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Consider an optical switch (transfer speed $10 \mathrm{Gb s^{-1}}$), whose output (after any necessary amplification) is used to illuminate the Moon. Thanks to the mirrors left behind by the Apollo mission, the signal comes back and can be used (after any necessary amplification) as an input to the switch. If we make sure the switch works reliably the transmitted data will circle in the system indefinitely. Thus we acquire a memory. What is its maximum capacity? Ignore any delays caused by the processing of the signal and any headers of the data.

Describe the imaging system of a microscope (consisting of two convex lenses) and that of a Keplerian telescope. Explain the difference in function and construction of a microscope and a telescope and sketch the rays passing through the systems. How can we usefully define magnification for these optical systems? Derive the equations for magnification.

Have you ever seen such a weird atmosphere? Up to a certain height the speed of light inside it is constant, $v_{0}$, but from that certain height the speed of light starts increasing linearly as $v(Δh)=v_{0}+kΔh$. At one point, exactly at the height where the speed of light starts changing, light beams are sent upward in all directions. Show that all these beams move along circular arcs and determine the radii of these arcs. Also find out the distance between the place where the the light was emitted and the point where the beams return to the original height.

Pikos' friend wears glasses. When she puts them on, her eyes seem to be smaller. Is she shortsighted or farsighted? Justify your answer.

Consider an optical system composed of three semitransparent mirrors placed behind each other along one axis. Every mirror by itself reflects half of incident light and lets the other half pass. Determine what fraction of light passes through the system of mirrors.

Bonus: Solve the problem for $n$ such mirrors.

Imagine that around the Sun on a circular orbit is a convex lens with a diameter that is equal to the diameter of the Sun, the focal point of which orbits with a sufficient precision on the orbit of Earth. Determine how the lens will burn the Earth during one of its orbit (ie. how much solar energy will be given to Earth by the lens), if it orbits at the distance of Mercury and compare it with the state where it will be as far as Venus.

Bonus: Consider the eclipse that the lens will cause during its orbit.

On the surface of water a thin biconcave lens made from a light-weight material is floating. The radii of both surfaces are $R=20\;\mathrm{cm}$. Determine the distance between the two focal points of the lens, if the index of refraction of the air above the lens is $n_{a}=1$, index of refraction of the lens is $n_{l}=1.5$ and index of refraction of water is $n_{w}=1.3$.

Bonus: Assume that it is a lens of width $T=3\;\mathrm{cm}$, and within it is symmetrically place an air bubble in the shape of a biconcave lens with the radii of curvature $r=50\;\mathrm{cm}$ and width $t=1\;\mathrm{cm}$.

Can you see the Moon from Mars with a naked eye.Ground your answer in calculations.

If an object is placed a distance $p$ from a thin glass lens (index of refraction $n_{s})$, we can see its image on a screen that is placed a distance $d$ from the lens. Without altering any distances, we immerse this system into a liquid (index of refraction $n)$. Under what conditions can we still observe the object's image on the screen, and how far from lens would this image be?

What would a rainbow look like if it was raining oil, sulfuric acid or glass instead of water?