• It is a dimensionless parameter that determines the number of modes a fiber can support. It is given by:

  $V= \frac{2π}λ a\sqrt{n_1^2-n_2^2} \\ \; \\ V = \frac{2πa}λ(N.A) \\ \; \\ V = \frac{2πan_1}λ(\sqrt{2∆}) \\ \; \\ $

  where,

  a = radius of the core

  λ = wavelength of ray

  N.A = numerical aperture

  $n_1$ and $n_2$ are the refractive indices of the core and cladding.

• Fiber with a V-parameter of less than 1.1505 only supports the fundamental mode, and is therefore a single mode fiber whereas fiber with a higher V-parameter has multiple modes.

• In single mode fiber, V is less than or equal to 1.1505, single mode fibers propagate the fundamental mode down the fiber core, while high order modes are lost in the cladding.

• For low V values, most of the power is propagated in the cladding material. Power transmitted by the cladding is easily lost at fiber bends.

• The value of the normalized frequency parameter (V) relates core size with mode propagation.

• The number of modes in an optical fiber distinguishes multimode optical fiber from single mode optical fiber.

• Graded Index fiber(Derivation):

1. Graded Index Fiber does not have a constant refractive index in the core. Due to this property they are also called inhomogeneous core fibers.

2. For guided modes we know the V number given as:

   $V= \frac{2π}λ a\sqrt{n_1^2-n_2^2 } \\ \; \\ V = \frac{2πa}λ(N.A) \\ \; \\ V =\frac {2πan_1}λ(\sqrt{2∆}) \\ \; \\ where, ∆ = relative \ \ refractive \ \ index \\ \; \\ Where, Δ=\frac{( n_1-n_2 )}{n_1} \\ \; \\ $

N.A = numerical aperture

Total number of guided modes is:

$M_g= \frac{α}{(α+2)}. (n_{1}. \frac{2π}λ.a)^2 . Δ$

$\alpha$ = profile parameter which gives the characteristics refractive index profile of the fiber core.

$but,n_{1} \frac{2π}λ.a.\sqrt{2.∆} =V$

$(n_1. \frac{2π}{λ}.a)^2 . \delta = V^2/2$

$∴ M_g = \frac{α}{α+2}. {V^2}2$

For a parabolic refractive index profile core fiber ($\alpha$=2),

$$M_g= \frac{V^2}4$$