Chapter 12 Lectures Physics
1230 Light & Color
Wave optics:
Interference and Diffraction
M. Goldman
I Interference
as a general wave phenomenon
a Slide
show of interference effects
b Examples
of interference
1
Laserdisk showing water wave interference in a ripple tank
o See
Fig. 12.7 in book
Notice
how a circular waves emanates from each
slit
Pattern
formed in space is due interference of the two circular waves
2
Physics 2000
/ [Atomic Lab] / [Interference Experiments]: examples of sinusoidal waves
interfering
o
Adding stationary waves of the same
wavelength but different amplitudes shows result of interference
In
this example have 2 overlapping plane
waves moving in the same direction but with a relative phase difference
In
phase wave
amplitudes add to give larger amplitude Đ brighter for light waves
Out
of phase wave
amplitudes add with one positive and the other negative so they cancel to give
zero or smaller amplitude darker for light
Easier
to visualize if draw one wave on top of
the other (crest on crest versus crest on trough)
Works
the same for waves of same wavelength moving with same velocity
Works
if the two waves are not plane waves for all of the spatial locations in which
they overlap (see water wave interference, next). Waves along different rays can be in different directions
and still overlap and interfere
o Water-wave interference
Here
the waves emanating from the slits are circular waves rather than plane
waves
They
only interfere where they overlap.
This forms an interference pattern
Interference
pattern is just the same for light waves entering two slits and interfering on
the other side of the double slit
If
the two (overlapping) circular waves are in phase at a point then the wave is higher (brighter
for light waves) at that point
If
they are out of phase at a point then
the wave is lower (darker for light waves) at that point
o Slits
and light cannon: interference of
moving light waves
Vary
slit separation and see what happens to the pattern on the screen a fixed
distance away.
We
will develop a formula for describing this behaviour quantitatively
3
Examples of sound wave interference
o Noise-cancelling
headphones can remove low-frequency sounds of plane while you listen to music
Outside
sounds are played back by the headphone,
with low frequency sounds out of phase with the direct sound from the outside which penetrates the earphones
This
cancels the low frequency sounds due to destructive
interference.
o Sound
enhancement on new TVs and CD players can simulate stereo or surround sound by changing the phases of
the sound waves at different wavelengths coming from two speakers.
c Ray optics is a fuller description of the way waves
interact than wave optics
1
These are two different ways in which classical physics treats
phenomena involving light waves
o Ray optics only pays attention to the rays of light (ray-tracing).
Ray optics is sometimes called geometric optics
Ray optics is enough to understand mirrors and lenses,
image formation, rainbows, mirages, etc
o
Wave optics pays attention to the
wave character of light along rays
It
is necessary to understand interference, diffraction, polarization and
scattering
It
is necessary to understand why oil slicks look colored, why the sky is blue,
what polarized sun glasses do, how LCD flat screen monitors work and what is a hologram
2
Wave optics takes into
account the way the tips of the electric fields of a monochromatic light wave
form a wave along a ray.
o
The
tips of the electic fields along a ray trace out the wiggly curve (sinusoidal
wave) we have been dealing with for light waves
o
Destructive interference if l/2 offset
of two waves of the same wavelength at the same place
The
2 waves are said to be out of phase at
that place
The
electric fields of the two waves cancel
at that place making the
The
net result is a dark spot at that place
Same
result if offset is 3l/2,
5l/2, 7l/2, .. -3l/2, -5l/2, etc
o
Constructive interference if 0 or l offset of
two such waves at the same place
The
2 waves are said to be in phase at that
place
The
electric fields of the two waves add at
that place
The
net result is a bright spot at that
place
Same
result if offset is 2l,
3l, 4l, ... -2l, -3l
o
Partial interference if phase difference
of 2 such waves at same place is a fraction of l not equal to l/2
o
Examples
(also see Mathematica file)
Two
waves almost in phase (crest on crest, trough on trough)
Two
waves out of phase (crest on trough, trough on crest)
o Examples
of waves which are at angles to each other and only interfere at one point,
labeled X
Two
waves almost in phase at X (crest on crest at X) at a given time will have
their electric fields add at X.
Two
waves almost out of phase at X (crest on trough at X) at a given time will have
their electric fields partly cancel.
d
Coherence
of light.
1
A coherent light wave
has a phase which remains the
same (is stable) in time and
space
2
Most light sources are incoherent- phase jumps around in time and space.
o Sketch
e
Two
ways to get interference effects from incoherent light sources.
1
Amplitude splitting
occurs as a result of transmission and reflection of light from thin films
2
Wave-front splitting
occurs in interference of coherent light incident on nearby slits
o Interference
occurs on the other side of the slits
II Interference due to light through slits
a Wavefront
splitting using slits
1 Phase is definite along
indicident wavefront before it is split by 2 or more slits.
o Source
may be incoherent, but phase jumps are common to light emerging from both slits
b Interference
of water waves OR light waves with monochromatic light (one wavelength) incident on a double slit.
1 General concepts and
observatrions (Figs. 12.7, 8, 9)
o Indicate
pattern of light and dark on a flat screen.
These
are defined as interference fringes (light or dark fringes)
Analogy
with water waves- position of high water wave is like position of bright light
wave, no water wave is like no light wave.
Central
fringe is always bright (12.7)
o
Demo
with laser on slits and/or Physics 2000 light cannon on slits
Change
slit spacing.
2
Fringe spacing formula deduced for monochromatic light of
wavelength l. (Figs. 12.11
& 12.12)
o Need
large distance from slits to screen for simplest formulas.
o Definitions
of symbols
Two
waves are in phase at slits where
they begin their jouney along rays to the screen. This is crucial to obtain the fringes as shown
At
large distance, D, from slit to screen, rays from s1 and s2 to screen are almost
parallel (idealization of Fig. 12.12, rather than as in 12.11).
q = angle of
observation measured from line along slit axis to screen. Since the rays are almost parallel, q is essentially the
angle the rays make to the y-axis.
Path
length of a ray
is distance from the slit it came from to any point along the ray.
e Ĺ path-length difference between two rays which
interfere. e varies, depending on the angle q.
The
wavelength of the monochromatic light incident on the slit is the same as the wavelengthof
the light emerging from the slits.
We call it l.
o
How
do the waves associated with rays at different angles, q, interfere to form fringes?
Central
fringe is always bright (e = 0 when q = 0) because path lengths are the same for
the two rays coming from each slits.
Crest is on crest (and trough is on trough)
Increase
q slightly until e = l/2. Crest on trough means dark fringe.
Increase
q more until e = l. Crest on crest means bright
fringe.
o
The
fringe spacing,
s, is defined as the distance along the screen from the centeral bright fringe
to the next bright fringe (it is roughly equal to the spacing between any two bright spots
The
fringe spacing, s, increases as the angle q increases
o What
is the effect on s of changing the distance, D, to the screen?
Keep
same angle but move screen away (increase D) increases fringe spacing, so s ľ D.
o Increase
slit separation, d. Fringes move
closer together (Fig
12.12), so s ľ 1/d.
o Increase
wavelength, l. This increases value of e (= l) at which 1st bright fringe occurs.
Larger angle q, means s ľ l.
o
Fringe
spacing, s = lD/d
Mathematical
derivation is in appendix.
If
know D & d (D >> d), you can measure s,
and deduce the very tiny size of l.
For
example, D = 1 m, and d - 1mm.
means, s will be one thousand times as large as l.
Try
it with the demo
c White
light incident on slits
1
White light is an additive mixture of all wavelengths.
2 Only waves of the same
wavelength can interfere
o Phase
relationship cannot be maintained unless wavelengths are the same.
3
Fig 12.13
o Central
fringe is white (pairs of waves at each wavelength interfere constructively).
o
As
move out from center shortest (blue) wavelength has its minimum closest to center.
o
This
leaves its complement, which is yellow.
o As
move out more, the blue maximum occurs (at about where the red minimum occurs).
o After
a few colored fringes, get white light because all colors are out of phase.
III
Interference due to transmission
or reflection of light off gratings
a
A grating is equivalent to a regularly-spaced large number of coherent
sources.
1
Transmission grating =
large number of slits.
2
Reflection grating =
large number of reflecting surfaces (grooves).
3
Such gratings are sometimes called diffraction gratings.
4
Spacing between slits or grooves is the grating constant, d.
b
A
grating produces bright fringes in the same places as a double slit (s = láD/d).
1
However, for a monochromatic light source, bright fringes are
narrower (sharper).
2
Regions in
between are darker (There is some variation in degree of darkness in between
bright fringes).
3
White light incident
on a grating gets broken up into its rainbow of colors by interference (not
dispersion). See Fig. 12.16.
c An
understanding of gratings is crucual for many applications of interference:
1
Holograms
o A
grating is, in fact, the simplest hologram
o The
pattern of bright and dark fringes formed by a grating is, in fact, the
simplest holographic image
2
Measuring wavelengths (spectroscopy)
o Will
have demo later using cheap plastic gratings
3
Iridescent materials.
4
Crystals are three dimensional gratings; crystallography uses X-rays to determine their structure
o The
first image of a DNA structure was
constructed by Rosalind Franklin using X-ray crystallography.
d
Interference
from more than 2 monochromatic coherent sources.
1
Fig. 12.14- constructive
interference from 4 in-phase sources
o Crests
are indicated by dots and labeled, 1, 2, 3, 4.
o For
angle shown, e = l (say).
All waves are in phase.
o Constructive
interference occurs at same values of e as
for two slits
o
Hence,
fringe spacing is again, s = lD/d, where d is the spacing between slits in the multiple-slit
configuration.
2
However, with multiple slits, there are many more regions of
(almost complete) destructive interference.
o Suppose
have 50 slits separated by, d = 0.1mm, and screen is at D = 1 M
o Then,
have minimum when 1 and 26 are out of phase, 2 and 27, etc.
o Effective
separation between these pairs is 25 slit spacings, or 2.5 mm.
o First
minimum is therefore not at half of s = lD/d
= 1/10-4 l = 10,000ál,
but at 1/2.5á10-3 =
400ál.
o
Next minimum occurs when 1 and 17 are out of phase,
together with 2 and 18, etc. This
one is at 600ál.
o
Net
result of all these minima between the first maxima is a sharper maximum
at the same place.
3
Illustrations of sharp interference maxima produced by
multiple slits.
o Demo
with laser and multiple slits going from 2 to 5. Note sharpening of maxima.
o Figure
12.15 shows 2 slits and 4 slits
o Viewgraph
of 6 slits.
o Characteristic
of interference pattern produced by monochromatic light incident on gratings.
e White
light incident on a grating.
1
Analyze using Fig. 12.16, which
shows spectrum at each order produced by transmission and by reflection
gratings.
2
Blue is closer than red to central fringe, just like for 2
slits
3
However now have black between central fringe & blue,
rather than yellow, because every color interferes destructively when there are
many slits.
4
Narrow maxima at each
wavelength produce a rainbow-like effect (compare with problem). NOT due to dispersion, as in a prism,
however.
5 Demo using plastic diffraction
gratings handed out to view white light source.
f
Iridescence.
1
Use mirror to flash light scattered off grating or CD into
into eyes- iridescence.
2
Show toy reflection gratings.
3
Insect gratings produce coloration (viewgraphs).
g Two
and three-dimensional gratings.
1
Toy transmission grating produces multiple spots.
2
Related to X-ray crystallography, in which crystal acts like a
3-D grating (viewgraph).
3
If know l
of x-rays, can find spacings of symmetric molecules in crystal.
h
Spectroscopy.
1
Fig. 12.23 shows locations of
maxima of monochromatic light of different color.
2
If know grating constant, can find wavelength (or vice-versa).
3
Grating demo using thin films handed out.
4 View Incandescent bulb, and
hydrogen through grating. Line
spectra will be are signatures of elements. Atomic physics.
Sunlight tells us what elements are present in the sun.
IV
Interference due to reflection from thin films
a Define
thin film: thickness comparable to wavelength.
1 A thin film can be a coating on a camera
lens, an oil slick in the street or a soap bubble
b Reflection
of MONOCHROMATIC LIGHT from a thin film.
1
Viewgraph of fig 12.5
2
Amplitude splitting.
o
Phase
of wave about to be reflected from front surface and phase of wave about to
enter film are same at front surface of film.
Phase
may jump around from incoherent source, but will be same for both waves at
front surface of film.
3 Hard vs soft reflections
o For
an incident wave pulse undergoing hard
reflection the crest becomes a trough when the pulse is reflected
See
Section 2.3, page 37 in book and Figures 2.13 and 2.14
o A sinusoidal (wiggling) wave incident from a medium with a low index of
refraction (e.g., air) reflecting from the
surface of a higher index of refraction medium (e.g., glass) always undergoes a hard reflection
For
such an incident wave undergoing hard reflection the reflected wave is shifted in phase by a half-wavelength relative to a the incident wave at the
reflection point (Fig 12.5)
o Illustrate
both pulse and wave hard reflection with hose tied to wall.
o A
wave incident from a medium with a high
index of refraction (e.g., glass)
reflecting from the surface of a medium (e.g., air) with a lower
index of refraction always undergoes a soft
reflection
In
a soft reflection, the reflected wave has the same phase as the incident wave at the point of
reflection. (Fig 12.5)
4 Is monochromatic light incident on a thin film brighter or darker when
reflected?
o Answer
depends on film thickness and type of reflection
Waves
of the same wavelength going in the same direction but differing in phase by a half-wavelength at a given pt. cancel out.
Two
such waves differing in phase by a
wavelength, l (or 2l, 3l,
etc.) add to make the total wave more intense (brighter)
Phase difference depends on hard vs
soft reflections, & the extra path length traveled by the wave reflected
from back surface,
o See
Fig. 12.5.
In
Fig. 12.5c, the two reflected waves to the left
of the film interfere constructively,
resulting in a brighter light than the incident wave
Coated
lenses such as in 12.5d do not reflect but
rather transmit all of the incident light since the two refelcted waves to the
left of the film interfere destructively
o Best
way to figure out whether reflected light is brighter or dimmer (Fig. 12.5b)
Consider
the images of the incident light source
at the two reflection points as effective sources of the reflected light
Take
into account the phase change produced by hard/soft reflection in determing the
phase at each of the those effective sources
The
images are each as far behind their
reflecting surfaces as the incident light source is in front (recall flat
mirrors)
They
are therefore separated by twice the thickness of the thin film.
Whatever
the phase difference is at s1 between the wave emanating s1
and the wave from s2 is (when at position s1) will
determine whether the reflected light is brighter or dimmer in front of the
thin film
c
Reflection of WHITE LIGHT from a
thin film can exhibit different colors.
1
Soap film- demo.
2 Don't get colors from nowhere
o Incident
light is white.
o White
"contains" all the wavlengths
Think
of the intensity distribution curve or
of
a prism which spreads out the wavelengths
3 Film thickness varies because
gravity is causing the film to settle.
4
Constructive or destructive interference depends on thickness
of film relative to each wavelength.
o At
certain special thicknesses get certain wavelengths reflected more than others
A
particular wavelength of light experiences constructive interference (in
phase rays from front and back surfaces of film meet on way out) OR
A
particular wavelength of light experiences destructive interference
(out-of-phase rays from front and back surfaces of film meet on way out)
The
phase difference is maintained all the way from the film back to your eye or
screen.
Out
of phase wavelength color is missing
(complement shows).
In
phase wavelength color is brighter
(desaturated by the rest of the wavelengths)
5 Examples
o Where
thickness is correct for waves in red range of wavelengths to be in phase get brighter red.
o Where
thickness is correct for waves in red range of wavelenths to be out of phase get less red, which means cyan appears due to
subtraction of red (C = W - R).
V Review of basic ideas of
interference so far
a We
have treated interference in which waves of the same wavelength and amplitude
but different phase can interfere to
produce light and dark spots.
1
In-phase means
constr