Mumbai University > Electronics Engineering > Sem7 > Optical Fiber Communication

Marks: 10M

Year: Dec2012, 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.

• 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

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

• 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.

• Differentiate equation 1 and 2 with respect to time t

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

• Taking curl of equation 1 and 2:

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

• Using the vector identity formula we get:

$$∇×(∇×E)= ∇(∇.E)-∇^2 E \\ \; \\ ∇×(∇×H)= ∇(∇.H)-∇^2 H$$

But, $∇(∇.E)=0 and ∇×(∇×H)=0$

• Equating curl and vector identity equations:

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

These equations represent standard wave equations.