Page 31 - Fiber Optic Communications Fund

12 Fiber Optic Communications

where c is the velocity of light in free space. Before Maxwell’s time, electrostatics, magnetostatics, and optics
were unrelated. Maxwell unified these three fields and showed that the light wave is actually an electromag-
netic wave with velocity given by Eq. (1.63).

1.5.4 Propagation in a Dielectric Medium
Similar to Eq. (1.63), the velocity of light in a medium can be written as
1
= √ , (1.64)

where = and = . Therefore,
0 r
0 r
1
. (1.65)
= √

0 0 r r
Using Eq. (1.64) in Eq. (1.65), we have
c
. (1.66)
= √

r r
For dielectrics, = 1 and the velocity of light in a dielectric medium can be written as
r
c c
= √ = , (1.67)
r n
√
where n = is called the refractive index of the medium. The refractive index of a medium is greater than
r
1 and the velocity of light in a medium is less than that in free space.

1.6 1-Dimensional Wave Equation
Using Eq. (1.64) in Eq. (1.62), we obtain

2
2
E 1 E
x = x . (1.68)
2
z 2 t 2
Elimination of E from Eqs. (1.55) and (1.58) leads to the same equation for H ,
x
y
2
H H
y 1 y
= . (1.69)
2
z 2 t 2
To solve Eq. (1.68), let us try a trial solution of the form
E (t, z)= f(t + z), (1.70)
x
where f is an arbitrary function of t + z.Let
u = t + z, (1.71)
u u
= , = 1, (1.72)
z t
E x E u E x
x
= = , (1.73)
z u z u