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4.1.5 VectorValued Extension
The preceding derivation of the scalar theory of Fourier optics
neglects the vector nature of light. For high numerical aperture systems,
e.g., for
NA = 0.5 and above, the approximations and
simplifications made throughout fail and
the oblique propagation of the waves becomes important.
The most obvious vector diffraction effects arise from polarization
in the illumination producing orientationdependent asymmetries in the
printed patterns. For example, in the case of phaseshifting masks
or offaxis illumination the aerial image strongly depends on
whether polarization parallel or perpendicular to the lines and spaces is
used. In this section we follow the imaging part of Yeung's key
work [114] and summarize the fundamental results therefrom using
our notation.
The scalar amplitude U(x, y) used so far
has to be replaced by vector functions
E(x, y) and
H(x, y) representing the electric and magnetic field,
respectively. Assuming again a periodic mask, the two field vectors
can be expanded into a plane wave superposition corresponding
to the Fourier series of (4.51):

(4.51) 
The lateral components i_{x, n} and i_{y, m} of the wavevectors
i_{nm} belonging to the image space
follow from (4.47) to

(4.52) 
Note that the magnification M is eliminated because
i_{nm} is
defined in the image space. The Fourier coefficients
E_{i, nm}^{pq}
and
H_{i, nm}^{pq} in the above plane wave expansion aresimilar
to (4.52)given by

(4.53) 
The right equation ensures that the electric, magnetic, and orientation vector
form a righthanded orthogonal triad.
The left equation is equivalent to the scalar
formula (4.52).
Instead of the scalar pupil function P(n, m) we have now
a vectorvalued pupil function
P(n, m : p, q)
that also depends on the source point position (p, q)
beside the diffraction order (n, m).
Following [114] we write
P(n, m : p, q) as

(4.54) 
Here,
(n, m : p, q) denotes the polarization vector.
For further calculations it is convenient to decompose the polarization vector
(n, m : p, q) into a transversalelectric (TE) and a
transversalmagnetic (TM) mode.
In case of TEillumination the ycomponent of the electric field vanishes
in the source space, for TMillumination the magnetic field in the image space
is transverse to the yaxis. The overall polarization state is composed like

(4.55) 
The two vectors
_{TE}(n, m : p, q) and
_{TM}(n, m : p, q) are orthogonal and the two angles
and
determine the actual polarization state. The illumination is said to
be linearly polarized in
(cos, sin)direction if
= 0,
circularly polarized for
= /4 and
= /2, and
elliptically polarized otherwise. In case of unpolarized light the images for
TE and TMmodes have to be calculated separately with
= /4 and
incoherently superposed afterwards.
Expressions for the two fundamental modes are found by tracing the
ray paths through the optical system as illustrated in
Figure 4.6.
For the sake of definiteness we first summarize the discretization of the
wavevectors occurring in the three relevant spaces:
As can be seen, the three vectors
s_{pq},
o_{nm}, and
i_{nm} lie in a common
plane usually called meridional plane. Due to singularities the
following two different cases have to be studied separately.
The derivation of the provided formulae can be found in [114].
 Vertical incident ray or
(n, m) = (0, 0).
The TE and TMpolarization vectors are given by
 Oblique incident rays or
(n, m)
(0, 0).
Here, the TE and TMpolarization vectors are further decomposed into a parallel
and perpendicular vector to the meridional plane, i.e.,
The two amplitudes
(n, m : p, q) and
(n, m : p, q) are
given by
and the two unit vectors
_{  }(n, m) and
_{}(n, m) equal to
Figure 4.6:
Ray path
through a schematic projection system. The three wavevectors
s_{pq},
o_{nm}, and
i_{nm} lie in a
common plane, the socalled meridional plane of the optical system.

The norm
P of the vectorvalued pupil function^{m}
will be of interest for
the energy normalization in case of more than one illuminating
point source (cf. Section 4.3.1).
It follows from (4.58) to

(4.57) 
whereby due to the orthogonality of the fundamental TE and TMmodes
the norm
 of the polarization vector obeys

(4.58) 
It can be further shown that the following relation holds:

(4.59) 
Finally, the image intensity
I_{i}^{pq}(x, y) due to one source point excitation
(p, q) equals to the vertical component of the Poynting vector
S_{i}^{pq} = E_{i}^{pq} x H_{i}^{pq}, i.e.,

(4.60) 
Footnotes
 ... function^{m}
 For
the sake of a compact notation we suppress the dependence on the indices
(n, m) and (p, q) for the moment.
Next: 4.2 Lens Aberrations and
Up: 4.1 Principles of Fourier
Previous: 4.1.4 Köhler Illumination of
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417