Laser communications calculators
Optimum transmit beam waist
A laser transmitter’s Gaussian beam fills a circular telescope whose centre is blocked by the secondary mirror. Make the beam too small and it spreads into a wide cone in the far field; make it too large and power spills past the rim and onto the obscuration. This finds the beam size between the two that puts the most power on axis at the receiver, after Klein and Degnan (1974).
Results
| Degnan–Klein fit | Exact optimum | ||
|---|---|---|---|
| α = a / w | 1.0470 | 1.0523 | |
| Beam radius w (1/e² intensity) | 47.754 | 47.517 | mm |
| Beam diameter at 1/e² intensity, 2w | 95.508 | 95.034 | mm |
| Beam diameter at 1/e intensity, √2 w | 67.534 | 67.199 | mm |
| Beam diameter at half maximum (FWHM), √(2 ln 2) w | 56.226 | 55.947 | mm |
| Gain efficiency g | 0.6560 | 0.6561 | |
| … in decibels | -1.831 | -1.831 | dB |
| Power through the aperture | 76.0 | 76.2 | % |
| On-axis antenna gain | 104.31 | 104.31 | dBi |
- Obscuration ratio γ = b / a = 0.2500.
- A uniformly lit aperture with the same obscuration would have g = 0.9375 (-0.280 dB); a Gaussian beam cannot reach that, because it must either spill past the rim or under-fill the aperture.
- Ideal gain of the full, uniformly lit, unobscured aperture: 106.14 dBi.
- The fit and the exact optimum agree: the difference in gain is under 0.001 dB.
The curve is the gain efficiency g for your obscuration ratio as the beam size changes; the marked point is the best one. Gain is relative to a uniformly lit, unobscured aperture of the same diameter.
How it is calculated
The beam’s field across the aperture is Gaussian, E(r) ∝ exp(−r²/w²), so w is the radius at which the intensity falls to 1/e² of its peak. With aperture radius a = D/2 and obscuration radius b = d/2, Klein and Degnan define
α = a / w (how hard the aperture truncates the beam), γ = b / a (the obscuration ratio),
and show that the on-axis gain in the far field is
G = (4πA / λ²) · g(α, γ), A = πa², g(α, γ) = (2 / α²) [ exp(−α²) − exp(−γ²α²) ]²,
where 4πA/λ² = (πD/λ)² is the gain of a uniformly lit, unobscured aperture, so g is the efficiency against that ideal. For an unobscured aperture the best is α = 1.12, g = 0.8145 (−0.89 dB). Klein and Degnan fitted the best α for an obscured aperture with the polynomial
αopt ≈ 1.12 − 1.30 γ² + 2.12 γ⁴ (for γ ≤ 0.4),
which gives the beam radius w = a / αopt. The beam’s intensity is I(r) = I0 exp(−2r²/w²), so its diameter is 2w at 1/e² of the peak, √2 w at 1/e, and √(2 ln 2) w ≈ 1.1774 w at half the peak (the full width at half maximum, FWHM). Lasers and beam profilers quote all three, so check which one you are given; the beam diameter converter turns any of them into the others. The exact optimum column finds the maximum of g numerically instead, which shows how close the fit is and still holds for larger obscurations. “Power through the aperture” is the fraction of the beam that passes between the obscuration and the rim, exp(−2γ²α²) − exp(−2α²); the rest is clipped, and lands on the telescope’s structure as heat or scattered light.
What it leaves out
- Spider vanes that hold the secondary mirror, and any other blockage: they lower the gain further.
- Wavefront error and defocus in the telescope: they lower the on-axis gain further (roughly by the Strehl ratio for small errors).
- The near field. The gain here is for a receiver in the far field, beyond about D²/λ.
- Pointing. The optimum is for the receiver on axis. With large pointing errors a smaller beam (a wider cone) can be the better choice; Klein and Degnan treat this too.
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 · NASA Technical Reports Server record.