Page 171 - Fiber Optic Communications Fund

152 Fiber Optic Communications

Suppose the optical field of the incident optical wave is

(t, 0) ≡ (t)= A exp (−i2f t). (4.36)
0
in
c
Using Eqs. (1.95), (4.34), and (4.35), the optical wave emerging from the LiNbO crystal is
3
(t, L)= A exp [−i(2f t − )], (4.37)
c
in
0
where ( 3 )
2 2L n r V
0 33
= nL = n − = −Δ. (4.38)
0
0
2d
0 0
Here, is the constant phase shift in the absence of the applied electric voltage, L is the length of the crystal,
0
and
3
Ln r V
Δ = 0 33 (4.39)
d
0
is the phase change. The required voltage to yield a phase change of is known as the half-wave voltage or
switching voltage V , and is given by

3
Ln r V 0 d
0 33
Δ = = or V = . (4.40)

3
d n r L
0
0 33
Substituting Eq. (4.40) in Eq. (4.37), we obtain
[ ( )]
V
(t, L)= A exp −i 2f t − + . (4.41)
in
0
0
c
V
Thus, we see that the phase change is directly proportional to the applied voltage. If V(t) is a message signal,
the phase of the optical carrier can be varied in accordance with the message signal. For the example, if
V(t)= V , in a bit interval 0 < t < T , the carrier phase is shifted by .If V(t)= 0, no phase shift is introduced.
b
Thus, the PSK or DPSK signal can easily be generated using a phase modulator.
Example 4.2
An electro-optic modulator operating at 1530 nm has the following parameters:
Thickness d = 10 μm
Length L = 5cm
Index n = 2.2
0
Pockel coefficient r 33 = 30 pm/V
Calculate the voltage required to introduce a phase shift of ∕2.
Solution:
From Eq. (4.39), we have
Δ d
0
V = 3 .
Ln r
0 33