Page 173 - Fiber Optic Communications Fund

154 Fiber Optic Communications

The optical beams in the upper and lower arms are recombined via a second y-branch. The optical fields at
the inputs of the second y-branch are
A 0
= √ exp [−i(2f t − )], j = 1, 2. (4.45)
j
j
c
2
The output of the second y-branch is
+ 2
1
out = √ . (4.46)
2
Substituting Eqs. (4.45) and (4.44) in Eq. (4.46), we find that the output of the MZM is [5], [6]
[ ]
A 0 exp (i ) exp (i )
1
2
out = √ exp (−if t) √ + √ = A out exp (−if t), (4.47)
c
c
2 2 2
where
exp [i( − )∕2]+ exp [−i( − )∕2]
1
2
2
1
A out = A exp [i( + )∕2]
0
1
2
2
= A exp [i( + )∕2] cos [( − )∕2]. (4.48)
1
2
0
1
2
From Eq. (4.48), we see that the power is conserved when = . When = + , optical fields coming
1 2 1 2
from two branches of the second y-branch do not excite a guided mode in the output waveguide; instead,
radiation modes are excited which go out of the waveguide [3]. From Eq. (4.48), it appears that conservation
of power is not satisfied when ≠ . But, if we take into account radiation modes, conservation of power
1 2
is always satisfied. Using Eq. (4.44) in Eq. (4.48), we obtain
{[ ] }
V (t)− V (t)
1
2
̄
A out = A exp (i) cos , (4.49)
0
2V
where
+ 2 [V (t)+ V (t)]
2
1
1
̄
= = − . (4.50)
0
2 2V

The instantaneous frequency shift or frequency chirp is given by (Eq. (2.165))
d ̄ ( dV 1 dV 2 )
=− = + . (4.51)
i
dt 2V dt dt

The output optical power is
{[ ] }
V (t)− V (t)
2
1
2
P = |A | = P cos 2 , (4.52)
out out 0
2V

2
where P = A . The frequency chirp combined with fiber dispersion could lead to pulse broadening (see
0 0
Example 2.18) and performance degradations. Therefore, it is desirable to have zero chirp. From Eq. (4.51),
we see that the chirp is zero if
dV (t) −dV (t)
1 = 2 (4.53)
dt dt
or
V (t)=−V (t)+ V bias , (4.54)
2
1