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. I. Grant, Barbara G. (Barbara Geri), 1957- II. Title. QD117.R3P35 2009 539.7'7--dc22 2009038491
Published 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.
Contents Foreword ................................................................................................... xi Preface .................................................................................................... xiii
Chapter 1 Introduction to Radiometry / 1 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 / 11 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 2.2.3 Irradiance .............................................................................. 28 2.2.4 Radiant intensity .................................................................... 29 2.3 Radiometric Approximations.............................................................. 30 2.3.1 Inverse square law ................................................................ 30 2.3.2 Cosine3 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 2.9.1 Intensities of a distant star and the sun ................................. 48 v
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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 / 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 / 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
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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 4.4.3.3 Electroluminescent sources ......................................... 117 4.4.3.4 LED sources ................................................................. 117 4.4.3.5 Lasers .......................................................................... 118 4.4.4 Natural sources ................................................................... 119 4.4.4.1 Sunlight ........................................................................ 119 4.4.4.2 Skylight, planetary, and astronomical sources ............. 120 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 / 127 5.1 Introduction...................................................................................... 127 5.2 Definitions ........................................................................................ 128 5.3 Figures of Merit ............................................................................... 131 5.4 #N$O%&I*S@E~^ ........................................................................... 133 5.4.1 Introduction to noise concepts............................................. 133 5.4.2 Effective noise bandwidth.................................................... 136 5.4.3 Catalog of most unpleasant noises ..................................... 137 5.4.3.1 Johnson noise .............................................................. 137 5.4.3.2 Shot noise .................................................................... 139 5.4.3.3 1/f 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 5.4.3.9 CCD noises .................................................................. 142 5.4.3.10 Amplifier noise .............................................................. 143 5.4.3.11 Quantization noise........................................................ 143
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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 5.5.1 Thermal circuit ..................................................................... 147 5.5.2 Thermoelectric detectors ..................................................... 150 5.5.2.1 Basic principles ............................................................ 150 5.5.2.2 Combinations and configurations ................................. 153 5.5.3 Thermoresistive detector: bolometer ................................... 155 5.5.4 Pyroelectric detectors .......................................................... 157 5.5.4.1 Basic principles ............................................................ 157 5.5.4.2 Pyroelectric materials ................................................... 160 5.5.4.3 Operational characteristics of pyroelectric detectors ... 162 5.5.4.4 Applications of pyroelectric detectors........................... 162 5.5.5 Other thermal detectors....................................................... 163 5.6 Photon Detectors ............................................................................. 164 5.6.1 Detector materials ............................................................... 164 5.6.2 Photoconductive detectors .................................................. 169 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 5.6.2.4 Applications of photoconductive detectors ................... 175 5.6.3 Photoemissive detectors ..................................................... 175 5.6.3.1 Basic principles ............................................................ 175 5.6.3.2 Classes of emitters....................................................... 176 5.6.3.3 Dark current ................................................................. 181 5.6.3.4 Noises in photoemissive detectors ............................... 182 5.6.3.5 Photoemissive detector types ...................................... 183 5.6.4 Photovoltaic detectors ......................................................... 185 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 5.6.4.4 Photovoltaic detector materials and configurations ...... 198 5.7 Imaging Arrays ................................................................................ 199 5.7.1 Introduction.......................................................................... 199 5.7.2 Photographic film................................................................. 199 5.7.2.1 History .......................................................................... 199 5.7.2.2 Physical characteristics ................................................ 201 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
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5.7.3.3 Readout mechanisms .................................................. 204 5.7.3.4 Comparison .................................................................. 207 5.7.4 Three-color CCDs ............................................................... 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 / 215 6.1 Introduction...................................................................................... 215 6.2 Instrumentation Requirements ........................................................ 215 6.2.1 Ideal radiometer .................................................................. 215 6.2.2 Specification sheet .............................................................. 215 6.2.3 Spectral considerations ....................................................... 216 6.2.4 Spatial considerations ......................................................... 217 6.2.5 Temporal considerations ..................................................... 217 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 6.3.7 Other methods for defining the field of view ........................ 224 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 6.4.1 Introduction.......................................................................... 228 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 6.4.7 Multiple slit systems ............................................................ 236 6.4.8 Filters................................................................................... 236 6.4.9 Interferometers .................................................................... 237 6.4.10 Fourier transform infrared.................................................... 237 6.4.11 Fabry-Perot ......................................................................... 238
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For Further Reading .............................................................................. 240 References ............................................................................................ 240
Chapter 7 Radiometric Measurement and Calibration / 241 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 / 269 Appendix A: Système 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 Postdetection Signal Processing ........................................................................ 357
Index / 361
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
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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 SPIE 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- [
2
Chapter 1
Figure 1.1 Classic vane radiometer, commonly called the Crooke radiometer.1 [Reprinted by permission from Webster’s Third New International® Dictionary, Unabridged ©1993 by Merriam-Webster, Incorporated (www.Merriam-Webster.com)].
The optical radiation spectrum will be treated in this text, including the ultraviolet, visible, and infrared regions. The visible portion of the optical spectrum covers a rather narrow band of wavelengths between 380 nm and 760 nm; the radiation between these limits, perceivable by the unaided normal human eye, is termed “light.” Measurements within this region may be called “photometric” if the instruments used incorporate the response of the eye. The short wavelength (ultraviolet) limit of radiometric coverage is about 200 nm, approximately the shortest wavelength that our atmosphere will transmit. The longest wavelength (infrared) treated in this book is about 100 μm. This wavelength range includes 99% of the energy (95% of the photons) from a thermal radiator at 0° C (273.16 K).
1.2 Why Measure Light? But why measure light in the visible, ultraviolet, or even infrared region? What are these measurements good for? Let's look at some historical perspectives: I 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
Figure 1.2 The electromagnetic spectrum.2 [Reprinted by permission of author from Optical Radiation Measurement series, Vol. 1, F. Grum and R. J. Becherer, Radiometry, p. 1 (1979)].
Introduction to Radiometry 3
4
Chapter 1
...nobody will object to an ardent experimentalist boasting of his measurements and rather looking down on the “paper and ink” physics of his theoretical friend, who on his own part is proud of his lofty ideas and despises the dirty fingers of the other. Max Born If you are really doing optics, you get photons under your fingernails. James M. Palmer Measurement is the point at which the rubber meets the road. Hypotheses, uncorroborated by measurement, cannot fulfill the same function. And if rubber doesn’t meet the road, the car cannot move. The measurement of light is often critical in transitioning from theory to the development of systems and techniques. Although instrument and system design may be based on theory, performance evaluation and system improvement require that accurate radiometric measurements be applied. When calibrated measurements are needed, that is, when field or laboratory measurements must be correlated with specific values presenting the relationship between measured phenomena and an absolute standard, radiometric measurements take on even greater significance.
1.3 Historical Background Scientists and engineers have been involved in the measurement of light since the early experiments and instruments described by P. Bouguer in 1729 and by J. H. Lambert in 1760. Exploration into other spectral regions began with the discovery of the infrared region by W. Herschel in 1800 and the ultraviolet region by J. W. Ritter the following year. Table 1.1 shows some of the significant events in the history of radiometry and photometry. Table 1.1 Some significant events in radiometry.
Year ? 1666 1729 1760 1800 1801 1830 1837 1839 1859
Event ... and then there was light! Investigation of the visible spectrum Inverse square law Cosine law, exponential absorption Discovery of the infrared region Discovery of the ultraviolet region Radiation thermopile (first practical detector) Calorimetric detector Photoelectric effect Relation between absorption and emission
Principal investigator from Genesis Newton Bouguer Lambert Herschel Ritter Nobile, Melloni Pouillet Becquerel Kirchhoff
Introduction to Radiometry
5 Table 1.1 (Continued.)
Year 1860
Event Standard lamp fueld by sperm whale oil
1879 1880 1892 1893 1900 1900 1905 1910 1931
Incandescent lamp (carbon filament) Bolometer Integrating sphere (theory) Absolute thermal detector (pyrheliometer) Blackbody radiation theory Integrating sphere (reduction to practice) Photoelectric effect Tungsten lamp Adoption of colorimetric standards
1936
Photomultiplier tube (multistage)
1938 1948
Pyroelectric detector (theoretical) Adoption of platinum blackbody for standard candela
1954
Silicon photodiode
1955 1960
Pyroelectric detector (reduction to practice) Invention of light amplification by stimulated emission of radiation (LASER) Tungsten-halogen lamp Laser calorimetry Fourier transform spectrometer Photometry relegated to subset of radiometry Self-calibrated silicon detector Trap detector Definitive measurement of the StefanBoltzmann constant Cryogenic absolute radiometer
1961 1970 1975 1977 1980 1983 1984 1985
Principal investigator British Metropolitan Gas Act Edison Langley Sumpner Angstrom, Kurlbaum Planck Ulbricht Einstein Collidge International Commission on Illumination (CIE) Zworykin, Morton & Malter Ta Consultative Committee on Photometry and Radiometry (CCPM) Chapin, Fuller & Pearson Chenowyth Maiman Zubler & Mosby West Vanesse CCPM Zalewski & Geist Zalewski & Duda Quinn & Martin Martin, Fox, & Key; Foukal
1.4 Radiometric Measurement Process This book describes the many facets of optical radiation measurement, from radiation sources to detectors and signal processing. To fully understand and appreciate a radiometric measurement, we must understand the processes of
6
Chapter 1
generation, transmission, and detection of optical radiation. In addition, we must possess a firm grasp of the underlying mathematics and what is loosely called “measurement science.” A generic radiometric configuration is shown in Fig. 1.3. A target, or object of measurement interest, can be either active, emitting radiation by virtue of its temperature or some form of atomic excitation, or passive, reflecting radiation from a different active or passive illuminator. Examples of active sources include the sun, tungsten or fluorescent lamps, lasers, and any nontransparent object with a temperature greater than 0 K. Passive sources include the entire range of natural and artificial reflective surfaces that make up our environment. An additional source of optical radiation can be classified as background, the radiation that may be in our instrumental field of view along with the target. Also included is the intervening medium, the atmosphere, which includes both radiation sources and sinks, acting through the mechanisms of absorption, emission, and scattering. The myriad of small arrows in Fig. 1.3 represent scattered, absorbed, emitted, and reflected radiation. After traversing the atmosphere, the rays from the target (and possibly the background as well) reach our instrument, whose parameters define the ranges over which the spatial, spectral, temporal, and radiometric characteristics of incoming radiation will be accepted. This is accomplished through the use of lenses and mirrors, choppers and apertures, prisms, gratings, filters, attenuators,
Figure 1.3 Generic radiometric configuration.2 [Reprinted by permission of author from Optical Radiation Measurement series, Vol. 1, F. Grum and R. J. Becherer, Radiometry, p. 7 (1979)].
Introduction to Radiometry
7
polarizers, optical fibers, etc. The optical radiation transmitted through the instrument is finally incident on one or an array of detectors, transducers which convert the incident optical radiation to a more tractable form of energy. Detectors may be thermal (thermoelectric, bolometric, and pyroelectric) or photon (photoemissive, photoconductive, and photovoltaic) in character; other viable detectors include the human eye and photographic film. The final block in the diagram involves signal processing. Most of the detectors in common use generate electrical signals. Postdetector processing may include filtering, linearization, and background suppression before the processed result is recorded and displayed. Even the eye and film detectors include processing steps, such as the filtering and interpretation of information by the brain and the photographic development process for film.
1.5 Radiometry Applications The fields in which radiometric instruments and techniques are used are very diverse. Table 1.2 lists some of the more common applications.
Table 1.2 Common applications of radiometry.
Appearance measurement Astrophysics Atmospheric physics Clinical medicine Colorimetry Diagnostic medicine Remote-sensing satellites Electro-optics Illumination engineering Laser measurements Materials science Meteorology Military systems Night-vision devices Optoelectronics Photobiology Photochemistry Photometry Radiative heat transfer Solar energy Television systems Visual displays Vision research
8
Chapter 1
Most books on radiometry begin with a vast and often confusing array of terms, definitions, etc. In this work, detailed listing of terminology is relegated to the glossary in Appendix E, and radiometric terms will be introduced as they are needed. Radiometry and photometry are applied to a variety of phenomena whose output occurs over many orders of magnitude. Tables 1.3 through 1.5 illustrate the ranges of illumination encountered. “Luminance” is power per unit area and unit solid angle weighted by the spectral response of the eye; its units are lumens per square meter per steradian (lm/m2/sr), or candelas per square meter (cd/m2). “Illuminance” is power per unit area weighted by the same function; its units are lumens per meter squared (lm/m2).*
Table 1.3 Luminances of astronomical sources.
Source Night sky, cloudy, no moon Darkest sky
10–4 4 × 10–4
Night sky, clear, no moon
10–3
Night sky, full moon
10–2
Clear sky 0.5 hr after sunset
0.1
Clear sky 0.25 hr after sunset
1
Cloudy sky at sunset
10
Gray sky at noon
102
Cloudy sky at noon
103
Moon
*
Luminance (cd/m2)
2.5 × 103
Average clear sky
8 × 103
Clear sky at noon
104
Solar disk
1.6 × 109
Lightning
8 × 1010
For more thorough discussions of photometry, see J. T. Walsh, Photometry, Dover, New York (1958); P. Moon, The Scientific Basis of Illuminating Engineering, Dover, New York (1961); and R. McCluney, Introduction to Radiometry and Photometry, Artech House, Boston (1994).
Introduction to Radiometry
9
Table 1.4 Luminances of practical sources.
Source Minimum visible level (human) Scotopic vision valid (human) Photopic vision valid (human) Green electroluminescent T8 cool white fluorescent Acetylene burner 60 W inside frosted lamp Candle Sodium vapor lamp High-pressure Hg vapor lamp Tungsten lamp filament Plain carbon arc crater Cored carbon arc crater Atomic fusion bomb
Luminance (cd/m2) 3 × 10–6 < 0.003 >3 25 104 105 1.2 × 105 6 × 105 7 × 105 1.5 × 106 8 × 106 1.6 × 108 109 1012
Table 1.5 Illuminances of various sources.
Source Absolute minimum Mv = 8 Typical minimum Mv = 6 0 Mv star outside atmosphere Venus Full moonlight Street lighting Recommended reading Workspace lighting Lighting for surgery
Illuminance (lm/m2) 1.6 × 10–9 10–8 2.54 × 10–6 1.3 × 10-4 1 10 102 102 to 103 104
References 1. Webster’s Third New International® Dictionary, Unabridged, MerriamWebster, Inc. (1993). 2. F. Grum and R. J. Becherer, Radiometry, Vol. 1 in Optical Radiation Measurements, F. Grum, Ed., Academic Press, New York (1979).
Chapter 2
Propagation of Optical Radiation 2.1 Basic Definitions 2.1.1 Rays and angles A ray is represented by a vector: a straight line indicating the magnitude and direction of propagation. A wavefront is a notional surface locally normal to a ray. Thus, a wavefront could be a plane (all rays parallel, as if from infinity) or a curved surface (indicating diverging rays, as if emanating from a point). Figure 2.1 illustrates rays and wavefronts for both cases. A beam can be defined by two separated elements of area as shown in Fig. 2.2. It is thus the locus of possible rays that pass through the two areas separated by distance d. As dA1 and dA2 approach zero, the beam approaches a single ray. Since there is a small, but nevertheless finite, cross-sectional area associated with a beam, it is capable of carrying power without the flux density (power per unit area) approaching infinity.
(b)
(a)
Figure 2.1 (a) Plane and (b) diverging wavefronts, with arrows indicating the direction of radiation propagation. 11
12
Chapter 2
d dA1
dA2 Figure 2.2 A beam between two area elements.
The speed of light is the rate at which light propagates through a vacuum. It is represented by c and is a constant equal to about 3.00 × 108 ms–1, a faster rate of travel than in any other medium. The index of refraction of a medium is the ratio of the speed of light in vacuum to the speed of light in the medium: n=
c , v
(2.1)
where n = index of refraction of the medium, and v = velocity of light in the medium. Because c must be greater than or equal to v in Eq. (2.1), a medium’s classical index of refraction is constrained to values equal to or greater than one. Examples of nominal indices of refraction of various media are shown in Table 2.1. Snell’s law allows one to calculate the angle change in propagation direction upon refraction: n1 sin θ1 = n2 sin θ2 . (2.2)
Note from Fig. 2.3 that the refracted ray is closer to the normal to the boundary between media when the index of refraction is higher; that is, when n2 > n1. 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 Al2O3 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
Propagation of Optical Radiation
13
θ1 θ2 n1=1
n2=2
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 normal to the surface’s unit vector. The differential form is dAp = cosθ dA, where θ is the angle between the line of observation and the local surface normal n. Integrating over the surface area we obtain Ap = cos θ dA .
(2.3)
A
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 θ, with l the arc length and r the circle radius. The solid angle ω extends the plane-angle concept to three dimensions. It is the ratio of the element of spherical area dAsph to the square of the sphere radius r2. Figure 2.6 illustrates a solid angle. The unit of the plane angle is the radian, defined as: The radian is the plane angle between two radii of a circle that cut off on the circumference an arc equal in length to the radius.1 Table 2.2 Shapes and corresponding surface- and projected-area formulas.
Shape
Surface area
Projected area
Rectangle
A = LW
Ap= LW cosθ
Flat circle
A = πr2 = [π(d2/4)]
Ap = πr2 cosθ = [π(d2/4)] cosθ
Sphere
A = 4πr2 = πd2
Ap= A/4 = πr2
14
Chapter 2
n
θ
Ap
Asph
Ap
(a)
(b)
Figure 2.4 Surface- and projected-area relationships for a (a) circle and (b) sphere.
l r θ
θ = l/r
Figure 2.5 Planar-angle relationships.
dAsph r ω = dAsph r2
Figure 2.6 Solid-angle relationships.
Propagation of Optical Radiation
15
rsinθdφ dφ
rdθ dAsph
θ
dθ
r
φ rsinθdφ rsinθ Figure 2.7 An element of solid angle in spherical coordinates.2
In other words, l = r defines one radian. Since there are 2π radians in a circle, the conversion between degrees and radians is 1 rad = (360/2π) = (180/π) degrees. The steradian (sr) is defined in an analogous manner: The steradian is the solid angle that, having its vertex in the center of a sphere, cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere.1 Dividing the entire surface area of a sphere by the square of its radius, we find that there are 4π sr of solid angle in a sphere, and 2π sr in a hemisphere. Figure 2.7 depicts the solid angle in terms of the planar angle θ and the rotational (azimuthal) angle φ, where r is sphere radius. The small element of area dAsph lies on the surface of the sphere. The element of solid angle subtended by dAsph is expressed as: dω =
dAsph r2
= sin θd θd φ .
(2.4)
To determine the value of the solid angle, integrate over θ and φ: ω = sin θd θd φ . φ θ
(2.5)
16
Chapter 2
Θ1/2
Figure 2.8 Right circular cone, with Θ1/2 the cone half angle.
In the most general case, a solid angle can subtend a surface of any shape. However, in optical systems, which typically have circular apertures, the envelope of the solid angle is a right circular cone, as shown in Fig. 2.8. In Fig. 2.8, Θ1/2 is the cone half angle. It is the plane angle between the centerline of the cone and anywhere on the edge of the cone. It is related to the solid angle of the cone ω by: 2π
Θ1/ 2
0
0
sin θd θ ,
(2.6)
ω = 2π(1 − cos Θ1/ 2 ) .
(2.7)
ω=
dφ
which results in
Application of a simple trigonometric identity provides the equivalent expression Θ ω = 4π sin 2 1/2 . (2.8) 2 A feel for the magnitudes of various solid angles can be obtained by inspecting Table 2.3. Table 2.4 facilitates conversion from steradians to other units. Table 2.3 Some objects and corresponding solid angles.
Object Dime at 1 km Jupiter (mean) Cone in human eye Sun, moon at earth’s surface Tennis ball at 1 m Plane Sphere (from the inside)
Linear subtense 0.065 arcsec 0.65 arcsec 1 arcmin 32 arcmin = 0.53 deg 3.7 deg — —
Solid angle ω 3.1 × 10–13 sr 3.1 × 10–11 sr 2.7 × 10–7 sr 6.8 × 10–5 sr 0.0033 sr
2π 4π
Propagation of Optical Radiation
17
Table 2.4 Steradian conversions.
1 sr = 1 rad2 1 sr = 3282.8 deg2 1 sr = 1.1818 × 107 arcmin2 1 sr = 4.2545 × 1010 arcsec2 Both plane angles and solid angles are dimensionless quantities, and their use can lead to confusion when attempting dimensional analysis. For example, the simple inverse square law of irradiance (to be discussed in detail in Sec. 2.3.1), E = I/d 2, appears dimensionally inconsistent. The left side has units W/m2, while the right side has W/m2sr. It has been suggested that this equation be written E = I Ωo/d2, where Ωo is the unit solid angle, 1 sr. Inclusion of the term Ωo will render the equation dimensionally correct, but Ωo will far too often be considered a free variable rather than a constant equal to 1, which leads to erroneous results. Current practice suggests that another type of solid angle, the projected (or weighted) solid angle, is more useful. The symbol for a projected solid angle is Ω, and the units are also steradians. It is defined as the solid angle ω projected onto the plane of the observer, as shown by the defining equation:
d Ω = d ω cos θ = sin θ cos θd θd φ .
(2.9)
It involves an additional cosine term. This is depicted graphically in Fig. 2.9.
Figure 2.9 Projected solid-angle relationships. [Adapted from Radiometric Calibration: 3 Theory and Methods, C. Wyatt, p. 21 (1978)].
18
Chapter 2
To determine the projected solid angle, integrate over θ and φ: Ω = sin θ cos θ d θ d φ .
(2.10)
φ θ
As before, the example most relevant to optical systems is the right circular cone depicted in Fig. 2.8, for which the integral may be expressed as 2π
Θ1/ 2
0
0
Ω = d φ
sin θ cos θ d θ ,
(2.11)
resulting in Ω = π sin 2 Θ1/ 2 ,
(2.12)
where Θ1/2 is the cone half angle, as before. The plane area subtended by the cone is Ωr2, and the spherical surface area subtended is ωr2. A couple of special cases are worthy of mention. For a hemisphere, Ω = π sr, while an entire sphere subtends 2π sr of a projected solid angle. In addition to the dimensional concern raised above, there is another good reason to employ two definitions of a solid angle. For many radiometric problems, the emitter or receiver is flat, and the projected solid angle Ω is the proper choice as it requires the inclusion of the cosθ term relating to the projected area of the surface. In other cases, the emitter or receiver approximates a point, emitting uniformly in all directions or responding to incoming radiation equally from all directions. The solid angle ω is appropriate under these conditions. For a right circular cone at a half angle Θ1/2 of 90 deg, the projected solid angle is π according to Eq. (2.12), and ω is 2π according to Eq. (2.7). At the other extreme, when Θ1/2 is equal to 0 deg, both ω and Ω are zero. For small angles, the solid angle and projected solid angle differ only by a cosine, are nearly identical, and are therefore interchangeable in numeric value, if not in concept. The error incurred by using ω rather than Ω is given in Table 2.5 (note that ω > Ω 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 ω and projected solid angle Ω interchangeably, incurring predictable confusion and potentially incorrect results.
Propagation of Optical Radiation
19
Table 2.5 Percentage error when not using a projected solid angle.
Θ1/2 (in deg)
Error using ω rather than Ω
10
< 1%
16
< 2%
25
< 5%
35
< 10%
48
< 20%
2.1.2 System parameters
So far, we have defined only the terms necessary to describe the angles important to a study of propagation. Now, let’s define quantities specific to optical systems. The f-number (f/#) of an optical element or system is most often seen as the ratio of the effective focal length f (object at infinity) of the optical element or system to the diameter D of the entrance pupil:* f #=
f . D
(2.12)
Good optical systems fulfill the Abbe sine condition with a spherical wavefront converging to the focal point, and the preferred definition of f/# is
f #=
1 . 2sin Θ1/2
(2.13)
The numerical aperture (NA) of a system is defined as the sine of the vertex angle (half angle) of the largest cone of meridional rays that can enter or leave an optical system or element, multiplied by the refractive index of the medium in which the vertex of the cone is located. It is generally measured with respect to an object or image point, and will vary as that point is moved. The defining equation is NA = n sin Θ1/2 . Both f/# and numerical aperture can be related to projected solid angle:
*
For an informative discussion of f/#, see D. Goodman, “The f-word in optics,” Optics & Photonics News, p. 38, April (1993).
(2.14)
20
Chapter 2
Ω=
π
,
(2.15)
π(NA) 2 , n2
(2.16)
n . 2NA
(2.17)
4( f #) 2
and Ω= and may be related to one another by f /# =
Throughput, also known as geometrical extent or étendue, is the product of the cross-sectional area of a beam and its projected solid angle
T = AΩ .
(2.18)
Figure 2.10 shows the interaction between two differential area elements dA1 and dA2 a distance d apart. The area dA1 subtends a solid angle ω1, having apex at dA2, while dA2 subtends a solid angle ω2, whose apex is at dA1. The surface normals are shown in the figure by boldface n. The throughput relationships become
T1→ 2 = d ω1dA2 cos θ 2 = T2→1 = d ω2 dA1 cos θ1 =
dA1 cos θ1dA2 cos θ 2 d
2
dA2 cos θ 2 dA1 cos θ1 d
2
, (2.19)
.
By inspection, T1→2 = T2→1 = T . The throughput is invariant. dA1
ω1
ω2 dA2
n1
d
n2
Figure 2.10 Area- and solid-angle relationships used to define throughput. (Adapted from a figure courtesy of William L. Wolfe.)
Propagation of Optical Radiation
21
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: Ωos is the angle the optics subtend at the source, Ωso the angle the source subtends at the optics, Ωdo the angle the detector subtends at the optics, and Ωod the angle the optics subtend at the detector. For this case, the following equalities hold: As Ωos = Ao Ω so = Ao Ω do = Ad Ωod .
(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 ΑΩ product to use. The author’s (Palmer) personal preference is the most often-used pair AoΩdo, inasmuch as the entrance aperture size Ao and the field of view of the system Ωdo are determinable characteristics of a radiometer. The next most often used pair is AdΩod, as the detector size and the f/# 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 n 2T = n 2 AΩ , and the relationship between medium 1 and medium 2 is
(2.21)
22
Chapter 2
n12T1 = n2 2T2 ,
(2.22)
where the subscripts denote the respective quantities in media 1 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 AΩ product. First, a spectrometer: Fig. 2.13 shows the area- and solid-angle product of the entrance slit (typically 1 × 10 mm) and the projected solid angle the collimating lens subtends at the slit, Ωls. The AΩ 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 f/# 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 f/#, 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.
Propagation of Optical Radiation
23
Slit, Area A
Lens
Ωls
Figure 2.13 Example of the correct area (A) and solid angle (Ωls) 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.† 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 intermediate pupil planes in a complex optical system.
Image
Marginal Ray
Object Chief Ray
Figure 2.14 Chief and marginal rays in an optical system, shown schematically.
†
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
NP CHIEF RAY
XP
MARGINAL RAY AS FS 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 (dQ/dt), has the symbol Φ, 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 come 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 W/m2sr. The defining equations are d 2Φ d 2Φ d 2Φ ΔΦ , L = lim = = = ΔAs , Δω→ 0 ΔA cos θΔω s dAs cos θd ω dAs d Ω dAp d ω
(2.23)
Propagation of Optical Radiation
25
where θ 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 and/or 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/m2. Integration over area yields radiant intensity expressed in W/sr. 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.
dω d θ n Figure 2.16 Radiance from area element dA, tilted at angle θ 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: Φ = L dAs d ωs cos θ = L dAs d Ω .
(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 (dΦ/dAs). The symbol for radiant exitance is M and the units are W/m2. Its defining equation is
ΔΦ d Φ M = lim . = ΔAs → 0 ΔA s dAs
(2.25)
The radiant exitance of a source is obtained by integrating radiance over the projected solid angle of a hemisphere:
M = L dΩ .
(2.26)
π
Integrating radiant exitance in W/m2 over area results in radiant power Φ in watts: Φ = M dAs .
(2.27)
A
Figure 2.17 further illustrates the concept of radiant exitance. The relationship between radiant exitance and radiance is complex, depending on the angular distribution of radiance L(θ,φ). This is illustrated in Fig. 2.18 and described mathematically in the following equations:
d2Φ =
L ( θ, φ ) dA1 cos θ dA2 r2
,
(2.28)
where r is the radius of the hemisphere, as shown in Fig. 2.18, dA1 is an element of the emitting area, dA2 an element of the receiving area, and θ is the angle between the surface normal of dA1 and the direction of propagation. Making a substitution for dA2 using polar coordinates, dA2 = r 2 sin θd θd φ results in d 2 Φ = L(θ, φ)dA1 sin θ cos θd θd φ .
(2.29)
Propagation of Optical Radiation
27
Figure 2.17 Radiant exitance from a source.
Applying limits for a hemisphere and constructing an integral results in 2π
π /2
0
0
Φ = dφ
L ( θ, φ ) A1 sin θ cos θd θ .
(2.30)
Assuming that L is independent of direction (that is, the source at A1 is Lambertian), then π2
sin 2 θ Φ = 2πLA1 . 2 0
(2.31)
Substituting the equality M = Φ/A and solving, we obtain M = πL .
(2.32)
The error most often committed in radiometry is in this conversion. Don’t make the frequent mistake of forgetting the sine and using 2π! φ r2sinθdθdφ=dA2 r
θ
dA1 Figure 2.18 Relationships between emitting and receiving areas used in deriving the relationship between radiance and radiant exitance.
28
Chapter 2
2.2.3 Irradiance
Irradiance (radiant incidence) is the opposite of radiant exitance, in that it is the power per unit area that is incident on a surface. Its symbol is E and its units are again W/m2: ΔΦ d Φ E = lim . = ΔAs → 0 ΔA s dAs
(2.33)
Irradiance on a surface is obtained by integrating radiance over the projected solid angle of a hemisphere:
E = Ld Ω . π
(2.34)
Integrating irradiance over area results in incident power Φ = EdA .
(2.35)
A
This is shown conceptually in Fig. 2.19. The most common example of irradiance is the solar “constant.” It is the irradiance from the sun incident upon the earth’s atmosphere at the mean earthsun distance. Its numeric value is nominally Eo = 1368 W/m2, but it varies slightly over the years, primarily due to sunspots. Also, the number 1368 is not exact, and other values are often quoted, usually within 0.1% or so. The solar irradiance at the earth’s surface, on the other hand, varies widely due to atmospheric effects; a nominal value of 1000 W/m2 is often used.
Figure 2.19 Irradiance upon a surface.
Propagation of Optical Radiation
29
2.2.4 Radiant intensity
Radiant intensity is power emitted per unit solid angle in a specific direction. Radiant intensity is frequently used when describing the radiation of an isotropic (same in all directions) point source. Its symbol is I and its units are W/sr. Figure 2.20 conceptually illustrates the radiant intensity of a point source and a surface illuminated by it. Equation (2.36) defines radiant intensity mathematically: ΔΦ d Φ I = lim . = Δωs → 0 Δω d ωs s
(2.36)
Intensity is derived from radiance by integrating over area I = LdA ,
(2.37)
A
and radiant power can be derived from intensity by integrating over solid angle ω. (Note that the definition of intensity does not include an area term; therefore, the integration is performed over solid angle, without a cosine projection.) Φ = Id ω .
(2.38)
ω
In the case of the isotropic point source, the radiant power is 4π times the intensity. Conversely, intensity may be found by dividing the total emitted power Φ by 4π. A word of warning: intensity is the most problematic radiometric quantity, because it means different things to different people in different but related fields. Laser physicists are prone to use it as the square of the electric field strength, with units W/m2. Atmospheric scientists and heat transfer engineers use the term “specific intensity,” frequently omitting the “specific,” to mean W/m2sr. Some scientists and engineers equate intensity simply to power, and a few use it to describe spectroscopic line strengths. Which usage is correct?
θ Figure 2.20 Radiant intensity from a point source.
30
Chapter 2
Figure 2.21 Illustration of the inverse square law.
In answering the question, we might invoke precedent. Atmospheric scientists’ usage goes back to Chandrasekhar’s Radiative Transfer (1950)4. The physics community usage goes back as far as Drude’s Lehrbuch der Optik (1900).5 Usage in photometry and radiometry dates back to at least 1760 (Bouguer). But such citing leads only to pointless argument. There is one compelling reason to standardize our usage: intensity is an SI base quantity. Any other use of the term intensity is incompatible with the SI and should be deprecated and abandoned.6,7,8
2.3 Radiometric Approximations 2.3.1 Inverse square law
There are a number of approximations in radiometry, most of them having to do with radiation geometry. The inverse square law of irradiance is perhaps the best known approximation. It states that the irradiance from an isotropic point source varies inversely with the square of the distance from the source. As shown Fig. 2.21, the rays are straight lines diverging from a source having intensity I. At distance d, they fill area A. At a distance 2d, the length of each side has doubled and the area subtended has increased by a factor of four. Assuming a lossless medium, the amount of radiant power at the second surface is the same as that at the first surface, due to the conservation of power. At the second surface, power is spread over a larger area, and irradiance decreases. The relationship between irradiance and distance is given by E=
Φ I = . A d2
(2.39)
The relationship expressed by Eq. (2.39) presumes that the area shown in Fig. 2.21 lies normal to the optical axis. If this is not the case and the surface is tilted at an angle θ to the optical axis, a cosine θ factor must be included in the equation: I cos θ E= . (2.40) d2 This situation is illustrated in Fig. 2.22.
Propagation of Optical Radiation
31
I
A θ
d n Figure 2.22 Area A tilted at angle θ from the axis.
Inverse square laws pervade all of physics, beginning with Newton’s universal law of gravitation. Our radiometric application works well for small sources observed at great distances. But how well may a real source approximate an isotropic point source? To avoid the question entirely, one might measure the source only in a welldefined, specified direction. This approach will allow application of the inverse square law, but results may not be repeatable if measurements are made in a different direction. (In photometric terminology, source output under this condition is termed “directional candlepower.”) Another approach involves use of a “small” source rather than a point source—once we determine what “small” is. If the maximum dimension of the source is less than d/10, i.e., one-tenth of the distance from the source (equivalent to about a 3-deg half angle), the inverse square law may be applied with an error of less than 1%. If this is not good enough, a stricter criterion may be applied. If the maximum source dimension is less than d/20 (1.5-deg half angle), the error in applying the inverse square law is less than 0.1%. 2.3.2 Cosine3 law
The cosine3 law is applied to calculate the irradiance on a plane surface from an isotropic point source, for example, the distribution of irradiance over a floor from a bare light bulb suspended above. It is often convenient to describe the law in terms of θ rather than distances along the surface, as the expression is somewhat simpler. Referring to Fig. 2.23, the irradiance at a point x directly below the source with intensity I may be calculated according to the inverse square law as Ex =
I . D2
At position y, the distance from the source has increased such that d=D/cosθ. If the target at y is perpendicular to the vector between source and target, the irradiance at y is: Ey =
I I cos 2 θ = . d2 D2
32
Chapter 2
Is
θ d
D
θ x
y Figure 2.23 The cosine3 law.
If the target is now rotated so that it is parallel to surface x-y (the floor), an additional cosθ term is introduced, resulting in Ey =
I cos3 θ , D2
E=
I cos3 θ , D2
or more generally, (2.41)
which is the cosine3 law. Note that two cosines were introduced to account for the increased distance from the source; the third accrued from the projected area of the target. 2.3.3 Lambertian approximation
Before discussing the final important approximation, another look at Lambertian radiation is needed. A Lambertian source is one whose radiance L is independent of direction: L (θ,φ) = constant.
Thus, in the equation d 2 Φ = L(θ, φ)dA cos θd Ω ,
the power received from a Lambertian radiator is proportional only to cosθ, the angle of observation, and by extension, the projected area of the source.
Propagation of Optical Radiation
33
The Lambertian approximation is frequently used to describe the angular distribution of radiation from a source. It is powerful because it significantly reduces the complexity of the mathematical solutions to radiative transfer problems, allowing them to be simplified so that calculation focuses only on radiation geometry. Fortunately, the Lambertian approximation is surprisingly good. Most conventional blackbody radiation simulators (often used as infrared sources) and integrating spheres (often used as visible sources) are very nearly Lambertian over a wide range of angles. Reflective sources, also called “flat” or matte reflectors (flat paint, matte white paper, etc.) are quite Lambertian out to nearly grazing angles. Most natural surfaces, with the exception of still water, are highly Lambertian. Surprisingly, polished metal surfaces, which are highly polarized in emission at angles away from normal, are Lambertian out to 50 deg or so when the total radiation is taken into account. By contrast, specular reflecting surfaces, such as glossy paints, mirrors, still water, etc., do not lend themselves to the Lambertian approximation. It is common practice to utilize the Lambertian approximation, but make sure to test the assumption with measurement whenever possible, and to keep track of its effect on radiometric calculations. 2.3.4 Cosine4 law
The cosine4 law of irradiance is similar to the cosine3 law, except that a small Lambertian source with radiance L replaces the isotropic point source. Such a source might be the combination of a single lamp with a flat diffuser panel oriented parallel to the ground, as shown in Fig. 2.24. Since the basic geometrical layout is the same as that in the previous example, start with the cosine3 law: I cos3 θ E= . D2 Ls
θ d D
θ x
y Figure 2.24 The cosine4 law.
34
Chapter 2
Recall that the projection of the target onto the floor added the third cosθ term; the projection of the source area in the direction of point y adds the fourth. For a Lambertian source, the relationship between intensity (point source) and radiance (extended source) is I = LAproj. Substituting, E=
LAp cos 4 θ D2
,
(2.42)
which is the cosine4 law. This fourth-power reduction in irradiance (falloff) applies to many, but not all, sources that are extended and Lambertian. The effect is a noticeable dimming of brightness near the image edges, often seen in images from projection systems and wide-angle cameras. The effects of cosine3 and cosine4 falloff are shown numerically in Table 2.6 for a variety of angles. There are some cases in which a source may be “too extended” for the cosine4 law to correctly predict irradiance. Consider the configuration shown in Fig. 2.25, in which a circular source (shown edge on) radiates downward to the observation plane. The angle θ is the angle of incidence to the surface, and angle Θ1/2 is the half angle of the system viewing the source, related to the system’s f/# by Eq.(2.13). Values of Θ1/2 for a variety of common f/#s are shown in Table 2.7. Table 2.6 Fractional irradiance at observation point as a function of cosine factor.
Cosine factor
0
10
20
30
40
50
60
70
80
cos θ
1
0.98
0.94
0.87
0.77
0.64
0.50
0.34
0.17
cos3θ
1
0.96
0.83
0.65
0.45
0.27
0.13
0.040
0.005
cos4θ
1
0.94
0.78
0.56
0.34
0.17
0.063
0.014
0.001
Angle θ (degrees)
Circular source viewed edge on
θ Θ 1/2
Figure 2.25 Foote’s formula geometry.
Propagation of Optical Radiation
35
Table 2.7 System half angles as a function of f/#.
f/# 16 8 4 2.8 2 1.4 1
Φ1/2 (deg.) 1.8 3.6 7.2 10.3 14.5 20.9 30
An expression for the irradiance at the sensor for this particular geometry has been derived by Foote (1915).9,10 Foote’s formula is shown below, and the corresponding plot using f/# as a parameter is shown in Fig. 2.26. For this particular geometry, irradiance is least likely to follow the cosine4 law in lower f/# systems. Note that at normal incidence (θ = 0), the equation reduces to E = πLsin2θ. 1 + tan 2 θ − tan 2 Θ1 2 πL E= . 1 − 1/ 2 4 2 2 4 2 tan θ + 2 tan θ(1 − tan Θ1 2 ) + sec Θ1 2
Figure 2.26 Foote’s formula results compared to Lambertian.
36
Chapter 2
2.4 Equation of Radiative Transfer The equation of radiative transfer is the most important equation in radiometry. All axioms discussed in the previous section depend on it. It is, in differential form: d 2 Φ1→2 =
L(θ, φ)dA1 cos θ1dA2 cos θ2 . d2
(2.43)
The physical situation described by the equation is shown in Fig. 2.27, where dA1 and dA2 are the differential area elements, d is the distance between them, and θ1 and θ2 are the angles between the normals to the area elements and the optical axis. As the equation indicates, the radiant power received at surface 2 depends directly on the radiance L(θ,φ), the area elements, the angles between area normals and the optical axis, and inversely on the square of the distance between the surfaces. If the distance squared is much larger than the largest area, i.e., d 2 >> (A1 or A2), the differential areas may be replaced with the actual areas, and Eq. (2.44) can be applied to calculate power: Φ1→2 =
L(θ, φ) A1 cos θ1 A2 cos θ2 . d2
(2.44)
Because d 2 is much greater than the size of any area element, the variations in θ and φ are small, and the term L(θ,φ) need be evaluated at only one particular set of angles. If either area is appreciable in relation to d 2, Eq. (2.44) cannot be used. The integral form of the equation of radiative transfer looks much more formidable:
Φ1− 2 =
L(θ, φ) cos θ1 cos θ2 dA1dA2 . d2 A2 A1
(2.45)
L θ1
d θ2 dA2
dA1 Figure 2.27 Radiation transfer geometry.
Propagation of Optical Radiation
37
To utilize this integral form, the following factors must be taken into account: (1) The angles θ1 and θ2 may vary from one part of area A1 or area A2 to another. (2) The distance d may also vary from one part of area A1 or area A2 to another. (3) The angular variation in radiance L(θ,φ) may be significant. An additional implied assumption is that the principle of superposition is applicable when adding up the small elemental contributions to radiant power; in other words, the source providing the radiation is incoherent and interference effects do not occur in the beam. If the source is Lambertian, the radiance is independent of θ and φ, and the equation of transfer becomes much simpler. It is cos θ1 cos θ2 dA1dA2 . d2 A2 A1
Φ1− 2 = L
(2.46)
In this case, the transfer equation can be thought of as the product of a radiance term L and the geometric term expressed by the double integral. The transfer equation can be simplified even further if certain assumptions can be made regarding the radiation geometry: first, that the square of the distance d 2 is much larger than either area A1 or A2, and second, that both areas are on axis, θ1 and θ2 are zero, and their cosines are therefore unity. If these assumptions can be made in conjunction with the Lambertian approximation, the result is the ultimate simplification: Φ1→2 = LAΩ .
(2.47)
A corollary expression to calculate flux density, power-per-unit area in a system, is E = LΩ . (2.48) These simple equations are the logical starting points for all radiometric engineering calculations, as they provide first-cut, back-of-the-envelope answers. In many instances, they are all we need. In order to fully understand a particular application, assumptions must be tested and the errors incurred by their use assessed. The Lambertian approximation is relatively simple to verify if we possess hard data about the directionality of the source. The on-axis assumption is also easy to verify. As discussed in Sec. 2.3, the distance between the two area elements must be at least 10× the maximum linear dimension of the largest
38
Chapter 2
element in order for the inverse square law to be good to 1%. If the distance between the two is increased to 20× the maximum dimension, the uncertainty in applying the law reduces to 0.1% or less.
2.5 Configuration Factors The concepts of radiative transfer are used not only in optics, but also in related fields, such as thermal and illumination engineering. Equation (2.46) shows the separation of the equation of radiative transfer into a radiance term and a geometry term, assuming the Lambertian approximation may be applied. Other names for the geometry term are view factor, interchange factor, shape factor, form factor, and the term we will utilize, configuration factor. The symbol F is used to denote this quantity, defined by Eq. (2.49): F=
Φ1→ 2 , Φ1
(2.49)
where Φ1 is the power leaving surface 1 and Φ1→ 2 is the power reaching surface 2 from surface 1. Both power terms are dimensionless. The radiant power terms are further defined as Φ1 = M 1 A1 , where M1 is the radiant exitance in W/m2 leaving surface 1, and A1 its area; and Φ1→ 2 =
M 1 cos θ1 cos θ2 dA1dA2 , d2 π
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 F12 =
cos θ1 cos θ2 1 dA1dA2 , πA1 d2
which is the configuration factor. The power transferred from surface 1 to surface 2 then becomes Φ1→ 2 = Φ1 F12 = M 1 A1 F12 = πL1 A1 F12 .
Propagation of Optical Radiation
39
Because the radiant power incident on surface 2 equals L1A1Ω21 by Eq. (2.47), Ω 21 = πF12 , and Φ1→2 = L1 A1Ω 21 . To restate: in engineering calculations, using the assumptions given above,
Φ1→2 = M 1 A1 F12 = L1 A1Ω 21 ,
(2.50)
E2 = L1Ω12 .
(2.51)
and
The advantage of using configuration factors is that numerous solved geometries appear throughout the literature. Some relevant to optics are shown in Fig. 2.28.11 Further information on configuration factors may be found in Refs. 12, 13, and 14.
Fd 1−2 =
1
(h r )
2
Fd 1− 2
+1
r = h
2
A2
A2 r r h h dA1 (a)
(b)
dA1
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: A1 and A2 are the areas of the stops; D is the distance from source to stop at A1; and S is the distance from A1 to A2 (at the detector d). The difference between the two configurations is solely the presence of the lens at A1 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): Ed =
Is , d2
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:
Φ d = Ed A2 .
(2.52)
Expressed in terms of intensity, the power is Φd =
I s A2 , ( D + S )2
(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 A1, the field stop, defines the detector’s field of view.
(a)
(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
Φ d = Ed A1 ,
(2.54)
Φ d = τlens E A1 A1 ,
(2.55)
or, more specifically, as
where τlens is the lens transmission. In terms of intensity, it is:
Φd =
τlens I s A1 . D2
(2.56)
Note that in this case, the aperture that limits the flux into the system is A1, 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=
τlens A1 , A2
(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):
Φ d = Ls AΩ . At area A = A2, the appropriate solid angle is subtended by area A1. This solid angle is expressed as in Eq. (2.12) by
Ω12 = π sin 2 Θ1/ 2 , where Θ1/2 is the cone half angle. Assuming that it is small, the approximation A1/S2 may be used for Ω12, so that AΩ =
A1 A2 , S2
(2.58)
42
Chapter 2
and
Φd =
Ls A1 A2 . S2
(2.59)
The same approach can be pursued with the other area–solid angle combination, that is, with A1 and Ω21. In that case, the solid angle is approximately A2/S2, and Φd is again obtained by Eq. (2.59). Inserting a lens at A1 limits the power by the transmission of the lens, so that
Φd =
τlens Ls A1 A2 . S2
(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 Ωsd, which may also be expressed by Eqs. (2.12) and (2.15) as
Ω sd = π sin 2 Θ1/2 =
π 4( f #) 2
,
(2.61)
where f/# is defined in Eq. (2.13). Considering the geometry in the figure, sin 2 Θ1/ 2 =
a2 . (a 2 + b 2 )
We can now substitute in Eq. (2.48) to provide several equivalent expressions for the irradiance at the detector: Ed = πL sin 2 Θ1/ 2 = LΩ sd = πL
a2 πL = = πL( NA) 2 . 2 2 (a + b ) 4( f #) 2
(2.62)
Propagation of Optical Radiation
43
a
Θ1/2
Ad
b Ls
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 πLa2b–2. The error incurred using a2b–2 rather than a2(a2 + b2)–1 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 πLa2 (the radiance times the area of the source) so that I s = LAs Ed =
Is . b2
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=0
Half-angle Θ1/2 (deg) very small 26.5 45 90
Irradiance Ed πLa2b–2=Ib–2 πL/5 πL/2 πL
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
a Θ1/2
Ad
Ls b 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 = Locosθ. In this case, the radiant exitance is
M s = Lo cos θd Ω ,
(2.63)
where dΩ is taken from Eq. (2.9). The resulting integral is then 2π
π /2
0
0
M s = Lo d φ
sin θ cos 2 θd θ ,
(2.64)
which results in Ms =
2πLo . 3
(2.65)
Assuming a lossless medium, the irradiance at the detector is Ed =
2πLo 1 − cos3 Θ1/ 2 ) . ( 3
(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 sin 2 Θ1/ 2 =
a2 a2 = , (a + b) 2 a 2 + 2ab + b 2
(2.67)
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=0
Half-angle Θ1/2 (deg) Very small 19.5 30 90
Irradiance Ed πLa2b–2=Ib–2 πL/9 πL/4 πL
and the expression for irradiance at the detector is Ed = πL
a2 . a 2 + 2ab + b 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 Ed = πL
a 2 LAs I s = 2 = 2. b2 b b
(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.
a Θ1/2
Ad
Ls b 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=0
Half-angle Θ1/2 (deg) very small 30 90 —
Irradiance Ed πLa2b–2=Ib–2 πL/4 πL —
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 dA1 and dA2, 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): d 2 Φ1→2 =
L(θ, φ)dA1 cos θ1dA2 cos θ2 . d2
By inspection, θ1 = θ2 = θ and cosθ = d(2R)–1. Also assume that dA1 = dA2 = dA. If the interior of the sphere is Lambertian, i.e., coated with material having Lambertian properties, then dE =
d Φ LdA = . dA 4 R 2
(2.70)
dA1 θ1
R
d θ2
dA2
R
Figure 2.33 The integrating sphere. (Adapted from Ref. 15 with permission from John Wiley & Sons, Inc.)
Propagation of Optical Radiation
47
This result means that irradiance within the sphere, for any area element dA, is independent of position θ 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 Φ 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=
Eρ , π
(2.71)
where ρ 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 dE =
EρdA . 4πR 2
(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 E=
ρΦ . 4πR 2 (1 − ρ)
(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-Lambertian 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 is16 L=
ρΦ , πAsph [1 − ρ(1 − 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 10–6 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 f/2 with a focal length of 1 m. The mirror diameter D is f/(f/#), or 0.5 m. The area of the mirror (assumed circular) is πD2/4. The irradiance on this (perfectly reflecting) mirror is Em =
Φ m 4 × 10−6 = = 5 × 10−6 W/m 2 . Am πD 2
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 I star = Em d 2 = 5 × 10−6 Wm −2 × (1012 m) 2 = 5 × 1020 W/sr .
I
DET
d = 1012
f/2, 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): L=
σT 4 , π
(2.75)
and its value is 2 × 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 × 109 m, so according to Table 2.2, its projected area is (πdsun2)/4, or 1.54 × 1018 m2. At earth-sun distance d = 1.5 × 1011 m, the solid angle subtended by the sun at the detector is Ωsd = 6.8 × 10–5 sr. Noting also that the sun subtends approximately 32 minutes of arc (arcmin), Ωsd may also be calculated as πsin2(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), is Ed = LΩ sd = 1360 W/m 2 .
(2.76)
Note that the diameter-to-distance ratio is substantially less than 0.1, and the inverse square law may be applied. Calculating intensity as in the example above: I sun = Ed d 2 = 3.06 × 1025 W/sr .
(2.77)
Ap A
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
Φ d = LAp Ω ds =
LAp A d2
,
(2.78)
where Ωds is the solid angle subtended by the detector, and that Ed =
Φ d LAp = 2 = 1368 W/m 2 . A d
(2.79)
Applying the inverse square law as in Eq. (2.77), we obtain Isun = 3.08 × 1025 W/sr. Note that though the numbers are not identical, they are very close. The value of 1368 W/m2 is referred to as the solar constant, and is specifically defined as the irradiance falling upon a 1 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 4π 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/m2 are incident upon the moon as well. If the moon is assumed to be Lambertian, with a reflectance of 0.2, its radiance is Lmoon =
Eo ρ = 87 W/m 2sr . π
(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 = πL Φ = MA Φ = 4πI I = Φ/4π I = Eod2
Value 6.16 × 1018 m2 6.28 × 107 W/m2 3.87 × 1026 W 3.87 × 1026 W 3.08 × 1025 W/sr 3.08 × 1025 W/sr
Propagation of Optical Radiation
51
At the earth’s surface, the angular subtenses of the moon and the sun are the same, approximately 32 arcmin. This means that ΩME, the solid angle subtended by the moon at the earth, is equal to Ωsd, above, with a value of 6.8 × 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 =
Φ M →E = Lmoon Ω ME = 5.9 × 10−3 W/m 2 . ApE
(2.81)
Note that in the above equation, ApE 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 Esun Eo = Emoon Eo , moon
1368 W/m 2 = 2.3 × 105 . 2 −3 5.9 10 W/m ×
Also, comparing radiances we find Lsun 2 × 107 W/m 2sr = = 2.3 × 105 . 2 87 W/m sr Lmoon 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 Eearth = τEo = 1026 W/m 2 , and assuming an earth reflectance of 0.2 along with the Lambertian approximation, the earth’s radiance is Learth =
Eearth ρ = 65 W/m 2sr . π
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 = τLmoon = 65 W/m 2 sr .
52
Chapter 2
This interesting result means that the radiance of an “average” sunlit scene‡ 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 × 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 retroreflective, 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 Et? 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 × 107 W/m2sr. The power from the sun to the collector is Φsc = LsAcΩsc, using the area of the collector and the solid angle the sun subtends at the collector. The irradiance at the collector is then Ec =
Φ s →c = Ls Ω sc , Ac
where Ωsc is the solid angle the sun subtends at the collector, 6.8 × 10–5 sr. Choosing a target diameter (or linear dimension) and system focal length f so that Ωsc = Ωtc, we have
Ωtc =
‡
At and At = Ωtc f 2 . f2
Eastman Kodak has shown through extensive research that the reflectance of an average scene is 18%; all exposure meters are calibrated using this assumption (J. M. Palmer, 2005).
Propagation of Optical Radiation
53
Ac
Ωtc
Ls = 2 x 107 Wm–2sr–1
At
Ωsc=Ωtc f Figure 2.36 The “solar furnace.”
Assuming no transmission loss between collector and target, the irradiance at the target Et is expressed as Et =
Φ c →t Ec Ac E A = = c c2 . At At Ωtc f
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 Ω ct =
π 4( f #) 2
,
so that Et =
πLs , 4( f #) 2
(2.82)
which provides a way of characterizing target irradiance in terms of both source radiance and the f/# system parameter, for a configuration such as this one. 2.9.4 Image irradiance for finite conjugates
The definition of f/# 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 “working f/#,” often symbolized as f/#′, is used.17
54
Chapter 2
As
Ac
At
L Figure 2.37 Finite conjugates.
A working f/# is defined as m f # ′ = f # 1 + m p
,
(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 of the 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. Substituting f/#′ for f/# in Eq. (2.82) we obtain Et =
πL
m 4 ( f # ) 1 + mp 2
2
.
(2.84)
If mp = 1, as it frequently does, Eq. (2.84) becomes the camera equation Et =
πL 4 ( f # ) (1 + m ) 2
2
.
Table 2.13 shows two important cases. Table 2.13 Target irradiance using the camera equation.
Case I – Object at infinity m=0 πL Et = 2 4 ( f #)
Case II – Equal conjugates m=1 πL πL Et = = 2 2 2 4 ( f # ) (1 + 1) 16 ( f # )
(2.85)
Propagation of Optical Radiation
55
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 Ωct 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/m2sr, somewhat less than the 65 W/m2sr 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 Eearth =
Φ sky →earth Aearth
2π
π/ 2
0
0
= Lsky Ω sky −earth = Lsky d φ sin θ cos θd θ
2
Eearth = πLsky sin (90 deg) Eearth = 157 W/m 2 . 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/m2 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 Φd on area Ad is calculated according to Φ d = Ls Ad Ωod = Ls Ad
Ao πDo 2 = Ls Ad , 2 f 4f2
which equates to Φd =
πLs Ad . 4( f #) 2
56
Chapter 2
Ls θ Ad d
f
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.
Figure 2.39 Two projection systems.
Propagation of Optical Radiation
57
The equation for illuminance on the screen resembles the camera equation, Eq. (2.85), with the addition of a cos4θ term to account for the off-axis angle to the screen as seen from the projector: Ev =
τo πLv cos 4 θ 4 ( f # ) (1 + m ) 2
2
,
(2.86)
where Ev is illuminance. The transmission of the optical system το appears also. To maximize irradiance for a given magnification, there are only two possibilities: minimize the f/# 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 cosn 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 τo as discussed above. A term to account for vignetting, fv, 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 Et =
τo πf v (1 − A2 ) L(θ, φ) cos n θ 4 ( f # ) (1 + m ) 2
2
.
(2.87)
Note that this expression does not take into account the spectral nature of the radiation, to be discussed in greater detail in Chapter 4, nor does it account for the transmission of the atmosphere.
58
Chapter 2
2.10.2 Point source
If we assume an isotropic source, apply the factors for vignetting, etc., mentioned above, and begin with Eq. (2.41) cosine3 law, we obtain τo f v (1 − A2 ) I cos n θ , d2
Et =
(2.88)
where n is 3 for an isotropic source.
2.11 Summary of Some Key Concepts This chapter has presented a number of concepts fundamental to an understanding of radiometry; a short summary of some appears below. First, the basic equation of radiative transfer in differential form: d 2 Φ1→2 =
L(θ, φ)dA1 cos θ1dA2 cos θ2 . d2
Second, the integral form of this equation: Φ1→2 =
L(θ, φ)cos θ1 cos θ2 dA1dA2 . d2 A2 A1
Handy simplifications can be utilized, provided that their underlying assumptions are met. These are and
Φ = LAΩ , E = LΩ .
Further, the Lambertian approximation is okay for most emitters, but angles must be considered when applying it to metals. It is alright for matte reflectors, and no good at all for specular reflectors. The choice of solid angle for radiometric calculations is an important one. When the source or receiver is isotropic, solid angle ω may be used. When the source or receiver is Lambertian, projected solid angle Ω is the correct choice. A final comment on sources deserves mention. These “notes” (lyrics by Jon Geist and Ed Zalewski, NIST, ca. 1982) are to be sung to the theme song from the ancient television series, “Mr. Ed:” A source is a source, of course, of course And no one can make a point of a source Unless, of course, it’s the sort of source That only exists in your head!
Propagation of Optical Radiation
59
For Further Reading F. C. Grum and R. J. Becherer, Radiometry, Vol. 1 in Optical Radiation Measurements series, F. Grum, Ed., Academic Press, New York (1979). R. McCluney, Introduction to Radiometry and Photometry, Artech House, Boston (1994). W. L. Wolfe, Introduction to Radiometry, SPIE Press, Bellingham, Washington (1998). E. F. Zalewski, “Radiometry and Photometry,” Chapter 24 in Handbook of Optics Vol. 2: Devices, Measurements, and Properties, Second Edition, M. Bass, Ed., Optical Society of America, Washington, D.C. (1994).
References 1. Guide for the Use of the International System of Units (SI), NIST SP811, U.S. Government Printing Office (1995). 2. F. E. Nicodemus, Self-Study Manual on Optical Radiation Measurements: Part I–Concepts, NBS Technical Note 910-1, p. 68, NBS, U.S. Government Printing Office, Washington, D.C. (1976). 3. C. Wyatt, Radiometric Calibration: Theory and Methods, Academic Press, New York (1978). 4. S. Chandrasekhar, Radiative Transfer, Clarendon Press, Oxford (1950), reprinted Dover (1960). 5. P. Drude, Lerhbuch der Optik (1900), translated into English and reprinted as The Theory of Optics, Longmans, Green, New York (1902). 6. B. N. Taylor, The International System of Units (SI), NIST SP330, U.S. Government Printing Office, Washington D.C. (1991). 7. J. M. Palmer, “Getting intense on intensity,” in Metrologia 30(4), pp. 371– 372 (1993). 8. J. M. Palmer, “Intensity,” in Optics & Photonics News, p. 6, February (1995). 9. P. D. Foote, “Illumination from a radiating disc,” Bulletin of the Bureau of Standards, NBS, 12, p. 583 (1915). 10. R. Kingslake, Optical System Design, Academic Press, New York (1983). 11. B. T. Chung and P. S. Sumitra, “Radiation shape factors from plane point sources,” J. of Heat Transfer 94(3), pp. 328–330 (1972). 12. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer, Brooks/Cole, Belmont, California (1970). 13. P. Moon, The Scientific Basis of Illuminating Engineering, McGraw-Hill, Dover, New York (1936).
60
Chapter 2
14. M. Donabedian, “Cooling systems,” Chapter 15 in The Infrared Handbook, W. L. Wolfe and G. L. Zissis, Eds., U.S. Government, Washington, D.C. (1978). 15. R. W. Boyd, Radiometry and the Detection of Optical Radiation, Wiley & Sons, New York (1983). 16. “A guide to integrating sphere theory and applications,” Labsphere Inc., at http://www.labsphere.com/tecdocs.aspx (2006). 17. R. Kingslake, Optical System Design, Academic Press, New York (1983). 18. P. N. Slater, Remote Sensing: Optics and Optical Systems, Addison-Wesley, Reading, Massachusetts (1980).
Chapter 3
Radiometric Properties of Materials 3.1 Introduction and Terminology When radiant flux is incident upon a surface or medium, three processes occur: reflection, absorption, and transmission. A fraction of the beam is reflected, another fraction is absorbed, and the remainder is transmitted. Transmittance τ is the ratio of transmitted power to incident power. Reflectance ρ is the ratio of reflected power to incident power. Absorptance α is the ratio of absorbed power to incident power. Figure 3.1 shows an ideal geometric case, where the transmitted and reflected components are either specular (regular, in the mirror direction) or diffuse (scattered into the hemisphere). Figure 3.2 shows the transmission and reflection for real surfaces. Both spectral and directional properties are important.
RETROREFLECTION
INCIDENT BEAM
DIFFUSE TRANSMISSION
INCIDENT BEAM
DIFFUSE REFLECTION SPECULAR REFLECTION
SPECULAR (REGULAR) TRANSMISSION
Figure 3.1 Idealized reflection and transmission. 61
62
Chapter 3
STRONG DIFFUSE REFLECTION
STRONG SPECULAR REFLECTION
STRONG RETROREFLECTION
DIFFUSE TRANSMISSION
REGULAR TRANSMISSION
Figure 3.2 Generalized reflection and transmission.
A continuing dialog over terminology has taken place, particularly over the suffixes -ance and –ivity. 1,2,3,4,5 The usage here reserves terms ending with -ivity (such as transmissivity, absorptivity, and reflectivity) for properties of a pure material, while the suffix –ance is used when the characteristics of a specific sample are described. One can then distinguish between the reflectivity of pure aluminum (as calculated from the complex index of refraction n and κ) and the reflectance of a particular specimen of 6061 aluminum with surface structure associated with rolling or machining and with a natural oxide layer. The adjective spectral refers to a characteristic at a particular wavelength and is indicated as a function of wavelength λ, i.e., τ(λ), ρ(λ) or α(λ). For example, spectral transmittance τ(λ) is often plotted against wavelength λ for a colored filter. The absence of “spectral” implies integration over all wavelengths, weighted by a source function.
3.2 Transmission Transmission is the process by which incident radiant flux leaves a surface or medium from a side other than the incident side (usually the opposite side). The spectral transmittance τ(λ) of a medium is the ratio of the transmitted spectral flux Φλt to the incident spectral flux Φλi: τ(λ) =
Φ λt . Φ λi
(3.1)
Total transmittance τ is the ratio of the total transmitted flux Φt to the total incident flux Φi:
Radiometric Properties of Materials
Φ τ= t = Φi
63
∞
0
τ ( λ ) Φ λ i dλ
∞
0
Φ λ i dλ
≠
τ(λ) dλ .
(3.2)
λ
Note particularly that the total transmittance is not the integral over wavelength of the spectral transmittance; it must be weighted by the incident source function Φλi. The transmittance may also be described in terms of radiance as follows: ∞
τ= 0 ∞
2 π sr
Ltλ dΩt dλ
0
2 π sr
Liλ dΩi dλ
,
(3.3)
where Lλi is the spectral radiance Lλi(λ;θi,φi) incident from direction (θi,φi), Lλt is the spectral radiance Lλt(λ;θt,φt) transmitted in direction (θt,φt), and dΩ is the elemental projected solid angle sinθ cosθdθdφ. Geometrically, transmittance can be classified as specular, diffuse, or total, depending upon whether the specular (regular) direction, all directions other than the specular, or all directions are considered. The bidirectional transmittance distribution function (BTDF, symbol ft and units sr–1) relates the transmitted radiance to the radiant incidence (irradiance) as f t ( θi , φi ; θt , φ t ) ≡
dLt ( θt , φt )
dEi ( θi , φi )
=
dLt ( θt , φt )
Li ( θi , φi ) d Ωi
.
(3.4)
This descriptor facilitates specification of the angular dependence of all (specular and diffuse) transmitted radiation.
3.3 Reflection In reflection, a fraction of the radiant flux incident on a surface is returned into the hemisphere whose base is the surface containing the incident radiation. The reflection can be specular (in the mirror direction), diffuse (scattered into the entire hemisphere), or a combination of both. Table 3.1 shows a wide range of materials that have different goniometric (directional) reflectance characteristics. Spectral reflectance is defined at a specific wavelength λ as ρ( λ ) =
Φ λr , Φ λi
(3.5)
64
Chapter 3 Table 3.1 Goniometric classification of materials.6 (Reprinted with permission of CIE.)
Material classification Exclusively reflecting materials
σ
Scatter
γ Structure (deg)
none
0
≅0
weak
≤0.4
≤27
strong
> 0.4 > 27
τ=0
none
Mirror
micro macro none
Matte aluminum Retroreflectors Lacquer & enamel coatings Paint films, BaSO4, PTFE Rough tapestries, road surfaces
micro macro
Weakly transmitting, strongly reflecting none materials
≅0
0
none micro
≤ 0.4 ≤ 27
weak τ ≤ 0.35 strong Strongly transmitting materials τ > 0.35
≅0
macro none micro macro none
≤ 0.4 ≤ 27
none micro macro
> 0.4 > 27
none micro macro
> 0.4 > 27
none
0
weak
strong
Example
Sunglasses, color filters, cold mirrors Matte-surface color filters Glossy textiles Highly turbid glass Paper Textiles Window glass Plastic film Ground glass Ornamental, prismatic glass Opal glass Ground opal glass Translucent acrylic plastic with patterned surface
while the total reflectance ρ is the ratio of the reflected flux Φr to the incident flux Φi: Φr ρ= = Φi
∞
0
ρ(λ) Φ λi d λ
∞
0
Φ λi d λ
≠
ρ( λ ) d λ . λ
(3.6)
Radiometric Properties of Materials
65
As in the case of transmittance, above, the integrated reflectance is not the integral over wavelength of the spectral reflectance; it must be weighted by the incident source function Φλi. Reflectance factor R or R(λ) is defined as the ratio of (spectral) radiant flux reflected by a sample to the (spectral) radiant flux which would be reflected by a perfect diffuse (Lambertian) reflector under the same irradiation conditions. While reflectance (in the absence of luminescence) cannot exceed unity, reflectance factor can assume values from zero to nearly infinity. Since the reference is a perfect diffuse reflector, reflectance factor is only useful as a descriptor for diffuse surfaces. Equations for nine types of reflectance factor appear in Table 3.2. Some notes on Table 3.1 are in order: (1) “Structure” refers to the nature of the surface. In a microscattering structure, the scatterers cannot be resolved with the unaided eye. The macrostructure scatterers can be readily seen. (2) Sigma, σ, is a diffusion factor, the ratio of the mean of radiance measured at 20 deg and 70 deg to the radiance measured at 5 deg from the normal, when the incoming radiation is normal. σ = [L(20) + L(70)] / [2L(5)]. It gives an indication of the spatial distribution of the radiance, and is unity for a perfect (Lambertian) diffuser. (3) Gamma γ is a half-value angle, the angle from the normal where the radiance has dropped to one half the value at normal. Its value is 60 deg for a perfect (Lambertian) diffuser. (4) It is suggested that the diffusion factor is appropriate for strongly diffusing materials and that the half-value angle is better suited for weakly diffusing materials. No single descriptor of reflectance suffices for the wide range of possible geometries. The fundamental geometric descriptor of reflectance is the bidirectional reflectance distribution function (BRDF) fr. It is defined as the differential element of reflected radiance dLr in a specified direction per unit differential element of irradiance dEi, also in a specified direction.7 It carries the unit of sr–1. f r ( θi , φi ; θ r , φ r ) ≡
dLr (θr , φr ) dLr (θr , φr ) −1 sr . = dEi (θi , φi ) Li (θi , φi )d Ωi
(3.7)
As shown in Fig. 3.3, the polar angle θ is measured from the surface normal, z. The azimuth angle φ is measured from an arbitrary reference in the surface plane, most often the plane containing the incident beam. The subscripts i and r refer to the incident and reflected beams, respectively.
66
Chapter 3
z dΩr
dΩi θr
θi
dA
y φi
φr
x
Figure 3.3 Geometrical definitions for BRDF (Adapted from Ref. 7).
Nicodemus et al.7 integrated over various solid angles and applied the earlier work of Judd8 to obtain nine goniometric reflectances and nine goniometric reflectance factors. These are listed in Tables 3.2 and 3.3. In these tables, the term “directional” refers to a differential solid angle dω in the direction specified by (θ,φ). “Conical” refers to a cone of finite extent centered in direction (θ,φ); the solid angle Ω of the cone must also be specified. The reflectances are illustrated in Fig. 3.4. Table 3.2 Nomenclature for nine types of reflectance factor.7
Bidirectional reflectance factor
R (θi , φi ; θr , φr )
= πf r ( θi , φi ; θr , φr )
Directional-conical reflectance factor
R(θi , φi ; ωr )
=
Directionalhemispherical reflectance factor
R(θi , φi ;2π)
= f r ( θi , φi , θr , φr )d Ω r
Conical-directional reflectance factor
R (ωi ; θr , φr )
=
π Ωi
Biconical reflectance factor*
R(ωi ; ωr )
=
π Ωi ⋅ Ω r
π Ωr
f r ( θi , φi , θr , φr )d Ω r
ωr
2π
ωi
f r ( θi , φi , θr , φr )d Ωi
ωi
ωr
f r ( θi , φi , θr , φr )d Ω r d Ωi
Radiometric Properties of Materials
67
Table 3.2 (Continued.)
Conicalhemispherical reflectance factor*
R (ωi ;2π)
=
Hemisphericaldirectional reflectance factor
R (2π; θr , φr )
= f r ( θi φi ; θ r φ r ) d Ω i
Hemisphericalconical reflectance factor*
R (2π; ωr )
=
1 Ωr
Bihemispherical reflectance factor
R (2π;2π)
=
1 f r (θi , φi ; θr , φr )d Ω r d Ωi π 2 π 2 π
*
1 Ωi
ωi
f r ( θi , φi ; θ r , φ r ) d Ω r d Ω i
2π
2π
f r ( θi , φi ; θ r , φ r ) d Ω r d Ω i
2 π ωr
Configurations that are measurable in practice. Table 3.3 Nomenclature for nine types of reflectance.7
Bidirectional reflectance Directional-conical reflectance Directionalhemispherical reflectance
dρ(θi , φi ; θr , φr )
= f r ( θi , φi ; θ r , φ r ) d Ω r
ρ(θi , φi ; ωr )
= f r ( θi , φi ; θ r , φ r ) d Ω r
ρ(θi , φi ;2π)
= f r (θi , φi ; θr , φr )d Ω r
Conical-directional reflectance
dρ(ωi ; θr , ϕr )
=
Biconical reflectance* Conicalhemispherical reflectance * Hemisphericaldirectional reflectance Hemisphericalconical reflectance* Bihemispherical reflectance
ρ(ωi ; ωr )
*
ωr
2π
d Ωr f r ( θi , φi ; θ r , φ r ) d Ω i Ωi ωi 1 = f r ( θi , φi ; θ r , φ r ) d Ω r d Ω i Ωi ωi ωr
ρ(ωi ;2π)
=
1 Ωi
dρ(2π; θr , φr )
=
d Ωr π
ρ(2π; ωr )
=
1 f r (θi , φi ; θr , φr )d Ω r d Ωi π 2 π ωr
ρ(2π;2π)
=
1 f r (θi , φi ; θr , φr )d Ω r d Ωi π 2 π 2 π
ωi
2π
2π
f r ( θi , φi ; θ r , φ r ) d Ω r d Ω i
f r (θi φi ; θr φr )d Ωi
Configurations that are measurable in practice.
In both Tables 3.2 and 3.3, configurations containing a directional term are considered theoretical, as dΩ→0.
Figure 3.4 Nine reflectance configurations.
68 Chapter 3
Radiometric Properties of Materials
69
3.4 Absorption Absorption is the process in which a fraction of the incident radiant flux is converted to another form of energy, usually heat. Absorptance is the fraction of incident flux that is absorbed. Spectral absorptance is defined at a specific wavelength λ as α (λ ) =
Φλa , Φλi
(3.8)
with the subscripts denoting absorbed and incident power, respectively. Total absorptance is defined as Φ α= a = Φi
∞
0
α ( λ ) Φ λi d λ
∞
0
Φ λi d λ
≠
α (λ ) d λ .
(3.9)
λ
Note the analogy with Eqs. (3.2) and (3.6). Absorption removes power from a beam; directional characteristics such as direct absorption and bulk scattering are not often taken into consideration.
3.5 Relationship Between Reflectance, Transmittance, and Absorptance Because energy is conserved, the sum of the transmission, reflection, and absorption of flux incident on a surface is unity, or τ + ρ + α =1.
(3.10)
The above statement assumes integration over all wavelengths and directions. In the absence of wavelength-shifting effects (such as luminescence or Raman scattering), this relationship is also valid for any specific wavelength: τ(λ) + ρ(λ) + α(λ) = 1.
(3.11)
3.6 Directional Characteristics 3.6.1 Specular transmittance and reflectance
The specular transmittance and reflectance for a single surface can be calculated via the Fresnel equations using the complex index of refraction n + iκ. The simplest case is that of no absorption, i.e., κ = 0. The general equations are
70
Chapter 3
n′ cos θ − n cos φ ρp = n′ cos θ + n cos φ n cos θ − n′ cos φ ρs = n cos θ + n′ cos φ
2
2
,
(3.12)
and 2
n′ cos φ 2n cos θ τp = n′ cos θ + n cos φ n cos θ 2
n′ cos φ 2n cos θ τs = n cos θ + n 'cos φ n cos θ
,
(3.13)
where the subscripts p and s represent the two polarization states, n and θ are on the incident side of the interface, and n′ and φ are on the transmitted (or reflected) side. The total transmittance and reflectance for unpolarized light is the average of the two polarized components ρ p + ρs ρT = 2 and (3.14) τ p + τs τT = . 2 Figure 3.5 shows reflection and transmission curves for a single surface of a nonabsorbing optical material with an index of refraction of 2. The three curves represent s- and p-polarization states as well as total polarization. To compute the curves for absorbing media, substitute n ± iκ for n in Eqs. (3.12) and (3.13), where κ = αλ 4π . Since refractive index n is wavelength dependent, the calculated reflectance and transmittance are also. In Fig. 3.6, we see a partially transparent plane slab of an optical material. Reflection, transmission, and absorption are all present. The Fresnel equations are greatly simplified at normal incidence, in which θ = φ = 0. For a nonabsorbing material, the reflectance and transmittance at a single surface are n′ − n ρss = n′ + n
2
τss =
4nn′
( n′ + n )
2
.
(3.15)
The internal transmittance τi of a piece of optical material describes only the absorption component and neglects the reflectance losses. The exponential absorption law, often referred to as the Lambert-Bouguer-Beer law, is
Radiometric Properties of Materials
71
(a)
SINGLE SURFACE REFLECTANCE
1 0.9 0.8 0.7 0.6 0.5 0.4
S
0.3
TOTAL
0.2
P
0.1 0 0
10
20
30
40 50 60 ANGLE FROM NORMAL (deg)
70
80
90
(b)
SINGLE SURFACE TRANSMITTANCE
1 0.9
P
TOTAL
0.8 0.7 0.6 0.5
S
0.4 0.3 0.2 0.1 0 0
10
20
30
40 50 60 ANGLE FROM NORMAL (deg)
70
80
90
Figure 3.5 (a) Reflection and (b) transmission of a single surface, n = 2.
τi (λ) = e −α′ ( λ ) x ,
(3.16)
where α′(λ) is the spectral absorption coefficient (cm–1) at the specified wavelength, and x is the thickness (cm). Different units can be used for thickness and absorption coefficient (i.e., m, km, mm, μm) as long as they are the same for both the absorption coefficient and the thickness. The product of the absorption coefficient and the thickness x is often called the optical thickness τo. These units and symbols are used differently in different fields; be careful! External transmittance is the quantity that is ordinarily measured, and includes the Fresnel reflection losses and the absorption. Equation (3.17) describes the transmittance of a parallel slab at normal incidence with singlesurface Fresnel reflection ρss, absorption coefficient α′, and thickness x:
72
Chapter 3
Φi n
Φr
Φt
Φa
Figure 3.6 Transmitted, reflected, and absorbed rays. −α′ x (1 − ρss ) τi . Φ t (1 − ρss ) e = = 2 −2 α′ x Φi 1 − ρss e 1 − ρss 2 τi2
2
τ=
2
(3.17)
If the slab is nonabsorbing (α = 0), then τi = 1, and Eq. (3.17) reduces to 2n . n2 + 1
τ=
(3.18)
Similar equations can be derived for reflectance and absorptance. For reflectance: ρ (1 − ρ ss ) e −2 α′ x ρss (1 − ρss ) τi2 Φr = ρss + ss = ρ + . ss Φi 1 − ρss 2 e −2 α′ x 1 − ρss 2 τi2
(3.19)
−α′ x Φ a (1 − ρss ) (1 − e ) (1 − ρss )(1 − τi ) = = . Φi 1 − ρss e −α′ x 1 − ρ ss τi
(3.20)
2
ρ=
2
For absorptance: α=
When the optical thickness is large enough, the material becomes opaque and the transmittance goes to zero. In this case, the reflectance ρ approximates the single surface reflectance ρss, and the absorptance α approaches 1 – ρ. For the opposite case, in which the optical thickness approaches zero, the material becomes transparent and the following relationships hold: ρ = ρss +
ρss (1 − ρss ) 1 − ρss 2
2
,
(3.21)
Radiometric Properties of Materials
73
(1 − ρss ) τ= 1 − ρss 2
2
.
(3.22)
As an example, an ordinary transparent glass (n = 1.5) has a single-surface reflectance ρss of 0.04. The total reflectance ρ is 0.077 (rather than 0.08), and the total transmittance τ is 0.923. 3.6.2 Diffuse transmittance and reflectance
A large part of the optical radiation in our environment is the result of reflections, with sources including sunlight and artificial tungsten and fluorescent lamps. For nonspecular reflection or transmission, the BRDF or its analog, the BTDF, may be characterized for any combination of incident and reflected beams. A full BRDF or BTDF specification is very complex, particularly when the reflectance and transmittance lie out of the plane that includes the incident and specular beams. Figure 3.7 illustrates this complexity.9 The set of polar diagrams in this figure for a partially diffuse surface is specific to just one incidence angle. A complete characterization would require data at other incidence angles as well, as the pattern is variable. Surfaces become more specular as the angle of the incoming beam increases towards grazing incidence. In addition, the diffuse scatter from rough surfaces diminishes with increasing wavelength, i.e., the surface becomes more specular. The BRDF characteristics of a surface are normally plotted as BRDF (sr–1) versus angle as measured from the specular beam. This method places the specular beam on the left ordinate, regardless of incidence angle. Positive and negative angles as measured from the specular beam are typically shown on the same graph, and angles as large as 175 deg are seen (for an 85-deg angle of incidence). The angle axis can be linear for diffuse surfaces, but a log plot is better suited for specular surfaces as shown in Fig. 3.8 For some materials, it has been found that the diffusely reflected radiation is symmetrical about the specular beam. In this case, a single plot suffices for each wavelength. The materials with this characteristic are spatially uniform (called isotropic in the literature, but different from isotropic as applied to a point source) and are either nearly specular or nearly perfectly diffusing. Insight into their BRDF characteristics can be gained through a plot of the natural log of the BRDF versus a special parameter, (β – βo), where β is the sine of the scattered beam angle and βo is the sine of the specular angle. This places the BRDF into direction cosine space. Figure 3.9 shows such a BRDF plot for a perfectly diffuse reflector (fr = 1/π) and for a very good specular reflector. The plot for the perfect specular reflector would be a delta function of infinite height at (β – βo) = 0, and would not be seen on this plot. What is actually seen is the instrument function that is primarily the convolution of the incident beam profile with the detected beam profile.
74
Chapter 3
Figure 3.7(a) BRDF of rough aluminum at incidence angle of 33 deg. (Reproduced from Ref. 9 with permission.)
Radiometric Properties of Materials
75
Figure 3.7(b) BRDF of rough aluminum at incidence angle of 33 deg. (Reproduced from Ref. 9 with permission.)
Figure 3.8 Conventional BRDF plot of perfect mirror (“instrument”) compared with a perfect diffuse reflector and two mirrors with scatter.
76
Chapter 3
Figure 3.9 Special parameter versus BRDF plot of perfect mirror (“instrument”) compared with a perfect diffuse reflector and two mirrors with scatter.
A perfect specular reflector has a BRDF equal to ρ/Ωi where ρ is the reflectance and Ωi is the solid angle subtended by the source. For example, a piece of opaque glass with a reflectance ρ of 0.05 in sunlight (Ωi = 6.8 × 10–5 sr) has a BRDF of 14706 × ρ = 735.3. To determine the radiance associated with this specular reflection, multiply the BRDF by the irradiance. The BRDF for a hemispherical source is ρ/π, the same as for a perfectly diffuse (Lambertian) reflector. In the confines of an integrating sphere, diffuse and specular samples having the same reflectance are indistinguishable.
3.7 Emission So far, we have considered the radiometric properties of materials with respect to incoming radiation. In fact, all materials above 0 K radiate, so the emission of radiation by a material is an important property, as well. The Infrared Handbook defines emissivity as “the ratio of the radiant exitance or radiance of a given body to that of a blackbody.”10 Its symbol is ε. Emissivity may be considered a “quality” factor, indicating the capability for thermal radiation by a material. It has both spectral and directional properties, it is dimensionless, and its values are between 0 and 1. As with the material properties already discussed, emissivity refers to the characteristics of a pure substance, while emittance refers to the properties of a specific sample. Spectral emittance ε(λ) is defined as emittance at a given wavelength, and it is not a derivative quantity. In the case that a radiator is neutral with respect to wavelength, with a constant spectral emittance less than 1, it is called a graybody. In that case, the spectral emittance is the ratio of the radiance of that source at that wavelength to the radiance of a blackbody at that wavelength:
Radiometric Properties of Materials
77
ε(λ ) =
Lλ , LλBB
(3.23)
where LλΒΒ is the value of the Planck function at that wavelength for a blackbody. Further discussion of this function and the radiometric characteristics of sources follows in Chapter 4. As reflectance, transmittance, and absorptance are related, as indicated by Eq. (3.10), so too are reflectance, transmittance, and emittance related. At equilibrium, the power emitted by a body to its surroundings must equal the power absorbed by the body from its surroundings. More succinctly, the body’s absorptance must equal its emittance: α = ε.
(3.24)
This is Kirchhoff’s law, to be discussed in more detail in Chapter 4. As a consequence, emittance may be substituted for absorptance in Eqs. (3.10) and (3.11), so that ε =1− τ − ρ ,
(3.25)
ε(λ) = 1 − τ(λ) − ρ(λ) .
(3.26)
and spectrally,
If the body is opaque (τ = 0), then
and
ε =1− ρ ,
(3.27)
ε(λ) = 1 − ρ(λ ) .
(3.28)
3.8 Spectral Characteristics The radiometric properties of materials of interest all share one spectral characteristic: the property of interest is not independent of wavelength. Many material characteristics may be “flat” over a portion of the spectrum, but at other wavelengths may differ significantly. Since these properties are weighting functions, the values are irrelevant in those spectral regions where the source function is insignificant. For example, ordinary window glass has a transmittance of about 0.92 (clean, normal incidence) in the visible part of the spectrum but drops to zero in the infrared where the eye is nonresponsive.
78
Chapter 3
Figure 3.10 Spectral emittance for several generic surfaces.11
These materials are often used for temperature control of spacecraft. Heating is by absorption of sunlight for wavelengths shorter than 3 μm, and cooling results from thermal radiation for wavelengths longer than 3 μm. If a material has high reflectance at shorter wavelengths, it will not absorb much of the incident radiation. If the reflectance is low at longer wavelengths, the absorption and consequently the thermal emission will be high. The surface will be cold. If the spectral regions are reversed, the surface will become hot. These surfaces are known as selective surfaces, and a wide range of surface temperatures have been achieved. Designers of these materials often utilize the ratio α/ε to describe the value of absorptance in one spectral region (usually solar) relative to the emittance value in another region (usually infrared.) Figure 3.10 shows the spectral emittance (1 – spectral reflectance) for several generic surfaces; some are selective. The Infrared Handbook provides detailed examples of the radiometric properties of both natural and artificial sources. An example of the spectral reflectances of several metals is shown in Fig. 3.11.
Figure 3.11 Spectral reflectance of films of silver, gold, aluminum, copper, rhodium, and 12 titanium.
Radiometric Properties of Materials
79
3.9 Optical Materials Checklist Finally, Table 3.4 provides an “optical materials checklist” that includes several properties useful to the designer selecting materials for an optical design effort. Table 3.4 Optical materials checklist.
Optical properties Transmission (function of wavelength, temperature, direction) Index of refraction (function of wavelength, temperature, direction) Dispersion, partial dispersion Surface reflectance Scatter (surface & bulk) Absorption (bulk) Homogeneity Birefringence, stress coefficient Fluorescence Anisotropy Electro-optic and/or acousto-optic coefficients Mechanical properties Young’s modulus Yield point Hardness Optical workability Coating compatibility Density, specific gravity Thermal properties Thermal conductivity Specific heat, heat capacity Coefficient of linear thermal expansion Softening point, melting point Environmental properties Solubility in H2O, other solvents Surface deterioration, devitrification Radiation susceptibility (UV, hard particle) Other factors Availability Safety factors, toxicity Cost Compiled by James M. Palmer 02/21/89
80
Chapter 3
For Further Reading “Standard practice for angle resolved optical scatter measurements on specular and diffuse surfaces,” ASTM Standard E1392-90, ASTM International, Philadelphia (1990). “Standard practice for goniometric optical scatter measurements,” ASTM Standard E2387-05, ASTM International, Philadelphia (2005). J. C. Stover, Optical Scattering: Measurement and Analysis, SPIE Press, Bellingham, Washington (1995).
References 1. A. G. Worthing, “Temperature, radiation, emissivities and emittances,” in Temperature: Its Measurement and Control in Science and Industry, Reinhold, New York (1941). 2. J. C. Richmond, “Rationale for emittance and reflectivity,” Applied Optics 21(1), pp. 1–2 (1982). 3. J. C. Richmond, W. N. Harrison, and F. J. Shorten, “An approach to thermal emittance standards,” in Measurement of Thermal Radiation Properties of Solids, NASA SP-31, J. C. Richmond, Ed., NASA, Washington, D.C. (1963). 4. W. L. Wolfe, “Proclivity for emissivity,” Applied Optics 21(1), p. 1 (1982). 5. J. C. Richmond, J. J. Hsia, V. R. Weidner, and D. B. Wilmering, Second Surface Mirror Standards of Spectral Specular Reflectance, NBS Special Publication SP 260-279, U.S. National Bureau of Standards, Washington, D.C. (1982). 6. Radiometric and Photometric Characteristics of Materials and Their Measurement, CIE Publication 38 (1977). 7. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160, U.S. National Bureau of Standards, Washington, D.C. (1977). 8. D. B. Judd, “Terms, definitions, and symbols in reflectometry,” J. Opt. Soc. Am. 57(4), pp. 445–450 (1967). 9. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Applied Optics 4(7), cover picture (1965.) 10. W. L. Wolfe, “Radiation theory,” Chapter 1 in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, Eds., pp. 1–28, U.S. Government, Washington, D.C. (1978). 11. J. C. Richmond, “Coatings for space vehicles,” in Surface Effects on Spacecraft Materials, F. J. Clauss, Ed., Wiley & Sons, New York (1960).
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81
12. W. L. Wolfe, “Optical materials,” Chapter 7 in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, Eds., pp. 7–81, U.S. Government, Washington, D.C. (1978).
Chapter 4
Generation of Optical Radiation 4.1 Introduction From a discussion in the previous chapter on the interaction of radiation with materials, we now turn to the subject of how radiation is generated, and the roles that emission, reflection, and other processes play. Sources of optical radiation can be classified in a variety of ways. Active sources emit optical radiation due to their temperature (thermal sources) or as a result of atomic transitions (luminescent sources). Passive sources reflect optical radiation from active sources or from other passive sources. Passive sources can also be classified as thermal or luminescent, depending upon the process that generated the radiation initially. Examples of thermal sources include blackbody radiation simulators, tungsten-filament lamps, gases, the sun, the moon, and you and I. Examples of luminescent sources include lasers, fluorescent lamps, mercury arcs, sodium lamps, electroluminescent panels, LEDs, and gases. Some sources combine both thermal and luminescent mechanisms, and some may be both active and passive, reflecting in one spectral region and emitting in another. Other means of classification have also been used. Some authors distinguish between artificial (man-made) and natural sources. Lamps are artificial sources, whereas the earth, the sun, and stars are natural sources. Still another practice is to divide sources according to their output spectral characteristics. Continuous sources have a spectral radiance that is slowly varying with wavelength, typical of thermal radiation, while line sources emit in narrow, well-defined spectral regions. Yet another attempt to distinguish sources is by their degree of spatial and/or temporal coherence. Thermal radiation has been extensively studied since the late nineteenth century. Stefan was the first to experimentally examine the relationship between radiation and temperature in 1879. He analyzed data from Tyndall and found that the total radiation is proportional to the fourth power of temperature T4. Boltzmann derived this T 4 relationship from the Carnot cycle in 1884. In 1891 Wien derived his displacement law, which relates the peak radiation to 83
84
Chapter 4
temperature, and the T 5 relationship for the magnitude of the peak. In 1896, Wien derived an equation for the spectral distribution of thermal radiation based on thermodynamic arguments. In 1900 Rayleigh derived another equation for spectral distribution based on equipartition, and Jeans in 1905 independently repeated this derivation. Planck in 1901 published an empirical equation involving the notion that energy exists as discreet “packets” that fit experimental data better than either the Wien or the Rayleigh-Jeans equation. This proved monumental, as it was later accepted as the birth of quantum mechanics. Verifications of the Planck equation and the physical constants continued up to 1982 with precise measurements of the spectral distribution of blackbody radiation, determination of the Stefan-Boltzmann constant, and confirmation of the thermodynamic temperature scale.
4.2 Radiation Laws 4.2.1 Planck’s law
The Planck expression for blackbody radiation is at the heart of all thermal radiation equations. Any object at a temperature above absolute zero (0 K) will radiate as a modified Planckian radiator. A full derivation of the Planck equation will not be presented here, as it can be found in many physics texts. A brief outline of the derivation will show its important features. It requires the determination of the number of discrete frequency modes in a cavity. The total mode density in a cavity (in vacuo) as a function of frequency ν is: N ν dν =
8πν 2 dν . c3
(4.1)
The average energy per mode looks like q=
hν e
hν / kT
−1
.
(4.2)
The energy density (energy per unit volume per unit frequency interval) is the product of the number of available modes and the average energy of each mode. It is expressed as: uν =
8πhν3 1 . c 3 (e hν / kT − 1)
(4.3)
The energy exits a cavity at velocity c into 4π sr. The behavior is described by Planck’s law, expressing radiance per unit frequency interval (spectral radiance):
Generation of Optical Radiation
85
Lν =
2hν 3 1 . hv / kT 2 c e −1
(4.4)
Frequency is fundamental (independent of the medium), but wavelength is preferred to frequency in radiometric applications due to ease of measurement. Making use of the identity Lλ d λ = Lν d ν and the relationship between wavelength and frequency λ = c/ν, the spectral radiance of blackbody radiation can be expressed in wavelength terms as 2hc 2 1 . hc / λkT 5 λ e −1
Lλ =
(4.5)
This is one form of Planck’s equation, whose units are W/m2sr·m. The above equation implies that the process is taking place in vacuum where the refractive index is unity. A more general expression utilizes n. The most frequently encountered and useful form of the Planck equation is Lλ =
c1 1 . 2 5 c 2 / nλT πn λ e −1
(4.6a)
Because blackbody radiation is Lambertian, the spectral radiant exitance may be expressed as Mλ =
c1 1 . 5 c 2 / nλT nλ e −1 2
(4.6b)
In this pair of important equations, 2
c1 = 2πhc = 3.74177107(29) × 10–16 Wm2 (7.8 × 10–8) (first radiation constant) c2 = hc/k = 1.4387752(25) 10–2 m·K (1.7 × 10–6) (second radiation constant) h = 6.62606876(52) ×10–34 J·s (7.8 × 10–8) (Planck constant) c = 299792458 m/s (exact) (velocity of light) k = 1.3806503(24) × 10–23 J/K (1.7 × 10–6) (Boltzmann constant) n = index of refraction (1 for vacuum, ≈1.00028 for air)
86
Chapter 4
The above notation is a conventional way to present uncertainties in the physical constants. The number in parenthesis that is attached to the value is the absolute uncertainty (one sigma, or 1σ) in the last digit(s) of the constant. For example, c2 = 0.014387752 ± 0.000000025 m·K. The last number in parenthesis is the relative uncertainty in the value of the constant. It is the absolute uncertainty divided by the value of the constant. The complete Planck equation as shown above is valid in any media having index of refraction n. In a vacuum (n = 1), the n can be dropped, λ = λo, and the Planck equation is most often seen this way. If in air (nair ≈ 1.00028), this correction is usually ignored except for extreme low-uncertainty applications. It is more convenient for optical measurement work to use micrometers (μm) or nanometers (nm) for wavelength to get W/m2sr·μm or W/m2sr·nm. The radiation constants then become c1 = 3.74177107 × 108 W·μm4/m2 (wavelength in μm), c1 = 3.741771 07 × 1020 W·nm4/m2 (wavelength in nm), c2 = 14387.752 μm·K = 1.438 7752 × 107 nm·K. The two curves in Fig. 4.1 show the form of Planck’s equation as a function of wavelength, with temperature as a parameter. They are strongly peaked, with the form governed by the λ–5 term for wavelengths longer than the peak and by the exponential term for shorter wavelengths. Note that only a limited range of temperatures can be shown on a single linear plot such as these, as the ordinates are highly nonlinear, varying over many orders of magnitude. The plots also show a dashed line, the locus of the wavelength of peak spectral radiance, having a characteristic hyperbolic shape. As will be seen, these dashed lines represent the Wien displacement law. If the spectral radiance curves are plotted on log-log axes as in Fig. 4.2, several interesting things are seen. First, this form allows for a wide range of temperatures and wavelengths on a single plot. Second, the locus of the wavelength of peak radiance is a straight line on a logarithmic plot, indicating hyperbolic behavior. Finally, note that all of the curves have an identical shape when the logs are plotted. Since the shape of the curve is independent of temperature, one could construct a nomogram by tracing a single curve and the straight line locus of maxima onto a transparent sheet and use it as an overlay. Slide the overlay along the straight line to display the blackbody radiation curve for any temperature. 4.2.2 Wien displacement law
The Wien displacement law describes this line, the locus of the peak wavelength of the blackbody curve as a function of temperature. It is determined by taking the derivative of Lλ with respect to wavelength (in the medium) and setting it to zero to find the maximum. The result is
Generation of Optical Radiation
87
nλmax T = 2.897 7686(51) × 10–3 m-K (1.7 × 10–6).
(4.7)
A rough but useful approximation for the peak wavelength (in μm, setting n = 1) is
λ maxT ≈ 3000 μm-K.
(4.8)
Table 4.1 lists the peak wavelengths for several common sources.
(a)
(b) Figure 4.1 Spectral radiance of blackbody radiation for (a) high temperatures and (b) lower temperatures.
88
Chapter 4
Figure 4.2 Log-log plot of blackbody spectral radiance as a function of wavelength and temperature.
Table 4.1 Peak wavelengths for several common sources.
Source
Temperature (K)
Peak wavelength (μm)
Sun
~6000
~0.5
Tungsten lamp
~3000
~1
Typical hot IR source
~1000
~3
Typical IR test source
~500
~6
Room temperature
~300
~10
Liquid nitrogen
77
~40
To find the value of Lλ at the peak wavelength, solve the blackbody equation using the peak wavelength (in the medium): Lλ max =
n 2 σ′ 5 T , π
(4.9)
Generation of Optical Radiation
89
where σ' = 1.286 × 10–11 W/m2K5μm. The radiance at the peak wavelength varies as the fifth power of the temperature. 4.2.3 Stefan-Boltzmann law
To determine the total radiance of blackbody radiation, integrate the Planck equation over all wavelengths. The result is the Stefan-Boltzmann law, ∞
L = Lλ d λ = 0
n2σ 4 T , π
(4.10)
where σ is the Stefan-Boltzmann constant, 2π5 k 4 15c 2 h3 = 5.670 × 10 –8 W/m 2 K 4 . The Stefan-Boltzmann law is most often seen in terms of radiant exitance with n = 1: M = σT 4 .
(4.11)
Insert room temperature (300 K) into the Stefan-Boltzmann equation to find that one square meter of a black surface emits ≈460 W. Therefore the walls, ceiling, and floor of a typical 6 × 6 × 2.5-m classroom have a total surface area of 132 m2 and emit over 60 kW into the room. For the moment, we ponder why we don’t bake under this onslaught. 4.2.4 Laws in photons
It is frequently advantageous to work with photons rather than watts, particularly in cases where light levels are low. Thus, a photon-based analysis may be helpful for applications in which photomultiplier tubes provide photon counting or CCDs are employed to image a faint target. The energy associated with a photon, in joules, is given by: Q = hν = h
c , λo
(4.12)
where h is again Planck’s constant. For energy in units of electron-volts (eV), divide by the electronic charge q (1.602 × 10–19 C).* At a (vacuum) wavelength λo of 1 μm, it takes 5.034 × 1018 photons per second to equal one watt, and more at longer wavelengths. Conversely, a single photon at a (vacuum) wavelength λo of 1 μm has an energy of 1.986 × 10–19 joules (W·s), or 1.239 eV, and has proportionally more energy at shorter wavelengths.
*
More accurate values for the constants that appear in this section are given in Appendix B; higher accuracy is sometimes required.
90
Chapter 4
The Planck equation rewritten for photons (n = 1) is Lqλ =
c1q 4
πλ e
1 c2 / λT
−1
,
(4.13)
where c1q = 2πc = 1.883 × 109 m/s. For wavelength in μm, c1q becomes 1.883 × 1027 m2/μm·s. Thus, Lqλ is expressed in photons per second per area per unit wavelength. Plots of spectral photon radiance are shown in Figs. 4.3 and 4.4. The curves appear similar to the previous radiance curves, but the range of ordinate values is not so extreme. The equation is subjectively the same; the exponential term is identical but the wavelength in the denominator is only raised to the fourth power.
(a)
(b) Figure 4.3 Photon spectral radiance versus wavelength for (a) high temperatures and (b) lower temperatures.
Generation of Optical Radiation
91
Figure 4.4 Log-log plot of spectral photon radiance as a function of wavelength and temperature.
The Wien displacement law for photons is derived as described earlier; the result is
λ q ,maxT = 3.6696986(62) × 10−3 m ⋅ K .
(4.14)
If the wavelength is expressed in micrometers, a useful approximation is λ q ,maxT ≈ 3700 μm ⋅ K .
(4.15)
The photon spectral radiance at the peak wavelength is Lqλ max =
σ′q π
T4 ,
(4.16)
where σ' q = 2.101 × 1011 s–1m–2K–4μm–1. The total photon radiance is the integral of the Planck equation for photons: ∞
Lq = Lqλ d λ = 0
σq π
T3,
(4.17)
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Chapter 4
where σq = 1.520 × 1015 s–1m–2K–3. At a temperature of 300 K, one square meter of a black surface emits about 4 × 1022 photons per second. Therefore there are more than 5 × 1024 photons per second being emitted by the walls, ceiling, and floor in our typical classroom. 4.2.5 Rayleigh-Jeans law
There are two common approximations to the Planck equation. The RayleighJeans equation for blackbody radiation was derived independently in 1900 by Rayleigh and in 1905 by Jeans from the thermodynamic principle of equipartition. It successfully predicted the shape of the spectral curve at long wavelengths, but was clearly inappropriate at short wavelengths. It predicted infinite energy at λ = 0, and the integral did not converge, giving rise to the labeling of this equation as the “ultraviolet catastrophe.” Assume that n = 1 in Eq. (4.6) and begin the derivation using the Planck equation for spectral radiance: Lλ =
c1 1 . 5 c 2 / λT πλ e −1
The exponential term can be expanded: e
c 2 / λT
=1+
c2 (c2 / λT ) 2 (c2 / λT )3 + + + ... . λT 2! 3!
If c2/λT << 1 (corresponding to a large value of λT), drop all of the higher-order terms. Then, e
c 2 / λT
−1 ≈
c2 . λT
Rearranging terms and substituting into the Planck equation, the result becomes: Lλ =
2ckT . λ4
(4.18)
This is the Rayleigh-Jeans law. This expression is valid with less than 1% error if λT > 0.778 m·K (i.e., long wavelengths and/or high temperatures). This approximation is not particularly useful, even for far-infrared work, as less than 0.1% of the output of a blackbody is located at λT values larger than 0.8 m·K, and the wavelength where it becomes valid is some 250× the peak wavelength. This means for sunlight (6000 K) and room temperature (300 K), the minimum wavelengths for which the approximation is valid are 130 and 2600 μm, respectively.
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93
4.2.6 Wien approximation
To derive the Wien approximation, begin with the Planck equation (n = 1). Assume that e( c2 / λT ) >> 1 . Then, the 1 can be dropped and the equation becomes: Lλ =
−c
c1 λT2 e πλ 5
(4.19)
The Wien approximation is valid with less than 1% error if λT < 3000 μm·K (short wavelengths and/or low temperatures). It is quite useful for a great deal of radiometric work as it is valid for blackbody radiation at all wavelengths shorter than the peak. Figures 4.5 and 4.6 depict the curves and the ranges of validity for the two approximations. 4.2.7 More on the Planck equation
A career can be spent fiddling with the Planck equation, presenting it for various frequencies, wavenumbers, etc. We will keep it simple and just do some normalization. Divide both sides of the Planck equation by T5: Lλ c1 1 = . T 5 π(λT )5 ec2 / λT − 1
(4.20)
(μm·K) Figure 4.5 Wien and Rayleigh-Jeans blackbody approximation curves.
94
Chapter 4
(μm-K) Figure 4.6 Errors associated with Wien and Rayleigh-Jeans approximations.
The result is an expression which is now a function of a new variable, (λΤ ). The result of this normalization is a single curve as shown in Fig. 4.7 for Planckian radiation for any wavelength and temperature. The Planck function can also be normalized by dividing by Lλmax such that the resulting curve peaks at unity. This curve is denoted as f(λT). An additional very useful curve shows the cumulative (integrated) radiance from 0 up to λT divided by the total radiance at temperature T. This cumulative curve is labeled F(λT). The defining equations for these two functions are: f (λ T ) = and
Lλ (λT ) L (λT ) = λ σ' 5 Lλ (λ maxT ) T π
λ
F (λT ) =
Lλ (λT )d λ
0 ∞
L (λT ) d λ λ
0
(4.21)
λ
L (λT ) d λ λ
=
0
σ 4 T π
.
(4.22)
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95
5E-12
RADIANCE/T^5
4E-12 3E-12 2E-12 1E-12 0 0
5000 10000 WAVELENGTH x TEMPERATURE (um-K)
15000
Figure 4.7 Normalization of the Planck curve.
Since these definitions are ratios, exitance M can readily be substituted for radiance L. These two functions are graphed in Fig. 4.8. To use these curves to determine radiance in a narrow wavelength interval (Δλ < 0.05λc), first select T for the blackbody radiation and the desired center wavelength λc. Determine the radiance at the peak using the equation for Lλ(max). Finally, use the function f(λT) from the graph and the wavelength interval Δλ to arrive at the result λ c +Δλ / 2
Lλ (λT )d λ = Lλ (λ maxT ) f (λ cT )Δλ .
(4.23)
λ c −Δλ / 2
If the wavelength interval is large, typically greater than 0.05× the center wavelength, use the other function F(λT) to determine the radiance in a finite wavelength interval. Again select T and the two desired wavelengths, λ1 and λ2. From the graph, read F(λ1T) and F(λ2T) and compute the total radiance using the Stefan-Boltzmann law. The result is λ2
σ L (λT ) d λ = π T [ F (λ T ) − F (λ T ) ] 4
λ
2
1
λ2 > λ1.
(4.24)
λ1
Figure 4.8 also shows the corresponding curves for photons, fq(λT) and Fq(λT). The defining equations are
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Figure 4.8 Curves of f(λ) and F(λ) for watts and photons.
f q ( λT ) =
Lqλ ( λT )
(4.25)
Lqλ ( λ maxT )
and λ
Fq (λT ) =
Lqλ (λT )d λ
0 ∞
L
qλ
0
(λT ) d λ
λ
L
qλ
=
0
(λ T ) d λ
σq π
. T
(4.26)
3
The application of the fq and Fq photon curves is identical to the curves for energy. In some applications, it is desirable to maximize the radiation contrast between a target and background of similar temperature. What wavelength might one choose for this task? The problem occurs regularly in the infrared where both the target and the background radiate near 300 K. Take the second derivative d2Lλ/(dTdλ), and set it to zero. The result is:
λ contrast T = 2411 μm ⋅ K .
(4.27)
This equation implies that the best “visibility,” or contrast with the background, occurs at a wavelength somewhat shorter than the peak wavelength, at λcontrast = 0.832λmax. Since the result is on the short wavelength side of the peak, the Wien approximation is valid and it simplifies the calculus considerably.
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Thus, if your target and background temperature were 305 K, the peak wavelength is 9.50 μm and the wavelength of maximum radiation contrast λcontrast is 7.9 μm. The wavelength for maximum photon contrast is 2898 μm·K, the same as the peak wavelength for energy. We often need to know how the spectral radiance Lλ changes with temperature; this can be determined by differentiating the Planck function with respect to temperature. The result, shown in differential form, is ΔLλ xe x ΔT = x , Lλ e −1 T
(4.28)
c hc = 2 . λkT λT xe x . For a small change in temperature ΔT such that the We define Z ≡ x e −1 change in x is also small, the change in Lλ with temperature at any wavelength is
where x =
ΔLλ ΔT =Z . Lλ T
(4.29)
If ex is significantly greater than unity (λT <3000 μm·K, the Wien approximation), then Z = x and ΔLλ/L = xΔT/T. Inspection of the blackbody curves and this equation shows that Z approaches infinity as the wavelength approaches zero, and Z approaches unity at longer wavelengths. Figure 4.9 shows Z plotted against λT; the curve is identical for watts and photons. The region where the Wien approximation is valid is λT < 3000 K (Z = 5). 4.2.8 Kirchhoff’s law
Blackbody radiation exists in any closed cavity at thermal equilibrium. As such, it is idealized, because whenever a hole is made into the cavity to allow radiation to exit, the conditions for blackbody radiation are altered. As noted in Chapter 3, emissivity is defined as the ratio of the radiance of an object to the radiance of a blackbody at the same temperature. The directional spectral emittance of a specific sample, ε(λ;θ,φ), is ε(λ; θ, φ) =
Lλ (θ, φ) . LλBB
(4.30)
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Figure 4.9 Parameter Z as a function of λT.
This equation differs slightly from Eq. (3.23), as it includes the directional component. Total emissivity is the integral of spectral directional emissivity over all angles and wavelengths, also weighted by the Planck function, which introduces a temperature dependence if the spectral emissivity is not uniform (that is, gray): ε=
(θ, φ; λ) L
λBB
sin θ cos θd θd φd λ
(σ / π)T 4
.
(4.31)
Since ε(λ;θ,φ) is not a derivative (per unit wavelength interval) quantity, it makes no sense to integrate it alone: ε ≠ (θ, φ; λ )d λ .
(4.32)
λ
Kirchhoff’s law was stated in Eq. (3.24): α = ε.
(4.33)
To illustrate, Fig. 4.10 shows a small body within a large isothermal enclosure. At equilibrium, in the absence of other sources or sinks, the source and enclosure must be the same temperature. The power absorbed by the small body (by the definition of absorptance) is Φα = αΦi. The power emitted by the body (by the definition of emittance) is Φe = εΦBB. At equilibrium, the power emitted
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by the body to the enclosure must equal the power absorbed by the body from the enclosure:
αΦ i = εΦ BB . As the incident and emitted power are the same (no other sources or sinks), the conclusion is α = ε. This seems too simple; is it really true? No! Look at the following equation for the equilibrium temperature of a flat plate in space, facing the sun and insulated on the back side. It is a simple statement of conservation of power, equating radiant incidence (absorbed) to radiant exitance (emitted): αEo = εσT 4 .
(4.34)
Here Eo is the solar constant (1368 W/m2). The units on both sides of the equation are W/m2. Solving for T, we find that it is a function of the ratio (α/ε). But Kirchhoff’s law says that α = ε, and these terms therefore cancel. The equilibrium temperature is therefore a function only of Eo. The implication is that a white (allegedly reflective) car and a black (supposedly absorptive) car have the same equilibrium temperature after sitting out in the sun all day. People from Arizona and Florida know better than that, and often buy white cars to keep cooler in the summer! So what’s wrong? The answer is that Kirchhoff's law does not apply in all situations. Table 4.2 indicates the applicability in terms of spectral and directional conditions. Specifically, α(λ;θ,φ) = ε(λ;θ,φ); the absorptance equals the emittance at a single wavelength in a single direction. After integrating over wavelength and geometry, test for the stated restrictions before applying Kirchhoff’s law. The
1 Figure 4.10 Illustration of Kirchhoff’s law. [Reprinted with permission of author from Optical Radiation Measurement series, Vol. 1, F. Grum and R. J. Becherer, Radiometry, p. 98 (1979).]
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simple equation above attempts to relate the absorbed energy from the sun, which is concentrated in the spectral range between 0.3 and 3 μm, to the thermal radiation located in the wavelength range between 3 and 30 μm. White and black paints have completely different reflectances in the solar region, but their emittances at 300 K are very similar. The thermal radiative properties of any material can be given in terms of its temperature T and its spectral directional emittance ε(λ;θ,φ). The spectral dependency is of primary importance, and the directional properties and temperature coefficients are usually of lesser concern. The blackbody equation is highly nonlinear with both wavelength and temperature, and is thus not particularly tractable. Fortunately, there are many computational and visual aids to help. The Infrared Handbook gives several calculator programs for now-obsolete HP and TI calculators, and some early calculators were available with plug-in cards to do blackbody calculations. The latest incarnation, The Infrared and Electro-Optical Systems Handbook, substitutes an extensive set of programs in BASIC to do the computations. There have been several slide rules (remember them?) that do calculations on blackbody radiation. The GE Radiation Calculator is a plastic rule that occasionally surfaces at a reasonable price. Cardboard knock-offs are often given away by vendors. For a little more money, Electro-Optical Industries at one time sold a high-quality metal rule. One side of my venerable GE calculator is shown in Fig. 4.11. Back when computers were scarce, tables generated by mainframes were commonly used to do precise blackbody calculations. Several tables may still be found in musty libraries.2,3,4 Today, simple computer programs can easily be written. Spreadsheets are sufficiently powerful to do these calculations with relative ease and provide superior graphics as well. Tools like Mathcad, MATLAB, and Mathematica also work well.
Figure 4.11 The venerable GE radiation calculator—J. M. Palmer’s own!
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1
Table 4.2 Summary of absorptance-emittance relations. [Adapted with permission of author from Optical Radiation Measurement series, Vol. 1, F. Grum and R. J. Becherer, Radiometry, p. 115 (1979).]
Directional spectral
α(λ;θ,φ,Τ ) = ε(λ;θ,φ,Τ )
Required conditions: None, other than thermal equilibrium Directional total
α(θ,φ,Τ ) = ε(θ,φ,Τ )
Required conditions: (1) Spectral distribution of incident energy proportional to blackbody at T, or (2) α(λ;θ,φ,Τ ) = ε(λ;θ,φ,Τ ) independent of wavelength Hemispherical spectral
α(λ,Τ ) = ε(λ,Τ )
Required conditions: (1) Incident radiation independent of angle, or (2) α(λ;θ,φ,Τ ) = ε(λ;θ,φ,Τ ) independent of angle Hemispherical total
α(Τ ) = ε(Τ )
Required conditions: (1) Incident energy independent of angle and spectral distribution proportional to blackbody at T, or (2) Incident energy independent of angle and α(λ;θ,φ,Τ ) = ε(λ;θ,φ,Τ ) independent of wavelength, or (3) Incident energy at each angle has spectral distribution proportional to blackbody at T and α(λ;θ,φ,Τ ) = ε(λ;θ,φ,Τ ) independent of angle, or (4) α(λ;θ,φ,Τ ) = ε(λ;θ,φ,T ) independent of angle and wavelength
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Spectral emittance was defined as the ratio between the spectral radiance of an object and the spectral radiance of a blackbody at the same wavelength and temperature. In general, it is also a function of angles θ and φ. Using this definition, any isothermal object has a spectral radiance given by 1 ε(λ; θ, φ)c1 Lλ (θ, φ) = c2 / λT , 5 −1 πλ e
(4.35)
and if ε(λ;θ,φ) and T are known, so is the spectral radiance. As ε(λ) increases towards unity, blackbody radiation is approached. Blackbody radiation simulators attempt to do just this, and many are commercially available. Since the spectral radiant exitance can be described by this simple equation, blackbody radiation simulators are most often used as standard sources of optical radiation in the calibration laboratory.
4.3 Emitter Types and Properties 4.3.1 Metals
An important factor in the behavior of a material is its electrical conductivity. Metals have a high conductivity, meaning large quantities of free electrons are available to interact directly with the radiation field. Since metals are generally good reflectors (a direct consequence of their high conductivity), they are poor emitters (conservation of energy). The emittance is a slowly varying function of wavelength, as shown by a simple empirical equation given by Hagen and Rubens:5 ε ≈ constant
resistivity (λ > 2 μm) λ
(4.36)
This equation, which is generally valid at wavelengths longer than 2 μm, shows that increasing resistivity (decreasing conductivity) increases emittance, and that emittance decreases at longer wavelengths. Figure 4.12 gives examples. The directional properties can be derived from Maxwell’s equations. The radiation is highly polarized at angles off normal, as shown in Fig. 4.13. Note, however, that the total radiation, the sum of both polarizations, is quite constant with angle (i.e., Lambertian) to within a few percent out to nearly 60 deg from specular. 4.3.2 Dielectrics
Dielectrics and gases have much lower conductivity than metals. Their electrons are more tightly bound to their parent nuclei and require specific atomic interactions with the radiation field. This implies that dielectrics tend to radiate in specific, fairly well-defined spectral regions, and not elsewhere.
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Figure 4.12 Spectral emissivity for some metals.6
The emitting properties of dielectrics are closely associated with the complex index of refraction n + iκ where κ, the extinction coefficient, is in turn related to the absorption coefficient α′, as discussed in Chapter 3. As noted in that chapter, the product of the absorption coefficient and the thickness of a material is the optical thickness τo. A material is considered optically thin when τo < 0.1 (transmission high) and optically thick when τo > 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.
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Figure 4.13 Directional emittance as a function of angle from specular.
The important species identified with our own atmosphere include “fixed” gases O2, N2, and CO2. Those species classified as variable include water, ozone, and methane (H2O, O3, and CH4.) It must be stressed that the energy absorbed by a gas is dependent upon the concentration and absorbing characteristics of the gas, and the energy emitted by the gas depends upon the temperature and spectral emittance of the gas. The processes of absorption and emission are not independent, but occur simultaneously.
4.4 Practical Sources of Radiant Energy 4.4.1 Two major categories
Table 4.3 presents a division of sources into the two categories of thermal and luminescent. Several of these sources will be discussed in this section. Natural sources of radiation, which may be thermal or luminescent, will also be discussed. NORMAL EMITTANCE
1
0.9
0.8
0.7
0.6 1
2
3
INDEX of REFRACTION
Figure 4.14 Normal emittance versus index of refraction.
4
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105
Table 4.3 Practical radiation sources.
Thermal sources Natural radiators Direct radiators Earth (longwave) Sun Stars Atmospheric gases Reflective natural objects Earth (shortwave) Moon, planets Skylight, clouds zodiacal light Atmospheric scatter Artificial radiators Incandescent substances Tungsten lamps Hot metals Silicon carbide rods Ceramic tubes Nernst glower Electrical transmission equipment Machinery Personnel Carbon arc Optical elements (lenses, mirrors, etc.) Open coal, wood, oil fires
Luminescent sources Natural sources Glow worms, fireflies Aurora, airglow Artificial sources Semiconductor devices LEDs, IREDs Diode lasers Phosphors CRT tube (TV, computer display, oscilloscope) Electroluminescent panels Lasers Gas (He-Ne, CO2, Ion, N2) Solid-state (Ruby, YAG) Metal vapor (He-Cd) Liquid (dye, cyanide, chelate) Chemical (HF, DF) Metal-vapor lamps Sodium vapor lamp Mercury vapor lamp Fluorescent lamp Gas discharge lamps Plasma display Neon lamp Glow discharge Xenon arc and flashtube Hot gases Welsbach mantle Bunsen burner Exhaust gases
4.4.2 Thermal sources 4.4.2.1 Tungsten and tungsten-halogen lamps
The most common household source of optical radiation (other than fire) has been the tungsten-filament lamp. Equation (4.35) may be used to describe the spectral radiance of a tungsten lamp; it is the equation for blackbody radiation with the emittance term included. Tungsten’s spectral emittance has been extensively studied. The most frequently used data are from DeVos, shown in
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Fig. 4.15. He measured the emittance of a flat ribbon of tungsten, but the results are also applicable to a round wire. Most lamps are made by drawing the tungsten into a round wire and then tightly coiling it, enhancing the emittance by making a partial cavity. If this tight coil is then further loosely coiled, emissivity is enhanced further. Tungsten lamps are designed to operate at a nominal voltage and current. The design compromise is between light and lifetime. If more light is needed, voltage (or current) may be increased, but not for very long. Figure 4.16 shows light (lumens), efficiency (lumens/watts), and lifetime as a function of operating voltage. Tungsten-filament lamps decay due to evaporation of the filament, leaving brown deposits on the inside of the lamp envelope. An uneven rate of evaporation creates “hot spots” which cause the lamp’s overall rate of decay to increase. Additionally, filaments may crystallize and become brittle particularly when the lamp is operated on dc, mechanically weakening the filament and making it susceptible to breakage from mechanical or thermal shock. Tungsten-halogen lamps are better suited than tungsten for most applications, as they either eliminate or delay the onset of both decay mechanisms. The addition of a halogen such as bromine or iodine creates a regenerative cycle, in which the evaporated tungsten combines with the halogen rather than plating on the envelope. A hot (minimum 250° C) envelope, usually of fused silica, is required. The resulting halide compound decomposes at a rate proportional to temperature. This decomposition occurs preferentially at the “hot spot,” causing tungsten to plate back onto the filament.
Figure 4.15 Spectral emittance of tungsten. (Reprinted from Ref. 7 with permission from Elsevier.)
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Figure 4.16 Tungsten lamp characteristics as a function of percent normal operating voltage.
The advantages of the tungsten-halogen lamp are many. They have a significantly longer lifetime—at least a factor of ten—than the tungsten filament lamp alone for a given filament temperature, and they can be operated at much higher temperatures. They exhibit superior long-term stability because the envelopes don’t darken. Because they require a high envelope temperature, they are physically of much smaller size than the simple tungsten lamp for a given wattage. There are disadvantages to this approach as well. The hot quartz envelope requires greater care, both in fixture design and in handling. Fingerprints are verboten; in calibration applications, fingerprints and grease on the lamp can severely impair measurement accuracy. If the temperature of the envelope is not hot enough, the regenerative cycle fails, and the lamp will fail sooner. The lamps have high internal pressure, and the increased ultraviolet content requires that users take safety measures. Finally, their purchase price is considerably higher than tungsten lamps alone. It should be noted that at the time of publication (2009), incandescent tungsten lamps have fallen out of favor, largely due to energy efficiency considerations in comparison to alternatives such as compact fluorescent bulbs and light-emitting diode (LED) lighting. In fact, sales of incandescent light bulbs
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are being phased out in the European Union, Australia, the United States, and other nations. 4.4.2.2 Other metallic sources
Metallic sources have some general properties worth noting. Since they are fairto-good conductors of electricity, they can be heated to the operating temperature by simply applying an electric current. The power dissipated is given by I2R. The resistance of metals increases with increasing temperature, meaning that a current surge can be expected when a device is turned on. This is particularly noticeable with tungsten, where the starting current required is an order of magnitude greater than the operating current. Other materials with lower operating temperatures show less of this effect. Refractory (high-temperature) metals that may find occasional application include tantalum and molybdenum. They have lower melting points than tungsten and higher vapor pressures. They are more reflective than tungsten, hence at normal operating temperatures their emittance is lower. They also are more susceptible to oxidation. Other than the above, they work well. Noble metals, those that remain in pure form, include platinum, palladium, and iridium. They also have lower melting points and lower emittances than tungsten, but are resistant to oxidation and will therefore operate in air. Certain alloys have been developed for use in heating applications. The most often-used alloy is 80Ni-20Cr (one trademark is Nichrome), which is the element in toasters, space heaters, and other devices. The resistance of Ni-Cr is essentially constant with temperature, simplifying the power supply design. To season it, run it at an elevated temperature in air to maximize the oxidation, which enhances the emittance. 4.4.2.3 Dielectric thermal sources
Nominal characteristics of dielectric sources include a high emittance in those spectral regions where they are optically thick, and a negative resistance versus temperature characteristic. The resistance at room temperature is usually sufficiently high that extreme voltages would be required to generate sufficient heat to get them started. Large amounts of ballast are needed to ensure an overall positive resistance versus temperature characteristic. Alternatively, they can be indirectly heated by a conventional resistive heater. The globar is a rod fabricated from silicon carbide (SiC). It is typically 5 mm in diameter and 50 mm long and requires about 200 W to reach the operating temperature of 1000 to 1500 K. Its emittance is approximately 0.75 out to a wavelength of 15 μm. This source can be directly heated, but requires an auxiliary ballast to overcome the slight negative resistance characteristics of the SiC. Establishing electrical contact is difficult and is usually accomplished with water-cooled silver electrodes. The Nernst glower is a small (1-mm diameter by 10 mm long) ceramic rod comprising mixed oxides of Zr, Y, Ce, Th, Be and the like. It dissipates about
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200 W and operates at 1500 to 2000 K. The emittance is about 0.75 out to a useful wavelength of 30 μm. It is rather fragile and has a lifetime of 200 to 1000 hours. Its resistance at room temperature is extremely high, requiring an auxiliary platinum heater coil to get it started. Nernst glowers are most often found in infrared spectrometers where their shape is ideally suited to image onto a narrow slit. A popular low-cost IR source consists of an aluminum oxide (Al2O3) ceramic tube heated with a coaxial nichrome wire. This source is often used in inexpensive infrared spectrophotometers and operates at about 1200 K with an emittance of about 0.8. The power source is a simple line-operated transformer. The Welsbach mantle is a woven fabric mesh impregnated with refractory oxides such as thorium oxide. It is heated with propane or white gas to a temperature up to 2400 K. The emittance is high for λ > 10 μm. This source is commonly known as the “Coleman” lantern. The carbon arc is a valuable source for announcing grand openings, detecting hostile aircraft, simulating solar radiation, etc. The radiation comes from the plasma-heated carbon at its sublimation temperature of 3800 K. By placing refractory oxides in the carbon rod, higher temperatures commensurate with the evaporating points of these oxides can be achieved, in the range 5000 to 8000 K. The high brightness comes from the small (about 10-mm diameter) size of the carbons. The disadvantages are lack of stability, mechanical issues involving continuous feeding of carbons, the power required (usually a noisy, smelly, unregulated motor generator) and the need for ventilation (hydrogen cyanide is generated). 4.4.2.4 Optical elements
The components in our optical systems also radiate in spectral regions where they are not perfectly transparent or reflective. Mirrors are coated with thin metallic films for high (ρ > 0.9) reflectance. Their emittance is consequently low in the infrared (ε = 0.02 is a reasonable approximation). Even a surface with this low emittance still generates copious quantities of photons per second unless cooled. Refractive elements are transparent at wavelengths where they are normally used; the emittance is low. At wavelengths where the material is absorptive, however, the emittance is high. If our detectors are responsive to wavelengths where the windows and lenses emit, they will respond to such radiation, contributing nothing but noise. Choppers are used to impart modulation to a beam. One must analyze what is being seen when a chopper is used. If the chopper is reflective, you must determine what the system sees in reflection. If the chopper is nonreflective (black), then it has self emission governed by its own temperature and emittance. 4.4.2.5 Miscellaneous thermal sources
Flames emit large quantities of radiation, principally in the infrared. There are two components to radiation from flames. The first is from particulate matter,
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like the hot soot from burning hydrocarbons with insufficient oxygen. Examples include the oxyacetylene torch before the oxygen is turned on, and flames from solid rocket motors. These particles are effectively “black,” emitting as small blackbody radiators. The other component is radiation from the gases, including impurities. For example, the yellow color from a Bunsen burner is due to atomic emission from sodium in the supply gas. Exhaust gases emit radiation in spectral regions where they are optically thick. The important gases (from a radiometric standpoint) are the hot combustion products CO, C2O, and H2O. Particulates, “black” if their size is > 30 μm, are often present, particularly from diesel exhaust and muzzle flash. If a gas both emits and absorbs at the same wavelength, how can we see it? Our initial reaction is that the absorption lines from these gases block the emission from being seen. In fact this is not the case. A phenomenon known as “line reversal” takes place wherein hot gases have slightly different spectral profiles than their unheated counterparts, broader because of increased temperature and pressure. Absorption will occur at the center wavelength of the profile, but the hot gas may be observed in two wavelength bands bordering the center. You and I make good infrared sources. Our temperature is about 300 K and our emittance is nearly unity for all wavelengths longer than about 5 μm. We are all black in the thermal infrared, regardless of skin pigmentation in the visible. Appliances and conveyances also make interesting sources. Cars, trucks, trains, tanks, and aircraft all have different temperatures than their surroundings when at work. Even such little things as insulators on a power transmission line get hot when leakage occurs, rendering them observable targets in the infrared. Table 4.4 lists the total (integrated over wavelength) directional (normal incidence) emittance for several materials at the indicated temperatures (in degrees kelvin). 4.4.3 Luminescent sources 4.4.3.1 General principles
The term “luminescence” refers to emission of light from materials that receive energy from various sources. Table 4.5 details some of the types of luminescence and the sources of energy for each. As we have seen, temperatures must be high to get significant radiation from thermal sources in the visible and ultraviolet regions of the spectrum. Luminescence can occur at much lower temperatures, with light generated as atoms decay to a lower energy level from a higher level. In this process, photon emission occurs at a specific wavelength, but emission lines may be spectrally broadened by Doppler (Gaussian) and pressure (Lorentzian) effects. In practice, this means that the emission occurs in a narrow band, rather than being confined to a single wavelength.
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Table 4.4 Emittances of some common materials.
Material Metals Aluminum, polished Aluminum, heavily anodized Brass, polished Chromium Copper, polished Copper, black oxidized Gold Nickel, polished Nickel, oxidized Silver Stainless steel, polished Stainless steel, oxidized Steel Tungsten Dielectric Materials Alumina Brick, red Carbon, lampblack Concrete, rough Glass, plate Ice Magnesium oxide Oil, 0.001 in. thick on nickel Oil, thick on nickel Paint, oil Paint, laqcuer Skin, human Snow Soil, dry Soil, wet Water
Temperature (K) — 300–900 373 373 310–1370 300 300 300 300 500–1500 300–800 300 1000 300 300–3000 — 300–1000 300 300 300 300 270 400–750 300 300 350 350 300 270 300 300 273–373
Emittance — 0.04–0.03 0.84 0.1 0.08–0.4 0.02 0.8 0.02 0.05 0.4–0.8 0.01–0.03 0.16 0.85 0.08 0.03–0.39 — 0.96–0.6 0.93 0.95 0.93 0.94 0.97 0.69–0.55 0.27 0.82 0.94 0.97 0.98 0.82 0.92 0.95 0.96
At low pressures (below 100 mbar), the Doppler effect predominates. The shape of the curve is Gaussian, and a typical equation takes the form: ( v − vo )2
0.47 S − ln 2 α2 k (ν ) = e , (4.37) α where k(ν) is an extinction coefficient at frequency ν, S is a line strength, α is an absorption coefficient related to the halfwidth of the line, and νo is the line
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Chapter 4 Table 4.5 Luminescence processes and associated energy sources.
Process Bioluminescence Cathodoluminescence Chemoluminescence Electroluminescence Photoluminescence Radioluminescence Sonoluminescence Thermoluminescence
Energy source Organic chemical reactions Electron beam Inorganic chemical reactions Electric field Optical radiation Charged particles Acoustic energy Mild heating below that required for incandescence Energy from friction or pressure
Triboluminescence
center frequency. This shape is characteristic of low-pressure discharge sources and atmospheric absorption and emission lines at elevations greater than 80 km in the stratosphere. The key element in determining this profile is a long meanfree path. At higher pressures, where the mean-free path is short, the lines are further broadened by collisions among the molecules. This pressure broadening gives a Lorentzian shape of the form k (ν ) =
Sα . π (ν − ν o ) 2 + α 2
(4.38)
The Lorentzian shape characterizes the emission of high-pressure discharge sources and atmospheric absorption and emission lines at elevations less than 50 km in the troposphere. At intermediate pressures, the line shape takes on a hybrid shape, a convolution of the Gaussian and the Lorentzian curves. Figure 4.17 compares the shapes of the two curves. The comparatively narrow, Doppler-broadened linewidth of low-pressure discharge is exploited in a number of sources. Because of the narrow width, there is little power in each of the lines; sources are limited to low-power applications involving sharp lines, such as wavelength calibration of spectrometers, excitation of fluorescence spectra, and interferometry. The means of excitation include alternating current and radio frequencies. In the latter case, unknown gases placed in a sealed tube are excited by an external radio frequency (RF) field and produce emission spectra which allow their identification. Common gases identified in this manner include mercury, helium, neon, sodium, potassium, zinc, cadmium, and cesium. These gases have further application in colorful signs and illumination. An example of the emission lines of mercury and argon appears in Fig. 4.18.
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Doppler Lorentz
Figure 4.17 Doppler and Lorentzian line shapes.
The high-pressure regime is entered when the pressure is increased to the point where a low-resistance arc is observed. The line shape becomes Lorentzian, with broad wings extending to either side of the line center. These wings combine to provide a continuum. Arc lamps are efficient and extremely bright. Common types include mercury, xenon, mercury-xenon, and sodium. They can be modulated and flashed, as in the photographic strobe lamp. They are useful for illumination, ultraviolet exposure for photoresist applications, projectors, searchlights, and solar radiation simulators. Figure 4.19 depicts the spectral irradiance of xenon arc sources.
8 Figure 4.18 Emission lines for Hg (< 600 nm) and Ar (> 600 nm). (Copyright held by Ocean Optics, Inc. Reproduced with permission.)
Figure 4.19 Spectral irradiance of xenon arc lamps. (Permission to use granted by Newport Corporation. All rights reserved.)
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Phosphors exhibit luminescent mechanisms and are highly temperature dependent. They represent a variety of colors and persistences (decay time constants greater than 20 ms). They are used in fluorescent lamps, cathode ray tubes (CRTs), x-ray and gamma-ray screens, UV detectors, charged particle detectors, and flat-panel displays, to name a few applications. Figure 4.20 depicts the spectral characteristics of several phosphors. 4.4.3.2 Fluorescent lamps
The modern fluorescent lamp is a combination of a moderate-pressure (about 1 atm) mercury discharge lamp and a phosphor. The phosphor is excited by radiation from the mercury discharge, predominantly from the 254-nm Hg line. The phosphor, coated on the inside of the cylindrical glass envelope, reradiates at longer wavelengths in the visible spectrum. Many phosphors are available with varying color characteristics. Several types of daylight simulation are available, from the rather harsh but inexpensive “daylight” to more subdued “warm white.” Several pale colors are available for special effects, and a plant growth phosphor is common for hothouse use. This latter phosphor emits in the red and blue portions of the visible spectrum, and lacks green, since plants reject green light by reflection. The mercury lines may be prominent, particularly when the more transparent phosphors are used. General characteristics of fluorescent lamps include a long lifetime and relatively high illumination efficiency, as their output is concentrated in the visible. They are a low-radiance extended source, principally suited for illumination. They operate with dc, ac, and RF excitation. The discharge needs help in getting started, as the mercury must be vaporized. Be careful in disposal of spent lamps, as both the mercury and some of the phosphors are toxic.
Figure 4.20 Phosphor spectral characteristics.9 (Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ.)
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The compact fluorescent lamp (CFL) finds increasing use in today’s environment, particularly as a replacement for tungsten incandescent bulbs common in household use. The efficiency of these lamps is high, primarily due to triphosphors and high-frequency solid-state electronic ballast. They are available in several colors, and their liftetime extends to 12,000 hours. Compact fluorescents are more expensive to buy than incandescent lightbulbs, but their longer lifetime has rendered them a cheaper choice than the latter for many uses. Figure 4.21 shows the components of a compact fluorescent lamp.
Figure 4.21 Components of the compact fluorescent lamp. (Reprinted with permission th from the IESNA Lighting Handbook, 9 Edition, by the Illuminating Engineering Society of 10 North America.)
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4.4.3.3 Electroluminescent sources
Electroluminescent sources are activated by an ac field, typically at 60 Hz. These lamps are constructed by layering the phosphor on a metallic electrode, covered by a semitransparent electrode and glass. They draw very little power and are quite efficient. Their radiance is low, and their geometrical characteristics are nearly Lambertian. The color selection is limited to available phosphors. Large areas and complex shapes are possible. Their output depends upon voltage, temperature, and excitation frequency. They are stable and exhibit long lifetimes. They find uses for nightlights, emergency signs, backlighting for portable LCD computer screens, and calibration sources for x-ray film. Other forms of excitation are available, including biological (fireflies and glowworms), chemical (plastic-encapsulated light sticks), and atomic (tritium-excited watch dials), for example. 4.4.3.4 LED sources
A forward-biased p-n junction is an excellent emitter of optical radiation. The emission comes from recombination radiation at the band edge excited by the high current density. The wavelength region of emission is narrow, approximately 5% of the center wavelength. These light-emitting diodes (LEDs) are small, consume little power, and have lifetimes in excess of 10,000 hours. The radiance from an LED is high due to the small emitting area, but the total radiated power is low. They may be modulated to frequencies up to 1 GHz. They are useful for displays and back-illuminating liquid crystal displays. They are also used extensively in airline illumination for exit signs, floor and overhead lighting, and cockpit lighting. Traffic and roadway lighting applications are increasingly benefiting from LED technology. Many older traffic lights are being replaced by LED sources in a matrix configuration, which allows the light to continue to function although several individual LEDs burn out. LEDs are also seen increasingly in architectural displays. Table 4.6 lists several materials used for LED lighting with their corresponding wavelengths or wavelength bands. OLEDs (organic light-emitting diodes) are a special class of LED. They are created from thin organic emitters sandwiched between a transparent anode and a metallic cathode. Enhancement of efficiency and color control are provided by doping emissive layers with highly fluorescent molecules. The structure of organic layers and the choice of anode and cathode materials are designed to maximize the recombination process in the emissive layer, thus maximizing light output. Due to the emitters’ thin width (a few micrometers), OLED display designs can replace LCDs in many applications, including television monitors and automotive displays. Figure 4.22 shows the OLED physical structure.
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Material
GaN SiC GaP ZnTe CdTe InP ZnCdTe GaAsP InGaAsP GaAlAsSb AlGaAs InGaAs InPAs
Wavelength (nm)
350 465 565 620 855 1000 530 – 830 550 – 900 550 – 3500 580 – 1800 620 – 900 850 – 3150 910 – 3150
4.4.3.5 Lasers
The LASER (light amplification by stimulated emission of radiation), was first demonstrated in 1960, and represented a radical departure from conventional sources known at the time. The combination of stimulated (rather than spontaneous) emission to provide gain and a frequency-selective feedback mechanism gives the laser its unique properties. The stimulated emission results from a population inversion, where there are more electrons at a higher energy level than at a lower energy level. This population inversion can be generated using one of several different energy sources, including electrical, optical, magnetic, chemical, or nuclear. The frequency-selective feedback mechanism
Figure 4.22 OLED structure.11 (Reproduced by permission of Silicon Chip.)
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most often takes the form of a resonant cavity with mirrors. For frequencies having overall gain greater than unity, oscillation is possible. Lasers differ from incoherent sources in many ways. Since the amplification process depends upon strict phase relationships, the radiation produced is coherent, i.e., the waves are in phase with each other. Since we are dealing with atomic transitions and a resonant cavity, the output is very nearly monochromatic, with a very narrow bandwidth. Because of the requirements placed on the resonant cavity, the beam diameter and the beam divergence angle are both small. Lasers come in all forms, shapes, and sizes. Some generate unmodulated light [continuous wave (CW)] with power levels from milliwatt to megawatt and above, while others generate pulses. Pulse widths can vary from several femtoseconds (10–15 s) to several milliseconds with repetition rates varying from gigahertz to millihertz. Size can be as small as a TO-18 transistor to as large as a full-size laboratory. A wide variety of materials have been found useful as gain media. Gas lasers include He-Ne and CO2. Examples of ion lasers are argon and krypton. Solidstate lasers are exemplified by ruby and Nd:YAG. A semiconductor laser is similar to an LED but incorporates the resonant cavity needed for laser operation within the semiconductor structure. Tunable lasers are available using organic dyes and special solid-state crystals such as alexandrite. Erbium can be doped into a fiber to provide in situ gain. 4.4.4 Natural sources 4.4.4.1 Sunlight
As Table 4.3 shows, natural sources appear in both thermal and luminescent categories. The most prominent natural source of radiation, indeed the most important, is our sun. The best estimate for the solar constant is 1368 W/m2 with an uncertainty and drift of about 0.2%. Since the orbit of the earth around the sun is elliptical, there is an additional diurnal variation of ±3.5%, with the maximum experienced in January. Things are somewhat different at the surface of the earth. Our atmosphere selectively attenuates by two primary means: scattering by molecules (Rayleigh scatter) and aerosols (Mie scatter), and absorption by molecules (H2O, CO2, and O3). The term “air mass” is frequently used to indicate how much atmosphere the solar radiation is traversing; it is approximately equal to the secant of the solar zenith angle.† Figure 4.23 shows extraterrestrial solar spectral irradiance in the visible and near infrared, along with transmitted solar spectral irradiance for several air masses.
†
At or near sea level. As the altitude increases significantly, and air pressure goes down, the air mass decreases significantly.
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12
Figure 4.23 Spectral characteristics of direct sunlight.
4.4.4.2 Skylight, planetary, and astronomical sources
Skylight is of some importance, as it contributes to both heating and illumination. It arises from both Rayleigh and Mie scattering. The sky appears blue during daytime because of the strong λ–4 wavelength dependence of Rayleigh scattering at shorter wavelengths. In the morning and evening, with airmass high, longerwavelength oranges and reds appear. Figure 4.24 shows some typical spectral irradiance curves. Note the dependence on solar zenith angle (z) and albedo (surface reflectance.)
Figure 4.24 Irradiance from skylight.13 [Reprinted from Daylight and its Spectrum, S. T. Henderson, p. 113 (1970).].
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Astronomers and planetary scientists are greatly interested in stellar and planetary radiation. Optical radiation from the planets in our solar system has two components: reflected solar radiation appearing like a 5700 K blackbody, and emitted thermal radiation corresponding to the temperature of the planet. Radiation from stellar bodies is even weaker, requiring ever larger telescopes to capture enough radiation to be measured. Astronomers use a number of systems to classify stars. Visual magnitude is a logarithmic system, wherein a difference of 1 magnitude equals an irradiance ratio of 2.512. It has been determined that a zero-magnitude star produces an illuminance of 2.65 × 10–6 lm/m2. The sun has a magnitude of about –27 (the scale is upside down.) Progressing to even weaker diffuse sources, we encounter the aurora, the airglow, and zodiacal light. Both the aurora and airglow arise from solar electrons and protons exciting atomic species in the upper regions of our atmosphere. The earth’s magnetic field dictates that most of this activity occurs in polar regions. The aurora is transient and related to intense solar activity, while the airglow is a constant background. Both are characterized by spectral line structure. Zodiacal light arises from solar radiation scattered by dust in the ecliptic plane. 4.4.4.3 Application: energy balance of the earth
The temperature of the earth is determined by the combination of the energy absorbed from the sun and the energy radiated to space. The globally averaged earth albedo weighted by the solar spectrum is 35%. This means we absorb 65% of the incident irradiance, or about 900 W/m2. To maintain thermal equilibrium, we must emit about 225 W/m2 to space (recall that the ratio of the total area to the projected area of a sphere is four). This irradiance corresponds to a blackbody equivalent temperature of about 250 K. The radiation from the earth does not appear as blackbody radiation because of the spectral characteristics of the atmosphere. In the atmospheric windows (regions of relative transparency), the radiation comes from the surface. In those regions where the atmosphere is opaque, the radiation comes from the cold upper atmosphere. Figure 4.25 shows the spectral radiance of a 272 K blackbody for reference (dashed curve) and the outward radiance of the earth as seen from space (solid curve). The dotted curve is the radiance seen when looking upward from the surface of the earth, which includes upwelling earth radiance and whose primary atmospheric component is emission at these wavelengths.
4.5 Radiation Source Selection Criteria Several factors must be considered when selecting a source of optical radiation for experimental or design use. Table 4.7 details selection criteria in the form of questions, the first eight of which were posed by A. G. Worthing in 1937.
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Figure 4.25 Earth radiance in the thermal infrared region. Dashed curve: blackbody at 272 K; solid curve: radiance of earth from space; dotted curve: radiance seen from earth’s 14 surface, looking up.
Table 4.7 Source selection criteria.
Does it supply energy at such a rate or in such an amount as to make measurements possible? Does it yield an irradiation that is generally constant or that may be varied with time as desired? Is it reproducible? Does it yield irradiations of the desired magnitudes over the areas of the desired extent? Has it the desired spectral distribution? Has it the necessary operating life? Has it sufficient ruggedness for the proposed problem? Is it sufficiently easy to obtain and replace, or is its purchase price/construction cost reasonable? What is its physical size? Will it fit into the allocated space? What are its operational limitations (cooling, shock & vibration, etc.)? Where do you plug it in?
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4.6 Source Safety Considerations Table 4.8 provides tips on the safe handling of light sources. Table 4.8 Necessary considerations for safe handling of light sources.
Ultraviolet eye damage is possible with UV sources; shield the lamp and wear protective goggles. There is an explosion hazard from high-pressure lamps. Use an approved housing, clean quartz envelopes thoroughly, handle lamps with gloves, and wear eye protection. A fire hazard exists with powerful lamps; keep flammable materials away from them. Ozone is generated by ultraviolet lamps; use ozone-free lamps (envelopes that do not transmit UV) or provide proper ventilation. There are electrical hazards from lamp power supplies; take normal precautions, particularly with high-voltage starters for arc sources and capacitor banks for pulsed lasers. Careful handling and disposal of CFLs is a must! See http://www.energystar.gov for more information.
4.7 Summary of Some Key Concepts All objects above 0 K radiate according to Planck’s law modified by the directional spectral emittance term, Eq. (4.35): 1 ε(λ; θ, φ)c1 Lλ (θ, φ) = c2 / λT . 5 e πλ − 1 For blackbody radiation (as opposed to blackbody radiation simulator and graybodies), the emittance term can be neglected. Metals are poor emitters. Opaque dielectrics are good emitters. Transparent dielectrics are poor emitters. Gases and glasses radiate only in absorption bands and are transparent elsewhere.
For Further Reading F. E. Carlson and C. N. Clark, “Light sources for optical devices,” Chapter 2 in Applied Optics and Optical Engineering, Vol. 1, R. Kingslake, Ed., Academic Press, New York (1975).
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J. E. Eby and R. E. Levin, “Incoherent light sources,” Chapter 1 in Applied Optics and Optical Engineering, Vol. 7, R. R. Shannon and J. C. Wyant, Eds., Academic Press, New York (1975). D. Kryskowski and G. H. Suits, “Natural sources,” Chapter 3 in Sources of Radiation, G. J. Zissis, Ed., Vol. 1 of The Infrared & Electro-Optical Systems Handbook, SPIE Press, Bellingham, Washington (1993). A. J. La Rocca, “Artificial sources,” Chapter 2 in Sources of Radiation, G. J. Zissis, Ed., Vol. 1 of The Infrared & Electro-Optical Systems Handbook, SPIE Press, Bellingham, Washington (1993). A. J. La Rocca, “Artificial sources,” Chapter 10 in Handbook of Optics, Vol. 1, Part 4, Optical Sources, McGraw Hill, New York (1995). H. Z. Malacara and A. Morales, “Light sources,” Chapter 5 in Geometrical and Instrumental Optics, Academic Press, New York (1988). J. B. Murdoch, Illumination Engineering—from Edison’s Lamp to the Laser, Macmillan, New York (1985). M. S. Rea, Ed., Lighting Handbook: Reference and Application, 9th Ed., Illuminating Engineering Society of North America, New York (2000). J. C. Richmond and F. E. Nicodemus, “Blackbodies, blackbody radiation, and temperature scales,” Chapter 12 in NBS Self-Study Manual on Optical Radiation Measurements, Part 1, U.S. Government (1985). R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corp., New York (1981). W. T. Silfvast, “Lasers,” Chapter 11 in Handbook of Optics, Vol. 1, Part 4, Optical Sources, McGraw Hill, New York (1995). R. H. Weissman, “Light emitting diodes,” Chapter 12 in Handbook of Optics, Vol. 1, Part 4, Optical Sources, McGraw Hill, New York (1995).
References 1. F. Grum and R. J. Becherer, Radiometry, Vol. 1 in Optical Radiation Measurements, F. Grum, Ed., Academic Press, New York (1979). 2. M. A. Bramson, Infrared: A Handbook for Applications, Plenum, New York (1966). 3. M. Pivivonsky and M. Nagel, Tables of Blackbody Radiation Functions, Macmillan, New York (1961). 4. M. Czerny and A. Walther, Tables of the Fractional Function of the Planck Radiation Law, Springer-Verlag, Berlin (1961). 5. E. Hagen and H. Rubens, “Uber die beziehung des reflexions und emissionsvermogens der metalle zu irhrem elektrischen leitvermogen,” Ann. d. Physik, 4(11), pp. 873–901 (1903).
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6. J. A. Jamieson et al, Infrared Physics and Engineering, McGraw-Hill, New York (1963). 7. J. C. DeVos, “A new determination of the emissivity of Tungsten ribbon,” Physica, 20, p. 690 (1954). 8. “HG-2 mercury argon calibration source,” Ocean Optics, Inc. http://www.oceanoptics.com/products/hg1.asp. 9. I. Csorba, Image Tubes, Howard W. Sams & Co, Inc., Indianapolis, (1985). 10. IES Lighting Handbook, 9th Ed., Illuminating Engineering Society of North America, New York (2000). 11. P. Smith, “OLED displays: better than plasma or LCD,” Silicon Chip Mag. 179 (1 Aug. 2003). http://www.siliconchip.com.au/cms/A_30650/article.html 12. S. L. Valley, Handbook of Geophysics and Space Environments, Office of Aerospace Research, USAF (1965). 13. S. T. Henderson, Daylight and its Spectrum, Elsevier, New York (1970). 14. F. Kneizys, et al., “Optical properties of the atmosphere,” Air Force Cambridge Research Laboratory (1972).
Chapter 5
Detectors of Optical Radiation Section 5.7.2 contributed by Robert C. Schowengerdt
5.1 Introduction Optical radiation detectors are transducers that transform optical radiant energy into a different form of energy that is more readily measured. Electrical energy is typically used for this purpose, as electrical measurement technologies are well established. Both thermal and photon detectors convert incident optical energy into electrical signals; in the thermal detector, the initial output takes the form of heat before conversion. Either detector type may be a “point” or an “area” detector. The former are single-element detectors, designed to respond to incident energy. The latter are one- or two-dimensional arrays used particularly for imaging, and include mechanisms to read out the signal on the array. Table 5.1 gives examples of photon and thermal detectors, while Table 5.2 lists differences between them. Table 5.1 Detector types.
Point Area
Thermal
Photon (quantum)
Bolometer Thermocouple Pyroelectric vidicon Linear thermoelectric array Microbolometer array
Photodiode Photoconductor CCD array CMOS array UV photodiode array
Table 5.2 Some important differences.
Thermal Low detectivity
Photon High detectivity
Slow response time
Fast response time
Do not require cooling
Typically require cooling for IR operation 127
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Figure 5.1 Characteristic response curves of (a) photon and (b) thermal detectors.1 [Reprinted from Optical Radiation Measurement series, Vol. 1, F. Grum and R. J. Becherer, Radiometry, p. 177 (1979).].
The parameters in Table 5.2 will be discussed in more detail later in this chapter. Figure 5.1 illustrates the generic differences in spectral response between photon and thermal detectors. Note that the thermal detector response is wavelength independent, and that the photon detector response is a function of several parameters including wavelength and quantum efficiency. A specific example of the latter is the commonly used silicon detector, whose wavelength of peak response is 950 nm and whose cutoff wavelength is near 1100 nm.
5.2 Definitions Several terms commonly used to describe detector parameters are not often used outside the field of optics. Recall that the term wavelength refers to the optical regime below 100 μm. The term frequency is used to describe the direct current (dc) to 1012 Hz audio/radio regime. The following defintions will aid our study of detectors. Spectral responsivity ℜ(λ) is the ratio of the output of a detector or radiometer to that of a monochromatic source of optical radiant power. The detector ouput is typically a current (amperes, A) or a voltage (volts, V), while the incoming optical quantity is power, measured in watts. It is also a function of wavelength, in that it is measured at specific wavelengths, and is correctly reported in A/W or V/W at a specific wavelength. It can also vary as a function of detector temperature and input power level. Responsivity ℜ is the ratio of the output of a detector or radiometer to the incoming optical radiant power, integrated over the spectral range which is common to the source and detector. The input and output signals are those described above. Since this parameter is the result of the quotient of two integrals as shown below, it is source dependent for nonmonochromatic or nonflat sources.
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ℜ=
R (λ )Φ (λ ) d λ . Φ (λ ) d λ
(5.1)
On occasion, one will also find responsivities related to other radiometric quantities, such as irradiance responsivity ℜE in amps (or volts) per W/m2, or radiance responsivity ℜL in amps (or volts) per W/m2sr. These responsivities are more likely to characterize radiometric instruments than detectors alone. Photon spectral responsivity ℜq(λ) is the ratio of the output of a detector or radiometer to the monochromatic optical radiant photon flux incident upon the detector or radiometer. The output is typically a current (A) or a voltage (V), while the incoming optical quantity is photon flux (photons/s). Therefore ℜq(λ) carries the units (A·s) or (V·s). This parameter is wavelength dependent for most detectors and also can vary with temperature and input power level. Like spectral responsivity, photon spectral responsivity is a function of wavelength. Photon responsivity ℜq is the ratio of the output of a detector or radiometer to the incoming photon flux, integrated over the spectral range which is common to the source and detector. The input and output signals are those described above. Since this parameter is the result of the quotient of two integrals, it is also source dependent for nonmonochromatic or nonflat radiation sources. The electrical signal that is output by a detector is governed by the electrical characteristics of the detector (resistance, capacitance, etc.) as well as its associated circuitry. A Bode plot is a convenient way to depict a system’s electrical frequency response, with gain or phase plotted on the ordinate and frequency on the abscissa. For our purposes, voltage or current gain are often depicted. Both axes are logarithmic, as seen from the generalized Bode plot in Fig. 5.2. With the proper choice of axes, slopes associated with single time constants (6 dB/octave, 20 dB/decade) plot as 45-deg lines, and the cut-on and/or cutoff frequencies are at the intercepts of the straight-line asymptotes.
Figure 5.2 Bode plot.
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Time constant τ is the time required for a signal to achieve 63% of its final output, given a step input. For a simple single-section resistance-capacitance (RC) circuit, τ = RC. The rise time is the time the signal takes to rise from 10% of its final value to 90%. For a single time-constant circuit, its value is 2.2τ. Figure 5.3 illustrates this concept. Cutoff frequency fc is the frequency at which the voltage or current response of a detector or circuit falls to 1 2 , or 0.707× its dc or midband maximum value. It is related to the time constant by fc = 1/(2πτ). In terms of power, it is the frequency where the power drops to half of the dc or midband maximum value. This is also called the 3-dB frequency. The root-mean-square (rms) value of a quantity (voltage in this example) is defined as T
vrms =
1 2 v (t )dt , T 0
(5.2)
where T is a single or integer multiple period for periodic waveforms. The rms values discussed in this chapter may be values of voltages or currents and are often referenced in descriptions of noise. Signal is the component of the output voltage or current from a detector that arises from a specific radiometric input. Signal is what you want to work with. It is an rms current or voltage and is denoted vs or is. The signal is the integral of the product of the spectral responsivity and the radiometric input such that: ∞
SIGNAL = R(λ)Φ (λ)d λ .
(5.3)
0
RISE TIME 1
OUTPUT
0.8
0.6
0.4
0.2
0
0
0.5
1
15 2 TIME (time constants)
2.5
Figure 5.3 Signal output as a function of time constant.
3
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Noise is the component of the output voltage or current from a detector that arises from random fluctuations in the detector circuit, in the incoming radiation, or from a number of other sources discussed in this chapter. It is characterized by an rms current or voltage and denoted vn or in. Signal-to-noise ratio (SNR) is the ratio of the rms signal current is to the rms noise current in. Voltage may be substituted for current when calculating SNR, which is dimensionless. Impedance Z is the slope of the current versus the voltage curve of a device at the designated operating point: Z=
dV (οhms) . dI
(5.4)
Linearity is the proportionality of output to input for a detector or an instrument. Linearity implies that the responsivity of a detector or instrument is constant over a defined range of input power, irradiance, etc. Expressed differently, it is the region in which the slope of a plot of output versus input is equal to a constant.
5.3 Figures of Merit Several figures of merit have been defined for optical radiation detectors. Among the more common are the following: Responsive quantum efficiency (RQE, η) is the number independent output events per incident photon. This definition eliminates gain terms that could mask the fundamental detection mechanism. It is frequently expressed as electrons per photon, which, in the case of a detector with internal gain (i.e., a photomultiplier or an avalanche photodiode) gives misleading information. RQE is between 0 and 1 and is often simply called “quantum efficiency.” Detective quantum efficiency (DQE) is the ratio of the square of the output SNR to the square of the input SNR. It is a measure of the SNR degradation caused by the detector and has value between 0 and 1. DQE may be expressed mathematically as DQE =
SNR 2 out . SNR 2 in
(5.5)
Noise-equivalent power (NEP) is the incoming signal, in watts, that produces a signal-to-noise ratio of 1. It is therefore the ratio of the rms noise current (or voltage) to the responsivity. Like responsivity ℜ, this term results from integration over wavelength.
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NEP =
in i = Φ n ; or ℜ is
(5.6) vn vn NEP = = Φ . vs ℜ Note that NEP is a useful figure of merit only when the limiting noise is inherent in the detector, and not due to the input signal. Spectral NEP(λ) is the NEP for a monochromatic single-wavelength input signal. Again, this is not a derivative quantity, per-unit wavelength, but merely a value at a specific wavelength. Noise-equivalent photon flux NEΦq is the incoming signal in photons/s that produces a signal-to-noise ratio of unity. It is therefore the ratio of the rms noise current (or voltage) to the photon responsivity. Detectivity D is the reciprocal of NEP, originally defined as a term because “bigger is better.” The unit is W–1. It is conveniently thought of as the SNR for a 1-W input. Multiply D by the input power to get the SNR. Like responsivity ℜ and NEP above, this term is a result of integration over wavelength: D=
1 SNR = . NEP Φinput
(5.7)
Specific (or normalized) detectivity D* is pronounced “dee-star.” Most detectors display noise that is proportional to (AdB)1/2, where Ad is the detector area and B is the noise bandwidth (to be defined later). D* permits a useful comparison of detectors of different materials, unequal areas, and different noise bandwidths. Its units are cm Hz1/2W–1. Derived from responsivity ℜ and NEP, this term is again a result of integration over wavelength. Its defining equations are D* = D Ad B =
Ad B NEP
=
Ad B vs cm·Hz1/2/W. Φ vn
(5.8)
In the expression above, signal and noise currents may replace signal and noise voltages. To obtain the SNR from D*, multiply D* by the input power and divide by (AdB)1/2. This term is called blackbody D* if the incident power comes from a blackbody radiation simulator. The notation used in this case is D*(T,fc,B), where T is the blackbody radiation simulator temperature in degrees kelvin (500 K is common), fc is the chopping frequency in Hz, and B is the effective noise bandwidth, by convention, 1 Hz. Spectral D*(λ) is D* at a specific wavelength. It is defined as
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D * (λ ) =
Ad B NEP (λ)
cm·Hz1/2/W.
(5.9)
A notation often seen is D*(λp,fc,B), where λp is the wavelength at the peak spectral responsivity (μm), fc is the chopping frequency (Hz), and B is the effective noise bandwidth. D**, pronounced “dee double star,” is a further normalization of D* to a π steradian, hemispheric, projected-solid-angle field of view. The term applies only to what we will later define as background-limited detectors. There is a spectral D** as well. D** is defined as D** = D*
Ω = D* sin Θ1 2 , π
(5.10)
since Ω = πsin2Θ1/2, where Θ1/2 is the system or detector’s half-angle field of view. Photon D* is the specific or normalized detectivity in terms of photons: D *q = D *
hc = λ
AdB is cm·Hz1/2/(phot/s), n in
(5.11)
where n-bar in the denominator is the photon flux in photons per second. BLIP is an acronym for background-limited infrared photodetector, the condition in which the limiting noise in a detector output arises from background photons. A BLIP detector’s internal noise has been reduced to the point where it is not significant. This ideal condition allows SNR to be easily calculated. RA product is the product of the detector resistance and area and is a constant for many materials. D* is proportional to (RAd)1/2 for many photovoltaic detectors; thus the RA product can be used as a figure of merit for material comparisons.
5.4 #N$O%&I*S@E~^ 5.4.1 Introduction to noise concepts
Noise places a fundamental limit on the detection process. Even if all noises intrinsic to the detector could be reduced to insignificance, the noise associated with the random arrival of incident photons will still limit the amount of power that can be detected. Noise is of three types: (1) intrinsic (i.e., noises from physical processes in detectors, associated preamplifiers, and signal processing circuits) (2) anthropomorphic (man-made noises, i.e. from motors, radio, TV, etc.) (3) environmental (i.e., lightning or ionospheric effects).
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Assuming that (2) and (3) can be minimized using good engineering practices (shielding, grounding, proper component and circuit layout), we concentrate on (1). Figure 5.4 is a plot of noise spectral density versus frequency that includes several of the above noise mechanisms. Noise comes in many forms and colors. The most prevalent model of noise is white noise, having a power spectrum independent of frequency. Of course real noise is never white, as that would imply infinite total power. Noise density actually goes to zero above 1012 Hz because of the finite mass of electrons. In Fig. 5.4, the region at frequencies greater than about 50 Hz is white (superimposed on spikes from fixed frequency sources). Pink noise has a power spectrum that increases with decreasing frequency at a nominal rate of 3 dB per octave, proportional to the inverse of the frequency (1/f ). It occurs at low frequencies, below about 5 Hz. Noise increasing at a rate of 6 dB per octave (proportional to 1/f 2) at low frequencies is referred to as red (or brown) noise. There are special names for relatively rare noises that increase with increasing frequency. Blue noise has a power spectrum that increases 3 dB per octave with increasing frequency (proportional to f ), and purple noise has a power spectrum that increases 6 dB per octave with increasing frequency (proportional to f 2).
Figure 5.4 Noise spectral density as a function of frequency. (Reproduced from Ref. 2 with permission of Wiley-Blackwell.)
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Table 5.3 Gaussian noise distribution characteristics.
Peak-to-peak value
2 × rms (±1σ) 4 × rms (±2σ) 6 × rms (±3σ) 6.6 × rms 8 × rms 10 × rms 12 × rms
Fraction of time p-p exceeded 0.32 0.046 0.0027 0.001 (1000 ppm) 60 ppm 0.6 ppm 2 × 10–9 ppm
Most noise can be accurately modeled as white. Also, many forms of noise, such as thermal noise within detectors, can be effectively modeled as Gaussian. There is no necessary correlation between the white noise model and the Gaussian noise model. The oscilloscope trace in Fig. 5.5 shows the characteristic appearance of Gaussian noise. White noise without a dc component has a zero average over time, and its peak is infinite (though with zero probability). The preferred way to characterize this noise is by its rms value denoted by σ, which is also called the standard deviation. Table 5.3 shows peak-to-peak values of Gaussian noise, along with the fraction of the time that the peak-to-peak (p-p) value is exceeded. A good estimate of the rms amplitude of the noise can be obtained by estimating the difference between the maximum and minimum amplitudes in an oscilloscope trace and dividing this difference by six: rms amplitude = (peakmax – peakmin)/6. This can also be done by looking at amplitudes in the data set. Noise can be expressed as a voltage, a current, or a power, and is best expressed as a spectral quantity if its magnitude depends upon frequency. The expression for a mean square noise voltage (equivalent to a noise power) looks like T 1 2 2 v = (vi − vavg ) = (vi − vavg ) 2 dt . (5.12) T0
Figure 5.5 Gaussian noise oscilloscope trace and accompanying probability distribution. (Reprinted from Ref. 3 with permission from John Wiley & Sons, Inc.)
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Its units are volts squared. In the expression, vi is the instantaneous voltage, vavg is the average voltage, and T is the observation time. The rms noise is vn = vn2 . Current may also be used according to Ohm’s law, v 2 = i 2 × R 2 . If the various noises in a system have different origins, they may be considered independent or uncorrelated. The noise powers then add algebraically, while the noise voltages (or currents) add in quadrature: vn2 = v12 + v22 + ... + vi2 , or vn = v12 + v22 + ...vi2 .
(5.13)
If the noises are partially correlated, emanating from the same source, a correlation coefficient must be included in the equation, and the resulting noise will be greater. In any case, the total noise will not exceed the algebraic sum of the noise voltages or currents. 5.4.2 Effective noise bandwidth
The effective noise bandwidth (ENB, but more often seen as B or Δf), first mentioned in the definition of D*, is defined by the equation ∞
1 B= G ( f )v( f ) 2 df , 2 G ( f o )vo 0
(5.14)
where G is the power gain and fo is the frequency where G is a maximum. Spectrally flat white noise is often assumed, simplifying the equation to B=
∞
1 G ( f )df . G ( f o ) 0
(5.15)
The ENB is an equivalent square-band bandwidth and differs from the conventional 3-dB bandwidth, as illustrated in Fig. 5.6. It is realized by an electrical filter operating in the audio- to low-radio-frequency range, 20 Hz to several MHz, or possibly by the detector itself. For a typical single-section (onetime constant) filter, the relationship between these two bandwidths is B = (π/2) × f3dB. A combination of white plus 1/f noise increases B, whereas pure 1/f noise or a combination of white plus generation-recombination (G-R) noise decreases B. These noises will be discussed in more detail later in this chapter. For multipole electrical filters, the ENB and the 3-dB bandwidth converge as the number of poles increases, and the filter response becomes more rectangular. For a common two-pole low-pass filter with equal time constants (Bessel filter), B = 1.22 × f3dB. For other filter types, e.g. Butterworth or Chebyshev, the
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137
3-dB bandwid h
Actual response Noise bandwidth
Frequency Figure 5.6 Effective noise bandwidth. (Adapted from Ref. 4.)
relationship differs and must be derived via integration. If the noise is nonwhite, integration must be performed. ENBs can also be specified in terms of time constants. The common two-pole low-pass filter with equal time constants (Bessel filter) has B = 1/(8τ), and the simple single-pole filter has B = 1/(4τ). Note that the ENB is always stated in frequency space at frequencies below 1012 Hz in the electronic realm. Do not confuse this bandwidth with the passband of an optical filter operating at frequencies greater than 1012 Hz, whose center and passband are specified in units of wavelength, nm or μm. 5.4.3 Catalog of most unpleasant noises
Most texts give the noise equations including the ENB term. The power spectral densities for each noise will be given first here, and then the total noise expression that results from integration over frequency will be presented. 5.4.3.1 Johnson noise
Johnson (Nyquist, thermal) noise arises from the random motion of carriers in any electrical conductor. Both the amplitude and frequency distribution are Gaussian. The power spectral density of Johnson noise is given by the equation S J ( f ) = 4kTR V2/Hz,
(5.16)
where k = Boltzmann's constant, 1.380658 × 10–23 J/K, R = the resistance of the conductor (Ω), and T = the absolute temperature (K). Note that since there is no explicit frequency term, Johnson noise is “white” at least to 1012 Hz. Integrating over frequency to obtain the mean-square noise voltage:
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v 2j = 4kTRB ,
(5.17)
where B is the effective noise bandwidth, defined above. The rms Johnson noise voltage is the square root of the mean square noise voltage, or v j = 4kTRB .
(5.18)
As an example, a 106 Ω resistor generates 4 μV rms for B = 103 Hz, T = 290 K. Since v2/R is power, the noise power per unit bandwidth is simply 4kT, independent of resistance. Invoking Ohm’s law, the mean-square Johnson noise current is given by i 2j =
4kTB . R
(5.19)
If the bandwidth is increased, it appears as if the noise voltage approaches infinity. Not so; at extremely high frequencies, a quantum correction must be applied. The energy per degree of freedom kT in the noise expressions must be replaced: kT →
hf e
.
(5.20)
x , e −1
(5.21)
( hf / kT )
−1
Then, v 2j = 4kTRB
x
where x = hf/kT. It is easy to show that Eq. (5.21) reduces to Eq. (5.17) by applying the approximation e s ≈ 1 + s when s is small, that is, much less than 1. This substitution is only necessary for frequencies greater than 1012 Hz, so it is not significant for most applications. This expression also indicates the linkage between thermal noise and blackbody radiation. In most practical applications, the noise bandwidth is established by an RC circuit time constant. Under these circumstances, the mean-square Johnson noise is ∞
S ( f ) df , 1 + (2πfRC ) 2 0
v 2j =
(5.22)
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which integrates to kT , or C v j = kT / C . v 2j =
(5.23)
Note that the expressions in Eq. (5.23) are independent of resistance R, but dependent upon circuit capacitance C. As an example, if T = 300 K and C = 1 picofarad (pf), then vj= 64 μV rms. Using the identity q = CV, the noise can also be expressed as 6.43 × 10–17 coulombs (C), or about 400 electrons. 5.4.3.2 Shot noise
Shot noise appears due to the discrete nature of electronic charge and occurs whenever current flows across a potential barrier. Its power spectral density is S s ( f ) = 2qI dc ,
(5.24)
where q equals the charge on an electron, 1.60217733 × 10–19 C, and Idc equals the direct current flowing across the potential barrier. The mean square shot noise current is 2 ishot = 2qI dc B .
(5.25)
Like Johnson noise, shot noise is also spectrally flat (“white”). As an example, if 2 × 10–5 amps flow across a barrier, then the noise current in a 103-Hz bandwidth is 8 ×10–11 amps, or 80 pA. If this noise current were to flow through a 105 Ω resistance, then the noise voltage is 80 μV rms. Contrary to some claims, a potential barrier must exist in order for shot noise to be present, as it is in photovoltaic detectors that will be discussed later in the chapter. In a purely resistive conductor, the flow of electrons is highly correlated, not independent. Shot noise follows Poisson statistics: the events all have the same amplitude at low photon-arrival rates. At high rates, the amplitude distribution becomes Gaussian. 5.4.3.3 1/f noise
1/f (contact, modulation, excess, flicker) noise is an empirical noise having various sources. A typical power spectral density equation is S( f ) =
KI α
dc
fβ
,
(5.26)
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where K = a “constant” for the particular technology/noise process, 1.25 < α < 4 (usually 2), and 0.8 < β < 3 (usually 1). Note the spectral frequency dependence of this noise, which has been demonstrated to occur at frequencies as low as 10–5 Hz and below. This is a particularly nasty noise, ultimately limiting dc and low-frequency detection and amplification. This form of fluctuation is ubiquitous, found throughout nature. A few of the manifestations of 1/f noise are seen in Table 5.4. The low-frequency form of 1/f noise is often called “drift.” Other names are “pink noise” (β = 1), “red” or “brown noise” (β = 2), “excess,” “flicker,” and “contact.” Causes in semiconductor detectors, for example, include nonohmic contacts and surface impurities. This noise is particularly insidious, as the noise power is constant in each frequency decade. The total noise is the integral of the power spectral density over frequency, which is proportional to ln(fhigh/flow) for pure 1/f noise. Evaluation is easy for ac-coupled systems, but in direct-coupled systems, we must choose some number other than zero for flow. Unless a system has a particularly long integration time, flow of 0.1 Hz is probably sufficient. While integration of “white” noise over time reduces white-noise effects, integration fails to decrease 1/f noise due to the decrease in flow with increased integration time. Table 5.4 Where does 1/f noise occur?
Waves on a beach Fluctuation in axon membrane Earthquakes Economic variables Ecological time series Self-organizing systems Fluctuations in human heart rate Photon counting Most music (not Metallica!) Frequency of rotation of earth (β = 2) Feedback controls in nuclear reactors Base arrangement of DNA sequences Traffic flow (both vehicular & network) Motion of man standing on one foot
5.4.3.4 Generation-recombination noise
Generation-recombination (G-R) noise arises from fluctuations in the rate at which charge carriers are generated and recombined in semiconductor devices. It
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includes variations in the carrier lifetime. The expression for G-R noise can be complicated, and the specific form depends upon the detector configuration. An example of the power spectral density of G-R noise for an extrinsic photoconductor (to be described later in this chapter) is SG - R ( f ) =
4 R 2i 2 τ , N (1 + 4π2 f 2 τl2 )
(5.27)
where N = the mean number of carriers, i = the average photocurrent due to N, R = the resistance, τl = the carrier lifetime, and f = frequency. Note that G-R noise is frequency dependent (nonwhite). Its Bode plot looks like that of a low-pass filter. 5.4.3.5 Temperature fluctuation noise
Temperature fluctuation noise is found only in thermal detectors. This noise has to do with microfluctuations in the temperature of thermal detectors and is the limiting noise for this detector class. The equation is related to statistical noise in blackbody radiation and is given by an equation of the form S ΔT 2 ( f ) =
4kT 2 KB , K 2 + 4π 2 f 2 H 2
(5.28)
where K = thermal conductance (W/deg), H = heat capacity (J/deg), and k = Boltzmann’s constant. Thermal conductance and heat capacity define a thermal time constant τT = H/K that limits the temperature fluctuations at higher frequencies (nonwhite). The effect of this noise must be determined via the responsivity of the particular detector. 5.4.3.6 Photon noise
Photon noise occurs because in a beam of light, photons do not propagate in an orderly fashion. They obey Poisson statistics, giving the result that the mean
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square fluctuations in the photon arrival rate are equal to the arrival rate. The rms photon noise equals the square root of the number of photons: (Δn) 2 = n ,
(5.29)
where n is the number of photons. The assumption of independence breaks down at extremely high photon arrival rates (“photon clumping”) and requires a quantum correction: (Δn) 2 = n
ex . ex − 1
(5.30)
There is still some controversy in the literature about the relationship between photon noise and other noises. The controversy stems from the carrier generation mechanism and its associated noise. If the generation of carriers in a photodetector is a random process, independent of the randomness of the photon arrival rate, then shot noise or G-R noise can exist independently from the noise in the photon stream. Careful measurements have not yet resolved this dilemma, but high-level measurements indicate that the noise is essentially that given by the shot or G-R equations. The implication is that the generation process is instantaneous, i.e., when a photon is absorbed, the associated carrier is generated instantaneously. 5.4.3.7 Microphonic noise
Microphonic noise, acoustically generated, has two primary manifestations. The most important is found in pyroelectric detectors, which are by nature piezoelectric and act like microphones. A more subtle form occurs when unanchored wiring is allowed to vibrate within a vacuum dewar, causing small but noticeable changes in circuit capacitance. 5.4.3.8 Triboelectric noise
Triboelectric noise is a curious electrostatic noise originating from charges that are built up in dielectrics, typically in coaxial cables used to prevent the entry of external noises. It is particularly nasty when the cable is permitted to flex, and the charge may take some time to dissipate. 5.4.3.9 CCD noises
CCD noises are those specific to the charge transport and readout mechanisms in charge-coupled device (CCD) arrays, including charge-transfer efficiency (CTE) variations, readout noise, KTC reset noise, and others. CCDs will be discussed in more detail in Sec. 5.7.
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5.4.3.10 Amplifier noise
Amplifier noise is a combination of the noises typically found in the low-level circuits that directly couple and signal-condition the detector. There are both shot noise components due to the flow of current through p-n junctions in diodes and transistors and Johnson noise components because of finite resistances in the circuit. There may also be “popcorn” noise, found in some integrated circuit amplifiers due to instabilities in the forward current gain. Amplifier noise is nasty in that it is a form of 1/f noise with a typical exponent f 2. Other forms of 1/f noise are also present in amplifier noise and may be seen in data sheets. 5.4.3.11 Quantization noise
Quantization noise occurs in systems requiring digitization of analog signals. It arises from the discrete nature of the digitization process and is related to the number of bits n in the digital word by LSB =
SIGNALmax , 2n
(5.31)
where n = number of bits, SIGNALmax = the full-scale signal (amps, volts, or electrons), and LSB = the magnitude of the least-significant bit. Since the quantization noise is proportional to the LSB, the larger the number of bits, the lower the quantization noise. Johnson noise, G-R noise, and 1/f noise are usually uncorrelated and therefore add in quadrature as shown in Fig. 5.7. This figure is typical for a photoconductive detector. The three frequencies labeled f1, f2, and f3 are the “corner” frequencies on the curve. In particular, since a plot of responsivity ℜ versus frequency looks like the G-R noise curve, the region where the D* is nominally flat extends from f1 to f3 and is the typical choice for operation. 5.4.4 Noise factor, noise figure, and noise temperature
Noise factor and noise figure are terms often seen in the electrical engineering literature. Noise factor F is the ratio of the actual noise to the theoretical (Johnson-limited) noise: F=
SNRin real noise power = = ideal (Johnson) noise power SNRout
1 DQE
.
(5.32)
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Figure 5.7 Combination of several significant noises.
Noise figure NF is the log (base 10) of the noise factor, given by
real noise power NF = 10 × log dB. ideal (Johnson) noise power
(5.33)
Noise figure is often seen in the specification of preamplifiers and is a valid measure of the additional noise introduced by such a device. One can attempt to artificially lower the noise figure by the simple maneuver of increasing the circuit resistance, thereby increasing the Johnson (theoretical) noise. Only one comment about this: don’t do it! Another quantity often encountered is noise temperature Tn. It is the temperature of a thermal source that provides a signal power level equal to the noise power level. It is defined as: Tn =
vn2 + in2 R 2 . 4kRB
(5.34)
5.4.5 Some noise examples
It is beneficial to see what some noise sources look like on an oscilloscope. The figures below are taken from Motchenbacher.3 Figure 5.8 shows scope traces of noise that is white over three finite bandwidths, listed in Table 5.5. As can be seen in Fig. 5.8, reducing bandwidth has an effect on the peak amplitude as well as the rms noise value. Figure 5.9 presents traces for 1/f noise within three different bandwidths, listed in Table 5.6.
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Figure 5.8 Noise that is white over three different bandwidths. (Reprinted from Ref. 3 with permission from John Wiley & Sons, Inc.) Table 5.5 White-noise bandwidths.
Curve Upper
Bandwidth 200 KHz
Center
20 KHz
Lower
2 KHz
Figure 5.9 1/f noise at three different bandwidths. (Reprinted from Ref. 3 with permission from John Wiley & Sons, Inc.)
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Chapter 5 Table 5.6 1/f noise bandwidths.
Curve Upper
Bandwidth 2 KHz
Center
200 Hz
Lower
20 Hz
Unlike the case for white noise, limiting bandwidth does not proportionately reduce peak amplitude for 1/f noise.* Finally, Fig. 5.10 presents a comparison of sinusoidal signals within white and 1/f noise, respectively. In both cases, the signal-to-noise ratio is approximately 1. As seen in the figure, white noise has a “furry” or “fuzzy” quality in the trace, while 1/f noise is “jumpy.”
Figure 5.10 (a) White noise and (b) 1/f noise for SNR = 1. (Reprinted from Ref. 3 with permission from John Wiley & Sons, Inc.) *
From the frequency plane perspective, the smoothing shown is the result of filtering out higherfrequency terms. From the time-plane perspective, rapidly changing signals are smoothed out by the energy storage elements in the filter (capacitors and inductors).
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5.4.6 Computer simulation of Gaussian noise
Those considering computer simulations of noise should note that the random number generator found in most programming languages and spreadsheets generates a uniform distribution rather than a Gaussian distribution. Consideration of the central limit theorem leads to a simple algorithm k
y=
xi − 0.5
i =1
k / 12
,
(5.35)
where x = RND(1). Use of this series generates random numbers y with mean = 0 and standard deviation σ = 1. The starting point is k uniformly distributed numbers xi, where x is between 0 and 1. A value of k = 12 is suggested, which will give maximum values for y of 6 at the 3σ point. If we wish to generate a new random variable y′ with mean m and standard deviation σ, we form the expression y′ = m + σy .
(5.36)
5.5 Thermal Detectors 5.5.1 Thermal circuit
Thermal detection of optical radiation is a two-step process. Incident optical radiation is absorbed by the receiving surface of the detector, giving rise to an increase in the temperature of the surface. The rise in temperature can then be detected by one of several means, which will be discussed in this section. All thermal detectors follow the same equation relating a change in temperature ΔT = Td – To to incident radiation and the thermal properties of the detector and its surroundings: ΔT = Td − To =
αΦRT 1 + ω2 RT2 H 2
,
(5.37)
where ΔT = the rise in temperature of the receiving element over a local reference temperature, Td = the detector temperature, To = the temperature of the heat sink, α = the absorptance of the receiver, RT = thermal resistance between the receiving element and the heat sink, H = the heat capacity of the receiving element (J/deg),
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ω = radian frequency of the incoming signal, and Φ = the incident radiant power, defined as Φoejωt to allow for either steadystate or modulated-input radiation. Because the thermal time constant τT may be written τT = RT H ,
(5.38)
Eq. (5.37) may also be written as ΔT = Td − To =
αΦRT 1 + ω2 τT 2
.
(5.39)
The thermal circuit is illustrated conceptually in Fig. 5.11. A receiver having α as close as possible to 1 (a perfect absorber) provides the best detector performance, and several black coatings provide good absorption characteristics. For example, Lampblack (carbon) is quite black in the visible and near IR, but becomes somewhat transparent at longer wavelengths. Metallic blacks (gold, platinum, etc.) formed by evaporation in a poor vacuum are of very low density (low RT) and very porous, and are black due to multiple reflections. They are also extremely fragile, sintering at modest temperatures (~65° C for gold black). Black paints can be effective absorbers, yet their heaviness yields rugged but slow (high RT) detectors. Newer etched and anodized coatings and conversion processes that yield extremely black coatings with less mass are also available. As seen from Eq. (5.37), the thermal resistance RT must be as high as possible for maximum sensitivity (high ΔT.) However, high values of RT come at the expense of speed and ruggedness. If speed and sensitivity are equally important, H must be small. This requirement necessitates a very small structure, a lightweight substrate, and a black coating. The ultimate limit of RT is achieved
HEAT SINK
RT
H
Ad
To Td Figure 5.11 Thermal circuit.
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when conduction and convection to the surroundings are minimized. One way to achieve this goal is to place the detector in a vacuum, support it with lowconductivity materials, and utilize small thin connecting wires. It can be shown5 that under these circumstances RT =
1 , 4ασAd Td 3
(5.40)
where Ad is the area of the detector. The thermal conductance K expressed in W/deg is the reciprocal of RT: K = 4ασAd Td 3 .
(5.41)
Under the conditions described, the K in Eq. (5.41) is the only conductance between the detector and its surroundings. The input power producing a temperature change ΔT may be expressed as Φ = K ΔT .
(5.42)
Setting the signal-to-noise ratio equal to one so that Φ=NEP, and assuming the limiting case in which the noise is due only to fluctuations in the incident power, we obtain: ΔT = ΔT 2 .
(5.43)
The mean square value of the temperature fluctuations in the incident beam may also be expressed as 4kTd 2 ΔT 2 = B, (5.44) K where B is the noise bandwidth.6 Equation (5.42) may be rewritten as Φ = 4kTd 2 KB .
(5.45)
Substituting for conductance from Eq. (5.41) and rearranging terms, we obtain NEP = ΔΦ 2 = 4
kTd 5 σAd B , α
(5.46)
bearing in mind that α= ε by Kirchhoff’s law for a system at thermal equilibrium.
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The maximum D* that can be achieved for a 300 K detector (Ad = 1 cm2, B = 1) is thus 1.8 × 1010 cm·Hz1/2/W. This is called the Havens limit for a thermal detector. If the detector element is cooled, this limit increases dramatically, reaching a D* of nearly 1015 cm·Hz1/2/W at 4 K and close to 1019 cm·Hz1/2/W at 0.1 K. At these low temperatures, other noises predominate; we cannot reach the theoretical maximum. 5.5.2 Thermoelectric detectors 5.5.2.1 Basic principles
Thermoelectric detectors operate via the thermoelectric effect, in which a temperature difference produces a voltage difference and vice versa. These two conditions are described below. The thermoelectric effect was first used for optical radiation measurements (solar radiometry) by Nobili and Melloni in 1835, with subsequent applications including early investigation of infrared spectra. Two dissimilar metals connected in series form a thermocouple, which may be used in a circuit as shown in Fig. 5.12. If the junctions between the wires are at different temperatures such that T2 > T1, a current will flow around the loop in the direction indicated. The current’s magnitude will be proportional to ΔT = (T2 – T1); its exact value depends upon the resistance of the circuit and the difference between the thermoelectric powers of the two metals. This phenomenon is called the Seebeck effect, named after its discoverer, T. J. Seebeck (1821). Opening the circuit, as shown in Fig. 5.13, and measuring the voltage results in the following relationship: S=
ΔV (V/deg), ΔT
(5.47)
where S = the Seebeck coefficient, or, alternatively, thermoelectric power. The open-circuit voltage is SΔT.
METAL 1
J1@T1
I
J2@T2
METAL 2 6 Figure 5.12 Thermoelectric circuit. [Reprinted from Optical Radiation Measurement series, Vol. 4, W. Budde, Physical Detectors of Optical Radiation, p. 101 (1983).]
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METAL 1
J1@T1
J2@T2
METAL 2
J3@T3
J4@T3
METAL 3
METAL 2
METAL 3 V 6
Figure 5.13 Open-circuit thermoelectric pair. [Reprinted from Optical Radiation Measurement series, Vol. 4, W. Budde, Physical Detectors of Optical Radiation, p. 101 (1983).]
In this case, an additional pair of metal wires represents a voltage-measuring instrument inserted into the circuit. These wires are typically made of copper. Two additional junctions, J3 and J4, are formed, whose thermoelectric contributions cancel out if they are at the same temperature, according to the law of intermediate materials commonly applied in mechanical engineering. In addition, if the wires are homogeneous, the voltage depends only on the temperatures of the junctions, and not on temperature distributions along the wires. In 1834, Jean C. A. Peltier experimented with thermoelectric circuits by passing currents through them. He noted that one of the junctions became warm, while the other cooled. The coefficient describing the magnitude of this effect is called the Peltier coefficient. It is expressed as: 1 dQ Π= , I dt
(5.48)
where dQ/dt is heat flow and I is current. This effect is exploited in thermoelectric coolers, which are used to cool detectors, laser diodes, small refrigerators, dew-point sensors, and many other things. The Peltier coefficient is related to the Seebeck coefficient by the second kelvin relationship, Π = T × S, where T is absolute temperature in degrees kelvin. This second-order effect occurs in opposition to the Seebeck effect; that is, a current flow due to a change in temperature causes a reduction in that temperature change. It is thus undesirable in radiation detectors, in which it is desirable to maximize ΔT for optimum sensitivity. Hence, such detectors are usually operated with little or no current flow. Signals from thermal detectors can be increased by placing several junction pairs in series, connecting alternate junctions to blackened receivers exposed to incoming radiation, and connecting the other junctions to heat sinks. Devices
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using multiple thermocouple junctions are called thermopiles and are used in most thermoelectric transducers. Some of the theory behind thermoelectric detectors can help to understand their operation. Going back to the thermal equations common to all detectors, we have ΔT = αΦRT
(dc case)
(5.49)
and ΔT =
α ΦRT
(ac case).
1 + ω2 τT 2
(5.50)
Neglecting Peltier cooling, the responsivities are given by ℜ = αSRT
(dc case)
(5.51)
and ℜ(ω) =
α SRT
1 + ω2 τT 2
(ac case).
(5.52)
If current is permitted to flow, the Peltier effect causes a reduction in sensitivity. In that case, ΔΤ is given by R S 2Td ΔT = αΦRT 1 − T , R
(5.53)
where unsubscripted R is the electrical resistance. Since thermocouples are purely resistive, the limiting noise is Johnson noise in the resistance R. Performance can thus be enhanced somewhat by cooling. It can also be enhanced by reducing the value of R; however, this results in decreased RT as well. The overall effect is to decrease thermal sensitivity ΔΤ. A better solution is to use materials having low electrical resistance and high thermal resistance, but for most metals this is not much of an option. Instead, we choose materials for their thermoelectric properties and then optimize either speed or sensitivity, depending upon application. Table 5.7 lists common thermoelectric materials, along with their thermoelectric power values.
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Table 5.7 Thermoelectric materials and thermoelectric power values.
Material
S (μV/°C)
Material
S (μV/°C)
Al Cu Ag Fe Constantan
-0.5 +2.7 +2.9 +16 38
Bi Sb Si Ge
-60 +40 -400* +300*
*
variable, depends on doping
5.5.2.2 Combinations and configurations
There are many material combinations that furnish useful thermoelectric pairs, the majority of which are designed for industrial thermometry over wide temperature ranges. Constantan, for example, is an alloy of copper and nickel developed specifically for thermoelectric measurement. It is paired with copper or iron to form the popular type J and T thermocouples, respectively. The type T thermocouple has an output voltage of approximately 40 μV for a 1° C difference between the hot and cold junction. Seebeck worked with bismuth-antimony (BiSb) pairs. These have the highest thermoelectric power for any of the commonly used metals, 100 μV/°C for the pair. Specially doped silicon and germanium yield the highest output but are difficult to fabricate. Four distinct configurations for thermocouple/thermopile radiation detectors are shown in Fig. 5.14. All early thermopiles were fabricated by soldering or welding fine wires (usually Bi-Sb), culminating with the Coblentz designs [Fig. 5.14(a)]. The linear models are still used for large spectroscopic detectors and the circular for laser power meters and radiometry. The Schwarz design [Fig. 5.14(b)] features two pins of doped Si and Ge, connected by a 0.3-μm-thick blackened gold foil. They are extensively used for small spectroscopic detectors. A wirewound thermopile [Fig. 5.14(c)] can be fabricated by winding many turns of Constantan wire onto a thin insulator and electroplating silver on half of it. The junctions are not very efficient, but they are easy to fabricate. More recent thermopile designs have been realized by vacuum evaporation of alternating layers of bismuth and antimony onto a substrate [Fig. 5.14(d)]. One example utilizes a thin Mylar™ substrate placed over an insulating channel. The thickness of the Mylar determines the speed/responsivity tradeoff for a given application. Another example utilizes micromachined silicon with a thin oxide layer (vanadium oxide for example) or aluminum to create monolithic structures used for thermal imaging. Typical characteristics for both wirewound and evaporated thermopile detectors are shown in Table 5.8.
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Figure 5.14 Thermopile configurations: (a) Coblentz, (b) Schwartz (c) wirewound, and (d) evaporated.7 [Reprinted from Semiconductors and Semimetals series, Vol. 5, N. B. Stevens, “Radiation Thermopiles,” pp. 300–304 (1970).] Table 5.8 Characteristics of wirewound and evaporated thermopile detectors.
Parameter
Wirewound
Evaporated
Active area
1 × 3 mm to 1 × 10 mm
0.5 × 0.5 mm to 4 × 4 mm
NEP
0.1 to 1 n·W/Hz1/2
0.3 to 1 n·W/Hz1/2
D*
108 to 109 cm·Hz1/2/W
1 to 3 × 108 cm·Hz1/2/W
Time constant
4 to 400 ms
25 to 100 ms
Responsivity
0.1 to 10 V/W
10 to 50 V/W
Resistance
10 Ω to 2 kΩ
2 to 20 kΩ
Spectral range
0.3 to 100 μm
0.3 to 30 μm
Window materials
SiO2, KRS-5
BaF2, CaF, KBr
Filter types
Long-wave pass, bandpass
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Table 5.9 Some applications of thermopile detectors.
Passive intrusion alarms Spectral gas analyzers Flame detection Radiometry—laser, spectral, and broadband IR thermometry Thermopile detectors are extremely versatile due to their small size, low cost, and wide wavelength range of operation at dc and room temperature. They can be ruggedized to survive space applications such as horizon sensing and earth radiation budget measurements. Some terrestrial uses of thermopile detectors are shown in Table 5.9. 5.5.3 Thermoresistive detector: bolometer
When a thermoresistive material absorbs incident radiation, it becomes warmer and its electrical resistance changes. The resistance change can be sensed using a device called a bolometer, first invented by S. P. Langley in 1880. The bolometer is a resistor that possesses a high temperature coefficient of resistance, often abbreviated as TCR and symbolized here as β, with units of K–1 . Bolometers are fabricated from metals (the classical approach) and semiconductors (modern). General characteristics of resistance as a function of temperature for both material types are shown in Fig. 5.15. The equation for resistance as a function of temperature is
R(T ) = Ro (1 + βΔT ) ,
(5.54)
where β = (1/R)(dR/dT) and Ro is the resistance at some nominal temperature, often 25° C.
R METAL
SEMI
T Figure 5.15 General characteristics of resistance as a function of temperature for metal and semiconductor materials used in bolometers.
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As can be seen from Eq. (5.54), the change in resistance ΔR is proportional to RoβΔT, where ΔT may be obtained using Eq. (5.37). Note that the resistances referred to above are electrical, not to be confused with RT, a material’s thermal resistance described earlier in the chapter. For metals, β is quite small, on the order of 0.5%/°C. These bolometers are rather insensitive. For the semiconducting materials, mixed oxides called thermistors, β is proportional to T–2, making them potentially quite sensitive. In order for a resistance to be measured, we must force a current through the circuit and measure a voltage drop. Although many circuit types may be used, the half-bridge circuit depicted in Fig. 5.16 is the most popular. A voltage source drives two resistors in series, with the upper resister the load resistor, and the lower resistor the bolometer itself. A coupling capacitor (C) is frequently used to block the dc signal across the sensitive component, RB. This is necessary due to the sensitivity of the component to dc voltage. As a result, modulated beams are required for successful operation of the circuit. The expression for signal voltage measured across the terminals is Vs = ΔV =
VB RL ΔRB . ( RB + RL ) 2
(5.55)
Taking a page from electrical engineering, we invoke the maximum power transfer theorem to set RL equal to RB. Then, Vs = ΔV =
VB RΔR VB ΔR VB RoβΔT = = ( R + R) 2 4 R 4 R
Figure 5.16 Bolometer half-bridge circuit.
.
(5.56)
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Substituting for ΔT from Eq. (5.50) and recognizing that for small ΔT, R~Ro, the bolometer voltage responsivity ℜv is ℜv =
Vs VB = Φ 4
αβRT 1 + ω2 τT 2
.
(5.57)
Inspection of this equation shows that to maximize the responsivity, we should make RT as large as possible. However, by Eq. (5.38), this also increases τT, making the device slower. Thus, we have a tradeoff between responsivity and speed. Because responsivity varies linearly with bias voltage VB, we may consider increasing the bias voltage. This also increases the current through the bolometer, which heats it. Increased heating is typically not a problem in metal bolometers, but can lead to thermal runaway and burnout in thermistor bolometers. A constant current bias can prevent burnout, but the additional electronics add noise to the circuit, making this approach unattractive unless wide variations in ambient temperature are encountered. A better solution is to make the load resistor a matched bolometer element and shield it from the incident radiation. Because the bolometer is a resistive device, Johnson noise predominates. If the electrical contacts are less than perfect, we can also get 1/f noise. The ultimate performance limit for the device, in which temperature fluctuation noise predominates, is rarely seen until the device temperature approaches 4 K. Typical room temperature bolometers exhibit D* values on the order of 109 cm·Hz1/2/W, with responsivities varying between 10 and 104 V/W depending on the material and temperature, with time constants varying between 1 and 100 milliseconds, depending upon the size of the bolometer element. (Uncooled microbolometer arrays, part of a relatively recent development in thermal imaging, feature thermal time constants of a few milliseconds.) Cooling the bolometer to cryogenic temperatures increases the D* to 1012 cm·Hz1/2/W at 2 K and 1016 cm·Hz1/2/W at 0.1 K. Bolometers used at these temperatures are typically fabricated from germanium (the Low bolometer) or composite materials. Another interesting variant is the superconducting bolometer, which operates at the superconductor transition temperature. It is extremely sensitive but has limited dynamic range, even with active bias control. 5.5.4 Pyroelectric detectors 5.5.4.1 Basic principles
The pyroelectric detector is unusual in that it is capable of high-speed operation and responds only to changing signals. The pyroelectric effect was first suggested for radiation detection by Yeou Ta in 1938. Certain ferroelectric materials with asymmetric crystal orientations display the pyroelectric effect, which is a change in surface charge (spontaneous polarization) with temperature. The effect is not
158
Chapter 5
observed at constant temperature as mobile charges within the material align to maintain neutrality. The pyroelectric coefficient p is the change in electric polarization per change in temperature: p=
dPs C/cm2K, dT
(5.58)
where dPs is the change in polarization. The pyroelectric coefficient increases with temperature as shown in Fig. 5.17 until the Curie temperature is reached, when it abruptly drops to zero with attendant loss of response. Fortunately, this is not usually a permanent condition, and the device can often be reactivated. The equivalent circuit is shown in Fig. 5.18(b) as a current generator in parallel with a capacitor and a shunt (or load) resistance RL. A change in temperature ΔT produces a charge Q such that
Q = pAd ΔT ,
(5.59)
where Ad = the sensitive area of the detector and p = the pyroelectric coefficient. The pyroelectric current ip is the product of radian frequency ω and charge: i p = ωQ = ωpAd ΔT .
(5.60)
6 Figure 5.17 Pyroelectric coefficient versus temperature. [Reprinted from Optical Radiation Measurement series, Vol. 4, W. Budde, Physical Detectors of Optical Radiation, p. 129 (1983).]
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Figure 5.18 (a) Initial circuit and (b) equivalent circuit of a pyroelectric detector, with the 8 current generator in parallel with a capacitor and load resistor. [Reprinted from Semiconductors and Semimetals series, Vol. 5, E. Putley, “The Pyroelectric Detector” (1970).]
The current responsivity for the pyroelectric detector may be found by inserting the expression for ΔT in Eq. (5.37) and substituting the expression for τT in Eq. (5.38): ℜi =
ip Φ
=
ωpAd ΔT ωpAd αRT ωpAd αRT = = , Φ 1 + ω2 RT 2 H 2 1 + ω2 τT 2
(5.61)
where τT is the thermal time constant. To determine the output voltage across a load resistor, RL in Fig. 5.18, recall that the output voltage signal is the product of current and impedance.9 In this case, the output voltage is given by v=
i p RL
1 + ω2 RL 2C 2
,
(5.62)
where RLC = the circuit’s electrical time constant τ and ip is given by Eq. (5.60). Therefore, the expression for voltage may be rewritten as v=
ωpAd ΔTRL
1 + ω2 τ2
.
(5.63)
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Figure 5.19 Voltage responsivity as a function of radian frequency.8 [Reprinted from Semiconductors and Semimetals series, Vol. 5, E. Putley, "The Pyroelectric Detector" (1970).].
Substituting for ΔT from Eq. (5.37) and applying the definition of thermal time constant, the voltage responsivity may be expressed as ℜv =
αωpAd RL RT
1 + ω2 τ2 1 + ω2 τT 2
.
(5.64)
In this case, the responsivity increases from zero to a flat region, then decreases at even higher frequencies. The width of the flat region depends upon the separation of the thermal and electrical time constants. The generic voltage responsivity behavior is shown in Fig. 5.19, while Fig. 5.20 depicts voltage responsivity as a function of electrical frequency with load resistance as the parameter. 5.5.4.2 Pyroelectric materials
Several pyroelectric materials and their properties are listed in Table 5.10. Tc is the Curie temperature and FM is a figure of merit defined as FM =
where
p , εC ′
p = the material’s pyroelectric coefficient, ε = the material’s dielectric constant, and C′ = the product of the material’s specific heat and density.
(5.65)
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Figure 5.20 Typical voltage responsivity curve for pyroelectric detectors.6 [Reprinted from Optical Radiation Measurement series, Vol. 4, W. Budde, Physical Detectors of Optical Radiation, p. 131 (1983).] Table 5.10 Pyroelectric materials and their properties.
Material
TGS
Tc (οC) 49
p (C/cm2·K) –8
4 × 10
10
9
Remarks
High D* Best D*
1500
6 × 108
Bulletproof
6 × 10–8
750
5 × 108
Fastest
2 × 10–8
1300
2 × 108
Cheap, flexible
60
LaTaO3
618
2 × 10–8
60 to 200 80
PVF2
5200
D* (cm·Hz1/2/W)
5 × 109
Doped TGS
SrBaNbO3
FM
TGS is the acronym for the organic compound triglycine sulfate. It depoles (loses its internal charge) readily, requiring the periodic or even continuous application of an electric field to maintain operation. It is also sensitive to moisture and needs protection. The D* can be enhanced by doping the material with L-alinine. SBN is strontium barium niobate, a mixture of the general form SrxBa1–xNbO3. Polyvinylidene fluoride, PVF2, is a plastic film (tradename Kynar)
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which can be cut and formed into custom configurations. Other pyroelectric materials include lead zirconate titanate (PZT), ceramic, barium titanate, and barium strontium titanate (BST). 5.5.4.3 Operational characteristics of pyroelectric detectors
The outstanding feature of the pyroelectric detector is its uniform wavelength response (depending on crystal absorption or external blackening) coupled with high sensitivity attainable without cooling. These detectors are small and require no external bias source. Several are available with built-in FET preamplifiers and matched load resistors. The impedance of the device is extremely high such that restoration of charge equilibrium is a relatively slow process. The crystal fitted with electrodes behaves like a capacitor with a variable dielectric. It can be used in either a current or voltage mode. Although the response of the pyroelectric is slow compared to photon detectors, the pyroelectric can be used in current mode for higher-speed applications when sufficient input power is available. The noises inherent in pyroelectric detectors include Johnson noise, thermal fluctuation noise, and microphonics. Pyroelectric detectors are piezoelectric and act as microphones; they must be isolated from the surrounding acoustic environment for successful operation. Preamplifiers provide an additional noise source. In terms of performance, D* values greater than 109 cm·Hz1/2/W have been achieved, as shown in Table 5.5 above, with voltage responsivities on the order of 104 V/W. If the Curie temperature is exceeded, the following procedure may be used to restore operation of the device: (1) Heat the pyroelectric element to a temperature slightly above the Curie temperature. (2) Apply a bias voltage across the electrodes. (3) Slowly lower the temperature back to ambient with the bias voltage applied. Some pyroelectric detectors will spontaneously depole, albeit slowly, at room temperature. These detectors require periodic application of a poling bias, or even a constant bias, to maintain proper poling. 5.5.4.4 Applications of pyroelectric detectors
Perhaps the most pervasive use of pyroelectric detectors is in infrared motiondetection systems, in which two detectors are placed behind a Fresnel lens array to monitor a designated area. At the equilibrium, or no motion condition, there is no output signal from either detector. Motion of any thermally radiating object across the detectors’ fields of view causes a change in received power in one or both detectors, setting off an alarm or triggering the lighting of an area. A separate photocell within the system inhibits operation during daylight hours. Pyroelectric detectors are also employed in:
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(1) tympanic thermometers (2) laser power meters and beam profilers (3) thermal imaging systems (4) fire detection (5) pollution monitoring/gas analysis systems. Pyroelectric detectors are available as single elements and in one- and twodimensional arrays. Some two-element arrays are differential, with the elements wired in opposition. This scheme enhances sensitivity for motion detection systems. Other linear arrays are available with 128 and 256 elements. Twodimensional arrays have been made to 320 × 240 elements. 5.5.5 Other thermal detectors
The Golay cell was invented by M. J. E. Golay in 1947 for use in what was at that time the relatively new field of infrared spectrophotometry. It remains one of the most sensitive thermal detectors ever built. Figure 5.21 shows its construction. In operation, incoming radiation is absorbed by the blackened membrane which heats the gas within the cell and distends the membrane, which is silvered on the opposite side. Light from a lamp illuminates a Ronchi screen which is imaged on the membrane. The reflected beam passes through another Ronchi screen, and the photocell detects the modulation when the membrane moves. The Crooke radiometer is an early example of an optomechanical detector. It is pictured in Fig. 1.1 and consists of a rotating element with four flat vanes. Each vane is absorptive (black) on one side and reflective (polished) on the other. Theoretically, the rotation would be such that the reflective side would recede when radiant energy is incident. The momentum of the incident photons would be absorbed on the black side, and would recoil from the shiny side. This mode is never observed, as a very hard vacuum and virtually perfect bearing would be required. In practice, a small amount of residual gas is present inside the glass envelope, and the gas is heated by the black side of a vane. The local gas pressure is slightly higher at the black face than at the shiny face, resulting in a slight pressure differential. This causes the black face to recede from the incoming radiation.
Figure 5.21 Golay-cell-detector schematic.6 [Reprinted from Optical Radiation Measurement series, Vol. 4, W. Budde, Physical Detectors of Optical Radiation, p. 133 (1983).]
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Other optomechanical detectors include the liquid-in-glass thermometer, a bimetallic strip, and a piezoelectric bimorph.
5.6 Photon Detectors This section describes the most common photon detection schemes: the photoconductive, photoemissive, and photovoltaic. First, we need a brief review of some simple semiconductor physics. 5.6.1 Detector materials
Intrinsic semiconductors are nearly pure materials with a minimum concentration of impurities. Most have a well-defined energy gap. Photons with energies less than the energy gap value are not absorbed, but are either reflected or transmitted. For a photon to be absorbed, it must have a minimum energy and will then elevate an electron from the valence band (bound state) up into the conduction band (free state) such that it is available for conduction under an applied electric field. Figure 5.22 demonstrates this concept schematically, while Eq. (5.66) expresses the high wavelength cutoff beyond which electrons will not have the energy to reach the conduction band: λc =
hc 1.2398 , = Eg Eg
(5.66)
where λc = the high wavelength limit (cutoff wavelength) in micrometers, and Eg = the gap energy, or the minimum energy required to elevate an electron into the conduction band. It is expressed in electron volts (eV). Thermalization loss Conduction band
Ep = Photon energy
Eg = Gap energy
Ef = Fermi level
Valence band Figure 5.22 Illustration of valence and conduction bands, with Eg the energy necessary to promote an electron from the former to the latter.
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Another way to think about the concept is simply to recall that when Ep = hc/λ is greater than or equal to Eg, for a given material, electrons generated by photons at wavelength λ will possess the energy necessary to elevate into the conduction band. In Fig. 5.22, the excess energy Ep – Eg appears as heat. In the case of a photovoltaic cell (to be discussed in greater detail later) this thermalization loss causes the cell’s voltage and power to decrease. It is one of the two primary loss mechanisms responsible for the fairly low peak theoretical efficiency of about 28% for simple photovoltaic cells. A number of interesting intrinsic semiconductor materials are candidates for optical radiation detection, as shown in Table 5.11. For example, silicon is seen to require a maximum wavelength of 1.1 μm, 1100 nm, and this leads to the dramatic falloff on the right side of the photon detector curve seen in Fig. 5.1. Most of these materials may be categorized in a straightforward manner, but the last combination deserves special mention. Mercury telluride (HgTe) is classified as a semimetal, with a small negative energy gap. This means that HgTe is a conductor at room temperature, albeit a rather poor one. Cadmium telluride (CdTe) is a semiconductor which has been exploited for visible radiation detection. When the two tellurides are combined as Hg1–xCdxTe, then the energy gap depends on x, the fraction of CdTe in the mix. A mixture where x = 2 is common and yields a detector with response out to about 12 μm. The longest wavelengths that are practical with this trimetal detector are about 25 μm, and cold temperatures are required for effective operation. Table 5.11 Semiconductor materials useful for optical radiation detection.
Material SiC CdS CdSe GaP GaAs InP Si Ge PbS PbSe InAs InSb CdTe HgTe Hg1–xCdxTe HgCdTe (x = 2)
Energy gap (eV) 3.0 2.4 1.74 2.25 1.4 1.25 1.12 0.68 0.37 0.26 0.33 0.23 1.6 –0.3 variable 0.1
Cutoff λ (μm) 0.41 0.52 0.71 0.55 0.89 0.99 1.1 1.8 3.35 4.8 3.8 5.4 0.78 — 1 to 24 12.4
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Chapter 5 Table 5.12 Intrinsic carrier concentrations for Si and InSb.
Material Si InSb InSb
T (K) 300 300 77
T3 2.7E7 2.7E7 4.6E5
Eg (eV) 1 23 23
e–Eg/kT 3.3–19 1.4–4 8.8–16
ni 1.3410 2.717 9.010
The relationship between intrinsic carrier concentration and temperature is due to thermal activity, and is expressed as: ni 2 = constant × T 3 × e
− Eg / kT
,
(5.67)
where ni = the concentration of carriers, constant ~ 2 × 1031 cm–6K–3 for most materials, and kT = 0.02585 at 300 K with units of eV.
Figure 5.23 shows the intrinsic carrier concentration plotted versus temperature, with energy gap as the parameter. As seen from the figure, an increase in temperature produces an increase in carrier concentration, and materials whose energy gap is large have lower carrier concentrations. Table 5.12 shows these effects for silicon and indium antimonide (InSb).
Figure 5.23 Intrinsic carrier concentration as a function of temperature for several Eg values.
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Looking at the two temperatures for InSb in Table 5.12, it is apparent that the T in the exponent in Eq. (5.67) has more influence on the calculated value of ni than does the T3 term. Since we want sufficient carriers available for photon excitation even at low values of ni, we must cool low Eg detectors that are designed for long-wavelength operation. This fact explains the need, for example, to cool HgCdTe detectors to temperatures of 77 K or lower. Detectors should be cooled so that kT <
Eg
25
or T <
600 , λc
(5.68)
with λc given by Eq. (5.66). Extrinsic semiconductors are those in which a small amount of a selected impurity (called a dopant) is introduced into an intrinsic material. Dopants create additional energy levels within the intrinsic energy gap. The most common host materials have been silicon (Si) or germanium (Ge), both from group IV of the periodic table of the elements. Germanium has historical precedence, and was for many years the only photon detector material available for wavelengths longer than 5 μm. Silicon-based detectors are used extensively in visible and nearinfrared applications; their ease of integration with other monolithic circuit components is a strong element in their favor. Donors are atoms from group V of the periodic table which have five electrons in their outer shell. Only four are needed to match up with Si or Ge, however. The fifth electron is rather loosely bound to the host atom. It resides at the donor level at 0 K, but may be thermally elevated to the conduction band at an elevated temperature. Figure 5.24 shows this schematically. Acceptors are atoms from group III which are short an electron when compared with the host material. These holes (absence of an electron) are at the acceptor level at 0 K. When heated, electrons are thermally elevated from the
Figure 5.24 Energy band structure for donor semiconductors.
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Chapter 5
Figure 5.25 Energy band structure for acceptor semiconductors.
valence band to acceptor levels, leaving behind a hole for conduction, as shown schematically in Fig. 5.25. Table 5.13 lists some of the impurity dopants and their levels and cutoff wavelengths. The maximum practical doping for extrinsic materials is about 1 ppm, or about 6 × 1017 atoms/cm3. At room temperature, they are nearly all used up. As the temperature increases, the material reverts to an intrinsic conductor. There are a number of other interesting semiconductor materials, among them PbxSn1–xTe, that have made good photovoltaic detectors. Many alloys can be formed from combinations of materials in groups III to V or II to VI in the periodic chart. The materials in Table 5.14 below have been successfully used in the fabrication of heterostructure alloys for solid-state sources and detectors. Some are useful in the ultraviolet, others in the visible, and many others in the infrared. Table 5.13 Donor and acceptor levels for germanium and silicon.
Dopant Au Cu Zn Hg Cd Ga B Al In S As Sb
in Germanium eV λc (μm) 0.15 A 8.3 0.041 A 30 0.035 A 35 0.087 A 14 0.055 A 22 0.011 A 112 — — — — 0.011 A — — — 0.013 D — 0.0096 D 129
in Silicon eV λc (μm) 0.54 A 2.3 0.24 A — 0.26 A — none — 0.3 A — 0.0723 A 17.8 0.045 A 27.6 0.0685A 18.4 0.155 A 7.4 0.187 D 6.8 0.054 D 23 0.039 D 32
Note: “A” denotes an acceptor and “D” denotes a donor.
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Table 5.14 Semiconductor materials used for solid-state sources and detectors.
AlP AlAs AlSb GaN GaP GaAs GaSb InP InAs InSb 5.6.2 Photoconductive detectors 5.6.2.1 Basic principles
Intrinsic and extrinsic photoconductive detectors have been discussed in the previous section. To recap, intrinsic photoconductors are typically found at shorter wavelengths than extrinsic photoconductors, whose doping with an impurity provides a longer-wavelength response. Figure 5.26 shows the general layout of a photoconductive detector, with L the distance between electrodes. The conductivity of a slab of semiconductor material is σe = q(nμ n + pμ p ) Ω–1cm–1,
(5.69)
where q = unit of electric charge on one electron or one hole, n = electron concentration (electrons/cm3), p = hole concentration (holes/cm3), μn = electron mobility [cm2/(Volt·s)], and μp = hole mobility [cm2/(Volt·s)]. INCIDENT PHOTONS
CONTACT
w L z
Figure 5.26 Photoconductive detector structure.
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Chapter 5
In an extrinsic semiconductor, one carrier is present. An n-type extrinsic semiconductor will be assumed to simplify the analysis, so that
σe = qμ n n .
(5.70)
In the presence of incoming light, the conductivity is σe = qμ n ( n + Δn) ,
(5.71)
where Δn = change in carrier concentration due to incident radiation. The relative change in conductivity due to incident radiation is Δσe qμ n Δn . = σe σe
(5.72)
To obtain an expression for Δn, recall the definition of responsive quantum efficiency, RQE (symbolized as η), from Sec. 5.3 as the ratio of independent output events per incident photon. In this case, the RQE is the number of electrons elevated to the conduction band per incident photon. The photon irradiance on the detector is Eq, expressed in photons/s·cm2, and the carrier lifetime is τl. The physical processes occur in a detector of thickness z. The change in carrier concentration due to incident radiation is Δn =
η Eq τ l z
.
(5.73)
Equation (5.72) can now be rewritten as Δσe qμ n ηEq τl . = σe σe z
(5.74)
The relative change in conductivity can also be written in terms of detector resistance R:10 Δσe −ΔR = , σe R
(5.75)
with the negative sign indicating that the relative change in resistance has opposite slope to the relative change in conductivity. We wish to derive an expression for the voltage responsivity ℜv at wavelength λ. To do so, we consider the placement of a photoconductive detector in a circuit, as shown in Fig. 5.27.
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Figure 5.27 Photoconductive detector bias circuit.
The signal voltage out Vs is VS = VB
RL , RL + RD
(5.76)
where VB = bias voltage, RL = load resistance, and RD = detector resistance.
In Fig. 5.27, C(opt) is an optional capacitor in the circuit. The optional capacitor placed in the circuit does not factor into the analysis, but is included because values of Vs can be very large, and VB can get as high as 200 V. Placing a capacitor in the circuit allows a modulated signal. Note that the device is symmetrical, and that the polarity of the applied bias in Fig. 5.27 is unimportant. We need an expression for ΔVs, the change in output signal voltage due to a change in resistance. Differentiating Eq. (5.76), we obtain ΔVs =
−VB RL × ΔRD . ( RL + RD ) 2
(5.77)
Note that VB /(RL+RD) is Idc, the dc current flowing through the detector, so that ΔVs =
− I dc RL × ΔRD . ( RL + RD )
(5.78)
Eq in Eq. (5.74) is the photon irradiance on the detector of area Ad. At a particular wavelength, it is Φλ – (hcAd)–1. Making this substitution in Eq. (5.74)
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Chapter 5
and applying the results of Eqs. (5.75) and (5.78) to the definition of voltage responsivity, we obtain ℜv =
ΔVs I R R ληqμ n τl . = dc L D Φ hczAd σe ( RL + RD )
(5.79)
According to the maximum power transfer theorem of electrical engineering, the output signal Vs is maximized when RL = RD. For small signals, we can call them both R. In that case, Eq. (5.79) becomes ℜv =
ΔVs I dc Rληqμτl , = Φ 2hczAd σe
(5.80)
where μ, the carrier mobility, may be μn as above, or more generally, μn + μp. For good photoconductivity, we want high values for η, τl, and μ, and low values for σe and z. As in most other practical applications, compromises and tradeoffs are required. If τl is large, the device will have a slow response. If the device is thin, small z, the RQE η will be reduced because of incomplete absorption. If a carrier in transit comes too close to a nucleus having a vacancy, it may recombine. In that case, an electron will return to the valence band or top the relevant donor or acceptor level. The carrier lifetime τl is the statistical time between generation and recombination. If a photogenerated carrier arrives at one of the electrodes without recombining, another carrier leaves the opposite electrode in order to maintain charge neutrality. In this way, more than one carrier may exist for each absorbed photon, a situation that constitutes a gain. The photoconductive gain G is defined as the ratio of the carrier lifetime to the carrier transit time G=
τl , τtr
(5.81)
where the transit time, in turn, is defined as τtr =
l2 . μVB
(5.82)
The gain is increased by decreasing the spacing between electrodes or applying a large electric field. The price paid here is an increase in response time. We can also derive an expression for voltage responsivity in terms of gain. The signal photocurrent is
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is = ηq
λΦ λΦ τl μVB ⋅ 2 . G = ηq hc hc l
(5.83)
Applying Ohm’s law to the circuit in Fig. 5.27, with RL = RD as above, we obtain λΦ τl μVB R , Vs = ηq 2 2 hc l
(5.84)
λ τ μV R ℜv = ηq l 2 B . hc l 2
(5.85)
and
Note that the current responsivity ℜi may be obtained directly from Eq. (5.83) by dividing the signal current by the power term, such that ℜI = ηq(λ/hc)G A/W. We can maximize ℜv by increasing VB, but if Joule heating occurs and the PC detector heats up, decreasing its resistance, we will burn it out! 5.6.2.2 Noises in photoconductive detectors
The noises commonly found in photoconductive detectors are Johnson, G-R, and 1/f. A typical noise expression is kT kT λ in 2 = 4q ηqΦ G 2 + qG 2 N ′ + + B. qRD qRL hc
(5.86)
The terms inside the bracket, according to their order, are (1) (2) (3) (4)
G-R noise from incident photons (signal and background), dark current noise due to N′ thermally generated carriers, Johnson noise in the detector resistance RD, and Johnson noise in the load resistor RL.
The limiting noise is G-R noise from radiation (signal plus background). When G-R noise overpowers all other noises, the rms signal current is λ is 2 = is = ηqΦ G , hc
while the rms noise current is expressed as
(5.87)
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Chapter 5
λ in 2 = in = 4q 2 ηΦ G 2 B , hc
(5.88)
where B is the effective noise bandwidth. Recall from Eq. (5.8) that D* may be expressed as i D* = s in
Ad B , Φ
where Ad is the detector area. Noting that Φ = EAd, and rearranging terms in Eqs. (5.87) and (5.88), D* for the photodetector becomes η λ D *BLIP (λ, f ) = . 4 E hc
(5.89)
This quantity is called D*BLIP because, as discussed in Sec. 5.3, the limiting noise arises from incident photons. Equation (5.89) expresses a quantity referenced to a particular wavelength and having a specific modulation frequency f. Expressing this irradiance in terms of photon incidence Eq we also obtain D *BLIP (λ, f ) =
λ η . 2hc Eq
(5.90)
The background can be reduced by appropriate shielding and filtering using cold filters when necessary. A detector can be background limited for a 300-K background, but may become Johnson-noise limited for cold backgrounds. Extrinsic detectors require more cooling for a given cutoff wavelength than do intrinsic detectors, and their physical thickness is typically much greater, due to the low absorption coefficient of the host material. A typical infrared photoconductor installation consists of a liquid nitrogen-cooled dewar with an appropriate window, and an optional cold shield and cold filter, as illustrated in Fig. 5.28. 5.6.2.3 Characteristics of photoconductive detectors
In the infrared, PbS, PbSe, InSb, and HgCdTe intrinsic photoconductors along with doped silicon and germanium extrinsic photoconductors are common. Each is optimized for a different wavelength region. HgCdTe is unique in that it is an alloy of CdTe (Eg ≈ 1 eV) and HgTe (Eg < 0 eV, a semimetal). By selecting the composition of the alloy, almost any bandgap and therefore almost any peak
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LIQUID NITROGEN
COLD SHIELD COLD FILTER WINDOW
DETECTOR VACUUM
Figure 5.28 Detector in vacuum dewar.
wavelength can be obtained. The doped silicon detectors are currently in favor for focal plane applications as on-chip signal processing can be accomplished with conventional silicon technology. The lead-salt detectors will operate at room temperature, albeit poorly, but do much better when cooled to –193° C. Most InSb and HgCdTe detectors prefer 77 K while the extrinsic detectors based on Si and Ge require even lower temperatures. The lead-salt detectors have somewhat slower response times than the others. 5.6.2.4 Applications of photoconductive detectors
In the visible portion of the spectrum, CdS, CdSe, and mixtures thereof are the most common photoconductive detectors. They are reliable and rugged if protected from the environment, fairly sensitive, though not very linear. They can handle large amounts of power. They are most often used in industrial and commercial photoelectric controls. In the infrared, photoconductors find use as point and array detectors, particularly beyond 5 μm where photovoltaic detectors are either unavailable or do not perform as well. 5.6.3 Photoemissive detectors 5.6.3.1 Basic principles
The process of photoemission relies on an external photoeffect, in which an electron receives sufficient energy from an incident photon to physically escape a photosensitive material called a photocathode. Photoemission can take place from virtually any surface given sufficient incident photon energy. It was first
176
Chapter 5
observed in metals in the UV by Heinrich Hertz (1887). Einstein received the Nobel Prize in physics in 1921 for his explanation of photoemission, originally published in 1905. Photoemission is a three-step process: (1) a photon is absorbed; the result is a “hot” electron; (2) the electron moves to the vacuum interface; and (3) the electron escapes over the surface barrier to the vacuum. In practical devices, the electron is attracted to and collected by a positively charged anode. The energy required for an electron to escape the surface barrier is known as the “work function,” symbolized as φ. It is material specific. In metals, it can be used to calculate the cutoff wavelength of an incident photon, beyond which an electron will not obtain the necessary kinetic energy to escape the barrier: λc =
hc 1239.8 , = φ φ
(5.91)
where λc is the cutoff wavelength in nm, and φ is in eV. 5.6.3.2 Classes of emitters
Figure 5.29 schematically depicts photoemission in metals. Because no two electrons may occupy the same energy state according to the Pauli exclusion principle, energy states within the metal are separated into a number of closely spaced levels.10 The lines to the left in the figure denote energy levels, the highest being Ef, the Fermi level. Typical work functions are shown in Table 5.15, along with the corresponding long-wavelength cutoffs. Metallic photocathodes are used in the ultraviolet as they are stable and “blind” to photons having lower energies. However, due to metallic reflection and internal absorption and electron scattering, the quantum efficiency is very low (η ≈ 0.001).
Figure 5.29 Photoemission in metals.
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177
Table 5.15 Work functions for metals.
Metal Pt W, Cu Mg Ca Na K Cs
φ (eV) 6.3 4.5 3.67 2.71 2.29 2.24 2.14
λc (nm) 200 275 340 460 540 555 580
Higher quantum efficiency and longer wavelength operation can be achieved with the “classical” photocathodes fabricated from alkali metals, chiefly cesium, and with semiconductor photocathodes. In semiconductors, the photon must impart enough energy to an electron in the valence band to reach the conduction band and have enough energy left to escape the material into the surrounding vacuum. The energy difference between the bottom of the conduction band and the vacuum level is called the electron affinity Ea. Recall that the energy required to elevate an electron from the valence band to the conduction band is called the gap energy Eg. Their sum defines the work function for semiconductors, φ = Eg + Ea. This is the total energy required to escape the potential barrier. In practice, Eg should be larger than Ea to minimize scattering losses within the material. As was the case for metals, the cutoff wavelength can be calculated according to Eq. (5.91). Figure 5.30 schematically depicts photoemission in semiconductors. The electron behavior described above, for both semiconductors and metals, may be described according to Fermi-Dirac statistics: P ( En ) =
1 1+ e
( En − E f )/ kT
,
Figure 5.30 Photoemission in semiconductors.
(5.92)
178
Chapter 5
FERMI-DIRAC FUNCTION 1
100K 400K
PROBABILITY
08
300K
0K
200K
0.6 0.4 02 0 0
05
1
1.5
2
ENERGY
Figure 5.31 Fermi-Dirac function, plotted with temperature as a parameter.
where En = the energy of the nth state, P(En) = the probability that state n is occupied, and k = Boltzmann’s constant.
At En = Ef, the probability of a state being occupied is 0.5. This statistical function is plotted as Fig. 5.31. With semiconductor materials, quantum efficiencies up to 0.3 can be achieved at wavelengths as long as 800 nm, and a quantum efficiency of 0.01 is found at wavelengths out to 1.2 μm. Table 5.16 gives the pertinent parameters. The higher quantum efficiencies result from lower reflection losses and less scattering. More recent materials based on photoconductive single-crystal semiconductors with a thin Cs-based surface demonstrate even higher quantum efficiency and response out to 1.65 μm. Table 5.17 shows some of those materials. Table 5.16 Photocathode energies and cutoff wavelengths.
Material
LiF CsI GaAs Si Ge Cs2Te K2CsSb (bi-alkali) CsSb (S-11) Ag-Bi-O-Cs (S-10) Na2KSb:Cs (S-20)
Eg (eV)
12 6.3 1.4 1.1 0.7 ~3.2 1 1.6 0.7 1
Ea (eV)
φ=Eg+Ea (eV)
1 0.1 4.1 4 4.2 ~0.3 1.1 0.45 0.9 0.55
13 6.4 5.5 5.1 4.9 3.5 2.1 2.05 1.6 1.55
λc (nm)
95 195 225 245 255 350 590 605 775 800
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10
Table 5.17 Nominal composition and characteristics of various photocathodes. (Reprinted by permission of Burle Technologies.)
Nominal composition
Ag-O-Cs Ag-O-Rb Cs3Sb Cs3Sb Cs3Sb Cs3Bi Ag-Bi-O-Cs Cs3Sb Cs3Sb Cs3Sb Cs3Sb Cs3Sb Na2KSb K2CsSb Rb-Cs-Sb Na2KSb:Cs Na2KSb:Cs Na2KSb:Cs Na2KSb:Cs Na2KSb:Cs GaAs:Cs-0 GaAsP:Cs-0 In.06Ga.94As:Cs-0 In.12Ga.88As:Cs-0 In.18Ga.82As:Cs-0 Cs2Te CSI Cul K-Cs-Rb-Sb S = semitransparent O = opaque
PC type
O O O O O O S S S S S O S S S S S S S S O O O O O S S S S
JETEC response designation
S-1 S-3 S-19 S-4 S-5 S-8 S-10 S-13 S-9 S-11 S-21 S-17 S-24 — — — S-20 S-25 ERMA II ERMA III — — — — — — — — —
Conversion (lumen/W at λmax)
92.7 285 1603 1044 1262 757 509 799 683 808 783 667 758 1117 767 429 428 276 220 160 116 310 200 255 280 — — — 672
Luminous responsivity (μA/lm)
25 6.5 40 40 40 3 40 60 30 60 30 125 85 85 120 150 150 160 200 230 1025 200 250 270 150 — — — 125
180
Chapter 5 Table 5.17 (Continued.)
Nominal composition
Ag-O-Cs Ag-O-Rb Cs3Sb Cs3Sb Cs3Sb Cs3Bi Ag-Bi-O-Cs Cs3Sb Cs3Sb Cs3Sb Cs3Sb Cs3Sb Na2KSb K2CsSb Rb-Cs-Sb Na2KSb:Cs Na2KSb:Cs Na2KSb:Cs Na2KSb:Cs Na2KSb:Cs GaAs:Cs-0 GaAsP:Cs-0 In.06Ga.94As:Cs-0 In.12Ga.88As:Cs-0 In.18Ga.82As:Cs-0 Cs2Te CSI Cul K-Cs-Rb-Sb
Wavelength of maximum response (nm)
800 420 330 400 340 365 450 440 480 440 440 490 420 400 450 420 420 420 530 575 850 450 400 400 400 250 120 150 440
Dark Responsivity Quantum emission efficiency at λmax at 25° C (mA/W) at λmax (%) (fA/cm2) 2.3 0.36 900 1.8 0.55 — 64 24 0.3 42 13 0.2 50 18 0.3 2.3 0.77 0.13 20 5.6 70 48 14 4 20 5.3 — 48 14 3 23 6.7 — 83 21 1.2 64 19 0.0003 95 29 0.02 92 25 1 64 19 0.4 64 19 0.3 44 13 — 44 10.3 2.1 37 8 0.2 119 17 92 61 17 0.01 50 15.5 220 69 21 40 42 13 75 25 12.4 0.0006 24 20 — 13 10.7 — 84 24 —
These conversion factors are the ratio of the radiant responsivity at the peak of the spectral response characteristic in amperes per watt (A/W) to the luminous responsivity in amperes per lumen (A/lm) for a tungsten lamp operated at a color temperature of 2856 K. A newer class of photoemitters known as negative electron affinity (NEA) materials feature a special surface treatment on a p-type semiconductor substrate to “bend” the band structure. In extreme cases, the vacuum level is below the bottom of the conduction band. The advantages of this NEA photocathode include longer wavelength operation and higher quantum efficiency. Figure 5.32 schematically depicts photoemission from these materials.
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Figure 5.32 Photoemission from NEA materials.
5.6.3.3 Dark current
Dark current is a limiting factor in photocathodes, and is indistinguishable from photocurrent. The principal source is thermionic emission from the photocathode, characterized by the Richardson equation: J = CT 2 e( −φ / kT ) ,
(5.93)
where J = current density in A/m2, C = a constant, approximately 1.2 × 106, and k = Boltzmann’s constant, 8.617385 × 10–5 eV/K.
The temperature dependence of dark current is illustrated in Fig. 5.33. This graph also shows that cooling is required to achieve best possible performance, especially with photocathodes designed for longer-wavelength operation (i.e., low work function). For intrinsic semiconductor photocathodes, substitute Ea + (Eg / 2) for φ in the Richardson equation. Other sources of dark current are thermionic emissions from the dynodes, leakage current between the anode and other structures, photocurrent from scintillation from the envelope or electrode supports, field-emission current, ionization from residual gasses in the envelope, and currents caused by cosmic rays, environmental gamma rays, and radioisotope radiation from the envelope and support structures.
182
Chapter 5
RICHARDSON EQUATION
1E-6
DARK CURRENT (A)
1E-8 1E-10 1E-12 1E-14 1E-16 1E-18 1E-20 1E-22 200
220
240
260
280
300
320
340
360
TEMPERATURE (K)
Figure 5.33 Richardson equation expressing dark current as a function of temperature.
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 is = ηqΦ q = ηqΦ
λ . hc
(5.94)
If signal current flows through load resistor RL, then the signal voltage is vs = RL ηqΦ
λ . hc
(5.95)
Applying Eq. (5.95) and the results from Eqs. (5.18) and (5.25), the noise voltage is λ 4kT vn = RL 2qid + 2q 2 ηΦ + B hc RL
The three terms in the inner bracket are as follows:
1/2
.
(5.96)
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183
(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
SNR =
λ η qΦ hc λ 4kT 2 2qid + 2q ηΦ + hc RL
1/ 2
B
.
(5.97)
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 =
ηλΦ . 2hcB
(5.98)
5.6.3.5 Photoemissive detector types Photomultiplier tubes. The impact of an electron onto a secondary emitting
material releases several secondary electrons. The gain is defined as the number of secondary electrons per incident electron; its symbol is δ. Values are ~8 in MgO, ~9 in Cs3Sb, 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 δn, 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 NF =
δ( n +1) . δ n (δ − 1)
For large values of δ, Eq. (5.99) becomes
(5.99)
184
Chapter 5
NF =
δ . (δ − 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 μm. 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-1)
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Figure 5.34 Photomultiplier dynode arrangements: (a) circular-cage type, (b) box-and-grid type, (c) linear-focused type, (d) venetian blind type, (e) fine mesh type, and (f) microchannel plate.11 (Reprinted by permission of Hamamatsu Photonics K. K.)
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
Figure 5.35 Typical spectral responses of common photocathode materials. by permission of Hamamatsu Photonics K. K.)
Chapter 5
11
(Reprinted
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187
During the process of junction formation, the following events occur: (1) Free electrons in the n region are attracted to the positive charge in the p region and drift over. (2) Free holes in the p region are attracted to the negative charge in the n region and they drift over. (3) Carrier drift leaves the n region with a net positive charge and the p region with a net negative charge. The crystal stays neutral with no net carrier gain or loss. In the n-type material (dopants are As, Sb, and P), the electrons are the majority carriers and the holes are the minority carriers. In the p-type material (dopants are Al, B, In, and Ga), the holes are the majority carriers, and the electrons are the minority carriers. Majority carriers are far more mobile than minority carriers, and they are the primary contributors to current flow. The barrier height depends upon the donor and acceptor levels and concentrations. This is shown schematically in Fig. 5.36. The region between the n and p regions is called the depletion region, and there is an electric field across it. The barrier height is calculated as φ≈
kT nn p p ln 2 , q ni
where nn = electron concentration in the n-region (majority carriers), pp = hole concentration in the p-region (majority carriers), and ni2 = the intrinsic carrier concentration.
Figure 5.36 Energy levels in a p-n junction.
(5.101)
188
Chapter 5
Figure 5.37 Application of forward bias to a p-n junction.
Since ni is an extremely strong function of temperature, so is φ. As T increases, φ decreases. In the equilibrium junction, the Fermi level is constant. If we apply an external bias across the junction, we can change the energylevel structure. Application of a forward bias to the p-type region reduces the barrier height by the amount of applied voltage (Vf in in Fig. 5.37). The positive terminal of the bias source attracts carriers from the other side of the junction (ntype) and vice versa. The consequence is a high current flow due to conduction by majority carriers. Lowering the barrier height reduces the depletion region. Application of a reverse bias (Vr in Fig. 5.38) to the n-type region increases the barrier height by the amount of applied voltage. The positive terminal of the
Figure 5.38 Application of reverse bias to a p-n junction.
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189
bias source repels carriers from the other side of the junction (n-type) and vice versa. The consequence is a low current flow due to conduction by minority carriers. Increasing the barrier height widens the depletion region. The equation expressing the I-V characteristic of a p-n junction diode is derived from a continuity equation: βqVkT I d = I o e − 1 ,
(5.102)
where q = electronic charge, k = Boltzmann’s constant, T = absolute temperature in degrees kelvin, V = applied voltage, β = a “constant” to make the equation fit the data, sometimes called the “ideality” factor, and Io = reverse saturation current.
At 300K, q/kT is equal to 38.7. The “constant” β varies with applied voltage. It is typically 1, but can be as high as 3. Curves for the I-V equation are shown in Fig. 5.39 for various values of Io. A large Io yields a large reverse current and a small forward voltage drop and vice versa.
Figure 5.39 Current-voltage curves for a p-n junction with various Io.
190
Chapter 5
The expression for the reverse saturation current Io is made up of terms involving the minority carrier concentrations, the Einstein diffusion constants, the minority carrier diffusion lengths, and the minority carrier lifetimes: n p Dn pn D p Io = q + Ln Lp
Ad ,
(5.103)
where q = electronic charge, np = minority carrier (electrons) concentration in the p-region, Dn= Einstein diffusion constant for electrons, Ln = minority carrier (electrons) diffusion length in the p-region, pn = minority carrier (holes) concentration in the n-region, Dp = Einstein diffusion constant for holes, Lp = minority carrier (holes) diffusion length in the n-region, and Ad = detector area.
The Einstein diffusion constant D is defined as D=
kT μ, q
(5.104)
and has units of cm2/s with μ being carrier mobility. Like D, it may be subscripted with n or p to specify electrons or holes. The minority carrier diffusion length is
L = D τl ,
(5.105)
REVERSE SATURATION CURRENT vs. TEMP 0 0001
1E-05
1E-06
1E-07
1E-08
1E-09 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 TEMPERATURE
Figure 5.40 Reverse saturation current versus temperature for a typical p-n junction.
Detectors of Optical Radiation
191
where τl is carrier lifetime. The reverse saturation current is strongly temperature dependent as shown in Fig. 5.40. Note that Figs. 5.39 and 5.40 combine to show that as the temperature increases, the saturation voltage decreases. Thermalization losses will heat the diode or PV cell. The result is that the voltage goes down, the cell current goes up slightly, and the power goes down. Incident optical radiation generates a current through the diode, which adds to the dark current and shifts the I-V curve downward, as seen in Fig. 5.41. The expression for the total current becomes qV I = I o e βkT − 1 − I g ,
(5.106)
where Ig is the photogenerated current, expressed as I g = η qΦ q = η q
λ Φ. hc
(5.107)
As seen from Eq. (5.107), the current generated is directly proportional to photon flux and reduced by the detector’s quantum efficiency. At a given wavelength, the current is also directly proportional to the incident power, and also directly reduced by quantum efficiency. A typical set of I-V curves for various incident power values is shown in Fig. 5.41. There are several ways in which one can operate a photovoltaic detector. The first, and one of the most important for radiometry and photometry, is called short-circuit-current mode. In a practical sense, this is accomplished by connecting the detector to a transimpedance amplifer, which is easy to make with
Figure 5.41 I-V curve for a photodiode at several light levels.
192
Chapter 5
Figure 5.42 Output voltage versus current for Ig >> Io.
an op amp and feedback resistor. This mode is represented mathematically by setting V = 0 in Eq. (5.106). The result is that I = –Ig, in which current generated is a linear function of incident radiant power. This linearity is easily demonstrated to seven decades in radiometric quality silicon photodiodes, though unsubstantiated claims place the number of decades at 14. Open-circuit voltage is another operating condition for a photovoltaic detector. This is represented mathematically by setting I = 0 in Eq. (5.106) and solving for V: Voc =
β kT I o + I g ln q Io
.
(5.108)
If Ig >> Io, which is almost always the case, then Voc is logarithmic with radiant power as seen in Fig. 5.42. If Ig << Io, then Voc becomes linear with incident power as shown in Fig. 5.43. If Io ~ Ig, operation is intermediate between linear and logarithmic. Photovoltaic detectors are also commonly used as power generators. Placing a load resistor RL directly across the detector causes the I-V curve to enclose an area, as seen in Fig. 5.44. Short-circuit current, graphically depicted as the location where the I-V curve crosses the current axis (V = 0), and open-circuit voltage, where the I-V curve crosses the voltage axis (I = 0), form the two extreme points of the (inverted) I-V curve; in between, both current and voltage are available simultaneously. This is the requisite condition for power generation, and devices operating in this region are commonly called solar cells. The lower curve in Fig. 5.44 is the power versus voltage characteristic for a particular cell, in which the maximum power is achieved at about 0.48 V.
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Figure 5.43 Output voltage versus current for Ig << Io.
The open circuit voltage Voc is a result of the forward-biased rolloff seen generically in Fig. 5.39. For silicon, this voltage is close to 0.6 V as in Fig. 5.44. The short-circuit current Ig is proportional to the incident radiation Φ as shown in Eq. (5.107), and the radiation is in turn proportional to the area of the cell. The cell in Fig. 5.44 with an Ig of 50 mA is quite small with an area of about 1 cm2. Cells for typical commercial solar panels or modules have an area of around 100 or more cm2, producing a short-circuit current of around 5 A. A typical panel has perhaps 72 cells connected in series to produce an open-circuit voltage near 40 V and a short-circuit current near 5 A. The ratio of the maximum power to the product VocIg is called the “fill factor,” (not to be confused with fill factor in CCDs, below) which is typically on the order of 0.75 or so. SOLAR CELL OPERATION
60 50
POWER x 2
MILLIAMPS
40 30 20 10 0 0
0.1
0.2
0.3 VOLTS
0.4
0.5
0.6
Figure 5.44 Photovoltaic solar cell operation (lower curve) and I-V characteristic (upper curve).
194
Chapter 5
Figure 5.45 Reverse-biased photodiode I-V behavior.
The photovoltaic detector may also be operated in reverse-bias mode, in which I = –Io – Ig. This mode of operation requires an external voltage source and a load resistor, both of which contribute to noise. An I-V curve for this mode of operation is seen in Fig. 5.45, showing the load line of the resistor and the bias source VB. In dark conditions, the diode resistance is very high, and most of the voltage drop occurs across it. With the addition of light, the diode resistance drops, and a higher fraction of the applied voltage is found across the load resistor. Note, however, that the curves in Fig. 5.45 are idealized, and that a real diode will eventually break down for sufficiently negative voltages due to the Zener effect. There are advantages and disadvantages of operating the photodiode in reverse-bias mode. Some of these are detailed in Table 5.19. Most manufacturers refer to reverse-bias operation as operation in “photoconductive mode.” This term is confusing, as photovoltaic and photoconductive detectors operate very differently. More accurate terms are “reverse-biased” or “back-biased.” Table 5.19 Advantages and disadvantages of reverse-bias operation.
Advantages Better long-wavelength response due to less recombination
Disadvantages Presence of Io and its temperature dependence
Increased speed due to E-field, which sweeps carriers out
Requires relatively stable, quiet voltage source and low-noise RL
Increased speed due to lower junction capacitance C=const (V–0.6)n, –1.2 < n < –1.3
Lower SNR due to noises from RL and VB
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A common approach to operating a photovoltaic detector in the short-circuit mode is to connect it to a transimpedence amplifier, which is easy to make using an op amp and a feedback resistor. 5.6.4.2 Responsivity and quantum efficiency
Figure 5.46 depicts the ideal and real current responsivities for a photovoltaic detector. Recalling that the current responsivity is equal to the current out of the detector divided by the input power, or equivalently ℜi = ηqλ(hc)–1, it is apparent that nonideal (less than 1) quantum efficiency η drives the nonideal response. There are a number of reasons why the quantum efficiency is less than 1, including: (1) Unabsorbed photons of wavelength λ > λc (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 photodiode) (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.
1
RESPONSIVITY (A/W)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
WAVELENGTH (um)
Figure 5.46 Ideal (straight line) and actual (curved line) current responsivity versus wavelength.
196
Chapter 5
5.6.4.3 Noises in photovoltaic detectors
Noise sources in photovoltaic detectors include the following: (1) shot noise from current flow across the potential barrier, (2) Johnson noise, and (3) 1/f noise. The most important noise source is shot noise. Shot noise current arises from the signal, background, and the device’s own dark current. The rms shot noise current is ishot = [ 2q (is + iback + id ) B ] . 1/ 2
(5.109)
Johnson noise can be minimized by choosing a detector with high dynamic resistance Rdyn which varies with voltage. Cooling the detector reduces Johnson noise due both to the decrease in temperature and the increase in Rdyn with decreasing temperature: iJ =
4kTB . Rdyn
(5.110)
The dynamic resistance is low at forward bias and high at reverse bias. It is calculated as Rdyn =
dV βkT − qV /( Ad kT ) = e . dI qIo
(5.111)
At the important zero-bias operating point it becomes Rdyn =
dV βkT = . dI qI o
(5.112)
Although it appears that Rdyn is linear with temperature, the exponential nature of I as a function of temperature predominates, and the dynamic resistance increases with decreasing temperature. Longer wavelength photodiodes have higher Io, lower Rdyn, and are noisier. The insidious 1/f noise is troublesome, as always. It cannot be avoided due to signal current, but can be minimized by operation at a higher frequency due to the following relation:
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197 1/2
i1 f
( constant )( I dc df ) = . f
(5.113)
To calculate D*BLIP for the photovoltaic detector, we need an expression for signal current. Recall that is = Φℜi where ℜi is the current responsivity. Expressed in terms of power, is = ηq
λ Φ. hc
(5.114)
Noise current, which is shot-noise dominated as stated above, may be expressed as a function of the signal current, in the visible portion of the spectrum: 1/ 2 ishot = ( 2qis B ) , (5.115) or as a function of the background, in the infrared portion of the spectrum: 1/ 2
ishot
λ = 2ηq 2 ΦB . hc
(5.116)
Inserting the expression for the signal current and the shot-noise current Eq. (5.116) (for the infrared) in Eq. (5.8) for D* results in * DBLIP (λ , f ) =
ηλ , 2 Eback hc
(5.117)
which is the background-limited expression, applicable in the infrared, where Eback refers to the background irradiance. Similar logic may be followed for photon flux, beginning with Eq. (5.114). In that case, * DBLIP (λ , f ) =
λ η . hc 2 Eq ,back
(5.118)
Comparing Eqs. (5.89) and (5.117), and (5.90) and (5.118), we see that the photovoltaic detector is better (higher D*) than the photoconductive detector by a factor of 2. The physical reason for the difference is the absence of G-R noise in the photovoltaic detector. Recombination takes place within the photoconductive detector itself; in the photovoltaic detector, it occurs in an external ohmic circuit where carriers are not statistically correlated. An exception to this rule arises when photoconductive detectors are operated in “sweepout”
198
Chapter 5
mode. In this case, carriers are swept out of the detector before they recombine, and the D* is equal to that of a photovoltaic detector. 5.6.4.4 Photovoltaic detector materials and configurations
Table 5.20 lists some materials that have proven useful for photovoltaic detectors. There are a number of configurations for photovoltaic detectors. For high-speed operation, a conventional photodiode requires a large reverse bias to decrease device capacitance and create a large drift field. Table 5.21 lists several common configurations; more detailed information about these may be found in the references at the end of the chapter. Moreover, photovoltaic materials are increasingly finding application in solar-cell technology. Some of these materials are listed in Table 5.22. Table 5.20 Some photovoltaic detector materials.
Material GaP GaAs Si InGaAs Ge InAs InSb HgCdTe PbSnTe
Bandgap (eV) 2.4 1.4 1.12 0.73 0.68 0.28 0.16 variable variable
Center wavelength (μm) 0.52 0.93 1.1 1.7 1.82 3.5 5.5 variable variable
Table 5.21 Common photodiode configurations.
Configuration p-i-n Avalanche Schottky barrier Inversion layer Ultraviolet Infrared Position sensing photodiodes (PSPD)
Comments Built-in depletion region, with an intrinsic layer of Si between p and n materials Operate under a large reverse bias; fabricated from Si, Ge, and InGaAs. Can be cooled to 77 K and biased beyond breakdown point. Created by depositing a thin semitransparent metal electrode on top of a semiconductor material. Particularly useful for large-area UV detectors. Created by doping top layer on p-type silicon with a material having a positive charge to form an n-type material. Quantum efficiency approaches 1 at short wavelengths. Include Schottky barrier, front-illuminated PtSi, and AlxGax–1N Include InGaAs, InAs, HgCdTe, and PbSnTe. Require cooling beyond 3 μm. Output as a function of position on the detector, often used in tracking applications.
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Table 5.22 Photovoltaic materials for solar cells.
Material Monocrystalline silicon
Comments n/a
Poly- or multicrystalline silicon
n/a
Ribbon silicon
Formed from flat thin films of molten silicon
New structures based on silicon
Special silicon arrangements to improve efficiency
Cadmium telluride (CdTe)
Large-scale manufacturing and delivery possible
CIGS
Copper indium gallium diselenide
CIS
CuInSe2
GaAs multijunction
High efficiency; have been used in satellites
Amorphous silicon
n/a
5.7 Imaging Arrays 5.7.1 Introduction
Many applications require spectral and spatial information that the use of a single-element detector cannot reasonably provide. In order to obtain this information, some imaging systems use single detectors along with scanning optics and other components. More commonly, linear or area (2D) arrays of detectors are used, with the area array in widest use. A comparison of the functional differences of single and multiple detectors is shown in Table 5.23. Array imagers are relatively recent attempts to emulate human vision. Other innovations throughout history are listed in Table 5.24. This section will review concepts important to array detection, including history, basic array parameters, device architecture, and specific array types along with their applications. Due to the large and growing amount of material on this subject, the reader is strongly encouraged to investigate the references at the end of the chapter. 5.7.2 Photographic film 5.7.2.1 History
One of the earliest man-made optical detectors was photographic film. The treatment here is brief because film has been supplanted in most scientific and consumer applications by solid-state electronic detectors, as described in this chapter. It is still used, however, in some specialized applications. For more information on photographic film, the reader is referred to two thorough books on the application of photography in science and engineering (Ray, 1999) and the theory and technology of photography (Stroebel et al, 2000).
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Chapter 5 Table 5.23 Comparison of single- and multiple-detector capabilities.
Single detectors Gather information from one wavelength or field of view at a time
Provide instantaneous measurement Require a simple electronic interface Need complex scanning scheme due to the presence of a single element One detector, one response
Multiple detectors Gather simultaneous measurements in multiple spectral bands, multiple fields of view, or both Utilize integration over time for all pixels in array Require more complex electronic interface (serial or parallel) Simple or no scanning possible; array with multiple elements can be utilized as a staring array Magnitude of response varies due to the presence of multiple detector elements and nonuniformities in the array
Table 5.24 Vision emulation efforts.
Imaging mechanism Film photography
Time period 19th to 20th century
Mechanical scanning systems Early television Remote sensing systems
20th century
Imaging vacuum tube devices Solid-state array imagers
1930s and onward 1970s and onward
Johann Heinrich Schulze discovered that silver nitrate (AgNO3) darkened when exposed to light and published a scientific paper describing the phenomenon in 1727. The technologies behind photographic film were developed over several decades in the 19th century. The earliest “permanent” recorded film image is generally credited to Joseph Nicéphore Niépce in 1827. The photograph required an eight-hour exposure to sunlight to form. Various pioneers, including Louis Jacques Mandé Daguerre and Henry Fox Talbot, further developed the technology, eventually allowing image recordings onto paper. Toward the end of the 19th century, George Eastman invented roll film and factory processing, thus commercializing the technology for routine consumer use. Photographic film was the most prevalent image recording technology for the next 100 years or so, when electro-optical detectors began to dominate in most fields of science and consumer applications.
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5.7.2.2 Physical characteristics
Film is a “sandwich” of materials, whose primary layers are a stable base of plastic or glass topped by a gelatin emulsion layer containing silver-halide crystals. The base thickness is in the range 50 to 100 μm, and the emulsion layer is about 5 μm thick for black and white film and about 15 μm thick for color film. Various refinements are included by additional layers to protect the emulsion, adhere the emulsion to the base, and reduce backscattering from the rear surface of the base. The film is exposed to light by a camera shutter, and subsequent chemical processing (development) converts the silver-halide crystals to silver, thus darkening the image in proportion to the amount of exposure (for a negative film). The spectral sensitivity characteristics are controlled by the chemical characteristics of the emulsion. The radiometric calibration characteristics of the final image are determined in part by the emulsion, but also by the chemical processing of the exposed film. 5.7.2.3 Spectral sensitivity
Black and white (panchromatic) films typically respond to light wavelengths of about 0.4 to 0.65 μm. The sensitivity for a particular type of film depends on the details of its construction, chemical compounds, and processing chemistry. Kodak has developed films sensitive to near-IR wavelengths out to about 0.9 μm; such films are able to record the high reflectance of vegetation in the near IR. Color films have three different sensitive layers in the emulsion that respond to red-, green-, and blue-light wavelengths. Development of the exposed film results in red, green, or blue transparent dye in the corresponding layer of a positive (slide) film, or cyan, magenta, or yellow in the corresponding layer of a negative film. The fairly broad spectral response of film cannot be easily refined by chemical modifications, so lens spectral filters are commonly used to limit the spectral response. For example, a yellow (haze) filter is used to remove blue scattered skylight, or a blue-green blocking filter is used to restrict IR film to the red and near-IR regions. 5.7.2.4 Radiometric calibration
A scientific procedure for calibration of film was established by Hurter and Driffield in 1890. They proposed a series of regular exposure steps (total exposure is irradiance multiplied by exposure time) onto film and measurement of the resulting optical density D, defined as the negative logarithm of the film transmittance T: D = − log(T ) .
(5.119)
A plot of D for each step versus the logarithm of the corresponding exposure log(E) is known as the “D-logE” or “H-D” (for Hurter Driffield) curve. The H-D
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curve essentially shows output (optical density) versus input (log exposure) calibration. It depends not only on film type, of course, but also on processing conditions (chemical temperatures, processing time), exposure spectral characteristics, density measurement procedures, and many of the variables associated with photography. The main characteristics of the H-D curve are a saturation toe at low exposure (the “base + fog” level, typically a D of about 0.1), a linear segment for moderate exposure, and a saturation shoulder at high exposure (typically a D of about 3.0). The equation of the linear segment is
D = γ log( E ) + D0 ,
(5.120)
where D0 is the projected intercept (usually negative) on the density axis (not the “base + fog” level), and gamma γ is the gain of the film, synonymous with its contrast; i.e. a low gamma means a low-contrast film. The H-D curve describes “macrocalibration” for relatively large-area measurements of a millimeter or more. If calibration of film is attempted for “micro” conditions of measurement of a tenth of a millimeter or less, other factors come into play, for example, the adjacency effect where nearby exposure affects the density at a point of interest. 5.7.2.5 Spatial resolution
The spatial resolution of film is not defined geometrically, as with electronic detector arrays. Film resolution is affected by light scattering within the emulsion, size of the developed silver grains, thickness of the emulsion, etc. Consequently, it has been traditionally measured by exposure of a specific target (for example, three high-contrast bars in a series of decreasing spacing and width) and determination of the smallest target that can be visually identified. A modulation transfer function (MTF) of film can be measured by exposing a series of sine wave targets of varying frequency and contrast. However, the intrinsic nonlinear characteristics of film are problematic for the application of MTF, which assumes a linear imaging system. For example, the adjacency effect causes the MTF to go above one for low- to midrange spatial frequencies. 5.7.2.6 Summary
Film remains a unique image recording mechanism in that the recorded and developed image is the archival medium itself, i.e. no additional processing is necessary to save the original image for long periods of time. It is also an efficient detector in terms of its combination of large format and high resolution. For example, a large-format aerial photograph 10 × 10 in. with a resolution of 10 μm contains some 645,160,000 resolution cells, or pixels. At the time of this writing, such large monolithic electronic detector arrays are not possible (although mosaics of individual arrays can achieve this size). However, the inconvenience, delay, and cost of chemical processing and conversion to a digital format by scanning and the associated quantitative difficulties in image measurements have seriously disadvantaged film relative to
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electronic detectors for applications requiring quantitative measurements. Steady improvements in the spatial density of detector arrays and computer processing capabilities continue to erode the role of film in scientific imaging. 5.7.3 Electronic detector arrays 5.7.3.1 History
One of the earliest array devices was the linear photodiode array marketed by EG&G Reticon. It used silicon photodiodes in the photon flux integrating mode coupled with field-effect transistor (FET) switches. These were driven by clocked shift registers and stored electrons in the device capacitance until readout. During the readout phase, electrons were transferred to an output signal amplifier. Similar linear-array devices today find application in spectroscopy, astronomy, grocery and department store scanners, and many other products. The first large-area imaging arrays were built in the late 1960s using x-y addressable photodiodes, phototransistors, and photoconductors. They were not particularly successful due to responsivity nonuniformities and spatial noise. In addition, only instantaneous readouts were employed, and the signal could not be integrated over time. Advantages were good area utilization and random access to pixels within the array. Charge-coupled devices (CCDs) allowed integration of photosensitive elements with complex readout mechanisms. Photodetectors within a CCD operate in integration mode, with the outputs serially clocked at high rates to a single readout circuit. CCDs having on the order of 100 megapixels (MP) have been fabricated; this number will increase with technology development over time. Some of the parameters characterizing imaging detector arrays are listed in Table 5.25. 5.7.3.2 Device architecture description and tradeoffs
Although CCD technology has achieved focal plane arrays with high pixel counts, high fill factors, high sensitivities, and charge-transfer efficiencies greater than 0.99999 (the fraction of stored charge transferred out of the array), complementary metal oxide semiconductor (CMOS) technology competes effectively with CCDs in the large-format array market. This is due primarily to the ability of CMOS manufacturing to take advantage of already-existing fabrication lines developed for commercial microprocessors. As a result, the cost of manufacture is lowered and the production yield of CMOS-based imagers is improved, relative to imagers based on CCD technology. CMOS devices can be active pixel sensors, employing signal processing functions, including amplification, on chip. This configuration requires more hardware, however, and reduces the fill factor from approximately 80% (in a passive-pixel CMOS) to between 30 and 50%. To increase the amount of light
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Chapter 5 Table 5.25 Array parameters.
Number and size of individual detector elements Overall size of array Format (aspect ratio) of array (e.g. 4:3, 16:9) Instantaneous field of view (IFOV) and array field of regard Fill factor (portion of an array sensitive to light) Scanning or staring array Spatial resolution (MTF) Responsivity Spectral characteristics Dark current Integration time Noise amount and type (fixed pattern, readout, etc.) Dynamic range Linearity Pixel data rate, frame rate, bandwidth Noise-equivalent temperature difference (NETD) for infrared arrays Pixel response uniformity Information capacity Charge transfer efficiency (for CCDs) Long-term storage potential of detector elements Auxiliary equipment necessary to provide maximum utilization of information detected, and minimize the possibility of light falling onto nonphotosensitive areas of the chip, arrays of microlenses that focus the incoming radiation are often used. Charge-injection devices (CIDs) are also finding increasing use in largeformat arrays. They’ve been around since the 1970s, but exploitation of their unique properties has only recently gained momentum. One of these properties is the capability to stare at scenes for long exposures without spillover of charge into adjacent pixels (“blooming”). CID pixels are individually addressable, with charge dumped (injected) to the substrate after each read. By contrast, CCD readouts involve transfer of charge from site to site during the readout process. 5.7.3.3 Readout mechanisms CCD. CCDs can utilize one of several readout mechanisms; the three described below are most common. In a full-frame architecture, the simplest to make and operate, serial and parallel shifts move the charge through the array to an output amplifier. This is shown schematically in Fig. 5.47. Rows of charge are shifted
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Serial clocks SERIAL REGISTER
Parallel clocks
Analog Output
FULL ARRAY
Parallel shift direction Figure 5.47 Full-frame readout.
upward to the serial-shift register in time with the parallel clock signals depicted at left. Upon reaching the serial-shift register, charge is read serially into the output amplifier. The process is repeated until all rows are read out. It requires some form of shuttering of the array, so that light collection does not occur simultaneously with charge transfer. Frame transfer architecture, depicted schematically in Fig. 5.48, is similar to the full-frame architecture with the inclusion of a storage array. As in the case of the full frame, rows of charge are shifted toward the serial register, where charge from each element is read out sequentially in accord with serial clock signals. The storage array seen in Fig. 5.48 is not photosensitive, and light collection and integration in the image array can occur simultaneously with integration in the storage array. As a result, there is no need for a shuttering scheme. A weakness of this mechanism is that integration within the image array is still occurring while transfer to the storage array takes place, resulting in image smear. Because twice the silicon is used as in the full-frame device, a frametransfer device costs more. On the other hand, higher data rates are achieved. In the interline transfer architecture, shown schematically in Fig. 5.49, photosensitive areas and readout registers are arranged in successive columns. After detection and integration takes place in the photosensitive area, charge is transferred to column registers, which are then clocked to the serial register as in the previous two architectural techniques. This architecture allows very fast response time, but is comparatively difficult to fabricate. Although image smear as in the frame transfer device is not entirely eliminated, the amount of smear is reduced.
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Serial clocks SERIAL REGISTER Parallel clocks (storage)
STORAGE ARRAY (MASKED)
Parallel clocks (image)
IMAGE ARRAY
Analog Output
Parallel shift direction Figure 5.48 Frame-transfer readout.
CID. Pixels are individually addressable in a CID. The charge remains intact in
each pixel after its signal level has been determined, making for a nondestructive readout. Row and column electrodes are shifted to ground to make way for collection of the next image frame, and the charge is “injected” to the substrate. This capability for individual pixel control is valuable in many imaging applications. For instance, long exposures to low-light-level sources can allow optimum exposure of a particular target, such as in astronomical use. If other objects of interest within the sensor’s field of view appear during the long integration, the pixels containing such images can be read out while the lowlight-level target continues to integrate on the array. At brighter target levels, blooming is less likely to occur than in the CCD, due to the fact that charge overload is confined to a single pixel.
Figure 5.49 Interline-transfer readout.
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CMOS. Like CIDs, CMOS devices are x-y addressable, allowing individual pixel
information to be read out without reading out the entire array. A disadvantage of CMOS is high readout noise unless an amplifier is part of the configuration of each pixel (that is, unless the device is an active-pixel sensor) which, as stated above, reduces the device fill factor. Additional signal processing circuitry providing functions such as thresholding, edge detection, and motion detection can be incorporated into the CMOS chip, allowing CMOS to compete on a performance level with scientific CCDs. In addition to the common monolithic CMOS structure (detector and readout together on one chip) CMOS devices are available in hybrid form (two chips bump-bonded together). The latter structure allows a 100% fill factor in the visible spectrum. 5.7.3.4 Comparison
When presented with a choice of a camera based on one of these three technologies—CCD, CID, or CMOS—one must consider one’s application relative to the technology and its cost. As in most radiometric system and measurement problems, there are tradeoffs to be made. While not purporting to offer the final word on technology selection, Table 5.26 presents a comparison of some of the characteristics of these technologies. This comparison is by necessity general; many specific tradeoffs can be made to enhance performance and reduce cost, depending upon application. 5.7.4 Three-color CCDs
CCD imagers employing three-color arrays having red, green, and blue filters can provide color imagery in the visible portion of the spectrum. Though this technique is not as effective as film photography in rendering true color images, it finds wide utility in digital photography, in which color images may be quickly available. Typical array formats are large, though not nearly as large as standard CCDs: Kodak, for example, has developed a 16.6-MP array camera, and larger arrays feature 25 MP.12 A four-square “checkerboard” filter pattern is often used on the focal plane to provide the red, green, and blue response channels. These schemes typically include two out of four pixels green, and one each blue and red. The sensor therefore gathers 50% of the green light over the area of the filter pattern, and 25% each of blue and red light. Substantial post processing is required to provide all three colors at all pixel locations; arguably the most famous technique employed for this purpose is the Bayer algorithm (U.S. patent No. 3971065) which may also be utilized in cheaper CMOS-based cameras in addition to other devices. As one might expect, radiometry performed on data that have undergone significant processing will be complex due to the number of error sources. At this time, absolute radiometry performed using three-color CCDs is problematic at
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Chapter 5 Table 5.26 CCD, CID, and CMOS comparisons.
CCD
CID
CMOS
Array fill factor
Can reach 100%
80% or better
Can range from 30% to 100% depending on configuration
Sensitivity
Highest
Not as high as CCD
Pixel-topixel uniformity
High
High when successive frames are integrated Not as high as CCD
Blooming control
Can be a problem
Excellent
Fabrication
Requires specialized processing
Uses standard CMOS processing
Advantages
Well-established technology, well characterized, many scientific uses
Takes advantage of existing semiconductor fab techniques Can be used in low- Many on-chip and high-lightsignal-processing level applications; features possible excellent blooming control
Problems
Blooming control
Noisy
Can be high if signal processing is on chip Needs improvement
Readouts can be noisy
best. The situation is changing, however, as digital cameras are being developed in competition with film cameras for photogrammetric purposes. Because of the application, these digital cameras would have to offer better radiometric sensitivity than that obtained using digitized film images and would provide the capability for self calibration.13 Alternatively, three separate CCDs may be used to provide color imaging, with filters designed to pass radiation in the red, green, and blue portions of the spectrum. This approach involves more hardware than the one discussed above, but will become more cost effective as the price of this technology comes down. 5.7.5 Ultraviolet photon-detector arrays
Ultraviolet detector arrays typically utilize hybrid architectures. They commonly employ arrays of photodiodes, though this is not always the case. Many arrays
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utilize aluminum gallium nitride (AlGaN) as the detector material due to its sensitivity in the UV and insensitivity at longer wavelengths. Flame detection, astronomy, and undersea communications are some applications that benefit from these devices. Current technical challenges include tailoring the cutoff wavelengths of AlGaN p-i-n photodiodes by controlling alloy composition, allowing AlxGa1–xN arrays to cut off between 227 nm and 365 nm. 5.7.6 Infrared photodetector arrays
Infrared photodetector arrays are utilized over all regions of the infrared spectrum, providing detection in the near infrared (NIR), short-wave infrared (SWIR), and long-wave infrared (LWIR). LWIR devices have been successfully used for many years in sensitive military applications, as have SWIR-based systems. Indium gallium arsenide (InGaAs) arrays are used in the NIR to provide alternatives to more traditional night-vision devices based on image intensifier technology, as well as many other applications. InGaAs arrays may be comprised of p-i-n photodiodes or avalanche photodiodes (APDs.) Varying the chemical composition of the alloy (InxGa1–xAs) allows optimization at a desired wavelength for maximum signal strength. Detectors made of lead selenide (PbSe) and lead sulfide (PbS) operate in the 1 to 3-μm and 3 to 5-μm portions of the spectrum; these detectors typically utilize a monolithic architecture. Another monolithic detector is platinum silicide (PtSi), operating in the 3 to 5-μm region. PtSi arrays are highly uniform, even if relatively low response compared to other detectors, and are often used to image objects with high spatial variation. Mercury cadmium telluride detectors (HgCdTe) are used in 3 to 5-μm and particularly in 8 to 12-μm regions. They typically employ a hybrid architecture and have high sensitivity and fill factor. They exhibit substantial nonuniformity when tailored for the LWIR and require cryogenic cooling to 77 K, increasing the size, weight, cost, and power consumption of devices. Indium antimonide (InSb) arrays are also used in hybrid devices and for scientific and defense applications in the SWIR and midwave infrared (MWIR). Signal processing in the element (SPRITE) detectors, commonly used in forward-looking infrared (FLIR) modules of British manufacture, utilize HgCdTe, but do not have the many leads and preamps that predominate in HgCdTe two-dimensional arrays. SPRITEs are typically linear arrays that allow integration of the incoming signal by scanning along the array at the same rate as the carrier drift. SPRITE-based FLIR imagers have been deployed on helicopters, aircraft, and other platforms for tactical military uses in the 8 to 12-μm region. Quantum-well infrared photodetectors (QWIPs) do not have the nonuniformity problems associated with HgCdTe. Based on GaAs and InGaAs technology, their manufacture takes advantage of established wafer productions processes.
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Chapter 5 Table 5.27 Initial approach to uncooled imaging detector array development.14
Staring focal plane arrays based on thermal detection mechanisms Focal plane response at video rates (30 Hz in the U.S.) Noise-equivalent temperature difference (NETD) of 0.1 K Resistive microbolometer staring arrays (Honeywell) and hybrid pyroelectric detector staring arrays (Texas Instruments) Use of modern integrated circuit electronic processing technology to enable development of low-cost multielement arrays 5.7.7 Uncooled thermal imagers
Arrays of thermal detectors operating according to the detection mechanisms discussed in Sec. 5.5 are finding increasing use in a wide variety of applications. They are cheaper to manufacture than infrared photon arrays and do not require cooling. In the early to mid-1990s, formerly classified research into uncooled thermal arrays became public. This research, sponsored by the U.S. Defense Advanced Research Projects Agency (DARPA), the U.S. Army’s Night Vision and Electronic Sensors Directorate (NVESD), and the U.S. Naval Air Warfare Center (NAWC) provided an approach to the development of uncooled thermal imaging detector array–based systems. Two U.S. companies were at the forefront of efforts to develop these systems: Honeywell Corporation in Plymouth, Minnesota, and Texas Instruments in Dallas, Texas. In addition, several organizations worldwide, including GEC Marconi of Great Britain, participated in developing uncooled infrared imaging detector arrays. Table 5.27 lists some factors important to the initial approach to uncooled detector development. Thermoelectric array development was also pursued; linear arrays based on this technology have been used in several radiometric applications including inmotion inspection of railroad train wheels. While early 2D arrays provided 320 × 240 focal planes with individual pixel elements of 50 μm in size and NETD of 0.1 K (f/1, 30 Hz), developments over the past decade have significantly improved performance. Sensitivity of the microbolometer arrays has been improved through improving pixel thermal isolation and array fill factor. Microbolometer arrays based on vanadium oxide detector material in a 640 × 480 format are now available, having NETDs of 50 mK or less (f/1, 30 Hz) with element sizes of approximately 25 μm. Packaging improvements are also in process. In fact, device-production strategy, effective use of existing fabrication methods, and development of low-cost fabrication techniques all contribute to how well a technology will be received in the marketplace. This reception, in turn, drives research. More “exotic” expensive technology is not always better. As in so many facets of radiometry, a technology user is wise to trade off costs, benefits, and level of technical performance required before purchasing a system or building a product. Table 5.28 lists some applications of uncooled thermalimaging devices.
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Table 5.28 Applications of uncooled thermal detector array-based systems.
Night-driving systems in automobiles Roofing inspection Leak detection from pipes and other fluid and gas delivery/storage structures Law-enforcement investigations of buildings, trucks, and other possible concealment locations Quality control along industrial process lines Detection of “hot spots” in electrical wiring to determine possible locations of malfunction Aircraft inspection to determine weak points in structure based on thermal information Border-patrol surveillance 5.7.8 Summary
The above descriptions of array detectors, their uses, and some comparisons are only an overview. Trading off parameters for a specific application is often necessary, and there is no particular “right” answer to many detector selection problems. A thermographer examining a housing structure for sources of heat leakage can purchase an uncooled thermal instrument for far less cost than a cooled device and yet receive adequate imagery for that task. An astronomer observing a distant galaxy or low-light-level astronomical source may find a cooled photodetector array to be his only reasonable option in the infrared, but might he choose a CID over a CCD for observations in the visible? Questions such as these are important for the student or professional wishing to become adept at systems analysis and systems engineering, two of the most important disciplines making use of the art of radiometry.
For Further Reading D. F. Barbe, Ed., Charge-Coupled Devices, Springer Verlag, Berlin (1980). J. D. E. Benyon and D. R. Lamb, Charge-Coupled Devices and Their Applications, McGraw-Hill, London (1980). L. Biberman and S. Nudelman, Photoelectronic Imaging Devices, Plenum, New York (1970). I. P. Csorba, Image Tubes, Howard W. Sams, Indianapolis (1985). G. C. Holst and T. S. Lonheim, CMOS/CCD Sensors and Camera Systems, JCD Publishing and SPIE Press, Bellingham, Washington (2007). R. J. Keyes, Ed., Optical and Infrared Detectors, Springer-Verlag, Berlin (1980). Photomultiplier Tubes: Principles and Applications, Philips Photonics (1994). Sidney F. Ray, Scientific Photography and Applied Imaging, Focal Press, Oxford (1999). G. H. Rieke, Detection of Light: From the Ultraviolet to the Submillimeter, Cambridge (1994).
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R. A. Smith, F. E. Jones, and R. P. Chasmar, The Detection and Measurement of Infra-red Radiation, Oxford, London (1957). A. H. Sommer, Photoemissive Materials, John Wiley & Sons, New York (1968). Reprinted by Krieger (1980). L. Stroebel, J. Compton, I. Current, and R. Zakia, Photographic Materials and Processes, Second Edition, Focal Press, Boston (2000). J. D. Vincent, Fundamentals of Infrared Detector Operation and Testing, John Wiley & Sons, New York (1990). A. T. Young, Photomultipliers, Their Causes and Cures, Chapter 1 in Methods of Experimental Physics: Astrophysics, Vol. 12, N. Carleton Ed., Academic Press, New York (1974). Several books in the series Semiconductors and Semimetals, R. K. Willardson and A. C. Beer, Eds., are relevant. Vols. 5 and 12 both treat infrared detectors, and Vol. 11 deals exclusively with solar cells. Vol. 47 deals with uncooled infrared imaging array systems. Similarly, the series Advances in Electronics and Electron Physics is rich in pertinent articles. Numerous volumes deal with imaging detector conference proceedings, while others have significant feature articles. Most notable are Vols. 34 and 55. A number of books have one or more chapters dealing with optical radiation detectors. These include the following: R. W. Boyd, Radiometry and the Detection of Optical Radiation, John Wiley & Sons, New York (1983). A. Chappell, Optoelectronics: Theory and Practice, McGraw-Hill, New York (1978). R. D. Hudson, Infrared System Engineering, John Wiley & Sons, New York (1969). D. Malacara, Physical Optics and Light Measurements, Academic Press, New York (1988). Several major handbooks offer articles and/or chapters dealing with optical radiation detectors. These include the following: J. S. Accetta, and D. L. Shumaker, Eds., Infrared and Electro-Optical Systems Handbook, Vol. 3, SPIE Press, Bellingham, Washington (1993). M. Bass, Ed., Handbook of Optics Vol. 1, McGraw-Hill, New York (1995). Parts 5, 6, and 7 contain Chapters 15 through 25, all pertinent to detectors and detection. W. L. Wolfe, Ed., Handbook of Military Infrared Technology, Chapters 11 and 12, Office of Naval Research, Washington D.C. (1965). W. L. Wolfe and G. Zissis, Eds., The Infrared Handbook, Chapters 11–16, ERIM and SPIE Press (1978).
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Another rich source for detector information, and probably the best for assessment of the current state of the art, is documented in the Proceedings of the various conferences of SPIE. SPIE sponsors several major conventions per year, each having one or more conferences on detectors. Compilations of the best Proceedings papers, along with other critical papers, are gathered in SPIE’s Milestones series http://spie.org/x649.xml.
References 1. F. Grum and R. J. Becherer, Radiometry, Vol. 1 in Optical Radiation Measurements series, F. Grum, Ed., Academic Press, New York (1979). 2. D. M. Munroe, “Signal to noise ratio improvement,” in Handbook of Measurement Science, Vol. 1, P. H. Sydenham, Ed., Wiley & Sons, New York (1982). 3. C. D. Motchenbacher and J. A. Connelly, Low Noise Electronic System Design, Wiley & Sons, New York (1993). 4. H. W. Ott, Noise Reduction Techniques in Electronic Systems, Wiley & Sons, New York (1976). 5. E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems, Wiley & Sons, New York (1996). 6. W. Budde, Physical Detectors of Optical Radiation, Vol. 4 in Optical Radiation Measurements series, F. Grum and C. J. Bartleson, Eds., Academic Press, New York (1983). 7. N. B. Stevens, “Radiation thermopiles,” in Infrared Detectors, Vol. 5 in Semiconductors and Semimetals series, R. K. Willardson and A. C. Beer, Eds., Academic Press, New York (1970). 8. E. Putley, “The pyroelectric detector,” in Infrared Detectors, Vol. 5 in Semiconductors and Semimetals series, R. K. Willardson and A. C. Beer, Eds., Academic Press, New York (1970). 9. E. L. Dereniak and D. G. Crowe, Optical Radiation Detectors, Wiley & Sons, New York (1984). 10. R. W. Engstrom, RCA Photomultiplier Handbook, Burle Technologies, Lancaster, Pennsylvania (1980). 11. Photomultiplier Tube: Principle to Application, H. Kume, Ed., Hamamatsu Photonics K. K. (1994). 12. A. Rogalski, Opto-Electron. Rev. 12(2), p. 235 (2004). 13. F. Leberl, R. Perko, M. Gruber, and M. Ponticelli, “Novel concepts for aerial digital cameras,” ISPRS Commission I Symposium, ISPRS, Denver (2002). 14. P. W. Kruse and D. D. Skatrud, Uncooled Infrared Imaging Arrays and Systems, Academic Press, New York (1997).
Chapter 6
Radiometric Instrumentation Adapted by William L. Wolfe
6.1 Introduction Radiometric instruments vary in what they are intended to measure, how they do it, how complicated and expensive they are, how rugged, and in a number of other ways. In this chapter, the simplest of radiometers is considered, the components of radiometers are described, and spectral radiometers are covered.
6.2 Instrumentation Requirements It surely comes as no surprise that different instruments have different requirements. For instance, the required accuracy and repeatability vary greatly with the application. A device meant for routine maintenance in the factory might have a radiometric uncertainty of about 20%, yielding a temperature uncertainty of a few degrees, whereas several different bureaus of standards have strived for uncertainties of 0.01%. 6.2.1 Ideal radiometer
It is hard to describe one ideal radiometer, but, in general, they should all measure the radiometric property accurately, precisely, and repeatedly. They should be simple to use, with easily interpreted outputs; of course, the ideal radiometer should be free of defects. Some need to be quite rugged for field use, but all should be able to withstand the environment or environments to which they are exposed. They should be sensitive and have a wide dynamic range, or they should at least be sensitive enough for the task at hand and have sufficient dynamic range for their use. 6.2.2 Specification sheet
In the fervent hope that you can obtain the ideal radiometer, the following specification sheet is presented. It may not be complete for all applications, and not all applications need all of these specifications, but it is a good start and a good guide. 215
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Radiometric properties Noise-equivalent irradiance, NEI Noise-equivalent temperature difference, NETD Accuracy, precision Linearity, dynamic range Spatial characteristics Entrance aperture size, A Instantaneous field of view, W Fixed or variable focus Single detector, linear array, or 2D array Spectral characteristics Overall bandwidth range Number of spectral channels Spectral bandwidth for each channel Continuous tuning or discrete wavelengths Optical characteristics Out-of-band rejection Out-of-field rejection Polarization sensitivity Temporal characteristics Spectral scan rate Spatial scan rate System response time and/or bandwidth Operational requirements Size, weight Power, cooling requirements Scanning requirements Motor drives or manual Environmental considerations Interfaces Mounting Output presentation
Values
6.2.3 Spectral considerations
These can be divided into three main categories: total radiation, broadband radiation, and narrowband radiation. The first category involves detecting as much of the total radiation as you can by using a spectrally flat detector. The applications are things like measuring the earth’s radiation budget and radiation temperatures.
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Broadband applications include making background measurements in the infrared spectral bands for the design of infrared cameras and similar devices—in the 3 to 5-μm and 8 to 12-μm bands, for instance. Some measurements are made in the visible spectrum over the photopic response of the eye; this is called photometry. Others are radiometric measurements over the visible region, while still others are trichromatic wherein the three colors are used to obtain color renditions. In the ultraviolet, both UV-A and UV-B are measured to assess such things as skin cancer susceptibility. In these cases, the spectral passband is usually defined by one or more absorption or interference filters, the detector, the window, or a combination of all of them. It is important to measure the spectral response of radiometers of this type. Narrowband measurements are usually made with some kind of spectrometer, or in this case, a spectroradiometer. Typically, a spectrometer is a device that measures the spectral distribution of radiation, but not its absolute magnitude, whereas a spectroradiometer does both. In some cases narrowband filters can be used to provide spectral selection. These measurements are usually made in situations where the radiation varies rapidly with wavelength or a certain amount of species analysis is desired. They also apply when brightness or ratio temperatures are to be determined. Since the bands are narrow, relatively intense sources are necessary to provide appropriate signal strength. 6.2.4 Spatial considerations
There are two primary types of spatial coverage—large area and small area. It seems fairly obvious that one designs or purchases a radiometer to cover the area of interest. This may be a spot on an electric generator or it may be an average over an entire house. The instantaneous field of view (IFOV) is determined by the angular subtense of the detector, that is, the size of the detector divided by the focal length, assuming the detector is at the focus. Its size is typically on the order of milliradians (mrad) for many IR instruments. If the instrument is a scanning radiometer, then the field of regard, the entire scanned field, should also be considered. Some instruments have zoom optics, a variable focus, or interchangeable optics. These allow both small and large fields. Some have linear or 2D arrays. These have the resolution of the individual pixels but the field of regard of the entire array. The array must be calibrated pixel by pixel and at different levels of irradiance. 6.2.5 Temporal considerations
One of the primary considerations in radiometric measurements is the fidelity of the data, and this relates directly to the signal-to-noise ratio (SNR) obtained by the instrument for a given source. In turn this is essentially the reciprocal of the relative uncertainty. So, to obtain a relative uncertainty of, say 5%, one needs an SNR of 20, and 1% requires an SNR of 100. These requirements are much more severe than simple detection and tracking.
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Aliasing is an artifact in the data that can be caused by inadequate sampling. To avoid these false signals, the bandwidth must be at least twice the highest frequency in the signal. These frequency components can be generated by natural fluctuations or by the use of a chopper, or perhaps turbulence. If a chopper or some other device is used to generate square pulses as the signal, then the bandwidth must be at least equal to the reciprocal of twice the pulse width. If just a chopper is used, the chopper frequency must be three times the highest frequency in the signal. For an uncertainty of 5%, the settling time must be two times the response time of the detector. Similarly, for 1%, five times, and for 0.1%, seven times. Since all radiometers have a finite bandwidth, there is noise in that bandwidth. Section 5.4 addresses noise in detail. To restate, white noise has a flat power spectrum, and effective noise bandwidth B is defined by Eq. (5.15). For 1/f noise, the expression in Eq. (5.14) is used. 6.2.6 Make or buy?
The decision to make or buy is one of availability of devices and of money. If there is a radiometer available that meets your needs or comes close enough and it fits within your budget, buy it. If the most useful unit needs too much adaptation, don’t buy it. If you need to train students, don’t buy it. If you do buy it and need to modify it, do so with as many commercial items as possible. For example, do not make the monochromator of a spectroradiometer; there are units for sale.
6.3 Radiometer Optics 6.3.1 Introduction
Chapter 2 introduced basic optical concepts relevant to radiometry as well as some simple radiometer configurations. They will be reviewed here in preparation for more detailed discussion of instrumentation. The main issues of radiometer optics are the placing of stops and pupils, possible reimaging, use of a field lens, and the reduction of scattered light. 6.3.2 Review of stops and pupils
There are two types of stops: aperture stop and field stop. The aperture stop controls the amount of light that is accepted; the field stop controls the field of view from which it is collected. Pupils are images of stops. The entrance pupil is the aperture stop as projected into object space (by all the optical elements that are between the aperture stop and object space). The exit pupil is the aperture stop projected into image space. The entrance and exit windows are the corresponding images of the field stop. These are illustrated in Fig. 6.1. Notice that the second lens L2 is the aperture stop because its image in object space is smaller than that of the first lens. The field stop is C, and its image out in object space is the entrance window.
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Figure 6.1 Stops and pupils.
6.3.3 The simplest radiometer: bare detector
A bare detector, shown in Fig. 6.2, can be used as a radiometer with its associated electronics. Its angular field of view is almost 180 deg, but it is probably not uniform over that range. Ideally it varies as the cosine of the angle, but most detectors have some other dependency as a result of coatings, surface imperfections, partial specular reflectivity, etc. If the detector has a polished specular window, the angular variation may be calculable from the standard Fresnel equations for reflection. Again, and it cannot be said enough, check it out! The bare detector can be used for relatively imprecise quasi-hemispherical measurements or for a distant source. 6.3.4 Added aperture
A slight improvement is the addition of an aperture, as shown in Fig. 6.3. With almost no expense, just a case with a hole in it, one can define the field of view. As a matter of possible interest, one can make a thermocouple by welding two dissimilar metals, then punch a hole in a number ten tin can to make the business end of a radiometer. The electronics could be one of those little multimeters you can buy at the electronics store for about five dollars. The detector could be a solar cell for visible spectrum use.
Figure 6.2 Bare detector.
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Figure 6.3 One-aperture system.
In this case the detector is the aperture stop, and the hole is the field stop. Since there are no lenses, there are no pupils. The flux on the detector from a point source is given by Φd =
I s Ad , d2
(6.1)
where Is is the source intensity, Ad is detector area, and d is the source–detector distance. For an extended source (one that is larger than the image of the detector), the flux on the detector is given by: Φd =
Ls Aap Ad d2
,
(6.2)
where Ls is the radiance of the source and Aap is the area of the aperture. The field of view is “fuzzy”: that is, different parts of the detector view somewhat different parts of the field of view because the detector is the field stop. Note that if you make the detector a little smaller, the field of view is a little smaller, so the outer portion of the detector views a different portion of the field than the inner portions. A reasonable improvement is obtained by the use of two apertures, as shown in Fig. 6.4. The detector is no longer the field stop. For a point source, and A2 smaller than the active area of the detector, A2 is the aperture stop, A1 is the field stop, and the flux on the detector is given by: I s A2 , d2
(6.3)
Ls A1 A2 , d2
(6.4)
Φd = and for an extended source, Φd =
where d is the distance between the two apertures. The field is still a little fuzzy.
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Figure 6.4 Two-aperture system.
One example of such a radiometer that uses a single detector and two apertures and baffles but no lenses, is a pyrheliometer (fire + sun, measuring the sun’s fire) used by the World Meteorological Organization and shown schematically in Fig. 6.5. The angles made with the detector by A, B, and C should be about 1, 2, and 4 deg, respectively. 6.3.5 Basic radiometer
Figure 6.6 shows a basic radiometer, just a lens and a detector. A mirror can be used in place of a lens, and often is, but the diagrams get more complicated than necessary. The lens is the aperture stop and the detector is the field stop. In this
Figure 6.5 Pyrheliometer.
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Lens
Detector
Figure 6.6 Basic radiometer.
arrangement, the field is well (not fuzzily) defined, but the detector still “sees” radiation coming at different angles from different parts of the field of view. This is the case as long as the detector is the field stop. The advantages are that this is a simple system—one lens, one detector, and associated electronics. The disadvantage is that the detector averages over different parts of the field of view. In this system the irradiance on the entrance pupil (the lens) from a point source is given by Eo =
I s τatm , d2
(6.5)
where Is is the source intensity, τatm is the atmospheric transmittance and d is the distance from the source to the entrance pupil. The flux on the entrance pupil is just the lens area Alens times this, and the flux on the detector is just the lens transmittance τlens, times that: Φd =
I s τatm τlens Alens . d2
(6.6)
The corresponding quantities for an extended source are Φd =
τatm τlens Ls Alens Ad . f2
(6.7)
If the source is extended, the distance from the instrument to the source is irrelevant; it is the focal length that counts.
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Aperture stop Field lens
Detector
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 Φ d = τo Ls Ad Ω fsd ,
(6.8)
where το is the transmission of the optics, Ls is the radiance of the source, Ad is the detector area, and Ωfsd is the solid angle the field stop subtends at the detector. Ω fsd =
Afs d2
,
(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 Φd =
τo Ls Ad Afs d2
.
(6.10)
If the image of the objective underfills the detector, then Φd =
τo Ls Ao A fs f2
.
(6.11)
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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.
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Detector
Mirror Eye Figure 6.8 Viewing with an obscured (Newtonian) optical system.
Detector Reflective chopper Reference
Mirror
Eye Figure 6.9 Viewing with a chopper.
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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 Gouffé1 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 1/f noise 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.
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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 θ 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.
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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(λ, t , x, y , z , α, β, 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,
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gratings, filters, by scattering, interference, and, in general, any phenomenon that is spectrally dependent. A typical spectroradiometer (or spectrometer) is shown in Fig. 6.11. 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, étendue, and AΩ 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.
Figure 6.11 Typical spectrometer.
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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 mλ = sin θi + sin θd ,
(6.13)
where m is the order number, an integer, λ is the wavelength of the light, and θi and θd 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 θd. The 0th order includes all wavelengths and is just regular reflection (or transmission). For a given incidence angle, the first order, m = 1 for a wavelength λ, has the same diffraction angle as the second order and half the wavelength. This is overlapping of orders and limits the free spectral range. The resolving power Q is given by Q=
λ = mN , dλ
(6.14)
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: T = AΩ =
Aslit Agrating cos θ f2
.
(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
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Q=
λ bdn = , dλ dλ
(6.16)
where b is the length of the base of the prism (actually the length of the beam near the base) and dn/dλ 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: T=
Aslit cos θAprism cos θ f2
.
(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.
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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.
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Figure 6.15 Czerny-Turner (laid out) configuration.
Figure 6.16 Alternate (cross) Czerny-Turner configuration.
Figure 6.17 Fastie-Ebert configuration.
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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
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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 μm. 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 μm! 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 μm. The answer was in the reference beam. The atmosphere is very absorbent at 4.3 μm 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 μm 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
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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=
Δλ Do , dλ
(6.18)
where Δλ is the spectral range, dλ 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 π. A representative filter has the following characteristics: the spectral range is 2.5 to 14.1 μm, Q is 67, FWHM is 1.5%, and resolution varies from 0.03 μm at
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2.5 μm to 0.2 μm at 14.1 μm. 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 μm. 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
Figure 6.20 Twyman-Green interferometer.
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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=
λ σ σ 5000 = = = . d λ d σ 2δ λδ
(6.19)
The expression σ/dσ is the wavenumber equivalent. The final expression incorporates the fact that the wavelength is usually given in μm while the wavenumber is in cm–1. 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 τ(λ) =
τo , 4ρ 2 φ 1+ sin (1 − ρ) 2
(6.20)
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239
and φ = 2πnd σ cos θi −
(φ1 + φ2 ) , 2
(6.21)
where τo is the transmission of the plates, ρ is their reflectivity, n is the refractive index of the medium between the plates, d is the plate separation, θi is the angle of inclination of the incident beam, which is usually 0, and the φi 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 Q=
ρ σ λ = = mπ , dσ dλ 1− ρ
(6.22)
the throughput is given by
T = τo AΩ ,
(6.23)
and the free spectral range is given by Δσ =
1 . 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.
Figure 6.21 Fabry-Perot Interferometer.
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For Further Reading P. Baumeister, Optical Coating Technology, SPIE Press, Bellingham, Washington (2004). Good section on filters. I. 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).
References 1. A. Gouffé, “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, “Multislit spectroscopy,” J. Opt. Soc. Amer. 39, pp. 437–444 (1949). 5. M. Harwit and N. Sloane, Hadamard Transform Optics, Academic Press, New York (1979).
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
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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
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243
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 1 (100%)
10
0.1 (10%)
100
0.01 (1%)
1000
0.001 (0.1%)
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, “±0.5% of reading ±0.2% 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 Measurement uncertainty =
1 . SNR
(7.1)
Thus, an SNR of 10 implies a 10% uncertainty (1σ) 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 ∝ N ). We then statistically analyze the data and make the following assumptions: (1) 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.
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Table 7.3 Mathematical tools for dataset analysis.
Tool
Formula
x
i
Mean
x =
i
, N = number of N measurements
Variance (sample)
σ 2=
( x
i
i
σx =
Standard deviation of mean
2
N −1
σ= σ
Standard deviation
− x)
2
σ N
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 (χ2) 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)1/2. 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 (1/f noise) 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: σ 2y ( τo ) =
2 1 N −1 ( yk +1 − yk ) . k =1 2( N − 1)
(7.2)
Compare with the classical variance σ 2( N ) =
2 1 N yk − y . k =1 N −1
(
)
(7.3)
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If σ2 ( N ) / σ2y (τo ) ≤ (1 + 1 / N ) , then white (classical) noise predominates over drift (1/f), and we can increase the sampling window τo 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 / N . The other three curves show the effects of varying drift rates. As the drift rate climbs (i.e., progressively higher 1/f noise 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 (A/D) converter. Quantization noise was defined in Eq. (5.31) and is LSB =
SIGNALmax , 2n
(7.4)
where LSB is the least significant bit and SIGNALmax 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.
Figure 7.3 Drift effects on data averaging for a number of cases.
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247
Table 7.4 Chauvenet’s criteria for data rejection.
Number of readings
Data point deviation Standard deviation
4 6 8 10 15 20 25 50 100 200 500 1000
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 ½N 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 1σ (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
248
Chapter 7 Table 7.5 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 (q = Ax)
Reciprocal (q = 1/x)
Power (q = xn)
Exponential q = e ax
Logarithmic q = ln(ax)
σq
σq
σq
σq
σq
q
=
σx x
q
=
σx x
q
=n
σx x
q
= aσ x
q
=
σx 1 a ln ( ax ) x
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Table 7.7 Error expressions for functions of several variables.
Addition and subtraction (q = x + y – z)
σq = σ x + σ y + σ z σq q
Multiplication and division (q = xy/z)
σq
Power (q = xn)
q
=
σx σ y σz + + x y z
=n
σx x
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 1σ level, then add the standard deviations in quadrature (that is, we use RSS) to get the combined 1σ uncertainty: σc = σ 2r + σ2s .
(7.5)
The 1σ 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 of 2 is 2σ, where σ 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 (1σ) uncertainty and a coverage factor k as follows: (45.26 ± 0.03) W/cm 2 (2σ) 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.2 Table 7.8 Root-sum-square formulas.
Addition and subtraction (q = x + y – z)
Multiplication and division (q = xy/z)
σq = σq
(σx )
2
+ ( σ y ) + ( σz ) 2
2
2
2
σ σy σ = x + + z q x y z
2
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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 ℜ to an input radiance L. In differential form, it is
dSIGNAL = ℜ(λ) world Lλ (λ) world dAd Ωd λ ,
(7.7)
which is not particularly useful. We must integrate over area, solid angle, and wavelength to get the integral form: SIGNAL = ℜ(λ ) Lλ (λ )dAd Ωd λ .
(7.8)
In both equations above, SIGNAL = the output radiometric signal, Lλ(λ) = the input spectral radiance, and ℜ(λ) = is the radiometric system’s power responsivity. The “world” (measurement environment) consists of: θ,φ x,y λ t,ν 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 ℜ(λ), it simplifies to: SIGNAL = ℜ ( λ ) [ Φ λ (λ) d λ ] ,
(7.9)
where Φλ(λ) = spectral radiant power incident on the detector and ℜ(λ) = the radiometer spectral power responsivity. For a detector with spectral responsivity ℜ(λ) and a narrow-band source or filter:
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251
SIGNAL = ℜ ( λ ) Φ λ (λ ) τ(λ ) Δλ ,
(7.10)
where τ(λ) = spectral transmittance of the filter and Δλ = bandwidth of the source or filter. For a laser line where the spectral linewidth is so small that the responsivity of the detector/radiometer can be considered constant, then
SIGNAL = ℜΦ .
(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 SIGNAL =
1 d2
ℜ ( λ ) I
λ
(λ) dAd λ ,
(7.12)
where SIGNAL = the radiometric signal, Iλ(λ) = the spectral radiant intensity of the source, d = the distance to the source, and ℜ(λ) = 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 R =
( Is
TERM 1
τa )
1
2
π Do τo 4 ( f #) Ω 2
1
2
( D *)
1
3
2
1 SNR B 4
1
2
(7.13)
where terms 1 through 4 are defined as: Term 1: Target parameters: intensity Is and atmospheric transmission τa Term 2: Optical system parameters: f/#, optics diameter Do, FOV Ω, and transmission τo Term 3: Detector parameter: D* Term 4: Signal processing parameters: required SNR and noise bandwidth B.
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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 in i = Φ n ;or is ℜ v v NEP = n = Φ n ;or vs ℜ Ad B NEP = . D* NEP =
(7.14)
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 =
v E NEP =E n = . SNR vs Ad
(7.15)
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 NEI =
4( f #)Ω1/ 2 B1/2 , πDo D * τo
(7.16)
which bears a striking resemblance to the range equation. The noise-equivalent temperature difference (NETD or NEΔT) 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 NETD =
4( f #) B λ2
∂M (λ, TB ) Ad τoptics (λ) D * (λ ) d λ ∂T λ1
,
(7.17)
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253
where TB 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
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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.
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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 (IEC), and the International Commission on Illumination (CIE), 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.
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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/CPM/GANTT 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).
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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.
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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, S2 ℜ E = ( SIGNAL) , (7.18) I where
ℜE = Irradiance responsivity in SIGNAL/(W/m2), S = distance from source to radiometer (m), and I = source intensity (W/sr).
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 ℜE 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 ℜ E = ( SIGNAL)
f2 , I
(7.19)
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Figure 7.5 Eccentric pupil parabola used in DSS calibration.
where f = focal length of the collimator (m) and I = source intensity (W/sr) = LAs. The result is again irradiance responsivity ℜE in SIGNAL/(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 focal length f (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 τ(λ) 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.
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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, ℜL =
SIGNAL , L
where ℜL = radiance responsivity in SIGNAL/(W/m2sr).
Figure 7.8 Off-axis reflective collimator.
(7.20)
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261
IMAGE FIELD STOP
SOURCE
FIELD ANGLE
APERTURE STOP
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 SIGNAL/(W/m2sr) 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
FIELD ANGLE APERTURE STOP Figure 7.10 NES calibration configuration (adapted from Wolfe and Zissis).6
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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 SIGNAL A ℜL = Ls As
,
(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 θo with the optical axis. The chief ray angle is also θo, which defines the field of view. This is simply a scaling of areas, and the radiometer is focused at infinity. The radiance responsivity ℜL has units SIGNAL/(W/m2sr). 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. 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 ℜΦ in SIGNAL/W. 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.
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263
INCOMING BEAM
C RCULAR FIELD STOP 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): Φ = LAΩ . The AΩ product (T, throughput, étendue) 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 Ω, we may convert from one form of responsivity to another using ℜΦ =
ℜE ℜL = . A AΩ
(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 1-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.
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1 m diameter T = 2000 K
200 km
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 =
Ad B D*
=
1cm 2 1 Hz 1010 cmHz1/ 2 /W
= 1 × 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
f/# of optics Detector active area Ad
f/3 1 cm2
Detector D* Nominal wavelength range of operation
1010 cm Hz1/2/W 8 μm to 12 μm
Electrical bandwidth B
1 Hz
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265
To do so, we determine the “diffraction-limited spot size” on the focal plane,* using a wavelength of 10 μm (center wavelength of our band of interest): Dblur = 2.44λ ( f # ) = 2.44(10 × 10−6 m)3 = 7.32 × 10−5 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
NEI =
NEP 1 × 10−10 W = = 1 × 10−10 W/cm 2 . Ad 1cm 2
(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 focal length f: f = ( f # ) × Do = 3 × 0.2 m = 0.6 m .
(7.26)
Next, the solid-angle field of view is determined as Ω=
Ad 1cm 2 1 = = = 2.78 × 10−4 sr . 2 2 (60 cm) 3600 f
(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 4(3) ( 2.78 × 10−4 ) 11/ 2 Hz 1/ 2
NEI =
*
10
1/2
π(20 cm)1 × 10 cmHz W
−1
= 3.183 × 10−13 W/cm 2 .
(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.
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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 1 × 10−10 W = = 3.185 × 10−13 W/cm 2 , Ao 314 cm 2
(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. (7.13). It is: SNR =
I s τa πDo τo D *
4 ( f #) ΩR2 B
.
(7.30)
It also requires that we know source intensity within the particular spectral band. This is obtained through I s = Ls Ap ,
(7.31)
where Ap is the source projected area. From a blackbody radiation calculation program, Ls = 5.04 × 103 W/m2sr in the 8- to 12-μm band. The projected area of the spherical source of 1-m diameter is 0.785 m2, so I s = 5.04 × 103 × 0.785 = 3956 W/sr . Substituting into Eq. (7.30), assuming unity transmittances and a 1-Hz electrical bandwidth, SNR =
3956 W/sr π rad(20cm)(1010 cmHz1/ 2 /W) 4(3) 2.78 × 10−4 sr (4 × 1014 cm 2 )1 Hz1/ 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 =
I s ..... . R 2 .....
Discussing irradiance in Chapter 2, we noted that E=I/R2 can sometimes be confusing, as to units, and that a different way to consider the expression is E=
I Ωo , R2
where Ωo 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 black/graybodies— these equations are basic, yet powerful enough to provide the engineer or analyst a starting point from which to develop solutions.
7.8 Final Thoughts I 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
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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).
Appendix A
Système International (SI) Units for Radiometry and Photometry Table A.1 SI* base units.
Base quantity Length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensity
Name 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 lm lx cd/m2 W/sr W/(m2sr)
Equivalent
N·m J/s s–1 A·s cd·sr lm/m2 lm/m2sr
* Complete SI information is available on the World Wide Web at www.bipm fr and at physics.nist.gov/pubs/sp811/sp811 html. 271
272
Appendix A Table A.3 Si prefixes.
Factor 1024 1021 1018 1015 1012 109 106 103 102 101
Prefix yotta zetta exa peta tera giga 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 μ 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)
Φ
dq/dt
watt [W]
Radiant intensity
I
dΦ/dω
W/sr
Radiant exitance
M
dΦ/da
W/m2
Irradiance
E
dΦ/da
W/m2
Radiance
L
d2Φ/(da cosθdω)
W/m2sr
Table A.5 Photon quantities.
Quantity
Symbol
Definition
Units
Photon power (flux)
Φq
dn/dt
/s
Photon intensity
I
dn/dω
/sr·s
Photon exitance
M
dn/da
/m2s
Photon irradiance
Eq
dn/da
/m2s
Photon radiance
Lq
d2n/(da cosθdω)
/m2sr·s
n = photon number.
Système International (SI) Units for Radiometry and Photometry
273
Spectral Quantities Spectral quantities are derivative, per unit wavelength with the additional dimension m–1, and are indicated by a subscript λ (e.g., spectral radiance Lλwith units W/m3sr). Nonspectral quantities that are wavelength dependant are indicated as (λ); e.g., transmission τ(λ). 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
lm
Luminous exitance
Φv Mv
lm/m2
Luminous incidance
Ev
lm/m2
Luminous intensity (SI base unit)
Iv
lm/sr = cd
Luminance
Lv
lm/m2sr = cd/m2
Appendix B
Physical Constants, Conversion Factors, and Other Useful Quantities Table B.1 1998 CODATA recommended values of the fundamental physical constants.
Relative uncertainty exact
Quantity
Symbol
Value
Units
Speed of light (vacuum) Permeability of vacuum Permittivity of vacuum Planck constant
c, co
299,792,458
m/s
μo
4π × 10–7
N/A2
exact
eo h
1/μoc2 = 8.854 187…×10–12 6.62606876 (52) × 10–34
F/m
exact 7.8 × 10–8
q, e k k σ
1.602176462 (63) × 10–19 1.3806503 (24) × 10–23 8.617342 (15) × 10–5 5.670400 (40) × 10–8
J· s C J/K eV/K W/m2K4
c1
3.741771 07 (29) × 10–16
Wm2
7.8 × 10–8
c1L
1.191042 722 (93) × 10–16
Wm2/sr
7.8 × 10–8
c2
1.4387752 (25) × 10–2
m·K
1.7 × 10–6
b
2.8977686 (51) × 10–3
m·K
1.7 × 10–6
Electronic charge Boltzmann constant Boltzmann constant Stefan-Boltzmann constant First radiation constant (2πhc2) First radiation constant for Lλ Second radiation constant Wien displacement law constant
3.9 × 10–8 1.7 × 10–6 1.7 × 10–6 7.0 × 10–6
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. Chem. 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
276
Appendix B
Here are some useful conversion factors: hc = 1.986445 × 10–25 J·m = 1.986445 × 10–19 J·μm = 1.986445 × 10–16 J·nm hc/q = 1.23984 eV·μm kT/q = 0.025852 V at 300 K 1 eV = 1.602176 462 × 10–19 J 1 astronomical unit (AU) = 1.495 × 1011 m λmaxT = 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). 1 new candle = 1 cd. 1 Carcel = 10 cd. 1 Carcel unit = 9.79613 cd (The measure of a Carcel 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. 1 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 of luminous flux (power) is the lumen (lm). 1 lm = 1 cd·sr. 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 lm. Symbol for light watt is Φv. 830
Φv = Km
Φ V (λ) dλ . v
360
Mechanical equivalent of light is 1/683 W/lm. Illuminance
The (derived) SI unit of illuminance is the lux (lx = lm/m2). 1 footcandle (fc) = 1 lm per square foot. 1 lux (lx) = 1 lm/m2 = 1 meter-candle. 1 phot (ph) = 1 lm/cm2= 104 lx. 1 milliphot (mph) = 10–3 lm/cm2. 1 nox = 1 millilux = 10–3 lx. 1 sea-mile candle = 1 cd @ 1 nautical mile (6,080 ft) = 2.9 × 10–7 lx. 1 pharosage = 1 lm/m2. Luminosity L is expressed in lm/ft2. Luminance
The (derived) SI unit of luminance is the nit (cd/m2). 1 nit = 1 candela per m2= π apostilb = 0.2919 foot-lambert (fL). 1 stilb (sb) = 1 candela per cm2. 1 nit = 104 Bougie-Hectomètre-Carré.
Antiquarian’s Garden of Sane and Outrageous Terminology
279
Note: Several luminance units are related to the illuminance units by assuming a perfect (ρ = 1) diffuse (Lambertian) reflector. This “simplification” leads to: 1 foot-candle (fc) of illumination 1 fL of luminance. 1 lambert (L)= 1 lm/cm2= (1/π) cd/cm2. 1 footlambert (fL) = (1/π) cd/ft2. 1 apostilb (asb) = (1/π) cd/m2 = (1/π) nit. 1 skot = 10–3 (1/π) cd/m2= 10–3 apostilb. 1 millilambert ≈ 1 fL. 1 equivalent phot = 1 L. 1 equivalent lux = 1 blondel = 1 asb. 1 equivalent footcandle = 1 fL. The unit bril is used to express the “brilliance” or subjective brightness of a source of light: bril =
log L + 100 . log 2
The scale is logarithmic: an increase of 1 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). 1 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 mm2, corrected for the Stiles-Crawford effect; formerly called the photon. 2. The external illuminance that produces retinal illumination of 0.002 lx.
280
Appendix C
Ultraviolet E-viton is erythemal effectiveness equivalent to 10 μW at 296.7 nm. 1 Finsen = 1 E-viton/cm2. 1 erythemal watt = 105 E-vitons. 1 EU = 1 E-viton = 1 erytheme. Floren is UV flux equivalent to 1 mW 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 × 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/m2, weighted by the standard erythemal action spectrum. For less-sensitive skin types, it rises to 1000 J/m2. 1 MED = 2 SED (standard erythemal dose). At the wavelength of maximum sensitivity for production of erythema (295 nm), the MED is 50 J/m2. [Br. J. Dermatol. 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(–log10T)/3. For example, if 1% of the light is transmitted, the shade
number is 4. Astronomy 1 Jansky (Jy) = 10–26 W/m2Hz (spectral irradiance). 1 W/cm2μm = 3 × 1016/λ2 Jy. 1 Solar flux unit (s.f.u.) = 104 Jy. Visual magnitude zero = 2.65 × 10–6 lx outside atmosphere (Infrared Handbook, pp. 3–23). Visual magnitude zero = 2.54 × 10–6 lx outside atmosphere (radiometry and photometry in astronomy). Visual magnitude zero = 2.09 × 10–6 lx outside atmosphere.
Antiquarian’s Garden of Sane and Outrageous Terminology
281
1 Rayleigh = 106 photons/cm2s. 1 Rayleigh = (1/4π) × 106 photons/c2s·sr. 1 S10 = 1.23 × 10–12 W/cm2sr·μm at 0.55 μm (equivalent to the number of 10th magnitude stars per square degree). 1 S10 = 1.899 × 106 photons/s·cm2sr·μm at 0.55 μm.
Color and Appearance Reciprocal megakelvin (MK)–1 = 106/Tc, where Tc is color temperature, also known as mirek or mired (microreciprocal kelvin or microreciprocal degree).
Miscellaneous 1 microeinstein (μE) = 6.022 × 1017 photons = 1 micromole. angstrom (Å) is obsolete unit of wavelength = 10–10 m. Kayser is waves per centimeter. Gillette is a measure of laser energy, sufficient to penetrate one standard razor blade. Microflick is a unit of spectral radiance μW/cm2sr·μm. Spectral lamprosity 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/m2s. Spherical photosynthetic photon flux density (SPPFD) is measured in mol/m2s. Photosynthetically active photons (PAP) is measured in mol/m2. Spherical photosynthetically active photons (SPAP) is measured in mol/m2. 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
1 pharos = 1 lumen. 1 helios = 1 blondel. 1 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 Θ (deg) 0.573 1.000 1.146 1.719 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
Θ (rad) .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 1.000 1.047 1.250 1.500 1.571
ω (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 1.906 2.377 2.888 3.142 4.269 5.839 6.283
Ω (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 1.617 1.928 2.224 2.356 2.830 3.126 3.142
ω/Ω 1.000 1.000 1.000 1.000 1.000 1.000 1.001 1.001 1.001 1.002 1.003 1.008 1.010 1.017 1.023 1.031 1.041 1.049 1.065 1.072 1.096 1.133 1.172 1.179 1.233 1.298 1.333 1.521 1.868 2.000
f/# 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 1.932 1.692 1.462 1.284 1.183 1.043 1.000 0.886 0.776 0.707 0.697 0.638 0.594 0.577 0.527 0.501 0.500
NA/n* 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 al., 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 1/f noise
Weird, ubiquitous noise from many familiar and strange sources, inversely proportional to frequency (pink or red noise). An approximation ( const ) ( I dcα B ) 2 is i1/ f = , where α is between 1.25 fβ and 4 (typically 2), and β is between 0.8 and 3 (typically 1). Also called flicker, contact, excess, modulation, etc. A major pain!
AΩ product
Symbol T, units m2sr; 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 ρ/π, while a perfectly specular reflector has a BRDF ρ/Ω; ρ is reflectance and Ω 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 = 1/3.
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–1.
D*
Specific or normalized detectivity. It is detectivity D = NEP–1 normalized for bandwidth and area. D*= (AB)1/2/NEP. Unit: cm·Hz1/2/W. It is the SNR per watt for a 1-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 (FOV). It is the SNR per incident watt for a 1-cm2 area, 1-Hz noise bandwidth, and π sr of a projected solid angle. D**= (ABΩ/π)1/2/NEP. Unit: cm·Hz1/2/W. A further normalization for field of view: D*(Ω/π)1/2.
D*BLIP
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. V I P dB = 20log10 2 = 20log10 2 dB = 10log10 2 V1 I1 P1 It is a relative measure. There are several ways of denoting absolute values. dBv refers to 1-V rms. dBm refers to 1 mW with stated impedance (75 Ω). 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, (SNRout)2/(SNRin)2, unitless, 0 < DQE < 1.
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 ∞ 2 1 ENB = ⋅ A f df . 2 0 ( ) A o The equivalent “brick-wall” rectangular passband. Alternate symbols are B and Δf, 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.
Glossary
289
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(1/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 lm/m2 = lx. Analagous to irradiance in W/m2.
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). vn2 = 4kTRB or in2 = 4kTB / R .
Jones calibration configuration
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 SIGNAL Aa ℜ= . L As
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 = lm/m2sr.
Luminous intensity
Measure of visible power per unit solid angle. Unit: candela (cd) = lm/sr. 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 = AΩ Lλ ℜ ( λ )d λ . 0
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 variance.
Noise-equivalent photon flux
The photon flux incident on a detector which gives rise to a signal-to-noise ratio of one. Units: s–1.
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, NEΔT)
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%, 1/e, 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(λ); 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 × cosθ, projected onto flat surface dΩ = dω cosθ. Symbol Ω, 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/m2sr.
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, ρ = 1).
Responsive quantum efficiency (RQE)
The number of independent output events per incident photon. Dimensionless, between 0 and 1. Symbols: η, RQE.
Responsivity
Ratio of the output of a detector to its input. Units: A/W or V/W. Symbol is ℜ. The result of an integral over wavelength: ℜ ( λ ) Φλ d λ . ℜ= Φλ d λ
Retroreflectance
A reflectance wherein the reflected beam retraces the path from the source back to the source.
Glossary
293
rms (root mean square)
A measure of the equivalent heating effect of a 1 T 2 voltage or current Vrms = v ( t )dt , where t T 0 is time, usually a single or integer multiple of periods for periodic waveforms. For a sine wave, the rms voltage is 1 / 2 times the peak voltage (measured from 0), or 1 2 2 times the peak-to-peak voltage. Electric power companies deliver 117 Vrms, which is 330 Vp-p. For a square wave, the rms amplitude is the same as the peak amplitude.
Saturation exposure
The exposure (product of irradiance and time) necessary to saturate an integrating detector. Used for CCD, CID, and CMOS array detectors.
Shot noise
Noise associated with current flow across a potential barrier, due to discrete nature of electrons. Power spectral density is independent of frequency (i.e., white). Mean-square shotnoise current, in2 = 2qI dc B .
Solid angle
Projected area ÷ (distance)2, given by 2π(1–cos θ1/2) for right circular cone. Symbol ω, units, sr.
Spectral directional emissivity
The ratio of radiance at a specified wavelength in a particular direction to that of a blackbody, at the same wavelength and in the same direction.
Spectral radiance
Symbol Lλ, watts per unit area per unit-projected solid angle per unit-wavelength interval; fundamental unit of radiometry.
Spectral radiant intensity
Symbol Iλ, units W/sr·μm; watts per steradian per unit wavelength, often from an isotropic “point” source. Ratio of reflected power at a specific wavelength to the incident power at the same wavelength.
Spectral reflectance
294
Appendix E
Spectral responsivity
Responsivity as a function of wavelength. Symbol ℜ(λ), units: amps (or volts, etc.) per watt. Not a derivative quantity, per-unit wavelength interval.
Specular reflectance
Ratio of radiation reflected in the mirror direction to the incident radiation. Excludes diffuse (scattered) component.
Throughput
Geometrical term relating to amount of power that can get through a system. The product of area and projected solid angle AΩ. Symbol T, units m2sr.
Time constant
A measure of the speed of response of a device. The time required to reach (1–1/e) or 0.632 of the final value in response to a step input. Symbol τ, units s. For a simple RC circuit, τ = RC.
Total hemispherical emissivity
Integral of spectral directional emissivity over the entire spectrum and over an entire hemisphere.
Transimpedance amplifier
Also, current-to-voltage converter, used to interface with current sources. R i
+
E=iR
Trap detector
A multiple detector array where detectors are placed in series optically and in parallel electrically. Has a quantum efficiency approaching unity.
Type A error
Older term: precision. Also, random error. A measure of the repeatability of a measurement. Comes from granularity, noise, etc. Determined by repeated measurements.
Type B error
Older term: accuracy. Also, systematic error or bias. A measure of the difference between the mean reading and “truth.” May be corrected by careful calibration and characterization.
Glossary
295
Uncertainty
The total estimated difference between a measurement and “truth”; includes both random and systematic error terms and a confidence parameter.
Wien’s displacement law
Product of wavelength and temperature constant.
Wien approximation
An equation representing blackbody radiation at short wavelengths and/or low temperatures. Form is: c M λ = 15 e − c2 λT . λ
Appendix F
Effective Noise Bandwidth of Analog RC Filters and the Selection of Filter Parameters to Optimize Signal-to-Noise Ratio By James M. Palmer Revised by L. Stephen Bell, May 2009
Introduction Engineering calculations involving noise and signal-to-noise ratio need to use the effective noise bandwidth (ENB) in order to calculate noise properly. Often the conventional (–3-dB voltage, –6-dB power) bandwidth is used, leading to erroneous results. When the difference between –3-dB bandwidth and ENB is recognized, it is often oversimplified by attempting to relate ENB to the –3-dB bandwidth. Table F.1 shows several of these relationships found in the open and corporate literature. These discrepancies, while not extremely serious, are disconcerting, particularly for the two-section filter, which is readily realizable with a single operational amplifier and a handful of R and C components. So I set off to find out which values are correct. This was accomplished by means of simple spreadsheet analysis and BASIC computer programs to do the necessary integrations. There are many multiple-pole filter types in the literature. This appendix is limited to simple Butterworth RC filters (maximally flat-frequency response), where each section has the same R and C and therefore the same cutoff frequency. The primary emphasis is limited to “white” noise with a uniform power spectral density. We conclude with a simple bandwidth optimization to maximize the SNR of a single-frequency signal in the presence of white noise. 297
298
Appendix F Table F.1 Ratio of ENB to –3-dB bandwidth for low-pass filters.
Sections 1 2 3 4 5
A, B 1.57 1.22 1.15 1.13 1.11
C, D 1.57 1.11 1.05 1.025
E 1.57 1.12 1.08 1.06 1.05
A: C. D. Motchenbacher and J. A. Connelly, Low Noise Electronic System Design, Secs. 1–4, John Wiley & Sons, New York (1993). B: H. W. Ott, Noise Reduction Techniques in Electronic Systems, Table 8-2, John Wiley & Sons, New York (1976). C: P. Horowitz and W. Hill, The Art of Electronics, Sec. 7.21, Cambridge Univ. Press, New York (1989). D: “Measuring noise spectra with variable electronic filters,” Ithaco Application Note IAN-102, Appendix 1, Ithaco Corp., Ithaca, New York (July 1983). E: E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems, Table 5.1, John Wiley & Sons, New York (1996).
Definitions
The signal bandwidth of a low-pass filter is defined as the frequency where the signal voltage falls to –3 dB (1 / 2 = 0.707) of the transmission at dc (typically unity). This frequency is also referred to as the half-power point, where the power transmission has dropped to –6 dB (0.5). For a bandpass filter, the same terms are used relative to the peak transmission of the filter. The bandwidth is often referred to as full-width at half-maximum (FWHM) for power. The signal bandwidth is not appropriate for characterization of noise. The effective noise bandwidth (ENB, more often seen as B or Δf) is defined as the equivalent brick-wall (rectangular) filter, having the same area under the power transmission curve. The equation is B=
1
∞
G ( f )v ( f ) df , G( f ) v 0
2 0
2
0
(F.1)
where G(f ) is the power gain as a function of frequency, and fo is the frequency where G(f ) is a maximum. ν ( f ) 2 is the power spectrum for the noise under consideration, and ν o 2 is the noise power at the peak frequency. For “white” (spectrally flat) noise, Eq. (F.1) simplifies to B=
∞ 1 G ( f ) df , G ( f0 ) 0
(F.2)
Effective Noise Bandwidth of Analog RC Filters…
299
and this format is the most often seen. In electro-optical systems, this bandwidth may be limited by the frequency response of a detector or its associated electronics, or by the insertion of an electrical filter operating in the audio to lowradio frequency range, 20 Hz to several MHz. The signal-to-noise ratio (SNR, S/N) of a system is the ratio of the rms signal current isig to the rms noise current in . Voltage may be substituted for current. SNR is dimensionless. Use of rms is indicated as it is the only measure appropriate for the characterization of random noise. Low-Pass Filters
The voltage transmission of a single-section (first-order) RC low-pass filter is A=
1 1 + ω 2 τ2
,
(F.3)
where ω is the radian frequency equal to 2πf in Hz, and τ is the circuit time constant, the RC product. At low frequencies (ωτ << 1), A is unity. The transmission curve drops to –3 dB in amplitude at the point where ωτ = 1, and falls at 6 dB/octave (20 dB/decade) at higher frequencies (ωτ >> 1). The cutoff frequency is defined as the –3 dB (0.707) point, although there is substantial transmission for higher frequencies. The –3 dB point is at 1/τ rad/s or 1/(2πτ) Hz. The ENB of this simple filter is readily found by integrating the square (power transmission) of Eq. (F.3) in closed form; the result is π/(2τ) in rad/s, or 1/(4τ) = (π/2)fo expressed in Hz. The ENB is therefore π/2 or 1.571 times the –3-dB bandwidth, and everybody in Table F.1 agrees! For higher-order filters, the closed-form integration becomes more complex. It must be noted that whenever two or more RC sections are cascaded, the –3-dB point for the new composite filter shifts to lower frequencies; thus, one does not use the –3-dB point for a single RC section. In addition, multiple sections are buffered (isolated from each other) so that subsequent sections present no loading to preceding sections). The results of the higher-order integrations are shown in Table F.2. Table F.2 Low-pass Butterworth filter characteristics, matched sections.
Sections 1 2 3 4 5 6
ω–3 dB
(rad/s) 1/τ 0.644/τ 0.511/τ 0.436/τ 0.386/τ 0.350/τ
BSF
1.0 0.644 0.511 0.436 0.386 0.350
F–3 dB (Hz) 1/2πτ 0.102/τ 0.081/τ 0.069/τ 0.061/τ 0.056/τ
ENB (rad/s) π/2τ π/4τ 0.589/τ 0.492/τ 0.430/τ 0.387/τ
ENB (Hz) 1/4τ 1/8τ 0.094/τ 0.078/τ 0.068/τ 0.062/τ
Ratio ENB/B–3 dB π/2 = 1.571 1.220 1.155 1.130 1.115 1.106
300
Appendix F
These results confirm that the values given by Motchenbacher and Connelly and by Ott in Table F.1 are correct. Note that the third column represents a bandwidth shrinkage as sections are cascaded. This is also termed bandwidth shrinkage factor (BSF) when normalized to 1.0 for a single section. The formula for the effective (shrunk) bandwidth ratio of cascaded identical sections is BSF = 2(1/ n ) − 1 ,
(F.4)
where BSF is the net bandwidth ratio due to shrinkage by cascading sections. For design purposes, the bandwidth of the individual stages must be set to the reciprocal of the BSF. Bandpass filters with matched time constants
The simple RC bandpass filter with matched time constants cascades a single high-pass filter with a single low-pass filter, each with the same time constant τ = RC. The voltage transmission of a single-section RC high-pass filter is ωτ
A=
1 + ω 2 τ2
,
(F.5)
where ω is the radian frequency, equal to 2πf in Hz, and τ is the circuit time constant, the RC product, in seconds. At high frequencies, A approaches unity, and the curve falls at 6 dB/octave (20 dB/decade) at lower frequencies. The cutoff frequency is defined as the –3-dB (0.707) point, even though there is substantial transmission at lower frequencies. The –3-dB point, where ω2τ2 = 1, is 1/τ rad/s or 1/(2πτ) Hz. The peak transmission for these simple bandpass filters is no longer unity, and the –3-dB-bandwidth points must be evaluated with respect to the peak transmission of the composite filter, not unity. For the singlesection bandpass filter, a peak transmission of 0.5 (–6 dB) is found, located at the coincidence of the –3-dB points of the high-pass and low-pass sections. The –3-dB passband of the bandpass filter is determined at 3 dB below this level, or at –9 dB = 0.3535. The closed-form integration of this single-stage bandpass filter gives an ENB of πfC. Results for filters with one to four sections are shown in Table F.3. Note that the bandwidth shrinkage factor (BSF) matches the same ratios as for cascaded Butterworth low-pass filters. It should also be noted that the cut-on and cutoff frequencies are disposed about the center frequency in a geometric sense, i.e., fLP × fHP = fC2, where fC is the center frequency of the filter. The location of fLP is given by f LP =
fC 2
(
)
B2 + 4 − B ,
where B is either the –3-dB bandwidth or the ENB. fH is then fC2/fL.
(F.6)
Effective Noise Bandwidth of Analog RC Filters…
301
Table F.3 Butterworth bandpass filter characteristics with matched cascaded sections.
Sections Peak transmission B–3 dB/fC =1/Q BSF fLP/fC fHP/fC ENB/fC fLP/fC
1 0.5 2 1.0 0.414 2.414 π 0.291
2 0.25 1.287 0.644 0.546 1.833 π/2 0.486
3 0.125 1.019 0.511 0.613 1.632 1.177 0.572
4 0.0625 0.871 0.436 0.655 1.526 0.982 0.623
fHP/fC
3.432
2.057
1.749
1.605
ENB/B–3 dB
π/2
1.220
1.155
1.127
Bandpass Filters with Different Time Constants Here we discuss bandpass filters constructed using a single high-pass section in series with a single low-pass section. Each section has a different time constant. First we note that the minimum bandwidth is achieved with the low-pass cutoff and high-pass cut-on frequencies matched. If the low-pass cutoff is chosen at a lower frequency than the high-pass cut-on frequency, the resulting bandwidth remains the same as the case where the two frequencies are identical. The only thing achieved is a reduction in filter transmission. A simple BASIC program was written to find the ENB, the peak transmission, the –3-dB points, and the –3-dB bandwidth for these filters. Again the integrations were done for several selections of low-pass cutoffs and highpass cut-on frequencies, and the –3-dB points were located with respect to the composite transmission. The results are shown in Table F.4 for a single-section bandpass filter. These calculations were done holding fHP constant at unity. Table F.4 ENB for bandpass filters where fLP ≥ fHP.
fLP/fHP 1 1.5 2 3 4 5 6 10 20
Tpeak 0.5 0.6 0.667 0.75 0.8 0.833 0.857 0.909 0.952
B–3 dB 2 2.5 3 4 5 6 7 11 21
ENB 3.14 3.91 4.70 6.26 7.82 9.39 10.96 17.20 32.78
302
Appendix F
From these data we find some surprisingly simple relationships. The maximum transmission is given by Tpeak =
f LP , f LP + f HP
(F.7)
and if fLP >> fHP, the peak transmission approaches unity as expected. The –3-dB bandwidth is given by
B−3 dB = f LP + f HP ,
(F.8)
so that if fLP >> fHP, the –3-dB bandwidth approaches that of the low-pass filter alone. The effective noise bandwidth is given by π ENB= B−3 dB . 2
(F.9)
Figure 7.63 in Horowitz and Hill shows a simple equation to relate the ENB for this case to the individual cut-on and cutoff frequencies: 2
f LP π . ENB = 2 f LP + f HP
(F.10)
If we were to use Eq. (F.10) to determine the ENB in the special case where fLP = fHP, it shows that the ENB is π/4 times the 3-dB bandwidth, which is less than the –3-dB bandwidth, a surprising result. This analysis shows that the ENB is (π/2) times the 3-dB bandwidth. It can be seen that when fL >> fH, the results of Eq. (F.10) converge to the correct solution, just that of the low-pass filter alone. A comparison between these new calculations for ENB and the results predicted using Eq. (F.10) is given in Table F.5. It is apparent that Eq. (F.10) gives erroneous results. Table F.5 Comparison with Horowitz and Hill.
fL/fH 1 2 5 10 20
ENB π 4.70 9.39 17.20 32.78
Eq. (F.10) π/4 2.09 6.55 14.28 29.9
Ratio 4 2.24 1.43 1.2 1.1
Effective Noise Bandwidth of Analog RC Filters…
303
Filter Selection to Optimize SNR We now address the selection of the optimum filter to maximize the signal-tonoise ratio. For many applications, we can ignore preservation of the signal waveform to achieve better SNR. We will recover just the fundamental with maximum SNR. Our discussion will be limited to “white” noise (uniform power spectral density). For the low-pass filter, the choice of cutoff frequency fC was determined using a short BASIC program to iterate fC for a fixed signal frequency of 100 (arbitrary units). The results show that for the single-section filter, the best SNR is obtained when fC is equal to the signal fundamental frequency. The results differ little for multisection low-pass filters. The double-section filter achieves its peak SNR at a frequency slightly higher (1.11 times the composite –3-dB point) than for the single-section filter. If we choose the composite –3-dB point, the SNR is 0.991 times the maximum achievable SNR. Three- and four-section filters showed similar results: the peak SNR was realized at about a 15% higher cutoff frequency, but the SNR at the composite –3-dB point was within 1% of the maximum SNR. Satisfactory results will be achieved by using the composite –3-dB frequency for one- to four-section low-pass filters. For the simple bandpass filter, where the high-pass and low-pass sections are set to the same frequency, the optimum SNR is achieved by setting both –3-dB points at the signal frequency, as that will maximize signal transmission. It was hypothesized that if the bandpass were increased somewhat, the transmission at the peak may increase faster than the noise, which is proportional to the square root of the ENB. To test this hypothesis, further spreadsheet work was done to implement the calculations. I again chose to keep the geometric mean of the lowpass and high-pass sections equal to the center frequency, i.e., fLP × fHP = fC2. The optimum SNR is achieved when the passband is defined where fL = fH. It can also be seen that setting the low-pass section to a lower frequency than the high-pass section is futile, yielding the same SNR, with both signal and noise attenuated equally. These above results are valid only for “white” noise. Every practical circuit has at a minimum a single RC time constant associated with it, defining an ENB and a corresponding SNR. If additional noise filtering is necessary, we can add a simple active filter using a single operational amplifier with two RC sections. It is then interesting to consider the simple bandpass filter (one section each matched high-pass and low-pass) and one- and two-section low-pass RC filters. The bandpass filter has an ENB of π and a peak gain of 0.5, whereas a single-section low-pass filter has an ENB of π/2 and a gain of unity. The noise transmission, the product of the peak transmission and ENB1/2, is 1.253 for the low-pass filter and 0.886 for the bandpass filter. The signal transmission for the low-pass filter is 0.707, whereas it is 0.5 for the bandpass filter. The SNR for each is therefore the same. However, for a twosection low-pass filter with the signal at the (composite) –3-dB frequency, the ENB is 1.220 × fC, the noise transmission is 1.105, the signal transmission is
304
Appendix F
0.707, and the SNR is therefore improved by some 13.5% over the simple bandbass and single-section filters.
Conclusions Several recommendations can be made: (1) A double-section (second-order) low-pass filter is sufficiently better in the reduction of white noise to warrant the use of two additional components. (2) In the absence of 1/f noise, a two-section low-pass filter will outperform a bandpass filter with the same number of components. (3) When cascading (identical) sections of Butterworth-type filters, the BSF must be taken into account. (4) When noise content of the (linear) signal channel is other than white noise, additional noise suppression techniques may be needed. Transient suppression and 1/f noise effects are not covered in this discussion. In the case of system nonlinearities (such as pulsed signals or in certain laser applications), judicious addition of certain types of noise can actually improve the SNR. This is called stochastic resonance, and it is beyond the scope of this book. (5) The above discussion relates only to the linear-circuit analog domain. Once the filtered signal becomes digitized, quantization noise and antialiasing effects must be taken into account. A good rule of thumb to follow is that use of a first-order filter requires a sample rate of at least five times the cutoff frequency of the filter. Only then will the filter fully meet the Nyquist criteria (to minimize aliasing in the digital output of the A/D converter).
Appendix G
Bandwidth Normalization by Moments General measurement equation: ∞
I = ℜ ( λ ) Φλ d λ . 0
(G.1)
If ℜ(λ) can be represented as a rectangle, λ2
I = ℜn Φ λ d λ . λ1
(G.2)
Let the source function Φλ be described as a second-degree polynomial: Φλ = A + Bλ + Cλ2.
(G.3)
Substitute Eq. (G.3) into Eq. (G.1), divide both sides by ℜ(λ)dλ, and multiply both sides by (λ2 – λ1) to get: ∞ ∞ λℜ ( λ ) d λ λ 2ℜ ( λ ) d λ 0 0 (λ − λ ) . = A+ B ∞ +C ∞ 1 ∞ 2 ℜ λ λ ℜ λ λ ℜ λ λ d d d ( ) ( ) ( ) 0 0 0
I (λ 2 − λ1 )
(G.4)
Next, integrate Eq. (G.3) between the limits λ1 and λ2:
λ2
λ1
B C Φ λ d λ = A + ( λ 2 + λ1 ) + ( λ 22 + λ 2 λ1 + λ12 ) ( λ 2 − λ1 ) . 2 3
(G.5)
Note the similarities between Eqs. (G.4) and (G.5). If the following conditions are applied,
305
306
Appendix G ∞
λℜ ( λ ) d λ , ℜ(λ ) dλ
λ 2 + λ1 = 2
0
∞
λ 22 + λ 2 λ1 + λ12 = 3
0
and
λ2
λ1
Φλ d λ =
I ( λ 2 − λ1 )
∞
0
ℜ(λ) dλ
∞
0
λ 2ℜ ( λ d λ ) ∞
0
ℜ(λ dλ)
.
(G.6)
(G.7)
Assume that area of response curve = area of equivalent rectangle, i.e. ∞
ℜn ⋅ Δλ = ℜ ( λ ) d λ .
(G.8)
0
Then,
λ2
λ1
Φλ d λ =
I , ℜn
(G.9)
and we have a band-limited power Φin-band. Now we proceed to determine λ1, λ2, and ℜn. Substitute: ∞
M1
λℜ ( λ ) d λ = ℜ(λ) dλ 0
∞
M2
=
∞
0
λ 2ℜ ( λ ) d λ ∞
0
0
ℜ(λ) dλ
.
(G.10)
Then M1 =
λ 2 + λ1 2
M2 =
λ 22 + λ 2 λ1 + λ12 . 3
(G.11)
M1 is the first moment divided by the area (0th moment) and is the centroid of the response curve, the effective or center wavelength λc. M2 is the second moment divided by the area, which is related to the square of the radius of gyration. Solution of simultaneous Eqs. (G.11) with the substitution M1 = λc yields λ1 = λ c − 3(M 2 − λ c 2 ) , λ 2 = λ c + 3(M 2 − λ c 2 ) ,
(G.12)
showing the bandpass limits λ1 and λ2 are symmetrically disposed about the center wavelength λc. The quantity (M2 – λc2) is recognized as the variance σ2. The bandwidth between wavelength limits λ1 and λ2 is:
Δλ = λ 2 − λ1 = 2 3 σ , and the short- and long-limit wavelengths are then
(G.13)
Bandwidth Normalization by Moments
307
λ1 = λ c − 3 σ λ 2 = λ c + 3 σ .
(G.14)
The bandwidth-normalized responsivity is
ℜn =
1 2
∞
ℜ(λ) dλ . 3σ
(G.15)
0
Now we have our three parameters, ℜn, λ1, and λ2. Note that the coefficients A, B, and C of the second-degree source polynomial [Eq. (G.3)] have vanished. The implication is significant: • Any source that can be represented by a second-degree polynomial can be characterized between the wavelength limits λ1 and λ2 (which are determined solely by the radiometer) without error. • There is no ambiguity in any of the normalization parameters; they are all uniquely determined from only the spectral responsivity curve. • The errors are related to the deviation of the source function from a quadratic.
Moments Normalization Summary This is the step-by-step procedure for accomplishing a moments normalization. The starting point is absolute spectral responsivity ℜ(λ). ∞
Zero’th moment
M 0 = ℜ(λ) dλ
First moment
M1 = λ ℜ ( λ ) d λ
Second moment
M 2 = λ2 ℜ ( λ ) d λ
Center wavelength (centroid) Variance
0
∞
0
∞
0
M1 M0 M σ2 = 2 − λ c2 M0 λc =
Short wavelength limit
λ1 = λ c − 3σ
Long wavelength limit
λ 2 = λ c + 3σ
Bandwidth
Δλ = 2 3σ M0 ℜn = 2 3σ
Normalized responsivity
Appendix H
Jones Near-Small-Source Calibration Configuration The following method is adapted from an obscure application note further describing the near-small-source method of radiometric calibration.
How to calibrate a radiometer A simple method of calibrating a radiometer using a small blackbody source close to the radiometer aperture has been described by Dr. R. Clark Jones of the Polaroid Corporation. The principles involved in this method are briefly reviewed here with special emphasis on the application to the optithermradiometer Cassegrain system. The essential point in the method is that a small radiation source close to the aperture of the radiometer will uniformly irradiate an area in the focal plane of the radiometer. The radiation on the detector (with the radiometer focused at infinity) is then given by
Φ = Ls Ad ωr ,
(H.1)
where Φ = radiation on detector (watts, Φ) L = source radiance (W/m2sr, Ls) Ad = detector area (m2) ωr = solid angle of radiometer field of view (sr) at the focal plane = source area / (focal length)2. Thus, the responsivity of a detector in a radiometer can readily be calibrated by dividing the signal voltage output of the instrument by the radiation on the detector. An important aspect of the method is that the radiation on the detector is independent of the radiometer aperture and source location, providing that a uniformly irradiated area covers the detector. This restriction places limits on the source size and location as given below.
309
310
Appendix H
Focal plane Image, I
Dpi (Entrance pupil of radiometer)
Df
Point source, S
f
P
Q Figure H.1 Point source near a radiometer. The detector surface is located at the focal plane.
Consider a point source near a radiometer (Fig. H.1). For a point source, the diameter of the uniformly irradiated disc Df in the focal plane is given by Eq. (H.2) (after using the thin-lens imaging formula):
Df =
f D pi , P
(H.2a)
where Df, f, P, and Dpi are defined in Fig. H.1. The size of the detector must be less than Df. It is easy to prove Eq. (H.1) for the configuration shown in Fig. H.1. For a small Lambertian source, flux at the entrance pupil of the radiometer is given by Φ D pi = Ls As AD pi / P 2 .
(H.3)
For a detector of size Df at the focal plane of the radiometer, the flux detected is given by
Φ D f = Ls AD f ωr .
(H.4)
But we know that Φ D pi = Φ D f . So, from Eqs. (H.2), (H.3), and (H.4), we get
ωr = As/f 2.
(H.5)
Asssuming uniform irradiance at the focal plane, Eq. (H.4) can be written as Eq. (H.1).
Jones Near-Small-Source Calibration Configuration
311
Similarly, if the radiometer optical system is of the Cassegrain type, an obscured area will result, and the uniformly irradiated area will then be an annular ring of width χ and mean diameter Dm:
χ=
1 f ( DP1 − DP 2 ) , 2P
(H.6a)
where DP2 = the diameter of an obscured disc produced by secondary mirror.
Dm =
1 f ( DP1 + DP 2 ) . 2P
(H.7)
If, as is always the case, the source has a finite diameter Ms (see Fig. H.2), the edges of the disc or ring will be vignetted by an amount (f Ms/P), divided equally on either side of the unblurred edge. The remaining uniformly irradiated disc or annular ring width is given by f ( DP1 − M S ) , P
(H.2b)
f 1 ( DP1 − DP 2 ) − M S . P 2
(H.6b)
Dx =
χ=
The mean diameter of the ring remains the same as given in Eq. (H.7).
Figure H.2 Thin-lens-equivalent layout for a Cassegranian optical system.
312
Appendix H
In calibrating a radiometer, a convenient source aperture and distance are first chosen. Then the size of the uniformly irradiated area must be checked to see whether it covers the detector. For example, if a Cassegrain system is used, the detector must fit in the ring of width z and mean diameter D as computed from Eqs. (H.6b) and (H.7). The source must be placed off axis such that the annular ring of width x falls on the detector. Several other precautions must be observed, particularly if a thermal detector is used. Since the calibration source fills only a small part of the field of view, the detector will “see” other radiation as well. If the source is chopped, the radiometer will respond only to it. However, in many cases the detector is chopped, and hence it responds to all radiation in its field of view. One way of separating the response to the calibration source from the background is to record the difference in response when the source aperture is opened and when it is closed.
Appendix I
Is Sunglint Observable in the Thermal Infrared? The short answer is yes. Orbiting spacecraft like GOES, AVHRR, ASTER, MODIS, and ATSR all have spectral observation windows in the thermal infrared. They all have to deal with sunglint at least some of the time. It is not just a minor nuisance, but a major effect that saturates their sensors. A saturated sensor gives no indication of the actual magnitude; all information from that direction is lost. The directional characteristics of sunglint have been adequately treated by Maurice Cox.1 This report deals with the radiometric aspects, the magnitude of the sunglint in comparison with the magnitude of normal target thermal emission.
Background Every object at a temperature above absolute zero emits electromagnetic radiation (EMR) as a result of molecular motion. This emission as a function of wavelength is described by the Planck equation 2hc 2 1 Lλ = ε ( λ ) 5 hc / λkT , −1 λ e
(I.1)
where Lλ is spectral radiance in W/m2sr, ε(λ) is spectral emissivity, a wavelengthdependent quality factor, h is Planck’s constant, c is the velocity of light, λ is wavelength, k is Boltzmann’s constant, and T is the absolute temperature. This equation gives the spectral radiance in W/m2sr·μm at any wavelength for an object at temperature T. If ε(λ) is 1 at all wavelengths, the emission is called blackbody radiation. For temperatures on the order of room temperature (~300 K, 80° F) this radiation peaks near 10 μm, and the region surrounding 10 μm is known as the thermal infrared (TIR). When a beam of EMR encounters an object, three things can happen. The object can: 1. absorb the EMR, 2. transmit the EMR, or 3. reflect the EMR. 313
314
Appendix I
In fact, all three processes happen simultaneously: an object reflects some of the incoming EMR, absorbs some of it, and transmits the balance. All of these three are relative quantities, the fraction reflected, absorbed, or transmitted, and have values between 0 and 1. Since EMR must be conserved, the sum of these three is 1:
A + T + R = 1.
(I.2)
This means that if an object is highly reflective (R→1, like a mirror), the sum of the absorption and transmission must necessarily be low. Similarly, if the object is transparent (T→1, like a window), it must have both low absorption and low reflection. Kirchhoff related the absorption and emission of an object, stating that if an object is a good absorber, it must also be a good emitter. With Kirchhoff’s law in mind, objects that are highly reflective or highly transmissive cannot be good emitters. The TIR is an interesting place to observe things. An infrared camera or forward-looking infrared (FLIR) “sees” a picture of the distribution of radiance in the observed scene, which usually includes an object of interest, a target. The observed radiance is comprised of three factors, the direct emission from the target, the radiance of whatever is behind the target multiplied by the transmittance of the target, and the radiance of whatever surrounds the target multiplied by the reflectance of the target. These cameras do not observe at all wavelengths, but are limited by the transmission of their optics and that of our atmosphere. Two “windows” are commonly used, between 3 and 5 μm and between 8 and 12 μm. In general, the 3- to 5-μm window is better suited for hotter targets and the 8- to 12-μm window is best for objects near room temperature. To determine the radiance within the wavelength region defined by an atmospheric window, we integrate Eq. (I.1) over the TIR wavelength range. The integration can be done with standard mathematical computer programs (i.e., MathCad, Mathematica, etc.), a custom computer program, or even brute force using a spreadsheet. Another issue with which we must deal is the atmosphere itself. It has a transmission less than one, and it also radiates in the thermal infrared according to Eq. (I.1). The problem is extraordinarily complex, and massive computer programs have been devised to determine the characteristics of our atmosphere. For this work, the program PCTRAN (www.ontar.com), a commercial version of the USAF program LOWTRAN7, has been used to determine the relevant parameters. To keep the analysis quite general, the preprogrammed atmosphere known as MID-LATITUDE SUMMER was chosen as characteristic of a broad range of characteristics in the CONUS over at least half of the year. The altitude was taken at sea level, and the solar zenith angle (as measured from directly overhead) was 25 deg. At this angle, the calculated transmission, averaged over the TIR (8- to 12-μm band), is 0.57, and the in-band radiance of the sky in this direction is 12.5 W/m2sr. The radiance was also determined over a range of
Is Sunglint Observable in the Thermal Infrared?
315
angles and integrated to determine the total in-band irradiance on the ground from the entire sky; it is 51.2 W/m2.
Direct Emission Most objects seen by FLIR devices are at ambient temperature, here taken as 300 K (80° F). A 300-K blackbody has an in-band (8- to 12-μm) radiance of 38.5 W/m2sr. For real objects, we multiply by the emissivity; for this analysis the emissivity is assumed to be constant over the band. We consider four materials: earth, water, aluminum, and glass. Table I.1 gives the radiative characteristics. The difference between the aluminum and the other materials is that aluminum is a good reflector and therefore a poor emitter. This is a characteristic of metals, which are good electrical conductors.
Reflected Radiation If these materials have an emissivity less than unity, the sum of their reflectance and transmittance must make up the difference. Since all of these materials are opaque (T = 0) in the TIR, the reflectance is (1 – emissivity). The directional properties of reflectance are highly varied, from perfectly diffuse (flat, matte) to perfectly specular (like a perfect mirror). We shall inspect the two limiting cases. Since the materials are reflecting something we must determine what that something is. For an object out-of-doors, lying horizontally on the ground, the background consists of three parts: the sun, the sky, and nearby objects that are above grade. Because of the variability of above-grade objects, we shall ignore them, assuming our object of interest is out in the open. The radiance of the sky, obtained via PCTRAN, was previously stated to be 12.5 W/m2sr at 25-deg zenith angle. The sun is assumed to be a blackbody at 6000 K and subtends a solid angle of 6.8 × 10–5 sr. The in-band radiance from a 6000 K blackbody is 19840 W/m2sr, so the irradiance outside the atmosphere is the product of the radiance and the solid angle and is 1.35 W/m2. After transmission through the atmosphere, the irradiance is reduced to 0.77 W/m2, a rather feeble amount.
Table I.1 Direct TIR emission from selected materials (300 K).
Material
Earth Water Aluminum Glass
Emissivity (8- to 12-μm band) 0.95 0.985 0.1 0/96
Radiance W/m2sr 36.6 37.9 3.85 37.0
316
Appendix I Table I.2 Reflected radiance from diffuse materials.
Material Earth Water Aluminum Glass
Diffuse reflectance 0.05 0.015 0.9 0.04
Reflected sun (W/m2sr) 0.012 0.004 0.22 0.010
Reflected sky (W/m2sr) 0.81 0.24 14.67 0.65
Diffuse reflectance
For diffuse objects, we invoke the Lambertian approximation, which states that radiance is independent of direction and is the product of the irradiance and the reflectance, divided by π, or ρ L= E. (I.3) π This equation may be used directly for a source like the sun. For a uniform hemispherical diffuse source like the sky, the irradiance E is π times the source radiance, so the reflected radiance is simply the source radiance multiplied by the surface reflectance. The sky is not really uniform, but for a diffuse reflector, we can use the total irradiance from the sky, multiply by the reflectance, and divide by π. Table I.2 shows the reflected radiance of each of these four materials from the direct sunlight and from the sky. Sunlight reflected from a diffuse surface in the 8- to 12-μm band is very small. The total radiance from all of these objects is the sum of the direct and reflected components, as shown in Table I.3. Note that with the exception of the aluminum, they are comparable, and will look much the same on a thermal image. The aluminum, being reflective rather than emissive, has a significantly lower radiance, and will appear dark on a typical FLIR display (“white-hot,” where a more radiant object appears white and a less radiant object appears black). Since we do not know a priori the nature of the objects in the scene, dark objects will be interpreted as cooler than white objects. Specular reflectance
Reflection from smooth surfaces is called specular, and straightforward laws of geometry apply. The incident and reflected beams and the normal to the reflecting surface all lie in the same plane, and the angles of both the incident and reflected beams with respect to the normal to the reflecting surface are equal. For an outdoor scene viewed from above, this means that at one specific angle, which must follow the above rules, the sun will be seen. At all other angles, the sky will be seen.
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Table I.3 Total radiance from diffuse materials.
Material
Earth Water Aluminum Glass
Direct radiance (W/m2sr) 36.6 37.9 3.85 37.0
Reflected sun (W/m2sr) 0.012 0.004 0.22 0.010
Reflected sky (W/m2sr) 0.81 0.24 14.67 0.65
Total radiance (W/m2sr) 37.42 38.14 18.74 37.66
Window method
One way to look at the problem is to treat the reflecting surface as a window through which the sun or the sky can be viewed. This window has the same size and orientation as the reflecting surface, and its transmission has the same value as the reflection of the reflecting surface. In this model, the effective radiance is the product of the surface reflectance and the radiance of whatever is seen in reflection. Table I.4 gives the effective radiance of the three materials that are capable of specular reflectance for both sunlight and skylight. Recall that the radiance of the sky in the 8- to 12-μm band (zenith angle 20 deg) is 12.5 W/m2sr, and the transmittance of the atmosphere is 0.57. The in-band radiance of the sun (before atmospheric losses) is 19840 W/m2sr, so the effective radiance, including the atmospheric transmission, is 11310 W/m2sr. The radiance of the object from a uniform diffuse sky is seen to be the same whether the reflector is diffuse or specular. When compared with the typical total radiances for diffuse surfaces, it can be seen that the specularly reflected sun (sunglint) is a factor of 5 to nearly 300 times the total from a diffuse surface (~37.5 W/m2sr). BRDF method
Another way of looking at reflection is through the concept of bidirectional reflectance distribution function (BRDF). This term is the ratio of the radiance of a surface, as a function of direction, to the irradiance incident upon the surface. It carries the units 1/sr. The BRDF is the surface reflectance divided by the projected solid angle of the irradiating source. The radiance of a surface is then the product of the irradiance E and the BRDF, or L = E × BRDF. For a diffuse Table I.4 Reflected radiance from specular materials.
Water
Specular reflectance 0.015
Reflected sun (W/m2sr) 170
Reflected sky (W/m2sr) 0.19
Aluminum
0.9
10180
11.25
Glass
0.04
452
0.50
Material
318
Appendix I Table I.5 Reflected radiance from specular materials via BRDF.
Water
Specular reflectance 0.015
BRDF from sun 221
Irradiance from sun (W/m2) 0.77
Reflected sun (W/m2sr) 170
Aluminum
0.9
13235
0.77
10190
Glass
0.04
588
0.77
453
Material
surface, the BRDF is ρ/π and the radiance L is then ρE/π. For a hemispherical diffuse source like the sky, the irradiance E is the source radiance Lsky multiplied by the projected solid angle of the source, which is π. Then the object radiance is just the source radiance Lsky times the reflectance ρ. For a specular surface, the BRDF is the surface reflectance divided by the projected solid angle of the source, or ρ/Ω. The sun has a projected solid angle of 6.8 × 10–5 sr, so the BRDF becomes 14706 ρ. Using these values, Table I.5 gives the reflected radiances. The feeble 0.77 W/m2 irradiance from the sun turns into an overwhelming radiance upon specular reflection. The results of the BRDF computation of the glint radiance are identical to the results obtained using the “window” method. Several factors can alter these results. First, the surface must be flat and clean. The presence of a scattering overlay, like a layer of dust, will remove a portion of the specularity and add a diffuse component. Second, the polish on the surface must be fairly good, though at these longer wavelengths, the surface roughness can be over ten times as great as that of a mirror designed for reflection in the visible. The effect of this large radiance on a FLIR depends upon several factors. First, our sensor must be positioned such that it is (1) located within the specular beam, and (2) looking in the proper direction. The instantaneous field of view (IFOV) of the sensor is the angle that a single detector subtends when observing a target and is measured in milliradians (mrad). Detector considerations (number and diffraction-limited size) and overall instrument constraints (aperture size, focal length, and these combined as f/#) place typical values of IFOV in the neighborhood of 0.1 to 0.2 mrad. If the detector is square, the footprint, or target area seen by the detector, is also nearly square, depending upon the viewing angle. The observed area of the target is then (IFOV × R)2 where R is the range, the distance from the target to the sensor. For example if the range is 1 km (3280 ft), the area seen by a detector with an IFOV of 0.1 mrad is 0.01 m2, and the detector footprint is a square 0.1 m (about 4 in.) on a side. Larger IFOV and/or larger range will cause an increase in the size of the detector footprint. If we are looking at an angle from nadir (which is vertical, straight down) then we divide the footprint by the cosine of the angle from nadir. The increase is small for low values of nadir angle; a 10% increase in effective area is seen if the nadir angle is 25 deg.
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The power on a detector from glint depends upon the size of the reflecting object compared with the detector footprint. The diameter of the sun subtends nearly 9 mrad, sufficiently large to overfill any properly oriented detector. The total radiance as seen by a detector is the sum of the earth radiance and the glint radiance, each weighted by its fraction of the subtended area. For example, a single square piece of glass (ρ = 0.04) that is 1 cm on a side has an area of 10–4 m2. If the detector footprint is 10–2 m2, the fraction subtended by the glass is 0.01 (1%). The remaining earth has a fractional area of 0.99 (99%). Taking the radiances from above and weighting them according to these fractional areas, we get 0.01 × 450 + 0.99 × 36.6 = 40.7 W/m2sr. Specular reflection of sunlight from a small piece of glass results in an 11% increase in radiance over the bare earth at 300 K (26.8° C, 80° F). This increase in radiance would be interpreted as a temperature difference of 12° F, a rise from 80° F to 92° F. An IFOV that includes the same size piece of shiny aluminum (square, 1 cm on a side) will have a total radiance of 0.01 × 10200 + 0.99 × 36.6 = 138.2 W/m2sr. This radiance is likely to saturate a detector. If a detector does have sufficient dynamic range to respond linearly to this radiance, it will indicate a temperature of 409 K (136° C, 276° F), which is totally unrealistic.
Conclusion We have demonstrated here that sunglint can be a significant factor in imaging systems in the thermal infrared. It will be observed if the sensor is looking at a target with a sunglint component, and the geometrical angles are such that the sun can be seen in reflection. The only way to avoid it is to look in some other direction.
Works Consulted 1. M. Cox, Sun Reflection Geometry, (1999). Unpublished. 2. A. W. Cooper, E. C. Crittenden, E. A. Milne, P. L. Walker, E. Moss, and D. J. Gregoris, “Mid- and far-infrared measurements of sunglint from the sea surface,” in Proc. SPIE 1749, 176–185 (1992). The first five authors are from the Naval Postgraduate School, Monterey, CA. Measurements were made with an AGA Thermovision 780 dual-band camera operating in the 2- to 5.6-μm and 8- to 12-μm bands.
Appendix J
Documentary Standards for Radiometry and Photometry Numerous agencies prepare and disseminate documentary (protocol) standards for radiometry and allied fields. Those listed here are either U.S. or international standards organizations. The European Committee for Standardization is called CEN,i whose mission is to promote voluntary technical harmonization in Europe in conjunction with worldwide agencies and its European partners. The purposes include the lowering of trade barriers and the promotion of common technical understanding. European standards (e.g., BSI from UK and DIN from Germany) may be accessed via CEN. The Japanese Standards Associationii also has extensive listings.
ANSI American National Standards Institute. The U.S. representative for ISO, the International Standards Organization. iii C78.40-1985 C78.180-1989 C78.375-1991 C78.386 C78.387 C78.388 C82.1-1985 C82.1(a-c) 1990 C82.3-1989 C82.4-1985
i
Specifications for mercury lamps Specifications for fluorescent lamp starters Guide for electrical measurements of fluorescent lamps Mercury lamps—measurement of characteristics Metal-halide lamps—measurement of characteristics High-pressure sodium lamps—measurement of characteristics Specifications for fluorescent lamp ballasts Specifications for fluorescent lamp ballasts (supplement to C82.1) Specifications for fluorescent lamp reference ballasts Specifications for intensity discharge and low-pressure sodium amp ballasts
www.cenorm.be/ www.jsa.or.jp iii (www.ansi.org) ANSI/IES RP-16, American National Standard Nomenclature and Definitions for Illuminating Engineering (1986). 321 ii
322
C82.5-1990 C82.6-1985 ANSI/NCSL 540-1-1994
Appendix J
Specifications for high-intensity-discharge lamp reference ballasts Methods of measurement of discharge lamp ballasts American National Standards for Calibration—calibration Laboratories and Measuring and Test Equipment—general requirements
ASTM The American Society for Testing and Materialsiv maintains an extensive collection of protocol or documentary standards, including practices, specifications, guides, procedures, and test methods for a large range of materials and instrumentation for their characterization. Some 10,000 standards are published annually in a 73-volume set, of which one volume is the index alone! The ASTM Book of Standards occupies nearly two meters of shelf space. These documents are subject to periodic review and revision by the committees that were responsible for their generation and maintenance. Always check at ASTM’s website to see if a later version is available. A number of these standards are the result of the efforts of committee E12 on Color and Appearance, committee E20 on Temperature Measurement and committee E37 on Thermal Measurement. Special collections in specific areas are also published, for example, ASTM Standards on Color and Appearance Measurement, 8th Ed. (2008). This book contains 130 ASTM standards as well as ISO and ISO/CIE standards used in appearance analysis for a variety of materials and products, including all the standards listed above. It includes a CD/ROM with even more information. The following listing shows a selection of relevant standards to radiometry, photometry, and colorimetry. Another useful guide is “Nomenclature and Definitions Applicable to Radiometric and Photometric Characteristics of Matter,” ASTM Special Technical Publication 475 (1971). “Guide for recommended uses of photoluminescent safety markings,” ASTM E2030 (1999). “Standard practice for preparation of pressed powder white reflectance factor transfer standards for hemispherical geometry and bi-directional geometries,” ASTM E259 (1998). “Recommended practice for goniophotometry of objects and materials,” ASTM E167 (1996). “Specification for daytime pedestrian visibility enhancement,” ASTM E1896 (1997). “Standard guide for describing and specifying the spectrometer of an optical emission direct-reading instrument,” ASTM E1507 (1998) .
iv
www.astm.org
Documentary Standards for Radiometry and Photometry
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“Standard guide for designing and conducting visual experiments,” ASTM E1808 (1996). “Standard guide for establishing spectrophotometer performance tests,” ASTM E1866 (1997). “Standard guide to evaluation of optical properties of powder coatings,” ASTM D5382 (1995). “Standard guide for examining electrical and mechanical equipment with infrared thermography,” ASTM E1934 (1999). “Standard guide for modeling the colorimetric properties of a visual display unit,” ASTM E1682 (2001). “Standard guide for preparation, maintenance, and distribution of physical product standards for color and geometric appearance of coatings,” ASTM D5531 (1999). “Standard guide for quality assurance of laboratories using molecular spectroscopy,” ASTM E924 (1994). “Standard guide for quantitative analysis by energy-dispersive spectroscopy,” ASTM E1508 (1998). “Standard guide for Raman shift,” ASTM E1840 (1996). “Standard guide for selection of geometric conditions for measurement of reflection and transmission properties of materials,” ASTM E179 (1996). “Standard guide for use of lighting in laboratory testing,” ASTM E1733 (1995). “Standard guide to evaluation of optical properties of powder coatings,” ASTM D5382 (1995). “Standard guide to properties of high visibility materials used to improve individual safety,” ASTM F923 (2000). “Standard method for calibration of reference pyranometers with axis vertical by the shading method,” ASTM E913 (1999). “Standard practice for angle resolved optical scatter measurements on specular or diffuse surfaces,” ASTM E1392 (1996). “Standard practice for calculating solar reflectance index of horizontal and lowsloped opaque surfaces,” ASTM E1980 (2001). “Standard practice for calculating yellowness and whiteness indices from instrumentally measured color coordinates,” ASTM E313 (2000). “Standard practice for calculation of photometric transmittance and reflectance of materials to solar radiation,” ASTM E971 (1996). “Standard practice for calculation of weighting factors for tristimulus integration,” ASTM E2022 (2001). “Standard practice for calibrating thin heat flux transducers,” ASTM C1130 (2001).
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Appendix J
“Standard practice for calibration of ozone monitors and certification of ozone transfer standards using ultraviolet photometry,” ASTM D5110 (1998). “Standard practice for calibration of the heat flow meter apparatus,” ASTM C1132 (1995). “Standard practice for calibration of transmission densitometers,” ASTM E1079 (2000). “Standard practice for calculating yellowness and whiteness indices from instrumentally measured color coordinates,” ASTM E313 (2000). “Standard practice for color measurement of fluorescent specimens,” ASTM E991 (1998). “Standard practice for computing the colors of fluorescent objects from bispectral photometric data,” ASTM E2152 (2001). “Standard practice for computing the colors of objects by using the cie system,” ASTM E308 (1999). “Standard practice for describing and measuring performance of dispersive infrared spectrometers,” ASTM E932 (1997). “Standard practice for describing and measuring performance of fourier transform mid-infrared (FT-MIR) spectrometers level zero and level one tests,” ASTM E1421 (1999). “Standard practice for describing and measuring performance of laboratory fourier transform near-infrared (FT-NIR) spectrometers: level zero and level one tests,” ASTM E1944 (1998). “Standard practice for describing and measuring performance of ultraviolet, visible, and near-infrared spectrophotometers,” ASTM E275 (2001). “Standard practice for describing and specifying inductively-coupled plasma atomic emission spectrometers,” ASTM E1479 (1999). “Standard practice for describing photomultiplier detectors in emission and absorption spectrometry,” ASTM E520 (1998). “Standard practice for describing retroreflection,” ASTM E808 (1999). “Standard practice for determining the steady state thermal transmittance of fenestration systems,” ASTM E1423 (1999). “Standard practice for electronic interchange of color and appearance data,” ASTM E1708 (2001). “Standard practice for establishing color and gloss tolerances,” ASTM D3134 (1997). “Standard practice for evaluating solar absorptive materials for thermal applications,” ASTM E744 (1996). “Standard practices for general techniques of ultraviolet-visible quantitative analysis,” ASTM E169 (1999). “Standard practice for goniophotometry of objects and materials,” ASTM E167 (1996).
Documentary Standards for Radiometry and Photometry
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“Standard practice for identification of instrumental methods of color or colordifference measurement of materials,” ASTM E805 (2001). “Standard practice for measuring colorimetric characteristics of retroreflectors under nighttime conditions,” ASTM E811 (2001). “Standard practice for measuring photometric characteristics of retroreflectors,” ASTM E809 (2000). “Standard practice for measuring practical spectral bandwidth of ultravioletvisible spectrophotometers,” ASTM E958 (1999). “Standard practice for near infrared qualitative analysis,” ASTM E1790 (2000). “Standard practice for obtaining bispectral photometric data for evaluation of fluorescent color,” ASTM E2153 (2001). “Standard practice for obtaining colorimetric data from a visual display unit using tristimulus colorimeters,” ASTM E1455 (1997). “Standard practice for obtaining spectrophotometric data for object-color evaluation,” ASTM E1164 (1994). “Standard practice for obtaining spectroradiometric data from radiant sources for colorimetry,” ASTM E1341 (2001). “Standard practice for preparation of pressed powder white reflectance factor transfer standards for hemispherical geometry and bi-directional geometries,” ASTM E259 (1998). “Standard practice for preparation of textiles prior to ultraviolet (uv) transmission testing,” ASTM D6544 (2000). “Standard practice for qualifying spectrometers and spectrophotometers for use in multivariate analyses, calibrated using surrogate mixtures,” ASTM E2056 (2000). “Standard practice for reducing the effect of variability of color measurement by use of multiple measurements,” ASTM E1345 (1998). “Standard practice for selecting and calibrating sources for the visual assessment of object colors,” ASTM Z6606Z. “Standard practice for selection and use of portable retroreflectomers for the measurement of pavement marking materials,” ASTM E1743 (1996). “Standard practice for solar simulation for thermal balance testing of spacecraft,” ASTM E491 (1999). “Standard practice for specifying and matching color using the colorcurve system,” ASTM E1541 (1998). “Standard practice for specifying and verifying the performance of colorimeters, spectrocolorimeters and goniospectrocolorimeters,” ASTM Z6899Z. “Standard practice for specifying color by the munsell system,” ASTM D1535 (2001). “Standard practice for specifying color by using the optical society of america uniform color scales system,” ASTM E1360 (2000).
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Appendix J
“Standard practice for specifying the geometry of observations and measurements to characterize the appearance of materials,” ASTM E1767 (1995). “Standard practice for the periodic calibration of narrow band-pass spectrophotometers,” ASTM E925 (1994). “Standard practice for testing fixed-wavelength photometric detectors used in liquid chromatography,” ASTM E685 (2000). “Standard practice for testing variable-wavelength photometric detectors used in liquid chromatography,” ASTM E1657 (2001). “Standard practice for transfer standards for reflectance factor for near-infrared instruments using hemispherical geometry,” ASTM E1791 (2000). “Standard practice for validation of multivariate process infrared spectrophotometers,” ASTM D6122 (1999). “Standard practice for visual appraisal of colors and color differences of diffusely-illuminated opaque materials,” ASTM D1729 (1996). “Standard practice for visual color evaluation of transparent sheet materials,” ASTM E1478 (1997). “Standard practice for visual evaluation of metamerism,” ASTM D4086 (1997). “Standard practices for general techniques of ultraviolet-visible quantitative analysis,” ASTM E169 (1999). “Standard practices for infrared multivariate quantitative analysis,” ASTM E1655 (2000). “Standard Practices for Internal Reflection Spectroscopy,” ASTM E573 (2001) “Standard Solar Constant and Zero Air Mass Solar Spectral Irradiance Tables,” ASTM E490 (2000). “Standard specification for infrared thermometers for intermittent determination of patient temperature,” ASTM E1965 (1998). “Standard specification for nighttime photometric performance of retroreflective pedestrian markings for visibility enhancement,” ASTM E1501 (1999). “Standard specification for photoluminescent (phosphorescent) safety markings,” ASTM E2072 (2000). “Standard specification for physical characteristics of nonconcentrator terrestrial photovoltaic reference cells,” ASTM E1040 (1998). “Standard specification for retroreflective sheeting for traffic control,” ASTM D4956 (2001). “Standard specification for silvered flat glass mirror,” ASTM C1503 (2001). “Standard specification for solar simulation for terrestrial photovoltaic testing.” ASTM E927 (1997). “Standard tables for references solar spectral irradiance at air mass 1.5: direct normal and hemispherical for a 37° tilted surface,” ASTM G159 (1998).
Documentary Standards for Radiometry and Photometry
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“Standard terminology of appearance,” ASTM E284 (2001). “Standard terminology relating to molecular spectroscopy,” ASTM E131 (2000). “Standard terminology relating to photovoltaic solar energy conversion,” ASTM E1328 (1999). “Standard terminology relating to solar energy conversion,” ASTM E772 (1993). Standard test method for 20-deg specular gloss of waxed paper,” ASTM D1834 (2000). Standard test method for 45-deg specular gloss of ceramic materials,” ASTM C346 (1998). Standard test method for 60-deg specular gloss of emulsion floor polish,” ASTM D1455 (1997). “Standard test method for calibration of a spectroradiometer using a standard source of irradiance,” ASTM G138 (1996). “Standard test method for calibration of heat transfer rate calorimeters using a narrow-angle blackbody radiation facility,” ASTM E638 (1992). “Standard test method for calibration of narrow- and broad-band ultraviolet radiometers using a spectroradiometer,” ASTM G130 (1995). “Standard test method for calibration of a pyranometer using a pyrheliometer,” ASTM G151 (2000). “Standard test method for color and color-difference measurement by tristimulus (filter) colorimetry,” ASTM E1347 (1997) (formerly E97). “Standard test method for determining the linearity of a photovoltaic device parameter with respect to a test parameter,” ASTM E1143 (1994). “Standard test method for estimating stray radiant power ratio of spectrophotometers by the opaque filter method,” ASTM E387 (1995). “Standard test method for field measurement of raised retroreflective pavement markers using a portable retroreflectometer,” ASTM E1696 (2001). “Standard test method for haze and luminous transmittance of transparent plastics,” ASTM D1003 (2000). “Standard test method for identifying fluorescence in object-color specimens by spectrophotometry,” ASTM E1247 (2000). “Standard test method for luminous reflectance factor of acoustical materials by use of integrating-sphere reflectometers,” ASTM E1477 (1998). “Standard test method for measuring and calculating emittance of architectural flat glass products using spectrometric measurements,” ASTM E1585 (1993). “Standard test method for minimum detectable temperature difference for thermal imaging systems,” ASTM E1311 (1999). “Standard test method for noise equivalent temperature difference of thermal imaging systems,” ASTM E1543 (2000).
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Appendix J
“Standard test method for obtaining colorimetric data from a visual display unit by spectroradiometry,” ASTM E1336 (1996). “Standard test method for obtaining colorimetric data from a visual display unit using tristimulus colorimeters,” ASTM E1455 (1997). “Standard test method for obtaining spectroradiometric data from radiant sources for colorimetry,” ASTM E1341 (1996). “Standard test method for photopic luminance of photoluminescent (phosphorescent) markings,” ASTM E2073 (2000). “Standard test method for radiation thermometer (single waveband type),” ASTM E1256 (1988). “Standard test method for reflection haze of high-gloss surfaces,” ASTM D4039 (1999). “Standard test method for reflectance factor and color by spectrophotometry using bidirectional geometry,” ASTM E1349 (1998). “Standard test method for solar absorptance, reflectance and transmittance of materials using spectrophotometers with integrating spheres,” ASTM E903 (1988). “Standard test methods for solar energy transmittance and reflectance (terrestrial) of sheet materials,” ASTM E424 (1993). “Standard test method for solar absorptance, reflectance, and transmittance of materials using integrating spheres,” ASTM E903 (1996). “Standard test method for spectral bandwidth and wavelength accuracy of fluorescence spectrometers,” ASTM E388 (1998). “Standard test method for specular gloss,” ASTM D523 (1999). “Standard test method for specular gloss of paper and paperboard at 75°,” ASTM D1223 (1998). “Standard test method for total luminous reflectance factor by use of 30/t integrating-sphere geometry,” ASTM E1651 (1999). “Standard test method for transfer of calibration from reference to field radiometers,” ASTM E824 (1994). “Standard test method for transmittance and color by spectrophotometry using hemispherical geometry,” ASTM E1348 (1996). “Standard test method for transparency of plastic sheeting,” ASTM D1746 (1997). “Standard test method for visual evaluation of gloss differences between surfaces of similar appearance,” ASTM D4449 (1999). “Standard test method for total hemispherical emittance of surfaces from 20 to 1400° C,” ASTM C835 (2000). “Standard test method for brightness of pulp, paper, and paperboard (directional reflectance at 457 nm),” ASTM D985 (1997).
Documentary Standards for Radiometry and Photometry
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“Standard test method for calculation of color differences from instrumentally measured color coordinates,” ASTM D2244 (2000). “Standard test method for calibration of pyrheliometers by comparison to reference pyrheliometers,” ASTM E816 (1995). “Standard test method for calibration of a pyranometer using a pyrheliometer,” ASTM G167 (2000). “Standard test method for calibration of a spectroradiometer using a standard source of irradiance,” ASTM G138 (1996). “Standard test method for calibration of narrow- and broad-band ultraviolet radiometers using a spectroradiometer,” ASTM G130 (1995). “Standard test method for calibration of primary non-concentrator terrestrial photovoltaic reference cells using a tabular spectrum,” ASTM E1125 (1999). “Standard test method for calibration of reference pyranometers with axis tilted by the shading method,” ASTM E941 (1999). “Standard test method for calibration of silicon non-concentrator photovoltaic primary reference cells under global irradiation,” ASTM E1039 (1999). “Standard test method for calorimetric determination of hemispherical emittance and the ratio of solar absorptance to hemispherical emittance using solar simulation,” ASTM E434 (1996). “Standard test method for coefficient of retroreflection of retroreflective sheeting utilizing the coplanar geometry,” ASTM E810 (2001). “Standard test method for color and color-difference measurement by tristimulus (filter) colorimetry,” ASTM E1347 (1997). “Standard test method for color of liquids using tristimulus colorimetry,” ASTM D5386 (2000). “Standard test method for conducting aqueous direct photolysis tests,” ASTM E896 (1997). “Standard test method for detecting delaminations in bridge decks using infrared thermography,” ASTM D4788 (1997). “Standard test method for determining solar or photopic reflectance, transmittance, and absorptance of materials using a large diameter integrating sphere,” ASTM E1175 (1996). “Standard test method for determining the linearity of a photovoltaic device parameter with respect to a test parameter,” ASTM E1143 (1999). “Standard test method for diffuse light transmission factor of reinforced plastics panels,” ASTM D1494 (1997). “Standard test method for electrical performance of photovoltaic cells using reference cells under simulated sunlight,” ASTM E948 (1995). “Standard test method for evaluating color image output from color printers and copiers,” ASTM F1206 (2000).
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Appendix J
“Standard test method for evaluation of color for thermoplastic traffic marking materials,” ASTM D4960 (1998). “Standard test method for evaluation of visual color difference with a gray scale,” ASTM D2616 (1996). “Standard test method for haze and luminous transmittance of transparent plastics,” ASTM D1003 (2000). “Standard test method for hiding power of paints by reflectometry,” ASTM D2805 (1996). “Standard test method for identifying fluorescence in object-color specimens by spectrophotometry,” ASTM E1247 (2000). “Standard test method for linearity of fluorescence measuring systems,” ASTM E578 (2001). “Standard test method for luminous reflectance factor of acoustical materials by use of integrating-sphere reflectometers,” ASTM E1477 (1998). “Standard test method for measuring total-radiance temperature of heated surfaces using a radiation pyrometer,” ASTM E639 (1996). “Standard test method for measurement and calculation of reflecting characteristics of metallic surfaces using integrating sphere instruments,” ASTM E429 “Standard test method for measurement of high-visibility retroreflective-clothing marking material using a portable retrorelectometer,” ASTM E1809 (1996). “Standard test method for measurement of retroreflective pavement marking materials with cen-prescribed geometry using a portable retroreflectometer,” ASTM E1710 (1997). “Standard test method for measurement of retroreflective signs using a portable retroreflectometer,” ASTM E1709 (2000). “Standard test method for minimum detectable temperature difference for thermal imaging systems,” ASTM E1311 (1999). “Standard test method for minimum resolvable temperature difference for thermal imaging systems,” ASTM E1213 (1997). “Standard test method for noise equivalent temperature difference of thermal imaging systems,” ASTM E1543 (2000). “Standard test method for normal spectral emittance at elevated temperatures,” ASTM E307 (1996). “Standard test method for normal spectral emittance at elevated temperatures of nonconducting specimens,” ASTM E423 (1996). “Standard test method for obtaining colorimetric data from a visual display unit by spectroradiometry,” ASTM E1336 (1996). “Standard test method for opacity of paper (15° diffuse illuminant A, 89% reflectance backing and paper backing),” ASTM D589 (1997).
Documentary Standards for Radiometry and Photometry
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“Standard test method for photoelastic measurements of birefringence and residual strains in transparent or translucent plastic materials,” ASTM D4093 (1995). “Standard test method for reflectance factor and color by spectrophotometry using hemispherical geometry,” ASTM E1331 (1996). “Standard test method for reflectance factor and color by spectrophotometry using bidirectional geometry,” ASTM E1349 (1998). “Standard test method for reflection haze of high-gloss surfaces,” ASTM D4039 (1999). “Standard test method for relative tinting strength of aqueous ink systems by instrumental measurement,” ASTM D6531 (2000). “Standard test method for relative tinting strength of white pigments by reflectance measurements,” ASTM D2745 (2000). “Standard test method for retroreflectance of horizontal coatings,” ASTM D4061 (2000). “Standard test method for solar absorptance, reflectance, and transmittance of materials using integrating spheres,” ASTM E903 (1996). “Standard test method for solar photometric transmittance of sheet materials using sunlight,” ASTM E972 (1996). “Standard test method for solar transmittance (terrestrial) of sheet materials using sunlight,” ASTM E1084 (1996). “Standard test method for spectral bandwidth and wavelength accuracy of fluorescence spectrometers,” ASTM E388 (1998). “Standard test method for specular gloss,” ASTM D523 (1999). “Standard test method for specular gloss of glazed ceramic whitewares and related products,” ASTM C584 (1999). “Standard test method for total luminous reflectance factor by use of 30/t integrating-sphere geometry,” ASTM E1651 (1999). “Standard test method for transmittance and color by spectrophotometry using hemispherical geometry,” ASTM E1348 (1996). “Standard test method for transparency of plastic sheeting,” ASTM D1746 (1997). “Standard test methods for continuous measurement of ozone in ambient, workplace, and indoor atmospheres (ultraviolet absorption),” ASTM D5156 (1995). “Standard test methods for measurement of gloss of high-gloss surfaces by goniophotometry,” ASTM E430 (1997). “Standard test methods for measuring and compensating for emissivity using infrared imaging radiometers,” ASTM E1933 (1999). “Standard test methods for measuring and compensating for reflected temperature using infrared imaging radiometers,” ASTM E1862 (1997).
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Appendix J
“Standard test methods for measuring and compensating for transmittance of an attenuating medium using infrared imaging radiometers,” ASTM E1897 (1997). “Standard test methods for measuring optical reflectivity of transparent materials,” ASTM E1682 (1996). “Standard test methods for measuring spectral response of photovoltaic cells,” ASTM E1021 (1995). “Standard test methods for measuring total-radiance temperature of heated surfaces using a radiation pyrometer” ASTM E639 (1990). “Standard test methods for measurement of gloss of high-gloss surfaces by goniophotometry,” ASTM E430 (1997). “Standard test methods for minimum detectable temperature difference for thermal imaging systems” ASTM E1311 (1993). “Standard test methods for minimum resolvable temperature difference for thermal imaging systems” ASTM E1213 (1992). “Standard test methods for noise equivalent temperature difference of thermal imaging systems” ASTM E1543 (1994). “Standard test methods for radiation thermometers (single waveband type),” ASTM E1256 (1995). “Standard test methods for solar energy transmittance and reflectance (terrestrial) of sheet materials,” ASTM E424 (1993). “Standard test methods for total normal emittance of surfaces using inspectionmeter techniques,” ASTM E408 (1996).
BIPM The BIPM (Bureau International des Poids et Mesures)v is an international institute operating under the supervision of the Comite International des Poids et Mesures (CIPM). It is charged with the establishment and maintenance of reference standards, the organization of international comparisons and carrying out of calibrations, and fundamental investigations that may result in better reference standards or measurement techniques. The prototype kilogram is located here. Some of their publications include: Principles Governing Photometry (1983). The International System of Units (SI), BIPM, 7th Edition (1998). “Radiation Thermometry,” Chapter 6 in Supplementary Information for the International Temperature Scale of 1990, BIPM (1990). International Vocabulary of Basic and General Terms in Metrology, joint BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML Standard, BIPM (1993).
v
www.bipm.fr
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Guide to the Expression of Uncertainty in Measurement, joint BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML Standard, BIPM (1993).
CIE The Commission Internationale de L’Eclairagevi has numerous technical committee reports that are relevant. Check into their web page for more information. Division 1 is involved with vision and color, while Division 2 deals with the measurement of light and radiation. The U.S. National Committee of the CIE is at http://www.cie-usnc.org. Some pertinent reports, primarily from Division 2, are: 13.3 Method of Measuring and Specifying Colour Rendering of Light Sources (1995). 15.2 Colorimetry, 2nd Ed. (1986). 17.4 International Lighting Vocabulary, 4th Ed. (Joint publication IEC/CIE) (1987). 18.2 The Basis of Physical Photometry, 2nd Ed. (1983). 38 “Radiometric and photometric characteristics of materials and their measurement,” (1977). 41 “Light as a true visual quantity: principles of measurement,” (1978). 44 “Absolute methods for reflection measurements,” (1979). 46 “A review of publications on properties and reflection values of material reflection standards,” (1979). 53 “Methods of characterizing the performance of radiometers and photometers,” (1982). 54 “Retroreflection: definition and measurement,” (2001). 59 “Polarization: definitions and nomenclature, instrument polarization,” (1984). 63 “The spectroradiometric measurement of light sources,” (1984). 64 “Determination of the spectral responsivity of optical radiation detectors,” (1984). 65 “Electrically calibrated thermal detectors of optical radiation (absolute radiometers)” (1985). 69 “Methods of characterizing illuminance meters and luminance meters: Performance, characteristics and specifications,” (1987). 70 “The measurement of absolute luminous intensity distributions,” (1987). 75 “Spectral luminous efficiency functions based upon brightness matching for monochromatic point sources, 20 and 100 fields,” (1988). vi
www.cie.co.at/ Write to Thomas Lemons, TLA- Lighting Consultants Inc., 7 Pond St., Salem, MA 01970 or contact Tel: (508) 745-6870, Fax: (508) 741-4420 for a listing of current publications and pricing.
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76 “Intercomparison on measurement of (total) spectral radiance factor of luminescent specimens,” (1988). 78 “Brightness-luminance relations: classified bibliography,” (1988). 81 “Mesopic photometry: history, special problems and practical solutions,” (1989). 84 “Measurement of luminous flux,” (1989). 85 “Solar spectral irradiance,” (1989). 86 “CIE 1988 2 spectral luminous efficiency function for photopic vision,” (1990). 87 “Colorimetry of self-luminous displays: a bibliography,” (1990). 96 “Electric light sources, state of the art—1991,” (1992). 105 “Spectroradiometry of pulsed optical radiation sources,” (1993). 114 CIE Collection in Photometry and Radiometry (1994). 121 “The photometry and goniophotometry of luminaires,” (1996). 125 “Standard erythema dose, a review,” (1997). 127 “Measurement of LEDs,” (1997). 130 “Practical methods for the measurement of reflectance and transmittance,” (1998). 141 “Testing of supplementary systems of photometry,” (2001).
IES The Illuminating Engineering Society of North America (IESNA)vii has an extensive list of publications dealing with illumination. Their IESNA Lighting Handbook, 9th Ed. is the definitive reference. The following are procedures dealing with photometric measurements of various lamps and luminaires: LM-9 LM-10 LM-11 LM-20 LM-31 LM-35 LM-41 LM-44 LM-45
vii
Electrical and Photometric Measurements of Fluorescent Lamps Photometric Testing of Outdoor Fluorescent Luminaires Photometric Testing of Searchlights Photometric Testing of Reflector-Type Lamps Photometric Testing of Roadway Luminaires Photometric Testing of Floodlights Using High-Intensity Discharge Lamps or Incandescent Filament Lamps Photometric Testing of Indoor Fluorescent Luminaires Method for Total and Diffuse Reflectometry (1985) Electrical and Photometric Measurements of General Service Incandescent Filament Lamps
www.iesna.org
Documentary Standards for Radiometry and Photometry
LM-46 LM-50 LM-51 LM-52 LM-54 LM-55 LM-58 LM-59 LM-63 LM-64 LM-66 LM-68 LM-70 LM-72 RP-16
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Photometric Testing of Indoor Luminaires Using HID Discharge or Incandescent Filament Lamps Photometric Measurement of Roadway Lighting Installations Electrical and Photometric Measurements of High-Intensity Discharge Lamps Photometric Measurement of Roadway Sign Installations Lamp Seasoning Measurement of Ultraviolet Radiation from Light Sources Spectroradiometric Measurements Electrical and Photometric Measurements of Low-Pressure Sodium Lamps Standard File Format for Electronic Transfer of Photometric Data Photometric Measurements of Parking Areas Electrical and Photometric Measurements of Compact Fluorescent Lamps Photometric Evaluation of Vehicle Traffic Control Signal Heads Near-Field Photometry Directional Positioning of Photometric Data Nomenclature and Definitions for Illuminating Engineering (ANSI Approved)
ISO The International Standards Organization.viii ISO 2470: Brightness for Fluorescent Materials (1999). ISO/CIE 10526: CIE standard illuminants for colorimetry (CIE S005/E-1998) (1999). ISO/CIE 10527: CIE standard colorimetric observers (CIE S002, 1986) (1991). ISO/CIE 15469: Spatial distribution of daylight—CIE standard overcast sky and clear sky (CIE S003, 1996) (1997). ISO/CIE 16508: Road traffic lights—Photometric properties of 200 mm round signals (CIE S006) (1999). ISO17166: Erythema reference action spectrum and standard erythema dose (CIE S007) (1999). ISO 11475: Paper and board—Determination of CIE whiteness, D65/10 degrees (outdoor daylight) (1999).
viii
www.iso.ch
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ISO 11476: ISO 8599: ISO 9845-1:
ISO 9022-9: ISO 9022-17: ISO 9050:
ISO 9059: ISO 9060:
ISO 9488: ISO 9846: ISO 9847: ISO/TR 9901: ISO 6: ISO 2240: ISO 8478: ISO 12232:
Appendix J
Paper and board—Determination of CIE-whiteness, C/2 degrees (indoor illumination conditions) (2000). Optics and optical instruments—Contact lenses—Determination of the spectral and luminous transmittance (1994). Solar energy—Reference solar spectral irradiance at the ground at different receiving conditions—Part 1: Direct normal and hemispherical solar irradiance for air mass 1,5 (1992). Optics and optical instruments—Environmental test methods— Part 9: Solar radiation (1994). Optics and optical instruments—Environmental test methods— Part 17: Combined contamination, solar radiation (1994). Glass in building—Determination of light transmittance, solar direct transmittance, total solar energy transmittance and ultraviolet transmittance, and related glazing factors (1990). Solar energy—Calibration of field pyrheliometers by comparison to a reference pyrheliometer (1990). Solar energy—Specification and classification of instruments for measuring hemispherical solar and direct solar radiation (1990). Solar energy—Vocabulary (1999). Solar energy—Calibration of a pyranometer using a pyrheliometer (1993). Solar energy—Calibration of field pyranometers by comparison to a reference pyranometer (1992). Solar energy—Field pyranometers—Recommended practice for use (1990). Photography—Black-and-white pictorial still camera negative film/process systems—Determination of ISO speed (1993). Photography—Colour reversal camera films—Determination of ISO speed (1994). Photography—Camera lenses—Measurement of ISO spectral transmittance (1996). Photography—Electronic still-picture cameras—Determination of ISO speed (1998).
IEC Founded in 1906, the International Electrotechnical Commission (IEC)ix is the world organization that prepares and publishes international standards for all ix
IEC, 3, Rue de Varembe, PO Box 131, 1211 Geneva 20, Switzerland. Tel: +41 22 919 02 11, Fax: +41 22 919 03 00, Website: http://www.iec.ch.
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electrical, electronic, and related technologies. The IEC was founded as a result of a resolution passed at the International Electrical Congress held in St. Louis, Missouri in 1904. The membership consists of more than 50 participating countries, including all the world’s major trading nations and a growing number of industrializing countries. Multimedia systems and equipment —Colour measurement and IEC management Part 2-1: Colour management—Default RGB 61966-2-1 colour space—sRGB Ed. l: 1999 IEC 61966-3 Multimedia systems and equipment —Colour measurement and Ed. l: 2000 management Part 3: Equipment using cathode ray tubes. IEC 61966-4 Multimedia systems and equipment —Colour measurement and Ed. 1: 2000 management Part 4: Equipment using liquid crystal display panels. IEC 61966-5 Multimedia systems and equipment —Colour measurement and Ed. 1: 2000 management Part 5: Equipment using plasma display panels. IEC 61966-8 Multimedia systems and equipment —Colour measurement and Ed. 1: 2001 management Part 8: Multimedia colour scanners. IEC 61966-9 Multimedia systems and equipment —Colour measurement and Ed. l: 2000 management Part 9: Digital cameras.
NVLAP The National Voluntary Laboratory Accreditation Program.x Established in 1976 and administered by the National Institute of Standards and Technology (NIST), NVLAP is an unbiased government-based third-party system for accrediting calibration laboratories and testing laboratories found competent to perform specific tests or calibrations. Criteria for NVLAP accreditation are published in the Code of Federal Regulations (Title 15, Part 285) and encompass the requirements of ISO/IES Guide 25 and the relevant requirements of ISO 9002. NVLAP accreditation is available to commercial laboratories, manufacturers’ in-house laboratories, university laboratories, federal, state, and local government laboratories, and foreign-based laboratories. “Procedures and General Requirements,” NIST Handbook 150, National Voluntary Laboratory Accreditation Program (NVLAP), C. D. Faison et al, Eds. (February 2006).
x
National Institute of Standards and Technology, National Voluntary Laboratory Accreditation Program, 100 Bureau Drive, MS 2140, Gaithersburg, Maryland 20899-2140, Telephone: 301-9754016, Fax: 301-926-2884, E-mail:
[email protected], Website: http://www.ts nist.gov/nvlap.
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“Energy Efficient Lighting Products,” NIST Handbook 150-1, National Voluntary Laboratory Accreditation Program (NVLAP), C. C. Miller and L. I. Knab, Eds. (March 2008). xi NVLAP identifies its accredited laboratories in a published directory, NIST Special Publication 810, and on their website.
SAE SAE International, formerly the Society of Automotive Engineers,xii is a nonprofit educational and scientific organization dedicated to advancing mobility technology to better serve humanity. Nearly 70,000 engineers and scientists who are SAE members develop technical information on all forms of self-propelled vehicles including automobiles, trucks and buses, off-highway equipment, aircraft, aerospace vehicles, marine, rail, and transit systems. SAE disseminates this information through meetings, books, technical papers, magazines, standards, reports, professional development programs, and electronic databases. Here is a selection of their relevant standards, mostly dealing with lighting and its measurement. HS-34 J387 J1330 J575 J 1383 J2217
SAE Ground Vehicle Lighting Standards Manual (1999). “Terminology—Motor Vehicle Lighting” (1995). “Photometry Laboratory Accuracy Guidelines” (1994). “Test Methods and Equipment for Lighting Devices and Components for Use on Vehicles Less than 2032 mm in Overall Width” (1992). “Performance Requirements for Motor Vehicle Headlamps” (1996). “Photometric Guidelines for Instrument Panel Displays that Accommodate Older Drivers” (1991).
TAPPI The Technical Association of the Pulp and Paper Industry.xiii TAPPI is the leading technical association for the worldwide pulp, paper, and converting industry. TAPPI provides its members rapid access to: (1) the largest international group of technically experienced people in the industry, (2) the most comprehensive collection of reliable technical information and knowledge in the industry, and (3) the highest quality products and services created to meet the needs of people who solve technical problems in the industry. Among their documentary standards are the following: T-425 Opacity of paper (15/d geometry, Illuminant A/2, 89% reflectance backing and paper backing) (1996). xi
Handbook 150 and 150-1 are available on the NVLAP website at http://www.ts.nist.gov/nvlap. www.sae.org xiii http://www.tappi.org xii
Documentary Standards for Radiometry and Photometry
T-452 T-480 T-560 T-562 T-1212 T-1213 T-1214 T-1215 T-1216 T-1217 T-1218
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Brightness of pulp, paper and paperboard (directional reflectance at 457 nm) (1998). Specular gloss of paper and paperboard at 75 degrees (1999). CIE whiteness and tint of paper and paperboard (d/0 geometry, C/2 illuminant/observer) (1996). CIE whiteness and tint of paper and paperboard (45/0 geometry, C/2 illuminant/observer) (1996). Light sources for evaluating papers, including those containing fluorescent whitening agents (1998). Optical measurements terminology (related to appearance evaluation of paper) (1998). Interrelation of reflectance; reflectivity; TAPPI opacity; scattering, s; and absorption, k (1998). The determination of instrumental color differences (1998). Indices for whiteness, yellowness, brightness, and luminous reflectance factor (1998). Photometric linearity of optical properties instruments (1998). Calibration of reflectance standards for hemispherical geometry (1998).
Appendix K
Radiometry and Photometry Bibliography In 1969 Fred Nicodemus authored a paper “Optical resource letter on radiometry” (JOSA 59, 243). It was reprinted in an AIP Radiometry—Selected Reprints collection in 1971. This bibliography is intended to update and supplement the 1969 version. The topic breakdown is more in line with the order of topics in this book than the original. Individual papers are for the most part bypassed in favor of books, significant book chapters, monographs, or reprint collections.
Symbols, Units, and Terminology American National Standard Nomenclature and Definitions for Illuminating Engineering. ANSI, Standard ANSI/IESNA RP-16 96 (1996). International Lighting Vocabulary, CIE Publication 17 (1970). “Quantities and units—Part 6. Light and related electromagnetic radiations,” ISO Standards Handbook, Quantities and Units 389.15 (1993). Symbols, Units and Nomenclature in Physics, International Union of Pure and Applied Physics (1987). C. L. Wyatt, V. Privalsky, and R. T. Datlu, Recommended Practice: Symbols, Units and Uncertainty Analysis for Radiometric Sensor Calibration, NIST Handbook 152 (1998).
General R. W. Boyd, Radiometry and the Detection of Optical Radiation, John Wiley & Sons, New York (1983). Good for the theoretical aspects of radiation geometry, along with several other topics. M. A. Bramson, Infrared: A Handbook for Applications, Plenum, New York (1966). F. C. Grum, and R. J. Becherer, Optical Radiation Measurements I. Radiometry, Academic Press, New York (1979). The best all-around book on radiometry to date. 341
342
Appendix K
A. Hadni, Essentials of Modern Physics Applied to the Study of the Infrared, Pergamon, Oxford Univ. Press, New York (1967). M. R. Holter, S. Nudelman, G. H. Suits, W. L. Wolfe and G. J. Zissis, Fundamentals of Infrared Technology, Macmillan, New York (1962). J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, and R. G. Richards, Infrared Physics and Engineering, McGraw-Hill, New York (1963). P. W. Kruse, L. D. McGlaughlin, and R. B. McQuistan, Elements of Infrared Technology, John Wiley & Sons, New York (1962). W. R. McCluney, Introduction to Radiometry and Photometry, Artech House, Boston (1994). A recent entry, again slightly slanted to the author’s (Palmer) specialty of solar energy applications. Both elementary and thorough. I. J. Spiro and M. Schlessinger, Infrared Technology Fundamentals, Marcel Dekker, New York (1989). W. L Wolfe and G. Zissis, The Infrared Handbook, Office of Naval Research, Washington, D.C. (1978). The most bang for your buck! At $60, everybody should have one! Available from SPIE. W. L. Wolfe, Introduction to Radiometry, SPIE Press, Bellingham, Washington (1998). Uneven and terse in spots, still recommended. He was my (Palmer) mentor! E. F. Zalewski, “Radiometry and Photometry,” Chapter 24 in Handbook of Optics, Vol. II, McGraw-Hill, New York (1995). An excellent review.
Geometric Radiation Transfer A. Gershun, “The Light Field,” Moscow, 1936, trans. P. Moon and G. Timoshenko, J. Math. Phys. 18, 51–151(1939). H. C. Hottel and A. F. Sarofim, Radiative Transfer, McGraw-Hill, New York (1967). J. R. Howell, A Catalog of Radiation Configuration Factors, McGraw-Hill, New York (1982). F. E. Nicodemus et al., Self-Study Manual on Optical Radiation Measurements, NBS Technical Note 910–xx. NIST, Washington (various dates). A series that is rather theoretical with increasing coverage. Some may be out of print. Check GPO for availability. R. Siegel and J. R. Howell, Thermal Radiation Transfer, Hemisphere, Washington, D.C. (1981). Intended for mechanical and heat transfer engineers (uses non-SI units like BTUs, degrees Rankine, and the wrong intensity) and highly detailed, this book is the best compendium on radiative transfer. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer, Brooks/Cole, California (1970). J. H. Taylor, Radiation Exchange, Academic Press, New York (1990).
Radiometry and Photometry Bibliography
343
Radiosity and Ray Tracing I. Ashdown, Radiosity: A Programmer’s Perspective, John Wiley & Sons, New York (1994). M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Syntheses, Academic Press, New York (1993).
Optical Radiation Sources F. E. Carlson and C. N. Clark, Light Sources for Optical Devices, Chapter 2 in Applied Optics and Optical Engineering, Vol. 1, R. Kingslake, Ed., Academic Press, New York (1975). K. L. Coulson, Solar and Terrestrial Radiation, Academic Press, New York (1975). J. E. Eby and R. E. Levin, Incoherent Light Sources, Chapter 1 in Applied Optics and Optical Engineering, Vol. 7, R. R. Shannon and J. Wyant, Eds., Academic Press, New York (1975). A. S. Green, Ed. The Middle Ultraviolet: Its Science and Technology, John Wiley & Sons, New York (1966). F. Grum and R. J. Becherer, Radiometry, Vol. 1 in Optical Radiation Measurements, Academic Press, New York (1979). H. Hewitt and A. S. Vause, Lamps and Lighting, Elsevier, New York (1966). M. Iqbal, An Introduction to Solar Radiation, Academic Press, New York (1983). P. A. Jacobs, Thermal Infrared Characterization of Ground Targets and Backgrounds, SPIE Press, Bellingham, Washington (1996). D. Kryskowski and G. H. Suits, “Natural sources,” Chapter 3 in Sources of Radiation, G. J. Zissis, Ed., Vol. 1 of The Infrared & Electro-Optical Handbook, SPIE Press, Bellingham, Washington (1993). A. J. LaRocca, “Artificial sources,” Chapter 2 in Sources of Radiation, G. J. Zissis, Ed., Vol. 1 of The Infrared & Electro-Optical Handbook, SPIE Press, Bellingham, Washington (1993). A. J. LaRocca, “Artificial sources,” Chapter 10 in Optical Sources, G. J. Zissis, Ed., Vol. 1, Part 4 Handbook of Optics, McGraw Hill, New York (1995). M. Luckiesh, Applications of Germicidal, Erythemal and Infrared Energy, Van Nostrand, New York (1946). D. Malacara, “Light sources,” Chapter 5 In Geometrical and Instrumental Optics, Academic Press, Boston (1988). P. Moon, The Scientific Basis of Illuminating Engineering, McGraw-Hill, New York (1936), reprinted by Dover, New York (1961). J. B. Murdoch, Illumination Engineering: From Edison’s Lamp to the Laser, Macmillan, New York (1985).
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Appendix K
RCA Electro-Optics Handbook, RCA, Lancaster PA (1974). Now available from Burle Industries. M. Rea, Ed., Lighting Handbook: Reference and Application, 8th Ed., Illuminating Engineering Society of North America (1993). J. C. Richmond and F. E. Nicodemus, “Blackbodies, blackbody radiation, and temperature scales,” NBS Self-Study Manual on Optical Radiation Measurements, Part 1, Chapter 12, GPO (1985). W. T. Silfvast, “Lasers,” Chapter 11 in Handbook of Optics, Vol. 1, Part 4. Optical Sources, McGraw-Hill, New York (1995). G. H. Suits, “Natural sources,” Chapter 3 in The Infrared Handbook, ERIM (1982). Distributed By SPIE. R. H. Weissman, “Light emitting diodes,” Chapter 12 in Handbook of Optics, Vol. 1, Part 4. Optical Sources, McGraw-Hill, New York (1995). G. J. Zissis and A. J. Larocca, “Optical Radiators and Sources,” Chapter 3 in Handbook Of Optics, McGraw-Hill, New York (1978).
Detectors A. Ambroziak, Semiconductor Photoelectric Devices, Gordon & Breach, New York (1969). D. F. Barbe, Ed., Charge-Coupled Devices, Springer-Verlag, New York (1980). C. J. Bartelson, F. C. Grum, Optical Radiation Measurements Vol. 5: Visual Measurements, Academic Press, New York (1984). J. D. E. Beynon and D. R. Lamb, Charge-Coupled Devices and their Applications, McGraw-Hill, New York (1980). L. Biberman and S. Nudelman, Photoelectronic Imaging Devices, Plenum, New York (1970). Two volumes. R. M. Boynton, Human Color Vision, Holt, Rinehart & Winston, New York (1979). W. Budde, Optical Radiation Measurements Vol 4: Physical Detectors of Optical Radiation, Academic Press, New York (1983). The best practical book on detectors and their characterization, slanted a bit towards the visible spectrum. C. Buil, CCD Astronomy, Willman-Bell, Richmond, Virginia (1991). A. Chappell, Optoelectronics: Theory and Practice, McGraw-Hill, New York (1978). I. P. Csorba, Image Tubes, Howard W. Sams Engineering-Reference Book Series, Indianapolis, Indiana (1985). P. N. J. Dennis, Photodetectors, Plenum, New York (1986). E. L. Dereniak and G. Boreman, Infrared Detectors and Systems, John Wiley & Sons, New York (1996).
Radiometry and Photometry Bibliography
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E. L. Dereniak and D. Crowe, Optical Radiation Detectors, John Wiley & Sons, New York (1984). Good general book, slanted towards the infrared. S. Donati, Photodetectors: Devices, Circuits and Applications, Prentice Hall, New Jersey (2000). Concise, accurate, highly recommended. M. J. Eccles, M. E. Sim, and K. P. Tritton, Low Light Level Detectors in Astronomy, Cambridge Univ. Press, New York (1983). R. W. Engstrom, RCA Photomultiplier Handbook, RCA (1980). Available from Burle Industries (http://www.burle.com/cgi-bin/byteserver.pl/pdf/Photo.pdf). G. C. Holst, CMOS/CCD Sensors and Camera Systems, SPIE Press, Bellingham, Washington (2007). R. D. Hudson and J. W. Hudson, Eds., Infrared Detectors, Hutchinson and Ross, Stroudsburg, Pennsylvania (1975). A collection of the more important papers to date. J. R. Janesick, Scientific Charge-Coupled Devices, SPIE Press, Bellingham, Washington (2001). R. J. Keyes, Ed., Optical and Infrared Detectors, Springer-Verlag, Bellingham, Washington (1980). I. S. McLean, Electronic Imaging in Astronomy, John Wiley & Sons, New York (1997). Photomultiplier Tube: Principle to Application, H. Kume, Ed., Hamamatsu Photonics K. K. (1994). S. O. Flyckt and C. Marmonier, Photomultiplier Tubes: Principles and Applications, Philips Photonics, Brive, France (2002). G. H. Rieke, Detection of Light from the Ultraviolet to the Submillimeter, Cambridge Univ. Press, New York (1994). A. Rose, Vision: Human and Electronic, Plenum, New York (1973). M. Ross, Laser Receivers, John Wiley & Sons, New York (1966). C. H. Séquin and M. F. Tomsett, Charge Transfer Devices, Academic Press, New York (1975). R. A. Smith, F. E. Jones, and R. P. Chasmar, The Detection and Measurement of Infra-red Radiation, Oxford Univ. Press, New York (1957). A. H. Sommer, Photoemissive Materials, John Wiley & Sons, New York (1968). Reprinted by Krieger, Huntington, New York (1980). J. D. Vincentm, Fundamentals of Infrared Detector Operation and Testing, John Wiley & Sons, New York (1990). Separate chapters on cryogenics, vacuum systems, and detector electronics. Answers many of those nagging questions. Highly recommended. A. T. Young, “Photomultipliers, their causes and cures,” Chapter 1 in Methods of Experimental Physics: Astrophysics, N. Carleton Ed., Vol. 12, Academic Press, New York (1974).
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Appendix K
The major handbooks offer one or more chapters dealing with optical radiation detectors. These include the following: J. S. Accetta and D. L. Shumaker, Eds., Infrared and Electro-Optical Systems Handbook, SPIE Press, Bellingham, Washington (1993). See Vol. 3, Chapter 4. M. Bass, Ed. Handbook of Optics, Vol. 1, McGraw-Hill, New York (1995). Parts 5, 6, and 7 contain chapters 15 through 25, all pertinent to detectors and detection. W. G. Driscoll and W. Vaughn, Handbook of Optics, McGraw-Hill, New York (1978). Chapter 4. W. L. Wolfe, Ed., Handbook of Military Infrared Technology, ONR, Washington (1965). Chapters 11 & 12. W. L. Wolfe and G. Zissis, Eds., The Infrared Handbook, ERIM and SPIE (1978). Chapters 11 through 16. Several books in the series Semiconductors and Semimetals, R. K. Willardson and A. C. Beer, Eds., are relevant. Volumes 5 and 12 both treat infrared detectors, and Vol. 11 deals exclusively with solar cells. Volume 47 deals with uncooled infrared imaging array systems.” Similarly, the series Advances in Electronics and Electron Physics is rich in pertinent articles. Numerous volumes deal with imaging detector conference proceedings, while others have significant feature articles. Most notable are Vols. 34 and 55. Another rich source for detector information, and probably the best for assessment of the current state of art, are the Proceedings of the various conferences of SPIE (see p. 355), available on SPIE’s digital library.i They have several major conventions per year, each having one or more conferences on detectors. Compilations of the best proceedings papers along with seminal papers are often gathered in their “Milestones” series. Several detector manufacturers publish manuals, collections of applications notes, etc. Particularly attractive are the photomultiplier tube books by Burle (formerly RCA), Hamamatsu, and Philips, and the CCD books by EG&G Reticon and Fairchild. Data books by Dalsa, EG&G Vactec, Texas Instruments, Thompson CSF, etc., are also quite helpful.
Noise, Electronics, and Signal Processing D. A. Bell, Noise and the Solid State, John Wiley & Sons, New York (1985). E. R. Davies, Electronics, Noise and Signal Recovery, Academic Press, New York (1983). P. J. Fish, Electronic Noise and Low Noise Design, McGraw-Hill, New York (1994). i
http://spiedigitallibrary.aip.org/
Radiometry and Photometry Bibliography
347
T. M. Frederiksen, Intuitive IC Op Amps, National Semiconductor (1984). J. G. Graeme, Photodiode Amplifiers: Op-Amp Solutions, McGraw-Hill, New York (1996). M. S. Gupta, Ed., Electrical Noise: Fundamentals and Sources, IEEE (1977). P. Horowitz and W. Hill, The Art of Electronics, Cambridge Univ. Press, New York (1989). The best single book available for the broad field of electronics, my personal favorite! W. G. Jung, IC Op-amp Cookbook, Howard W. Sams (1980). More tutorial than Stout, with fewer useful circuits. C. D. Motchenbacher and J. A. Connelly, Low-Noise Electronic System Design, John Wiley & Sons, New York (1993). H. W. Ott, Noise Reduction Techniques in Electronic Systems, 2nd Ed., John Wiley & Sons, New York (1988). R. A. Pease, Troubleshooting Analog Circuits, Butterworth-Heinemann, Newton, Massachusetts (1991). D. F. Stout and M. Kaufman, Handbook of Operational Amplifier Circuit Design, McGraw-Hill, New York (1976). Excellent single-purpose reference book. A. Van der Ziel, Noise in Measurements, John Wiley & Sons, New York (1976). C. A. Vergers, Handbook of Electrical Noise: Measurement and Technology, 2nd Ed., Tab Books, Inc., Pennsylvania (1987). J. Williams, Analog Circuit Design: Art, Science and Personalities, ButterworthHeinemann, Newton, Massachusetts (1991). J. Williams, The Art and Science of Analog Circuit Design, ButterworthHeinemann, Newton, Massachusetts (1995). T. H. Wilmshurst, Signal Recovery, Adam Hilger, Bristol (1990). A. Yariv, Optical Electronics in Modern Communications, Oxford Univ. Press, New York (1997). Chapter 10 treats noise in optical detection and generation.
Radiometric Instruments B. N. Begunov, N. P. Zakaznov, S. I. Kiryushinand V. I. Kuzichev, Optical Instrumentation: Theory and Design, MIR, Moscow (1988). R. D. Hudson, Infrared System Engineering, John Wiley & Sons, New York (1969). Dated but still valuable. J. F. James and R. S. Sternberg, Design of Optical Spectrometers, Chapman & Hall, London (1969). K. Seyrafi, Ed., Engineering Design Handbook on Infrared Military Systems, United States Army Materiel Command, Pamphlet AMCP 706-127 (1971). Similar to Hudson but in MIL-format. Later modified and self published as Electro-Optical Systems Analysis, Electro-Optical Research Co, Los Angeles (1985).
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Appendix K
W. L. Wolfe, Introduction to Infrared System Design, SPIE Press, Bellingham, Washington (1996). W. L. Wolfe, Introduction to Imaging Spectrometers, SPIE Press, Bellingham, Washington (1997). W. L. Wolfe, Infrared Design Examples, SPIE Press, Bellingham, Washington (1999). C. L. Wyatt, Radiometric Systems Design, Macmillan, New York (1987). More for the designer than the calibrator, but very useful. Later available as ElectroOptical System Design for Information Processing, McGraw-Hill, New York (1990). More for the system designer, but very useful.
Radiometric Measurements and Errors D. C. Baird, Experimentation: An Introduction to Measurement Theory and Experiment Design, Prentice-Hall, New York (1962). 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 Ed., McGraw-Hill, New York (1992). Calibration: Philosophy in Practice, 2nd Ed., Fluke Corporation, Everett, Washington (1994). Available directly from them. J. P. Holman, Experimental Methods for Engineers, McGraw-Hill, New York (1978). 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). S. Rabinovich, Measurement Errors: Theory and Practice, AIP Press (1995). B. N. Taylor & C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297 (1994). Available on the web at http://physics.nist.gov/Pubs/guidelines/contents.html. J. R. Taylor, An Introduction to Error Analysis, 2nd Ed., University Science (1997). H. D. Young, Statistical Treatment of Experimental Data, McGraw-Hill, New York (1962).
Measurement of Radiometric Quantities G. Bauer, Measurement of Optical Radiations, Focal Press, London (1965). W. E. Forsythe, Ed., Measurement of Radiant Energy, McGraw-Hill, New York (1937).
Radiometry and Photometry Bibliography
349
H. J. Kostkowski, Reliable Spectroradiometry, Spectroradiometry Consulting, La Plata, Maryland 20646. Self-published lessons learned over some 40 years. Authoritative! D. Malacara, Physical Optics and Light Measurements, Academic Press, New York (1988). Chapter 6.
Photometry W. E. Barrows, Light, Photometry and Illuminating Engineering, McGraw-Hill, New York (1951). P. Bouguer, Essai d’optique sur la gradation de la lumiere, Paris (1729). It all started here! Translated into English by W. E. K. Middleton, The Gradation of Light, Toronto (1961). C. DeCusatis, Handbook of Applied Photometry, AIP Press (1997). Authoritative, with pertinent chapters written by technical experts at BIPM, CIE, and NIST. Primarily for visible radiation, mostly excellent. C. Fabry, “Photometrie,” Revue d’Optique, Paris (1927). H. A. E. Keitz, Light Calculations and Measurements, Cleaver Hume, London (1955), MacMillan, New York (1971). J. H. Lambert, Photometrie, Augsberg (1760). A beautiful English translation from the original Latin with copious notes by D. DiLaura was published in 2001, available from IESNA. Y. Ohno, “Radiometry and Photometry Review for Vision Optics,” Chapter 14 in Handbook of Optics III, McGraw-Hill, New York (2001). J. M. Palmer, “Radiometry and Photometry: Units and Conversions,” Chapter 7 in Handbook of Optics III, McGraw-Hill, New York (2001). A. Stimson, Photometry and Radiometry for Engineers, John Wiley & Sons, New York (1974). J. W. T. Walsh, Photometry, 3rd Ed., Constable (1958), reprinted by Dover, New York (1965). The standard reference since 1924.
Astronomical Measurement of Light I choose not to include these in the photometry section to avoid confusion with the strict definition of the word, the measurement of light. A. A. Henden and R. H. Kaitchuk, Astronomical Photometry, Van Nostrand Reinhold, New York (1982). J. L. Hopkins, Zen and the Art of Photoelectric Photometry, HPO Desktop Publishing (1990).
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Appendix K
Color and Appearance A. Berger-Schunn, Practical Color Measurement, John Wiley & Sons, New York (1994). F. W. Billmeyer and M. Salzman, Principles of Color Technology, WileyInterscience, New York (1981). R. W. Burnham, R. M. Hanes, and C. J. Bartleson, Color: A Guide to the Basic Facts and Concepts, John Wiley & Sons, New York (1963). G. J. Chamberlin and D. G. Chamberlin, Colour: Its Measurement, Computation and Application, Heyden & Sons, London (1980). Committee on Colorimetry, The Science of Color, OSA, Crowell (1953). A second edition is due any day now! R. M. Evans, An Introduction to Color, John Wiley & Sons, New York (1948). B. Fortner and T. E. Meyer, Number by Colors, Springer-Verlag, New York (1997). F. C. Grum, and C.J. Bartelson, Optical Radiation Measurements II: Color Measurement, Academic Press, New York (1981). A. C. Hardy, Handbook of Colorimetry, MIT Press, Cambridge (1936). R. S. Hunter, The Measurement of Appearance, John Wiley & Sons, New York (1987). G. Kortüm, Kolorimetrie, Photometrie und Spektrometrie, Springer-Verlag, Berlin (1962). D. B. Judd, Contributions to Color Science, D. L. MacAdam, Ed., NBS Special Publication SP-545 (1979). A collection of the significant papers by Judd, the Deane of colorimetry! D. B. Judd and G. Wysczecki, Color in Business, Science and Industry, John Wiley & Sons, New York (1975). W. D. Wright, The Measurement of Colour, Macmillan, New York (1969). G. Wyszecki, and W. S. Stiles, Color Science, John Wiley & Sons, New York (1967); 2nd revised Ed. John Wiley & Sons, New York (1982). This classic has been reprinted in a low-cost paperback format by John Wiley & Sons, New York (2000). A must!
Radiometric Properties of Materials H. H. Blau and H. Fischer, Radiative Transfer from Solid Materials, Macmillan, New York (1962). C. Burgess and K. D. Mielenz, Advances in Standards and Methodology in Spectrophotometry, Elsevier, New York (1987). F. J. Clauss, Material Effects in Spacecraft Thermal Control, John Wiley & Sons, New York (1960).
Radiometry and Photometry Bibliography
351
G. G. Gubareff, J. E. Janssen, and R. H. Torberg, Thermal Radiation Properties Survey, Honeywell, Minneapolis (1960). B. Hapke, Theory of Reflectance and Emittance Spectroscopy, Cambridge Univ. Press (1993). J. Kamler, Luminescent Screens: Photometry and Colorimetry, Illife, London (1969). S. Katzoff, Ed., Symposium on Thermal Radiation of Solids, NASA SP-55 (1965). T. G. Kyle, Atmospheric Transmission, Emission and Scattering, Pergamon, Oxford (1991). E. P. Lavin, Specular Reflection, Elsevier, New York (1971). R. Mavrodinuenu, J. I. Schultz, and O. Menis, Accuracy in Spectrophotometry and Luminescence Measurements, NBS Special Publication 378 (1973). K. D. Mielenz, Ed., Optical Radiation Measurements III: Measurement of Photoluminescence, Academic Press, New York (1982). J. C. Richmond, Ed., Measurement of Thermal Radiation Properties of Solids, NASA SP-31 (1963). G. H. Schenk, Absorption of Light and Ultraviolet Radiation: Fluorescence and Phosphorescence Emission, Allyn & Bacon, Boston (1973). J. C. Stover, Optical Scattering: Measurement and Analysis, SPIE Press, Bellingham, Washington (1995). R. Tilley, Colour and the Optical Properties of Materials, John Wiley & Sons, New York (2000).
Spectroscopy R. B. Barnes, R. C. Gore, U. Liddel, and V. Z. Williams, Infrared Spectroscopy: Industrial Applications and Bibliography, Reinhold, New York (1944). W. R. Brode, Chemical Spectroscopy, John Wiley & Sons, New York (1943). J. R. Edisbury, Practical Hints on Absorption Spectrometry, Plenum, New York (1967). N. J. Harrick, Internal Reflection Spectroscopy, Harrick Scientific, Ossining, New York (1967). G. R. Harrison, R. C. Lord, and J. R. Loofbourow, Practical Spectroscopy, Prentice-Hall, New Jersey (1948). J. F. James and R. S. Sternburg, The Design of Optical Spectrometers, Chapman & Hall, London (1969). G. Kortüm, Reflectance Spectroscopy, Springer-Verlag, Berlin (1969). J. R. Lakowicz, Principles of Fluorescence Spectroscopy, Plenum, New York (1983). G. F. Lothian, Absorption Spectrophotometry, Hilger & Watts, London (1958).
352
Appendix K
E. J. Meehan, Optical Methods of Analysis, Interscience, New York (1964). R. A. Sawyer, Experimental Spectroscopy, Prentice-Hall, New Jersey (1951). Stearns, E. I., The Practice of Absorption Spectrophotometry, John Wiley & Sons, New York (1969). J. E. Stewart, Infrared Spectroscopy, Marcel Dekker, New York (1970). W. W. Wendlandt and H. G. Hecht, Reflectance Spectroscopy, Interscience, New York (1966). J. Workman and A. Springsteen, Applied Spectroscopy: A Compact Reference for Practitioners, Academic Press, New York (1998).
Laser Power and Energy H. G. Heard, Laser Parameter Measurements Handbook, John Wiley & Sons, New York (1968).
Temperature D. P. DeWitt and G. D. Nutter, Theory and Practice of Radiation Thermometry, John Wiley & Sons, New York (1988). H. Kaplan, Practical Applications of Infrared Thermal Sensing and Imaging Equipment, SPIE Press, Bellingham, Washington (1999). J. M. Lloyd, Thermal Imaging Systems, Plenum, NewYork (1975). T. Quinn, Temperature, 2nd Ed., Academic Press (1990). J. C. Richmond and D. P. Dewitt, Eds., Applications of Radiation Thermometry, ASTM (1985). W. P. Wood and J. M. Cork, Pyrometry, McGraw-Hill, New York (1941).
Radiometric Standards and Calibration F. Hengtsberger, Absolute Radiometry, Academic Press, New York (1989). Broad coverage with emphasis on room-temperature electrically calibrated thermal radiometers. C. L. Wyatt, Radiometric Calibration: Theory and Methods, Academic Press, New York (1978). Good, fairly thorough treatment, somewhat slanted towards the author’s (Palmer) specialty, which is far-infrared cryogenic radiometry.
Compilations A. J. Drummond, Ed., Precision Radiometry, Vol. 14 in Advances in Geophysics, Academic Press, New York (1970). H. K. Hammond III and H. L. Mason, Selected NBS Papers on Radiometry and Photometry, in Precision Measurement and Calibration, NBS Special Publication 300, Vol. 7 (1971).
Radiometry and Photometry Bibliography
353
R. B. Johnson and W. L. Wolfe, Selected Papers on Infrared Design, 2 vols. SPIE Press, Bellingham, Washington (1985). I. Nimeroff, Selected NBS Papers on Colorimetry, in Precision Measurement and Calibration, NBS Special Publication 300, Vol. 9 (1972). I. J. Spiro, Ed., Selected Papers on Radiometry, SPIE Press, Bellingham, Washington (1990).
Optical Design R. E. Fischer and B. Tadic-Galeb, Optical System Design, McGraw-Hill, New York (2000). R. Kingslake, Lens Design Fundamentals, Academic Press, New York (1978). R. Kingslake, Optical System Design, Academic Press, New York (1983). P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design, Oxford Univ. Press, New York (1997). D. C. O’Shea, Elements of Modern Optical Design, John Wiley & Sons, New York (1985). R. R. Shannon, The Art and Science of Optical Design, Cambridge Univ. Press, New York (1997). Bob was my (Palmer) professor for optical design. G. H. Smith, Practical Computer-Aided Lens Design, Willmann-Bell, Richmond, Virginia (1998). W. J. Smith, Modern Optical Engineering, 3rd Ed., McGraw-Hill, New York (2000).
Miscellaneous P. C. D. Hobbs, Building Electro-Optical Systems: Making It All Work, John Wiley & Sons, New York (2000). This publication is a pretty useful book. C. A. Poynton, A Technical Introduction to Digital Video, John Wiley & Sons, New York (1996).
Technical Organizations in Radiometry and Photometry CIE
The Commission Internationale de L’Eclairage (CIE) has numerous technical committee reports that are relevant. Contact Thomas Lemons, TLA-Lighting Consultants Inc., 7 Pond St., Salem, MA 01970, Tel: (508) 745-6870, Fax: (508) 741-4420, for a listing of current publications and pricing. Check into their web page (www.cie.co.at) for more information. Some pertinent reports, primarily from Division 2, are listed in Appendix H.
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Appendix K
CORM
The Council for Optical Radiation Measurements (CORM) was founded over twenty years ago to promote optical radiation measurement science and engineering and foster cooperation among the many government agencies, industrial firms, and universities, and to formulate and transmit national needs to NIST. They meet annually in May and publish Optical Radiation News biannually. Contact: CORM Treasurer, 1043 Grand Ave. #312, St. Paul, Minnesota 55105 (www.corm.org). CORM documents are listed in Appendix H. NEWRAD
The NEWRAD conference series is an outgrowth of a meeting organized by Peter Foukal and the papers were published by Cambridge University Press, Massachusetts, in 1985. The proceedings of the first meeting were a private publication. The second meeting took place at the National Physical Laboratory in London in 1988, and the proceedings were published as New Developments and Applications in Radiometry, N. Fox and D. Nettleton, Eds., by IOP Publishing, London (1989). The next four meetings were held in Davos (1990), Baltimore (1992), Berlin (1995) and Tucson (1997). The proceedings of these meetings were published as special issues of Metrologia (Elsevier), Volumes 28(3), 30(4), 32(6), and 35(4), respectively. The seventh conference was held October 25–27, 1999, in Madrid. Check out http://newrad.metrologia.csic.es. NIST
The U.S. National Institute of Science and Technology (NIST, formerly National Bureau of Standards, NBS) has a number of valuable special publications that describe their calibration services and procedures. Appropriate ones include the following. They may be purchased from NTIS.
NBS Measurement Services: Spectral Radiance Calibrations, J. H. Walker, R. D. Saunders, and A. T. Hattenburg, NBS, Spec. Publ. 250-1 (1987). NBS Measurement Services: Far Ultraviolet Detector Standards, L. R. Canfield and N. Swanson, NBS, Spec. Publ. 250-2 (1987). NBS Measurement Services: Radiometric Standards in the Vacuum Ultraviolet, J. Z. Klose, J. M. Bridges, and W. R. Ott, NBS, Spec. Publ. 250-3 (1987). NBS Measurement Services: Regular Spectral Transmittance, K. L. Eckerle, J. J. Hsia, K. D. Mielenz, and V. R. Weidner, NBS, Spec. Publ. 250-6 (1987). NIST Measurement Services: Spectral Reflectance, P. Y. Barnes, E. A. Early, and A. C. Parr, NIST Spec. Publ. 250-8 (1987, revised 1997). Photodetector Spectral Response Calibration Transfer Program, E. F. Zalewski, NBS, Spec. Publ. 250-17, 45 (1988). NIST Measurement Services: Photometric Calibrations, Y. Ohno, NIST, Spec. Publ. 250-37 (1997).
Radiometry and Photometry Bibliography
355
NIST Measurement Services: Radiance Temperature Calibrations, NIST, Spec. Publ. 250-43 (1997). NBS Measurement Services: Spectral Irradiance Calibrations, J. H. Walker, R. D. Saunders, J. K. Jackson, and D. A. McSparron, NBS, Spec. Publ. 250-20 (1987). NIST Measurement Services: Spectroradiometric Detector Measurements: Parts I and II—Ultraviolet and Visible to Near Infrared Detectors, T. C. Larason, S. S. Bruce, and A. C. Parr, NIST, Spec. Publ. 250-41, (1998). NIST Measurement Services: Spectroradiometric Detector Measurements: Part III—Infrared Detectors, A. L. Migdall and G. Eppeldauer, NIST, Spec. Publ. 250-42, (1998). SPIE
Proceedings of SPIE are published unrefereed papers as presented at conferences of SPIE. Several conferences and resulting proceedings have been devoted to radiometry, photometry, and closely allied topics. These Proceedings volumes are available on SPIE’s digital library: http://spiedigitallibrary.org. They include: 196 499 888 1109 2161 2815 7298 7299 7300
Measurements of Optical Radiations (1979) Optical Radiation Measurements I (1984) Laser Beam Radiometry (1988) Optical Radiation Measurements II (1989) Photometry (1993) Optical Radiation Measurements III (1996) Infrared Technology and Applications XXXV (2009) Thermosense XXXI (2009) Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XX (2009) 7410 Optical Modeling and Measurements for Solar Energy Systems III (2009) 7419A Infrared Detectors and Focal Plane Arrays X (2009)
Open Literature Numerous archival and trade journals also offer significant papers on radiometry and detection of optical radiation. They include:
Applied Optics (OSA) IEEE Transactions on Electron Devices (IEEE) Infrared Physics and Technology (Elsevier) Journal of the Optical Society of America A (OSA)
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Journal of Scientific Instruments (IOP) Laser Focus World (PennWell Corp.) Lightwave Technology (IEEE) Optical Engineering (SPIE) Photonics Spectra (free) Review of Scientific Instruments (AIP)
Publications Available on the Internet All you ever wanted to know about the SI is contained at BIPM (www.bipm.fr) and at NIST (physics.nist.gov/cuu/). Available publications (highly recommended) include:
The International System of Units (SI) 7th Ed. (1998), direct from BIPM. The official document is in French; the English translation is available in PDF format. NIST Special Publication SP330, “The International System of Units (SI).” The U.S. edition of the above BIPM publication. Available in PDF format. NIST Special Publication SP811, “Guide for the Use of the International System of Units (SI).” Available in PDF format. Papers published in NIST Journal of Research since 1995 are also available on the web in PDF format.
Appendix L
Reference List for Noise and Postdetection Signal Processing Supplied by L. Stephen Bell May 7, 2009
Books and articles D. P. Blair and P. H. Sydenham, “Phase sensitive detection as a means to recover signals buried in noise,” J. Phys. E: Sci. Instrum. 8, p. 621 (1975). E. R. Davies, Electronics, Noise and Signal Recovery, Academic Press, London (1993). A. De Sa, Principles of Electronic Instrumentation, Arnold, London (1981). See Chapter 10, “Principles of signal recovery.” H. Doherty, “Techniques of low level light measurement,” Lasers & Applications, pp. 41–45, July (1983). P. J. Fish, Electronic Noise and Low-Noise Design, McGraw-Hill, New York (1994). J. Graeme, Photodiode Amplifiers, McGraw-Hill, New York (1995). Fundamentals of photodetection circuits. Good coverage of noise-reduction techniques for circuits and discusses effects from power supplies and external sources. P. Horowitz and W. Hill, The Art of Electronics, 2nd Ed., Cambridge, New York (1989). One of the best single books on electronics. Look in Chapter 7 on “Precision Circuits and Low-noise Techniques” and Chapter 15 on “Measurements and Signal Processing,” which has sections on reducing bandwidth and spectral analysis. G. Lawday, D. Relann, and G. Edlund, A Signal Integrity Engineer’s Companion, Prentice Hall, New Jersey (2008). Real-time test and measurement and designsimulation techniques to systematically eliminate signal integrity problems. 357
358
Appendix L
M. L. Meade, “Advances in lock-in amplifiers,” J. Phys. E: Sci. Instrum. 15, p. 395 (1982). C. D. Motchenbacher and J. A. Connelly, Low-Noise Electronic System Design, John Wiley & Sons, New York (1993). D. M. Munroe, “Signal-to-noise ratio improvement,” Chapter 11 in Handbook of Measurement Science, Vol. 1, P. H. Sydenham, Ed., John Wiley & Sons, New York (1982). Includes sections on noise and noise bandwidth, signals and SNR, preamp selection, grounding and shielding, bandwidth reduction, lock-in amplifiers, signal averaging, correlation, and photon counting. H. W. Ott, Noise Reduction Techniques in Electronic Systems, 2nd Ed., John Wiley & Sons, New York (1988). R. A. Pease, Ed., Analog Circuits-World Class Designs, Elsevier, Burlington Massachusetts (2008). A top-notch, thorough coverage of analog amplifiers, filters, and analog-to-digital converters. The filter sections cover low pass, high pass, and bandpass, along with the operational amplifier requirements to make the filters meet their desired specifications. R. G. Lyons, Understanding Digital Signal Processing, Prentice Hall, New Jersey (2004). This is another easy-to-understand introduction to DSP. D. C. Smith, High Frequency Measurements and Noise in Electronic Circuits, Kluwer Academic Publishers, New York (1993). J. Williams, Ed., Analog Circuit Design—Art, Science, and Personalities, Butterworth-Heineman, Newton, Massachusetts (1991). Per reviewer Phil Hobbs: “Design ... is an art; there’s always more than one way to do it, and the individuality of the designer has a strong influence on the way the design turns out: hence Art, Science, and Personalities. What it means is that for designs that are not routine, the designer’s personality has a lot to do with how it comes out.” I recommend this book to anyone who wants to become a better analogue designer and is not easily put off by whimsy in technical writing. J. Williams, Ed., The Art and Science of Analog Circuit Design, ButterworthHeineman, Woburn, Massachusetts (1998). This is Jim William’s second book. Per reviewer Phil Hobbs: “The emphasis of this volume is growing good analog engineers, by teaching the rhythm of the insight, design, prototype, debug iteration as practiced by the best. If you have circuits to design, this book will pay for itself in about 5 minutes, and you’ll be a more confident and adventurous designer.” T. H. Wilmshurst, Signal Recovery from Noise in Electronic Instrumentation, 2nd Ed., Adam Hilger, Bristol (1990).
Reference List for Noise and Postdetection Signal Processing
359
Software and Commentary, Downloadable Resources, and E-books Numerical Recipes: The Art of Scientific Computing, 3rd Ed., Cambridge Univ. Press, New York (2007). The book includes commented full listings of more than 400 unique C++ routines that can be downloaded in machine-readable form for inclusion in users’ programs. Information is available from www.nr.com. Matlab® software has extension packs (“toolkits”) for signal processing, image processing, and wavelets. It is available from www.mathsoft.com. Academic and student versions are available for discounted pricing. R. Pratap, “Getting Started with Matlab® 7,” A quick introduction for scientists and engineers, Oxford Univ. Press, New York (2006). V. Ingle and J. Proakis, Digital Signal Processing Using Matlab®, 2nd Ed., CLEngineering (2007). This book is part of a large library of Matlab® information via: http://engineering.thomsonlearning.com. ISBN-13: 978-0495-07311-6. This is the big list of books for Matlab® signal processing: http://www.mathworks.com/support/books/index_by_categorytitle.html?category =11&sortby=title The main Matlab® repository of contributed code is found at: http://www.mathworks.com/ matlabcentral. Wavelab is a software library of Matlab® routines for wavelet and other signal analyses. The library is available free of charge over the Internet from Stanford University: http://www-stat.stanford.edu/~wavelab/. MathCad® software has extension packs (“toolkits”) for signal processing, image processing, and wavelets. It is available from: PTC (formerly Mathworks): http://www.ptc.com/products/mathcad/. Academic and student versions are available for discounted pricing. Mathematica® software also has capabilities for signal processing, image processing, and wavelets via its “Wavelet Explorer.” It is available from www.wolfram.com. Academic (discounted) versions are available. Stéphane Mallat, A Wavelet Tour of Signal Processing, Academic Press, New York (1999). Information is available from the site: http://www.cmap. polytechnique.fr/~mallat/book.html. This is the wavelet toolbox called “LastWave,” which is written in the “C” language. Available from http://www.cmap.polytechnique.fr/~bacry/LastWave. http://www.amara.com/current/wavesoft.html This site also includes links to wavelet code for use with Mathematica®. FFTW is a “C” subroutine library for computing the discrete Fourier transform (DFT) in one or more dimensions, of arbitrary input size, and of both real and complex data (as well as of even/odd data, i.e. the discrete cosine/sine
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transforms or DCT/DST). FFTW is free (source code in “C”) software available from: www.fftw.org. S. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, California Technical Pub. (1997). Available as a book and freely distributed over the internet in electronic form: http://www.dspguide.com/ Other online resources for signal processing and/or FFT-related questions are available from http://www.dsprelated.com/ and from Usenet: comp.dsp (digital signal processing), sci.math.num-analysis (numerical analysis and scientific computing). R Project for Statistical Computing. This is free GNU-licensed software from http://www.r-project.org. This website provides a peer-reviewed encyclopedia for many topics in science: http://www.scholarpedia.org/. A specific discussion on the phenomena of stochastic resonance regarding SNR is available from http://www.scholarpedia.org/article/Stochastic_resonance. D. L. Instruments (Ithaco) App Note IAN-102, “Measuring noise spectra with variable electronic filters,” Available from the website: http://www.dlinstruments.com/. 233 Cecil A. Malone Drive, Ithaca, New York 14850. The site has many other “Technotes” on use of lock-in amplifiers.
Index background, 6, 96 test chambers, 259 baffles, 227 bandpass filter, 301 simple, 303 bandwidth shrinkage factor, 300 barrier height, 187 basic radiance, 25 radiometer, 221 Bayer algorithm, 207 bias voltage, 157 bidirectional reflectance distribution function, 65, 73, 317 bidirectional transmittance distribution function, 63, 73 blackbody, 77, 121 curve, 86 equation, 100 radiation, 87, 95, 138 simulator, 33, 83, 132, 226, 242, 255, 261 source, 49 spectral radiance, 88 BLIP, 133 blooming, 204 Bode plot, 129 bolometer, 155, 157 Boltzmann’s constant, 85, 178, 181, 189, 313 Bouguer, 4, 30 Butterworth RC filters, 297
1/f, 139 1/f noise, 143, 145, 157, 196, 226, 245, 246, 304 2D array, 216 3-dB bandwidth, 136 3-dB frequency, 130 absorptance, 69, 72, 77, 98 spectral, 69 absorption, 6, 61, 69, 79, 104, 314 absorption coefficient, 71 accuracy, 242, 243 acousto-optical tunable filters, 239 active sources, 6 additive dispersion, 235 air mass, 119 airglow, 121 aliasing, 218 Allan variance, 245 Amici, 231 amorphous silicon, 199 amplifier noise, 143 angle of observation, 32 aperture stop, 23, 24, 40, 41, 218, 220 apparent radiance, 51, 52 area array, 199 array 2D, 127 detector, 175 imagers, 199 atmosphere, 6, 28, 104, 235, 314 atmospheric transmission, 51 loss, 49 aurora, 121
calibrate, 247 a radiometer, 309 361
362
calibration, 102, 253, 256 configurations, 241, 256, 257 philosophy, 257 camera equation, 54, 57 carbon arc, 109 carrier lifetime, 170, 172 CCD, 89, 193 noise, 142 central obscuration, 57, 260 charge injection device, 204, 208, 211 charge-coupled device, 203, 204, 208, 211 charge-transfer efficiency, 142, 203 Chauvenet criteria, 247 chief ray, 23 chopper, 109, 218, 225, 235 reflecting, 224 circular variable filter, 236 classical variance, 245 collimators, 259 color films, 201 compact fluorescent lamp, 107, 116, 123 complementary metal-oxide semiconductor, 207 conduction band, 164, 177 cone half angle, 16, 18, 41 configuration factor, 38 cosine3 law, 31, 32, 33, 58 cosine4 law, 33–35 Crooke radiometer, 163 cross-sectional area, 20 Curie temperature, 158, 160 current responsivity, 195 cutoff frequency, 130 wavelength, 128, 177 Czerny-Turner, 232
D*, 132, 150, 154, 162, 174, 197 photon, 133 D**, 133 D*BLIP, 174, 197
Index
dark current, 181, 191 detection, 5 detective quantum efficiency, 131 detectivity, 132 detector, 7 array, 236 D*, 264 field of view, 55 noise-equivalent power, 264 silicon, 128 thermal, 128 dielectrics, 102, 123 sources, 108 diffuse, 61, 227 materials, 317 reflector, 75 surface, 65, 73, 316 diffusers, 224 dimensional analysis, 17 direct-method approach, 262 distant extended source configuration, 260 distant small source configuration, 258 Doppler Gaussian, 110 effect, 111 dynamic resistance, 196 earth projected area, 51 radiance, 122 reflectance, 51 effective focal length, 19 effective noise bandwidth, 136, 137, 174, 218, 297 Einstein diffusion constants, 190 electrical resistance, 152 time constant, 159 electroluminescent sources, 117 electron-hole pairs, 195 electro-optical instrument calibration, 257
Index
system, 263, 299 emission, 6, 76, 83, 104, 314 emissivity, 76 emittance, 76, 98 spectral, 76, 78 emitter, 18 energy gap, 164, 166, 167 entrance pupil, 19, 23, 24, 54, 218, 222, 263, 266 entrance slit, 22 equation of radiative transfer, 36 basic form, 58 differential form, 46 integral form, 36 error, 241, 244 assessment, 256 étendue, 20, 229, 263 exhaust gases, 110 exit pupil, 23, 24, 54, 218 extended source, 34, 222, 223, 264 extrinsic photoconductor, 169 semiconductor, 167 eye damage, 123
f/#, 19, 21, 22, 35, 264 system parameter, 53 Fabry-Perot, 237, 238 Fastie Ebert, 232 Fermi level, 176 field of view, 6, 220, 222, 251, 263 instantaneous, 217, 318 of the detector, 55 field stop, 23, 24, 218, 220, 258 filters, 236 flat-plate solar collectors, 55 FLIR devices, 315 fluorescent lamp, 115 flux density, 11 Foote’s formula, 35 forward-looking infrared, 209, 318 devices, 315 Fourier transform infrared spectrometer, 237
363
frame transfer, 206 architecture, 205 free spectral range, 230, 237, 239 frequency, 134 Fresnel equation, 103 reflection losses, 71 full frame, 205 architecture, 204 gain of power, 41 gases, 103, 123 Gaussian, 110, 111, 112, 137, 139 distribution, 147, 244 noise, 135 GE radiation calculator, 100 generation, 5 generation-recombination noise, 141, 143, 173, 197 geometrical extent, 20 glint, 319 Golay cell, 163 half-power point, 298 Havens limit, 150 H-D curve, 202 heat capacity, 141, 147 hybrid architecture, 208 illegitimate errors, 248 illuminance, 8, 57 illumination engineering, 38 image irradiance, 55 impedance, 131 incandescent light bulbs, 107 incoherent radiation, 250 sources, 119 index of refraction, 12, 79 infrared, 2 photodetector array, 209 sources, 110
364
Infrared Handbook, The 76, 78, 226 instantaneous field of view, 217, 318 integrating sphere, 5, 46, 47, 224, 261 intensity, 31, 40, 48, 266 interferometer, 237, 238 interline transfer, 206 architecture, 205 International Bureau of Weights and Measures, 254 intrinsic photoconductor, 169 semiconductor, 164 invariance of throughput, 21 invariant, 20 inverse square law, 37, 45, 48, 258 of irradiance, 17, 30 inversion layer, 198 irradiance, 25, 28, 30, 31, 121, 201, 267, 316 at the detector, 42, 48 inverse square, law of, 17, 30 on the detector, 43, 171 reduction in, 57 responsivity, 129 isotropic source, 58 I-V curve, 191, 194 I-V equation, 189 Johnson noise, 137, 143, 152, 157, 162, 173, 182, 196 Jones method, 262 Jones, R. Clark, 262, 309 Kirchhoff, 4 law, 77, 98, 99, 149, 314 KTC reset noise, 143
Lambert, 4 Lambert-Bouguer-Beer law, 70
Index
Lambertian, 27, 37 approximation, 33, 37, 316 disc, on-axis, 43 disc radiance, 42 source, 32, 310 sphere, 44 sphere, on-axis, 45 Langley, S. P., 155 large-area blackbody radiation simulator, 261 imaging array, 203 laser, 118, 262 laser-power meter, 262 least-significant bit, 143 lens transmission, 41 light-emitting diode (LED), 83, 105, 117 linear array, 216 photodiode array, 203 linearity, 131, 247 Littrow, 231, 234 long-wave infrared, 209 Lorentzian, 110, 112, 113 lossless medium, 44 optical system, 42 LOWTRAN7, 314 lumens, 8 luminance, 8 luminescence, 69, 110 luminescent sources, 83 Lyot stop, 228 magnification, 54, 57 majority carriers, 187 Marcel Golay, 236 marginal ray, 24 matte reflectors, 58 maximum power transfer theorem, 156 of electrical engineering, 172
Index
mean earth–sun distance, 50 square noise voltage, 135, 137 measurement, 4 measurement equation, 250 metallic sources, 108 metals, 102, 123, 156, 176 Michelson, 237 microchannel plates, 184 microphonic, 162 noise, 142 Mie scatter, 119 minority carriers, 187 concentrations, 190 model validity, 248 modulation transfer function, 202 moments normalization, 307 monochromator, 218, 229, 231, 234 moon and sun angular subtense, 51 multiple reflections, 47 muzzle flash, 110 narrowband measurements, 217 national standard, 255 near infrared, 209 near-extended-source, 55 configuration, 261 near-small-source calibration, 262 negative electron affinity material, 180 Nernst glower, 108 Nicodemus, 66 noise, 131, 134, 243 1/f, 139, 143, 145, 157, 196, 226, 245, 246, 304 amplifier, 143 bandwidth, 132, 149 CCD, 142 factor, 143, 184 Gaussian, 135 generation-recombination, 141, 142, 143, 173, 197
365
Johnson, 137, 143, 157, 162, 173, 182, 196 microphonic, 142 power, 298 quantization, 143, 246 shot, 139, 182, 196 temperature, 144 fluctuation, 141 thermal, 135 thermal fluctuation, 162 triboelectric, 142 white, 135, 146, 246, 303 noise-equivalent flux density, 252 irradiance, 216, 252, 265 photon flux, 132 power, 154, 252 temperature difference, 210, 216 normal distribution, 245 incidence, 35, 103 numerical aperture, 19 object at infinity, 53 off-axis collimator, 260 Ohm’s law, 136, 138, 173 OLEDs, 117 optical axis, 24, 36 element, 19 radiant power, 128 radiation, 73 detectors, 127 system, 21, 23, 48 systems, 16, 19 thickness, 71, 72 transmission, 265 organic light-emitting diodes. See OLEDs overlapping of orders, 230
366
passive sources, 6 Pauli exclusion principle, 176 PCTRAN, 314 Peltier coefficient, 151 effect, 152 phosphors, 115 photocathode, 181, 185 photoconductive, 7, 164 detector, 169, 171, 173, 175 gain, 172 mode, 194 photodiode, 5 photoemission, 176 photoemissive, 7, 164 photoemissive detector, 182 photoemitter, 180 photographic film, 199, 200 photometric, 2 photometry, 191, 217 photomultiplier tubes, 89, 183 photon, 128 D*, 133 detectors, 127 flux, 129 irradiance, 170 noise, 142 radiance spectral, 90 responsivity, 129 photopic response of the eye, 217 photovoltaic, 7, 164, 191 cell, 165 detector, 175, 185, 191, 192, 195, 198 solar cell, 193 physical standards, 253 Planck, 5, 84 constant, 89, 313 equation, 84, 86, 89, 91, 92, 93 expression for blackbody radiation, 84 function, 77 plane angle, 13, 16, 17 p-n junction, 185, 187
Index
point detector, 175 source, 29, 30, 31, 34, 220, 222, 252, 310 polarization, 248 sensitivity, 216 states, 70 polarized, 102, 103 polished metal surfaces, 33 power gain, 136, 298 generator, 192 responsivity, 262 spectrum, 134 precision, 243 pressure (Lorentzian), 110 primary standard, 254 principle of superposition, 37 projected area, 13, 32 solid angle, 17, 18, 19, 20, 26 projection systems, 56 pure material, 62 pyrheliometer, 221 pyroelectric coefficient, 158, 160 detector, 157, 159, 161 quantization noise, 143, 246 quantum efficiency, 176, 178, 195 quantum-well infrared photodetectors, 209 RA product, 133 radian, 13, 15 radiance, 24, 26, 27, 32, 52, 63, 314 apparent, 52 of the source, 223 responsivity, 129 spectral, 63, 87 radiant energy, 24 exitance, 25, 26, 27, 38, 44
Index
flux, 61, 63 intensity, 25 power, 24, 25, 29, 36, 39 radiation contrast, 96, 97 geometry, 30 radiative transfer, 36, 38 basic equation, 58, 263 radiometer, 21, 215, 241 optical system, 311 radiometric calibration, 17, 241 configuration, 6, 251 equation, 228 instruments, 215, 228, 250 measurements, 217 system, 252 radiometry, 1, 24, 191, 211, 223, 248 Raman scattering, 69 random errors, 243 noise, 244 uncertainty, 249 range equation, 251, 252, 266 ray, 11 Rayleigh, 84 scatter, 119 Rayleigh-Jeans, 93 equation, 84, 92 law, 92 RC bandpass filter, 300 circuit, 130 low-pass filter, 299 receiver, 18 reflectance, 47, 64, 70, 315 factor, 65, 67 spectral, 63, 78 reflecting chopper, 224 reflection, 61, 63, 83 reflective natural objects, 105 refractive collimator, 259 elements, 109
367
index, 19, 85, 231, 239 resolving power, 229, 230, 238 responsive quantum efficiency, 131, 170 responsivity, 128, 132, 157 reverse saturation current, 189, 190, 191 reverse-bias mode, 194 Richardson equation, 181, 182 right circular cone, 18 root mean square, 130 root sum square, 249 rough aluminum, 75 scattering, 6 Schottky barrier, 198 secondary standard, 254 Seebeck coefficient, 151 effect, 150 semiconductor detectors, 140 photocathodes, 177 Seya-Namioka, 234 short-wave infrared, 209 shot noise, 139, 182, 196, 244 signal processing, 7 signal-to-noise ratio, 131, 149, 217, 244, 266, 297, 303 silicon amorphous, 199 detector, 128 simple bandpass filter, 303 simple equation for transfer of radiant power, 263 single-element detectors, 127 Snell’s law, 12, 25 solar cell, 192, 198, 219 collectors flat-plate, 55 constant, 28, 50 irradiance, 49 panels, 193
368
spectrum, 121 zenith angle, 119, 314 solid angle, 15–17, 25, 66, 223 field of view, 265 source image, 21 spatial resolution, 202 specification sheet, 215 spectral absorptance, 69 absorption coefficient, 71 bandwidth, 216 directional emissivity, 98 emittance, 78, 102, 106 linewidth, 251 photon radiance, 90 radiance, 63, 87, 250, 313 blackbody, 88 radiometer, 215 reflectance, 63, 78 response, 201 of the eye, 8 responsivity, 128, 130, 250 sensitivity, 201 transmittance, 62, 251 spectrometer, 22, 217, 230, 235 spectroradiometer, 217, 228, 241 specular, 61, 227, 316 reflecting surfaces, 33 reflection, 319 reflectors, 58 surfaces, 73 speed of light, 12 sphere coatings, 47 SPRITE detector, 209 standard, 254–255 detector, 257 source, 257 statistical analysis, 243 Stefan-Boltzmann constant, 84 Stefan-Boltzmann law, 89, 95 steradian, 15 stray light, 227, 228, 260 subtractive dispersion, 235 sun projected area, 51
Index
sunglint, 313 sunlit scene, 52 surface normal, 13 reflectance, 317 systematic error, 241, 247 uncertainty, 249 systems analysis, 211 engineering, 211 target, 96 temperature coefficient of resistance, 155 temperature fluctuation noise, 141 thermal, 7, 104, 127 conductance, 141, 149 detection of optical radiation, 147 detector, 128, 141, 147, 312 emission, 313 fluctuation noise, 162 infrared, 319 noise, 135 resistance, 147, 148 time constant, 141, 148, 159 thermistors, 156 thermocouple radiation detectors, 153 thermoelectric detector, 150 effect, 150 thermopile radiation detectors, 153 three-color CCDs, 207 throughput, 20, 229, 230, 239, 263 time constant, 130, 154 total emissivity, 98 traceability, 254 transmission, 5, 61, 62, 79 loss, 40, 48, 53 of the lens, 42 of the optics, 223
Index
transmittance, 70, 72, 315 total, 62 triboelectric noise, 142 tungsten lamps, 105, 258 tungsten-filament lamp, 105–106 tungsten-halogen lamp, 106–107 Twyman-Green, 237 type A errors, 243 type B errors, 241 Ulbricht, R., 5, 46 ultraviolet, 2 detector array, 208 uncertainty, 241, 244, 253 uncooled thermal-imaging devices, 210 uniform radiance source, 47 unit solid angle, 17 unpolarized light, 70
369
valence band, 168, 177 vector, 11 vignette, 41 vignetting, 57 visible, 2 voltage responsivity, 160 Wadsworth, 231, 234 wavefront, 11 Welsbach mantle, 109 white diffuse panel, 261 noise, 134, 135, 146, 246, 297, 303 Wien, 84, 93 approximation, 93, 97 displacement law, 86 work function, 176 working standard, 254 Zener effect, 194 Zodiacal light, 121
James M. Palmer (1937-2007) was a research professor emeritus in the College of Optical Sciences, University of Arizona. He received his AB in physics from Grinnell College in 1959, and his MS and PhD degrees in optical sciences in 1973 and 1975, respectively, from the University of Arizona, specializing in radiometry and infrared systems. Prior to attending the University of Arizona, he worked in industrial positions at Hoffman Electronics Corporation and Centralab, Semiconductor Division of Globe Union, Inc. Over a career spanning more than 40 years, he authored or coauthored more than 60 technical papers on many aspects of radiometry and photometry, and he was named Fellow of SPIE in 2003. Other awards include a NASA Group Achievement award for his work on the Pioneer Venus Mission (1979), a Tau Beta Pi Teacher of the Year Award (1992), and a Non-Traditional Student Teaching Award from the University of Arizona (1993). He taught numerous short courses at SPIE conferences, CIE meetings, and conferences of the Optical Society of America. He served as a consultant on commercial and government projects. Dr. Palmer was a brilliant lecturer whose former students, worldwide, have expressed gratitude for the knowledge they gained under his tutelage. Barbara G. Grant received her BA in mathematics from San Jose State University in 1983, and her MS in optical sciences from the University of Arizona in 1989, where her graduate research focused on the absolute radiometric calibration of spaceborne imaging sensors. She was subsequently employed at Lockheed Missiles and Space Company, Sunnyvale, California, where supported by excellent management, she pursued problems in infrared sensor calibration and postflight data analysis of electro-optical payloads. She also worked as a NASA contractor, overseeing integration and test of imager and sounder payloads on the GOES weather satellite, for which she received two NASA awards. She is the author of two book-length volumes of market research for process spectroscopy instruments. Since 1995, her consultancy, Lines and Lights Technology, has addressed problems in systems engineering, infrared imaging and data analysis, UV measurement, and spectroradiometry, among other areas.
