Comment on "Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes" [J. Chem. Phys. 120, 10871 (2004)]

In this Comment I discuss two incorrect statements which were made in the paper "Silver nanoparticle array structures that produce remarkably narrow plasmon line shapes" [J. Chem. Phys.120, 10871 (2004)] by Zou, Janel, and Schatz (ZJS). The first statement is about the use of quasistatic approximation in my earlier work on the similar subject, and the second statement concerns the possibility of exact cancellation of radiative relaxation in periodical chains of nanoparticles. The relationship between the quasistatic approximation, the dipole approximation, and the approximation due to Doyle [Phys. Rev. B39, 9852 (1989)] which was used by ZJS is clarified. It is shown that the exact cancellation of radiative relaxation cannot take place in the particular geometry considered by ZJS.Comment: 3 pages, no figure

Can photonic crystals be homogenized in higher bands?

We consider conditions under which photonic crystals (PCs) can be homogenized in the higher photonic bands and, in particular, near the $\Gamma$-point. By homogenization we mean introducing some effective local parameters $\epsilon_{\rm eff}$ and $\mu_{\rm eff}$ that describe reflection, refraction and propagation of electromagnetic waves in the PC adequately. The parameters $\epsilon_{\rm eff}$ and $\mu_{\rm eff}$ can be associated with a hypothetical homogeneous effective medium. In particular, if the PC is homogenizable, the dispersion relations and isofrequency lines in the effective medium and in the PC should coincide to some level of approximation. We can view this requirement as a necessary condition of homogenizability. In the vicinity of a $\Gamma$-point, real isofrequency lines of two-dimensional PCs can be close to mathematical circles, just like in the case of isotropic homogeneous materials. Thus, one may be tempted to conclude that introduction of an effective medium is possible and, at least, the necessary condition of homogenizability holds in this case. We, however, show that this conclusion is incorrect: complex dispersion points must be included into consideration even in the case of strictly non-absorbing materials. By analyzing the complex dispersion relations and the corresponding isofrequency lines, we have found that two-dimensional PCs with $C_4$ and $C_6$ symmetries are not homogenizable in the higher photonic bands. We also draw a distinction between spurious $\Gamma$-point frequencies that are due to Brillouin-zone folding of Bloch bands and "true" $\Gamma$-point frequencies that are due to multiple scattering. Understanding of the physically different phenomena that lead to the appearance of spurious and "true" $\Gamma$-point frequencies is important for the theory of homogenization.Comment: Accepted in this form to Phys. Rev. B. Small addition in Sec.V (Discussion) relative to previous version. The title to appear in PRB has been changed to "Applicability of effective medium description to photonic crystals in higher bands: Theory and numerical analysis" per the journal policy not to print titles in the form of question

Nonasymptotic Homogenization of Periodic Electromagnetic Structures: Uncertainty Principles

We show that artificial magnetism of periodic dielectric or metal/dielectric structures has limitations and is subject to at least two "uncertainty principles". First, the stronger the magnetic response (the deviation of the effective permeability tensor from identity), the less accurate ("certain") the predictions of any homogeneous model. Second, if the magnetic response is strong, then homogenization cannot accurately reproduce the transmission and reflection parameters and, simultaneously, power dissipation in the material. These principles are general and not confined to any particular method of homogenization. Our theoretical analysis is supplemented with a numerical example: a hexahedral lattice of cylindrical air holes in a dielectric host. Even though this case is highly isotropic, which might be thought as conducive to homogenization, the uncertainty principles remain valid.Comment: 11 pages, 5 figure

Solution of the inverse scattering problem by T-matrix completion. II. Simulations

This is Part II of the paper series on data-compatible T-matrix completion (DCTMC), which is a method for solving nonlinear inverse problems. Part I of the series contains theory and here we present simulations for inverse scattering of scalar waves. The underlying mathematical model is the scalar wave equation and the object function that is reconstructed is the medium susceptibility. The simulations are relevant to ultrasound tomographic imaging and seismic tomography. It is shown that DCTMC is a viable method for solving strongly nonlinear inverse problems with large data sets. It provides not only the overall shape of the object but the quantitative contrast, which can correspond, for instance, to the variable speed of sound in the imaged medium.Comment: This is Part II of a paper series. Part I contains theory and is available at arXiv:1401.3319 [math-ph]. Accepted in this form to Phys. Rev.

Nonlinear inverse problem by T-matrix completion. I. Theory

We propose a conceptually new method for solving nonlinear inverse scattering problems (ISPs) such as are commonly encountered in tomographic ultrasound imaging, seismology and other applications. The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant formalism. We formulate the ISP as a problem whose goal is to determine an unknown interaction potential $V$ from external scattering data. Although we seek a local (diagonally-dominated) $V$ as the solution to the posed problem, we allow $V$ to be nonlocal at the intermediate stages of iterations. This allows us to utilize the one-to-one correspondence between $V$ and the T-matrix of the problem, $T$. Here it is important to realize that not every $T$ corresponds to a diagonal $V$ and we, therefore, relax the usual condition of strict diagonality (locality) of $V$. An iterative algorithm is proposed in which we seek $T$ that is (i) compatible with the measured scattering data and (ii) corresponds to an interaction potential $V$ that is as diagonally-dominated as possible. We refer to this algorithm as to the data-compatible T-matrix completion (DCTMC). This paper is Part I in a two-part series and contains theory only. Numerical examples of image reconstruction in a strongly nonlinear regime are given in Part II. The method described in this paper is particularly well suited for very large data sets that become increasingly available with the use of modern measurement techniques and instrumentation.Comment: This is Part I of a paper series containing theory only. Part II contains simulations and is available as arXiv:1505.06777 [math-ph]. Accepted in this form to Phys. Rev.