Bragg spectroscopy is used to measure excitations of a trapped, quantum-degenerate gas of 87 Rb atoms in a 3-dimensional optical lattice. The measurements are carried out over a range of optical lattice depths in the superfluid phase of the Bose-Hubbard model. For fixed wavevector, the resonant frequency of the excitation is found to decrease with increasing lattice depth. A numerical calculation of the resonant frequencies based on Bogoliubov theory shows a less steep rate of decrease than the measurements.PACS numbers: 03.75.Kk, 03.75.Lm, 05.30.Jp, 32.80.Pj Quantum-degenerate atoms in optical lattices form a strongly interacting many-body system whose parameters can be readily controlled. As first pointed out by Jaksch et al., [1] bosonic atoms in an optical lattice constitute a nearly ideal realization of the Bose-Hubbard model [2].This model predicts a superfluid to Mott insulator quantum phase transition that has been observed by Greiner et al. [3]. Since then, this field has attracted great interest due to its potential for the realization of quantum computation and quantum simulation of strongly-correlated manybody systems [4].A key property of a quantum gas is its excitation spectrum. Excitations of a Bose-Hubbard gas by a gradient of magnetic field [3] or a modulated optical lattice depth [5,6] have been previously observed. However, neither of these techniques directly probes the linear excitation spectrum of the gas, since a tilted lattice perturbs the gas only at zero frequency, and a modulated optical lattice only at zero quasi-momentum. The latter case results in a nonlinear excitation spectrum that has been analyzed only very recently [7]. Bragg spectroscopy has been demonstrated as a probe of the linear excitation spectrum of a Bose-Einstein condensate [8][9][10][11], and has also been proposed as a method to study the Mott insulator phase of the Bose-Hubbard

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The low-k 1 domain of immersion lithography tends to result in much smaller depths of focus (DoF) compared to prior technology nodes. For 28 nm technology and beyond it is a challenge since (metal) layers have to deal with a wide range of structures. Beside the high variety of features, the reticle induced (mask 3D) effects became non-negligible. These mask 3D effects lead to best focus shift. In order to enhance the overlapping DoF, so called usable DoF (uDoF), alignment of each individual features best focus is required. So means the mitigation of the best focus shift.This study investigates the impact of mask 3D effects and the ability to correct the wavefront in order to extend the uDoF. The generation of the wavefront correction map is possible by using computational lithographic such Tachyon simulations software (from Brion). And inside the scanner the wavefront optimization is feasible by applying a projection lens modulator, FlexWave TM (by ASML). This study explores both the computational lithography and scanner wavefront correction capabilities.In the first part of this work, simulations are conducted based on the determination and mitigation of best focus shift (coming from mask 3D effects) so as to improve the uDoF. In order to validate the feasibility of best focus shift decrease by wavefront tuning and mitigation results, the wavefront optimization provided correction maps are introduced into a rigorous simulator. Finally these results on best focus shift and uDoF are compared to wafers exposed using FlexWave then measured by scanning electron microscopy (SEM).