Eugene Hecht
`
`Adelphi University
`
`PEARSON
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`Addison
`Wesley
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`Library of Congress Cataloging-in-Publication Data
`
`Hecht, Eugene
`Optics / Eugene Hecht; —-— 4th ed.
`p. cm.
`
`Includes bibliographical
`I. Optics. 1. Title.
`
`references and index.
`
`QC355.3.H43
`535-——dc21
`
`2002
`
`2001032540
`
`ISBN 0-8053-8566-5
`
`
`
`Copyright ©2002 Pearson Education, Inc., publishing as Addison Wesley, 1301 Sansome St., San Francisco, CA 94111.
`All tights reserved. Manufactured in the United States of America. This publication is protected by Copyright and permission
`should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in
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`PEARSON
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`23 24 25—DOl-I-18 l7 16 15
`
`

`
`
`
`
`
`5
`
`]_
`
`1
`1 A Brief Hietory
`]__1 Pmlegomenon
`1
`1.2 in the Beginning
`1.3 From the Seventeenth Century
`1,4 The Nineteenth Century 4
`1.5 Twentieth-Century Optics
`
`7
`
`7
`
`3
`
`2
`
`_
`
`10
`
`10
`2 Waile M0110"
`2.1 0neDimensional Waves
`2.2 Harmonic Waves
`14
`17
`2.3 Phase and Phase Velocity
`2.4 The Superposition Principle 20
`2.5 The Complex Representation
`21
`2.6 Phasors and the Addition of Waves
`2-7 P'a"9 WaVe~°j
`24_
`g
`’
`2.8 we Ti1Ere<-:~if'imenséit;na| Differential
`
`ave q”a'°”
`
`4.2 Rayleigh Scattering 86
`4.3 -Refiection
`95
`4-4 Refraction
`100
`106
`4-5 Fefmafs Principle
`4.6 The Electromagnetic Approach
`4.7 TotaiinternaiRefiection
`122
`127
`4.8 Optical Properties of Metals
`4.9 Familiar Aspects of the interaction of
`Light and Matter
`131
`4.10 The Stokes Treatment of Reflection and
`Refraction
`136
`4.11 Photons, Waves, and Probability
`Problems
`141
`
`137
`
`111
`
`23
`
`5 Geometricaioptics 149
`54 Introductory Remarks
`149
`
`5.2 Lenses
`
`150
`
`.
`
`y
`
`“Ca
`
`aVe$
`
`P'°b'e"‘5 32
`3 Electromagnetic Theory, Photons,
`3.1 Basic Laws of Electromagnetic Theory
`3.2 Eiectromagnetic Waves 44
`3.3 Energy and Momentum 47
`3.4 Radiation
`58
`3.5 Light in Bulk Matter 66
`3.6 The Electromagnetic-Photon Spectrum 73
`3.7 Quantum Field Theory 80
`Problems 82
`
`37
`
`'
`4 The Propagation of Light 86
`4.1 introduction
`86
`
`5.4
`
`186
`5.5
`5'6 F‘b‘?’°p“°$
`193
`Gfavfimgtnalfplng
`P bl
`23:nS'ng
`m ems
`
`'
`
`222631
`
`6 More on Geometrical Optics 243
`6.1 Thick Lenses and Lens Systems 243
`6.2 Analytical RayTracing
`246
`6.3 Aberrations
`253
`6.4 GRiN Systems 273
`5.5 Concluding Remarks
`Pf0b1€mS
`277
`
`276
`
`

`
`vi’ Contents
`
`'
`
`—
`
`'
`
`10 Diffraction 443
`10.1 Preliminary Considerations 443
`10.2 Fraunhofer Diffraction 452
`10.3 1 Fresnel Diffraction . 485
`_
`_
`_.
`10.4 Kirchhoff’s Scalar Diffraction Theory" 510
`10.5 Boundary Diffraction Waves. 512
`problems 514
`'
`-
`A
`
`'
`
`_
`
`x
`
`.-
`
`—
`
`_
`
`_
`
`3
`
`-'
`.
`
`302
`
`11 Fourier Optics 519
`11.1 introduction
`519
`11.2 FourierTransforms 519
`11.3 Optical Applications 529
`problems 555
`
`550
`
`12 Basics of Coherence
`Theory 560
`121 inmducfion
`12.2 Visibility 562
`12.3 The Mutual Coherence Function and the
`Degree Ofcoherence 555
`12.4 Coherence and Stellar interferometry 573
`problems 573
`
`13 Modern Optics: Lasers and
`other Topics 531
`.
`581
`13.1 Lasers and Laserllght
`13.2 Imagery —— The Spatial Distribution of
`Optical information 606
`13“-3 Holography
`623
`13_4 Noniinemgptgcs
`P,-Ob|ems 644
`p
`Appendix 1 549
`.
`:::::f!x6:3652
`Solutions to Selected Problems 658
`Bibliography 685
`index 539
`
`539
`
`y
`
`_
`
`-.
`
`'
`
`7 The Superposition of
`Waves 281
`7.1 The Addition of Waves of the Same
`_Frequency 282
`A
`7-'2 ghe Addition £3;/1VaVeS 01‘ Different
`fe<1UenCY
`_
`-
`-
`» 7.3.Anharmonic Periodic Waves
`7.4 Nonperiodic Waves 308
`Problems 320
`_
`_
`8 P°'3”zat'°" 325
`8.1 The Nature of Polarized Light 325
`8.2 Polarizers
`331
`
`~
`
`4
`
`8.3 Dichroism 333
`_
`8-4 Birefringence 335_
`8.5 Scattering and Polarization 344
`8.6 Polarization by Reflection 348
`8-7 Retarders 352
`
`_
`357
`3-3 Clrcularmarlzels
`_
`8.9 Polarization of Polychromatlc Light 358
`8-10 Optical/\CtiVity 350
`8.11 induced Optical Effects—-Optical
`Modulators 365
`3-12 Uqmd C'V5*?'5
`370_ .
`8.13 .[l3\0hl/iaE:1h§’:'_T(l)atlCa5l7l32eSCr|ptl0n of
`Probiems 379
`
`I2
`
`l n
`
`.
`38.5
`9 Interference
`386
`9.1 General Considerations
`9.2 Conditions for interference 390
`393
`9.3 Wavefrontsplitting interferometers
`9.4 Amplitude-splittinginterferometers 400
`9.5 Types and Localization of Interference
`Fringes 414
`9.6 MultipleBeamlnterference 416
`9.7 Applications of Single and Multilayer
`Film? 425
`9.8 Applications of interferometry 431
`Problems 438
`
`

`
`
`
`l tics?
`
`5.1 Introductory Remarks
`
`
`The surface of an object that is either self~luminous or exter-
`nally illuminated behaves as if it consisted of a very large
`number of radiating point sources. Each of these emits spheri-
`cal waves; rays emanate radially in the direction of energy
`flow, that is, in the direction of the Poynting vector. In this
`case, the rays diverge from a given point source S, whereas if
`the spherical wave were collapsing to a point, the rays would
`of course be converging. Generally, one deals only with a
`small portion of a wavefront. A pointfrom which a portion of
`a spherical wave diverges, or one toward which the wave seg-
`ment converges, is known as afocas of the bundle of rays.
`Figure 5.1 depicts a point source in the vicinity of some
`arrangement of reflecting and refracting surfaces representing
`an optical system. Of the infinity of rays emanating from S,
`generally speaking, only one will pass through an arbitrary
`point in space. Even so, it is possible to arrange for an infinite
`number of rays to arrive at a certain point P, as in Fl g. 5.1. If
`for a cone of rays coming from S there is a corresponding cone
`of rays passing through P, the system is said to be stigmatic
`for these two points. The energy in the cone (apart from some
`inadvertent losses due to reflection, scattering, and absorption)
`reaches P, which is then referred to as a perfect image of S.
`The wave could conceivably arrive to form a finite patch of
`light, or blur spot, about P; it would still be an image of S but
`no longer a perfect one.
`It follows from the Principle of Reversibility (p. 110) that
`a point source placed at P would be equally well imaged at S,
`and accordingly the two are spoken of as conjugate points.
`In an ideal optical system, every point of a three-dimensional
`region will be perfectly (or stigmatically) imaged in another
`
`region, the former being the object space, the latter the
`image space.
`Most commonly, the function of an optical device is to col—
`lect and reshape a portion of the incident wavefront, often with
`the ultimate purpose of forming an image of an object. Notice
`that inherent in realizable systems is the limitation of being
`unable to collect all the emitted light; a system generally
`accepts only a segment of the wavefront. As a result, there will
`always be an apparent deviation from rectilinear propagation
`even in homogeneous media——the waves will be diffracted.
`The attainable degree of perfection of a real imaging optical
`system will be diffraction-limited (there will always be a blur
`spot, p. 467). As the wavelength of the radiant energy decreas«
`es in comparison to the physical dimensions of the optical sys-
`tem, the effects of diffraction become less significant. In the
`conceptual limit as A0 -> 0, rectilinear propagation obtains in
`homogeneous media, and we have the idealized domain of
`Geometrical Optics?“ Behavior that is specifically attribut-
`able to the wave nature of light (e.g., interference and diffrac-
`tion) would no longer be observable. In many situations, the
`great simplicity arising from the approximation of Geometri»
`cal Optics more than compensates for its inaccuracies. In
`short, the subject treats the controlled manipulation of wave
`fronts (or rays) by means of the interpositiontng of reflecring
`and/or refracting bodies, neglecting any diffraction effects.
`
`..............................................
`
`‘Physical Optics deals with situations in which the nonzero wavelength of
`light must be reckoned with. Analogously, when the de Broglie wavelength
`of a material object is negligible, we have Classical Mechanics; when it is
`not, we have the domain of Quantum Mechanics.
`
`

`
`150 Unapterb tjeometricaloptics
`
`lb)
`
`5
`
`
`
` Figure 5.1 Conjugate foci. (a) A point source 8
`
`sends out spherical waves. A cone of rays enters
`an optical system that inverts the wavefronls,
`causing them to converge on point P. (b) in cross
`section rays diverge. from S, and a portion of
`them converge to P. If nothing stops the light at
`P, it continues on.
`
`\\5v».
`
`5.2 Lenses
`.-r.aamwnmxcamb~mwmu ;,1emm.».;:suc~:s::as.m\a><2xrn§o:>s.i:=t¢'enx1x:.:s-xztsawmaex5r—=*>;c~=ma~2.i»
`
`The lens is no doubt the most widely used optical device, and
`that’s not even considering the fact that we see the world
`through a pair of them. Human—made lenses date back at least
`to the burning-glasses of antiquity, which, as the name
`implies, were used to start fires long before the advent of
`matches. In the most general terms, a lens is a refracting
`device (i .e., a discontinuity in the prevailing medium) that
`reconfigures a transmitted energy distribution. That much
`is true whether we are dealing with UV, lightwaves, IR,
`microwaves, radiowaves, or even sound waves.
`
`The configuration of a lens is determined by the required
`reshaping of the wavefront it is designed to perform. Point
`sources are basic, and so it is often desirable to convert diverg-
`ing spherical waves into a beam of plane waves. Flashlights,
`projectors, and searchlights all do this in order to keep the
`beam from spreading out and weakening as it progresses. In
`just the reverse, it’s frequently necessary to collect incoming
`parallel rays and bring them together at a point, thereby focus-
`ing the energy, as is done with a burning—glass or a telescope
`lens. Moreover, since the light reflected from someone’s face
`
`scatters out from billions of point sources, a lens that causes
`each diverging wavelet to converge could form an image of
`that face (Fig. 5.2).
`
`5.2.1 Aspherical Surfaces
`
`To see how a lens works, imagine that we interpose in the path
`of a wave a transparent substance in which the wave’s speed is
`different than it was initially. Figure 5.3a presents a cross-sec-
`tional view of a diverging spherical wave traveling in an inci-
`
`Figure 5.2 A person's face, like
`everything else we ordinarily see in
`reflected light, is covered with count-
`
`less atomic scatterers.
`
`

`
`
`
`5.2 Lenses 151
`
`of time to get there, no matter what [h_E3Ctllal route taken by
`any particular ray. In other words, F1A/)t,-.(the number of
`wavelengths along the arbitrary ray. from F1 to A) plus X5/A,
`V, (the number of wavelengths along the ray from A to D) must
`be constant regardless of where on the interface A happens to
`be. Now, adding these and multiplying by A0, yields
`n,-' (EX) + n, (X5) = constant
`
`(5.1)
`
`Each term on the left is the length traveled in a medium mul~
`tiplied by the index of that medium, and, of course, each repre~
`sents the optical path length, OPL, traversed. The optical path
`lengths from S to 5.07 are all equal. If Eq. (5.1) is divided by c,
`the first term becomes the time it takes light to travel from S to
`A and the second term, the time from A to D; the right side
`remains constant (not the same constant, but constant). Equa-
`tion (5.1) is equivalent to saying that all paths from S to -55’
`must take the same amount of time to traverse.
`
`Let’s return to finding the shape of the interface. Divide Eq.
`(5.1) by n,-, and it becomes
`
`Figure 5.3 A hyperbolic interface between air and glass. (a) The
`wavefronts bend and straighten out. (b) The rays become parallel. (c)
`The hyperbola is such that the optical path from S to A to D is the same
`no matter where A is.
`
`dent medium of index I1; impinging on the curved interface of
`a transmitting medium of index n,. When n, is greater than n,~,
`the wave slows upon entering the new substance. The central
`area of the wavefront travels more slowly than its outer
`extremities, which are still moving quickly through the inci~
`dent medium. These extremities overtake the midregion, con-
`tinuously flattening the wavefront. If the interface is properly
`configured, the spherical wavefront bends into a plane wave.
`The alternative ray representation is shown in Fig. 5.31;; the
`rays simply bend toward the local normal upon entering the
`more dense medium, and if the surface configuration is just
`
`right, the rays emerge parallel.
`To find the required shape of the interface, refer to Fig.
`5.3c, wherein point A can lie anywhere on the boundary. One
`Wavefront is transformed into another, provided the paths
`along which the energy propagates are all “equal," thereby
`maintaining the phase of the wavefront (p. 26). A little spheri-
`cal surface of constant phase emitted from S must evolve into
`a flat surface of constant phase at 55’. Whatever path the light
`takes from S to fifl it must always be the same number of
`Wavelengths long, so that the disturbance begins and ends in-
`phase. Radiant energy leaving S as a single wavefront must
`arrive at the plane 5-D’, having traveled for the same amount
`
`7‘:
`
`EX + (—~) /T5 = constant
`
`"1
`
`(5.2)
`
`This is the equation of a hyperbola in which the eccentricity
`(e), which measures the bending of the curve, is given by
`(n,/n,») > 1; that is, e = n,,- > 1. The greater the eccentricity,
`the flatter the hyperbola (the larger the difference in the
`indices, the less the surface need be curved). When a point
`source is located at the focus F1 and the interface between the
`two media is hyperbolic, plane waves are transmitted into the
`higher index material. It’s left for Problem 5.3 to establish that
`when (rt,/n,-) < 1, the interface must be ellipsoidal. In each
`case pictured in Fig. 5.4, the rays either diverge from or con-
`verge toward a focal point, F. Furthermore, the rays can be
`reversed so that they travel either way; if a plane wave is inci~
`
`
`
`Figure 5.4 (a) and (b) Hyperbololda! and (c) and (d) ellipsoidal
`retracting surfaces (n2 > n1) in cross section.
`
`.
`
`

`
`152 Chapterfi Geometrical Optics
`
`dent (from the right) on the interface in Fig. 5.4c, it will con-
`verge (off to the left) at the farthest focus of the ellipsoid.
`The first person to suggest using conic sections as surfaces
`for lenses and mirrors was Johann-Kepler (161.l),.but he
`wasn’t able to go very far with the idea without Sn‘el'l’s' Law.
`Once that relationship was discovered, Descartes (1637) using
`his Analytic Geometry coulddevelop the theoretical founda-
`tions of the optics of aspherical surfaces. The analysis present-
`ed here is in essence a gift from Descartes.
`It‘s an easy matter now to construct lenses such that both
`the object and image points (or the incident and emerging
`light) will be outside of the medium of the lens. In Fig. 5.5a
`diverging incident spherical waves are made into plane waves
`at the first interface via the mechanism of Fig. 5.4a. These
`plane waves within the lens strike the back face perpendicular-
`ly and emerge unaltered: 0, == 0 and B, = 0. Because the rays
`are reversible, plane waves incoming from the right will con-
`verge to point F1, which is known as the focal point of the lens.
`Exposed on its flat face to the parallel rays from the Sun, our
`rather sophisticated lens would serve nicely as a buming~g1ass.
`In Fig. 5.5!), the plane waves within the lens are made to
`converge toward the axis by bending at the second interface.
`Both of these lenses are thicker at their midpoints than at their
`edges and are therefore said to be convex (from the Latin con-
`vexus, meaning arched). Each lens causes the incoming beam
`to converge somewhat, to bend a bit more toward the central
`axis; therefore, they are referred to as converging lenses.
`In contrast, a concave lens (from the Latin concavus,
`meaning hollow-and most easily remembered because it
`contains the word cave) is thinner in the middle than at the
`edges, as is evident in Fig. 5.5c. It causes the rays that enter as
`a parallel bundle to diverge. All such devices that turn rays
`outward away from the central axis (and in so doing add diver-
`gence to the beam) are called diverging lenses. In Fig. 5.50,
`parallel rays enter from the left and, on emerging, seem to
`diverge from F2; still, that point is taken as a focal point.
`When a parallel bundle of rays passes through a converg-
`ing lens, the point to which it converges (or when passing
`through a diverging lens, the point from which it diverges)
`is a focal point of the lens.
`_
`If a point source is positioned on the central or optical axis
`at the point F, in front of the lens in Fig. 5.51), rays will con-
`verge to the conjugate point F2. A luminous image of the
`source would appear on a screen placed at F2, an image that is
`therefore said to be real. On the other hand, in Fig. 5.5c the
`point source is at infinity, and the rays emerging from the sys-
`
`A
`
`tem this time are diverging. They appear to come from a point
`F2, but no actual luminous image would appear on a screen at
`that location. The image here is spoken of as virtual, as is the
`familiar image generated b.y a plane mirror.
`Optical elements (lenses and .mirrors) of the sortwe have
`talked about, with one or both surfaces neither planar nor‘
`spherical, are referred to asaspherics. Although their opera- _
`tion is easy to understand and they perform certain tasks
`exceedingly well, they are still difficult to manufacture with
`‘great accuracy. Nonetheless, where the costs are justifiable or
`the required precision is not restrictive or the volume produced
`is large enough, aspherics are being used and will surely have
`an increasingly‘ important role. The first quality glass aspheric
`
`Figure 5.5 (a), (b), and (c) Several hyperbolic lenses seen in cross
`section. To explore an ellipsoidal lens, see Problem 5.4. (d) A selection
`of aspherical lenses. (Photo courtesy Melles snot.)
`
`

`
`3 manufactured in great quantities (tens of millions) was a
`rob
`lens for the Kodak disk camera (1982). Today aspherical lens-
`es are frequently used as an elegant means of correcting imag-
`ing errors in complicated optical systems. Aspherical eyeglass
`lenses are‘ flatter and lighter than regular spherical ones. As
`such, they’re well suited for strong prescriptions. Furthermore.
`they minimize the magnification of the 'wearer’s eyes as seen
`by other. people.
`_
`'
`A new generation of computer-controlled machines,
`aspheric generators, is producing elements with tolerances
`(i.e., departures from the desired surface) of better than 0.5
`‘um (0.000 020 inch). This is still about a factor of 10 away
`from the generally required tolerance ofA/4 for quality optics.
`After grinding, aspheres can be polished magnetorheological-
`1y, This technique, used to figure and finish the surface, mag-
`netically controls the direction and pressure applied to the
`workpiece by the abrasive particles during polishing.
`Nowadays aspherics made in plastic and glass can be found
`in all kinds of instruments across the whole range of quality,
`including telescopes, projectors, cameras, and reconnaissance
`. devices.
`
`5.2.2 Refraction at Spherical Surfaces
`
`Consider two pieces of material, one with a concave and the
`other a convex spherical surface, both having the same radius.
`It is a unique property of the sphere that such pieces will fit
`together in intimate contact regardless of their mutual orienta-
`tion. If we take two roughly spherical» objects of suitable cur-
`vature, one a grinding tool and the other a disk of glass,
`
`5.2 Lenses 153
`
`
`
`Figure 5.6
`_
`Refraction at a
`spherical interface.
`Conjugate foci.
`
`separate them with some abrasive, and then randomly move
`them with respect to each other, we can anticipate that any
`high spots on either object will wear away. As they wear, both
`pieces will gradually become more spherical (see photo). Such
`surfaces are commonly generated in batches by automatic
`grinding and polishing machines.
`Not surprisingly, the vast majority of quality lenses in use
`today have surfaces that are segments of spheres. Our intent
`here is to establish techniques for using such surfaces to simul-
`taneously image a great many object points in light composed
`of a broad range of frequencies. Image errors, known as aber-
`rations, will occur, but it is possible with the present technol-
`ogy to construct high-quality spherical lens systems whose
`aberrations are so well controlled that image fidelity is limited
`
`only by diffraction.
`Figure 5.6 depicts a wave from the point source S imping-
`ing on a spherical interface of radius R centered at C. The
`point V is called the vertex of the snrfaceflie length so = W
`is known as the object distance. The ray SA will be refracted
`at the interface toward the local normal (tr; > n,) and there
`fore toward the central or optical axis. Assume that at some
`
`
`
`Polishing a spherical lens. (Photo courtesy Optical Society of America‘)
`
`Figure 5.7 Rays incident at the same angle.
`
`

`
`154 Chapter 5 Geometrical Optics
`
`point P the ray will cross the axis, as will all other ra_y_s inci-
`dent at the same angle 6; (Fig. 5.7). The length s, = VP is the
`image distance. Ferrnat’s Principle maintains that the optical
`‘path length 0_PL will be stationary; that is, its derivative with
`_ respect to the position variable will be zero. For the ray in
`question.
`.
`_
`V
`
`= nleo + 7128;
`
`(5-3)‘ .
`
`Using the law of‘cosines in triangles SAC and ACP alongwith
`the fact that cos cp = -cos(l8()° - cp), we get
`
`6,, = {R2 + (so + R)2 ~ 2R(s,, + R) cos or/2
`
`and
`
`e, = [R2 + (s,- ~ R)2 + 2R(s,- ~— R) cos or/2
`
`The OPL can be rewritten as
`
`0191. = mm? + (s,, + R)2 ~ 2R(s0 + R) cos or/3
`
`+ n2[R2 + (s; — R)2 + 2R(s,- - R) cos or/2
`
`All the quantities in the diagram (s,~, so, R, etc.) are positive
`numbers, and these form the basis of a sign convention which
`is gradually unfolding and to which we shall return time and
`again (see Table 5.1). Inasmuch as the point A moves at the
`end of a fixed radius (i.e., R = constant), <p is the position vari-
`able, and thus setting d(0PL)/do = 0, via Fermat’s Principle
`
`—:—gx
`
`TABLE 5.1 Sign Convention for Spherical
`Refracting Surfaces and Thin Lenses*
`(Light Entering from the Left)
`
`
`
`smfo
`
`x0
`
`.9,-, f,
`
`x,«
`
`R
`
`+ left of V
`
`+ left of F0
`
`+ right of V
`
`+ right of F,-
`
`+ifCisrightofV
`
`
`
`yo, y,- + above optical axis
`
`‘This table anticipates the imminent introduction of a tew quanti
`
`ties not yet spoken of.
`
`we have
`
`n1R(s,, + R) sin cp _ n2R(s,~ - R) sin ip = 0
`260 .
`I
`.
`'
`-'
`
`(5.4)
`.
`
`g
`
`from which it follows that
`
`I
`
`-— + —— = — —- -—
`
`'11
`(3,,
`
`V12
`_£’,-
`
`"25."
`1
`R i e,
`
`'l1So)
`E0
`
`(5.5)
`
`This is the relationship that must hold among the parameters
`for a ray going from S to P by way of refraction at the spheri-
`cal interface. Although this expression is exact, it is rather
`complicated. If A is moved to a new location by changing (p,
`the new ray will not intercept the optical axis at P. (See Prob-
`lem 5.1 concerning the Cartesian oval which is the interface
`configuration that would bring any ray, regardless of (p, to P.)
`The approximations that are used to represent 80 and 6,, and
`thereby simplify Eq. (5.5), are crucial in all that is to follow.
`Recall that
`
`__
`cosgo—l
`
`~
`_
`sin rp —— go
`
`and
`
`2
`5
`4
`,,§?..
`.?1._$..
`2! + 4!
`6! +
`3
`5
`7
`-59..
`i’._._.‘P_
`3! + 5!
`7! +
`
`(5.6)
`
`(5.7)
`
`If we assume small values of 4: (i.e., A close to V), cos (,0 == 1.
`Consequently, the expressions for 4, and €,~ yield €,,
`== so,
`6’, z s,-, and to that approximation
`
`fig
`I1]
`__+_.=
`S0
`3;
`
`"2‘"i
`
`R
`
`(5.8)
`
`We could have begun this derivation with Snell’s Law rather
`than Fermat’s Principle (Problem 5.5), in which case small
`values of go would have led to sin cp ~ go and Eq. (5.8) once
`again. This approximation delineates the domain of what is
`calledfirst-order theory; we’ll examine third—order theory (sin
`cp z (p - cp3/3!) in the next chapter. Rays that arrive at shal—
`low angles with respect to the optical axis (such that cp and f1
`are appropriately small) are known as paraxial rays. The
`emerging wavefront segment corresponding to these paraxial
`rays is essentially spherical and will form a “perfect” image
`at its center P located at s,-. Notice that Eq. (5.8) is indepen—
`dent of the location of/l over a small area about the symmetry
`axis, namely, the paraxial region. Gauss, in 1841, was the first
`to give a systematic exposition of the formation of image5
`
`

`
`
`
`5.2 Lenses 155
`
`
`
`Figure 5.8 Plane waves propagating beyond a spherical interface-
`the object focus.
`
`Figure 5.10 A virtual image point.
`
`the second or image focus is the axial point F1, where the
`image is formed when so = 00; that is,
`
`11
`
`0°
`
`11
`
`S,‘
`
`12""?!
`
`R
`
`Defining the second or image focal length f,- as equal to s,» in
`this special case (Fig. 5.9), we have
`
`"2
`f~=——--~12
`’l2‘"1
`
`(5.10)
`
`Recall that an image is virtual when the rays diverge from
`it (Fig. 5.10). Analogously, an object is virtual when the rays
`converge toward it (Fig. 5.11). Observe that the virtual object
`is now on the right-hand side of the vertex, and therefore s,,
`will be a negative quantity. Moreover, the surface is concave,
`and its radius will also be negative, as required by Eq. (5.9),
`since fa would be negative. In the same way, the virtual image
`distance appearing to the left of V is negative.
`
`
`
`under the above approximation, and the result is variously
`known as first-order, paraxial, or Gaussian Optics. It soon
`became the basic theoretical tool by which lenses would he
`designed for several decades to come. If the optical system is
`well corrected, an incident spherical wave will emerge in a
`form very closely resembling a spherical wave. Consequently,
`as the perfection of the system increases, it more closely
`approaches first-order theory. Deviations from that of paraxi-
`al analysis will provide a convenient measure of the quality of
`an actual optical device.
`If the point F0 in Fig. 5.8 is imaged at infinity (s,~ == 00), we
`have
`
`"1
`
`so
`
`"2_ n2“‘fl[
`00
`R
`
`That special object distance is defined as the first focal length
`or the object focal length, so $12, so that
`
`(5.9)
`
`
`
`Figure 5.9 The reshaping of plane into spherical waves at a spherical
`interface-—the image focus.
`
`Figure 5.11 A virtual object point.
`
`

`
`A lens for short-wavelength radiowaves. The disks serve to retract
`these waves much as rows of atoms retract light. (Photo courtesy Optical
`Society of America.)
`
`CONVEX
`
`CONCAVE
`
`convex
`
`R2>0
`
`Planar convex
`
`Planar concave
`
`Meniscus
`
`Meniscus
`concave
`
`Figure 5.12 Cross sections of various centered spherical simple
`lenses. The surface on the left is #1 since it is encountered first.
`its radius is R1. (Photo courtesy of Meiles Griot.)
`
`156 Chapter5 Geometricaioptics
`
`5.2.3 Thin Lenses
`
`Lenses are made in a wide range of forms; for example, there
`- are acoustic and microwave 1ense_s.’Some_of the latter are .
`made of glass or wax in easily recognizable shapes, whereas -
`others are far more subtle in appearance (see .photo). Most
`often a lens has two or more refracting interfaces, and at least
`one of these is curved. Generally, the nonplanar surfaces are
`centered on a common axis. These surfaces are most frequent-
`ly spherical segments and are often coateciwith thin dielectric
`films to control their transmission properties (see Section 9.9).
`A lens that consists of one element (i.e., it has only two
`refracting surfaces) is a simple lens. The presence of more than
`one element makes it a compound lens. A lens is also classi-
`fied as to whether it is thin or thick—that is, whether or not its
`thickness is effectively negligible. We will limit ourselves, for
`the most part, to centered systems (for which all surfaces are
`rotationally symmetric about a common axis) of spherical sur-
`faces. Under these restrictions, the simple lens can take the
`
`forms shown in Fig. 5.12.
`Lenses that are variously known as convex, converging, or
`positive are thicker at the center and so tend to decrease the
`radius of curvature of the wavefronts. In other words, the wave
`converges more as it traverses the lens, assuming, of course,
`that the index of the lens is greater than that of the media in
`which it is immersed. Concave, diverging, or negative lenses,
`on the other hand, are thinner at the center and tend to advance
`that portion of the wavefront, causing it to diverge more than
`it did upon entry.
`
`Thin-Lens Equations
`
`Return to the discussion of refraction at a single spherical
`interface, where the location of the conjugate points S and P is
`
`given by
`
`[58]
`
`RH
`
`712__..__.
`
`S;
`
`111
`
`50
`
`When so is large for a fixed (I12 - nl)/R, s,- is relatively small.
`As so decreases, .9, moves away from the vertex; that is, both 6,-
`and 6, increase until finally so = jg, and s,- = 00. At that point,
`It; /so = (222 ~ nl)/R, so that if s,, gets any smaller, s,- will have
`to be negative, if Eq. (5.8) is to hold. In other words, the image
`becomes virtual (Fig. 5.13).
`
`

`
`5.2 Lenses 157
`
`"1
`
`(a)
`
`(b)
`
`(c)
`
`Figure 5.13 Refraction at a spherical interface between two transpar-
`ent media shown in cross section.
`
`Let’s now locate the conjugate points for a lens of index n,
`surrounded by a medium of index n,,,, as in Fig. 5.14, where
`another end has simply been ground onto the piece in Fig.
`5.l3c. This certainly isn’t the most general set of circum~
`stances, but it is the most common, and even more cogently, it
`is the simplest.* We know from Eq. (5.8) that the paraxial rays
`issuing from S at so] will meet at P’, a distance, which we now
`call Sn, from V,, given by
`
`711
`n,,,
`___+___..___
`Sol
`Sn
`
`In — n,,,
`R1
`
`5.11
`
`(
`
`)
`
`Thus, as far as the second surface is concerned, it “sees” rays
`coming toward it from P’, which serves as its object point a
`............................................ ..
`
`‘See Jenkins and White, Fundamentals of Optics, p. 57, for a derivation
`containing three different indices.
`
`Figure 5.14 A spherical lens. (a) Rays in a vertical plane passing
`through a lens. Conjugate loci. (b) Refraction at the interfaces. The
`radius drawn from C1 is normal to the first surface, and as the ray
`enters the lens it bends down toward that normal. The radius from C2 is
`normal to the second surface; and as the ray emerges, since n, > na,
`the ray bends down away from that normal. (c) The geometry.
`
`distance S02 away. Furthermore, the rays arriving at that sec-
`ond surface are in the medium of index n,. Thus the object
`space for the second interface that contains P’ has an index n,.
`
`

`
`158 Chapters Geometrical Optics
`
`Note that the rays from P’ to that surface are indeed straight
`lines. Considering the fact that
`
`isa2i f‘. isni
`
`‘o
`
`'
`
`since so; is on the left and therefore positive, so; = lsogl, and s,-,
`is also on the left and therefore negative, -sir = Is,-1|, we have '
`
`S02 3 "‘ S,'1A+ d
`
`Thus at the second surface Eq. (5.8) yields
`
`”’
`(‘Sn ‘*7 fl)
`
`+ 35% = ”"‘ _ "’
`Siz
`R2
`
`(5.13)
`
`Here n, > no, and R2 < 0, so that the right-hand side is posi—
`tive. Adding Eqs. (5.11) and (5.13), we have
`
`£"~'-+-nl=(n)-n)(--——)+—-—-—---
`
`n,d
`
`d)
`(srl
`"" Sn
`
`(5.14)
`
`So:
`
`3:2
`
`1
`
`R1
`
`m
`
`1
`
`R2
`
`
`
`The actual wavefronts of a diverging llghtwave partially focused by a
`lens. The photo shows five exposures, each separated by about 100 ps
`(i.e., 100 x 10"” 5), of a spherical pulse 10 ps long as it swept by
`and through a converging lens. The picture was made by using a holo-
`graphic technique. (Courtesy of N.H. Abramson.)
`
`If the lens is thin enough (d -9 0), the last term on the right is
`effectively zero. As a further simplification, assume the sur-
`rounding medium to be air (i.e., no, -~ 1). Accordingly, we
`have the very useful Thin-Lens Equation, often referred to as
`the Lensmake:r’s Formula:
`
`_L+_1_:(,,I_,)(_1_-_1_)
`So
`5:
`R1
`R2
`
`5.15
`
`(
`
`)
`
`which is the famous Gaussian Lens Formula (see photo).
`As an example of how these expressions might be used,
`let’s compute the focal length in air of a thin planar-convex
`lens having a radius of curvature of 50 mm and an index of
`1.5. With light enterin