Amplified and Directional Spontaneous Emission from Arbitrary Composite

PHYSICAL REVIEW B 93, 125415 (2016)

Ampliﬁed and directional spontaneous emission from arbitrary composite bodies: A self-consistent treatment of Purcell effect below threshold

Weiliang Jin,1 Chinmay Khandekar,1 Adi Pick,2 Athanasios G. Polimeridis,3 and Alejandro W. Rodriguez1 1Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 3Skolkovo Institute of Science and Technology, Moscow, Russia (Received 19 October 2015; revised manuscript received 22 December 2015; published 9 March 2016)

We study amplified spontaneous emission (ASE) from wavelength-scale composite bodies—complicated arrangements of active and passive media—demonstrating highly directional and tunable radiation patterns, depending strongly on pump conditions, materials, and object shapes. For instance, we show that under large enough gain, PT symmetric dielectric spheres radiate mostly along either active or passive regions, depending on the gain distribution. Our predictions are based on a recently proposed fluctuating-volume-current formulation of electromagnetic radiation that can handle inhomogeneities in the dielectric and fluctuation statistics of active media, e.g., arising from the presence of nonuniform pump or material properties, which we exploit to demonstrate an approach to modeling ASE in regimes where Purcell effect (PE) has a significant impact on the gain, leading to spatial dispersion and/or changes in power requirements. The nonlinear feedback of PE on the active medium, captured by the Maxwell-Bloch equations but often ignored in linear formulations of ASE, is introduced into our linear framework by a self-consistent renormalization of the (dressed) gain parameters, requiring the solution of a large system of nonlinear equations involving many linear scattering calculations.

DOI: 10.1103/PhysRevB.93.125415

I. INTRODUCTION latter stems primarily from changes to atomic decay rates, which can be either enhanced or suppressed through the Noise in structures comprising passive and active materials Purcell effect (PE) [19]. Because PE is sensitive to the gain can lead to important radiative effects [1], e.g., spontaneous and geometry of the objects, such a dependence manifests as emission (SE) [2], superluminescence [3], and fluorescence a nonlinear and nonlocal interaction (or feedback) between [4]. Although large-etalon gain amplifiers and related devices the atomic medium and the optical environment [20], which have been studied for decades [5–8], there is increased interest we model within the stationary-phase approximation [21] in the design of wavelength-scale composites for tunable via a self-consistent renormalization of the (dressed) atomic sources of scattering and incoherent emission [9,10], or for parameters. We show that this leads to a system of nonlinear realizing perfect absorbers [11] at midinfrared and visible equations, involving as many degrees of freedom as there wavelengths. In this paper, we extend a recently developed fluctuating are volumetric unknowns, which in principle require many volume-current (FVC) formulation of electromagnetic (EM) scattering problems (radiation from dipoles) to be solved fluctuations [12–15] to the problem of modeling spontaneous simultaneously, but which thanks to the low-rank nature of emission and scattering from composite, wavelength-scale volume-integral equation (VIE) scattering operators [14] can structures, e.g., metal-dielectric spasers [16–18], subject to be accurately obtained with far fewer scattering calculations inhomogeneities in both material and noise properties. We than there are unknowns. Our predictions indicate that under begin by studying amplified spontaneous emission (ASE) significant PE, composite objects can exhibit a high degree of from piecewise-constant composite bodies, showing that dielectric gain enhancement/suppression and inhomogeneity, their emissivity can exhibit a high degree of directionality, affecting power requirements and emission patterns. depending sensitively on the gain profile and shape of the Gain-composite structures are the subject of recent the- objects. For instance, we find that under large enough gain, oretical and experimental work [1], and have been studied the directivity of parity-time (PT ) symmetric spheres can in a variety of different contexts, including spasers (com- be designed to lie primarily along active or passive regions, binations of metallic and gain media) with low-threshold depending on the presence or absence of centrosymmetry, characteristics [16–18,22,23], random structures with special respectively. Such composite micron-scale emitters act as absorption properties [24–28], and nanoscale particles with tunable sources of incoherent radiation, forming a special highly tunable emission and scattering properties [29,30]. PT - class of infrared/visible antennas exhibiting polarization- and symmetric structures have received special attention recently direction-sensitive absorption and emission properties. An as they shed insights into important non-Hermitian physics, important ingredient for the design of directional emission is such as design criteria for realizing exceptional points [31], the ability to tune the gain profile of the objects, which can be symmetry breaking [31], unidirectional scattering [32–34], far from homogeneous in realistic settings. Here, we consider and lasing thresholds [35]. Until recently, most studies of two important sources of inhomogeneities affecting population radiation/scattering from PT structures remained confined inversion of atomically doped media: inhomogeneous pump to 1D and 2D geometries [9,33,34,36–40]. In such low- profiles and modifications stemming from changes to the dimensional systems, it is common to employ scattering matrix emitters’ local radiative environment. Below threshold, the formulations [33,36,37] to solve for the complex eigenmodes

2469-9950/2016/93(12)/125415(10) 125415-1 ©2016 American Physical Society JIN, KHANDEKAR, PICK, POLIMERIDIS, AND RODRIGUEZ PHYSICAL REVIEW B 93, 125415 (2016) and scattering properties of bodies, leading to many analytical gain medium [44,47,61,62]. Such approximations, however, insights. For instance, while the introduction of gain violates ignore nonlinearities stemming from the induced radiation rate energy conservation, a generalized optical theorem can be ∼E · P present in the MB equations, which captures feedback obtained in 1D, establishing conditions for unidirectional on the atomic medium due to amplification or suppression transmission of light [32–34]. Other studies focus on 2D of noise from changes in the local density of states (also high-symmetry objects such as cylindrical or spherical bodies known as Purcell effect) [45,63,64]. Here, we show that PE [38,39] or particle lattices [40], demonstrating strong asym- can be introduced into the linearized framework via a self- metric and gain-dependent scattering cross sections, while consistent renormalization or dressing of the gain parameters, 3D structures such as ring resonators have been studied an approach that was recently suggested [20] but which has within the framework of coupled-mode theory [41,42]. With yet to be demonstrated. In particular, working within the few exceptions [9], however, most studies of gain-composite scope of the linearized MB equations and stationary-inversion bodies have focused on their scattering rather than emission approximation, we capture the nonlinear feedback of ASE on properties. gain, i.e., the steady-state enhancement/suppression of gain Furthermore, while these systems are typically studied and atomic decay rates due to PE, via a series of nonlinear under the assumption of piecewise-constant [9,33,36]or equations involving many coupled, linear, classical scattering linearly varying [10] gain profiles, in realistic situations, calculations—local density of states or far-field emission inhomogeneities in the pump or material parameters (e.g., due to electric dipole currents. Since the gain profile can arising from PE [20], hole burning [43,44], and gain saturation become highly spatially inhomogeneous, it is advantageous [44]) result in spatially varying dielectric profiles which alter to tackle this problem using brute-force methods, e.g., finite SE. For example, the highly localized nature of plasmonic differences [65], finite elements [66], or via the scattering VIE resonances in spasers results in strongly inhomogeneous framework described below. Our FVC method is particularly pumping rates [17] and orders-of-magnitude enhancements advantageous in that it is general and especially suited for in atomic radiative decay rates [45]. In random lasers [24], handling scattering problems with large numbers of degrees partial pumping plays an important role in determining the of freedom (defined only within the volumes of the objects), lasing threshold [26,46] and directionality [27,28]. Rigorous in contrast to eigenmode expansions [59] which become descriptions of lasing effects in these systems commonly resort inefficient in situations involving many resonances [48,59]or to solution of the full Maxwell-Bloch (MB) equations, in near-field effects [67,68]. which the electric field E and induced (atomic) polarization field P couple to affect the atomic population decay rates II. FORMULATION [47]. However, the MB equations are a set of coupled, time-dependent, nonlinear partial differential equations [48] Active medium. To begin with, we review the linearized which, not surprisingly, prove challenging to solve except in description of a gain medium consisting of optically pumped simple situations involving high-symmetry [45,49]orlow- four-level atoms, as shown in Fig. 1: a dense collection dimensional structures [1]. Despite their overhead, brute- of active emitters (e.g., dye molecules [44] or quantum force FDTD methods have been employed to study the dots [69,70]) embedded in a passive (background) dielectric transition from ASE to lasing in 1D random media [25,50], medium r . (Note that our choice of four-level system here is 2D metamaterials [49,51], photonic crystals [52], and, more merely illustrative since the same approach described below recently, nanospasers [45]. also applies to other active media.) Below threshold where A more recent, general-purpose method that is applicable stimulated emission (and effects such as hole burning) can be to arbitrary structures is the steady-state ab initio laser safely ignored [44], the effective permittivity of such a medium theory (SALT), an eigenmode formulation that exploits the can be well approximated by a simple two-level Lorentzian stationary-inversion approximation to remove the time de- gain profile [44,61,62], pendence and internal atomic dynamics of the MB equations 2 4πg(x) γ⊥D (x) [21,53,54], yet captures important nonlinear effects such as ω,x = ω + 0 , ( ) r ( ) − + (1) hole burning and gain saturation [21] through effective two- γ⊥ ω ω21 iγ⊥ level polarization and population equations [55]. The resulting g (x,ω) nonlinear eigenvalue equation can be solved via a combination of Newton-Raphson [48,56], sparse-matrix solver [57], and where g depends explicitly on the frequency ω12, polariza- nonlinear eigenproblem [58] techniques, exploiting either tion decay rate γ⊥ (or gain bandwidth), coupling strength 3 2 = √3 c r = − spectral “CF” basis expansions (especially suited for structures g 3 γ21 [61,62], and inversion factor D0 n2 n1 2 r ω21 with special symmetries) [59] or brute-force methods [60] that associated with the 2 → 1 transition. Under the adiabatic or can handle a wider range of shapes and conditions. Although stationary-inversion approximation [21] and assuming that the this formulation can describe many situations of interest, it system is pumped at ω30, the steady-state population inversion nevertheless poses computational challenges in 3D or when is given by: applied to structures supporting a large number of modes [48]. − γ21 P Furthermore, the impact of noise below or near threshold has 1 /γ21 D = γ10 n. (2) 0 γ21 γ21 yet to be addressed, though recent progress is being made 1 + A + + 1 P/γ21 along these directions [48,56]. γ32 γ10 Below and near the lasing threshold, stimulated emission (Note that in a dense medium, γ⊥ γ21 is dominated by is often negligible, enabling linearized descriptions of the collisional and dephasing effects [44].) Here, A = 1(2) for

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exploit a recently developed FVC formulation based on the |3 VIE method which captures all of the relevant physics. γ b FVC formulation. In order to obtain the individual and/or 32 cumulative radiation from all dipoles within a given object, |2 Φ γ n we exploit the FVC formulation introduced in [14] and 21 ω ധ 21 a summarized here. The starting point of FVC is the VIE ω |1 formulation of EM scattering [66], describing scattering of an 30 γ 1010 incident, six-component electric (E) and magnetic (M) field = |0 φinc (E; H) from a body described by a spatially varying c 6 × 6 susceptibility tensor χ(x). Given a six-component a ε =ε (γ ) γ g,ng,n g 21,n21,n loss 21 electric (J) and magnetic (M) dipole source σ = (J; M), the incident field is obtained via a convolution ( ) with the b Φ n=Φ=Φ n(ε(εg) gain 6 × 6 homogeneous electric and magnetic Green’s function γ =Φ=Φ /Φ/Φ γ r,0+γ nr c 21,n21,n n 0 21 2121 (GF) of the ambient medium (x,y), such that φinc = σ = d3 y (x,y)σ (y). Exploiting the volume equivalence FIG. 1. Schematic of a gain-loss composite body consisting of principle [66], the unknown scattered fields φsca = ξ can a dense (active) collection of four-level atoms (left inset) optically also be expressed via convolutions with , except that here pumped via monochromatic light at frequency ω30 and pump rate P, ξ =−iωχφ are the (unknown) bound currents in the body, resulting in population inversion and spontaneous emission (SE) at related to the total field inside the body φ = φinc + φsca through ω21. The nonlinear feedback of Purcell effect on the SE rate of the χ. Writing Maxwell’s equations in terms of the bound currents, object is captured via a system of nonlinear equations illustrated on the we arrive at the so-called JM-VIE equation [12]: right schematic, in which (a) the dielectric constant at the location of [ +(iωχ)−1] ξ =−( σ), (3) the nth emitter depends on the corresponding atomic decay rate γ21,n through (1), (b) the radiative emission n of a classical dipole at the Z location of the nth emitter, described by (7), is altered, e.g. enhanced whose solution can be obtained by a Galerkin discretization of or suppressed, by the surrounding ε(x), and (c) the radiative decay = = rate γ r of the emitter is dressed by the Purcell effect through (8), the currents σ (x) n snbn(x) and ξ(x) n xnbn(x)ina 21,n { } which in turn affects the dielectric response in (1). convenient, orthonormal basis bn of N six-component vec- tors, with vector coefficients s and x, respectively. The resulting matrix expression can be written in the form x + s = Ws, −1 defining the VIE matrix (W )m,n =bm,bn + iωχ( bn) incoherent (coherent) pump source; c denotes the speed of = with , denoting the standard conjugated inner product. Direct light, ni the population (per unit volume) of level i, n application of Poynting’s theorem = 1 Re d3x (E∗ × H) = r + nr 2 i ni the overall atomic population, γij γij γij the decay → yields the following expression for the far-field radiation flux rate from level i j, consisting of radiative and nonradia- from σ (here, a single dipole source embedded within the P = √ σc | |2 tive terms, respectively, and (x) E(x,ω30) is r (ω30) ω30 volume) [78]: the position-dependent pump rate from 0 → 3, which is often =−1 ∗ =−1 + ∗ + the main source of spatial dispersion. σ 2 Re ξ φ 2 Re(ξ σ ) (ξ σ ), (4) The SE properties of such a medium are described by =−1 (x + s)∗ sym G(x + s), (5) the fluctuation-dissipation theorem (FDT) [71–73]. In par- 2 =−1 ∗ ∗ ticular, local thermodynamic considerations imply that the 2 s W sym GWs, (6) presence of absorption or amplification must be accompa- =−1 Tr [DW∗ sym GW], (7) nied by fluctuating polarization currents σ whose corre- 2 ∗ = 4 − ∗ lation functions, σi (x,ω)σj (y,ω) Im g(x,ω)n2/(n1 where D = s s and G are N × N matrices, with Gmn = − π n2)δ(x y)δij , depend on the corresponding macroscopic gain bm, bn . − profile Im g and population inversion, n2/(n1 n2)[74]. The If {bn} is chosen to be a localized basis of unit am- latter is often described in terms of an effective (local) tem- plitude, i.e., bm,bn =δnm, volume elements, then the flux perature T defined with respect to the Planck spectrum [71], contribution from a given dipole source in the volume − ω/kBT − 1 (e 1) , whose value turns negative [75] for systems (including different polarizations) bn is precisely the diagonal undergoing population inversion, n >n, where T → 0− − 1 ∗ 2 1 element 2 (W sym GW)n,n. In contrast, the overall ASE in the limit of complete inversion and T →−∞when the is given by an ensemble average over all such fluctuating system is in the ground state [72]. In the particular case dipole sources =σ , in which case the elements of of a steady-state four-level system, the relative populations the matrix D, which encodes information about the current n2/n1 depend only on the relative decay rates and one finds amplitudes, are given by the current-current correlations n1 γ10 3 3 ∗ ∗ that kB T = ω ln( ) = ω ln( ) depends implicitly on the = n2 γ21 D mn d x d y bm(x) σ (x)σ (y) bn(y), determined by pumping rate only through changes in the atomic decay the FDT above. Direct computation of (7) is expensive due rates (Purcell effect); in contrast, the effective temperature to the large dimensionality N of the problem, but it turns out of three-level systems depends explicitly on the pumping rate. that the Hermitian, negative-semidefinite, and low-rank nature The presence of inhomogeneities in the dielectric function and of sym G (since it is associated with the smooth, imaginary fluctuation statistics can be a hurdle for calculations of SE that part of the Green’s functions) enables reexpressing the trace rely on scattering-matrix formulations [76,77], but here we as the Frobenius norm of a low-rank matrix. Specifically,

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=− ∗ decomposing sym G Ur Sr Ur via a fast approximate SVD The gain profile of a body subject to an incident pump = ∗ P [79], where r N [80], and further decomposing Sr LS LS , rate can be obtained by enforcing that (1) and (8)be we find that the product W ∗ sym GW can be written in the satisfied simultaneously. Such systems of nonlinear equations ∗ ∗ form QQ , with Q = W Ur LS , reducing the calculation of are most often solved iteratively using one or a combination the diagonal elements to a small series r N of scattering of algorithms, ranging from simple fixed-point iteration [86] calculations (matrix-inverse operations) [12]. Similarly, as to more sophisticated approaches like Newton-Raphson and shown in [14], the same follows for the overall ASE which nonlinear Arnoldi methods [48,87]. Essentially, as illustrated is just a weighted sum of the diagonal elements. For example, in Fig. 1, starting with the bare parameters, dressed decay while the calculations below require N 403 basis functions rates are computed via (8) from the radiation equation (7) to obtain accurate spatial resolution, we find that generally after which, having updated the gain-medium equation (1), r 20. the entire process is repeated until one arrives at a fixed point Purcell effect. Although often assumed to be uniform below of the system. In principle, this requires hundreds of thousands r threshold, the atomic radiative decay rates γij entering (1) N of scattering calculations (flux from each dipole source in are in fact position dependent due to PE [45,63], leading to the active region) to be solved per iteration, which becomes changes in the dipole coupling g and population inversion prohibitive in large systems, but the key here is that the { } factor D0, either enhancing or suppressing (quenching) gain entire spatially varying flux bn throughout the body can be [63]. In what follows, we only consider modifications to the computed extremely fast, requiring far fewer ( N) scattering r radiative decay rate at the lasing transition γ21; unlike the pump calculations (as described above). Note that these large systems r which is incident at a nonresonant frequency, changes in γ21 of nonlinear equations have many fixed points and hence can have a significant impact on SE and must therefore be convergence to the correct solution is never guaranteed, treated self-consistently. depending largely on the initial guess and algorithm employed r The impact of PE on the radiative decay rate γ21 of an atom [88]. However, a convenient and effective approach is to begin at some position bn is captured by the coupling of the atomic by first solving the system in the fast-converging (passive) polarization and electric fields ∼P · E, or the induced radiation regime Pn/γ12 1, and then employing this solution as an term in the MB equations, in the presence of the noise and initial guess at larger pumps. surrounding dielectric environment [81]. While technically this requires abandoning the linear model above, the weak III. RESULTS nature of noise (ignoring stimulated emission) implies that the latter can also be obtained (perturbatively) from a linear, We begin this section by showing that wavelength-scale classical calculation: the radiative flux bn from a classical composite bodies can exhibit highly complex, tunable, and dipole at bn. Specifically, the renormalized or dressed decay directional radiation patterns. Although it is not surprising rate of an atom at position x can be expressed as [63] that objects undergoing ASE (once known as “mirrorless lasers”) exhibit highly directed radiation patterns [89], few r = F r,0 γ21(x) (x)γ21 , (8) studies have gone beyond large-etalon Fabry-Perot´ cavities where F(x) denotes the Purcell factor of a dipole at x, and the [90] or fiber waveguides [91], often modeled via ray-optical superscript “0” denotes the decay rate of the atomic population or scalar-wave equations [1], which miss important effects in the lossy (background) medium. It follows that the decay present in wavelength-scale systems [48]. Further below, we rate associated with a given bn and entering (1)isgivenby show that dielectric inhomogeneities arising from the pump = nr + F r,0 and/or radiation process can also introduce important changes γ21(bn) γ21 bn γ21 , where the Purcell factor to the ASE patterns. In particular, we apply the renormalization F = 0 bn bn / . (9) approach described above to consider the nonlinear impact of ∼ ∗ Purcell effect on the gain medium, and show that while in [Note that all bn (W sym GW)n,n can be computed very F many cases a homogeneous approximation leads to accurate fast, as explained above, and that bn does not depend on the amplitude of the fluctuations since it is measuring the relative results, there are situations where these can fail dramatically. These calculations serve only as a proof of principle and radiation rate of a classical emitter at this position.] Here, 3+ we assume that the bulk (background) medium only has involve highly doped ( Er and rhodamine) dielectric objects, r allowing faster computations but requiring very large values a significant impact on γ nr,0 (obtained either experimentally 21 of Im to achieve significant gain. Similar results follow, or theoretically by accounting for atomic interactions within g however, in systems subject to smaller or smaller doping the bulk) [44] but not on the radiative decay rate γ r,0,in g 21 densities, at the expense of larger pump powers or by 0 = ω4/ π c3 which case 12 0 is the emission rate of the exploiting resonances with greater confinement or smaller atom in vacuum (assuming a unit-amplitude dipole). Note radiative loss rates (e.g., compact bodies of larger dimensions that in a lossy medium, e.g., in metals, the bare γ 0 will 21 and/or refractive indices, or more complex structures such as be dominated by nonradiative processes [82,83], leading to photonic-crystal resonators). r,0 0 small quantum yields (QY) γ21 /γ21 1. Technically, the Tunable radiation patterns. We begin by exploring SE calculation of the Purcell factor requires integration over the from piecewise-constant PT -symmetric spheres (Fig. 2 insets) F = γ⊥/π bn (ω) gain bandwidth, bn dω 2 2 0 , but here we γ⊥+(ω−ω21) (ω) consisting of a background dielectric medium, e.g., nanocom- make the often-employed and simplifying assumption that posite polymers [92,93] or semiconductors [41,94], doped with γ⊥ γ21 and γ⊥ spectral radiative features [84,85], so as active materials to realize different (N) regions of equal gain to only consider radiation at ω21. (orange) or loss (gray). Figure 2(a) shows the SE flux

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N=2

N=1 N=2 N=3 N=4 N=3

N=4

N=1 0.5 1.5 2 2.5 3 3.5 Radius R(c/ω)

FIG.2.(a)SEflux from PT -symmetric spheres consisting of N ={1,2,3,4} (black, green, blue, and red lines) regions of equal gain (orange) and loss (gray), with permittivities = 4 ± i and gain temperature T → 0− K (corresponding to complete population inversion), 2 2 as a function of sphere radius R (for a fixed vacuum wavelength 2πc/ω). (ω) is normalized by the flux |BB(ω)|=R (ω/c) ω from a “blackbody” of the same surface area and temperature. Insets show angular radiation intensities (θ,φ) for different N at selective radii R ={2.6,3,2.7,2.1}(c/ω) (corresponding to increasing N), with orange/gray colormaps denoting regions of gain/loss. (b) Selected (θ,φ) of various PT -symmetric shapes, including “beach balls,” “magic” cubes, and cylinders, showing complex directivity patterns that depend strongly on the centrosymmetry of the objects. Here the color white (black) refers to the maximum (minimum) flux value. (c) Peak SE flux (right axis, dashed lines) and directivity G/ (left axis, solid lines), or the ratio of the flux emitted along gain direction G (see text) to the total flux , near the third resonance of the N = 1 sphere, as a function of the gain/loss tangent δ =|Im |/ Re and for multiple values of Re ={2,4,12} (red, black, and blue lines), where the vertical dashed black line denotes δ for (a). The plots show increased directivity attained as the system approaches the lasing threshold (marked in the case of Re = 12, where diverges). from spheres of varying N ={1,2,3,4} (black, green, blue, cylinders—as before, the presence/absence of centrosymmetry red lines) and gain/loss permittivities = 4 ± i, at a fixed results in high/low gain directivity. frequency ω and gain temperature T → 0− K (corresponding To understand the features and origin of these emission to complete inversion), as a function of radius R (in units patterns, Fig. 2(c) explores the dependence of the peak of the vacuum wavelength c/ω). (ω) is normalized by the (dashed lines) and G/ (solid lines) on the gain/loss tangent 2 2 flux |BB(ω)|=R (ω/c) ω from a “blackbody” of the same δ =|Im |/ Re of the N = 1 sphere, near the third resonance surface area and temperature. As expected, shows peaks and for multiple values of Re ={2,4,12}. As shown, there at selected R (c/ω) corresponding to enhanced emission is negligible ASE in the limit Im → 0, yet the localization at Mie resonances. (Note that the amplitude of the peaks in of fluctuating dipoles to the gain-half of the (increasingly continue to increase with increasing R due to decreased uniform) sphere leads to a small (though observable) amount radiative losses, with the bandwidth of the resonances nar- of directionality, favoring emission toward the loss direction. rowing as the system reaches the lasing transition, at which The tendency of dipoles within a sphere to emit in a preferred point our linear approach breaks down.) As spontaneous direction has been studied in the context of fluorescence [95] emission increases, so too does the directivity, illustrated by in the ray-optical limit R c/ω, which as shown here is the radiation patterns (θ,φ) shown on the insets of Fig. 2(a) exacerbated in the presence of gain [96]: essentially, dipoles at selected R, whose high directionality contrasts sharply with within a sphere tend to emit in the direction opposite the the emission profile of passive particles. (With few exceptions nearest surface, which explains why spheres having/lacking [15], the latter tend to emit quasi-isotropically, as can be centrosymmetry tend to emit along directions of gain/loss. verified by decreasing the gain of the spheres.) We find that Moreover, in order to achieve large directivity, there needs the direction of largest ASE changes drastically with respect to be a significant amount of mode confinement and gain, to N, with radiation coming primarily from either active or as illustrated by the negligible ASE and directivity of the passive regions depending on whether the spheres exhibit or Re = 2 sphere. Finally, we find that for large enough Re , lack centrosymmetry, respectively (insets). In particular, the the directivity increases with increasing Im , peaking at a ratio G/ of the flux emitted from the gain surfaces, critical δ, corresponding to the onset of lasing. Such a transition is marked by a diverging near the threshold along with

N −1 π(2k+1)/N a corresponding narrowing of the resonance linewidth (not 2 G = dθ sin (θ) dφ (θ,φ), shown). (Note that our predictions close to and above this k=0 2πk/N critical gain are no longer accurate since they neglect important effects stemming from stimulated emission [97]. For instance, to the total flux , is generally 0.5 for odd N (centrosym- at and above the critical gain, the resonance linewidth goes metric) and 0.5 otherwise. For instance, near the peak of to zero and then broadens with increasing Im , while it is the third resonance at R ≈ 3(c/ω), N = 1(2) spheres exhibit well known that nonlinear gain saturation results in a finite G/ ≈ 0.1(0.75). The sensitive dependence of emission laser linewidth [53,54].) Nevertheless, our results demonstrate pattern on geometry and gain profile is not unique to spherical that a significant amount of directivity can be obtained below structures, as illustrated in Fig. 2(b), which shows (θ,φ)for the onset of lasing, where the linear approximation is still various shapes, including “magic” cubes, “beach balls,” and valid.

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Pump inhomogeneity. Next, we employ the four-level gain model of (1) to illustrate the impact of gain inhomogeneities P z on ASE. While both and PE are simultaneous sources of 8 spatial dispersion, for the sake of comparison we consider each independently of one another. We begin by studying the impact of pump on the N = 2 sphere (above) for an φ=-π/4 6 x y active region consisting of a background medium r = 4 that φ is doped with rhodamine 800 dye molecules with atomic y | -Im ε = × 15 −1 ≈ = parameters: ω21 2.65 10 s (λ 711 nm), γ⊥/ω21 BB 1 −7 −3 0.04, γ21/ω21 = 7.5 × 10 , γ32/ω21 = γ10/ω21 = 10 ,QY 4 x 19 −3 of 20%, and concentration n = 40 mM (2.4 × 10 cm ) Φ/|Φ φ=0 [61]. Note that since ω30 ω21, it is safe to neglect feed- 0 φ π back due to PE and hence P is determined from a single 2 = /4 scattering calculation [14]. For these parameters [61], a pump rate P/γ21 ≈ 3 results in − Im g(ω21) ≈ 1. We consider illumination with z-polarized plane waves incident from two 0 opposite directions along the x-y plane, shown schematically 1 1.5 2 2.5 3 3.5 (a) in Fig. 3. The insets depict z = 0 cross sections of the Radius R(c/ω21) resulting g proﬁle [Fig. 3(a)]atR = 3.4(c/ω21) along with = φ π emission patterns (θ,φ) [Fig. 3(b)]atR 4(c/ω21), under 0.7 = /4 three incident conditions, corresponding to different directions of incidence, xˆ cos φ + yˆ sin φ and −xˆ cos φ − yˆ sin φ, with φ ={0,±π/4}; in each case, the incident power is chosen 0.65 φ=0 such that max{−Im[g]}=1. As shown, the gain proﬁles vary dramatically with respect to position and incident angle, with Φ / Φ/Φmax

Im g changing from 0 → 1 on the scale of the wavelength. G 1 These spatial variations lead to correspondingly large changes Φ 0.6 in the overall ASE [Fig. 3(a)] and directivity [Fig. 3(b)]. More importantly, we find that these features cannot be explained by 0 naive, uniform-medium approximations (UMA). For instance, 0.55 = 1 replacing g with the average gain g V V g in the case φ =−π/4, we find that UMA predicts an emission rate φ=-π/4 ≈ /BB 30 that is three times larger than that predicted by 0.5 exact calculations. Differences in illumination angle also result 1 1.5 2 2.5 3 3.5 (b) R c ω ) in different angular radiation patterns (θ,φ). For instance, Radius ( / 21 we find that φ =−π/4 leads to much more isotropic radiation FIG.3.(a)SEflux and (b) gain directivity /, or the ratio than φ ={0,π/4}, a consequence of the larger Im g near the G center of the sphere and the fact that dipoles near the center of the flux emitted along the gain directions G (see text) to the total = tend to radiate more isotropically and efficiently than those flux , from the N 2 sphere (inset) of Fig. 2 at frequency ω21, farther away. corresponding to the transition frequency of an active region consisting of Rhodamine 800 dye molecules. The gain medium is excited Purcell effect. We now consider inhomogeneities arising by plane waves propagating in opposite directions, x cos φ + y sin φ from PE, assuming a uniform pump and doping concentra- and −x cos φ − y sin φ, for three different orientations, φ = 0 (black tion. In particular, we apply the self-consistent framework line), π/4 (blue line), and −π/4 (red line), leading to significant described in Sec. II to study ASE from the N = 1 sphere spatial inhomogeneities. is normalized by BB as in Fig. 2 and above, but with an active region consisting of a background → = 3+ plotted as a function of radius R, in units of the dye 2 1 transition medium r 4 that is doped with Er atoms [47,98], with wavelength c/ω (see text). Insets in (a) show z = 0 cross sections = × 14 −1 ≈ = 21 parameters ω21 6.28 10 s (λ 2.8 μm), γ⊥/ω21 of the resulting gain profiles −Im at R = 3.4(c/ω ), while those = × −5 = = g 21 0.03, γ21/ω21 5 10 , γ32/ω21 γ10/ω21 1, bare QY in (b) show angular radiation intensities (θ,φ) normalized by the of 50%, and concentration n = 1019 cm−3. (Further below maximum intensity max at R = 4(c/ω21) for the different pump we also consider a different geometry, a metal-dielectric orientations. spaser consisting of similar gain parameters but passive −4 metallic regions.) Here, a pump rate P/γ21 = 10 results − ≈ in Im g(ω21) 1 in the absence of PE. As discussed above, slope) [86]. The convergence also depends on the degree of we employ fixed-point iteration to solve (7) and (8) and nonlinearity in the system, which in the case of our four-level hence obtain consistent values of g and , starting with the system can be significant under small QY (γ r /γ 1), large bare (F = 1) atomic parameters and iterating until the gain 21 21 P,orγ21/γ10 1 (in which case there is significant gain parameters converge to the nearest fixed point. Generally, saturation). Nevertheless, in practice we find that for a wide the convergence rate of the fixed-point algorithm depends range of parameters, a judicious combination of fixed-point sensitively on the chosen parameter regime, requiring larger iteration and Anderson acceleration [88] ensures convergence ∂g number of iterations with decreasing r (decreasing local ∂γ21 within dozens of iterations. The bottom/top insets of Fig. 4(c)

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2 0.5 (i) 10 Φ/Φmax UMA QY=0.5 1 40 10 iteration 1 fixed-point 0.4 30 10 1 (iv) (ii) -Im ε (ii) 0 | 20 | ε (iii) 2 P.E 10 1 BB QY=0.1

bare Φ BB y /

G 0.3 0 (iv) 1 0 510 Φ ε1 Φ/|Φ 20 Φ/|Φ x (iii) Iteration 10 P.E 0 10 (i) 0.2 0 10 -2 0 51015 20 iteration 1 fixed-point 0.1 Iteration 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 10 -5 10 -4 (a) R c ω ) (b) R c ω ) (c) / 0 Radius ( / 21 Radius ( / 21 Pump rate ധ γ 21

FIG. 4. Self-consistent treatment of Purcell effect (PE) based on solution of (1)and(8) using fixed-point iteration. (a) SE flux normalized by BB as in Fig. 2 and (b) gain directivity G/ from the N = 1 sphere of Fig. 2, as a function of radius R (in units of c/ω21), with an active 3+ P 0 = −4 0 → medium consisting of Er atoms and subject to a uniform pump rate /γ21 10 ,whereω21 and γ21 denote the bulk (bare) 2 1 transition frequency and decay rate of the atoms, respectively. Top/bottom insets in (a) show z = 0 cross sections (contour plots) of the gain profiles −Img obtained during the first and final (fixed-point) iteration of the algorithm at two separate radii R ={2,2.5}(c/ω21), with black/white denoting min , max −[Im g] ≈{1.9,1.3} and {1.9,1.5}, respectively. Insets in (b) show the angular radiation intensities (θ) at selected radii = P 0 (black dots), normalized by the maximum intensity max.(c) at a fixed radius R 2(c/ω21) as a function of /γ21, with insets illustrating P 0 = −4 the evolution of as a function of iteration at a fixed /γ21 10 and for two different values of quantum yields (QY) (see text). All plots compare quantities in the presence (solid blue lines) or absence (dashed black lines) of PE, and under a uniform-medium approximation (UMA) F = 1 F (solid red lines) described in the text; the solid green line in (a) denotes the average Purcell factor V V at each radius. demonstrate the iterative process at a fixed R = 2(c/ω21) and and correspondingly changes in emission patterns (insets) at 19 −3 for two different sets of concentrations n ={1,5}×10 cm selective R ={2,2.5}(c/ω21). and quantum yields ≈{10,50}%. Figure 4(c) also explores the dependence of on P Figure 4 illustrates the impact of PE on the emission of at a fixed R = 2(c/ω21), showing that peaks at a finite P 0 −4 the sphere, showing variation in (a) SE flux (ω21) and (b) value of /γ12 10 and then decreases with increasing gain directivity G/ of the sphere with respect to radius P; the same is true for F and g (not shown). Such a P 0 = −4 R at a fixed /γ21 10 , or with respect to (c) pump nonmonotonicity stems from the fact that near the critical rate P at a fixed R = 2(c/ω21), both including (solid blue pump rate, Im g ≈ Re g, causing the resonance frequency lines) and excluding (dashed black lines) PE. As before, is to shift to smaller radii, a trend that is observed both in normalized by BB. Shown as insets in Fig. 4(a) are z = 0 cross the presence and absence of PE. Surprisingly, however, we sections of Im g for the first and final (fixed-point) iteration find that while PE causes large inhomogeneities in g,in of the algorithm, at two different radii R ={2,2.5}(c/ω21) both scenarios the peak emission is approximately the same, (black dots), demonstrating large gain enhancement and spatial suggesting the possibility that one could explain the impact variations. As expected, is either enhanced or suppressed of PE by a simple UMA. In what follows, we exploit a UMA depending on the average PE (green line) which we have that not only greatly simplifies the calculation of PE but also defined as F = 1, where for convenience we have defined leads to accurate results over a wider range of parameters. In particular, we consider a UMA in which the otherwise 1 1 inhomogeneous gain profile of the object is replaced with F = F = /0 → V N bn that of a uniform medium g(x) g (assuming a uniform V n r → r =F r,0 pump rate) given by (1) but with γ21 γ21 γ21 , 1 ∗ corresponding to a homogeneous broadening/narrowing of the =− Tr W sym GW. 2N0 gain atoms throughout the sphere. Within this approximation, the system of nonlinear equations above is described by r (As discussed above, F ∼ turns out to be the Frobenius a single (as opposed to N 1) degree of freedom γ21 , norm of a low-rank matrix and is therefore susceptible to fast enabling faster convergence and application of algorithms computations.) As shown, at small R 1.2(c/ω21), or in the especially suited for handling low-dimensional systems of absence of resonances, F < 1 and hence is suppressed equations [99]. Ignoring other sources of inhomogeneity (e.g., with respect to predictions that neglect PE. Conversely, F > induced by density or pump variations), such an approximation 1 near resonances and hence is enhanced. Note that for our allows calculation of via scattering formulations best suited choice of parameters, the gain profile scales linearly with the for handling piecewise-constant dielectrics, including SIE ∝ = r [100] and related scattering matrix [76] methods. The solid quantum yield, i.e., g QY γ21/γ21, such that in the limit F →∞ r → red lines in Fig. 4 are obtained by employing the UMA, as (ignoring quenching occurring as γ21 γ10), − Im g → 2. (For smaller QY 1, g can be many times demonstrating its validity over a wide range of parameters. larger than the bare permittivity with increasing F , saturating Surprisingly, we find that this holds even in regimes marked at much larger values of PE.) In addition to changing the by strong gain saturation (e.g., γ10 γ21). It follows that in overall SE rate, PE also modifies the sphere’s directivity. This this geometry, the effect of PE on radiation can be attributed is illustrated in Fig. 4(b), which shows enhancements in G/ primarily to the presence of a larger average gain or pump

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exact (solid blue lines) and UMA (dashed red line) predictions, 6 -Im ε one finds that the presence of multiple nodes in leads to P.E g 2 a dramatic failure for UMA (with a peak error of ≈50%). 5 T(K) Despite the different radiation rates, however, we find that 0.4 -450 UMA effectively captures the main features of the far-field radiation pattern (inset).

| 4 UMA -650

BB IV. CONCLUDING REMARKS Φ

Φ/| 3 We have shown that wavelength-scale, active-composite Φ/Φmax bodies can lead to complex radiative effects, depending 1 sensitively on the arrangement of gain and loss. By exploiting a 2 general-purpose formulation of EM fluctuations, we quantified 0 the non-negligible impact that dielectric and noise inhomogeneities can have on emission in these systems. Furthermore, 1 0.8 0.85 0.9 0.95 1 1.05 we introduced an approach that captures feedback from Purcell Radius R(c/ω21) effect (i.e., the optical environment) on the gain medium. We note that in situations where ASE is dominated by FIG. 5. Self-consistent treatment of Purcell effect (PE) in the relatively few leaky resonances, it is possible and practical metal-dielectric composite shown on the inset, with orange and blue to perform a similar procedure by expanding the fields in denoting Er3+-doped polymer and Au regions, described via full terms of eigenmodes, in which case the problem boils down solution of (1)and(8) (solid blue lines) or via the uniform-medium to solution of a linear generalized eigenvalue problem for approximation (dashed red line) described in the text. The solid line the leaky modes. (In VIE as in FDFD or related brute-force shows the SE flux from the particle as a function of radius R (in methods, leaky modes can be computed via the solution of a = units of c/ω21). Insets show z 0 cross sections of the gain profile generalized eigenvalue problem of the form Z(ω)ξ = 0, for a − Img (top) and active temperature (right), along with the angular complex frequency [101].) However, the FVC approach above = radiation intensity (θ,φ) (bottom) at a selected R 0.95(c/ω21), is advantageous in that it casts the problem in the context of normalized by the maximum intensity max. solutions of relatively few ( degrees of freedom) scattering calculations. More importantly, FVC can handle structures supporting many modes or situations where near-field effects are of interest and contribute to PE [68]. The latter is especially rate in the sphere, whereas the actual spatial variation in g is important when the relevant quantity is the energy exchange largely unimportant. between two nearby objects, a regime that motivated initial There are geometries and situations where such a UMA development of these and related scattering methods [67]. Note is expected to fail, e.g., structures subject to even large that above we mainly explored structures with small Re ≈ 4 dielectric inhomogeneities (as in Fig. 3). Such conditions and large gain concentration n, leading to large Im g Re arise in large objects (supporting higher-order resonances) or even for relatively weakly confined resonances. However, in metal-dielectric composites (supporting highly localized similar effects can be obtained with smaller n and Im g in surface waves). Figure 5 shows for one such structure structures with larger Re and dimensions, or supporting (bottom inset): a dielectric sphere with the same gain medium highly localized fields (e.g., spacers), where there exists (orange) of Fig. 4 but partitioned into three metallic (red) larger Purcell enhancement. Finally, we note that micron-scale =− + regions along the azithmutal direction, given by (φ) 2 particles like the spheres explored above lie within the reach ∈ + = i for φ [2nπ/3,2nπ/3 π/8], where n 0,1,2. (Note that of current experiments [102]. our choice of for the metal does not lead to a strong plasmonic resonance, but still yields significant subwavelength ACKNOWLEDGMENTS confinement.) A z = 0 cross section of the steady state g and temperature distributions at a fixed R = 0.95c/ω21 are We would like to thank Li Ge, Hakan Tureci, Steven G. shown on the top inset, demonstrating significant and complex Johnson, Zin Lin, and Jacob Khurgin for useful discussions. variations in both quantities. In particular, while population This work was partially supported by the Army Research inversion is suppressed near the metal (black regions), εg Office through the Institute for Soldier Nanotechnologies attains its maximum value (white regions) close to the metal under Contract No. W911NF-13-D-0001 and by the National surface, decaying rapidly within the dielectric. Comparing the Science Foundation under Grant No. DMR-1454836.

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