Transmit far field

The beam a laser transmitter sends out through a telescope with a central obscuration: how bright it is at the receiver, how wide it spreads, and where its power goes. The aperture’s field is propagated to the target numerically, so wavefront error, defocus, the near field and a pointing offset are all included.

Calculating…

Power

At the receiver

Gain and losses

The beam

On-axis intensity against range

At the target direction, with everything above included. The dashed lines mark your range and the far-field distance 2D²/λ; nearer than that, the beam is still close to its own diameter and the on-axis intensity rises and falls (the Fresnel zones of the obscured aperture).

The pattern at your range

Drag to turn it round.

Cut through the beam

Intensity along a line through the beam’s peak, against the angle from the target direction: how fast it falls with pointing error, and where the side lobes are.

Wavefront error across the aperture

How it is calculated

The beam leaving the fibre is Gaussian, with 1/e² intensity radius w, carrying the power left after the optical loss. The aperture (diameter D) passes the annulus between the obscuration and the rim; the power lost on each is exact: P(1 − e^−2b²/w²) on an obscuration of radius b, and P e^−2a²/w² past the rim of radius a.

Across the aperture the beam’s phase carries a random wavefront error, defocus W₀₂₀(r/a)² (W₀₂₀ being the peak-to-valley defocus, in waves), and the pointing offset as a tilt. The field at range z and angle θ is the Fresnel diffraction integral

E(θ) = (1/λz) ∫ E(ξ) exp(iπ|ξ|²/λz) exp(−i2πθ·ξ/λ) d²ξ, I = |E|²,

evaluated over a 256 × 256 grid across the aperture with a matrix Fourier transform. The quadratic term is what makes the near field differ from the far field; well beyond 2D²/λ it no longer matters. The gain is G = 4πz²I/P, against the ideal (πD/λ)² of a uniformly lit, unobscured aperture; its losses are separated by switching each effect on in turn: truncation and obscuration (Klein and Degnan’s g), then wavefront error and defocus, then the finite range, then pointing.

The wavefront-error map is white noise smoothed by a Gaussian whose standard deviation is the correlation length, with its piston, tip and tilt removed and scaled to the RMS you enter, over the clear aperture. Tilt is left out because a tilt of the whole beam is a pointing error, which you enter separately; so with no pointing offset the beam stays centred on the target. The pattern’s middle sample is always exactly the target direction. A short correlation length scatters light to wide angles (the core keeps its shape but loses power, the Strehl ratio); a long one bends and distorts the core. Each “New random screen” is another example with the same statistics.

What it leaves out

Spider vanes, and any other blockage besides the central obscuration.
The atmosphere: turbulence, absorption and scattering (this is a vacuum or space link).
Pointing jitter: the offset is fixed. A random jitter lowers the average intensity further.
Polarization, and the beam quality of the source (it is an ideal Gaussian, M² = 1).

Reference

B. J. Klein and J. J. Degnan, “Optical antenna gain. 1: Transmitting antennas,” Applied Optics, vol. 13, no. 9, pp. 2134–2141 (1974). doi:10.1364/AO.13.002134. J. W. Goodman, Introduction to Fourier Optics (Fresnel and Fraunhofer diffraction). R. Soummer et al., “Fast computation of Lyot-style coronagraph propagation,” Optics Express 15, 15935 (2007) (the matrix Fourier transform).