Point-Ahead Angle

Point-ahead angle 2v⊥/c from the transverse relative velocity, compared with the diffraction half-angle, with the miss distance and light time for LEO and GEO laser links.

Model: Textbook approximation

Link Range (km)Transverse Relative Velocity v⊥ (km/s, inertial)Wavelength (nm)TX Aperture Diameter (cm)

Presets:

Point-ahead angle 2v⊥/c

46.70 µrad

independent of range

Diffraction half-angle 1.22λ/D

18.9 µrad

first null of a uniformly filled aperture

Point-ahead / half-angle

2.47

the beam must be steered ahead

Miss distance without point-ahead

46.7 m

θ_PA × R at the far terminal

One-way light time R/c

3.34 ms

round trip 6.67 ms

Point-ahead angle vs transverse velocity

The beacon from the far terminal shows where it was R/c ago, and the transmitted beam arrives R/c later, so the transmit direction must lead the received beacon by θ_PA= 2v⊥/c. Use the transverse velocity in an inertial frame: a GEO satellite is fixed in the sky, yet relative to the stars it moves 2.61 km/s faster than a station beneath it, which gives 17.4 µrad. The cited chapter takes a LEO satellite's orbital velocity, 7.61 km/s at 500 km (51 µrad); the LEO presets subtract the station's own rotation, 0.465 km/s eastward at the equator, as a vector: √((v cos i − ω_ER_E)² + (v sin i)²) for inclination i. A prograde equatorial pass gives 47.7 µrad, a polar one 50.9 µrad, and a retrograde one 53.9 µrad.

Point-ahead matters when θ_PA is a sizeable fraction of the beam half-angle; a 10 cm aperture at 1550 nm has 1.22λ/D = 18.9 µrad, so a LEO downlink (≈ 51 µrad) needs a point-ahead mirror.

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

θ = 2v⊥/c to first order in v/c, with v⊥ the inertial transverse velocity (GEO preset: ω_E(r_GEO − R_E) = 2.61 km/s; LEO presets: orbital velocity less the equatorial station's rotation, 7.15 km/s prograde, 7.63 km/s polar). Straight-line light path, no refraction; uniform-aperture 1.22λ/D for comparison.

Carrasco-Casado A., Mata-Calvo R. (2020). Space optical links for communication networks. In Springer Handbook of Optical Networks, 1057–1103. Eq. (8.4), point-ahead angle 2v/c with the orbital velocity; ≈ 51 µrad for LEO 500 km, 18 µrad for GEO (§8.1.2; preprint arXiv:2012.13166). doi:10.1007/978-3-030-16250-4_34
Petit G., Luzum B. (eds.) (2010). IERS Conventions (2010). IERS Technical Note 36, Table 1.1: GM_E and ω_E. iers.org

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