PC 20312 Wave Optics Section 4 Diffraction HuygensFresnel

PC 20312 Wave Optics Section 4: Diffraction

Huygens-Fresnel Principle I • Fresnel combined ideas of Huygens’ wavelets & interference • Postulated in 1818: “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. . . ” Hecht, p 444 Augustin-Jean Fresnel 1788 -1827 Image from Wikipedia

Huygens-Fresnel Principle II • 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 Gustav R. Kirchhoff 1824 -1887 Image from Wikipedia

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

Huygens-Fresnel Principle IV http: //www. acoustics. salford. ac. uk/feschools/waves/diffract 3. htm Divide an aperture into elemental areas each of which is a source of a spherical wavelet Image from Wikipedia

The Huygens-Fresnel Integral Q s r Observation point, P Source, S s 0 R Spherical wavefront

Fraunhofer diffraction The case of small, linear phase variation, i. e. : • r R + r , y d • r << R • r x, y x aperture Satisfied when s, r >> d Hence, “Far-field diffraction” Joseph von Fraunhofer 1787 -1826 Image from Wikipedia

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

Analysis of Fraunhofer diffraction Observation point, P(X, Y) r Q(x, y) s R Source, S s 0 Z Aperture, A(x, y)

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

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

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

Circular aperture II I=0. 0175 I(0) kaθD=3. 83

The diffraction limit If there was no diffraction: • parallel rays focused to a point • images would be perfectly sharp BUT, diffraction from instrumental apertures : f • 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. f Image from Google Images

Radius of the Airy disc Fraunhofer diffraction patterns also formed in focal plane of a lens¶ D f ¶ e. g. Radius, RA= f D = 1. 22 f /d see ‘Modern Optics’ by R Guenther Appendix 10 -A

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

Point spread function Images courtesy of A Pedlar & from Wikipedia

The diffraction grating A periodic structure designed to diffract light • Rittenhouse 1785: • fine threads between screws – 100 threads/inch • Fraunhofer 1821: • thin wires • Henry Augustus Rowland: • curved gratings David Rittenhouse 1732 -1796 • spectrocopy • Henry Joseph Grayson 1899: • developed precise ‘ruling engine’ Henry Augustus Rowland 1848 -1901 • 120, 000 lines/inch Images from Wikipedia

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

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

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

Gratings in nature Nacre Butterfly wings Peacock feathers Images from Wikipedia

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

General diffraction (again) Q s r Observation point, P Source, S s 0 R Spherical wavefront

Half-period zones s S rm+1 rm P rm+1 S rm P

Area of the mth zone s d d s sin s si n s s sin d S P s S rm P s+R

Zone plates

Arago’s spot François Jean Dominique Arago (1783 -1856) Merde ! Siméon Denis Poisson (1781 -1840) http: //demo. physics. uiuc. edu/Lect. Demo/scripts/demo_descript. idc? Demo. ID=749

Fresnel diffraction from straight edges y Q(x, y) r s x S P s 0 R