## Interference and diffraction

Reading: Chapters 35, 36, INTERFERENCE and DIFFRACTION

Study guide: Chapter 35

1. Waves and Particles

2. Superposition of waves

3. Interference patterns

### Interference patterns

We can pictorially calculate how this pattern should come out by marking up a diagram of the undisturbed waves from the two sources with
• "X" at places where crests meet crests (or troughs meet troughs), and
• "O" at places where a crest from one meets a trough from the other.
An observer standing at the right side of the pool would see large waves rolling in at points marked with an X, and no waves motion at points marked O.

### Young's double-slit

Have the waves from just one source hit a screen with two openings in it. Each opening acts like a "source" in its own right. But now the two "sources" are always in synch with each other, and are guaranteed to have the same frequency (that of the original plane wave coming at the wall).

If $d$ is the distance between $A$ and $B$, and $\Delta theta$ is the angular separation between the lines of constructive interference, it appears that
$$\Delta \theta \propto \frac{1}{d}$$.

Switching from wavefronts to rays...

• Assume each slit acts like a point source of waves of wavelength $\lambda$,
• Both slit point sources are in-phase,
• So, at any location equidistant from both point sources, the waves are always exactly in-phase, and interfere constructively

Consider different directions ($\theta$) from the two slits towards a distant screen...

Waves from the two slits in the straight-ahead $\theta=0$ direction are always in-phase.

In another direction $\theta$, a wavefront from the top source travels a distance $d\sin\theta$ longer than the other. If this difference is an integer, $m$, number of wavelengths, then the two 'rays' interfere constructively: $$m\lambda=d\sin\theta$$ $\Rightarrow$ maxima whenever $$\sin\theta=m\frac{\lambda}{d}$$

So if $\lambda\ll d$, then the angular separation between maxima is approximately:$$\Delta \theta\approx\frac{\lambda}{d}.$$

Pretty much like....

[Petey, Jacob, Tek]

Probably of finding one photon, $P_1$, in the apparatus at any one time *should have been* (time of flight of one photon)/(average time between counts):

Time of flight=0.5m/c=0.5m/3e8 m/s=1.7e-9 s

average time between counts $\approx$ 1 sec/ 400 counts = 0.0025 s

So $P_1\aprox6.8e-7 = 7\times 10^{-5}$%.

Single-slit:

Is this shape also a diffraction effect?

At what angle will we get destructive interference?

#### Circular aperture

Photo image of two sources...
a) $\alpha \gt 1.22\lambda/D$,
b) $\alpha \approx 1.22\lambda/D$.

Diffraction grating

Used for separating colors (spectrometer)

### Bragg x-ray diffraction

Animation at Stony Brook

Path difference for reflection from adjacent rows of atoms separated by distance $d$ in terms of $\lambda$ of x-ray source is $2d\sin\theta$, so constructive interference happens when this is an integer number of wavelengths:$$m\lambda=2d\sin\theta$$.

#### Lab preparation:

1. Copy the IMPORT-able parameters below (highlight all the lines and "Copy" to the clipboard).
2. Open up Paul Falstad's ripple tank simulator.
3. Choose "Import/Export",
4. In the window that opens up, Paste the parameters over top of the ones that are there, and click Import.
5. You should see a plane wave advancing towards a rough "wall" with two openings in the wall. After the pattern has run for a while, note the spacing between the lines of destructive interference.
6. Now, choose "Clear Walls". Set the Mouse option to "Edit Walls".
7. Draw with your mouse a new wall that has two openings that are closer together than in the previous situation. Are the lines of destructive interference closer together or further apart than they were before?? Show your lab assistant your results...

There is a formula for the angular distance $\Delta \theta$ between destructive interference lines, (or between constructive interference lines, for that matter) in terms of the distance $d$ between the two slits (holes in the wall) which is:

$$\lambda=2d\sin\Delta\theta$$

The wavelength of the waves, $\lambda$ is a constant for what you've been looking at. Is this formula compatible with what you observed?

Initial parameters:

\$ 0 189 10 2 true false 8 25 477 1
s 20 21
s 208 21
c 13833 0
w 16 0
l 29
w 61 0
l 111
w 13 0
c 15 0
w 20 0
l 10
c 59 0
l 1
w 30 0
l 62
w 5 0
l 10
w 7 0
c 43 0
l 2
w 4 0
l 5
c 139 0
w 14 0
c 5 0
w 6 0
l 4
c 152 0
w 48 0
c 37737 0