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Young’s Experiment Coherence Two sources to produce an interference that is stable over time, if their light has a phase relationship that does not change with time: E(t)=E0cos(wt+f) Coherent sources: Phase f must be well defined and constant. When waves from coherent sources meet, stable interference can occur － laser light (produced by cooperative behavior of atoms) Incoherent sources: f jitters randomly in time, no stable interference occurs － sunlight 35- 2 Intensity and phase E t E 0 sin w t E 0 sin w t f ? E 2 E 0 cos E 4 E 0 cos 2 I I0 2 E 2 E 2 0 2 E 0 cos 12 f 2 1 2 f 4 cos 2 1 2 f I 4 I 0 cos 2 1 2 f Eq. 35-22 phase path length difference difference 2 f Fig. 35-13 phase 2 path length difference difference f 2 d sin Eq. 35-23 35- 3 Intensity in Double-Slit Interference E2 E1 E 1 E 0 sin w t I 4 I 0 cos m axim a w hen: 1 2 2 1 2 f 1 2 2 d f m for m 0,1, 2, d sin m for m 0,1, 2, m inim a w hen: f f m 1 2 an d E 2 E 0 sin w t f sin f 2 m 2 d sin (m axim a) d sin m 1 2 for m 0,1, 2, (m inim a) 35- 4 Intensity in Double-Slit Interference I avg 2 I 0 Fig. 35-12 35- 5 Ex.11-2 35-2 wavelength 600 nm n2=1.5 and m=1→m=0 Interference form Thin Films Reflection Phase Shifts n1 n1 n1 > n2 n1 < n2 n2 n2 Reflection Reflection Phase Shift Off lower index 0 Off higher index 0.5 wavelength Fig. 35-16 35- 8 Phase Difference in Thin-Film Interference Three effects can contribute to the phase difference between r1 and r2. 1. Differences in reflection conditions 2 Fig. 35-17 0 2. Difference in path length traveled. 3. Differences in the media in which the waves travel. One must use the wavelength in each medium ( / n), to calculate the phase. 35- 9 Equations for Thin-Film Interference ½ wavelength phase difference to difference in reflection of r1 and r2 2L odd num ber w avelength = odd num ber 2 2 n 2 (in-phase w aves) 2 L integer wavelength = integer n 2 (out-of-phase waves) n 2 n2 2L m 2L m 1 2 for m 0,1, 2, (m axim a-- bright film in a ir) n2 for m 0,1, 2, (m inim a-- dark film in air) n2 35- 10 Color Shifting by Paper Currencies,paints and Morpho Butterflies weak mirror soap film better mirror looking directly down ： red or red-yellow tilting ：green 35- 11 大 藍 魔 爾 蝴 蝶 雙狹縫干涉之強度 Ex.11-3 35-3 Brighted reflected light from a water film thickness 320 nm n2=1.33 m = 0, 1700 nm, infrared m = 1, 567 nm, yellow-green m = 2, 340 nm, ultraviolet Ex.11-4 35-4 anti-reflection coating Ex.11-5 35-5 thin air wedge Michelson Interferometer L 2d1 2d 2 Lm 2 L (in terferom eter) (slab of m aterial of thickness L placed in front of M 1 ) Fig. 35-23 35- 17 Determining Material thickness L Nm= 2L m = 2 Ln (num ber of w avelengths in slab of m a terial) Na= 2L (num ber of w avelengths in sam e thickness of air) N m -N a = 2 Ln 2L = 2L n-1 (difference in w avelengths for paths w ith and w ithout thin slab) 35- 18 Problem 35-81 In Fig. 35-49, an airtight chamber of length d = 5.0 cm is placed in one of the arms of a Michelson interferometer. (The glass window on each end of the chamber has negligible thickness.) Light of wavelength λ = 500 nm is used. Evacuating the air from the chamber causes a shift of 60 bright fringes. From these data and to six significant figures, find the index of refraction of air at atmospheric pressure. 35- 19 Solution to Problem 35-81 φ1 the phase difference with air ； 2 ：vacuum f1 f2 n 1g L 2 n 2 O 4 b L 2 LM P N Q b g 2 N 4 n 1 L N fringes n 1 N 2L 1 c 2c 5.0 10 60 500 10 h 1.00030 . mh 9 2 m 35- 20 11-3 Diffraction and the Wave Theory of Light Diffraction Pattern from a single narrow slit. Side or secondary maxima Light Central maximum Fresnel Bright Spot. Light Bright spot These patterns cannot be explained using geometrical optics (Ch. 34)! 36- 21 The Fresnel Bright Spot (1819) Newton corpuscle Poisson Fresnel wave Diffraction by a single slit a sin st (1 minima) a sin 2 (2 nd minima) 單 狹 縫 繞 射 之 強 度 雙狹縫與單狹縫 Double-slit diffraction (with interference) Single-slit diffraction Diffraction by a Single Slit: Locating the first minimum a 2 sin 2 a sin (first minimum) 36- 26 Diffraction by a Single Slit: Locating the Minima a 4 sin a sin 2 (second minimum) 2 a sin m , for m 1, 2, 3 (minima-dark fringes) 36- 27 Ex.11-6 36-1 Slit width Intensity in Single-Slit Diffraction, Qualitatively phase 2 path length difference difference N=18 =0 2 f x sin small Fig. 36-7 1st min. 1st side max. 36- 29 Intensity and path length difference sin f 1 2 E 2R E I Im Fig. 36-9 f Em 2 E 2 Em R Em 1 2 f sin 12 f sin I I m 2 f a sin 2 36- 30 Intensity in Single-Slit Diffraction, Quantitatively Here we will show that the intensity at the screen due to a single slit is: sin I I m w here 1 f 2 a 2 (36-5) sin (36-6) In Eq. 36-5, minima occur when: m , for m 1, 2, 3 If we put this into Eq. 36-6 we find: a m sin , for m 1, 2, 3 Fig. 36-8 or a sin m , for m 1, 2, 3 (m inim a-dark fringes) 36- 31 Ex.11-7 36-2 1 m , m 1, 2, 3, 2 Diffraction by a Circular Aperture Distant point source, e,g., star d lens sin 1.22 (1st m in.- circ. aperture) Image is not a point, as expected from geometrical optics! Diffraction is responsible for this image pattern d a Light Light sin 1.22 a (1st m in.- single slit) a 36- 33 Resolvability Rayleigh’s Criterion: two point sources are barely resolvable if their angular separation θR results in the central maximum of the diffraction pattern of one source’s image is centered on the first minimum of the diffraction pattern of the other source’s image. sm all Fig. 361 R sin 1.22 1.22 11 d d R (R ayleigh's criterion) 36-34 11-4.9 Diffraction – (繞射) Why do the colors in a pointillism painting change with viewing distance? Ex.11-8 36-3 pointillism D = 2.0 mm d = 1.5 mm Ex.11-9 36-4 d = 32 mm f = 24 cm λ= 550 nm The telescopes on some commercial and military surveillance satellites Resolution of 85 cm and 10 cm respectively D L R 1.22 d = 550 × 10–9 m. (a) L = 400 × 103 m ， D = 0.85 m → d = 0.32 m. (b) D = 0.10 m → d = 2.7 m. 36- 38 Diffraction by a Double Slit Single slit a~ Two vanishingly narrow slits a<< Two Single slits a~ I sin I m cos 2 2 (double slit) d a sin sin 36- 39 Ex.11-10 36-5 d = 32 μm a = 4.050 μm λ= 405 a sin d sin m 2 for m 0,1, 2, nm Diffraction Gratings Fig. 36-18 Fig. 36-19 d sin m for m 0,1, 2 Fig. 36-20 (m axim a-lines) 36- 41 Width of Lines Fig. 36-21 N d sin hw , hw sin hw hw (half w idth of central line) Nd Fig. 36-22 hw N d cos (half w idth of line at ) 36- 42 Grating Spectroscope Separates different wavelengths (colors) of light into distinct diffraction lines Fig. 36-24 Fig. 36-23 36- 43 Compact Disc Optically Variable Graphics Fig. 36-27 36- 45 全像術 Viewing a holograph A Holograph Gratings: Dispersion D (dispersion defined) D m d cos (dispersion of a grating) (36-30) Angular position of maxima d sin m Differential of first equation (what change in angle does a change in wavelength produce?) d co s d m d For small angles d an d d d co s m m d cos 36- 49 Gratings: Resolving Power R avg (resolving pow er defined) R N m (resolving pow er of a grating) (36-32) Rayleigh's criterion for halfwidth to resolve two lines hw Substituting for in calculation on previous slide hw N d cos m N R Nm 36-50 Dispersion and Resolving Power Compared 36- 51 X-Ray Diffraction X-rays are electromagnetic radiation with wavelength ~1 Å = 10-10 m (visible light ~5.5x10-7 m) X-ray generation X-ray wavelengths to short to be resolved by a standard optical grating Fig. 36-29 sin 1 m d sin 1 1 0.1 nm 0.0019 3000 nm 36- 52 Diffraction of x-rays by crystal d ~ 0.1 nm → three-dimensional diffraction grating 2 d sin m for m 0,1, 2 (B ragg's law ) Fig. 36-30 36- 53 X-Ray Diffraction, cont’d 5d 5 4 a 2 0 or d a0 20 0.2236 a 0 Fig. 36-31 36-54 Structural Coloring by Diffraction