (SI) Units for Radiometry and Photometry ......... .............................................................. 271 Appendix B: Physical Constants, Conversion Factors, and Other Useful Quantities ........... .............. .. ............................................... 275 Appendix C: Antiquarian's Garden of Sane and Outrageous Terminology ...................................................................... 277 Appendix D: Solid-Angle Relationships ............................ .. ................... 283 Appendix E: Glossary ............................................................................ 285 Appendix F: Effective Noise Bandwidth of Analog RC Filters and the Selection of Filter Parameters to Optimize Signal-to-Noise Ratio ................................................................................. 297 Appendix G: Bandwidth Normalization by Moments ........ .. ................... 305 Appendix H: Jones Near-Small-Source Calibration Configuration ........ 309 Appendix I: Is Sung lint Observable in the Thermal Infrared? .............. 313 Appendix J: DocumentaiY Standards for Radiometry and Photometry 321 Appendix K: Radiometry and Photometry Bibliography ........................ 341 Appendix L: Reference List for Noise and Postdetection Signal Processing ......... .... .................. .. ...................................... . 357

Index 1361

Foreword The material for this book grew out of a first-year graduate-level course, "Radiometry, Sources, Materials, and Detectors," that Jim Palmer created and taught at the University of Arizona College of Optical Sciences for many years. The book is organized by topic in a similar manner, with the first five chapters presenting radiation propagation and system building blocks, and the final two chapters focusing on instruments and their uses. Chapter 1 provides an overview and history of the subject, and Chapter 2 presents basic concepts of radiometry, including terminology, laws, and approximations. It also includes examples that will allow the reader to see how key equations may be used to address problems in radiation propagation. Chapter 3 introduces radiometric properties of materials such as reflection and absorption, and Chapter 4 extends that discussion via a detailed consideration of sources. Point and area detectors of optical radiation are considered in Chapter 5, which also includes thermal and photon detection mechanisms, imaging arrays, and a discussion about film. In Chapter 6, the focus shifts to instrumentation. Concepts introduced in Chapter 2 are here applied to instrument design. Practical considerations relating to radiometer selection are detailed, and a "Make or Buy?" decision is explored. Several monochromator configurations are examined, and spectral instruments are discussed. Proceeding from instruments to their uses, Chapter 7 details types of measurements that may be made with radiometric systems and provides a discussion of measurement error. The philosophy of calibration is introduced, and several radiometric calibration configurations are considered. The material in the appendices covers a variety of topics, including terminology, standards, and discussions of specific issues such as Jones source calibration and consideration of solar glint. Due to Jim's attention to detail and the length of time over which he accumulated material, the long lists he provided here may be viewed as comprehensive, if not current by today's standards. The level of discussion of the material is suitable for a class taught to advanced undergraduate students or graduate students. The book will also be useful to the many professionals currently practicing in fields in which radiometry plays a part: optical engineering, electro-optical engineering, imagery analysis, and many others. In 2006, Jim Palmer was told that he was terminally ill, and he asked me to complete this work. I was humbled and honored by the request. I'd met Jim as a graduate student in optical sciences in the late 1980s, and he had served on my thesis committee. My career after graduation had focused on systems engineering and analysis, two areas in which radiometry plays a significant role. For nearly the last ten years of Jim's life, I'd been able to receive mentoring from the master simply by showing up at Jim's office door with a question or topic for discussion, but I never anticipated that our discussions would one day come to an end. Upon Jim's death, I sought to weave his collection of resources and narrative together xi

xii

Forward

with newer material and discussion in a manner I hope will be both informative to read and valuable to reference. The preface that follows was written by Jim before he died and has been left as he wrote it. I am grateful for the assistance of many. First is William L. Wolfe, Jim's professor and mentor, who offered helpful comments on each chapter and adapted Chapter 6 on radiometric instrumentation. Others for whose help I am grateful, all from or associated with the University of Arizona College of Optical Sciences, are Bob Schowengerdt, who contributed the narrative on film; Anurag Gupta of Optical Research Associates, Tucson, Arizona, who adapted the appendix material; and L. Stephen Bell, Jim's close friend and colleague, who revised the signal processing discussion that appears in that section and provided a complete bibliography on the subject. A special note of thanks goes to Eustace Dereniak, who provided office space for me, helpful discussions, and hearty doses of encouragement. I also wish to thank John Reagan, Kurt Thome (NASA Goddard Spaceflight Center, Greenbelt, Maryland), Mike Nofziger, and Arvind Marathay for review, discussion, and helpful insights. In addition, I am grateful for the assistance of Anne Palmer, Jim's beloved sister, and University of Arizona College of Optical Sciences staff members Trish Pettijohn and Ashley Bidegain. Gwen Weerts and Tim Lamkins of SPIE Press have my gratitude for the special assistance they provided to this project. I also gratefully acknowledge Philip N. Slater, my professor in optical sciences, who selected me as a graduate student and trained me in remote sensing and absolute radiometric calibration from 1987 to 1989, and Michael W. Munn, formerly Chief Scientist at Lockheed Martin Corporation, who instilled the value of a systems perspective in the approach to technical problems. Finally, I am grateful to my family for providing financial support; to Ralph Gonzales, Arizona Department of Transportation, and Sylvia Rogers Gibbons for providing professional contacts; and my friends at Calvary Chapel, Tucson, Arizona, whose donations and prayers sustained me as I worked to complete this book. Barbara G. Grant Cupertino, California October 2009

Preface This volume is the result of nearly twenty years of frustration in locating suitable material for teaching the subject of radiometry and its allied arts. This is not to say that there is not a lot of good stuff out there-it's just not all in one place, consistent in usage of units, and applicable as both a teaching tool and as a reference. I intend this book to be all things to all people interested in radiometry. The material comes from teaching both undergraduate and graduate-level courses at the Optical Sciences Center of the University of Arizona, and from courses developed for SPlE and for industrial clients. I have unabashedly borrowed the tenor of the title from the superb text The Art of Electronics by Paul Horowitz in the hope that this volume will be as useful to the inquisitive reader. I gratefully acknowledge the contributions of my mentor, William L. Wolfe, Jr., and the hundreds of students whose constant criticism and occasional faint praise have helped immeasurably. This book is dedicated to the memory of my mother, Candace W. Palmer (1904-1996) and my father, James A. Palmer (1905- 1990). She was all one could wish for in a Mom, and he showed me the path to engineering.

James M Palmer 1937-2007

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Chapter 1

Introduction to Radiometry 1.1 Definitions Consider the following definitions a starting point for our study of radiometry: radio- [ n j. Also note that a cone of light becomes narrower in a higher-index medium. Table 2.1 Indices of refraction of various media.

Medium Vacuum Air Water Quartz Glasses CaF2 Ab 0 3 ZnSe Si Ge

n 1 1.0003 1.33 1.45 1.5- 1.9 1.42 1.75 2.4 3.4 4.0

13

Propagation of Optical Radiation

_.. _.. _..

_.

Figure 2.3 Illustration of Snell's law.

Projected area is defined as the rectilinear projection of a surface of any shape onto a plane nonnal to the surface's unit vector. The differential fonn is dA p = cose dA , where e is the angle between the line of observation and the local surface normal n. Integrating over the surface area we obtain (2.3)

Some common examples of projected area are shown in Table 2.2. Figure 2.4 depicts the relationships between surface and projected area for a circle and a sphere. Plane angles and solid angles are both derived units in the SI system. Figure 2.5 depicts the plane angle e, with I the arc length and r the circle radius. The solid angle Q for angles greater than 0 deg). The definitions and symbols presented here have not been universally applied in the past. One must be very cautious when reading the literature, as different investigators use the terms and symbols solid angle ro and projected solid angle Q interchangeably, incurring predictable confusion and potentially incorrect results.

19

Propagation of Optical Radiation

Table 2.5 Percentage error when not using a projected solid angle. @1/2

(in deg)

Error using

0)

10

< 1%

16

< 2%

25

< 5%

35

< 10%

48

2

=T2--->1 = T.

(2.19)

2

The throughput is invariant. W2

~ ............... .

................................

..................................... d Figure 2.10 Area- and solid-angle relationships used to define throughput. (Adapted from a figure courtesy of William L. Wolfe.)

F

21

Propagation of Optical Radiation

Q 05

A5

L

.................................. Ao Figure 2.11 Invariance of throughput for a case in which the source image fills the detector.

So far, our discussion has focused on theoretical constructs. Now, let's introduce some system elements as we proceed with demonstrating the invariance of throughput in an optical system. Consider Fig. 2.11. It represents a special case in which the image of the source exactly fills the detector. This configuration consists of a source, a lens, and a detector. In the figure, As is the area of the source, Ao the area of the optics, and Ad the area of the detector. The projected solid angles are defined thus: Qos is the angle the optics subtend at the source, Q so the angle the source subtends at the optics, Q do the angle the detector subtends at the optics, and Q od the angle the optics subtend at the detector. For this case, the following equalities hold: (2.20) Since these pairings are equal, any of the above pairs can be chosen for calculation purposes. If the image of the source does not exactly fill the detector area, care must be taken to determine the proper AQ product to use. The author's (Palmer) personal preference is the most often-used pair AoQdo , inasmuch as the entrance aperture size Ao and the field of view of the system Q do are determinable characteristics of a radiometer. The next most often used pair is AdQod, as the detector size and the jl# of the optics are also measurable characteristics of the radiometer. Basic throughput is the name given to the quantity conserved across a lossless boundary between two media having different indices of refraction. It can be written as (2.21) and the relationship between medium 1 and medium 2 is

22

Chapter 2

(2.22) where the subscripts denote the respective quantities in media and 2. Like throughput, basic throughput is invariant. Most optical systems have both the object and the image located in the same medium, typically air, so basic throughput is not often used. Figure 2.12 depicts the appropriate and inappropriate area- and solid-angle pairings used to define throughput. The correct area-solid angle pair is shown in Fig. 2.12(a), the incorrect angle pair is shown in Fig. 2.12(b). Because the definition of throughput includes two (projected) areas and the distance between them, a correct pairing has the apex of its solid angle located at the correct area. The incorrect pairing uses one area twice and ignores the other. The maxim "no ice cream cones" should be applied. Let's look at some examples of the An product. First, a spectrometer: Fig. 2.13 shows the area- and solid-angle product of the entrance slit (typically 1 x 10 mm) and the projected solid angle the collimating lens subtends at the slit, n r,. The An product of a spectrometer is usually very small, and narrow spectral bandwidths are typically employed. Therefore, it is difficult to get much light through a spectrometer. A different example of throughput may be found in the common camera. It is related to the j/# of the lens and the size of the film. In this case, the detector (film) size is not the overall dimension of the image, but the size of an individual film grain. The smaller the j/#, the "faster" the camera and the greater the throughput. Similarly, "fast" film has a larger grain area, permitting a higher throughput and a shorter exposure time than "slow" film with a smaller grain area.

(a)

(b)

Figure 2.12 Right and wrong area-solid angle combinations for throughput determination .

--23

Propagation of Optical Radiation

Slit, Area A

Lens

Figure 2.13 Example of the correct area (A) and solid angle (n/s ) product used to determine throughput in a spectrometer. (Adapted from a figure courtesy of William L. Wolfe.)

2.1.3 Optical definitions Some optical quantities are relevant to a study of radiometry, and they are defined here. For more detailed treatment, the reader is advised to consult a text on geometrical optics. t Figure 2.14 depicts the location of object and image planes, along with some key rays. In Fig. 2.15, the chief ray in an optical system originates at the edge of the object and passes through the center of the entrance pupil (NP). It passes through the center of the aperture stop (AS), the edge of the field stop (FS), and the center of the exit pupil (XP), to the edge of the image, defining the image size (height) and the lateral (transverse) magnification. There may be several intennediate pupil planes in a complex optical system.

Image

Figure 2.14 Chief and marginal rays in an optical system, shown schematically.

t Several excellent texts exist; a recent one is E. L. Dereniak and T. D. Dereniak, Geometrical and Trigonometric Optics, Cambridge University Press, Cambridge (2008).

24

••••••••••••

Chapter 2

CHIEF RAY

..........................

.-

Figure 2.15 Optical system stops and pupils.

The marginal (rim) ray in an optical system is the ray from the object that originates at the intersection of the object and the optical axis and passes through the edge of the entrance pupil. It touches the edge (rim) of the aperture stop, proceeds through the center of the field stop, and the edge of the exit pupil. The marginal ray intersects the optical axis at the center of the image, defining the location of the image and the longitudinal magnification. There may be several intermediate image planes in a complex optical system. The entrance pupil is the image of the aperture stop in object space (as seen from the object), while the exit pupil is the image of the aperture stop in image space (as seen from the image). Of particular importance to radiometry are the aperture and field stops. The aperture stop determines how much light may enter the system, while the field stop determines the system's angular field of view. In a simple system consisting of a lens and detector at the rear focal point, the lens is the aperture stop, and both the entrance and exit pupils are located at the lens, with the same size as the lens. In more complicated systems, the stops may be internal and separated from the pupils, as shown in Fig. 2.15.

2.2 Fundamental Radiometric Quantities 2.2.1 Radiance The study of radiometry begins with fundamental units. Radiant energy has the symbol Q and has as its unit the joule (J). Radiant power, also known as radiant flux, is energy per unit time (dQldt), has the symbol , and is measured in watts (W). These definitions give no indication of the spatial distribution of power in terms of area or direction. Radiance is the elemental quantity of radiometry, power per unit area, and per unit projected solid angle. It is a directional quantity; it can corne from many points on a surface that is either real or virtual; and because it is a field quantity, it can exist anywhere. The symbol for radiance is L and the units are WIm 2sr. The defining equations are (2.23)

25

Propagation of Optical Radiation

where e is the angle between the normal to the source element and the direction of observation as shown in Fig. 2.16. Radiance is also associated with a source, either active (thermal or luminescent) or passive (reflective), as discussed further in Chapter 4. Because radiance may be evaluated at any point along a beam, it is associated with specific locations within an optical system, including image planes and pupils. Other radiometric units may be derived from radiance by integrating over area andlor solid angle. Integration over solid angle yields irradiance (arriving at a location, such as a sensor) or radiant exitance (leaving a location, such as a source), both of which are expressed in W /m 2 • Integration over area yields radiant intensity expressed in W Isr. Integration over both area and solid angle yields radiant power in watts. If rays are traced across a lossless boundary between two materials having different indices of refraction as shown in Fig. 2.3 , the solid angle changes according to Snell's law. Taking this change into account, the quantity Ln-2 is seen to be invariant across the boundary. This quantity is called basic radiance. It is useful for calculations when an object and its image are located in spaces with different indices of refraction. In the absence of sources or sinks along the path of a beam, power along a beam is conserved. Since it was previously demonstrated that throughput is conserved in an optical system, the radiance must also be invariant in order for conservation of power (energy per unit time) to be obeyed. The results of this invariance of radiance are significant: (1) The radiance of the image at the detector plane of a camera (film or array device) is the same as the radiance of the scene if there are no transmission losses due to atmosphere and optics; and (2) The radiance at the focal plane of a radiometer (imaging or point) is the same as that of the target, if there are no transmission losses due to atmosphere and optics. Note that the transmission of atmosphere and optics is not likely to be unity (perfect transmission); however, results (1) and (2) greatly simplify radiometric calculations.

dW\

d

----- ---

---

--- ---

--------------------fn Figure 2.16 Radiance from area element dA, tilted at angle

e from surface normal, n.

26

Chapter 2

The defining equation for radiance can be inverted and integrated over area (and in the most general case, a projected solid angle) to determine the power in an optical system: = JJL dAsdroscos8 = JJLdAs dO..

(2.24)

2.2.2 Radiant exitance

Radiant exitance is radiation that exits a source. It is defined as power per unit area radiated into a hemisphere (dldAs). The symbol for radiant exitance is M and the units are W/m 2• Its defining equation is M

= lim (L\ 2 is the power reaching surface 2 from surface 1. Both power terms are dimensionless. The radiant power terms are further defined as

where MI is the radiant exitance in W1m2 leaving surface 1, and A 1 its area; and

1--->2

= MI 1t

flcOS 9 1 cos 9 2 dA dA d2 1 2,

where the radiance term outside the integral is obtained from Eq. (2.32), itself dependent upon the Lambertian approximation . The fraction of radiant power leaving surface 1 that arrives at surface 2 is F. 12

= _1_ ficas 9 1 cos 92 dA dA 1tA

d2

1

2'

1

which is the configuration factor. The power transferred from surface 1 to surface 2 then becomes 1 --->2

= 1~ 2

=MIAI~ 2

=rrLIAI~ 2·

pu

39

Propagation of Optical Radiation

Because the radiant power incident on surface 2 equals LIA 10.21 by Eq. (2.47), and

To restate: in engineering calculations, using the assumptions given above, (2.50) and (2.51) The advantage of using configuration factors is that numerous solved geometries appear throughout the literature. Some relevant to optics are shown in Fig. 2.28. II Further information on configuration factors may be found in Refs. 12, 13, and 14.

=

F d1 - 2

1

(hi r)2 + 1

r

2

Fd1- 2 - ( h)

h

1 (a)

(b)

Figure 2.28 Configuration factor examples: (a) Planar element parallel to circular disk, and (b) planar element to sphere. (Adapted from Ref. 11.)

40

Chapter 2

2.6 Effect of Lenses on Power Transfer Radiometer configurations will be discussed in detail in Chapter 6, but the effects of lenses on power transferred to a detector will be introduced here. Look first at Fig. 2.29. Two configurations are shown: Al and A2 are the areas of the stops; D is the distance from source to stop at A I; and S is the distance from A I to A2 (at the detector d). The difference between the two configurations is solely the presence of the lens at A I in the second. Expressions for power at the detector in each case, for both point and extended sources, will be formulated. In both cases, it will be assumed that the transmittance of the atmosphere between the source and the detector is unity. For the point source case, the irradiance at the detector in Fig. 2.29 is expressed by the inverse square law, Eq. (2.39):

where d is the distance between source and detector. Without a lens, [Fig. 2.29(a)], assuming no transmission losses in the intervening medium, the power at the detector is simply the irradiance multiplied by the available sensitive area: (2.52) Expressed in terms of intensity, the power is (2.53)

where (D + S) is the source-detector distance. Note that in this configuration, A2 acts as the aperture stop, defining how much flux is collected, while AI , the field stop, defines the detector's field of view.

__

o

~___ A2_~ .I~ (a)

5

.1 (b)

Figure 2.29 Configurations (a) without and (b) with a lens.

....

Propagation of Optical Radiation

41

Adding a lens, as in Fig. 2.29(b), yields a different set of equations. In this case, the power at the detector may be expressed as (2.54) or, more specifically, as (2.55) where 'tlens is the lens transmission. In terms of intensity, it is: (2.56) Note that in this case, the aperture that limits the flux into the system is A}, the aperture stop, and that A2 is the field stop, limiting the detector's field of view. The size of A2 is unimportant as long as it does not vignette the source's image at the detector. The difference in received power between the two expressions in terms of irradiance is expressed as G = 'ttensAJ

~'

(2.57)

where G is the gain of power on the detector. To maximize G, make the lens transmission and the area ratio as large as possible, while not vignetting the source Image. To determine the effect of a lens on the same instrument configuration with an extended source, begin with Eq. (2.47):

At area A = A 2, the appropriate solid angle is subtended by area A I. This solid angle is expressed as in Eq. (2.12) by

where 81/2 is the cone half angle. Assuming that it is small, the approximation AIls'- may be used for n 12 , so that (2.58)

42

Chapter 2

and (2.59) The same approach can be pursued with the other area-solid angle combination, that is, with A I and 0 21 • In that case, the solid angle is approximately A2/S, and dis again obtained by Eq. (2.59). Inserting a lens at Al limits the power by the transmission of the lens, so that (2.60) The radiance-area-solid angle relationship holds true regardless of whether the first or second area-solid angle combination is used to calculate throughput. Inserting a lens yields no net gain in detector power for an extended source. Rather, the power is less due to the nonunity (in the real world) transmission of the lens.

2.7 Common Radiative Transfer Configurations 2.7.1 On-axis radiation from a circular Lambertian disc

This case is shown in Fig. 2.30. Assuming a lossless optical system, the flux transferred from source to detector is given by Eq. (2.47), where L is the Lambertian disc radiance. The area-solid angle pair we will use in this case is the area of the detector Ad (in the figure) and the solid angle the source subtends at the detector Osd, which may also be expressed by Eqs. (2.12) and (2.15) as no :''',d

.

. 28 1t =1tsm ~ 112 = / 2' 4(f #)

(2.61)

wherefl# is defined in Eq. (2.13). Considering the geometry in the figure,

. 28

SIn

~ 1/ 2

= (a

a 2

2

2'

+b )

We can now substitute in Eq. (2.48) to provide several equivalent expressions for the irradiance at the detector: 2

Ed

=nLsin 2 8 1/ 2 =LO d =nL (a 2a+b) 2 = nL =nL(NA)2 . 4(f/ #)2 S

(2.62)

Propagation of Optical Radiation

43

Figure 2.30 On-axis Lambertian disc, irradiance measured at detector of area Ad.

If the distance b is far greater than 2a, the linear dimension of the source, then the inverse square law holds and Ed may be approximated as rrLa 2b-2 • The error incurred using a 2b-2 rather than a 2(a 2 + b2r l is less than 1% when the diameter-to-distance ratio is less than 0.1. Under these conditions, source intensity Is may be substituted for TtLa 2 (the radiance times the area of the source) so that

Table 2.8 summarizes the relationships between source-detector distance and irradiance at the detector for a variety of cases. Table 2.8 Irradiance at detector as a function of source distance for a Lambertian disc.

Distance b» 2a b =2a b=a b=O

Half-angle 9 112 (deg) very small 26.5 45

90

Irradiance Ed TtLa 2b-2=Ib -2 rrL/5 TtL/2 rrL

To determine the irradiance on the detector from an annulus (ring) rather than a disc, calculate irradiances from discs having both outer and inner radii, and subtract the latter from the former.

2.7.2 On-axis radiation from a non-Lambertian disc In this case, source radiance is not independent of observation direction, and an integration must be performed. The source's radiant exitance may be obtained by

44

Chapter 2

Figure 2.31 On-axis Lambertian sphere, irradiance measured from surface.

integrating Eq. (2.26) over a hemisphere, and the irradiance at the detector calculated as a function of the half angle. To illustrate, take the relatively simple example in which Ls = LacosS. In this case, the radiant exitance is (2.63) where dO. is taken from Eq. (2.9). The resulting integral is then (2.64) which results in

= 2rrLa

M s

3

.

(2.65)

Assuming a lossless medium, the irradiance at the detector is (2.66) In general, closed-form solutions are not readily available, and numerical methods must be employed. 2.7.3 On-axis radiation from a spherical Lambertian source

If the disc is replaced with a Lambertian sphere of the same radius, as in Fig. 2.31, Eq. (2.62) may still be used, except that the sine squared of the half angle now becomes (2.67)

F

Propagation of Optical Radiation

45

Table 2.9 Irradiance at the detector as a function of source distance for a Lambertian sphere, measured from a surface.

Distance b » 2a b = 2a b= a b=O

Half-angle @112 (deg) Very small 19.5 30

Irradiance Ed TtLa 2 b- 2 =Ib-2 nL/9 nLl4 TtL

------------------=----=~~~-------

~

============================== and the expression for irradiance at the detector is 2

E =TtL d

a a + 2ab + b 2 2

(2.68) •

Table 2.9 summarizes irradiance at the detector for a variety of cases. Note that when b » 2a, the inverse square law applies and the irradiance from the sphere is the same as that from the disc, above. If the source-detector distance is measured from the center of the sphere, as shown in Fig. 2.32, the sine of the half angle is always a/b. The irradiance at the detector is therefore (2.69) Equation (2.69) reveals an interesting result: the inverse square law holds for any sphere and at any source-detector distance, as long as the surface is Lambertian and the distance is measured from the center of the sphere. This counterintuitive result simplifies calculation; results are shown in Table 2.10.

Figure 2.32 On-axis Lambertian sphere, irradiance measured from the center.

46

Chapter 2

Table 2.10 Irradiance at the detector as a function of source distance for a Lambertian sphere, measured from the center.

Distance b»2a b=2a b=a b=O

Half-angle @1/2 ( deg) very small 30 90

Irradiance Ed rtLa2b-2=Ib-2 rtLI4 rtL

2.8 Integrating Sphere The integrating sphere, invented by British scientist W. E. Sumpner in 1892, and fully described by German scientist R. Ulbricht a few years later, is a device that provides a spatially uniform source of radiance. It is depicted in Fig. 2.33, with two elements of area inside the sphere labeled dA I and dA 2 , the linear distance between them d, and the sphere radius R. To analyze the sphere's behavior, we begin with the differential form of the equation of radiative transfer, Eq. (2.43):

By inspection, 8 1 = 8 2 = 8 and cos8 = d(2Rr 1• Also assume that dA I = dA 2 = dA. If the interior of the sphere is Lambertian, i.e., coated with material having Lambertian properties, then (2.70)

Figure 2.33 The integrating sphere. (Adapted from Ref. 15 with permission from John Wiley & Sons, Inc.)

pa

47

Propagation of Optical Radiation

This result means that irradiance within the sphere, for any area element dA, is independent of position e within the sphere and is dependent only on sphere radius and radiance L. In other words, irradiance is constant over the sphere. This fact makes the integrating sphere useful as a uniform radiance source. If a source with power is placed into the sphere (through a "port" in the sphere), the radiance of the sphere wall L (assumed to be Lambertian) can be determined as L=Ep

,

(2.71)

1t

where p is a property of the sphere coating material called its reflectance (to be discussed in detail in Chapter 3). Combining Eqs. (2.70) and (2.71) and solving for dE, we obtain (2.72) This is the irradiance on an infinitesimal element of sphere area dA. Integrating over the area to produce sphere irradiance is complex, as it must take into account multiple reflections within the sphere. The bottom line is (2.73)

This result is interesting because as the reflectance approaches unity, the irradiance approaches infinity, as all the input power remains in the sphere. Real sphere coatings are nonideal, however, with non-Lamberti an surfaces and reflectances less than one. Real spheres are fitted with ports and baffles, the purpose of the latter to prevent "first pass" (unreflected) radiation from reaching the detector. A useful equation for the radiance in a practical sphere is 16 L

=

p 1tA.p h [I - p(l- f)] ,

(2.74)

in which f is the ratio of the total port area to that of the sphere. (A sphere may have several ports.) Thus, real spheres are not particularly efficient unless reflectance is high and the total port area is kept small. Table 2.11 details some of the many uses of integrating spheres.

48

Chapter 2

Table 2.11 Some integrating sphere applications.

Uniform light sources

Uniform detection systems

Measurement of transmission

Measurement of reflectance

Depolarization

Cosine receiver

Light source mixing

Color mixing

2.9 Radiometric Calculation Examples 2.9.1 Intensities of a distant star and the sun

Figure 2.34 depicts a simplified configuration in which a distant star is viewed by a telescope. Assuming that there are no atmospheric or optical system transmission losses (i.e., that the mirror reflects 100% of the incident radiation), that all power collected by the mirror is relayed to the detector, and that 10-{i W are incident on the detector, we can determine the irradiance at the detector. We can then use detector irradiance to calculate the star's intensity. As seen in Fig. 2.34, the system is j12 with a focal length of 1 m. The mirror diameter D isfl(f/#), or 0.5 m. The area of the mirror (assumed circular) is 7tIY14. The irradiance on this (perfectly reflecting) mirror is

The inverse square law applies due to the source distance, and as the source is on axis, no cosine term is required. Inverting Eq. (2.39) to calculate intensity, we have

A

DET

"Q............. _.......... __ .......... _..... _............../ \ / .................................. ~

fl2, f= 1

Figure 2.34 Hypothetical distant star and system used to measure irradiance.

Propagation of Optical Radiation

49

Now, the intensity of our sun can be approximated by a spherical blackbody source at 5750 K. Its radiance is given by the following expression (which will be discussed in detail in Chapter 4): (2.75) and its value is 2 x 107 W/m2sr. Consider the geometry in Fig. 2.35, where Ap is the projected area of the sun. The sun's diameter is 1.4 x 109 m, so according to Table 2.2, its projected area is (1tdsun 2 )/4, or 1.54 x 10 18 m2 • At earth-sun distance d = 1.5 x 10 II m, the solid angle subtended by the sun at the detector is nsd = 6.8 5 x 10- sr. Noting also that the sun subtends approximately 32 minutes of arc (arcmin), n sd may also be calculated as 1tsin2(16 arcmin), which produces the same result. The irradiance at the detector, area A in the figure (assumed to be placed at the top of the atmosphere, therefore no atmospheric transmission loss), 1S

(2.76) Note that the diameter-to-distance ratio is substantially less than 0.1, and the mverse square law may be applied. Calculating intensity as in the example above: (2.77)

I·

d

Figure 2.35 Source-detector geometry for solar irradiance calculation .

50

Chapter 2

Irradiance may also be obtained in another way. Consider that the power delivered to the detector with area A may be represented by (2.78) where O

ds

is the solid angle subtended by the detector, and that E d

A

LA =1368 W /m 2 . d

(2.79)

=_d = - _ P 2

Applying the inverse square law as in Eq. (2.77), we obtain I sun = 3.08 X 1025 W Isr. Note that though the numbers are not identical, they are very close. The value of 1368 W/m 2 is referred to as the solar constant, and is specifically defined as the irradiance falling upon a I m2 unit surface (hypothetical surface) at the mean earth- sun distance. The solar constant has wide application in fields including remote sensing. It is often given the symbol Eo. Note that the total intensity of the sun has to do with the power radiated into 41t sr, the solid angle of a sphere as referenced in Table 2.3. Table 2.12 provides relevant calculations related to solar power and intensity.

2.9.2 Lunar constant 2.9.2.1 Calculation

This concept is analogous to that of the solar constant, whose 1368 W /m 2 are incident upon the moon as well. If the moon is assumed to be Lambertian, with a reflectance of 0.2, its radiance is

Lmi)()n

2 = EaP 1t =87 W /m sr.

(2 .80)

Table 2.12 Solar quantities and their values.

Quantity

Solar area Total solar radiant exitance Total solar power Total solar power (alternative) Intensity Intensity (alternative)

Formula Asun = 4Ap M=1tL = MA = 41tI 1= /41t 1 = Eaef

Value 6.16 x 10 18 m2 6.28 x 107 W/m2 3.87 x 1026 W 3.87 x 1026 W 3.08 x 1025 W /sr 3.08 x 1025 W /sr

51

Propagation of Optical Radiation

At the earth' s surface, the angular subtenses of the moon and the sun are the same, approximately 32 arcmin. This means that QME, the solid angle subtended by the moon at the earth, is equal to Qsd , above, with a value of 6.8 x 10- 5 sr. Neglecting the relatively minimal distance between the top of earth's atmosphere and its surface, the irradiance produced by the moon at the top of earth's atmosphere is

Emoon .TOA

=

E

=

L moonQ ME

3

= 5.9 x 10- W /m

2.

(2 .81)

pE

Note that in the above equation, A pE is the projected area of the earth, analogous to the projected area of the sun discussed earlier. 2.9.2.2 Moon-sun comparisons

Comparing irradiances from the sun and moon, we have

E sun

=(~J[

Emoon

E o, moon

'!;lm 22) =2.3 x l05. W/m

1368 5.9xl0

Also, comparing radiances we find

The numbers are the same because the solid angles subtended are the same. Assuming an atmospheric transmission of 0.75, the solar irradiance at the earth's surface is this factor multiplied by the solar constant E eort"

=

'tE o

= 1026 W /m

2 ,

and assuming an earth reflectance of 0.2 along with the Lambertian approximation, the earth 's radiance is L

eorth

= E earth P = 65 W/m 2 sr 7t .

Applying an atmospheric transmittance of 0.75 to the moon' s radiance at the top of the atmosphere, we obtain the moon's apparent radiance; that is, its radiance when viewed from the ground: L'moon

= 'tLmoon = 65 W /m 2 sr

52

Chapter 2

This interesting result means that the radiance of an "average" sunlit scenet is the same as the apparent radiance of the moon. It also means that in photography, the same exposures can be used to photograph both. Exposure parameters should be set during the daytime and applied to night photography. If the moon is photographed through a telescope, exposures should be increased to compensate for transmission losses within the instrument. In addition, given the factor of 2.3 x 105 difference between solar and lunar irradiances, photographing a moonlit scene requires significantly longer exposures than are needed for daylight illumination. A point about assumptions should be made, specifically, that the moon is not a strict Lambertian surface. It is somewhat retrorefiective, as though covered with Scotchlite.™ Simple measurements made by Palmer with a silicon detector indicate that the apparent intensity of the full moon is more than ten times that of the quarter moon. When viewed with a telescope or binoculars, the edge appears a bit brighter than the rest. The lunar surface is dusted with small glassy spheroids, ejecta from meteorite collisions. Its reflectance is approximately 0.08, somewhat less at shorter wavelengths and somewhat more at longer wavelengths.

2.9.3 "Solar furnace" This example concerns a "solar furnace" operated in space, delivering power to a collector just outside the earth's atmosphere, but the equations are valid for any source located at a large distance from a collector. What is the irradiance delivered to the target E t ? Consider Fig. 2.36, in which the sun is represented by the vertical bar at the left-hand side with radiance [Eq. (2.75)] of 2 x 107 W/m 2sr. The power from the sun to the collector is se =LsAcn se, using the area of the collector and the solid angle the sun subtends at the collector. The irradiance at the collector is then E

c

= As->c = Lsn

SC'

C

where n se is the solid angle the sun subtends at the collector, 6.8 x 10-5 sr. Choosing a target diameter (or linear dimension) and system focallengthfso that nsc = n/e, we have

t Eastman Kodak has shown through extensive research that the reflectance of an average scene is

18%; all exposure meters are calibrated using this assumption (1. M. Palmer, 2005).

F

Propagation of Optical Radiation

53

Ac

0 tc

........... ... ......... I ' " ...~ -- ~~-(- 2 10 W -2 -1 ••• 6............ " ··i········~.~.

LS

-

X

7

m sr

.'

.......

osc--

I

tc

;,; ;

;

~

t

f -.... ~I Figure 2.36 The "solar furnace."

Assuming no transmission loss between collector and target, the irradiance at the target E t is expressed as

Therefore, the target irradiance is the product of the source radiance and solid angle the collector subtends at the target. That solid angle may also be characterized [Eq. (2.15)] as

=

Q ct

n

4(/ / #)2 '

so that

E t

= nLs 4(/ / #)2 '

(2.82)

which provides a way of characterizing target irradiance in terms of both source radiance and thejl# system parameter, for a configuration such as this one.

2.9.4 Image irradiance for finite conjugates The definition of jl# presented earlier was for an object at infinity; however, many systems operate at finite conjugates. Figure 2.37 depicts such a system, in which neither image nor object is at infinity. In such cases, a "workingjl#," often symbolized as jl#', is used. 17

54

Chapter 2

. .:. . . -. . - . - . - . -~. .:-:=.::-=:::.=+-=f:=-=::::.=-~.::.:-

. - . - . - ."- . . . ,.:"!

L

Figure 2.37 Finite conjugates.

A workingjl# is defined as

(2.83)

where magnification m is the ratio of image height to object height, and has values between 0 and infinity. The term mp is pupil magnification, the ratio ofthe diameter of the exit pupil to the diameter of the entrance pupil, and has values between 0.5 and 2. For a single lens or mirror, it is always 1. Substitutingjl#' for jl# in Eq. (2.82) we obtain (2.84)

If mp = 1, as it frequently does, Eq. (2.84) becomes the camera equation (2.85)

Table 2.13 shows two important cases. Table 2.13 Target irradiance using the camera equation.

Case I - Object at infinity m=O E=

TtL

I 4(/ /#)2

Case II - Equal conjugates m= 1 E= I

TtL

=

TtL

4(/ /#/ (1 + 1)2 16(/ /#/

.... 55

Propagation of Optical Radiation

The expressions for irradiance at the target show us that image irradiance decreases as the in-focus object is moved closer to the camera. In order to maintain focus, the detector must be moved backward, which decreases the solid angle of the lens as seen from the detector n et by a factor of four. 2.9.5 Irradiance of the overcast sky

A reasonable value for the radiance of the overcast sky is 50 W /m 2 sr, somewhat less than the 65 WIm 2sr calculated above for a typical sunlit scene. Assuming that the sky radiance is constant, with no brightening at the horizon, the irradiance from the sky at the earth' s surface is

E ear' h

=

21[

sky4 earlh

= L Sky n

sky-ear,h

A earlh

1[/ 2

J Jsin acos ada

= L sky d 0

2

E earlh

=rrLsky sin

E earlh

= 157 W /m 2 •

0

(90 deg)

This is a factor of 6 or 6.5 less than the irradiance received from the sun on a clear day (1000 W /m2), which explains why flat-plate solar collectors continue to function well on a cloudy day (provided that the clouds are "conservative" scatterers.) By comparison, on a clear day, the diffuse solar irradiance (excluding the direct beam) can be as high as 50 to 100 W /m 2 due to scattering. 2.9.6 Near extended source

A near extended source such as the one shown in Fig. 2.38 may be found in the laboratory. It provides a nice way to calibrate a radiometer, because: (1) If the image of the extended source overfills the field of view of the detector with area Ad, the distance d is unimportant; (2) If the source is Lambertian, the angle between source and optical axis is unimportant; and (3) If the detector or radiometer with area Ad is not placed exactly at the focal distance f, it doesn't matter.

The power don area Ad is calculated according to

which equates to

56

Chapter 2

8

~--------- d------------~+---

, - -..·1

Figure 2.38 Near extended source.

2.9.7 Projection system

Figure 2.39 depicts two different designs for projection systems. The Abbe projector was invented first, and has significant disadvantages. As can be seen from the diagram, the source is imaged onto the slide, which is then imaged onto the screen. Hot spots can occur at the slide, resulting in smoke. The Koehler system is superior. The source is imaged into the projection lens, a pupil location rather than an image location. The slide is positioned in an area of relatively uniform brightness, allowing for a more uniform image on the screen.

CONDENSER

PROJECTION LENS

.-::':"'6 J'-"-"_o, ........

CR

........ SCREEN

ABBE PROJECTOR

CONDENSER

PROJECTION LENS

,'-:' -t'[oo_"-,, CR -- -SCREEN

KOEHLER PROJECTOR Figure 2.39 Two projection systems.

,..

57

Propagation of Optical Radiation

The equation for illuminance on the screen resembles the camera equation, Eq. (2.85), with the addition of a cos 4 e term to account for the off-axis angle to the screen as seen from the projector:

=

E v

'tartLv cos

4

e

(2.86)

4(//#)2(1+m)2'

where Ev is illuminance. The transmission of the optical system 'to appears also. To maximize irradiance for a given magnification, there are only two possibilities: minimize the j/# or maximize the radiance of the source. Candidate sources with high radiance values include tungsten lamps, tungsten-halogen lamps, carbon arcs, xenon arcs, metal-halide lamps, and high-brightness phosphor screens.

2.10 Generalized Expressions for Image-Plane Irradiance 2.10.1 Extended source To provide a more general expression for the irradiance at the image plane from an extended source, several factors must be added to the expression in Eq. (2 .85). First is a cos n term, accounting for the reduction in irradiance as we look off axis. Its value is typically 4 to account for projections of the source and target areas, but good optical designers can reduce this factor to 3. 18 Next, losses in the optical system due to transmission, reflection, and scattering may be combined into the general term 'to as discussed above. A term to account for vignetting, lv, the reduction in the cross-sectional area of the beam as the off-axis angle is increased, applies as well. Finally, to account for the presence of a central obscuration in a system such as a Cassegrain, the factor (1 - A2) is applied, where A is the ratio of the diameter of the central obscuration to the diameter of the primary mirror. (If there is no central obscuration, this factor can be eliminated.) Considering the above terms and using the most general expression for source radiance, the expression for image-plane irradiance from an extended source becomes

E I

= 't o1tfv(1- A2)L(e, 2 (transmission low). Optically thin materials approach transparency and have low emittance; for optically thick materials, the normal emittance is (1 - reflectance) and depends on the index of refraction as determined by the Fresnel equation at normal incidence as illustrated in Fig. 4.14. Emittance at other angles also comes from the Fresnel equations and is polarized. 4.3.3 Gases

Gases are optically thin over wide wavelength ranges and may be transparent over long paths. Therefore, their emittance is essentially zero at these wavelengths. However, there are specific spectral regions where absorption, and therefore emission, occur. Each species has its own absorption characteristics, correlated with its atomic and molecular structure and energy levels. These characteristics take the form of a series of spectral lines at regular locations in the spectrum. They are occasionally seen as discrete lines, but more often as a series of overlapping lines called bands.

Chapter 4

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TEMPERATURE (K)

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5.6.3.4 Noises in photoemissive detectors

Noise sources in photoemissive detectors include the following: (1) (2) (3) (4)

shot noise from signal photocurrent, shot noise from background photocurrent, shot noise from dark current, and Johnson noise from the load resistor.

A noise expression may be developed by considering the quantities that make up these sources. The signal current from a photoemissive detector is i = T!q = T\ q!:.. . s q he

(5.94)

If signal current flows through load resistor RL , then the signal voltage is (5.95) Applying Eq. (5.95) and the results from Eqs. (5.18) and (5.25), the noise voltage is

(5.96)

The three terms in the inner bracket are as follows :

183

Detectors of Optical Radiation

(1) shot noise due to dark current id • (2) Shot noise due to signal + background current, (3) Johnson noise in load resistor RL . The signal-to-noise ratio is therefore

(A)

11q he

SNR= . [(

2q l d

(5.97)

4kT) ]1/2. + 2q 11 he + RL B A

2

The ultimate limit is achieved when the dark current shot noise and the Johnson noise from the load resistor can be reduced, leaving only the signaldependent shot noise. Under these conditions, the SNR is SNR =

~11A

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.

(5.98)

5.6.3.5 Photoemissive detector types

Photomultiplier tubes. The impact of an electron onto a secondary emlttmg

material releases several secondary electrons. The gain is defined as the number of secondary electrons per incident electron; its symbol is Values are ~8 in MgO, ~9 in Cs3 Sb, and variable at ~ 50/keV for GaP:Cs. Special structures called electron multipliers arrange a series of these secondary emitting materials such that electrons can be accelerated towards the next electrode (dynode) which has a more positive potential. The total electron multiplier gain is On, where n is the number of dynodes. The gain also depends upon applied voltages. In a photomultiplier tube (PMT), a photosensitive photocathode is combined with an electron multiplier. In operation, a photoelectron is ejected from the photocathode and accelerated towards the first dynode. Several electrons are released and accelerated towards the second dynode, the third, and so on. There are many interesting designs for electron multiplier structures, yielding up to 14 stages of gain. There is some additional noise introduced in the multiplication process. A noise factor (NF) may be calculated as

o.

(5.99)

For large values of 0, Eq. (5.99) becomes

Chapter 5

184

NF=_O_. (0-1)

(5.100)

This noise factor is quite small, typically less than 1.2. The gain of the electron multiplier is essentially noise free. Table 5.18 lists some of the positive and not-so-positive characteristics of photomultiplier tubes. Photomultiplier tubes have found a number of different uses in areas including photon counting, spectroradiometry, and imaging. In the latter, many PMT-based devices have been replaced with solid-state imagers. Microchannel plates. Microchannel plates (MCPs), useful in many UV, visible,

and x-ray applications, are disks built up from millions of microchannels, small glass tubes whose diameters may range from 10 to 40 !lm. They provide an electron multiplication function and form the core of many image intensifier systems, with each channel of the disk (plate) contributing one picture element (pixel) to the resulting image. Typical MCP disk sizes range from 18 to 75 mm in diameter with lengths between 0.5 and 1 mm. Table 5.18 Photomultiplier tube characteristics.

PMT characteristics (good) Large number of photocathode spectral sensitivities

PMT characteristics (not so good) FRAGILE! Most are made of glass

Detectors with S-numbers are "classical" photocathodes

Require stable high-voltage power supply (~1 kV)

Newer NEA photocathodes described by base semiconductor material

Voltage divider string required

Very fast, limited by transit time

Require shielding from electrostatic and magnetic fields

Crossed-field version confines electron paths via a magnetic field

May require light shielding to prevent photons from getting to dynodes

Quantum efficiencies from 0.01 to 0.5

Residual response to cosmic rays, radioactive materials in tube

Can be physically large

Can be physically large Phosphorescence in window Photocathode memory and fatigue Photocathode spatial nonuniformity Photocathode stability (particularly S-l)

F

Detectors of Optical Radiation

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The principle of operation of the microchannel plate is very similar to that of the photomultiplier tube, with the difference that the microchannel replaces a series of dynodes as the vehicle for amplification. The microchannel's inner surface is coated with a high-resistivity material having good secondary emission characteristics. In operation, a primary electron entering from a photocathode strikes the wall and causes secondary emission; this process continues until a high number of electrons have been accelerated toward the positive electrode at the other end of the tube. 7 Fig. 5.34 shows the dynode arrangement for several photomultipliers. There are a number of photocathode spectral sensitivities from which to choose; a representative sample is shown in Fig. 5.35.

5.6.4 Photovoltaic detectors 5.6.4.1 Basic principles

The photovoltaic detector is a popular detector whose operation relies upon an internal potential barrier with an electric field applied. A p-n junction in a semiconductor material is typically used to provide this condition. The potential barrier is formed by doping adjacent regions such that one is an n-type (donor) region and the other a p-type (acceptor).

186

ChapterS

REFLECTION MODE PHOTOCATHODE 100 80 60

s
Ac (i.e. photons at longer wavelength than the cutoff wavelength) (2) Unutilized electron-hole pairs created beyond diffusion length (depletion region can be widened via reverse biasing the photo diode ) (3) Surface recombination of carriers (can be reduced with a dielectric coating) (4) Optical losses due to transmission and reflection (can reduce with antireflection coating) (5) Heating of the device due to the fact that most photons have more energy than needed to create an electron-hole pair (6) All efficiencies are less than one.

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The corresponding quantities for an extended source are (6.7)

If the source is extended, the distance from the instrument to the source irrelevant; it is the focal length that counts.

IS

223

Radiometric Instrumentation

Aperture stop Field lens

Figure 6.7 Improved radiometer.

6.3.6 Improved radiometer The improvement includes the addition of a field lens, as shown in Fig. 6.7. The field lens images the aperture A onto the detector D, thereby making the flux on the detector uniform. Although it is not yet obvious, this also helps control stray light. The radiometry of this radiometer is not much different, but it must be separated, as before, into the point-source case and the extended-source case. For an extended source, where the detector is overfilled with the object, the expression for the flux on the detector is (6.8) where 'to is the transmission of the optics, Ls is the radiance of the source, Ad is the detector area, and nftdis the solid angle the field stop subtends at the detector. (6.9)

where d is the distance from the detector to the field stop. Thus, one can also write the power on the detector as (6.10) If the image of the objective underfills the detector, then (6.11 )

224

Chapter 6

If the image of the objective exactly fills the detector, either of Eqs. (6.10) or (6.11) is accurate.

6.3.7 Other methods for defining the field of view The two other principal methods for defining the field of view are the use of diffusers and of integrating spheres. One may think of an integrating sphere as a special diffuser. A simple diffuser is just a good scattering material. In the choice of such a material, the grit and the spectral characteristics need to be considered. Usually a good absorbing material, a diffuse black, is preferred. And remember, just because it looks black does not mean it is black in the infrared or ultraviolet. The black can be spread on the detector by an appropriate method, or it could be used to reflect to the detector, in which case, it should be a white! There is no substitute for measuring its spectral and angular responses. Such a diffuser is usually a good depolarizer as well as a spatial averager. The integrating sphere is coated with a highly reflecting diffuse material. It has an entrance aperture (hole) and an exit aperture. The light enters the first and exits the second to the detector. The multiple reflections inside the sphere are what make the exiting light both uniform and unpolarized. Again, the spectral characteristics need to be considered. There are several good references on the calculations of the performance of integrating spheres. Try entering the search criteria "integrating spheres" on the Internet. There are several good references that give the throughput equations and offer a variety of types and sizes for sale. Spheres provide an excellent cosine response, which is nice for hemispherical measurements, but they are notoriously inefficient.

6.3.8 Viewing methods It is usually very helpful to be able to see exactly what is being measured. Thus, people have developed a variety of viewing schemes to accompany the measurement instrumentation. One simple scheme is to put a telescope on top or to the side of the radiometer. This scheme is simple, straightforward, and removable, but it suffers from parallax: the two fields of view will not coincide at all distances. Coaxial methods are preferable from this standpoint. One way to do this with obscured systems is to use the folding mirror, as shown in Fig. 6.8. Another popular way is to use a reflecting chopper. Then the fields are coaxial and alternating in time. The persistence of the eye takes care of the interruptions. Such a scheme is shown in Fig. 6.9. The mirror of Fig. 6.9 could be a large chopper in the incoming beam, and then the eye and the reference can be in either position. Other schemes may incorporate a fiber-optic pickoff someplace in the system, but these do not usually work out very well, mostly because they are not coaxial.

.....

225

Radiometric Instrumentation

.,.

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226

Chapter 6

6.3.9 Reference sources

Although all radiometers must be calibrated before and after use, it is also useful to have an internal reference source. In radiometers with choppers, covered in the next section, the measurement signal is the difference between the two detector outputs: when the detector views the object and when the detector views the reference source. The ideal situation is when there is no difference between the signals. So the reference source should be designed to give approximately the same flux on the detector as the object to be measured. This is a "null" reading. The fundamental requirement is that the reference must be known. Sometimes it is a blackened chopper, but this can be difficult since the spinning chopper may vary in temperature and may have temperature gradients. Of course the chopper temperature and emissivity must be known at the time of the measurement. A good alternative is for the chopper to be highly reflective. Then it reflects an internal reference to the detector. The internal reference can then be a wellcontrolled source with high emissivity. Its temperature can be measured continuously, and, since it is stationary, it is likely to be more stable. Most of these internal references are blackbody simulators. They are usually conical in shape with a relatively small aperture. Extensive research has been done on the shapes and sizes to get the highest possible emissivity. The critical thing is that the interior length should be much larger than the aperture. However, the ultimate is not necessary for these internal references since they are secondary calibration sources. Other shapes have included waffles, cylinders, spheres, and the like. Basic design techniques have been published by Gouffe' and DeVos2 and are also described in The Infrared Handbook. 3 Many commercial units can be found by surfing or using the Photonics Spectra catalogs (on disk or in hard copy). These are not usually designed to fit into equipment but to be laboratory standards. You can surf for "blackbody simulators," and find many commercial devices. 6.3.10 Choppers

Most good radiometers use choppers, or radiation modulators. Of course there are advantages and disadvantages. The ac signal from a chopped radiometer provides a discriminant against a static background. It allows the use of drift-free ac amplifiers. It avoids the low-frequency part of the lifnoise region. It provides the ability to use a synchronous detector, and it provides an a-b type of comparison measurement. On the other hand, choppers reduce the available flux by a little over 50%; they can be noisy both electrically and acoustically, they can have reliability problems, and there can be phase noise if things are not exactly right. Most choppers are just spinning blades, driven by an electric motor. In this form they do not use much energy. There are many types available on the market. Some can be resonant devices that oscillate in and out of the beam, perhaps tuning forks. These, too, are readily available from companies listed on the Internet. Browse for "optical choppers"; both types are advertised.

... Radiometric Instrumentation

227

6.3.11 Stray light Stray light on the detector is a false signal. Therefore, stray light should be eliminated or reduced to a level that makes it insignificant. There are basically two types of stray-light problems: those that come from stray light that is in the field of view of the radiometer (background light) and those that come from outside the field of view. The latter can be the sun or another intense source that is near the edge of the field of view. It can be greatly reduced by careful design of stops and baffles. The proper use of baffles can attenuate these out-of-field sources by many orders of magnitude. The stray light from out-of-field sources can be reduced by the following procedures. First, place the field stop or its image as close to the front of the system as possible. Place the aperture stop or its image as close to the detector as possible. Avoid obscured systems like Newtonians and Cassegrains (and this may make a viewer more difficult). Make any baffles angled and black. There is still an argument as to whether they should be specular or diffuse. The diffuse baffles have better attenuation; the specular ones control the direction of the light better. Obscured systems will provide their own scatter, but they can be used. Figure 6.10 shows how to design a simple baffle. The baffle is designed to protect against a source, like the sun, that is at an angle e off axis. The first step is to make the baffle long enough to achieve this. Then draw line ab at that angle from the tip of the baffle to past the mirror tip. Then draw line cd parallel to ab and to the end of the baffle tube. Then draw ef to intersect that line; that is where the first vane is inserted, as shown. Then draw j to intersect at f Place the next vane where f intersects the line of baffle tips. This line is parallel to the tube at the edge of the mirror. Now repeat the process for the other vanes.

Figure 6.10 Baffle design.

228

Chapter 6

The final step is to analyze the system with one of several programs that can calculate the amount of radiation on the detector from out-of-field sources as a function of their angle out of field. And there is no substitute for a measurement if the application warrants it. A number of stray-light-analysis systems are listed on the Internet. Search for "stray light optical techniques." The reduction of stray light is an iterative technique that is shortened by experience. The analysis programs tell you how you are doing. The Lyot stop is a special kind of stop useful in controlling scatter. It was used by Lyot to measure the solar corona. He imaged an interior disk inside the system to the front. The image of the disk matched the solar disk, allowing him to see the light outside the main disk of the sun-the corona and flares. The image of the disk became the entrance pupil, an image of the aperture stop. This is one application of using an entrance pupil to control stray light.

6.3.12 Summing up The design and use of radiometric instruments is simple in principle, but difficult in practice. It is a science of precision and care. One of the maxims is to think of everything. One way to do this is to write the radiometric equation for the radiometer:

v = R(A, t, x,y,z, a, ~, moon, RH, T, ... ).

(6.12)

This means that the output voltage (or other electrical signal) is a function of wavelength, time, spatial coordinates, angular coordinates, the phase of the moon, the relative humidity, the temperature (of the radiometer and the source and the background), and everything else you can think of. Then test the radiometer against all of these variables. The devil is in the details-and so are good results!

6.4 Spectral Instruments 6.4.1 Introduction Spectral instruments include both those that make relative measurements and those that make absolute measurements. The first type measures how much radiation there is in each part of the spectrum, but only on a relative basis. They are useful for chemical analysis, for instance. The absolute instruments, generally called spectroradiometers, measure the spectrum, but also give information about how many watts (or equivalent radiometric quantity) are at each wavelength. They are calibrated and more difficult to use. This section is about various types of spectroradiometers. Another way to view them is that they are radiometers with several, or many, narrow spectral bands. The different types vary from each other according to the way they obtain the spectrum. That can be with prisms,

F'"

229

Radiometric Instrumentation

gratings, filters, by scattering, interference, and, in general, any phenomenon that is spectrally dependent. A typical spectroradiometer (or spectrometer) is shown in Fig. 6.1l. Foreoptics on the left bring the light to the entrance slit of a monochromator (the device that generates the spectrum). The output of the monochromator goes to a detector, and then to electronics, and then to some kind of display or recording. The monochromator singles out a specific, usually narrow, wavelength band at a time, and usually then sequences through several bands. With multiple detectors all wavelengths can be sensed at the same time. The monochromator consists of an entrance slit, collimating optics, a disperser (usually a prism or grating that spreads the light of different wavelengths), focusing optics, and an exit slit. Although the optics shown here are lenses, most monochromators use mirrors to avoid problems with chromatic aberration. Spectroradiometers are characterized by how much light they collect, and therefore their sensitivity, and by how narrow their spectral bands are. The amount of light they collect is specified by their throughput (also called optical extent, etendue, and An product.) The narrowness of the spectral band is usually specified by the resolving power, that is, the central wavelength divided by the spectral width of the band. This is the same as the Q of an electronic filter. In some instances the descriptor is the resolution, which is just the width of the spectral band, usually specified as the full width at half maximum (FWHM). The free spectral range is a description of the spectral region over which there is no interference of other spectra, perhaps by overlapping of orders. Other descriptors include the multiplex advantage and the throughput advantage. These latter two are also called the Felgett and the Jacquinot advantages. The throughput advantage relates to how much light you can get through the system. The multiplex advantage refers to the measurement of many wavelengths of light at one time, rather than sequentially. If there is a multiplex advantage, the bandwidth can be narrower and the sensitivity therefore greater.

Foreoptics

Entrance slit

Disperser

Exit slit

~~w' Source Collimating lens

Condensing lens

Figure 6.11 Typical spectrometer.

Focusing lens

230

Chapter 6

To get the optimum response from a spectrometer, one should make the width of the entrance and exit slits the same. One should also just fill the entrance slit with the foreoptics. Underfilling robs you of input; overfilling gives no advantage and even increases stray light.

6.4.2 Prisms and gratings Spectroradiometers based on prisms and gratings are probably the most common. Each has its advantages and disadvantages. The resolution of a grating spectrometer is constant across the spectrum and usually more than that of a prism system. The throughput of the grating is also a little larger than that of the prism. The resolving power of a grating is generally higher than that of a prism. The grating equation is (6.13) where m is the order number, an integer, A is the wavelength of the light, and ai and ad are the angles of incidence and diffraction, respectively. For a given wavelength and angle of incidence there are many maxima at various diffraction angles that correspond to different values for m and ad. The oth order includes all wavelengths and is just regular reflection (or transmission). For a given incidence angle, the first order, m = 1 for a wavelength A, has the same diffraction angle as the second order and halfthe wavelength. This is overlapping of orders and limits the free spectral range. The resolving power Q is given by

A

(6.14)

Q= dA =mN ,

where m is the order number and N is the number of grating lines illuminated. The throughput is the slit area times the projected area of the used portion of the grating divided by the focal length of the optics:

a

Astir A graring cos T=An=------"----::-"--j 2

(6.15)

The free spectral range of the prism is unlimited, while that of the grating is limited by multiple orders, the different maxima corresponding to different values of m. Gratings can be transmissive, reflective, and even concave to incorporate some of the focusing properties of the system. The resolving power of a grating is approximately constant across the spectrum, whereas the prism's Q varies. The resolving power of a prism is given by

pas

Radiometric Instrumentation

231

(6.16) where b is the length of the base of the prism (actually the length of the beam near the base) and dn/d)... is the change in refractive index of the prism with respect to wavelength. The throughput is dictated by the slit area and the focal ratio of the focusing optics: (6.17)

Although it is a generalization, gratings are generally better: they have greater throughput and resolving power than prisms. 6.4.3 Monochromator configurations

Since the monochromator is the heart of generating spectra and measuring them, several variations are described here. They can be divided generally into prism devices and grating devices, according to the type of disperser that is used. Prism systems consist of several different mounts: Littrow, Wadswoth, and Amici. There are more, but these are the main ones. The Littrow, as shown in Fig. 6.12, has a mirror at an angle behind the prism. Thus the light enters the prism, exits to the mirror, and returns through the prism, thereby creating a double pass for twice the dispersion. The mirror can be adjusted for the degree of separation of the input and exit beams. The Wadsworth, as shown in Fig. 6.13, is a single pass. The light is refracted at minimum angle and exits the mirror parallel to the input beam. The mirror is an extension of the prism base. The Amici, which is shown in Fig. 6.14, allows the light with the central (or any chosen wavelength) ray undeviated and undisplaced (with careful design). It requires the use of prisms of at least two different materials (with different dispersions). More than three prisms can be used at the designer's discretion.

Figure 6.12 Littrow mount.

232

Chapter 6

Figure 6.13 Wadsworth prism mount.

Figure 6.14 The Amici prism.

Mirror systems have more variability, but generally there is an entrance slit, an exit slit, one or two mirrors, and a grating. The Czerny-Turner, Fig. 6.15, was developed by Marianus Czerny and Francis Turner, a onetime student of Czerny's and later a researcher at Eastman Kodak and professor at the College of Optical Sciences, The University of Arizona. There are two versions; one might be called straight and the other crossed, as in Fig. 6.16. The crossed version uses the two paraboloids closer to on axis and should therefore have smaller aberrations. Of course, if aberrations are not a problem, the paraboloids can be replaced by spheres. In that case, and depending upon the speed of the optics, spherical aberration rather than coma will be dominant. It is shown with a reflective grating, but it is not hard to imagine how this setup can be used with a transmissive grating. The essence of the design is that both the collimating optic and the focusing optic are off-axis paraboloids. However, they are off axis in opposite directions so that the comatic aberrations offset each other. The Fastie Ebert mount is shown in Fig. 6.17. It has the advantage of using only one mirror, although the mirror has to be larger than either of the mirrors of the Czerny-Turner system. Again, the mirror can be either a sphere or a paraboloid, and again the comatic aberrations tend to offset each other. Obviously, this cannot be used with a transmissive grating.

z

ps

233

Radiometric Instrumentation

Entrance slit

Grating

Exit slit Figure 6.15 Czerny-Turner (laid out) configuration.

Grating Figure 6.16 Alternate (cross) Czerny-Turner configuration . Mirror

Entrance slit

Grating

Exit slit

Figure 6.17 Fastie-Ebert configuration.

Chapter 6

234

Concave grating

-

Entrance slit

Exit slit

-

Figure 6.18 Seya-Namioka configuration.

The Seya-Namioka is shown in Fig. 6.18. It has the advantage of having no mirrors at all. Thus, it is very compact and usually cheap. The focusing properties are all in the convex reflective grating. However, this grating can be difficult to make so that is has enough concavity and also enough lines. These configurations are also discussed in the first of the online references below.

6.4.4 Spectrometers These devices come in several different varieties. They can be single or double pass through the dispersers, like the Wadsworth (single) or the Littrow (double). They can be single or double beam, as shown below. The double beams were developed to get an automatic referencing system. In some older single-beam systems, one had to wrap a cord around the screw that rotated the prism in just the right way. This was to "program" the slit to account for the variation in intensity from the source. One also had to run a calibration run, and then a sample run, and manually do the ratioing. The double-beam system eliminates all this. There are many ingenious variations on the several simple systems shown here. One beam passes through the sample, whatever it may be. The other is the reference beam and is just in air in the spectrometer. The output of the sample beam is divided by the output of the reference beam to give the transmission of the sample. There are several ways to do this, but the best way is to use the same detector for both beams, perhaps with an appropriate chopper. This avoids the obvious problem of detector matching, initially and repeatedly. The generic basic two-beam instrument is shown in Fig. 6.19. The output coming from the exit slit of the monochromator is collimated and divided into two beams by a divider. It can be a (semitransparent) beamsplitter or a bladed chopper. The one beam goes straight through; the other is diverted by two mirrors and then combined with the

235

Radiometric Instrumentation

Mirror

Detector Monochromator exit slit

Beam divider

Beam combiner

Figure 6.19 Generic double-beam spectrometer.

first beam by a beam combiner just like the beam divider-a synchronized chopper or a plane-parallel plate. If the divider and combiner are choppers, the electronics can just take the ratio of signals. If beamsplitters are used, some technique must be used to "tag" each of the beams. The beams need not be collimated, but could be relayed by lenses for compactness. Other commercial instruments can be found on the internet by entering the search term "spectrometers." Many will show up with diagrams, prices, and advertising! There is one subtlety about double-beam systems that I (Wolfe) encountered quite by accident. I was participating in a military study devoted to the detection of land mines. Someone suggested that soil was transparent around 4 /--lffi. I scoffed, but I went home to prove my point. I took one inch of certified backyard dirt into our double-beam spectrometer. Lo and behold, there was a transmission peak at 4.3 Ilm! This could not be! So I took one inch of aluminum plate and made the same measurement. Same result. I pondered this for a while. Clearly aluminum is not transparent at 4.3 /--lm. The answer was in the reference beam. The atmosphere is very absorbent at 4.3 /--lm due to carbon dioxide. What I was measuring was the absorption in the reference beam that translated to apparent transmission in the overall measurement. I have since seen this phenomenon in other double-beam measurements. If there is extra transmission at 4.3 /-!ill in the scan of an optical material you have ordered, be skeptical. This illustrates at least one principle of radiometry: do not simply believe whatever you measure; make sure it makes sense. By the way, this problem can be obviated by filling the reference tube with nitrogen.

6.4.5 Additive versus subtractive dispersion Additive dispersion is just that: two prisms or two gratings operating in series to increase the dispersion of the light. The Littrow-mounted prism gives additive dispersion. Additive dispersion can be accomplished with multiple dispersers or multiple passes or both. Subtractive dispersion combines the dispersions in

Chapter 6

236

opposite directions, thereby combining the light and eliminating or canceling the spread of the spectrum.

6.4.6 Arrays Monochromators can also be used with detector arrays. The arrangement includes the foreoptics described above, an entrance slit, collimating optics, and the array, which functions as a set of exit slits. If the array is linear, the system can be viewed simply as a monochromator with many exit slits, each one sensing a different wavelength. This system obviously has the multiplex advantage, but does not have the throughput advantage. The detector elements can usually be sized correctly for the proper operation, but apertures can also be used with a concomitant loss of signal.

6.4.7 Multiple slit systems Several different schemes have been developed to give a multiplex advantage by using several entrance slits and several exit slits at the same time. Perhaps the first of these was invented by Marcel Golay;4 perhaps the most popular, the Hadamard transform, was developed by Harwit. 5 Although these are interesting designs that incorporate the multiplex advantage, they have not supplanted the other prism, grating, and interferometric spectrometers.

6.4.8 Filters Filters come in a variety of types. They can be based on absorption, interference, and even scattering. Absorption and interference filters are the main candidates in spectroradiometry. Almost any kind of bandpass can be generated by a proper thin-film design. They can be narrowband, broadband, angle tolerant, multiple bandpass, etc. Then they can be put in a filter wheel to obtain a spectrum of a sort, although not a continuous spectrum. They have good throughput but do not have as good resolving power as prisms and gratings. They can come in segmented wheels or circular or linear variable filters. These latter devices are interference filters with layer thickness variation around the circumference or along the length. They therefore have a spectral band that varies with either angle or position. Their characteristics vary by design. The circumference of a circular variable filter (CVF) is given by

c = /).'A D d'A

0'

(6.18)

where /).'A is the spectral range, dA is the resolution and Do is the diameter of the aperture stop, where it is placed. The diameter of the CVF is this value divided by n. A representative filter has the following characteristics: the spectral range is 2.5 to 14.1 ~m, Q is 67, FWHM is 1.5%, and resolution varies from 0.03 ~m at

p

237

Radiometric Instrumentation

2.5 /lm to 0.2 /lm at 14.1 /lm. The CVF is divided into three segments that cover from 2.5 to 4.3, 4.3 to 7.7, and 7.7 to 14.1 /lm. It can be operated at the focal plane, as the manufacturer recommends, but it should be operated at a field stop rather than the focal plane. Since it would then be in converging light, the resolution will be affected. Of course other varieties are available, semicircles for instance. Typically the spectral range is an octave, the resolving power is about 50, and the diameters are about 4 cm. Specific devices can be found on the Internet by searching for "optical filters" or "variable optical filters." If you leave out "optical," all sorts of electrical filters will appear in the search results. Optical filters are optically and mechanically simple, cheap, rugged, and easy to automate, but they have poor stray-light suppression and limited resolving power. 6.4.9 Interferometers The main interferometers used for spectral analysis are the Michelson and the Fabry-Perot, or as some insist, the Perot-Fabry. The former is a two-beam instrument, the latter a multiple-beam device. The mUltiple-beam system has greater resolving power, but smaller free spectral range. 6.4.10 Fourier transform infrared The Fourier transform infrared spectrometer is a Michelson, or actually a Twyman-Green, as shown in Fig. 6.20. (The Twyman Green is a Michelson with collimated light). The light from the source is collimated by the first lens. The beam is then divided into two by the beamsplitter. One beam goes up, the other continues to the right. The mirrors then reverse the direction of both beams and they are combined by the same beamsplitter, which is now acting as a combiner. The beams interfere, and the intensity of the interfering radiation is sensed by the Mirror

Beam divider/combiner

Source

Mirror

. . . . . . . . . . . . Lens

Figure 6.20 Twyman-Green interferometer.

Chapter 6

238

Detector, whose operation can be understood by considering first a single monochromatic input, then, successively, added inputs of different wavelengths. When the interferometer is set so that the two arms have exactly the same optical length, light from the two beams will interfere constructively. Then, as one mirror is moved, the beams gradually go out of phase until they reach destructive interference, and as the mirror is further moved, they gradually reach constructive interference again. So there will be a sinusoidal output based on the motion of the scanning mirror. If a beam of somewhat different wavelength is inserted it will do the same, but for somewhat different positions of the mirror. A third beam, a fourth, and so on will add to the complexity, but each will contribute a sinusoidal component to the output signal. At any position of the scanning mirror, the output is the sum of all waves, each at a different point in phase. The sum total of the scan is called an interferogram-the interference pattern of whatever the input beam was. The Fourier transform of the interferogram is the spectrum of the input beam. Because this device is used mostly in the infrared, it is called the Fourier transform infrared spectrometer. It has good throughput and the so-called multiplex advantage; i.e., it senses all wavelengths of light at the same time (as opposed to prism and grating instruments that see one small sample of the spectrum at a time). The resolving power is given by Q _ 'A _ a _ a _ 5000 - d'A - da - 20 - ~ .

(6.19)

The expression a/da is the wavenumber equivalent. The final expression incorporates the fact that the wavelength is usually given in 11m while the wavenumber is in cm- l . These systems are complex, susceptible to vibration, have a limited spectral range, and require considerable computing power to perform the transform operation, but they have the multiplex advantage and good throughput. They are limited to the infrared, but they are very useful for what they do.

6.4.11 Fabry-Perot This interferometer is generally used for high-resolution spectroscopy. It is essentially a pair of plane-parallel plates, as shown in Fig. 6.21. The light from the source is collimated by the first lens, and passes through the first plate after a little lost reflection. Light reaches the second plate and is reflected back to the first, which reflects it back to the second, which reflects it back to the first. This interferometer does have high resolution because of the multiple-beam interference. The basic equation for the transmission of the Fabry-Perot is (6.20)

jiiii"

239

Radiometric Instrumentation

and (6.21) where 'to is the transmission of the plates, p is their reflectivity, n is the refractive index of the medium between the plates, d is the plate separation, 8 i is the angle of inclination of the incident beam, which is usually 0, and the i are the phase changes on reflection from the plates. It can be shown that the maximum transmission is when the reflectance is highest (in the limit, 1), which is a rather strange result. The resolving power is given by 0"

A

JP

dO"

dA

I-p

Q=-=-=--m1t

,

(6.22)

the throughput is given by (6.23) and the free spectral range is given by 1 dO"=-. 2d

(6.24)

The Fabry-Perot has a relatively poor throughput because of the requirement for collimation between the plates. But it is better than a prism or grating. Final note: Acousto-optical tunable filter (AOTF) devices for spectroscopy of all

kinds are relatively new. "Classical" spectrometers employing prisms and gratings have seen many improvements. There are pros and cons for both. They are a tool that should be in the radiometrist's kit and are described further in Wolfe and in Chang, below.

Source

Detector

Figure 6.21 Fabry-Perot Interferometer.

240

Chapter 6

For Further Reading Baumeister, Optical Coating Technology, SPIE Press, Bellingham, Washington (2004). Good section on filters. 1. C. Chang, "Acousto-optic devices and applications," Chapter 12 in Handbook of Optics, Vol. II, M. Bass, Ed., McGraw-Hill, New York (1999). D. S. Goodman, "Basic optical instruments," Chapter 4 in Geometrical and Instrumental Optics, D. Malacara, Ed., Academic Press, New York (1988). F. Grum and R. Becherer, Radiometry, Sec. 7.3, Vol. 1 in Optical Radiation Measurements series, F. Grum, Ed., Academic Press, New York (1979). G. R. Harrison, R. C. Lord, and J. R. Loofbourow, Practical Spectroscopy, Prentice-Hall, New York (1948). R. Kingslake, "Dispersing prisms," Chapter 1 in Applied Optics and Optical Engineering, Vol. 5, R. Kingslake, Ed., Academic Press, New York (1969). R. Meltzer, "Spectrographs and monochromators," Chapter 3 in Applied Optics and Optical Engineering, Vol. 5, R. Kingslake, Ed., Academic Press, New York (1969). W. J. Potts, and A. L. Smith, "Optimizing the operating parameters of infrared spectrometers," Appl. Opt. 6, 257 (1967). D. Richardson, "Diffraction gratings," Chapter 2 in Applied Optics and Optical Engineering, Vol. 5, R. Kingslake, Ed., Academic Press, New York (1969). R. A. Sawyer, Experimental Spectroscopy, Dover, New York (1963). R. Willey, Practical Design and Production of Optical Thin Films, Marcel Dekker, New York (2002). Good reference on filters. W. L. Wolfe, Introduction to Imaging Spectrometers, SPIE Press, Bellingham, Washington (1997). J. Workman and A. W. Springsteen, Applied Spectroscopy, Academic Press, New York (1998). P.

References 1. A. Gouff6, "Corrections d'ouvertures des corp-nois artificels compte tenu des diffusions multiples internes," Revue d' optique 24( 1) (1945). 2. J. C. DeVos, "Evaluation of the quality of a blackbody," Physica 20, p. 669 (1945). 3. A. J. LaRocca, "Artificial sources," Chapter 2 in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, Eds., u .S. Government, Washington, D.C. (1978). 4. M. Golay, "Multi slit spectroscopy," J Opt. Soc. Amer. 39, pp. 437-444 (1949). 5. M. Harwit and N. Sloane, Hadamard Transform Optics, Academic Press, New York (1979).

'S

«

F'

Chapter 7

Radiometric Measurement and Calibration 7.1 Introduction This chapter deals with the numerous measurements for which we use radiometers and spectroradiometers. This is the true meaning of the word radiometry, the measurement of radiant energy. First, we describe the types of measurements made. Next is a discussion on errors, their sources, and treatment. The generalized measurement equation and several derived range equations follow. An introduction to the philosophy of radiometric calibration is presented next, and a discussion of calibration configurations completes the chapter.

7.2 Measurement Types Radiometric measurements may be classified into four general types. They are: (1) detector and radiometer characterization, (2) optical radiation source measurement, (3) material properties measurement, and (4) temperature measurement. A fifth measurement type is calibration, which will be discussed later. Table 7.1 subdivides the first four categories into what is not an exhaustive list.

7.3 Errors in Measurements, Effects of Noise, and Signal-toNoise Ratio in Measurements A measurement of any kind is incomplete unless accompanied with an estimate of the uncertainty associated with that measurement. The term error implies a difference or deviation from a "true" value, while the term uncertainty means an estimate characterizing the range of values within which the true value of the measured quantity lies, including all sources of error. Errors come in two primary flavors, random and systematic, as we will see through consideration of Fig. 7.1. Systematic (type B) errors are readings that vary in a predictable, hopefully detectable, way. Systematic errors are repeatable and consistent, with a fixed bias, the difference between the measured value of x (mean of N measurements) and the true value of x. 241

242

Chapter 7

Table 7.1 Types of radiometric measurements.

Detector and radiometer characterization Relative spectral responsivity Absolute spectral responsivity Noise properties Detective properties (NEP, D*, D**) Field of view, out-of-field response Linearity Frequency response Polarization response Wavelength characterization (for spectroradiometers) Passband characterization Measurement of optical radiation sources Active (self-radiating) and passive (reflective) sources Source intensity Source radiance or brightness Source power or total flux Light (photometry) Ultraviolet and infrared sources Source temperature Collimated (laser) sources Measurement of radiometric properties of materials Specular reflectance Diffuse reflectance Transmittance Scattering properties, BRDF, and BTDF Indirect measurements of absorptance and emittance Direct measurement of emittance Color Measurement of temperature Radiation temperature using entire spectrum Brightness temperature using one wavelength Ratio temperature using two wavelengths Color temperature using chromaticity Temperature using multiple wavelengths Distribution and correlated color temperature Examples of a systematic error are an incorrect setting of a calibration potentiometer of a voltmeter, or a slipped or improperly installed temperature dial on a blackbody radiation simulator. Systematic errors are not revealed by repeated measurements. If detected, they may often be corrected or at least taken into account. The term accuracy is often applied to systematic errors, implying small systematic errors. This term should be replaced by inaccuracy, inasmuch as

243

Radiometric Measurement and Calibration

f(x) Systematic error

Random error L-----------~--------~~----~------~------~x

True value of x

Measured value ofx

Figure 7.1 Systematic and random errors.

a voltmeter with only 1% accuracy is far less desirable than one with 99% accuracy. The systematic error in a measurement is closely related to calibration of the apparatus used to conduct the measurement. Random (type A) errors are those that vary in an unpredictable manner when the same quantity is repeatedly measured under identical conditions. They are revealed only by multiple measurements. Precision is a term often associated with random errors; a measurement is considered precise if it is repeatable. We may employ several methodologies to reduce random errors, enhancing the measurement precision. First, we can use a finer scale division, i.e., more bits, to reduce granularity in the measurement. Second, we can reduce some of the inherent noise in the measurement process by filtering, cooling, shielding, etc. The most important tool to reduce random errors is statistical analysis. We take multiple readings and perform analysis to reduce the effects of "noise" and increase our confidence in the measurement. To understand the total uncertainty in a measurement, we must consider both the systematic and the random error components. Figure 7.2 shows a set of "measurements" with various combinations of accuracy and precision. The average (x and y) of the third pattern lies very close to the center of the target. Errors can be specified as absolute, the magnitude of the error in the appropriate engineering units, or as a relative or fractional error, usually in percent.

HIGH PRECISION

HIGH PRECISION

LOW PRECISION

LOW PRECISION

HIGH ACCURACY

LOW ACCURACY

HIGH ACCURACY

LOW ACCURACY

Figure 7.2 Precision and accuracy possibilities.

244

Chapter 7

Table 7.2 Signal-to-noise ratios and corresponding uncertainties.

SNR (single measurement) 1

Uncertainty

10

0.1 (10%)

100

0.01 (1%)

1000

0.001 (0.1 %)

1 (100%)

Errors may also be categorized as multiplicative or additive. A multiplicative error, often referred to as a scale or gain error, is proportional to an instrument reading. An additive error, often referred to as an offset, yields an absolute uncertainty that is independent of reading. The error on a typical instrument specification might read, "±O.5% of reading ±O.2% full scale." These are the multiplicative and additive errors, respectively. Note that the symbol ± is redundant; a 0.5% deviation can clearly go in either direction. The fundamental error limit in a measurement is random noise. If all other error sources were reduced to zero, the remaining noise would be Gaussian, most likely Johnson or shot noise associated with our detector. We certainly desire to have the limit to uncertainty in a system dependent upon noise, as this indicates that the systematic errors are understood and under control. In such a system where noise is the predominant error term, the measurement uncertainty is inversely related to the signal-to-noise ratio (SNR), or . 1 M easurement uncertamty =- - .

SNR

(7.1)

Thus, an SNR of 10 implies a 10% uncertainty (lcr) in a single measurement, i.e. , 1. Table 7.2 shows uncertainties for several values of SNR. We wish to take repeated measurements to enhance the SNR. With multiple measurements, the signal is additive, and the average of the noise tends towards zero. The resultant SNR is proportional to the square root of the number of independent measurements, (SNR DC IN). We then statistically analyze the data and make the following assumptions: (I) The data possess a "normal" (Gaussian) distribution of random errors. (2) The individual measurements are statistically independent (uncorrelated). (3) The quantity being measured is stationary.

We then apply the analytical tools listed in Table 7.3.

245

Radiometric Measurement and Calibration

Table 7.3 Mathematical tools for dataset analysis.

Tool

Formula

Mean

X

=

LXi -i-,N=numberof N

measurements Variance (sample)

L(xi-xY

a 2= -",-._ __ N-l

Standard deviation

a=N

Standard deviation of mean

a ax = ..IN

We further explore the three assumptions listed above. (1)

(2)

(3)

Assurance of a normal distribution is approached by taking a large amount of data (central limit theorem). Verification of the normal distribution can be accomplished using a Chi-square (X 2 ) test. The measurements must be independent (uncorrelated). In reality, it is impossible to achieve complete independence because of the exponential nature of signal changes with time. Spacing the data sampling interval by one time constant improves the SNR by (N/2)1I2. Note that there is no prohibition to faster sampling, but the correlation of such measurements reduces the apparent gain in SNR. The maximum SNR improvement is proportional to the square root of the observation time and is independent of the sampling rate. 1 If there is a low frequency drift (lifnoise) in the data, normal statistical analysis may be compromised and further analysis is mandatory. You may be able to low-pass filter the data to reduce the random noise, then use regression analysis (linear, exponential, power, etc.) to fit a curve to the remaining low-frequency (drift) component. A more sophisticated method is the analysis of Allan variances, developed for drift assessment of atomic clocks. Data sets are analyzed over different time frames. The Allan variance is given by: (7.2)

Compare with the classical variance (7.3)

246

Chapter 7

If (J2 (N) / (J~('to) ~ (1 + 1/

IN),

then white (classical) noise predominates

over drift (lij), and we can increase the sampling window 'to and therefore the number of samples N. Several examples of this situation are shown in Fig. 7.3. The bottom curve is the white-noise case, where the SNR improves as 1/ The other three curves show the effects of varying drift rates. As the drift rate climbs (i.e., progressively higher lifnoise than white noise) the sampling or integration time shortens, resulting in a lower SNR. An important contribution to random error (precision) is the granularity of the instrumentation. If the markings on the analog meter movement are too widely spaced, interpolation between divisions may be difficult. Reading a ruler or a micrometer may be difficult for the same reason. Most electrical measurements are now taken with digital instrumentation, where quantization takes place via an analog-to-digital (AID) converter. Quantization noise was defined in Eq. (5.31) and is

IN .

LSB = SIGNALmax

(7.4)

2n

'

where LSB is the least significant bit and SIGNAL max is the full-scale reading. If this quantization error is significant in a measurement, the use of more bits (larger n) is indicated. If the presence of sharp quantization levels is apparent in your data and you cannot get more bits, you can artificially smooth the data by adding noise. This is called dithering, used to "smear" and mask the appearance of discrete levels. EFFECT OF DRIFT ON DATA AVERAGING

- --- - -

- - - - --: ,

,

-

- - - --:-- - - ,

, 0 . 8H---+---~---r ' --~----r ' ---+--~----~--+---~

~

06 0.4

f-+_\--__~~__ -___

~_--~

_-~_ :-: _ _-__--+-~ _. _-7 -__:_-__ .="" ..

--+-_1 ____ --+-_ ___ __ --_.__ __;_-__ :__--+--r _ ____-:

: : : -~: : ~ : ___ f..--"'T" - - - --~ - - - - - - - - -- --;- - - - - - -- - - - -'- - - --- - - - - - -' - --

-:-

0.2 -- -- -

-- -- ---

-

-

OL---~--~--~--~----L---~--~--~--~--~

o

20

40

60

80

100

NUMBER OF DATA POINTS Figure 7.3 Drift effects on data averaging for a number of cases.

247

Radiometric Measurement and Calibration

Table 7.4 Chauvenet's criteria for data rejection.

Number of readings 4 6

8 10 15 20 25 50 100 200 500 1000

Data point deviation Standard deviation 1.54 1.73 1.86 1.96 2.13 2.24 2.33 2.57 2.81 3.02 3.29 3.48

One is often tempted to reject data that appears to be "out of line." The general recommendation is DON'T. The data has warts, leave them alone. If you really must, there are reasonable guidelines for data rejection. Don't make the mistake of expunging data from your data set, just set the outliers aside. After all, there might be something really interesting there. The most popular tool is the Chauvenet criteria for rejection of outliers. Here we reject a data point if the probability of observation of the suspect point is less than y~ where N is number of observations. Then we can recompute mean and standard deviation. The criteria are shown in Table 7.4 above. Application of Chauvenet's criteria affects standard deviation more than mean. It is interesting to test the dataset with and without data rejection to see if it really matters! Usually it doesn't. A final note: even if tempted, do not apply this technique more than once. If you do, the data will probably be skewed. If one keeps rejecting data, eventually one will have only a single data point remaining. To minimize systematic errors, calibrate frequently with suitable apparatus, as noted above. Table 7.5 lists several sources of systematic error. The evaluation of systematic error is, in practice, a judgment call, based upon an investigator's familiarity with previous data, instrument behavior, manufacturers' specifications, handbook reference data, and calibration. Current recommended practice is to estimate systematic errors at the 10' (68% confidence) level. Systematic errors in optical radiation measurements tend to be large because the quantities involved are a function of everything in the world; the following factors are the most common: (1) Wavelength (broadband or monochromatic) (2) Power or energy level (3) Linearity

Chapter 7

248 Table 7.S Sources of systematic error.

Environmental factors (temperature, humidity, pressure, electromagnetic interference, dust and dirt, power line fluctuation) Dynamic (response time and slew rate) errors Offset error (additive) Gain or scale error (multiplicative) Hysteresis Nonlinearity Faulty procedures Personal bias Calibration frequency Calibration standards (4) Modulation frequency or pulse characteristics (5) Position, direction (6) Polarization, diffraction, coherence, "phase of the moon." There are two secondary error categories: illegitimate errors and model validity. Illegitimate errors include blunders, mistakes, computational errors (roundoff, etc.), and chaotic errors. Model validity has to do with our preconceived ideas about the results of a measurement, which influences the way we conduct the measurement. While these errors may be significant or even overwhelming in a particular situation, none are essential contributors to the limits of radiometry and can be eliminated with proper care. When all of the various error sources have been identified, they must be combined into an overall uncertainty estimate. Errors propagate according to well-known formulae as shown in the following tables. Table 7.6 presents the formulas for single-valued functions. For combinations of functions, the worst-case ultraconservative treatment is to directly add the errors. This method would be correct if all of the errors were correlated. Table 7.7 presents the formulas for functions in which independent variables are combined. Table 7.6 Error expressions for single-valued functions.

Constant

Reciprocal

Power

(q = Ax)

(q = lIx)

(q =

aq

aq

a Iql

_

ax

jqf-g

_

ax

jqf-g

Exponential

xn)

a

=Inll~

q=e

wc

aq jqf=aax

Logarithmic q = In(ax)

aq

jqf

_

1

aln(ax)

ax

Ixl

.... Radiometric Measurement and Calibration

249

Table 7.7 Error expressions for functions of several variables.

Addition and subtraction (q = x + y - z)

Multiplication and division (q = xy/z)

Most often, the errors are independent or uncorrelated. Under these conditions, we use the root sum square (RSS) technique shown in Table 7.8. When all of the random and systematic uncertainties have been assessed, we combine them into a single uncertainty estimate using the propagation formulas above. The combined standard uncertainty is the standard deviation of the measurement. We assess the random and systematic uncertainties independently at the lcr level, then add the standard deviations in quadrature (that is, we use RSS) to get the combined lcr uncertainty: (7.5)

The 1cr level implies a confidence level that is only 68%. To give more confidence in the measurements, we multiply sigma by a coverage factor k. A k of2 is 2cr, where cr is the standard deviation of the measurement. Use a suggested confidence factor k of 2, which gives a confidence of approximately 95%. A recommended format for reporting results shows the measured value with the standard (1cr) uncertainty and a coverage factor k as follows: (45.26 ± 0.03) W/cm 2 (2cr) or (k =2) .

(7.6)

In standards work, authors frequently split the random and systematic uncertainty components and report both. For further details, consult Taylor? Table 7.8 Root-sum-square formulas .

Addition and subtraction (q

=

Multiplication and division (q

x + Y - z)

= xy/z)

250

Chapter 7

7.4 Measurement and Range Equations The general form of the measurement equation relates the observed SIGNAL from a radiometric instrument or detector having a responsivity 9\ to an input radiance L. In differential form, it is (7.7) which is not particularly useful. We must integrate over area, solid angle, and wavelength to get the integral form: SIGNAL = ffJ9\(A)L" (A)dAdndA.

(7.8)

In both equations above, SIGNAL = the output radiometric signal, L;...(A) = the input spectral radiance, and 9\(1..) = is the radiometric system's power responsivity.

The "world" (measurement environment) consists of:

e, x,y A t, v s,p

angular dependences position dependences wavelength time, frequency polarization components diffraction nonlinearities "phase of the moon"

These equations are valid only for incoherent radiation; another layer of complexity is added when speckle and interference effects are added. There are many alternate forms of the measurement equation. For a single detector with spectral responsivity 9\(1..), it simplifies to: (7.9) where ",(1..) = spectral radiant power incident on the detector and 9\(A) = the radiometer spectral power responsivity. For a detector with spectral responsivity 9\(1..) and a narrow-band source or filter:

d

po

251

Radiometric Measurement and Calibration

(7.10) where 'teA) = spectral transmittance of the filter and J:..A = bandwidth of the source or filter. For a laser line where the spectrallinewidth is so small that the responsivity of the detector/radiometer can be considered constant, then

SIGNAL =~cI> .

(7.11 )

Different radiometric configurations have different measurement equations. For example, the signal from a distant small object that underfills the radiometer field of view is described as

(7.12) where

SIGNAL = the radiometric signal, h(A) = the spectral radiant intensity of the source, d = the distance to the source, and ~(A) = the radiometer spectral power responsivity. Many other forms can be developed depending upon the chosen measurement configuration. Similar to the measurement equation is the range equation, which gives the distance at which a source will generate a specified SNR. It is useful to visualize which system parameters are important in particular applications. A general form is provided by Hudson: 3

{D*)li TERM

1

2

3

1 )li ( SNRJB

(7.13)

4

where terms I through 4 are defined as: Term 1: Target parameters: intensity Is and atmospheric transmission 'ta Term 2: Optical system parameters:j7#, optics diameter Do, FOV fl, and transmission 'to Term 3: Detector parameter: D* Term 4: Signal processing parameters: required SNR and noise bandwidth B.

252

Chapter 7

This range equation can take many specific forms to determine SNR or range for various source/radiometer spatial and temporal configurations. Note, also, that the term intensity implies a point source. There are several alternate means of assessing the performance of a radiometric system based on what are called "noise equivalent" quantities, whose input values produce an SNR of 1 (signal equals noise). Noise-equivalent power (NEP), for example, was defined in Chapter 5 as

i i NEP =...!!.... = ...!!...·or 9\ is '

v

v

NEP = ~ = ~. or 9\ vs'

(7.14)

fAB ll NEP =_,,_ fldd - . D* It is usually applied to detectors, but may also be useful for systems designed to measure radiant power. We can also define a noise-equivalent irradiance or flux density (NEI or NEFD), which is frequently used to characterize systems for detection of distant small sources:

NEI=~=Evn =NEP. SNR

(7.15)

Ad

Vs

Note that the term NEI does not reflect our symbol for irradiance E. For an unresolved target that underfills the FOV, Hudson3 gives the noiseequivalent irradiance (W/m2) as 4(f /#)0. 1/2 BI /2 NEI = ---"'---'---''-----

(7.16)

rcDoD * 'to

which bears a striking resemblance to the range equation. The noise-equivalent temperature difference (NETD or NELll) is used to predict the performance of thermal radiometers and infrared cameras. It characterizes a system by its ability to distinguish a small temperature difference between a resolved target (underfills FOV) and the background. For a single small detector Holst gives: 4

4(f/#)~

NETD= ~

C4 S- 't

"fld

A,

. OpllCS

(A) aM (A, TB ) D * (A)dA

aT

,

(7.17)

pas

Radiometric Measurement and Calibration

253

where Tn in the equation is the temperature of the background. Numerous variations of these system parametric equations can be found in the literature, each specific for a given application. Each experimenter must model his or her own system to generate one of these equations for accurate assessment. Once the basic equation has been generated and tested, we can do further modeling to include the effects of other variables, such as scene parameters, environmental influences, degradation, etc.

7.5 Introduction to the Philosophy of Calibration What is calibration and why do we do it? The format of the raw data from a radiometric instrument is usually in the form of a digital count or data number for digital instruments or a voltage, current, or resistance for analog instruments. These numbers are quite meaningless, inasmuch as the units of radiance are not volts, and irradiance is not given in digital counts. Therefore, the primary purpose of calibration is to assign absolute values in engineering units to measured data according to an accepted standard. A secondary but still important purpose of calibration is to estimate uncertainties of the acquired data. Several formal definitions of calibration are provided below. calibration n: (1) The set of operations which establish, under specified conditions, the relationship between values indicated by a measuring instrument and the corresponding known values of a standard (NASA EOS). (2) The measurement of some property of an object that yields as an end result a number that indicates how much of the property the object has (Webster New Collegiate). (3) The comparison of a measurement standard or instrument of known accuracy with another standard or instrument to detect, correlate, report, or eliminate by adjustment any variation in the accuracy of the item being compared (MILSTD-45662A.) (4) A set of operations, performed in accordance with a definite documented procedure that compares the measurements performed by an instrument to those made by a more accurate instrument or standard for the purpose of detecting and reporting, or eliminating by adjustment, errors in the instrument tested (Fluke, Calibration). (5) The process of assigning engineering units and uncertainties to meter deflections, digital counts, etc., such that an instrument reading conforms to a recognized standard (Palmer's definition).

The last of these five definitions is the most useful, as it adds reality to the measurements that we conduct. The common thread in the definitions of calibration is the involvement of a standard. A valid measurement is inextricably linked to the calibration process and therefore to physical standards. How well we can measure is closely related to the quality of the standards we employ, and

254

Chapter 7

future improvements in our measurements will necessitate better standards. A measurement or physical standard can be defined as an accepted object, artifact, material, instrument, experiment, or system that stores or provides a physical quantity that serves as the basis for measurements of the quantity. It is used as a reference for establishing a unit for the measurement of the physical quantity. There are several types of standards used by the community. A primary standard is one that has the highest metrological qualities. It may be realized from first principles, calculable, or built to plan with no other measurements required. This type of primary standard is also known as an intrinsic standard. An example is the degree, which is based upon the triple point of water. Primary standards may also be established by international agreement as an artifact standard. An example is the kilogram, a particular artifact stored at the International Bureau of Weights and Measures (BIPM) in Paris. A secondary standard is designed to carry and transport a calibration scale. It must be as repeatable and stable as possible, and is calibrated with reference to a primary standard. Secondary standards are used to disseminate a scale for widespread distribution. Finally, a working standard is similar to a secondary standard but is one generation further removed from the primary standard. These are the generation that we usually purchase from secondary suppliers of calibration equipment and standards and use for routine calibrations of our instrumentation. Note that as one gets further from the primary standard, the uncertainties increase due to the inevitable errors present in the transfers. Other terminology is also applied to standards. An international standard is one which has been adopted based upon an international agreement. A transport standard is one which has been designed to maintain its calibration through the rigors of transport via common carrier. A consensus standard is one used by consenting parties when no suitable standard is available. u.S. Department of Defense requirements for the use of physical standards are spelled out in MIL-STD-45662A, which states that they must be: (1) certified as traceable to the National Bureau of Standards [now National Institute of Standards and Technology (NIST)), or (2) derived from accepted values of natural physical constants, or (3) derived by ratio type of self-calibration techniques. Traceability is defined as the ability to relate individual measurements to national standards or nationally accepted measurement systems through an unbroken chain of comparisons (MIL-STD-45662A). We have a problem here with requirement (1) because NIST says: NIST does not define nor enforce traceability except in its NVLAP laboratory accreditation program. Moreover, NIST is not legally required to comply with traceability requirements of other federal agencies; nor do we determine what must be done to comply with another party's contract or regulation calling for such traceability.

-Radiometric Measurement and Calibration

255

However, NIST can and does provide technical advice on how to make measurements consistent with national standards. 5 There are several possible solutions to this dilemma: (1) We can keep on doing what we have been doing for many years, claiming our calibrations and measurements are traceable to an agency that disavows the word. Standards maintained in this way are called Type I standards. (2) There have been other attempts to define traceability, and as the above quotation implies, they have not been entirely successful. Perhaps another try is in order: Traceability n: the demonstration that an instrument or artifact standard has been either calibrated by NIST (or equivalent) at appropriate intervals or has been calibrated against other designated standards via an unbroken chain of comparisons. The designated standard may be a national standard, an international standard, or a standard based upon fundamental physical constants. This new definition allows us to: (3) Go offshore to another recognized national laboratory with a reputation for low-uncertainty measurements. (4) Purchase or generate a Type II standard given sufficient expertise, time, and funding. Example: freezing-point blackbody radiation simulator (requires certified pure material), or electrical substitution radiometry (requires measurement of electrical power). Here we generate our own standards from first principles (i.e., standard of length by counting fringes). NIST is moving in this direction as well, conducting research and providing information that will allow our self determination. (5) Implement a Type III standard using ratio and self-calibration techniques. There is another type of standard in regular use called a procedural or documentary standard, also called a protocol in Europe. It is a document outlining operations and processes to be performed in order to achieve a particular end. These are generated and maintained by several organizations, including the American Society for Testing and Materials (ASTM), the International Electrotechnical Commission (lEC), and the International Commission on Illumination (ClE), to name a few . A prime example is the aforementioned MIL-STD-45662A. Over the years I (Palmer) have developed a calibration philosophy which may be of interest, consisting of the principles in Table 7.9.

Chapter 7

256

Table 7.9 The Palmer philosophy of radiometric calibration.

(1) Calibration is the process of assigning absolute engineering units (i.e., radiance, temperature, etc.) to acquired data (volts, digital counts); i.e., it is the determination of the instrument transfer function. (2) Calibration requirements are driven by science and engineering goals. (3) For any instrument, include calibration requirements and methodology in the initial design phase. Among these considerations is the feasibility of including an on-board calibrator. (4) Apply the following general principles in the design of the calibration exercise: A. make the calibration independent of the specific instrument; B. calibrate the sensor in the configuration it will be used; C. take into account every factor that may influence the calibration: the more that Principle B is violated, the more important Principle C becomes; D. calibration involves a comparison with primary or secondary standards; select appropriate standards. (5) Conduct an error assessment during the calibration planning phase to allow estimation of uncertainties on acquired data. Give special attention to model error. (6) An end-to-end calibration is preferable to summation of individual component-level calibrations. (7) Vary relevant external environmental parameters (temperature, pressure, humidity, etc.) to determine their influence on the transfer function. (8) Determine the transfer function over the entire dynamic range of the instrument. (9) To maximize confidence in the calibration, use several calibration configurations and compare the results for consistency. (10) Prior to the final calibration of a flight instrument, conduct the entire calibration procedure on a dummy, prototype, engineering model, or whatever is available, to uncover and fix any problems with the calibrator and/or procedures: (11) Inspect and interpret the results early, while the device undergoing calibration is still in the test position; this allows for an immediate reality check and timely fix if needed. (12) Calibration is the last step on the PERT/CPMlGANTT chart prior to delivery. Because of this precarious position, it is most susceptible to the old squeeze play, so plan ahead. (13) Above all, adhere to the KISS (Keep It Simple, Smarty!) principle. -This was done with imager and sounder prior to the launch of GOES-8 and proved invaluable (BGG).

Radiometric Measurement and Calibration

257

Statement 4 is a response to the "de-embedding problem"; i.e., that no system, device, or component has a unique identity outside its environment. Embedded items will always differ from their de-embedded counterparts due to mutual interaction/parasitics among various components. Following the principles listed in the statement will help minimize, although never completely eliminate, these interactions. An example: one of the authors (Grant) had the experience of witnessing electro-optical instrument calibration and test results and comparing them with results of similar tests after the instrument was integrated to its platform. Different results were observed due to the different environments. One take-away lesson is that when developing instrument specifications at the beginning of a design process, do not fail to include consideration of the manner/platform in which the instrument will ultimately be used! Add a little extra margin for postintegration performance, if you can, or at least identify the factors involved. Further, in order to approach the requirements of calibration philosophy Statement 4, Principle B, one must choose between a standard source and a standard detector. If at all possible, choose a standard source when the unknown quantity is radiance (extended source) or intensity (point source). Place the standard next to the unknown and view them sequentially. Examples of standard sources include blackbody radiation simulators, calibrated tungsten or deuterium lamps, and calibrated integrating sphere sources. Choose a standard detector when the unknown quantity is irradiance or a power. Interchange the standard detector with the radiometer being calibrated. Examples of standard detectors include electrical substitution radiometers and light-trapping quantum detectors. The selection of an appropriate configuration to conduct a calibration is governed both by the desired measurements and by the availability of appropriate standards. There are several calibration configurations that parallel the measurement configurations shown in Chapter 6. One of them should suffice for any radiometric instrumentation calibration. If at all possible, use two or more configurations to gain additional confidence in the calibration.

7.6 Radiometric Calibration Configurations 7.6.1 Introduction

The selection of an appropriate configuration to conduct a calibration is governed both by the desired measurements and by the availability of appropriate facilities and standards. There are a number of different ways to set up a calibration or comparison source and a radiometer. The five basic calibration configurations that follow are the most common combinations of radiometer aperture-stop and field-stop considerations and source distance and size. One of them should suffice for any radiometric instrumentation calibration. If at all possible, use two or more configurations to gain additional confidence in the calibration.

258

Chapter 7

----------

----- ----- ---

I I I I

FIELD ANGLE

8

0

I I

... "". . - - - - - - - - - - S

I

- - - - - - - -...... APERTURE STOP

Figure 7.4 DSS calibration configuration.

7.6.2 Distant small source In the distant small source (DSS) configuration shown in Fig. 7.4, the source is placed at a distance where the inverse square law is valid. The radiometer need not be focused, but the image of the source must be entirely contained within the field stop. Then,

9\ E -- (SIGNAL) S2 , I

(7.18)

where

9\£ = Irradiance responsivity in SIGNALI(W/m2 ), S = distance from source to radiometer (m), and = source intensity (WI sr).

I

The primary advantage of this configuration is that almost any calibration source can be used as long as the distance is sufficient to meet the inverse square law. There are several disadvantages: (1) the signals are typically small, (2) there may be an intervening atmosphere, (3) one must know S (the error in 9\£ is twice the error in S due to the square term), and (4) a background is present because the source image does not typically fill the field stop. Examples of DSS sources include small blackbody radiation simulators and small tungsten lamps at suitable distances, and the sun and stars. The DSS configuration can be significantly improved if the small source is placed at the focal point of a collimator, as shown in Fig. 7.5. As before, the image of the source must fall within the field stop. The size of the image is equal to the size of the collimator source multiplied by the ratio of the radiometer focal length to the collimator focal length. Then

9\ E -- (SIGNAL) j2 , I

(7.19)

259

Radiometric Measurement and Calibration

COLLIMATED BEAM

Figure 7.5 Eccentric pupil parabola used in DSS calibration.

where f = focal length of the collimator (m) and 1 = source intensity (W /sr) = LAs. The result is again irradiance responsivity 9\E in SIGNALI(W/m2 ). The advantages to this approach include: (1) a controllable atmosphere (vacuum chamber, if necessary), (2) a controllable background, (3) the distance S is not in the equation, (4) you can use almost any small source, and (5) the radiometric signals tend to be larger. The disadvantages include (1) the need to know the focallengthf(typically a one-time measurement), and (2) the collimator/chamber hardware can get very expensive. Examples include laboratory calibrators and low-background test chambers. As to collimators, their basic types are refractive and reflective. The basics of a refractive collimator are shown in Fig. 7.6. Its advantages are: (1) relatively simple alignment due to unfolded path (in the visible, only), and (2) a simple setup, particularly if off-the-shelf components are used. The disadvantages include: (1) the wavelength range is limited by the 'teA) of the lens material, (2) reflection losses occur due to refractive index, (3) ghost images arise due to reflections from optical elements, (4) antireflection coatings are wavelength dependent, (5) chromatic aberration from refractive optical elements occurs, and (6) difficulties in alignment occur if the lens is not transmissive. However, most of these disadvantages can be dismissed if operating at a wavelength for which the collimator components are optimized.

Figure 7.6 Basics of a refractive collimator.

260

Chapter 7

Figure 7.7 On-axis reflective collimator.

Disadvantages of the reflective on-axis collimator include: (1) a central obscuration, (2) diffraction from the secondary mirror mount, (3) a direct path for stray light from the source, and (4) difficulties in baffling. On the other hand, an advantage is that wide fields of view are possible with this collimator, shown schematically in Fig. 7.7. Several of these disadvantages can be eliminated by use of an off-axis collimator, shown schematically in Fig. 7.8. For example, stray light is minimized and an off-axis parabola minimizes aberrations. The reflective offaxis design has a narrow field of view, however. As in so many other topics mentioned in this book, tradeoffs and choices must be made. 7.6.3 Distant extended source

In the distant extended source (DES) configuration shown in Fig. 7.9, the distant source subtends a larger angle than the radiometer field of view, overfilling it. For this configuration, 9\ L =

SIGNAL L

where 9\L = radiance responsivity in SIGNALI(W/m 2sr).

Figure 7.8 Off-axis reflective collimator.

(7.20)

261

Radiometric Measurement and Calibration

FIELD STOP

8 SOURCE

FIELD ANGLE

APERTURE

Figure 7.9 DES calibration configuration (adapted from Wolfe and Zissis) .6

The advantages of this configuration include: (1) the distance between the source and the radiometer is not important, and (2) there is no background due to the fact that the source overfills the field of view. Disadvantages include: (1) an intervening atmosphere, and (2) need for a large uniform source. Source examples include White Sands, New Mexico, and a lake of known surface temperature for remote-sensing applications, and a large integrating sphere, blackbody radiation simulator, or a white diffuse panel in the laboratory. 7.6.4 Near extended source

In the near-extended-source (NES) configuration shown in Fig. 7.10, an extended source is placed directly in front of the radiometer undergoing calibration. Radiation from the source (out of focus) must completely fill the field stop. In this configuration, Eq. (7.20) applies and the radiance responsivity is in SIGNALI(W/m 2 sr) as before. Advantages of this configuration include: (1) the distance between the source and the radiometer are not important, (2) there is no background, and (3) there is minimal atmosphere. On the down side, you need a rather large uniform source. Examples include a large-area blackbody radiation simulator, an integrating sphere, or a transmission or reflection diffuser used with a standard lamp. SOURCE FIELD STOP

ANGLE APERTURE STOP

Figure 7.10 NES calibration configuration (adapted from Wolfe and Zissis).6

262

Chapter 7

z Optic Axis

Lens and

Aperture Stop

Figure 7.11 NSS (Jones) calibration configuration (adapted from Wolfe and Zissis).6

7.6.5 Near small source Also called the "Jones method" (as seen in Fig. 7.11) after its ubiquitous inventor, R. Clark Jones, the near-small-source (NSS) calibration provides radiance responsivity calculated according to (7.21 )

where A = aperture area (m2 ) and As = source area (m2 ). In this approach, the source must be contained within the region bounded by XZ and YZ; both segments make the angle with the optical axis. The chief ray angle is also which defines the field of view. This is simply a scaling of areas, and the radiometer is focused at infinity. The radiance responsivity 9\L has units SIGNALI(W/m2 sr). The advantages of the Jones method include (1) minimal atmosphere, and (2) the possibility of using a small calibration source. The primary disadvantage is that you must account for background radiation. An example is the use of a small blackbody radiation simulator that provides radiation to a system having a large entrance aperture. Appendix H provides additional information on this method.

eo,

eo

7.6.6 Direct method In the direct-method approach, seen in Fig. 7.12, we use a small narrow beam that underfills both aperture and field stops (for example, a laser). The beam power is measured with a calibrated detector or laser-power meter. The beam is then pointed toward the radiometer and the output is measured. The result is a power responsivity 9\ in SIGNALIW. The primary advantage of this method is its extremely simple setup. The disadvantages include: (1) visible background radiation, and (2) no accounting for aperture and field-stop nonuniformities. In addition, lasers can be quite noisy.

...

Radiometric Measurement and Calibration

263

INCOMING BEAM

LENS AND APERTURE STOP

Figure 7.12 Direct-method calibration configuration.

To mInImIZe effects from laser drift and noise, use a beam-power stabilizer (expensive) or a beamsplitter and another stable detector to characterize the beam-power fluctuations during the measurement. In addition, you must ensure that saturation of either detector does not occur.

7.6.7 Conclusion The above calibration configurations yield different types of responsivities, but under many circumstances we can use the simple equation for transfer of radiant power in an optical system, Eq. (2.47): = LAn .

The An product (T, throughput, etendue) of a radiometer is usually characterized by the area of its entrance pupil and its field of view. If the radiometer has both a well-defined aperture A and field of view n, we may convert from one form of responsivity to another using

= 9\£ = 9\L

9\

A

An

(7.22)

These conversions permit the use of such a well-defined radiometer to measure one quantity using a calibration derived from a different calibration configuration.

7.7 Example Calculations: Satellite Electro-optical System An example will help to illustrate some of the equations presented in this chapter. Consider a satellite in space located 200 km from a spherical source of I-m diameter that radiates as a blackbody of 2000 K against a background of cold black space. This is depicted in Fig. 7.13. The satellite contains an electro-optical system having parameters listed in Table 7.10.

Chapter 7

264

o

mdiameter~

1 T~2000K

~

200km

~ Figure 7.13 Source-satellite configuration.

(1) What is the detector noise-equivalent power, NEP?

As we are not given specific information regarding voltage or current, it is best to use the third of Eqs. (7.14):

NEP=

~AdB = D*

2

.Jlcm 1Hz 1010 cmHzl/2/W

=1

X

10- 10 W.

(7.23)

(2) Does the source represent a point source for this configuration?

The first thing to figure out is if we are dealing with a point source or an extended source, as we do not wish to use system performance equations indiscriminately. We know that the source is "small," given its distance to the sensor, but we need to determine its relationship to our detector's size.

Table 7.10 Satellite system parameters.

Primary mirror diameter, Do

0.2 m

jl# of optics

jl3

Detector active area Ad

1 cm2

Detector D*

10 10 cm Hz1l2/W

Nominal wavelength range of operation

811m to 12 11m

Electrical bandwidth B

I Hz

265

Radiometric Measurement and Calibration

To do so, we determine the "diffraction-limited spot size" on the focal plane: using a wavelength of 10 f..lm (center wavelength of our band of interest): D b1ur = 2.441.,(//#) = 2.44(10xl0-6 m)3 = 7.32xl0-5 m.

(7.24)

Because the diameter of the (diffraction-limited) source image is less than the dimension of our detector, 1 cm or 0.01 m, this source qualifies as a point source. (3) What is the system's noise-equivalent irradiance, NEI?

Noise-equivalent irradiance, as stated above, is often used to characterize a system for its ability to detect distant small sources. From Eq. (7.15), the NEI of this system may be calculated as

(7.25)

(4) How does this value compare to the NEI obtained from Eq. (7.16), above?

First, we have to determine the solid-angle field of view of the sensor. We need to calculate system focallengthf / = (//#)xDo =3x0.2 m =0.6m.

(7.26)

Next, the solid-angle field of view is determined as A

n=_d =

/2

1 cm 2 1 4 =--=2.78xl0- sr. (60cm)2 3600

(7.27)

If we assume that the transmittance is unity (there is no atmospheric component, and we will assume unity optical transmission), then NEI becomes

NEI =

112 4(3) ( 2.78xl0-4 ) 1112Hz 10

1/2

1t(20cm)lxlO cmHz W-

I

2 = 3.183 X 10- 13 W/cm .

(7.28)

• Note, although this is not a text on system design, the size of the diffraction-limited "blur" plays a role in sizing a system's detector(s). Further information on system design is obtained in references, some listed below and others in the appendices.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - _ ._..._. .. _.... _-

266

Chapter 7

This result differs from that in Eq. (7.25) by three orders of magnitude. Rechecking calculations, we find no error causing a discrepancy this large; what could the problem be? To answer, look again at Eq. (7.25). NEI was calculated according to the area of the detector-not the area of the system entrance pupil Ao. If we repeat that calculation using the correct surface, we obtain NEI

= NEP = lxlOAo

1O

W 314cm 2

= 3.185xlO- 13 W/cm2

(7.29)

which is much better. Conclusion: Make sure you are addressing problems with the mathematical expression which corresponds to the setup/configuration you are analyzing. (5) What is the expected signal-to-noise ratio?

The answer to this question requires an inversion of the range equation, Eq. (7.l3). It is: (7.30)

It also requires that we know source intensity within the particular spectral band. This is obtained through

(7.31) where Ap is the source projected area. From a blackbody radiation calculation program, Ls = 5.04 x 103 W /m 2sr in the 8- to 12-llm band. The projected area of the spherical source of I-m diameter is 0.785 m2 , so I .. = 5.04xl0 3 xO.785 = 3956 W /sr.

Substituting into Eq. (7 .30), assuming unity transmittances and a I-Hz electrical bandwidth, SNR

= 3956 W /sr It rad(20cm)(lOlO cmHz"2 /W) 4(3).J2.78 x 10-4 sr(4xlOl4 cm 2)l HZl/2

:=::

31.

(7.32)

Whether or not a SNR of 31 is adequate depends, of course, on the particular application.

Radiometric Measurement and Calibration

267

Equation (7.32) may appear, on first glance, to be dimensionally inconsistent, as it appears to reduce to units of steradians in the denominator. But look at Eq. (7.30) again, in this way:

SNR= \ .... . R .....

Discussing irradiance in Chapter 2, we noted that E=IIR2 can sometimes be confusing, as to units, and that a different way to consider the expression is

where no may be thought of as the "unit solid angle," having value 1 sr. Applying this notion to Eq. (7.32) allows for dimensional consistency. In conclusion, while real-world problems may be different from those described in this section-for example, they may include source/target parameters that change with time and sources that differ from blacklgraybodiesthese equations are basic, yet powerful enough to provide the engineer or analyst a starting point from which to develop solutions.

7.8 Final Thoughts 1 often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be. - Lord Kelvin A measurement of any kind is incomplete unless accompanied with an estimate of the uncertainty associated with that measurement. - James M. Palmer

268

Chapter 7

For Further Reading Y. Beers, Introduction to the Theory of Error, Addison-Wesley, Reading, Massachusetts (1957). P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd Edition, McGraw-Hill, New York (1992). A. Daniels, Field Guide to Infrared Systems, SPIE Press, Bellingham, Washington (2006).

Calibration: Philosophy in Practice, 2nd Edition, Fluke Corporation, Everett, Washington (1994). J. Mandel, The Statistical Analysis of Experimental Data, Dover, New York (1984). S. L. Meyer, Data Analysis for Scientists and Engineers, John Wiley & Sons, New York (1975). J. R. Taylor, An Introduction to Error Analysis, 2nd Edition, University Science, Mill Valley, California (1997). J. D. Vincent, Fundamentals of Infrared Detector Operation and Testing, John Wiley & Sons, New York (1990). H. D. Young, Statistical Treatment of Experimental Data, McGraw-Hill, New York (1962).

References 1.

S. J. Wein, "Sampling theorem for the negative exponentially correlated output of lock-in amplifiers," Appl. Opt. 28,4453 (1989).

2.

B. N. Taylor and C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, (1994). Available at http://physics.nist.gov/Pubs/guidelines/contents.html.

3.

R. D. Hudson, Infrared Systems Engineering, Wiley & Sons, New York (1969).

4.

G. C. Holst, Common Sense Approach to Thermal Imaging, JCD Publishing, Winter Park, Florida and SPIE Press, Bellingham, Washington (2000).

5.

NIST SP-250 Appendix, U.S. Government Printing Office, Washington D.C. (1998).

6.

G. J. Zissis, "Radiometry," Chapter 20 in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, Eds., U.S. Government, Washington, D.C. (1978).

Table of Appendices Adapted by Anurag Gupta

Appendix A: Systeme Internationale (SI) Units for Radiometry and Photometry .................................................................... 271 Appendix B: Physical Constants, Conversion Factors, and Other Useful Quantities ....................................................................... 275 Appendix C: Antiquarian's Garden of Sane and Outrageous Terminology ................................................................... 277 Appendix D: Solid-Angle Relationships .............................................. 283 Appendix E: Glossary ......................................................................... 285 Appendix F: Effective Noise Bandwidth of Analog RC Filters and the Selection of Filter Parameters to Optimize Signal-to-Noise Ratio .............................................................................. 297 Appendix G: Bandwidth Normalization by Moments .......................... 305 Appendix H: Jones Near-Small-Source Calibration Configuration ..... 309 Appendix I:

Is Sunglint Observable in the Thermal Infrared? ........... 313

Appendix J: Documentary Standards for Radiometry and Photometry .................................................................... 321 Appendix K: Radiometry and Photometry Bibliography ..................... 341 Appendix L: Reference List for Noise and Signal Processing, by L. Stephen BelL ......................................................... 357

269

--

Appendix A

5ysteme International (51) Units for Radiometry and Photometry Table A.1 SI* base units.

Base quantity

Name

Length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensity

meter kilogram second ampere kelvin mole candela

Symbol m kg s A K mol cd

Table A.2 Selected SI-derived units.

Quantity Plane angle Solid angle Energy Power Frequency Electric charge Luminous flux Illuminance Luminance Radiant intensity Radiance

Name radian steradian joule watt hertz coulomb lumen lux candela per square meter watt per steradian watt per square meter steradian

Symbol rad sr J W Hz C 1m Ix cd/m2 W/sr W/(m2 sr)

Equivalent

N·m J/s S-l

A-s cd·sr Irnlm2 lrnlm2sr

* Complete SI information is available on the World Wide Web at www.bipm.fr and at physics.nist.gov/pubs/sp81Ilsp811.html.

271

272

Appendix A

Table A.3 Si prefixes.

Factor 10 1021 10 18 10 15 10 12 109 106 103 102 10 1

Prefix yotta zetta exa peta tera glga Mega kilo hecto deka

Symbol Y Z E P T G M k h d

Factor 10- 1 10- 2 10- 3 10-6 10-9 10- 12 10- 15 10- 18 10-21 10- 24

Prefix deci centi milli micro nano pico femto atto zepto yocto

Symbol d c m ~

n P f a z Y

The following tables show radiometric and photometric quantities, symbols, definitions, and units. Table A.4 Radiometric quantities.

Quantity

Symbol

Definition

Units

Radiant energy

Q

joule [J]

Radiant power (flux)

dqldt

watt [W]

Radiant intensity

I

dldO)

W/sr

Radiant exitance

M

dlda

W /m2

Irradiance

E

dlda

W/m 2

Radiance

L

d2/(da cosadO))

W /m2 sr

Table A.5 Photon quantities.

Quantity

Symbol

Definition

Units

Photon power (flux)

q

dn/dt

Is

Photon intensity

I

dnldm

ISf"s

Photon exitance

M

dnlda

Im2 s

Photon irradiance

Eq

dnlda

Im2 s

Photon radiance

Lq

ctn/(da cosadO))

Im2sf"s

n = photon number.

Systeme International (SI) Units for Radiometry and Photometry

273

Spectral Quantities Spectral quantities are derivative, per unit wavelength with the additional dimension m- I , and are indicated by a subscript A (e.g., spectral radiance LAwith units W/m3 sr). Nonspectral quantities that are wavelength dependant are indicated as (A); e.g., transmission 'teA). Photometry is the measurement of light (optical radiant energy as above, but weighted by the response function of the human eye). The symbols used are the same as radiometric quantities with the subscript v (for visual) added. Table A.6 Spectral quantities.

Quantity

Symbol

Units

Luminous power

v

1m

Luminous exitance

Mv

lmlm2

Luminous incidance

Ev

lmlm2

Luminous intensity (SI base unit)

Iv

lmlsr = cd

Luminance

Lv

lmlm2sr = cd/m2

------------------------------------------------------------------------------------------------

..

Appendix B

Physical Constants, Conversion Factors, and Other Useful Quantities Table 8.1 1998 CODATA recommended values of the fundamental physical constants.

Quantity

Symbol

Value

Units

Speed of light (vacuum) Permeability of vacuum Permittivity of vacuum Planck constant

C, Co

299,792,458

m/s

Relative uncertainty exact

/lo

4n x ]0-7

N/A2

exact

eo h

1I1.V2 = 8.854 187 ... xlO- 12 6.62606876 (52) x ]0-34

F/m

exact 7.8 X ]0-8

Electronic charge Boltzmann constant Boltzmann constant Stefan-Boltzmann constant First radiation constant

(J

1.602176462 (63) x ]0- 19 1.3806503 (24) x 10- 23 8.617342 (15) x ]0-5 5.670400 (40) x 10-8

CI

q,e

k k

J·s C

3.9 X ]0-8 1.7 x ]0-6

J/K eV/K W/m2K4

1.7 x 10-6 7.0xl0-6

3.74177107 (29) x ]0-16

Wm2

7.8 x 10- 8

CIL

1.191042722 (93) x 10- 16

Wm /sr

2

7.8 x ]0-8

C2

1.4387752 (25) x 10-2

m-K

1.7 x 10- 6

b

2.8977686 (51) x ]0-3

m·K

1.7 x ]0-6

(2nhc 2)

First radiation constant for L). Second radiation constant Wien displacement law constant

These are the 1998 CODATA recommended values of the fundamental physical constants. Adapted in part from P. J. Mohr & B. N. Taylor, "The fundamental physical constants," J. Phys. Chern. Ref Data 28, 1713 (1999), and Rev. Mod. Phys. 72, 351 (2000). These constants are also available in Physics Today, 54 (Part 2), BG6 (2001), reprinted yearly, and from http://physics.nist.gov/constants. 275

Appendix B

276

Here are some useful conversion factors: hc = 1.986445

X

10-25 J·m = 1.986445 x 10- 19 J·llm = 1.986445 x 10- 16 J·nm

hc/q

=

1.23984 eY·llm

kT/q

=

0.025852 Y at 300 K

1 eY = 1.602176462 x 10-19 J 1 astronomical unit (AU) = 1.495 x lOll m

"'-maxT= c2/4.96511423 ...

Appendix C

Antiquarian's Garden of Sane and Outrageous Terminology Perhaps the most difficult task in both teaching and learning about radiometry and photometry is learning and conveying an appropriate and sensible system of symbols, units, and nomenclature. This can be a formidable task because of the enormous extent of these found in the literature. I have attempted to be consistent with the accepted units in this text and have addressed the situation with regard to intensity as well . The following is a collection of terms, symbols, and units that I have gathered with little effort. Perhaps you can add some more to this list. Some are still current and some are long obsolete.

Photometry Perhaps in no scientific field is the language more obtuse than in photometry. This is in large measure because of the tortuous path of the development of suitable standards. Luminous intensity

The SI base unit of luminous intensity is the candela (cd). 1 Bougie decimale = 1.02 cd. 1 Bougie nouvelle = 1 cd. 1 International candle = 1.01937 cd (Average of candle standards of the U.S., U.K., and France). I new candle = I cd. 1 Carcel = 10 cd. 1 Carcel unit = 9.79613 cd (The measure of a Careel lamp burning calza oil). 1 hefnerkerze = 0.903 cd (German measure of luminous intensity from 1884 to 1940 = 0.903 cd or 0.92 cd. (Replaced by the candela). 1 violle = 20.4 cd. 1 Pentene candle = 1 cd. I English sperm candle = 1 cd. 277

278

Appendix C

Table C.1 Some interesting numbers from Phillips Lighting Company.

Bicycle headlamp without reflector, in any direction Bicycle headlamp with reflector, center of beam Incandescent reflector lamp PAR38E Spot 120 W, center of beam Lighthouse, center of beam

2.5 cd 250 cd 10,000 cd 2,000,000 cd

Luminous power The (derived) SI unit ofluminous flux (power) is the lumen (1m). 1 1m = 1 cd·sf. A light watt is a unit of radiant power weighted by human-eye response. One light watt is the power required to produce a perceived brightness equal to that of light at a wavelength of 555 nm and luminous power of 683 1m. Symbol for light watt is v. 830

v =Km

f Y(A)dA .

360

Mechanical equivalent of light is 1/683 W11m.

Illuminance The (derived) SI unit of illuminance is the lux (Ix = Imlm 2). 1 footcandle (fc) = 1 1m per square foot. 1 lux (Ix) = 1 Irnlm2 = 1 meter-candle. 1 phot (ph) = llrnlcm2= 104 Ix. 1 milliphot (mph) = 10- 3 Imlcm2. 1 nox = 1 millilux = 10- 3 Ix. 1 sea-mile candle = 1 cd @ 1 nautical mile (6,080 ft) 1 pharosage = 11mlm2.

=

2.9 x 10-7 Ix.

Luminosity L is expressed in Imlfe.

Luminance The (derived) SI unit of luminance is the nit (cd/m2). 1 nit = 1 candela per m2= 1t apostilb = 0.2919 foot-lambert (fL). 1 stilb (sb) = 1 candela per cm2. 1 nit = 104 Bougie-Hectometre-Carre.

Antiquarian's Garden of Sane and Outrageous Terminology

279

Note: Several luminance units are related to the illuminance units by assuming a

perfect (p = 1) diffuse (Lambertian) reflector. This "simplification" leads to : I foot-candle (fc) of illumination ~ 1 fL ofluminance. 1 lambert (L)= 1 Imlcm2= (lin) cd/cm2 . 1 footlambert (fL) = (l In) cd/ft2. 1 apostilb (asb) = (lin) cd/m2 = (lin) nit. 1 skot = 10-3 (lin) cd/m2 = 10-3 apostilb. 1 millilambert '" 1 fL. 1 equivalent phot = 1 L. I equivalent lux = 1 blondel = I asb. 1 equivalent footcandle = 1 fL. The unit bril is used to express the "brilliance" or subjective brightness of a source of light: bril = 10gL + 100 . log2 The scale is logarithmic: an increase of I bril means doubling the luminance emitted by the source. A luminance of 1 lambert (L) is defined to have a brilliance of 100 brils. Luminous energy

Luminous energy is radiant energy weighted by the visual response of the eye. The (derived) SI unit of luminous energy is the talbot (lm·s). I talbot = 107 lumergs. 107 erg = 1 W·s. 1 phos = 1 talbot.

Vision Research troland: 1. Retinal illuminance produced by luminance of 1 cd/m2 if

entrance pupil of eye is 1 mm 2, corrected for the Stiles-Crawford effect; formerly called the photon. 2. The external illuminance that produces retinal illumination of 0.002 Ix.

280

Appendix C

Ultraviolet E-viton is erythemal effectiveness equivalent to 10 IlW at 296.7 nm.

1 Finsen = 1 E-vitonlcm2 • 1 erythemal watt = 105 E-vitons. 1 EU = 1 E-viton = 1 erytheme. Floren is UV flux equivalent to 1 m W between 320 and 400 nm. Bactericidal microwatt is weighted by bactericidal action spectrum. Ultraviolet microwatt or UV watt is evaluated at 253.7 nm. MPE (minimum perceptible erythema) = 0.025 erythemal W/cm2 • 1 MPE = 2500 finsens = 2.5 x 105 erg/cm2 at 296.7 nm. One minimum erythemal dose (MED) is the dose required to produce a minimum redness on sun-sensitive skin. Its value is dependent on skin type. For the most sensitive skin type it is 200 J/m 2, weighted by the standard erythemal action spectrum. For less-sensitive skin types, it rises to 1000 J/m 2 • 1 MED = 2 SED (standard erythemal dose). At the wavelength of maximum sensitivity for production of erythema (295 nm), the MED is 50 J/m 2 . [Br. J. Dermato!' 82,584 (1970).]

Shade number is a unit of light transmission for the protective glasses used in welding. If T is the fraction of visible light transmitted, the shade number is 1 + 7(-loglOn/3. For example, if 1% of the light is transmitted, the shade number is 4.

Astronomy 1 Jansky (Jy) = 10-26 W/m 2Hz (spectral irradiance).

1 W/cm2 f.1m

= 3 x 10 16/1..1 Jy.

1 Solar flux unit (s.f.u.) Visual magnitude zero

= 104 Jy.

=

2.65 x 10-6 Ix outside atmosphere (Infrared

Handbook, pp. 3-23). Visual magnitude zero

= 2.54 x 10-6 Ix outside atmosphere (radiometry

and photometry in astronomy). Visual magnitude zero = 2.09 x 10-6 Ix outside atmosphere.

....

Antiquarian's Garden of Sane and Outrageous Terminology

1 Rayleigh

281

106 photons/cm2 s.

=

I Rayleigh = (1I4n) x 106 photonslc2 s·sr. I SIO = 1.23

X

10- 12 W/cm2sq.tm at 0.55 /lm (equivalent to the number of

10th magnitude stars per square degree). 1 SIO = l.899

X

106 photons/s· cm2 sr-/lm at 0.55/lm.

Color and Appearance Reciprocal megakelvin (MKr l = 106/Tc , where Tc is color temperature, also known as mirek or mired (microreciprocal kelvin or microreciprocal degree).

Miscellaneous 1 microeinstein (/lE) = 6.022 x 10 17 photons = 1 micromole. angstrom (A) is obsolete unit of wavelength = 10- 10 m. Kayser is waves per centimeter. Gillette is a measure oflaser energy, sufficient to penetrate one standard razor blade. Microflick is a unit of spectral radiance /lWlcm2sr·/lm. Spectrallamprosity is in youngs per watt. Lamprosity (y) is in youngs per radiated watt. 1 = 1 lumens per input watt. Photosynthetic photon flux density (PPFD) is measured in mol/m2 s. Spherical photosynthetic photon flux density (SPPFD) is measured in mol/m 2s. Photosynthetically active photons (PAP) is measured in mol/m2 . Spherical photosynthetically active photons (SPAP) is measured in mollm2. Irradiation = radiant pharosage = radiant incidance (W/m2). Radiosity = radiant pharosage = radiant exitance (W /m2). Phengosage = spectral pharosage. Radiant pharos = radiant power (W). 1 W/m2 = 0.317 BTU/ft2hr. 1 langley = 1 gm·cal/cm2. 1 langley/minute = 697.3 W/m2. 1 pyron = 1 calorie/cm2min = 697.633 J/m2s used to measure heat flow from solar radiation. Radiant phos is exposure (W·s). Radiant helios is radiance (hershel).

282

Appendix C

I pharos = I lumen. I helios = I blondel. I heliosent = blondel/m. Radiant heliosent = path radiance (hershel/m). Luminous efficiency is luminous flux/radiant flux. Luminous efficacy is luminous flux/electrical input power.

•

Appendix D

Solid-Angle Relationships e (deg)

e (rad)

0.573 1.000 1.146 l.719 2.000 2.292 2.865 3.000 4.000 5.000 5.730 10.00 11.46 15.00 17.19 20.00 22.92 25.00 28.65 30.00 34.38 40.11 45 .00 45.84 51.57 57.30 60.00 71.63 85.95 90.00

.0100 .0175 .0200 .0300 .0349 .0400 .0500 .0524 .0698 .0873 .1000 .1745 .2000 .2618 .3000 .3491 .4000 .4363 .5000 .5236 .6000 .7000 .7854 .8000 .9000 l.000 l.047 1.250 1.500 1.571

ro (sr) .00031 .00096 .00126 .00283 .00383 .00503 .00785 .00861 .0153 .0239 .0314 .0955 .1252 .2141 .2806 .3789 .4960 .5887 .7692 .8418 1.097 1.478 1.840 l.906 2.377 2.888 3.142 4.269 5.839 6.283

o (sr) .00031 .00096 .00126 .00283 .00383 .00502 .00785 .00861 .0153 .0239 .0313 .0947 .1240 .2104 .2744 .3675 .4764 .5611 .7221 .7854 1.002 1.304 1.571 l.617 1.928 2.224 2.356 2.830 3.126 3.142

mlO 1.000 1.000 l.000 1.000 l.000 l.000 1.001 l.001 l.001 l.002 l.003 1.008 l.01O 1.017 l.023 l.031 l.041 l.049 1.065 l.072 1.096 l.l33 1.172 1.179 1.233 1.298 1.333 1.521 1.868 2.000

jl# 50.00 28.65 25.00 16.67 14.33 12.50 10.00 9.554 7.168 5.737 5.008 2.880 2.517 l.932 l.692 1.462 1.284 1.183 1.043 l.000 0.886 0.776 0.707 0.697 0.638 0.594 0.577 0.527 0.501 0.500

NAln* 0.010 0.017 0.020 0.030 0.035 0.040 0.050 0.052 0.070 0.087 0.100 0.174 0.199 0.259 0.296 0.342 0.389 0.423 0.479 0.500 0.565 0.644 0.707 0.717 0.783 0.841 0.866 0.949 0.997 1.000

"To obtain the numerical aperture NA, numbers in this column must be multiplied by the index of refraction n of the local media. Adapted from F.E. Nicodemus et at., Self-Study Manual on Optical Radiation Measurements, NBS Technical Note 910-01, National Institute of Standards and Technology, Washington, D.C. (1976). 283

Appendix E

Glossary lifnoise

Weird, ubiquitous noise from many familiar and strange sources, inversely proportional to frequency (Pink or red noise). An approximation

. -:z (const )( I:C B ) . = ~ , where a IS between 1.25 f

IS III f

and 4 (typically 2), and ~ is between 0.8 and 3 (typically 1). Also called flicker, contact, excess, modulation, etc. A major pain!

An product

Symbol T, units m 2sr; geometrical term relating to amount of power that can get through a system; Also throughput, etendue.

Bandwidth normalization

Determination of an equivalent responsivity using a rectangle; the areas under the curve and rectangle are set equal.

Blackbody radiation simulator

An object that simulates blackbody (Planckian) radiation via careful cavity design and temperature measurement.

Background-limited infrared photodetector (BLIP)

One whose noise is predominantly due to the noise in the incident photon stream, not intrinsic to the detector.

Bode plot

Plotting log(signal or noise) versus log(frequency); shows many orders of magnitude, asymptotes of linear plots map to straight lines on Bode plot.

285

286

Appendix E

Bidirectional reflectance distribution function (BRDF)

A directional quantity that denotes output radiance as a function of direction and irradiance. A perfectly diffusing reflector has a BRDF of p/1t, while a perfectly specular reflector has a BRDF p/Q; P is reflectance and Q is the projected solid angle of the source. Units: sr- 1•

Brightness temperature

The brightness temperature of an object is the temperature of blackbody radiation that has the same spectral radiance as the object.

Bidirectional transmission distribution function (BTDF) Charge transfer efficiency (CTE)

The angular distribution of transmitted radiance around the normal transmitted beam. Units: sr- 1•

Chopping factor

Ratio of the rms amplitude of the fundamental frequency component of a modulated signal to the peak-to-peak amplitude of the unmodulated signal. Equal to 0.45 for square wave chopping.

CIE chromaticity diagram

A horseshoe-shaped diagram showing the gamut of all possible colors in terms of hue and saturation. The horseshoe is the spectrum locus and white is at the center where x = y = z = 113.

Cold filter

A filter that passes desired bandpass and is cooled to minimize self radiation at other wavelengths.

Cold stop

An aperture placed in front of a detector to limit the field of view and cooled to minimize the stop radiance. Improves SNR.

Collimator

An optical system designed to make a near small source appear as if it were located at infinity.

Color temperature

The color temperature of an object is the temperature of blackbody radiation that has the same chromaticity (color) as the object.

Fraction of charge that is successfully transferred from one CCD charge storage element (potential well) to the adjacent charge storage element.

Glossary

287

Conduction calorimeter

A device to measure laser power and energy by calorimetric means, i.e., heating of an absorber.

Contrast sensitivity function

The visual acuity of the eye as a function of both spatial frequency and contrast. Determined by looking at variable frequency cos2 wave patterns that have contrast decreasing from bottom to top.

Correlated color temperature

The temperature of a blackbody having a chromaticity (color) as close as possible to the chromaticity of the source in question.

Cosine response

The response curve desired for an instrument designed to measure irradiance from a hemisphere. Related to projected area.

D

Detectivity, the reciprocal of noise equivalent power (NEP), the input power for which the SNR is 1. Unit: W- I .

D*

Specific or normalized detectivity. It is detectivity D = NEP- 1 normalized for bandwidth and area. D*= (AB)1 /2INEP. Unit: cm·Hz I/21W. It is the SNR per watt for a l-cm2 detector with a bandwidth of 1 Hz. Allows a fair comparison between detector types.

D**

A normalized detectivity, taking into account detector area, noise bandwidth, and field of view (FOY). It is the SNR per incident watt for a l-cm2 area, I-Hz noise bandwidth, and n sr ofa projected solid angle. D**= (ABQ./n) II2 INEP . Unit: cm'Hz 1/21W. A further normalization for field of view: D*(Q.ln) 1/2,

D*BLlP

Background-limited infrared photodetector, the best SNR you can get when photon noise from the background limits detection.

Decade, octave

Frequency ratio of 10 and 2, respectively,

288

Decibel (dB)

Appendix E

A ratio of two voltages, currents, or powers.

Pz dB = 2010g lo -~ = 2010g lo -12 dB = lOloglO~ ~ ~ It is a relative measure. There are several ways of denoting absolute values. dBv refers to I-V rms. dBm refers to 1 m W with stated impedance (75 Q). To compare unlike waveforms, such as a sine wave to Gaussian noise, use the power formulation. Detective quantum efficiency (DQE)

A relative measure of the amount of noise added by a detector. Detective quantum efficiency is like RQE but includes noise, (SNR oui l(SNR;n)2, unitless, 0 < DQE < I .

Diffuse reflectance

Ratio of radiation reflected into a hemisphere (whose base is the reflector) to the incident radiation. Excludes specular component.

Distribution temperature

The distribution temperature of an object is the temperature of a blackbody radiator that has the same (or nearly the same) relative spectral distribution over a substantial portion of the spectrum as the object.

Effective noise bandwidth

The equivalent square-band power bandwidth, used for evaluation of noise. Given for "white" noise by ENB=_1_2' [ [ A (/)Td/ .

IAol The equivalent "brick-wall" rectangular passband. Alternate symbols are Band 11f, units are Hz. Electrical substitution radiometer

A radiometer based upon a thermal detector with provisions for injecting a known power via electrical means for the purpose of calibration.

289

Glossary

Full well capacity

The number of electrons (signal + noise + dark current) that a potential well in a CCD structure can hold.

Generation-recombination nOise

Noise due to the generation of carriers by photon absorption and by recombination random motion of carriers (electrons) in a resistive material. Spectral power density depends on frequency.

H-D curve

The characteristic curve of photographic film (named after Hurter and Driffield) which plots density log(l/t) versus log of the exposure (product of irradiance and time).

Hemispherical reflectance

Directional reflectance integrated over an entire hemisphere.

Iluminance

Luminous flux per unit incident on a surface from a hemisphere; units Imlm2 = Ix. Analagous to irradiance in W1m2 •

Intensity

Symbol I, watts per unit solid angle, often from an isotropic "point" source W/sr.

Irradiance

Symbol E, units W /m2, watts per unit area incident on a surface.

Isotropic point source

Small (relative to distance) source where intensity is independent of direction.

Johnson noise

Noise due to random thermal agitation of carriers (electrons) in a resistive material. Spectral power density independent of frequency (i.e.,

-

white). v~ Jones calibration configuration

=4kTRB

-

or

i~

= 4kTB I R .

Also known as near small source. The radiometer is focused at infinity. A small calibration source is placed within a cone whose base is the entrance aperture and whose half angle is the chief ray angle. It fills a fraction of the entrance aperture. The calibration equation is

9\ = ( SIG;AL )(

~: ).

290

Appendix E

Lambertian source

Source where radiance is independent of direction.

Laser calorimeter

A device to measure laser energy, particularly for short pulses, by calorimetric means, i.e., heating of an absorber. Output proportional to time integral of power, i.e., energy.

Luminance

Photometric equivalent of radiance. Measure of visible power per unit-projected area per unit solid angle. Unit: candela m- 2 = Imlm2sr.

Luminous intensity

Measure of visible power per unit solid angle. Unit: candela (cd) = Imlsr. One of the seven SI base units. Analogous to radiant intensity, W/sr.

Measurement equation

An equation that relates the output signal from a detector or radiometer to a function of the receiver and source spectral parameters. An example:

SIGNAL = An

r

LA ~(A)dA.

Moments normalization

A normalization based on a moments analysis of a spectral responsivity. The center wavelength is the centroid and the bandwidth, and cut-on and cutoff wavelengths are computed from the vanance.

Noise-equivalent photon flux

The photon flux incident on a detector which gives rise to a signal-to-noise ratio of one. Units: s- I .

Noise-equivalent power (NEP)

The power incident on a detector which gives rise to a signal-to-noise ratio of 1. Units: W.

Noise-equivalent temperature difference (NETD, NEllI)

The temperature difference between the target and the background that produces an rms signal equal to the rms noise.

Passband normalization

Normalization method wherein the band limits of the equivalent rectangle are assigned at fixed response points (50%, 10%, lie, etc.). The normalized responsivity is then related to the area under the response curve.

Glossary

291

Peak normalization

A normalization where the band responsivity is set to the peak of the actual response curve. The bandwidth is calculated by matching the area of the equivalent rectangle to the area under the response curve.

Photoconductive gain

The ratio of the transit time to the carrier lifetime for a photoconductive detector. A measure of the number of electrons a single absorbed photon can generate.

Photon noise

Noise due to the random arrival of photons; manifest as shot noise or the G component of GR noise.

Photopic

Pertaining to light-adapted vision.

Photopic visibility curve

The relative spectral responsivity of the standardized light-adapted human eye (cones). Symbol is V(A); dimensionless.

Precision

A measure of the repeatability of a measurement. Comes from granularity, noise, etc. Determined and enhanced by repeated measurements.

Projected solid angle

Solid angle x cos9, projected onto flat surface dO. = dro cos9. Symbol 0., units sr.

Quantum efficiency

see Responsive quantum efficiency.

Quantum trap detector

A multiple-detector array where detectors are placed in series optically and in parallel electrically. Has a quantum efficiency approaching unity.

Radiance

Fundamental quantity of radiometry, "brightness." Symbol L, units W/m 2 sr.

Radiance temperature

The radiance temperature of an unknown object is the temperature of a blackbody that has the same spectral radiance as the unknown object.

Radiant exitance

Radiant power per unit area leaving a source into a hemisphere. Symbol M, units W/m2 •

292

Appendix E

Radiant intensity

Radiant power per unit solid angle. Symbol I, units W/sr.

Radiation reference

A comparison source for a radiometer; zero-based for short-wave radiometers, a known thermal source for long-wave radiometers.

Radiation temperature

The radiation temperature of an object is the temperature of blackbody radiation that has the same total (integrated over all wavelengths) radiance as the object.

Range equation

An equation that gives the distance from a source that one can detect with a stated SNR.

Ratio temperature

The ratio temperature of an object is the temperature of blackbody radiation that has the same ratio of spectral radiances at two wavelengths as the object.

Reflectance factor

Ratio of flux reflected from a sample to the flux that would be reflected from a perfect diffuse reflector (Lambertian, p = 1).

Responsive quantum efficiency (RQE)

The number of independent output events per incident photon. Dimensionless, between 0 and 1. Symbols: 11, RQE.

Responsivity

Ratio of the output of a detector to its input. Units: NW or V /W. Symbol is 9\. The result of an integral over wavelength: 9\ = f9\(A )