2002 OPTICS LETTERS / Vol. 38, No. 12 / June 15, 2013 Magneto-optical spaser D. G. Baranov,1,2 A. P. Vinogradov,1,2 A. A. Lisyansky,3,* Yakov M. Strelniker,4 and David J. Bergman5 1Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudniy, Moscow 141701, Russia 2Institute for Theoretical and Applied Electromagnetics, 13 Izhorskaya, Moscow 125412, Russia 3Department of Physics, Queens College of the City University of New York, Queens, New York 11367, USA 4Department of Physics, Bar-Ilan University, Ramat-Gan IL-52900, Israel 5Raymond and Beverly Sackler School of Physics and Astronomy, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv IL-69978, Israel *Corresponding author: [email protected] Received December 26, 2012; revised April 28, 2013; accepted May 8, 2013; posted May 8, 2013 (Doc. ID 182465); published June 3, 2013 We present an electrodynamical model of a quantum plasmonic device—the magneto-optical (MO) spaser. It is shown that a spherical gain nanoparticle coated with a metallic MO shell can operate as a spaser amplifying circularly polarized surface plasmons. The MO spaser may be used in design of an optical isolator in plasmonic transmission lines as well as in spaser spectrometry of chiral molecules. © 2013 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (140.3410) Laser resonators; (310.6628) Subwavelength structures, nanostructures; (350.4238) Nanophotonics and photonic crystals; (230.3810) Magneto-optic systems. http://dx.doi.org/10.1364/OL.38.002002 Recently, nanosources of coherent light have attracted Maxwell’s equation, while the dynamics of the polariza- significant attention [1]. After [2], different plasmonic- tion and the population inversion of active atoms n are based light sources have been studied theoretically governed by the equations following from the density [3–5] and realized experimentally [6–8]. In most cases, matrix formalism (see [19] for detailed derivation). such a light source is a pumped optical emitter (a quan- The gain atoms embedded into the host medium are mod- tum dot or an active molecule) placed inside or near a eled as two-level systems with transition dipole moment plasmonic resonator. For the case of a metallic nanopar- μ spread in the host matrix. Assuming harmonic time ticle, such a coupled system represents a spaser [2,9–12] dependence of the electric field, e−iωt, we obtain the or dipole nanolaser [13], whose properties have been relation between the polarization P and the electric field intensively studied in detail during the last decade. How- E inside the medium, resulting in the following expres- ever, to the best of our knowledge, radiation from all sion for nonlinear permittivity of a gain medium with suggested sources is linearly polarized. the anti-Lorentzian profile: Today, interaction of circularly polarized light with matter is becoming an attractive field of research ω2−ω2 ω −i 0 [14–18]. The increasing interest in this topic stems from ε ωε 0 2ωΓ gain 0 D0 2 ; (1) the fact that sources of circularly polarized light find a ω ω2−ω2 1 βj ωj2 0 number of intriguing applications in molecular sensing E 2ωΓ [16,17] and quantum information [18]. Sources most commonly used to generate circularly polarized light are 2 where D0 4πμ τ n0∕ℏ is proportional to the population either vertical cavity surface-emitting lasers or spin- p inversion n in active atoms in absence of plasmonic polarized diodes [15]. The major feature of such sources 0 metal, and, therefore, it describes the pumping rate, is that they are diffraction limited. 2 2 β μ τ τ ∕ℏ , and Γ 1∕τ Here τ and τ are relaxa- In this Letter, we suggest a magneto-optical (MO) n p p p n tion times for polarization and inversion, respectively. spaser—a subdiffractive source of circularly polarized The dielectric properties of MO metal, which is as- light operating in the near field. Employing the descrip- sumed to be cobalt, are described with the permittivity tion of the gain medium in terms of negative losses, we tensor having off-diagonal elements accounting for the study its spasing modes. We establish the spaser gener- MO effect: ation condition and show that for some values of pump intensity, only one of two modes can be realized. A suggested MO spaser consists of an amplifying core, e.g., a quantum dot or active molecules, coated by a met- allic layer exhibiting MO response at optical frequencies [see a schematic drawing in Fig. 1(a)]. The whole core- shell nanoparticle is of subwavelength size. The dielec- tric permittivity of the gain medium is approximated by the Lorentzian profile with “negative” losses [19,20]. The simplest expression for the permittivity suitable for our purposes may be deduced from the Maxwell–Bloch equations [21]. In the framework of this approach, the Fig. 1. (a) Schematic drawing of the suggested composite evolution of electric field E is related to the macroscopic core-shell MO spaser. (b) Amplitudes of spasing modes of the polarization P of a gain subsystem via the classical MO spaser versus gain near the bifurcation point. 0146-9592/13/122002-03$15.00/0 © 2013 Optical Society of America June 15, 2013 / Vol. 38, No. 12 / OPTICS LETTERS 2003 0 1 ε ig 0 spaser [24], their dependence on gain exhibits the Hopf @ A bifurcation. Indeed, the right-hand side of Eq. (5) does εˆMO −ig ε 0 : (2) 00ε not depend on Ein and D0 and hence has a nonvanishing imaginary part for all frequencies in case of lossy metal. At the same time, ε has an arbitrary small imaginary For values of the diagonal and off-diagonal elements gain part for vanishing D0. Threshold gain for spasing Dth, of the permittivity tensor, we use experimental data which is the smallest possible gain D for which obtained in [22]. 0 Eq. (5) is satisfied, is achieved for E 0. We start our consideration by finding steady-state in To model the gain medium, we use values of parame- solutions of an MO spaser with no applied electric field ters that are typical for organic dyes [25]: ω0 2 eV, within the quasi-static approximation. As a rule, in the ε0 2, and γ 0.05 eV. The electric field is measured absence of the applied field, there is no field (and no di- −1∕2 in the units of β , so its value is of no importance. pole moment) inside the core-shell particle. An exception The radii of the gain core and MO shell are r 20 nm is the plasmon resonance at which the response of an and R 1.1r. inclusion to the external field is infinite. For a spherical Our calculations show that the two spasing modes inclusion, this occurs when εincl −2εhost, where εincl and have different spasing frequencies ω and different εhost are permittivities of the inclusion and the host spasing thresholds D. In Fig. 1(b) we depict the mode matrix, respectively. Note that for lossy metal, a gain amplitudes (the field inside the amplifying core Ein) matrix with Im εhost < 0 is required [23]. In what follows, versus gain D0 for an MO spaser. One can see that in we find such a solution with no incident field for an MO 4 <D < 5 spaser with gain core and lossy shell. the interval of gain values 0 the two spasing Assuming a homogenous nonzero field inside the core, modes arise with square-root-like dependence of ampli- tudes on gain. The calculated spasing frequencies are Ecore r ≡ E, the field inside the shell may be sought as ω− 1.976 eV and ω 1.977 eV. E Bˆ E − Cˆ E∕ρ3 3 Cˆ E; nn∕ρ3 and the field outside shell Figure 1(b) shows that there exists a range of gain D0 for E − ˆ E∕ρ3 3 ˆ E nn∕ρ3 ˆ nanoparticle as ext A A ; , where A, which only a left or right circularly polarized spasing ˆ ˆ B, and C are some unknown antisymmetric tensors, and ρ mode is emitted. Which particular mode is realized is the distance from the center of the particle. Employing depends on the direction of the magnetization vector. the usual boundary conditions, we can find the unknown The difference between the thresholds of the modes is tensors Aˆ , Bˆ , and Cˆ and write the condition of the exist- simply the consequence of the circular dichroism of the ence of a nonzero internal field inside the nanoparticle at MO shell. Indeed, in the circular polarization basis the per- zero incident field in the form of an eigenvalue problem mittivity tensor of MO metal is εˆMO diagε g; ε − g; ε, T (see [24]): so that ε g eigenvalue corresponds to E− 1; −i; 0 T polarization and ε − g to E 1;i;0 , respectively. This ˆ ˆ 3 ˆ ˆ ˆ 3 ˆ εgainE ε B 2C∕r G B − C∕r E ME; (3) means that the effective permittivity of MO metal differs for right and left polarized modes, leading to different Q ˆ εˆ − εˆT ∕2 factors for these modes. As a result, the threshold for one where G MO MO stands for the nondiagonal part of the permittivity tensor of the MO shell. Due to the of the two modes is smaller. 2 The behavior of the MO spaser above the highest dependence of εgain on E and pumping D0, this equation can be treated as the condition for spasing. We only con- threshold D is strongly affected by the structure of dipole transitions of the gain medium [26]. If the transi- sider solutions with Ez 0, for which the polarization is perpendicular to the magnetization vector of the MO tion dipole moments between the excited and ground shell.