Principles of Optics
`
`Electromagnetic Theory of Propagation,
`Interference and Difliraction of Light
`
`by
`
`MAX BORN
`
`M.A., Dr.Phi1., F.R.S.
`Nobel Laureate
`
`Formerfy Professor at the Universities of Goltirrgen and Edfnburgh
`and
`
`EMIL WOLF
`
`Ph.D., D.Sc.
`
`Professor of Physics, University of Rochester, N. Y.
`
`with contributions by
`
`A. B. BHATIA, P. G. CLEMMOW, D. Gsnon, A. R. STOKES,
`A. M. TAYLOR. P. A. WAYMAN and W. L. WILGOCK
`
`SIXTH (CORRECTED) EDITION
`
`@
`
`PERGAMON PRESS
`
`OX_FORD ‘ NEW YORK - BEIJING - FRANKFURT
`SAO PAULO - SYDNEY - TOKYO - TORONTO
`
`Capella 2016
`Capella 2016
`JDS Uniphase v. Capella
`JDS Uniphase V.
`IPR2015-00731
`IPR2015-00731
`
`0001
`
`

`
`U.K.
`
`U.S.A.
`
`PF.OP‘I.F.'S REPUBLIC
`OF CHINA
`FEDERAL REPUBLIC
`or GERMANY
`BRAZIL
`
`AUSTRALIA
`
`JAPAN
`
`CANADA
`
`Pergamon Press. Headingmn Hill Hall.
`Oxford OX3 OBW. England
`Pergamon Press, Maxwell House, Fairview Park,
`Elrnsford, New York H1523, U.S.A.
`
`Pergamon Press, Qianmen Hotel, Beijing,
`People's Republic of China
`Pergamon Press, Hammerweg 6,
`D-13242 Kronberg, Federal Republic of Germany
`Pergamon Editora. Rua Ega de Quciros, 346,
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`
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`
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`Pergarnon Press Canada. Suite 104.
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`
`Copyright © 1980 Max Born and Emil Wolf
`All Rights Rescued. No part of this pubiicatian may be reproduced,
`stored in :1 retrietxal system or transmitted in any form or J5}; any means:
`eiectranir.
`electrostatic. magnetic
`tape, mechanical,
`photocopying,
`recording or
`ritherwire,
`uiithout permission
`in writing from the
`pubbhhers.
`First edition 1959
`Second (revised) edition 1954
`Third (revised) edition 1965
`Fourth (revised) edition 1970
`Fifth (revised) edition 1975
`Reprinted l975. 1977
`Sixth edition 1980
`Reprinted (with corrections) 1983
`Reprinted 1984
`Reprinted (with corrections) 1986
`Reprinted I987
`
`Library of Congress Cataloging in Publication Data
`Born, Max
`Principles of optics—6th ed. [with corrections).
`1. Optics
`H. Wolf, Emil
`1. Title
`QC351
`80-41470
`535
`ISBN 0~{)8—02fi482—-1 Hardcover
`ISBN 0—08—{}254Bl—6 Flexicover
`
`Printed in Great Britain by A. Wheaten C9‘ Co. Ltd., Exeter
`
`0002
`
`

`
`13.2]
`
`OPTIOS or METALS
`
`615
`
`The “penetration depth” d for copper for radiation in three familiar regions of
`the spectrum, calculated with the static conductivity cr on 5-14 . 10“ sec‘1aud it ..—_ 1.
`
`TABLE XXVI
`
`Radiation
`
`Infra-red
`
`Microwaves
`
`i Long radio waves
`I
`
`10 cm
`{i-1 .10"-" cm
`
`I000 m = 105 cm
`6-] .104 cm
`
`A perfect conductor is characterized by infinitely large conductivity (cr—+ 00).
`Since according to (16), e/o = (1 — .-:3)/vx, we have in this limiting case :<2—1- 1, or
`by (163,), n —» 00. Such a conductor would not permit the penetration of an electro-
`magnetic wavc to any depth at all and would reflect all the incident light (cf. § 13.2
`below).
`Whilst the refractive index of transparent substances may easily be measured from
`the angle of refraction, such measurements are extremely difficult to carry out for
`metals, because a specimen of the metal which transmits any appreciable fraction
`of incident light has to be exceedingly thin. Nevertheless KUND'r* succeeded in con-
`structing mctal prisms that enabled direct measurements of the real and imaginary
`parts of the complex refractive index to be made. Usually, however, the optical
`constants of metals are determined by means of katoptric rather than dioptric
`experiments, i.e. by studying the changes which light undergoes on reflection from a
`metal, rather than by means of measurements on the light transmitted through it.
`
`13.2. REFRACTION AND REFLECTION AT
`A METAL SURFACE
`
`We have seen that the basic equations relating to the propagation of a plane time-
`harmonic wave in a conducting medium differ from those relating to propagation in a
`transparent dielectric only in that the real constants c and is are replaced by complex
`constants E and E. It follows that the formulae derived in Chapter I, as far as they
`involve only linear relations between the components of the field vectors of plane
`monochromatic waves, apply also in the present case.
`In particular, the boundary
`conditions for the propagation of a Wave across a surface of discontinuity and hence
`also the formulae of § 1.5 relating to refraction and reflection remain valid.
`Consider first the propagation of a plane Wave from a dielectric into a conductor,
`both media being assumed to be of infinite extent, the surface of contact between
`them being the plane z = 0. By analogy with § 1.5 (8) the law of refraction is
`1
`sin 6, a: g sin 8,-.
`
`(1)
`
`Since ii. is complex, so is 0,, and this quantity therefore no longer has the simple
`significance of an angle of refraction.
`Let the plane of incidence be the 2:2-plane. The space-dependent part of the
`phase of the Wave in the conductor is given by ic(r.s‘”) where (of. § 1.5 (4))
`
`s,,‘” = sin 0,,
`
`8,,” = O,
`
`3,”) = cos 3,.
`
`(2)
`
`" A. KUNDT, Aim. at. Physik. 3i (13331. 469.
`
`0003
`
`

`
`616
`
`PRINCIPLES or OPTICS
`
`From (1) and (2) and §13.1 (15)
`
`sin 9-
`_
`3:3“ = sin 6, = ?-—-'—
`‘n(l + ikl
`
`_
`1 — ix
`-— B111 8,,
`:3 n.(l -1- K2)
`
`s,“’ = cos 6, = \/T_— sin“ 6:
`_j.w_é_
`
`=~/1 _
`
`sin” a,+=sfi12:KB)$ sin” 3;.
`
`It is convenient to express 3,,”’ in the form
`
`3,“) = cos 8, = gs‘?
`
`(3b)
`
`(4)
`
`(g. 32 real). Expressions for q and y in terms of n, K and sin 8; are immediately obtained
`on squaring (31)) and (4) and equating real and imaginary parts. This gives
`
`q”cos2y=-1-
`
`gz sin 2-}: =
`
`It follows that
`
`Hr . 3*”) = E:-'n(l + £n:)(:rs_,_,‘" + zs,‘“}
`
`'n(l -1- ix) I:
`
`a:(l — ix)
`
`'n(l + K”)
`
`sin 6, + z(q cos y + {q sin 32)]
`
`= [27 sin 9; -5- mg (cos y — K sin y} —|— 1':rzzg(.-.2 cos -y + sin 72)].
`
`(6)
`
`We see that the surfaces of constant amplitude are given by
`
`z = constant,
`
`(7)
`
`and are, therefore, planes parallel to the boundary. The surfaces of constant real
`phase are given by
`
`x sin 3‘ + znq (cos y — K sin y) = constant,
`
`(8)
`
`and are planes whose normals make an angle 6; with the normal to the boundary,
`where
`
`cos I9; -=
`
`92q(cos y —— re sin -y)
`1/sin” 63,- + n9g*(oos y —— 1: sin 3;)’
`
`_
`8,
`sin 9,
`em = »
`l
`\/sin“ 6, —|— n“q’(cos y — K sin )2)“
`
`(9)
`
`Since the surfaces of constant amplitude and the surfaces of constant phase do not in
`general coincide with each other, the Wave in the metal is an inhomogeneous wave.
`If we denote the square root in (9) by n’, the equation for sin 6; may be written in
`the form sin 9' = sin 85/n’, i.e. it has the form of S1~TELL's law. However, 11.’ depends
`
`0004
`
`

`
`13.2]
`
`orrxos or METALS
`
`61'?
`
`now not only on the quantities that specify the medium, but also on the angle of
`incidence 9,.
`We may also derive expressions for the amplitude and the phase of the refracted
`and reflected waves by substituting for 3, the complex value given by (1) in the
`FBESNEL formulae (§ 1.5.2). The explicit expressions will be given in § 13.4.1 in con-
`nection with the theory of stratified conducting media. Here we shall consider how
`the optical constants of the metal may be deduced from observation of the reflected
`wave.
`
`Since we assumed that the first medium is a dielectric, the reflected wave is an
`ordinary (homogeneous) wave with 3. real phase factor. As in § 1.5 (2111) the amplitude
`components A ,, A L of the incident wave and the corresponding components R” R;
`of the reflected wave are related by
`
`‘R
`
`__ tan (9, — 9,)
`' E tan (6; + 6,)
`
`All
`
`RL =
`
`sin (6,- — 8,)
`'' 3W???)
`
`A L.
`
`(103
`
`Since 3, is now complex, so are the ratios RJA. and Ri/Al, i.e. characteristic
`phase changes occur on reflection;
`thus incident
`linearly polarized light will
`in
`general become elliptically polarized on reflection at the metal surface. Let <35. and
`95_,_ be the phase changes, and p. and p J_ the absolute values of the refleetion
`coeflicients, i.e.
`
`Suppose that the incident light is linearly polarized in the azimuth an i.e.
`
`A1.
`tan 9;, = A ,
`I
`
`(11)
`
`(12)
`
`and let at, be the azimuthal angle (generally complex) of the light that is reflected.
`Then‘
`
`Rl
`tana.,=—=—R]
`
`cos (B,- — 6,)
` :—B3 tan I1; = P8 _‘fi tan Ct.-,
`
`where
`
`=&
`PI»
`
`=
`
`9%
`
`_ ,
`¢l
`
`*3
`
`(14)
`
`We note that ac, is real in the following two cases:
`
`incidence (6. m 0);
`(1) For normal
`1 * tan My
`
`then P = 1 and A m -11-, so that tan or,
`
`(2) For grazing incidence (6,; = 7r/2);
`=‘ tan 3‘.
`
`then P = 1 and A = 0, so that tan ac,
`
`It should ‘be remembered that in the case of normal incidence the directions of the
`
`incident and reflected rays are opposed;
`
`thus the negative sign implies that the
`
`“ We write — €A rather than —}- 15A in the exponent on the right.-hand side of (13) to facilitate
`comparison with certain results of § 1.5.
`
`0005
`
`

`
`618
`
`raincirmis or OPTICS
`
`[13.2
`
`azimuth of the linearly polarized light is unchanged in its absolute direction in space.
`It is also unchanged in its absolute direction when the incidence is grazing.
`Between the two extreme cases just considered, there exists an angle 5, called
`principal angle of incidence which is such that A = — -ir/2. ‘At this angle of incidence,
`linearly polarized light is, in general, reflected as elliptically polarized light, but as
`may be seen from §l.4 (311)) (with 6 = «[2], the axes of the vibration ellipse are
`parallel and perpendicular to the plane of incidence.
`If, moreover, P tan 0:, = 1,
`then according to (13) tan on, = — i, and the reflected light is circularly polarized.
`Suppose that with linearly polarized incident light an additional phase diiference
`A is introduced between R, and R, by means of a suitable compensator (of. § 14.4.2).
`The total phase difierence is then zero, and,
`according to (13) the reflected light is linearly
`polarized in an azimuth oi; such that
`
`tan 0:; = P tan c:,.
`
`(15)
`
`_
`_
`_
`_
`_
`Fig’ 1&1‘ vibration ellipse of light re‘
`flected from a. metal at the principal
`angle of ;,,,._.,-,d,.,,,,,,,_
`
`The angle oi; is, for obvious reasons, called the
`angle of restored polarization,
`though it
`is
`usually defined only with incident light that is
`3'
`I
`lc'd'
`th
`'
`th
`,-=45°*.
`'lth:a:a?lu:Js0ofr:feandnP reflasdimililo th: rinci al
`_
`_
`1'
`_ g
`P
`P“
`angle of incidence 6, = 5,
`be denoted by at,
`and P respectively. If we imagine a rectangle
`to be circumscribed round the vibration ellipse of the (uncompensated) reflected
`light obtained from light that is incident at the principal angle, with its sides parallel
`and perpendicular to the plane of incidence, then the sides are in the ratio P tan cc,
`and the angle between a diagonal and the plane of incidence is oi; (see Fig. 13.1).
`For the purpose of later calculations it is useful to introduce an angle to such that
`
`tango: P;
`
`(16)
`
`the value of 1p corresponding to the principal angle of incidence will be denoted by 1,5.
`Using (10) and (1) we can compute the quantities P (= tan ip) and A in terms of 6,,
`if the constants n and x of the metal are known. Fig. 13.2 (a) shows their dependence
`on B, in a typical case.
`In Fig. 13.2 (b) analogous curves relating to reflection from
`a transparent dielectric are displayed for comparison. The sudden discontinuity from
`-11- to O in the value of A which occurs when light is reflected from a transparent
`dielectric at the polarizing angle is absent when light is reflected from a metal
`surface. The sharp cusp when tan zp becomes infinite is likewise absent, and the curve
`is replaced by a smooth curve with a comparatively broad maximum. The angle of
`incidence at which this maximum occurs is sometimes called the guard-polarizing
`angle;
`it is nearly equal to the principal angle of incidence 5,».
`It is commonly
`assumed that this maximum is actually at 5,, which is almost exactly true if
`‘n2(l —)— K2) >> 1, as is usually the case [cf. Table XVII]. In general the two angles are,
`however, different;
`for example, in the case of silver at the ultraviolet wavelength
`3280 A;
`the quantity n’(1 —|— K2) is small;
`then 9, = 47-8° and iv = 31-8°, whereas
`rpm“ = 295° and occurs at 8, = 40“, approximately.
`Generally speaking the problem is not to find ip and A from known values of ii and
`K, but to determine at and K from experimental observations of the amplitude and phase
`of light reflected from the metal.
`
`" Then an; is equal to the angle up introduced in (16).
`
`0006
`
`

`
`13.2]
`
`orrros or METALS
`
`619
`
`ip and A are all functions of 9,-, and of a and K,
`As the quantities RI, R_I_, qil, ¢>J_,
`measurement of any two of these quantifies for a specific value of the angle of
`incidence 9‘ will in general permit the evaluation of n and 1:. Since in many experi-
`ments one determines the last two of these quantities, we shall derive the fundamental
`expressions for n and K in terms of 1p and A. From (13) and (1)
`
`cos 8; cos 3, __
`sin 3, sin 6, _
`
`N/‘ti.’ —~ sin‘ 6’,»_
`sin 9; tan 3,-
`
`(17)
`
`-4
`
`11"
`
`1 ~— Pe“"5
`1 + Pe“""-" =
`-A
`
`‘IT
`
`E2
`
`'
`
`0
`
`L5’ 30“ 45° 60’ ?|5° 90’
`
`9.
`"
`
`'
`
`0
`
`I5‘ 30‘ 45° 60' 7.5’ 9
`on
`
`"ion [0
`I05
`r-04
`
`t-cal
`mal-
`1-0!‘
`H30
`
`0
`
`I
`J_.-_.1-. _L_,_:_
`F5‘ 30‘ 45' 6'0’ 7.5’ 95"
`
`_._
`
`_
`9‘
`
`-
`.
`:00 L5’ 50° £5‘ 50" .75’ 90'
`
`_
`
`_
`9‘
`
`la}
`
`{b}
`
`Fig. 13.2. The quantities --A = q5_,_ —~ 99, and P = tan tp = PJ_[P" which
`characterize the change in the state of polarization of light on reflection from
`a typical metal {3} and from a transparent dielectric (13).
`
`Since P = tan go, the left-hand side of equation (17) may be expressed in the form
`
`I — P345 __ 1 —e‘'mta.ng1 “ cos2gu+£siJi21psi,n.*l
`l+ Pe“"A _1—{—e‘mtan1p _ 1 —}—sin21pcoszk
`
`From (17) and (13),
`
`____
`‘Vii.’ — sinamflg _
`sin 9; tan 3; —
`
`cos 21p+isin 21,9 sin
`1 —}— sin 2tp cosfi
`
`Now if, as is usually the case in the visible region,
`
`-nF(1 + K2) >- 1,
`
`sin? 19,- may be neglected in comparison with at? and we obtain
`
`n(l + ix)
`ii.
`sin 8‘ tan 0, _ sin 6, tan 6; N
`
`_* cos 21;: + isin 2:}: sin A
`1 + SiI12tp cos A
`
`Equeting the real parts, we obtain
`
`find:
`
`sin 5, tan 9,- cos 21;:
`1 + sin 21,0 cos A
`
`Equating the imaginary parts and using (223) we find that
`
`K as tan 21p sin A.
`
`(13)
`
`(19)
`
`(20)
`
`(21)
`
`(22a)
`
`(22b)
`
`0007