Physics 214 Lecture 4

1
Applications of Interference and Diffraction
2
Overview

• Circular Diffraction (foreshadowing of quantum
  uncertainty)
• Angular resolution (Rayleighs criterion)
• Minimum spot size
• Interferometers
• Michelson
• Applications

3
Diffraction-limited Optics

• Diffraction has important implications for
  optical instruments
• Lens-making is a craft. Even for a perfectly
  designed lens, however, the image of a point
  source will be a little blurry due to
  diffraction in passing through the circular
  aperture of the lens.

The image of a point source through a circular
aperture is like a single-slit diffraction
pattern. But note the difference
4
Transmission of light through slits and circular
apertures
Observation screen
Slit, width a
Observation screen
Pinhole, diameter D
Image Plane
Lens, diameter D
Laser with pinholes
Circular-aperture diffraction pattern the Airy
disk. Central lobe contains 84 of power.
5
Exercise 1 Expansion of a Laser beam

• In 1985, a laser beam with a wavelength of l
  500 nm was fired from the earth and reflected off
  the space shuttle Discovery, in orbit at a
  distance of L 350 km away from the laser.
• If the (circular) aperture of the laser was D
  4.7 cm, what was the beam diameter d at the space
  shuttle?

6
Exercise 1 Expansion of a Laser beam - Solution

• In 1985, a laser beam with a wavelength of l
  500 nm was fired from the earth and reflected off
  the space shuttle Discovery, in orbit at a
  distance of L 350 km away from the laser.
• If the (circular) aperture of the laser was D
  4.7 cm, what was the beam diameter d at the space
  shuttle?

84 of power is in central lobe.
7
Act 1 Expansion of a Laser beam

• In 1985, a laser beam with a wavelength of l
  500 nm was fired from the earth and reflected off
  the space shuttle Discovery, in orbit at a
  distance of L 350 km away from the laser.
• To make a smaller spot on the shuttle, what
  should we do to the beam diameter at the source?
• a. reduce it
• b. increase it
• c. cannot be made smaller

8
Act 1 Expansion of a Laser beam - Solution

• In 1985, a laser beam with a wavelength of l
  500 nm was fired from the earth and reflected off
  the space shuttle Discovery, in orbit at a
  distance of L 350 km away from the laser.
• To make a smaller spot on the shuttle, what
  should we do to the beam diameter at the source?
• a. reduce it
• b. increase it
• c. cannot be made smaller

Counter-intuitive as this is, it is correct you
reduce beam divergence by using a bigger beam.
(Note this will work until D d)
We will see soon that this can be understood as a
consequence of the uncertainty principle Dx
Dpxgt??
9
Exercise 2 Focusing of a laser beam

• There are many times you would like to focus a
  laser beam to as small a spot as possible.
  However, diffraction limits this.
• The (circular) aperture of a laser (l 780 nm)
  has Dlaser 5 mm. What is the spot-size d of the
  beam after passing through a (perfect) lens with
  focal length f5mm, diameter Dlens 6 mm? (Hint
  light passing through lens center is
  undeflected.)

Dlens
d
Dlaser
f
10
Exercise 2 Focusing of a laser beam - Solution

• There are many times you would like to focus a
  laser beam to as small a spot as possible.
  However, diffraction limits this.
• The (circular) aperture of a laser (l 780 nm)
  has Dlaser 5 mm. What is the spot-size d of the
  beam after passing through a (perfect) lens with
  focal length f5mm, diameter Dlens 6 mm? (Hint
  light passing through lens center is
  undeflected.)

Dlens
d
Dlaser
f
Light at this angle will intercept the focal
plane at d/2 f qo
11
Act 2 Focusing of a laser beam

• There are many times you would like to focus a
  laser beam to as small a spot as possible.
  However, diffraction limits this.
• Which of the following will reduce the spot
  size?
• a. increase l
• b. decrease l
• c. increase Dlens
• d. decrease Dlens

Dlens
d
Dlaser
f
12
Act 2 Focusing of a laser beam - Solution

• There are many times you would like to focus a
  laser beam to as small a spot as possible.
  However, diffraction limits this.
• Which of the following will reduce the spot
  size?
• a. increase l
• b. decrease l
• c. increase Dlens
• d. decrease Dlens

Dlens
d
Dlaser
f
13
Angular Resolution

• Diffraction also limits our ability to resolve
  (i.e., distinguish) two point sources. Consider
  two point sources (e.g., stars) with angular
  separation a viewed through a circular aperture
  or lens of diameter D.

Rayleighs Criterion define the images to be
resolved if a ³ ac , where
At ac the central max of one image falls on the
first minimum of the second image
14
Exercise 3 Angular resolution

• Car headlights in the distance
• What is the maximum distance L you can be from an
  oncoming car at night, and still distinguish its
  two headlights, which are separated by a distance
  d 1.5 m? Assume that your pupils have a
  diameter D 2 mm at night, and that the
  wavelength of light is l 550 nm.

15
Exercise 3 Solution

• Car headlights in the distance
• What is the maximum distance L you can be from an
  oncoming car at night, and still distinguish its
  two headlights, which are separated by a distance
  d 1.5 m? Assume that your pupils have a
  diameter D 2 mm at night, and that the
  wavelength of light is l 550 nm.

Rayleighs Criterion
16
Act 3 Resolving Stars
Halleys Comet
1. Assuming diffraction-limited optics (best
possible), what is the minimum angular separation
of two stars that can be resolved by a D 5 m
reflecting telescope using light of l 500
nm? a. 0.1 mrad b. 1 mrad c. 10 mrad
2. If the two points are not quite resolved
at screen 1, will they be resolved at screen 2?
17
Act 3 Resolving Stars - Solution
Halleys Comet
1. Assuming diffraction-limited optics (best
possible), what is the minimum angular separation
of two stars that can be resolved by a D 5 m
reflecting telescope using light of l 500
nm? a. 0.1 mrad b. 1 mrad c. 10 mrad
The real limit of earth-bound telescopes is about
an order of magnitude larger due to atmospheric
effects (\ the Hubble).
2. If the two points are not quite resolved
at screen 1, will they be resolved at screen 2?
NO! Only the angle counts.
18
Example Problem Camera resolution (Discussion
next week)
A modern-day digital camera basically looks
something like this

• If the distance between adjacent pixels is less
  than the minimum resolvable separation due to
  diffraction, then the image can look blurry.

• The f-number of a lens is defined as f/D. To
  minimize diffraction, you want a small f-number,
  i.e., a large aperture.

http//www.cambridgeincolour.com/tutorials/diffrac
tion-photography.htm

• This assumes a perfect lens. In practice,
  lens aberrations limit the resolution if D is too
  big.

19
Everyday Interferometers! Thin Films!

• Why do soap bubbles appear colored? Oil films on
  water?
• Interference -- light reflected from the front
  and back surfaces interferes.
• However, light that reflects off a higher-index
  layer gets an extra p phase-shift (from
  Maxwells equations).
• For a film of thickness d, viewed at an angle q,
  the path length difference is d 2dsinq and the
  phase difference between the light reflected from
  the front and back surfaces is f 2pd/l p.

Destructive interference 2dsinq
ml Constructive interference 2dsinq (m1/2)l
20
Optical Interferometers

• Interference arises when there are two (or more)
  ways for something to happen, e.g., two slits for
  the light to get from the source to the screen.
• I 4I1 cos2(?/2), with ? 2p d/l, and
  path-length difference d.
• An interferometer is a device using mirrors and
  beam splitters (half light is transmitted, half
  is reflected) to give two separate paths from
  source to detector.
• Two common types
• Mach-Zehnder Michelson

beam- splitter
mirror
21
Michelson Interferometer

• Michelson interferometer works by varying the
  relative phase of the light waves for the two
  paths light can take
• One possibility is to vary the lengths L1 or L2
• Makes possible very accurate measurements of
  displacements

mirror
Total Path length L1
Path-length difference d L2 - L1
2I1
2I1
Total Path length L2
4I1
2I1
mirror
2I1
I1
I1
beam- splitter
I 4I1 cos2(?/2), with ? 2p d/l
22
ACT 4
d

• Consider the following Michelson interferometer.
  Assume that for the setup shown, all the light
  (with l 500 nm) comes out the bottom port.
• 1. How much does the top mirror need to be
  moved so that none of the light comes out the
  bottom port?

2. Where does the light then go? a. down b.
up c. left d. right
23
ACT 4 - Solution
d

• Consider the following Michelson interferometer.
  Assume that for the setup shown, all the light
  (with l 500 nm) comes out the bottom port.
• 1. How much does the top mirror need to be
  moved so that none of the light comes out the
  bottom port?

We need to go from complete constructive to
complete destructive interference ? ?f 180 ? ?
l/2 Howeverwhen we move the mirror by d, we
change ? by 2d. Therefore, d ?/2 l/4 500/4
125 nm.
2. Where does the light then go? a. down b.
up c. left d. right
24
ACT 4 - Solution
d

• Consider the following Michelson interferometer.
  Assume that for the setup shown, all the light
  (with l 500 nm) comes out the bottom port.
• 1. How much does the top mirror need to be
  moved so that none of the light comes out the
  bottom port?

2. Where does the light then go? a. down b.
up c. left d. right
The light goes out the way it came in. Energy
is conserved --the light cant just disappear!
(This is still true in quantum mechanics see
later)
The Michelson interferometer is perhaps most
famous for disproving the hypothesis that EM
waves propagate through an aether this result
helped stimulate the Special Theory of Relativity
25
Michelson Interferometer

• Another possibility is to vary the phase by
  changing the speed of the waves in the two arms
• Recall vc/n where n index of refraction
• Using l vf, f 2pL(1/l1 1/l2) (where L
  L1 L2)
• Makes possible very accurate measurement of
  changes in the speed of light in the two arms

mirror
Total Path length L, l1 (c/n1)f
Vary index of refraction n in one arm
Phase difference ? 2p L (f/c) (n1 n2)
2I1
2I1
Total Path length L, l2 (c/n2)f
4I1
2I1
mirror
2I1
I1
I1
beam- splitter
I 4I1 cos2(?/2), with ? 2pL(f/c) (n1 n2)
26
FYI Application Optical Coherence Tomography

• One mirror of the Michelson is replaced by
  human tissue. The type of tissue controls the
  amount of reflection, and the phase shift.

• By sending in many colors, one can learn about
  the density, composition, and structure of the
  tissue.
• Used for medical diagnostics like a
  microscope, but you dont have to excise the
  sample from the body!
• Used to study
• skin cancer
• cardiovascular disease (detect bad plaques)
• glaucoma and macular degeneration (incurable eye
  disease)

27
FYI Application Gravity Wave Detection

• Einstein predicted that when massive objects
  accelerate, they produce time-dependent
  gravitational fields gravity waves that
  propagate as warpings of spacetime at the speed
  of light. (EM radiation from accelerating e)
• The effect is very tiny E.g., estimated DL/L of
  10-21 for in-spiraling binary neutron stars.
  How to detect this???

28
FYI Application Gravity Wave Detection
LIGO Laser Interferometric Gravitational wave
Observatory -Worlds largest interferometers
4-km -2 in Hanford, WA 1 in Livingston, LO -
gt400 scientists -Projected sensitivity 3 x 10-23
? DL 10-19 m (10-9 Ang.) -Real searches now
underway!
29
ACT 5

• Consider the following Sagnac sahn-yack
  interferometer. Here the two possible paths are
  the clockwise and counter-clockwise circuits
  around the fiber loop.

fiber loop
1. If we insert an extra piece of glass as shown,
how does the relative path length change?
2. How could we change the relative path-length
difference, and thereby change how much light
exits the bottom port?
30
ACT 5

• Consider the following Sagnac sahn-yack
  interferometer. Here the two possible paths are
  the clockwise and counter-clockwise circuits
  around the fiber loop.

fiber loop
1. If we insert an extra piece of glass as shown,
how does the relative path length change?
It doesnt! Because the interference paths
completely overlap, the Sagnac is a remarkably
stable interferometer, e.g., to temperature
fluctuations in the fiber.
2. How could we change the relative path-length
difference, and thereby change how much light
exits the bottom port?
31
ACT 5

• Consider the following Sagnac sahn-yack
  interferometer. Here the two possible paths are
  the clockwise and counter-clockwise circuits
  around the fiber loop.

fiber loop
1. If we insert an extra piece of glass as shown,
how does the relative path length change?
It doesnt! Because the interference paths
completely overlap, the Sagnac is a remarkably
stable interferometer, e.g., to temperature
fluctuations in the fiber.
2. How could we change the relative path-length
difference, and thereby change how much light
exits the bottom port?
Rotate the entire interferometer (in the plane
of the paper). For example, if we rotate it
clockwise, the light making the clockwise circuit
will have farther to go (the beamsplitter is
running away), while the counterclockwise path
will be shortened. It is not difficult to show
that
Monitor output intensity ? determine f ? rate of
rotation w ? laser ring gyroscope!
32
The origins of quantum mechanics

• 1900 Planck solves the blackbody problem by
  postulating that the oscillators that emit light
  have quantized energy levels.
• Until after some weeks of the most strenuous
  work of my life, light came into the darkness,
  and a new undreamed-of perspective opened up
  before methe whole procedure was an act of
  despair because a theoretical interpretation had
  to be found at any price, no matter how high that
  might be.
• 1905 Einstein proposes that light energy is
  quantized with quanta called photons - waves
  behave like particles
• Photoelectric electric effect for which he got
  the Nobel Prize
• 1913 Bohr proposes that electron orbits are
  quantized
• Idea that electrons act like waves - explained
  H atom, but wrong in crucial ways
• 1923 de Broglie proposes that particles behave
  like waves
• The step that paved the way for understanding all
  of nature
• 1925 Pauli introduces exclusion principle
  only 2 electrons/orbital
• The step that leads to understanding of electrons
  in atoms, molecules, solids
• 1926 Schrödinger introduces the wave-formulation
  of QM
• The fundamental equation that predicts the nature
  of matter
• 1927 Heisenberg uncertainty principle
• The principle that shows the fundamental
  uncertainty in any one measurement
• 1928 Dirac combines quantum mechanics and special
  relativity
• The step that made QM the most successful theory
  in the history of physics description of
  atoms, nuclei, elementary particles, prediction
  of antiparticles, . . .

Note in 1921 Lunn proposed the wave equation and
solved it for H atom (rejected by journal!)