An optical signal is an electromagnetic signal that consists of different wavelengths. Each wavelength components travels through the fiber at different speeds depending on the refractive index of the material. These individual wavelengths reach the other end of the fiber at different time. It can cause the wave to spread out and become distorted over time. This broadening of pulses over time is called Dispersion. The distortion of optical signals affects the quality of the signals. It may be observed that each individual pulse within the signal broadens and overlaps with its neighbours, eventually becoming indistinguishable at the receiver input. 

The amount of pulse broadening depends on several factors, including the length of the fiber, the wavelength of the light, and the properties of the fiber itself, such as its refractive index profile and material composition. Pulse broadening can limit the maximum data rate that can be transmitted over long distances in optical communication systems.

Dispersion in optical fiber includes intramodal, intermodal and polarization mode dispersions.

Intermodal dispersion

In optical fibers, modes refer to the different paths that light can take as it travels through the fiber core. The fiber can be classified into two types based on the number of propagation modes they support: single-mode (only one propagation mode) and multimode fibers (simultaneous propagation of multiple distinct light modes).

Some of these light rays will travel in a straight path through the center of the fiber called the axial mode. The remaining light rays will repeatedly bounce off the core-cladding boundary as they travel through the waveguide. Each mode propagating within the fiber follows a different angle and paths. As these modes travel at different speeds, it reaches the fiber output at different time. The differences in the propagation time for each of these modes results in the spreading out of signals between the modes. This is called inter-modal dispersion. The spreading out causes signal overlapping which makes it difficult to distinguish them separately. As the path length increases, there will be a corresponding increase in the dispersion of the modes. The pulse width at the output depends on the transmission times of the slowest and fastest modes.

Modal dispersion is absent in single mode fibers as there is only one mode that travels in the fiber.

The pulse broadening due to intermodal dispersion is given as:

Pulse broadening Δt is 

Δt = L x (dispersion/km)

N.A is the numerical aperture

L is the fiber length

n₁ is the refractive index of the core

Intramodal dispersion

An optical signal can have different wavelengths that travels at a slightly different speed and arrive at the receiver at different times within the mode causing the signal to spread out and become distorted. This type of dispersion is called Intramodal dispersion and occurs in both single and multimode fibers. 

Intramodal dispersion can be further divided into material dispersion and waveguide dispersion. The delay difference due to the waveguide materials' dispersive properties is termed as material dispersion, and due to the geometry of the waveguide leads to waveguide dispersion. 

Material dispersion

Material dispersion is caused by the variation in the refractive index of the material used in the optical fiber. Since the refractive index of the material determines the speed at which light propagates through the fiber, different wavelengths of light travel at different speeds, causing the signal to spread out and become distorted. The sign of material dispersion can be positive or negative depending on the material properties and the wavelength of the transmitted light. The sign of dispersion refers to the direction of the change in refractive index with respect to the change in wavelength.

In materials with normal dispersion, the refractive index decreases as the wavelength of light increases. This means that the longer wavelengths of light (such as red) travel faster than the shorter wavelengths (such as blue), causing the pulse to spread out over time. In this case, the sign of dispersion is negative.

On the other hand, in materials with anomalous dispersion, the refractive index increases as the wavelength of light increases. This means that the shorter wavelengths of light (such as blue) travel faster than the longer wavelengths (such as red), causing the pulse to also spread out over time. In this case, the sign of dispersion is positive.

These transmitted signals will have different delay times and cause a spread in arrival times Δt.

The pulse broadening due to material dispersion is given by:

D_M is the material dispersion coefficient,

Total pulse spreading by material dispersion is given by Δt_mat :

In multimode fibers, intermodal dispersion is much larger than material dispersion. 

Waveguide dispersion

Waveguide dispersion, on the other hand, is caused by the geometry of the fiber optic cable itself. This type of dispersion occurs because the light waves that propagate through the fiber are confined to a narrow waveguide structure, which can cause the different wavelengths of light to travel at different speeds due to the geometrical variations. It can cause the signal to spread out and become distorted, and is typically more prominent at shorter wavelengths. Waveguide dispersion can be reduced by using fibers with a wider core diameter or by using specialized waveguide structures to improve the uniformity of the waveguide geometry. It occurs in both single and multi-mode optical fiber.

In an optical fiber, light propagates through the core of the fiber, which is surrounded by a cladding layer with a lower refractive index. Light at shorter wavelengths stays more in the core, while longer wavelengths is distributed in the cladding. The longer wavelengths travels at a higher propagation speed than the shorter wavelengths due to the lower refractive index of the cladding. This time delay results in the spreading out of the signals.

Waveguide dispersion is always negative in optical fibers due to the difference in effective refractive index for different modes of light, which is a result of the fiber's geometry and the way the refractive index of the fiber changes with frequency. The higher-order modes experience a lower effective refractive index than the lower-order modes.

Pulse spreading due to waveguide dispersion is given as:

D_W is the waveguide dispersion coefficient,

Waveguide dispersion can be reduced using fibers with larger cores, which allow more propagation modes and reduce the difference in the effective refractive index between the modes. 

Waveguide dispersion, however, may be significant in single-mode fibers where the effects of the various dispersion mechanisms are difficult to distinguish.

Intramodal dispersion is expressed in terms of the chromatic dispersion parameter D, which is related to β, the mode propagation constant, a function of size of the fiber’s core relative to the wavelength of operation.

Both material and waveguide dispersion are measured in picoseconds per nanometer per kilometer. This cause an increase in magnitude of source linewidth and an increase in dispersion with fiber length.

Polarization mode dispersion 

Light has two different polarization modes that travel at different speed inside the core due to the variations in refractive index. This phenomenon is called birefringence, where the refractive index of the fiber is different for light polarized in different directions. This arise due to the asymmetry of the fiber structure. The difference in speed results in the delay in arrival of signal and the pulse broadening. This kind of dispersion due to the light distributed over different polarizations is polarization mode dispersion.

Single mode fiber consists of one propagation mode comprising two orthogonal polarization modes.

Polarization mode dispersion is given as follows:

D_PMD is the polarization mode dispersion coefficient.

Total fiber dispersion

For a given fiber length, total fiber dispersion is calculated as:

Other types of dispersions include Residual chromatic dispersion and Kerr effect dispersion.

• Residual dispersion occurs because of imperfections in the manufacturing process or variations in the fiber's physical properties, distorting the signal.
• Kerr effect dispersion occurs because the refractive index of the fiber core can be modified by the intensity of the light passing through it, causing different wavelengths to travel at different speeds.

Applications of Dispersion

The dispersion can also have useful applications such as fiber optic sensors, dispersion compensation systems, optical sensing, nonlinear optics, pulse shaping, wavelength division multiplexing, biomedical applications such as optical coherence tomography (OCT), gemology, imaging, and pulsar emissions.

It can also be used to design integrated optical circuits, devices that manipulate light at a small scale. Dispersion is also used to analyze the frequency response of waveguide structures, such as filters and resonators.