- The pulse get 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 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.

### Information Capacity Determination

** **

- Dispersion and attenuation of pulse travelling along the fiber is shown in Fig. 6.1.

- 2.6.1 shows, after travelling some distance, pulse starts broadening and overlap with the neighbouring pulses. At certain distance the pulses are not even distinguishable and error 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.

### 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,

… (2.6.1)

- Thus different frequency components in a signal will 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 δτ.

… (2.6.2)

where, = Wavelength difference between upper and lower sideband (spectral width)

= Dispersion coefficient (D)

Then, where, L is length of fiber.

and considering unit length L = 1.

Now

... (2.6.3)

- Dispersion is measured in picoseconds per nanometer per

### Material Dispersion

** **

- Material dispersion is also called as chromatic dispersion. Material dispersion exists due to change in index of refraction for different A light ray contains components of various wavelengths centered at wavelength λ
_{10}. The time delay is different for different wavelength components. This results in time dispersion of pulse at the receiving end of fiber. Fig. 2.6.2 shows index of refraction as a function of optical wavelength.

- The material dispersion for unit length (L = 1) is given by

… (2.6.4)

where,

c = Light velocity

λ = Center wavelength

= Second derivative of index of refraction w.r.t wavelength

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

- A plot of material dispersion and wavelength is shown in 2.6.3

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

**Example 2.6.1 **: An LED operating at 850 nm has a spectral width of 45 nm. What is the pulse spreading in ns/km due to material dispersion? **[Jan./Feb.-2007, 3 Marks]**

**Solution : Given : **λ = 850 nm

σ = 45 nm

R.M.S pulse broadening due to material dispersion is given by,

σ_{m} = σ LM

Considering length L = 1 metre

For LED source operating at 850 nm, = 0.025

M = 9.8 ps/nm/km

\ σ_{m} = 45 x 1 x 9.8 = 441 ps/km

σ_{m} = **441 ns/km … Ans.**

** **

** **

**Example 2.6.2 : **What is the pulse spreading when a laser diode having a 2 nm spectral width is used? Find the the material-dispersion-induced pulse spreading at 1550 nm for an LED with a 75 nm spectral width **[Jan./Feb.-2007, 7 Marks]**

**Solutions : Given : **λ = 2 nm

σ = 75

σ_{m} = 2 x 1 x 50 = **100 ns/km … Ans.**

** **

For LED

σ_{m} = 75 x 1 x 53.76

σ_{m} = **4.03 ns/km … Ans.**

### 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
- Waveguide dispersion is significant only in fibers carrying fewer than 5-10 modes. Since multimode optical fibers carry hundreds of modes, they will not have observable waveguide
- The group delay (τ
_{wg}) arising due to waveguide dispersion.

… (2.6.5)

Where, b = Normalized propagation constant k = 2π / λ (group velocity)

Normalized frequency V,

\ ... (2.6.6)

The second term is waveguide dispersion and is mode dependent term..

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

### Chromatic Dispersion

** **

- The combination of material dispersion and waveguide dispersion is called chromatic 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

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

### 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 o 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
- Modal dispersion is given by,

where ∆t_{modal} = Dispersion

n_{1} = Core refractive index Z = Total fiber length

c = Velocity of light in air

∆ = Fractional refractive index

Putting in above equation

- The modal dispersion ∆t
_{modal}describes the optical pulse spreading due to modal effects optical pulse width can be converted to electrical rise time through the

### Signal distortion in Single Mode Fibers

** **

- The pulse spreading σ
_{wg}over range of wavelengths can be obtained from derivative of group delay with respect to λ.

where,

- This is the equation for waveguide dispersion for unit

**Example 2.6.3 **: For a single mode fiber n_{2} = 1.48 and Δ = 0.2 % operating at A = 1320 nm, compute the waveguide dispersion if

Solution : n_{2} = 1.48 Δ = 0.2

λ = 1320 nm Waveguide dispersion is given by,

### = -1.943 picosec/nm . km.

** **

**Higher Order Dispersion**

** **

- Higher order dispersive effective effects are governed by dispersion slope

where, D is total dispersion.

Also,

where,

β_{2} and β_{3} are second and third order dispersion parameters.

- Dispersion slope S plays an important role in designing WDM system

### Dispersion Induced Limitations

** **

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

### Basic Propagation Equation

** **

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

where,

β_{1}, β_{2} and β_{3} are different dispersion parameters.

### Chirped Gaussian Pulses

** **

- A pulse is said to b e chirped if its carrier frequency changes with time.
- For a Gaussian spectrum having spectral width σ
_{ω}, the pulse broadening factor is given by,

where, V_{ω} = 2σ_{ω} σ_{0}

### 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,

where, S is dispersion slope.

- Limiting bit rate a single mode fibers as a function of fiber length for σ
_{λ}= 0, a and 5nm is shown in fig. 2.6.5.

### Polarization Mode Dispersion (PMD)

** **

- Different frequency component of a pulse acquires different polarization state (such as linear polarization and circular polarization). This results in pulse broadening is kno
**w 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
### 2.1 Pulse Broadening in GI Fibers

- The core refractive index varies radially in case of graded index fibers, hence it supports multimode propagation with a low intermodal delay distortion and high data rate over long distance is possible. The higher order modes travelling in outer regions of the core, will travel faster than the lower order modes travelling in high refractive index region. If the index profile is carefully controlled, then the transit times of the individual modes will be identical, so eliminating modal dispersion.
- The m.s pulse broadening is given as :

where,

σ

_{intermodal}– R.M.S pulse width due to intermodal delay distortion.σ

_{intermodal}– R.M.S pulse width resulting from pulse broadening within each mode.- The intermodal delay and pulse broadening are related by expression given by

… (2.7.2)

Where τ

_{g}is group delay.From this the expression for intermodal pulse broadening is given as:

- The intramodal pulse broadening is given as :

… (2.7.4)

Where σ_{λ}is spectral width of optical source. Solving the expression gives :…(2.7.5)