thermal lensing, depolarization and beam quality degradation. The ray tracing algorithm is based on alternatively evaluating the axial and radial gradient and thereby finding the trajectory in thermally influenced media. This enables to find the focal length distribution of the system. Additionally, to the position of the rays, its phase is also determined, which enables to reconstruct the wavefront of the beam after passing through the crystal. This wavefront is used for a Zernike polynomial analysis to determine spherical aberration, which is linked to the beam quality of the passing beam. Furthermode the total depolarization is obtained by finding the change of polarization for each ray separately. The simulation for thermal lensing is compared with a single-pass Nd:YVO4 system, the beam degradation is compared with a Nd:YVO4-MOAP system. Both show good agreement with the simulation data, as long as the gain of the system is homogeneous.

d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension d. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension $d$. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.

Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X

Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension $d$. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.

Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X

Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension $d$. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.

Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X

To fill the gap, this work proposes a novel variant of a Talbot-Lau interferometer. Instead of a single phase grating, it uses two phase gratings, each consisting of tiny prisms. The height of the prisms is an additional factor in the measurement range, which allows to fill the gap.

The potential is a dose-reduction by a factor of 5.4 compared to Talbot-Lau setups of same post-patient length. Simulation results indicate a polychromatic visibility of up to 20%.

2] Ẽ(x, y, z) = 0 is transformed into a solvable 3D eigenvalue problem by separating out the propagation factor exp(-ikz) from the phasor amplitude E(x, y, z) of the time-harmonic electrical field. For cavities with parabolic optical elements the new approach has successfully been verified by the use of the gaussian mode algorithm. For a DPSSL with a thermally lensing crystal inside the cavity the expected deviation between gaussian approximation and numerical solution could be demonstrated clearly.},
author = {Altmann, K. and Pflaum, Christoph and Seider, D.},
faupublication = {no},
journal = {OSA trends in optics and photonics},
note = {UnivIS-Import:2015-03-09:Pub.2004.tech.IMMD.c3sim.comput},
pages = {497-502},
peerreviewed = {unknown},
title = {{Computation} of laser cavity eigenmodes by the use of a {3D} finite element approach},
url = {http://www.opticsinfobase.org/viewmedia.cfm?uri=ASSP-2004-497&seq=0},
year = {2004}
}
@article{faucris.117925104,
author = {Pflaum, Christoph and Berneker, Britta},
doi = {10.1016/j.cam.2009.08.042},
faupublication = {yes},
journal = {Journal of Computational and Applied Mathematics},
note = {UnivIS-Import:2015-04-14:Pub.2009.tech.IMMD.lsinfs.conver},
peerreviewed = {Yes},
title = {{Convergence} analysis of non-conforming {Trigonometric} {Finite} {Wave} {Elements}},
year = {2009}
}
@article{faucris.213679991,
abstract = {his paper demonstrates 2-stack and 3-stack white organic light-emitting diodes (WOLEDs) with fluorescent blue and phosphorescent yellow emissive units. The 2-stack and 3-stack WOLED comprises blue-yellow and blue-blue-yellow (blue-yellow-blue) combinations. The position of the yellow emitter and possible cavity lengths in different stack architectures are theoretically and experimentally investigated to reach Commission Internationale de L’Eclairage (CIE) coordinates of near (0.333/0.333). Here, a maximum external quantum efficiency (EQE) of 23.6% and current efficiency of 62.2 cd/A at 1000 cd/m2 as well as suitable CIE color coordinates of (0.335/0.313) for the blue-blue-yellow 3-stack hybrid WOLED structure is reported. In addition, the blue-yellow-blue 3-stack architecture exhibits an improved angular dependence compared to the blue-blue-yellow structure at a decreased EQE of 19.1%.

Analysis with the coupling of arbitrary sets of rate equations. Those arbitrary sets include the representation

of gain media, which state population is influenced by interionic mechanisms. Especially the state population of

Er:YAG is strongly influenced by upconversion and cross relaxation, which impacts the population inversion and

the generated heat load. Therefore, the simulation of a resonator based on 50%-doped Er:YAG is performed and

compared to experimental results. The accuracy of the presented technique is pointed out by a fine agreement

between simulation and experiment with respect to gained cw output power and slope efficiency. Moreover, a

finite element analysis of the heat load and the 3-dimensional population inversion in the crystal is illustrated.

}, author = {Pflaum, Christoph and Springer, Ramon}, booktitle = {SPIE Photonics Europe Proceedings}, doi = {10.1117/12.2306394}, faupublication = {yes}, peerreviewed = {unknown}, title = {{Dynamic} {Mode} {Analysis} with {Arbitrary} {Rate} {Equations}}, year = {2018} } @inproceedings{faucris.119713704, abstract = {We derive a new model to simulate passively Q-switched intracavity frequency-doubling solid-state laser. By introducing a nonlinear loss term caused by frequency-doubling crystal into the rate equations ,we can express the effect of second-harmonic generation (SHG). We apply a finite volume discretization on gain medium, saturable absorber and frequency-doubling crystal. "Dynamic Multimode Analysis (DMA)" and several Gaussian modes are utilized. At the end, numerical results of passively Q-switched intracavity frequency-doubling solid-state laser are presented. In the case of large pump radius,chaoic phenomenon can be observed numerically. In order to realize the 3D simulation, we mainly use two technics: One is that common rate equations are extended to a set of 3D multimode rate equations, which calculate photon number for different modes separately. The other is to take into account a finite volume discretization. © 2012 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).}, address = {Brüssel}, author = {Feng, Fan and Pflaum, Christoph}, booktitle = {Nonlinear Optics and Applications VI}, date = {2012-04-16/2012-04-18}, doi = {10.1117/12.921852}, edition = {943404}, faupublication = {yes}, isbn = {9780819491268}, keywords = {3D-Laser modeling; Chaotic; Frequency-doubling; Multimode; Passively Q-switched}, note = {UnivIS-Import:2017-12-18:Pub.2012.tech.IMMD.lsinfs.dynami{\_}5}, pages = {1-13}, peerreviewed = {No}, publisher = {SPIE}, title = {{Dynamic} multimode analysis ({DMA}) of passively {Q}-switched intracavity frequency-doubling solid state laser}, url = {http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1316438}, venue = {Brüssel}, volume = {8434}, year = {2012} } @inproceedings{faucris.119711944, abstract = {Passive Q-switched lasers are constructed using saturable absorbers (SA). One characteristic of these lasers is that they are built with small dimensions. There are difficulties in designing lasers with a given pulse repetition rate or pulse energy using saturable absorbers. Numerical simulation of Q-switches facilitates the design and production of such lasers and helps to reduce development time and cost. This paper presents a new simulation method which calculates beam quality, maximal output power, pulse-width and pulse energy. © 2012 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).}, address = {San Francicso}, author = {Pflaum, Christoph and Rahimi, Zhabiz and Feng, Fan}, booktitle = {Laser Resonators, Microresonators, and Beam Control XIV}, date = {2012-01-22/2012-01-25}, doi = {10.1117/12.905201}, faupublication = {yes}, isbn = {9780819488794}, keywords = {dynamic mode analysis; laser modeling; Passively Q-switched}, note = {UnivIS-Import:2017-12-18:Pub.2012.tech.IMMD.lsinfs.dynami}, pages = {1-8}, peerreviewed = {No}, publisher = {SPIE}, title = {{Dynamic} multi-mode analysis of passive {Q}-switched lasers}, url = {http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1277227}, venue = {San Francicso}, volume = {8236}, year = {2012} } @article{faucris.120177684, abstract = {We derive a new model and simulation technique called "Dynamic Multimode Analysis (DMA)" to simulate the 3-dimensional dynamic behavior of a laser. A Gaussian mode analysis is used to obtain resonator eigenmodes taking into account thermal aberrations. These modes are coupled by a set of rate equations to describe the dynamic behavior of the individual modes for cw and Q-switched lasers. Our approach analyzes mode competition and provides a detailed description of the laser beam in terms of output power, beam quality factor M(2), and pulse shape. Comparison of experimental data with our simulation results provides new insight into the effect of mode competition and the operation of Q-switched lasers.}, author = {Wohlmuth, Matthias and Pflaum, Christoph and Altmann, Konrad and Paster, Martin and Hahn, C}, doi = {10.1364/OE.17.017303}, faupublication = {yes}, journal = {Optics Express}, pages = {17303-17316}, peerreviewed = {Yes}, title = {{Dynamic} multimode analysis of {Q}-switched solid state laser cavities}, volume = {17}, year = {2009} } @inproceedings{faucris.116893084, address = {Erlangen}, author = {Härdtlein, Jochen and Pflaum, Christoph}, booktitle = {18th Symposium Simulationstechnique ASIM 2005 Proceedings}, faupublication = {yes}, isbn = {3-936150-41-9}, note = {UnivIS-Import:2015-04-16:Pub.2005.tech.IMMD.c3sim.effici}, pages = {748-753}, publisher = {SCS Publishing House}, title = {{Efficient} and {User}-friendly {Computation} of local {Stiffness} {Matrices}}, url = {https://www10.informatik.uni-erlangen.de/Publications/Papers/2005/Haerdtlein{\_}ASIM05.pdf}, venue = {Erlangen}, year = {2005} } @inproceedings{faucris.118021684, abstract = {Semi-unstructured grids consist of a large structured grid in the interior of the domain and a small unstructured grid near the boundary. It is explained how to implement differential operators and multigrid operators in an efficient way on such grids. Numerical results for solving linear elasticity by finite elements are presented for grids with more than 108 grid points. © Springer-Verlag 2002.}, address = {Berlin}, author = {Pflaum, Christoph and Seider, David}, booktitle = {Computational Science - ICCS2002}, date = {2002-04-21/2002-04-24}, faupublication = {no}, note = {UnivIS-Import:2015-04-16:Pub.2002.tech.IMMD.c3sim.effici}, pages = {622-531}, peerreviewed = {No}, publisher = {Springer-verlag}, series = {Lecture Notes in Computer Science}, title = {{Efficient} {Implementation} of {Operators} on {Semi}-{Unstructured} {Grids}}, url = {http://link.springer.com/content/pdf/10.1007%2F3-540-47789-6{\_}65.pdf}, venue = {Amsterdam}, volume = {2331}, year = {2002} } @article{faucris.119003984, author = {Feng, Fan and Pflaum, Christoph}, doi = {10.1002/pamm.201210316}, faupublication = {yes}, journal = {Proceedings in Applied Mathematics and Mechanics}, note = {UnivIS-Import:2015-03-09:Pub.2012.tech.IMMD.c3sim.effici}, pages = {655-656}, peerreviewed = {Yes}, title = {{Efficient} numerical methods for initial-value solid-state laser problems}, url = {http://onlinelibrary.wiley.com/doi/10.1002/pamm.201210316/abstract}, volume = {12}, year = {2012} } @inproceedings{faucris.234986368, abstract = {Optical simulations are a crucial part in designing and understanding new solar cell devices to help with the transition to more environmentally friendly energy generation. We present a multicore wavefront diamond blocking scheme with multi-dimensional cache block sharing for our hybrid-parallel (MPI+OpenMP) optical solar cell simulation code utilizing the Time Harmonic Inverse Iteration Method (THIIM) to discretize Maxwell's equations. This approach allows to decouple the code from the main memory bandwidth bottleneck and achieves a single node speed-up of up to three compared to an optimal implementation with pure spatial blocking and a multi node speed-up of up to two even with large numbers of compute nodes.}, author = {Hornich, Julian and Hager, Georg and Pflaum, Christoph}, doi = {10.1117/12.2312545}, faupublication = {yes}, keywords = {Simulation;Time Harmonic InverseIteration Method;Maxwell's Equations;High Performance Computing;Optical Simulation}, month = {Jan}, peerreviewed = {unknown}, title = {{Efficient} optical simulation of nano structures in thin-film solar cells}, volume = {10694}, year = {2018} } @inproceedings{faucris.203728452, address = {Online}, author = {Hartmann, Rainer and Pflaum, Christoph}, booktitle = {PAMM}, doi = {10.1002/pamm.201710348}, edition = {1}, faupublication = {yes}, keywords = {sparse grid; variable coefficients, Ritz-Galerkin; Finite Elements}, note = {UnivIS-Import:2018-09-06:Pub.2018.tech.IMMD.c3sim.fficie}, peerreviewed = {unknown}, publisher = {Wiley}, title = {{Efficient} {Ritz}-{Galerkin} {Discretization} of {PDEs} with {Variable} {Coefficients} in arbitrary {Dimensions} using {Sparse} {Grids}}, venue = {Weimar}, volume = {17}, year = {2018} } @phdthesis{faucris.201247609, author = {Hartmann, Rainer}, faupublication = {yes}, peerreviewed = {automatic}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Effiziente} {Verfahren} zur {Lösung} partieller {Differentialgleichungen} mit variablen {Koeffizienten} auf dünnen {Gittern}}, year = {2018} } @article{faucris.114180044, author = {Altmann, K. and Pflaum, Christoph and Seider, D.}, faupublication = {no}, journal = {Applied Optics}, note = {UnivIS-Import:2015-03-09:Pub.2004.tech.IMMD.c3sim.erratu}, pages = {3178}, peerreviewed = {unknown}, title = {{Erratum}: {Third}-dimensional finite element conputation of laser cavity eigenmodes}, url = {http://www.opticsinfobase.org/view{\_}article.cfm?gotourl=http://www.opticsinfobase.org/DirectPDFAccess/123C7986-FF41-C70A-DB1AE38246E9B841{\_}79912/ao-43-15-3178.pdf?da=1&id=79912&seq=0&mobile=no∨g=Friedrich-Alexander-UniversitÃ¤t Erlangen-NÃ¼rnberg Bibliothek}, volume = {43}, year = {2004} } @inproceedings{faucris.110654324, address = {Montreal, Quebec, Canada}, author = {Yan, Shuai and Pflaum, Christoph and Xu, Xiaoyu and Ren, Zhuoxiang}, booktitle = {Proceedings of Compumag 2015}, faupublication = {no}, note = {UnivIS-Import:2015-10-26:Pub.2015.tech.IMMD.c3sim.errore}, pages = {1-2}, peerreviewed = {No}, publisher = {IEEE}, title = {{Error} {Estimation} of {Discrete} {Geometry} {Method} on {Plasmonic} {Structures}}, url = {http://hal.upmc.fr/hal-01152726}, venue = {Montreal, Quebec, Canada}, year = {2015} } @article{faucris.110869044, author = {Pflaum, Christoph}, faupublication = {no}, journal = {Proceedings in Applied Mathematics and Mechanics}, note = {UnivIS-Import:2015-03-09:Pub.2000.tech.IMMD.c3sim.estima}, pages = {841-842}, peerreviewed = {Yes}, title = {{Estimation} of the {Condition} {Number} of {Additive} {Preconditioners} by the {Constant} in the {Strengthened} {Cauchy}-{Schwarz} {Inequality}}, url = {https://www10.informatik.uni-erlangen.de/Publications/Papers/2000/Pflaum{\_}GAMM99.pdf}, volume = {80}, year = {2000} } @inproceedings{faucris.119931724, address = {Berlin}, author = {Pflaum, Christoph}, booktitle = {Proceedings of the Sixth European Multigrid Conference Held in Gent, Belgium, September 27-30, 1999}, faupublication = {no}, note = {UnivIS-Import:2015-04-16:Pub.2000.tech.IMMD.c3sim.estima{\_}6}, pages = {207-213}, peerreviewed = {No}, publisher = {Springer}, series = {Lecture Notes in Computer Science and Engineering}, title = {{Estimation} of the {Condition} {Number} of {Additive} {Preconditioners} on {Tensor} {Product} {Grids}}, url = {https://books.google.de/books?hl=de&lr=&id=XYsQXIyIjycC&oi=fnd&pg=PA1&dq=Proceedings+of+the+Sixth+European+Multigrid+Conference+Held+in%09Gent,+Belgium,+September+27-30,+1999&ots=OBoz6k4q7q&sig=u2xfasgCSSkjEvfdj0cFAjHSfDM}, venue = {Gent}, volume = {14}, year = {2000} } @article{faucris.120520004, abstract = {The implementation of numerical algorithms for solving partial differential equations costs a lot of time and the optimization of such codes is difficult. These problems can be reduced by expression templates for partial differential equations. Using this concept, one can implement numerical algorithms for PDE's in a language which is close to the mathematical language. Expression templates give the compiler global informations on an expression such that the optimization of the code can be left to the compiler. Numerical results for the Stokes operator on general domains in 3D are presented. © 2001 Springer-Verlag.}, author = {Pflaum, Christoph}, doi = {10.1007/s007910100051}, faupublication = {no}, journal = {Computing and Visualization in Science}, note = {UnivIS-Import:2015-03-09:Pub.2001.tech.IMMD.c3sim.expres}, pages = {1-8}, peerreviewed = {unknown}, title = {{Expression} {Templates} for {Partial} {Differential} {Equations}}, url = {http://link.springer.com/content/pdf/10.1007%2Fs007910100051.pdf}, volume = {4}, year = {2001} } @article{faucris.268264675, abstract = {Far field calculations of beams, such as laser beams, are often applied in optical engineering. Current beam propagation methods fail in certain range parameters due to high storage requirements of the algorithms. This paper presents a new beam propagation method for far field calculations of distorted Gaussian beams in a homogeneous medium including optical elements, such as lenses. The method works even in the case of a large distance from the observation plane to the beam waist and can be applied to larger divergence angles. This new simulation technique factors out the phase of the Gaussian TEM00 beam and solves the resulting partial differential equation by suitable finite difference or finite element discretization methods.

4 and Nd:GdVO

work, a maximum pulse energy of 3.1 mJ is observed for 0.9 kW continous wave end-pumping at 785 nm. The

simulation results demonstrate that interionic mechanisms such as upconversion and energy transfer significantly

influence the population of states and consequently, the amplification. In detail, the most dominant mechanisms

are identified by introducing the rate term kxNiNm as a quantity to compare the strength of all occuring

interionic mechanisms. It can then be shown that the energy transfer mechanism E6512 between Holmium and

Thulium ions is the greatest source of population loss for the upper lasing state 5 I7 in Holmium. In summary,

the presented model represents an efficient tool to characterize the influence of interionic mechanisms on the

extractable energy in solid-state media under pulsed operation.

}, author = {Springer, Ramon and Pflaum, Christoph}, booktitle = {SPIE Photonics Europe Proceedings}, doi = {10.1117/12.2306235}, faupublication = {yes}, peerreviewed = {unknown}, title = {{Influence} of interionic energy transfer mechanisms in {Tm},{Ho}:{YAG} on the maximum extractable energy in regenerative amplifiers}, url = {https://www.spiedigitallibrary.org/conference-proceedings-of-spie/10683/106832K/Influence-of-interionic-energy-transfer-mechanisms-in-TmHo-YAG-on/10.1117/12.2306235.full?SSO=1}, year = {2018} } @article{faucris.202050908, abstract = {A novel way to build arrays of X-ray lenslets is proposed for use in medical imaging, in particular for X-ray phase contrast imaging. Focusing on Talbot-Lau interferometers, this work is about patient dose reduction, especially for design energies above 50 keV.

A low dose poses a fabrication problem, because it requires an analyzer grating which is both fine and high: It has to be fine for a good angular sensitivity. It has to be high to absorb well. However, gratings can currently be built either fine or high.

The proposed solution is to use a fine novel lens grating in front of a high analyzer grating: The lens grating uses lenslets to combine fine fringes into wider strips. This coarser pattern is then analyzed by a high grating. Regular binary production processes are sufficient to build lens gratings. Simulation-based results show that lens gratings can save dose with no impact on reconstructed image},
author = {Preusche, Oliver},
doi = {10.1364/OE.24.026161},
faupublication = {yes},
journal = {Optics Express},
keywords = {Phasenkontrast;Röntgen-Interferometrie;Linsengitter;Optik;Gitter;Talbot-Effekt},
pages = {26161-26174},
peerreviewed = {Yes},
title = {{Lens} gratings for dose optimization of medical {X}-ray phase contrast imaging},
url = {https://www.osapublishing.org/oe/abstract.cfm?uri=oe-24-23-26161},
volume = {24},
year = {2016}
}
@inproceedings{faucris.241957956,
abstract = {We present a numerical simulation technique for short laser pulse
amplification in solid-state crystals. It takes into account the full
spatial pumping profile by solving the 3-dimensional photon transport
and allows to calculate pulse energy accuratel},
author = {Pflaum, Christoph and Springer, Ramon},
booktitle = {CLEO 2020},
date = {2020-05-12/2020-05-14},
doi = {10.1364/cleo{\_}at.2020.jw2b.5},
faupublication = {yes},
peerreviewed = {No},
title = {{Mode} {Dependent} {Laser} {Pulse} {Amplification}: {A} {Computational} {Approach} in {3D}},
url = {https://www.cleoconference.org/home/schedule/?day=Wednesday#JW2B},
venue = {San Jose, USA und online},
year = {2020}
}
@inproceedings{faucris.116436584,
abstract = {For analyzing the properties of a laser, it is helpful to calculate the eigenmodes of the laser cavity. A new approach for modeling these eigenmodes is to approximate them by a finite element discretization of a 3D two-wave eigenvalue problem. In this paper, we analyze the properties of this model. First, we show that the two-wave eigenvalue problem is equivalent to an Helmholtz eigenvalue problem. Second, we analyze the stability of the finite element discretization and explain how to construct a suitable iterative solver. Numerical results are presented.},
address = {Bellingham, Washington, USA},
author = {Pflaum, Christoph and Seider, David and Altmann, Konrad},
booktitle = {Solid State Lasers and Amplifiers},
date = {2004-04-27/2004-04-29},
doi = {10.1117/12.544778},
editor = {SPIE},
faupublication = {no},
isbn = {0-8194-5382-X},
keywords = {laser; simulation},
note = {UnivIS-Import:2015-04-16:Pub.2004.tech.IMMD.c3sim.modeli{\_}5},
pages = {204-211},
peerreviewed = {No},
publisher = {International Society for Optical Engineering; 1999},
series = {Proceedings of SPIE},
title = {{Modeling} and {Computation} of {Laser} {Cavity} {Eigenmodes}},
url = {http://spie.org/Publications/Proceedings/Paper/10.1117/12.544778},
venue = {Strasbourg},
year = {2004}
}
@inproceedings{faucris.241957334,
abstract = {Solid‐state lasers are widely used for different industrial applications
ranging from material processing to medical applications. For designing
and optimization of a solid‐state laser an accurate simulation tool is
need. Such a simulation tool has to be able to simulate different kind
of physical effects like thermal effects, depolarization caused by
stress inside a laser crystal, non‐linear gain, and mode competition. In
order to calculate output power and beam quality, we apply an extension
of the dynamic mode analysis concept (DMA) (see [1]). This model is
based on a 3‐dimensional discretization of population inversion and a
system of complex rate equations. The resulting mathematical equation
system is a large system of ODE's, which has to be solved by a suitable
method including time step control. The difficulty in such a simulation
is the extreme change of parameters. As a consequence the output signal
might be a pulsed signal with high output power of pulse width 1 ns and
time period 1 ms. In order to solve such a system of ordinary
differential equations accurately, we present a suitable numerical
method. Simulation results with industrial applications are presente},
author = {Pflaum, Christoph},
booktitle = {GAMM 2018},
date = {2018-03-19/2018-03-23},
doi = {10.1002/pamm.201800215},
faupublication = {yes},
keywords = {Laser simulation, adaptive time stepping, rate equations},
peerreviewed = {unknown},
publisher = {PAMM},
title = {{Modeling} and {Simulation} of {Solid}‐{State} {Lasers}: {From} {Mathematics} to {Industrial} {Application}},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/pamm.201800215},
venue = {München TUM},
year = {2018}
}
@inproceedings{faucris.110394504,
abstract = {Ultra-short pulses with high average power are required for a variety of technical and medical applications. Single, multi-pass, and regenerative amplifiers are used, in order to increase the power of ultra-short lasers. Typical laser crystals for such amplifiers include Ti:Sapphire or Yb:YAG laser crystals. Difficulties in the amplification of ultra-short pulses include gain narrowing effects and dispersion effects in the laser crystal. In particular, these complications arise, when a pulse stretcher is needed before amplification of the laser beam. We present a technique to model and simulate the amplification of ultra-short pulses. This technique allows to model both gain narrowing effects and decrease of beam quality caused by amplification of the laser beam. This requires a detailed 3-dimensional simulation of population inversion. Gain narrowing effects are taken into account by analyzing the gain of the spectrum of the laser beam. It is important to distinguish amplifiers with one or only two passes and a regenerative amplifier. These two different kind of amplifiers are modeled by different approaches. A regenerative amplifier is modeled by a set of time dependent rate equations. However, a single pass amplifier is modeled by a set of spatial dependent rate equations. In both cases, a system of rate equations arises from spectral discretization of the laser beam. Detailed simulation results are presente},
address = {USA, Bellingham},
author = {Pflaum, Christoph and Hartmann, Rainer and Rahimi, Zhabiz},
booktitle = {Solid State Lasers XXV: Technology and Devices},
date = {2016-02-15/2016-02-18},
doi = {10.1117/12.2211784},
faupublication = {yes},
isbn = {9781628419610},
keywords = {laser simulation},
note = {UnivIS-Import:2017-07-10:Pub.2016.tech.IMMD.c3sim.modeli},
peerreviewed = {No},
publisher = {SPIE},
series = {SPIE Proceedings},
title = {{Modeling} and simulation of ultra-short pulse amplification},
url = {http://spie.org/Publications/Proceedings/Paper/10.1117/12.2211784},
venue = {San Francisco},
volume = {9726},
year = {2016}
}
@inproceedings{faucris.118700604,
abstract = {Semiconductor saturable absorber mirrors (SESAMs) are used to produce passively Q-switched ultrashort pulsed lasers. Numerical modeling of physical effects of SESAM is required to effectively design this type of lasers. For this purpose, simulations are performed to study the dynamic behavior of Gauss modes, gain of modes and saturation of the saturable absorber mirror. The laser beam quality has to be good enough in order to avoid chaotic laser behavior. We extended our dynamic mode analysis (DMA) algorithm to calculate laser beam quality. This simulation technique is based on rate equations for a set of Gauss modes and population inversions. Gain of each mode can be calculated separately by solving the corresponding set of rate equations. We have assumed that the reflectivity of the mirror is spatially invariant in the SESAMs model. An additional rate equation is required to include the saturation of SESAM. This equation considers parameters such as modulation depth, saturation fluence and relaxation time. Simulation results show that our model can predict pulse energy and non-chaotic behavior of the laser. © 2013 Copyright SPIE.},
author = {Rahimi, Zhabiz and Pflaum, Christoph},
booktitle = {Laser Resonators, Microresonators, and Beam Control XV},
date = {2013-02-03/2013-02-07},
doi = {10.1117/12.2004005},
faupublication = {yes},
keywords = {laser modeling; Semiconductor saturable absorber mirrors (SESAM)},
note = {UnivIS-Import:2015-04-16:Pub.2013.tech.IMMD.c3sim.modeli},
pages = {1-8},
peerreviewed = {No},
publisher = {International Society for Optical Engineering; 1999},
title = {{Modeling} of semiconductor saturable absorber mirrors using dynamic mode analysis},
url = {http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1656000},
venue = {San Francisco, USA},
year = {2013}
}
@inproceedings{faucris.203845199,
abstract = {New simulation methods for solid-state lasers are presented: We describe a dynamic multimode analysis to model mode competition and Q-switching. Furthermore, we propose a 3D finite element analysis of the electric field without mode decomposition. © 2009 Optical Society of America.},
author = {Wohlmuth, Matthias and Altmann, K. and Pflaum, Christoph},
booktitle = {Conference Program and Technical Digest OSA ASSP 2009},
date = {2009-02-01/2009-02-04},
faupublication = {yes},
isbn = {978-1-55752-864-3},
note = {UnivIS-Import:2018-09-11:Pub.2009.tech.IMMD.c3sim.newapp{\_}7},
pages = {1-3},
peerreviewed = {unknown},
publisher = {Optical Society of America},
title = {{New} {Approaches} for the {Dynamic} {3D} {Simulation} of {Solid}-{State} {Lasers} .},
url = {http://www.opticsinfobase.org/viewmedia.cfm?URI=ASSP-2009-TuB20&seq=0∨igin=search},
venue = {Denver},
year = {2009}
}
@inproceedings{faucris.234986925,
abstract = {We present a simulation tool to model performance of a bulk solid state laser and propose several ways how this tool can be used to enhance educational experience of the student studying laser technology. In one of the possible approaches, the ASLD software can complement available educational laser kits to provide the students with more universal practical training. In the second approach, which is the primary focus of this contribution, the ASLD software can be used as a development tool that allows students to verify their understanding of the subject as well as to propose and verify their own design ideas. The software can be extremely helpful if an experimantal setup has to be built from already present optical components in order to reduce the cost of training or if specific design objectives have to be attained.},
author = {Alexeev, Ilya and Pflaum, Christoph and Schmidt, Michael},
doi = {10.1117/12.2264777},
faupublication = {yes},
keywords = {Laser resonator modeling;ASLD software;ultrafast lasers},
month = {Jan},
peerreviewed = {unknown},
publisher = {SPIE-INT SOC OPTICAL ENGINEERING, PO BOX 10, BELLINGHAM, WA 98227-0010 USA},
title = {{New} approaches in teaching laser engineering classes - modeling and building up a laser},
volume = {10452},
year = {2017}
}
@article{faucris.115197104,
abstract = {We present a new method to simulate optical waves in large geometries. This method is based on newly developed finite elements, so-called trigonometric finite wave elements (TFWEs). They are constructed by linear elements as well as by trigonometric functions such that the one-dimensional Helmholtz equation is exactly solved unde r certain conditions. In comparison with the transfer matrix method, the TFWE method offers equally good results, but it can be extended to higher dimensions and applied to time-dynamic problems. The analysis of TFWEs shows that these elements approximate functions with certain oscillation properties more accurately than standard finite elements. Thus, a finite element discretization with TFWEs leads to a smaller system of equations which eases the solving process. Numerical results obtained by applying the TFWE method to the simulation of the optical wave in distributed feedback lasers are presented. © 2008 Society for Industrial and Applied Mathematics.},
author = {Pflaum, Christoph and Berneker, Britta and Steinle, Günther},
doi = {10.1137/070692224},
faupublication = {yes},
journal = {SIAM Journal on Scientific Computing},
keywords = {Computational optics; Distributed feedback laser; Finite elements; Helmholtz equation; Simulation of semiconductor lasers; Wave equation},
note = {UnivIS-Import:2015-04-14:Pub.2008.tech.IMMD.c3sim.newfin{\_}8},
pages = {1063-1081},
peerreviewed = {Yes},
title = {{New} {Finite} {Elements} for {Large}-{Scale} {Simulation} of {Optical} {Waves}},
url = {http://epubs.siam.org/doi/abs/10.1137/070692224},
volume = {31},
year = {2008}
}
@phdthesis{faucris.201215804,
author = {Yan, Shuai},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Numerical} {Analysis} and {Application} of {Time}-harmonic {Inverse} {Iterative} {Method} on {Simulation} of {Plasmonic} {Effects} in {Silver} {Nanowire} {Films}},
url = {https://www10.cs.fau.de/publications/dissertations/Diss{\_}2014-Yan.pdf},
year = {2014}
}
@article{faucris.117337484,
abstract = {Discrete geometric method (DGM) is implemented for solving Maxwell's equations on plasmonic structures. Surface plasmons introduce non-derivable field components at the metallic/dielectric interface that can influence the accuracy of DGM. An analysis shows that the proper setting of material parameters at the interface can increase the accuracy of numerical solutions. Numerical examples with a structured mesh and an unstructured mesh provide further evidence for the fact that a proper interface treatment is important to obtain accurate results with relatively less degrees of freedom for the simulation of plasmonic effects with DGM. Coupling with its flexibility and explicit formulation, DGM can be used as an accurate and fast solver for problems involving complex plasmonic structures.},
author = {Yan, Shuai and Pflaum, Christoph and Xu, Xiaoyu and Zhuoxiang, Ren},
doi = {10.1109/TMAG.2015.2490278},
faupublication = {no},
journal = {IEEE Transactions on Magnetics},
keywords = {Computational electromagnetics; convergence of numerical methods; discrete geometric method; surface plasmons},
note = {UnivIS-Import:2016-06-01:Pub.2016.tech.IMMD.c3sim.numeri},
pages = {1-4},
peerreviewed = {unknown},
title = {{Numerical} {Analysis} of {Discrete} {Geometric} {Method} on {Plasmonic} {Structures}},
url = {http://ieeexplore.ieee.org/document/7297862/},
volume = {PP},
year = {2016}
}
@article{faucris.203833366,
abstract = {We provide a-priori bounds with improved domain dependency for the solution of Stokes equations and the numerical error of an approximation by conforming finite element methods. The domain dependency appears primarily in terms of the LBB-constant , and several previous works have shown that degenerates with the aspect ratio of the domain. We explain the LBB dependency of common a-priori bounds on and and improve most of these estimates by avoiding a global inf-sup condition and assuming locally-balanced flow, which is in particular satisfied if g = 0. In this case, all error bounds on u -uH and p - p h except for ||p - Ph||L^{2}(ω), prove to be completely independent of L.},
author = {Wohlmuth, Matthias and Dobrowolski, M.},
faupublication = {yes},
journal = {Electronic Transactions on Numerical Analysis},
keywords = {A-priori estimates; Finite elements; Inf-sup condition; LBB-constant; Stokes equations},
note = {UnivIS-Import:2018-09-11:Pub.2008.tech.IMMD.lsinfs.numeri{\_}9},
pages = {173-189},
peerreviewed = {Yes},
title = {{Numerical} {Analysis} of {Stokes} {Equations} with {Improved} {LBB} {Dependency}},
url = {http://etna.mcs.kent.edu/vol.32.2008/pp173-189.dir/pp173-189.html},
volume = {32},
year = {2008}
}
@article{faucris.117604344,
abstract = {The interaction between light and silver nanowires (Ag NWs) in a thin film is simulated by solving Maxwells equations numerically. Time-harmonic inverse iterative method is implemented to overcome the problem of negative permittivity of silver, which makes the classical finite-difference time-domain iteration unstable. The method is validated by showing the correspondence between the plasmonic resonance of an Ag NW from a two dimensional simulation and the analytical solution. In agreement with previous experimental studies, the simulation results show that the transmissivity of the Ag NW films is higher than expected from the geometric aperture. The cause of this phenomenon is studied using TE/TM modes analysis for Ag NW films with different surface coverage of parallel-aligned Ag NWs. Furthermore, 3D simulation of Ag NW films with randomly arranged Ag NWs is performed by parallel computation on high performance computers. A binder layer is taken into account for a preliminary comparison between the simulation and experimental results. The agreements and disagreements between the simulated and measured spectra are discussed. © 2013 AIP Publishing LLC.},
author = {Yan, Shuai and Krantz, Johannes and Forberich, Karen and Pflaum, Christoph and Brabec, Christoph},
doi = {10.1063/1.4801919},
faupublication = {yes},
journal = {Journal of Applied Physics},
keywords = {Finite difference time domains; High performance computers; Negative permittivity; Parallel Computation; Plasmonic resonances; Silver nanowires; Surface coverages; Two-dimensional simulations Engineering controlled terms: Computer simulation; Electric resistance measurement; Finite difference time domain method; Inverse problems; Iterative methods Engineering main heading: Silver},
note = {UnivIS-Import:2015-03-09:Pub.2013.tech.IMMD.c3sim.numeri},
pages = {1-6},
peerreviewed = {Yes},
title = {{Numerical} {Simulation} of light propagation in silver nanowire films using time-harmonic inverse iterative method},
url = {http://jap.aip.org/resource/1/japiau/v113/i15/p154303{\_}s1},
volume = {113},
year = {2013}
}
@inproceedings{faucris.234987193,
author = {Pflaum, Christoph and Heubeck, Britta},
booktitle = {Solid State Lasers and Amplifiers II},
doi = {10.1117/12.661514},
faupublication = {yes},
isbn = {0819462462},
peerreviewed = {unknown},
title = {{Numerical} simulation of multiple modes in solid state lasers},
venue = {Strasbourg},
volume = {6190},
year = {2006}
}
@article{faucris.213486050,
abstract = {Within this work, the numerical solution of the photon transport equation for pulse amplification is presented. Several discretization schemes are introduced that enable the calculation of the coupling between the transport equation and population inversion. It is demonstrated that the presented discretization schemes are convergent with respect to the analytic Frantz-Nodvik solution. Specifically, the application of a prediction-correction approach based on Heun's method leads to an improvement in accuracy compared to the pure explicit approach. Finally, novel discretization schemes are applied to simulate different regenerative amplifiers based on Ti:Sapphire and Ho:YAG. Moreover, bifurcation of the Ho:YAG system is analyzed, which results in the determination of stable operating regimes for pulsed amplification. (C) 2019 Optical Society of America},
author = {Springer, Ramon and Alexeev, Ilya and Heberle, Johannes and Pflaum, Christoph},
doi = {10.1364/JOSAB.36.000717},
faupublication = {yes},
journal = {Journal of the Optical Society of America B-Optical Physics},
note = {CRIS-Team WoS Importer:2019-03-15},
pages = {717-727},
peerreviewed = {Yes},
title = {{Numerical} simulation of short laser pulse amplification},
volume = {36},
year = {2019}
}
@article{faucris.119911264,
abstract = {We are studying the influence of spherical silver nanoparticles (AgNP) in absorbing media by numerically solving the Maxwell's equations. Our simulations show that the near-field absorption enhancement introduced by a single AgNP in the surrounding medium is increasing with the growing particle diameter. However, we observe that the relative absorption per particle volume is on a similar level for different particle sizes; hence, different numbers of particles with the same total volume yield the same near-field absorption enhancement. We also investigate the effect of non-absorbing shells around the AgNP with the conclusion that even very thin shells suppress the beneficial effects of the particles noticeably. Additionally, we include AgNP in an organic solar cell at different vertical positions with different particle spacings and observe the beneficial effects for small AgNP and the scattering dependent performance for larger particles.},
author = {Hornich, Julian and Pflaum, Christoph and Brabec, Christoph and Forberich, Karen},
doi = {10.1063/1.4962459},
faupublication = {yes},
journal = {Journal of Applied Physics},
peerreviewed = {unknown},
title = {{Numerical} study of plasmonic absorption enhancement in semiconductor absorbers by metallic nanoparticles},
volume = {120},
year = {2016}
}
@inproceedings{faucris.118767484,
abstract = {We present an edge-pumped Yb:YAG /YAG trapezoid-shape thin disk laser with slanted faces of 30 degree. The crystal consist of a 0.2-mm-thick Yb:YAG crystal as a gain medium and a 1.3-mm-thick un-doped YAG crystal bonded on the gain medium. The crystal is pumped from four sides in such a way that pump light trapped inside the crystal after total reflections. To study this configuration, we performed a detailed simulation of our delivery system using our laser simulation code ASLD^{TM} software. The simulation include Monte-Carlo ray tracing of pump light and 3-dimentional opto-mechanical analysis. Furthermore, we calculate the optical path difference of the crystal respect to the output power and relating beam quality using Dynamic Multimode Analysis (DMA) method. Finally, we demonstrated that the OPD inside the crystal increased from -3613.74mm to -542.85mm in the output power range of 33W to 250W. © 2014 SPIE.},
address = {Bellingham},
author = {Aminpour, Ahmed and Pflaum, Christoph},
booktitle = {Solid State Lasers XXXIII: Technology and Devices},
date = {2014-02-02/2014-02-04},
doi = {10.1117/12.2034282},
faupublication = {yes},
keywords = {Diode pumped laser; Optical path difference; Thermal lensing; Thin disk laser},
note = {UnivIS-Import:2015-04-16:Pub.2014.tech.IMMD.c3sim.optica},
pages = {1-9},
peerreviewed = {No},
publisher = {SPIE},
title = {{Optical} path difference of slanted edge diode-pumped {Yb}:{YAG}/{YAG} thin-disk laser},
url = {http://spie.org/Publications/Proceedings/Paper/10.1117/12.2034282},
venue = {San Francisco, CA},
volume = {8959},
year = {2014}
}
@article{faucris.241956413,
abstract = {A new ray tracing method for the propagation of electrical fields in
inhomogeneous and weakly anisotropic media is presented. With this
method, we can efficiently simulate the propagation of laser beams in
solid-state laser amplifiers, which suffer from high thermal loads. As a
result of this method, we can find optical compensation setups that
significantly reduce the depolarization losses in high-power solid-state
amplifiers. To the best of our knowledge, we theoretically demonstrate
for the first time that [100]-cut YAG crystals reduce the depolarization
by more than a magnitude in comparison to [111]-cut crystals while
achieving an overall better beam quality at the same tim},
author = {Rall, Phillip Lino and Pflaum, Christoph},
doi = {10.1364/JOSAB.391216},
faupublication = {yes},
journal = {Journal of the Optical Society of America B-Optical Physics},
keywords = {ray tracing, polarization, laser crytal, thermal lensing},
peerreviewed = {Yes},
title = {{Polarization} ray tracing in thermally loaded solid-state laser crystals},
url = {https://www.osapublishing.org/josab/abstract.cfm?uri=josab-37-7-1933},
volume = {Vol. 37},
year = {2020}
}
@article{faucris.106823904,
abstract = {Building integrated semitransparent thin-film solar cells is a strategy for future eco-friendly power generation. Organic photovoltaics in combination with dielectric mirrors (DMs) are a potential candidate as they promise high efficiencies in parallel to the possibility to adjust the color and thus the transparency of the whole device. A fully solution processed and printable DM with an easily adjustable reflection maximum is presented that can be facilely attached to solar cells. The DM is optimized via optical simulations to the particular needs of the device with regard to photocurrent enhancement. The excellent agreement between experimental and theoretical results confirms the high optical quality of the printed layers with respect to homogeneity and surface roughness. The used inks are organic-inorganic nanocomposites with a large refractive index contrast of ≈0.7. The short-circuit current is enhanced by up to ≈24% for a semitransparent polymer solar cell.},
author = {Bronnbauer, Carina and Hornich, Julian and Gasparini, Nicola and Guo, Fei and Hartmeier, Benjamin and Luechinger, Norman A. and Pflaum, Christoph and Brabec, Christoph and Forberich, Karen},
doi = {10.1002/adom.201500216},
faupublication = {yes},
journal = {Advanced Optical Materials},
keywords = {Dielectric mirrors; Organic photovoltaics; Printing; Refractive indices; Semitransparent},
pages = {1424-1430},
peerreviewed = {Yes},
title = {{Printable} {Dielectric} {Mirrors} with {Easily} {Adjustable} and {Well}-{Defined} {Reflection} {Maxima} for {Semitransparent} {Organic} {Solar} {Cells}},
volume = {3},
year = {2015}
}
@article{faucris.115931684,
abstract = {In this paper, we present a new approach to construct robust multilevel algorithms for elliptic differential equations. The multilevel algorithms consist of multiplicative subspace corrections in spaces spanned by problem dependent generalized prewavelets. These generalized prewavelets are constructed by a local orthogonalization of hierarchical basis functions with respect to a so-called local coarse-grid space. Numerical results show that the local orthogonalization leads to a smaller constant in strengthened Cauchy-Schwarz inequality than the original hierarchical basis functions. This holds also for several equations with discontinuous coefficients. Thus, the corresponding multilevel algorithm is a fast and robust iterative solver.},
author = {Pflaum, Christoph},
doi = {10.1007/s00607-002-1464-y},
faupublication = {no},
journal = {Computing},
keywords = {Generalized prewavelets; Multilevel methods},
note = {UnivIS-Import:2015-03-09:Pub.2002.tech.IMMD.c3sim.proble},
pages = {339-352},
peerreviewed = {Yes},
title = {{Problem} {Dependent} {Generalized} {Prewavelets}},
url = {https://www10.informatik.uni-erlangen.de/Publications/Papers/2002/Pflaum{\_}Computing{\_}Vol69Nr4.pdf},
volume = {69},
year = {2002}
}
@inproceedings{faucris.267188367,
abstract = {High-power laser beams are usually imaged with the help of suitable optics. The propagation of such beams, through optical systems, often result in a focus shift. This is mainly due to the deposited power in the beam acting as a heat source in the optical elements. Finally, the resulting change in temperature leads to both, a deformation of the lens surface, and a change of the refractive index in the lens. Ray tracing and the finite element method are needed in order to analyze the above effects. The accuracy and the computational time of focus shift calculations, in both simulation techniques, are strongly influenced by the interpolation of the data. This paper will present, a newly proposed interpolation method, allowing a fast and accurate interpolation of data using an unstructured finite element grid. This method will specifically focus on the accurate calculation of laser beam thermal load inside an optical lens.

**Method:
**The basic idea is to use a refracting lens to combine N>1
fringes into a single fringe (see Fig. 1). An array of such lenses
(called 'lens grating') replaces the analyzer grating G2. A new G2
(the period now is N times larger) is put into the focal plane of the
lens-array. To this end, several problems have to be solved.

Problem 1 – Refraction strongly depends on photon energy. Since G2 is now coarse, the refraction angle is allowed to change by a factor of 3.

Problem 2 – Production of strong prisms for a fresnel-style lens. [4] inclines the light source during lithography by rotation around a grating bar (around the y-axis), this delivers strong alternating prisms. We incline by rotating within a grating bar (around the x-axis) and use a y-periodic contract in the grating layout to produce differently strong prisms – this allows to encode a complete fresnel lens in a simple binary layout.

Problem 3 – The lenses must analyze the fringe phase. To do so, two arrays of fresnel lenses are interleaved into each other. The even lenses (black) focus on G2 grating bars, the odd lenses (magenta) focus on G2 grating slits.

Problem 4 – This does not work with a regular source grating G0. An additional coherence constraint defines a set of planes in which G2 can be placed. Also required is a grouping: In each group of Q slits, only q < Q/2 slits may stay open – we need hierarchical coherence in the form of a darker G0 (see orange lines in Fig. 1).

The resulting setup allows analyzer grating periods as large as in the source grating G0. A lens grating may consist of very narrow strips, allowing higher angular sensitivities (refraction is proportional to aspect ratio).

**Results:**
We present simulation results of a setup of 180 mm post-patient
length for a design energy of 59 keV. The setup uses a modified
binary phase grating G1 to operate in second talbot order. This
increases sensitivity. The lens grating has a strip width of 0.9 µm
('period' 1.8 µm) and a gold height of 54 µm (aspect ratio 73). It
combines 5 fringes to one, resulting in a period of 9.3 µm for G2.
The simulation (see Fig. 2) is based on Fresnel-Kirchhoff
diffraction. It shows that the lens grating keeps 92% of the
visibility created by the first two gratings.

**Conclusion:
** The proposal solves the height-problem in the analyzer grating
G2. However, G0 will become the critical component of the setup,
because of the additional coherence requirements. One option to solve
this might be the combination with the glacing angle [3] approach.

**References:**

[1] T. Donath, F. Pfeiffer, O. Bunk, W. Groot, M. Bednarzik, C. Grünzweig, E. Hempel, S. Popescu, M. Hoheisel and C. David, "Phase-contrast imaging and tomography at 60 keV using a conventional x-ray tube source", Review of Scientific Instruments 80, 053701 (2009), DOI 10.1063/1.3127712

[2] T. Thüring, M. Abis, Z. Wang, C. David, and M. Stampanoni, "X-ray phase-contrast imaging at 100 keV on a conventional source", Scientific Reports 4 : 5198 (2014), DOI: 10.1038/srep05198

[3] A. Sarapata, J. W. Stayman, M. Finkenthal, J. H. Siewerdsen, F. Pfeiffer, and D. Stutman, "High energy x-ray phase contrast CT using glancing-angle grating interferometers", Med Phys. Feb 2014; 41(2): 021904, Doi: 10.1118/1.4860275