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 X-ray diffraction from crystal Diffraction/single slit Last lecture Today’s lecture

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

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

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

Preflight 21.1 L 1 2 3 θ d θ 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

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

Three slit interference

Multiple Slit Interference (Diffraction Grating) Peak location depends on wavelength! For many slits, maxima are still at Region between maxima gets suppressed more and more as no. of slits increases – bright fringes become narrower and brighter. 2 slits (N=2) intensity l 2l 10 slits (N=10) intensity l 2l Here do demo using diffraction grating for red and green laser

X-Ray Diffraction: A technique to study crystal structure θ d Crystal solid such as sodium Constructive interference: Mention that this is used by solid state physicists to determine lattice spacing in crystals in NaCl 1st maximum will be at 100 For =0.017nm X-ray Measure , determine d

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

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!

Central maximum 1st minima

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

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

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 (m = 1, 2, 3, …) All together… THIS FORMULA LOCATES MINIMA!! Narrower slit => broader pattern Note: interference only occurs when w > l

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? (1) (2) (3) (4)

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

Intensity from Circular Aperture First diffraction minima

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

Resolving Power To see two objects distinctly, need qobjects > qmin qobjects is angle between objects and aperture: qmin qobjects ≈ tan (d/y) qmin is minimum angular separation that aperture can resolve: demo 752; 2 point sources and different size slits D sin qmin ≈ qmin = 1.22 l/D y d Improve resolution by increasing qobjects or decreasing qmin

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

Recap Interference: Coherent waves Multiple Slits Single Slit: 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 opposite!