# 13.fresnel diffraction.ppt

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• Geometrical opticslight cant turn a corner.IDiffraction

Physical opticsactually, it can.IFrancesco Maria Grimaldi(1618 - 1663)Diffraction

every point on a wavefront may be regarded as a secondary source of spherical waveletsThe propagated wave follows the periphery of the wavelets.Huygens, just add the wavelets considering interference! Huygens-Fresnel principle

If one perturbs a plane wavefront, the Huygens wavelets will no longer constructively interfere at all points in space. Adding the wavelets by physical optics explains why light can turn corners and create fringes around images of objects.Huygens-Fresnel principle

Obliquity factor(oblique obscure?)- wavelets propagate isotropicallyin forward and reverse directions- to use the Huygens approach, modify amplitude of wavefront as a function of q:

Calculating the diffracted waveobliquity factorphase shiftspherical waves:In general, not an easy task. Lucky for you, Fresnel made it simpler.Fresnel-Kirchhoff diffraction integral:

Diffraction dudes: Fresnel and FraunhoferContemporaries, but not collaborators (nor competitors).

• Fraunhofer: both incident and diffracted waves may be considered to be planar (i.e. both S and P are far from the aperture)Fresnel: occurs when either S or P are close enough to the aperture that wavefront curvature is not negligiblePSFresnel vs. Fraunhofer diffraction

Fresnel diffraction from an edge

• irradiance vs. position, just after a slit illuminated by a laserFresnel diffraction from a slit

• Incident plane waveFrom Fresnel to Fraunhofer diffraction

Fresnel vs. Fraunhofer diffractionview from source:view from point of interest:near field where d represents p or q (=distance from source or point to aperture)

A is aperture area

Fresnel number, FFresnel diffraction occurs when: Fraunhofer diffraction occurs when:whereh = aperture or slit sizel = wavelengthd = distance from the aperture (p or q)source:point:

• one of the few Fresnel diffraction problems that can be solved analytically beam pattern alternates between two different fringe patternsFresnel diffraction from an array of slits: the Talbot effecthttp://www.flickr.com/photos/gaeloso/4256791024/

• Rolling out the optical carpet: Talbot effecthttp://physicsworld.com/cws/article/print/133

Getting into the zoneFresnels approach to diffraction from circular aperatureszone spacing = l/2:r1 = r0 + l/2r2 = r0 + lr3 = r0 + 3l/2 rn = r0 + nl/2these are called the Fresnel zones

as we draw a phasor diagram where each zone is subdivided into 15 subzonesStay in the zone123455 half-period zones obliquity factor shortens successive phasors circles do not close, but spiral inwards amplitude a1 = A1 : resultant of subzones in 1st half-period zone composite amplitude at P from n half-period zones:

Adding up the zonesindividual phasorscomposite phasorsfor large N, resultant amplitude= half that of zone 1

Implications of Fresnel zonesFor N contributing Fresnel zones,

If N is small, a1 ~ aNfor odd N, AN ~ a1for even N, AN ~ 0

If N is large (i.e. huge aperture) aN 0 for any N, AN ~ a1

Implications of Fresnel zonesstrangeA circular aperture is matched in size with the first Fresnel zone:AP = a1

Now open the aperture wider to also admit zone 2: AP ~ 0 !

Now remove aperture, allowing all zones to contribute: AP = a1 !!!

(Irradiance only !)What is amplitude of the wavefront at P?. Part 1

Implications of Fresnel zonesstrangeA circular disk is matched in size with the first Fresnel zone: all zones except the first contribute first contributing zone is the secondAP = a2

irradiance at center of shadow nearly the same as without the disk present!

What is amplitude of the wavefront at P?How absurd!Simon Denis Poisson (1781-1840). Part 2

Poisson/Arago spot

The Fresnel zone plateIf the 2nd, 4th, 6th, etc. zones are blocked, then:16 zonesAmplitude at P is 16 times the amplitude of a1 /2Irradiance at P is (16)2 times!

An alternative to blocking zonesFresnel vs. plano-convexlenslensphases of adjacent Fresnel zones changed by p

Kewaunee, Wisconsin

Fresnels treatment of straight edgesedge view:cylindrical wavefronts diffracted by rectangular aperture:consider source to be a slitzones are now rectangular strips

Fresnels treatment of straight edgesagain, zone spacing = ladding the phasors gives the endpoints of a Cornu spiral

Phasors trace a Cornu spiralareas of Fresnel strip zones decrease rapidly with nsuccessive phasor amplitudes of zones are much shorterhalf circle never reachedphasors continue to spiral to limit point E (eye)zones of lower half produce twin spiral in 3rd quadrant2 zones:

Applications of the Cornu spiralstraight edge:wire:

• Cornu spiral pendant\$34.99http://www.mboot.netCornu aspersum, garden snailCornucopiaRoman Cornu hornOther Cornu spirals

• ExercisesYou are encouraged to solve all problems in the textbook (Pedrotti3).

The following may be covered in the werkcollege on 13 October 2010:Chapter 13:1, 2, 3, 5, 6, 7, 10, 18

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