1. Lecture 27-1 Thin-Film Interference-Cont’d Path length difference: (Assume near-normal incidence.) 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:  =  ’+180.

2. Lecture 27-2 Two (narrow) slit Interference 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 According to Huygens’s principle, each slit acts like a wavelet. The the secondary wave fronts are cylindrical surfaces. Young’s double-slit experiment

3. 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, 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 y ~ D*tan(θ)~ D*mλ/d

4. 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  = (2m+1) , i.e. half cycle + any number of cycle.

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

6. Lecture 27-6 Phasor Diagram 11 22

7. Lecture 27-7 Phasor Diagram for Single-Slit Diffraction total phase difference: 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

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

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

10. Lecture 27-10 Young’s Double-Slit Experiment Revisited If each slit has a finite width a (not much smaller than ), single-slit diffraction effects must be taken into account! Intensity pattern for an ideal double-slit experiment with narrow slits (a<< ) slit separation 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. where I 0 is the intensity if one slit were blocked a

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

12. 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

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

14. 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, …

15. 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, …  Standard gratings cannot be used as X ray spectrometers. (Slit separation must be comparable to the wavelength!)  Von Laue discovered the use of crystals as 3-dimensional diffraction gratings. Nobel 1914