Project Cyclops, A Design... - Department of Earth and Planetary ...

APPENDIX
D
SQUARE LAW DETECTION THEORY
A square-law detector is a square-law device followed
by a filter that transmits a desired range of frequencies
as the output. Thus the output of a square-law detector
is a filtered measure of the instantaneous power incident
on the square-law device. When the incident power is in
the radio frequency band, the output filter usually
excludes harmonics of the radio frequencies, and thereby
averages the incident power over a time comparable
to one RF cycle. At optical frequencies, the square-law
device is a photon detector whose output is generally
taken to be proportional to the product of the amplitude
of the incident wave with its complex conjugate.
This product contains no double frequency component,
and output filtering may be omitted. In either the radio
frequency or optical detector, the postdetection filtering
may be much narrower in bandwidth than any predetection
filtering so that the output is averaged over times
long compared with the reciprocal of the input bandwidth,
let alone the center frequency of this band.
Shot noise is usually negligible in radio-frequency
detectors, but may be dominant in photodetectors.
Fluctuation noise may or may not be important in
photodetection. We shall include both sources of noise
in order to derive more generally applicable results.
Although we shall use a photodetector as a model of a
square-law device, the analysis applies equally well to
radio frequency devices, provided only one polarization
mode is assumed.
Assume that a coherent monochromatic power Pr of
frequency _o in a particular polarization falls on a
photodetector of quantum efficiency rj. Assume that
a total incoherent power Po in a band of frequencies
centered at Vo also falls on the detector. The total
power Po may comprise a part Ps representing an incoherent
signal received from space, which we wish to
measure, and a part Pn representing receiver noise or
dark current-so that Po = Ps + In, We represent the
amplitude of the coherent wave as x_Ae i21r_o t and the
amplitudes of the components of the incoherent wave as
x,/2[al(t) + az(t) + ibl (t) + ib2(t)] ei2rrv°t, where the
subscript 1 indicates a component of the same polarization
as the coherent wave and the subscript 2 indicates a
component of orthogonal polarization. The a's and b's are
gaussian variables of zero mean and zero cross correlation.
The instantaneous power falling on the photodetector
is half the sum of the products of the total amplitudes
for each polarization, and each time phase, with
their complex conjugates, and is given by
P = [A +ax(t)l 2 +a22(t)+b12(t)+b22(t)
= A 2 + 2Aa, (t) + [a, 2(t) + a22(t) + b, _(t) + b22(t)]
= A 2 + 2Aal(t)+P i (D1)
where .P_Lis the instantaneous incoherent noise power;
that is, Pi = Po -
We take the probability per unit time of the emission
of a photoelectron to be proportional to P. The photo
current will thus contain shot noise, which we will
include later. For the present, while we are considering
fluctuation noise, we will deal in expected values and
write i = a_P, where t_ = rlq/hv. The average photocurrent
is thus from equation (D1)
i = Ir+I o = ot(Pr+Po) = _(Pr+Ps+Pn )
(D2)
If we are trying to detect the coherent power, only
Preceding pageblank ,8,