Optical Antenna Gain
Telescope transmit gain (Klein–Degnan, clipped and obscured Gaussian beam), receive gain, far-field divergence, beam waist and Rayleigh range, with beam quality M² up to 10 as a Gaussian Schell-model beam.
Inputs
For a coherent beam (M² = 1) the transmit gain peaks at α = 1.094 for γ = 0.15 (g = 0.7530, -1.23 dB).
Transmit gain (on axis)
103.22 dBi
g = 0.6812 (-1.67 dB); coherent g(α, γ) = 0.7439
Receive gain
104.79 dBi
(πD/λ)² (1 − γ²) η
Directivity limit (πD/λ)²
106.14 dBi
uniform, unobscured
Beam waist W₀ = a/α (unclipped Gaussian)
50.00 mm
Far-field half-angle (1/e²)
14.43 µrad
unclipped Gaussian M²λ/(πW₀): 10.85 µrad
Rayleigh range πW₀²/(M²λ) (unclipped Gaussian)
4606 m
Model
G_tx = (πD/λ)² · g · η, g(α, γ) = (2/α²)[e^(−α²) − e^(−γ²α²)]² at M² = 1 (Klein & Degnan 1974)
A Gaussian beam of 1/e² radius W₀ = a/α fills the telescope. Filling it more (small α) spills power past the rim; filling it less puts more of it on the secondary mirror and leaves the outer aperture dark, which widens the beam. The best compromise without obscuration is α = 1.12, g = 0.8145 (−0.89 dB). As a receiver the same telescope collects a plane wave over its clear area, so the obscuration costs only 1 − γ².
The far-field half-angle is where the Fraunhofer pattern of the clipped beam falls to 1/e² of its peak. At α = 1 the clipping makes it 1.44× the untruncated Gaussian λ/(πW₀); the two agree from α ≈ 2. The Rayleigh range is that of the Gaussian beam W₀.
A beam with M² > 1 is modelled as a Gaussian Schell-model source (Gori 1980): Gaussian intensity of radius W₀ and Gaussian degree of coherence of width σ_g = W₀/√(M⁴ − 1). Such a beam is an incoherent sum of coherent Gaussians tilted by a Gaussian spread of angles, so its far field is the clipped pattern above averaged over those tilts. Unclipped it is a Gaussian of half-angle M²λ/(πW₀) and on-axis g = 2/(α²M⁴); clipping trims the wide angles, so a clipped multimode beam is narrower and brighter on axis than M² alone suggests (at α = 1, M² = 2: 1.6× the gain, 0.8× the width). Real multimode beams (top-hat fibre modes, laser-diode stacks) are not Gaussian Schell; treat M² near 10 as indicative. Spider vanes and wavefront errors go in η.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Klein–Degnan gain of a clipped, obscured Gaussian; M² ≤ 10 as a Gaussian Schell-model beam (Gori's tilted coherent copies); 1/e² half-angle from the far field; receive gain (πD/λ)²(1 − γ²)η.
- Klein B. J., Degnan J. J. (1974). Optical antenna gain. 1: Transmitting antennas. Appl. Opt. 13(9), 2134. doi:10.1364/ao.13.002134
- Degnan J. J., Klein B. J. (1974). Optical antenna gain. 2: Receiving antennas. Appl. Opt. 13(10), 2397. doi:10.1364/ao.13.002397
- Siegman A. E. (1990). New developments in laser resonators. Proc. SPIE 1224, 2. Beam quality M². doi:10.1117/12.18425
- Gori F. (1980). Collett–Wolf sources and multimode lasers. Opt. Commun. 34(3), 301. Gaussian Schell-model beam as an incoherent superposition of tilted coherent Gaussians. doi:10.1016/0030-4018(80)90382-x