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Optical Sources, Detectors, and Systems by Robert H. Kingston

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124 Chapter
6
Heterodyne or Coherent Detection and Optical Amplification
mined by the diffraction-limited field of view of the receiver aperture.
This should be contrasted with direct detection where the angular field
of view is determined by the ratio of the detector dimension to the focal
length and is independent of the aperture size.
In our pursuit of high-quantum-efficiency detection systems that are
limited only by signal or photoevent shot noise, we have found that the
photomultiplier and to a limited extent the silicon APD meet this cri-
terion, but are unfortunately limited to wavelengths shorter than about
0.8
ixm.
Both these devices utilize photoelectron multiplication to over-
come the noise in the following amplifier stage. An alternative to this
approach is to go backward in the detection process and amplify the
optical signal
before
photodetection. This may be accomplished using a
laser medium such as an injection-pumped semiconductor or an opti-
cally pumped impurity-doped solid such as erbium-doped glass. The
optical amplification process is also coherent since, as discussed in
Chapter 2, the stimulated wave is both a directional and a phase replica
of the incident wave. Because of this coherent behavior, we find it
advantageous to use heterodyne concepts in our treatment of optical am-
plifiers. In fact, we find similar behavior to the extent that optical
amplifiers operate most efficiently with a diffraction-limited
field-of-
view and a signal spectral bandwidth matched to the system optical
bandwidth.
6.1 Heterodyne: Simple Plane Wave Analysis
We first treat a simple heterodyne system involving codirectional
plane waves, then we extend our treatment to arbitrary wave shapes to
establish the directional sensitivity of a heterodyne detection system.
We consider two aligned plane waves normally incident on a flat
photodetector surface, the local oscillator, with power P^^ and electric
field E^Q, and the other a weak signal with power P^ and electric field
E^ The detected current is given by i(t) = riqP(t)/hv, where P(t) is in
turn given by E^(t)A/z^, where A is the photodetector area and z^is the
wave impedance of free space. The local oscillator power at radian
frequency
CO^Q
will "beat" with the signal power at
(O^
yielding a
sinusoidal power variation at the intermediate or difference frequency,
ft).^ Mathematically, we may write
6.1 Heterodyne: Simple Plane Wave Analysis 125
E^(t) =
[ESCOS(
(Dst + 0) -^
Eiocoscoiot]
7
7
^--Elll-h
cosKcost +
0)J
+
-
EIO(1
+
coslcoiot)
2 2 (6.1)
+
EiQEs{C0S[((0st 4- 0) -/-
0)2^0^-'
+
COS[(COst
+ 0) - COiot]}
7
7
=>
^
E|
+ 2 ^^o
+ EioEsCos(cOift +
0)
where in the last line we have dropped all terms at optical frequencies
such as Icos, ^^^v since the detector cannot respond at such high fre-
quencies. Since fi>..«
co^
we can use a mean frequency, v ~ co^ln, and
write
id)
=
is
+
I'LO
+ Ay
^^^ ^(f ^ ^^'^
is=mPs/hv
ko =mPLolhv
and we note that is« iu « iiQand the i.f. current is proportional to
the
square root
of the signal current, that is, to the optical electric field.
We have here assumed that the local oscillator is
single frequency
with no
phase or
amplitude fluctuations so that the i.f. current becomes a replica of
the signal field in amplitude and phase. This local oscillator criterion re-
quires a laser source and if the laser frequency or phase fluctuates, so
does that of the output signal. Ideally, however, we have performed a
coherent measurement of the incoming optical signal. The noise current
produced in the detection process is again shot noise but it is now pro-
duced by the local oscillator current and can be increased to such a level
that it overwhelms the noise in the following resistor/amplifier system.
Specifically we may write
in
=
^cjko^^ iff
=
(2^|kois
)^
(^os^
^if^) = ^koh
S^ _ is' _ 2iiois _ is _TjPs ^^-^^
<N)p i^2 ZqiioB qB hvB
and we find that the heterodyne NEP, the required signal power for

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