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
• Crystal X-Ray Diffraction

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
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
20
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
21
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
22
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
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?

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
24
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 v/f, 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/(n1f)
Vary index of refraction n in one arm
Phase difference ? 2p L (f/c) (n1 n2)
2I1
2I1
Total Path length L, l2 c/(n2f)
4I1
2I1
mirror
2I1
I1
I1
beam- splitter
I 4I1 cos2(?/2), with ? 2pL(f/c) (n1 n2)
25
Michelson Interferometer, cont.

• Its actually the wavelength that changes.
• However, its often easier to think/calculate in
  terms of the effective length of the arms L
  Ln.
• Then f (2p/l)(L1 L2 ), where l c/f is
  the vacuum wavelength.

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
Crystal diffraction How do we know the atomic
scale structure of matter around us?

• A crystal is a very large number of atoms or
  molecules arranged in a periodic fashion
• Acts like a grating with an extremely large
  number (Avagadros number) of units that
  diffract waves coherently
• Every crystal has its own signature of the
  spacings between atoms that act like gratings
• By measuring the diffraction, we can determine
  the atomic scale structure

NaCl structure

• Figure from http//www.cmmp.ucl.ac.uk/kpm/people/
  keith.htm

33
Crystal Diffraction (2)

• Typical distances between atoms are of order
  0.1-0.3 nm.What are characteristic wavelengths
  needed to study crystals?
• We need waves with wavelength l10-10m
• X-rays! e-m waves with much smaller wavelength
  than visible light -- i.e., for x-rays l10-10m

34
Historical Note X-ray Crystallography
The Braggs made so many discoveries that Lawrence
described the first few years as like looking
for gold and finding nuggets lying around
everywhere

• showed that the sodium and chloride ions were not
  bonded into molecules, but arranged in a lattice

• could distinguish different cubic lattices
• discovered the crystal structure of diamond
• Lawrence Bragg was the youngest Laureate ever
  (25) to receive a Nobel Prize (shared with his
  father in 1915)
• now standardly used for all kinds of materials
  analysis, even biological samples!
• The same multi-layer interference phenomenon is
  now used to make highly wavelength-specific
  mirrors for lasers
• (distributed Bragg feedback DBF)

35
X-ray scattering Modern Example

• X-rays remain the primary methods for
  establishing the atomic scale structures of
  complex molecules
• Example of rabbit liver carboxylesterase (one
  molecule showing atomic groups and attached
  large scale structures with atoms not shown)

Alternative strategies to improve theantitumor
efficacy have concentrated upon the design of
novel camptothecin analogs. To effect this, we
have determined the x-ray crystal structures of
the rabbit liver carboxylesterase Source
St. Jude Children Research Hospitalhttp//www.stj
ude.org/faculty/0,2512,407_2030_4278,00.html
36
FYI Diffraction from Crystals
The structure of the crystal can be found using
almost the same law we have for optical
gratings! Bragg Law for constructive
interference 2d sinq ml d lattice spacing,
l x-ray wavelength q x-ray angle (with
respect to plane of crystal)
Each crystal has many values of d - the distances
between different planes. For a known wavelength
l the observed angles q can be used to determine
the crystal structure
37
FYI Diffraction from Crystals
Why is there a factor of 2? (The grating
law is d sinq ml) Bragg Law for constructive
interference 2d sinq ml It is the same
idea applied to different cases!
In Bragg scattering from a crystal q x-ray
angle (with respect to plane of crystal) - two
terms d sin q in the path length difference
In the grating the light was at normal incidence
only one term d sin q in the path length
difference
The change in direction of the light is q
The change in direction of the x-rays is 2q
38
FYI Diffraction from Crystals

• How do we know the structures of DNA, proteins
  and other biological molecules? X-ray Bragg
  diffraction!
• The molecules are crystallized to create a
  crystal in which the molecules are arranged in a
  periodic lattice. By using the sharp Bragg
  diffraction from many molecules, the structure of
  each molecule is determined - the positions of
  thousands of atoms

39
FYI 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