126 Chapter 6 Heterodyne or Coherent Detection and Optical Amplification
(S/N)p = 1, becomes
This is one-half the value we
derived for direct or incoherent detection in the signal or photon-noise-
limited case. This may be understood if we realize that the if. signal
bandwidth, B, for any signal waveform is twice the bandwidth required
at baseband in direct detection, because of the upper and lower side-
bands about the center
frequency in the heterodyne case. In general
a pulse envelope of width
at radio or optical frequencies produces a
signal with a bandwidth given by
As we noted earlier, the the-
oretical heterodyne NEP may be achieved by sufficient local oscillator
power, such that the detection system is shot noise limited. Because the
i.f. power increases at the same rate as the noise j)Ower with increasing
iiQ, the noise of the amplifier stage can be made negligible.
6.2 Heterodyne Detection with Arbitrary Wavefronts
Thus far, we have treated a uniform plane wave and have also
neglected the relative polarizations of the signal and local oscillator
waves. We now treat the general case, using the configuration of Figure
with arbitrary waves producing electric fields on the detector plane
defined by
Es(x.y.t) = Rt[Es(x,y)eJ^st] ^^^^
ELo(^.y.t) = Rt[Eio(x,y)ei''^ot]
where the bold type indicates a spatial vector, and the fields
complex. Using this formulation, we may write the signal, local
oscillator, and intermediate frequency currents as
6.2 Heterodyne Detection with Arbitrary Wavefronts 127
I S^fV)
Figure 6.1 Detector plane with incident signal and local oscillator fields
which are the space-variant vector forms of Eqs. (6.1) and (6.2), with z^
the wave impedance of free space and t] assumed to be independent of
position in the detector plane. A more general derivation can be found
(6.3), we can then write
and using
we obtain the result of
(6.3) for the simple plane wave case.

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