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Scaling trends in the semiconductor industry towards smaller technology nodes and feature sizes are continuing and first consumer products manufactured with the help of EUV technology are already on the market. Major industrial players have introduced EUV lithography into their production at the 7nm technology node and with the 5nm node being on its way [1], the amount of EUV lithographic layers is expected to rise significantly and implementation of EUV double patterning is anticipated. These developments lead to more strict technological requirements especially for the corresponding EUV but also for the used high-end DUV photomasks in terms of minimum feature sizes and acceptable Edge Placement Errors (EPE). Moreover, photomask defectivity increases dramatically with shrinking feature sizes. This creates significant challenges to the industry, as in particular the most cost intensive EUV photomasks possess the highest numbers of defects. The current industry standard for high-end photomask repair tools is the MeRiT neXT [2]. To face the upcoming challenges an efficient and reliable way to repair future high-end photomasks is inevitable. A corresponding repair tool must address decreased minimum feature sizes and increased pattern complexity on high-end photomasks. In this paper we present our latest results of high-end EUV repairs carried out on the next generation photomask repair tool MeRiT LE. The tool shows improved system dynamics, makes use of a new electron beam column, which operates at a low electron beam voltage down to 400V and enables the repair of next generation ultra-small defects.

We use the one-dimensional Burgers equation to illustrate the effect of replacing the standard Laplacian dissipation term by a more general function of the Laplacian -- of which hyperviscosity is the best known example -- in equations of hydrodynamics. We analyze the asymptotic structure of solutions in the Fourier space at very high wave-numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing Reynolds numbers from algebraic recursion relations involving iterated integrations. We give a detailed analysis of their analytic structure for two different types of dissipation: a hyperviscous and an exponentially growing dissipation term. Our results, obtained in the limit of vanishing Reynolds numbers, are validated by high-precision numerical simulations at non-zero Reynolds numbers. We then study the bottleneck problem, an intermediate asymptotics phenomenon, which in the case of the Burgers equation arises when ones uses dissipation terms (such as hyperviscosity) growing faster at high wave-numbers than the standard Laplacian dissipation term. A linearized solution of the well-known boundary layer limit of the Burgers equation involving two numerically determined parameters gives a good description of the bottleneck region., Comment: 23 pages; 6 figures

The Lagrangian complex-space singularities of the steady Eulerian flow with stream function $\sin x_1 \cos x_2$ are studied by numerical and analytical methods. The Lagrangian singular manifold is analytic. Its minimum distance from the real domain decreases logarithmically at short times and exponentially at large times., Comment: 19 pages, 6 figures, revised version

Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this procedure works already for Reynolds numbers of the order of 100 (or even lower when combined with a suitable extended self-similarity technique). At moderate Reynolds numbers, for the accurate determination of scaling exponents, it is crucial to use higher than double precision. Similar issues are likely to arise for three-dimensional Navier--Stokes simulations., 5 pages, 2 figures, Published in Phys. Rev E (Rapid Communications)

We consider a modification of the three-dimensional Navier--Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as $\ue ^{|k|/\kd}$ at high wavenumbers $|k|$. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than $\ue ^{-C(k/\kd) \ln (|k|/\kd)}$ for any $C, 29 pages, 3 figures, Comm. Math. Phys., in press

It is shown that the use of a high power $\alpha$ of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid \textit{conservative} dynamics with a finite range of spatial Fourier modes. Those at large wavenumbers thermalize, whereas modes at small wavenumbers obey ordinary viscous dynamics [C. Cichowlas et al. Phys. Rev. Lett. 95, 264502 (2005)]. The energy bottleneck observed for finite $\alpha$ may be interpreted as incomplete thermalization. Artifacts arising from models with $\alpha > 1$ are discussed., Comment: 4 pages, 2 figures, Phys. Rev. Lett. in press

For a class of complex-valued initial conditions, we construct solutions for the inviscid Burgers equations of the same type as those recently introduced by Li and Sinai. Obtained by very simple considerations, these solutions correspond to stable distributions with 1 < α ≤ 2. We furthermore show that solutions of non-Gaussian type can be constructed for the viscous Burgers equation.

This paper considers the two-dimensional Euler equation for complex spatial variables and two complex modes in the initial condition. A hierarchy of third-order ordinary differential equations (ODEs) is used to study the location and structure of the complex singular manifold for short times. The system has two key parameters, the ratio ? of the wave numbers of the two modes, and the angle between the two wave vectors. Using this hierarchy for the case =?/2 the results of earlier authors (Pauls et al 2006 Physica D 219 40?59) are reproduced numerically. To make analytical progress, the paper considers the limit ? 0 in which the wave vectors become parallel, rescaling time also. By considering the limiting behaviour of the ODE hierarchy, an asymptotic framework is set up that describes the geometry of the singular manifold and local behaviour of vorticity in this limiting case =0 of parallel modes. In addition, the hierarchy of ODEs can be solved analytically, order by order, in the parallel case using computer algebra. This is used to confirm the asymptotic theory and to give evidence for a scaling exponent ?=1 for the blow-up of vorticity on the singular manifold, ?=O(s? ?) in this case of vanishing angle .