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Integral Equation Methods In Scattering Theory

We report room temperature lasing in two-dimensional diffractive lattices of silver and gold plasmon particle arrays embedded in a dye-doped polymer that acts both as waveguide and gain medium. As compared to conventional dielectric distributed feedback lasers, a central question is how the underlying band structure from which lasing emerges is modified by both the much stronger scattering and the disadvantageous loss of metal. We use spectrally resolved back-focal plane imaging to measure the wavelength- and angle dependence of emission below and above threshold, thereby mapping the band structure. We find that for silver particles, the band structure is strongly modified compared to dielectric reference DFB lasers, since the strong scattering gives large stop gaps. In contrast, gold particles scatter weakly and absorb strongly, so that thresholds are higher, but the band structure is not strongly modified. The experimental findings are supported by finite element and fourier modal method calculations of the single particle scattering strength and lattice extinction.

Lasing at the band edges of plasmonic lattices

We study the plane wave scattering and absorption by a flat grating of thin silver nanostrips located in free space, in the visible-light range. The formulation involves generalized boundary conditions imposed on the strip median lines. We use an accurate numerical solution to this problem based on the dual-series equations and the method of analytical regularization. This guarantees fast convergence and controlled accuracy of computations. Reflectance, transmittance, and absorbance as a function of the wavelength and the grating parameters are analyzed. In addition to well-known surface-plasmon resonances, sharp resonances are revealed in the H-polarized scattering near but not equal to the Rayleigh wavelengths of nonzero diffraction orders; in the E-polarized scattering these resonances are not visible. Asymptotic formulas for the frequencies and natural fields of the grating resonances are presented.

Accurate Analysis of Light Scattering and Absorption by an Infinite Flat Grating of Thin Silver Nanostrips in Free Space Using the Method of Analytical Regularization

This article reviews the nature and history of the discovery of high-quality natural modes existing on periodic arrays of many subwavelength scatterers; such arrays can be viewed as specific periodically structured open resonators. These grating modes (GMs), like any other natural modes, give rise to the associated resonances in electromagnetic-wave scattering and absorption. Their complex wavelengths are always located very close to (but not exactly at) the well-known Rayleigh anomalies (RAs), determined only by the period and the angle of incidence. This circumstance has long been a reason for their misinterpretation as RAs, especially in the measurements and simulations using low-resolution methods. In the frequency scans of the reflectance or transmittance, GM resonances usually develop as asymmetric Fano-shape spikes. In the optical range, if a grating is made of subwavelength-size noblemetal elements, then GMs exist together with better-known localized surface-plasmon (LSP) modes. Thanks to high tunability and considerably higher Q-factors, the GM resonances can potentially replace the LSP-mode resonances in the design of nanosensors, nanoantennas, and solar-cell nanoabsorbers.

Periodicity Matters: Grating or lattice resonances in the scattering by sparse arrays of subwavelength strips and wires.

We consider the two-dimensional (2-D) scattering of the H- and E-polarized plane waves by several discrete configurations made of M> > 1 periodically arranged circular cylindrical silver wires. To find the scattered field, we use the field Fourier expansions in local coordinates and addition theorems for cylindrical functions. Resulting M × M block-type matrix equation is cast to the Fredholm second-kind form that guarantees convergence of numerical solution when each block is truncated to finite dimensions and truncation order is taken larger. The scattering and absorption cross-sections and the near-field patterns are computed. The interplay of plasmon and grating-type resonances is studied for finite in-line and stacked arrays, corners, and crosses made of nano-size silver wires in the visible range of wavelengths, with the refractive index of silver taken from the experimental data.

Effect of Periodicity in the Resonant Scattering of Light by Finite Sparse Configurations of Many Silver Nanowires

We study numerically the effect of periodicity on the plasmon-assisted scattering and absorption of visible light by infinite and finite gratings of circular silver nanowires. The infinite grating is a convenient object of analysis because of the possibility to reduce the scattering problem to one period. We use the well-established method of partial separation of variables however make an important improvement by casting the resulting matrix equation to the Fredholm second-kind type, which guarantees convergence. If the silver wires have sub-wavelength radii, then two types of resonances co-exist and may lead to enhanced reflection and absorption: the plasmon-type and the grating-type. Each type is caused by different complex poles of the field function. The low-Q plasmon poles cluster near the wavelength where dielectric function equals -1. The grating-type poles make multiplets located in close proximity of Rayleigh wavelengths, tending to them if the wires get thinner. They have high Q-factors and, if excited, display intensive near-field patterns. A similar interplay between the two types of resonances takes place for finite gratings of silver wires, the sharpness of the grating-type peak getting greater for longer gratings. By tuning carefully the grating period, one can bring together two resonances and enhance the resonant scattering of light per wire by several times.

Periodicity-induced effects in the scattering and absorption of light by infinite and finite gratings of circular silver nanowires.

Results from the electromagnetic modeling of the threshold conditions of hybrid plasmon modes of a laser based on a silver nanotube with an active core and covered with an active shell are presented. We study the modes of such a nanolaser that have their emission wavelengths in the visible-light range. Our analysis uses the mathematically grounded approach called the lasing eigenvalue problem (LEP) for the set of the Maxwell equations and the boundary and radiation conditions. As we study the modes exactly at the threshold, there is no need to invoke nonlinear and quantum models of lasing. Instead, we consider a laser as an open plasmonic resonator equipped with an active region. This allows us to assume that at threshold the natural-mode frequency is real-valued, according to the situation where the losses, in the metal and for the radiation, are exactly balanced with the gain in the active region. Then the emission wavelength and the associated threshold gain can be viewed as parts of two-component eigenvalues, each corresponding to a certain mode. In the configuration considered, potentially there are three types of modes that can lase: the hybrid localized surface plasmon (HLSP) modes of the metal tube, the core modes, and the shell modes. The latter two types can be kept off the visible range in thin enough configurations. Keeping this in mind, we focus on the HLSP modes and study how their threshold gain values change with variations in the geometrical parameters of the nanotube, the core, and the shell. It is found that essentially a single-mode laser can be designed on the difference-type HLSP mode of the azimuth order m = 1, shining in the orange or red spectral region. Furthermore, the threshold values of gain for similar HLSP modes of order m = 2 and 3 can be several times lower, with emission in the violet or blue parts of the spectrum.

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Electromagnetic analysis of the lasing thresholds of hybrid plasmon modes of a silver tube nanolaser with active core and active shell.

We consider the two-dimensional problem of a nonattenuating H-polarized electromagnetic field that can exist in the presence of a circular cylindrical nanowire (core) made of silver and coated with a concentric layer of active material (shell). Assuming that the natural-mode frequency is real-valued, we search for it together with the associated threshold value of material gain in the coating. Using the separation of variables, we arrive at a set of independent characteristic equations for each azimuthal index and seek their roots numerically. Here, it is important to use an adequate description of the metal core dielectric function. We observe that the well-known Drude formula not only leads to predictable errors in the lasing wavelengths but additionally, fails to reveal the complete composition of the modal spectrum. Known examples of sophisticated Drude-like formulas help overcome these drawbacks; however, these formulas suffer from negative absorption in silver in part of the visible range. Therefore, the use of experimental data for the metal dielectric function is mandatory. Our analysis reveals that, in each azimuthal order, the considered nanowire laser can emit light on several localized surface plasmon modes and the shell modes.

Modes of core–shell silver wire plasmonic nanolaser beyond Drude formula

We consider the H -polarized plane wave scattering by a finite linear grating of circular silver wires using the angular field expansions in local coordinates and addition theorems for cylindrical functions. The study is focused on the influence of the plasmon and the grating resonances on each other. It demonstrates that the scattering per one silver cylinder can be dramatically enhanced if the grating period is tuned to the plasmon-resonance wavelength value.

Denys M. Natarov

Papers

Periodicity-induced effects in the scattering and absorption of light by infinite and finite gratings of circular silver nanowires.

Effect of Periodicity in the Resonant Scattering of Light by Finite Sparse Configurations of Many Silver Nanowires

Electromagnetic analysis of the lasing thresholds of hybrid plasmon modes of a silver tube nanolaser with active core and active shell.

Modes of core–shell silver wire plasmonic nanolaser beyond Drude formula

Periodicity-Enhanced Plasmon Resonances in the Scattering of Light by Sparse Finite Gratings of Circular Silver Nanowires