**.**The pulse gets distorted as it travels along the fiber lengths. Pulse spreading in fiber is referred as dispersion. Dispersion is caused by difference in the propagation times of light rays that takes different paths during the propagation. The light pulses travelling down the fiber encounter dispersion effect because of this the pulse spreads out in time domain. Dispersion limits the information bandwidth. The distortion effects can be analyzed by studying the group velocities in guided modes.

__1. Information Capacity Determination__

**.**Dispersion and attenuation of pulse travelling along the fiber is shown in Fig. 1.

**.**Fig. 1 shows, after travelling some distance, pulse starts broadening and overlap with the neighbouring pulses. At certain distance the pulses are not even distinguished and erros will occur at receiver. Therefore the information capacity is specified by bandwidth distance product (MHz . km). for step index bandwidth distance product is 20 MHz . km and for graded index it is 2.5 MHz . km.

__2. Group Delay__

**.**Consider a fiber cable carrying optical signal equally with various modes and each mode contains all the spectral components in the wavelength band. All the spectral components travel independently and they observe different time delay and group delay in the direction of propagation. The velocity at which the energy in a pulse travels along the fiber is known as group velocity. Group velocity is given by,

**.**Thus different frequency components in a signal with travel at different group velocities and so will arrive at their destination at different times, for digital modulation of carrier, this results in dispersion of pulse, which affects the maximum rate of modulation. Let the difference in propagation times for two side bands is δ

_{Ԏ}

_{}.

**.**Dispersion is measured in picoseconds per nanometer per kilometer.

__3. Material Dispersion__

**.**Material dispersion is also called as chromatic dispersion. Material dispersion exists due to change in index of refraction for different wavelengths. A light ray contains components of various wavelength centered at wavelength λ

_{0}. The time delay is different for different wavelength components. This results in time dispersion of pulse at the receiving end of fiber. Fig. 2 shows index of refraction as a function of optical wavelength.

**.**The material dispersion for unit length (L =1) is given by,

Negative sign shows that the upper sideband signal (lowest wavelength0 arrives before the lower sideband (highest wavelength).

**.**A plot of material dispersion and wavelength is shown in Fig. 3.

**.**The unit of dispersion is : ps/nm . km. the amount of material dispersion depends upon the chemical composition of glass.

__4. waveguide Dispersion__

**.**Waveguide dispersion is caused by the difference in the index of refraction between the core and cladding, resulting in a 'drag' effect between the core and cladding portions of the power.

**.**Waveguide dispersion is significantly only in fibers carrying fewer than 5-10 modes. Since multimode optical fibers carry hundreds of modes, they will not have observable waveguide dispersion.

**.**The group delay (Ԏ

_{wg}) arising due to waveguide dispersion.

where, b = Normalized propagation constant

k = 2π/λ (group velocity)

Normalized frequency V,

**.**As frequency is a function of wavelength, the group velocity of the energy varies with frequency. This produces additional losses (waveguide dispersion). The propagation constant (b) varies with wavelength, the causes of which are independent of material dispersion.

__5. Chromatic Dispersion__

**.**The combination of material dispersion and waveguide dispersion is called chromatic dispersion. These losses primarily concern the spectral width of transmitter and choice of correct wavelength.

**.**A graph of effective refractive index against wavelength illustrates the effects of material, chromatic and waveguide dispersion.

**.**Material dispersion and waveguide dispersion effects vary in opposite sense as the wavelength increased, but at an optimum wavelength around 1300 nm, two effects almost cancel each other and chromatic dispersion is at minimum. Attenuation is therefore also at minimum and makes 1300 nm a highly attractive operating wavelength.

__6. Modal Dispersion__

**.**As only a certain number of modes can propagate down the fiber, each of these modes carries the modulation signal and each one is incident on the boundary at a different angle, they will each have their own individual propagation times. The net effect is spreading of pulse, this form of dispersion is called modal dispersion.

**.**Modal dispersion takes place in multimode fibers. It is moderately present in graded index fibers and almost eliminated in single mode step index fibers.

**.**Modal dispersion is given by,

**.**The modal dispersion describes the optical pulse spreading due to modal effects optical pulse width can be converted to electrical rise time through the relationship,

__7. Signal Distortion in Single Mode Fibers__

**.**The pulse spreading σ

_{w }over range of wavelength can be obtained from derivative of group delay with respect to λ

**.**This is the equation for waveguide dispersion for unit length.

__8. Higher Order Dispersion__

**.**Higher order dispersive effective effects are governed by dispersion slope S.

Where,

ᵦ

_{2}and ᵦ

_{3}are second and third order dispersion parameters.

**.**Dispersion slope S plays an important role in designing WDM system.

__9. Dispersion Induced Limitations__

**.**The extent of pulse broadening depends on the width and the shape of input pulses. The pulse broadening is studied with the help of wave equation.

__9.1 Basic Propagation Equation__

**.**The basic propagation equation which governs evolution in a single mode fiber is given by,

ᵦ

_{1}, ᵦ

_{2}, and ᵦ

_{3 }are different dispersion parameters.

__9.2 Chriped Gaussian Pluses__

**.**A pulse is said to be chirped if its carrier frequency changes with time.

**.**For a Gaussian spectrum having spectral width σ

_{w}, the pulse broadening factor is given by,

__10. Limitations of Bit Rate__

**.**The limiting bit rate is given by,

4B σ ≤ 1

**.**The condition relating bit rate-distance product (BL) and dispersion (D) is given by,

**.**Limiting bit rate a single mode fibers as a function of fiber length for σ

_{λ}= 0, 1 and 5 nm is shown in Fig. 5.

__11. Polarization Mode Dispersion (PMD)__

**.**Different frequency component of a pulse acquires different polarization states (such as linear polarization and circular polarization). This results in pulse broadening is known as polarization mode dispersion (PMD).

**.**PMD is the limiting factor for optical communication system at high data rates. The effects of PMD must be compensated.

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