Principles of Optics
`
`.
`
`ELECTROMAGNETIC THEORY OF PROPAGATION
`INTERFERENCE AND DIFFRACTIOM OF LIGHT
`Sixth Edition
`MAX BORN & EMIL WOLF
`
`Pergamon' Press
`
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`..
`
`Principles of Optics
`
`Electromagnetic Theory of Propagation,
`Interference and Diffraction of Light
`by
`
`MA X BORN
`M.A., Dr.Phil., F.R.S.
`Nobel Laureate
`Formerly Prof essor at the Universities of GIJttingen and Edinburgh
`and
`EMIL WOLF
`Ph.D., D.So.
`Professor of Physics, University of Rochester, N. Y.
`
`'
`with contributions by
`A. B. BHATIA, P. C. Cr.EMMow, D. GABOR, A. R. STOKES,
`A. M. TAYLOR, P . A. WAYMAN and w. L. WILCOCK
`
`SIXTH EDIT~ON
`
`PERGAMON PRESS
`
`OXFORD · NEW YORK · TORONTO · SYDNEY · PARIS · FRANKFURT
`
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`•
`
`U.K.
`
`AUSTRALIA
`
`FRANCE
`
`FEDERAL REPUBLIC
`OF GERMANY
`
`Pergamon Press Ltd., Headington Hill Hall,
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`Pergamon Press GmbH, 6242 Kronberg-Taunus,
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`
`Copyright© 198Q M<!X ~om and Emil Wolf
`All Fi/gl,ts ~eser'ved. No par,t oj this publication may be
`reproduce<!;, ~towd in, a retrieval system or transmitted in
`anJ_ f.orm or by. qn:v, 111,eaTJ,<: electrqnfc, electrostatic,
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`in writi"ng from
`the
`copyrigh{.hql¢e,rs
`
`:first e\}ition 19,1,) 9, ·
`Sec;:?nd (re~d) edition 196~
`Thfrd (revised) edition -1965
`:fo,urth (revised) edition 1970
`:fifth (re~cl) ~dition 1975
`R.~J?i:W.\~!;l; ~975.i 19,7,'?
`S.ixtl:i. ~di1ti9n_ ~ ~B.O.
`Lil;miry o{ Cc;mgress. Cataloging in Pul>Iication Data
`Bo;r:n, Max
`J.?ri,nciples of optics. - 6th ed.
`l. Optics - Collec;:ted works
`~· ~i.He
`I~. Wolf, ~mil
`535
`(lC351
`80-41470
`IS:SN Oc08-026482-4 hardcover
`~SBN 0-0(!~026481-9 ~«~.co\'.er
`
`Prt'nted in Great Bri~ain by A. Wheaton (I Co. Ltd., Exeter
`
`I
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`THE ii$
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`follows'
`additiol
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`j
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`,,
`
`156
`PlUNOIPLES OF OPTICS
`Let c be the distance between the focal ·points F~ and F 1 • Since the image space o
`the first transformation coincides with the object space of the second,
`Z1 = Zo - c,
`Y1 = Yo·
`Elimination of the coordinates of the intermediate space from (22) by means of (23)
`gives
`
`Y' = Z{Y1 = Z{Yo = ZifoYo = fof1Yo.
`' )
`Ji
`Ji
`1
`!ofo - cZo
`f{Zo
`Z' = f1fi = . f1fi =
`Z0 - c
`Z1
`1
`
`.
`fif{Zo
`f ofo - cZo
`
`.
`
`fifi =
`fofo
`- - c
`·
`Zo
`z = Zo _ fofo,}
`
`Let
`
`Y= Y 0 ,
`
`Z' = z; +/,!I.
`
`where
`
`Equations (25) express a change.of'Coordinates, the origins of the two systems being
`fif{/c respectively in the Z-direction. In terms of
`shifted by distances f~f0/c and -
`these variables, the equations of the combined transformation become
`f
`Y'
`Z'
`-v=z=r·
`r =101:.
`I = _!0!1,
`c
`The distance between the origins of the new and the old systems of coordinates, i.e.
`the distances d = F 0F and d' = F~F' of the foci of the equivalent transformation
`from the foci of the individual transformations are seen from (25) to be
`d = !of~.
`d' = _/iii.
`c
`c
`If c = 0, then f = f' = c6 so that the equivalent collineation is telescopic.
`equations (24) then reduce to
`
`c
`
`/1
`,
`Y1 = J{, Yo,
`
`} .
`
`!if{ z
`Z I
`1 -:- I ofo. 0 ;
`the constants <:1. and· p"in (18) of the equivalent transformation, are therefore
`{J _ !iii.
`/1
`fo
`-
`fofo
`
`( ) ( - -
`
`-
`The angular magnification is now
`Jo
`tan y'
`tan y =ff= J{
`If one or both of the transformati9ns are telescopic, the above considerations must
`be somewhat modified.
`
`,
`
`<:1.
`
`(20)
`
`(30)
`
`(31)
`
`, ,4)
`
`We shall now
`tioilB· In ~his
`lie in the 1mn
`powers of off·
`be neglected.
`
`4.4.l R efra•
`Consider ape
`two homogex
`ro.ye in both
`origin will be
`symmetry.
`Let P 0(Xo,
`respectively.
`§ 4.1 (40), at
`the two rayE
`
`where, a.cco
`
`r being the
`Let us ex
`to lie in th
`P1 will ther:
`rays, so tbE
`Now fror
`
`and substit
`
`• As befo
`used. The v
`the Teaohinf
`
`I
`
`\
`I
`I
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`1 space of
`
`IS of (23)
`
`(24)
`
`(25)
`
`as being
`;erms of
`
`(26)
`
`(27)
`
`.tea, i.e.
`·ma ti on
`
`(28)
`
`c. The
`
`(29)
`
`(30)
`
`(31)
`
`1 must
`
`4.4]
`
`GEOMETRIOAL THEORY OF O PTIC AL IMAGING
`
`157
`
`4.4 GAUSSIAN OPTICS
`
`We shall now study the elementary properties of lenses, mirrors, and their combina(cid:173)
`tions. In this elementary theory only those points and rays Will be considered which
`lie in the immediate neighbourhood of the axis; terms involving squares and higher
`powers of off-axis distances, or of the angles which the rays make with the axis, will
`be neglected. The resulting theory is known as Gaussian optics.*
`
`4.4.1 Refracting surface of revolution
`Consider a pencil of rays incident on a refracting surface of revolution which separates
`two homogeneous media of refractive indices n0 and ?ti· To begin with, points and
`rays in both media will be referred to the same Cartesian referen® system, whose
`origin will be to.ken at the pole 0 of the surface, with the z-direction along the a.xis of
`symmetry.
`Let P 0(x0 , y0 , z0) and P1(x1 , y1, z,,) be points on the incident and on t he refracted ray
`respectively. Neglecting terms of degree higher than first, it follows from § 4.1 (29),
`§ 4.1 (40), and § 4.1 (44) that the coordinates of these points and the components of
`the two rays are connected by the relations
`2
`Zo = -..,- = 2.Plpo + <(pl>
`0T<
`>
`0Po
`oT<2>
`dJlx = - 2fip1 - <(po,
`
`}
`
`(l a)
`
`Xo -
`
`Po
`-
`no
`.
`P
`X1 - n: Z1 = -
`
`Yo -
`
`"C = -
`
`ClT121
`q0
`- Zo = -"- = 2.Plqo +<(qi,
`no
`oqo
`q
`o~l'm
`Y1 - ~zx = - - - =
`n,
`Oql
`where, according to § 4.1 (45),
`1
`r
`r
`'
`.
`d = fJ = -
`'
`?ti - n 0
`2 ?ti - n0
`r being the para.rial radius of curvature of the surface.
`Let us examine under what conditions all the rays from P0 (which may be assumed
`to lie in the plane x = 0) will, after refraction, pass through P1• The coordinates of
`P1 will then depend only on the coordinates of P0 and not on tho components of the
`rays, so that when q1 is eliminated from (lb), q0 must also disappear.
`Now from the first equation '(lb}
`q1 = ~ {Yo - qo(2.PI + ~ zo)),
`and substituting this into the second equation; 've obtain
`
`}
`
`- 2f'q1 - <(go,
`
`(l b)
`
`(2)
`
`(3)
`
`Y1 = -
`
`( 2f4 - :~ Z,,) ~Yo + [ ~ ( 2f4 - ~ Zi) ( 2.PI + 1~0 Zo) - ~] qo.
`
`(4)
`
`• As before, tho usual sign convention of analytical geometry (Cartesian sign convention) is
`used. The various sign convent.ions employed in practice are very fully discussed in a Report on
`the Teaching of Geometrical Optics published by tbe Physical Societ y (London) in 1934.
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`158
`Hence Pi will be 'a stigmatic image of P0 if
`
`PRINCIPLES OF ' OPTIOS
`
`or, on substituting from (2), if
`
`(6) may- be written in the form
`
`It is seen that, within the present degree of approximation, every point gives rise to
`stigmatic image; and the distances of the conjugate planes from the pole 0 of th
`surface are related by (7). Moreover, equation (4), subject to (5), implies thait th
`imaging is a projective transformation.
`The expressions on either side of (7) are known as Abbe's (refraction) invariant a
`play an important part in the theory of optics.I' imaging. (7) may also be written '
`the form
`
`?ti
`-
`Zi
`
`?ti - no
`no
`- -= ---·
`zo
`r
`
`Th~ quantity (?ti - n 0)/r is known as the power of the refracting surface and will b
`denoted by f!JJ,
`
`According to (4) and (5) the lateral magnification Yi/Yo is equal to unity whe
`Zi/?ti = 2fJ + ~. But by (2), 2&/ + ~ = 0. Hence the unit points U0 and Ui are
`given by z0 = z1 = 0, i.e. the unit points coincide with the pol,e of the surface. Further,
`from (8),
`
`and
`
`asz0 ~ -
`
`co,
`
`so that the abscissae of the foci F 0 and F 1 are - n0r/(ni - n 0) and n,,r/(n1
`respectively. The focal lengths / 0 = F0U0,f1 = F 1 U1 a.re therefore given by
`
`- n 0)
`
`and ·
`
`fo = _!!:L,
`n1 -no
`
`}
`
`n 1r
`Ji = - - - - ;
`?ti - no
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`[4.4
`
`(5)
`
`(6)
`
`(7)
`
`s rise to a
`0 of the
`that the
`
`riant and
`Titten in
`
`(8)
`
`:l will be
`
`(9)
`
`·Y when
`ul a.re
`further,
`
`(10)
`
`4.4]
`or, in terms of the power gJ of the surface,*
`
`GEOMETRICAL THEORY 01!' OPTICAL IMAGING
`
`159
`
`no __ n1_/1Jl
`Ji - ;;r.
`Jo -
`Since f 0 and f 1 hav:e different signs, the imaging is concurrent (cf. § 4.3.3). If the surface
`is convex towards the incident light (r > 0) and if n0 < n 1 then Jo > 0, J1 < 0, and
`the iJnaging is convergent. If r > 0 and n0 > ni it is divergent (Fig. 4.12). When the
`surface is concave towards the incident light the situation is reversed.
`
`(11)
`
`Fig. 4.12. Position of the cardinal points for refraction ate. surface of revolution.
`
`(12)
`
`Using the expressions (10) for the focal lengths, (8) becomes
`~+b=- 1 ·
`Zi
`'
`Zo
`and the coefficients (2) may be written in the form
`go=-. h_, ~ = _ _ Jo= J1.
`.JJ1 = h_,
`n0
`2n0
`2n1
`n1
`Let us introduce separate coordinate systems in the two spaces, with the origins at
`the foci, and with the axes parallel to those at 0:
`Xo = Xo,
`X1 = X1,
`Yi = Yi·
`Yo= Yo•
`Z1 = Zi + f1·
`Zo = Zo +Jo,
`• It will be seen later that the relation n 0/f0 = - n1/f 1 is not restricted to e. single refra.cting
`eurface, but holds in general for any centred system, the quantities with suffix zero referring to tbs
`object space, those with the suffix 1 to the image space. (11) may therefore be regarded a.a· defining
`the power of a. genera.I centred system. The practice.I unit of power is a dioptre; it is the reciprocal
`of the focal length, when the focal length is expressed in meters. The power is positive when .the
`eystem is convergent (/0 > 0) o.nd negative when it is divergent (/0 < 0).
`
`(13)
`
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`160
`
`PRINOIPLES OF . OPTICS
`
`Equations (12) and (4), subject to (5), then reduce to the standard form §.4.3 (10)
`
`lo
`Z1
`Y1
`Yo= Zo = !1.
`
`4.4.2 Reflecting surface .of revolution
`It is seen that with a notation strictly analogous to that used in the previous section
`equations of the form (la) and (lb) also hold when reflection takes place on a sun
`of revolution, the coefficients d, !?,band <I now being replaced by the correspon ·
`coefficients d', !?.b' and~· of§ ~.1.5. It has been shown in § 4.1.5 thatd', !?.b' and
`can be obtained from d, !?.band~ on setting n0 = - ni = n, n denoting the refr
`tive index of the medium in which the rays are situated. Hence the appropria.
`
`-~~~~~~~c="-=c~__,.,c,---z
`ro,r,
`
`Fig. 4.13» Position of the cardinal points for refiectioo :in a. mirror of revolution.
`
`f'<O
`
`formulae for reflection may be ~mediately written down by making this substitutio
`in the .preceding formulae. tn: particular, (7) gives
`.1
`1
`l
`1
`- +·-,
`-
`-
`- =
`· z1
`z0
`r
`r
`
`-
`
`the expression on either side of (15) being Abbe's (reflection) invariant.
`be written as
`·
`
`1
`1
`2
`-+- =-·
`z1
`z0
`r
`The focal lengths fo and f 1-are now given by
`
`r
`lo = f1 = - 2•
`
`and the power f!JJ is
`
`2n
`f!JJ = - - · r
`Sincef0J1 > 0, the imaging is contracurrent (katoptric). When the surface is convex
`·towards the incident light (r > 0) Jo < 0, and the imaging is then divergent; when it
`i!J: concave towards the incident light, (r < 0), Jo> 0 and the imMing is convergent'.
`(Fig. 4.13).
`.
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`10):
`
`(14)
`
`GEOMETRICAL THEORY OF OPTIOAL IM.A.GINO
`
`161
`
`.i.4]
`4A.3 The thick lens
`Next we derive the Gaussian formulae relating to imaging by two surfaces which are
`rotationally symmetrical about the ,same axis.
`Let n 0, ni and n 2 be the refractive indices of the three regions, taken in the order
`in which light passes through them, and let r1 and r2 be the radii of curvature of the
`surfaces at their axial points, measured positive when the surface is convex towards
`the incident light.
`By (10), the focal lengths of the first surface are given by
`!~ =- ~· 1"1 - no
`
`(19)
`
`,y: '
`
`u
`
`U'
`
`:F'
`
`d
`
`c
`
`Fa
`
`.:Ff
`J'{,' ~· U,=U1
`Ua=Uo
`Fig. 4.14. The ca.rdina.l points of a. combined system (thick Jens).
`
`and of the second surface by
`f -
`f' -
`nzr2
`nir2
`1 -
`1 - n2 - ni'
`na -
`1"1
`According to § 4.3 (27) the focal lengths of the combination are
`f' =.f~f~,
`f = _Jof1
`c '
`c
`where c is the distance between the foci F~ and F 1 • Lett be the axial thickness of the
`lens, i.e. the distance between the poles of the two surfaces; then (see Fig. 4.14)
`c = t + !~ - Ji.
`On substituting into (22) for f~ and ·for / 1 we obtain
`D
`'
`c=
`(ni - no).(nz ·- ni)
`
`-
`
`~20)
`
`(21)
`
`(22)
`
`(23)
`
`(24)
`
`where
`
`The required expressions for the focal lengths of the combination a.re now obtained
`on substituting for / 0, / 1 , f~, f~ and c into (21). This gives
`
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`162
`PRINCIPLES OF OPTIOS
`Since ff' is negative, the imaging is concurrent. The power fJ' of the lens is
`fJ' = n0 = _nz= _ _!_ D
`!'
`f
`n,. r1r2
`t
`= gi>1 + fJ'z -
`fJ'1 fJ'z •
`-
`n,
`where fJ'1 and fJ'2 are the powers of the two surfaces.
`By§ 4.3 (28), the distances <5 = F 0F and&' = F~F' are seen to be
`nz - n,. ,.12
`n,. - no r22
`'
`6 = n,.n2 - -- D.
`<5 = - non, - - - D'
`n, - no
`nz - n,.
`The distances d and d' of the principal planes o/I and o//' from the poles of the surfa.c
`are (see Fig. 4.14)
`
`d = <5 + f - fo = - n 0(n2 - n1) ;;.}
`
`rt
`d' = <5' + f' - f~ = n 2(n1 - n 0).J; ·
`Of particular importance is the case where the media on both sides of the lens
`of the same refractive index, i.e. when n 2 = n0. If we set nJn0 = n,./n2 = n, th
`formulae then reduce to
`
`r,_t
`d' = (n -1)-X'
`
`where
`
`6 = (n -
`(n -
`l )t].
`r 2) -
`l)[n(rJ -
`Referred to axes at the foci F and F', the absoissa.e of the unit points are Z =fan
`Z' = f', and the nodal points are given by Z = - f', Z = - f. Since f = - f' th
`unit points and the nodal points now coincide. The formula§ 4.3 (16) which.relate&
`the distances ' and ,, of conjugate planes from the unit planes becomes
`
`1
`1
`1
`, - - r= - f '
`The lens is converge!l't (f > 0) or divergent (f < 0) according as
`
`f = - ny?: 0,
`
`1'11'2
`
`i.e. according as
`
`1 n-1 t
`1
`- - - § - - - .
`n
`"2
`"1
`r1r2
`When f = co, we have the intermediate case of telescopic imaging.
`n-1
`r 1 = - - t.
`n
`
`r 1 -
`
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`

`f 4.4
`
`(26)
`
`(27)
`
`surfaces
`
`(28)
`
`lens are
`= n, the
`
`(29)
`
`(30)
`
`=/and
`-!'the
`i relates
`
`(31)
`
`(32)
`
`(33)
`
`= 0, i.e.
`
`(34)
`
`163
`4.4]
`GEOMETRIOAL THEORY OF OPTIOAL IMAGING
`The three caaes ma.y be illustrated by considering a double convex lens, r 1 > 0,
`r < O (cf. Fig. 4.15). If, for simplicity, we assume that both radii a.re numerically
`6~ua.l, i.e. r 1 = -
`r2 = r, the imaging will be convergent or divergent according
`as t ~ 2nr/(n - 1) and telescopic when t = 2nr/(n - 1).
`
`UU'
`I
`I
`
`a
`
`U U'
`I
`I
`
`UU'
`I
`
`I !«
`m
`
`I
`I
`
`I
`,
`b
`
`Conver9enl;
`
`c
`
`UU'
`I I
`I I
`
`UV:
`I I
`I I
`
`I I
`a
`
`c
`
`Divergent;
`
`UU'
`I I
`
`11 ([ I
`
`I
`! I
`f
`
`Fig. 4.15. CoD'.ll'.llon types oflenses:
`(a.) Double-convex;
`(b) Plano-convex;
`(c) Convergent meniscus;
`(d) Double·con~a.ve; (e) Plano-concave; (f) Divergent meniscus.
`"II and If/' a.re the unit planes, light being assumed to be incident from the left.
`
`4.4.4 The thin lens
`The preceding formulae ta.ke a. particularly simple form when the lens is so thin that
`the a.x:ia.l thickness t ma.y be neglected. Then, according to (28), d = d' = 0, so that
`the unit planes pass through the axial point of the (infinitely thin) lens. Consequently,
`the rays which go through the centre of the lens will not suffer any deviation; this
`implies that imaging by a thin lens is a central projection from the centre of the lens.
`From (26) it follows, on setting t = 0, that
`
`(35)
`
`i.e. the power of a thin lens is equal to the sum of the powers of the surfaces forming it.
`If the media on the two sides of the lens are of equal refractive indices (n0 = n 2),
`we have from (25)
`
`- ~ = (n - 1) (!. - ~)·
`!_ =
`I
`f
`r2
`r1
`where as before, n = nJn0 = niJn2• Assuming that n > 1, as is usually the case, f is
`seen to be positive or negative according as the curvature l /r1 of the fust surface is
`greater or smaller than the curvature l /r2 of the" second surface (an appropriate sign
`being associated with the curvature). This implies that thin lenses whose thickness
`decreases from centre to the edge are convergent, and those whose thickness increMes
`to the edge a.re divergent.
`·
`For later purposes we sha.11 write down an expression for the focal lengths f and f' of
`a system consisting of two centred thin lenses, situated in air. According to§ 4.3 (27)
`we ha.ve, since / 0 = - f~. !1 = -
`fi,
`
`(36)
`
`c
`I
`1
`-= - -= - - ·
`f'
`fo/1
`f
`
`(37)
`
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`

`164
`a being the dista.nee between the foci F~ and F 1 (see Fig. 4.16).
`between the two lenses, then
`Z= fo + a+f1·
`
`l'RINOIPLES OF OPTICS
`
`~~~~~~~ i~~~~~~~~
`
`Fig. 4.16. System formed by two centred thin lenses.
`
`Hence.
`
`1
`1
`1
`1
`l
`l = - p =To +ii - !0!1.
`When the lenses a.re in contact (Z = 0), (39) ma.y also be written in the for
`f!J> = f!l'1 + f!l'2, so that the power oft.he combination is then simply equal to sumo
`the powers of the two lenses.
`
`4.4.5 The general centred system
`Within the approximations of Gaussian theory, a refraction or a reflection at a surfi
`of revolution was seen to give rise to a projective relationship between the object
`the image spaces.• Since according to § 4.3 successive applications of projeoti
`transformations a.re equivalent to a single projective transformation, it follows tha.
`imaging by a. centred system is, to the present degree of approximation, also
`transformation of this type. The cardinal points of the equivalent transform~tio
`may be found by the application of the formulae of § 4.4.1, § 4.4.2 and § 4.3.4.
`shall ma.inly con.fine our discussion to the derivation of an important invariant relatio
`valid (within the present degree of accuracy) for any centred system.
`
`y
`
`s,
`s,
`Fig. 4.17. Illustrating the SMITR- HELllaJOLTZ formula.
`Let 8 1, 8 2, • • • 8,,. he the successive surfaces of the system, / 0 , fv . . . f.,._1 tho
`corresponding focal lengths, and n0, 1'1, . . . nm the refractive indices of the successive
`spaces (Fig. 4.17). Further, let P 0,P: be two points in the object space, situated in a
`• As in the case of a single surfa.oe, the object and the image spaces a.re regarded as superposed
`on. to ea.oh other and extending indefinitely in all directions. The part of the object spa.oe which
`lies in front of the first surface (counted in the order in which the light traverses the system) is
`said to· form the real portion of the object space and the portion of the image spa.oe which follow•
`the last surface is called the real portion of the.image space. The remaining portions of the two
`spaces are said to be virtual. Iu a similo.r mo.nner we may define the real and virtuo.l parts of any
`of the intermediate spac~ of the system.
`
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`

`

`~ distance
`
`(38)
`
`F.' 1
`::l
`
`(39)
`
`he form,
`;o sum of
`
`a. surface
`1ject and
`rojective
`)WS that
`1, also a.
`1rmation
`3.4. We
`relation
`
`(
`- z
`\ s,
`
`~m-l the
`ccessive
`ted in a
`
`perposed
`~e which
`rstem) is
`l follows
`the two
`;s of any
`
`GEOMETRICAL THEORY Ol!' Ol'TIOAL IMAGING
`
`165
`1Deridional plane, and let P1,Pf, P 2,pt . .. , be their images in the successive surfaces.
`Referred to axes at the foci of the first surface, the coordinates of P0,P: and P 1,.Pf
`are related by eq. (14), viz.
`
`Hence
`
`and
`
`Let
`
`z: - Z0
`fo
`
`Then from (~) and (43),
`
`Y 1Y: - Y 0Yt
`Y1 Yf
`.
`
`Y 0
`Y:
`= Yf - Yi =
`z: - Z 0 = D.Z0,}
`Zt - Z1 = D.Z1.
`
`Now by (10), / 0/fi =
`
`foYoY:
`/1Y1Yt
`D.Z1 ..
`LlZ
`= -
`0
`.....:. n 0/n,., so that this equation may be written as
`n 0 Y 0r:
`n,.Y1 Yt
`.
`D.Zo
`= D.Z1
`
`(40)
`
`(41)
`
`(42)
`
`(43)
`
`(44)
`
`(45)
`
`(46)
`
`and generally
`
`=
`
`-
`
`(l < i < m).
`
`Similarly, for refraction at· ~he second surface,
`n 2 Y2Yt
`niY1 Yt
`,
`D.Z1
`D.Z2
`n,_1 Y,_1 Yt_1 _ n,Y,Yf'
`D.Z,_1
`D.Z,
`Hence n,Y,Yrf D.Z1 is an invariant in the successive transfomiations. This result plays
`an important part in the geometrical theory of image formation. If we set Yf' / D.Z, =
`tan y, (see Fig. 4.17), (48) be!)omes
`n 1_ 1 Y1_ 1 tan y,_1 _: n,Y, tan y,.
`Since to the present degree of accuracy, tan y and tan y' may be repl~ced by y and y'
`respectively, we obtain the Smithr-Helmholtz formula*
`n,_1 Y,_1yH = n,Y i'Yi·
`The quantity n,Y,.y, is known as the Smithr-Helmholtz invariant.
`From (48) and (49) a number of important conclusions may. be drawn. As one is
`usually interested only in relations bet.ween quantities pertaining to the first and the
`last medium (object and image space), we shall drop the suffixes and denote quantities
`which refer to these two spaces by unprimed and primed symbols respectively.
`
`(47)
`
`(48)
`
`(49)
`
`• This formula. ie a.tao a.eeooia.ted with the names of LAGRANGE and Cuusros. It was a.otuo.lly
`known in more restricted forms to earlier writers, e.g. HUYGENS and COTES. (Cf. Lord RAYLEIGU,
`Phil. Mag., (5) 21 (1886), 466).
`PQ6U, Ed .... Q
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`

`166
`PRINCIPLES OF OPTIOS
`Let ( Y,Z) a.nd (Y + oY, Z + oZ) be two neighbouring points in the object sp
`a.nd (Y',Z') a.nd (Y' + oY', Z' + oZ') the conjugate points. The SMITH- HELMROL
`formula. gives, by successive application, the following relation:
`nY{Y + oY)
`n'Y'(Y' + oY')
`o·z
`oZ'
`In the limit as o Y -+- 0 and oz-+- 0, this reduces to
`dZ'
`n' Y' 2
`dZ = n Y2 •
`
`According to § 4.3 (11), we may write Y'/Y = (dY'/dY)z-comt.• and (51) becomes
`n' (dY')a
`
`dZ'
`dZ = ;: dY z -comt.'
`
`known a.s Maxwell's elongation formula. It implies that the longituclinal magnificat'
`is equal to the square ·Of the lateral magnification multiplied by the ratio n' Jn of t
`refractive indices. Now in§ 4.3 we derived an a.na.logous·formula [(13)) connecting t
`magnifications a.nd the ratio of the focal lengths. On comparing these two formul
`it is seen that
`
`f'
`n'
`-=--,
`f
`n
`i.e. the ratio of the focal lengths of the instrument is equal to the ratio n'/n of the refractiu
`indices, taken with a negative sign.
`From the S:t>nTH-HELMROLTZ formula it also follows, that
`
`dY' Y'
`n
`dY Y = n'
`showing that the produd of the lateral .and the angular magnifu:ation is independent a
`the ·choice of the conjugate planes.
`It has been assumed so far that the system comists of refracting surfaces only.
`one of the surfaces (say the ith one) is a. mirror, we obtain in pla-0e of (48),
`Y,_1Yt,1
`Y,Y'f
`tiz, '
`AZ,_1 = -
`the negative sign a.rising from the fact that for reflection f,_Jf; = 1, · whereas fo
`refraction f1-1/f, = - n 1_ 1/n1• In consequence n' must be replaced by - n' in th
`finail formulae. More generally n' must be replaced by - n' when the system conta'
`an odd number of mirrors; when it contains an even number of mirrors, the fi
`formulae remain unchanged.
`
`4.5 STIGMATIC IMA GING WITH WIDE-ANGLE PENCILS
`
`The laws of Gaussian Optics were derived under the assumption that the size of the
`object and the angles which the rays make with the axis are sufficiently small. Often
`one also has to consider systems wher.e the object is of small linear dimensiom, b
`where the inclination of the rays is appreciable. There are two simple criteria., known
`
`the .ri
`-
`mauoh i
`Let Oa
`~yon
`let 0 1 an
`Let(%
`referred
`taken al
`t,llepath
`the poin
`
`<Po"'• qo
`correap
`P0 and
`them
`Two
`'4, = 0
`The t.wQ
`
`4.5.1
`Withou
`(:ra =
`
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`
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`

`

`[4.ts
`ject space
`ELMROLTZ
`
`(50)
`
`(51)
`
`ecomes
`
`(52)
`
`mification
`'/n of the
`~ctingthe
`formulae
`
`(53)
`
`ref radive.
`
`(54)
`
`. ereas for
`
`ILS
`ze of the
`ll. Often
`ions, but
`i, known
`
`167
`4.5)
`GEOMETRICAL TR»ORY OF O:PTIOAL IMAGING
`88 the sine condition* and the Herschel conditiont relating to the stigma.tic imaging
`in such instruments.
`Let 0 0 be an axial object point and P0 any point in its neighbourhood, not neces(cid:173)
`sarily on the axis. Assume that the system images these two points stigma.tioally, and
`let 0 1 and P1 be the stigma.tic images.
`, y0 , z0) and {x1, y1 , Zi) be the coordinates of P 0 and P 1 respectively, P0 being
`Let (x0
`referred to rectangular axes at 0 0 a.nd P1 to parallel axes at 0 1, the z-directions being
`ta.ken a.long the axis of the system (Fig. 4.18). By the principle of equal optical path,
`the pa.th lengths of all the rays joining P0 a.nd P1 are the same. Hence, if V denotes
`the point chara.cteristic of the medium,
`
`(1)
`
`Fig. 4.18. Illustrating the sine condition and the IIEnaOBEL condition.
`
`F being some function which is independent of the ra.y components. Using the basic
`relations § 4.1 (7) which express the ra.y component in terms of the point characteristic,
`we have from (1), if terms above the first power in distances a.re neglected,
`(P110>x1 + q110,Y1 + mi<0>Zi) - <Po101Xo + qo<0>yo + mo<0>zo) = F(xo, Yo• zo; Xi• Y1• Zi), (2)
`(p010), q0<o>, m0<0>) a.nd (p1<oi, q1CO>, mi<O>) being the ray components of a.ny pair of
`corresponding rays through 0 0 and 01• It is to be noted that, although the points
`P0 and P1 are assumed to be in the neighbourhood of 0 0 and 0 1 , no restriction as to
`the magnitude of the ray components is imposed.
`Two cases are of particular interest, namely when (i) P0 and P1 lie in the planes
`z0 = 0 and z1 = 0 respectively, and (ii) when P0 and P 1 are on the axis of symmetry.
`The two cases will be considered separately .
`
`4.5.1 The sine condition
`Without loss of generality we may again consider only points in a meridional plane
`(x0 = x1 = 0). If P0 lies in the plane z0 = 0 and P1 in the plane z1 = 0, (2) becomes
`(3)
`
`• The sine condition was first derived by R. Cr.Ausros (Pogg. Ann., 121 (1864), 1) and by
`R. HELMHOLTZ (Pogg. Ann. Jubelband (18'74), 557) from thermodynamical considerations. Its
`importance was, however, not recognized until it was rodiscoverod by E. ABBE (Jena.iscl~. Gu.
`Mul. Naturw. 1879), 129, also Carl. Repert. Phys., 16 (1880), 303).
`The derivations given bore are essentially due to C. HoCKIN, J. R oy. Micro. Soc. (2) 4 (1884), 337.
`t J. F. W. HEBSOB'EL, Phil. Tram. Roy. Soc., ill (1821), 226.
`
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`

`

`PRINOIPLES OF OPTIOS
`
`168
`[4.5
`This relation holds for each pair of conjugate rays. In particular it must, therefore
`hold for the axial pair p 0<0l = q0<oi = 0, p 1<0l = q1<0i = 0. Hence
`'
`F(O, y 0 , O; 0, y1, 0) = 0.
`
`Relation (3) becomes
`
`(4)
`
`or, more explicitly,
`
`n1Y1 sin Y1 = noYo sin Yo•
`y0 and y1 being the angles which_ the corresponding ray~ through 0 0 and 01 make with
`the z-axis, and n0 and n1 being the refractive indices of the object and image spaces.
`(6) is known as the sine condition, and is the required condition under which a small
`region of the object plane in the neighbourhood of the axis is imaged sharply by a.
`pencil of any angular divergence. If the angular divergence is sufficiently small,
`sin Yo and sin y1 may be replaced by y 0 and y1 respectively, and the sine condition
`reduces to the SMITH-HELMHOLTZ formula, § 4.4 (49).
`
`ho
`
`Fig. 4.19. The sine condition, when the object ia a.t infinity.
`
`If the object lies at infinity, the sine condition takes a different form. Assume
`first that the axial object point is at a great distance from the first surface. If Z0
`is the abscissa of this point referred now to axes at the first focus, and h0 is the
`height above the axis at which a ray from the axial point meets the first surface, then
`sin y0 ,...., - h0/Z 0 ; more precisely Z 0 sin y0/h0 -+ -1 as Z 0 -+ -
`co whilst h0 is kept
`fixed. Hence, if Z 0 is large enough, (6). may be written~
`Yi z · Y
`no
`- ho = -
`-
`Yo
`n1
`But. by § 4.3 (10), y1Z0/y0 = f 0, and by § 4.4 (53) n0/ni = -
`(6) reduces to (see Fig. 4.19)
`
`/ 0//1, so that in the limit
`~
`
`o ai.µ 1·
`
`,,6)
`otf-a.xis dista
`ciroular coma.
`Since the~
`in terms of t~
`
`4.5.l The~
`Next conside
`s1 =Yi= 0)
`
`or, in terms
`
`In particula1
`
`Hence (10) 1
`
`This is one
`aseuroed to
`
`so that the
`
`'
`
`I
`
`Whenth1
`neighb,outl\
`angular div
`Itistol
`aimultaneo·
`lateral ma~
`object and
`
`ho
`sin Y1 = f1·
`
`This implies that each ray which is incident in the direction parallel to the axis inter·
`sects its conjugate ray on a sphere of radius f 1, which is centred at the focus F 1 •
`Axial points which are stigmatic images of each other, and which, in addition, have
`the property that conjugate rays which pass through them satisfy the sine condition,
`are said to form an aplanatic pair. We have already encountered. such point pairs
`when studying the refraction at a spherical surface (§ 4.2.3).
`In the terminology of the theory of aberrations (cf. Chapter V), axial stigmatiem
`implies the absence of all those terms in the expansion of the characteristic function
`which do not depend on the off-axis distance of the object, i.e. it implies the absence
`of spherical aberration of all orders. If, in addition, the sine condition is satisfied,
`then all terms in the characteristic function which depend on the first power of the
`
`Rectilineal!
`The associi
`eeotion. I1
`In gene11
`It will the1
`rays . .
`
`4.6.1 Foe
`Lets den<
`rays, and ·
`and denof
`
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`

`

`[4.5
`
`~herefore,
`
`(4)
`
`(5)
`
`(6)
`lake with
`~e spaces.
`:ha small
`rply by a
`;ly sma.II,
`condition
`
`--r,
`
`Assume
`~e. If Z0
`h0 is the
`'ace, then
`i0 is kept
`
`. xis ~nter­
`s F 1•
`lion, have
`iondition,
`>int pairs
`
`igmatiem
`• function
`e absence
`satisfied,
`rer of the
`
`GEOMETRICAL THEORY OF OPTIOAL IMAGING
`
`6.6)
`off-a.xis distance must also vanish; these terms represent aberrations known as
`circular coma.
`Since the sine condition gives information about the quality of the off-axis image
`in terms of the properties of axial pencils it is of great importance for optical design.
`
`169
`
`4,5.2 The Herschel condition
`Next consider the case when P 0 and P 1 lie on the axis of the system (x0 = y 0 = 0,
`Si = y1 = 0). The condition (2) for sharp imaging reduces to
`m1<0>Zi - m0<0>z0 = F(O, 0, z0 ; 0, 0, Zi),
`
`(9)
`
`(10)
`
`or, in terms of y0 and y1,
`?iiZi cos y1 - n0z0 cos y0 = F(O, 0, z0 ; 0, 0, Zi)·
`In particular for the axial ray t his gives
`F(O, 0, z0 ; 0, 0, Zi) = ?iiZi - noZo.
`Bence (10) may be written as
`nizt sin2(y1/2) = noZo sin2(y0/2).
`This is one form of the Herschel condition. Since the distances from the origin are
`aaeumed to be smaJ.l, we have, by § 4.4 (52),
`
`(11)
`
`(12)
`
`(13)
`
`so that the HERSCHEL condition may also be written in the form
`
`?iiYt sin (y1/2) = noYo sin (y0/2).
`When the IlEBsoHEL condit ion is satisfied, an element of the axis in the immediate
`neighbourhood of 0 0 will be imaged sharply by a pencil of rays, irrespective of the
`angular divergence of the pencil.
`It is to be noted that the sine condition and the liEBsoHEL condition cannot hold
`simultaneously unless y1 = y0• Then y1/y0 = zt/z0 = n 0/ni, i.e. the longitudinal and
`la.tera.J. magnifications must then be equal to the ratio of the refractive indices of the
`object and image space.
`
`(14)
`
`4.6 ASTI GMATIC PENCILS OF RAYS
`Rectilinear rays which have a. point in common are said to form a homocentric pencil .
`The associated wave-fronts a.re then spherical, centred on their common point of inter(cid:173)
`section. It was with such pencils that we were concerned in the preceding sections.
`In general, the homocentricity of a pencil)s destroyed on r efraction or reflection.
`It will therefore be useful to study the properties of more general pencils of rectilinear
`rays.
`
`4.6.1 Focal properties of a thin pencil
`Let 8 denote one of the orthogonal trajectories (wave-fronts)