1. EXAM II
Tonight April :00 – 10:00 pm
Loeb Playhouse, Stewart Center
Covers up through and including Chapter 24 (all), with emphasis on “new” material since Exam 1
You are encouraged to create your own crib sheets – two sheets, (that’s four sides). Safer not to rely on us to give absolutely all necessary equations on the sheet supplied with the exam.

2. Exam II will have one optical focusing question, relating object and image distances and the focal length of a lens or curved mirror. Quick review:
1/so + 1/si = 1/f where positive f means “converging”
Examples: if object distance is 2f, then so is image distance, since ½ + ½ = 1
If object distance is f, then image distance is 1/0 = ∞ (remember the definition of the focal point).
Similarly if image distance is f, then object was “at infinity”, or approximately very distant compared to f.

3. If f is positive, and the object is closer to mirror or lens than a distance f, then the image distance is Negative, meaning a virtual image: behind the mirror, (or on the incoming side of the lens, not the side that the light goes to.)
Example: let so = 15 cm, f = 10 cm, what is si?
1/15 + 1/si = 1/10
1/si = 1/10 – 1/15 = (15-10)/150 = 5/150 = 1/30
si = 30 cm
Note: so < 2f hence si >2f, useful reality check.

4. We covered a lot of ground in the last lecture
We covered a lot of ground in the last lecture. Now let’s see some demonstrations of interference and diffraction of laser light, the highly coherent, monochromatic, kind of radiation that works so well for such phenomena.
Soap film interference in air (using white light)
Single slit, various widths
Double slit, various separations
Diffraction gratings
Various atomic optical emission spectra (electrical discharge lamps using gaseous elements such as H, He, Hg, Na, etc.)

5. Thin-Film Interference, Case 2
Assume the soap bubble film is surrounded by air
There is a phase change at the front of the film but no phase change at the back
Since only one wave undergoes a phase change, it effectively shifts by half a wavelength. Then the interference conditions are
***
Section 25.3

6. Soap film liquid tends to “drain” towards the bottom of the film.
Gradient => horizontal colored bands
Top of film eventually becomes thin enough to approximate m = 0 in the air-film-air case.
That’s where phase flip, non-flip at the two film surfaces give destructive interference even with almost negligible path length (in terms of the wavelength of ANY of the white light.)

7. Double Slit Analysis Assume W is very large
slits are separated by a distance d
ΔL = d sin θ
A set of equally spaced bright and dark bands
constructive interference, d sin θ = m λ
m = 0, ±1, ±2, … m is any integer
destructive interference, d sin θ = (m + ½) λ
Section 25.5

8. Spacing Between TWO Slits
Notation:
d is the distance between the slits
W is the distance between the slits and the screen
h is the distance between the adjacent bright fringes
For m = 1,
Since the angle is very small, sin θ ~ θ and θ ~ λ/d
Using the approximations, h = W θ = W λ / d
Decrease d, increases h
Section 25.5

9. Single-Slit Fringe Locations
Here’s the geometry of the first dark fringe.
Destructive interference occurs at the angle θ where ΔL is one-half of a wavelength.
ΔL = w/2 sin(θ) = m/2 λ
Smaller width, larger θ
“Angle-position uncertainty principle”
Section 25.6

10. Diffraction Grating, final
The condition for bright fringes from a diffraction grating is identical to the condition for constructive interference from a double slit
But the overall intensity pattern depends on the number of slits
The larger the number of slits, the narrower the peaks --- more precise
Section 25.7

11. First look at what a diffraction grating does to white light
Next, let’s show the (few) colors coming from certain atoms, like Hydrogen gas, Helium gas, Mercury vapor, and Sodium vapor

12. Crystal Diffraction of X-rays
Diffraction effects occur with other types of waves
The atoms of a crystal are arranged in a periodic array, forming planes which can reflect X-rays
Some X-ray wavelengths are comparable to atomic spacings --- leads to:
Interference of reflected rays in certain directions
Section 25.7

13. X-Ray Diffraction, cont.
The effective slit spacing is the distance between atomic planes
Typically 3 x m
Compared to 10-4 m or 10-5 m for a grating
Some X-rays have the appropriate wavelength
The planes give bright dots instead of fringes
By measuring the angles that give constructive interference, the distance between the planes can be measured
Section 25.7

14. Optical Resolution
For a circular opening of diameter D, the angle between the central bright maximum and the first minimum is
The circular geometry leads to the additional numerical factor of 1.22
(Bessel functions, fancy 2-D versions of trig functions)
Section 25.8

15. Telescope Example Assume you are looking at a star through a telescope
Diffraction at the opening produces a circular diffraction spot
Assume there are actually two stars
The two waves are incoherent and do not interfere
Each source produces its own diffraction pattern
Section 25.8

16. Rayleigh Criterion
If the two sources are sufficiently far apart, they can be seen as two separate diffraction spots (A)
If the sources are too close together, their diffraction spots will overlap so much that they appear as a single spot (C)
Section 25.8

17. Rayleigh Criterion, cont.
Two sources can be resolved as two distinct sources of light if their angular separation is greater than the angular spread of a single diffraction spot
This result is called the Rayleigh criterion for the angular resolution:
Two objects will be resolved when viewed through an opening of diameter D if the light rays from the two objects are separated by an angle at least as large as θmin
This puts the bright spot of one object onto the first dark ring of the other object. Just happens to be approximately where discrimination is possible
Section 25.8

18. Limits on Focusing
A perfect lens will focus a narrow parallel beam of light to a precise point at the focal point of the lens
The ray optics approximation ignores diffraction
The real focus is spread over a disc of angular radius θ = 1.22 λ/D and disc radius r = f λ /D
Section 25.8

19. Limits on Focusing, final
The wave nature of light limits the focusing qualities of even a perfect lens
It is not possible to focus a beam of light to a spot smaller than approximately the wavelength
The ray approximation of geometrical optics can be applied accurately enough at size scale much greater than the wavelength
When a slit or a focused beam of light is made so small that its dimensions are comparable to the wavelength, diffraction effects become important
Section 25.8

20. Scattering
When the wavelength is larger than the reflecting object, the reflected waves travel away in all direction and are called scattered waves
The amplitude of the scattered wave depends on the size of the scattering object compared to the wavelength
Blue light is scattered more than red
Called Rayleigh scattering
Section 25.9

21. Blue Sky
The light we see from the sky is sunlight scattered by the molecules in the atmosphere
The molecules are much smaller than the wavelength of visible light
They scatter blue light more strongly than red
This gives the atmosphere its blue color
Section 25.9

22. Scattering, Sky, and Sun Blue sky is scattered sunlight
Sun near horizon
Looking towards or near the sun, you see the unscattered light, which is depleted in blue, that is:
Most of the blue is scattered away, leaving the red
There is also a long path length through the atmosphere in this direction, so there’s plenty of scattering of the light on its way to your eye, or to sunset clouds that receive the reddened sunlight.
Section 25.9

23. Color Vision
Color vision is due to light detectors in the eye called cones
The three types of cones are sensitive to light from different regions of the visible spectrum
Particles of light, photons, carry energy that depends on the frequency of the light

24. There is evidence that our mammalian ancestors, small nocturnal critters, had lost one of the three color receptors. These mammals managed to survive the K-T extinction of the dinosaurs (and most other life on earth) 65 Million years ago after a 10 km asteroid hit Chixulub at the Northern tip of Yucatan, leaving a 110 mile wide crater, a worldwide rain of fire, and a “nuclear winter” of dust and air pollution
Luckily, some mammals have regained a third color of cone receptor. Including us.
Chickens (and some chicks, a.k.a. women) have four types of color receptors. And generally better color sense than those of us with only 3 color vision, it would seem.