Download Chapter 12. Wave Optics Optics can be divided into two categories: I

Chapter 12. Wave Optics
Optics can be divided into two categories:
I. Geometrical optics: describes processes that can be understood by ray tracing. This
applies when the size of objects and lenses is much greater than the wavelength of light.
Applies to: Optical instruments using mirrors and lenses.
II. Wave optics: takes into consideration the wavelength and phase of light waves. This
is necessary when the size of objects is comparable to the wavelength.
Applies to: Thin films (oil slicks), holograms.
How do waves add?
Recall that waves have crests and troughs.
Waves add constructively when the crests are on top of crests and troughs are on top of
troughs. These waves are in phase.
Waves add destructively (they cancel) when the crests of one wave overlap the troughs of
another wave. These waves are out of phase.
How is phase measured?
Points on a wave can be labeled in degrees like the points around a circle. On full cycle
of up and down is 360 degrees.
12WavePhase.gif
As on a circle, 360 degrees is at the same place as 0 degrees, and 370 degrees is at the
same place as 10 degrees. Then –90 degrees is the same as …?
The two waves below are in phase and their sum is a wave with twice the amplitude:
12AddInPhase.gif
The waves below are out of phase by 180 degrees because the 0 degree places on one
wave lie above the 180 degree places on the other wave.
12AddOutOfPhase.gif
More complicated cases:
If the waves are of different amplitude, say 3 and 1, and they are out of phase, the sum
wave has amplitude 3 – 1 = 2. If they are in phase, the sum is 4.
If the phase difference is 90 degrees, the sums and differences are harder to find. You can
draw the waves and add the amplitudes at each point measured with a ruler or you can
use trigonometric identities to find sin (A) + sin (A+90° ) = ?
Coherent and incoherent sources:
A coherent source emits a single continuous frequency (and single wavelength, since the
two are related by an equation.)
An incoherent source emits many frequencies (or one frequency that is broken into
waves with unrelated phases)
A good example is sound waves from loudspeakers:
12Coherent.gif
An incoherent wave can all be at the same frequency, but with a phase that varies. For
example, hitting the piano key for middle C over and over again. This wave is not
coherent because it is broken up into sections that don’t "line up."
12BongBongBong.gif
Note that the phase "jumps" from one wave to the next. If the phases line up, you cannot
tell where one wave ends and the other begins.
Adding coherent sources:
In the case where sources are coherent, you can easily predict what the sums will be.
Incoherent sources do not add in a way that is easy find.
The big idea (again):
Waves combine to give greater amplitude where they add in phase (crests overlap crests
and troughs overlap troughs). Waves cancel where they add out of phase.
When there are two sources, the phase of the waves locally depends on the distances
from the sources.
Simplest case: Sources in phase, distance to sources may vary.
Sources in phase, distances are the same:
12TwoSpeakers.gif
There is a symmetric pattern of loudness and softness called a diffraction pattern.
Below the sources are in phase, but the listener is off center so that the distances from the
sources are not the same:
12DiffPattern.gif
The level paths are seven wavelengths long and the paths at an angle are 7 ½ wavelengths
long so the waves arrive out of phase.
General rules:
If the sources are in phase, the waves amplitudes add in phase if the distances differ by
nλ, where λ is the wavelength and n is a whole number (-1, 0, 1, etc.).
If the sources are in phase, the wavelengths add out of phase if the wavelengths differ by
(n+1/2)λ.
Diffraction patterns:
Consider a speaker that emits in all directions. Here the black areas are wave crests and
the white areas are troughs:
12Bullseye.gif
The pattern that two in phase speakers makes is shown below:
The speakers are displaced 2 ½ wavelengths vertically, so there is cancellation along a
line in the vertical direction and there is addition in the horizontal direction. In the spaces
between the loud areas the sound is softer.
Suppose the speakers are 180 degrees out of phase:
In this case, the line that is equidistant from the speakers (the "axis") is a place where the
sound cancels. On either side of this there is a maximum, and a little further on each side
is a minimum. In the drawing above, all the LOUD locations would become silent
locations and all the silent locations become LOUD.
Huygens’ principle applied to slits:
When a wavefront encounters a slit, what happens on the other side can be found by
putting a row of sources across the slit with the same phase as the incident wave.
12TwoSlits.gif
The screen with the diffraction pattern is brightest where the difference between the
distances is an even
number of wavelengths:
The pattern is darkest where the difference in distances is an odd number of
wavelengths.
12wavelength.gif
Note that the lemon and lime triangles (must be lunchtime) are similar.
That means that the ratio of the shortest side to the longest side is the same for both.
So, the formula relating D, X, and λ is λ / X = S / D.
[I have cheated here and used the next-to-longest side (D instead of D + λ) because the
two longs sides are very nearly the same in length.]
Now I can find the wavelength of light using the spacing of the diffraction pattern S, the
distance from the slit to the screen (approximately) D, and the slit separation X.
Wavelength of light , λ = S X /D.
The answer is in the same units that S, X, and D are measured in.
Gratings:
Gratings are many parallel lines which can be viewed as many slits side by side.
These give a brighter pattern than two slits because more light can come through many
slits. The spacing is the same as for two slits.
Gratings with horizontal and vertical lines:
The pattern for this is rows and columns of dots.
12Patterns.gif
The spacing between the dots is found from the same formula as the spacing between the
lines.
Finding the wavelength of light with pantyhose and a laser:
Finely woven cloth, such as ladies stockings, is a grating. Shine a laser beam through
cloth and you get a dot pattern like the one above. If you know the spacing between
threads (X in our formula), the distance between dots (S in our formula) and the distance
from the cloth to the wall (D in our formula), you can calculate the wavelength of light.
Demo: Wavelength of light. A cloth is used with 10 threads per millimeter. The distance
to the board is 1600 mm and the distance between the dots in the pattern is 10 mm. So the
formula gives us 1/1600 mm for the wavelength of the red light from the laser which is
about 625 nm. The laser we used has a known wavelength of 633 nm so our experiment
was pretty close.