# Poster Session and Reception

Tuesday, December 13, 2016 - 4:30pm - 6:00pm

Lind 400

**Consistency of Dirichlet Partitions**

Todd Reeb (The University of Utah)

A Dirichlet k-partition of a domain U is a collection of k pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs and used in data analysis, while the continuum problem has been used to model mixtures of distinct species of Bose-Einstein condensates. We extend results of N. Garcia Trillos and D. Slepcev to show via Gamma-convergence that there exist solutions of the continuum problem arising as limits to solutions of a sequence of discrete problems. Our results imply the statistical consistency statement that Dirichlet partitions of geometric graphs converge to partitions of the sampled space in the Hausdorff sense. This is joint work with Braxton Osting.**Opening Band Gaps in Periodic Media**

Robert Viator (University of Minnesota, Twin Cities)

The location and size of band gaps in periodic acoustic and photonic crystals have been topics of intense study for many decades. Band gap materials have a wide array of applications, ranging from the design of microwave ovens to the optimization of solar panel efficiency. Although a vast amount of numerical and experimental studies have been done to calculate the size and location of band gaps in periodic materials, work regarding how and why these band gaps occur remains sparse. Here will be presented a mathematical criteria to determine the lower bound necessary to open band gaps in the frequency spectrum of a certain class of periodic acoustic and 2-d photonic crystals, as well as estimates on the size and location of these gaps, all in terms of spectral information garnered entirely from the geometry of the crystal.**Wavefront shaping and random matrix theory in disordered photonics**

Chia Wei Hsu (Yale University)

By controlling the many degrees of freedom in the incident wavefront, one can manipulate wave propagation in complex disordered systems and violate the typical diffusive behaviors. While such “coherent control” is powerful, it is also challenging because multiple scattering couples the incident wave to a large number of outgoing modes (speckles), necessitating the simultaneous control of many speckles. We show that because speckles at different angles are correlated with each other, one can control far more speckles than what would be possible if correlations were negligible. Theoretical predictions using random matrix theory are in excellent agreement with experiments on ZnO powders. In addition, for broadband light, we show that correlations between speckles at different frequencies cause the spectral degrees of freedom to scale as the square root of the bandwidth rather than the bandwidth itself, opening the door for broadband control of transport.**Perfectly-matched-layer Boundary Integral Equation Method for Wave Scattering in a Layered Medium**

Wangtao Lu (Michigan State University)

For scattering problems in a layered medium, standard BIE methods based on the Green’s function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Green’s function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. Our PML-based BIE method uses the Green’s function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Green’s function of the PML-transformed free space is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy. This work is jointly with Ya Yan Lu and Jianliang Qian.**An Efficient High Order Accurate Discretization Solution Technique for high Frequency Helmholtz Problems**

Adrianna Gillman (Rice University)

In this poster, we present a high-order accurate discretization technique designed for scattering problems in heterogeneous media. The method is based on local spectral collocation with a hierarchical merge procedure which glues the local problems together. The result is a discretization technique that naturally has an efficient direct solver which has the same asymptotic scaling as the nested dissection method but with a constant that does not grow with the order of the discretization.

For example, applying the proposed method tor free-space scattering problems with locally varying media, a problem approximately 100 wavelengths in size can be solved to nine digits of accuracy in a a few minutes on a workstation. For each new incident wave, the solution can be found in 3 seconds. Numerical results will illustrate the performance of the proposed method in both the forward and inverse scattering setting.**Phase Retrieval and Phase Contrast Tomography**

Thorsten Hohage (Georg-August-Universität zu Göttingen)

As opposed to conventional computed tomography, phase contrast tomography does not only yield information on the imaginary part, but also the real part of the refractive index of the sample. The former gives rise to attenuation, whereas the latter causes phase shifts of the x-ray beam. Phase contrast tomography is particularly well-suited for imaging optically thin objects, e.g. soft tissue of small length scale since phase contrast for x-rays is typically several orders of magnitude larger than absorption contrast. In this talk we only consider propagation based contrast tomography: Here phase variations are turned into measurable intensity variations simply by propagation, i.e. the intensity is not measured immediately behind the sample, but at some distance to the sample. For point sources this also provides magnification.

At each direction of the incident beam a phase retrieval problem has to be solved: The missing phase information of the field in the detector plane have to be reconstructed from the measurable intensities using a-priori information. Then the exit field (i.e. the field in a fictious plane parallel to the detector plane immediately behind the sample) can be computed by back-propagation. The values of the exit field are functions of line integrals over the refractive index. Surprisingly, for compactly supported objects the complex-valued field in the exit field is uniquely determined by the amplitude of the field in the detector plane. Even more, we show that the corresponding linearized inverse problem is well-posed. However, the condition number grows exponentially with the Fresnel number. For real-valued refractive indices one has a much more favorable linear growth with the Fresnel number.

Finally, we discuss the solution of phase retrieval problems by regularized Newton method. In particular we show that joint reconstruction of phase and amplitude is possible for moderate Fresnel numbers. Moreover, we present three-dimensional reconstructions from experimental tomographic data using a Newton-Kaczmarcz method.**Immersed FEM-multigrid Method for Two Phase Flows in Porous Media**

Gwanghyun Jo (Korea Advanced Institute of Science and Technology (KAIST))

Multiphase flows in porous media arise in various disciplines including petroleum engineering, contamination analysis etc. One of the difficulties in solving the porous media problems is that material properties such as permeability, porosity, capillary pressure are changing abruptly along the material interface. Conventional methods such as FEM, DG, Control-volume methods use fitted grids to solve the problem. However, fitted grid makes complex data structure which makes the geometric multigrid methods unavailable.

There were some new developments in FEM. Immersed FEM (IFEM) uses uniform grid for the interface problem. The one of the big advantage of IFEM is the availability of multigrid solver for the interface problem. I develop the IFEM based method for the two phase flows in heterogeneous porous media for the first time. The method is based on IMPES procedure where pressure and saturation equations are solved sequentially (independently). The numerical experiments show that my method has optimal convergence rates for the pressure and velocity, and suboptimal convergence rate for the saturation. To enhance the efficiency, the multigrid solver was applied to the pressure equations. The numerical experiments show that the multigrid has optimal scalability even for the permeability constrast (for example to 1 versus 1000).**Finite Element Methods for Metamaterial Cloaking Simulation**

Jichun Li (University of Nevada)

In this poster, we present our recent work on time-domain cylindrical and carpet cloaks. Well-posedness of these models are established, and finite element methods are developed, analyzed, and implemented. Numerical simulations of cloaking phenomena obtained with these models are presented.**Transformation Optics using Electrostatic and Magnetostatic Analogies**

Jensen Li (University of Birmingham)

Transformation optics is a unique approach in designing optical devices based on an intuitive transformation of the electromagnetic fields with a coordinate map as the usual starting point. Here, we explore how a numerical solution of an analog problem can simplify and generate transformation optical devices. In one example, we show that an electrostatic solution can be used to generate a quasiconformal carpet cloak by skipping the map generation step. In another example, we show that a pseudo magnetic force for photon, in analogy to that for electron motion, and its corresponding material representation can be established by considering magnetostatic solutions. A cylindrical cloak with pseudo magnetic force and the generation of one-way edge states are investigated.**Modeling and Designing Graphene Metasurfaces with New Electromagnetic Functionalities**

Christos Argyropoulos (University of Nebraska)

The conductivity of graphene at THz and infrared (IR) frequencies is dominated by intraband transitions and can be characterized by the Drude model. Strong surface plasmons can be excited at the surface of graphene in this frequency regime. Even stronger resonance conditions in the transmission or reflection spectrum can be obtained when graphene is patterned to periodic rectangular patches or strips forming new planar graphene metasurfaces. We will demonstrate that the strong coupling between the resonances excited by different polarized incident waves can lead to the design of ultrathin, broadband, and tunable polarization converters. In addition, we will employ the enhanced nonlinear properties of graphene metasurfaces to boost several relative weak THz and IR nonlinear optical processes, such as third-harmonic generati**Fully Discrete Energy Stable Compatible Numerical Methods for Maxwell’s Equations in Nonlinear Polarization Media**

Vrushali Bokil (Oregon State University)Puttha Sakkaplangkul (Oregon State University)

The propagation of electromagnetic waves in general media is modeled by time-dependent Maxwell’s equations coupled with constitutive laws that describe the response of the media. In this work, we consider a nonlinear dispersive model in its first order PDE-ODE form, where the nonlinearity comes from the instantaneous electronic Kerr response and the residual Raman molecular vibrational response, together with the single resonance linear Lorentz disperson. To design efficient, accurate, and stable computational methods, we apply two compatible discretizations: (1) high order discontinuous Galerkin discretizations in space, and (2) high order staggered finite difference methods in space. The resulting semi-discrete methods are proved to be stable. The challenge to achieve provable stability for fully-discrete methods lies in the temporal discretizations of the nonlinear terms. To overcome this, novel modifications are proposed for both the second-order leap-frog and implicit trapezoidal temporal schemes. The performance of the proposed methods are further demonstrated through numerical experiments.**Spectra of quantum trees and orthogonal polynomials**

Zhaoxia Wang (Louisiana State University)

I investigate the spectrum of regular quantum trees of increasing length through a relation to orthogonal polynomials depending on two variables. For Robin (real Dirichlet-to-Neumann) vertex conditions, the behavior of the low eigenvalues is analyzed through the interlacing property of the roots of orthogonal polynomials. The spectrum approaches a band-gap structure as the length of the quantum tree increases. The branching numbers and Robin constants of the vertices determine the weight associated with the orthogonal polynomials, and from this, one can compute the spectral bands. Part of the lowest band is negative (except in the Neumann case), and there emerge two isolated eigenvalues below the bands. The high eigenvalues approach the regular band-gap structure for the Neumann vertex conditions.**Bound states in the continuum in periodic structures**

Chia Wei Hsu (Yale University)

Bound states in the continuum are solutions of wave equations that remain spatially localized with no radiation, even though their frequencies are embedded in the continuous spectra of spatially extended states. We find bound states in the continuum on the surface of bulk photonic crystals and in photonic crystal slabs. We observe, both numerically and experimentally, that at discrete k points on certain bands, leaky guided resonances above the light line become perfectly confined within the slab. These bound states are vortex centers in the polarization direction of the far-field radiation, thus they carry topological charges and generally emerge or annihilate in pairs.