1. Chapter 35
Interference
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2. Learning Goals for Chapter 35
What happens when two waves combine (interfere) in space.
Understand interference pattern formed by interference of two coherent light waves.
Calculate intensity at locations of interference pattern.
Understand how interference occurs when light reflects from two surfaces of a thin film.
Interferometry, & how interference enables measurement of extremely small distances.
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3. Introduction
Why soap bubbles show vibrant color patterns, even though soapy water is colorless!
What causes the multicolored reflections from CDs & DVDs?
Look at optical effects that depend on wave nature of light.
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4. Principle of superposition
Interference occurs in any situation where two or more waves overlap in space.
Total wave at any point at any instant of time is governed by principle of superposition:
When two or more waves overlap, resultant displacement at any point & at any instant is found by adding instantaneous displacements that would be produced by individual waves if each were present alone.
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5. Wave fronts from a disturbance
Interference effects most easily seen when combine sinusoidal waves with a single frequency & wavelength.
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6. Constructive and destructive interference
Two identical sources of monochromatic waves, S1 and S2.
Two sources permanently in phase vibrating in unison.
Constructive interference occurs at point a equidistant from both sources.
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7. Conditions for Constructive Interference
Distance S2 to point b is exactly two (2) wavelengths greater than distance S1 to b.
Both waves arrive in phase
Waves “reinforce” each other.
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8. Conditions for Destructive Interference
Distance S1 to point c is half-integral # of wavelengths greater than distance S2 to c.
Both waves cancel or partly cancel each other.
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9. Constructive and Destructive interference
Start with 2 identical sources of monochromatic waves, S1 & S2, in phase.
Red curves show all positions where constructive interference occurs (antinodal curves).
Nodal curves would show where destructive interference occurs.
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10. Constructive and destructive interference
Concepts of constructive & destructive interference apply to water waves as well as to light & sound waves.
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11. Young’s Double-Slit Experiment
If light is a wave, interference effects will be seen, where one part of a wave front can interact with another part.
One way to study this is to do a double-slit experiment:
Figure (a) Young’s double-slit experiment. (b) If light consists of particles, we would expect to see two bright lines on the screen behind the slits. (c) In fact, many lines are observed. The slits and their separation need to be very thin.

12. Young’s Double-Slit Experiment
If light is a wave, interference effects will be seen, where one part of a wave front can interact with another part.
One way to study this is to do a double-slit experiment:
Figure (a) Young’s double-slit experiment. (b) If light consists of particles, we would expect to see two bright lines on the screen behind the slits. (c) In fact, many lines are observed. The slits and their separation need to be very thin.

13. Young’s Double-Slit Experiment
If light is a wave, interference effects will be seen, where one part of a wave front can interact with another part.
One way to study this is to do a double-slit experiment:
Figure (a) Young’s double-slit experiment. (b) If light consists of particles, we would expect to see two bright lines on the screen behind the slits. (c) In fact, many lines are observed. The slits and their separation need to be very thin.

14. Interference – Young’s Double-Slit Experiment
If light is a wave, there should be an interference pattern.
Figure If light is a wave, light passing through one of two slits should interfere with light passing through the other slit.

15. Two-source interference of light
Young’s Experiment (1801)
Interference of waves from slits S1 & S2 produces a pattern on the screen. (Check out weblink!)
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16. Two-source interference of light
Thomas Young
“The last man who knew everything”
Dr. of Medicine (haemodynamics)
Wave Theory of Light
Young’s Modulus (Stress/Strain)
Founder of physiological optics
Music (Young temperament)
Translator of Rosetta Stone…
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17. Young’s Double-Slit Experiment
Interference occurs because each point on screen is NOT same distance from both slits.
Depending on the path length difference, wave can interfere constructively (bright spot) or destructively (dark spot).
Figure How the wave theory explains the pattern of lines seen in the double-slit experiment. (a) At the center of the screen the waves from each slit travel the same distance and are in phase. (b) At this angle θ, the lower wave travels an extra distance of one whole wavelength, and the waves are in phase; note from the shaded triangle that the path difference equals d sin θ. (c) For this angle θ, the lower wave travels an extra distance equal to one-half wavelength, so the two waves arrive at the screen fully out of phase. (d) A more detailed diagram showing the geometry for parts (b) and (c).

18. Young’s Double-Slit Experiment
Between the maxima and the minima, the interference varies smoothly.
Figure (a) Interference fringes produced by a double-slit experiment and detected by photographic film placed on the viewing screen. The arrow marks the central fringe. (b) Graph of the intensity of light in the interference pattern. Also shown are values of m for Eq. 34–2a (constructive interference) and Eq. 34–2b (destructive interference).

19. Young’s Double-Slit Experiment
Interference pattern lines.
(a) Will there be an infinite number of points on the viewing screen where constructive and destructive interference occur, or only a finite number of points? (b) Are neighboring points of constructive interference uniformly spaced, or is the spacing between neighboring points of constructive interference not uniform?
Solution. a. Due to the fact that sin θ cannot be greater than 1, the maximum value of m is equal to the (truncated) value of d/λ.
b. The spacing increases with θ, although for small θ it is close to being uniform.

20. Constructive Interference from two slits
Constructive interference (reinforcement) occurs at points where path difference is integral number of wavelengths
path difference d = mλ
Bright regions on screen occur at angles θ for which
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21. Destructive Interference from two slits
Destructive interference (cancellation) occurs, forming dark regions on screen, at points for which path difference is a half-integral number of wavelengths.
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22. Line spacing for double-slit interference.
A screen containing two slits mm apart is 1.20 m from the viewing screen. Light of wavelength λ = 500 nm falls on th slits from a distant source. Approximately how far apart will adjacent bright interference fringes be on the screen?
Figure Examples 34–2 and 34–3. For small angles θ (give θ in radians), the interference fringes occur at distance x = θl above the center fringe (m = 0); θ1 and x1 are for the first-order fringe (m = 1), θ2 and x2 are for m = 2.
Solution: Using the geometry in the figure, x ≈ lθ for small θ, so the spacing is 6.0 mm.

23. Example: Changing wavelength
(a) What happens to interference pattern in previous example if incident light (500 nm) is replaced by light of wavelength 700 nm?
(b) What happens if wavelength stays at 500 nm but slits are moved farther apart?
Check out
Solution: a. As the wavelength increases, the fringes move farther apart.
b. Increasing the slit spacing causes the fringes to move closer together.

24. Young’s Double-Slit Experiment
Since position of maxima (except central one) depends on wavelength, first- and higher-order fringes contain a spectrum of colors.
Figure First-order fringes are a full spectrum, like a rainbow.

25. Example : Wavelengths from double-slit interference.
White light passes through two slits 0.50 mm apart, and an interference pattern is observed on a screen 2.5 m away. First-order fringe resembles a rainbow with violet and red light at opposite ends. The violet light is about 2.0 mm and the red 3.5 mm from the center of the central white fringe.
Estimate wavelengths for the violet and red light.
Solution: Using the geometry of the previous examples and the small-angle approximation, the wavelength for the first-order fringe is given by dx/l = 400 nm for the violet light and 700 nm for the red light.

26. Two-source interference of light
If distance R to screen is much greater than distance d between slits….
Use approximate geometry (c).
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27. Electric field in interference patterns
Find intensity at any point in two-source interference pattern, by combining two sinusoidally varying fields.
If two sources are in phase, then waves arriving at point P differ in phase by amount ϕ proportional to difference in path lengths
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28. Electric field in interference patterns
Assuming amplitudes of two waves are both approximately equal to E at point P, combined amplitude is:
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29. Phasor diagram for superposition
To add two sinusoidal functions with a phase difference, use phasor representation for simple harmonic motion & voltages & currents in AC circuits
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30. Phasor diagram for superposition
Each sinusoidal function is represented by rotating vector (phasor)
Projection on horizontal axis at any instant represents instantaneous value of sinusoidal function.
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31. Intensity in Double-Slit Interference Pattern
The two waves can be added using phasors, to take the phase difference into account:
Figure Phasor diagram for double-slit interference pattern.

32. Intensity in interference patterns
Intensity at any point in two-source interference pattern:
I0 is maximum intensity = 2 e0cE2 = four times larger than intensity from each individual source.
Averaged over all phase differences, I = I0/2, just twice intensity of each source.
Total energy isn’t changed – just redistributed in space.
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33. Intensity in Double-Slit Interference Pattern
The electric fields at the point P from the two slits are given by .
Figure Determining the intensity in a double-slit interference pattern. Not to scale: in fact l >> d and the two rays become essentially parallel.
where

34. Intensity in Double-Slit Interference Pattern
The time-averaged intensity is proportional to the square of the field:

35. Intensity in Double-Slit Interference Pattern
Intensity as a function of angle.
Figure Intensity I as a function of phase difference δ and position on screen y (assuming y << l).

36. Intensity in interference patterns
Phase difference is:
Must use wavelength in medium!
l = l0 / n
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37. Interference in thin films
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38. Phase shifts during reflection
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39. Phase shifts during reflection
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40. Interference in thin films: No relative shift
For light of normal incidence on thin film wavelength λ in film, where neither (or both) reflected waves have a half-cycle phase shift:
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41. Interference in thin films: Half-wave shift
For light of normal incidence on a thin film wavelength λ in film, when only one of reflected waves has a half-cycle phase shift:
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42. Nonreflective coatings
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43. Sketch of a Michelson Interferometer
Laser
Beam Splitter
End Mirror
Screen
Viewing

44. Michelson interferometer
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45. LIGO

46. LIGO sites LIGO (Washington) LIGO (Louisiana) (4km and 2km) (4km)
Funded by the National Science Foundation; operated by Caltech and MIT; the research focus for more than 670 LIGO Scientific Collaboration members worldwide.

47. The LIGO Observatories
LIGO Hanford Observatory (LHO) H1 : 4 km arms H2 : 2 km arms
10 ms
LIGO Livingston Observatory (LLO) L1 : 4 km arms
Adapted from “The Blue Marble: Land Surface, Ocean Color and Sea Ice” at visibleearth.nasa.gov
NASA Goddard Space Flight Center Image by Reto Stöckli (land surface, shallow water, clouds). Enhancements by Robert Simmon (ocean color, compositing, 3D globes, animation). Data and technical support: MODIS Land Group; MODIS Science Data Support Team; MODIS Atmosphere Group; MODIS Ocean Group Additional data: USGS EROS Data Center (topography); USGS Terrestrial Remote Sensing Flagstaff Field Center (Antarctica); Defense Meteorological Satellite Program (city lights).

48. Sensing the Effect of a Gravitational Wave
Gravitational wave changes arm lengths and amount of light in signal
Change in arm length is
10-18 meters,
or about
2/10,000,000,000,000,000 inches
Laser
signal

49. Gravitational Wave Detection
Suspended Interferometers
Suspended mirrors in “free-fall”
Fabry-Perot cavity
4km
g.w. output
port
power recycling
mirror
LIGO design length sensitivity: 10-18m

50. Core Optics Suspension and Control
Optics suspended as simple pendulums
Shadow sensors & voice-coil actuators provide damping and control forces
Mirror is balanced on 30 micron
diameter wire to 1/100th degree of arc

51. Seismic Isolation – Springs and Masses
damped spring cross section

52. How Small is 10-18 Meter? One meter, about 40 inches
Human hair, about 100 microns
Wavelength of light, about 1 micron
Atomic diameter, meter
Nuclear diameter, meter
LIGO sensitivity, meter

53. Gravitational Waves
Transverse distortions of spacetime due to motion of massive astronomical bodies.
Expected sources:
Inspiraling neutron stars/black holes
(Asymmetric) supernovae
Rotating pulsars
Cosmic gravitational-wave background
Expected properties:
Quadrupole polarization
Propagating at speed of light
Strains of ΔL/L = or less

54. Binary: a typical GW source
Strength of GWs
For binary system shown in the figure, strain h=2δl/l becomes h~R1R2/Dd
For neutron stars of 1.4M⊙out at Virgo cluster (15Mpc): h~10-21
Crazy but within current reach!
Binary: a typical GW source

55. Hulse-Taylor Binary Pulsar
17 / sec
~ 8 hr
PSR , measured in 1975
System should lose energy through gravitational radiation
Stars get closer together
Orbital period gets shorter

56. Why Are We Looking? “Chirp Signal”
We can use weak-field gravitational waves to study strong-field general relativity.

57. Fabry-Perot Michelson Interferometer
Uses light interference to measure path length difference between two arms
Each arm is a Fabry-Perot cavity, effectively increasing arm length
Geometry ideally suited for quadrupole radiation

58. Fabry-Perot Michelson Interferometer
Each arm is Fabry-Perot cavity, effectively increasing arm length
FP cavities increase path length by ~280 times, making equivalent length 1120 km long!

59. Latest Detection: Neutron Star Collision