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Chapter 3
Signal Degradation in Optical Fibers
 3.1 Attenuation
 3.1.2 Absorption
 3.1.3 Scattering Losses
 3.1.4 Bending Losses
 3.2 Signal Distortion in Fibers
 3.2.1 Information Capacity Determination
 3.2.2 Group Delay
 3.2.3 Material Dispersion
 3.2.4 Waveguide Dispersion
 3.2.5 Signal Distortion in SMFs
 3.2.6 Polarization-Mode Dispersion
 3.2.7 Intermodal Distortion
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光纖通訊實驗室 黃振發教授 編撰
3.1 ATTENUATION
 Attenuation mechanisms:
 Absorption -- related to the fiber material;
 Scattering -- associated with fiber material and
waveguide structure;
 Radiative losses -- originates from perturbations
of the fiber geometry.
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光纖通訊實驗室 黃振發教授 編撰
3.1 ATTENUATION
 The power P(z) at a distance z down the fiber is
P(z) = P(0) exp(-apz)
(3-1a)
where
ap = (1/z)ln[P(0)/P(z)]
(3-1b)
is the fiber attenuation coefficient.
 In units of dB/km, the attenuation coefficient a can
be expressed as
a(dB/km) = (10/z)log[P(0)/P(z)]
= 4.343 ap (km-1)
(3-1c)
 This parameter is referred to as the fiber attenuation.
It is a function of the wavelength, as is illustrated in
Fig. 3-1.
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光纖通訊實驗室 黃振發教授 編撰
3.1 ATTENUATION
Figure 3-1. Fiber attenuation as a function of wavelength yields nominal
values 0.5dB/km at 1300nm and 0.3dB/km at 1550nm for standard singlemode fiber (solid curve). This fiber shows an attenuation peak around 1400
nm resulting from absorption by water molecules. The dashed curve is for a
water-free AllWave® fiber.
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光纖通訊實驗室 黃振發教授 編撰
3.1.2 Absorption
 Absorption mechanisms:
1. Absorption by atomic defects in the glass composition.
2. Extrinsic absorption by impurity atoms in the glass material.
3. Intrinsic absorption by constituent atoms of the fiber material.
 The basic response of a fiber to ionizing radiation is an
increase in attenuation owing to the creation of atomic
defects, or attenuation centers, that absorb optical
energy.
 The higher the radiation level, the larger the attenuation,
as Fig. 3-2a illustrates. The attenuation centers will relax
or anneal out with time, as shown in Fig. 3-2b.
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光纖通訊實驗室 黃振發教授 編撰
3.1 ATTENUATION
Figure 3-2. Effects of ionizing on fiber attenuation.
(a) Loss increase during steady irradiation.
(b) Subsequent recovery after radiation has stopped.
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3.1.2 Absorption
 Extrinsic absorption :
 Impurity absorption results predominantly from
transition metal ions and from OH (water) ions. The
transition metal impurities cause losses from 1 ~ 10
dB/km. The impurity levels in vapor-phase deposition
processes are usually 1 ~ 2 orders of magnitude lower.
 Early optical fibers had high levels of OH ions which
resulted in large absorption peaks occurring at 1400,
950, and 725 nm. These are the 1st, 2nd, and 3rd
overtones, respectively, of the fundamental absorption
peak of water near 2.7 mm, as shown in Fig. 1-7.
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3.1.2 Absorption
 By reducing the residual OH content of fibers to
around 1 ppb (part per billion), SMFs have nominal
attenuations of 0.5 dB/km in the 1300-nm window
and 0.3 dB/km in the 1550-nm window, as shown by
the solid curve in Fig. 3-1.
 An effectively complete elimination of water
molecules from the fiber results in the dashed curve
shown in Fig. 3-1.
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3.1.2 Absorption
 Intrinsic absorption :
1). Electronic absorption bands in the ultraviolet region;
2). Atomic vibration bands in the near-infrared region.
 The ultraviolet edge of the electron absorption
bands follows the Urbach's rule:
auv = C exp(E/Eo) .
(3-2a)
Here, C and Eo are empirical constants and E is the
photon energy.
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光纖通訊實驗室 黃振發教授 編撰
3.1.2 Absorption
 The characteristic decay of the ultraviolet absorption
is shown in Fig. 3-3.
 Since E is proportional to 1/l, ultraviolet absorption
functions with wavelength l and the mole fraction x
of GeO2 as
(3-2b)
 The ultraviolet loss is small compared with scattering
loss in the near-infrared region.
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3.1.2 Absorption
Figure 3-3. Optical fiber attenuation characteristics
and their limiting mechanisms for a GeO2-doped
low-loss low-OH silica fiber.
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3.1.2 Absorption
 In the near-infrared region above 1.2mm, the
infrared absorption for GeO2-SiO2 glass is
(3-3)
 Figure 3-4 indicates that for operation at longer
wavelengths GeO2-doped fiber material is the most
desirable.
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3.1.2 Absorption
Figure 3-4. A comparison of the infrared absorption
induced by various doping materials in low-loss silica
fibers.
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3.1.3 Scattering Losses
 Rayleigh scattering loss at a wavelengths l resulting
from density fluctuations can be approximated by
(3-4)
 Here, n is the refractive index, kB is Boltzmann's
constant, bT is the isothermal compressibility of the
material, and Tf is the temperature at which the
density fluctuations are frozen into the glass.
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3.1.3 Scattering Losses
 Rayleigh scattering follows a characteristic l-4
dependence, it decreases dramatically with
increasing l, as is shown in Fig. 3-3.
 For l < 1 mm, ultraviolet absorption is the dominant
loss in a fiber and gives the ascat versus l plots the
characteristic downward trend with increasing l.
 At l > 1 mm, infrared absorption effects tend to
dominate optical signal attenuation.
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3.1.3 Scattering Losses
 Combining the infrared, ultraviolet, and scattering
losses, we get the results shown in Fig. 3-5 for MMFs
and Fig. 3-6 for SMFs.
 Multimode fibers are subject to higher-order-mode
losses owing to perturbations at the core-cladding
interface.
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3.1.3 Scattering Losses
Figure 3-5. Typical spectral attenuation range
for production graded-index multimode fibers.
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3.1.3 Scattering Losses
Figure 3-6. Typical spectral attenuation range
for production-run single-mode fibers.
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3.1.4 Bending Losses
 Fibers can be subject to two types of bends:
(a) Macroscopic bends having larger radii compared
with the fiber diameter;
(b) Microscopic bends of fiber axis arise when fibers
are incorporated into cables.
 The amount of optical radiation from a bent fiber
depends on the field strength at xc and on the radius
of curvature R.
 Since higher-order modes are bound less tightly to
the fiber core than lower-order modes, the higherorder modes will radiate out of the fiber first .
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3.1.4 Bending Losses
 The total number of modes that can be supported by a curved
fiber is less than in a straight fiber.
 Gloge has derived the following expression for the effective
number of modes Neff that are guided by a curved multimode
fiber of radius a:
(3-7)
 where a defines the graded-index profile, D is the core-cladding
index difference, n2 is the cladding refractive index, k = 2p/l is
the wave propagation constant, and
(3-8)
is the total number of modes in a straight fiber.
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3.1.4 Bending Losses
Figure 3-8. Small-scale fluctuations in the radius of curvature
of the fiber axis leads to microbending losses. Microbends can
shed higher-order modes and can cause power from loworder modes to couple to higher-order modes
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3.1.4 Bending Losses
 Microbends are repetitive small-scale fluctuations in
the radius of curvature of the fiber axis, as is illustrated
in Fig. 3-8.
 An increase in attenuation results from microbending
because the fiber curvature causes repetitive coupling
of energy between the guided modes and the leaky
modes in the fiber.
 One method of minimizing microbending losses is by
extruding a compressible jacket over the fiber. When
external forces are applied to this configuration, the
jacket will be deformed but the fiber will tend to stay
relatively straight, as shown in Fig. 3-9.
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3.1.4 Bending Losses
Figure 3-9. A compressible jacket extruded over
a fiber reduces microbending resulting from
external forces.
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3.1.4 Bending Losses
 For a multimode graded-index fiber having a core
radius a, outer radius b, and index difference D, the
microbending loss aM of a jacketed fiber is reduced
from that of an unjacketed fiber by a factor
(3-9)
Here, Ej and Ef are the Young's moduli of the jacket
and fiber, respectively.
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3.2 Signal Distortion in Fibers
 Intramodal dispersion (Chromatic dispersion) is pulse
spreading that arises from the finite spectral emission
width of an optical source.
 This is also known as group velocity dispersion (GVD),
since the dispersion is a result of the group velocity
being a function of the wavelength.
 Because intramodal dispersion depends on the
wavelength, its effect on signal distortion increases
with the spectral width of the optical source.
 It is normally characterized by the rms spectral width
al (see Fig. 4-12).
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3.2 Signal Distortion in Fibers
Figure 4-12. Spectral emission pattern of a
Ga1-xAlxAs LED with x = 0.08.
The width of the spectral pattern is 36 nm.
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3.2 Signal Distortion in Fibers
 For LEDs the rms spectral width is approximately
5% of a central wavelength.
 A typical LED source spectral width is ~45-nm when
the peak emission wavelength is 850-nm.
 Laser diode optical sources have much narrower
spectral widths, with typical values being 1-2 nm for
multimode lasers and 10-4 nm for single-mode lasers.
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3.2 Signal Distortion in Fibers
Main causes of intramodal dispersion:
1. Material dispersion, which arises from the variation
of the refractive index of the core material as a
function of wavelength.
 This causes a wavelength dependence of the group
velocity of any given mode; that is, pulse spreading
occurs even when different wavelengths follow the
same path.
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光纖通訊實驗室 黃振發教授 編撰
3.2 Signal Distortion in Fibers
2. Waveguide dispersion, which occurs because a singlemode fiber confines only about 80% of the optical
power to the core.
 Dispersion thus arises, since the 20% of the light
propagating in the cladding travels faster than the
light confined to the core.
 The amount of waveguide dispersion depends on the
fiber design, since the modal propagation constant b
is a function of a/l.
 The other factor giving rise to pulse spreading is
intermodal delay, which is a result of each mode
having a different value of the group velocity at a
single frequency.
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3.2.1 Information Capacity
Determination
 A result of the dispersion-induced signal distortion is
that a light pulse will broaden as it travels along the
fiber.
 As shown in Fig. 3-10, this pulse broadening will
cause a pulse to overlap with neighboring pulses.
 The dispersive properties determine the limit of the
information capacity of the fiber, specified by the
bandwidth-distance product in MHz.km.
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3.2.1 Information Capacity
Determination
Figure 3-10. Broadening and attenuation of two adjacent pulses as
they travel along a fiber. (a) Originally, the pulses are separate;
(b) the pulses overlap slightly and are clearly distinguishable;
(c) the pulses overlap significantly and are barely distinguishable;
(d) eventually, the pulses strongly overlap and are indistinguishable.
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3.2.1 Information Capacity
Determination
 For a step-index fiber the various distortion effects
tend to limit the bandwidth-distance product to
about 20-MHz.km.
 In graded-index fibers the radial refractive-index
profile can be carefully selected so that pulse
broadening is minimized at a specific operating
wavelength. This had led to bandwidth-distance
products as high as 2.5-GHz.km.
 A comparison of the information capacities of
various optical fibers with the capacities of typical
coaxial cables used for UHF and VHF transmission is
shown in Fig. 3-11. The curves are shown in terms of
signal attenuation versus data rate.
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3.2.1 Information Capacity
Determination
Figure 3-11. A comparison of the attenuation as
a function of frequency or data rate of various
coaxial cables and several types of high-bandwidth
optical fibers.
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3.2.2 Group Delay
 As the optical signal propagates along the fiber, each spectral
component can be assumed to travel independently, and
to undergo a group delay per unit length given by
(3-13)
 Here, L is the distance traveled by the pulse, b is the
propagation constant along the fiber axis, k = 2p/l, and
the group velocity
(3-14)
is the velocity at which the energy in a pulse travels along
a fiber.
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3.2.2 Group Delay
 If the spectral width of the optical source is not too
wide, the delay difference per unit wavelength along
the propagation path is approximately dtg/dl.
 For spectral components which are dl apart and
which lie dl/2 above and below a central wavelength
l0, the total delay difference dt over a distance L is
(3-15a)
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3.2.2 Group Delay
 In terms of the angular frequency w, this is written
as
(3-15b)
The factor b2 ≡ d2b/dw2 is the GVD parameter, which
determines how much a light pulse broadens as it
travels along an optical fiber.
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3.2.2 Group Delay
 If the spectral width sl of an optical source is characterized by
its rms value (see Fig. 4-12), then the pulse spreading can be
approximated by the rms pulse width,
(3-16)
 The dispersion factor
(3-17)
defines the pulse spread as a function of wavelength and is
measured in ps/(nm.km).
 To a good approximation, D can be written as the sum of the
material dispersion Dmat and the waveguide dispersion Dwg.
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3.2.2 Group Delay
Figure 4-12. Spectral emission pattern of a Ga1-xAlxAs LED
with x = 0.08. The width of the spectral pattern is 36-nm.
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光纖通訊實驗室 黃振發教授 編撰
3.2.3 Material Dispersion
 Exemplified in Fig. 3-12 for silica, various spectral
components of a given mode travel at different speeds.
 Material dispersion is an intramodal dispersion effect,
and is of importance for single-mode waveguides and
for LED system (since an LED has a broader output
spectrum than a laser diode).
 Consider a plane wave propagating in a dielectric
medium having refractive index n(l) equal to that of
the fiber core. The propagation constant b is given as
b = 2pn(l) / l
(3-18)
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3.2.3 Material Dispersion
 Substituting this b into Eq. (3-13) with k = 2p/l yields
the group delay tmat resulting from material
dispersion:
tmat = (L/c)(n - l.dn/dl)
(3-19)
 Using Eq. (3-16), the pulse spread smat for a source of
spectral width sl is found by differentiating tmat with
respect to l and multiplying by sl to yield
(3-20)
where Dmat(l) is the material dispersion.
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3.2.3 Material Dispersion
 A plot of the material dispersion for unit length L
and unit optical source spectral width sl is given in
Fig. 3-13 for the silica material.
 Material dispersion can be reduced either by
choosing sources with narrower spectral output
widths (reducing sl) or by operating at longer
wavelengths.
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3.2.3 Material Dispersion
Figure 3-12. Variations in the index of refraction
as a function of the optical wavelength for silica.
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3.2.3 Material Dispersion
Figure 3-13. Material dispersion as function of
optical wavelength for pure silica and 13.5%
GeO2 / 86.5% SiO2.
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3.2.4 Waveguide Dispersion
 Group delay is the time required for a mode to travel
along a fiber of length L.
 The group delay can be expressed in terms of the
normalized propagation constant b defined as
(3-21)
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3.2.4 Waveguide Dispersion
 For small values of the index difference D = (n1 - n2)/n1,
Eq. (3-21) can be approximated by
b = (b/k - n2) / (n1 - n2)
(3-22)
Solving Eq. (3-22) for b, we have
b = n2k(bD + 1)
(3-23)
 With the above expression for b and with n2 not a
function of wavelength, the group delay twg arising
from waveguide dispersion can be expressed as
(3-24)
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3.2.4 Waveguide Dispersion
 Using the approximation to write the group delay in
Eq. (3-24) in terms of V instead of k will yield
(3-25)
The factor d(Vb)/dV is plotted in Fig. 3-14 as a
function of V for various LP modes.
 For MMFs the waveguide dispersion is generally very
small compared with material dispersion and can be
neglected.
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3.2.4 Waveguide Dispersion
Figure 3-14. The group delay arising from waveguide dispersion
as a function of the V number for a step-index optical fiber. The
curve numbers jm designate the LPjm modes.
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3.2.5 Signal Distortion in SMFs
 For SMFs, waveguide dispersion can be of the same
order of magnitude as material dispersion.
 The pulse spread swg is obtained from the derivative
of the group delay with respect to wavelength:
(3-26)
where Dwg (l) is the waveguide dispersion.
 For the lowest-order mode (i.e., the HE11 mode or,
equivalently, the LP01 mode), the normalized
propagation constant is
(3-27b)
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3.2.5 Signal Distortion in SMFs
 Figure 3-15 shows plots of this expression for b and
its derivatives d(Vb)/dV and Vd2(Vb)/dV2 as functions
of V.
 Figure 3-16 gives examples of material and
waveguide dispersion for a SMF having V = 2.4.
 Waveguide dispersion is important around 1320-nm.
At this point, the two dispersion factors cancel to
give a zero total dispersion.
 Material dispersion dominates waveguide dispersion
at shorter and longer wavelengths; for example, at
900-nm and 1550-nm.
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3.2.5 Signal Distortion in SMFs
Figure 3-15. The waveguide parameter b and its
derivatives d(Vb)/dV and Vd2(Vb)/dV2 plotted as
a function of the V number for the HE11 mode.
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3.2.5 Signal Distortion in SMFs
Figure 3-16. Examples of the material and waveguide
dispersions as a function of optical wavelength for a
single-mode fused-silica-core fiber.
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3.2.6 Polarization-Mode Dispersion
 As shown in Fig. 3-17, two orthogonal polarization
modes travel at a slightly different velocity and the
polarization orientation rotate with distance.
 The resulting difference in propagation time between
the two orthogonal polarization modes will result in
polarization-mode dispersion (PMD).
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3.2.6 Polarization-Mode Dispersion
 If the group velocities of the two orthogonal
polarization modes are vgx and vgy, then the
differential time delay Dtpol between the two
polarization components during propagation
of the pulse over a distance L is
Dtpol = |L/vgx - L/vgy|
(3-28)
 In contrast to chromatic dispersion, PMD varies
randomly along a fiber. Thus, Dtpol given in Eq.
(3-28) cannot be used directly to estimate PMD.
Instead, statistical predictions are needed to
account for its effects.
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3.2.6 Polarization-Mode Dispersion
 A useful means of characterizing PMD for long
fiber lengths is in terms of the mean value of the
differential group delay
<Dtpol> = DPMDL½
(3-29)
 where DPMD measured in ps/, is the average PDM
parameter. In contrast to the instantaneous value
Dtpol, the mean value does not change from day to
day or from source to source.
國立成功大學 電機工程學系
光纖通訊實驗室 黃振發教授 編撰
3.2.6 Polarization-Mode Dispersion
Figure 3-17. Variation in the polarization states of
an optical pulse as it passes through a fiber with
varying birefringence along its length.
國立成功大學 電機工程學系
光纖通訊實驗室 黃振發教授 編撰
3.2.7 Intermodal Distortion
 Intermodal distortion is a result of different values
of the group delay for each individual mode at a
single frequency.
 For the step-index fiber in Fig. 2-12, the steeper the
angle of propagation of the ray congruence, the
higher is the mode number and the slower the axial
group velocity.
國立成功大學 電機工程學系
光纖通訊實驗室 黃振發教授 編撰
3.2.7 Intermodal Distortion
 The maximum pulse broadening arising from
intermodal distortion is the difference between the
travel time Tmax of the longest ray congruence paths
(the highest-order mode) and the travel time Tmin of
the shortest ray congruence paths (the fundamental
mode):
dTmod = Tmax – Tmin
= n1DL/c
(3-30)
國立成功大學 電機工程學系
光纖通訊實驗室 黃振發教授 編撰