4752 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 12, DECEMBER 2006
In this paper, we will present new results for the fundamental
limits of the vacuum photon-counting channel. These results are
motivated by the notion that the relevant channel capacity of in-
terest to the system designer is the capacity-per-unit transmitter
energy at a target data rate (and not the capacity per photon
received at low rates). Typically, the capacity of the photon-
counting channel is expressed in terms of rate per photon, and
using that metric, the capacity limit indeed is infinite. However,
in that limit, the photon energy also has to be infinite. We
will show that the capacity-per-unit transmitter energy (not per
photon) at a given rate is finite (albeit still very large) and
not infinite. In practical systems, only a small fraction of the
carrier frequency is available for communications due to device
limitations. The capacity in that case is naturally also finite.
We will also discuss the use of spatial, temporal, frequency,
and polarization diversity to mitigate clear-air-turbulence ef-
fects on free-space optical links and networks. We will suggest
how coherent systems may provide significant performance
gains for turbulence mitigation as well as for multiple-access
optical aggregation. These technology and architecture devel-
opments are critical for the economical realization of free-
space optical-communication systems and networks, and their
success will have a profound transforming effect on free-space
system architectures and applications.
II. U
LTRAHIGH-EFFICIENCY OPTICAL-LINKS
PHOTON-COUNTING CHANNELS
A. Potential System Improvements With Highly Efficient
Optical Links (for the Vacuum-Deterministic Channel)
In [1] and [2], we suggested that if the optical-
communication link sensitivity is increased, substantial sys-
tem improvements are possible. The complexity of the
optomechanical–thermal system can be reduced by using
smaller telescopes thereby reducing the telescope fabrication
cost. The power consumption of the thermal-control system
can be reduced as a result of less complex hardware to be
kept within a close temperature range. The spatial-tracking
system bandwidth can be lowered due to the broader beam
divergence and, thus, more tolerant to tracking errors (perhaps
to the point that even the use of one steering mirror, instead
of a cascade of coarse and fine tracking mirrors, is sufficient).
Spatial-acquisition times can be reduced also since the beam
divergences are larger.
B. Fundamental Limit of Photon-Counting Channels
In her 1968 Massachusetts Institute of Technology (MIT)
doctoral thesis [66], Liu has derived the capacity of the photon-
counting channel for M-ary “orthogonal” signaling (these
signals, such as pulse-position modulation (PPM), are not
orthogonal-quantum mechanically), and it was evident that the
noiseless channel is singular and has infinite capacity per pho-
ton. Later, in the early 1970s, Yuen et al. [67], [68] found the
necessary and sufficient conditions for the quantum optimum
receiver (OC) for “orthogonal” signals and specifically ana-
lyzed the PPM case. All the above works [66]–[68] treated the
general case of communication in the presence of background
noise, and the noiseless case is merely a limiting case of the
general problem. In 1980, the author also treated the problem
of soft-decision decoding for the semiclassical photon-counting
channel with arbitrary (including none) amount of background
noise [63]. This paper was later extended in [60]–[62]. In 1981,
Massey treated the simplified noiseless-PPM photon-counting
channel specifically and explicitly considered the channel as
an erasure channel, [69]. The optical channel is different from
any classical channel in the sense that quantum effects are often
the performance-limiting mechanisms. Thus, we cannot accept
all established classical communication and information theory
results as being true in optical channels but have to reexamine
each case with quantum mechanics in mind. There is a long-
standing result for communication in white Gaussian noise
with unlimited bandwidth that capacity is achieved with block-
orthogonal signaling (simplex signal is a variant using slightly
less energy) and the reliability function for large symbol-size
block-orthogonal signaling gives the capacity of the channel.
Though the solution of the general quantum M-ary-detection
problem is still outstanding, there is no reason to believe, in the
unlimited bandwidth case, “orthogonal” signals such as PPM
are not near optimum if not optimum signal sets. For example,
in [65], the capacity of the noiseless multiPPM is shown to
have a capacity less than that for the orthogonal single PPM-
signal set.
When the optical-communication system is not staring at a
strong noise source, such as the sun and earth, the channel is
not background-noise limited and exhibits quantum behavior
[1], [2]. For direct detection, photon counting is the realization
of the number-state operator or the quantum-energy observable
[66]–[68]. In the past, it has been very difficult to implement a
photon-counting receiver in space using photo-multiplier tubes.
Recent development at the MIT campus and MIT Lincoln
Laboratory in solid-state photon-counting detectors, such as the
Geiger-mode avalanche photodetectors and super-conducting
solid-state detectors, have made photon-counting receivers vi-
able in space applications [5], [70]–[74]. While a coherent
detection system behaves like a classical additive white-
Gaussian-noise channel (but the noise is quantum in origin), the
photon-counting receiver, as we have alluded to, is singular and
is known to have “infinite” capacity per photon [2]–[6]. While
that limit is correct in principle, it is achieved when very large
symbol-size PPM is used at the expense of very low spectral
efficiency. One of the most important free-space optical-system
parameters is the average power of the transmitter-power am-
plifier, and thus, the relevant quantity to optimize upon is not
the capacity per photon but the capacity-per-unit energy. The
solutions to these two optimization problems are different, be-
cause as the bandwidth of the modulation increases, the center
frequency of the optical field must also increase and induces a
corresponding increase in the energy per photon. Thus, infinite
capacity per photon is attained when the photon energy also
goes to infinity. Nonetheless, arbitrary high efficiency can still
be achieved, if we allow the transmission data rate to become
vanishing small. However, for a specific target transmission
rate, the optimum efficiency of the direct-detection photon-
counting channel is not known. In the following, we will
formulate the optimization problem under the aforementioned