Page 416 - Fiber Optic Communications Fund

Channel Multiplexing Techniques 397

Consider an optical wave with propagation constant k = 2n∕ , where n is the refractive index of the MZ
1
1
interferometer. If
k ΔL =(2m + 1), m = 0, ±1, ±2, (9.41)
1
we have
2
P out,1 = A ,
0
P = 0. (9.42)
out,2
So, all the input power appears in port 1. Given another optical wave with k = 2n∕ and if
2
2
k ΔL = 2l, l = 0, ±1, ±2, (9.43)
2
we find that all the input power appears in port 2. Therefore, if we choose the wavelengths and such
1
2
that
nΔL (2m + 1)
= , (9.44)
2
1
nΔL
= l, (9.45)

2
the optical fields with wavelengths and appear in ports 1 and 2, respectively. From Eqs. (9.44) and
1 2
(9.45), we obtain
′
(2m + 1)
1 2
′
− = , m = m − l = 0, ±1, ±2. (9.46)
2
1
2nΔL
Since f = c∕ , j = 1, 2, from Eqs. (9.41) and (9.43), we obtain
j
j
′
(2m + 1)c
Δf = , (9.47)
2nΔL
where Δf = f − f is the channel spacing. For example, wavelengths = 1540 nm and = 1540.4nm are
2
2
1
1
multiplexed in a WDM system. At the receiver, these wavelengths can be separated if
(2m + 1)
1 2
ΔL = , m = 0, ±1, ±2, … . (9.48)
( − )2n
2 1
If we choose m = 0, we find ΔL = 2 mm.
A1 × N demultiplexer can be constructed by cascading the 1 × 2 demultiplexer of Fig. 9.6. Fig. 9.8 shows
a schematic of a 1 × 4 demultiplexer. Suppose the input consists of four channels with wavelengths , ,
1
2
Port

1 2 1 1
1, 3
1 Demux 2 2
1 2 2
1, 2, 3, 4
Demux 1 3 3
2 1 2
Demux 3 4 4
2, 4
Figure 9.8 1 × 4 wavelength demultiplexer.