# Light Propagation in Optical Fibers Study Notes

By Chandani Prakash|Updated : December 1st, 2021

Fibre Optic Communications: Fibre optic communication is a method of transmitting information from one place to another by sending pulses of light through an optical fiber. The information-carrying capacity of a cable or radio channel is directly proportional to its bandwidth.

• A transducer basically converts information from a source into an electrical signal.
• An optical source (LASER or LED) converts an electrical signal into an optical (light) signal.
• The optical channel is an optical fiber.
• An optical detector converts the optical signals into an electrical signals. Avalanche photo-diode is used as an optical detector.

Key Points

• If there is an interaction between a material and light then we go for photon theory.
• When we deal with the transmission of optical signal or reception of optical signal then we go for wave theory.
• When we have to compare the behavior of light in two mediums then we go for ray theory.
• The primary application of fiber optic communications is in long-distance telephone systems.
• Because of the great attenuation of light in a fiber optic cable, repeater units are used to amplify and regenerate the signals over long distances.
• Because of the very high frequency of light compared to typical information signals, tremendous bandwidth is easily available.
• Light is an electromagnetic signal like a radio wave but is much higher in frequency. It can be used as a carrier for information signals.
• The angle at which light strikes a surface is called the angle of incidence. The angle at which light is reflected from a surface is called the angle of reflection. The angle of incidence is equal to the angle of reflection.
• When a light ray passes from one medium to another, it is bent. This is called refraction.
• The amount of refraction is called the index of refraction n and is the ratio of the speed of light in air to the speed of light in another medium, such as water, glass, or plastic. [n = 1 in air, n = 1.3 in water, n = 1.5 in glass]. n=c/v, the Value of the refractive index is always greater than or equal to 1.

Total Internal Reflection: By Snell's law n1 sin φ1 = n2 sin φ2

if φ2 = 90; internal reflection takes place

Total internal reflection will take place if the angle of incidence φ1 will be greater than or equal to the critical angle.

• The critical angle is the angle of incidence that causes the refracted light to travel along the interface between two different media.
• All the information in optical fiber is carried out by the principle of total internal reflection and all the information is carried in the core of the optical fiber.
• Cladding does not support any transmission of information.

Key Points

• A popular fiber optic cable with a glass core and plastic cladding is called Plastic Clad Silica (PCS).
• The cladding surrounding the core protects the core and provides an interface with a controlled index of refraction.

Acceptance Angle a): Acceptance angle is the maximum angle to the fiber axis at which the light may enter in order to propagate.

n1, n2 = Refractive indices of core and cladding respectively

If the angle of incidence to the fiber will be greater than θa, then the total internal reflection will not take place in the optical fiber and some information will be lost.

Numerical Aperture (NA): It is a measure of the light-collecting ability of the fiber. It establishes the relationship between the acceptance angle and refractive indexes of the different mediums involved.

n0 sin θa = n1sin(90 – θc) = n1cos θc

Relative Refractive Index Difference (Δ)

NA = n1(2Δ)1/2

Meridional Rays: Meridional rays are those rays that pass through the core axis.

Skew Rays: Skew rays are those rays that do not pass through the core axis.

where, θas = Acceptance angle for skew says.

γ = The angle between the angle of incidence and normal at the point of incidence.

Key Points

• θas is always greater than θa.
• Skew rays will be in large numbers inside fiber.

Normalized Frequency (v)

λ = Wavelength of operation

It is a dimensionless quantity.

Step Index Fibre: The step index means there is a sharp difference in the index of refraction between the core and cladding.

n(r) = n1 r < a

n2 r ≥ a

Mode: Number of modes or mode volume . Some modes depend on energy and phase equivalence is given below.

Key Points

• A single-mode cable is very small in diameter and essentially provides only a single path for light.
• Multimode cores are large and provide multiple paths for light.
• Multiple light paths through a step-index core cause a light pulse to be stretched and attenuated. This is called modal dispersion and it limits the upper pulse repetition rate and thus the information bandwidth.
• If we reduce the radius such that a single mode is only transmitted through the core so no dispersion and so no different delays

Graded Index Fibre: Graded index means that the index of refraction of the core varies over its cross-section, highest in the center and lowest at the edges.

n2 = n1[1 – 2Δ]1/2 ; r ≥ a

where α is called as the profile of the fiber.

• As a increases, graded index fiber will tend toward step-index fiber behavior.
• Multiple light paths in a graded index core are controlled so that they converge at multiple points along the cable. Modal dispersions do occur, but it is not as serve as that caused by a step-index core.
• Modal dispersion does not occur in single-mode cores.
• The three most widely used types of fiber-optic cables are multimode step-index, single-mode step-index, and multimode graded-index.
• The number of modes or mode volume is
• If we want to transmit only a single mode, then the normalized frequency must be in the range 0 ≤ v ≤ vc, where vc = 2.405.
• For single-mode transmission minimum value of wavelength, λ is

Losses in Optical Fibre: The primary specification of a fiber optic cable is attenuation which is usually expressed as the loss in decibels per kilometer. Light loss in a fiber optic cable is caused by absorption, scattering, and dispersion.

Absorption Losses: Absorption losses can be classified as

• Intrinsic [due to interaction of one or major components of glass]
• Extrinsic [due to OH ion or due to transition element impurity]

Key Points

• Cable attenuation is directly proportional to its length.
• Cable losses at range from 1db/km in glass single-mode step-index cable to 100 dB/km for plastic multi-mode step-index cable.
• Fiber-optic cables can be spliced by gluing.

For multimode

For single mode

Optical Sources: Fibre-optic systems use Light Emitting Diodes (LEDs) and semiconductor lasers as the main light sources. Laser is the source of monochromatic and coherent light. LED is the source of monochromatic and non-coherent light. Light-emitting diodes are used in short-distance, low-speed systems. Injection Laser Diodes (ILDs) are used in long-distance, high-speed systems.

Key Points

• In the case of the He-Ne laser for the same output optical power loss, input power is required compared to the Ruby laser.
• Most LEDs and ILDs emit light in the invisible near-infrared range (0.82 to 1.55μm)
• In the case of LED total recombination rate Rt is directly proportional to forward biased current and is given by Rt = Rnr + Rr

Internal Quantum Efficiency of LED: The internal quantum efficiency of LED can be calculated as

where e is a charge on an electron

Internal power Fint = Photon rate × hf

Pint = Rr × hf

External power is somewhat lesser than internal power and is given by

where, F is the transmission coefficient and lies between (0, 1), 0 for opaque and 1 for transparent medium.

n = Refractive index of the medium

nx = Refractive index of the crystal

• External quantum efficiency is equal to  , where P is input electrical power.

and

Coupling Efficiency: Coupling efficiency is a parameter that defines the amount of optical power coupled to the optical fiber from LED.

for I(θ) = I0 cos θ

Optical Detector: The most commonly used light sensor is a photodiode or PIN diode. PIN junction diodes are faster and more sensitive than conventional photodiodes. The fastest and most sensitive light detector is the Avalanche Photo Diode (APD). The optical detector must have high sensitivity, high fidelity, short response time, and stability within the range of operation.

Quantum Efficiency of Optical Detector: The quantum efficiency of the optical detector can be calculated as

where re and rp are electron and photon generation rates respectively.

Responsivity (R): It gives the transfer characteristic of the detector and is equal to

where, Ip = Photon current, P0 = Incident photon power

P0 = Photon rate × hf

P0 = rp × hf

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