1. Chapter 7: Interference of light Chapter 7: Interference of light

2. in·ter·fer·ence
1. Life. Hindrance or imposition in the concerns of others.
2. Sports. Obstruction of an opponent, resulting in penalty.
3. Physics. Superposition of two or more waves, resulting in a new wave pattern.
constructive
destructive 2

3. HeNe laser

4. Radio City Rockettes, New York, NY

5. rood blauw oranje paars oranje blauw groen rood blauw paars groen rood oranje blauw rood groen paars oranje rood blauw groen rood blauw paars oranje blauw rood groen paars oranje rood blauw
J.R. Stroop "Studies of interference in serial verbal reactions" Journal of Experimental Psychology 18: (1935). 5

6. Peacock

7. Soap bubbles

8. from superposition principle:
2-beam interference
initial phase (at t=0)
propagation distance from source of disturbance
from superposition principle:

9. Measuring interference
- Electric fields are rapidly varying (n ~ 1014 Hz)
- Quickly averages to 0
- Instead of measuring E directly, measure radiant power density
= irradiance, Ee (W/m2)
= time average of the square of the electric field amplitude
- Note: to avoid confusion, Pedotti3 now uses the symbol I instead of Ee

10. Irradiance at point P I = I1 + I2 + I12
- when E1 and E2 are parallel, maximum interference
- when orthogonal, dot product = 0; no interference

11. The interference term I12
dot product of electric fields:
simplify by introducing constant phases:
use trigonometry: 2cosAcosB = cos(A+B) + cos(B-A) and consider again the time average:
w kills it

12. The interference term I12
simplify by introducing d:
to yield the interference term of the irradiance:

13. Irradiance formula if E1║ E2, then -where d is the phase difference
-for parallel electric fields

14. constructive interference destructive interference
mutually incoherent beams (very short coherence time)
mutually coherent beams (long coherence time)
maximum when cos d = 1
constructive interference
d = (2mp)
minimum when cos d = -1
destructive interference
d = (2m+1)p

15. Interference fringes
maximum when I1 = I2 = I0
1 + 1 = 4 !?!

16. Interference in time and space Michelson interferometer
Young’s experiment
wavefront division
Michelson interferometer
amplitude division

17. The double slit experiment (first performed in 1803)

18. Double slit experiment with electrons

19. Criteria for light and dark bands
- approximate arc S1Q to be a straight line
- optical path difference D = a sinq
conditions for interference:
constructive
destructive
m = 0, 1, 2, 3, …

20. Interference from 1 source: reflection
Lloyd’s mirror
part of the wavefront is reflected; part goes direct to the screen
Fresnel’s mirrors
part of the wavefront is reflected off each mirror

21. Fresnel’s mirrors as solar collectors

22. Interference from 1 source: refraction
Fresnel’s biprism
part of the incident light is refracted downward and part upward

23. Interference via amplitude division
- thin films
- oil slicks
- soap bubbles
- dielectric coatings
- feathers
- insect wings
- shells
- fish
- …

24. Interference intermezzo Interference intermezzo

26. The Dancing Couple-1663-Jan Steen

28. Anatomy of a soap bubble

29. Soap bubble interference

30. Thin film interference: normal incidence
optical path difference: D = nf(AB + BC) = nf (2t)

31. Thin film interference: non-normal incidence
optical path difference: D = nf(AB + BC) – n0(AD)
= 2nf t cosqt
D = ml: constructive interference
D = (m + ½)l: destructive interference
where m = 0,1,2,…

32. Keep in mind the phase
“hard”
reflection
“soft”
reflection
Simple version: phase of reflected beam shifted by p if n2 > n1
0 if n1 > n2
Correct version: use Fresnel equations!

33. Summary of phase shifts on reflection
air
glass
external reflection
n1 < n2
TE mode
TM mode n1 n2
air
glass
internal reflection
n1 > n2
TE mode
TM mode n1 n2

34. Back to the bubbles

35. Colors indicate bubble thickness
How thick here (red band)? t
n>1
180o phase change
0o phase change
Constructive interference for
2t ~ (m + ½)l
At first red band m = 0
t ~ ¼ (700 nm)

36. Dark, white, and bright bands
pop!
Bright: Colored “monochromatic” stripes occur at (1/4)l for visible colors
White: Multiple, overlapping interferences (higher order)
Dark: Super thin; destructive interference for all wavelengths
(no reflected light)

37. Multiple beam interference
r, t : external reflection
r’, t’ : internal reflection
Note: thickness t !
where d is the phase difference
geometric series

38. Multiple beam interference
Introduce Stokes relations: r’=-r and tt’=1-r2 and simplify to get:
Irradiance:

39. Multiple beam interference
Working through the math, you’ll arrive at:
where Ii is the irradiance of the incident beam
Likewise for transmission leads to:

40. Fabry-Perot interferometer (1897)
This simulation was performed for the two sodium lines described above, with reflectivity                 and the separation of the mirrors increasing from 100 microns to 400 microns.
Fabry-Perot interferometer (1897) d
simulation of two sodium lines:
l1 = mm
l2 = mm
mirror reflectivity r = 0.9
mirror separation: mm

41. Fabry-Perot interferometer
see chapter 8
where F is the coefficient of finesse:

42. Fabry-Perot interferometer: fringe profiles
Michelson d
- transmission maxima occur when d = 2pm
as r approaches 1 (i.e. as F increases), the fringes become very narrow
see Chapter 8 for more on Fabry-Perot:
fringe contrast, FWHM, finesse, free spectral range

43. Fringes of equal thickness
Constructive reflection
2d = (m+1/2)λ m=0, 1, 2, 3...
Destructive reflection
2d = mλ m=0, 1, 2, 3...

44. Newton’s rings
white-light illumination
pattern depends on contact point: goal is concentric rings

45. Oil slick on pavement Constructive reflection 2d = mλ m=0, 1, 2, 3...
Destructive reflection
2d = (m+1/2)λ m=0, 1, 2, 3...

46. Thin film coatings: anti-reflective
Glass: n = 1.5 MgF2 coating: n = To make an AR coating for l = 550 nm, how thick should the MgF2 layer be?

47. Broadband anti-reflective films

48. Multilayer mirrors
thin layers with a high refractive index n1,interleaved with thicker layers with a lower refractive index n2
path lengths lA and lB differ by exactly one wavelength
each film has optical path length D = l/4: all reflected beams in phase
ultra-high reflectivity: % or better over a narrow wavelength range

49. Anodized titanium

50. Natural multi-layer reflectors

51. Exercises
You are encouraged to solve all problems in the textbook (Pedrotti3).
The following may be covered in the werkcollege on 5 October 2011:
Chapter 7:
1, 2, 7, 9, 15, 16, 24