They use optical amplifiers, which take light at one wavelength and use it to intensify light at another wavelength. They're much like lasers (technically I think they count as optically-pumped lasers?), and they turn on from a very small input signal, effectively reenforcing it.
This can happen across multiple signals, on different wavelengths, in parallel. Like a broadband radio amplifier, it boosts everything across a large working bandwidth. There are even optical compressors (also powered by light), which speed up the baud rate of signals. That way a slow electronic system can produce the original pulses, and then they can be compressed to faster than electronics can work with, and then multiplexed with many other signals at different wavelengths, and this whole composite thing is sent down the line, amplified without decoding along the way, and then finally the whole thing is reversed at the other end.
This is the trick behind how fibre links are so fast, considering there are no electronics that can handle data serially at those speeds.
That is certainly not the case, the pump light is generated from electricity right where the laser amplifier sits in the fiber. No real amounts of energy are sent optically down the fiber. To power the amplifier, a high voltage DC line is designed right into the submarine fiber cable. And those things carry a lot of power, a long fiber cable will draw tens of kilowatts of DC for all the optical repeaters.
The reason is, of course, that thousands of miles of cable has a pretty insane optical attenuation, no matter what you do, because optical attenuation rises exponentially with length. The electrical resistance of a high voltage DC power line only rises linearily, on the other hand.
These are typically used for short submarine connections to e.g. connect an island. As it is much cheaper than running a full repeatered system.
I believe the attenuation is stated in dB/km, therefore rises linearly (or even logarithmically if you look at it from the uncommon energy-wise point of view) with distance. Why should exponential be the case?
Time division multiplexing and frequency or amplitude division multiplexing preclude most things because they are so cheap and simple these days. Polarity is also another knob to mess with when you need to squeeze more.