Phase. E = A sin(2p f t+f) (wave in time) or E = A sin(2p x/l +f) (wave in space)

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1 Interference When two (or more) waves arrive at a point (in space or time), they interfere, and their amplitudes may add or subtract, depending on their frequency and phase. 1

2 Phase E = A sin(2p f t+f) (wave in time) or E = A sin(2p x/l +f) (wave in space) E= A sin(2p (x/l nt) + f) (traveling wave) = A sin(2p (x-vt)/l + f ) since v=l n 2

3 In - phase Amplitudes add Constructive Interference 3

4 180 degrees out of phase Amplitudes subtract Destructive Interference 4

5 90 degrees out of phase Both constructive and destructive interference 5

6 Different frequencies Complicated beat patterns 6

7 Coherence For two sources to interfere, they must have some amount of coherence Coherence: a well-defined phase relationship in time or in space 7

8 Coherence is related to bandwidth A single frequency (monochromatic) is fully coherent A wave composed of many frequencies does not have a well-defined phase 8

9 Coherence length and times Coherence length: l = c/dn white light - Dn ~ 3X10 14 Hz, l ~ 10-6 m laser light - Dn ~ 3X10 8 Hz, l ~ 1 m Coherence time t = c/dl white light - Dl ~ 300 nm, t ~ s laser light - Dn ~ 0.03 nm, t ~ s 9

10 Interference with light is not that common requires very small path differences Soap films laser light 10

11 Thin Film Interference I When the light is reflected from a thin film, there are two reflections: from the front surface and the back one. The two reflections have a fixed phase relation and interfere coherently. On the front surface, the reflection is hard (low index to high index) and there is a 180 degrees of phase difference. The second reflection is soft (high index to low index), yielding no phase difference. 11

12 Two coherent waves from reflection 12

13 13

14 Reflection Glass n=1.5 Air=1.0 R=(.5/2.5) 2 = 0.04 = 4% (Two surfaces for a plate of glass so ~8%) If we put a layer of in the middle (.225/2.25) 2 + (.275/2.75) 2 = = 2% 14

15 Thin Film Interference II If d is the thickness of the film, 2d is the extra distance the second reflection travels. If 2d is an odd half integer of wavelengths, the two reflections interfere constructively: very little transmission - lots of reflection If 2d is equal to an integer number of wavelengths, two reflected waves interfere destructively: 100% transmission. Remember different phase shifts on front and back! 15

16 2d is a half-integer wavelength 16

17 If both reflections are low-high 17

18 Thin Film Reflection Incident white light Light reflecting from the top surface of the oil has 180 o phase change Thickness of the oil film is ¼ l of red light Light reflected from the bottom of the oil has no phase change, but travels an extra l/2 so it interferes constructively with the light from the top surface so the film looks red. Other colors interfere mostly destructively. 18

19 If both reflections are low-high Incident white light Light reflecting from the top & bottom surface of the coating has 180 o phase change Thickness of the oil film is ¼ l of green light Light reflected from the bottom surface travels an extra l/2 so it interferes destructively with the light from the top surface so almost all light from the middle of the spectrum gets through. Some red and violet are reflected so the coating looks purple. 19

20 Thin Film Interference III The thickness of a film may vary, from which we can find interference patterns Applications: Camera lens coated with a thin film to reduce the reflection. Can only do for one wavelength. 20

21 Multicoating 21

22 Diffraction a wave phenomenon 22

23 Diffraction from a slit 23

24 Light travels in something other than straight lines?!? A property of waves 24

25 Diffraction zeroes slit b screen D Diffraction angle = (sin x) 2 x 2 b D b 25

26 Smaller slits - wider diffraction pattern 26

27 Huygen s principle Any wavefront can be replaced by many sources located uniformly all over the wavefront, radiating a spherical wave in phase Can be used for any shape opening 27

28 Circular aperture diffraction 28

29 Babinet s principle The diffraction pattern of an aperture and its complement are the same! 29

30 Fresnel vs. Fraunhofer diffraction IMMEDIATELY beyond slit, image of slit FAR from slit, Fraunhofer pattern IN BETWEEN, Fresnel diffraction (complicated) 30

31 The Spot of Arago 1818 Fresnel submits paper to French Academy contest - light is a wave Poisson (judge) - using Fresnel s theory, shows ridiculous prediction of a bright spot behind a circular obstruction Arago(judge) - tested Poisson s absurd result It was true! Fresnel wins the prize Poisson gets the spot named Poisson s spot 31

32 Resolving Power Rayleigh - maximum of one pattern aligned w/ minimum of the other q Rayleigh =1.22 (in radians) ~ 70 (in degrees) b b 32

33 Lenses - Diffraction limit Any lens is a finite aperture Minimum spot size is determined by diffraction - even for the PERFECT lens. The human eye: b = 3 mm, l ~ 550 nm q Rayleigh = 1/80 degree Measured ~ 1/60 degree - almost diffraction limited resolvable spots on retina ~ 4 µm separation Distance between cones in fovea ~ 1.5 µm 33

34 Diffraction a wave phenomenon can limit optical resolution can be used to measure sizes of small objects 34

35 Diffraction zeroes slit b screen D Diffraction angle = (sin x) 2 x 2 b D b 35

36 Diffraction from Multiple Slits 36

37 Diffraction from Multiple Slits 37

38 Diffraction from 6 Slits Features get narrow(1/n) Get stronger (N 2 ) 38

39 Diffraction Gratings a many-slit pattern produces sharp diffraction peaks angle depends on wavelength 3 kinds: Transmission Reflection Phase 39

40 Diffraction grating: interference of multiple sources l If l = n l, all waves add in phase - only true for specific q 40

41 Iridescence Diffraction gratings in Nature 41

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