Mumbai University > Electronics and Telecommunication > Sem7 > Optical Communication and Networks

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:

  $\triangledown × E = -\frac{\delta B}{\delta t } …. (1)$

  $\triangledown × H = \delta D/\delta t …. (2)$

  $\triangledown . D = 0 …. (3)$

  $\triangledown . B = 0 …. (4)$

  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)

• 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:

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

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

• 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.