Table of Contents

Preface

Acknowledgements

1 Astronomy

Introduction

Blackbody Temperature of the Sun

Number of Stars as a Function of Wavelength

Simple Model of Stellar Populations

Number of Infrared Sources Per Square Degree

Number of Stars above a Given Irradiance

Direct Lunar Radiance

Atmospheric Seeing

Comparison of Resonant Fluorescence and Rayleigh Guide Stars

Number of Actuators in an Adaptive Optic

Bandwidth Requirement for Adaptive Optics

Photon Rate at a Focal Plane

Reduction of Magnitude by Airmass

Night-Sky Exposure Time with a Fixed Camera

2 Atmospherics

Introduction

Vertical Profiles of Atmospheric Parameters

Visibility Distance for Rayleigh and Mie Scattering

Atmospheric Effects at 10.6 Microns

C_n² as a Function of Weather

C_n² Estimates

Impact of Weather on Visibility

Bufton Vertical Profile of Wind Speed

Index of Refraction of Air

Fried Parameter

Horizontal-Path Fried Parameter

Phase Error Estimation

Day vs. Night Scintillation Models for Laser Beams

Resolution Looking Down

Isoplanatic Angle

Strehl Ratio of the Atmosphere

Aperture Averaging

Adaptive Optics Influence Function

Shack–Hartmann Noise

Laser Beam Wander Variance Is Approximately Proportional to About the Cube of the Pathlength

Pulse Stretching in Scattering Environments

Optimal Truncation of a Gaussian Beam Propagating in the Atmosphere

Increased Requirement for Rangefinder SNR to Overcome Atmospheric Effects

Free-Space Link Margins

Summary of Phase Modulators for Adaptive Optics

Telescope Seeing Created by a Dome Floor

Telescope Seeing Due to Still or Ventilated Air

3 Acquisition, Tracking, and Pointing

Introduction

Correct Measure of Detection Performance

Tracker vs. Detection

Detection Criteria

Signal-to-Noise Ratio Requirements

Psychometric Function

Optical Blur Should Be Oversampled by FPA Pixels (Don't Overdo It!)

Dwell in Cell

Probability of Detection Estimation

Limits of Position Estimation

Multisensor Tracking

Johnson Criteria

Extension of Johnson Criteria to Other than 50 Percent

Identification and Recognition Improvement for Interpolation

Resolution Requirement

Resolution Required to Read a Letter

Detection Nomograph

Correcting for Probability of Chance

National Image Interpretability Rating Scale

4 Backgrounds

Introduction

Clutter and Signal-to-Clutter Ratio

Clutter Power Spectral Density

Infrared Clutter Behavior

Frame Differencing Gain

Earth's Emission and Reflection

Illuminance at Earth's Surface from Various Sources

Illuminance Changes during Twilight

Emissivity Approximations

Reflectivity of a Wet Surface

Effective Sky Temperature

Sky Irradiance

Zodiacal Light

Backgrounds from Asteroids

5 Cost and Economics

Introduction

Moore's Law

Metcalfe's Law

Englebart's Law

The Value of Early Investment

Cost Reduction Techniques

Learning Curves

Learning Curves for Optics

Optics Cost

Cost Function of a Lens

Price of a Custom vs. Off-the-Shelf Optic

Telescope Component Costs

Impact of Tolerances on the Cost of Optics

Stahl Segmented Cost Rule

Tolerancing Guidelines for Glass Spherical Optics

Tolerancing Guidelines for Plastic Optics

Cost of Digital Image vs. Film

Small Pixels Reduce Cost

System Percentage Costs

Length of a Job Search

Photolithography Yield

6 Degraded Visual Environments

Introduction

Basic Attenuation and Visibility: Beer's Law

Atmospheric Attenuation Curves

Atmospheric Visibility Curves

Attenuation vs. Particle Size

Visibility in Smoke

Equations for Empirical Visibility

Penetration vs. Resolution

Mandatory Mitigation for Sensor Blindness

Deep Wells Are Good

7 Focal Plane Arrays

Introduction

Infrared Detector Characterization

Responsivity and Quantum Efficiency

ROIC Charge Capacity

Low Quantum Efficiency Detectors Are Useful

Silicon Quantum Efficiency

HgCdTe x Concentration

Quantum Dot Fundamentals

Avalanche Photodiode Performance

Responsivity of Avalanche Photodiodes

Peak vs. Cut-Off

CMOS Depletion Scaling

Focal Plane Array Noise Sources

Rule 07

Law 19

Radiative Estimate of Dark Current

Defining Background-Limited Performance for Detectors

The Concepts of D and D*

Ideal D* and View Angle

Dependence on R₀A

Shot Noise Rule

Infrared Detector DC Pedestal

Digitizer Sizing

Noise as a Function of Temperature

Noise Bandwidth of Detectors

Noise Equations for CMOS

Specifying 1/f Noise

Nonuniformity Effects on SNR

Correlated Double Sampling

8 Human Vision

Introduction

Retinal Illumination

Diffraction Effects in the Human Eye

Energy Flow into the Eye

Pupil Size

Quantum Efficiency of Rods and Cones

Rod and Cone Response Models

Cone Density of the Human Eye

Rod Density Peak

Eye Resolution

Optical Fields of View

Contrast Performance

Simplified Optical Transfer Functions for Eye Components

Eye Motion during the Formation of an Image

Visual Performance as a Function of Age

Old-Age Rules

Superposition of Colors

Dyschromatopic Vision

Eye Adaptation Time

Eat Your Vegetables

Stereograph Distance

Assorted Eye Facts

Head-Mounted-Display Latency

9 Lasers

Introduction

Lidar Basic Equations

Laser Brightness

Laser Beam Quality

Gaussian Beam Radius

On-Axis Intensity of a Beam

Aperture Size for Laser Beams

Laser Beam Divergence

Laser Beam Spread vs. Diffraction

Types of Lidars

Laser Radar Range Equation

Lidar Bidirectional Reflectance Distribution Function

Thermal Focusing in Laser Rods

Cross-Section of a Retroreflector

Air Breakdown

10 Materials and Structures

Introduction

Diameter-to-Thickness (Aspect) Ratio

The Influence of the Mounting Method on Plate Deflection

Self-Weight Deflection of Mounted Mirrors

Mirror Support Criteria

Fundamental Frequency of a Vibrating Plate

Natural Frequency of a Deformable Mirror

Design Guidelines for Pressure Windows

Dome Collapse Pressure

Glass Does Not Flow

Allowable Stress in an Optic

Relationship between Tensile and Compressive Stress

Estimation of Preload Torque

Stress Birefringence Induced by an Applied Load

Maximum Stress on an Optic Due to a Metal Retainer

A Bonded Mirror Is Three Times More Stable in Tension or Compression than Shear

Mechanical Stability Rules

Mass Is Proportional to Element Size Cubed

Deflection of a Mirror at the End of a Beam

Scan Mirror Deflection

Figure Change of Metal Mirrors

Foam Core Mirrors

Spin-Cast Mirrors

Serrurier Truss

Spacecraft Issues Related to Space Optics

Damage Mechanisms Associated with Micrometeoroids

Black Coatings

Index of Refraction Resources

Carbon-Silicon Carbide Coefficient of Thermal Expansion

Properties of Aluminum as a Function of Temperature

Permeability of Gases through Thin Films of Aluminum

Time to Uniform Temperature

Temperature Dependence of the Verdet Constant

Modeling Cryo Multilayer Insulation

11 Miscellaneous

Introduction

Position of the Sun

Distance to Horizon

Contrast

Digital Pixel Equivalent of Chemical Film

Common Image and Video Compression Formats

The Power of Dimensional Analysis

Scissor Integration

Number of Seconds in a Year

Solid Angles

Speed of Light

Water Weighs a Ton

Avoid Galling Metal

Failure of Cylinders and Spheres under External Pressure

Defining Screw Threads

Friction-Induced Pointing Error after Rate Reversal

Shipping Environments

Clean Room Classifications

Converting Transmission and Optical Density

Safety Factors for Optics

Use Speckle to Focus

90 Percent of Anything Is Plumbing

Arrhenius Equation

Miller's Rule of Test Failure

Cooling with Liquid or Solid Cryogen

Joule-Thomson Cool-Down Time

Low-Earth-Orbit Thermal Changes

Crickets as Thermometers

Image Intensifier Resolution

Photomultiplier Tube Power Supply Noise

Quantization Error

12 Ocean Optics

Introduction

Index of Refraction of Seawater

Absorption Coefficient

Absorption of Ice at 532 nm

Absorption Caused by Chlorophyll a

Bathymetry

f-stop under Water

Underwater Detection

Underwater Glow

Ocean Reflectance

Wave Slope

13 Optical Design and Analysis

Introduction

Small-Angle Approximation

Effects from Light Passing Through a Plane Parallel Plate

Beam Deviation Due to a Thin Wedge Prism

Impacts of Optical Element Motion

Defocus for a Telescope Focused at Infinity

Hyperfocal Distance

Focal Length and Field of View

Limit on FOV for Reflective Telescopes

Maximum Useful Pupil Diameter

Minimum f-Number

f-Number for Circular Obscured Apertures

Why Light Won't Refract in a Cube

Aberration Scaling

Spherical Aberration and f-Number

Blur vs. Field-Dependent Aberrations

Reducing Optical Design Sensitivities

Separate the Centers of Curvature

Reduce the Ray Angles of Incidence

Efficient Reflective Triplet Layout for Feasibility Checks

Properties of Visible Glass

Per Pixel Resolution of a Spectrometer

Smith's Modern Optics Design Rules of Thumb

Diffraction Graph

Diffraction Is Proportional to Perimeter

Diffraction Principles Derived from the Uncertainty Principle

Why There Are Spikes Radiating from Only Some Objects Seen in Astronomical Images

Estimating Surface Scatter

Power Spectral Density of Surface Roughness

In-Field Source Contribution to Stray Light in the Focal Plane

Fest's Stray Light Rules of Thumb

Performance Budgeting Using the Hopkins Ratio

Linear Approximation for the Optical Modulation Transfer Function

Telescope Aberrations Not Caused by the Atmosphere

Strehl for Obscured Apertures

Total Error Using the Root-Sum-Square Approach

Optical Performance of a Telescope

Visible Imaging System Resolution

Optimal Telescope Resolution for the Human Eye

Peak-to-Valley Approximates Four Times the Root Mean Square

Ritchey-Chretien Telescope Aberrations

Spectral Bandwidth and Resolution of Acousto-optical Tunable Filters

Circular Variable Filters

Blazed Grating Performance

Fabry-Perot Etalons

Pulse Broadening in a Fabry-Perot Etalon

Hollow Waveguides

Inflated Mirrors

Handheld Binocular Efficiency

Stop Down Two Stops

Anti-reflection Coating Index

Coating Shift with Angle of Incidence

Coating Shift with Temperature

Grating Blockers

Far-Field Model of a Light Source, or the "Rule of 5"

Lambertian Source Illumination of a Detector

Detecting Linear Polarization

Modeling an Optical System Using the Fast Fourier Transform

A Collection of Optical Engineering Rules of Thumb

Use a "Pencil Bounce" to Determine Image Orientation

Thermal Gradients in a Primary Mirror

Thermal Lensing

14 Optical Manufacture and Test

Introduction

Progress in Optical Fabrication of High-Quality Mirrors

Caution while Cleaning Optics

Thickness of a Doublet Bond

Sag of an Optic

Scratch-Dig Specifications

Oversizing an Optical Element for Producibility

Mind Your Karow Factor

Cyanoacrylate Usage

Surface Tilt Is Typically the Worst Error

Diamond-Turned Mirror Substrate Design

Diamond-Turned Mirror Figure Error

Accuracy of Figures

Fringe Movement

Approximations for the Foucault Knife-Edge Test

Effect of Surface Irregularity on the Wavefront

Hartmann Test for Pinhole Size

Maximum Size of the Opening for an Integrating Sphere

Choosing an Eyepiece for Star Tests

Collimator Design

Temperature Control Is Critical for Accurate Inhomogeneity Testing

Optical Table Displacement Requirements

Detection of Flatness by the Human Eye

Lesser-Known Lab Tools

Yoder's Rules of Thumb

15 Photogrammetry

Introduction

Basic Optical Equations

Triangulation vs. Model-Based Pose Estimation

Stereo vs. Triangulation

Basic Triangulation Equations

Two-Camera Triangulation Accuracy

Triangulation Error Tree

Sensor Placement for Triangulation

Maximum Triangulation Range

Triangulation Equations for Cameras

Azimuth Corrections for Euclidean Coordinate Systems

Model-Based Pose Estimation: Number of Points and Spatial Distribution

From What Point in an Optical System Is Range Measured?

16 Radiometry

Introduction

The Electromagnetic Spectrum

Photons-to-Watts Conversion

Brightness of Common Sources

The Blackbody Equation

Logarithmic Blackbody Functions

Narrow-band Approximation to Planck's Law

Peak Wavelength of Wien's Displacement Law

Choice of Waveband

Lambert's Law

Etendue

In-Band Solar Irradiance at the Top of the Atmosphere

Rule of 4(f/#)²

Relationship between Minimum Resolvable Temperature and Noise-Equivalent Temperature Difference

Ideal NEΔT Simplification

Cavity Emissivity

Incorrectly Sizing Radiometric Areas

No Ice Cream Cones

Calibrate under User's Conditions for Best Results

Radiometry of a Spherical Cow

Dependence on R₀A

Quick Test of NEΔT

17 Systems

Introduction

Pick Any Two

Divide by the Number of Visits

Dawes Limit

BLIP Limiting Rule

Rayleigh Criterion

Focal Length and Resolution

Diffraction Limit in LWIR

Procedures to Reduce Narcissus Effects

System Off-Axis Optical Rejection

Signal-to-Noise Ratio for Different Targets

Simplified Range Equation

General Image Quality Equation

Mechanical Shock Response

Estimating rms Acceleration Response due to Random Vibrations

Typical Values of Electro-optical System Parameters

Vibration Isolation

Wind Loading

18 Target Phenomenology

Introduction

Emissivity Approximations

Solar Reflection Always Adds to the Signature

Lambertian vs. Specular

Bidirectional Reflectance Distribution Function

Hagen-Rubens Relationship for the Reflectivity of Metals

Causes of White Pigment's Color

Human Body Signature

Infrared Skin Characteristics

Jet Plume Phenomenology

Plume Thrust Scaling

Rocket Plume Rules

Temperature as a Function of Aerodynamic Heating

Laser Cross-Section

Chlorophyll Absorptance

Normalized Difference Vegetation Index

Appendix

Glossary

Tables of Numerical, Physical, and Material Properties (or Other Information)

Properties of Infrared Materials

Thermal and Structural Properties of Materials

CIE Chromaticity Diagram

Basic Equations

Blackbody Nomograph

Nomoscope of Telescope Performance

Azimuth and Elevation Conventions

Photonic Noise Sources

Guidelines for Writing SI Units

Derivation of the Third Equation in the Rule "The Relation of Ideal D* to View Angle"

Index

Preface

The evolution of the photonic sciences parallels, and feeds from, developments in a number of somewhat unrelated fields, including astronomy, satellite and remote sensing technology, materials science, electronics, biomedical sciences, optical communications, military developments, and many others. The common thread of all of this effort, which was defined in the 1950s, is that scientists and engineers have been able to combine highly successful electronic technologies with the more ancient concepts and methods of optics and electromagnetic wave propagation. The merging of these fields has provided an unprecedented capability for instruments to "see" targets and communicate with them in a wide range of wavelengths for the benefit of security systems, science, defense, and (more recently) consumers. In the future, we see the rise of autonomous systems as a sea change that will drive a significant increase in the need for sensing systems to allow the autonomous system to sense and understand its environment.

Major departments at universities are now devoted to producing new graduates with specialties in this field. There is no end in sight for the advancement of these technologies, especially with the continued development of electronics and computing as increasingly integral parts of photonic instrumentation. One of the disturbing trends in this technology is the constant narrowing of the role of engineers. As the technology matures, it becomes more difficult for anyone working in an area of photonics to understand all that is being done in the related sciences and engineering. This book has been assembled to make a first, small step to expose anyone working in the optics and photonics community to a wide range of critical topics through simple calculations and explanations.

There is no intent to compete with classic texts or the many journals or conferences devoted to the photonics field, all of which provide considerable detail in every area. Rather, this book is intended to allow any engineer or scientist, regardless of specialty, to make rapid and accurate guesses at solutions in a wide range of topics that might be encountered in system design, modeling, or fabrication, as well as to provide a guide for choosing which details to consider more diligently. This book will help any electro-optics (EO) team to make quick assessments, generally requiring no more than a calculator, so that they quickly find the right solution for a design problem.

The book is also useful for managers, marketeers, and other semi-technical folks who are new to the optics industry (or are on its periphery) to develop a feel for the difference between the chimerical and the real. Students may find the same type of quick-calculation approach valuable, particularly in the case of oral exams in which the professor is pressuring the student to do a complex problem quickly. Using these assembled rules, you can keep your wits about you and provide an immediate and nearly correct answer, which usually will save the day. But after the day is saved, you should go back to the question and perform a rigorous analysis. These rules are useful for quick sanity checks and basic relationships. Being familiar with the rules allows one to rapidly pinpoint trouble areas or ask probing questions in meetings. They aid in thinking on your feet and in developing a sense of what will work and what won't. Another potential application of the contents is to provide a checklist for reviewers asked to assess the completeness of a design or resolve trade studies early in the development of a system. But they are not, and never will be, the last word.

Dear reader, it is fully recognized that errors may still be present, and for that we apologize in advance to you and to those from whom the material was derived. We try to the best of our abilities to remove errors we inherit from the references we used and in new material we created. To assist us in this endeavor, we have solicited the cooperation of as many experts as would agree to help. Their input gives us a wide variety of experience from many different technical points of view. Alas, technology advances, and all of us wonder how we can possibly keep up. Hopefully, this book will not only provide some specific ideas related to photonics technology, it will also suggest some ways of thinking about things that will lead to a whole new generation of such rules and ideas.

As we discovered with the previous editions of this book, not everyone has the same thing in mind when considering "a rule of thumb." To qualify for our definition of a rule of thumb, a rule should be useful to a practitioner and possess at least most of the following attributes:

• * It should be easy to implement.
• * It should provide roughly the correct answer.
• * The main points should be easy to remember.
• * It should be simple to express.
• * It should highlight the important variables and diminish the role of generally unimportant variables.
• It should provide useful insight to the workings of the subject matter.

As in earlier editions of books in this series, we found it valuable to create a detailed standard form and stick to it as closely as possible. We did in this edition as well. References are provided whenever possible. In addition, reference material is mentioned that can be considered as recommended reading for the reader with a desire for more detail than could be presented in the "rule" and "discussion." The reader should note that each rule "stands on its own," so the abbreviations and terminology may not be entirely consistent throughout. This is intentional; we use the notation of the reference whenever we can so that if you read the original material you will recognize what that author defined. Some rules are derived from the laws of physics, and some represent existing technology trends. Many derive from observations made by researchers in the field, augmented by curve fitting that results in polynomial approximations.

The authors of the previous versions of this book arrived at the same place by very different paths. John (now retired, but busy consulting) spent some of his career in infrared astronomy before joining the aerospace industry to work on infrared sensors for space surveillance. He later worked on tactical sensors for search-and-rescue, self-driving cars, active imaging and enhanced vision systems. Ed (now retired, but very busy teaching) spent most of his career working on remote sensing technologies applied to Earth, its atmosphere and oceans, and, more recently, astronomical instruments and advanced pointing systems. John and Ed met in Denver in 1985, both working for a major government contractor on exotic electro-optical systems.

Those were halcyon days, with money flowing like water, and contractors winning billions of dollars for some concepts that were overly optimistic or barely possible at best. In the center of the whole fray were bureaucrats, politicians, and managers who were demanding that we design systems that would be capable of the impossible. We saw many requirements and goals being levied on our systems that were far from realistic, often resulting from confusing (and poorly understood) interpretations of the capabilities of optical and electro-optical systems and the properties of targets or backgrounds. We found a common ground when managers discovered that many co-workers, in an attempt to outdo the competition, were promising to perform sensor demonstrations that violated many rules of engineering, if not physics. On one multibillion-dollar program, after some consistent exposure to neophytes proposing all sorts of undoable things, we decided to try to educate everyone by creating a half-serious, half-humorous posting for the local bulletin board (this was before websites were ubiquitous) called "Dr. Photon’s Rules of Thumb." It was a list of basic rules that apply when optics or electro-optics are used. Figure 1 shows the first version that we found all across the company, and even among competitors.

For the current version, John and Ed invited (maybe some would say tricked) three other younger authors to modernize, enhance, and yield a new perspective. These luminaries are Jack Sanders-Reed,Brian McComas, and Katie Schwertz, whose bios appear in the back matter of this book.

Katie is a hybrid optical and optomechanical designer, drawing from her time at both the University of Rochester and University of Arizona optics programs. She primarily works on commercial and industrial optical subsystems. Her time spent under the tutelage of Jim Burge in Arizona taught her the value of a good estimation, which she didn’t fully appreciate until her time spent in industry. Her Master’s work involved optomechanics "rules of thumb." Jack has 40 years of experience ranging from medical imaging, to surface science, to pilot vision systems and atmospheric phenomenology, to target detection, tracking, and photogrammetry and covering the electromagnetic spectrum from hard x-ray through visible and infrared to millimeter-wave with both passive and active imaging. Brian has over 30 years of experience working on EO systems for military, astronomical, remote sensing, and industrial use. His Ph.D. was developed under the direction of Ed Friedman.

To summarize, this collection of rules and concepts represents an evolving, idiosyncratic, and eclectic toolbox. The rules, like tools, are neither good nor bad; they can be used to facilitate the transition of whimsical concepts to mature hardware or to immediately identify a technological path worth pursuing. Conversely, misused, they can obfuscate the truth and, if improperly applied, derive incorrect answers. Our job was to refine complex ideas to simplified concepts and present these rules to you with appropriate cautions. However, it is your responsibility to use them correctly. Remember, it is a poor worker who blames their tools and we hope you will find these tools useful.

John L. Miller
Edward Friedman
Jack Sanders-Reed
Katie Schwertz
Brian McComas
March 2020