The equation (C-5) can be
separated in a set of two first order differential equations
|
(C-6) |
In absence of the grating, the modes
propagate without affecting each other. Otherwise, the modes will couple to
each other through the quantity D(z). The grating index perturbation can be
expressed as
|
(C-7) |
where the design period Ld is chosen in order to guaranty a slowly varying phase function q(z). The
functions Deac and Dedc are real and slowly varying function much smaller than n2core.
The quantity D(z) can also be expressed as a quasi-sinusoidal function
|
(C-8) |
where k(z) is complex, slowly
varying with z and s(z) is real, also slowly varying and represents the contribution of Dedc. The forward and backward components b± are written as
|
(C-9) |
The new variables u and v can be
treated as the fields themselves once the reference planes have been fixed
since they only differ from b± by constant, frequency independent
phase factors. Starting from (eq. (C-6), using equations (C-8) and
(C-9) and neglecting the rapidly oscillating terms that contribute little
to the energy coupling we obtain the coupled-mode equations
|
(C-10) |
where d = b-p/Ld is called the wavenumber detuning and where q(z) is called the
coupling coefficient and is defined as
|
(C-11) |
We note that the function u, v and q
are slowly varying with z compared to the period Ld because b is close to p/Ld when the wavelength is close to the
Bragg wavelength (2neffLd).
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