The coupling coefficient is a complicated function of many parameters like, the width of the waveguide, refractive index of the waveguide, separation between the waveguide, the mode, and wavelength of operation.
The two coupled equations can be solved by differentiating one equation with respect and substituting from the other. We therefore get differential equation for the fields on the two waveguides as
And
General solutions of these equations can be written as
And
Where are the arbitrary constants to be evaluated from the boundary conditions. And we have defined
Average propagation constant ,
And
.
For obtaining the arbitrary constants we apply the boundary conditions.
Since we excite waveguide A at say z = 0, and no excitation to waveguide B at z = 0, We have boundary conditions,
At z = 0, (normalized)
The fields on two waveguides can the be written as
and substituting from the other. We therefore get differential equation for the fields on the two waveguides as 
are the arbitrary constants to be evaluated from the boundary conditions. And we have defined 
, 
. 
(normalized) 
