3.1 Fiber Attenuation
103
Rayleigh scattering follows a characteristic λ
−4 dependence, so it decreases
dramatically with increasing wavelength, as is shown in Fig. 3.3. For wavelengths below about 1 μm it is the dominant loss mechanism in a fiber and gives
the attenuation-versus-wavelength plots their characteristic downward trend with
increasing wavelength. At wavelengths longer than 1 μm, infrared absorption effects
tend to dominate optical signal attenuation.
3.1.4 Fiber Bending Losses
Radiative losses occur whenever an optical fiber undergoes a bend of finite radius of
curvature [9, 10]. Fibers can be subject to two types of curvatures: (a) macroscopic
bends having radii that are large compared with the fiber diameter, such as those that
occur when a fiber cable turns a corner, and (b) random microscopic bends of the
fiber axis that can arise when the fibers are incorporated into cables.
Large-curvature radiation loss is known as macrobending loss or simply bending
loss. For slight bends the excess loss is extremely small and is essentially unobservable. As the radius of curvature decreases, the loss increases exponentially until at a
certain critical bend radius the curvature loss becomes observable. If the bend radius
is made a bit smaller once this threshold point has been reached, the losses suddenly
become extremely large.
The amount of optical radiation from a bent fiber depends on the field strength
outside of the fiber core and on the bending radius of curvature R. Because higherorder modes in a multimode fiber are bound less tightly to the fiber core than lowerorder modes, the higher-order modes will couple more strongly into the cladding
region when the fiber is bent and thus will radiate out of the fiber first. Thus the total
number of modes that can be supported by a curved fiber is less than in a straight
fiber. The following expression [11] has been derived for the effective number of
modes M eff that are guided by a curved multimode fiber of radius a:
M e f f = M ∞
1 −
α + 2
2αα
2a
R
+
3
2n 2 k R
2/3
(3.11)
where α defines the graded-index profile, is the core-cladding index difference, n 2
is the cladding refractive index, k = 2π/λ is the wave propagation constant, and
M ∞ =
α
α + 2
(n 1 ka)
2
(3.12)
gives the total number of modes in a straight fiber [see Eq. (2.81)].
Example 3.6 Consider a graded-index multimode fiber for which the index profile
α = 2.0, the core index n 1 = 1.480, the core-cladding index difference = 0.01,
103
Rayleigh scattering follows a characteristic λ
−4 dependence, so it decreases
dramatically with increasing wavelength, as is shown in Fig. 3.3. For wavelengths below about 1 μm it is the dominant loss mechanism in a fiber and gives
the attenuation-versus-wavelength plots their characteristic downward trend with
increasing wavelength. At wavelengths longer than 1 μm, infrared absorption effects
tend to dominate optical signal attenuation.
3.1.4 Fiber Bending Losses
Radiative losses occur whenever an optical fiber undergoes a bend of finite radius of
curvature [9, 10]. Fibers can be subject to two types of curvatures: (a) macroscopic
bends having radii that are large compared with the fiber diameter, such as those that
occur when a fiber cable turns a corner, and (b) random microscopic bends of the
fiber axis that can arise when the fibers are incorporated into cables.
Large-curvature radiation loss is known as macrobending loss or simply bending
loss. For slight bends the excess loss is extremely small and is essentially unobservable. As the radius of curvature decreases, the loss increases exponentially until at a
certain critical bend radius the curvature loss becomes observable. If the bend radius
is made a bit smaller once this threshold point has been reached, the losses suddenly
become extremely large.
The amount of optical radiation from a bent fiber depends on the field strength
outside of the fiber core and on the bending radius of curvature R. Because higherorder modes in a multimode fiber are bound less tightly to the fiber core than lowerorder modes, the higher-order modes will couple more strongly into the cladding
region when the fiber is bent and thus will radiate out of the fiber first. Thus the total
number of modes that can be supported by a curved fiber is less than in a straight
fiber. The following expression [11] has been derived for the effective number of
modes M eff that are guided by a curved multimode fiber of radius a:
M e f f = M ∞
1 −
α + 2
2αα
2a
R
+
3
2n 2 k R
2/3
(3.11)
where α defines the graded-index profile, is the core-cladding index difference, n 2
is the cladding refractive index, k = 2π/λ is the wave propagation constant, and
M ∞ =
α
α + 2
(n 1 ka)
2
(3.12)
gives the total number of modes in a straight fiber [see Eq. (2.81)].
Example 3.6 Consider a graded-index multimode fiber for which the index profile
α = 2.0, the core index n 1 = 1.480, the core-cladding index difference = 0.01,