Spaser and optical amplification conditions in gold-coated active nanoparticles
Nicolás Passarelli, Raúl A. Bustos-Marún, Eduardo A. Coronado
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Spaser and optical amplification conditions in gold-coated active nanoparticles.
Nicolás Passarelli , Raúl A. Bustos-Marún , , Eduardo A. Coronado INFIQC-CONICET, Departamento de Fisicoquímica,Facultad Ciencias Químicas, UNC, Ciudad Universitaria, 5000 Córdoba,Argentina, IFEG-CONICET, Facultad de Matemática Astronomía y Física,UNC, Ciudad Universitaria, 5000 Córdoba, Argentina ∗ Due to their many potential applications, there is an increasing interest in studying hybrid systemscomposed of optically active media and plasmonic metamaterials. In this work we focus on aparticular system which consists of an optically active silica core covered by a gold shell. We findthat the spaser (surface plasmon amplification by stimulated emission of radiation) conditions canbe found at the poles of the scattering cross section of the system, a result that remains validbeyond the geometry studied. We explored a wide range of parameters that cover most of the usualexperimental conditions in terms of the geometry of the system and the wavelength of excitation.We show that the conditions of spaser generation necessarily require full loss compensation, but theopposite is not necessarily true. Our results, which are independent of the detailed response of theactive medium, provide the gain needed and the wavelength of the spasers that can be produced by aparticular geometry, discussing also the possibility of turning the system into optical amplifiers andSERS (surface enhanced Raman spectroscopy) substrates with huge enhancements. We believe thatour results can find numerous applications. In particular, they can be useful for experimentalistsstudying similar systems in both, tuning the experimental conditions and interpreting the results.
I. INTRODUCTION
Currently there is a great interest in the interaction ofplasmonic nanoparticles (NPs) and nanostructures withelectromagnetic fields to optimize and increase as muchas possible the magnitude of the evanescent field gen-erated around their surfaces . These investigations aretriggered by the plethora of applications arising from thisproperty in enhanced spectroscopies such as SERS ,TERS (Tip Enhanced Raman Spectroscopy), able toreach the single molecule level , light focusing andimaging in the subdifraction limit of light as wellas in non linear effects such as SHG (Second Har-monic Generation), and metamaterials with novel opti-cal properties . In particular, it has been recentlydemonstrated that incorporating a gain or active mediato a plasmonic nanostructure (NE) give rise to a newset of possibilities and opportunities. In this way, theoptical behavior in such systems can considerably im-prove the performance of plasmonic devices. Theoret-ical studies of this phenomena could be found in dif-ferent systems such as semi-shells , multishells , V-shape arrays , core-shell nanorods , nanotubes and itsdimers , nanoparticle chains , dimer emitter coupledto a metal nanoparticle , as well in metamaterials .Also examples of experimental works on several systemscan be found such as silica-core gold-shell (the systemstudied here) , gold-core silica-shell , gold-core sodiumsilicate-shell , mesocapsules , silver aggregates in gainmedia or even metamaterials .On the other hand, these hybrid systems can generatenew optical phenomena, such as a spaser which are nano-metric sources of evanescent and propagating electromag-netic fields with a high level of wavelength tunability .From the theoretical point of view, the selection anddesign of suitable systems where this kind of behavior is likely to be observed constitutes a topic of paramount im-portance. However, most of previous studies on this topichave focused either on small systems where simple ex-pressions can be obtained , or on very demand-ing numerical calculations where a systematic variationsof the geometry of the system is precluded .In this respect, recently Arnold et al. studied the min-imal spaser threshold for spheroidal and spherical core-shell NPs including retardation effects .In the present work we study spherical core-shell NPswhere the core is made of doped-silica, providing the ac-tive medium, and the shell is made of gold, giving theplasmonic material. In this geometry one can effectivelymake use of the huge evanescent electromagnetic fieldaround the plasmonic structure, for example for sensingpurposes. The electromagnetic properties of this systemcan be calculated analytically by means of the gener-alized Mie’s theory. This allow us to perform a deepphysical analysis of the system and to relate the far andnear fields in a rigorous way. Additionally, the computa-tion of electromagnetic properties in these structures arevery fast, allowing us to explore a wide range of possibleexperimental conditions. We should mention that thematerials chosen as well as the dimensions of the coreand the shell are feasible to be fabricated by chemicalmethods. . For these reasons, it is our hope that thepresent work could be a benchmark for experimentalistworking on this or similar systems. II. THEORYA. Active media
The interaction of photons with the conduction elec-trons of a plasmonic metamaterial gives rise to opticallosses, which for visible light can be significant. Thereare two main sources of optical losses: ohmic heatingand radiative losses . Several strategies have beenproposed to overcome optical losses on particular exam-ples, but probably the most promising one is the use ofactive media.Active media, or gain materials, are made of dyemolecules, semiconductors nanocrystals, or doped di-electrics, where there is a population inversion, cre-ated optically or electrically, that sustains the stimulatedemission of radiation. This stimulated emission is usedto compensate the intrinsic optical losses of plasmonicmaterials. Depending on how strong is this stimulatedemission of radiation compared with the optical losses,the hybrid system can be undercompensated, fully com-pensated or overcompensated. These conditions can bedistinguished by the value of the extinction coefficient Q ext , where Q ext > corresponds to undercompensa-tion, Q ext = 0 to full compensation, and Q ext < toovercompensation. We adopted the term full compensa-tion to distinguish it from Q abs = 0 which correspond tothe situation were only omhic losses are compensated.If the system is undercompensated it just behaves as aregular plasmonic material but with an increased inten-sity of the electromagnetic fields around the plasmonicstructure and narrower resonances in general. When thesystem is overcompensated at frequencies far from its res-onances it behaves as an optical amplifier. For frequen-cies close to some resonance and when losses are fullycompensated, the hybrid system behaves as the nanoplas-monic counterpart of a laser, known as spaser . Thisbehaviors can be readily understood by drawing a par-allelism with conventional macroscopic lasers. The maindifferences are that in a spaser the optical cavity is re-placed by plasmonic resonances and the electromagneticfields are composed of propagating as well as evanescentwaves.When the system is not so close to its spaser condi-tions, its effect can be modeled phenomenologically onthe basis of classical electrodynamics without taking intoaccount explicitly the quantum dynamics of the groundand excited states. This is usually done by consider-ing the medium as a dielectric with an additional neg-ative imaginary part added to its refractive index n , n = n + iκ (with κ ≤ ). This approxima-tion may seem too simple for the proper description ofcomplicated systems such as active plasmonic structures.However, as soon as the dye is diluted enough in the basematerial that support the active media, the applicationof an effective medium theory to the system results ina real part of n almost identical to that of the base mate-rial Re( n ) and an imaginary part given by Im( n ) + κ ,or in other words n ≈ n + iκ . Note that, although thisapproximation seems reasonable under the appropriateconditions, it does not strictly fulfill the Kramers-Kronigrelation for the active media. Thus, our calculationsshould be taken in general as approximate results, usefulto guide future experiments and calculations on this is- sue. However, when the resonances of the active mediumand the spaser are close enough our results should beexact, assuming a model for the gain media as in ref. .In this work we are only taking into account the effectof the active media at precisely the frequency of exci-tation λ exc and neglecting the effect of the real part ofthe refractive index of the active media. This is equiv-alent to consider an active media with Lorentzian wave-length dependence at resonance with λ exc and with aFWHM (full width at half maximum) small enough thatthe spaser condition is not reached at a different λ . Thesituation is analogous to use κ ( λ ) = κδ ( λ exc − λ ) where λ exc is the wavelength of excitation and δ is the Kro-necker delta function. Note that this is the opposite ofthe usual approximation of considering κ in a wide bandlimit κ ( λ ) = κ . We will retake this issuelater on, but we will see that, thanks to the superposi-tion principle of linear electrodynamics, our results canbe readily reinterpreted within the context of the wideband approximation or even considering more compli-cated wavelength dependences of κ .There are of course some issues related to not tak-ing into account the dynamic of the population inversionof the active media . A consequence of that appearswhen the system approaches the spaser condition. Thisresults in singularities in the electromagnetic fields aswell as in the extinction, aborption and scattering crosssections. This fact is indeed used as a simple numer-ical way of finding the spaser condition. The behavior ofthe system very close to the spaser condition, i.e. trueintensity of the electromagnetic fields and cross sections,is beyond the scope of the present work.
B. Singularities in Mie’s theory
According to Mie theory, the electric field E s ( ~r ) outsidea core-shell nanoparticle (CSNP) illuminated by a planewave is given by : E s ( ~r ) = ∞ X n =1 E n [ ia n N (3) e n ( ~r ) − b n M (3) o n ( ~r )] , (1)where the coefficients a n and b n are obtained by conti-nuity conditions, the vectorial functions M o m and N e n are the well known vector spherical harmonics of order n ,and the the E n = i n E (2 n +1) /n ( n +1) functions are theprojections of the plane wave in the n -th harmonic. Thecoefficients a n and b n depend on the relative refractionindexes of the core, m , and the shell, m (equations 2),but also on the core radius ( r ), shell thickness ( D ), andon the wavenumber k . Close expressions for them canbe found in ref. . No surface damping corrections wereperformed because they are not significant for achievinga good spectral correlation with experiments as shownin ref. for example. The relative refractive indexes aregiven by: m ( λ ) = n core + iκ core n med m ( λ ) = n shell + iκ shell n med (2)They depend on the core ( n core , κ core ) the shell ( n shell and κ shell ), and the non absorbing media ( n media ) com-plex refractive indexes as well as the incident wavelength( λ ) As the refractive indexes are assumed to be only afunction of λ , they are unambiguously specified by the il-lumination wavelength λ . The active media is given by adoped silica. We will assume that its real part is the samethan in the pure bulk material, unchanged by the pres-ence of the dopant, and only its imaginary part ( κ core )change with the gain strength. Therefore, once we setwhich materials are the core and shell of the NE, thereare only four independent variables ( r , D , λ and κ core )to be evaluated, to systematically investigate gain effectsat different geometries and illumination conditions. Forsimplicity in the following we will refer to κ core as just κ .
1. Poles of Q sca and Γ( ~r ) For a given core shell nanostructure, its complex field(eq 1) at each point of the space ~r , and the correspond-ing scattering cross section (eq 3 below) are describedin terms of the same set of coefficients, a n and b n . Weare interested in the magnitude of the near electric fieldenhancement, | Γ( ~r ) | = | E s ( ~r ) /E | , but as will becomeclear soon there is a direct correspondence between C sca and | Γ | ( ~r ) , as both quantities depends on the the squaremodulus of a n and b n . C sca = 2 πk ∞ X n =1 (2 n + 1)( | a n | + | b n | ) (3)Even though we are not giving the expressions for ratio-nal functions a n and b n it is enough to mention that fora given set of values of the independent variables ( r , D , λ and κ ), there is a condition under which the expressionin the denominators of a n and b n vanishes totally (realand imaginary parts). This condition leads to a diver-gency, or a pole, of C sca as well as Γ( ~r ) . As we approacha divergency, one of the multipoles dominates the expan-sions in eqs. 3 and 1, therefore the electric field E s andthe scattering cross section ( C sca ) can be approximatedby: E s ( ~r ) ≃ i n +1 E a n n + 1 n ( n + 1) N (3) e n ( ~r ) , (4)and C sca = π ( r + D ) Q sca ≃ πk (2 n + 1) | a n | = λ πn med (2 n + 1) | a n | , (5)where Q sca is the scattering efficiency (the ratio between C sca and the geometric area of the nanostructure). Tak-ing the square modulus of eq 4, and considering eq 5,one can readily obtain an analytical expression for the squared field enhancement, Γ ( ~r ) , at each point outsidethe NE, | Γ( ~r ) | ≃ π ( r core + D ) λ n med (4 n + 2)( n ( n + 1)) × Q sca (cid:12)(cid:12)(cid:12) N (3) e n ( ~r ) (cid:12)(cid:12)(cid:12) (6)This result is general and valid for any system of concen-tric spheres in the surroundings of their poles or singular-ities. For nonspherical nanoparticle one can still expandthe fields in eigenmodes and obtain equations equivalentto eqs. 1, 3 and 6. In this case, eq. 6 will have a differentprefactors multiplying Q sca and different eigenfunctions.However, the connection among | Γ | and Q sca will stillbe valid close to a pole.Eq. 6 indicates that finding the poles of the scatteringefficiency is equivalent to search for the poles of the fieldenhancement. This has the advantage that one does notneed to care about the specific regions in space outsidethe CSNP where the enhancement occurs.The physical origin of these divergences are interpretedas the spaser conditions, which implies full loss compen-sation at a resonant frequency. In the next section wewill see that singularities are always located at the fullloss compensation condition.
2. Poles of Q ext and Q abs Now, let us consider the expressions for C ext and C abs : C ext = 2 πk ∞ X n =1 (2 n + 1) Re [ a n + b n ] (7) C abs = C ext − C sca = 2 πk ∞ X n =1 (2 n + 1)( Re [ a n + b n ] − | a n | − | b n | ) (8)As it has been shown for C sca , there are sets of valuesof r core , D , λ and κ that make zero or almost zero thedenominator of a particular term a n (or b n ) which thendominates the summation. Taking this approach we canwrite this dominant coefficient as the ratio of two com-plex functions f = f ′ + if ′′ and g = g ′ + ig ′′ , where wedenote the real and imaginary parts as f ′ , g ′ and f ′′ , g ′′ respectively, i.e. a n = f a n ÷ g a n (or b n = f b n ÷ g b n ).Then eqs. 7 and 8 can be rewritten as: C ext ≃ πk (2 n + 1) f ′ g ′ + f ′′ g ′′ | g | (9) C abs ≃ πk (2 n +1) (cid:18) f ′ g ′ + f ′′ g ′′ | g | − | a n | (cid:19) (10)Eq. 9 presents a divergence in the limit when g van-ishes and therefore the absolute value of C ext goes toinfinity, although its sign changes at this point. Fora fixed geometry at resonant wavelength, the changeof sign of C ext can be readily understood if we ex-pand the function g around κ pole , where g ( κ pole ) = 0 , g ( κ ) ≈ dgdκ (cid:12)(cid:12)(cid:12) κ pole ( κ − κ pole ) . Then, the change of sign of g , and thus of C ext , requires only a nonzero dg/dκ whichis our case. This feature is interesting because a changeof sign of C ext matches the full loss compensation condi-tion. This implies that, similarly to that found in ref. ,the conditions of spaser generation necessarily requiresfull loss compensation in our system, even beyond thequasi-static limit. C ext is the total power deflected from the incidentplane wave by scattering and absorption, thus its nega-tive value implies an overall energy release outside theparticle. We call this an optical amplifier due to itspotential applications for information transport at thenanoscale. The value of κ at which this occurs will becalled κ flc , where f lc stands for full loss compensation.As discussed above, poles should always fall over the κ flc curves, but due to the discreteness of the pole’s condi-tions in finite systems, full loss compensation not nec-essarily implies a pole. In the case of periodic infinitesystems, where discrete resonances turn into bands, itis still possible to have full loss compensation withoutreaching the spaser condition, outside the bands.Around a pole, the last term of eq 10 (which comesfrom C sca ) dominates. As a consequence, C abs ≈ − C sca .Then the singularity of | Γ | can be found either as apositive singularity of C sca or a negative singularity of C abs . The other alternative, using C ext , is also possiblebut cumbersome to apply in practice due to the changeof sign. III. RESULTS AND DISCUSSIONA. System studied
The system studied (shown in fig. 1) consists of acore and shell of radius r and thickness D respectively,where the core is made of silica and contains the ap-propriate dopant while the shell is made of gold. Thissystem was chosen mainly by its experimental feasibilityand the possibility to control precisely the core and shellgeometries .The real part of the refractive index of the silicacore, n , was taken from ref. . As in many previousworks, , its gain character is emulated by addinga negative imaginary part to n , n = n + iκ , where κ ≤ .The complex refractive index of gold is given by a cubicinterpolation of the experimental data of ref. The par-ticle was assumed to be immersed in an aqueous mediumwith a refractive index of 1.33. Q sca , Q ext and Q abs , werecalculated by using standard Mie theory. The values of κ that make Q ext = 0 , i.e full loss compensation condi-tion, will be denoted as κ flc , while the values of κ that FIG. 1. System studied. make Q abs = 0 , ohmic loss compensation condition, willbe denoted as κ olc . As mentioned , in our calculation weare only considering the effect of κ at the wavelength ofexcitation, which is equivalent to consider κ ( λ ) = 0 for λ s different from the wavelength of observation and/orexcitation. B. General behavior
In order to understand the problem of finding and de-scribing the conditions under which singularities are en-countered, we will start with a representative fixed ge-ometry. For no special reason, we selected r core = 75 nm and D = 5 nm for this purpose. For this geometry weperformed a systematic study of the variation of Q sca , Q ext , and Q abs in the space generated by the remainingvariables, λ and κ . The values of Q sca obtained as a func-tion of κ and λ are displayed, in logarithmic scale, in thethe 3D plot shown in fig. 2 a). There, the poles discussedin section II B can be clearly seen. For the sake of a morecomprehensive analysis, fig. 2 also depicts a contour plotof this 3D figure, in panel b). In this panel, the greenand orange continuous lines represent the values of κ inwhich Q ext and Q abs present a change on their signs, κ flc and κ olc respectively. Similarly to that found byStockman in ref. in the quasi-static limit, here we cansee that the spaser condition, given by the divergences of Q sca , implies always the full loss compensation condition.Note that the opposite is not necessarily true, due to thediscreteness of the resonances. Three poles or singular-ities are easily distinguished in this figure, they are lo-cated at: ( λ = 1085 . nm , κ = − . ) ; ( λ = 835 . nm , κ = − . ) and ( λ = 724 , nm , κ = − . ). Panelsc) and d) of figure 2 show Q ext and Q abs projected forthe same range of λ and κ . In these panels also the con-tinuous and dotted red lines correspond to κ flc and κ olc respectively.The orange and green lines define three distinctive re-gions corresponding to three different regimes of energylosses. For small κ , between the bottom axis and the or-ange line, the values of Q ext , Q abs are both positive indi-cating that the radiative and dissipative energy losses arenot compensated. For intermediate values of κ , betweenthe orange line and the green line, Q abs is negative while FIG. 2. Scattering cross sections of a gold-coated active NP of r = 75 nm and D = 5 nm . a) Q sca vs κ and λ . b) Contour plot of the left figure. c) contour plot of Q ext vs κ and λ . d) contour plot of Q abs vs κ and λ . The color scale islogarithmic except panels c) and d) between -1 and 1, where it is linear. Green continuous line marks κ flc , where “ flc ” standsfor full loss compensation condition. Orange line marks κ olc , where “ olc ” stands for Ohmic loss compensation condition. Graylines marks the contours of the positive and negative powers of ten starting from − . Q ext is still positive (figs 2 c) and d)). This implies thatthe gain media is able to compensate dissipative lossesbut not radiative ones. Under these conditions, moreradiation is going out of the system than the incidentone. However the system still looks under-compensated.The intensity of the forward radiation is lower than theincident one. For large values of κ , beyond the continu-ous green line, Q ext is negative indicating that the active medium is able to fully compensate both radiative anddissipative losses. There, the system is in principle ableto amplify the incoming radiation producing an outgoingwave of the same or more intensity than the incomingone. This region must be taken with caution because ifthis occurs at the wavelength of resonance ( λ = λ pole ),the system will start to increase its energy with time. Inthe present formalism, as we are not taking into account -200 -100 0 100 200 x ( nm ) -200-1000100200 z ( n m ) Γ ( x, , z ) ~p~k a -200 -100 0 100 200 x ( nm ) -200-1000100200 z ( n m ) Γ ( x, , z ) ~p~k b -200 -100 0 100 200 x ( nm ) -200-1000100200 z ( n m ) Γ ( x, , z ) ~p~k FIG. 3. Electric field enhancement in the xz plane, Γ ( x, , z ) , for a core-shell NP with r = 75 nm and D = 5 nm at threedifferent λ and κ (near the three poles encountered). Γ is shown in arbitrary scale. The orange and blue arrows representsthe directions of propagation ~k and polarization ~p respectively. a) λ = 1085 . nm and κ = − . . b) λ = 835 . nm and κ = − . . c) λ = 724 . nm and κ = − . the dynamic of the excited and ground states of the dyemolecules or the dopants, this will cause the divergenceof the electromagnetic fields that allows us to identify thespaser condition.Fig. 3 shows the near field at conditions very closeto the first three poles shown on fig 2. For each pole,the near field enhancement was calculated with the BH-FIELD program . We performed a exploration of the xz plane through the middle of the nanoparticle withpolarization and propagation along the x and z axis re-spectively within a square of side 400 nm with 2 nm widegrid. The first pole, which corresponds to the highest λ ,clearly can be assigned to a dipole mode while the sec-ond one corresponds to a quadrupole and the third to anoctupole.Finally we want to mention that, as fig. 2 shows, thepoles are always blue shifted with respect to the maxi-mum at κ = 0 . It is important to consider this pointwhen studying plasmonic active systems, as one couldmake the mistake of trying to find the lasing conditionat the wavelength of the maximum at κ = 0 . The blueshift of the poles as κ increases, is expected recalling thebehavior of damped harmonic oscillators. There, intro-ducing energy losses give rise to a red shift of the resonantfrequencies, along with a spectral broadening and a de-crease of the peak intensities. Consequently, increasingthe gain, which decreasing the damping, should producea blue shift of the peaks together with a narrowing oftheir width and an increment of their height. C. Effect of the wavelength dependence of the gain
Up to this point, our treatment has neglected the wave-length dependence of active media, which in our case wasequivalent to consider κ ( λ ) as a Kronecker delta function at the wavelength of excitation. This allowed us to keepour analysis independent of the particularities of the ac-tive medium and to focus only on its general effects. How-ever, the problem of adding a λ dependence of κ can bereadily treated. Essentially, due to the principle of super-position, the behavior of systems with frequency depen-dent gains can be obtained by the appropriate weightingof our results. Fig. 4 shows two examples of wavelengthdependences of κ . Left panels (a, c, and e) assume thedye is rodhamine B while right panels (b, d, and f) as-sume a flat dependence, or a wide band approximation.The values of Q ext and Q sca at each frequency were calcu-lated using the values of κ ( λ ) shown in the top panels (aand b). Two things are interesting to note in the figure.First, the wide band approximation should be used withcare when two poles are not close enough. As shown inthe example, blindly using this approximation can lead towrong estimations of the observed κ pole and λ pole . Sec-ond, the non flat dependence of κ on λ is what enablesthe system to go from a regular plasmonic structure toa spaser by passing first through an optical amplifier be-havior. A gradual increase of the gain strength without a λ dependence will preclude the observation of the opticalamplifier behavior. D. Effect of the geometry of the system
In this section we discuss the dependence of the posi-tions of the different poles with the geometric parame-ters that describe the morphology of the system: r and D . This analysis may result specially useful for exper-imentalists, as it gives the conditions that will produceactive-nanoplasmonics systems with lasing activity in adesired wavelength.In order to find the poles for a given geometry, we first − κ wavelength (nm) 580 600 620wavelength (nm)a ba b048 580 600 620 Q e x t wavelength (nm) 580 600 620wavelength (nm) d d −
580 600 620 Q s c a wavelength (nm) 580 600 620wavelength (nm)e fe f FIG. 4. Top panels (a and b) show the wavelength dependenceof κ for different strengths of the active medium, in increasingorder κ/κ pole = 0 . , 0.5, 0.7, 0.9, 0.98, and 0.99. Left panels(a, c, and e) used a realistic wavelength dependence assum-ing the dye is rhodamine B, whose emission spectrum wastaken form ref. . Right panels (b, d, and f) assume a wideband approximation. Full loss compensation, Q ext ( λ, κ ) = 0 ,is indicated by a green continuous line while the spaser con-ditions ( λ pole , κ pole ) are indicated by black circles. Middle (cand d) and bottom (e and f) panels show the extinction andscattering coefficients respectively calculated with the valuesof κ ( λ ) shown in the top panels. The direction of the arrowsindicate increasingly higher values of κ . We used r = 65 nmand D = 12 nm. made numerical calculations of Q sca as a function of λ and κ , similarly to section III B. All local maximum in Q sca were recorded. In order to distinguish true diver-gences from simple maxima, we performed a simplex optimization starting from each maximum, using Q − sca as the cost function and a quadratic interpolation of thevalue of the shell refractive index. We repeat this proce-dure varying systematically r and D . The parameter r was varied from 50nm to 150nm at steps of 5nm and theparameter D was changed from 5nm to 30nm at steps of1nm. The intervals in λ and | κ | taken were 350-1500nmand 0-3 respectivelyClose to a pole, two conditions should be fulfilled.First, the contribution of the dominant mode to Q sca should approach 1, and second the value of Q sca shouldgo to infinity. We considered that a set of values κ pole and λ pole corresponds to a true pole when Q sca > and the contribution of the dominant mode to Q sca washigher than . . Some representative examples of κ pole and λ pole versus D (for fixed r ) are shown in fig.5. The complete calculations are provided as supportinginformation (SI).We should mention, that the methodology describedabove should be equivalent to directly finding the zerosof the denominators of the a n coefficients as in ref. .However, this last method requires the knowledge of theanalytical expressions for a n . Our proposal, instead,could be readily adapted to almost any currently usedmethod for the numerical calculation of scattering crosssections. Therefore, it can be applied to NPs of arbitraryshape.Fig. 5 shows the typical behavior of varying D forsmall, medium and large cores ( r = 50 , , nm ). Inthe top panels (a, b, and c) it can be observed the lasingconditions in terms of λ and κ separately. The bottompanels (d, e, and f) summarize the information sown inthe above panels (a, b, and c) by showing the values D as a function of both κ and λ . The value of D is encodedas a color index and the mode that produce the lasingcan be distinguished as different dot types. The domi-nant mode indicated in the figures was labeled with thename of the coefficients of eq. 1. Note that several polescan correspond to the same a n coefficient. This occursjust because a pole is a zero in the denominator of thecoefficients of eq. 1 and this condition can in principle befulfilled for several pairs of values of λ and κ , dependingon the dielectric constant of the materials involved. Allthis poles will share the same electromagnetic field profileoutside the NP, as shown in eq. 6, but will differ in thedistribution of the electromagnetic field inside the NP.The inner fields are not relevant for sensing applicationsand generally not accessible experimentally. Therefore,for the purposes of the present work we will skip the dis-cussion regarding this issue.For small cores the systems behave as shown in theleft column of fig. 5. Typically two branches of the samemode and opposite λ (or κ ) dependence on D are ob-served. These branches are denoted as “first” and “sec-ond” in this figure. Note that there is a critical value forwhich the poles collapse into exceptional points. Thisphenomenon has been previously reported in other activeplasmonic systems. In the lower panel of the left columnof fig. 5, one can notice that, for both branches, | κ pole | decreases as λ pole increases. This behavior is in agree-ment with ref. and it is consistent with the dependenceof the intrinsic losses of the materials with λ , larger λ scorrespond to smaller losses. For small cores, this effectalways overcome the increase in the total amount of ab-sorbing material with D . This can be clearly seen in thecase of the second branches, where increasing D givesanyway a smaller | κ pole | (see panel d).For intermediate cores, central column of fig. 5, modesof higher order appear but they follow the same λ vs D (or κ vs D ) trend discussed above. The difference is λ ( n m ) (cid:28) rst s e o nd r = 50 nm r = 80 nm r = 140 nm (key)-2.5-1.5-0.5 10 20 30 κ D(nm) (cid:28) rst s e o nd r = 50 nm r = 80 nm r = 140 nm (key)-2.5-2-1.5-1-0.50 500 800 1100 κ λ ( nm ) (cid:28) rst s e o nd r = 50 nm r = 80 nm r = 140 nm (key)5008001100 third r = 50 nm r = 80 nm r = 140 nm (key)-2.5-1.5-0.5 10 20 30D(nm)third r = 50 nm r = 80 nm r = 140 nm (key)-2.5-2-1.5-1-0.50 500 800 1100 λ ( nm ) t h i r d r = 50 nm r = 80 nm r = 140 nm (key)5008001100 r = 50 nm r = 80 nm r = 140 nm (key)-2.5-1.5-0.5 10 20 30D(nm) r = 50 nm r = 80 nm r = 140 nm (key)-2.5-2-1.5-1-0.50 500 800 1100 λ ( nm ) modeD ( nm ) r = 50 nm r = 80 nm r = 140 nm (key)a b d e fa b d e fa b d e fa b d e fa b d e fa b d e fa b d e fa b d e f a a a a a a FIG. 5. Poles positions of gold-coated active NPs of three different radius (in columns). The first row shows the dependenceof λ coordinate of the pole with the geometrical parameter D . The second row shows the dependence of κ coordinate of thepole with D . The third row summarizes the two previous dependences on κ and λ in a single plot. All graphics share the samecolor key and the same symbols, which depend on D and the mode respectively. the appearance of a third pole corresponding also to a a mode, labeled in fig. 5 as “third”. As mentioned, thenumber of poles result from equating the denominatorof the a n coefficients to zero which can gives more thantwo solutions. The appearance of a third pole has beenrecently reported, even for small particles in the quasi-static limit . Probably only due to numbers, we did notfind this third solution for particles with small cores inthe range of parameters studied. Notice that this extramode does not follow the trend described for the firstand second branches, that is, increasing λ pole is not cor-related with smaller | κ pole | . Lets recall that poles labeledwith the same a n index should present the same electro-magnetic field profile outside the NP. Then, at a givenradius and shell thickness, the radiative losses must bethe same for all poles corresponding to the same order.Their different κ pole values must arise form differences inthe ohmic losses, which we are not able to explain sys-tematically for the third pole. Note that this pole notalways follows the same κ pole vs λ pole trend, see panels e and f. For larger cores (c and f panels) the main differ-ence is the appearance of additional higher order modesas expected. In general as r increases, and for the samemultipolar order, there is a shift of the poles to smaller D values. For a given multipole, this is accompanied bya red shift of λ pole with r . IV. CONCLUSIONS
We have thoroughly studied the wavelength and gaindependence of the response of core-shell nanospheresmade of silica and gold, and where the silica core acts asan optically active medium. The system studied is feasi-ble experimentally and has the advantage that hot spotsaround the plasmonic structure are in principle physicallyaccessible for sensing purposes. We have demonstratedanalytically that for CSNP with gain, the magnitude ofthe field enhancement is proportional to the scatteringcross section. We have used this result to find the spaserconditions directly from the poles of the scattering crosssections. As discussed in section II B, this method canalso be applied to NPs of other shapes. We have foundthat the spaser conditions always fall on the curves givenby κ flc vs λ , where κ flc stands for the value of κ at whichfull loss compensation condition is reached. The curvesof κ flc vs λ also determine the condition for optical am-plification which can be troublesome for this last appli-cation. However, due to the discreteness of the spaserconditions, it is possible to tune the system to act as anspaser or as an optical amplifier, provided the frequencyresponse of the active medium is narrow enough. Wereport the different spasing conditions for each multipo-lar mode, available for the set of geometrical parametersthat define the morphology of the system. Our system-atic study have covered a wide range of possible experi-mental conditions, which can result especially useful forexperimentalists working on similar systems. We believeour results will be useful for many applications, includ-ing optical amplification, but especially for sensing as it is known that the near fields produced by spasers arehuge, even higher than those of normal metallic NPs. V. ACKNOWLEDGEMENTS
The authors acknowledge the financial support fromCONICET, SeCyT-UNC, ANPCyT, and MinCyT-Cordoba.
VI. ASSOCIATED CONTENT
Supporting Information Available: Full list of val-ues of the core radius Rcore, shell thickness D, Polewavelength λ and the imaginary part of the active me-dia refractive index κ and the nature of the mode a n that give rise to the spaser conditions of spher-ical nanoparticles made of an active silica core anda gold shell. Supporting Information available at:http://pubs.acs.org/doi/abs/10.1021/acs.jpcc.6b05240. ∗ [email protected] A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin,Phys. Rep. , 131 (2005). E. J. Blackie, E. C. Le Ru, and P. G.Etchegoin, J. Am. Chem. Soc. , 14466 (2009),http://dx.doi.org/10.1021/ja905319w. N. L. Gruenke, M. F. Cardinal, M. O. McAnally,R. R. Frontiera, G. C. Schatz, and R. P. Van Duyne,Chem. Soc. Rev. , 2263 (2016). K. A. Stoerzinger, J. Y. Lin, and T. W. Odom,Chem. Sci. , 1435 (2011). E. C. Le Ru, E. Blackie, M. Meyer, and P. G. Etchegoin,J. Phys. Chem. C , 13794 (2007). R. Beams, L. G. Cançado, A. Jorio, A. N. Vamivakas, andL. Novotny, Nanotechnology , 175702 (2015). C. Chen, N. Hayazawa, and S. Kawata,Nat. Commun. , 1 (2014). M. D. Sonntag, E. A. Pozzi, N. Jiang, M. C. Hersam, andR. P. V. Duyne, J. Phys. Chem. Lett. , 3125 (2014). E. A. Pozzi, A. B. Zrimsek, C. M. Lethiec,G. C. Schatz, M. C. Hersam, and R. P. V.Duyne, J. Phys. Chem. C , 21116 (2015),http://dx.doi.org/10.1021/acs.jpcc.5b08054. S. Nie, Science , 1102 (1997). L. Wu and B. M. Reinhard,Chem. Soc. Rev. , 3884 (2014). H. Cang, A. Salandrino, Y. Wang, and X. Zhang,Nat. Commun. , 1 (2015). I. I. Smolyaninov, HFSP J , 129 (2008). N. Esteves-López, H. M. Pastawski, and R. A. Bustos-Marún, J. Phys.: Condens. Matter , 125301 (2015),arXiv:arXiv:1502.01196v1. K. B. Eisenthal, Chem. Rev. , 1462 (2006). W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Kr-ishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck,Nano Lett. , 1027 (2006). S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W.Kim, Nature , 757 (2008). M. Kim and J. Rho, Nano Convergence , 22 (2015). A. Boltasseva and H. a. Atwater, Science , 290 (2011). N. Arnold, B. Ding, C. Hrelescu, and T. a. Klar,Beilstein J. Nanotechnol. , 974 (2013). Opt. Express , 2622 (2007), 0612192 [physics]. D. J. Bergman and M. I. Stockman,Phys. Rev. Lett. , 027402 (2003). S.-Y. Liu, J. Li, F. Zhou, L. Gan, and Z.-Y. Li,Opt. Lett. , 1296 (2011). H. Yu, S. Jiang, and D. Wu,J. Appl. Phys. , 153101 (2015). I. L. Rasskazov, S. V. Karpov, and V. a. Markel,Opt. Lett. , 4743 (2013). D. S. Citrin, Opt. Lett. , 98 (2006). B. S. Nugroho, V. A. Malyshev, and J. Knoester,Phys. Rev. B , 1 (2015), arXiv:1508.04197. a. Fang, T. Koschny, M. Wegener, and C. M. Souk-oulis, Phys. Rev. B: Condens. Matter , 8 (2009),arXiv:0907.0888. J. M. Hamm, S. Wuestner, K. L. Tsakmakidis,and O. Hess, Phys. Rev. Lett. , 1 (2011),arXiv:arXiv:1109.4411v1. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm,and O. Hess, Phil. Trans. R. Soc. A , 3525 (2011),arXiv:1012.1576. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M.Hamm, and O. Hess, Phys. Rev. Lett. , 1 (2010),arXiv:1006.5926. a. De Luca, R. Dhama, a. R. Rashed, C. Coutant,S. Ravaine, P. Barois, M. Infusino, and G. Strangi,Appl. Phys. Lett (2014), 10.1063/1.4868105. A. De Luca, M. Ferrie, S. Ravaine, M. La Deda, M. In-fusino, A. R. Rashed, A. Veltri, A. Aradian, N. Scara-muzza, and G. Strangi, J. Mater. Chem. , 8846 (2012). M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M.Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Sutee-wong, and U. Wiesner, Nature , 1110 (2009). M. Infusino, A. De Luca, A. Veltri, C. Vázquez-Vázquez,M. A. Correa-Duarte, R. Dhama, and G. Strangi,ACS Photonics , 371 (2014). M. a. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. Small,B. a. Ritzo, V. P. Drachev, and V. M. Shalaev,Appl. Phys. B: Lasers Opt. , 455 (2007). W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T.Co, M. R. Wasielewski, G. C. Schatz, andT. W. Odom, Nat. Nanotechnol. , 506 (2013),arXiv:NNANO.2013.99 [10.1038]. M. I. Stockman, J. Opt. , 1 (2009). R. A. Bustos-Marún, A. D. Dente, E. A. Coronado, andH. M. Pastawski, Plasmonics , 925 (2014). N. M. Lawandy, Appl. Phys. Lett. , 5040 (2004). Z.-Y. Li and Y. Xia, Nano Lett. , 243 (2010). N. Arnold, C. Hrelescu, and T. Klar, Ann. Phys. (Berlin) , 295–306 (2016). S. L. Westcott, S. J. Oldenburg, T. R. Lee, and N. J.Halas, Langmuir , 5396 (1998). N. Phonthammachai, J. C. Y. Kah, G. Jun, C. J. R. Shep-pard, M. C. Olivo, S. G. Mhaisalkar, and T. J. White,Langmuir , 5109 (2008). C. F. Bohren and D. Huffman,
Absorption and scattering of light by small particles ,Wiley science paperback series (Wiley, 1983). S. Maier,
Plasmonics: Fundamentals and Applications (Springer US, 2007). N. Calander, D. Jin, and E. M. Goldys,J. Phys. Chem. C , 7546 (2012). Y. Huang, J. J. Xiao, and L. Gao,Opt. Express , 8818 (2015). D. G. Baranov, E. S. Andrianov, A. P. Vinogradov, andA. A. Lisyansky, Opt. Express , 10779–10791 (2013). F. Hao and P. Nordlander,Chem. Phys. Lett. , 115 (2007). M. I. Stockman, Phys. Rev. Lett. , 156802 (2011),arXiv:1011.3751. S. Oldenburg, R. Averitt, S. Westcott, and N. Halas,Chem. Phys. Lett. , 243 (1998). J. Razink and N. Schlotter,J. Non-Cryst. Solids , 2932 (2007). N. R. Jana, L. Gearheart, and C. J. Murphy,Langmuir , 6782 (2001). I. H. Malitson, J. Opt. Soc. Am. , 1205 (1965). P. B. Johnson and R. W. Christy,Phys. Rev. B , 4370 (1972). H. Suzuki and I.-y. S. Lee, Int. J. Phys. Sci. , 038 (2008). X. Gao, J. He, L. Deng, , and H. Cao, Opt. Mater. ,1715 (2009). W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery,
Numerical Recipes 3rd Edition: The Art of Sci-entific Computing , 3rd ed. (Cambridge University Press,2007). I. Rotter, J. Phys. A: Math. Theor.42