Chain models and the spectra of tridiagonal k-Toeplitz matrices

Chain models can be represented by a tridiagonal matrix with periodic entries along its diagonals. Eigenmodes of open chains are represented by spectra of such tridiagonal $k$-Toeplitz matrices, where $k$ represents length of the repeated unit, allowing for a maximum of $k$ distinct types of elements in the chain. We present an analysis that allows for generality in $k$ and values in $\mathbb{C}$ representing elements of the chain, including non-Hermitian systems. Numerical results of spectra of some special $k$-Toeplitz matrices are presented as a motivation. This is followed by analysis of a general tridiagonal $k$-Toeplitz matrix of increasing dimensions, beginning with 3-term recurrence relations between their characteristic polynomials involving a $k^{th}$ order coefficient polynomial, with the variables and coefficients in $\mathbb{C}$. The existence of limiting zeros for these polynomials and their convergence are established, and the conditioned $k^{th}$ order coefficient polynomial is shown to provide a continuous support for the limiting spectra representing modes of the chain. This analysis also includes the at most $2k$ eigenvalues outside this continuous set. It is shown that this continuous support can as well be derived using Widom's conditional theorems (and its recent extensions) for the existence of limiting spectra for block-Toeplitz operators, except in special cases. Numerical examples are used to graphically demonstrate theorems. As an addendum, we derive expressions for $O(k)$ computation of the determinant of tridiagonal $k$-Toeplitz matrices of any dimension

Collective eigenstates of emission in an N-entity heterostructure and the evaluation of its Green tensors and self-energy components

Our understanding of emission from a collection of emitters strongly interacting among them and also with other polarizable matter in proximity has been approximated by independent emission from the emitters. This is primarily due to our inability to evaluate the self-energy matrices and the collective eigenstates of emitters in heterogeneous ensembles. A method to evaluate the self-energy matrices that is not limited by the geometry and the material composition is presented here to understand and exploit such collective excitations. Numerical evaluations using this method are used to highlight the significant differences between independent and the collective modes of emission in heterostructures. A set of n emitters driving each other and m other polarizable entities, where N=m+n, is used to represent the coupled system of a generalized geometry in a volume integral approach. Closed form relations between the Green tensors of entity pairs in free space and their correspondents in a heterostructure are derived concisely. This is made possible for general geometries because the global matrices consisting of all free-space Green dyads are subject to conservation laws. The self-energy matrix of the emitters can then be assembled using the evaluated Green tensors of the heterostructure, but a decomposition of its components into their radiative and non-radiative decay contributions is non-trivial. This is accomplished using matrix decomposition identities applied to the global matrices containing all free-space dyads. The relations to compute the observables of the eigenstates (such as quantum efficiency, power/energy of emission, radiative and non-radiative decay rates) are presented. We conclude with a note on extension of this method to collective excitations that also include strong interactions with a surface in the near-field

Error estimators and their analysis for CG, Bi-CG and GMRES

We present an analysis of the uncertainty in the convergence of iterative linear solvers when using relative residue as a stopping criterion, and the resulting over/under computation for a given tolerance in error. This shows that error estimation is indispensable for efficient and accurate solution of moderate to high conditioned linear systems ($\kappa>100$), where $\kappa$ is the condition number of the matrix. An $\mathcal{O}(1)$ error estimator for iterations of the CG (Conjugate Gradient) algorithm was proposed more than two decades ago. Recently, an $\mathcal{O}(k^2)$ error estimator was described for the GMRES (Generalized Minimal Residual) algorithm which allows for non-symmetric linear systems as well, where $k$ is the iteration number. We suggest a minor modification in this GMRES error estimation for increased stability. In this work, we also propose an $\mathcal{O}(n)$ error estimator for A-norm and $l_{2}$ norm of the error vector in Bi-CG (Bi-Conjugate Gradient) algorithm. The robust performance of these estimates as a stopping criterion results in increased savings and accuracy in computation, as condition number and size of problems increase

Radiative and non-radiative effects of a substrate on localized plasmon resonance of particles

Experiments have shown strong effects of some substrates on the localized plasmons of metallic nano particles but they are inconclusive on the affecting parameters. Here we have used Discrete Dipole Approximation in conjunction with Sommerfeld integral relations to explain the effect of the substrates as a function of the parameters of incident radiation. The radiative coupling can both quench and enhance the resonance and its dependence on the angle and polarization of incident radiation with respect to the surface is shown. Non-radiative interaction with the substrate enhances the plasmon resonance of the particles and can shift the resonances from their free-space energies significantly. The non-radiative interaction of the substrate is sensitive to the shape of particles and polarization of incident radiation with respect to substrate. Our results show that plasmon resonances in coupled and single particles can be significantly altered from their free-space resonances and are quenched or enhanced by the choice of substrate and polarization of incident radiation.Comment: 14 pages, 4 figure