3.2 Optical Signal Dispersion Effects
113
(b) From Eq. (3.21)
σ s
L
≈
n 1
2
20
√
3 c
=
1.458(0.01)
2
20
√
3 × 3 × 10 8 m/s
= 14.0 ps/km
In graded-index fibers, careful selection of the radial refractive-index profile can
lead to bit rate-distance products of up to 1 Gb/s km.
3.2.3 Factors Contributing to Dispersion
This section briefly examines the various factors contributing to dispersion.
Sections 3.2.4–3.2.8 and Sect. 3.3 describe these factors in more detail.
As Sect. 2.3.2 notes, the z component of the wave propagation constant β is a
function of the wavelength or, equivalently, of the angular frequency ω. Because β
is a slowly varying function of this angular frequency, one can see where various
dispersion effects arise by expanding β in a Taylor series about a central frequency
ω 0 . Inserting such an expansion into the waveform equation, for example Eq. (2.1),
then shows the effects of variations in β due to modal dispersion and delay effects
on the frequency components of a pulse during its propagation along a fiber.
Expanding β to third order in a Taylor series yields
β(ω) ≈ β 0 (ω 0 ) + β 1 (ω 0 )(ω − ω 0 ) +
1
2
β 2 (ω 0 )(ω − ω 0 )
2
+
1
6
β 3 (ω 0 )(ω − ω 0 )
3
(3.22)
where β m (ω 0 ) denotes the mth derivative of β with respect to ω evaluated at ω =
ω 0 ; that is,
β m =
∂
m
β
∂ω m
ω=ω 0
(3.23)
Now consider the different components of the product βz, where z is the distance
traveled along the fiber. The resulting first term β 0 z describes a phase shift of the
propagating optical wave. From the second term of Eq. (3.22), the factor β 1 (ω 0 )z
produces a group delay τ g = z/V g , where z is the distance traveled by the pulse and
V g = 1/β 1 is the group velocity [see Eqs. (3.27) and (3.28)]. Assume β 1x and β 1y
are the propagation constants of the polarization components along the x-axis and
y-axis, respectively, of a particular mode. If the corresponding group delays of these
two polarization components are τ gx = z β 1x and τ gy = z β 1y in a distance z, then the
difference in the propagation times of these two modes
τ PMD = z
β 1x − β 1y
(3.24)
113
(b) From Eq. (3.21)
σ s
L
≈
n 1
2
20
√
3 c
=
1.458(0.01)
2
20
√
3 × 3 × 10 8 m/s
= 14.0 ps/km
In graded-index fibers, careful selection of the radial refractive-index profile can
lead to bit rate-distance products of up to 1 Gb/s km.
3.2.3 Factors Contributing to Dispersion
This section briefly examines the various factors contributing to dispersion.
Sections 3.2.4–3.2.8 and Sect. 3.3 describe these factors in more detail.
As Sect. 2.3.2 notes, the z component of the wave propagation constant β is a
function of the wavelength or, equivalently, of the angular frequency ω. Because β
is a slowly varying function of this angular frequency, one can see where various
dispersion effects arise by expanding β in a Taylor series about a central frequency
ω 0 . Inserting such an expansion into the waveform equation, for example Eq. (2.1),
then shows the effects of variations in β due to modal dispersion and delay effects
on the frequency components of a pulse during its propagation along a fiber.
Expanding β to third order in a Taylor series yields
β(ω) ≈ β 0 (ω 0 ) + β 1 (ω 0 )(ω − ω 0 ) +
1
2
β 2 (ω 0 )(ω − ω 0 )
2
+
1
6
β 3 (ω 0 )(ω − ω 0 )
3
(3.22)
where β m (ω 0 ) denotes the mth derivative of β with respect to ω evaluated at ω =
ω 0 ; that is,
β m =
∂
m
β
∂ω m
ω=ω 0
(3.23)
Now consider the different components of the product βz, where z is the distance
traveled along the fiber. The resulting first term β 0 z describes a phase shift of the
propagating optical wave. From the second term of Eq. (3.22), the factor β 1 (ω 0 )z
produces a group delay τ g = z/V g , where z is the distance traveled by the pulse and
V g = 1/β 1 is the group velocity [see Eqs. (3.27) and (3.28)]. Assume β 1x and β 1y
are the propagation constants of the polarization components along the x-axis and
y-axis, respectively, of a particular mode. If the corresponding group delays of these
two polarization components are τ gx = z β 1x and τ gy = z β 1y in a distance z, then the
difference in the propagation times of these two modes
τ PMD = z
β 1x − β 1y
(3.24)