Coherent Detection

9006-009

9006-009a

The Carrier-To-Noise Ratio (CNR) in a coherent (heterodyne) detection system with a 100% mixing efficiency, is given by:

                                (MRi)2PrPo
CNR  =  ---------------------------------------------------------
        [e{Ri(Pr + NbBo + Po) + Id}M(2+x) + 2(MRi)2NbPo + 2kTF/RL]Be

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

9006-009b

where:

M = avalanche gain (1 for PIN detector),
x = excess noise factor (0 for a PIN photodetector),
R_i = current responsivity (0.26 mA/mW),
P_r = received signal power (1.60 x 10^-15),
P_o = local oscillator power (1 mW),
N_b = background radiation (= 0),
e = electronic charge (1.6 x 10^-19 C),
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 kohm),
B_o = optical bandwidth (1 Hz),
B_e = electrical output bandwidth (B_I.F. = 1 Hz).

9006-009c

Because of the "gain" of coherent receivers, we assume a PIN photodetection system (9008-029). There is no point in using the more expensive APD device as this would only reduce sensitivity and dynamic range, and would be difficult to implement in two dimensional array form. For a small PIN photodetector (M = 1, x = 0, I_d ~ 0) and P_o >> P_b and P_r, 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
Po > ----
     eRiRL

For the 10 kohm front-end with with the values given above in parentheses:

Po > 100 µW

9006-009d

As a rule of thumb for optical receivers, we need about a milliwatt of power on the photodetector to ensure shot-noise limited detection for a 1 k load. At that point the CNR will be 3 dB less than the ultimate.

Using the parameter values given in parentheses for a 1 kW SETI transmitter at 10 light years distance, the received power P_r = 1.60 x 10^-15 W, i.e., -118 dBm, we find that:

CNR = 34 dB re 1 Hz

This is significantly greater (by more than 27 dB) than is possible with power- starved incoherent receivers. The other very important thing to notice here is that the Planck spectral density in starlight from distances greater than several light years is very low (P_b << P_o). Even if the entire spectrum of the star falls upon the photodetector, P_b (N_bB_o) does not affect the shot-noise level. Contrast this with the case of incoherent detection, where an optical bandwidth B_o = 1.6 kHz would cause P_b = P_r, and essentially double the shot-noise level (if x = I_d = 0). Indeed, in the coherent system, the Planck spectrum from many stars could be allowed to simultaneously fall onto the photodetector and still not increase the noise floor.

9006-009e

Only the 2(MR_i)2N_bP_o component of the Planck continuum spectrum that falls within the electrical output (I.F.) bandwidth B_e is important. This noise is produced by the background beating with the local oscillator. This is independent of the optical bandwidth. Noise in the optical spectrum covering a bandwidth 2B_e will be down converted to an output bandwidth B_e, so the effective Planck noise spectral density referred to the output is twice (3 dB) as large as indicated in the spectral graphs. Note that as we increase the electrical output bandwidth B_e (B_I.F.), the quantum noise floor increases at the same rate as the Planck radiation (background) noise, since both sources of noise power are proportional to B_e. Thus, as long as the Planck noise spectral density is less than the quantum noise-floor, the Planck noise is not detected.

Whatever the value of Planck continuum spectral density, as long as P_b << P_o, the continuum cannot affect the recovered SNR. If all other sources of noise can be eliminated, i.e., excess photodetector noise, dark current noise and background radiation noise is negligible, a perfect quantum (shot) noise limited optical heterodyne receiver has a Carrier-To-Noise Ratio given by:

9006-009f

       Pr
CNR = ----
      hfBe

This is 3 dB more than obtainable with an ideal incoherent (direct detection) optical receiver. To obtain near 100% mixing efficiency and this value, the signal and local oscillator beams must be colinear and overlap on the photodetector surface (9008-030).

The CNR for optical homodyne receivers is another 3 dB greater. For practical considerations, e.g., the large Doppler shifts and chirp, heterodyne reception would be preferable as homodyne receivers have a zero Intermediate Frequency. Also, heterodyne detection allows for the use of existing parallel multiplexing radio frequency techniques to reduce the "search" time. Thus the subject of homodyne detection has been ignored in this study. This is not to imply that homodyne detection is ruled out. Indeed, it might be found that the ability to frequency lock the local oscillator to the received signal after "detection" of the signal, may be desirable. This will be dependent on the modulation format chosen by the aliens.

The Columbus Optical SETI Observatory
Copyright (c), 1990