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PC20312 Wave Optics Section 4: Diffraction

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Huygens-Fresnel Principle I Image from Wikipedia Augustin-Jean Fresnel “Every unobstructed point of a wavefront… serves as a source of spherical secondary wavelets … The amplitude of the optical field at any point beyond is the superposition of all these wavelets...” Fresnel combined ideas of Huygens’ wavelets & interference Postulated in 1818: Hecht, p444

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Huygens-Fresnel Principle II Gustav R. Kirchhoff Fresnel’s postulate (1818) predates Maxwell’s equations (1861) Formally derived from the scalar wave equation by Kirchoff in 1882 Worked with Schuster for year at the University of Heidelberg Image from Wikipedia

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Huygens-Fresnel Principle III Total area, A P dA 1 dA 3 dA 2 Optical field at P depends on the superposition of contributions from each elemental area dA of the total area A

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Huygens-Fresnel Principle IV Divide an aperture into elemental areas each of which is a source of a spherical wavelet Image from Wikipedia

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The Huygens-Fresnel Integral Source, S Observation point, P Q Spherical wavefront s r s0s0 R

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Fraunhofer diffraction Joseph von Fraunhofer The case of small, linear phase variation, i.e.: r R + r, r << R r x,y Satisfied when s,r >> d Hence, “Far-field diffraction” Image from Wikipedia y x aperture d

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Far-field diffraction S s0s0 s 0 >> d wavefront plane at aperture s s 0 P d D R >>d const. set K( ) 1 D >> d R

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Analysis of Fraunhofer diffraction Source, S Observation point, P(X,Y) Q( x,y ) s r R s0s0 Z Aperture, A( x,y )

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Single slit diffraction y x a/2 -a/2 Image from Wikipedia

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Rectangular aperture y x a/2 -a/2 b/2 -b/2 Image from Wikipedia

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Circular aperture I y x a u Image from Wikipedia Airy disc Airy rings The Airy Pattern

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Circular aperture II kaθ D =3.83 I= I(0)

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The diffraction limit f f Image from Google Images If there was no diffraction: parallel rays focused to a point images would be perfectly sharp BUT, diffraction from instrumental apertures : produce rays at a range of angles which are focused at different points image is thus smeared out. Even for a perfect optical system, diffraction limits resolution.

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Radius of the Airy disc f DD Radius, R A = f D = 1.22f /d Fraunhofer diffraction patterns also formed in focal plane of a lens ¶ ¶ e.g. see ‘Modern Optics’ by R Guenther Appendix 10-A

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Two finite slits E 2 (X) E 1 (X) E 1 (X) E 2 (X) d R x X a a Image courtesy of A Pedlar

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Point spread function Images courtesy of A Pedlar & from Wikipedia

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The diffraction grating David Rittenhouse Rittenhouse 1785: fine threads between screws – 100 threads/inch Fraunhofer 1821: thin wires Henry Augustus Rowland: curved gratings spectrocopy Henry Joseph Grayson 1899: developed precise ‘ruling engine’ 120,000 lines/inch Henry Augustus Rowland A periodic structure designed to diffract light Images from Wikipedia

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Grating structure Ruled grating Blazed grating – enhances diffraction in one direction Gratings: central to modern spectrometers reflection or transmission amplitude or phase Phase grating

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Analysis of diffraction from gratings d d 1 2 Path length difference for incident rays:Path length difference for diffracted rays: d 1 2

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Modern gratings Reflection gratings Transmission gratings CDs / DVDs Images from Wikipedia

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Gratings in nature Nacre Peacock feathers Butterfly wings Images from Wikipedia

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Grating based spectrometers The Czerny-Turner monochromator. A – input light B – entrance slit C – collimating mirror D – diffraction grating E – focusing mirror F – exit slit G – output light Image from Wikipedia

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General diffraction (again) Source, S Observation point, P Q Spherical wavefront s r s0s0 R

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Half-period zones s S P r m+1 rmrm rmrm S P

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Area of the m th zone S dd s sdsd s sin P s sin d S s P rmrm s+R

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Zone plates

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Arago’s spot François Jean Dominique Arago ( ) Siméon Denis Poisson ( ) Merde !

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Fresnel diffraction from straight edges y x Q( x,y ) S s P r s0s0 R

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