Receiver FOV vs Background Noise
Electrical SNR of a PIN or APD receiver versus field of view, with background light from clear sky, clouds or the Sun, the signal clipped by the field stop (diffraction and turbulence), and the FOV at which background noise overtakes amplifier noise.
Electrical SNR
21.7 dB
(M I_s)² / σ²
Background power at the detector
-31.6 dBm
6.91e-7 W
Signal power at the detector
-31.5 dBm
η_opt × field-stop fraction × signal at the aperture
Signal through the field stop
99.99 %
long-exposure encircled energy at the FOV half-angle
FOV passing 90 % of the signal
0.0238 mrad
half-angle
Fried parameter r₀
14.1 cm
seeing λ/r₀ = 0.0110 mrad (spherical wave)
Noise variance from background
34.6 %
amplifier 30.4 %, excess noise F = 5.14
FOV where background = amplifier noise
18.7 mrad
half-angle; wider, background noise exceeds amplifier noise
Diffraction half-angle 1.22λ/D
9.45e-3 mrad
first dark ring: 83.8 % of a turbulence-free spot
SNR vs FOV half-angle
SNR vs signal power
Model
P_bg = L_λ · (πD²/4) · π sin²θ · Δλ · η_opt, SNR = (M R η_opt P_s)² / [2qM²F (I_s + I_bg + I_d) B + i_n² B]
R = QE·qλ/(hc), F = kM + (1 − k)(2 − 1/M) (McIntyre 1966). Background radiances are typical clear-sky and cloud values (Giggenbach 2022) or a 5772 K solar disc above the atmosphere.
EE(θ) = 2v ∫₀¹ T(s) J₁(2vs) ds, v = πDθ/λ, T(s) = (2/π)[arccos s − s√(1 − s²)] · exp[−3.44 (sD/r₀)^(5/3)]
The field stop passes the long-exposure encircled energy EE of the signal spot: the aperture's transfer function times Fried's (1966) long-exposure turbulence factor, with r₀ = (0.423 k² Cn² L)^(−3/5) for a plane wave or (0.423 · 3/8 · k² Cn² L)^(−3/5), 1.8× larger, for the spherical wave from a point source at the far end of a horizontal path. The SNR uses the mean (long-exposure) fraction; the short-exposure spot wanders, and the receiver's central obscuration is ignored. It includes image motion (tilt); a tip-tilt tracker would make the spot smaller. Without turbulence EE is Rayleigh's 1 − J₀²(v) − J₁²(v); for D ≫ r₀ the spot is ≈ λ/r₀ wide instead of λ/D. A wider FOV eases pointing and turbulence-induced spot motion, but once the background shot noise passes the amplifier noise every doubling of the half-angle quadruples it for an extended sky (for the Sun, only until the FOV covers the 4.65 mrad half-angle disc).
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
Shot noise with McIntyre's F plus amplifier noise; background L·A·Ω·Δλ·η, typical sky radiances. Field stop: long-exposure encircled energy (aperture OTF × Fried's, spherical or plane r₀, tilt included).
- Giggenbach D. (2022). Free-space optical data receivers with avalanche detectors for satellite downlinks regarding background light. Sensors 22(18), 6773. Table 1: sky, cloud and sun radiances at 850, 1064 and 1550 nm; eq. (22). doi:10.3390/s22186773
- McIntyre R. J. (1966). Multiplication noise in uniform avalanche diodes. IEEE Trans. Electron Devices 13(1), 164–168. Excess noise factor F = kM + (1 − k)(2 − 1/M). doi:10.1109/T-ED.1966.15651
- Agrawal G. P. (2010). Fiber-Optic Communication Systems, 4th ed. Wiley. §4.4, receiver noise and SNR of PIN and APD receivers.
- Fried D. L. (1966). Optical resolution through a randomly inhomogeneous medium for very long and very short exposures. J. Opt. Soc. Am. 56(10), 1372. Long-exposure OTF exp[−3.44 (λν/r₀)^(5/3)]. doi:10.1364/josa.56.001372