Wave Equation in Planar Waveguides

Wave Equation in Planar Waveguides

We will now discuss the electromagnetic theory of light applied to a planar waveguide,
because it is the simplest structure to be analysed from the point of view of
its mathematical description, and from it the general features related to more complex
waveguide geometry can be understood. Starting from Maxwell’s equations and from
the constitutive relations, we will obtain the wave equations for TE and TM propagation
that govern light behaviour in planar waveguides. These wave equations will be solved
for the general case of asymmetric step-index planar waveguides, and later on we
will discuss some methods for solving the wave equations in more complex planar
waveguides, as is the case of graded-index planar waveguides. Finally, we will discuss
the problem in the modelling of channel waveguides, and examine some approximate
methods that can be applied to calculate the propagation modes in 2D structures, such
as the effective index method and Marcatili’s method.
Assuming that the light is propagating through a dielectric (conductivity σ = 0),
non-magnetic (magnetic permeability μ = μ0), isotropic and linear medium (D = εE),


These two vectorial equations indicate that for an inhomogeneous medium the Cartesian
components of the electric field vector Ex ,Ey and Ez (and the components of the magnetic
vector) are coupled, and therefore we cannot establish a scalar equation for each
component as we did in the case of a homogeneous medium. Only in light propagation
in a homogeneous medium, in which the refractive index is constant (∇n2 = 0), the
second terms in equations (3.8) and (3.9) vanish, and each of the Cartesian components
for the fields E and H satisfy the scalar wave equation (2.24).
If the refractive index of the inhomogeneous medium depends only on two Cartesian
coordinates, for instance x and y, so that n = n(x, y), and we choose the third
coordinate (z) as the propagation direction of the radiation, the solutions for the inhomogeneous
wave equations (3.8) and (3.9) for monochromatic


E(r, t) = E(x, y)ei(ωt−βz) (3.10)
H(r, t) = H(x, y)ei(ωt−βz)


ω being the angular frequency and β the propagation constant of the wave. These
two expressions determine the electromagnetic field for a propagating mode, which is
characterised by its propagation constant β. This solution is found in light propagation
in straight channel waveguides or in optical fibres, because in both cases the structure,
defined by the spatial dependence of the refractive index, is invariant with the
z-coordinate.
Assuming now that the refractive index depends only on a single Cartesian coordinate,
for instance n = n(x), which is the case of planar optical waveguides, the spatial
part of the complex exponential function in the expressions (3.10) and (3.11) takes the
form −i(γy + βz). If we further assume propagation along the z-axis, the wave has
no dependence on the y-axis, thus γ = 0, and the electric and magnetic fields take
the form:
E(r, t) = E(x)ei(ωt−βz) (3.12)
H(r, t) = H(x)ei(ωt−βz) (3.13)
Therefore, given a refractive index distribution n(x) that defines the planar waveguide,
the solutions for the electromagnetic fields that support that waveguide are reduced to
find out the solutions for the complex field amplitudes E(x) and H(x) as well as for
the propagation constants β. We will show that for a particular propagation constant β,
whether corresponding to a confined mode or a radiation mode, the field distributions
are completely determined. Thus, providing that the polarisation character of the light
has been initially established, a mode is one-to-one defined by its propagation constant.
In order to find the propagation modes in a planar waveguide we will study two
independent situations: in the first case, the electric field associated with the mode has
only a transversal component, and so its solutions are the TE modes; the second case
involves the situation in which the electric field has only a parallel component, and
the solutions are called TM modes.


TE modes In this case we must find the general solution for the complex amplitudes
E(x) and H(x) when the electric field vector has only perpendicular component
(referred to the incident plane, as discussed in Section 2.2). Following the geometry
on Figure 3.14, the perpendicular component of the electric field corresponds to Ex,
and thus Ey = Ez = 0. On the other hand, the magnetic field satisfies Hy = 0.


(i) If the propagation constant â is lower than k0nj (orN <nj ) then the parameter ãj
is a real number, following the definition given in (3.24), and the general solution
postulated by (3.23) will correspond to a sinusoidal function.
(ii) By contrast, if the propagation constant satisfies that â > k0nj (or N >nj ), the
parameter ãj is a pure imaginary number, and therefore the solution given by
(3.23) should be described by exponential functions.