INTERFERENCE & DIFFRACTION II

1.

A certain fluid has a density of 0.8 g/cm³, an index of refraction of 1.2 and is immiscible with water. Suppose that 1 cm³ of this fluid is spilled into a circular pool of water 2.2 m in diameter and spreads evenly over the entire surface of the pool, which then calms to a mirror-like smoothness. You now kneel by the edge of the pool and look down at it from a height of 50 cm. The pool is illuminated with diffuse white light [all wavelengths coming from all directions].

(a)

What colour does the edge of the pool appear directly beneath you? (Give the exact wavelength as well as a qualitative description.)

(b)

Draw a sketch of the appearance of the pool's surface indicating the dominant wavelengths (colours) of the light reflected from different (specified) regions.

Note: Don't forget about refraction.

2.

In fairly bright light the pupil of your eye will contract to a diameter of about 4 mm. Under these conditions, assuming that you have ``perfect'' vision,

(a)

how far from your eye can you hold a book and still be able to resolve two lines separated by 70 $\mu$m using green light with $\lambda = 546$ nm?

(b)

Describe what you would see at this distance when observing the same two lines with white light (all wavelengths).

3.

A laser operating at a wavelength of 0.6 $\mu$m emits a beam which is a perfectly plane wave (apart from diffraction effects) as it leaves the aperture of a 1.5 m diameter telescope coupled to the laser. The beam is aimed at a 0.75 m diameter retro-reflector array sitting on the surface of the Moon. The reflected pulse is detected by the same Earth-based telescope that emitted the original beam.

(a)

If the laser emits an average power of 20 mW [milliwatts], what is the average power in the detected return signal?

(b)

How many photons per second does this represent for the system described above?

Optional: The solar light flux in the vicinity of the Earth is about $1.3 \times 10^3$ W/m². If the Earth's albedo (reflectivity) is 0.1, estimate the light flux reflected from the Earth as it arrives at the Moon's surface: this ``Earthshine'' is what faintly illuminates the Moon when it is ``new'' (i.e. dark) as seen from the Earth. If the albedo of the Moon is 0.3 and our telescope looks at 1% of the lunar hemisphere when seeking the laser signal, how much light (in watts) does it intercept from the Earthshine-lit Moon? How does this compare with the intensity of the returning laser signal?

4.

An x-ray beam of wavelength 1 Å [Angstroms -- 1 Å $\equiv 10^{-10}$ m] is incident upon a crystal as shown below (viewed ``end-on'' along an orthogonal symmetry axis). At what angle $\Theta$ will the beam undergo its first reflection maximum from the indicated ``[3,1,0]'' planes?