Norbert Kerwien

Normal vector method for convergence improvement using the RCWA for crossed gratings

The rigorous coupled wave analysis (RCWA) is a widely used method for simulating diffraction from periodic structures. Since its recognized formulation by Moharam [J. Opt. Soc. Am. A12, 1068 and 1077 (1995)], there still has been a discussion about convergence problems. Those problems are more or less solved for the diffraction from line gratings, but there remain different concurrent proposals about the convergence improvement for crossed gratings. We propose to combine Popov and Nevières formulation of the differential method [Light Propagation in Periodic Media (Dekker, 2003) and J. Opt. Soc. Am. A18, 2886 (2001)] with the classical RCWA. With a suitable choice of a normal vector field we obtain a better convergence than for the formulations that are known from the literature.

Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio

The signal-to-noise ratio required to obtain 10-nm accuracy in the measurement of lateral position is studied with an interference microscope. Evaluations are performed using the rigorous coupled-wave analysis (RCWA) modal approach and Hopkins image formation theory.

Diffraction-induced coherence levels

We examined the influence of complex diffraction effects on low-coherence fringes created for high-aspect depth-to-width ratio structures called trenches. The coherence function was analyzed for these micrometer-wide trenches and was registered with a white-light interference microscope. For some types of surface structure we observed that additional low-coherence fringes that do not correspond directly to the surface topology are formed near the sharp edges of the structures. These additional coherence fringes were studied by rigorous numerical evaluations of vector diffractions, and these simulated interference fields were then compared with experimental results that were obtained with a white-light interference microscope.

Vector simulations of dark beam interaction with nanoscale surface features

In earlier publications, it was shown that scanning of surfaces by dark beams can be exploited for sub-wavelength feature analysis. In this work, we present vector simulations based in Rigorous Coupled-Wave Analysis with the purpose to estimate the expected resolution of the method, both lateral (feature size) and axial (height). The dark beam used in this study has a line singularity generated by a π-phase step positioned in a Gaussian beam. Various combinations of the illumination and detection nuFmerical apertures (from NA=0.2 to NA=0.8) and different surface features were studied. Polarization effects which become significant at high numerical apetures, were considered as an additional source of information for the analysis. In the case of a sub-wavelength feature on an ideal surface, the resolution of the method is limited only by the electronics noise. In particular, under a reasonable assumption of a 105 signal to noise ratio, it is possible to detect a 0.2 nm step.

Vectorial thin-element approximation: a semirigorous determination of Kirchhoff's boundary conditions

A semirigorous model is presented that bridges the gap between classical scalar diffraction theory on the one hand and fully rigorous simulation models on the other. It falls back on the basic ideas of scalar diffraction theory, especially the Kirchhoff approximation. In contrast to classical techniques, however, the boundary values are determined by rigorous methods of the stratified medium theory in the scope of a fully vectorial formulation. By this means the proposed approach takes vertical rigorous coupling effects inside the grating into account while neglecting the lateral ones. We therefore call this method semirigorous and use the name vectorial thin-element approximation. A direct comparison with rigorous coupled-wave analysis as a fully rigorous simulation model allows a detailed discussion of its range of validity and demonstrates a reduction of computation time of the order of 3 magnitudes. In addition, it also reveals deeper insight into the details of the electrodynamics inside diffracting structures. Some examples will demonstrate this benefit.

Resolution enhancement technologies in optical metrology

In history any field of activity shows periods that are characterized by evolutionary, revolutionary and stagnating moments, respectively. Especially in optics the 17th, 19th, and 20th century can be considered as revolutionary periods that changed our insight dramatically. However, the often observed opinion that a discipline is already completed was always revised by new facts that broadened our knowledge about the certain field. For instance, in modern optics the discovery of the wave front reconstruction principle (holography) and the implementation of the light amplification by stimulated emission in the laser pushed the further development of the field in an outstanding way. Nevertheless, a lot of questions are looking for an answer. Particularly, in order to justify its role as an enabling technology for most of the key technologies optics has to follow strictly the enormous requirements in spatial resolution, robustness, reliability and traceability. Since it is far beyond the claim of this contribution to address all these problems we concentrate our view on modern approaches of resolution enhancement only.

Ray selection for optimization of rotationally symmetric systems

Abstract Efficient performance assessment is essential during the design of systems involving complex aspheres. We present new classes of pupil sampling schemes that, with a reduced number of rays, yield accurate estimates of the RMS wavefront aberration over a circular pupil. It turns out that the number of samples in the pupil can be reduced by a factor of about 0.7, and these ideas can also be expected to lead to a similar additional reduction factor when averaging over the field and color. Beyond that, analysis of a patented lens system is used to establish the path to further significant reductions.

Rapid quantitative phase imaging using phase retrieval for optical metrology of phase-shifting masks

The accuracy of a wave front generated by a lithography phase-shifting mask is essential for the performance of a lithography system. The main task of mask inspection is therefore to detect and quantify phase distortions caused by defects on the mask. There are three different classical ways to get the phase information generated by an object: First, interferometric techniques such as Linnik- or Mach-Zehnder-interferometry, second the method of spatial filtering, as is done in quantitative Zernike phase contrast microscopy, and third the use of phase-retrieval. The first two methods need a highly adapted set-up and are difficult to adjust. Phase retrieval, however, just needs a set of intensity images captured under different focus conditions to reconstruct the phase information. Therefore, this method is ideally suited for fast inspection in an industrial environment. We present phase measurements on MoSiN phase masks reconstructed on the basis of the transport of intensity equation. The effect of low-pass filtering inherent in this method is quantitatively investigated by numerical simulations done with our microscope simulation tool MicroSim. By systematic variation of imaging conditions as well as object-parameters we give insight to the applicability and the limits of the method. Comparison to Mach-Zehnder-interferometry allows the evaluation of the method also for partially coherent illumination conditions.

Inspection of subwavelength structures and zero-order gratings using polarization interferometry

The low-pass characteristic of the optical imaging limits severely a quantitative measurement of structure-sizes below the optical wavelength and leads to measurement errors. On the other hand, small structures show different optical characteristics for different polarizations. A fact that corresponds to the form-birefringence of microstructures. It is described for zero-order gratings by polarization dependent dielectric constants in the effective medium theory. The birefringence can be measured accurately by use of polarization interferometry where two orthogonal polarizations interfere so that their phase-difference can be determined. To that end an electro-optic modulator is inserted into the optical path of a polarization microscope to provide the well-defined phase steps for an evaluation according to phase-shifting interferometry. From the phase difference we can conclude on the optical path-difference for both polarizations and from this - using the structure thickness - on the birefringence. Waveguide-models are applied for image interpretation. For an estimation of the width of the structure we compare polarization-interferometrical measurements with rigorous numerical simulations.

Coherence effects in narrow trench measurements using white light interferometry

White light interferometry (WLI)1 and optical coherence tomography (OCT) for micro-roughness and profile measurements are both based on the principle of low coherence peak sensing. Additional coherence peaks not corresponding to the surface topology exist due to diffraction from narrow structures; these additional coherence levels must be properly interpreted by WLI and OCT algorithms. We observed additional coherence peaks for a silicon surface with etched trenches only a few microns wide and investigated the amplitude variation of the coherence peaks versus the trench width and depth. Effects from changes in the optical system such as the numerical aperture (NA), magnification, polarization and wavelength spectra were also studied. For example, for an objective with a given numerical aperture the amplitude of the additional coherence level increases with the decreasing trench width. The amplitude of the fringes at the bottom of trench decreases making the amplitude of the fringes representing the real surface much weaker than the additional coherence levels created from strong complex diffraction effects. The experimental data obtained with optical profiler NT 8000 was compared with simulated results obtained from rigorous coupled-wave analysis2 (RCWA) and Hopkins image-formation theory that traces different polarization components in the diffraction image. We have found very good agreement between experimental and simulation results for the case of circular polarization and object structure in the form of a trench 5 microns wide and about 20 microns deep. The additional coherence level can be a problem in i.e. MEMS measurement if not interpreted correctly.