Page 410 - Satellite Communications, Fourth Edition

390 Chapter Twelve

Here, wavelength rather than frequency is used in the equation as this
is the quantity usually specified for a laser, and of course r and l must
be in the same units.
The intensity distribution of a laser beam generally follows what is
termed a Gaussian law, for which the intensity falls off in an exponen-
tial manner in a direction transverse to the direction of propagation. The
beam radius is where the transverse electric field component drops to
1/e of its maximum value, where e ≈ 2.718. The diameter of the beam
(twice the radius) gives the total beamwidth. The on-axis gain (similar
to the antenna gain defined in Sec. 6.6 is given by (Maral et al., 2002)

32
G (12.64)
T

2 T
where
is the total beamwidth.
T
On the receive side, the telescope aperture gain is given by:
2
a D b
G R (12.65)
l
where D is the effective diameter of the receiving aperture.
The optical receiver will receive some amount of optical power P . The
R
energy in a photon is hc/l, where h is Plank’s constant (6.6256 10 34
8
J-s) and c is the speed of light in vacuum (approximately 3 10 m/s).
For a received power P the number of photons received per second is
R
therefore P l/hc. The detection process consists of photons imparting
R
sufficient energy to valence band electrons to raise these to the con-
duction band. The quantum efficiency of a photo-diode is the ratio (aver-
age number of conduction electrons generated)/(average number of
photons received). Denoting the quantum efficiency by , the average
number of electrons released is P l/hc and the photo current is:
R
q P l
R
(12.66)
I ph
hc
where q is the electron charge. The responsivity of a photodiode is defined
as the ratio of photo current to incident power. Denoting responsivity
by R and evaluating the constants in Eq. (12.66) gives
0
l
R with l in m
0
1.24 (12.67)
The energy band gap of the semiconductor material used for the pho-
todiode determines the wavelengths that it can respond to. The require-
ment in general is that the bandgap energy must be less than the photon