Multi-layer approximation

Multi-layer approximation

This method is based on the solutions obtained for asymmetric step-index planar waveguides,
in such a way that the graded-index waveguide with an inhomogeneous core is
decomposed into a finite number (as large as necessary) of homogeneous layers having
constant refractive indices.
The operating method in multi-layer approximation is the following: first, the gradedindex
region, forming the core waveguide, is sectioned into p thin layers parallel to
the planar interface, each of them having a constant refractive index nj , as shown in
Figure 3.25. In these conditions, the waveguide structure with graded refractive index
n(x) is defined by (p + 1) layers of constant refractive index nj (j = 0, 1, . . . , p),
where n0 = nc and np = ns . In addition, if the left boundary of the j th layer is situated
at x = xj , the individual layer thickness is automatically determined by dj = xj −
xj−1. Finally, the number of boundaries between two adjacent media is given by p.
If we restrict the solution of the graded waveguide to TE polarised modes, the
non-vanishing electric field component will be expressed

being nj the refractive index of the j th layer and β the propagation constant of the
mode. The effective index of a mode is calculated, as usual, by N = β/k0.
Considering the expressions given for the electric field amplitude Ej and the parameter
γj given by (3.51) and (3.52) respectively, in the region where its refractive index
nj is higher than the effective refractive index of the mode N, the solution for the
electric field is a sinusoidal function, while in a layer having a refractive index nj
lower than N the parameter γj is a pure imaginary quantity, and therefore the electric
field will show exponential behaviour.