Abstract: Disclosed is a method for reconstructing a parameter of a lithographic process. The method comprises the step of designing a preconditioner suitable for an input system comprising the difference of a first matrix and a second matrix, the first matrix being arranged to have a multi-level structure of at least three levels whereby at least two of said levels comprise a Toeplitz structure. One such preconditioner is a block-diagonal matrix comprising a BTTB structure generated from a matrix-valued inverse generating function. A second such preconditioner is determined from an approximate decomposition of said first matrix into one or more Kronecker products.

Abstract: Disclosed is a method for reconstructing a parameter of a lithographic process. The method comprises the step of designing a preconditioner suitable for an input system comprising the difference of a first matrix and a second matrix, the first matrix being arranged to have a multi-level structure of at least three levels whereby at least two of said levels comprise a Toeplitz structure. One such preconditioner is a block-diagonal matrix comprising a BTTB structure generated from a matrix-valued inverse generating function. A second such preconditioner is determined from an approximate decomposition of said first matrix into one or more Kronecker products.

Abstract: A method of calculating electromagnetic scattering properties of a structure, the structure including materials of differing properties and the structure being periodic in at least one lateral direction and extending in a vertical direction, comprises: numerically solving a volume integral equation for electromagnetic scattering for a plurality of modes in the at least one lateral direction, by performing, for each respective mode, integration (1350) of a pseudo-spectral polynomial (Chebyshev) expansion in the vertical direction multiplied by a ID Green's function using the same sample points in the orthogonal direction for all of the plurality of modes. The integration is performed by solving a regularized linear system of equations between first (1116) and second (1120) discrete transformation steps to compute (1118) values of a regularized Chebyshev expansion coefficient vector (?). Electromagnetic scattering properties of the structure are calculated using the results of the numerical solution.

Abstract: A method of calculating electromagnetic scattering properties of a structure, the structure including materials of differing properties and the structure being periodic in at least one lateral direction and extending in a vertical direction, comprises: numerically solving a volume integral equation for electromagnetic scattering for a plurality of modes in the at least one lateral direction, by performing, for each respective mode, integration (1350) of a pseudo-spectral polynomial (Chebyshev) expansion in the vertical direction multiplied by a ID Green's function using the same sample points in the orthogonal direction for all of the plurality of modes. The integration is performed by solving a regularized linear system of equations between first (1116) and second (1120) discrete transformation steps to compute (1118) values of a regularized Chebyshev expansion coefficient vector (?). Electromagnetic scattering properties of the structure are calculated using the results of the numerical solution.

Abstract: Algorithm for reconstructing grating profile in a metrology application is disclosed to numerically solve a volume integral equation for a current density. It employs implicit construction of a vector field that is related to electric field and a current density by a selection of continuous components of vector field being continuous at one or more material boundaries, so as to determine an approximate solution of a current density. The vector field is represented by a finite Fourier series with respect to at least one direction, x, y. Numerically solving volume integral equation comprises determining a component of a current density by convolution of the vector field with a convolution operator, which comprises material and geometric properties of structure in the x, y directions. The current density is represented by a finite Fourier series with respect to x, y directions. Continuous components are extracted using convolution operators acting on electric field and current density.

Abstract: Improved convergence in the volume-integral method (VIM) of calculating electromagnetic scattering properties of a structure is achieved by numerically solving a volume integral equation for a vector field, F, rather than the electric field, E. The electric field, E, is determined from the vector field, F, after solving of the volume integral equation. The vector field, F, may be related to the electric field, E, by a change of basis, and may be continuous at material boundaries where the electric field, E, has discontinuities. Convolutions of the vector field, F, are performed using convolution operators according to the finite Laurent rule, which allows for efficient matrix-vector products using Fast Fourier Transforms. An invertible convolution-and-change-of-basis operator, C, is configured to transform the vector field, F, to the electric field, E, by performing a change of basis according to material and geometric properties of the periodic structure.

Abstract: A projection operator framework is described to analyze the concept of localized normal-vector fields within field-material interactions in a spectral basis, in isotropic and anisotropic media. Generate a localized normal-vector field n in a region of the structure defined by the material boundary, decomposed into sub-regions with a predefined normal-vector field and possibly corresponding closed-form integrals. Construct a continuous vector field F using the normal-vector field to select continuous components ET and Dn. Localized integration of normal-vector field n over the sub-regions to determine coefficients of, C. Determine components Ex, Ey, Ez of the electromagnetic field by using field-material interaction operator C to operate on vector field F. Calculate electromagnetic scattering properties of the structure using the determined components of the electromagnetic field.

Abstract: Numerical calculation of electromagnetic scattering properties and structural parameters of periodic structures is disclosed. A reflection coefficient has a representation as a bilinear or sesquilinear form. Computations of reflection coefficients and their derivatives for a single outgoing direction can benefit from an adjoint-state variable. Because the linear operator is identical for all angles of incidence that contribute to the same outgoing wave direction, there exists a single adjoint-state variable that generates all reflection coefficients from all incident waves that contribute to the outgoing wave. This adjoint-state variable can be obtained by numerically solving a single linear system, whereas one otherwise would need to solve a number of linear systems equal to the number of angles of incidence.

Abstract: In order to guide electromagnetic waves in the terahertz range over long distances of several meters with low bending losses and large bandwidth, a device, a system and a method are provided such that electromagnetic waves in the terahertz range can be coupled into a wire having a core structure and at least one confinement structure, wherein the confinement structure extends continuously along a length of the wire.

Abstract: Algorithm for reconstructing grating profile in a metrology application is disclosed to numerically solve a volume integral equation for a current density. It employs implicit construction of a vector field that is related to electric field and a current density by a selection of continuous components of vector field being continuous at one or more material boundaries, so as to determine an approximate solution of a current density. The vector field is represented by a finite Fourier series with respect to at least one direction, x, y. Numerically solving volume integral equation comprises determining a component of a current density by convolution of the vector field with a convolution operator, which comprises material and geometric properties of structure in the x, y directions. The current density is represented by a finite Fourier series with respect to x, y directions. Continuous components are extracted using convolution operators acting on electric field and current density.

Abstract: Improved convergence in the volume-integral method (VIM) of calculating electromagnetic scattering properties of a structure is achieved by numerically solving a volume integral equation for a vector field, F, rather than the electric field, E. The vector field, F, may be related to the electric field, E, by a change of basis, and may be continuous at material boundaries where the electric field, E, has discontinuities. Convolutions of the vector field, F, are performed using convolution operators according the finite Laurent rule (that operate according to a finite discrete convolution), which allow for efficient matrix-vector products via 1D and/or 2D FFTs (Fast Fourier Transforms). An invertible convolution-and-change-of-basis operator, C, is configured to transform the vector field, F, to the electric field, E, by performing a change of basis according to material and geometric properties of the periodic structure.

Abstract: Numerical calculation of electromagnetic scattering properties and structural parameters of periodic structures is disclosed. A reflection coefficient has a representation as a bilinear or sesquilinear form. Computations of reflection coefficients and their derivatives for a single outgoing direction can benefit from an adjoint-state variable. Because the linear operator is identical for all angles of incidence that contribute to the same outgoing wave direction, there exists a single adjoint-state variable that generates all reflection coefficients from all incident waves that contribute to the outgoing wave. This adjoint-state variable can be obtained by numerically solving a single linear system, whereas one otherwise would need to solve a number of linear systems equal to the number of angles of incidence.

Abstract: A projection operator framework is described to analyze the concept of localized normal-vector fields within field-material interactions in a spectral basis, in isotropic and anisotropic media. Generate a localized normal-vector field n in a region of the structure defined by the material boundary, decomposed into sub-regions with a predefined normal-vector field and possibly corresponding closed-form integrals. Construct a continuous vector field F using the normal-vector field to select continuous components ET and Dn. Localized integration of normal-vector field n over the sub-regions to determine coefficients of, C. Determine components Ex, Ey, Ez of the electromagnetic field by using field-material interaction operator C to operate on vector field F. Calculate electromagnetic scattering properties of the structure using the determined components of the electromagnetic field.