An optical fibre, by definition, is a dielectric waveguide that operates at optical frequencies. This fibre waveguide is normally in cylindrical form and it guides the light in a direction parallel to its axis. The structural characteristics determines the transmission properties of the optical waveguide, which is how an optical signal is affected as it propagates along the fibre. The propagation of light along an optical waveguide can be described in terms of a set of guided electromagnetic waves called the modes. Therefore, a mode is a pattern of electric and magnetic field lines that is repeated along the fibre at interval equal to the wavelength. Not every modes can propagate along the waveguide, only those modes which satisfy the homogeneous wave equation –Maxwell’s equations can be guided along the core of the fibre. Therefore, it is essential to solve Maxwell’s equations that give the relationships between the electric and magnetic fields to understand the optical power propagation mechanism in a fibre. From Maxwell’s equation to Bessel function Assuming a linear, isotropic dielectric material having no currents and free charges, these equations take the form 1 ∇ × = − (1) ∇ × H = (2) ∇ ∙ = 0 (3) ∇ ∙ = 0 (4) D =∈ E and = 𝜇 . The parameter ∈ is the permittivity (or dielectric constant) and 𝜇 is the permeability of the medium. (Keiser, 1991)
∇ × 𝑉 is called the curl of the vector filed V which is a measure of the angular velocity of the fluid in the neighbourhood of that point .
∇ ∙ is called the divergence of the vector field V which is can be considered as a quantitative measure of how much a vector field diverges (spread out) or converges at any given point. (K.F.RILEY, 2006)
A relationship defining
∇ × (∇ × ) = ∇ × (−
) = −∇ × (
) = − 𝜎 𝜎
(∇ × ) (5)
Substitute Eq.2 into Eq.5 yields
2 2
𝜎 𝜎
𝜎 𝜎
𝜎 𝜎
∈ E
(∇ × ) = −
(
) = −
(
) = −∈
−
(6)
And using Eq.6 and this vector identity ∇ × (∇ × ) = ∇(∇ ∙ ) − ∇ 2 E
Since ∇(∇ ∙ ) = 0
15
Made with FlippingBook - professional solution for displaying marketing and sales documents online