Threshold conditions for transversal modes of tunable plasmonic nanolasers shaped as single and twin graphene-covered circular quantum wires.

We implement the lasing eigenvalue problem (LEP) approach to study the electromagnetic field in the presence of a circular quantum wire (QW) made of a gain material and wrapped in graphene cover and a dimer of two identical graphene-covered QWs, at the threshold of stationary emission. LEP delivers the mode-specific eigenvalue pairs, namely the frequencies and the threshold values of the QW gain index for the plasmon and the wire modes of such nanolasers. In our analysis, we use quantum Kubo formalism for the graphene conductivity and classical Maxwell boundary-value problem for the field functions. The technique involves the resistive boundary conditions, the separation of variables in the local coordinates, and, for the dimer, the addition theorem for the cylindrical functions. For single-wire plasmonic laser, we derive approximate engineering expressions for the lasing frequencies and threshold values of the gain index that complement the full-wave computations. For the dimer, we derive separate determinantal equations for four different classes of symmetry of the lasing supermodes and solve them numerically. Our investigation of the mode frequencies and thresholds versus the graphene and QW parameters shows that plasmon modes or, for the dimer, plasmon supermodes have lower frequencies and thresholds than the wire modes provided that the QW radius is smaller than 10µm, however in thicker wires they are comparable. Only the plasmon-mode characteristics are well-tunable using the graphene chemical potential. In the dimer, all lasing supermodes form closely located quartets, however, they quickly approach the single-wire case if the inter-wire separation becomes comparable to the radius. These results open a way for building essentially single-mode plasmonic nanolasers and their arrays and suggest certain engineering rules for their design.

Seeing the order in a mess: optical signature of periodicity in a cloud of plasmonic nanowires.

We consider the two-dimensional (2-D) problem of the H-polarized plane wave scattering by a linear chain of silver nanowires in a cloud of similar pseudo-randomly located wires, in the visible range. Numerical solution uses the field expansions in local coordinates and addition theorems for cylindrical functions and has a guaranteed convergence. The total scattering cross-sections and near- and far-zone field patterns are presented. The observed resonance effects are studied and compared with their counterparts in the scattering by the same linear chain of wires in free space.

Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes.

The lasing spectra and thresholds of a selectively pumped photonic molecule composed of two microdisks is investigated using effective index approximation and full-wave 2-D electromagnetic equations. The lasing eigenvalue problem formulation is used to find modal frequencies and threshold values of material gain. The influence of the optical coupling between active and passive microdisks on the lasing eigenvalues and directionalities of emission is studied. It is shown that for strong coupling the effect of making one of the resonators passive leads to the doubling of the threshold.

Refractive-index sensitivities of hybrid surface-plasmon resonances for a core-shell circular silver nanotube sensor.

We study the scattering and absorption of an H-polarized plane electromagnetic wave by a circular silver nanotube in the visible range of wavelengths using the separation of variables. The computed spectra of the extinction cross section display several hybrid localized surface-plasmon resonances of the dipole and multipole type. Analytical equations are derived for their resonance wavelengths. Bulk refractive-index sensitivities of nanotube-based sensors are determined, showing higher values for multipole resonances.

Spoiling of tunability of on-substrate graphene strip grating due to lattice-mode-induced transparency.

We report a prediction of the optical effect apparently not discussed earlier. As known both from theory and experiment, the gratings of flat graphene strips lying on dielectric substrates display moderate-Q resonances on the strip plasmon modes in the H-polarization case. In the plasmon resonances, high reflectance and absorbance are observed. These characteristics are tunable with the aid of the graphene chemical potential, which controls the plasmon-mode frequency. However, if this frequency coincides with the high-Q lattice-mode frequency, a narrow-band regime of electromagnetically induced transparency (EIT) appears. A new point in our finding is that, in the EIT regime, the tunability of the reflectance and absorbance of a grating of narrow graphene strips get spoiled profoundly. This is established using a full-wave meshless code based on the method of analytical regularization, which leads to a Fredholm second-kind matrix equation that guarantees the code convergence. Numerical results are presented for the strip width and period, having the microsize dimensions so that all resonances lie in the THz range. However, the same effect takes place in the infrared range for narrower strips and smaller periods. The lattice modes are caused by the periodicity and can have ultra-high Q-factors; however, they do not exist if the substrate is absent. The loss of tunability at EIT is explained by the lattice-mode field pattern, which has deep E-field minima at the strips.

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

Subwavelength terahertz spin-flip laser based on a magnetic point-contact array.

We present a theoretical design for a single-mode, truly subwavelength terahertz disk laser based on a nanocomposite gain medium comprising an array of normal-metal/ferromagnetic (FM) point contacts embedded in a thin dielectric layer. Stimulated emission of light occurs due to spin-flip relaxation of spin-polarized electrons injected from the FM side of the contacts. Ultrahigh electrical current densities in the contacts and a dielectric material with a large refractive index, neither condition being achievable in conventional semiconductor media, enables the thresholds of lasing to be overcome for the lowest-order modes of the disk, making single-mode operation possible.

Low-threshold lasing eigenmodes of an infinite periodic chain of quantum wires.

We study the lasing eigenvalue problems for a periodic open optical resonator made of an infinite grating of circular dielectric cylinders standing in free space, in the E- and H-polarization modes. If possessing a "negative-absorption" refractive index, such cylinders model a chain of quantum wires made of the gain material under pumping. The initial-guess values for the lasing frequencies are provided by the plane-wave scattering problems. We demonstrate a new effect: the existence of specific grating eigenmodes that have a low threshold of lasing even if the wires are optically very thin.

Optical fields of the lowest modes in a uniformly active thin subwavelength spiral microcavity.

A numerical study is presented of several lowest in frequency modes in a spiral microlaser. The modes in an arbitrarily shaped active cavity are considered as solutions to the two-dimensional eigenproblem for the Muller boundary-integral equations. After discretization using the Nyström-type algorithm, the eigenvalues are found in terms of frequency and material-gain threshold.

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

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.

Exact off-resonance near fields of small-size extended hemielliptic 2-D lenses illuminated by plane waves.

The near fields of small-size extended hemielliptic lenses made of rexolite and isotropic quartz and illuminated by E- and H-polarized plane waves are studied. Variations in the focal domain size, shape, and location are reported versus the angle of incidence of the incoming wave. The problem is solved numerically in a two-dimensional formulation. The accuracy of results is guaranteed by using a highly efficient numerical algorithm based on the combination of the Muller boundary integral equations, the method of analytical regularization, and the trigonometric Galerkin discretization scheme. The analysis fully accounts for the finite size of the lens as well as its curvature and thus can be considered as a reference solution for other electromagnetic solvers. Moreover, the trusted description of the focusing ability of a finite-size hemielliptic lens can be useful in the design of antenna receivers.

Integral equation analysis of an arbitrary-profile and varying-resistivity cylindrical reflector illuminated by an E-polarized complex-source-point beam.

A two-dimensional reflector with resistive-type boundary conditions and varying resistivity is considered. The incident wave is a beam emitted by a complex-source-point feed simulating an aperture source. The problem is formulated as an electromagnetic time-harmonic boundary value problem and cast into the electric field integral equation form. This is a Fredholm second kind equation that can be solved numerically in several ways. We develop a Galerkin projection scheme with entire-domain expansion functions defined on an auxiliary circle and demonstrate its advantage over a conventional moment-method solution in terms of faster convergence. Hence, larger reflectors can be computed with a higher accuracy. The results presented relate to the elliptic, parabolic, and hyperbolic profile reflectors fed by in-focus feeds. They demonstrate that a partially or fully resistive parabolic reflector is able to form a sharp main beam of the far-field pattern in the forward half-space; however, partial transparency leads to a drop in the overall directivity of emission due to the leakage of the field to the shadow half-space. This can be avoided if only small parts of the reflector near the edges are made resistive, with resisitivity increasing to the edge.

Lasing frequencies and thresholds of the dipole supermodes in an active microdisk concentrically coupled with a passive microring.

The lasing spectra and threshold values of material gain for the dipole-type supermodes of an active microdisk concentrically coupled with an external passive microring are investigated. TE polarized modes are treated accurately using the linear electromagnetic formalism of the 2-D lasing eigenvalue problem (LEP) with exact boundary and radiation conditions. The influence of the microring on the lasing frequencies and thresholds is studied numerically, demonstrating threshold reduction opportunities. This is explained through the analysis of the mode near-field patterns and the degree of their overlap with the active region, as suggested by the optical theorem applied to the LEP solutions.

Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder.

Our objective is the assessment of the accuracy of a conventional finite-difference time-domain (FDTD) code in the computation of the near- and far-field scattering characteristics of a circular dielectric cylinder. We excite the cylinder with an electric or magnetic line current and demonstrate the failure of the two-dimensional FDTD algorithm to accurately characterize the emission rate and the field patterns near high-Q whispering-gallery-mode resonances. This is proven by comparison with the exact series solutions. The computational errors in the emission rate are then studied at the resonances still detectable with FDTD, i.e., having Q-factors up to 10(3).

Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes.

Lasing modes in cyclic photonic molecules (CPMs) composed of several identical thin semiconductor microdisks in free space are studied in a linear approximation. Maxwell's equations with exact boundary conditions and the radiation condition at infinity are considered as a specific eigenvalue problem that enables one to find natural frequencies and threshold gains. It is demonstrated that careful tuning of the distance between the disks in CPMs is able to drastically reduce the lasing thresholds of the whispering-gallery modes having small azimuth indices.

Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization.

A fast and accurate method is developed to compute the natural frequencies and scattering characteristics of arbitrary-shape two-dimensional dielectric resonators. The problem is formulated in terms of a uniquely solvable set of second-kind boundary integral equations and discretized by the Galerkin method with angular exponents as global test and trial functions. The log-singular term is extracted from one of the kernels, and closed-form expressions are derived for the main parts of all the integral operators. The resulting discrete scheme has a very high convergence rate. The method is used in the simulation of several optical microcavities for modern dense wavelength-division-multiplexed systems.