Schedule for: 19w5209 - Herglotz-Nevanlinna Theory Applied to Passive, Causal and Active Systems

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This will be a survey of known results, with emphasis on constructive aspects, rational solutions, comparison between several integral transforms which are related to the Herglotz-Nevanlinna class of functions in classical domains.

We give a survey of the theory of reproducing kernel spaces of the kind defined by de Branges and Rovnyak and how they can be used to study Nevanlinna functions. In particular we give a proof of the Bargmann-Schiffer lemma. The advantages of the approach are that one can consider the matrix-valued case and also the case of generalized Nevanlinna functions, i.e. when the underlying kernel has a finite number of negative squares. We will also present a unified setting, which contains in particular the case of Schur functions and Nevanlinna functions in a unified framework.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Performance limits for electromagnetic devices can be determined from sum rules and optimization over equivalent sources. Sum rules are derived by identifying a passive system with a Herglotz function and using integral identities to relate the dynamic response with its low and high-frequency asymptotic expansions. These bounds and identities are of great interest in many areas of physics and engineering partly because of their simplicity and close form expressions. Optimization over equivalent sources complements the sum rule limits by providing single frequency bounds and a flexibility to incorporate different types of information as constraints in the optimization problems. Here, an overview of Herglotz function and optimization-based bounds in EM is presented with examples from metamaterials, antennas, and scattering. We also compare the theoretical results with state-of-the-art designs. We also discuss different ways to combine the two approaches.

Under the physically realizable conditions that the susceptibilities of dipolar media approach zero as the frequency approaches infinity, one can prove from the causality and passivity relations for the susceptibilities of spatially nondispersive continua that the susceptibilities are equal to or greater than zero as the frequency approaches zero. Since this result presents an apparent paradox for diamagnetic materials and metamaterials, it appears that diamagnetic susceptibilities cannot be represented by the usual Herglotz functions. This apparent paradox is discussed in the talk and positive semidefinite expressions are derived, directly from Maxwell’s microscopic equations, for the time-domain macroscopic energy densities in passive, spatially nondispersive paramagnetic and diamagnetic continua.

Strongly nonreciprocal structures and photonic topological insulators are emerging as an important class of material platforms that support the propagation of robust unidirectional surface waves. In this talk, we discuss our recent work on topological, continuous, electromagnetic media, with particular focus on complex scenarios that include dissipation, spatial dispersion (nonlocality), and nonlinearities. As a model system, we consider a magnetized plasmonic material, which exhibits a typical gyrotropic nonreciprocal response. We discuss how nonlocality and dissipation affect the unidirectional nature of the supported modes, the topological transitions that may emerge, and some apparent paradoxes. We also clarify that, while nonreciprocal topological media break temporal symmetries for wave propagation, allowing to overcome some limitations of conventional electromagnetic systems, other relevant physical bounds and limits are unaffected since they depend on causality and passivity. Our study reveals limitations and potential of unidirectional/topological material platforms, and may pave the way to novel applications of topological wave-guiding systems.

In this talk we will show an application of the theory of Herglotz-Nevannlina functions for the linear viscoelastic problem, the dielectric problem and the conductivity problem in the time domain. Specifically, by using the analyticity of the Dirichlet-to-Neumann map which relates the applied field on the boundary to the corresponding measured field on the boundary one can determine bounds on the response of the body for any moment of time. Such bounds are tighter the more information regarding the body is incorporated. By tailoring the time-dependent applied field so that the bounds incorporating the volume of the inclusion are extremely tight at specific moments of time, one can then use them in an inverse fashion to determine the size of the inclusion.

Operator functions with an algebraic dependence on a spectral parameter have applications in electromagnetic field theory, where the $\lambda$-dependent material properties are of Drude-Lorentz type. Let $M_\ell$, $\ell=1,2,\dots,L$ denote bounded linear operators in a Hilbert space $\mathcal H$ and denote by $A$ a self-adjoint operator with compact resolvent that is bounded from below. In this talk we consider operator functions of the form \[ \mathcal S(\lambda)=A-\lambda-\lambda\sum_{\ell=1}^L\dfrac{M_\ell}{c_\ell+id_{\ell}\sqrt{\lambda}-\lambda}, \quad \text{dom} \,\mathcal S(\lambda) = \text{dom} \,A, \quad \lambda \in \mathbb{C} \setminus P, \] where $P$ is the set of poles of the rational function and $c_\ell$, $d_{\ell}$ are non-negative constants. The operator function is self-adjoint if the damping $d_{\ell}$ is set to zero. Then we establish variational principles and provide optimal two-sided estimates for all the eigenvalues of $\mathcal S$. In the second part of the talk, I briefly present extensions to the non-self-adjoint case $d_{\ell}>0$ and ongoing work. The first part of the talk is based on a joint work with Heinz Langer and Christiane Tretter and the second part of the talk is based on a joint work with Axel Torshage.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Nanophotonics is developing at a rapid pace, with ever more materials, form factors, and structural degrees of freedom now available. These large design spaces offer transformative technological possibility, but optimization subject to the nonconvex Maxwell-equation constraint is difficult. And incorporation of extraneous constraints such as fabrication tolerance further stymie various analytical approaches. I will discuss optimization concepts and physical insights that lead to global bounds, and structures that nearly achieve them, for three applications: (1) spontaneous emission, where causality and convexity can be united to identify power-bandwidth limits, (2) superresolution, where various physical relaxations lead to an analytically tractable quadratically constrained quadratic program, and (3) minimal mode volume, where we leverage Lagrangian duality for computational bounds that seamlessly incorporate fabrication-tolerance and multi-frequency-design constraints.

In this problem we consider a matching problem with applications to integrated antennas. We study the matching between a given 22 GHz amplifier and an antenna, that is seen as a load. At these high frequencies, it is essential that the matching is as close to optimal as possible, both in efficiency and in bandwidth. Fano-matching is based on a sum-rule for the reflection coefficient, and tend to over-estimate the available bandwidth. In this talk, we will study a current-based optimization approach to this type of matching.

We present a method for imaging the conductivity of a body by heating it in certain locations and measuring thermal noise currents at its boundary. The connection between thermal noise currents, temperature and the conductivity was discovered by Johnson and Nyquist in 1927 and relies on the Fluctuation Dissipation Theorem that we exploit here for imaging purposes. We show how the problem can be transformed into a well-known hybrid inverse problem and discuss possible applications.

We extend the classical boundary values for (general, three-coefficient) regular Sturm-Liouville operators on compact intervals to the singular case as long as the associated minimal operator is bounded from below, utilizing principal and nonprincipal solutions of the underlying differential equation. We derive the singular Weyl-Titchmarsh-Kodaira $m$-function and illustrate the theory with the examples of the Bessel, Legendre, and Laguerre (resp., Kummer) operators. This is based on joint work with Lance Littlejohn and Roger Nichols.

The Herglotz class is defined as the class of holomorphic functions mapping the $d$-fold Cartesian product of the right half-plane with itself into the right half-plane. The Herglotz-Agler class is defined as the space of such holomorphic functions such that (under any reasonable functional calculus) maps a $d$-tuple of commuting accretive operators to an accretive operator (accretive operator here meaning having positive-definite real part). Integral representation formulas and transfer-function realization formulas for the Herglotz class are rather involved as soon as $d > 2$ while such formulas are more like the single-variable case for the Herglotz-Agler class. A routine observation is that the Herglotz-Agler class is a sub-class of the Herglotz class with equality for $d=1,2$ and strict inclusion for $d>2$. Also of interest are the homogeneous Herglotz class and the homogeneous Herglotz-Agler class where here homogeneous means "homogeneous of degree 1": $f(\lambda z_1, \dots, \lambda z_d) = \lambda f(z_1, \dots, z_d)$ for all complex scalars $\lambda$. This class has motivation from electrical circuit theory and has been studied at length by V. Bessmertnyi and D. Kaliuzhnyi-Verbovetskyi. A parallel picture exists for the homogeneous Herglotz and homogeneous Herglotz-Agler classes: there is a better structural understanding of the homogeneous Herglotz-Agler class than for the general homogeneous Herglotz class, and it is an easy observation that the homogeneous Herglotz-Agler class is a subclass of the homogeneous Herglotz class. However there remains an open question: for what values of $d$ (the number of variables) is it the case that the homogeneous Herglotz class and the homogeneous Herglotz-Agler class are actually the same? We make some observations on this question and discuss a possible approach to resolving this question.

Certain Borel measures in the plane appear as representing parameters for classes of holomorphic functions with non-negative real or imaginary part. Such measures must satisfy a positivity condition which constitutes a severe restriction within the set of Borel measures. $$ $$ In this talk, we will give a characterization of all product measures satisfying the positivity condition that characterizes representing measures of holomorphic functions of two variables with non-negative imaginary part.

Polar sea ice is a key component of Earth’s climate system. As a material it exhibits complex composite structure on length scales ranging from tenths of millimeters to tens of kilometers. A principal challenge in modeling sea ice and its role in climate is in relating behavior and structure on small scales to effective or homogenized behavior on larger scales, and in estimating parameters controlling small scale processes from large scale observations. In this lecture we address fundamental questions in sea ice homogenization, and the development of Stieltjes integral representations and rigorous bounds for effective parameters in the sea ice system. In particular, we consider fluid, thermal, and electromagnetic transport through sea ice, advection enhanced diffusion processes, and the propagation of ocean waves through the sea ice pack. This work is helping to advance how sea ice is represented in climate models, and to improve projections of the fate of Earth’s sea ice packs and the ecosystems they support.

Permeability and tortuosity play a crucial role in the fluid transport and viscodynamics of porous/poroelastic media. In this talk, recent results on IRF for permeability and tortuosity, both in the form of Stieltjes functions, will be presented. The application of the results on numerical treatment of the fractional time-derivatives will also be explained. This is joint work with Dr. Chuan Bi (NIH/NIA, USA) and Mr. Jiangming Xie (Chongqing University, China).

The need for analytic continuation arises frequently in many applications, such as the extrapolation of complex electromagnetic permittivity from a given band of frequencies or the determination of geometric features of microstructure of a composite based on measurements of its effective properties. In a joint work with Yury Grabovsky we consider a large class of such problems where analytic continuation exhibits a power law precision deterioration as one moves away from the source of data. We introduce a general Hilbert space-based approach for determining these exponents. The method identifies the "worst case" function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus, an ellipse or an upper half-plane the solution of the integral equation and the corresponding exponent can be found explicitly. In more general geometries numerical solution of the integral equation supports the power law precision decay.

Effective properties of infinite periodic structures are governed by the dispersion law of the Bloch waves. A modeling of the dispersion law is proposed for all frequencies and wave vectors. This modeling is based on the complex analytic structure of the Bloch spectrum and on the Kramers-Kronig expression extended to spatially dispersive media. Several explicit formulas for effective parameters will be given for multilayer stacks, for all frequencies and wave vectors, in situations where a perturbation approach can be applied. The relevance of the obtained effective parameters will be discussed.

In this survey talk I will recall how the familiar integral representation for Herglotz--Nevanlinna functions can be recast in operator theoretic or rather system theoretic terms, namely as a realization of the function as the transfer function of an input/state/output linear system. There are similar results for Schur functions, i.e., functions taking values in the unit disc (as opposed to the upper or the right halfplane). After discussing the one variable case, I will move to the case of several complex variables. It turns out that there is a very good generalization of the classical realization theory provided one considers possibly more restricted classes of functions. These classes are defined by testing the values of the function not merely on tuples of scalars but on tuples of commuting matrices of all sizes. Time permitting I will wrap up with the most recent incarnation of these ideas, where one tests the values on tuples of not necessarily commuting matrices.

The response of many passive linear physical systems are governed by Herglotz or Stieltjes functions. Typically associated with these functions is some sort of Hilbert space, or in the case of composites, a subspace collection. Operations on Herglotz or Stieltjes function often have a parallel operation on subspace collections. Here we will explore this correspondence. Also we give representation formulas for the effective (conductivity, elastic, piezoelectric, etc.) properties of two-dimensional composites of two anisotropic phases.

Sum rules can be derived using properties of Herglotz functions. Typically, they can be put in the form of physical bounds in the frequency domain, as the product of a performance metric (for instance, minimum extinction cross section) and bandwidth being limited by the static polarizability of the scatterer. In this talk, we will see how the sum rule can also, at least in some cases, be interpreted in the time domain, as a moment of the impulse response, corresponding to the average delay of the system

Metamaterials may be composed of not only space-varying but also time-varying particles. We generically refer to such media as “spacetime metamaterials.” Spacetime metamaterials involve unprecedented diversity, with up to 36 bianisotropic constitutive parameters potentially each depending each on direct spacetime (spatial and temporal modulations) and inverse spacetime (spatial and temporal dispersions). Given their time-variance, spacetime metamaterials break typical fundamental bounds associated with time-invariant passive systems, but they are still subject to fundamental laws of physics such as causality and uncertainty. We present here an overview of our recent works in this area with the double purpose to share knowledge on this fascinating emerging research field and to stimulate discussions on fundamental physical and mathematical aspects arising from it.

In this talk, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media, that is in particular media such as metamterials for which the dielectric permittivity $\epsilon$ and magnetic permeability $\mu$ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to Herglotz-Nevanlinna functions. We consider successively the cases of so-called local media and then of general passive media. In particular, we will discuss connections between mathematical properties of models (e.g. stability, “energy conservation”, ...), physical assumptions (e.g. passivity) and the existence of two Herglotz-Nevanlinna functions of the frequency: $\omega\epsilon$ and $\omega\mu$ that determine the dispersion of the material. We will also present dispersion and spectral analysis of this rather general class of electromagnetic media. This is joint work with Patrick Joly (INRIA Poems) and Maryna Kachkanovska (INRIA Poems).

We know that the Nevanlinna (Herglotz) functions arise in physics in a fundamental way as an alternative representation of causality. We have discovered recently that they also relevant to the amplification phenomenon occurring in traveling wave tubes (TWTs). The Nevanlinna (Herglotz) functions represent there the so-called slow wave structure - a significant a component of the TWT. Though an ideal TWT system is perfectly conservative its energy functional turns out to be not positive definite. This explains the existence of exponentially growing in space eigenfunctions manifesting the amplification. The corresponding theory is intimately related to the Krein spectral theory.

In this talk, I will discuss the Cherkaev-Gibiansky variational method for developing bounds on effective operators from the perspective of the abstract theory of composites and then apply the method to a large class of operators arising in the constitutive relations which include dissipative, coercive, and sectorial operators. The important points that will be made in this talk are: 1) the bounds are derived from variational minimization principles even for operators which are non-self-adjoint; 2) they apply to Schur complements of such operators as they are effective operators; 3) effective operators can be seen as a generalization of the Schur complement concept; 4) effective operators derived from Herglotz functions are themselves Herglotz functions and the variational bounds apply to such effective operators. To demonstrate the applicability of the method, we will discuss two important examples, the effective complex conductivity in the theory of composites and the Dirichlet-to-Neumann (DtN) map for the conductivity equation with a non-symmetric conductivity tensor on a Lipschitz bounded domain in two- or -three dimensions. This is joint work with Maxence Cassier (Institut Fresnel).

Complex electromagnetic permittivity functions are functions of frequency that have analytic extension into the upper half-plane. Their positive imaginary parts describe the absorption of EM radiation of a given frequency by materials, while their real parts describe the refractive properties. This function can be measured in a band of frequencies, and one wants to use its analyticity to extrapolate to a wider band of frequencies. A fundamental question is how reliable such extrapolation algorithms can possibly be. In a joint work with Narek Hovsepyan we have been able to recast the problem in terms of stability of analytic continuation of Hardy functions. In another joint work with Narek the latter problem is reduced to a solution of a linear integral equation of Fredholm type, which can be solved numerically, leading to a quantification of uncertainty of any extrapolation procedure.

Almost four decades ago, David Bergman and Graeme Milton derived a set of bounds on the complex permittivity of two-phase composites. These bounds, which are now known as the Bergman-Milton bounds, confine the set of effective permittivities to lens shaped regions in the complex plane. In this talk, we discuss the relation of the Bergman-Milton bounds to bounds on the complex polarizability derived recently by Miller, Hsu, Reid, Qiu, DeLacy, Joannopoulos, Soljačić, and Johnson. Furthermore, we discuss a refinement of the Bergman-Milton bounds for isotropic composites and present our recent results on new families of laminates attaining the bounds. This is joint work with Owen Miller and Graeme Milton.

The set of quasi-Herglotz functions is introduced as a natural extension of the convex cone of Herglotz functions. The new class of functions consists of differences of Herglotz functions and we demonstrate that it has properties that are useful in the modeling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions and we will illustrate that several of the important properties and modeling perspectives are inherited by the new set of quasi-Herglotz functions. In this presentation, we will focus on the approximation theory and the formulation as a convex optimization problem where the generating measure is modeled by using a finite expansion of B-splines and point masses. Numerical examples are included to demonstrate the modeling of a non-passive gain media.

In this overview talk will deal with generalizations of Herglotz-Nevanlinna functions . We will start with some updates concerning functions in several variables and then consider generalizations (still in only one variable though) that are not any more related to passive systems. Particular focus will be put on quasi-Herglotz functions, these are complex linear conbinations of usual Herglotz functions and they exhibit interesting phenomena. (Part of the talk is based on joint work with Mitja Nedic).

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.