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The Breakthrough Starshot concept is to send a swarm of tiny spacecraft on a flyby mission to Proxima Centauri b. These are tiny spacecraft (on the scale of centimeters). Given how miniscule the arriving signal is/has been from any of the much larger probes sent to the outer planets, how could we possibly receive what must surely be a much less powerful signal from a vastly greater distance? The Voyager spacecraft, for example, could transmit at several watts from about 40 or 50 light-minutes away, requiring a big dish to capture a usable signal; the tiny Starshot probes would surely be transmitting not more than a few milliwatts from over 4 light-years away.

Is the idea for them to form up into a phased array to send their data back to Earth? And even so, would that really help the situation much? How has the project planned for data downlink?eh

EDIT: Seems this question has already been asked before: How are the tiny spacecraft proposed by Project Starshot supposed to communicate with Earth?

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This is discussed on the project website. Essentially each probe would carry a small laser and use its lightsail to focus that beam back towards the solar system, where a huge and very sensitive telescope array would be needed to detect it.

Images of the target planet could be transmitted by a 1Watt laser onboard the nanocraft, in a ‘burst mode’ which uses the energy storage unit to rapidly draw power for the power-intensive laser communications mode. Upon approach to the target, the sail would be used to focus the laser communication signal.

For a 4m sail, for example, the diffraction limit spot size on Earth would be on order of 1000m. A kilometer-scale receiving array would intercept 10-14 of the transmitted signal. The main challenge is to use the sail as diffraction limited optics for the laser communication system. This would be achieved by shaping the sail into a ‘Fresnel lens’ upon approach to the target. The sail structure could be different at the launch and communication phases. In order to maintain a high transmission through the Earth’s atmosphere, the communication would need to operate at a wavelength shorter than that used by the launch laser system, due to the Doppler shift of the nanocraft relative to the Earth.

That "1000m" is obviuosly a typo. A 4m sail at 400nm wavelength (blue) gives a beam width of about $10^{-7}$ radians, and so a spot at 4ly of about $4\times 10^9m$. A kilometer sized receiving array (they propose to adapt the laser array used for propulsion would receive about $10^{-14}$ of the beam, which should allow a data rate of a a few kb/s.

The probe would need to transmit in short bursts, then wait to refill its power storage.

Another possibility is to reflect the propulsion laser back towards Earth and modulate that by moving the lightsail.

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