Reflection and transmission coefficients: reflectance and transmittance

Reflection and transmission coefficients: reflectance
and transmittance

Now we will focus on the relations between the electric field amplitude for the incident,
reflected and transmitted waves. In order to do this, we will use the appropriate
boundary conditions (2.90)–(2.93) that should be fulfilled by the fields at the interface.
We will consider two basic types of linearly polarised incident waves separately: the
first deals with an EM wave in which the associated electric field vector lies on the
incident plane; in the second the electric field vector is perpendicular to that plane. In
the general case of an incident wave with an arbitrary polarisation state, the procedure
is to decompose it into the two basic polarisations, treating them separately, and finally
to re-compose the electric field by adding the two mutually orthogonal components.
Using appropriate boundary conditions at the interface, it can be demonstrated that if
the wave has its electric field parallel to the incident plane, the reflected and transmitted
waves will also have their electric field in that plane. In the same way, if the
electric wave associated to the incident wave is perpendicular to the incident plane,
the electric fields of the reflected and transmitted waves will also be perpendicular to
the incident plane.

Let us consider the first case in which the electric field vector associated with
the incident monochromatic plane wave lies on the incident plane, as depicted in
Figure 2.4. As the wavevector is also on this plane, and the magnetic field vector is
perpendicular to both vectors, it is deduced that the magnetic field vector must be
perpendicular to the incident plane: this is the reason why this case is called transverse
magnetic incidence (TM incidence). In this situation, the electric and magnetic fields
are given by:
Ei ≡ E||
i
≡ [Eix, 0,Eiz] (2.111)
Hi ≡ H⊥
i
≡ [0,Hiy, 0] (2.112)
where the symbols || and ⊥ denote vectors parallel and perpendicular to the incident
plane, respectively. As the electric field vector is parallel to the incidence plane, the TM
incidence is also called parallel incidence. Applying the condition of the continuity of
the tangential component of the electric field at the interface given by (2.92) we obtain:
Eiz + Erz = Etz (2.113)
and in terms of the incident, reflected and transmitted angles, using the geometry shown
in Figure 2.4, we obtain:
[Eiei(ωi t−ki r) cos θi − Erei(ωr t−kr r) cos θr ]x=0 = [Etei(ωt t−kt r) cos θt ]x=0



where the three expressions should be evaluated at the interface (x = 0). As we have
seen before, the temporal and spatial dependences of the exponentials are equal (at
x = 0), and therefore it follows that:
Ei cos θi − Er cos θi = Et cos θt (2.115)
On the other hand, the condition of continuity of the normal component of the
dielectric displacement vector (2.90) is expressed in this geometry as:
Dix + Drx = Dtx (2.116)
and taking into account the constitutive relation (2.13), we can express this relation as
a function of the electric fields:
ε1 Ei sin θi + ε1 Er sin θi = ε2 Et sin θt


The continuity of the tangential component of the electric field across the boundary
(2.92) is expressed in this case as:
Eiy + Ery = Ety (2.131)
To obtain the reflection and transmission coefficients it is necessary to find a second
relation between the electric field amplitudes. This is obtained by imposing the condition
of continuity of the tangential component of the magnetic field vector (2.93) at
the interface:
Hiz + Hrz = Htz (2.132)
and by relating the magnetic field vectors with the electric field vectors by using
equation (2.51). After straightforward calculations, the boundary condition (2.132)
becomes:
kix(Eiy − Ery) = ktxEty