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

Abstract

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.

Spaser and optical amplification conditions in graphene-coated active wires

“…Moreover, it can help to estimate the minimum gain required to observe an important enhancement of near and far fields, see, for example, Refs. [25,30]. The gain-loss compensation condition discussed in our work, also referenced as the gain threshold [26], does not necessarily mean that most emitted photons are the result of stimulated (coherent) emission, since, as calculations including saturation show, spontaneous (incoherent) emission can still be very important (especially if Purcell effects are relevant in the system studied).…”

“…Indeed, this property can be used for finding gain thresholds as divergences of properties such as the scattering coefficient [26,31]. Instead, here, we use another strategy, which is to find the critical value of the imaginary part of ε 1 , for which the modal eigenfrequency ω n is real [24][25][26]. Despite its limitations, the strategy used in the present work is still useful to correctly predict the mode that will be lasing (or spasing) and the frequency at which this occurs.…”

“…In a first stage, we use the eigenmode approach [22], particularly well-suited for finding the criticalgain values [Imε 1 ] c of the imaginary part of the permittivity of the active medium where the spaser condition is fulfilled. In the context of the eigenmode approach, these critical-gain values can be rigorously obtained by requiring simply that the imaginary part of a modal frequency be zero [24][25][26]. To do so, we use fully retarded methods in all of the examples presented here.…”

Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes

“…As is easy to see, a presence of both active regions with gain material and lossy regions with absorptive material can be taken into account, in LEP, without any difficulty. Therefore, recently on-threshold mode analyses were published for the plasmonic nanolasers based on a silver strip in a quantum wire [18] and a silver tube in such a wire [19], assumed to be made of gain material. In today's laser engineering, the direction of research associated with periodic arrays of metal or dielectric nanoparticles and nanowires, placed inside or on top of the quantum well (active layer) is very interesting and much promising.…”

Mathematical and numerical modeling of a drop-shaped microcavity laser

“…Spectral characteristics of disk-like microcavity lasers were numerically analyzed by the Lasing Eigenvalue Problem (LEP) starting with the pioneering works [Smotrova et al, 2005], [Smotrova, Nosich, 2004]. Lately, LEP has been effectively used also for computer simulations of plasmonic nanolasers [Shapoval et al, 2017], [Natarov et al, 2019]. LEP has two real-valued eigenvalues: the frequency of lasing and the mode threshold gain.…”