Receive: fiber coupling

A receive telescope collects the incoming beam and focuses it into a single-mode fiber. This works out how much of the power falling on the aperture reaches the fiber: what the obscuration blocks, what the optics lose, and how well the focused spot, blurred by wavefront error, matches the fiber’s mode. A fast steering mirror is assumed to remove any tilt, so only the higher-order wavefront error counts.

Calculating…

Power

Losses

Coupling

Coupling efficiency against focal length

With the wavefront error entered. Too short a focal length makes a spot smaller than the fiber’s mode; too long, larger.

Wavefront error across the aperture

How it is calculated

The beam arrives as a plane wave of intensity I, so the full aperture collects IπD²/4. The obscuration blocks the fraction γ² of it (γ = d/D); the optical loss takes its share of the rest. The wavefront error is a random map made like the transmit one (white noise smoothed by a Gaussian of the correlation length), with its tip and tilt removed, as the steering mirror would, and scaled to the RMS entered.

The telescope compresses the beam to a collimated diameter D_c at the coupling lens. Seen back through a lens of focal length f, the fiber’s mode (a Gaussian of radius w_f = MFD/2) is a Gaussian of radius w_p = λf/(πw_f) at the lens, and the coupling efficiency is the overlap of the received field E with it:

η = |∫ E G dA|² / ( ∫ |E|² dA · ∫ |G|² dA ), G(r) = exp(−r²/w_p²).

This equals the overlap of the focused spot with the fiber mode in the focal plane. For an unobscured, perfect wavefront the best is 81.45 % (−0.89 dB), when D_c/(2w_p) = 1.12, the same number as the transmit beam’s Degnan–Klein optimum, by reciprocity. The coupling efficiency here is relative to the power that reaches the lens, so the obscuration counts twice: once as the light it blocks, and again in the coupling, because a ring of light matches a Gaussian mode less well than a full disc.

What it leaves out

Pointing and tracking error: the steering mirror is taken as perfect.
Spider vanes; atmospheric turbulence; polarization; the fiber’s own losses and Fresnel reflections at its face.
The plane-wave approximation: the receive aperture is assumed much smaller than the arriving beam.

Reference

S. Shaklan and F. Roddier, “Coupling starlight into single-mode fiber optics,” Applied Optics 27(11), 2334–2338 (1988). B. J. Klein and J. J. Degnan, “Optical antenna gain. 1: Transmitting antennas,” Applied Optics, vol. 13, no. 9, pp. 2134–2141 (1974).