Physical Optics

Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences Volume 2 Edited by

I. GRATTAN-GUINNESS

LONDON AND NEW YORK

9.1 Physical optics N. KIPNIS

The term 'physical optics' was introduced by Thomas Young in 1802,and initially it meant all optics exceptfor geometricaloptics and vision (Young 1802),with vision accommodatedafterwards (Young 1807).Nowadays,it usually refers to such phenomena as interference, diffraction, double refraction, polarization and dispersion (seealso 99.2 on the velocity of light). This article looks at how physical optics becamea mathematical science,using selectedexamplesfrom its history. By 'mathematicalscience' is meant a sciencequantified by any mathematicalmeansso as to allow a comparisonwith a quantitative experiment. 1 HUYGENS AND NEWTON The mathematicalapproachto physicaloptics was pioneeredby Christiaan Huygensand IsaacNewton. Huygensfocusedon a theory of double refraction, while Newton tried to build a theory of all phenomenaof colours. In his Traitd de Ia lumiire (1690), Huygens assumedthat light was a wave processin a special medium, the ether, which permeatedall bodies. He describedthe propagationof a wavefront as follows: to locate a wavefront at any moment I giventhe wavefront at the precedingmoment, t - Lt, consider eachpoint on the old wavefront to be a centreof secondaryspherical wavesof radius R: c X Al spreadingforward, and draw a surfacetangential to them all. This hypothesis later became known as the Huygens principle. With this principle, Huygensexplaineddouble refraction in Iceland spar by the propagation of two different waves.The ordinary wavefront was a sphere, and propagated at the same velocity in all directions, while the extraordinary wavefront was an ellipsoid of revolution, the velocity of which dependedon the direction of propagationat eachpoint and could be representedby a vector drawn from the centre of the wave to the chosen point. To determine the ratio of the two unequal axes of the ellipsoid, Huygens used the indices of refraction of the two rays measuredin the

tt43

PHYSICSAND MATHEMATICAL PHYSICS

principal plane. He did not provide any experimental data, but stated instead that the experimentfully confirmed his theory. Newton developedthe first version of his theory of light and colours around 1668,whenhe was 26yearsold, and it wassupposedto show'how valuable mathematicsis in natural philosophy'. Inhis Opticks, first published in 1704,he did not mention mathematicsin his opening statement: 'my design in this book is not to explain the properties of light by hypotheses,but to propose and prove them by reason and experiments'. 'reason' becauseit However, mathematicsmight have beenincluded in the that he did try to mathematicizephysicaloptics is clear from other passages using geometricaloptics as a model (Newton 1730: 131,240,244). Actually, Newton's mathematicalapproachaffectednot only the form of his theory, but alsoits content. Indeed,he claimedthat: (a) thereweremany kinds of light, which differed in their refraction; (b) lieht of each kind produced a specific simple colour; (c) simple colours mixed together producecompoundlight, and in particular, white light; and (d) a prism (or another device) produced colours not by modifying white light, but by separatingdifferent rays from one another. This part of the theory could be demonstratedby qualitative experiments.This was not true of his claim that colorific ability is an innate and immutable property of light. Proving this was equivalentto proving that a colour which appearsto be the samewhen observedin different phenomenais produced by the same kind of light. To achievethis, Newton had to describelight quantitatively. 'refrangibility' and the For this purposehe introducedtwo parameters,the 'reflexibility', which accountedfor a differencein refraction (by a prism) or reflection (by a thin film), respectively,given the sameincidence.Newton then demonstrateda constantrelation betweencolour and either parameter. For the colours of thin films, he found that dark and bright rings were produced by specificthicknessesof a film which formed an arithmetical progression.To explain this he postulatedthe existencein light particlesof periodical changes('fits'), which made them susceptibleto either reflection or refraction. He characterizedthis spatial periodicity by a separate parameter,the 'interval of fits', which was different for different colours (it also dependedon the angleof incidence,which meansthat Newton's concept of periodicity differed from the modern one). Later, Young found that the interval of fits at normal incidence of light was equal to half the wavelength. Newton extendedthe concept of intervals of fits to colours of thick plates, and obtained very good agreementwith experiment.For prismatic colours, he showedthat the distancesfrom each part of a spectrumto a specificpoint werein a constantproportion taken from an optico-acoustical analogy. In doing so he drew an analogy with acoustics,designingthe

tt44

PHYSICAL OPTICS

$9.I

theory so that the proportions of prismatic colours were the sameas those of periodic colours in thin films or thick plates (Newton 1730: 127, 212, 225-7, 284, 295, 305). The conclusionabout a constantproportion for prismatic colours turned out to be erroneous,and it delayedthe invention of achromaticlenses.On the other hand, without it Newton could not have supported his theory of colours. Demonstrating that a quantitative relation between any two colours is the samein different optical phenomenawas the next thing to finding a universal'measure'for coloured light, which was provided in the nineteenthcentury by the wavelength.Whether Newton's proof was valid is beyondthe scopeof this article, but at leasthe understoodthe importance of such a quantitative parameterin a theory of colour. Thus, Huygensand Newton demonstratedthat optical phenomenacould be mathematicizedin either the wave or the corpusculartheory of light. However, neither their contemporariesnor the following generationsof physicists shared their concern with quantifying physical optics. Consequently,they rejectedHuygens'theory of doublerefraction and Newton's theory of fits as mechanicallyunsound, but offered nothing in their place. As a result, throughout the eighteenthcentury physical optics remaineda qualitativescience.The new era of mathematicalphysicalopticsbeganearly in the nineteenthcentury, and it started with the rediscoveryof Huygens' and Newton'stheories. 2 YOUNG AND FRESNEL In 1801,at the age of 28, Thomas Young revitalizedthe old wave theory of light by adding to it the principle of interference.According to this principle, under certain conditions two rays of light can destroy one another. The mathematicalconceptbehind it was the principle of superpositionof waves,which Young obtainedby generalizingthe conceptof superposition of forces and vibrations. He considereda superpositionof only two vibrations, which had either the sameor oppositephase.Initially, he appliedthe superpositionof wavesto acousticsand explainedbeats of sound (1799), then he modified it so as to make it applicable to light (1801). Young believedthat everyphenomenonof alternatecolours required the principle of interference(together with other hypotheses)for its explanation, and that the difficulty was only in selectinga suitablepair of interfering waves (Kipnis I99l: Chap. 5). He found that using only two interfering waves simplified the mathematical part, reducing it to simple geometry, while still providing suftciently precise explanations of the coloured fringes producedby thin films, thick platesand 'mixed plates', additional rainbows

tt45

PHYSICS AND

MATHEMATICAL

PHYSICS

sometimesobservedinside the primary or outside the secondarybow, and diffraction (Young 1802: 108-14; 1807: 434-46). In France,mathematicalphysicistsled by Pierre Simon Laplacepreferred analytical methods to geometry. Thus, when Etienne Malus decided to check William Hyde Wollaston's (1802) claim about verifying Huygens' theory, he beganby rewritingit in an analyticalform (1810).When Malus concludedthat his experimentssupported it, Laplace suggestedusing the theory but replacingits wave foundation with a corpuscularone (Chappert 1977').Huygens' theory was compatiblewith Fermat's principle, which could be replaced with the principle of least action by substituting the velocity of light with its inverse.According to Laplace,the principle of least action implied the existencein Iceland spar of short-rangeattractive and repulsiveforces,which actedon particlesof light. In Young'sview (1809), this conclusionwas totally unfounded. Another important piece of work done on the basis of the corpuscular theory was Jean-BaptisteBiot's theory of chromaticpolarization(1812l4). In l8l I , FrangoisArago discoveredthat a thin plateof mica displayed different colours when viewed through Iceland spar. By using Newton's theory of fits, Biot developeda quantitative theory in which the colour of a crystal plate dependson its thicknessand the orientation of its optical axis about the plane of polarization of polarizerand analyser.Biot's precise measurementsconfirmed his theoretical predictions of colours, and his theory receiveda favourable response. The next breakthrough in physical optics was linked with the wave theory. In 1815,at the ageof 27, Augustin Fresnelrediscovered the principle of interferenceand offered a theory of diffraction very similar to Young's.In l8l8 he improvedthe theoryand presentedit in the mathematical contest announcedby the Paris Academy of Sciences.By that time Fresnelhad discoveredhow to add two vibrations with an arbitrary phase differenceand how to add more than two vibrations. This enabledhim to considera diffraction fringe asthe result of interferenceof secondarywaves coming from all points on an open wavefront AMI (Figure l). Thus, the intensity of light at the point of observationsP is

'* r). '. o_.]l il |1 .[a."",(r.'tt [[a.,i"ftd-@: " ' \ cu^ /J ab\, l)' \ , LJ

(r)

where a and b arethe distancesof the diffractor from the sourceC and the screenDB, respectively,and X is the wavelength.This integral could be evaluatedonly numerically. To Fresnel,mathematicswasa tool rather than an end in itself. Wherever possiblehe usedsimplemathematicalmeans,as, for instance,in his theories of chromatic polarization and of the reflection and refraction of polarized lt46

PHYSICAL OPTICS

$9.I

DPB FigureI Fresnel'smodelof a diffractionfringe light. There he introduced the very important conceptof the transversality of light waves.He conceivedof this ideain l8l6 after discovering,together with Arago, the non-interferenceof light beamspolarizedin perpendicular directions;however,only in 1821did he becomeconvincedthat light waves do not have a longitudinal component. Assuming that all refracted and reflectedwaveswere transverse,Fresnelderivedthe laws for their intensity and confirmedthem experimentally(Kipnis I99l: Chap.7). The transversality of light waves becamethe physical foundation of Fresnel'stheory of double refraction. To explain it mechanically,Fresnel assumedthat the aether'sresistanceto compressionis much greaterthan to distortion; the resistanceto distortion implied that the aether somehow resembleda solid body. Incidentally, in this analogyFresnelcould not draw upon the theory of elasticity, which had yet to be developed($8.6); in contrast, Augustin Louis Cauchy, one of the founders of the theory of elasticity, was to be influencedin some of his ideas by Fresnel. Fresnel showedthat in a solid body there are three orthogonal directions('elasticity axes') in which the displacementof a particle producesa force parallel to the displacement,and that an arbitrary displacementproduces a force which has its componentsalong the elasticity axes. He proposedto determine the velocity of light in a crystal by meansof the elasticity ellipsoid, the axesof which coincided with x, /, e (Figure 2). If the ellipsoid is cut through its centreperpendicularlyto the direction of light, the sectionwill be an ellipse,the main axesof which are the directionsof vibrations in the two waves,the semi-axesrepresentingthe magnitudesof their ray velocities.

tt47

PHYSICS AND MATHEMATICAL

PHYSICS

\

optical axes

-x---

"/ Figure2 Fresnel's'elasticityellipsoid' If the section is a circle, both velocitiesare equal, which meansthat the direction of propagation of light coincides with an optical axis of the crystal. From suchconsiderationsFresnelconcludedthat in a biaxial crystal neitherwave obeysSnel'slaw (Buchwald 1989). 3 MATHEMATICAL

THEORIES OF THE AETHER

Fresnel'sdeathin 1827endedthe erain whichnew advances in opticsbegan with physical discoveries,and mathematicswas subordinate to physics. Now, mathematicians had taken the lead, and they were more concerned with the generalityof their equationsand solutionsthan with physicallimitationsof the results.While adoptingFresnel'slaws for doublerefraction, reflection and refraction, and other phenomena,mathematiciansrejected their derivation as lacking generalityand being mechanicallyinconsistent. Cauchy suggestedbasing physical optics on the wave equation borrowed from the theory of elasticity(Whittaker I9I0: Chap. 5). In this way he obtainedin 1830his first theoryof doublerefractionand that of reflection. Franz Ernst Neumann, JamesMacCullagh and GeorgeGreen followed his lead and produced a number of theories of different optical phenomena. The new approachto physicaloptics had its own difficulties,as is illustrated by the theory of the reflection and refraction of polarized light. Although all theoriesstartedfrom the samewave equation and aimed to reproduceFresnel'slaws of reflection,they differed in severalpoints. First, it had to be decidedwhetherto assumethe aethervibrations to be perpendicular to the planeof polarization or parallel to it. The former assumption was consistentwith the hypothesisthat the distortional elasticity of the aetherwasconstantand its densitywasvariable,while the latter assumption was consistentwith constant density of the aether and variable elasticity. Cauchy(1830,1836)and Green(1837)usedconstantelasticity,while Carl Neumann(1835)and MacCullagh(1835)preferredconstantdensity.The l 148

PHYSICAL OPTICS

$9.I

seconddecisionto be madewas what to do with longitudinal waves,which coexistedwith transversewavesin a solid body. MacCullagh (1835) and Neumann(1835)avoidedthem, while Cauchy(1830,1836,1839)and Green (1837) tried to deal with them. The final decision was how closely the models of the aether should imitate real solids. Cauchy, for instance,was satisfied to select any boundary conditions which led to Fresnel's experimentallaws. On the other hand, Neumann and MacCullagh wanted somesimilarity with the boundary conditions of elasticsolids, so they kept the continuity of three componentsof velocity; anothercondition being the conservationof vis viva (or energy).To achievethis they had to opt for the hypothesisof constant density. Sincethis hypothesiswas also used in the theory of double refraction, choosing it had the advantageof a unified approachto two different phenomena.Yet, their aetherwas not a true solid (Whittaker 19I0). Green (1837) went even further and took oll the boundary conditions from elasticsolids: continuity of three componentsof displacementand of three componentsof stress.He abandonedCauchy'smolecular aetherand replacedit with a continuous model, which he representedwith a suitable potential function. This 'truly dynamic' theory, however, had its own difficulties,in particular the longitudinal waves.Neumannand MacCullagh had only four boundary conditions, which could be satisfiedby transverse wavesalone: thus they postulated that all wavesin the aether were transverse.Green, on the other hand, had six boundary conditions, and he had to retain longitudinal waves. To prevent them from carrying away any energy,he postulatedtheir velocity to be much higher than that of transversewaves.Yet the theory contradictedexperiment,and it was rejected. Green'sfailure to make the aethersimilar to an ordinary solid body gave rise to some unusual models. MacCullagh (1S39)invented an aether in which elasticitywascausedsolelyby rotation of its volume elements,which eliminatedlongitudinal waves.The sameyear Cauchy offered another way to ban thesewaves:he attributed to them zero velocity. This led to a negative compressibilitywhich madethe aetherunstable.Although both theories provided 'natural' boundary conditions and agreed with Fresnel,slaws, they weretoo distant from reality to be accepted.In the secondhalf of the nineteenth century, many mathematicians, including George Stokes, William Thomson (Lord Kelvin) and Lord Rayleigh, tried their hand at aether theories. Successeluded them, however, and they resolvedneither the problem of the direction of the vibrations nor that of the existenceof longitudinal waves.Those whose aether imitated ordinary solids came to conflicting theories of different phenomena,while those who manageda more-or-lessunified theory found their model of the aether to be too different from real bodies.

n49

PHYSICS AND MATHEMATICAL PHYSICS

Jamesclerk Maxwell was the first to develop mathematicallythe idea that optical, electricaland magneticphenomenaall result from disturbances of the sameaether.This enabledhim to give a new explanationof the transversality of light and of magneto-opticaland electro-opticalphenomena (Everitt Ig74). However,the electromagnetictheory of light did not resolve all the problems of wave optics becausemodelling the electromagnetic aetherturned out to be no easierthan modelling the solid aether-This does not mean that all the efforts of mathematicianswere in vain. Apart from modelling the aether,they were successful:they made advancesin explaining dispersion,selectiveabsorption, diffraction and other phenomena.It is important to note that mathematical theories confirmed by experiment retained their validity even after their interpretation changed. The first change was when the solid aether gave way to the electromagneticaether. For instance,in 1878,GeorgeFitzgerald discoveredthat by identifying e with magnetic force, where e is the displacementvector in MacCullagh's theory, and curle with dielectric displacement,he could obtain the sameexpressionsfor kinetic and potential energyin Maxwell's theory as in MacCullagh's, which made MacCullagh's theory of reflection and refraction of light correct in the electromagneticfield free of charges and conduction currents. Cauchy's 1839theory of the unstableaetherwas also resurrected:JosiahWillard Gibbs showedthat its boundary conditions could be transferred into electromagnetictheory (Stein /981)' Another changeoccurredwith the removal from physicsof the aether:the equations of physicaloptics continuedto describephenomenacorrectly, eventhough the electromagneticwaveswere no longer connectedwith any carrier. 4 THE ORIGIN OF RELATIVITY AND QUANTUM THEORY Besidesdifhcultieswith the mechanicalmodelling of the aether, there was justifying another powerful reasonfor abandoningit: the impossibility of physicists centuries For two its existenceas an absolutereferencesystem. relaEarth the had tried to discoversomeevidenceof a uniform motion of tive to the aether, first by optical and later by electromagneticmeans,and had failed. By 1900,the experimentshad improved so much that there was no longer any doubt about their result. To explain the negativeoutcome, JosephLarmor, Hendrik Lorentz and Henri Poincar6 developedbetween 1900and 1905linear transformationsof coordinatesand time which made electromagneticphenomenain a systemindependentof the system'suniform motion. Poincar{ treated thesetransformations as a group of rotations in a four-dimensionalspace.In 1905,Albert Einstein brought forth 'Lorentz transformations' were his theory of relativity, which assertedthat I 150

PHYSICAL

OPTICS

$9.1

not limited to Maxwell's theory, but representedgeneralpropertiesof space and time (Miller l98l; seealso $9.13). Another revolutionary theory of the twentieth century - the quantum theory - originatedin the study of black-body radiation ($9.15).Max Planck applied the conceptof entropy and Ludwig Boltzmann's statistics to a set of electromagneticresonators.To obtain a spectral distribution compatible with experimentallaws, he assumed(1900)that the energyof radiationis madeup of discrete'bundles'.However,it wasonly after 1905 that physicistsbeganto interpret this mathematicalhypothesisas expressing a physicaldiscontinuityof energy(Kuhn 1978).Einstein'sdiscovery(1905) that the entropy of radiation in a given frequencyinterval is the samefunction of volume as that of gas was instrumental in this change.Following Planck, Arnold Sommerfeld(1912)suggestedthat the fundamentalconcept in the theory must be the quantum of action rather than the energy quantum, and that at every act of absorption or emissionthe time integral of the Lagrangian of the systemmust be equal to Planck's constant. While such optical phenomenaas the photoelectriceffect, characteristic X-rays and the Compton effect lent their support to the quantum theory of radiation, spectroscopyprovided necessarydata for the first quantum theoryof the hydrogenatom, proposedby NielsBohr (1913)and improved by Sommerfeld(1915). The rise of the quantum theory of radiation did not meanrejection of the wavetheory; it wasjust that the latter wasinapplicablefor radiation of very short wavelength and extremely low intensity ($9.15). Interestingly, Einstein's1916paper on quantumradiationand the Bohr atom, in which he introduced the concept of stimulated emission, eventually led to the invention of lasers,which in turn stimulatedthe developmentof waveoptics (holography). BIBLIOGRAPHY Buchwald,J. Z. 1980,'Optics andthetheoryof thepunctiformether', Archivefor History of Emct Sciences,2l,U5-78. [On the wavetheoryof dispersion.] 1989,The Riseof the ll/ave Theoryof Light, Chicago,IL: Universityof ChicagoPress.[An excellent mathematical treatmentof opticaltheoriesfrom Malusto Fresnel.l Chappert,A. 1977,EtienneLouisMalus(1775-1812) et la thdoriecorpuscalaire de la lumidre,Paris:Vrin. [Onthetheoriesof doublerefractionandpolarization.] Everitt, C. W. F. 1974,'Maxwell,JamesClerk', in Dictionaryof Scientific Biography, Vol,9, NewYork:Scribner's, pp. 209-14on 198-230.[Especially the electromagnetic theoryof light.l Garding,L. 1989,'History of themathematics of doublerefraction',Archivefor Historyof ExoctSciences,40, 355-85. I 151

PHYSICS AND MATHEMATICAL PHYSICS

' '], 3 vols' Grattan-Guinness,I. 1990,conyolutions in French Mathematics [' Basel:Birkhduser,ChaPs7,15, 17' of Light' Basell. Kipnis, N. 1991, History'of the Principle of Interference Birkhduser. Kuhn,T.]978,Block-BodyTheoryandtheQuqntumDiscontinuity,lS94_1912' New York: Oxford UniversitYPress' Reading, MA: Miller, A. 1981,Albert Einsteii's special Theory of Relativity, background'] optical I on Chap. Addison-Wesley.[See used: 1952'New York: Newton, l. 1730,Optickst,4thedn, London: Innys' [Edition Dover.l Lectures' -1984, The Optical Popers of IsaacNewton' Vol' l' The Optical 167o_]672(ed.A.E'.Shapiro),Cambridge:CambridgeUniversityPress. 'Newtonis "achromatic" dispersionlaw [" '1" Archive for Shapiro, A. E. 1980, History of Exact Sciences,2l' 92-128' Cantor and J' S' Hodge Stein,H. 1981,"'subtler forms of matter" [' ' ']', in G' N' (eds), Conceptions of Aether: Studiesin the History of Aether Theories' 1740-lgm, Cambriige: Cambridge University Press' 309-40' [Electromagneticmodels of the aether'l Stuewer,R.tgTs,TheComptonEffect,NewYork:ScienceHistoryPublications. theoriesof light, 1897-1925.1 [The debatebetweenthi wave and corpuscular whittaker,E.T.Igrc,AHistoryoftheTheoriesofAetherandElectricity,lstedn, London:Longmans,Green.[2ndedn195l:repr'1960'NewYork:Harper'] Young,T.ts02,ASyllabusofaCourseofLectures'London:TheRoyal Institution. l' London: -1807, A Courseof Lectureson Naturol Philosophy ["']' Vol' Johnson.

tl52