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In the preceding section we have presented a powerful formalism to analyze
the field propagation phenomenon within one homogeneous planar layer.
The results are now extended to a stack of such homogeneous planar layers
as shown in the schematic of Figure C.2.
Onto the medium a plane wave with amplitude
E_{i} is incident.
Due to the different refraction indices a wave
E_{r}
is reflected. At the bottom only an outgoing wave
E_{s} leaves the
stack traveling into the infinitely extended substrate.
Inside the stack within each of the
L layers two waves propagating downwards and upwards,
E_{l}^{+}
and
E_{l}^{}, respectively, occur.
The height of one layer is denoted by h_{l},
the interfaces are located at z_{l}, and the refraction index of
the layer materials is n_{l}.
Figure C.2:
A stratified medium consists of a
stack of
homogeneous planar layers. Above we have an incident and a reflected wave,
E_{i} and
E_{r}, respectively, below only an outgoing wave
E_{s} travels into the infinitely extended substrate, and in
between the waves
E^{+}_{l} and
E^{}_{l} travel downwards and
upwards the L layers.

We start with the description of the boundary conditions valid at any interface
between two adjacent layers l  1 and l. Elementary results of EM theory
postulates that in the absence
of a surface current density both the tangential electric and tangential
magnetic field components are continuous across the
surface [11, p. 6]. Hence we obtain in our notation at any
interface position z_{l  1}

(C.20) 
Using the relation (C.19) yields the lateral field components
at the next interface located at
z_{l} = z_{l  1} + h_{l}

(C.21) 
whereby
is the propagation matrix of layer l.
This relation is of fundamental importance as it propagates the field
from one interface to the next. Now a recursive evaluation connects the
lateral field components at any vertical position z to
the field occurring on top of the interface at z = 0 by

(C.22) 
At the interface to the substrate, i.e., at the bottom
z_{L} of the stack, we thus obtain

(C.23) 
whereby
is simply called the propagation matrix of the
stratified
medium. Obviously the lateral field components at the top and at the
bottom of the stack are related by this equation.
Thus also the wave amplitudes occurring in the air and in the substrate
can be related. From (C.19) and (C.14)
we obtain
u_{air}(h) = e_{air} and
u_{sub}(h) = e_{sub} and
the relation looked for writes to

(C.24) 
Finally, we again leave off the matrixvector notation used so far to account
for the physical situation depicted in Figure C.2. As can be
seen the two amplitude vectors
e_{air} and
e_{sub} are given by (cf. (C.9))
because in the substrate only outgoing light has to be considered.
By explicitly writing the third and fourth rows of the matrix equation
(C.26) we obtain the two necessary conditions
that have to be satisfied by the incoming and the reflected light
E_{i} and
E_{r}, respectively.
These two equations relate the lateral components of the field amplitudes
occurring above the stack. They establish the boundary conditions needed
for the differential method in case of multiple planar homogeneous layers
below the simulation domain (cf. Section 6.3).
For the simple exposure simulation method commonly called transfer matrix
algorithm
(cf. Section 5.2.3) one last step is missing. Clearly
only the incoming amplitudes
E_{i} are known. They are
calculated by the aerial image tool using for example the vectorvalued
formulation of the Fourier optics as described in Section 4.1.5.
Hence the unknown reflected light
E_{r} has to be expressed by the
incident light
E_{i}.
This can easily be done by inverting (C.28) resulting in

(C.25) 
The overall transfer matrix algorithm now consists of two
steps:

 1.
 Firstly, the propagation matrix
in (C.26) is
calculated and the reflected amplitudes are
derived from the incident ones by (C.29).
The lateral field
u_{air}(0) occurring at the
upmost interface between air and stack is thus determined.
 2.
 Secondly, the recursion formula (C.24) is evaluated at
any desired vertical position z. From the lateral
field components
u(z) the vertical ones can readily
be calculated from (C.4) and the overall EM field is
obtained.
Next: D. Nonplanar Material Interface
Up: C. Stratified Medium
Previous: Propagation Matrix
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417