LLLLLLLL
`Eugene Hecht
`
`'Ade|phi University
`
`PEARSON
`/~""""'"'°'"”"“"‘*-\
`
`Addison
`
`Wesley _
`
`San Francisco Bosgon New York
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`
`Library of Congress Cataloging-in-Publication Data
`
`.
`.,
`Hecht, Eugene
`Optics I Eugene Hecht; -—- 4th ed.
`
`Includes bibliographical
`1. Optics; I.,'_lTi‘t1e.’ 5
`
`references and-i;index.’
`-
`
`‘
`
`QC355.3.H43
`535———-dc2l
`
`2002
`
`'
`
`’
`
`2001032540
`
`ISBN 0-8053-8566-5
`
`
`
`Copyright @2002 Pearson Education, Inc., publishing as Addison Wesley, 1301 Sansome St., San Francisco, CA 94111.
`All rights 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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`material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1900 E. Lake
`Ave., Glenview, IL 60025. For information regarding permissions, call 847/486/2635.
`‘
`
`Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where .
`those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed 6
`in mrtial caps or all caps.
`
`23 24 25-DOH—18 17 16 15
`
`ii
`
`WWW-aW-bc.-com/physics
`
`‘
`
`-
`
`

`
`
`
`1
`
`1
`1 A Brief History
`1.1 Prolegomenon
`_
`1
`1.2 Inthe Beginning
`1.3 From the Sell/enteenth Century
`A
`1.4 The Nineteenth Century" 4
`1.5 Twentieth’-Century Op"ti'jc_s’
`._ 7 "
`
`2
`
`y
`
`10 N
`'2 Wave Motion
`2.1 One-Dimenslonalwaves _ 10
`2.2 Harmonic Waves
`’1'4'"'-
`_
`2.3‘ Phase and Phase Velocityv" '17
`_
`2.4 The Superposition_Principle
`20 '
`‘
`3
`f
`2.5 The Complex Repres'entation_'__ 21
`2.6’ Phasors andthe Addition_',of-‘lll/_aves 23
`2.7 Plane Waves _24‘
`b’
`'
`2.8 The Three-Dimensional Differential _
`
`V
`
`Wave Equation 27
`_2.9 Spherical Waves 28 _
`2.10 Cylindrical Waves
`'31
`Problems
`32'
`'
`
`'
`
`3 Electromagnetic Theory, Photons,
`and Light 36 S
`.
`.
`,
`L
`3.1 Basic Laws of Electromagnetic Theory_- 37 .
`3.2 Electromagnetic Waves" 44
`A
`'
`3.3 Energy and.Mo"mentum 47
`3.4 Radiation
`58
`
`'
`
`1
`3.5 Light in Bulk Matter 66
`3.6 The Electromagnetic-Photon Spectrum 73
`3.7 Quantum Field Theory 80- F
`-
`Problems-
`82-
`
`-
`
`s
`
`4 The Propagation of Light 86, ‘L
`'
`4.1 Introduction 86.
`
`T
`
`4.2
`
`4.3
`g 4.4
`
`4.5
`
`4.6
`
`4.75
`
`4.8
`
`4.9
`
`4.10
`
`4.11
`
`Rayleigh Scattering _ 86
`Reflection .95 ’
`Refraction
`100
`
`Fermat’s Principle
`106
`The Electromagnetic Approach ‘_111
`Total Internal Reflection
`122
`127
`Optical Properties of Metals
`Familiar Aspects of the Interaction of
`Light and Matter
`131
`The Stokes Treatment of Reflection and
`Refraction
`136
`'
`Photons, Waves, and Probability
`Problems
`141’
`'
`'
`
`137 «
`
`3
`
`6
`
`r 5 Geometrical Optics 149
`Introductory Remarks 149
`Lenses
`150
`
`5.2
`
`5.1
`
`‘ 5.3
`
`5.4
`
`5.5
`
`5.6
`
`5.7
`
`5.8 4
`
`5.9
`
`Stops
`Mirrors
`
`171
`175
`
`Prisms
`
`186
`
`V
`
`'
`Fiberoptics
`193
`V
`201
`Optical Systems
`Wavefront Shaping 226
`Gravitational Lensing
`231
`Problems 234"
`
`‘
`
`6 More on Geometrical Optics 243
`Thick Lenses and Lens Systems
`243
`6.1
`Analytical Ray Tracing
`246.
`'
`Aberrations 253
`GRIN Systems 273
`.C;onc_|uding Remarks
`Problems . 277
`
`6.5
`
`276
`
`6.2
`
`6.3
`
`6.4
`
`

`
`vi Contents
`
`. 2
`
`<
`
`.8
`
`.
`
`'
`
`_
`
`.
`
`:'
`7 The Superpositionuof
`:
`Waves 281
`7.1 The Addition of Waves of the Sam
`‘Frequency 282
`7.2 The Addition of Waves of Different
`Freciuenci/, 294
`8
`7.3 Anharmonic Periodicvwaves,‘ 302
`7.4 Nonperiodic Waves _308
`Q
`P'°b'emS .320_’
`’ _
`. 1
`_
`_
`.
`_.
`-_.
`.‘
`_
`'
`_
`8 P°'3”Z3t'°" 325
`8-1 The Nature of Polarized Light 1.325 ,
`8.2 Polarizers
`331_ P
`_
`b
`'
`'
`8.3 Dichroism 333‘
`«
`’.
`3
`3-4 Birefringence 335
`'
`'
`8.5 Scattering and Polarization. 344}
`8.6. Polarization by Ref_le_CtiO.n
`348_ _§ _ v_
`.3-7 Re-leldele
`352
`_[
`s
`it
`.
`3-8 Ci’C”'aV.P°'a’lZe’5 357
`8.9 Polarization of Polychromatic Light
`3-10 0P’ClCalAC’IlViiY 350 -‘
`T "
`8.11 Induced Optical Effects-—-Optical -‘
`Modulators .365
`.
`~
`8-12 .L'q”'d C’Y5tf3'5
`370_ ,'
`8.13 AMathematical Description of
`Polarization 372
`-
`..
`Problems 379
`
`1
`
`.358
`
`‘
`
`_.
`
`= ..
`
`,.
`
`.
`
`.
`
`3
`9 interference 385
`.
`9.1 General Considerations .386
`9.2 Conditions for Interference. 3950,
`9.3 Wavefront-splitting interferometers 393
`9.4 Amplitude-splitting interferometers
`400.’.
`9.5 Types and Llocalizationfof Interference
`.
`Fringes 414
`'
`"
`9.6 Multiple—BeamInterference 416‘
`'_:
`9.7 Applications of Single and Multilayer *
`Films 425
`,
`'
`"A
`9.8 Applications of interferometry‘/“"431 '
`Problems 438
`
`‘
`
`‘
`
`-
`
`'
`
`_
`
`.
`
`10 _.Diffraction 443
`10.1 Preliminary Considerations .443
`10.2 Fraunhofer Diffraction 452'
`10.3 Fresnel Diffraction 485
`10.4 Kirchhoff’s Scalar Diffraction Theory 510
`10.5 Boundary Dirfractionwaves, 512
`problems 514
`5
`'-
`'
`'
`11’Fourie.r Qptips .519 V
`_ ‘_
`11.1 lntrodu'ct’ion' V519
`11.2 FourierTransforms 519
`11.3 Optical App_liC'ations 529
`problems 556
`
`.
`
`P
`
`12 Basics of ‘Coherence
`Theory 550,.
`’
`V
`12_1
`introductm 550
`_
`I
`9
`1
`12.2 Visibility" 562-’
`12.3 Th,e.Mut_ Coherenee Function and-the
`[)'"e§r'eé gfcdherence 3556 _
`12.4 Coherence and Stellar interferometry: _7573
`p‘r‘Ob|'em$ '573_
`'
`.
`
`.
`
`'
`
`13 Modern Optics: Lasers and
`Qther Topics’ 531
`-
`'
`581
`13.1 Lasers and Laserlight
`13.2.
`|m.38erye;— _T.h_e.SpatialDistribution of,
`_
`Optical lnformation 606,
`'
`C
`13.3 Holography 623
`’13_4 N¢n”‘nearQ'p-figs 539
`prob|ems 544
`»
`
`'
`
`=
`
`'
`
`Appendixl 549
`Appendix 2 _ 652
`*
`Tam“ 653-?
`Solutions to Selected Problems 658
`Bibliography 685
`Index 5891 "
`
`'
`
`1
`
`

`
`
`
`Tfie intricate color patterns shimmering across an oil slickvon
`3 wet asphalt pavement (see photo) result from one of the
`more common manifestationsof the phenomenon of interfer-
`ence.* on a macroscopic scale we might consider the related
`problem of the ipnteraction of surface ripples on aipool of
`water. Our everyday experience with this kind’ of situation
`allows us to envision acomplex distribution of disturbances
`(as shown, e.g.,' in Fig." 9.1). There might be regions Where two
`.(or more) waves have overlapped, partially or even complete-
`1y canceling each other. Still other regions might exist in the
`pattern, where the resultant troughs and crests are even more
`pronounced than those of any of the constituent waves. After
`being superimposed, the individual waves separate and con-
`tinue on, completely unaffected by their previous encounter.
`Although the subject could be treated from the perspective
`of QED (p. 139), we’ll take a much simpler approach. The
`wave theory of the electromagnetic nature of light provides a
`natural basis from which to. proceed. Recall that the expression
`describing theroptical disturbance is a second-order, homoge-
`neous, linear, partial, differential equation [Eq. (3.22)]. As we
`have seen, it therefore obeys the important Superpositiglt
`Principle. -Accordingly, the resultant electric-field intensity E,
`at a point in space where two or more lightwaves overlap, is
`equal to the vectorrsum of the individual constituent distur-
`bances. Briefly then, optical interferen.ce‘corresponds to the
`interaction of two or more lightwaves yielding ti resultant
`irradiance that deviates from the sum of the component
`irradiances.
`I
`nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnu
`
`‘The layer of water on the asphalt allows the oil film to assume the
`shape of a smooth planar surface. The black asphalt absorbs the
`, transmitted light, preventing back reflection, which would tend to
`obscure the fringes.
`
`385
`
`3 8 5
`
`These roughly circular interference fringes are due to an oil film on wet
`pavement. They are fringes of equal thickness (see p..404) and so don't
`change when viewed at different angles. Of course, they appear in a
`rainbow of colors.
`_
`._
`-
`_
`*
`.
`.
`'
`
`Out of themultitude of optical systems that produce inter-
`ference, we will choose a few of the more important to exam-
`ine. Interferometric devices will be divided, for the sake of
`discussion, into two groups: wavefront splitting and amplitude
`splitting. In the first instance, portions of the primary wave-
`front are used either directly as sources to emit secondary
`waves or in conjunction with optical devices to’produce.virtu-
`al sources of secondary waves. These secondary waves are
`then brought together, thereupon to interfere. In the caseof
`amplitude splitting, the primary wave itself is divided into two
`segments, which travel different paths before recombining and
`interfering.
`'
`'
`Z
`'
`’
`
`

`
`386 Chapter9 ‘Interference
`
`Max, ‘4
`
`
`
`Min Max Min Mflx Min Max Mm Max Min
`.
`i
`
`
`
`Figure‘ 9.1 Water waves from two in-phase point sources in a ripple
`tank. In the middle of the pattern the wave peaks (thin bright bands), a
`and troughstthin black bands) lie within long wedge-shaped areas -
`(maxima) separated by narrow dark regions of calm (minima). Although
`the nodal lines look straight, they're really hyperbolic. The optical equiva-
`lent is the electric field distribution depicted in Fig. 9.3c. (Photo courtesy
`PSSC College Physics, 1968, @ 1965 Educational Development Center, Inc.)
`
`by
`
`E=E+E+~
`
`(9.1)
`
`,
`
`The optical disturbance, or light field E, varies in time at an
`exceedingly rapid rate, roughly
`
`4.3>< 1014 Hz
`
`to
`
`7.5 x 10” Hz
`
`making the actual field an impractical quantity to detect. On
`the other hand, the irradiance I can be‘ measured directly with
`a wide variety of sensors (e.g., photocells, bolometers, photo-
`graphic emulsions, or eyes). The study of interference is there-
`fore best approached by way of the irradiance.
`1
`Much of the analysis to follow can be performed without
`specifying the particular shape ofthe wavefronts, and the
`results are therefore quite general (Problem 9.1). For the sake
`of simplicity, however, consider two point sources, S1 and S2,
`emitting monochromatic waves of the same frequency in a
`homogeneous medium. Let their separation at be much greater
`than A. Locate the point of observation P far enough away
`from the sources so that at P the wavefronts will be planes
`(Fig. 9.2). For the moment, consider only linearly polarized
`waves of the form
`‘
`
`E1(?,t) = E01 cos (E -F - wt + 81)
`
`(9.261)
`
`and
`
`V Egg, U 7: E02 COS
`
`‘ -11 ". wt +9 £2)
`
`We saw in'Chaptcr 3 that the irradiance at P is given by
`
`V
`
`I = €U<-E>2 >1‘
`
`‘91 General Considerations
`
`Inasmuch as we will be concerned only with relative irradi'
`ances within the same medium, we will, for the time being at
`
`We have already examined the problem of the superposition of
`two scalar waves» (Section 7.1), and in many respects those
`results will again be applicable. But light is, of course, a vec-
`tor phenomenon; the electric and magnetic fields are vector
`fields. An appreciation of this fact is fundamental to any kind
`of intuitive understanding of interference. Still, there are many
`situations in which the particular optical system can be so con-
`figured that the vector nature of light is of little practical sig-
`nificance. We will derive the basic interference equations
`within the context of the vector model, thereafter delineating-
`the conditions under which the scalar treatment is applicable.
`In accordance with the Principle of Superposition, the elec-
`tric field int_<;nsi£yE, at a point in space, arising from the sep-
`arate fields E1, E2, .. of various contributing sources is given
`
`least, simply neglect the constants and set
`
`I = <fi2)'r
`
`What is meant by <-fi2>T is of course the time average 01:,the
`magnitude of the electric-field intensity squared, or (E ~ E)?
`Accordingly
`
`‘I-32 = ‘E’ . is’
`
`_
`F‘? = (113 + E) - (E + ii)
`
`where now
`
`and thus
`
`it? = E’? + ’1':’§+ 213’, -‘E’,
`
`i
`
`(93?
`
`

`
`(a)
`
`£1/k1
`
`%viwa
`‘ r'_
`-‘
`if""""""" P
`‘nym-
`4.»
`
`122/k2
`
`a>)t
`
`|=igure.9.2,,Waves from two: pointsourc.es_ ovetrlapning in 5930.9-
`
`Taking the time average of both sides, we find that the irradi-
`ance becomes
`—
`
`provided that
`
`
`'
`
`,I=I1+I2+I12
`,
`I
`I1 = <'I1:’%>T
`
`12 = <'I':’%>T
`
`and
`
`112 ‘‘ 2<E1'E2>T
`
`(9.4)
`
`(9.5)
`
`(9.6)
`
`(9-7)
`
`The latter expression is known as the interference term. To
`evaluate it in this specific instance, we form
`«-->
`-9
`->
`->
`—>
`_,
`E1'E2= E01'E02COS (k1'l’ “ (1)l"'i‘81)
`X cos (it); -i" - cot + 32)
`
`(9.8)
`
`4
`
`or equivalently
`
`‘E1 ‘E2 =
`
`fi01~fi02 [cos
`
`-3‘ + e1)‘cos wt + sin (E1-ii + 51) sin wt]
`
`9.1 General Considerations 387
`
`
`
`The period 1' of the harmonic functions is 27r/co, and for our
`present concern T >> r. In that case the 1_/T coefficient in
`front of the integral has a dominant effect. After multiplying
`out and averaging Eq. (9.9) We have
`
`—>
`-->
`1-)
`-_—>
`-)
`_,
`—->
`_)
`(E1°E2)T = EE01‘E02 003 (k1‘l‘ fl‘ 8.1‘ k2'1‘ “ 82)
`
`where use was made of the fact (p. 49) that (_c_os2 wt)T = §,
`(sing a1t)T = %, and (cos wt sin cot)T f—‘ O. The interference term
`is then
`
`112 = E0, -"1330, cos 5
`
`(9.11)
`
`and 6, equal to (i’, -? - ii; -9? + 51 - 82), is the phase di]j”er-
`ence arising from a combined path length and initial phase-
`anglgdifference. Notice that if T501 and E02 (and therefore E1‘
`and E2) are perpendicular, 112 = O and I = 11 + 12. Two such
`orthogonal 9}’-states will combine to yield an 911:, £84, 9-, or %-
`state, but the flux—density distribution will be unaltered.
`The most) common situation in the work to follow corre-
`sponds to E01 parallel to E02.» In that case, the irradiance
`reduces to the value found in the scalar treatment of Section
`7.1. Under those conditions
`
`‘
`
`>< [cos (:2 -i" + 82) cos cot + sin (E2 -i" + 82) sin wt]
`
`(9.9)
`
`112- ‘-3 E01E()2 COS 5
`
`Recall that the time average of some function f(t), taken over
`
`an interval T, is.
`
`t+T
`
`This can be written in a more convenient way by noticing that
`
`‘
`
`1
`
`<f(t)>T = $1 f(t’) dt’
`
`(9.10) —
`
`1, = (E19 T = ~—-—
`
`(9.12)
`
`

`
`388 Chapter 9.
`
`interference,
`
`and
`
`1, = (iii), = E53
`
`(9.13)
`
`Equation. (9. 14) holds equally well for the spherical waves
`emitted by S1 and S2. Such waves can be expressed as A
`
`The interference term becomes
`
`if1(r,, t) = i301(r,) exp [i(kr1 — cut + 21)]
`
`(9.l8a)
`
`1
`
`112 = 2\/111-2 cos 8
`
`whereupon the totalirradiance is
`
`"At various points in space, the resultant irradiance can be
`greater, less than, or equal to 11 + 12, depending on the value
`of I12, that is, depending on 8. A maximum irradiance is
`obtained when cos 8 = 1, so that
`
`Imax = 11+ 12 + 2 V 1112
`
`when
`
`8 = 0, i'2'7r, ;':47r,...
`
`In this case of total iconstructive interference,’ the‘ phase" dif-
`ference between the two waves is an integer multiple of 277,
`and the disturbances are in-phase. When __0 < cos8 < l the
`waves are out-of-phase,‘ 11 + 12 < I '< Im,’,,1, and the result is
`constructive interference. At 5 = 77/2, cos 6 = 0, the optical
`disturbances are 90°‘ out-ofgphase, and I »= 11’ + '12: For 0 >
`cos 6 > ~1 we have the condition of destructive interference,
`11 + 12 > I > I,,,1,,. A minimum irradiance results when the
`waves are 180° out~of—phase, troughs overlap crests, cos 6 =
`~ 1, and
`
`and
`
`ii,(r,, :) = E’0,(r,) exp [i(kr2 —- wt + 52)]
`
`(9.18b)
`
`The terms r1 and 2, are thegradii of the spherical wavefronts
`overlapping at P; they specify the distances from the sources
`to P. In this case
`
`5=Mn*m%Hn*a)
`
`flw)
`
`The flux density in the region surrounding S1 and S2 will
`certainly vary from point to point as (r1 - r2) varies. None-
`theless, from the principle of conservation of energy, we
`"expect the spatial average of I to remain constant and equal to
`the average of 11 + 12. The space average of 112 must therefore
`be zero, a property verified by Eq. (9.11), since the average of
`the cosine term is, in fact, zero. (For further discussion of this
`point, see Problem 9.2.)
`’
`Equation (9.17) will be applicable when the separation
`between S1 and S2 is small in comparison with r1 and r2 and -
`when the interference region_is also s_r_pal1 in the same sense.
`Under these circumstances, E111 and E02 may be considered
`independent of position, that is, constant over the small region
`gxamingd. If the emitting sources are of equal strength,
`E01 = E02, 11 : I2 = [0 and We have
`
`1 = 41,, cos2 %[k(r1 j— r2) + (£1 - 82)].
`
`I,,,1,,== 11+ 1, - 2\,_/T11‘,
`
`1
`
`(9.16)
`
`Irradiance maxima occur when
`
`This occurs when 8 = in‘, ‘$317, —: 57r,. . ., and it is referred to
`as total destructive interference.
`1
`Another somewhat special yet very important case arises
`when the amplitudes of both waves reaching P in Fig. 9.2 are
`equal.-(i.e., E01 = E112). Since the irradiance contributions from
`both sources are then equal, let 11 2 I2 = 10. Equation (9.14)
`can now be written as
`.
`.
`’
`«
`
`8 = 217m
`
`provided that m = o, : 1, :2,.‘.. Similarly," minima, for which
`I = O, arise when
`'
`
`8==7rm’
`
`where m’ = :1, :3, i5,..., or if you like, m’ = 2m +1-
`Using Eq. (9.19) these two expressions for 8 can be rewritten
`such that maximum irradiance occurs when
`
`1%‘2I11(;1‘fii*;Ct)Sgh§)”§e4I11={:os?'7§f
`
`_
`
`(9.17)
`
`A
`
`(F: * r2) = [2'ITm + (Ia * 81)l/k '
`
`(9-203)
`
`from which itfollows that 1,,,;, = 0 and r,,,,,1 = 41,. For an
`analysis in terms of the angle between the two beams, see
`Problem 9.3.
`
`and minimum when
`
`(r1_ - r2) = [7m’ + (61 - 81)]/k
`
`(9.20b)
`
`

`
`9.1 General Considerations 389
`
`Figure 9.3 (a) Hyperboloidal surfaces of maximum
`irradiance for two point sources. The quantity
`m is positive where r1,> r2. (b) Here we" see
`how the irradiance maxima are distrib‘uted'ori‘a
`plane containing 81 and 82. (c) The electriofieid
`distribution in the plane shown in part (b)-. The tall
`peaks, are the point sources 81 and 82. Note
`that the spacing of the sources is different in (b)
`and (C)- (Photo courtesy,‘ Ttie"Optics Project,’ Mississippi
`State University.)
`'
`i
`
`
`
`
`
`at
`
`Either one of these equations defines a family of surfaces, each
`of whichsis a hyperboloid of revolution. The vertices of the
`hyperboloids are separated by distances equal to the right-
`hand sides of Eqs. (9.20a) and (9.20b). The foci are located at
`S1 and S2.-If the waves are in-phase at the emitter, £1 - £2 = 0,
`and Eqs. (9.20a) and (9.20b) can be simplified to
`
`[maximal
`
`(r1 - r2) '= 277m/k 5 m)t
`
`(9.21a) ‘
`
`[minimal
`
`(r1 - r2) = vrm’/k =-= %m’)t
`
`»(9.2lb)
`
`for maximum and minimum irradiance, respectively. Figure
`9.3a shows a few of the surfaces over which there are irradi—_
`ance rnaxima. The dark and light zones that would be seen on
`a screen placed in the region of interference are known as
`interference fringes (Fig. 9.31)). Notice that the central bright
`
`band, equidistant from the two sources, is.the so—called zeroth-
`order fringe (m = O), which is straddled by the m’ : ipl min-
`’ ima, and these, in turn, are bounded by the first-order (m =
`.4: 1) maxima, which are straddled by the m’ = :23 minima,
`and so forth.
`_
`,
`_
`V
`_
`_
`Since the wavelength A for light is very small, a large num-
`ber of -surfaces corresponding to the lower values of m will
`exist close to, and on either side of, the plane m = O. A num-
`-ber of‘fairly straight parallel fringes will therefore appear on a
`screen placed perpendicular to that,_(m ‘F 0) plane and in the
`vicinity of it, andfor this case the approximation r1 '~= r2 will
`hold. If S1 and S; are then displaced normal to the_§T§‘; line,
`the fringes will merely be displaced parallel to themselves.
`Two narrow slits will generate a large number of exactly over-
`lappingfringes, thereby increasing the irradiance, leaving the
`central region of the two-point source pattern otherwise
`essentially unchanged.
`T
`
`

`
`390 Chapter9 Interference
`
`9. ditins folner
`
`......................................_.
`..
`........,
`)5,.-
`
`
`If two beams are to interfere to produce a stable pattern, they
`must have very nearly the same frequency. A significant "fre-
`quency difference would result in a rapidly varying, time-
`dependent phase difference, which in turn would cause I12 to
`average to zero during the detection interval (see Section 7.1).
`Still, if the sources both emit white light, the component reds
`will interfere with reds, and the blues with blues. A great many
`fairly similar, slightly displaced, overlapping monochromatic
`patterns will produce one total white-light pattern. It will not
`be as sharp or as extensive as a quasimonochromatic pattern,,
`but white light will produce’ observable interference.
`The clearest patterns exist when the interfering waves have
`equal or nearly equal amplitudes. The central regions of the
`dark and light fringes then correspond to complete destructive
`and constructive interference, respectively, yielding maximum
`contrast.
`‘
`
`For a fringe pattern to be observed, the two sources need
`not be in—phase with each other. A somewhat shifted but oth-
`erwise identical interference pattern will occur if there is some
`g initial phase difference between the sources, as long as it
`remains constant. Such sources (which may or may not be in
`step, but are always marching together) are coherent?“
`
`‘
`
`9.2.1 Temporal and Spatial Coherence
`
`Remember that because of the granular nature of the emission
`process, conventional quasimonochromatic sources produce
`light that is a mix of photon wavetrains. At each illuminated
`point in space there is a net field that oscillates nicely (through
`roughly a million cycles) for less than 10 ns or so before it ran-
`domly changes ph'ase.'This interval over which the lightwave
`resembles a sinusoid is a measure of its temporal coherence.
`The average time interval during which the lightwave oscil-
`lates in a predictable way we have already designated as the
`coherence time of the radiation. The longer the coherence
`time, the greater the temporal coherence of the source.
`As observed from a fixed point in space, the passing light-
`wave appears fairlysinusoidal for some number of oscilla-
`tions between abrupt changes of phase. The corresponding
`spatial extent over which the lightwave oscillates in a regular,
`
`uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu n
`
`*Chapter 12‘ is devoted to the study of coherence, so here we'll merely
`touch on those aspects that are immediately pertinent.
`
`predictable way is the coherence length [Eq. (7.64)]. Once
`again, it will be convenient to picture the light beam as a pro.
`gression of well-defined, more or less sinusoidal, wavegroups
`of average length Ale, whose phases are uncorrelated to one
`A another. Bear in mind that temporal coherence is a manifes-
`tation ofspectral purity. If the light were ideally monochro-
`matic, the wave would be a perfect sinusoid with an infinite
`coherence length. All real sources fall short of this, and all
`actually emit a range of frequencies, albeit sometimes quite
`narrow. For instance, an ordinary laboratory discharge lamp
`has a coherence length of several millimeters, whereas certain
`kinds of lasers routinely provide coherence lengths of tens of
`kilometers.
`
`Figure 9.4 summarizes some of these ideas. In (a) the
`wave, which arises from a point source, is monochromatic and.
`has complete temporal coherence. What happens at P1 will, a
`moment later, happen at P5 and still later at P§——all totally pre-
`dictably. In fact, by watching Pf, we can determine what the
`wave will be doing at Pi at any time. Every point on the wave
`is correlated; its coherence time is unlimited. By contrast, Fig,
`9.4b shows a point source that changes frequency from
`moment to moment. Now there’ s no correlation of the wave at
`
`points that are far apart like P3 and P2,. The waves lack the
`total temporal coherence displayed in (a), but they’re not com-
`pletely unpredictable; the behavior at points that are close
`together such as P5 and Pg are somewhat correlated. This is an.
`instance ofpartial temporal coherence, a measure of which is;
`the coherence length—-the shortest distance over which the ‘
`disturbance issinusoidal, that is, the distance over which the
`phase is predictable.
`g
`Notice, in both parts of Fig. 9.4, that the behavior of the
`waves at points P1, P2, and P3 is completely correlated. Each
`of the two wave streams arises from a single point source and
`P1, P2, and P3 lie on the same wavefront in both cases; the dis-
`turbance at each of these laterally separated points is in—phase
`and stays in-phase. Both waves therefore exhibit complete
`spatial coherence. By contrast, suppose the source is broad,
`that is, composed of many widely spaced point sources
`(monochromatic ones of period 1'), as is Fig. 9.5. If we could
`take a picture of the wave patternin Fig. 9.5 every 1' seconds,
`it would be the same; each wavefront would be replaced by an
`identical one, one wavelength behind it. The disturbances‘ at
`P1, Pg, and Pg are correlated, and the wave is temporally
`coherent.
`
`Now to insert a little realism; suppose each point source
`changes phase rapidly and randomly, emitting l0—ns long
`sinusoidal wavetrains. The waves in Fig. 9.5 would randomly
`
`

`
`9.2 Conditions for interference 391
`
`advent of the laser, it was a working principle that no two indi-
`' vidual sources could ever produce an observable interference
`pattern. The coherence time of lasers, however, can be appre-
`ciable, and interference via independent lasers has been
`observed and photographed.* The most common means of
`overcoming this problem with ordinary thermal sources is to
`make one source serve to produce two coherent secondary
`sources.
`
`9.2.2 The Fresnel-Arago Laws _
`
`In Section 9.1 it was assumed that the two overlapping optical
`disturbance vectors were linearly polarized and parallel.
`Nonetheless, the formulas apply as well to more complicated
`situations; indeed, the treatment is applicable regardless of the
`polarization state of the waves. To appreciate this‘, recall that
`
`
`
`(b) _
`
`Figure 9.4 Temporal and spatial coherence. (a) Here the waves-display
`both forms of coherence perfectly. (b)~Here there is complete spatial
`coherence but only partial temporal coherence.
`
`change phase, shifting, combining, and recombining in a fren-
`zied tumult. The disturbances at P1, P5, and Pg would only be
`correlated for a time less than 10 ns. And the wave field at
`
`two, even modestly spaced, lateral points such as P1 and P2
`would be almost completely uncorrelated depending on the
`size of the source. The beam from A candle flame or a shaft of
`
`sunlight is a multi-frequency mayhem much like this.
`Two ordinary sources, two lightbulbs, can be expected to
`maintain a constant. relative phase for a time no greater than
`Arc, so the interference pattern they produce will randomly
`shift around in space at an exceedingly rapid rate, averaging
`out and making it quite impractical to observe. Until the
`
`Figure 9.5 With multiple (here four) widely spaced point sources, the
`resultant wave is still coherent. But if those sources change phase rapid-
`ly and randomly, both the spatial and temporal coherence diminish
`accordingly.
`'
`»
`
`nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn --
`
`* G. Magyar and L. Mandel, “Interference fringes produced by superpo-
`sition of two independent maser light beams,” Nature 198, 255 (1963).
`F. Louradour, F. Reynaud, B. Colombeau, and C. Froehly, ‘flnterference
`fringes between two separate lasers,” Am. J. Phys. 61, 242 (1993). L.
`Basano and P. Ottonello, “lnterference fringes from stabilized diode
`lasers,” Am. J. Phys. 68, 245 (2000). E. C. G. Sudarshan and T.
`Rothman, “The two-slit interferometer reexamined,” Am. J. Phys. 59,
`592 (1991).
`,
`'
`
`

`
`392 Chapter9 Interference
`
`Figure 9.6 Interfer-
`ence of polarized light.
`
`any polarization ‘state can be synthesized out of two orthogo-
`nal QP-states. For natural light these 9}’-states are mutually
`incoherent, but that represents no particular difficulty.
`Suppose that every wave has its propagation vector in the
`same plane, so that we can label the constituent orthogonal 9}’-
`states, with respect to that plane, for example, in and El,
`which are parallel andlperpendicular to the plane, respectively
`(Fig. 9_.6a). Thus any plane wave; whether polarized or not,
`can be written inthe for_r>n (E, + EL). Imagine that the waves
`(Em + E,1) and (EH2 + Eiz) emitted from two identical.eoher-
`ent sources superimpose in some region of space. The result-
`ing flux-density ’distribution will consist of two indeperident,
`precisely, overlapping interference patterns ((33.1 -_I- E,,2)2)T
`and «I3:L1 + E_[_2)2)-T. Therefore, although we derived the
`equations of the previous section specifically for linear light,
`they are applicable to any polarization state, including natur-
`al light.
`*
`'
`‘
`'
`A
`Notice thgt even_t>hough ll,1 and _I§_L2 are always parallel to
`each other, E,” and Eng, which are in the reference plane, need
`not be. They will be parallel only when the two beams are
`‘themselves parallel’(i;e.s;s‘k_1s’;==“I;>2)."Thesinherents vector nature‘ ““ i’
`of the interference process as manifest in the dot—product rep-
`
`resentation [Eq. (9.1 1)] of [12 cannot be ignored. There are
`many practical situations in which the beams approach being
`parallel, and in these cases the scalar theory will do nicely.
`Even so, (b) and (c) in Fig. 9.6 are included as an urge to cau-
`tion. They depict the imminent overlapping of two coherent
`linearly polarized waves. In Fig. 9.6b the optical vectors are
`parallel, even though the beams ar_en’t, and interference would
`nonetheless "result. In Fig. 9.6c the optical vectors are perp611'
`dicular, and I12 = 0, which would be the case here ‘even if the
`beams were parallel.
`_
`Fresnel and Arago made an extensive study of the cond1'
`tions under which the interferenceof polarized light occurs-
`and their conclusions summarize some of the above consider‘
`ations. The Fresnel—Arago Laws are as follows:
`‘
`
`1. Two orthogonal, coherent 9}’-states cannot interfere in the
`sense that 112 é 0 and no fringes result.
`2. Two parallel, coherent 9’-states will interfere in the same
`way as will natural light.
`,
`_
`_
`3. The two constituent orthogonal 9}?-states, of natural light
`-cannot interfere to form a readily observable fringe pattern
`even"ifrotatedssinto“ali'gnment."Thisilast*point is under
`_ -standable, since these 9l?~states are incoherent.
`
`

`
`9.3 Wavefront-splitting,
`
`
`
`The main problem in producing interference is the sources:
`they must be coherent. And yet separate, independent, ade-
`quately coherent sources, other than the modern laser, don’t
`exist! That dilemma was first solved two hundred years ago by
`Thomas Young in his classic double-beam experiment. He
`- brilliantly took a single wavefront, split off from it two coher-
`ent portions, and had them interfere.
`
`9.3.1 Young’s_Experiment-
`In 1665 Grimaldi described an experiment he had performed
`to examine the interaction between two beams of light. He
`admitted sunlight into a dark room through two close-together
`pinholes in an opaque screen. Like a camera obscura (p. 215),
`each pinhole cast an image of the Sun on a distant white sur-
`face. The idea was to show that where the circles of light over-
`’ lapped, darkness could result. Although at the time he couldn’t ‘
`possibly understand why, the experiment failed because the
`primary source, the Sun’s disk (which subtends about 32 min-
`utes of arc), was too large and therefore the incident light
`didn’t have the necessary spatial coherence in order to proper-
`ly simultaneously illuminate the two pinholes. To do that, the
`Sun would have "had to subtend only a few seconds of arc.
`A hundred and forty years later, Dr. Thomas Young (guid-
`ed by the phenomenon of beats, which was understood to be
`produced by two overlapping sound waves) began his efforts
`to" establish the wave nature of light. He redid Grimaldi’s
`experiment, but this time the sunlight passed through an initial
`pinhole, which became the primary source (Fig. 9.7). This had
`the effect of creating a spatially coherent beam that could
`identically illuminate the two apertures. In this way Young
`succeeded in producing a system_of alternating bight and dark
`bands-—interference fringes. Today, aware of the physics
`involved, we generallyreplace the pinholes with narrow slits
`that let through much more light (Fig. 9.8a).
`_
`Consider a hypothetical monochromatic plane waveillumi-f
`nating a long narrow slit. From that primary slit light will be
`diffracted out at all angles in the forward direction and a cylin-
`drical wave will emerge. Suppose that thiswave, in turn, falls
`on two parallel, narrow, closed spaced slits, S1 and S2. This is
`shown in a three-dimensional View in Fig. 9.8a. When sym-
`metry exists, the segments of the primary wavefront arriving
`at the two slits will be exactly in-phase, and the slits will con-
`
`9.3 Wavefront-splitting interferometers 393'
`
`stitute two coherent secondary sources. We expect that wher-
`ever the two waves coming from S1 and S2 overlap, interfer-
`ence will occur (prov