Physics 102 Lecture 21 Diffraction Gratings Resolving Power

Physics 102: Lecture 21 Diffraction, Gratings, Resolving Power James Scholar Honors Credit article proposals due this Thursday, April 7! Physics 102: Lecture 21, Slide 1

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 Last lecture – Thin films – Multiple slit – X-ray diffraction from crystal Today’s lecture – Diffraction/single slit Physics 102: Lecture 21, Slide 2

ACT: Double slit review Which condition gives destructive interference? y θ θ dsin( ) L 1) dsin( ) = m m = 0, 1, 2 Physics 102: Lecture 21, Slide 3 2) dsin( ) = (m+1/2)

Multiple Slits: (Diffraction Grating – N slits with spacing d) Assume screen is very far away (L>>d): L 1 θ 2 3 4 θ Path length difference 1 -2 Path length difference 1 -3 Path length difference 1 -4 = d sinq =l = 2 d sinq =2 l = 3 d sinq =3 l Constructive interference for all paths when Physics 102: Lecture 21, Slide 4 dsin( ) = m m = 0, 1, 2

Multiple Slits: (Diffraction Grating – N slits with spacing d) Assume screen is very far away (L>>d): L 1 θ 2 3 4 θ Constructive: dsin( ) = m m = 0, 1, 2 Same condition as Young’s double slit! Holds for arbitrary N Physics 102: Lecture 21, Slide 5

Preflight 21. 1 L 1 θ 2 3 θ All 3 rays are interfering constructively at the point shown. If the intensity from ray 1 is I 0 , what is the combined intensity of all 3 rays? 1) I 0 2) 3 I 0 3) 9 I 0 7% 63% 31% Physics 102: Lecture 21, Slide 6 Each slit contributes amplitude Eo at screen. Etot = 3 Eo. But I a E 2. Itot = (3 E 0)2 = 9 E 02 = 9 I 0 = Imax

ACT/Preflight 21. 2 L 1 θ 2 3 θ this one is still there! these add to zero When rays 1 and 2 are interfering destructively, is the intensity from the three rays a minimum? 1) Yes 2) No 46% 54% Physics 102: Lecture 21, Slide 7 Rays 1 and 2 completely cancel, but ray 3 is still there. Expect intensity Itot = I 0 = 1/9 Imax

Three slit interference 9 I 0 Physics 102: Lecture 21, Slide 8 - -

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 10 slits (N=10) 0 l Physics 102: Lecture 21, Slide 9 2 l d sin(θ) 0 l 2 l d sin(θ)

X-Ray Diffraction: A technique to study crystal structure θ θ d Crystal solid such as sodium d sin(θ) Constructive interference: 2 d sin(θ) = mλ d ≈ 0. 5 nm in Na. Cl For =0. 017 nm X-ray Physics 102: Lecture 21, Slide 10 1 st maximum will be at 100 Measure , determine d

Single slit interference? Monochromatic light travels through a screen with opening Shadow Bright spot This is not what is actually seen! Physics 102: Lecture 21, Slide 11

Diffraction/Huygens’ principle Huygens: Every point on a wave front acts as a source of tiny wavelets that move forward. • • • Physics 102: Lecture 21, Slide 12 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 1 st minima Physics 102: Lecture 21, Slide 13

Single Slit Diffraction 1 θ w θ θ 2 1 2 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. Physics 102: Lecture 21, Slide 14 1 st minimum at sin q = l/w

Single Slit Diffraction θ 1 2 1 2 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. Physics 102: Lecture 21, Slide 15 2 nd minimum at sin q = 2 l/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 All together… (m = 1, 2, 3, …) THIS FORMULA LOCATES MINIMA!! Narrower slit => broader pattern Note: interference only occurs when w > λ Physics 102: Lecture 21, Slide 16 Preflight 21. 3

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) Physics 102: Lecture 21, Slide 17 (2) (3) (4)

Diffraction from Circular Aperture Central maximum 1 st 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 Physics 102: Lecture 21, Slide 18

Intensity from Circular Aperture I First diffraction minimum 1. 22λ/D Physics 102: Lecture 21, Slide 19

Demo: Resolving Power Not resolved Just resolved These objects are just resolved Two objects are just resolved when the maximum of one Physics 102: Lecture 21, Slide 20 pattern is at the minimum of the other. diffraction

Resolving Power To see two objects distinctly, need qobjects > qmin qobjects is angle between objects and aperture: qobjects ≈ tan-1(d/y) qmin is minimum angular separation that aperture can resolve: sin qmin ≈ qmin = 1. 22 /D D d y Improve resolution by increasing qobjects or decreasing qmin Physics 102: Lecture 21, Slide 21

ACT: Resolving Power sin qmin ≈ qmin = 1. 22 /D How does the maximum resolving power of your eye change when the brightness of the room is decreased? 1) Increases 2) Constant 3) Decreases When the light is low, your pupil dilates (D can increase by factor of 10!) But actual limitation is due to density of rods and cones, so you don’t notice an effect! Physics 102: Lecture 21, Slide 22

Recap • Interference: Coherent waves – Full wavelength difference = Constructive – ½ wavelength difference = Destructive opposite! • Multiple Slits – Constructive d sin( ) = m (m=1, 2, 3…) – Destructive d sin( ) = (m + 1/2) 2 slit only – More slits = brighter max, darker mins • Single Slit: – Destructive: – Resolution: Physics 102: Lecture 21, Slide 23 w sin( ) = m (m=1, 2, 3…) Max from 1 at Min from 2

See you Monday! Physics 102: Lecture 21, Slide 24