Next: 4.1.2 Phase Transformation Properties
Up: 4.1 Principles of Fourier
Previous: 4.1 Principles of Fourier
Light is an electromagnetic wave with coupled electric and magnetic fields
traveling through space. In a homogeneous isotropic medium, like free space
or a lens with constant refractive index, the electric and magnetic
field vectors form a righthanded orthogonal triad with the direction of
propagation. Disregarding polarization, the field can be described
by a scalar function
(x;t) representing either the
electric or the magnetic field amplitude. As the light used in microlithography
exhibits a strong monochromacity (cf. Section 2.3.1) the time
dependence of the field is a harmonic one and can thus be explicitly written as

(4.1) 
Here,
denotes the angular frequency of the light, and the
complexvalued amplitude or phasor
U(x) depends on the
spatial coordinates
x = (x y z)^{T} only. If
both quantities
(x;t) and
U(x) represent
an optical wave, they must satisfy the
Helmholtz equation^{a}

(4.2) 
The Helmholtz equation directly follows from the Maxwell equations
under the condition of a homogeneous medium and in absence of sources.
The quantity
k is termed the wavenumber or propagation constant of the medium and
is related to the light velocity c, the angular frequency , and
the vacuum wavelength
by^{b}

(4.3) 
whereby
is the refractive index of the medium, e.g., in vacuum
= 1.
The complex disturbance
U(x) at any observation point
x_{o} in space can be computed from the Helmholtz
equation (4.2) with the help of
Green's theorem.^{c} If a unitamplitude spherical wave expanding
about the observation point
x_{0} is chosen as Green's function, i.e.,

(4.4) 
the socalled integral theorem of Helmholtz and Kirchhoff is
obtained [11, p. 377]:

(4.5) 
This relation plays a central role in the derivation of the scalar theory
of diffraction, for it allows the field
U(x) at an observation point
x_{o} to be expressed in terms of the ``boundary values''
on any closed surface
surrounding
x_{o}.
The integral theorem of (4.5)
can readily be used to study diffraction effects occurring at
a plane object. Therefore the integration surface
is segmented
into three disjoint parts, i.e.,
=
.
The boundary
is chosen across the object location in the screen
plane,
is the opaque part of the screen, and
is a halfsphere containing the observation point
x_{o}. A graphical illustration of this specific choice is given
in Figure 4.1. The solution of the diffraction problem
is found by specifying Kirchhoff boundary conditions on
and
, and the Sommerfeld radiation condition
on
:
whereby
U_{s}(x) is the field incident on the
screen. The Kirchhoff boundary conditions are an idealization
to the real field distribution on the screen by assuming that the field
and its derivative across
is exactly
the same as they would be in the absence of the screen.
In the geometrical shadow across
the field is simply set to zero.
Although these assumptions are reasonable, they fail in the immediate
neighborhood of the rim of the opening. Thus, the range of validity is
related to the wavelength
and the geometrical dimension w of the
opening (cf. Figure 4.1). The Sommerfeld radiation condition
on
is satisfied, if the disturbance
U(x)
vanishes at least as fast
as a spherical wave. Since the illuminating light
U_{s}(x)
invariably consists of a
linear combination of spherical waves, this requirement is always fulfilled.
Figure 4.1:
Range of validity
of scalar diffraction theories. Depending on the distance z_{o} between
observation point
x_{o} and screen,
several approximations can be
employed to describe diffraction. For aerial image simulation the
farfield theory of Fraunhofer valid for
z_{o}
w^{2}/
is
adequate since in M : 1reduction projection systems the patterns to
be imaged are of size
w
M
and we thus get
z_{o}
M^{2}
0.1 mm for M = 10 and
400 nm.

Inserting the boundary conditions of (4.6) into
the integral theorem (4.5) shows that only the
illuminating light
U_{s}(x) across the opening
contributes to the
integral, i.e.,

(4.6) 
We further assume that the aperture is illuminated with a single expanding
spherical wave arising from a point source located in
x_{s},
a distance r_{s} away from the screen (cf. Figure 4.1),

(4.7) 
Hence, both Green's function in (4.4) and the incident
disturbance in (4.8) have the shape of spherical waves.
A further simplification is
obtained by noting that the distances from the screen to the observation
point and
the location of the point source, r_{o} and r_{s}, respectively, are
many optical wavelengths. Therefore the following
approximations hold for
r_{o}, r_{s}:^{d}

(4.8) 
Insertion of these approximations into (4.7)
yields together with (4.4) and (4.8)
the FresnelKirchhoff diffraction formula

(4.9) 
The term in square brackets is called obliquity or inclination factor and
was first introduced by Fresnel based on heuristic arguments,
when he combined Huygens'
envelope construction with Young's principle of interference. The mathematical
sound derivation described above was later developed by Kirchhoff.
An interesting property of the diffraction formula (4.10)
is its symmetry with respect to the source
and observation point, a result that is sometimes referred to as the
reciprocity theorem of Helmholtz.
For our purposes we rewrite (4.10) by introducing
a general linear relationship between the exciting field U_{s}(x, y)
at the screen and the resulting
diffraction pattern
U(x_{o}, y_{o})^{e}

(4.10) 
Note that we are always interested in field values lying in certain
planes parallel to the screen. Therefore, only the dependence on the
lateral x and ycoordinate is explicitly written, whereas the vertical
position is indicated by subscripts.
The integration kernel
h(x, y;x_{o}, y_{o}) of the superposition integral
in (4.11) is given by (cf. (4.8)
and (4.10))

(4.11) 
Depending on the distance z_{o} between the observation point
x_{o} and the screen, the kernel
h(x_{o}, y_{o};x, y) can be simplified.
The denominator r_{o} is approximated by
z_{o} provided that z_{o} is much larger than the lateral extent w
of the object. For the same reason and by assuming that the source as well as
the observation point are located near the optical axis, the inclination factor
is approximately given by
cos
(cf. Appendix A).
As shown in Figure 4.1
the angle
lies between the line from source to observation point
and the normal vector
n.
The approximation of the exponential factor is more involved,
since r_{o} is multiplied by the very large wavenumber k_{0}.
Hence, a truncation of Taylor series expansions is employed,
 Fresnel approximations are obtained by truncating the Taylor
series after the first term, i.e.,

(4.12) 
The resulting integral relation (4.11) involves Fresnel's
integrals.^{f} Due to
the square dependence on the integration variables (x, y),
no analytical solution exists. Tabulated values can be found in,
e.g., [115, ch. 7].
The approximation (4.14) holds for
(2z_{o})^{3}
k_{0}[(x  x_{o})^{2} + (y  y_{o})^{2}]^{2}_{max} or simply z_{o}
w.
 Fraunhofer approximations additionally neglect the square
dependence on the integration variables (x, y) resulting in

(4.13) 
The integral relation (4.11) corresponds now to a
Fourier transform, i.e., the dependence on the integration variables
is a linear one. Efficient numerical techniques
like FFT can be used for evaluation. The range of validity
of (4.15) is given by
2z_{o}
k(x^{2} + y^{2})_{max} or
z
w^{2}/.
Footnotes
 ... ^{a}
 The symbol
denotes the
Laplace operator, i.e.,
=
+
+ .
 ... by^{b}
 The quantity c_{0} is the vacuum
light velocity and has a value of
c_{0}
2.998 10^{8} m/s.
 ... theorem.^{c}
 Green's theorem states that if a complexvalued
function
U(x) is continuous and has continuous singlevalued first
and second partial derivatives inside a volume
surrounded by a
closed surface
, the following relation holds:
whereby the socalled Green's function
G(x) is any other function
with the same continuity requirements as
U(x) and
= n^{ . }grad
signifies a partial derivative in
the inward normal direction
n at each point on
.
 ...:^{d}
 For
r and
thus
1/r
k_{0} we get
e^{jk0r}/r
= cos(n, x)jk_{0}  1/re^{jk0r}/r
jk_{0}cos(n, x)e^{jk0r}/r.
 ...)^{e}
 All integrations and sums
go from 
to
unless specified otherwise.
 ...
integrals.^{f}
 Integrals of the type
C(w) = cos(/2) d
and
S(w) = sin(/2) d
are referred to as Fresnel's integrals [115, ch. 7].
Next: 4.1.2 Phase Transformation Properties
Up: 4.1 Principles of Fourier
Previous: 4.1 Principles of Fourier
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417