,
Here we describe the transfer matrix method to obtain the solution of the wave-equation for light propagating in planar structures. Most frequently, bases of tangential components of electric and magnetic fields and amplitudes of light waves propagating in positive and negative directions are considered. In these two bases the transfer matrix looks different. Note, that a variety of similar techniques is used to solve Maxwell or Schroedinger equations in planar structures, in particular, the scattering matrix method is a common encounter. We do not use this method in the paper and do not describe it here.
Consider a light-wave propagating along z-direction in a media characterized by a refractive index n homogeneous in the xy plane but possibly z-dependent. The wave equation reads in this case:
|
,
| (A1) |
where k0 is the wave-vector of light in vacuum. We will chose two linearly independent solutions of Equation A1 y1(z), y2(z) subject to the set of boundary conditions:
 ,
| (A2) |
where k = k0n.
The transfer matrix
across the layer of width a
is according to our definition a 2x2 matrix
with the following elements:
 ;
 ,
| (A3) |
If n is a constant across the layer a,
,
and
 .
| (A4) |
It is easy to check that for a vector
 ,
| (A5) |
where E(z), H(z) are the amplitudes of electric and magnetic field of any light wave propagating in z direction in the structure under study, the condition if fulfilled:
 .
| (A6) |
Note that
is continuous at any point of the structure that follows from Maxwell boundary
conditions. In particular it is continuous at all interfaces where n is
changing abruptly. Thus a transfer matrix across a structure composed by
m layers can be found as
 ,
| (A7) |
where
is the transfer matrix across the i-th layer. The product's order
in (A7) is essential.
The transfer matrix across a layer can be expressed via reflection r and trasmission t coefficients of this layer. If reflection and transmission coefficients for light incident from right-hand side and left-hand side on the layer are the same (symmetric case realized, in particular, in a QW embedded in a cavity) condition (A6) yields:
 
| (A8) |
In (A8) the first equation is written for the light incident from the left
and the second equation is written for the light incident from the right. n
is the refractive index of the surrounding media. It allows to express the
matrix
as:
 .
| (A9) |
and z=-.A solution of Equation A1 can be formally represented as:
| E(z) = E+(z) + E-(z) , | (B1) |
where E+(-)(z)
is the complex amplitude of light-wave propagating in positive (negative)
direction. One can define the transfer matrix
by its property:
. | (B2) |
Consider a few particular cases.
If the refractive index n is constant across the layer a, the transfer matrix has a simple form:
 .
| (B3) |
The transfer matrix across an interface between a media having the refractive index n1 and a media having the refractive index n2 is
 .
| (B4) |
It can be obtained with use of the condition (B2) applied to the light waves incident from the left side and right side of the interface and bearing in mind the well-known expressions for the reflection and transmission coefficients of interfaces. In a similar way, the transfer matrix across a symmetric object (QW embedded in the cavity, for instance) can be written as:
 ,
| (B5) |
where r and t are amplitude reflection and transmission coefficients of the object.
Consider an infinite structure whose refractive index is homogeneous in the
(xy)-plane and its dependence on the coordinate z is a periodical
function having a period d. The shape of this function is not essential,
we will only assume that a transfer matrix
across the period of the structure can be written as a product of a finite
number of matrices of the type (A4). Let an electromagnetic wave propagate
along the z-direction. For this wave
 ,
| (C1) |
where
is defined by (Equation A5). According to Bloch's theorem, it can be written as:
 ,
| (C2) |
where UE,H(z)
have the periodicity of the structure and Q is, in general, a complex
number. Note that the factor eiQz
is the same for electric (E) and magnetic (H) fields in a light
wave because in the normal incidence case they are linked by:
.
Substitution of (C2) into (C1) yields
 ,
| (C3) |
thus eiQd is
an eigen-value of the matrix
and, therefore
 ,
| (C4) |
where
is an identity matrix. Resolving equation C4 we will use an important
property of the matrix
which
follows from equation A4 and equation A7:
 .
| (C5) |
| 1 - (T11 + T22) eiQd + e2iQd = 0, | (C6) |
and Tij are elements of the matrix
.
Multiplying all terms by e-iQd we
finally arrive to:
| cosQd = (T11 + T22)/2. | (C7) |
|(T11 + T22)/2|  1
| (C8) |
are allowed photonic bands. In them Q is purely real, and the light wave can propagate freely without attenuation. Conversely, the bands for which
| |(T11 + T22)/2| > 1 | (C9) |
are usually called stop-bands or optical gaps. In them Q has a
non-zero imaginary part which determines the decay of propagating light-waves.
All this is completely analogous to conventional crystals. Eqs. ( C7, C8, C9)
are valid also in the oblique incidence case while the form of the matrix
is sensitive to the angle of incidence, and band boundaries shift as one
changes the incidence angle.
© 2003 The Materials Research Society
