|Figure 1: Irradiance cross-section of beam
after propagating 1500 m.
Interest in high-data-rate
free-space optical (FSO) laser communication systems has grown significantly
in recent years because of the advantages offered by FSO systems over
radio frequency (RF) systems. Most advantages are simple consequences
of the short wavelengths associated with optical waves. However, there
are some drawbacks that arise from the shorter wavelengths used in FSO
systems. Among the advantages and disadvantages are the following:
Atmospheric factors are the most serious drawback to FSO systems. Optical
turbulence resulting from small temperature variations gives rise to
power losses from spreading of the beam beyond that due to diffraction
alone, and to temporal and spatial fluctuations of the laser beam known
as scintillation. In addition, we may have to account for beam-wander-induced
scintillation in a focused beam on a horizontal path or in a collimated
beam on an uplink path to space.
In Fig. 1 we illustrate what the irradiance cross-section of a laser
beam may look like after propagating 1500 m through extended atmospheric
turbulence along a horizontal path close to the ground. The dark spots
correspond to a potential fade that may exist for a receiver placed
in this position.
An optical wave propagating through atmospheric turbulence will experience
irradiance fluctuations (called scintillation) due to small index of
refraction fluctuations (i.e., optical turbulence). Theoretical and
experimental studies of irradiance fluctuations generally center around
the scintillation index
where the quantity I denotes irradiance (intensity) of the optical wave
and the angle brackets < > denote an ensemble average, or equivalently,
a long-time-average. For constant values of the refractive-index structure
parameter, it is known that the scintillation index increases with increasing
path length until it reaches a maximum value greater than unity in the
regime characterized by random focusing (see Fig. 2). With increasing
path length, the focusing effect is weakened by multiple scattering
and the fluctuations slowly begin to decrease, saturating at a level
for which the scintillation index approaches unity from above. Saturation
occurs because multiple scattering causes the optical wave to become
increasingly less coherent as it propagates. The theory that we use
in this paper is based on recently developed theory contained in Refs.
 and . Comparisons of this theory with outdoor experimental data
and simulation results justify its use under all conditions of optical
|Figure 2: Scintillation index.
|Figure 3: Direct detection system.
Optical receivers are broadly divided into two basic
typesXdirect (or power) detecting receivers and coherent receivers.
The simplest type of optical receiver for implementation is a power
detecting receiver (see Fig. 3). The optical field is always photodetected
in the presence of various noise sources, e.g., background radiation,
detector noise, and circuit and electronic thermal noise. Increasing
the collecting lens aperture diameter beyond the irradiance correlation
width of the received optical wave not only increases the average signal
level, but decreases the irradiance fluctuation level through a process
called aperture averaging.
|Figure 4: Probability of fade curves.
An important parameter used to characterize system performance is the
receiver signal-to-noise ratio (SNR). False alarm and fade probability
both involve the notion of SNR. The reliability of an FSO system operating
in the atmosphere can be deduced from the probability density function
(PDF) of the irradiance signal. Two models commonly used for this purpose
are the lognormal PDF and the recently developed gamma-gamma PDF. However,
the lognormal model is limited to weak fluctuations, and even then it
generally predicts overly optimistic results.
In Fig. 4 we show the probability of fade versus the mean SNR for the
case of a spherical wave with specified Rytov variance . In this case
we have included noise in the receiver and used conditional statistics.
Also, is a normalized aperture diameter and the false alarm rate (FAR)
per bandwidth is specified at .
In digital transmission the desired message is converted to binary symbols
(bits) and transmitted as a modulated optical field. The performance
measure in such systems is commonly provided by the probability of error,
also called the bit error rate (BER). The most basic form of pulsed
modulation in binary direct detection receivers is on-off keying (OOK).
|Figure 5: Mean BER for on-off keying (OOK).
In the presence of optical turbulence, the probability of error is considered
a conditional probability that must be averaged over the PDF of the
random signal to determine the unconditional BER. We show the mean BER
in Fig. 5 as a function of mean SNR for the case of a spherical wave
under various conditions. The case corresponds to weak irradiance fluctuations
and we have taken the receiver normalized aperture diameter .For wavelength
m and , this corresponds to a path length just over 300 m and receiver
aperture = 3.2 cm. For the moderate irradiance fluctuation case , we
plot the probability of fade for two normalized aperture sizes corresponding
to and . Under the conditions specified above, the path length would
be roughly 1 km and aperture diameters and 40 cm, respectively.
|Figure 6: Scintillation index of a focused
beam on a horizontal path. The dashed curve is based on conventional
first-order Rytov theory.
Along a horizontal path, beam wander-induced scintillation may not be
a problem for a collimated or divergent beam. However, beam wander effects
have to be accounted for if the beam is focused. Recent modeling 
has shown that beam wander causes an effective pointing error in the
scintillation index of an untracked beam that leads to larger values
of the scintillation index than that predicted by the Rytov theory.
The amount of reduction in the scintillation index when the beam is
tracked depends on the tracking method. For example, in Fig. 6 we show
recent simulation data (courtesy of G. J. Baker and R. Parenti) of a
beam focused at the receiver for both the untracked beam and tilt-removed
beam along with theoretical results based on weak fluctuation theory.
|Figure 7: Uplink scintillation index.
|Figure 8: Probability of fade curves.
For an uplink collimated beam to space we must account for beam wander
effects when the beam is untracked. In Fig. 7 we illustrate recent simulation
results (courtesy of G. J. Baker) for the on-axis scintillation index
as a function of beam radius . Also shown are theoretical curves based
on newly developed beam wander theory (solid curve) and conventional
Rytov theory (dashed curve). If the beam is tracked, the on-axis scintillation
index more closely matches that of the Rytov theory. Lastly, in Fig.
8 we illustrate the effect of beam wander on the fade probability of
an uplink collimated beam to a satellite in geostationary orbit. The
beam radius at the transmitter is 10 cm and the wavelength is 1.55 microns.
Both tracked and untracked beam fade probabilities are illustrated at
zenith angles of 0 deg and 45 deg. Detector noise is neglected.
 L. C. Andrews and R. L. Phillips, Laser Beam Propagation through
Random Media, 2nd ed. (SPIE Press, 2005).
 L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation
with Applications (SPIE Press, 2001).
 L. C. Andrews, et al., “Beam wander effects on the scintillation
index of a focused beam,” SPIE 5793 (2005), to appear.