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.