Handbook of optical engineering

Single-Mode Fiber Optics: Principles and Applications, Second Edition, Re vised and Expanded, Luc B.. Electron and Ion Microscopy and Microanalysis: Principles and Applications, Second E

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Founding Editor

Brian J Thompson

University of Rochester Rochester, New York

Imperial College of Science,

Technology, and Medicine

London, England

Bahram Javidi

University of ConnecticutStorrs, Connecticut

Lawrence E Murr

2 Acousto-Optic Signal Processing: Theory and Implementation, edited by Nor

man J Berg and John N Lee

3 Electro-Optic and Acousto-Optic Scanning and Deflection, Milton Gottlieb, Clive

L M Ireland, and John Martin Ley

4 Single-Mode Fiber Optics: Principles and Applications, Luc B Jeunhomme

5 Pulse Code Formats for Fiber Optical Data Communication: Basic Principles and

Applications, David J Morris

6 Optical Materials: An Introduction to Selection and Application, Solomon

Musikant

7 Infrared Methods for Gaseous Measurements: Theory and Practice, edited by

Joda Wormhoudt

8 Laser Beam Scanning: Opto-Mechanical Devices, Systems, and Data Storage

Optics, edited by Gerald F Marshall

9 Opto-Mechanical Systems Design, Paul R Yoder, Jr.

10 Optical Fiber Splices and Connectors: Theory and Methods, Calvin M Miller with

Stephen C Mettler and Ian A White

11 Laser Spectroscopy and Its Applications, edited by Leon J Radziemski, Richard

W Solarz, and Jeffrey A Paisner

12 Infrared Optoelectronics: Devices and Applications, William Nunley and J Scott

Bechtel

13 Integrated Optical Circuits and Components: Design and Applications, edited by

Lynn D Hutcheson

14 Handbook of Molecular Lasers, edited by Peter K Cheo

15 Handbook of Optical Fibers and Cables, Hiroshi Murata

16 Acousto-Optics, Adrian Korpel

17 Procedures in Applied Optics, John Strong

18 Handbook of Solid-State Lasers, edited by Peter K Cheo

19 Optical Computing: Digital and Symbolic, edited by Raymond Arrathoon

20 Laser Applications in Physical Chemistry, edited by D K Evans

21 Laser-Induced Plasmas and Applications, edited by Leon J Radziemski and

David A Cremers

22 Infrared Technology Fundamentals, Irving J Spiro and Monroe Schlessinger

23 Single-Mode Fiber Optics: Principles and Applications, Second Edition, Re vised

and Expanded, Luc B Jeunhomme

24 Image Analysis Applications, edited by Rangachar Kasturi and Mohan M Trivedi

25 Photoconductivity: Art, Science, and Technology, N V Joshi

26 Principles of Optical Circuit Engineering, Mark A Mentzer

27 Lens Design, Milton Laikin

28 Optical Components, Systems, and Measurement Techniques, Rajpal S Sirohi

and M P Kothiyal

29 Electron and Ion Microscopy and Microanalysis: Principles and Applications,

Second Edition, Revised and Expanded, Lawrence E Murr

30 Handbook of Infrared Optical Materials, edited by Paul Klocek

31 Optical Scanning, edited by Gerald F Marshall

32 Polymers for Lightwave and Integrated Optics: Technology and Applications,

edited by Lawrence A Hornak

33 Electro-Optical Displays, edited by Mohammad A Karim

34 Mathematical Morphology in Image Processing, edited by Edward R Dougherty

35 Opto-Mechanical Systems Design: Second Edition, Revised and Expanded, Paul

R Yoder, Jr.

36 Polarized Light: Fundamentals and Applications, Edward Collett

37 Rare Earth Doped Fiber Lasers and Amplifiers, edited by Michel J F Digonnet

38 Speckle Metrology, edited by Rajpal S Sirohi

S Weiss

40 Photonic Switching and Interconnects, edited by Abdellatif Marrakchi

41 Design and Fabrication of Acousto-Optic Devices, edited by Akis P Goutzoulis

and Dennis R Pape

42 Digital Image Processing Methods, edited by Edward R Dougherty

43 Visual Science and Engineering: Models and Applications, edited by D H Kelly

44 Handbook of Lens Design, Daniel Malacara and Zacarias Malacara

45 Photonic Devices and Systems, edited by Robert G Hunsberger

46 Infrared Technology Fundamentals: Second Edition, Revised and Expanded,

edited by Monroe Schlessinger

47 Spatial Light Modulator Technology: Materials, Devices, and Applications, edited

by Uzi Efron

48 Lens Design: Second Edition, Revised and Expanded, Milton Laikin

49 Thin Films for Optical Systems, edited by Francoise R Flory

50 Tunable Laser Applications, edited by F J Duarte

51 Acousto-Optic Signal Processing: Theory and Implementation, Second Edition,

edited by Norman J Berg and John M Pellegrino

52 Handbook of Nonlinear Optics, Richard L Sutherland

53 Handbook of Optical Fibers and Cables: Second Edition, Hiroshi Murata

54 Optical Storage and Retrieval: Memory, Neural Networks, and Fractals, edited by

Francis T S Yu and Suganda Jutamulia

55 Devices for Optoelectronics, Wallace B Leigh

56 Practical Design and Production of Optical Thin Films, Ronald R Willey

57 Acousto-Optics: Second Edition, Adrian Korpel

58 Diffraction Gratings and Applications, Erwin G Loewen and Evgeny Popov

59 Organic Photoreceptors for Xerography, Paul M Borsenberger and David S.

Weiss

60 Characterization Techniques and Tabulations for Organic Nonlinear Optical

Materials, edited by Mark G Kuzyk and Carl W Dirk

61 Interferogram Analysis for Optical Testing, Daniel Malacara, Manuel Servin, and

Zacarias Malacara

62 Computational Modeling of Vision: The Role of Combination, William R Uttal,

Ramakrishna Kakarala, Spiram Dayanand, Thomas Shepherd, Jagadeesh Kalki, Charles F Lunskis, Jr., and Ning Liu

63 Microoptics Technology: Fabrication and Applications of Lens Arrays and

De-vices, Nicholas Borrelli

64 Visual Information Representation, Communication, and Image Processing,

edited by Chang Wen Chen and Ya-Qin Zhang

65 Optical Methods of Measurement, Rajpal S Sirohi and F S Chau

66 Integrated Optical Circuits and Components: Design and Applications, edited by

Edmond J Murphy

67 Adaptive Optics Engineering Handbook, edited by Robert K Tyson

68 Entropy and Information Optics, Francis T S Yu

69 Computational Methods for Electromagnetic and Optical Systems, John M.

Jarem and Partha P Banerjee

70 Laser Beam Shaping, Fred M Dickey and Scott C Holswade

71 Rare-Earth-Doped Fiber Lasers and Amplifiers: Second Edition, Revised and

Expanded, edited by Michel J F Digonnet

72 Lens Design: Third Edition, Revised and Expanded, Milton Laikin

73 Handbook of Optical Engineering, edited by Daniel Malacara and Brian J.

Thompson

74 Handbook of Imaging Materials: Second Edition, Revised and Expanded, edited

by Arthur S Diamond and David S Weiss

75 Handbook of Image Quality: Characterization and Prediction, Brian W Keelan

77 Optical Switching/Networking and Computing for Multimedia Systems, edited by

Mohsen Guizani and Abdella Battou

78 Image Recognition and Classification: Algorithms, Systems, and Applications,

edited by Bahram Javidi

79 Practical Design and Production of Optical Thin Films: Second Edition, Revised

and Expanded, Ronald R Willey

80 Ultrafast Lasers: Technology and Applications, edited by Martin E Fermann,

Almantas Galvanauskas, and Gregg Sucha

81 Light Propagation in Periodic Media: Differential Theory and Design, Michel

82 Handbook of Nonlinear Optics, Second Edition, Revised and Expanded,

Richard L Sutherland

Additional Volumes in Preparation

Optical Remote Sensing: Science and Technology, Walter Egan

This comprehensive and cohesive work includes all the relevant data to allow opticalengineers worldwide to meet present and upcoming challenges in their day-to-dayresponsibilities The thrust of the Handbook of Optical Engineering is towardengineering and technology rather than theoretical science

The book has 26 chapters that cover most but not all topics in optics, beginningwith a few chapters describing the principles of optics elements These are followed

by more technical and applied chapters

All authors prepared their chapters with the following criteria in mind:Descriptions are restricted to explaining principles, processes, methods, andprocedures in a concise and practical way so that the reader can easilyapply the topics discussed Fundamental descriptions and a how-to-do-itapproach are emphasized

Useful formulas are provided wherever possible, along with step-by-step,worked-out examples, as needed, to illustrate applications and clarify calcu-lation methods Formulas are arranged in the best sequence for use on acomputer or calculator

The book is replete with tables, flow charts, graphs, schematics and line drawings inthe tradition of useful reference books and major handbooks National and ISOstandards are included where appropriate, and permitted, in suitable abridgementfor useful reference Overlapping among different chapters has been avoided unlessabsolutely necessary

Daniel MalacaraBrian J Thompson

iii

3.5 Ultrashort Optical Pulses 105

Daniel Malacara and Duncan T Moore

7.8 Indirect Ophthalmoscopes and Fundus Camera 201

Gonzalo Paez and Marija Strojnik

11.2 Coherence of Light Sources for Interferometers 340

11.10 Talbot Interferometer and Moire´ Deflectometry 36111.11 Foucault Test and Schlieren Techniques 364

12.7 Extended Range Fringe Pattern Analysis 40812.8 Applicability of Fringe Analysis Methods 419

13 Optical Methods in Metrology: Point Methods 427

H Zacarı´as Malacara and Ramo´n Rodrı´guez-Vera

16.4 Wavelet Transform Processing 578

18.2 Standard Terminology for Radiometric Quantities 651

18.5 Radiative Power Transfer in a Three-Dimensional Space 663

21 Spatial and Spectral Filters 743Angus Macleod

Luis Efraı´n Regalado and Daniel Malacara

23.7 Optical Coatings and Other Amorphous Materials 844

24.5 Application of Electric Fields: Induced Birefringence

25 Light-Sensitive Materials: Silver Halide Emulsions,

Sergio Calixto and Daniel J Lougnot

25.3 Silver Halide Emulsions for Holography 890

Vicente Aboites Centro de Investigaciones en Optica, Leo´n, Mexico

Sofia E Acosta-Ortiz Centro de Investigaciones en Optica, Aguascalientes, MexicoDavid Anderson Rayleigh Optical Corporation, Tucson, Arizona

Glenn D Boreman University of Central Florida, Orlando, Florida

Jim Burge The University of Arizona, Tucson, Arizona

Sergio Calixto Centro de Investigaciones en Optica, Leo´n, Mexico

Alberto Cordero-Davila Centro de Investigaciones en Optica, Leo´n, MexicoAlejandro Cornejo-Rodriguez Centro de Investigaciones en Optica, Leo´n, MexicoDennis H Goldstein Air Force Research Laboratory, Eglin AFB, Florida

Mohammad A Karim The City College of the City University of New York, NewYork, New York

Malgorzata Kujawinska Institute of Precise and Optical Instruments TechnicalUniversity, Warsaw, Poland

Daniel J Lougnot UMR CNRS, Mulhouse, France

Daniel Malacara Centro de Investigaciones en Optica, Leo´n, Mexico

Daniel Malacara, Jr Centro de Investigaciones en Optica, Leo´n, Mexico

xiii

H Zacarias Malacara Centro de Investigaciones en Optica, Leo´n, Mexico

Angus Macleod Thin Film Center, Inc., Tucson, Arizona

Duncan T Moore University of Rochester, Rochester, New York

Gonzalo Paez Centro de Investigaciones en Optica, Leo´n, Mexico

Luis Efrain Regalado Centro de Investigaciones en Optica, Leo´n, Mexico

Ramon Rodriguez-Vera Centro de Investigaciones en Optica, Leo´n, MexicoManuel Servin Centro de Investigaciones en Optica, Leo´n, Mexico

Cristina Solano Centro de Investigaciones en Optica, Leo´n, Mexico

A N Starodumov Centro de Investigaciones en Optica, Leo´n, Mexico

Orestes Stavroudis Centro de Investigaciones en Optica, Leo´n, Mexico

Marija Strojnik Centro de Investigaciones en Optica, Leo´n, Mexico

Chandra S Vikram The University of Alabama in Huntsville, Huntsville, AlabamaWilliam Wolfe The University of Arizona, Tucson, Arizona

Francis T S Yu The Pennsylvania State University, University Park, Pennsylvania

The basic elements of geometrical optics are rays and wavefronts: neither exist,except as mathematical abstractions A ray can be thought of as a beam of light with

an finitesimal diameter However, to make a ray experimentally by passing lightthrough a very small aperture causes diffraction to rear its ugly head and the lightspreads out over a large solid angle The result is not a physical approximation to aray but a distribution of light in which the small aperture is a point source Awavefront is defined as a surface of constant phase to which can be attributeddefinite properties such as principal directions, principal curvatures, cusps, andother singularities But, like the ray, the wavefront cannot be observed Its existencecan only be inferred circumstantially with interferometric methods

However there is in geometrical optics an object that is observable and surable: the caustic surface [1] It can be defined in distinct but equivalent ways: As the envelope of an orthotomic system of rays; i.e., rays ultimately from asingle object point

mea-1

As the cusp locus of a wavefront train, or, equivalently, the locus of pointswhere the element of area of the wavefront vanishes.

I think the most useful definition is that the caustic is the locus of principal centers ofcurvature of a wavefront In general, every surface has two principal curvatures ateach of its points This definition then shows clearly that the caustic is a two-sheetedsurface

1.2 GAUSSIAN OPTICS A` LA MAXWELL

Usually the formulas of Gaussian optics are derived from paraxial optics, a systembased on approximations to the equations for ray tracing These we will encounter in

a subsequent section Maxwell, on the other hand, took a global approach He used

a model of a perfect optical instrument and from that model, in a very elegant butstraightforward way, deduced its properties, defined its parameters, and derived thevarious equations associated with Gaussian optics Gauss actually found the equa-tions for paraxial optics from the first-order terms of two power series expansions.While this is not a forum appropriate for a detailed discussion of the methodMaxwell used, I will present an outline of his argument

Maxwell [2] began by assuming that a perfect lens maps each point in objectspace into one and only one point in image space Since a lens turned around is still alens, the inverse of this mapping has to have exactly the same mathematical struc-ture Included in this mapping and its inverse are points at infinity whose images arethe focal points of the instrument

The mapping that Maxwell chose is the linear fractional transformation,

in the following way Suppose a plane in object space is given by the equation,

into which we substitute ðx; y; zÞ from Eq (1.2) The result is

ðpA1þ qA2þ rA3þ sAÞx0þ ðpB1þ qB2þ rB3þ sBÞy0

þ ðpC1þ qC2þ rC3þ sCÞz0þ ðpD1þ qD2þ rD3þ sDÞ ¼ 0; ð1:4Þclearly the equation of a plane in image space that is evidently the image of the plane

in object space

This transformation, therefore, maps planes into planes Since a straight linecan be represented as the intersection of two planes, it follows that this transformmaps straight lines into straight lines

From Eq (1.1) we can see that the plane in object space, ax þ by þ cz þ d ¼ 0

is imaged at infinity in object space; from Eq (1.2), infinity in object space is imagedinto the plane Ax0þ By0þ Cz0þ D ¼ 0, in image space

We have established coordinate systems in both object and image space Now

we impose conditions on the coefficients that bring the coordinate axes into spondence First we look at a plane through the coordinate origin of object spaceperpendicular to the z-axis, Eq (1.3), with r ¼ s ¼ 0, as its equation From this, and

corre-Eq (1.4), we obtain the equation of its image,

ðpA1þ qA2Þx0þ ðpB1þ qB2Þy0þ ðpC1þ qC2Þz0þ pD1þ qD2¼ 0:

For this plane to pass through the image space coordinate origin and be cular to the z0-axis, the coefficient of z0and the constant term must vanish identi-cally, yielding

Again using Eq (1.3), by setting q ¼ 0, we get the equation of a plane perpendicular

to the y-axis whose image, from Eq (1.4), is

ðpA1þ rA3þ sAÞx0þ ðpB1þ rB3þ sBÞy0þ ðrC3þ sCÞx0þ rD3þ sDÞ ¼ 0:For this to be perpendicular to the y0-axis the coefficient of y0 must equal zero,yielding

The final step in this argument involves a plane perpendicular to the x-axis, obtained

by setting p ¼ 0 in Eq (1.3) Its image, from Eq (1.4), is

ðqA2þ rA3þ sAÞx0þ ðqB2Þy0þ ðrC3þ sCÞz0þ rD3þ sD ¼ 0:

Now the coefficient of x0must vanish, yielding the last of these conditions,

These conditions assure that the coordinate axes in image space are the images

of those in object space Nothing has been done to change any of the optical ties of this ideal instrument

proper-Substituting these, from Eqs (1.5), (1.6), and (1.7), into Eq (1.2) yields

The plane perpendicular to the z-axis, given by the equation cz þ d ¼ 0, has, asits image, the plane at infinity, as can be seen from Eq (1.12) Therefore, zf ¼
d=c

is the z-coordinate of the focal point of the instrument in object space In exactly thesame way, we can find the z0-coordinate of the focal point in image space is

zf0¼ c3=c, from Eq (1.13) To summarize, we have shown that

Finally, let e equal the distance of an axial point in object space to the firstprincipal point and let e0 be the distance between its conjugate and the secondprincipal point Then it follows that

com-From these results we can find object–image relationships using a graphicmethod In Fig 1.1 the points zf and zf0are the instrument’s foci and zp and zp0 itsprincipal planes Let O be any object point Let OP be a ray parallel to the axis,passing through P Let its extention pass through P0 Since P and P0 lie on theconjugate principal planes the ray in image space must pass through P0 Since thisray is parallel to the axis in object space its image must pass through zf0 These twopoints determine completely this ray in image space Now take a second ray, OzfQ,through the object point O Since it passes through zf it must emerge in image spaceparallel to the axis Since it passes through Q on the principal plane it must also passthrough its image Q0 These two points determine this ray in image space Where thetwo rays cross is I , the image of O.

With this concept we can find a most important third pair of conjugates forwhich the instrument’s angular magnification is unity Then a ray passing throughone of these points will emerge from the instrument and pass undeviated through theother These are the nodal points

Refer now to Fig 1.2 Suppose a ray passes through the axis at z0, at an angle

, and intersects the principal plane at yp After passing through this ideal instrument

it intersects the axis in image space at z00, at an angle
0

, and passes through the

Figure 1.1 Graphical construction of an object–image relationship The points zfand zf0arethe instrument’s foci and zpand zp0its principal planes From object point O, ray OP is parallel

to the axis Since P is on the object principal plane, its image P0 must be at the same height.Image ray must therefore pass through P0zf0 A second ray, Ozf0Q, passes through the objectfocus and therefore must emerge in image space parallel to the axis It must also pass through

Q0, the image of Q The two rays cross at I , the image point

principal plane at yp0 Newton’s formula, Eq (1.20) provides a relationship between

z0 and z00,

Moreover, y1 and y10 are equal, since they represent the heights of conjugate points

on the principal planes From Fig 1.2 we can see that

:With the aid of Eq (1.25) this becomes

Figure 1.2 Graphical construction of the nodal points The points zfand zf0are the two focalpoints and zpand zp0its two principal planes; z0and z00 are the nodal points, where
¼
0and fand f0 are the front and rear focal lengths

With this in mind we make another change of variables: a translation of the axes to place the origins at the two nodal points The new z-coordinates will be g and

z-g0 The change is realized by

no mention of wave fronts, velocities, or refractive indices These characteristics will

be introduced in subsequent sections of this chapter

1.3 THE EIKONAL FUNCTION AND ITS ANTECEDENTS

This subject has a rather odd pedigree It was first discovered by Hamilton, whocalled it the characteristic function Then it was rediscovered by Bruns who dubbed itthe eikonal [3] Its origins lie much earlier The law of refraction, discovered byWillebrord Snell using empirical methods, after his death, came into the hands ofDescartes who derived for it what he claimed to be an analytic proof Fermat dis-agreed In his opinion, Snell’s law was only an approximation and Descartes’ proofwas erroneous He then set out to find the exact formula for refraction But, to hissurprise, Snell’s law was indeed exact

The approach to the derivation that he used has come down to us as Fermat’sprinciple: light consists of a flow of particles, termed corpuscles, the trajectories ofwhich are such that their time of transit from point to point is an extremum, either amaximum or a minimum These trajectories are what we now call rays Fermat’sjustification for this principle goes back to observations by Heron of Alexandria, butthat is the subject of an entirely different story What does concern us here is theinterpretation of his principle in mathematical terms: its representation in terms ofthe variational calculus which deals specifically with the determination of extrema offunctions

To set the stage, let us consider an optical medium in which a point is sented by a vector P ¼ ðx; y; zÞ and in which the refractive index is given by a vectorfunction of position: n ¼ nðPÞ We will represent a curve in this medium by the vectorfunction PðsÞ, where the parameter s is the geometric distance along the curve Itfollows that dP=ds ¼ P0 is a tangent vector to the curve It can be shown that P0is aunittangent vector

repre-Note that

so that

P0 dP ¼ P02ds ¼ ds: ð1:30ÞThe velocity of light in this medium is c=n where c is its velocity in vacuo Thetime of transit between any two points on a ray is therefore given by

iso-It is useful to look at this differential equation from the point of view of thedifferential geometry of space curves We define the unit tangent vector to the ray t,the unit normal vector, n; and the unit binormal, b, as follows: [6]

of n By squaring the expression in Eq (1.37) we obtain a formula for the ray’scurvature,

1

¼ 

2

where ðrnÞ0 represents the derivative of the gradient with respect to s

Figure 1.3 The sliding tetrahedron As a point moves along the space curve the vectors t, n,and b slide along with it The Frenet–Serret equations (Eq (1.38)) describe their rates ofchange

To show how this works, consider Maxwell’s fish eye [8] in which the refractiveindex function is given, in idealized form, by

¼ d

ds P  n

dPds

ds Z  P  n

dPds

where n0¼ nðP0Þ, n1¼ nðP1Þ, P00 ¼ P0jP0, and P10 ¼ P0jP1 The function I , given by

Eq (1.52), is known as Hamilton’s characteristic function or, more simply, the nal, while the equations in Eq (1.53) are Hamilton’s characteristic equations [10] Bysquaring either of the expressions in Eq (1.53) we obtain the eikonal equation,

The eikonal equation can be derived from the Maxwell equations in severaldifferent ways By assuming that a scalar wave equation represents light propaga-tion, along with an application of Huygens’ principle, Kirchhoff obtained a harmo-nic solution (discussed in Chapter 2) for light intensity at a point He showed that theeikonal equation was obtained as a limit as wavelength approached zero Kline andKay [11] give a critique as well as a detailed account of this method

Luneburg, [12] on the other hand, took a radically different approach Hestarted with an integral version of the Maxwell equations, regarded a wave front

as a singularity in the solution of these equations and that radiation transfer sisted of the propagation of these singularities Then he used Huygens’ principle toobtain what we have called the eikonal equation

con-This has led to speculation that geometric optics is a limiting case of physicaloptics as frequency becomes small Perhaps Suffice it to say that the eikonal equa-

tion, since it can be derived from sundry starting points remains a crucial point inoptical theory.

Now suppose the optical medium is discontinuous: that there is a surface Swhere the index of refraction function has a jump discontinuity To fix ideas, assumethat light travels from left to right Choose a point P0 to the left of S and a second,

P1, to its right Unless P0and P1 are conjugates they will be connected by a uniqueray path determined by Fermat’s principle and, moreover, the segments of the raypath will be solutions of Eqs (1.32) or (1.33) Let
PP be the point where this ray pathcrosses the discontinuity S and let n
and nþbe the left- and right-hand limits of therefractive index function along the ray path at
PP For convenience let S ¼ dP=dsrepresent a ray vector and let S
and Sþrepresent the left and right limits of S at thepoint
PP

To best describe the consequences of a discontinuity with mathematical rigorand vigor one should apply the Hilbert integral [13] of the calculus of variations Forour purposes a schoolboy explanation is more appropriate

Joos [14] uses the definition of the gradient in terms of limits of surface grals to define a surface gradient as a gradient that straddles a surface of disconti-nuity so that

be the tive indices

refrac-It turns out that for P and P0to be perfect conjugates the optical path lengthalong any raymust be constant [16] In other words,

½ðn2
n02Þð
xx2þ
yy2þ
zz2Þ þ 2
zzðn2t þ n02t0Þ2

4nn0ðnt þ n0t0Þ½ðn0t þ nt0Þð
xx2þ
yy2þ
zz2Þ þ 2tt0ðn
n0Þ
zz ¼ 0: ð1:60Þ

a quartic surface in the shape of an oval This is shown in Fig 1.4

Kepler found this numerically early in the 17th century Descartes, a tion later, found the mathematical formula And it has been rediscovered over andover again, even by Maxwell, ever since

genera-An interesting (indeed, fascinating) consequence obtains when the object pointapproaches infinity Divide this formula by t2t02 and then let t become large Theresult is

n2
zz2
n02ð
xx2þ
yy2þ
zz2Þ
2n0

t0ðn
n0Þ
zz ¼ 0; ð1:61Þwhich can be rearranged in the form

n0> n and a hyperboloid when n0< n This can be seen in Fig 1.5

Finally we come to the aplanatic surfaces of a sphere [17] Let

t ¼
kð1 þ n0=nÞ; t0¼ kð1 þ n=n0Þ; ð1:63Þand substitute into Eq (1.60) Since nt þ n0t0¼ 0, this degenerates into the equation

of a sphere that passes through the origin and has a radius of k,

Since the refracting sphere has central symmetry, t and t0can be taken as theradii of two other spheres, that are perfect conjugates These are the aplanaticsurfaces and they are shown in Fig 1.6 The aplanatic surfaces were the basis for

a system for tracing meridional rays before the advent of computers

Figure 1.4 The Cartesian oval Point P is imaged perfectly on point P0 Rays coming fromthe left are from the real object P For rays coming from the right, P is a virtual object

1.4 RAY TRACING AND ITS GENERALIZATION

Now we take up the very practical problem of tracing rays in a homogeneous,isotropic medium, a medium in which the refractive index, n, is constant It followsthat its gradient, rn, is zero so that Eq (1.32) becomes

d2P

a second-order differential equation whose solution is a linear function of s; fore, a ray in this medium must be a straight line Note here that in media of constantrefractive index,

where S and S0are the direction cosine vectors of a ray before and after refraction,respectively; where N is the unit normal vector to the refracting surface at the point

of incidence; and where n and n0 are the refractive indices of the media before andafter the refracting surface Note that in the preceding section we used the primesymbol (0) to denote differentiation with respect to the parameter s; here we use it tosignal refraction or (subsequently) transfer

By taking the absolute value of Eq (1.67) we get the more familiar form ofSnell’s law

where i and i0are the angles of incidence and refraction, respectively This statement,unlike its vector form, does not tell the whole story Equation (1.67) provides theadditional information that the vectors S0, S, and N are coplanar and determines theplane of incidence

In what follows we will develop the equations for ray tracing The form ingeneral use today, developed by T Smith over a period of several decades, [18] will

be cast in vector form here [19] An earlier scalar version designed particularly forcomputer use, is by Feder [20] If we rearrange the terms of Eq (1.67) to get

Figure 1.6 The aplanatic surfaces of a sphere Here k is the radius of the refracting sphere;

t ¼ kð1 þ n0=nÞ is the radius of the object surface; t ¼ kð1 þ n=n0Þ, that of the image surface

Note that cos i ¼ S  N and cos i0¼ S0 N, so that by taking the scalar product of Eq.(1.71) with N we find that is given by

¼ n0ðS0 NÞ
nðS  NÞ ¼ n0

The convention for reflecting surfaces is only a convention, quite divorced fromany physical reality One sets n0¼
n in Eqs (1.71) and (1.72) Since i0¼ i, it followsthat

This takes care of the refraction or reflection operation It involves only localproperties of the refracting surface: the location of the point of incidence and the unitnormal vector N, at that point On the other hand, the transfer operation, by means

of which the point of incidence and the surface normal are found, involves the globalproperties of the surface

Suppose the surface is given by a vector function of position,

With the value of

normal to a surface is best given by the gradient of its equation, rf, so that the unitnormal vector is found from

N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffirf

ðrfÞ2

calculated, of course, at the point of incidence

But there is a problem here Equation (1.76) may have multiple roots For thesphere, or for that matter for any other conic section, f is a quadratic and will havetwo solutions: either two real roots, indicating that the ray intersects the surface attwo points; or two complex roots, in which case the ray misses the surface comple-tely More complicated surfaces produce more complicated solutions A torus, aquartic surface, may have up to four real roots, corresponding to four pointswhere ray and surface intersect Deciding which is which is a daunting problem

A particularly useful method is to identify that region of the surface that is ofinterest, then do a translation of coordinates to a point in that region, and then solvethe equation for the reformulated f function and choose that solution that lieswithin that region or is nearest to the chosen point Thus, the transfer operationbecomes a two-step process

To illustrate this, consider a rotationally symmetric optical system consisting ofspherical refracting surfaces Let each surface have a local coordinate system withthe z-axis as the axis of symmetry and the x- and y-axes tangent to the sphere where

it is intersected by the z-axis This x, y-plane is called, for reasons that I do notunderstand, the vertex plane Suppose P is the point of incidence of a ray with a

refracting surface whose coordinates are relative to the local coordinates associatedwith that surface; suppose, further, that the distance along the z-axis, between thissurface and the next succeeding surface, is t; then
PP, the point of intersection of theray with the next vertex plane, is given by Eq (1.75), which in scalar form is

a quadratic equation in

2 P
rZÞ  S þ ð
PP
rZÞ2
r2¼ 0; ð1:82Þwhose solution is

where

This constitutes the second part of this two-part transfer operation

The ambiguity in sign in Eq (1.80) has an easy explanation In general, a raywill intercept a sphere at two points We are almost always interested in the point ofincidence nearest the vertex plane and therefore choose the appropriate branch of thesolution

The unit normal vector is easily obtained from the expression for the gradient

in Eq (1.77) and that for the sphere in Eq (1.80), and is

N ¼1

rðP
rZÞ ¼1

rðx; y; z
rÞ ¼ ðcx; cy; cz
1Þ; ð1:85Þwhere c ¼ 1=r is the sphere’s curvature

For surfaces more complicated than the sphere, we need only substitute theirformulas into Eq (1.75) and proceed

The skewness invariant, shown in Eq (1.50), takes a slightly different form Inmedia of constant refractive index,

dP

ds ¼ S ¼ ð; ; Þ;

so that

Since P ¼ ðx; y; zÞ, the skewness invariant becomes

valid for both the refraction and transfer operations

Geometric wavefronts (we exclude wavefronts that arise from diffraction) aredefined in several equivalent ways [21] As a surface of constant phase, it is the locus

of points that have the same optical path length from some object point A system ofrays originating from some common object is termed an orthotomic system or anormal congruence In these terms a wavefront can be thought of as a transversalsurfaceorthogonal to each of the rays in the system A third definition is based onHuygens’ principle, in which the wavefront is taken to be the envelope of a family ofspherical wavelets centered on a preceding wavefront

However they are defined, wavefronts have structures and properties that arebest described using the language of the differential geometry of surfaces [22] As weshall see, wavefronts are smooth surfaces that may possess cusps but which havecontinuous gradients

In general, a smooth surface, at almost every point, possesses two uniquedirections, called the principal directions, that may be indicated by a pair of ortho-gonal vectors tagent to the surface They have the property that curvatures of arcsembedded in the surface in these directions have curvatures that are extrema; the arccurvature in one principal direction is a maximum relative to that of all other arcsthrough the same point The arc curvature in the other principal direction is aminimum These two maximum and minimum curvatures are called the principalcurvatures Obvious exceptions are the plane, in which curvature is everywhere zero,and the sphere, where it is everywhere constant In both cases principal directionscannot be defined Another exception is the umbilical point, a point on a surface atwhich the two principal curvatures are equal There the surface is best fit by a sphere.What follows is a method for determining the changes in the principal direc-tions and principal curvatures of a wavefront in the neighborhood of a traced ray.These calculations depend on and are appended to the usual methods of tracing rays

I have called them generalized ray tracing [23]

Consider now a ray traced through an optical system Through each of itspoints passes a wavefront that has two orthogonal principal directions and twoprincipal curvatures As before, let S be a unit vector in the direction of raypropagation and therefore normal to the wavefront Suppose one of these principaldirections is given by the unit vector T so that the other principal direction is

T  S Let the two principal curvatures be 1=1 and 1=2 The quantities 1=1,1=2, and T are found using general methods of differential geometry These willnot be treated here

Suppose this ray is intercepted by a refracting surface that has, at the point ofincidence, a unit normal vector N, a principal direction
TT and as principal curva-tures, 1=
1and 1=
2 Through this point passes one of the incident wavefronts, withparameters defined as above

The equations for refraction, Eqs (1.71) and (1.72), define the plane of dence The unit normal vector P to this plane is defined by

inci-P ¼N  S

sin i ¼N  S

0

where we have used Eqs (1.67) and (1.68) Note that P is invariant with respect torefraction From this we may define three unit vectors lying in the plane of incidence:

Q ¼ P  S; Q ¼ P  N
Q ; Q0¼ P  S0: ð1:89ÞTaking the vector product of P and Eq (1.71) gives us the refraction equations forthe Q vectors,

The next step is to find
0

, the angle between P and one of the principaldirections of the wavefront after refraction,

tan 2
0¼ 2

0 1

0 q

1

0 p

þsin 2
0

0 ;1

where

Note that the first and third equations in Eq (1.95) are exactly the Coddingtonequations [24] If a principal direction of the wavefront lies in the plane of incidence,then
and therefore 1= are zero The same is true for the principal directions of therefracting surface at the point of incidence:


and 1=
both vanish If both occur, ifboth the wavefront and the refracting surface have a principal direction lying in theplane of incidence, then Eqs (1.92), (1.94) and (1.98) become ephemeral, in whichcase Eq (1.95) reduces to the two Coddington equations However, generally speak-ing, the surface principal directions will not lie in the plane of incidence Indeed thiswill happen only if the refracting surface is rotationally symmetric and the plane ofincidence includes the axis of symmetry

In a rotationally symmetric system a plane containing the axis of symmetry iscalled a meridional plane; a ray lying entirely in that plane is a meridional ray A raythat is not a meridional ray is a skew ray What we have shown here is that theCoddington equations are valid only for meridional rays in rotationally symmetricoptical systems where a principal direction of the incident wavefront lies in themeridional plane No such restriction applies to the equations of generalized raytracing

This concludes the discussion on generalized ray tracing except for a fewobservations If the incident wavefront is a plane or a sphere, or if the traced ray

is at an umbilical point of the wavefronts, then the principal curvatures are equal andthe principal directions are undefined In that case, as a modus operandi, the incident

T vector may be chosen arbitrarily as long as it is perpendicular to S

It is no secret that rays are not real and that the existence of wavefronts can beinferred only by interferometry The only artifact in geometric optics that can beobserved directly is the caustic surface [25] Unlike waves and wavefronts the caustic