Boundary conditions at the interface

EM Waves at Planar Dielectric Interfaces

 Boundary conditions at the interface

Up to now we have described the propagation of EM waves in free space or through
a material medium. Another important aspect in the study of light propagation is the
behaviour of EM waves passing from one medium to another. We will analyse this
by studying the behaviour of an EM monochromatic plane wave travelling through a
homogeneous medium, incident on a second homogeneous medium, separated from the
former by a planar interface. We will see that, besides the existence of a transmitted
wave in the second medium, the incident wave partially reflects at the interface, giving
rise to a reflected wave. The equations that determine the reflection and transmission
coefficients can be studied separately in two groups: in one situation, the electric field
of the incident EM wave has only a parallel component with respect to the incident
plane (the magnetic field being perpendicular to that plane); the other group refers to
incident EM waves in which the electric vector has only the component perpendicular
to the incident plane, and therefore the magnetic vector is perpendicular to that plane.
These two cases are mutually independent, and can be treated separately: from them
it is possible to deduce the equations that govern reflection and transmission for any
plane wave with arbitrary polarisation state.


The relations between the incident, reflected and transmitted waves are obtained by
setting the adequate boundary conditions for the fields at the planar interface, which
are derived directly from Maxwell’s equations. Because the E, D, H and B fields are
not independent, but related by Maxwell’s equations and the constitutive relations of
the media, only some of the boundary conditions should be taken into account.
From equations (2.9) and (2.10), one obtains respectively that the normal components
of the fields D and B should be kept across the boundary, that is:
(DNormal)Medium 1 = (DNormal)Medium 2 at the interface (2.90)
(BNormal)Medium 1 = (BNormal)Medium 2 at the interface (2.91)
On the other hand, by using Maxwell’s equations (2.11) and (2.12) respectively, the
conditions of continuity across the interface of the tangential components of the E and
H fields are obtained:
(ETangential)Medium 1 = (ETangential)Medium 2 at the interface (2.92)
(HTangential)Medium 1 = (HTangential)Medium 2 at the interface


Let us consider an EM monochromatic plane wave, characterised by its angular
frequency ωi and wavevector ki, incident from a homogeneous medium (1) to a planar
frontier separating a different homogeneous medium (2). The dielectric media are
characterised by their optical constant (ε1,μ1) and (ε2,μ2), where the subscript denotes
the medium (1 or 2). If the two media are isotropic and homogeneous, the electric field
vectors, using complex notation, corresponding to the incident, reflected and transmitted
(or refracted).