Incoherent Detection

9006-008

9006-008a

The Carrier-To-Noise Ratio (CNR) in an incoherent (direct) detection system with an Avalanche Photodetector (APD), is given by:

                         (MRiPr)2
CNR  =   ----------------------------------------
         [2e{Ri(Pr + NbBo) + Id}M(2+x) + 4kTF/RL]Be

               ^     ^       ^             ^
               .     .       .             .
               .     .       .             .
               .     .       .             .
           Signal    .   Dark Current      .
       Quantum Noise .    Shot Noise       .
                     .                     .
                     .                     .
                Background              Thermal
                Shot Noise               Noise

9006-008b

where:

M = avalanche current gain (400),
x = excess avalanche noise factor (0.25),
R_i = current responsivity (0.26 mA/mW),
P_r = received signal power (1.60 x 10^-15),
N_b = background radiation (= 0),
I_d = dark current at unity gain (0.3 nA),
k = Boltzmann's constant (1.38 x 10^-23 J/K),
T = temperature (300 K),
F = noise figure of amplifier (6 dB),
R_L = effective load resistance (10 Mohm),
B_o = optical bandwidth (1 Hz),
B_e = electrical output bandwidth (1 Hz).

9006-008c

Shot noise limited detection is reached when the shot noise due to left hand side of the denominator exceeds the thermal noise of the right hand side of the denominator, viz

                2kTF
RL > -------------------------
     e{Ri(Pr + NbBo) + Id}M(2+x)

In order to maintain shot noise limited detection when high bandwidths are required, a high avalanche gain M and/or a transimpedance front-end amplifier can be employed (9008-028). The combination of avalanche photodetector and transimpedance amplifier generally produces the most sensitive front-end. If M²R_L = 1000, the shot noise threshold is reached when the received power is about 1 mW. Note that no account has been taken of 1/f noise which may be substantial at very low frequencies, i.e., below 1 kHz.

In an ideal photon-counting receiver, the dark current and thermal noise sources may be ignored. If the background radiation received P_b = N_bB_o is negligible, then the sensitivity of the receiver is ideal.

9006-008d

For a front-end with the values given in parentheses, the shot-noise limit is reached in an avalanche photodetector system with received power P_r = -90 dBm (10^-12 W), when:

R_L > 1 Mohm

Using the values given in parentheses:

CNR = 7 dB re 1 Hz

Under shot-noise limited conditions, the expression for CNR may be simplified to:

                  (MRiPr)2
CNR = ------------------------------
      [2e{Ri(Pr + NbBo) + Id}M(2+x)]Be

9006-008e

Since P_r is very small, depending on the values of dark current, avalanche gain and excess noise factor, the recovered CNR will be less than optimum. The background radiation N_bB_o (P_b), which hopefully is mainly starlight from the alien star, can be minimized by choosing a very narrow band optical filter. At 10 light years range for a received signal power of -118 dBm, the shot noise due to P_b is only about 32 dB below the shot noise due to P_r alone if the optical bandwidth is 1 Hz. This is the relative level of Planckian starlight at 10 light years if the transmitter and star are not separately resolved in a 10 meter diameter telescope. This ratio increases to about 72 dB for diffraction limited telescopes.

If the optical filter has a bandwidth as low as 1.6 kHz, the CNR will fall by 3 dB, because P_b = P_r, and the shot noise would double. The occurs irrespective of the electrical post-detection bandwidth. Such high Q optical filters are very difficult to produce and tune for tracking of Doppler shifts and drifts (chirp). Incoherent detection also means that the modulation format is restricted to be only one of two basic forms; intensity or polarization modulation. A 1.6 kHz optical filter is virtually impossible to construct, so we can eliminate this approach to filtering for achieving the ultimate SNR. For a diffraction limited telescope, the corresponding optical bandwidth is 16 MHz; a reasonable bandwidth for a Fabry-Perot interferometer.

9006-008f

A way to obtain increased sensitivity in incoherent receivers is to operate an APD in the photon-counting mode. In this situation, dark current has a minor effect. As before, if P_b is equal to P_r the recovered SNR will be degraded by 3 dB.

If all other sources of noise can be ignored, i.e., excess photodetector noise, dark current noise and background radiation noise, the above equation may be further simplified to:

       eta.Pr
CNR  = ------
       2hfBe

This is 3 dB less than obtainable with a perfect optical heterodyne receiver.

The Columbus Optical SETI Observatory
Copyright (c), 1990