Dispersion:
The spreading of light pulse as it propagates down the fiber is called Dispersion.
• Occurs due to the dependence of phase velocity of a wave on its frequency or wavelength
• The pulse broadening causes adjacent pulses to get overlapped and limit the maximum number of pulses sent per second, thus reducing the information carrying capacity of the fiber.
• To avoid overlapping of optical pulses the bit rate (BT) must be less than the reciprocal by broadened duration 2τ.
BT ≤ 1/2τ

• Two types of dispersion are,
1. Intramodal dispersion.
2. Intermodal dispersion.
1. Intramodal Dispersion
• Also known as ‘chromatic dispersion’ or ‘group velocity dispersion’ (GVD).
• When an EM wave travels through a medium of RI, ‘n’ the speed of wave is reduced from speed of light.
That is, v = c/n.
Hence the speed of light in a material depends on its RI which inturn is a frequency dependent parameter. As a result different spectral components of the light pulse travels at slightly different group velocities which causes group velocity dispersion or chromatic dispersion.
• Group Velocity and Phase Velocity:
Group Velocity (V_{g}) is the speed at which energy in a particular mode travels along the fibre.
Phase Velocity (V_{p}) of a wave is the rate at which the phase of a wave propagates.
V_{p} = ω/β
Where, β – propagation constant
• V_{p }and V_{g} depends on the frequency and the medium.
• Group delay is the time delay experienced by the spectral component of a signal as it propagates through the fibre.
τ_{g} = L/V_{g}
Where, L = fiber length
• If Δω is the spectral width of the pulse, the extend of pulse broadening for a fiber of length ‘L’ can be given as,
ΔT = dτ/dω. Δω
Substitute for τ,
We have, ΔT = d/dω(L/V_{g}). Δω
Substitute for V_{g},
ΔT = d/dω(L/(dω/dβ)). Δω
= L d^{2}β/dω^{2}. Δω
ΔT = L β_{2}. Δω
Where, β_{2} = d^{2}β/dω^{2} = GVD parameter
• GVD parameter determines how much an optical pulse will broaden while propagating inside the fiber.
• Two types of intramodal dispersion are,
a. Material Dispersion
b. Waveguide Dispersion
Material Dispersion:
• Caused by variation of RI as a function of ‘optical wavelength’
• Since the group velocity of a mode is a function of RI, the various spectral components of a given mode will travel at different speeds depending on wavelength.
• The larger the spectral width the higher will be the pulse broadening. Since all wavelength in the spectral width (λ ± Δλ) of the optical source propagate with different speed.
• Prominent for LED Sources due to broader spectrum
Derivation:
We know, β = nk
β – propagation constant
k – wave vector = 2 π/ λ
diff w.r.to k,
dβ/dk = n
Group delay, τ_{g} = L/V_{g} = L/(c/n)
τ_{g} = L/c/( dβ/dk)
τ_{g} = L/c x dβ/dk
=L/c (dβ/dλ x dλ/dk)
We have, k = 2π/ λ
Differentiating we have,
dk = 2π/ λ^{2} dλ
dλ/dk =  λ^{2}/2π
Therefore, τ_{g} =  λ^{2}L/2πc . dβ/dλ  (1)
This is the time delay experienced by the wave of wavelength λ and wave vector k as it travels through the fiber of length ‘L’.
Pulse spread,
σ_{mat} = dτ_{g}/dλ. σ_{λ}  (2)
where, σ_{λ} is the RMS value of dλ
we know, β = 2πn(λ)/ λ
Substitute β in eq(1) we have,
τ_{g} =  λ^{2}L/2πc . d(2πn(λ)/ λ)/dλ
Therefore, τ_{g} = L/c[n(λ) λ dn(λ)/dλ]
Substitute τ_{g} in eq(2)
σ_{mat} = [d(L/c[n(λ) λ dn(λ)/dλ])/dλ]. σ_{λ}
= σ_{λ} L/c λ d^{2}n/dλ^{2}
= σ_{λ} L D_{mat}(λ)
where, D_{mat} is the optical fibre material dispersion.
D_{mat} = λ/c. d^{2}n/dλ^{2}

For pure silica at ‘λ = 1.276, D_{mat} = 0’. Hence this wavelength is known as “Zero Dispersion Wavelength, λ_{zθ}”
Wave Guide Dispersion:
• Results from the variation in group velocity with wavelength for a particular mode
• One mode in the single mode fiber or each mode in a multimode fiber can have its own waveguide dispersion.
• For a mode distribution of light in the fiber varies for different wavelength. Shortr wavelengths are more confined to the core, whereas for larger wavelength, larger portion of power propagates through the cladding.
• Due to lower RI in the cladding that portion of light travels faster. Also since refractive index depends on wavelength different spectral components in a single mode have different propagation speeds. Thus difference in corecladding spatial power distribution together with speed variation of various wavelengths causes change in propagation velocity for each spectral component resulting in dispersion.
• They are significant in single mode fiber.
Equation:
• Normalized propagation constant, b = a^{2}ω^{2}/v^{2}
Where, a = coradius
ω = decay parameter in cladding
v = v number
Sub. ω^{2} = β^{2} – K_{2}^{2}
b = a^{2}(β^{2} – K_{2}^{2})/ a^{2} K^{2}(n_{1}^{2} – n_{2}^{2})
b = β^{2}/ K^{2}  n_{2}^{2}/ (n_{1}^{2} – n_{2}^{2})
for small values of RI difference,
ie, for a weekly guided condition,
Δ = (n_{1} – n_{2})/n_{1} << 1
and b = (β/ K – n_{2})/(n_{1} – n_{2})
Therefore, β = (b(n_{1} – n_{2}) + n_{2})K
= Kb(n_{1} – n_{2}) + Kn_{2}
= Kb(n_{1} – n_{2}).n_{1}/n_{1} + Kn_{2}
= Kb Δn_{1} + Kn_{2}
= Kn_{2}[(bΔn_{1}/n_{2}) + 1]
Since, n_{1}/n_{2} = 1
β = Kn_{2}[bΔ+ 1]
Group Delay, τ_{waveguide} = L/C dβ/dK
Sub for β,
τ_{waveguide} = L/C d(Kn_{2}[bΔ+ 1])/dk
= L/C d(Kn_{2}bΔ+ Kn_{2})/dk
= L/C [n_{2}ΔdKb/dk + n_{2}]
Expressing in terms of V number, V
τ_{waveguide} = L/C[n_{2} +n_{2}ΔdV_{b}/dV]

Pulse spread due to τ_{waveguide}
σ_{wg} = d τ_{waveguide}/dλσ_{λ}
= d (L/C[n_{2} +n_{2}ΔdV_{b}/dV])/dλσ_{λ}
= n_{2} L Δ σ_{λ}/ Cλ . V. d^{2}(V_{b})/ dV^{2}
^{
}
σ_{wg} = LD_{wg}(λ) σ_{λ}

where, D_{wg}(λ)= n_{2 }Δ/Cλ . V . d^{2}(V_{b})/ dV^{2}
Also, D_{wg}(λ)= waveguide dispersion factor
2. Intermodal Dispersion:
• Also known as modal delay or modal dispersion
• Occur due to each mode having different group velocity at a single frequency
• Input light is made of a number of modes
• Different modes travel through fibre at different speeds in different directions. Hence, distance travelled by each mode is different. Thus time taken by each mode to cover the same distance is different. This results in pulse broadening.
• Increases with distance travelled through the fiber.
• Does not exist in single mode fibers
• Major source of dispersion in multimode fibers
Equation:
Let, T_{min} = Time taken by axial ray to travel a distance ‘L’.
ie, T_{min} = L/V = L/(C/n_{1}) = L n_{1}/C
Let, T_{max} = Time taken by meridional ray to travel distance ‘L’.
ie, T_{max} = (L/Cos θ)/V = (L/Cos θ)/(C/n_{1}) = L n_{1}/ (C x Cos θ)
Cos θ = Sin θ = n_{2}/n_{1}
Therefore, T_{max} = L n_{1}^{2}/ C n_{2}
Time Delay = T_{max}  T_{min}
= L n_{1}^{2}/ C n_{2}  L n_{1}/ C
= L n_{1}^{2}/ C n_{2} (1  n_{2}/ n_{1})
= L n_{1}^{2}/ C n_{2} . Δ
= L/Cn_{2} . (NA)^{2}/2 .............. Since, NA = n_{1}√(2Δ)
Delay = δ Ts = L(NA)^{2}/2Cn_{2 }
Polarisation Mode Dispersion:
• The fibre birefringence on the polarisation states of the optical signal can cause pulse broadening. Polarisation is the electric field orientation of the signal. This can vary along the length of the fiber.
• Birefringence is the optical property of a material having a RI that depends on the polarisation and propagation direction of the light.
• Birefringence can be due to intrinsic factors (geometric irregularities, internal stress etc) and external factors (bending, twisting etc).
• Signal energy at a given wavelength has two orthogonal polarisation modes.
• Varying birefringence along the length of the fiber will cause each polarisation mode to travel at slightly different velocity. The resulting difference in propagation (Δτ_{PMD}) time between the two orthogonal polarization modes will result in pulse spreading. This is known as polarisation mode dispersion (PMD).
• Differential Time Delay,
Δτ_{PMD} = L/V_{gz} – L/V_{gy}
Where, V_{gz} and V_{gy} = group velocities of two orthogonal polarization modes.
• PMD varies randomly along the fiber, since causes of birefringence may vary with temperature, stress dynamics etc. Hence Δτ_{PMD} cannot be used directly to measure PMD. It limits the performance of the fiber in longhaul optical communication system operating at high bit rate.
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