The simulation, design and tolerancing of optical systems for wafer inspection is a challenging task due to
the different feature sizes, which are involved in these systems. On the one hand light is propagated through
macroscopic lens systems and on the other hand light is diffracted at microscopic structures with features in the
range of the wavelength of light. Due to this variety of scale plenty of different physical effects like refraction,
diffraction, interference and polarization have to be taken into account for a realistic analysis of such inspection
systems. We show that all of these effects can be included in a system simulation by field tracing, which
combines physical and geometrical optics. The main idea is the decomposition of the complex optical setup in a
sequence of subdomains. Per subdomain a different approximative or rigorous solution of Maxwell’s equations is
applied to propagate the light. In this work the different modeling techniques for the analysis of an exemplary
wafer inspection system are discussed in detail. These techniques are mainly geometrical optics for the light
propagation through macroscopic lenses, a rigorous Fourier Modal Method (FMM) for the modeling of light
diffraction at the wafer microstructure and different free-space diffraction integrals. In combination with a
numerically efficient algorithm for the coordinate transformation of electromagnetic fields, field tracing enables
position and fabrication tolerancing. As an example different tilt tolerance effects on the polarization state and
image contrast of a simple wafer inspection system are shown.
The optimization of multi-parameter resonators requires flexible simulation techniques beyond the scalar approximation.
Therefore we generalize the scalar Fox and Li algorithm for the transversal eigenmode calculation to a
fully vectorial model. This modified eigenvalue problem is solved by two polynomial-type vector extrapolation
methods, namely the minimal polynomial extrapolation and the reduced rank extrapolation. Compared to other
eigenvalue solvers these techniques can also be applied to resonators including nonlinear components. As an
example we show the calculation of an azimuthally polarized eigenmode emitted by a resonator containing a
discontinuous phase element and a nonlinear active medium. The simulation is verified by experiments.
The energy-efficient use of LED light requires the development of compact illumination systems for the customized homogenization and shaping of partially-coherent LED light. Therefore a design concept which is based on arrays of aperiodic micro structures, namely cells, for primary or secondary optics is introduced. Each cell of the array deflects locally the light into predefined directions and results in a light spot in the target plane. The light spots of all array cells together form the desired light pattern. The performance of three different cell geometries (linear gratings, micro prisms andmicromirrors) on the homogenization and shaping ofmonochromatic as well as white light LEDs is demonstrated. For the realistic evaluation of the illumination system an LED model including power spectrum, polarization, spatial and temporal coherence is chosen. Furthermore wave-optical effects like diffraction at the cell apertures are taken into account. For the grating cells arrays a rigorous analysis of the diffraction efficiencies is included.
Crystals are widely applied in laser systems and they play important roles as polarization manipulators. A simple uniaxial crystal plate with a certain thickness may serve as a linear/circular polarizer. A uni/biaxial crystal with a proper orientation may work as a polarization beam splitter/combiner. A biaxial crystal at conical refraction may be used to generate radially polarized light or help form a Bessel beam. To design a laser system including crystal component(s), or to optimize the performance of such a system, simulation techniques which models the light propagation through crystals is required. We present a full-vectorial and electromagnetic method that enables the modeling of crystal components with general laser ﬁelds, based on the angular spectrum of plane waves. Propagation of general ﬁeld within anisotropic media and the reﬂection and transmission at a planar interface between isotropic/anisotropic and anisotropic/anisotropic media are discussed. We demonstrate examples on how crystals are used to manipulate the polarization state of laser beams.
Nowadays lasers cover a broad spectrum of applications, like laser material processing, metrology and communications. Therefore a broad variety of different lasers, containing various active media and resonator setups,
are used to provide high design flexibility. The optimization of such multi-parameter laser setups requires powerful simulation techniques. In literature mainly three practical resonator modeling techniques can be found:
Rigorous techniques, e.g. the finite element method (FEM), approximated solutions based on paraxial Gaussian
beam tracing by ABCD matrices and the Fox and Li algorithm are used to analyze transversal resonator modes.
All of these existing approaches have in common, that only a single simulation technique is used for the whole
resonator. In contrast we reformulate the scalar Fox and Li integral equation for resonator eigenmode calculation
into a fully vectorial field tracing operator equation. This allows the flexible combination of different modeling
techniques in different subdomains of the resonator. The work introduces the basic concepts of field tracing in
resonators to calculate vectorial, transversal eigenmodes of stable and unstable resonators.
The fast and accurate propagation of general optical fields in free space is still a challenging task. Most of the
standard algorithms are either fast or accurate. In the paper we introduce without further physical approximations three new algorithms for the fast propagation of non-paraxial vectorial optical fields containing smooth
but strong phase terms. Dependent on the shape of the smooth phase term different propagation operators are
The first method for the efficient propagation of fields, which are containing smooth spherical phase terms,
is based on Mansuripur's extended Fresnel diffraction integral1 using fast Fourier Transformations. This concept
is improved by Avoort's parabolic fitting technique2 and the parameter space, for which the extended Fresnel
operator is numerically feasible, is discussed in detail. Furthermore we introduce the inversion of the extended
Fresnel operator for the fast propagation of non-paraxial fields into the focal region.
Secondly we discuss a new semi-analytical spectrum of plane waves (SPW) operator for the quick propagation of fields with smooth linear phase terms. The method is based on the analytical handling of the linear phase
term and the lateral offset, which reduces the required computational window sizes in the target plane.
Finally we generalize the semi-analytical SPW operator concept to universal shapes of smooth phases by de-
composing non-paraxial fields into subfields with smooth linear phase terms. In the target plane, all propagated
subfields are added coherently where the analytical known smooth linear phase terms are handled numerical
efficient by a new inverse parabasal decomposition technique (PDT).
Numerical results are presented for examples, demonstrating the efficiency and the accuracy of the three new
propagation methods. All simulations were done with the optics software VirtualLabTM.3
This article introduces an efficient tilt operator for harmonic fields. In optical modeling and design, a field tilting
operation is often needed, e.g., the propagation of a harmonic field between non-parallel planes, since most of
the existing propagation operators only deal with the case of propagation between parallel planes. Such operator
enables the modeling of various optical components, like the case of prisms and tolerancing with tilted components.
The tilt operator is a rigorous method to calculate vectorial harmonic fields on tilted planes. The theory
applies a non-equidistant sampling in the k-space of the field before rotation in order to obtain an equidistant
sampling of the rotated field. Different interpolation techniques are employed for the non-equidistant sampling
in the k-space of the initial field and their performances are evaluated.
Besides the tilt operator, the propagation method of harmonic fields through planar interface is proposed as well.
The application of both methods makes it possible to model a sequence of tilted optical interfaces, e.g., prisms.
At the end of this article, a dispersive prisms example is presented. All simulations are done with the optics
The propagation of harmonic fields through homogeneous media is an essential simulation technique in optical
modeling and design by field tracing, which combines geometrical and physical optics. For paraxial fields the
combination of Fresnel integral and the Spectrum of Plane Waves (SPW) integral solves the problem. For non-
paraxial fields the Fresnel integral cannot be applied and SPW often suffers from a too high numerical effort. In
some situations the far field integral can be used instead, but a general solution of the problem is not known.
It is useful to distinguish between two basic cases of non-paraxial fields: 1) The field can be sampled without problems in the space domain but it is very divergent because of small features. A Gaussian beam with large
divergence is an example. 2) The field possesses a smooth but strong phase function, which does not allow its
sampling in space domain. Spherical or cylindrical waves with small radius of curvature are examples. We refer
to such fields as fields with a smooth phase term. The complete phase is the sum of the smooth phase term and
For both cases we present a parabasal field decomposition, in order to propagate the field. In the first case
we perform the decomposition in the Fourier domain and in the second case in the space domain. For each of the
resulting parabasal fields we separate a linear phase factor which has not to be sampled. In order to propagate
the parabasal fields we present a rigorous semi-analytical SPW operator for parabasal fields, which can handle
the linear phase factors without sampling it at any time. We show that the combination of the decomposition
and this modified SPW operator enables an ecient propagation of non-paraxial fields.
All simulations were done with the optics software VirtualLab™.
The propagation of harmonic fields between non-parallel planes is a challenging task in optical modeling. Many
well-known methods are restricted to parallel planes. However, in various situations a tilt of the field is demanded,
for instance in case of folded setups with mirrors and tolerancing with tilted components. We propose a rigorous
method to calculate vectorial harmonic fields on tilted planes. The theory applies a non-equidistant sampling
in the k-space of the field before rotation in order to obtain an equidistant sampling of the rotated field. That
drastically simplifies the interpolation challenge of the tilt operation. The method also benefits from an analytical
processing of linear phase factors in combination with parabasal field decomposition. That allows a numerically
efficient rotation of any type of harmonic fields. We apply this technique to the rigorous propagation of general
harmonic fields through plane interfaces. This propagation can be based on a plane wave decomposition of the
field. If the field is known on the interface a fast algorithm results from the decomposition. However in general,
the field is not known on the interface. Then a rotation operator must be applied first. All simulations were
done with the optics software VirtualLab™.