Modeling and Simulation

The Institute concentrates extensively on simulation methods and their application in optical metrology and optical design. This includes rigorous simulation methods as well as raytracing or scalar diffraction. In addition to the RCWA (Rigorous Coupled Wave Analysis) as the main application, also finite element (FEM) or finite difference time-domain methods (FDTD) are used for rigorous calculations. Since the late 90’s, the simulation tool MICROSIM based on RCWA was set up and continuously improved.

MICROSIM is a software package for the numericalsimulation of Maxwell equations, without physicalapproximation in the diffraction problem domain.In addition to its simulation features, the calculation andvisualization of the corresponding near-fields and microscopicfar-field images are also possible. The applicationsof MICROSIM range from rigorous treatment ofscatterometry and diffractometry, e.g. for semiconductorindustry, to near-field calculations and systematicinvestigations of microscopic imaging techniques.Due to MICROSIM’s modularstructure, extensions and adoptions of the simulationkernel to different project related demands can be performedquickly and easily.The diffraction spectrum is the basis for the calculationof the corresponding near and far-fields. It providescomplete information on the grating in the pupil-planeof an imaging system. Diffraction calculus is based on rigorouscoupled wave analysis (RCWA), in combination withthe enhanced-transmittance-matrix approach. MICROSIM provides different classes of gratingsin 2D and 3D. The implemented 2D structure typesconsist of line-gratings with multiple layers, where ineach layer the refraction index is piecewise constant butotherwise arbitrary.In the 3D case different cross-sections are implemented.These include circles, squares, ellipses andrectangles with different grating-periods in the x- andy-directions, as well as structures within any given userdefined boundary in the framework of RCWA.In order to compute near- or far-fields for arbitraryillumination conditions, the diffraction spectrum has tobe calculated for a group of different plane waves whichare defined by a set of polarisation-states, wavelengthsand angles of incidence. When the complete rigorousdiffraction spectrum in the pupil-plane becomesaccessible, different microscopic imaging techniquescan be simulated by appropriate filtering in the pupilplane.These include bright field microscopy under fullconsideration of polarisation dependent aberrationsand apodisations, dark-field imaging, Zernike phasecontrast, interference microscopy and different types ofpolarisation microscopy.The computation of the near-fields in particularoffers a deep insight into the interaction of the lightwith the structure and thus also into the optical imageformation by the consideration of evanescent fieldcomponents. These modes carry the complete highresolution information of the structure. Because oftheir exponentially short range they don’t contributeto the far field, so they don’t become directly apparentin the microscopic far-field image

RCWA is based on a Fourierseries expansion. In order to save computationtime as few Fourier coefficients (modes) as necessaryto obtain a sufficiently accurate result are retained.The complexity of RCWA applied to crossed gratingswith the truncation order n is O(n6); thus it is agreat benefit to reduce the number of required Fouriermodes for crossed grating structures.The key point in this formulation is tofind a normal vector (NV) field which is orthogonalto the material boundary and which contains thelocal orientation of the material boundary. Thisinformation has to be transferred to Fourier space,in order to correctly form the product D=ε·E asa convolution in Fourier space. However, theNV has to be continued throughout the completeelementary cell. This continuation is notunique and different continuations can lead to differentconvergence behavior.

 A large aperture, such as a DOE,is divided into equally spaced sub-apertures. Thesesub-apertures overlap on the left and right edge. Theyare treated as local gratings in the following rigorouscalculation. The overlap is necessary to minimize theerror coming from the "wrong" continuation at theedges of the local grating periods. The bigger theoverlap, the smaller the error. We havecomputed the nearfields of the sub-apertures (withoverlap) with MICROSIM to show that theydo not deviate significantly from the nearfields of thewhole structure computed without field-stitching.The obtained diffraction orders of the local calculationsare taken as Rayleigh coefficients at thetop (or the bottom) of the local structure. The fieldexpressed in these Fourier-coefficients is equatedwith the field at top of the whole structure. Then theRayleigh coefficients and subsequently the diffractionorders of the whole DOE can be obtained.The field-stitching method offers the possibilityof combining faster (e.g. scalar) methods tosimulate areas of a structure showing feature sizeswell above the considered wavelength with rigorouscalculations of areas of the structure showing subwavelengthfeatures. It is, however, only useful ifsufficient overlap can be taken into account.