Mumbai University > Electronics and telecommunication > Sem 7 > optical communication and networks

Marks: 10

Years: MAY 2015

The electromagnetic wave theory provides an improved model for the propagation of light in optical fibers. The basis for the study of electromagnetic wave propagation was provided by Maxwell.

$\Rightarrow$ To analyze optical waveguide, Maxwell’s equations give relationship between electric and magnetic fields. Assuming a linear, isotropic dielectric material having no current and free charges, these equations take the form:

$∇ × E = -∂ B/∂ t …. (1) \\ ∇ × H = ∂D/∂t …. (2) \\ ∇ . D = 0 …. (3) \\ ∇ . B = 0 …. (4) $

Where ∇ is a vector operator.

E - Electric field

B - Magnetic field

D - Electric flux density

H - Magnetic flux density

$\Rightarrow$ Equations (1) and (2) are known as the curl equations and equations (3) and (4) are known as the divergence equations.

$\Rightarrow$ The four field vectors are related to each other by the relation:

$D = εE …. (5) \\ B = μH …. (6) $

where ε - the dielectric permittivity

and μ - the magnetic permeability of the medium.

$\Rightarrow$ Differentiate equation 1 and 2 with respect to time t

$\dfrac ∂{∂t} (∇×E)= -µ{∂^2 H}{∂t^2 } \\ \dfrac {∂}{∂t} (∇×H)= ε \dfrac {∂^2 E}{∂t^2} $

Taking curl of equation 1 and 2:

$∇×(∇×E)= -µ \dfrac {∂}{∂t} (∇×H)= -µε \dfrac {∂^2 E}{∂t^2} \\ ∇×(∇×H)=ε \dfrac {∂}{∂t} (∇×E)= -µε \dfrac {∂^2 H}{∂t^2} $

$\Rightarrow$ Using the vector identity formula we get:

$∇×(∇×E)= ∇(∇∙E)-∇^2 E \\ ∇×(∇×H)= ∇(∇∙H)-∇^2 H \\ But, ∇(∇∙E)=0\space\space and \space\space ∇×(∇×H)=0$

$\Rightarrow$ Equating curl and vector identity equations:

$∇^2 E = µε \dfrac {∂^2 E}{∂t^2 } \\ ∇^2 H = µε \dfrac {∂^2 H}{∂t^2 } $

These equations represent standard wave equations.