316
8 Digital Optical Fiber Links
where the parameter q ranges between 0.5 and 1, and B 0 is the bandwidth of a 1-km
length of cable. A value of q = 0.5 indicates that a steady-state modal equilibrium has
been reached, whereas q = 1 indicates little mode mixing. Based on field experience,
a reasonable estimate is q = 0.7.
Another expression for N concatenated fibers that has been proposed for B M based
on curve fitting of experimental data, is
1
B M
=
N
n=1
1
B n
1/q
q
(8.8)
where the parameter q ranges between 0.5 (quadrature addition) and 1.0 (linear
addition), and B n is the bandwidth of the nth fiber section.
The next step is to find the relation between the fiber rise time and the 3-dB
bandwidth. First assume that the optical power emerging from the fiber has a Gaussian
temporal response described by
g(t) =
1
√
2πσ
e
−t
2 /2σ
2
(8.9)
where σ is the rms pulse width. The Fourier transform of this function is
G(ω) =
1
√
2π
e
−ω
2 σ
2 /2
(8.10)
From Eq. (8.9) the time t 1/2 required for the pulse to reach its half-maximum
value, that is, the time required to have
g
t 1/2
= 0.5g(0)
(8.11)
is given by
t 1/2 = (2 ln 2)
1/2
σ
(8.12)
Defining the time t FWHM as the full width of the pulse at its half-maximum
(FWHM) value, then yields
t FWHM = 2t 1/2 = 2σ (2 ln 2)
1/2
(8.13)
The 3-dB optical bandwidth B 3dB is defined as the modulation frequency f 3dB at
which the received optical power has fallen to 0.5 of the zero frequency value. Thus,
setting Eq. (8.10) equal to 0.5G(0) to find the 3-dB frequency and using Eq. (8.13),
the relation between the FWHM rise time t FWHM and the 3-dB optical bandwidth is
f 3d B = B 3d B =
0.44
t FW H M
(8.14)
8 Digital Optical Fiber Links
where the parameter q ranges between 0.5 and 1, and B 0 is the bandwidth of a 1-km
length of cable. A value of q = 0.5 indicates that a steady-state modal equilibrium has
been reached, whereas q = 1 indicates little mode mixing. Based on field experience,
a reasonable estimate is q = 0.7.
Another expression for N concatenated fibers that has been proposed for B M based
on curve fitting of experimental data, is
1
B M
=
N
n=1
1
B n
1/q
q
(8.8)
where the parameter q ranges between 0.5 (quadrature addition) and 1.0 (linear
addition), and B n is the bandwidth of the nth fiber section.
The next step is to find the relation between the fiber rise time and the 3-dB
bandwidth. First assume that the optical power emerging from the fiber has a Gaussian
temporal response described by
g(t) =
1
√
2πσ
e
−t
2 /2σ
2
(8.9)
where σ is the rms pulse width. The Fourier transform of this function is
G(ω) =
1
√
2π
e
−ω
2 σ
2 /2
(8.10)
From Eq. (8.9) the time t 1/2 required for the pulse to reach its half-maximum
value, that is, the time required to have
g
t 1/2
= 0.5g(0)
(8.11)
is given by
t 1/2 = (2 ln 2)
1/2
σ
(8.12)
Defining the time t FWHM as the full width of the pulse at its half-maximum
(FWHM) value, then yields
t FWHM = 2t 1/2 = 2σ (2 ln 2)
1/2
(8.13)
The 3-dB optical bandwidth B 3dB is defined as the modulation frequency f 3dB at
which the received optical power has fallen to 0.5 of the zero frequency value. Thus,
setting Eq. (8.10) equal to 0.5G(0) to find the 3-dB frequency and using Eq. (8.13),
the relation between the FWHM rise time t FWHM and the 3-dB optical bandwidth is
f 3d B = B 3d B =
0.44
t FW H M
(8.14)