Question

Find the maximum possible core diameter which allows single_mode operation at a wavelength of 1.3 μm in a graded-index fiber having a parabolic refractive index profile core with core refractive index at the core axis of 1.5 and relative index difference of 1%.

1. 2.5 μm
2. 3.3 μm
3. 5 μm
4.  6.6 μm

Answer 1

Correct Answer - Option 4 :  6.6 μm

Concept:

• Optical fiber is a dielectric waveguide that operates in optical frequencies, normally in cylindrical form.
• The inner cylindrical structure is core surrounded by the solid dielectric called cladding.
• Variation in material composition gives rise to two commonly used fibers.
• If the refractive index of fiber is uniform throughout and undergoes an abrupt change at the cladding boundary is called step-index fiber.
• If the refractive index of fiber is varied as a function of radial distance from the center of the fiber is called graded-index fiber.

V number

It is a dimensionless number that is related to wavelength and numerical aperture and determines how many modes fiber can support and is given by.

\(V = \frac{{2\pi a}}{\lambda }{\left( {n_1^2 - n_2^2} \right)^{\frac{1}{2}}}\)

\(\\V = \frac{{2\pi a}}{\lambda }NA\)

\( \\V= \frac{{2\pi a}}{\lambda }{n_1}\sqrt {2{\rm{\Delta }}} \)

Where,

n₁ = refractive index of core

n₂ = refractive index of cladding

a = radius of the core

λ = operating wavelength

Δ  = Relative index difference of fiber and is given by

\(\Delta = \frac{n_1-n_2}{n_1}\)

NA = numerical  aperture of the fiber and is defined as

The numerical aperture of an optical fiber is a numerical value 

it is the sine of the maximum possible launching angle of the optical fiber and is given by

\(NA = \;nsin\left( \theta \right) \)

\(\\NA= \;{\left( {n_1^2 - n_2^2} \right)^{\frac{1}{2}}} \)

\(\\NA= {n_1}\sqrt {2{\rm{\Delta }}}\)

n = refractive index of medium from which light ray enter the fiber core

For lowest order or single-mode operation, V number is given by

\(V = 2.405\sqrt {1 + \frac{2}{\alpha }} \)

α = various profile parameter of the fiber

α = 1 for triangular profile

α = 2 for parabolic profile

Calculation:

Given that

Operation wavelength (λ) = 1.3 μm

Core refractive index (n₁) = 1.5

Relative index difference (Δ) = 0.01

α = 2 for parabolic profile

then

\(V = 2.405\sqrt {1 + \frac{2}{\alpha }} \)

\(V = 2.405\sqrt {1 + \frac{2}{2 }} \)

V = 2.405 × √2 = 3.401

We know that

\(V = \frac{{2\pi a}}{\lambda }{n_1}\sqrt {2{\rm{\Delta }}} \)

The above equation can be rearranged as

\(a = \frac{{V\lambda }}{{2\pi {n_1}\sqrt {2{\rm{\Delta }}} }}\)

Putting all values in the above equation

\(a = \frac{{3.401 \;\times\; 1.3 \;\times\; {{10}^{ - 6}}}}{{2\pi \;\times \;1.5\;\sqrt {2\; \times\; 0.01} }}\)

a = 3.316×10^-6  meter

a = 3.316 μm

diameter of core (d) = 2a

d = 2 × 3.316 μm

d = 6.632 μm

The number of modes for graded-index fiber is given by

\({M_g} = \frac{\alpha }{{\alpha + 2\;}}\frac{{\;{V^2}}}{2}\)

α = various profile parameter of the fiber

α = 1 for triangular profile

α = 2 for parabolic profile