Page 272 - Fiber Bragg Gratings

6.3 The Michelson interferometer band-pass filter 249

depends on the overlap of the electric fields E-^ and E 2 of the coupled
modes,

where n co (x, y) is the transverse refractive index profile of the waveguides,
n cl is the cladding index and, P 0 is the total power.
For a small difference in the propagation constants, A/3 is small com-
pared to the coupling coefficient, K. Under these conditions, very nearly
all the power can be transferred across from one fiber to the other [38].
With A/3 «= 0 and (n co — n cl)/n co < 1, the following expression for the
coupling coefficient can be use [391:

a is the core radius, and the normalized waveguide parameters u, v, and
w are defined in Chapter 4, h is the distance between the core centers,
and the modified Bessel functions of order 0 and 1, K 0 and KI, are due
to the evanescent fields of the modes in the cladding.
Assuming that there is only a single field at the input port 1 of the
coupler, JBj = 0. Introducing gratings in ports 3 and 4 with amplitude
reflectivities and phases, p± expti^U)] and p 2 exp[i<£ 2(A)] described by
Eq. (4.3.11), the normalized field amplitudes at the input to the coupler
in ports 3 and 4 are

in which the path lengths from the coupler to the gratings are L^ and
Lf2- The additional factors cr^ and cr 2 include detrimental effects due to
polarization and loss in arms 3 and 4. The output fields, R l and S 2, at
ports 1 and 2 of the interferometer can be written down by applying Eq.
(6.3.la) again, with the input fields from Eq. (6.3.2). Therefore,

In Eq. (6.3.3) the normalized field amplitudes R l and S t fully describe
the transmission transfer function of the Michelson interferometer band-