Lecture 27 1 ThinFilm InterferenceContd Assume nearnormal incidence

Lecture 27 -1 Thin-Film Interference-Cont’d (Assume near-normal incidence. ) Path length difference: destructive constructive where • ray-one got a phase change of 180 o due to reflection from air to glass. • the phase difference due to path length is: • then total phase difference: f = f’+180.

Lecture 27 -2 Two (narrow) slit Interference Young’s double-slit experiment • According to Huygens’s principle, each slit acts like a wavelet. The the secondary wave fronts are cylindrical surfaces. • Upon reaching the screen C, the two wave interact to produce an interference pattern consisting of alternating bright and dark bands (or fringes), depending on their phase difference. Constructive vs. destructive interference

Lecture 27 -3 Interference Fringes For D >> d, the difference in path lengths between the two waves is • A bright fringe is produced if the path lengths differ by an integer number of wavelengths, y ~ D*tan(θ)~ D*mλ/d • A dark fringe is produced if the path lengths differ by an odd multiple of half a wavelength, y ~ D*tan(θ)~ D*(m+1/2)λ/d

Lecture 27 -4 Intensity of Interference Fringes Let the electric field components of the two coherent electromagnetic waves be The resulting electric field component point P is then Intensity is proportional to E 2 I=0 when f = (2 m+1)p , i. e. half cycle + any number of cycle.

Lecture 27 -5 Dark and Bright Fringes of Single-Slit Diffraction

Lecture 27 -6 Phasor Diagram f 2 f 1

Lecture 27 -7 Phasor Diagram for Single-Slit Diffraction The superposition of wavelets can be illustrated by a phasor diagram. If the slit is divided into N zones, the phase difference between adjacent wavelets is total phase difference:

Lecture 27 -8 Intensity Distribution 1 maxima: central maximum because or minima:

Lecture 27 -9 Intensity Distribution 2 for small • Fringe widths are proportional to /a. • y ~ D*θ • Bright fringe: D*(m+1/2)λ/a • Dark fringe: D*mλ/a • Width: D*λ/a except central maximum • Width of central maximum is twice any other maximum. • Width = D*λ/a – D*(-1)λ/a = 2 D*λ/a • Intensity at first side maxima is (2/3 )2 that of the central maximum. y

Lecture 27 -10 Young’s Double-Slit Experiment Revisited • Intensity pattern for an ideal double-slit experiment with narrow slits (a<< ) Light leaving each slit has a unique phase. So there is no superimposed single-slit diffraction pattern but only the phase difference between rays leaving the two slits matter. slit separation a where I 0 is the intensity if one slit were blocked • If each slit has a finite width a (not much smaller than ), single-slit diffraction effects must be taken into account!

Lecture 27 -11 Intensity Distribution from Realistic Double-Slit Diffraction double-slit intensity replace by single-slit intensity envelope

Lecture 27 -12 Diffraction by a Circular Aperture • The diffraction pattern consists of a bright circular region and concentric rings of bright and dark fringes. • The first minimum for the diffraction pattern of a circular aperture of diameter d is located by geometric factor • Resolution of images from a lens is limited by diffraction. • Resolvability requires an angular separation of two point sources to be no less than R where central maximum of one falls on top of the first minimum of the other: Rayleigh’s criterion

Lecture 27 -13 Diffraction Gratings • Devices that have a great number of slits or rulings to produce an interference pattern with narrow fringes. • One of the most useful optical tools. Used to analyze wavelengths. D Types of gratings: • transmission gratings • reflection gratings up to thousands per mm of rulings Maxima are produced when every pair of adjacent wavelets interfere constructively, i. e. , mth order maximum

Lecture 27 -14 Spectral Lines and Spectrometer • Due to the large number of rulings, the bright fringes can be very narrow and are thus called lines. • For a given order, the location of a line depends on wavelengths, so light waves of different colors are spread out, forming a spectrum. Spectrometers are devices that can be used to obtain a spectrum, e. g. , prisms, gratings, …

Lecture 27 -15 X Ray Diffraction • X rays are EM radiation of the wavelength on the order of 1 Å, comparable to atomic separations in crystals. • X rays are produced, e. g. , when core electrons in atoms are inelastically excited. They are also produced when electrons are decelerated or accelerated. • Vacuum tubes, synchrotrons, … Ø Standard gratings cannot be used as X ray spectrometers. (Slit separation must be comparable to the wavelength!) Ø Von Laue discovered the use of crystals as 3 -dimensional diffraction gratings. Nobel 1914