Comparison of Step Index and Graded Index Fibres

Comparison of Step Index Fibres and Graded Index Fibres(GRIN)→

S.No.	Step Index Fibre	Graded Index Fibre
1.	In a step-index fibre, the refractive index of the core a constant value.	In graded-index fibre, the refractive index in the core decreases continuously in a nearly parabolic manner from a maximum value at the centre of the core to a constant value at the core-cladding interface.
2.	For a step-index fibre, the variation of refractive index is mathematically expressed as, $\begin{cases} & \mu(r)=\mu_{1} \qquad 0 < r < a \quad for (core)\\ & \mu(r)=\mu_{2} \qquad r >a \quad for(Cladding)\\ \end{cases} \\ Where \: \mu_{1} > \mu{2} $	Parabolic refractive index variation in GRIN fibre is mathematically expressed as, $ \begin{cases} & \mu^{2}(r)=\mu^{2}_{1} \left[ 1- \left(\frac{r}{\alpha} \right)^{2} \right] \qquad 0 < r < a \quad for (core) \\ & \mu(r)=\mu^{2}_{2} \qquad \qquad \qquad \qquad r > a \quad for (Cladding) \end{cases} $
3.	In the step-index fibre, the propagating light rays reflect abruptly from the Core cladding boundary.	In graded-index fibre, the propagating light rays bend smoothly as they approach the cladding.
4.	for given fibre diameter, the numerical aperture of step-index fibre is large.	For the same fibre diameter, the numerical aperture of graded-index fibre is small.
5.	In the step-index fibre, there may be some irregularities at the interface between the core and cladding.	In the graded-index fibre, there are no such irregularities at the interface between core and cladding.
6.	The step-index fibre has higher attenuation.	The graded-index fibre has lower attenuation.
7.	For a step-index fibre of a given physical size, with a loss of power of the order of $12 \frac{dB}{km}$, the numerical aperture is of the order of $0.2$ to $0.35$.	For a graded-index fibre of the same physical size, with an attenuation between $5$ to $10 \frac{dB}{km}$, the numerical aperture tends to run between $0.16$ and $0.2$
8.	In step index fibre, the time interval at the output end or pulse dispersion is expressed as, $\Delta \tau = \frac{\mu_{1} l}{c} \left ( \frac{\mu_{1}}{\mu_{2}} - 1 \right)=\frac{\mu_{1} l}{c} \Delta$ Where $l$ → The length of the fibre.	In a graded index fibre, the time interval at the output end or pulse dispersion is expressed as, $\Delta \tau = \frac{\mu_{2} l}{2c} \left ( \frac{\mu_{1} - \mu_{2}} {\mu_{2}} \right)^{2}=\frac{\mu_{2} l}{2c} \Delta^{2}$ Where $l$ → The length of the fibre.
9.	Pulse dispersion in multimode step-index fibre is large.	Pulse dispersion in a graded-index fibre is small.
10.	A good quality step-index fibre may have a bandwidth of $50 MHz km$	The equivalent graded-index fibre can have $200$, $400$, or $600 MHz km$ bandwidth.

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Comparison of Single Mode and Multimode Index Fibres

Comparison of Single-Mode Index Fibres and Multimode Index Fibres→

S.No.	Single Mode Index Fibre	Multimode Index Fibre
1.	In single mode index fibre, the diameter of the core is very small and is of the same order as the wavelength of light to be propagated. It is in the range $5\mu m - 10 \mu m$. The Cladding diameter is about $125 \mu m$.	In multimode index fibre, the diameter of the core is large. It is in the range $30\mu m - 100 \mu m$. The Cladding diameter is in the range $125 \mu m - 500 \mu m$.
2.	The difference in refractive indices of the core and cladding material is very small.	The difference in the refractive indices of the core and the cladding materials is large.
3.	In single-mode fibre, only a single mode is propagated.	In multi-mode fibre, a large number of modes can be propagated.
4.	Single mod fibre does require a much more sophisticated light source in order to launch enough light into the tiny core.	Multi-mode fibre does not require any sophisticated light source.
5.	Single-mode fibre is more expensive but more effective.	Multimode fibre is less expensive.
6.	The acceptance angle and the size of the acceptance cone of single-mode fibre are small.	The acceptance angle and the size of the acceptance cone of multimode fibre are large.
7.	The numerical aperture of single-mode fibre is small.	The numerical aperture of multimode fibre is large.
8.	Single-mode fibre has a very high information-carrying capability.	Multimode fibre has low information carrying capability.
9.	Single-mode fibre is used when sort distance communication is required.	It is used for long-distance communication.
10.	Model dispersion in single-mode fibre is almost nil.	Model dispersion in multimode fibre is the dominant source of dispersion.
11.	Material dispersion in single-mode fibre is low.	Material dispersion in multimode fibre is large.
12.	When a transmission has a very large bandwidth, single-mode fibre is used Example: Under Sea Cables.	When the system bandwidth requirement is low, multimode fibres are used Example: Datalink

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Numerical Aperture and Acceptance Angle of the Optical Fibre

Angle of Acceptance →

The angle of incident of light on the core for which the angle of incidence on the core-cladding interface equals the critical angle is known as the "Angle of Acceptance".

Transmission of light when the incident angle is less than the acceptance angle

For $\theta_{i}>\theta_{0}$, the angle of incidence on the core-cladding interface is less than the critical angle due to which part of incident light is transmitted as shown in the figure below

Transmission of light when the incident angle is greater than the acceptance angle

Hence the light will be transmitted through the fibre for angles of incidence on the core which are less than the acceptance angle. 

The acceptance angle is the maximum angle of incidence on the core for which total internal reflection takes place inside the core.

The light entering the core in a cone of semi-vertical angle $\theta_{0}$ is transmitted in the core through total internal reflections. This cone is known as the acceptance cone.

Numerical Aperture→

"The sine of the angle of acceptance of the optical fibre is known as the numerical aperture of optical fibre."

The numerical aperture determines the light-gathering ability of the fibre. It measures the amount of light that can be accepted by a fibre. The numerical aperture depends upon the refractive index of the core and cladding material and does not depend on the physical dimension of the fibre. It is a dimensionless quantity that is less than unity. The value of the numerical aperture range from $0.13$ to $0.15$. A large numerical aperture implies that a fibre accepts a large amount of light from the source. It varies due to variations of refractive index in the core and it has zero value after the core-cladding boundary. The number of propagation modes to multimode graded-index fibre depends upon the parameter of numerical aperture and hence upon the relative refractive index difference $\Delta n$

Derivation of Angle of Acceptance and Numerical Aperture →

Let us consider, step-index optical fibre into which light is launched from one end at angle $\theta_{i}$. Let the refractive index of core and cladding is $n_{1}$ and $n_{2}$.

Transmission of light when the incident angle is equal to the acceptance angle

Let us consider that light rays enter the fibre at an angle $\theta_{i}$ to the axis of the fibre refracted at an angle $\theta_{r}$ and strike the core-cladding interface at an angle $\phi$.

When ray incident at point $A$ on the core then According to Snell's law

$\frac{sin \theta_{0}}{sin \theta_{r}}= \frac{n_{1}}{n_{0}} $

Where $n_{0}$ → refractive index of air and vacuum

$sin\theta_{0}=\frac{n_{1}}{n_{0}} sin \theta_{r} \qquad(1)$

Now the refracted ray incident at point $B$ at the interface of core and cladding. So for critical angle condition

$n_{1}\: sin\phi=n_{2} \: sin90^{0} $

$n_{1}\: sin(90-\theta_{r})=n_{2} \: sin90^{0} \qquad (\because \phi=90-\theta_{r})$

$n_{1}\: cos \theta_{r}=n_{2}$

$cos\theta_{r}=\frac{n_{2}}{n_{1}} \qquad(2)$

$sin\theta_{r}=\sqrt{1-cos^{2}\theta_{r}}$

Now substitue the value of $cos\theta_{r}$ from equation$(2)$ to above equation then we get

$sin\theta_{r}=\sqrt{1- \left ( \frac{n_{2}}{n_{1}} \right)^{2}}$

$sin\theta_{r}=\frac{1}{n_{1}}\sqrt{n_{1}^{2}- n_{2}^{2}}$

Now substitue the value of $sin\theta_{r}$ in equation$(1)$ then we get

$sin\theta_{0}=\frac{n_{1}}{n_{0}}\frac{1}{n_{1}}\sqrt{n_{1}^{2}- n_{2}^{2}}$

$sin\theta_{0}=\frac{\sqrt{n_{1}^{2}- n_{2}^{2}}}{n_{0}}\qquad(3)$

This $sin\theta_{0}$ is known as Numerical Aperture. i.e.

$N.A.=\frac{\sqrt{n_{1}^{2}- n_{2}^{2}}}{n_{0}}$

If fibre is in the air then $n_{0}=1$ so the above equation can be written as

$N.A.=\sqrt{n_{1}^{2}- n_{2}^{2}}$

The equation $(3)$ also can be written as

$\theta_{0}=sin^{-1}\frac{\sqrt{n_{1}^{2}- n_{2}^{2}}}{n_{0}}$

The angle $\theta_{0}$ is known as the Angle of Acceptance.

The light is transmitted through the fibre when

$\theta_{i} < \theta_{0}$

i.e. $sin \theta_{i} < sin \theta_{0}$

$sin \theta_{i} < N.A.$

The light will be transmitted through the fibre with multiple total internal reflections when the above condition is satisfied.

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