Chapter 28 Physical Optics: Interference and Diffraction

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Chapter 28Physical Optics Interference and
Diffraction

• Certain physical phenomena are observed which are
  most easily explained by invoking the wave nature
  of light, rather than the particle nature. These
  topics are
• interference
• diffraction
• polarization

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Jefferson Lab Open House

• Saturday April 16, 2005
• 10-2pm
• Info at www.jlab.org
• Extra Credit 5points (out of 200) on HW score.
• Sign in at ODU-Physics Booth in CEBAF Center
  Building

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Constructive and Destructive Interference
Two waves (top and middle) arrive at the same
point in space. The total wave amplitude is the
sum of the two waves. The waves can add
constructively or destructively
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Coherence

• If the phase of a light wave is well defined at
  all times (oscillates in a simple pattern with
  time and varies in a smooth wave in space at any
  instant), then the light is said to be coherent.
• If, on the other hand, the phase of a light wave
  varies randomly from point to point, or from
  moment to moment (on scales coarser than the
  wavelength or period of the light) then the
  light is said to be incoherent.
• For example, a laser produces highly coherent
  light. In a laser, all of the atoms radiate in
  phase.
• An incandescent or fluorescent light bulb
  produces incoherent light. All of the atoms in
  the phosphor of the bulb radiate with random
  phase. Each atom oscillates for about 1ns, and
  produces a coherent wave about 1 million
  wavelengths long. But after several ns, the next
  atom radiates with random phase.

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Interference

• Recall last semester we discussed interference of
  sound waves. Light waves also display
  constructive and destructive interference.
• For incoherent light, the interference is hard to
  observe because it is washed out by the very
  rapid phase jumps of the light.
• Soap films are one example where we can see
  interference effects even with incoherent light.

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Youngs Double Slit Expt

• Interference of light waves was first
  demonstrated by Thomas Young in 1801.
• When two small apertures are illuminated with
  coherent light, an interference pattern of light
  and dark regions is observed on a distant screen

Light
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Path Difference

• We can understand the interference pattern as
  resulting because light from the two apertures
  will, in general, travel a different distance
  before reaching a point on the screen. The
  difference in distance is known as the path
  difference.

P
Light
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Two Slit Diffraction
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Two Slit Interference
An incoherent light source illuminates the first
slit. This creates a nearly-uniform but coherent
illumination of the second screen (from
side-to-side on the screen, the light wave has
the same oscillating phase). The two waves from
the two slits S1 and S2 create a pattern of
alternating light and dark fringes on the third
screen.
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Interference of waves from double slit

• Each slit in the previous slide acts as a source
  of an outgoing wave.
• Notice that the two waves are coherent
• The amplitude of the light wave reaching the
  screen is the coherent sum of the wave coming
  from the two slits.

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Why did Young (1800s) use single slit before the
double slit?

• The first slit forces the wave to be coherent all
  the time
• From moment to moment (after many oscillations of
  wave) the wave is still incoherent, but at each
  moment in time, the wave has the same phase at
  the two slits.
• He was too cheap to buy a 19th century laser.

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• If the two slits are separated by a difference d
  and the screen is far away then the path
  difference at point P is Dl ? dsinq
• If we put a lens of Focal Length fL, then the
  expression Dl dsinq is exact.
• If Dl l, 2l, 3l, etc, then the waves will
  arrive in phase and there will be a bright spot
  on the screen.

L
P
q
Light
dsinq
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Fringes

• Consider apertures made of tall, narrow slits.
  If at point P the path difference yields a phase
  difference of 180 degrees between the two beams a
  dark fringe will appear. If the two waves are in
  phase, a bright fringe will appear.

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Interference Conditions

• For constructive interference, the path
  difference must be zero or an integral multiple
  of the wavelength
• For destructive interference, the path difference
  must be an odd multiple of half wavelengths
• m is called the order number

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Example
If the distance between two slits is 0.050 mm and
the distance to a screen is 2.50 m, find the
spacing between the first- and second-order
bright fringes for yellow light of 600 nm
wavelength.
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Two Slit Diffraction

• When green light (l 505 nm) passes through a
  pair of double slits, the interference pattern
  shown in (a) is observed. When light of a
  different color passes through the same pair of
  slits, the pattern shown in (b) is observed.
• Wavelength of the second color is greater than
  505 nm
• Wavelength of the second color is smaller than
  505 nm

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Walker Problem 18, pg. 942
When green light (l 505 nm) passes through a
pair of double slits, the interference pattern
shown in (a) is observed. When light of a
different color passes through the same pair of
slits, the pattern shown in (b) is observed. (a)
Is the wavelength of the second color greater
than or less than 505 nm? Explain. (b) Find the
wavelength of the second color. (Assume that the
angles involved are small enough to set sinq
tanqq.)
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Walker Problem 18, pg. 942
green light (l 505 nm) 4.5 orders of green
light 5 orders of mystery light 4.5 (505 nm)
(5) l l lt 505 nm, l (4.5/5)(505 nm) 454 nm
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Lloyds Mirror

• An interference pattern is also observed with the
  Lloyds Mirror setup
• The pattern, however, is found to be reversed
  from the Youngs setup because the light
  undergoes a 180 degree phase shift upon reflection

P
q
q
Mirror
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Phase of wave reflected by interface between two
media
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Interference in Thin Films

• We have all seen the colorful patterns which
  appear in soap bubbles. The patterns result from
  an interference of light reflected from both
  surfaces of the film

180o phase change
0o phase change
t
ngt1
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Thickness of film selects wavelength

• Each outgoing ray
• has two contributions
• 1) reflection from the top surface
  phase p
• 2) reflection from the bottom surface (almost
  normal incidence)
• phase 0 2p (distance / wavelength)
• phase 2p (2 t) / (l/n) (2nt) / l
• Phase difference p 2p 2nt / l
• 2nt/l m, ( m0, 1, 2,) Destructive
  invisible
• 2nt/l m1/2, ( m0, 1, 2,) Constructive
  bright color

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Soap Film Interference
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Dark Water

• Just before the soap film pops, it goes dark.
• If thickness t ltlt l, then
• 2nt/l ltlt 2
• Destructive interference for all wavelengths
• No reflected light

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Interference from non-parallel surfaces
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Diffraction

• The bending of light around objects into what
  would otherwise be a shadowed region is known
  as diffraction. Diffraction occurs when light
  passes through very small apertures or near sharp
  edges.

geometrical
diffracted
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Single Slit Diffraction

• We have seen how we can get an interference
  pattern when there are two slits. We will also
  get an interference pattern with a single slit
  provided its size is approximately l (neither
  too small nor too large)

Light
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• To understand single slit diffraction, we must
  consider each point along the slit (of width a)
  to be a point source of light. There will be a
  path difference between light leaving the top of
  the slit and the light leaving the middle. This
  path difference will yield an interference
  pattern.
• Path difference of rays to P from top and bottom
  edge of slit
• DL a sinq ? destructive if DL ml, m1,2,

P
Light
q
(a/2) sinq
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Single Slit Diffraction
Notice that central maximum is twice as wide as
secondary maxima Sinq m l / W, Destructive Dark
Fringes on screen y L tanq ? L (ml/W) Maxima
occur for y 0 and, y ? L (m?1/2)(l/W)
m1
m-1
L
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Diffraction from a pinhole
See photo 28-21 28-22.
Dark fringes occur at zeros of Bessel function,
(2-D geometry). First dark fringe Sinq 1.22
(l/D) D diameter of pinhole
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Single-Slit Diffraction from a large aperture
(telescope, microscope, camera).

• A lens images parallel rays to a point at the
  focal distance f.
• All parallel rays experience the same phase
  change from a incident plane wave to the focus.
• A image formed by a lens of diameter a is fuzzed
  out by the single slit diffraction pattern, whose
  central maximum is of width dq ?1.22 l/a

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Rayleigh CriterionDiffraction limited Resolution

• Two objects can be resolved (barely) if the
  diffraction maximum of one object lies in the
  diffraction minimum of the second object.
• qmin 1.22 wavelength/diameter of lens or
  mirror

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Diffraction Limit of spy telescope

• A telescope with a 1 m aperture orbits the earth
  at an altitude of 400km. What is the diffraction
  limit for the smallest distance it can resolve on
  the surface of the earth, using blue light (l400
  nm)?
• Resolution ? dq 1.22 l/D
• dy L dq 1.22 (400.e3 m) (400.e-9 m) / (1m)
  0.2 m
• The only ways to improve this are
• Bigger diameter telescope mirror (very expensive,
  or aperture synthesis)
• Shorter wavelength (but atmosphere is semi-opaque
  to UV)
• Get closer to surface (airplanes, rather than
  satellite)
• Technology can exploit laws of physics, not evade
  them.

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Angular Resolution of Hubble Telescope

• Green light l 550 nm Hubble aperture D2.4
  m
• qmin 1.22 (550 10-9 m) / (2.4 m) 2.810-7
  radians
• At 200 106 km, y 56 km

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Angular Resolution of Human eye

• Pupil diameter D 4mm (typical)
• Green light l 500 nm
• dq 1.22 l/D 1.22 (0.50e-6 m) / (4.00e-3 m)
• dq 0.14 milli-rad
• minimum usefull spacing of rods cones
• dy f dq
• f diameter of eyeball
• dy (2cm)(0.14e-3) 3mm

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Aperture Synthesis

• There is a trick to make the effective aperture
  of a telescope as large as possible.
• If the phase of the waves arriving at distant
  telescope can be recorded and/or interfered, then
  the Synthesized Aperture is equal to the
  separation of the two telescopes
• Aperture Synthesis with Radio telescopes
• l 21 cm line in atomic H (nuclear spin flip).
• D 5000 km (telescopes on opposite ends of N.
  America)
• qmin 1.22 (0.21 m) / (5.e6 m) 5 10-8
  radians
• Aperture synthesis with optical telescopes
• Much harder, D 10m is best to date.

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Diffraction from a Grating

• Each slit is a source of a wave
• Observe the outgoing wave at an angle q, the
  contributions from all slits add up coherently if
  
• d sinq m l, (m0, 1, 2,)
• If the incident wave uniformly and coherently
  excites N slits, then the contribution from all
  of the slits will exactly cancel if
• d sinq (m1/N) l, (m0, 1, 2,)
• By virtue of using many slits, the diffraction
  grating reduces the width of each maximum by a
  factor 1/N.

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Sharpening of Diffraction Pattern

• Diffraction pattern with N5
• Width of each principal maximum is dq ?l/(Nd)
• d spacing of grating
• N number of slits illuminated by source.

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Resolution of Diffraction Grating

• A grating can be used to measure the wavelength
  of a spectral line from an atomic or molecular
  transition.
• A grating has 5000 rulings/cm,
• Our light source makes a spot 5mm across on the
  grating.
• We observe the diffraction pattern in 3rd order.
• With what precision can we measure the wavelength
  of incident light?
• d sinq (m1/N) l,
• N (5000/cm)(0.5cm) 2500
• Consider two wavelengths l1 and l2 such that
• Sinq (m) l1/d (m1/N)l2/d
• l1 l2 measurement precision

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Resolution of Diffraction Grating (2)

• l1 sinq (d/m) l2 sinq d/(m1/N)
• (l1 l2) /l1 relative precision

Precision improves with larger values of either N
or m, But diffraction maxima get weaker and
weaker as m increases
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Diffraction Grating Resolution

• N 2500, m 3
• Relative precision 1/(25003) 1.3e-4
• Red light l 800 nm
• Absolute precision (800 nm) 1.3e-4 0.1nm
• (One atomic diameter!!!!!)

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4/11/05 Attendance

• Are you here?
• Yes
• No

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Single Multiple Slit Diffraction

• Single slit of width a
• Diffraction Minima a sinqm ml,
  m?0, m?1, ?2,
• Diffraction Grating, spacing d, N slits
  illuminated
• Diffraction Maxima d sinqn nl, n0,
  ?1, ?2,
• Diffraction Minima d sinqn (n1/N)l, n0,
  ?1, ?2,