406
10 Wavelength Division Multiplexing (WDM)
Fig. 10.15 Layout of a basic 2 × 2 Mach–Zehnder interferometer
the following analysis does not take into account waveguide material losses or bend
losses.
The propagation matrix M coupler for a coupler of length d is
M coupler =
cos κd j sin κd
j sin κd cos κd
(10.29)
where κ is the coupling coefficient. Because the devices of interest here are 3-dB
couplers that divide the power equally, then 2κd = π/ 2, so that
M couplex =
1
√
2
1 j
j 1
(10.30)
In the central region, when the signals in the two arms come from the same light
source, the outputs from these two guides have a phase difference given by
=
2π n 1
λ
L −
2π n 2
λ
(L +
(10.31)
Note that this phase difference can arise either from a different path length (given
by or through a refractive index difference if n 1 = n 2 . Here, assume both arms
to have the same index and let n 1 = n 2 = n eff (the effective refractive index in the
waveguide; see Sect. 2.5.4). Then Eq. (10.31) can be rewritten as
= k
(10.32)
where k = 2πn eff /λ.
For a given phase difference the propagation matrix M for the phase shifter
is