Library of Congress Cataloging-in-Publication Data

Smith, Daniel Gene.

Field guide to physical optics / Daniel G. Smith.

pages cm. – (The field guide series ; FG17)

Includes bibliographical references and index.

ISBN 978-0-8194-8548-9

1. Physical optics. I. Title.

QC395.2.S65 2013

535'.2–dc23

2013000915

Published by

SPIE

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Bellingham, Washington 98227-0010 USA

Phone: +1.360.676.3290

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The content of this book reflects the work and thought of the author. 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. For the latest updates about this title, please visit the book’s page on our website.

Printed in the United States of America.

First printing

## Introduction to the Series

Welcome to the * SPIE Field Guides*—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the

*is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.

The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at

**fieldguides@SPIE.org.**

John E. Greivenkamp, **Series Editor**

College of Optical Sciences

The University of Arizona

## The Field Guide Series

Keep information at your fingertips with all of the titles in the Field Guide Series:

* Adaptive Optics, Second Edition*, Robert Tyson & Benjamin Frazier

* Atmospheric Optics*, Larry Andrews

* Binoculars and Scopes*, Paul Yoder, Jr. & Daniel Vukobratovich

* Diffractive Optics*, Yakov Soskind

* Geometrical Optics*, John Greivenkamp

* Illumination*, Angelo Arecchi, Tahar Messadi, & John Koshel

* Image Processing*, Khan M. Iftekharuddin & Abdul Awwal

* Infrared Systems, Detectors, and FPAs*,

*, Arnold Daniels*

**Second Edition*** Interferometric Optical Testing*, Eric Goodwin & Jim Wyant

* Laser Pulse Generation*, Rediger Paschotta

* Lasers*, Rüdiger Paschotta

* Lens Design*, Julie Bentley & Craig Olson

* Microscopy*, Tomasz Tkaczyk

* Optical Fabrication*, Ray Williamson

* Optical Fiber Technology*, Rudiger Paschotta

* Optical Lithography*, Chris Mack

* Optical Thin Films*, Ronald Willey

* Optomechanical Design and Analysis*, Katie Schwertz & James Burge

* Physical Optics*, Daniel Smith

* Polarization*, Edward Collett

* Probability, Random Processes, and Random Data Analysis*, Larry Andrews

* Radiometry*, Barbara Grant

* Special Functions for Engineers*, Larry Andrews

* Spectroscopy*, David Ball

* Terahertz Sources, Detectors, and Optics*, Creidhe O’Sullivan & J. Anthony Murphy

* Visual and Ophthalmic Optics*, Jim Schwiegerling

## Field Guide to Physical Optics

Physical optics is a broad subject that has been in vigorous and continuous development for more than a century. It can be thought of as encompassing all optics, except possibly ray optics, but it may also be regarded as a subset of physical phenomena described by electromagnetic optics.

This Field Guide is a practical overview of the subject area, with specific emphasis on information most useful in the field of optical engineering. Within this Field Guide, the reader will find formulae and descriptions of basic electromagnetic wave phenomena that are fundamental to a wave theory of light. Tools are provided for describing polarization. And, although vector diffraction theory and electromagnetic methods (e.g., FDTD and RCWA) are not treated here, emphasis is placed on scalar diffraction and imaging theory, which are essential in solving most practical optical engineering problems.

I owe thanks to the various professors who first taught me these subjects: Jack Kasher and Daniel Wilkins at the University of Nebraska at Omaha; and then later at the University of Arizona College of Optical Sciences: Arvind S. Marathay, Jack D. Gaskill, Roland V. Shack, James C. Wyant, and John E. Greivenkamp. I also owe thanks to Tom D. Milster, who graciously allowed me to review his soon-to-be published text and adopt portions of his notation. Thanks also go to Kerry McManus Eastwood for her extensive help in preparing this volume and to Eric P. Goodwin for his help in reviewing the material.

Finally, I dedicate this Field Guide to my wife, Jenny, who is always there to keep me coherent and in phase.

Daniel G. Smith

Nikon Research Corporation of America

January 2013

## Table of Contents

**Glossary of Terms and Acronyms x**

**Electromagnetic Waves 1**

Maxwell’s Equations and the Wave Equations 1

Principle of Linear Superposition and Complex Notation 2

The Helmholtz Equation 3

Plane Wave 4

Spherical and Cylindrical Waves 5

Speed of Light 6

The Beer—Lambert Law 7

Increasing versus Decreasing Phase Convention 8

**Polarization 9**

Linear Polarization 9

Right- and Left-Hand Circular Polarization 10

Elliptical Polarization 11

Elliptical Polarization Handedness 12

Poincaré Sphere 13

Jones Vectors 14

Jones Matrices and Eigenpolarizations 15

Jones Rotation and Reflection Matrices 16

Retardance 17

Dichroism and Diattenuation 19

Polarizers and Malus’ Law 20

Optical Activity 21

Stokes Vectors and Degree of Polarization 22

Mueller Matrices 23

Mueller Matrices and Rotation 24

**Interference 25**

Poynting Vector, Irradiance, and Optical Admittance 25

Plane of Incidence 26

Stokes Relations 27

Airy Formulae and Airy Function 28

Airy Function and Finesse 29

Fresnel Equations 30

Coefficient of Reflection 31

Reflectance and Transmittance 32

Laws of Refraction and Reflection 33

Phase Difference between Parallel Reflections 34

Characteristic Matrix of Thin Films 35

Reflectance and Transmittance of Thin Films 36

Superposed Plane Waves 37

Interference of Two Plane Waves (Different Frequency) 38

Interference of Two Plane Waves (Same Frequency) 39

Phase Velocity and Group Velocity 40

Interference of Two Plane Waves (3D) 41

Fringe Visibility / Modulation / Contrast 42

Interference of Two Polarized Plane Waves 43

Grating Equation 44

Interference of Two Spherical Waves 45

**Scalar Theory of Diffraction 46**

Huygens’ and Huygens—Fresnel Principles 46

Fresnel Diffraction 47

On-Axis Irradiance behind a Circular Aperture 48

Fresnel Zone Plate 49

Integral Theorem of Helmholtz and Kirchhoff 50

Sommerfeld Radiation and Kirchhoff Boundary Conditions 51

Fresnel—Kirchhoff Diffraction Integral 52

Rayleigh—Sommerfeld Diffraction Integral 53

Boundary Conditions and Obliquity Factors 54

Fresnel Diffraction Formula 55

Fresnel Diffraction between Confocal Surfaces 56

Fraunhofer Diffraction Formula 57

Huygens’ Wavelet 58

Angular Spectrum of Plane Waves 59

Transfer Function of Free Space 60

Method of Stationary Phase 61

Talbot Images 62

Babinet’s Principle 63

Fresnel Diffraction by a Rectangular Aperture 64

Cornu Spiral 65

**Imaging 67**

Propagation through a Lens 67

Airy Disk 68

Double-Telecentric Imaging System 69

Linear and Shift-Invariant Imaging System 70

Coherent and Incoherent Point Spread Function 71

PSF for Rectangular and Circular Apertures 72

Transfer Function 73

Coherent Transfer Function (CTF) 74

Incoherent Transfer Function and the Optical Transfer Function 75

Strehl Ratio 76

Properties of the OTF and MTF 77

CTF and OTF of a Circular Aperture 78

CTF and OTF of a Rectangular Aperture 79

Coherent and Incoherent Cutoff Frequency 80

Rayleigh Criterion 81

**Gaussian Beams 82**

Rotationally Symmetric Gaussian Beams 82

Gaussian Beam Size 84

Rayleigh Range and Sister Surfaces 85

Gouy Shift and Wavefront Curvature 86

ABCD Method for Gaussian Beams 87

**Coherence Theory 88**

Young’s Double Pinhole 88

Mutual Coherence Function 89

Spatial Coherence: Mutual Intensity 90

Van Cittert—Zernike Theorem 91

Temporal Coherence 92

Coherence Length 93

Coherence Length for Simple Spectra 94

Fabry—Pérot Interferometer 95

Fabry—Pérot Spectrometer 96

**Appendix 97**

Special Functions and Fourier Transforms 97

**Equation Summary 98**

**Bibliography 110**

**Index 111**

#### Glossary

* a_{x}*,

*,*

**a**_{y}

**a**_{z}Field amplitudes in * x*,

*, and*

**y***directions*

**z****A**

Scalar optical field amplitude

**A**

Vector optical field amplitude

**A _{c}**

Coherence area

* A*,

*,*

**B***,*

**C**

**D**Elements of a lens system matrix

**B**

Magnetic induction

**c**

Speed of light

CTF

Coherent transfer function

* C*(

*),*

**x***(*

**S***)*

**x**Fresnel cosine and sine integrals

**D**

Diattenuation

**D**

Electric displacement

DOP

Degree of polarization

**e**

Base of the natural logarithm

**e**

Eccentricity

**E**

Electric field magnitude (scalar quantity)

**E**

Electric field

**E**

Jones vector

**f**

Lens focal length

**F**

Coefficient of finesse

𝔉

Finesse

F/#

* F-*number

FWHM

Full-width at half-max

G

Green’s function

**h**

Point spread function

**h**

Gaussian beam half-width at half-max

${h}_{z}^{F}$

Fresnel’s wavelet

${h}_{z}^{H}$

Huygens’ wavelet

**H**

Transfer function

**H**

Magnetic field strength

HLP

Horizontal linear polarization

HWP

Half wave plate

**i**

Imaginary number $\sqrt{}-1$

**I**

Irradiance

j

Electric current density

**J**

Jones matrix

**J _{n}**

N^{th} order Bessel function of the first kind

**k**

Wavenumber

**k**

Wavevector

**K**

Obliquity factor

**L**

Linear operator

LCP

Left-hand circular polarization

LP

Linear polarization/polarizer

LSI

Linear and shift invariant

m

Order of diffraction

**M**

Mueller matrix/characteristic matrix

**M**_{j}

Characteristic matrix of the **j**^{th} film layer

MTF

Modulation transfer function

**n**

Index of refraction (real)

**n**

Surface normal

**N**

Complex refractive index

NA

Numerical aperture

**N _{j}**

Number densities

N_{f}

Fresnel number

OFZ

Open Fresnel zone

OPD

Optical path difference

OTF

Optical transfer function

**P**

Pupil function

PER

Polarization extinction ratio

PSF

Point spread function

**q**

Complex curvature of a Gaussian beam

**q**

Grating vector

QWP

Quarter wave plate

**r**

Scalar position coordinate

* r_{s}*,

**r**_{p}Coefficient of reflection for * s* and

*polarization*

**p****r**

Vector position coordinate

R

Reflectance

ℜ

Resolving power

RCP

Right-hand circular polarization

**s**_{1}, **s**_{2}, **s**_{3}

Object and image distance

**S**_{0}

Magnitude of the Stokes vector

**S**

Stokes vector

**S**

Poynting vector

SR

Strehl ratio

* t_{s}*,

**t**_{p}Coefficient of transmission for * s* and

*polarization*

**p****t _{A}**

Amplitude transparency function

**T**

Transmittance

TE

Transverse electric

TIR

Total internal reflection

TM

Transverse magnetic

**U**

Scalar optical field in complex notation

**U**

Vector optical field in complex notation

**v**

Fringe direction

**V**

Fringe visibility

VLP

Vertical linear polarization

**w**

Gaussian beam 1/**e**^{2} radius

**w**_{0}

Gaussian beam waist radius

* x*,

*,*

**y**

**z**Position coordinates

$\widehat{x},\widehat{y},\widehat{z}$

Unit vectors in the * x*,

*, and*

**y***directions*

**z****y**

Optical admittance

**z**_{0}

Rayleigh range

α

Absorption coefficient

α, β, γ

Direction cosines

γ(**r**_{1}, **r**_{2}, * τ*)

Complex degree of coherence

Γ(**r**_{1}, **r**_{2}, * τ*)

Mutual coherence function

δ

Path difference or a small angle

∇

Laplacian operator

Δ**l**

Coherence length

Δλ

Range of or change in wavelengths

Δ**v**

Range of or change in frequency

Δ**t**

Coherence time

ε, ε_{0}

Permittivity and permittivity of free space

η

Tilted or effective optical admittance

θ

An angle

κ

Extinction coefficient

λ

Wavelength

$\overline{\text{\lambda}}$

Mean wavelength

λ_{1}, λ_{2}

Amplitude transmission of eigenpolarizations

Λ

Grating period

μ, μ_{0}

Permeability and permeability of free space

ν

Optical frequency

ξ,η

Spatial frequency components

ρ

Electric charge density

σ

Electric conductivity

Σ

Aperture transmission function

τ

Delay time

ϕ

Lens power

ϕ

Phase angle

Φ

Radiant flux

ψ_{1}, ψ_{2}

Eigenpolarizations

ω

Angular optical frequency

ϖ

Mean angular optical frequency