Page 533 - Fiber Optic Communications Fund

514 Fiber Optic Communications

Data, x
ψ x,in
MZM Fiber-optic channel ψ x,out
PBS PBS Coherent
Laser Rx. front
end
ψ y,out
MZM ψ y,in

Data, y
Figure 11.18 Polarization-multiplexed fiber-optic system. PBS = polarization beam splitter, MZM = Mach–Zehnder
modulator.

effects, the output of the coherent receiver front end can be written as (see Section 5.6.5)
[ ] [ ][ ]
̃
̃
̃ () H () H () ̃ ()
x,out = F xx xy x,in , (11.87)
̃
̃
̃ () H () H () ̃ ()
y,out yx yy y,in
where F is a scalar that represents the loss in the fiber-optic channel. In the absence of the polarization-
dependent loss (PDL) or polarization-dependent gain (PDG), total power should be conserved, which implies
that the determinant of the matrix in Eq. (11.87) should be unity. Eq. (11.87) may be written as
̃
̃
̃ ()= F[H () ̃ ()+ H () ̃ ()], (11.88)
x,out xx x,in xy y,in
̃
̃
̃ y,out ()= F[H () ̃ x,in ()+ H () ̃ y,in ()]. (11.89)
yy
yx
After taking the inverse Fourier transform and discretizing, Eqs. (11.88) and (11.89) become
N
∑
x,out [m]= F {H [k] x,in [m − k]+ H [k] y,in [m − k]}, (11.90)
xy
xx
k=−N
N
∑
[m]= F {H [k] [m − k]+ H [k] [m − k]}. (11.91)
y,out yx x,in yy y,in
k=−N
Let
[ ]
x,in [k]
[k]= , (11.92)
in y,in [k]
[ ]
x,out [k]
[k]= . (11.93)
out

y,out [k]
[ ]
H [k] H [k]
xy
xx
H[k]= F . (11.94)
H [k] H [k]
yx yy
Now, Eqs. (11.90) and (11.91) may be rewritten as
N
∑
[m]= H[k] [m − k]. (11.95)
out
in
k=−N