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## Light Interference Continued…

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**+1**t -1 +1 t -1 +2 t -2 Superposition ConstructiveInterference + In Phase 5**+2**t -2 Superposition Destructive Interference +1 t -1 + +1 Out of Phase 180 degrees t -1 7**Superposition**+ Different f 1) Constructive 2) Destructive 3) Neither 10**Interference Requirements**• Need two (or more) waves • Must have same frequency • Must be coherent (i.e. waves must have definite phase relation) 12**hmmm… I’m just far enough away that l2-l1=l/2, and I**hear no sound at all! l1 l2 Interference for Sound … For example, a pair of speakers, driven in phase, producing a tone of a single f and l: But this won’t work for light--can’t get coherent sources 15**Observe Laser Light Through…**One Slit: Two Slits: Multiple Slits:**Observe Laser Light Through…**One Slit: Broad Central Maximum… Two Slits: Central Bright Spot with symmetric dark fringes. Multiple Slits: Central Bright Spot. Narrowor bright spots, brighter maximums, darker minimums.**Double Slit**Interference Only Interference + Diffraction**How do we predict the locations of the bright and dark**fringes produced by a single slit? double slit? Multiple slit?**Young’s Double Slit #1**Light waves from a single source travel through 2 slits before meeting on a screen. The interference will be: • Constructive • Destructive • Depends on L d The rays start in phase, and travel the same distance, so they will arrive in phase. Single source of monochromatic light L 2 slits-separated by d Screen a distance L from slits 23**½ l shift**Young Double Slit #2 The experiment is modified so that one of the waves has its phase shifted by ½ l. Now, the interference will be: • Constructive • Destructive • Depends on L d The rays start out of phase, and travel the same distance, so they will arrive out of phase. Single source of monochromatic light L 2 slits-separated by d Screen a distance L from slits 25**At points where the difference in path**length is the screen is dark. (destructive) 2 slits-separated by d Young’s Double Slit Concept At points where the difference in path length is 0, l,2l, …, the screen is bright. (constructive) d Single source of monochromatic light L Screen a distance L from slits 27**Young’s Double Slit Key Idea**L Two rays travel almost exactly the same distance.(screen must be very far away: L >> d) Bottom ray travels a little further. Key for interference is this small extra distance. 30**Constructiveinterference**Destructive interference where m = 0, or 1, or 2, ... Young’s Double Slit Quantitative d d Path length difference = d sin q Need l < d 32**y**Young’s Double Slit Quantitative L d A little geometry… sin(q) tan(q) = y/L Constructive interference Destructive interference where m = 0, or 1, or 2, ... 33**Young’s Double Slit #3**L y d When this Young’s double slit experiment is placed under water. The separation y between minima and maxima 1) increases 2) same 3) decreases Under water l decreases so y decreases 35**d**1) Which condition gives destructive interference? 2) where m = 0, or 1, or 2, ... Double Slit #4 L d = d sinq Path length difference d sin() 8**1**2 3 4 d d Path length difference 1-3 = 2d sinq = 3d sinq Path length difference 1-4 Constructive interference for all paths when Multiple Slits: (Diffraction Grating – N slits with spacing d) L d =l = d sinq Path length difference 1-2 =2l =3l 13**Constructive Interference Maxima are at:**Same as for Young’s Double Slit ! Diffraction Grating N slits with spacing d q * screen VERY far away**Three slit interference**9I0 I0 19**For many slits, maxima are still at**2 slits (N=2) 10 slits (N=10) intensity intensity 0 l 0 2l l 2l Multiple Slit Interference (Diffraction Grating) Peak location depends on wavelength! Region between maxima gets suppressed more and more as no. of slits increases – bright fringes become narrower and brighter. 22**Wall**shadow bright Diffraction Rays This is not what is actually seen! Screen with opening (or obstacle without screen) 26**Diffraction/ Huygens**Every point on a wave front acts as a source of tiny wavelets that move forward. • Light waves originating at different points within opening travel different distances to wall, and can interfere! • We will see maxima and minima on the wall. 30**Central maximum**1st minima**1**2 1 2 When rays 1 and 1 interfere destructively. Single Slit Diffraction W Rays 2 and 2also start W/2 apart and have the same path length difference. Under this condition, every ray originating in top half of slit interferes destructively with the corresponding ray originating in bottom half. 1st minimum at sin q = l/w 33**2**1 2 1 When rays 1 and 1 will interfere destructively. Single Slit Diffraction w Rays 2 and 2also start w/4 apart and have the same path length difference. Under this condition, every ray originating in top quarter of slit interferes destructively with the corresponding ray originating in second quarter. 2nd minimum at sin q = 2l/w 35**(m=1, 2, 3, …)**Single Slit Diffraction Summary Condition for halves of slit to destructively interfere Condition for quarters of slit to destructively interfere Condition for sixths of slit to destructively interfere All together… THIS FORMULA LOCATES MINIMA!! Narrower slit => broader pattern l Note: interference only occurs when w > l 38**opposite!**Recap. • Interference: Coherent waves • Full wavelength difference = Constructive • ½ wavelength difference = Destructive • Multiple Slits • Constructive d sin(q) = m l (m=1,2,3…) • Destructive d sin(q) = (m + 1/2) l 2 slit only • More slits = brighter max, darker mins • Huygens’ Principle: Each point on wave front acts as coherent source and can interfere. • Single Slit: • Destructive: w sin(q) = m l (m=1,2,3…) • Resolution: Max from 1 at Min from 2 50