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
Ÿ Notice how a circular waves emanates from each slit
Ÿ
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
Ÿ In this example have 2 overlapping plane waves moving in the same direction but with a relative phase difference
Ÿ
In
phase wave
amplitudes add to give larger amplitude Ð brighter for light waves
Ÿ
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
Ÿ Easier to visualize if draw one wave on top of the other (crest on crest versus crest on trough)
Ÿ
Works
the same for waves of same wavelength moving with same velocity
Ÿ
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
Ÿ Here the waves emanating from the slits are circular waves rather than plane waves
Ÿ They only interfere where they overlap. This forms an interference pattern
Ÿ Interference pattern is just the same for light waves entering two slits and interfering on the other side of the double slit
Ÿ If the two (overlapping) circular waves are in phase at a point then the wave is higher (brighter for light waves) at that point
Ÿ 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
Ÿ
Vary
slit separation and see what happens to the pattern on the screen a fixed
distance away.
Ÿ 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
Ÿ 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
Ÿ 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).
Ÿ Ray optics is sometimes called geometric optics
Ÿ 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
Ÿ It is necessary to understand interference, diffraction, polarization and scattering
Ÿ 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
Ÿ The 2 waves are said to be out of phase at that place
Ÿ The electric fields of the two waves cancel at that place making the
Ÿ The net result is a dark spot at that place
Ÿ 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
Ÿ The
2 waves are said to be in phase at that
place
Ÿ The
electric fields of the two waves add at
that place
Ÿ The net result is a bright spot at that place
Ÿ 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)
Ÿ Two waves almost in phase (crest on crest, trough on trough)
Ÿ 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
Ÿ Two waves almost in phase at X (crest on crest at X) at a given time will have their electric fields add at X.
Ÿ 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.
Ÿ
These
are defined as interference fringes (light or dark fringes)
Ÿ Analogy with water waves- position of high water wave is like position of bright light wave, no water wave is like no light wave.
Ÿ Central fringe is always bright (12.7)
o
Demo
with laser on slits and/or Physics 2000 light cannon on slits
Ÿ 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
Ÿ
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
Ÿ 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).
Ÿ 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.
Ÿ
Path
length of a ray
is distance from the slit it came from to any point along the ray.
Ÿ
e Å path-length difference between two rays which
interfere. e varies, depending on the angle q.
Ÿ 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?
Ÿ 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)
Ÿ Increase q slightly until e = l/2. Crest on trough means dark fringe.
Ÿ 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
Ÿ 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?
Ÿ Keep same angle but move screen away (increase D) increases fringe spacing, so s µ D.
o Increase slit separation, d. Fringes move closer together (Fig 12.12), so s µ 1/d.
o Increase wavelength, l. This increases value of e (= l) at which 1st bright fringe occurs. Larger angle q, means s µ l.
o
Fringe
spacing, s = lD/d
Ÿ Mathematical derivation is in appendix.
Ÿ
If
know D & d (D >> d), you can measure s,
and deduce the very tiny size of l.
Ÿ For example, D = 1 m, and d - 1mm. means, s will be one thousand times as large as l.
Ÿ 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.
Ÿ 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
Ÿ 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
Ÿ 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
Ÿ 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
Ÿ Waves of the same wavelength going in the same direction but differing in phase by a half-wavelength at a given pt. cancel out.
Ÿ Two such waves differing in phase by a wavelength, l (or 2l, 3l, etc.) add to make the total wave more intense (brighter)
Ÿ
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.
Ÿ 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
Ÿ
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)
Ÿ Consider the images of the incident light source at the two reflection points as effective sources of the reflected light
Ÿ Take into account the phase change produced by hard/soft reflection in determing the phase at each of the those effective sources
Ÿ The images are each as far behind their reflecting surfaces as the incident light source is in front (recall flat mirrors)
Ÿ They are therefore separated by twice the thickness of the thin film.
Ÿ 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
Ÿ Think of the intensity distribution curve or
Ÿ 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
Ÿ A particular wavelength of light experiences constructive interference (in phase rays from front and back surfaces of film meet on way out) OR
Ÿ A particular wavelength of light experiences destructive interference (out-of-phase rays from front and back surfaces of film meet on way out)
Ÿ The phase difference is maintained all the way from the film back to your eye or screen.
Ÿ Out of phase wavelength color is missing (complement shows).
Ÿ 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 constructive interference or "bright" spots.
2 Out of phase means destructive interference (phase diifference of a half wavelength) or "dark place.
b A grating is a device which produces a regularly-spaced large number of effective coherent sources.
1 Light transmitted through or reflected from a grating has narrow bright fringes in the same locations where a double slit would produce broad bright fringes.
2 Fringe separation is proportional to wavelength (Blue closer to center than red).
3 Spectrum produced from white light which contains all wavelengths incident on a grating.
4 See different effects (e.g., color) at different angles- this is a primitive hologram.
VI Interference can also produce standing waves.
a Previous interference effects have been between waves going in SAME direction.
1 Now look at interference between
waves going in opposite directions.
Counterpropagating.
b Describe time-dependent wave motion
1 Fig. 12.31- interference of counter-propagating waves produced by in-phase sources s1 and s2.
2 concept of standing wave: equivalent to 2
counter-propagating waves of equal wavelength & amplitude.
3 Use one picture OR the other.
4 Location of nodes and anti-nodes.
5 Time-dependence.
c One of the two sources can be a virtual image:
1 Mirror with hard reflection at surface.
2 Incident and reflected waves interfere.
3 Node at mirror.
4 Can think of in terms of counterpropagating waves- one from real source, other from its virutal image.
5 Same as rope waves, in which incident and reflected waves interfere to form a standing wave.
6 Demo of rope waves.
d Lippmann plate
1 Put photographic film over mirror and time expose.
2 Film not exposed at nodes, so dark at nodes.
3 Film brightest at anti-nodes, so brightest there.
4 Get layers of dark and light, as in Fig. 12.33.
5 Use special film which has different index of refraction in dark places from bright places.
6 another example of a primitive hologram. Will get iridescent effect if view directly
7 See Fig. 12.28- interference from stacked films of alternating index of refraction
VII Huygens' principle
a
Any
wavefront of a traveling wave can be replaced, as far as effects further along
in the propagation direction are concerned, by a lot of sources located
uniformly all over the wavefront, radiating in phase.
1 Plane waves in water can be thought of as produced by paddle motion even where there is no paddle.
2 Only need up and down motion- can be produced by adjacent water molecules or by paddle. No difference.
3 Fig. 12.43- propagation of wave front can be understood in terms of wavelets emanating from effective point sources along wavefront.
4 Envelope of wavelets equivalent to
original wavefront propagation.
5 Equivalent pictures of wave propagation, but need to consider one or the other picture.
VIII
Diffraction effects are interference
effects within a single beam of light
a
Diffraction
from a single slit. (Fraunhofer
diffraction, if screen is very distant.)
1 Review predictions of ray optics- rays go in straight lines and cannot be bent except by lenses.
2 Will now find that rays get bent
by diffraction when passing through thin slits or holes.
3 Fig. 12.42 illustrates Huygens principle
applied to a plane wave incident on a slit.
4 Expect interference pattern when
replace slit by large number of coherent sources (assumed to be regularly
spaced).
5 Demo
b Derive equation for distance from central bright fringe to the first intensity zero.
1 Fig 12.46 shows mental subdivision of a slit of width b into 100 subslits
o Each subslit is an effective coherent source of wavelets from a coherent monochromatic light source.
2 Imagine sequence of more and more subslits (1000, 10000, etc).
3 First maximum occurs at sbright = lD/d, where d = b/100.
o As go to more and more slits , sbright goes from 100álD/b, to 1000álD/b, to 10,000álD/b, so the 1st maximum is always off the screen (in the limit)
4 First minimum occurs where slit 1 interferes destructively with slit 51, slit 2 with slit 52, etc.
o Effective slit width for first minimum is d = b/2, which is the distance between any of the destructively interfering pairs of slits.
5 The effective dark fringe spacing we know from 2-slit interference theory to be sdark = lD/d = 2lD/b.
o Note this result does not depend on the number of effective sources introduced in the slit.
o This is the distance from, say, the dark fringe just to the left of the central bright fringe to the one just to the right.
6 Hence, the Distance from the
central bright fringe to the first intensity zero = lD/b.
o Beyond the first minimum there are other minima, just as in the interference pattern of a grating.
c
How
does the diffraction pattern depend on the slit width?
1 Fig. 12.47.
2 The distance from the central bright fringe to the first intensity zero decreases as the slit width b is increased.
o Hence, for wide slits, we get the expected image of a point source predicted by ray optics. (Think of the pinhole camera).
o
Ray
optics says that rays travel in straight lines unless a lens or mirror, etc. causes them to bend or
bounce.
o Hence, the image formed on a distant screen by parallel rays incident on a slit according to ray optics is just the straight-line continuation of the parallel rays through the slit- all else is shadow (dark).
3 Diffraction is a wave-optics
effect, in which the image on the screen is spread out into a diffraction
pattern whose first intensity zero is beyond the image size predicted by
ray optics.
o
the
diffraction pattern may only be evident for relatively narrow slits.
4 The narrower the slit, the wider the diffraction pattern.
5 Demo with laser and various slit
sizes.
d Diffraction from circular apertures.
1 Light beam blocked in two directions (transmitted through a 2-D aperture) produces interference pattern in 2 directions
2 Fig. 12.48 from book
3 Small diameter holes produce
more diffraction than wide diameter holes, just as for slits.
4 Small diameter holes let through less light than wide diameter holes.
5 Intensity pattern on screen depends on distance from the center.
6 Demo with laser and a hole
e Coronas and glories.
1 Consider effect of many random holes in a screen.
o Each one produces a diffraction pattern.
2 If the holes are randomly placed, (rather than regularly placed), there is no additional interference pattern formed by light from one hole interfering with light from another hole.
o
Pattern
is simply the single-hole diffraction pattern made brighter by having many
holes. (Narrow light-beam Þ all holes contribute to central bright spot).
o Same effect produced by many randomly placed opaque disks or particles (read about babinet's principle- bright ray-optics image is only difference between holes & disks)
3 Demo using laser through lycopodium powder slide, to show same diffraction pattern as for single hole.
4 Condensed moisture on windshield
shows street lights with diffraction pattern as result of tiny water droplets
randomly placed.
o
Demo?
using glass plate
o
this
is a corona.
o size of rings depends on size of drops (µ 1/b) .
5 Another situation- tiny water
droplets or ice crystals suspended in air.
6 Slides or pictures?
7 May have drop size such that
refraction, reflection and diffraction combine to produce colored rings- a glory.
o Demo using point white light source and lycopodium powder slide to pass around.
f Resolving power.
1 Important practical application of diffraction.
2 Every lens (telescope, microscope, camera, etc.) has a finite aperture.
3 Diffraction (from small apertures) can make it difficult to see very fine detail in an image produced by a lens or pinhole.
4 Resolving power is defined as follows:
o 2 point sources are distinguishable (can be resolved) if the central fringes of their respective diffraction patterns can be seen as separate, rather than merged.
o Fig. 12.49 shows resolved, just resolved and unresolved sources.
o best to give criterion for resolution in terms of the angle, q, between rays from point sources to lens. Angular separation of sources.
o Read about Rayleigh criterion.
5 Large aperture b means large lens, so better angular resolution.
6 Fig. 12.50 sequence of pictures taken through smaller aperture.
o Start with fuzziest and guess how many objects there are. Then look at larger lens pictures. Diffraction-limited resolution.