1. Chapter 3 Signal Degradation in Optical Fibers
3.1 Attenuation
Absorption
Scattering Losses
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
Intermodal Distortion

2. 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.

3. 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)]
= 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.

4. 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 single-mode 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.

5. 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.

6. 3.1 ATTENUATION Figure 3-2. Effects of ionizing on fiber attenuation.
Loss increase during steady irradiation.
Subsequent recovery after radiation has stopped.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 3.1.2 Absorption
Figure 3-4. A comparison of the infrared absorption induced by various doping materials in low-loss silica fibers.

14. 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.

15. 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.

16. 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.

17. 3.1.3 Scattering Losses
Figure 3-5. Typical spectral attenuation range for production graded-index multimode fibers.

18. Scattering Losses
Figure 3-6. Typical spectral attenuation range for production-run single-mode fibers.

19. 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 higher-order modes will radiate out of the fiber first .

20. 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.

21. 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 low-order modes to couple to higher-order modes

22. 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.

23. 3.1.4 Bending Losses
Figure 3-9. A compressible jacket extruded over a fiber reduces microbending resulting from external forces.

24. 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.

25. 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).

26. 3.2 Signal Distortion in Fibers
Figure Spectral emission pattern of a Ga1-xAlxAs LED with x = 0.08.
The width of the spectral pattern is 36 nm.

27. 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.

28. 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.

29. 3.2 Signal Distortion in Fibers
2. Waveguide dispersion, which occurs because a single-mode 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.

30. 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.

31. 3.2.1 Information Capacity Determination
Figure 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.

32. 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 The curves are shown in terms of signal attenuation versus data rate.

33. 3.2.1 Information Capacity Determination
Figure 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.

34. 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.

35. 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)

36. 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.

37. 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.

38. Group Delay
Figure Spectral emission pattern of a Ga1-xAlxAs LED with x = The width of the spectral pattern is 36-nm.

39. Material Dispersion
Exemplified in Fig 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)

40. 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.

41. Material Dispersion
A plot of the material dispersion for unit length L and unit optical source spectral width sl is given in Fig 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.

42. Material Dispersion
Figure Variations in the index of refraction as a function of the optical wavelength for silica.

43. Material Dispersion
Figure Material dispersion as function of optical wavelength for pure silica and 13.5% GeO2 / 86.5% SiO2.

44. 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)

45. 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)

46. 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 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.

47. Waveguide Dispersion
Figure 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.

48. 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)

49. 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.

50. 3.2.5 Signal Distortion in SMFs
Figure 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.

51. 3.2.5 Signal Distortion in SMFs
Figure Examples of the material and waveguide dispersions as a function of optical wavelength for a single-mode fused-silica-core fiber.

52. 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).

53. 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.

54. 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.

55. 3.2.6 Polarization-Mode Dispersion
Figure Variation in the polarization states of an optical pulse as it passes through a fiber with varying birefringence along its length.

56. 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.

57. 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)