Q:

Power requirement for transmitting communications further than 1 ly?

The StarShot project from Breakthrough Initiatives aims to send a paper light spacecraft to alpha centauri using pressure from a laser located on earth at 20,000kms. 
What are the options for a transmitter that can be received from 1-4 lyrs away, if the transmitter weighs the same as a few jellybeans (<10 grams)? What wavelength and power range can they work with for that distance using that weight of equipment?

A:

A partial answer to your question emerged today on arXiv. Hippke (2017) analyses this exact problem - dealing with power requirements for the transmitter, arrangements for detection, the information transfer rate, extinction etc. etc.
Their conclusion is that there is no problem in principle with getting something to work at about 1 bit/s per Watt of power that the probe can obtain from photovoltaic power and if a dedicated 39-m telescope is observing the probe using a coronagraphic imager. They also conclude that the best wavelength to work at is 400nm for a ground-based receiving telescope and 300 nm if a 10-m space based telescope is available.
The basic transmission device is a $\sim 1$m diameter telescope, firing a coherent laser beam back towards Earth. The data rate scales as the area of the transmitting telescope. They do not discuss how such a thing could be built or sent to $\alpha$ Cen. A promised follow-up paper is to discuss the use of gravitational lensing to improve signal-to-noise.

