Page 170 - Fiber Bragg Gratings

4A Codirectional coupling 147

The cross-coupling constant K OC remains the same as for contradirec-
tional coupling as

The amplitude of the input mode evolves as

Notice that the dc coupling constants may be different for the evolu-
tion of the input and coupled modes. To resolve this problem, we introduce
new variables, R and S as before, but slightly modified, to result in a
common coupling factor:

The subscripts //, and v on the dc coupling constants K dc are specific
to each mode and is defined by Eq. (4.2.5) for identical modes. Differentiat-
ing R and S, collecting terms, and substituting into Eqs. (4.4.1) and (4.4.6)
leads to

The phase-mismatch factor A/3 is now proportional to the difference
in the propagation constants of the two modes as shown in Eq. (4.4.2).
The cross-coupling constant K ac is defined by Eqs. (4.2.6) and (4.4.5)
as K ac>IM} for identical or nonidentical modes. Note that the coupling con-
stant is real so that K ac>JLLV = K^ >vfJL = /c ac.
The grating transmission function comprises two modes — in the
simplest case, two orthogonal modes of the same order. However, the
general case includes nonidentical modes (including a radiation mode)
with the same or orthogonal polarization. The details of the coupling
constants K ac and K dc need to be evaluated numerically. Radiation modes
are considered in Section 4.7, while coupling between different polariza-
tions is presented in Section 4.5.