ELEGANT CONNECTIONS IN PHYSICS Diffraction, Par t 3 CURVED WAVEFRONTS AND FRESNEL DIFFRACTION Fig. 1. The geometry of the Fresnel diffraction analysis. Fig. 1. The geometry of the Fresnel diffraction analysis. When Thomas Young presented his wave theory of Please note the distances ρ = , r = , ro , the angles by DwightdiffractionWhen E.Thomas Neuenschwander, before Young Great presented Brita Southernin’s his Royal wave Nazarene Society theory on ofUniversity 16 January φPlease and θ note, and the the distances annular ring ρ = of area, r = da =, (2ro πρ sinφ, )(theρ d anglesφ). 1800,diffraction it was before cooly Great received. Brita Onein’s Royalhistorian Society of physics on 16 January φ and θ, and the annular ring !"of area da!" = (2πρ= sin!"φ)(ρ dφ). !" !" = !" writes1800, [1]it was cooly received. One historian of physics writes[1] When“…the Thomas wave Youngtheory presentedmight have hissuffered wave sterility theory and of diffraction oblivion before Great Britain’s Royal Society on January 16,1800, it had“…the not wave sounder theory critics might revived have sufferedit in France. sterility In 1815and oblivion was coolly received. One historian of physics writes[1] Aughad unotstin sounder Frensel, critics a brilliant revived young it in militaryFrance. engineer In 1815 and mathematiciaAugustin Frensel,n, submitted a brilliant to theyoung Acad militaryemy a engineerpaper on and “…thediffraction,mathematicia wave which,theoryn, submitted asmight was have theto the custo suffered Acadm ofemy sterility that a papelearned andr on oblivion body, had not sounder critics revived it in France. In 1815 Augustin Fresnel,wasdiffraction, reported a brilliant which, upon youngasby wastwo military themembers custo engineerm— ofFrançois that and learned Aragomathematician, body, and submitted to the Academy a paper on diffraction, which, as was reported upon by two members—François Arago and wasLouis the Poinsot. custom ofThe that former learned took body, up the was matter reported with upon great by two members—François Arago and Louis Poinsot. The former Louis Poinsot. The former took up the matter with great tookenthusiasm up the matter and drew with Fresnel’s great enthusiasm attention and to the drew almost Fresnel’s attention to the almost identical views of Young published identicalenthusiasm views and of drew Young Fresnel’s published attention fifteen to years the almost previously. fifteen years previously. Although this was the year of the battle of Waterloo, Fresnel paid a generous tribute to Young and Althougidenticalh viewsthis was of Youngthe year published of the battle fifteen of yearsWaterloo, previously. Fresnel theypaidAlthoug corresponded a generoush this was tribute frequentlythe year to Young of until the battle andthe yearthey of Waterloo,corresponded1827 when Fresnel the death of the former put an end to a career full of great promise.” frequentlypaid a generous until the tribute year to 1827 Young when and the they death corresponded of the former Partsputfrequently an 1 andend until2to of a careerthisthe yearseries[2,3] full 1827 of great when considered promise.” the death diffraction of the former produced by plane wave fronts—Fraunhofer diffraction. Its signature phenomena put an end include to a Young’scareer full famous of great double-slit promise.” experiment, which demonstrated the wave nature of light. But of course there is more Parts 1 and 2 of this series[2,3] considered diffraction to the story.Parts 1 An and unobstructed 2 of this series[2,3] wave front considered radiating diffraction from a point source forms an expanding spherical surface. Fresnel diffraction takes into producedaccount this by sphericalplane wave shape, fronts, and Fraunhofer I would be diffraction. remiss to not Its discuss it, and show how to derive it, in this summary of the basic ele- signatureproduced phenomenby plane wavea include fronts,s Young’s Fraunhofer famous diffraction. double- slitIts mentsexperimentsignature of diffraction phenomen that theory.demonstrateda include s theYoung’s wave naturefamous of double light. - slitBut of Ascourseexperiment in the there plane that is morewave demonstrated toparadigm, the story. the Huygens’ Anwave unobstructed nature principle of light. wformsave But front the of working tool for understanding Fresnel diffraction. The principle holdsradiatingcourse that each there from infinitesimal is amore point to source the patch story. forms of Ansurface an unobstructed expanding on a primary spherical wave wave front front may be considered the source of a secondary wave. The signal subsequentlysurface.radiating Fresneldetectedfrom a pointdiff at araction pointsource Ptakes formsbeyond into an the accountexpanding surface this isspherical spherical the superposition of those secondary waves that reach P. surface. Fresnel diffraction takes into account this spherical Considershape, and a pointI would source be remiss S that toradiates not discuss a monochromatic it in this summary wave of angular frequency ω, so that the signal leaving S is proportional shape, and I would be remiss to not discuss it in this summary to cos(of ωtthe). basic It is sufficientelements of to diffraction consider a theory.monochromatic harmonic wave because, so long as the wave equation is linear, any wave can of As the i nbasic the plane elements wave of paradigm, diffraction Huygens theory. ’ Principle forms the As working in the plane tool inwave Fresnel paradigm, diffraction Huygens. The’ Principleprinciple holdsforms thatthe working each infinitesimal tool in Fresnel patch diffraction of surface. on The a primaryprinciple wave holds frontthat each may infinitesimal be considered patch the sourceof surface of a on secondary a primary wave. wave The signalfront may subsequently beAS considered IN THEdetected the PLANE source at a point of a PWAVE secondary beyond thePARADIGM, wave. surface The is HUYGENS’ PRINCIPLE FORMS thesignal superposition subsequently of detectedthose secondary at a point waves P beyond that reach the surface P. is the Consider superposition a point of source those Ssecondary that radiates waves a monochromatic that reach P. wave Considerthe of angular a point workingfrequenc sourcey S ω that, so radiatesthat the asignal monochromatictool leaving S isin Fresnel diffraction proportionalwave of angular to frequenc. yIt ω is, sosufficient that the to signal consider leaving a S is monochromaticproportional to harmonic. Itwave is sufficient because, to so consider long as thea wave equationmonochromatic is linear,cos harmonic any(!" wave) wave can because,be written so as long a superposition as the wave equation is linear,cos any(!" wave) can be written as a superposition be writtenof harmonics. as a superposition Picture the of spherical harmonics. wave Picture front whenthe spherical it has of harmonics. Picture the spherical wave front when it has waveexpanded front when to radiusit has expandedρ, to form to a radiusspheri calρ, andsurface formed σ. Becausea spheri - ofexpanded energy conservation,to radius ρ, to theform amplitude a spheri calof asurface spherical σ. Becausewave cal surface σ. Because of energy conservation, the amplitude of dropsof energy off asconservation, the inverse distancethe amplitude from theof a source. spherical The wave a sphericaldrops off wave as thedrops inverse off as distance the inverse from of the the source. distance The from the spherical wave front at σ carries time-dependence source.spherical The spherical wave front wave at σfront carries at σ time carries-dependence time-dependence cos [ω(t-ρ/c)] with with amplitude amplitude ~ ~1/ 1/ρ,ρ, as as described described by by the the wave wavecos funct function [!ion! − ! with amplitude ~ 1/ρ, as described by the wavecos funct [!ion! − ! ! ] ! ψ](ρ,t) = (1) (1) ψ(ρ,t) = ℇ! (1) ℇ! ! cos !" − !" wherewhere k = 2π/λ k = ! 2denotesπcos/λ denotes!" the− !"wavenumber, the wavenumber, and λwith the λwavelength. the The wavelength.wherecoefficient k = 2 π /,Theλ determined denotes coefficient the by wavenumber, the, determinedsource’s withluminosity by λ thethe source’s L, will wavelength. o The coefficient , determined by the source’s be consideredluminosity known; L, will inbe particular, considered for known a light; in wave, particular, for a luminosity ℇL, will be consideredℇ! known; in particular, for a = √(lightL/2 wave,πϵ c) .[4] .[!4] o light wave,o ℇ.[4] On On another another spherical spherical surface surface σ’ σcentered’ centered on onS but S but having having On anotherℇ !spherical= !/2 surface!!!! σ’ centered on S but having ℇ radius ρ + ro (see! Fig. 1), i.e.! a surface that includes point P, radiusradius ρ + rρo (see+ rℇ (seeFig.= Fig.1), ! i.e./ 1),2! a !i.e. surface! a surface that includesthat includes point point P, Eq. P, (1) Eq. (1) sayso that for the wave front passing over P, Fig.FIG. 1.1: The geometry of the Fresnel diffraction analysis. says Eq.that (1) for says the wavethat for front the passing wave front over passing P, over P, When Thomas Young presented his wave theory of Please note the distances ρ = , r = , ro , the angles diffraction ψ(ρ + ro ,tbefore) = Great Britain’s Royal Society. on(2) 16 January φ and θ, and the annular ring of area da = (2πρ sinφ)(ρ dφ). ψ(ρ + ro ,t) = ℇ! . (2) (2) 1800, it was cooly received. One historian of physics !" !" = !" ℇ! ! ! writes [1] !!! cos [!(! + ! ) − !"] !!!! cos [!(! + !!) − !"] The SPS Observer / Fall 2013 25 “…the wave theory might have suffered sterility and oblivion had not sounder critics revived it in France.