Transmission Cross Coefficient Research Articles

Symmetry-preserving modeling for lithographic imaging.

In computational imaging and lithography, it has been a challenge for a numerical model to faithfully preserve symmetries in the physical imaging system. In this Letter, we present a project-to-symmetry-subspace (PTSS) method to prevent symmetry loss during the iterative generation of optical kernels. Essentially, PTSS is to project iterative vectors onto a predefined symmetric subspace when decomposing the transmission cross coefficient (TCC). Simulation results demonstrate the PTSS-generation of a truncated set of optical kernels that are substantially free of symmetry error, regardless of the order of truncation.

Accelerating extreme ultraviolet lithography simulation with weakly guiding approximation and source position dependent transmission cross coefficient formula

Accelerating extreme ultraviolet lithography simulation with weakly guiding approximation and source position dependent transmission cross coefficient formula

Spin-Dependent Nonlinear Contrast Transfer in Transmission Electron Microscopy.

In this study, the spin-dependent nonlinear contrast transfer in transmission electron microscopy is derived from the eikonal expansion of the Dirac equation. The transmission cross-coefficient of the standard imaging theory is amended by a spin-dependent factor, whose effect is investigated for single scattering in the object by an electrical field under polarized and unpolarized illumination, and it is illustrated with numerical results and plots for a kinetic energy of 80 keV. The resulting image displacement and image convolution increase with decreasing kinetic energy but are always smaller than a wavelength. General features of the cross-coefficient are discussed to identify favorable conditions for the measurement of the small spin effects, possibly in an unmodified instrument.

Gradient-based source mask and polarization optimization with the hybrid Hopkins–Abbe model

Source mask and polarization optimization (SMPO) is a promising extension of the widely used resolution enhancement technology, source mask optimization (SMO), to further enhance chip manufacturability beyond 28-nm node. Our work is aimed to develop an efficient gradient-based SMPO method by employing the hybrid Hopkins–Abbe imaging model to fulfill the goal. In addition to source and mask variables, the model is adapted to also include polarization variables to realize the optimization. Compact formulas for forward and backward model application are derived. The computation benefits from precomputed transmission cross coefficients and features high efficiency. Validity of the method is confirmed by case studies. For dense array pattern case, the optimal source and polarization can be found analytically. SMPO optimized results match well with the theoretical expectations. In addition, process window, mask error enhancement factor, and normalized image log-slope for the studied cases all get improved over the counterpart SMO results, which employ commonly used polarization. Runtime analysis shows the method is computationally efficient. Our work provides a valid way to optimize polarization together with source and mask.

Comparison of different imaging models handling partial coherence for aberration-corrected HRTEM at 40–80 kV

Partial coherence of the electron waves leaving the specimen is taken into account in the high-resolution transmission electron microscopy (HRTEM) image simulation by mainly three methods - the incoherent summation approach, the transmission cross-coefficient (TCC) model, and the quasi-coherent model. In the incoherent summation approach, coherent images are calculated for each point in the effective source and summed up incoherently. The TCC is the transfer function of the microscope obtained based on the incoherent summation approach. An analytical form of the TCC can be derived by assuming a Gaussian distribution for the source radiation and for the variation of the focal length caused by the energy spread of the effective source. In the quasi-coherent model, the partial coherence effect is simplified by multiplying the wave function at the diffraction plane with the envelope functions. Envelope functions suppress the contributions to the image contrast from waves which do not propagate along the optical axis. The quasi-coherent model is usually sufficient for the image simulation of weak phase objects. This model is more computationally efficient than both the incoherent summation approach and the TCC model. For the Cs−corrected and Cc/Cs−corrected microscopes operating at 80 kV, we have compared images simulated by using the three models with the experimental images. The comparison shows that the quasi-coherent model also provides a sufficient approximation for the image simulation of high-Z materials if chromatic aberration is corrected and the samples comprise only several atomic layers. In the case of only Cs-correction, the incoherent summation approach or the TCC model needs to be employed for modelling the imaging of high-Z samples even though it is more computationally consuming.

Analytical transmission cross-coefficients for pink beam X-ray microscopy based on compound refractive lenses

Analytical expressions for the transmission cross-coefficients for x-ray microscopes based on compound refractive lenses are derived based on Gaussian approximations of the source shape and energy spectrum. The effects of partial coherence, defocus, beam convergence, as well as lateral and longitudinal chromatic aberrations are accounted for and discussed. Taking the incoherent limit of the transmission cross-coefficients, a compact analytical expression for the modulation transfer function of the system is obtained, and the resulting point, line and edge spread functions are presented. Finally, analytical expressions for optimal numerical aperture, coherence ratio, and bandwidth are given.

Impact of partial coherence on the apparent optical transfer function derived from the response to amplitude edges.

We present an investigation of the impact of partial coherence on optical imaging systems with the focus on whole slide imaging (WSI) systems for digital pathology. The investigation is based on the analysis of the edge response of the optical system, which gives rise to an apparent optical transfer function (OTF) that can be linked to two elementary complex functions Q and U. The function Q is directly related to the transmission cross coefficient (TCC) and can be identified with the performance function first introduced by Kintner and Sillitto. The function U depends on the TCC in a more involved way. When there are no aberrations the Q-function corresponds to the real part of the apparent OTF and the U function to the imaginary part of the apparent OTF. Close to the incoherent limit the effect of the U function is a mere shift of the edge compared to the fully incoherent case. We propose a new expression for the dependence of the depth of focus (DOF) on spatial frequency and on the partial coherence factor σ, and validate it by simulation. Partial coherence effects are investigated experimentally on a WSI system with a compact LED-based Köhler illumination unit with variable condenser NA. This unit incorporates a top hat diffuser for providing a reasonably uniform illumination field, with variations below 10% across the imaged field of view. The measurements of the apparent through-focus OTF derived from edges on a custom resolution chart for different σ were substantially in agreement with the simulations. Finding an optimal value for σ is not straightforward as lateral resolution and the level of edge ringing improve with increasing σ, whereas edge contrast and DOF improve with decreasing σ. We assess that the trade-off for the particular application of WSI systems for digital pathology is optimized for a σ value in the range of 0.55-0.75.

Why Do We Need to Use Three-Dimensional (3D) Fourier Transform (FT) Analysis to Evaluate a High-Performance Transmission Electron Microscope (TEM)?

The resolution of high-resolution transmission electron microscopes (TEM) has been improved down to subangstrom levels by correcting the spherical aberration (Cs) of the objective lens, and the information limit is thus determined mainly by partial temporal coherence. As a traditional Young's fringe test does not reveal the true information limit for an ultra-high-resolution electron microscope, new methods to evaluate temporal coherence have been proposed based on a tilted-beam diffractogram. However, the diffractogram analysis cannot be applied when the nonlinear contribution becomes significant. Therefore, we have proposed a method based on the three-dimensional (3D) Fourier transform (FT) of through-focus TEM images, and evaluated the performance of some Cs-corrected TEMs at lower voltages. In this report, we generalize the 3D FT analysis and derive the 3D transmission cross-coefficient. The profound difference of the 3D FT analysis from the diffractogram analysis is its capability to extract linear image information from the image intensity, and further to evaluate two linear image contributions separately on the Ewald sphere envelopes. Therefore, contrary to the diffractogram analysis the 3D FT analysis can work with a strong scattering object. This is the necessary condition if we want to directly observe the linear image transfer down to a few tens of picometer.

Two models for partially coherent imaging

This paper discusses the physical difference between two image formation approaches for partially coherent imaging. In one approach, the pupil function is shifted according to illumination condition as in the transmission cross coefficient (TCC) approach, whereas in the other approach, the object spectrum is shifted. Although the two approaches result in identical images, they are built on distinct physical models. Eigenfunction analysis reveals that the TCC approach is built on an artificial optical model only for image calculation. Therefore, the two approaches are not interchangeable except for image calculation. Such an example is found in calculating the entropy in an imaging system.

Fast aerial image simulations for partially coherent systems by transmission cross coefficient decomposition with analytical kernels

Aerial image simulation is one of the most critical components in the model-based optical proximity correction (OPC), which has become a necessary part of resolution enhancement techniques used to improve the performance of subwavelength optical lithography. In this paper, a fast aerial image simulation method is proposed for partially coherent systems by decomposing the transmission cross coefficient (TCC) into analytical kernels. The TCC matrix is projected onto a function space whose basis is analytical circle-sampling functions (CSFs) and converted into a much smaller projected matrix. By performing singular value decomposition (SVD) to the projected matrix, its eigenvectors together with the CSFs are used to generate a set of analytical TCC kernels. The proposed method avoids directly performing SVD to the large TCC matrix, making it much more runtime efficient than the conventional SVD method. Furthermore, the grid size of the kernels can be flexibly set to any desired value in aerial image simulations, which is not realizable with the conventional SVD method. The comparison of aerial image intensity errors and edge placement errors calculated by the proposed method and the conventional SVD method has confirmed the validity of the proposed method. An OPC example is also provided to further demonstrate its efficiency.

Fast imaging algorithm for DMD-based photolithography with partially coherent illumination

In the paper, the characteristic of imaging with digital micro-mirror device (DMD) illuminated by a partially coherent light is discussed using Abbe's method. When a mask is sampled by DMD, the intensity in the substrate is modulated by sinc function and cut off by pupil function, according to oblique illumination theory and Fourier transform property of exponential function, a fast partially coherent imaging algorithm for DMD-based dynamic projection photolithography is proposed to combine Hopkins's method with Abbe's method. In the algorithm, the transmission cross coefficient (TCC) matrix of DMD projection photolithography is constructed as two-matrix multiplication according to illuminate source points. The new development presented in this paper utilizes matrix multiplication technique to speed up the computation of TCC matrix by tenfold on an average, which is beneficial to real time mask shift and mask optimization in DMD-based gray-tone photolithography.

Fast Evaluation of Aberration-Induced Intensity Distribution in Partially Coherent Imaging Systems by Cross Triple Correlation

We propose a method suitable for the fast calculation and evaluation of aberration-induced intensity distribution in partially coherent imaging systems, such as projection lithographic tools. The method is based on transmission cross coefficient (TCC) decomposition by a general operator, namely cross triple correlation (CTC). By expanding the aberrated pupil function into a Taylor series, the TCC is decomposed into different terms. Each term is further represented as a weighted sum of several CTCs. By exploring the properties of CTC, the aerial image intensity induced by wavefront aberration is calculated quickly and separated clearly from that without aberration. Simulation results and discussion are presented.

Three-dimensional imaging by partially coherent light under nonparaxial condition

In this paper, we present the theory of three-dimensional (3D) imaging using partially coherent light under the nonparaxial condition. Using the linear system approach, we derive the image intensity in terms of the 3D nonparaxial transmission cross coefficient (TCC) and the transmission function defined in this paper. We present that the 3D TCC can be calculated by multiple applications of the 3D fast Fourier transform instead of the six-dimensional integral in the original formula. Using the simplified formula, we simulate phase contrast and Nomarski differential interference contrast (DIC) imaging of a transparent 3D object. Within our knowledge, the 3D model for the DIC based on the 3D nonparaxial TCC is the most rigorous approach that has been suggested. It demonstrates clearly the optical sectioning effect of DIC.

Expansion of image interaction by the eigenfunction of the transmission cross coefficient to show the interaction between features

This paper presents a method for evaluating the image interaction of two object patterns of arbitrary shape in partially coherent imaging. The interaction is shown to be the interference of the eigenfunctions of the transmission cross coefficient at the image plane convolved with each of the object patterns. For two arbitrary patterns, the ith eigenfunction convolved with one object pattern interferes only with the ith eigenfunction convolved with the second object pattern. This method is useful for understanding how object feature shapes influence image interactions, and examples are simulated to evaluate the method. A method to determine constructive and destructive interference areas is also introduced.

Fast fine-pixel aerial image calculation in partially coherent imaging by matrix representation of modified Hopkins equation

A fast computation algorithm for fine-pixel aerial images is presented that modifies the transmission cross coefficient approach of Hopkins as a product of two matrices. The spatial frequency of the image is calculated by the sum of diagonal and off-diagonal elements of the matrix. Let N, N(F), and M be the number of the point sources, the sampling number for fast Fourier transform, and the sampling number in the spatial frequency domain ranging twice the pupil size, respectively. The calculation time of this method is proportional to BN[(M-1)/2](4), while that of a conventional source integration method is 2ANN(F)log(2)N(F), where A and B are constants and generally B < A. If N(F) is sufficiently greater than M or M is small enough, which is the fine-pixel condition, this method runs faster than the source integration method. If the coherence factor is 0.9 and M < or = 55, this method runs faster than the source integration even under the Nyquist sampling condition.

Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients

This paper physically compares two different matrix representations of partially coherent imaging with the introduction of matrices E and Z, containing the source, object, and pupil. The matrix E is obtained by extending the Hopkins transmission cross coefficient (TCC) approach such that the pupil function is shifted while the matrix Z is obtained by shifting the object spectrum. The aerial image I can be written as a convex quadratic form I = <phi|E|phi> = <phi|Z|phi>, where |phi> is a column vector representing plane waves. It is shown that rank(Z) < or = rank(E) = rank(T) = N, where T is the TCC matrix and N is the number of the point sources for a given unpolarized illumination. Therefore, the matrix Z requires fewer than N eigenfunctions for a complete aerial image formation, while the matrix E or T always requires N eigenfunctions. More importantly, rank(Z) varies depending on the degree of coherence determined by the von Neumann entropy, which is shown to relate to the mutual intensity. For an ideal pinhole as an object, emitting spatially coherent light, only one eigenfunction--i.e., the pupil function--is enough to describe the coherent imaging. In this case, we obtain rank(Z) = 1 and the pupil function as the only eigenfunction regardless of the illumination. However, rank(E) = rank(T) = N even when the object is an ideal pinhole. In this sense, aerial image formation with the matrix Z is physically more meaningful than with the matrix E. The matrix Z is decomposed as B(dagger)B, where B is a singular matrix, suggesting that the matrix B as well as Z is a principal operator characterizing the degree of coherence of the partially coherent imaging.

Tensorial Model for Photolithography Aerial Image Simulation

In this paper, we propose to adapt the multilinear algebra tools to the tensor of Transmission Cross-Coefficients (TCC) values for aerial image simulation in order to keep the data tensor as a whole entity. This new approach implicitly extends the singular value decomposition (SVD) to tensors, that is, Higher Order SVD or TUCKER3 tensor decomposition which is used to obtain lower rank- tensor approximation (LRTA ). This model requires an Alternating Least Square (ALS) process known as TUCKALS3 algorithm. The needed number of kernels is estimated using two adapted criteria, well known in signal processing and information theory. For runtime improvement, we use the fixed point algorithm to calculate only the needed eigenvectors. This new approach leads to a fast and accurate algorithm to compute aerial images.

Optimal approximation of transfer cross coefficient for system transmission based on tensorial signal methods

In photolithography, aerial image simulation of the mask has become mandatory. To compute aerial images, transmission cross coefficients (TCCs), drawn from Hopkins optical system transmission function, are arranged as a four-way array (four-entry table) called a fourth-order tensor. To estimate the kernels using the linear algebra-based methods, the existing algorithms unfold this tensor into a matrix. To reduce the computational load, this matrix is approximated by lower rank order matrix owing to the singular value decomposition (SVD). We propose to adopt the multilinear algebra tools to the tensor of TCC values in order to keep this data tensor as a whole entity. For runtime improvement, we use a fixed point algorithm to estimate only the needed eigenvectors. To estimate the optimal number of needed eigenvectors, two well-known criteria of signal processing and information theory are adopted. This tensorial approach leads to a fast and accurate algorithm to compute aerial images.

Aerial image back propagation with two-dimensional transmission cross coefficient

We present the basic concept of a fast mask optimization method that utilizes target-intensity back propagation. This method decomposes the target-intensity using a two-dimensional (2-D) transmission cross coefficient. After applying normal incidence approximation to the decomposed target intensity, the spectrum of the mask is optimized in the pupil plane. Since the optimization is performed in the pupil plane, one can use relatively small sampling points, leading to a fast optimization. By setting a high-contrast target intensity, we can obtain an approximation of a phase shifting mask, whereas a low-contrast target intensity approximates a binary intensity mask or attenuated phase shifting mask.

ELIAS: An Accurate and Extensible Lithography Aerial Image Simulator With Improved Numerical Algorithms

Lithography simulators have been playing an indispensable role in process optimization and design for manufacturability (DFM). The ever smaller feature sizes demand higher numerical accuracy and faster runtime on these lithography simulators. Aerial image simulation is the first key step in lithography simulation, and the method using transmission cross coefficient (TCC), which is a two-dimensional integral, is the most commonly used technique for full-chip aerial image simulation. In this paper, we present a very accurate, yet efficient and extensible aerial image simulator, ELIAS. We find that the majority of the numerical error during the TCC computation is due to the discontinuous boundaries of the support of the TCC integrand. We reduce the error dramatically by using a recursive integration algorithm. Because TCC is usually computed on uniform grids, we further speed up the algorithm without increasing the errors. Given the same accuracy, our new algorithm can speed up the runtime by 100 times - 1000times. Our algorithm also provides smooth tradeoff between accuracy and runtime. It can be used to benchmark other lithography aerial simulators. In addition, ELIAS provides an open-source, flexible software framework to incorporate different lithography settings.