Next: Bibliography
Up: PhD Thesis Heinrich Kirchauer
Previous: C.2 Stack of Homogeneous
D. Nonplanar Material Interface
The error introduced by the truncation (6.27) of
the Fourier expansion of the EM field is now investigated for the
stepped topography of Figure 8.16. A sketch of the
relevant region of the geometry is replotted in Figure D.1.
All derivations are performed in two dimensions only since the results apply
to three dimensions as well.
Figure D.1:
Schematic of the investigated nonplanar material interface (left)
and lateral dependence of the permittivity along a horizontal cut
(right). The cut positions Z and Z^{} are
chosen within the nonplanar region of the resist.

The lateral dependence
(x) of the permittivity is simply given by
(cf. Figure D.1)

(D.1) 
with
=

as the difference of the resist and step material
permittivities. The intersection
of the cutline
with the step slope is denoted by x_{s},
and (x) refers to Heaviside's step function.^{a}
The differential method requires the Fourier coefficients
of the
permittivity function
(x) of (D.1),

(D.2) 
We restrict our investigations to TEpolarized light. In this case
the Fourier coefficients
E_{y, n}(z) of the ycomponent of the electric field
satisfy^{b}

(D.3) 
The impact of the truncation of the Fourier series can best be understood
by explicitly studying one specific mode
E_{y, i}(z) at an
arbitrary vertical cutposition z_{c}, e.g., at
as indicated in Figure D.1. For the analysis the amplitude of
E_{y, i}(z_{c}) can be chosen arbitrarily.
We conveniently set it equal to 1/k_{0}^{2} so that

(D.4) 
Inserting this relation into (D.3) gives

(D.5) 
For simplicity we consider only the case when
k_{x, n}^{2}
vanishes,
i.e., the propagation of the vertical mode i = 0 or, equivalently, the energy
spread of an arbitrary order i into the zerothorder n = 0 is studied. Since
the vertical propagating mode usually carries most of the energy, this
restriction gives still valuable information.
The impact of the truncation of the Fourier expansions (6.27)
is now quantified by the following measure
describing the ratio between ``lost'' and ``total'' energy spread:

(D.6) 
This quantity
can be interpreted as truncation
error since it describes the error made in the calculation of the second
derivative (cf. (D.5)).
Note that the measure is restricted to n = 0 or i = 0 since only then
the first term
k_{x, n}
vanishes in (D.5).
The extension to the general situation is selfevident.
We will now derive an upper bound for
.
For that we first evaluate the denominator occurring
in (D.6) with the
help of Parseval's theorem [131, p. 224] to (cf. (D.1))

(D.7) 
Next we consider the numerator of (D.6).
An upper bound for N1 is given by (cf. (D.2))

(D.8) 
with

(D.9) 
The bound is obtained by replacing
 sin(nx_{s}/a) with unity, i.e.,

(D.10) 
The righthand side equals

(D.11) 
since
n^{2} = /6.
Inserting (D.7)
and (D.8) into (D.6) yields

(D.12) 
with

(D.13) 
(D.12) shows that the bound consists of a geometrical factor
_{xs,} and a numerical factor
_{N}.
The geometrical factor
_{xs,} (cf. (D.13))
is proportional to the squared difference
^{2} = 
 ^{2}
of the resist and step material permittivities.
The truncation error
is thus governed by the strength of the inhomogeneity, which is in
accordance with physical considerations. The dependence of the numerical factor
_{N} (cf. (D.9)) on the truncation
frequency N is plotted for a constant
_{xs,} in Figure D.2. In this figure
also an exact expression for the special situation x_{s} = a/2 is shown, i.e.,
the case when the intersection point x_{s} lies in the middle of the interval
[0, a]. In this case the truncation error
of (D.6) can be exactly calculated like
whereby
(2n  1)^{2} = /8 is used.
Both expressions (D.12) and (D.14) show that for
increasing N the error decreases quadratically.
Figure D.2:
Truncation error
of the second derivative
(cf. (D.6)) as a function of the cutoff frequency N.
The geometrical factor is set equal to unity. Both the
bound (D.12) and the exact expression (D.14)
obtained for x_{s} = a/2 are shown in linear and logarithmic
scale, upper and lower graph, respectively. For increasing
N the error decreases quadratically.

As a final remark note that (D.6) describes the error of the
second derivative at one
single vertical point z_{c}. The overall discretization error depends
on the number N_{z} of vertical grid points.
Footnotes
 ... function.^{a}
 Heaviside's step
function is defined as
(x) = 0 for x < 0 and 1 otherwise.
 ...
satisfy^{b}
 For TEpolarization the three field components E_{x}(z),
E_{z}(z), and
H_{y, nm}(z) vanish [11, p. 52]. Taking the
remaining three field components E_{y}(z), H_{x}(z), and H_{z}(z)
in (6.15) and (6.18) and eliminating the
latter two yields (D.3).
Next: Bibliography
Up: PhD Thesis Heinrich Kirchauer
Previous: C.2 Stack of Homogeneous
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417