Fibre optic cables experience losses from attenuation and dispersion. Attenuation includes material absorption from intrinsic effects like electronic and molecular vibrations, and extrinsic effects like impurities. Scattering losses occur from irregularities causing Rayleigh, Brillouin, Raman, waveguide and Mie scattering. Nonlinear losses include micro/macro bending, leaky modes and mode coupling. Proper fibre design and manufacturing can minimize these losses to support high bandwidth signal transmission.

1. OPTOMETRY – Part XII
OPTICAL FIBRE LOSSES
ER. FARUK BIN POYEN
DEPT. OF AEIE, UIT, BU, BURDWAN, WB, INDIA
FARUK.POYEN@GMAIL.COM

2. Contents:
1. Fibre Losses
2. Attenuation Types
3. Dispersion
4. Material Absorption
5. Scattering Losses
6. Nonlinear/ Radiative Losses
2

3. Fibre Losses:
 Fibre Losses: Optical fibre cables suffer few losses. They are classified as Attenuation
and Dispersion. These two are further classified into several other losses.
 Attenuation Coefficient: Signal attenuation or transmission loss is defined as the ratio of
the input transmission optical power 𝑃𝑖𝑛 into a fibre to the output (received) optical
power 𝑃𝑜𝑢𝑡 from the fibre. This ratio is a function of the operating wavelength.
 The symbol α 𝑑𝐵 is commonly used to express the attenuation in decibels (dB) per
kilometre (L).
𝜶 𝒅𝑩 =
𝟏
𝑳
[𝟏𝟎 𝐥𝐨𝐠 𝟏𝟎
𝑷𝒊𝒏
𝑷 𝒐𝒖𝒕
]
3

4. Attenuation Types:
 Material Absorption
 Scattering Losses
 Nonlinear/ Radiative Loses
4

5. Material Absorption:
 Material Absorption
 Intrinsic Absorption
1. Electronic Absorption
2. Molecular Absorption
3. Ultra Violet Absorption
4. Infra Red Absorption
 Extrinsic Absorption
1. Ion – Resonance Absorption
2. Impurity Absorption
5

6. Scattering Losses:
 Scattering Losses
1. Rayleigh Scattering
2. Brillouin Scattering
3. Raman Scattering
4. Wave Guide Scattering
5. Mie Scattering
6

7. Nonlinear/ Radiative Loses:
 Nonlinear/ Radiative Loses
1. Bending Loss
A. Micro Bending
B. Macro Bending (Constant Radius Bending)
C. Leaky Mode (Skew Ray)
D. Mode Coupling Loss
7

8. Dispersion:
 Modal Dispersion
 Material Dispersion
 Waveguide Dispersion
 Non Linear Dispersion
8

9. Attenuation – Material Absorption:
 Material absorption is a loss mechanism related to both the material composition and the
fabrication process for the fibre.
 The optical power is lost as heat in the fibre.
 The light absorption can be intrinsic (due to the material components of the glass) or
extrinsic (due to impurities introduced into the glass during fabrication).
 Pure silica-based glass has two major intrinsic absorption mechanisms at optical
wavelengths:
1. Fundamental UV absorption
2. Fundamental IR or Far-IR absorption
9

10. Material Absorption: Fundamental UV Absorption
 In fundamental UV absorption edge, the peaks are centred in the UV region.
 Fused silica valence electrons absorb light and can be ionized to conduction electrons.
 This gives rise to an energy loss in the light field contributing to transmission loss.
 The absorption loss increases with the decrease of wavelength.
 The UV edge of electron absorption band in both crystalline and amorphous materials
follows Urbach’s Rule
𝜆 𝑢𝑣 = 𝐶𝑒
𝐸
𝐸0
 Here C and E0 are empirical constants. E is the photon energy.
 𝜆 𝑢𝑣 = attenuation constant in the UV region
10

11. Material Absorption – Fundamental IR Absorption:
 Fundamental IR and Far-IR absorption edge is due to the molecular vibrations (Si-O).
 The tail od these absorption peaks may extend into the longer wavelengths.
 IR absorption occurs because the photons are absorbed by atoms within the glass
molecules and converted to random mechanical vibrations typical of heating.
11

12. Material Absorption – Intrinsic:
 Electronic Absorption: The bandgap of fused silica is about 8.9 eV (~140 nm). This
causes strong absorption of light in the UV spectral region due to electronic transitions
across the band gap.
 An amorphous material like fused silica generally has very long bandtails. These
bandtails lead to an absorption tail extending into the visible and infrared regions.
Empirically, the absorption tail at photon energies below the bandgap falls off
exponentially with photon energy.
 Molecular Absorption: In the infrared region, the absorption of photons is
accompanied by transitions between different vibrational modes of silica molecules. The
fundamental vibrational transition of fused silica causes a very strong absorption peak at
about 9 μm wavelength.
 Nonlinear effects contribute to important harmonics and combination frequencies
corresponding to minor absorption peaks at 4.4, 3.8 and 3.2 μm wavelengths.
 A long absorption tail extending into the near infrared, causing a sharp rise in absorption
at optical wavelengths longer than 1.6 μm.
12

13. Material Absorption – Extrinsic:
 Ion – Resonance Absorption: Major extrinsic loss mechanism is caused by absorption
due to water (as the hydroxyl or OH- ions) introduced in the glass fiber during fiber
pulling by means of oxyhydrogen flame. This leads to Ion – Resonance Absorption.
 The lowest attenuation for typical silica-based fibers occur at wavelength 1.55 μm at
about 0.2 dB/km, approaching the minimum possible attenuation at this wavelength.
 Impurity Absorption: Most impurity ions such as OH-, Fe2+ and Cu2+ form
absorption bands in the near infrared region where both electronic and molecular
absorption losses of the host silica glass are very low.
 Near the peaks of the impurity absorption bands, an impurity concentration as low as
one part per billion can contribute to an absorption loss as high as 1 dB km-1.
 Today, impurities in fibers have been reduced to levels where losses associated with
their absorption are negligible, with the exception of the OH- radical.
13

14. Scattering Loss:
 Scattering results in attenuation (in the form of radiation) as the scattered light may not
continue to satisfy the total internal reflection in the fiber core.
 The scattered ray can escape by refraction according to Snell’s Law.
 Scattering is due to irregularity of materials.
 When a beam of light interacts with a material, part of it is transmitted, part it is
reflected, and part of it is scattered.
 Mainly there are five such losses viz.
1. Rayleigh
2. Brillouin
3. Raman
4. Wave Guide
5. Mie
14

15. Scattering Loss – Rayleigh:
 Rayleigh scattering results from random inhomogeneities that are small in size
compared with the wavelength. It takes place due to the variations in the refractive
index in glass. The glass used is amorphous one, prepared by allowing glass to cool
from molten state at high temperature until it freezes.
 During this transition two defects may arise.
1. Glass being amorphous is composed to randomly connected network of molecules.
Ans therefore it may contain regions in which the molecular density is higher or lower
than the average density in the glass.
2. Since the glass is made up of several oxides, such as SiO2, GeO2 and P2O5,
compositional fluctuations may occur.
 For a single component glass, the Rayleigh scattering coefficient is given by
𝜏 𝑅 =
8𝜋3
3𝜆4 𝑛1
8
𝑝2
𝐵 𝐶 𝐾 𝐵 𝑇 𝐹
15

16. Scattering Loss – Rayleigh:
 For a single component glass, the Rayleigh scattering coefficient is given by
𝜏 𝑅 =
8𝜋3
3𝜆4 𝑛1
8
𝑝2
𝐵 𝐶 𝐾 𝐵 𝑇 𝐹
 Where 𝜏 𝑅 = Rayleigh scattering coefficient, λ = wave length of optical radiation
 n1 = refractive index of the medium, p = average photo elastic coefficient,
 BC = isothermal compressibility at fictive temperature TF and KB = Boltzmann constant.
 The fictive temperature of glass is defined as the temperature at which glass can reach a state
of thermal equilibrium and closely related to the anneal temperature.
 Sub microscopic variations in the glass density and doping impurities are frozen into glass
during manufacture and they act as the reflecting and refracting facets to scatter a small
portion of light through the glass.
 These defects may be in the form of trapped bubbles, unreacted starting materials and
crystallized regions in the glass.
16

17. Scattering Loss – Brillouin:
 It may be regarded as the modulation of light through thermal molecular vibration
within the fibre.
 The incident photons of light undergo nonlinear interaction to produce vibrational
energy or phonons in the glass as well as the scattered light or photons.
 The scattered light is found to be frequency modulated by the thermal energy and both
upward and downward frequency shifts are observed.
 The amount of frequency shift and the strength of scattering vary as the function of the
scattering angle maximum occurring at the backward direction and the minimum or zero
being observed in the forward direction.
 Thus Brillouin scattering mainly occurs in the backward direction which directs the
power to the source and the power of the receiver is reduced.
17

18. Scattering Loss – Brillouin:
 The optical power level at which Brillouin scattering becomes significant in a single
mode fibre is given by an empirical formula. The threshol2d power level PB is given by
𝑃 𝐵 = 10−3
𝑑2
𝜆2
𝛼 𝑑𝑏∆𝜏
Where d and λ are the core diameter and the operating wavelength respectively, αdb is the
fibre attenuation in dB/km. ατ is the source bandwidth in GHz.
18

19. Scattering Loss – Raman:
 The non-linear interaction in Raman scattering produces a high frequency phonon and a
scattered photon, where as low frequency phonons are produced in Brillouin scattering.
 In Raman scattering, light is predominantly in the forward direction and thus the power
is not reduced in the receiver.
 The threshold power level for the significant Raman scattering to occur is given by
𝑃 𝑅 = 5.9 ∗ 10−2
𝑑2
𝜆𝛼 𝑑𝐵
 Where d is the diameter of the fibre in μm, λ is the wavelength emitted by the source in
μm, 𝛼 𝑑𝐵 is the fibre loss in dB/km and PR is the threshold optical power.
19

20. Scattering Loss – Wave Guide:
 Imperfections in the waveguide structure of a fiber, such as non-uniformity in the size
and shape of the core, perturbations in the core-cladding boundary, and defects in the
core or cladding, can be generated in the manufacturing process.
 Environmentally induced effects, such as stress and temperature variations, also cause
imperfections.
 The imperfections in a fiber waveguide result in additional scattering losses.
 They can also induce coupling between different guided modes.
20

21. Scattering Loss – Mie:
 Linear scattering may occur at inhomogeneities which are comparable in size with the
guided wavelength.
 When the size of scattering inhomogeneity is greater than λ/10, the scattering intensity
has an angular dependence and can be quite large.
 The scattering occurring due to such inhomogeneity is mainly in the forward direction
and is known as Mie Scattering.
 Depending on the fibre material, design and manufacture, Mie scattering can cause
considerable power loss. The inhomogeneity can be minimized by
1. Reducing imperfection during glass manufacturing process
2. Careful controlled extrusion and coating of the fibre
3. Increasing the fibre guidance by increasing the relative refractive index between
core and cladding.
21

22. Non linear/Radiative Losses:
 As light is confined over long distances in an optical fiber, nonlinear optical effects can
become important even at a relatively moderate optical power.
 Nonlinear optical processes such as stimulated Brillouin scattering and stimulated
Raman scattering can cause significant attenuation in the power of an optical signal.
 Other nonlinear processes can induce mode mixing or frequency shift, all contributing
to the loss of a particular guided mode at a particular frequency.
 Nonlinear effects are intensity dependent, and thus they can become very important at
high optical powers.
 Radiative losses occur whenever an optical fibre undergoes a bend of finite radius of
curvature.
 Fibres can be subject to two types of bends viz. Micro bending and Macro bending or
Constant Radius Bending.
22

23. Non linear/Radiative Losses - Micro Bending:
 It is a microscopic bending with repetitive changes in the axis of the core and it takes
place due to the slightly different contraction rate between the core and the cladding
materials.
 It occurs due to non uniform lateral pressure created during cabling.
 Losses in the micro bending take place because the small bends act as the scattering
facets and these facets cause mode coupling to occur.
 Energy from the guided modes is cross coupled to the leaky mode and is lost through
the cladding.
 Micro bending are randomly distributed over the length of the fibre.
23

24. Non linear/Radiative Losses - Micro Bending:
 Careful precaution in manufacturing and handling of fibres will reduce the loss.
 One method to minimize is done by extruding a compressible jacket over the fibre
which will be able to take on external tension without deforming the core.
 Potential micro bending losses may be minimized by
1. Designing fibres with large relative refractive index differences between the core
and the cladding.
2. Operating at the shortest possible wavelength.
24

25. Non linear/Radiative Losses - Macro Bending:
 It is also called Constant Radius Bending.
 Bends are introduced while installing cable ducts to join corners.
 Sometimes these bends are quite sharp.
 These large radius bends introduce losses in the fibre.
 The bending may provide incidence angles less than the critical angle thereby allowing
a part of the light energy to escape from the fibre through the cladding.
 It is therefore necessary to ensure that no sharp bends are introduced in the path of the
fibre.
25

26. Critical radius of Bend:
 Critical radius of Bend: The relationship between the radius of curvature of the bend
and radiation attenuation coefficient 𝜆 𝑟 is given by
𝜆 𝑟 = 𝐶1exp(−𝐶2 𝑅)
R = radius of curvature; C1 and C2 are constants independent of R.
 Large bending losses tend to occur in multi mode fibre at a critical radius of curvature
R 𝐶 given by 𝑅 𝐶 =
3𝑛1
2 𝜆
4𝜋(𝑛1
2−𝑛2
2)
3
2
26

27. Non linear/Radiative Losses – Skew Rays
 At each reflection of a ray propagating in helical type of path (skew mode) the angle of
incidence 𝜃1 at the core – cladding surface is more than the critical angle 𝜃 𝐶 and the
mode will propagate through the fibre.
 But at some higher order modes, the 𝜃 𝐶 may be less than 𝜃 𝐶 and a part of the
propagation will escape the core by refraction.
 Successive such refractions will make the intensity weak and eventually will be lost.
 It is desirable to remove this leaky mode from the core and cladding as rapidly as
possible to reduce signal dispersion.
 This is accomplished by surrounding thin cladding by a third party layer of pure silica
having refractive index more than cladding but less than core.
 This provides mechanical strength to the fibre and acts to remove the partially refracted
ray from the leaky mode and possesses rays from the cut off modes by total refraction.
27

28. Non linear/Radiative Losses: Mode – Coupling
 Power may be launched successfully into a propagating mode but it may be coupled into
a leaky or radiating mode at some point further down the fibre.
 This type of improper coupling can occur for several reasons.
 Small imperfections in the core glass or in the core – cladding interface may occur due
to small variations in the core diameter, cross sectional shape or bubbles in the glass and
these are responsible for causing the energy to be coupled into one of the leaky modes.
 Losses from these sources will be uniform along the length of the fibre.
 Another source of mode coupling is the imperfectly formed splices or imperfectly
aligned connectors.
 These are discrete losses which can be reduced by decreasing the number of splices or
connectors in a given fibre.
28

29. DISPERSION:
 Dispersion is the primary cause of limitation on the optical signal transmission
bandwidth through an optical fiber.
 Dispersion is referred to widening the pulse as the light travels through the fiber optics.
 When a short pulse of light travels through an optical fiber its power is “dispersed” in
time so that the pulse spreads into a wider time interval.
 There are four sources of dispersion in optical fibers: modal dispersion, material
dispersion, wave guide dispersion and nonlinear dispersion.
 Both material dispersion and waveguide dispersion are examples of chromatic
dispersion because both are frequency dependent.
 Waveguide dispersion is caused by frequency dependence of the propagation constant β
of a specific mode due to the wave guiding effect.
 The combined effect of material and waveguide dispersions for a particular mode alone
is called intra mode dispersion.
29

30. DISPERSION: Modal
 Modal dispersion occurs in multimode fibers as a result of the differences in the group
velocities of the modes.
 A single impulse of light entering an M-mode fiber at z = 0 spreads into M pulses with the
differential delay increasing as a function of z.
 For a fiber of length L, the time delays encountered by the different modes are
𝜏 𝑞 = 𝐿/𝑣 𝑞
q = 1, ....., M, where vq is the group velocity of mode q.
 If vmin and vmax are the smallest and largest group velocities, the received pulse spreads over
a time interval are
𝐿
𝑣 𝑚𝑖𝑛
−
𝐿
𝑣 𝑚𝑎𝑥
.
 Since the modes are generally not excited equally, the overall shape of the received pulse is a
smooth profile.
 An estimate of the overall rms pulse width is
𝜎 𝜏 =
1
2
( 𝐿
𝑣 𝑚𝑖𝑛
− 𝐿
𝑣 𝑚𝑎𝑥
)
 This width represents the response time of the fiber.
30

31. DISPERSION: Modal
 In a step-index fiber with a large number of modes,
𝑣 𝑚𝑖𝑛 ≈ 𝑐1 1 − ∆ 𝑎𝑛𝑑 𝑣 𝑚𝑎𝑥 ≈ 𝑐1
 Since (1 − ∆)−1
≈ 1 + ∆, the response time is
𝜎 𝜏 ≈
𝐿
𝑐1
∆
2
(response time for multi mode step – index fibre)
i.e., it is a fraction ∆/2 of the delay time 𝐿 𝑐1.
 Modal dispersion is much smaller in graded-index fibers than in step-index fibers since the
group velocities are equalized and the differences between the delay times
𝜏 𝑞 = 𝐿/𝑣 𝑞
of the modes are reduced.
 In a graded-index fiber with a large number of modes and with an optimal index profile,
𝑣 𝑚𝑖𝑛 ≈ 𝑐1(1 − ∆2
/2) 𝑎𝑛𝑑 𝑣 𝑚𝑎𝑥 ≈ 𝑐1
 The response time is therefore 𝜎 𝜏 ≈
𝐿
𝑐1
∆2
4
(𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑡𝑖𝑚𝑒 𝑓𝑜𝑟 𝑔𝑟𝑎𝑑𝑒𝑑 𝑖𝑛𝑑𝑒𝑥 𝑓𝑖𝑏𝑟𝑒)
which is a factor of ∆/2 smaller than that in a step index fibre.
31

32. DISPERSION: Modal
 The pulse broadening arising from modal dispersion is proportional to the fiber length L
in both step-index and graded-index fibers.
 This dependence, however, does not necessarily hold when the fibers are longer than a
certain critical length because of mode coupling.
 Coupling occurs between modes of approximately the same propagation constants as a
result of small imperfections in the fiber (random irregularities of the fiber surface, or
inhomogeneities of the refractive index) which permit the optical power to be
exchanged between the modes.
 Under certain conditions, the response time στ of mode-coupled fibers is proportional to
L for small L and to L1/2 when a critical length is exceeded, so that pulses are broadened
at a slower rate.
32

33. DISPERSION: Material
 Glass is a dispersive medium; i.e. its refractive index is a function of wavelength.
 An optical pulse travels in a dispersive medium of refractive index n with a group
velocity 𝑣 = 𝑐0
𝑁 where N = n – λ0, dn/d λ0.
 Since the pulse is a wavepacket, composed of a spectrum of components of different
wavelengths each travelling at a different group velocity, its width spreads.
 The temporal width of an optical impulse of spectral width σλ (nm), after travelling a
distance L, is
𝜎 𝜏 = 𝑑
𝑑𝜆0
𝐿/𝑣 𝜎 𝜆 = 𝑑
𝑑𝜆0
(𝐿𝑁/𝑐0) 𝜎 𝜆
from which
𝝈 𝝉 = 𝑫 𝝀 𝝈 𝝀 𝑳 (response time for material dispersion)
Where
𝐷 𝜆 =
𝜆0
𝑐0
𝑑2
𝑛
𝑑𝜆2
0
is the material dispersion coefficient.
33

34. DISPERSION: Material
 The response time increases linearly with the distance L.
 Usually L is measured in km, 𝜎 𝜏 in ps and 𝜎 𝜆 in nm, so that 𝐷 𝜆 has units of ps/km-nm.
 This type of dispersion is called material dispersion (as opposed to modal dispersion).
34

35. DISPERSION: Wave guide
 The group velocity of the modes depends on the wavelength even if material dispersion
is negligible.
 This dependence, known as waveguide dispersion, results from the dependence of the
field distribution in the fiber on the ratio between the core radius and the wavelength
(a/λ0).
 If this ratio is altered, by altering λ0, the relative portions of optical power in the core
and cladding are modified.
 Since the phase velocities in the core and cladding are different, the group velocity of
the mode is altered.
 Waveguide dispersion is particularly important in single-mode fibers, where modal
dispersion is not exhibited, and at wavelengths for which material dispersion is small
(near λ0 = 1.3 μm in silica glass).
35

36. DISPERSION: Wave guide
 The group velocity 𝑣 = (𝑑𝛽/𝑑𝜔)−1
and the propagation constant β are determined
from the characteristic equation which is governed by the fibre V parameter
𝑉 = 2𝜋
𝑎
𝜆0
𝑁𝐴 = (𝑎. 𝑁𝐴/𝑐0)𝜔
 In the absence of material dispersion (i.e. when NA is independent of ω), V is directly
proportional to ω, so that
1
𝑣
=
𝑑𝛽
𝑑𝜔
=
𝑑𝛽
𝑑𝑉
𝑑𝑉
𝑑𝜔
=
𝑎. 𝑁𝐴
𝑐0
𝑑𝛽
𝑑𝑉
 The pulse broadening associated with a source of spectral width 𝜎 𝜆 is related to the time
delay 𝐿/𝑣 by
𝜎 𝜏 = 𝑑/𝑑𝜆0 (𝑙/𝑣) 𝜎 𝜆
 Thus
𝜎 𝜏 = 𝐷 𝑤 𝜎 𝜆 𝐿
𝐷 𝑤 =
𝑑
𝑑𝜆0
1
𝑣
= −
𝜔
𝜆0
𝑑
𝑑𝜔
1
𝑣
36

37. DISPERSION: Wave guide
 𝐷 𝑤 is the waveguide dispersion coefficient.
 Thus we obtain
𝐷 𝑤 = (
1
2𝜋𝑐0
)𝑉2 𝑑2
𝛽
𝑑𝑉2
 Thus the group velocity is inversely proportional to
𝑑𝛽
𝑑𝑉
and the dispersion coefficient is
proportional to 𝑉2 𝑑2
𝛽
𝑑𝑉2.
 Since β varies nonlinearly with V, the waveguide dispersion coefficient 𝐷 𝑤 is itself a
function of V and is therefore also a function of the wavelength.
 The dependence of 𝐷 𝑤 on λ0 may be controlled by altering the radius of the core or the
index grading profile for graded-index fibres.
37

38. DISPERSION: Combined Material & Wave guide
 The combined effect of material dispersion and waveguide dispersion is also known as
“Chromatic Dispersion”.
 Chromatic Dispersion may be determined by including the wavelength dependence of
the refractive indices n1 and n2 and therefore NA, when determining
𝑑𝛽
𝑑𝑉
from the
characteristic equation.
 Although generally smaller than material dispersion, wavelength dispersion does shift
the wavelength at which the total chromatic dispersion is the minimum.
 Since chromatic dispersion limits the performance of single – mode fibres, more
advanced fibre designs aim at reducing this effect by using graded – index cores with
refractive – index profiles selected such that the wavelength at which waveguide
dispersion compensates material dispersion is shifted to the wavelength at which the
fibre is to be used.
38

39. DISPERSION: Combined Material & Wave guide
 Dispersion – shifted fibres have been successfully made by using a linearly tapered
core refractive index and a reduced core radius.
 This technique can be used to shift the zero – chromatic – dispersion wavelength from
1.3 μm to1.55 μm where the fibre has its lowest attenuation.
 However the process of index grading itself introduces losses since dopants are used.
 Other grading profiles have been developed for which the chromatic dispersion vanishes
at two wavelengths and is reduced for wavelengths between.
 These fibres, called dispersion – flattened, have been implemented by using a
quadruple – clad layered grading.
39

40. DISPERSION: Combined Material and Modal
 The effect of material dispersion on pulse broadening in multimode fibres may be
determined by returning to the original equations for the propagation constants βq of the
modes and determining the group velocities
𝑣 𝑞 = (𝑑𝛽 𝑞/𝑑𝜔)−1
with n1 and n2 being functions of ω.
 Although n1 and n2 are dependent on ω, it is reasonable to assume that the ratio
∆= (𝑛1 − 𝑛2)/𝑛1
is approximately independent of ω.
40

41. DISPERSION: Combined Material and Modal
 Using this approximation and evaluating
𝑣 𝑞 = (𝑑𝛽 𝑞/𝑑𝜔)−1
we obtain
𝑣 𝑞 =
𝑐0
𝑁1
1 −
𝑝 − 2
𝑝 + 2
(
𝑞
𝑀
)
𝑝
(𝑝+2)
∆
Where 𝑁1 = 𝑑
𝑑𝜔 𝜔𝑛1 = 𝑛1 − 𝜆 𝑜(𝑑𝑛1/𝑑𝜆0) is the group index of the core material.
p is the profile parameter and q is the mode of fibre.
 For a step – index fibre (p = ∞), the group velocities of the modes vary from 𝑐0 𝑁1 to
𝑐0 𝑁1 (1 − ∆), so that the response time is
𝝈 𝝉 ≈
𝑳
𝒄 𝟎 𝑵 𝟏
∆
𝟐
(response time for multi mode step index fibre with material dispersion).
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42. DISPERSION: Non-linear
 Another dispersion effect occurs when the intensity of light in the core is sufficiently
high, since the refractive indices then become intensity dependent and the material
exhibits nonlinear behaviour.
 The high-intensity parts of an optical pulse undergo phase shifts different from the low-
intensity parts, so that the frequency is shifted by different amounts.
 As because of material dispersion, the group velocities are modified, and consequently
the pulse shape is altered.
 Under certain conditions, nonlinear dispersion can compensate material dispersion, so
that the pulse travels without altering its temporal profile.
 The guided wave is then known as a solitary wave, or a soliton.
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