Thin-Film Interference-Cont’d

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

2. 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 Two (narrow) slit Interference

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

4. Intensity is proportional to E2 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 I=0 when f = (2m+1)p , i.e. half cycle + any number of cycle.

5. Dark and Bright Fringes of Single-Slit Diffraction

6. Phasor Diagram f2 f1

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

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

9. Intensity Distribution 2 for small q y • 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. • y ~ D*θ • Bright fringe: D*(m+1/2)λ/a • Dark fringe: D*mλ/a • Width: D*λ/a except central maximum

10. 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 I0 is the intensity if one slit were blocked Young’s Double-Slit Experiment Revisited • Intensity pattern for an ideal double-slit experiment with narrow slits (a<<) • If each slit has a finite width a (not much smaller than ), single-slit diffraction effects must be taken into account!

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

12. 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 dis located by geometric factor Rayleigh’s criterion Diffraction by a Circular Aperture • 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:

13. D Types of gratings: • transmission gratings • reflection gratings up to thousands per mm of rulings 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. Maxima are produced when every pair of adjacent wavelets interfere constructively, i.e., mth order maximum

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. 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 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 coreelectrons in atomsare inelastically excited. They are also produced when electrons are decelerated or accelerated. • Vacuum tubes, synchrotrons, …