Guided Modes in Step-index Planar Waveguides

Guided Modes in Step-index Planar Waveguides

The general solution discussed in the previous paragraph can easily be applied to the
case of guided modes supported by asymmetric step-index planar waveguides, considering
the geometry given in Figure 3.17. The three media have refractive indices nc
(cover), nf (film) and ns (substrate), and are separated by planar boundaries perpendicular
to the x-axis, the light propagation being along the z-axis. We further assume
that nf < ns < nc, and that the plane x = 0 corresponds to the cover–film boundary.
Therefore, if the film thickness is d, the film–substrate interface is located at the plane
x = −d.
Guided TE-modes Although step-index planar waveguides are structures inherently
inhomogeneous, within each of the three region the refractive index is constant. Thus,
considering each region separately, the wave equation for TE modes.


Bearing in mind this result, the wave equation (3.17) in each homogeneous region
can be written as:
d2Ey/dx2 − γ 2
c Ey = 0 x ≥ 0 (Cover) (3.29)
d2Ey/dx2 + κ2
fEy =0 0> x > −d (Film) (3.30)
d2Ey/dx2 − γ 2
s Ey = 0 x ≤ −d (Substrate)


The electric field in the cover also admits an additional solution of the form of
A
 exp(γcx), but as an increasing exponential function for x > 0 has not physical
meaning for a confined mode, we have A
 = 0. A similar reasoning has been used to
eliminate the term D
 exp(−γsx) corresponding to the substrate region.
The boundary conditions require that Ey and dEy/dx must be continuous at the
cover–film interface (x = 0) and at the film–substrate frontier (x = −d), giving place
to four equations that relate the constant parameters A, B, C and D and the propagation
constant β. Therefore, we have five unknown quantities to be determined from only
a set of four equations. Indeed, one of the constant parameters cannot be determined
and should remain free (for instance, the parameter A), and it will be determined once
the energy carried by the propagating mode is settled. By solving this set of equations,
and after cumbersome calculation.


The electric fields in the cover and in the substrate are indeed evanescent waves, a
particular case of inhomogeneous wave as discussed in the previous chapter, where the
directions of propagation and attenuation are perpendicular. The modulus of the attenuation
vector at defined in Section 2.1.6 is now given by γc in the cover region and by
γs in the substrate. Therefore, the evanescent wave penetrations are determined by 1/γc
and 1/γs . As it can be observed in Figure 3.20, for a particular mode the field penetration
in the cover is lower than in the substrate, and this is so because nc < ns , from
(3.32) and (3.34) it follows that γc > γs . Another important feature related to evanescent
fields is that as the mode number m increases, the wave penetration in a particular
region is deeper (see the different modes in Figure 3.20). This behaviour is due to the
fact that as the mode order increases, the propagation constant of the modes decreases,
thus lowering the value of γc, which implies an increase in the field penetration.
Once the electric field component Ey of a particular guided mode has been established
(the only non-vanishing electric field component for TE modes), the determination
of the magnetic field associated with the mode is straightforwardly obtained
by using the equations (3.11) and (3.12), which relate the Hx and Hz components of
the magnetic field to the Ey component. In this way, the waveguide mode is fully
characterised, leaving only the parameter A to be determined from the energy carried
by the mode.
Guided TM modes In this case we are interested in the determination of the electromagnetic
field structure within the planar waveguide based on the magnetic field,
because in TM polarisation the magnetic field has a single component (Hy). The wave
equation established for TM propagation


In a similar way seen for TE modes, the solution for guided TM modes has exponentially
decreasing behaviour in the cover and substrate, and a sinusoidal solution in the
film region. At variance to that found for TE modes, in TM polarised modes, there
exists a discontinuity in the first derivative of the magnetic field component Hy(x) at
x = 0 and x = −d, coming from the fact of the continuity condition of (1/n2)dHy/dx
at the interfaces.
The electric field associated with TM modes can now be obtained from
equations (3.18) and (3.19), having thus completely characterised the electromagnetic
field pattern of the guided TM mode.
Cut-off An important aspect concerning waveguides is to know what should be the
minimum film width necessary for the waveguide support of a specific mode of order m,
at a given wavelength. In this situation, the effective refractive index of this particular
mode N should be very close to the substrate refractive index ns , as it is shown
schematically in Figure 3.21. In this case, it yields:
N ≈ ns ⇒ b = (N2 − n2
s )/(n2
f
− n2
s ) ≈ 0 (3.46)
In this situation, the mode is said to be at cut-off. If the film width decreases, the
effective refractive index decreases, and the mode is not longer a guided mode, but a
substrate radiation mode, giving rise to a leaky mode.
The normalised film thickness V for TE and TM modes at the cut-off is given by:
V TE
C
= tan−1(a1/2) + mπ TE modes (3.47)
V TM
C
= tan−1(a1/2/γ2) + mπ TM modes (3.48)
where V TE
C and V TM
C express the film width of the core waveguide, relative to the wavelength,
necessary to support the mth mode in TE and TM propagation, respectively.
From these relations, two important conclusions can be deduced:
(i) As nc must be lower than nf , it follows that γ2 = (nc/nf )2 < 1, and consequently
it holds that V TM
C > VTE
C . This inequality implies that if a waveguide supports a
TM mode of mth order, the waveguide also supports a TE mode of the same order.
The reciprocal situation does not apply in general.