Optical Fiber Waveguides

In free space light ravels as its maximum possible speed i.e. 3 x 10⁸ m/s or 186 x 10³ miles/sec. When light travels through a material it exnibits certain behavior explaned by laws of reflection, refraction.

Electromagnetic Spectrum

The radio waves and light are electromagnetic waves. The rate at which they alternate in polarity is called their frequency (f) measured in hertz (Hz). The speed of electromagnetic wave (c) in free space is approximately 3 x 10⁸ m/sec. The distance travelled during each cycle is called as wavelength (λ)

In fiber optics, it is more convenient to use the wavelength of light instead of the frequency with light frequencies, wavlengfth is often stated in microns or

1 micron (µ) = 1 Micrometre (1 x 10^-6)

1 nano (n) = 10^-9 metre

Fiber optics uses visible and infrared light. Infrared light covers a fairly wide range of wavelengths and is generally used for all fiber optic communications. Visible light is normally used for very short range transmission using a plastic

Ray Transmission Theory

Before studying how the light actually propagates through the fiber, laws governing the nature of light m ust be These was called as laws of optics (Ray theory). There is conception that light always travels at the same speed. This fact is simply not true. The speed of light depends upon the material or medium through which it is moving. In free space light travels at its maximum possible speed i.e. 3 x 108 m/s or 186 x 103 miles/sec. When light travels through a material it exhibits certain behavior explained by laws of reflection, refraction.

Reflection

The law of reflection states that, when a light ray is incident upon a reflective surface at some incident angle f₁ from imaginary perpendicular normal, the ray will be reflected from the surface at some angle f₂ from normal which is equal to the angle of

Refraction occurs when light ray passes from one medium to another e. the light ray changes its direction at interface. Refractio occurs whenever density of medium changes.

E.g. refraction occurs at air and water interface, the straw in a glass of water will appear as it is bent.

The refraction can also observed at air and glass interface.

When wave passes through less dense medium to more dense medium, the wave is refracted (bent) towards the Fig. 1.6.3 shows the refraction phenomena.
The refraction (bending) takes place because light travels at different spped in different The speed of light in free space is higher than in water or glass.

Refractive Index

The amount of refraction or bending that occurs at the interface of two materials of different densities is usually expressed as refractive index of two materials. Refractive index is also known as index of refraction and is denoted by
Based on material density, the refractive index is expressed as the ratio of the velocity of light in free space to the velocity of light of the dielectric material (substance).

The refractive index for vacuum and air os 1.0 for water it is 1.3 and for glass refractive index is 1.5.

Snell’s Law

Snell’s law states how light ray reacts when it meets the interface of two media having different indexes of refraction.
Let the two medias have refractive indexes n1 and n₂ where n₁ >n₂.

f1 and f2 be the angles of incidence and angle of refraction respectively. Then according to Snell’s law, a relationship exists between the refractive index of both materials given by,

A refractive index model for Snell’s law is shown in 1.6.4.

The refracted wave will be towards the normal when n₁ < n₂ and will away from it when n₁ > n₂.

Equation (1.6.1) can be written as,

This equation shows that the ratio of refractive index of two mediums is inversely proportional to the refractive and incident angles.

As refractive index and substituting these values in equation (1.6.2)

Critical Angle

When the angle of incidence (f₁) is profressively increased, there will be progressive increase of refractive angle (f₂). At some condition (f₁) the refractive angle (f₂) becomes 90o to the normal. When this happens the refracted light ray travels along the interface. The angle of incidence (f₁) at the point at which the refractive angle (f₁) becomes 90^o is called the critical angle. It is denoted by f_c.
The critical angle is defined as the minimum angle of incidence (f₁) at which the ray strikes the interface of two media and causes an agnle of refraction (f₂) equal to 90^o.

Hence at critical angle f₁ = f_c and f₂ = 90^o Using Snell’s law : n₁ sin f₁ = n₂ sin f₂

Therefore,

The actual value of critical angle is dependent upon combination of materials present on each side of boundary.

Total Internal Refleciton (TIR)

When the incident angle is increase dbeyond the critical angle, the light ray does not pass through the interface into the other medium. This gives the effect of mirror exist at the interface with no possibility of light escaping outside the medium. In this condition angle of reflection (f₂) is equal to angle of incidence (f₁). This action is called as Total Internal Reflection (TIR) of the beam. It is TIR that leads to the propagation of waves within fiber-cable TIR can be observed only in materials in which the velocity of light is less than in air.

The two conditions necessary for TIR to occur are :

The refractive index of first medium must be greater than the refractive index of second
The angle of incidence must be greater than (or equal to) the critical

Example 1.6.1 : A light ray is incident from medium-1 to medium-2. If the refractive indices of medium-1 and medium-2 are 1.5 and 1.36 respectively then determine the angle of refraction for an angle of incidence of 30^o.

Solution : Medium-1 n1 = 1.5

Medium-2 n2 = 1.36 Angle of incidence f₁ = 30^o. Angle of incident f₂ = ?

Angle of refraction 33.46^o from normal.

Example 1.6.2 : A light ray is incident from glass to air. Calculate the critical angle (f_c).

Solution : Refractive index of glass n₁ = 1.50 Refrative indes of air n₂ = 1.00

From definition of critical angle, f₂ = 90^o and f₁ = f_c.

Example 1.6.3 : Calculate the NA, acceptance angle and critical angle of the fiber having n₁ (Core refractive index) = 1.50 and refractive index of cladding = 1.45.

Soluiton : n₁ = 1.50, n₂ = 1.45

Optical Fiver as Waveguide

An optical fiber is a cylindrical dielectric waveguide capable of conveying electromagnetic waves at optical frequencies. The electromagnetic energy is in the form of the light and propagates along the axis of the The structural of the fiver determines the transmission characteristics.
The propagation of light along the waveguide is decided by the modes of the waveguides, here mode means path. Each mode has distict pattern of electric and magnetic field distributions along the fiber length. Only few modes can satisfy the homogeneous wave equation in the fiver also the boundary condition a waveguide surfaces. When there is only one path for light to follow then it is called as single mode propagation. When there is more than one path then it is called as multimode
A single fiber structure is shown in Fig. 1.6.6. It consists of a solid dielectric cylinder with radius ‘a’. This cylinder is called as core of fiber. The core is surrounded by dielectric, called cladding. The index of refraction of core (glass fiber) is slightly greater than the index of refraction of

If refractive index of core (glass fiver) = n₁ and refractive index of cladding = n₂

then n₁ > n₂.

Propagation in Optical Fiber

To understand the general nature of light wave propagation in optical fiber. We first consider the construction of optical fiber. The innermost is the glass core of very thin diameter with a slight lower refractive index n2. The light wave can propagate along such a optical fiber. A single mode propagation is illustrated in Fig. 1.6.7 along with standard size of fiber.

Single mode fibers are capable of carrying only one signal of a specific wavelength.
In multimode propagation the light propagates along the fiber in zigzag fashion, provided it can undergo total internal reflection (TIR) at the core cladding
Total internal reflection at the fiber wall can occur only if two conditions are

Condition 1:

The index of refraction of glass fiber must be slightly greater than the index of refraction of material surrounding the fiber (cladding).

If refractive index of glass fiber = n₁ and refractive index of cladding = n₂ then n₁ > n₂.

Condition 2 :

The angle of incidence (f₁) of light ray must be greater than critical angle (f_c).

A light beam is focused at one end of cable. The light enters the fibers at different angles. 1.6.8 shows the conditions exist at the launching end of optic fiber. The light source is surrounded by air and the refractive index of air is n₀ = 1. Let the incident ray makes an angle f₀ with fiber axis. The ray enters into glass fiber at point P making refracted angle f₁ to the fiber axis, the ray is then propagated diagonally down the core and reflect from the core wall at point Q. When the light ray reflects off the inner surface, the angle of incidence is equal to the angle of reflection, which is greater than critical angle.
In order for a ray of light to propagate down the cable, it must strike the core cladding interface at an angle that is greater than critical angle (f_c).

Acceptance Angle

Applying Snell’s law to external incidence angle. n₀ sin f₀ = n₁ sin f₁

But f₁ = (90 - f_c)

sin f₁ = sing (90 - f_c) = cos f_c

Substituting sin f₁ in above equation. n₀ sin f₀ = n₁ cos f_c

Applying Pythagorean theorem to ΔPQR.

The maximum value of external incidence angle for which light will propagate in the

fiber.

When the light rays enters the fivers from an air medium n₀ = 1. Then above equation

reduces to,

The angle f₀ is called as acceptance angle and defines the maximum angle in which the light ray may incident on fiber to propagate down the fiber.

Acceptance Cone

Rotating the acceptance angle around the fiber axis, a cone shaped pattern is obtained, it is called as acceptance cone of the fiber input. Fig 1.6.10 shows formation of acceptance cone of a fiber cable.

The Cone of acceptance is the angle within which the light is accepted into the core and is able to travel along the fiber. The launching of light wave becomes easier for large acceptance
The angle is measured from the axis of the positive cone so the total angle of convergence is actually twice the stated value.

Numerical Aperture (NA)

The numerical aperture (NA) of a fiber is a figure of merit which represents its light gathering Larger the numerical aperture, the greater the amount of light accepted by fiber. The acceptance angle also determines how much light is able to be enter the fiber and hence there is relation between the numerical aperture and the cone of acceptance.

Numerical aperture (NA) = sin

For air n_o = 1

Hence acceptance angle = sin^-1 NA

By the formula of NA note that the numerical aperture is effectively dependent only on refractive indices of core and cladding material. NA is not a function of fiber dimension.

The index difference (Δ) and the numerical aperture (NA) are related to the core and cladding indices:

Also

Example 1.6.5 : Calculate the numerical aperture and acceptance angle for a fiber cable of which n_core = 1.5 and n_cladding = 1.48. The launching takes place from air.

Solution :

NA = 0.244

Acceptance angle = sin^-1 0.244

f₀ = 14.12^o

Types of Rays

If the rays are launched within core of acceptance can be successfully propagated along the fiber. But the exact path of the ray is determined by the position and angle of ray at which it strikes the

There exists three different types of rays.

Skew rays ii) Meridional rays iii) Axial
- The skew rays does not pass through the center, as show in Fig. 1.6.11 (a). The skew rays reflects off from the core cladding boundaries and again bounces around the outside of the core. It takes somewhat similar shape of spiral of helical

The meridional ray enters the core and passes through its axis. When the core surface is parallel, it will always be reflected to pass through the enter. The meridional ray is shown in fig. 1.6.11 (b).
The axial ray travels along the axis of the fiber and stays at the axis all the time. It is shown in fig. 1.6.11 (c).

Modes of Fiber

Fiber cables cal also be classified as per their Light rays propagate as an electromagnetic wave along the fiber. The two components, the electric field and the magnetic field form patterns across the fiber. These patterns are called modes of transmission. The mode of a fiber refers to the number of paths for the light rays within the cable. According to modes optic fibers can be classified into two types.

i) Single mode fiber ii) Multimode

Multimode fiber was the first fiber type to be manufactured and commercialized. The term multimode simply refers to the fact that numerous modes (light rays) are carried simultaneously through the waveguide. Multimode fiber has a much larger diameter, compared to single mode fiber, this allows large number of modes.
Single mode fiber allows propagation to light ray by only one path. Single mode fibers are best at retaining the fidelity of each light pulse over longer distance also they do not exhibit dispersion caused by multiple modes.

Thus more information can be transmitted per unit of time.

This gives single mode fiber higher bandwidth compared to multimode fiber.

Some disadvantages of single mode fiber are smaller core diameter makes coupling light into the core more difficult. Precision required for single mode connectors and splices are more

Fiber Profiles

A fiber is characterized by its profile and by its core and cladding
One way of classifying the fiber cables is according to the index profile at fiber. The index profile is a graphical representation of value of refractive index across the core
There are two basic types of index

Step index ii) Graded index fiber. Fig. 1.6.12 shows the index profiles of fibers.

Step Index (SI) Fiber

The step index (SI) fiber is a cylindrical waveguide core with central or inner core has a uniform refractive index of n₁ and the core is surrounded by outer cladding with uniform refractive index of n₂. The cladding refractive index (n₂) is less than the core refractive index (n₁). But there is an abrupt change in the refractive index at the core cladding Refractive index profile of step indexed optical fiber is shown in Fig.

1.6.13. The refractive index is plotted on horizontal axis and radial distance from the core is plotted on vertical axis.

The propagation of light wave within the core of step index fiber takes the path of meridional ray e. ray follows a zig-zag path of straight line segments.

The core typically has diameter of 50-80 µm and the cladding has a diameter of 125 µm.

The refractive index profile is defined as –

Graded Index (GRIN) Fiber

The graded index fiber has a core made from many layers of glass.
In the graded index (GRIN) fiber the refractive index is not uniform within the core, it is highest at the center and decreases smoothly and continuously with distance towards the The refractive index profile across the core takes the parabolic nature. Fig.

1.6.14 shows refractive index profile of graded index fiber.

In graded index fiber the light waves are bent by refraction towards the core axis and they follow the curved path down the fiber length. This results because of change in refractive index as moved away from the center of the
A graded index fiber has lower coupling efficiency and higher bandwidth than the step index fiber. It is available in 50/125 and 62.5/125 sizes. The 50/125 fiber has been optimized for long haul applications and has a smaller NA and higher 62.5/125 fiber is optimized for LAN applications which is costing 25% more than the 50/125 fiber cable.
The refractive index variation in the core is giver by relationship

where,

r = Radial distance from fiber axis a = Core radius

n₁ = Refractive index of core

n₂ = Refractive index of cladding α = Shape of index profile.

Profile parameter α determines the characteristic refractive index profile of fiber The range of refractive index as variation of α is shown in Fig. 1.6.15.

Comparison of Step Index and Graded Index Fiber

Sr. No.	Parameter	Step index fiber	Graded index fiber
1.	Data rate	Slow.	Higher
2.	Coupling efficiency	Coupling efficiency with fiber is higher.	Lower coupling efficiency.
3.	Ray path	By total internal reflection.	Light ray travels in oscillatory fashion.
4.	Index variation
5.	Numerical aperture	NA remains same.	Changes continuously with distance from fiber axis.
6.	Material used	Normally plastic or glass is preferred.	Only glass is preferred.
7.	Bandwidth efficiency	10 – 20 MHz/km	1 GHz/km
8.	Pulse spreading	Pulse spreading by fiber length is more.	Pulse spreading is less
9.	Attenuation of light	Less typically 0.34 dB/km at 1.3 µm.	More 0.6 to 1 dB/km at 1.3 µm.

10.

Typical light source

LED.

LED, Lasers.

11.

Applications

Subscriber local network

communication.

Local and wide area

networks.

Optic Fiber Configurations

Depending on the refractive index profile of fiber and modes of fiber there exist three types of optical fiber configurations. These optic-fiber configurations are -

Single mode step index
Multimode step index fiber.

Multimode graded index fiber.

Single mode Step index Fiber

In single mode step index fiber has a central core that is sufficiently small so that there is essentially only one path for light ray through the cable. The light ray is propagated in the fiber through reflection. Typical core sizes are 2 to 15 µm. Single mode fiber is also known as fundamental or monomode

Single mode fiber will permit only one mode to propagate and does not suffer from mode delay differences. These are primarily developed for the 1300 nm window but they can be also be used effectively with time division multiplex (TDM) and wavelength division multiplex (WDM) systems operating in 1550 nm wavelength region.
The core fiber of a single mode fiber is very narrow compared to the wavelength of light being used. Therefore, only a single path exists through the cable core through which light can Usually, 20 percent of the light in a single mode cable actually travels down the cladding and the effective diameter of the cable is a blend of single mode core and degree to which the cladding carries light. This is referred to as the ‘mode field diameter’, which is larger than physical diameter of the core depending on the refractive indices of the core and cladding.

The disadvantage of this type of cable is that because of extremely small size interconnection of cables and interfacing with source is difficult. Another disadvantage of single mode fibers is that as the refractive index of glass decreases with optical wavelength, the light velocity will also be wavelength dependent. Thus the light from an optical transmitter will have definite spectral width.

Multimode step Index Fiber

Multimode step index fiber is more widely used type. It is easy to manufacture. Its core diameter is 50 to 1000 µm e. large aperture and allows more light to enter the cable. The light rays are propagated down the core in zig-zag manner. There are many many paths that a light ray may follow during the propagation.
The light ray is propagated using the principle of total internal reflection (TIR). Since the core index of refraction is higher than the cladding index of refraction, the light enters at less than critical angle is guided along the fiber.

Light rays passing through the fiber are continuously reflected off the glass cladding towards the centre of the core at different angles and lengths, limiting overall
The disadvantage of multimode step index fibers is that the different optical lengths caused by various angles at which light is propagated relative to the core, causes the transmission bandwidth to be fairly small. Because of these limitations, multimode step index fiber is typically only used in applications requiring distances of less than 1 km.

Multimode Graded Index Fiber

The core size of multimode graded index fiber cable is varying from 50 to 100 µm The light ray is propagated through the refraction. The light ray enters the fiber at many different angles. As the light propagates across the core toward the center it is intersecting a less dense to more dense medium. Therefore the light rays are being constantly being refracted and ray is bending continuously. This cable is mostly used for long distance communication.

The light rays no longer follow straight lines, they follow a serpentine path being gradually bent back towards the center by the continuously declining refractive index. The modes travelling in a straight line are in a higher refractive index so they travel slower than the serpentine modes. This reduces the arrival time disparity because all modes arrive at about the same
Fig 1.6.19 shows the light trajectory in detail. It is seen that light rays running close to the fiber axis with shorter path length, will have a lower velocity because they pass through a region with a high refractive

Rays on core edges offers reduced refractive index, hence travel more faster than axial rays and cause the light components to take same amount of time to travel the length of fiber, thus minimizing dispersion losses. Each path at a different angle is termed as ‘transmission mode’ and the NA of graded index fiber is defined as the maximum value of acceptance angle at the fiber axis.
Typical attenuation coefficients of graded index fibers at 850 nm are 2.5 to 3 dB/km, while at 1300 nm they are 0 to 1.5 dB/km.
The main advantages of graded index fiber are:

Reduced refractive index at the centre of
Comparatively cheap to

Standard fibers

Sr. No.	Fiber type	Cladding diameter (µm)	Core diameter (µm)	Δ	Applications
1.	Single mode (8/125)	125	8	0.1% to 0.2%	1. Long distance 2. High data rate
2.	Multimode (50/125)	125	50	1% to 2%	1. Short distance 2. Low data rate
3.	Multimode (62.5/125)	125	62.5	1% to 2%	LAN
4.	Multimode (100/140)	140	100	1% to 2%	LAN