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Diffraction, Gratings, Resolving Power

Physics 102: Lecture 21. Diffraction, Gratings, Resolving Power. Recall. Interference (at least 2 coherent waves) Constructive (full wavelength difference) Destructive (half wavelength difference) Light (1 source, but different paths) Young’s double slit Thin films Multiple slit

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Diffraction, Gratings, Resolving Power

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  1. Physics 102:Lecture 21 Diffraction, Gratings, Resolving Power

  2. Recall • Interference (at least 2 coherent waves) • Constructive (full wavelength difference) • Destructive (half wavelength difference) • Light (1 source, but different paths) • Young’s double slit • Thin films • Multiple slit • X-ray diffraction from crystal • Diffraction/single slit Last lecture Today’s lecture

  3. y ACT: Double slit review Which condition gives destructive interference? θ θ d L dsin(q) 1) dsin(q) = ml 2) dsin(q) = (m+1/2)l m = 0, 1, 2

  4. Multiple Slits: (Diffraction Grating – N slits with spacing d) Assume screen is very far away (L>>d): L 1 2 θ θ 3 d d d 4 =l = d sinq Path length difference 1-2 =2l = 2d sinq Path length difference 1-3 =3l = 3d sinq Path length difference 1-4 Constructive interference for all paths when dsin(q) = ml m = 0, 1, 2

  5. Multiple Slits: (Diffraction Grating – N slits with spacing d) Assume screen is very far away (L>>d): L 1 2 θ θ 3 d d d d d d 4 m = 0, 1, 2 Constructive: dsin(q) = ml Same condition as Young’s double slit! Holds for arbitrary N

  6. Preflight 21.1 L 1 θ 2 θ 3 d d All 3 rays are interfering constructively at the point shown. If the intensity from ray 1 is I0 , what is the combined intensity of all 3 rays? 1) I0 2) 3 I0 3) 9 I0

  7. ACT/Preflight 21.2 L 1 θ 2 θ 3 d d When rays 1 and 2 are interfering destructively, is the intensity from the three rays a minimum? 1) Yes 2) No

  8. Three slit interference 9I0 I0

  9. For many slits, maxima are still at 2 slits (N=2) 10 slits (N=10) intensity intensity l l 2l 0 0 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.

  10. d Constructive interference: in NaCl For =0.017nm X-ray X-Ray Diffraction:A technique to study crystal structure θ θ Crystal solid such as sodium 1st maximum will be at 100 Measure , determine d

  11. Single slit interference? Monochromatic light travels through a screen with opening Shadow Bright spot This is not what is actually seen!

  12. Diffraction/Huygens’ principle 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!

  13. Central maximum 1st minima

  14. 1 2 1 2 When rays 1 and 1 interfere destructively. Single Slit Diffraction W Rays 2 and 2also 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

  15. 2 1 2 1 When rays 1 and 1 will interfere destructively. Single Slit Diffraction w Rays 2 and 2also 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

  16. (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 Note: interference only occurs when w > l

  17. (1) (2) (3) (4) ACTS/Preflights 21.4, 21.5 A laser is shined onto a screen through a very small hole. If you make the hole even smaller, the spot on the screen will get: (1) Larger (2) Smaller Which drawing correctly depicts the pattern of light on the screen?

  18. Central maximum 1st diffraction minimum First diffraction minimum is at q Diameter D light Diffraction from Circular Aperture Maxima and minima will be a series of bright and dark rings on screen

  19. Intensity from Circular Aperture I First diffraction minima

  20. These objects are just resolved Two objects are just resolved when the maximum of one is at the minimum of the other.

  21. Resolving Power To see two objects distinctly, need qobjects>qmin qobjects qobjectsis angle between objects and aperture: qmin qobjects≈ tan (d/y) qmin is minimum angular separation that aperture can resolve: D sin qmin≈qmin = 1.22 l/D y d Improve resolution by increasing qobjectsor decreasingqmin

  22. ACT: Resolving Power How does the maximum resolving power of your eye change when the brightness of the room is decreased. 1) Increases 2) Constant 3) Decreases

  23. 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 • Single Slit: • Destructive: w sin(q) = m l (m=1,2,3…) • Resolution: Max from 1 at Min from 2

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