Mode volume, energy transfer, and spaser threshold in plasmonic systems with gain
Tigran V. Shahbazyan




Department of Physics, Jackson State University, Jackson, MS
39217 USA



 
We present a unified approach to describe spasing in  plasmonic systems modeled by quantum emitters interacting with resonant plasmon mode. We show that spaser threshold implies detailed energy transfer balance between the gain and  plasmon mode and derive explicit spaser condition valid for arbitrary plasmonic systems. By defining carefully the plasmon mode volume relative to the gain region, we show that the spaser condition represents, in fact, the standard laser threshold condition extended to plasmonic systems with dispersive dielectric function. For extended gain region, the saturated mode volume depends solely on the system parameters that determine the lower bound of threshold population inversion.


Mode volume, energy transfer,  and spaser threshold in plasmonic systems with gain
    Tigran V. Shahbazyan
    January 27, 2017
==================================================================================









§ INTRODUCTION


The prediction of plasmonic laser (spaser) <cit.> and its  experimental realization in various systems <cit.> have been among the highlights in the rapidly developing field of plasmonics during the past decade <cit.>. First observed in gold nanoparticles (NP) coated by dye-doped silics shells <cit.>, spasing action was reported in hybrid plasmonic waveguides <cit.>, semiconductor quantum dots on metal film <cit.>,  plasmonic nanocavities and nanocavity arrays <cit.>,  and metallic NP and nanorods <cit.>, and more recently, carbon-based structures <cit.>. Small  spaser size well below the diffraction limit gives rise to wealth of applications  <cit.>. 


The spaser feedback mechanism is based on the near-field coupling between  resonant plasmon mode and gain medium, modeled here by an ensemble of  pumped two-level quantum emitters (QE) with  excitation frequency  tuned to the plasmon   frequency. The spaser threshold condition has been suggested as  <cit.> 


    4πμ^2τ_2/3ħN_21/ V  Q  ≃ 1,


where μ and τ_2 are the QE dipole matrix element and polarization relaxation time,  respectively, N_21=N_2-N_1 is the ensemble population inversion (N_2 and N_1 are, respectively, the number of excited and ground-state QEs), Q is the  mode quality factor, and V is the mode volume.  Equation (<ref>) is similar to the standard laser condition <cit.> that determines the threshold value of N_21, but with  the cavity mode  quality factor and volume replaced by their plasmon counterparts in metal-dielectric system characterized by dispersive dielectric function ε(ω,r).  While the plasmon quality factor Q is  well-defined   in terms of  the metal dielectric function ε(ω)=ε'(ω)+iε”(ω), there is an active debate on mode volume definition  in plasmonic systems <cit.>.  Since  QEs are usually distributed outside the plasmonic structure, the standard expression for cavity mode volume, ∫ dV ε(r) |E(r)|^2 /max[ε(r) |E(r)|^2], where E(r) is the mode electric field,  is ill-defined for open systems <cit.>. Furthermore, defining the  plasmon mode volume in terms of   field intensity at a specific point <cit.> seems impractical due to very large local field variations near the metal surface caused by particulars of system geometry, for example, sharp edges or surface imperfections: strong  field fluctuations would grossly underestimate the mode volume that determines  spasing threshold for gain distributed in an extended region. At the same time, while spasing was theoretically studied for several  specific systems <cit.>, the general spaser condition was derived, in terms of  system parameters such as   permittivities and   optical constants, only for two-component systems <cit.> without apparent relation to the  mode volume in Eq. (<ref>).  Note that the actual spasing systems can be comprised of many components, so that the extension of the laser condition (<ref>) to plasmonics implies some procedure, valid for any nanoplasmonic system, to determine the plasmon mode volume.


On the other hand, the steady state spaser action implies detailed balance of  energy transfer (ET) processes between the QEs and the plasmon mode (see Fig. <ref>). Whereas the energy flow between  individual QEs and plasmon  can go in either direction depending on the QE quantum state,  the net gain-plasmon ET rate is determined by  population inversion N_21 and, importantly,   distribution of plasmon states in the gain region. Since  individual QE-plasmon ET rates are proportional to the plasmon local density of states (LDOS), which can vary in a wide range depending on  QEs' positions and system geometry <cit.>, the net   ET rate  is obtained by    averaging the plasmon LDOS over the gain region. Therefore, the plasmon mode volume should relate, in terms of average  system characteristics, the  laser condition (<ref>) to the microscopic gain-plasmon ET picture. The goal of this paper is to establish such a relation.







First, we derive the general spaser condition for any multicomponent nanoplasmonic system  in terms of individual ET rates  between QEs, constituting the gain, and resonant plasmon mode,  providing the feedback. Second, we introduce the plasmon mode volume 𝒱 associated with a region of volume V_0 by relating 𝒱 to the plasmon LDOS, averaged over that region, and establish that the spaser condition does have the general  form (<ref>).  We then demonstrate, analytically and numerically, that a sufficiently extended region outside the plasmonic structure  can saturate the plasmon mode volume, in which case 𝒱 is independent of the plasmon field distribution  and  determined solely by the system parameters,


    V/V_0 
     =Q ε_d ε”(ω_pl) /|ε' (ω_pl)|,


where ε_d is the gain region dielectric constant and  ω_pl is the plasmon frequency. With saturated mode volume (<ref>), the laser condition (<ref>) matches the spaser condition for two-component systems <cit.> and, in fact, defines the lower bound of threshold N_21. Finally, we demonstrate that, in realistic systems, the threshold  N_21 can  significantly exceed its minimal value.





§ SPASING AND GAIN-PLASMON ENERGY TRANSFER BALANCE 




We consider N_0 QEs  described by pumped two-level systems, located at positions r_j near a plasmonic structure, with excitation energy ħω_21=E_2-E_1, where E_1 and E_2 are, respectively, the lower and upper level energies. Within the  density matrix approach, each QE is described by  polarization ρ_21^(j) and occupation n_21^(j)≡ρ_22^(j)-ρ_11^(j), so that  N_21= N_2-N_1=∑_jn_21^(j) is the ensemble population inversion. In the rotating wave approximation, the steady-state dynamics of QEs coupled to  alternating electric field ℰ(r)e^-iω t is described by the Maxwell-Bloch equations 


    (ω - ω_21 +i/τ_2) ρ_21^(j) =μ/ħ  n_21^(j) n_j· ℰ(r_j),
    
        n_21^(j) -n̅_21 =-  4μτ_1/ħ Im[ρ_21^(j)* n_j· ℰ(r_j)],


where τ_2 and  τ_1 are the time constants characterizing polarization and population relaxation, μ and  n_j are, respectively, the QE dipole matrix element and orientation, and n̅_21 is the average population inversion per QE due to the pump.  The  local field ℰ(r_j) at the  QE position is generated by all QEs' dipole moments p_j=μn_jρ_21^(j) and, within semiclassical approach, has the form <cit.> 



    ℰ(r_j ) =4πω^2/c^2∑_kG̅(ω;r_j,r_k) ·p_k,


where G̅(ω;r,r') is the electromagnetic Green dyadic in the presence of metal nanostructure and c is the speed of light. For nanoplasmonic systems, it is convenient to adopt  rescaled Green dyadic that has  direct near-field limit, D̅(ω;r,r')=-(4πω^2/c^2)G̅(ω;r,r'). Upon eliminating the electric field, the system (<ref>) takes the form


    Ω_21p_j
    +μ^2/ħ  n_21^(j)n_j∑_kn_j·D̅(ω;r_j,r_k)·p_k
    =0,
    
       δ n_21^j/τ_1 -4/ħ Im∑_k[p_j^*·D̅(ω;r_j,r_k)·p_k]
    = 0,


where we use shorthand notations Ω_21=ω-ω_21+i/τ_2  and δ n_21^j=n_21^(j)-n̅_21. The first equation in  system (<ref>), being homogeneous in p_j, leads to the spaser threshold condition. Since the Green dyadic includes contributions from all electromagnetic modes, the spaser threshold in general case can only be determined numerically.  However, for QEs coupled to a resonant plasmon mode, that is, for ω_21 close to the mode frequency ω_pl, the contribution from  off-resonance modes is relatively small <cit.> and, as we show below, the spaser condition can be obtained explicitly for any nanoplasmonic system.



 §.§ Gain coupling to a resonant plasmon mode



For QE frequencies ω_21   close to the  plasmon   frequency ω_pl,  we can adopt the  single mode  approximation for the Green dyadic  <cit.>


    D̅(ω;r,r') = ω_pl/4 UE(r)⊗E^*(r')/ω-ω_pl+i/τ_pl,


where E(r) is the slow envelope of plasmon field satisfying the Gauss  law ∇· [ε' (ω_pl,r) E(r) ]=0  and 1/τ_pl is the plasmon   decay rate. In  nanoplasmonic systems, the  decay rate is dominated by the Ohmic losses  and  has the form


    1τ_pl=W2U,


where 


    U 
    = 1/16π∫  dV    |E (r) |^2∂ [ω_plε'(ω_pl,r) ]/∂ω_pl


is the  mode stored energy, and 


    W=ω_pl/8π∫ dV   |E(r) |^2ε”(ω_pl,r)


is the mode dissipated power <cit.>. The volume integration in U and W takes  place, in fact, only over the metallic regions with dispersive dielectric function.  For systems with a single metallic region, one obtains the standard plasmon decay rate: 



    1/τ_pl=ε”(ω_pl)/∂ε'(ω_pl)/∂ω_pl.


The Green dyadic (<ref>)  is valid for a  well-defined plasmon mode (ω_plτ_pl≫ 1) in any nanoplasmonic system, and its consistency is ensured by the optical theorem <cit.>.


With  the plasmon Green dyadic (<ref>),  the system (<ref>) takes the form


    Ω_21p_j + μ^2/ħω_pln_21^(j)/4 UΩ_pl n_j[n_j·E(r_j)]∑_kE^*(r_k)·p_k
    =0,
    
       δ n_21^j/τ_1 
    - Im [ω_pl/ħ UΩ_pl  [p_j^*·E(r_j)]
    ∑_kE^*(r_k)·p_k ]
    =0,


where  Ω_pl=ω-ω_pl+i/τ_pl. Multiplying the first equation by E^*(r_j) and summing up over j, we obtain the  spaser condition


    Ω_21Ω_pl+ μ^2/ħω_pl/4 U∑_jn_21^(j) |n_j·E(r_j) |^2=0.


The second term in Eq. (<ref>) describes coherent coupling between the QE ensemble and plasmon mode. Below we show that spasing implies  detailed ET balance between the gain and plasmon mode.





 §.§ Energy transfer and spaser condition


Let us now introduce, in the standard manner, the individual QE-plasmon ET rate as <cit.>





    1/τ 
    =-μ^2/ħIm [
    n·D̅(ω_pl;r,r)·n ]
    
    
    =4πμ^2/ħ|n·E(r)|^2/∫dV ε”|E|^2,


where we used Eqs. (<ref>) and  (<ref>), and implied ε≡ε(ω_pl,r) under the integral. The condition (<ref>) can  be recast as


    (ω-ω_21 +i/τ_2 )  (ω-ω_pl+i/τ_pl )+1/τ_ gτ_pl =0,


where we introduced  net gain-plasmon ET rate, 


    1/τ_g
    =∑_jn_21^(j)/τ_j
    = 4πμ^2/ħ∑_j  n_21^(j) |n_j·E(r_j)|^2/∫dV ε”|E|^2,


which represents the sum of individual QE-plasmon ET rates 1/τ_j, given by Eq. (<ref>), weighed by  QE occupation numbers. Since n_21^(j) is positive or negative for   QE  in the  excited or ground state, respectively, the direction of energy flow between the QE and the plasmon mode depends on  the QE  quantum state. Note that the main contribution to 1/τ_g comes from  the regions with  large  plasmon LDOS, that is,  high  QE-plasmon ET rates (<ref>). The imaginary part of Eq. (<ref>)  yields the spaser frequency <cit.>


    ω_s=ω_plτ_pl+ω_21τ_2/τ_pl+τ_2,


while its real part, with the above ω_s, leads to  


    1/τ_gτ_pl=1/τ_2τ_pl +  (ω_pl-ω_21)^2τ_2τ_pl/(τ_pl+τ_2)^2.


In  the case when the QE  and plasmon spectral bands overlap well,  that is.,  |ω_pl-ω_21|τ_pl≪ 1 or |ω_pl-ω_21|τ_2≪ 1 depending on relative magnitude of the respective bandwidths 1/τ_2 and 1/τ_pl, the last term in Eq. (<ref>) can be disregarded,  and we arrive at the spaser condition in the form   1/τ_g=1/τ_2, or


    ∑_jn_21^(j)/τ_j
    =1/τ_2.


Equation (<ref>) implies that   spaser threshold is reached when   energy transfer balance between gain and  plasmon mode is established. 




 §.§ System geometry and QE-plasmon ET rate 









Individual ET rates in the spaser condition (<ref>) can vary in a wide range depending on the QE position and system geometry. In Fig. <ref>, we show the ET rate (<ref>) for a QE located at  distance d from  a tip of   gold nanorod, modeled here by  prolate spheroid with semiaxes a and b  (see Appendix). In all numerical calculations, we use the experimental dielectric function for gold <cit.>. To highlight the role of system geometry, the ET rate 1/τ for nanorod is normalized by the ET rate 1/τ_sp for  sphere of radius a. The latter ET rate has the form  


    1/τ_sp=12μ^2/ħε”(ω_sp)a^3/(a+d)^6,


and experiences a sharp decrease for d>a. With changing nanoparticle shape,  three degenerate dipole modes of a sphere split into longitudinal and two transverse modes. The latter move up in energy to get damped by  interband transitions in gold with their onset just above the plasmon energy in spherical particles, while the longitudinal mode moves down in energy away from the transitions onset, thereby gaining in the oscillator strength <cit.>. This sharpening of plasmon resonance together with condensation of plasmon states near the tips (lightning rod effect)   results in up to 100-fold rate increase with reducing b/a ratio, as shown in Fig. <ref>, indicating that spasing is dominated by QEs  located in the large plasmon LDOS regions. Large variations of 1/τ magnitude imply that the plasmon mode volume, which characterizes spatial extent of the gain region with sufficiently strong QE-plasmon coupling, is determined by the average plasmon LDOS  in that region, as we show in the next section.




§ PLASMON MODE VOLUME AND SPASER THRESHOLD





 §.§ General spaser condition


The form (<ref>) of spaser condition reveals the microscopic origin of spaser action as the result of cooperative ET between  gain  and resonant plasmon mode, with each QE contribution depending on its position and quantum state. Below we assume that QEs  are distributed within some region of volume V_0 and that  population inversion distribution follows, on average, that of QEs. After averaging   over   QEs' dipole orientations, the gain-plasmon  ET rate (<ref>) takes the form


    1/τ_g= 4πμ^2/3ħ∫ dV_0 n_21 (r)|E(r) |^2/∫ dV ε”(ω_pl,r)| E(r)|^2,


where n_21 (r) is  population inversion density, yielding the spaser threshold condition 


    4πμ^2τ_2/3ħ∫ dV_0 n_21 (r)|E(r) |^2/∫ dV ε”(ω_pl,r)| E(r)|^2 =1,


which is valid for any multicomponent system supporting a well-defined surface plasmon. In the case of uniform gain distribution,  n_21=N_21/V_0, and a single metallic component with volume V_m [e.g., a metal particle with dye-doped dielectric shell (see Fig. <ref>)], the threshold condition (<ref>) takes the form 


    4πμ^2τ_2/3ħ n_21/ε”(ω_pl) ∫ dV_0  |E|^2/∫ dV_m |E|^2 =1.


The threshold  value of n_21  is determined by   ratio of plasmon field integral intensities in the gain and metal regions. Evidently, the spaser threshold does depend on the gain region size and shape, which prompts us to revisit the mode volume definition for plasmonic systems in order to ensure its consistency with the general laser condition (<ref>).



 §.§ Plasmon LDOS   and associated mode volume


Here, we show that the mode volume in plasmonic systems  can be accurately defined in terms of  plasmon LDOS.  The LDOS of a single plasmon mode is related  to the  plasmon Green dyadic (<ref>) as ρ (ω,r)
=-(2π^2ω_pl)^-1 Im Tr D̅(ω;r,r), and  has the Lorentzian form <cit.>,


    ρ (ω,r)
    
    =τ_pl/8π^2  U |E(r) |^2/(ω-ω_pl)^2τ_pl^2+1,


where U is given by Eq. (<ref>).  The plasmon LDOS (<ref>) characterizes the distribution of plasmon states in the unit volume and frequency interval. Consequently, its frequency integral, ρ (r)=∫ dωρ (ω,r), represents the  plasmon mode density that describes the plasmon states' spatial distribution:


    ρ  (r)= |E (r) |^2/8π U 
    = 2 |E (r) |^2/∫ dV [∂ (ω_plε')/∂ω_pl]|E|^2.


Introducing the mode quality factor Q=ω_pl U/W, the mode density can be written as 


    ρ  (r) 
    = 1/Q |E (r) |^2/∫ dV  ε” |E|^2.


Note that, in terms of ρ  (r), the gain-plasmon ET rate (<ref>) takes the form


    1/τ_g= 4πμ^2/3ħ  Q   
    ∫ dV_0 n_21 (r) ρ(r),


implying that the largest contribution to 1/τ_g comes from QEs located in the regions with high plasmon density.

We now relate the plasmon mode volume V associated with region V_0 to the average  mode density in that region:


    1/𝒱
    = 1/V_0∫ dV_0ρ(r) 
    = 1/V_02∫ dV_0 |E |^2/∫ dV [∂ ( ω_plε')/∂ω_pl]|E|^2,


or, equivalently,


    𝒱/V_0 
     =Q  ∫ dV  ε” |E|^2/∫ dV_0 |E|^2.


The expressions (<ref>) or (<ref>) are valid for nanoplasmonic systems of any size and shape and with any number of metallic and dielectric components.

It is straightforward to check that, for uniform gain distribution with n_21=N_21/V_0, the spaser threshold condition (<ref>) coincides with the laser condition (<ref>) with associated mode volume 𝒱 given by Eq. (<ref>). Equivalently, for uniform gain distribution, the gain-plasmon ET rate (<ref>) takes the form


    1/τ_g= 4πμ^2/3ħ N_21/𝒱  Q,


and the laser condition (<ref>) follows from the ET balance condition 1/τ_g=1/τ_2. 

For systems with  single metal  component, the plasmon mode volume takes the form [compare to Eq. (<ref>)]


    𝒱/V_0 
    =ω_pl/2∂ε'(ω_pl)/∂ω_pl∫ dV_m  |E|^2/∫ dV_0 |E|^2
     =Q  ε”(ω_pl) ∫ dV_m  |E|^2/∫ dV_0 |E|^2,


where the plasmon quality factor has the form 



    Q  = ω_pl ∂ε'(ω_pl) /∂ω_pl/2ε” (ω_pl) =ω_plτ_pl/2.


Note that, for a well-defined plasmon  with Q≫ 1, the plasmon mode volume is independent  of Ohmic losses in metal.



 §.§ Mode volume saturation and lower bound of  spaser threshold


Since the QE-plasmon ET rate rapidly falls outside the plasmonic structure (see Fig. <ref>), spasing is dominated by QEs located sufficiently close to the metal surface. In the case when a metal nanostructure of volume V_m is surrounded by an extended gain region V_0, so that the plasmon LDOS spillover beyond V_0 is negligible,  the  plasmon mode volume is saturated by the gain, leading to constant value of V/V_0 that is independent of the plasmon field distribution. To demonstrate this point, we note  that, in the quasistatic approximation,  the integrals in Eq. (<ref>)  reduce  to surface terms, 


    ∫ dV_0  |E|^2= ∫ dS Φ ^*∇_nΦ  + ∫ dS_1Φ^*∇_nΦ ,
    
       ∫ dV_m  |E |^2=∫ dS Φ ^*∇_nΦ ,


where S is the common interface separating the metal and dielectric regions, S_1 is the outer  boundary of the dielectric region, Φ is the potential related to the plasmon field as   E=-∇Φ, and ∇_nΦ is its normal derivative relative to the interface.  The potentials in the first and second equations of system (<ref>) are taken, respectively, at the dielectric and metal sides of the interface S. Since the plasmon fields rapidly fall  away from the metal,  the  contribution from the outer interface S_1 can be neglected for extended dielectric regions (see below). Then, using the standard boundary conditions at the common  interface S, we obtain from Eqs. (<ref>)  and (<ref>)   the saturated mode volume:


    𝒱/V_0 
    =ω_plε_d/2|ε'(ω_pl)| ∂ε'(ω_pl) /∂ω_pl 
     =Q ε_d ε”(ω_pl) /|ε' (ω_pl)|.


Remarkably, the saturated mode volume depends on system geometry  only via the plasmon frequency ω_pl in the metal dielectric function. Combining Eqs. (<ref>) and (<ref>), we arrive at the spaser condition for saturated case,


    4πμ^2τ_2/3ħε_d|ε'(ω_pl)|/ε”(ω_pl)   n_21 =1,


which matches the spaser condition obtained previously for two-component systems, that is, with gain region extended to infinity <cit.>. We stress  that the condition (<ref>) provides the lower bound for  threshold value of n_21, while in real systems, where plasmon field distribution can extend beyond the gain region, the threshold   can be significantly higher, as we illustrate below.






 
In Fig. (<ref>), we show the change of threshold n_21 with expanding gain region in nanorod-based spaser modeled by composite spheroidal particle with gold core and QE-doped silica shell.  Calculations were performed using  Eq. (<ref>) for confocal spheroids (see Appendix for details), and  QE   frequency ω_21 was tuned to  resonance with  longitudinal dipole  mode  frequency ω_pl. Note that the gain optical constants enter the spaser condition (<ref>) through a single parameter


    n_0=  3ħ/4πμ^2τ_2,


which represents characteristic gain concentration and sets the overall scale of threshold n_21 for a  specific gain medium. The ratio n_21/n_0, plotted in Fig. <ref>,  depends only on  plasmonic system parameters  and, with expanding gain region,  decreases prior reaching plateau corresponding to the saturated mode volume regime described by Eq. (<ref>).  Note that, in nanorods, the rapid mode volume saturation seen in Fig. <ref>, as compared to spherical particles, is caused by   condensation of plasmon states near the tips (lightning rod effect), leading to the much larger plasmon LDOS  and, correspondingly, the QE-plasmon ET rate (see Fig. <ref>).





§ CONCLUSIONS


In summary, we have  developed a unified approach to spasing in a system of pumped quantum emitters interacting with a plasmonic structure of arbitrary shape in terms of energy transfer processes within the system. The threshold value of population inversion is determined from the condition of detailed energy transfer balance between quantum emitters, constituting the gain, and resonant plasmon mode, providing the feedback. We have shown that, in plasmonic systems, the mode volume should be defined relative to a finite region, rather than  to a point of maximal field intensity, by averaging the plasmon local density of states over that region. We demonstrated that, in terms of  plasmon mode volume, the spaser condition has the standard form of the laser threshold condition, thus,  extending the latter to plasmonic systems with dispersive dielectric function. We have also shown that, for extended gain region, the saturated plasmon mode volume is determined solely by the system permittivities, which define the lower bound of threshold  population inversion.


This work was supported in part by NSF grants No. DMR-1610427 and No. HRD-1547754.






§ CALCULATION OF QE-PLASMON ET RATE FOR SPHEROIDAL NANOPARTICLE





The ET rate between a plasmon mode in metal nanoparticle with frequency ω_pl and a QE  located at the point r distanced by d from the metal surface and  polarized along the normal n to the  surface is given by


    1/τ
    =
    4πμ^2/ħ|n·E(r )|^2/∫dV ε”|E|^2
    =
    4πμ^2/ħε”(ω_pl)|∇_nΦ(r )|^2/∫dS  Φ^*∇_nΦ,


where ∇_n=n·∇ stands for the normal derivative, and real part of the denominator is implied.


Consider  a QE  at distance d from the tip of a spheroidal particle with semiaxis a along the symmetry axis and semiaxis  b in the symmetry plane (a>b). The potentials have the form Φ∝ R_lm(ξ)Y_lm(η,ζ), where ξ is the radial (normal) coordinate and the pair (η,ζ) parametrizes the surface (Y_lm are spherical harmonics). The surface area element is dS=h_ηh_ζ dη dζ, and normal derivative is ∇_n=h_ξ^-1(∂/∂ξ), where h_i are the scale factors (i=ξ,η,ζ)  given by 


    h_ξ=f√(ξ^2-η^2/ξ^2-1),
      
    h_η=f√(ξ^2-η^2/1-η^2),
    
       h_ζ=f√((ξ^2-1)(1-η^2)),


f =√(a^2-b ^2) is half distance between the foci, and spheroid surface corresponds to ξ_1=a/f. 

For QE located at point z=fξ on the z-axis (η=1) outside the spheroid, the radial potentials for  dipole longitudinal plasmon mode (lm)=(10) have the form  R(ξ)=P_1(ξ) for ξ<ξ_1 and R(ξ)=Q_1(ξ)P_1(ξ_1)/Q_1(ξ_1) for ξ>ξ_1,  where P_l and Q_l are the  Legendre functions of first and second kind, given by 


    P_1(ξ)=ξ,
      
    Q_1(ξ)=ξ/2ln (ξ+1/ξ-1 )-1,
    
       Q'_1(ξ)=1/2ln (ξ+1/ξ-1 )-ξ/ξ^2-1.


Using h_ξ=f  along the z-axis, the ET rate equals


    1/τ=
    3μ^2/ħε”(ω_pl)R^' 2(ξ)/f^3ξ_1(ξ_1^2-1)
     = 
    3μ^2/ħ ab^2ε”(ω_pl) [Q'_1(ξ)ξ_1/Q_1(ξ_1) ]^2,


with ξ=(a+d)/f, where  the plasmon frequency ω_pl determined by the boundary condition ε'(ω_pl)=ε_d Q'_1(ξ_1). In the limit of spherical particle of radius a, that is, f→ 0 and ξ→∞ as b→ a, we have Q(ξ)≈ 1/3ξ^2, yielding 


    1/τ_sp=
    12μ^2/ħε”(ω_sp)a^3/(a+d)^6,


where ω_sp is surface plasmon resonance frequency for a sphere determined by ε'(ω_sp)+2ε_d=0. The normalized ET rate τ_sp/τ has the form


    τ_sp/τ
    =
    a^2/4b^2 (1+d/a )^6ε”(ω_sp)/ε”(ω_pl) [Q'_1(ξ)ξ_1/Q_1(ξ_1) ]^2,


with ξ=ξ_1+d/f.







§ CALCULATION OF POPULATION INVERSION DENSITY IN SPHEROIDAL CORE-SHELL NANOPARTICLE


We consider a core-shell nanoparticle with dielectric functions ε_c, ε_s, and ε_d in the core, shell, and outside dielectric, respectively, with inner and outer interface S_1 and S_2. The integrals over  core ans shell regions in the condition (<ref>) reduce to surface terms


    ∫ dV_c |E|^2=∫ dS_1Φ^* E_n^c,
    
       ∫ dV_s |E|^2=∫ dS_2Φ^* E_n^s - ∫ dS_1Φ^* E_n^s,


where E_n^j(S_i)=-∇_jnΦ(S_i) is the field outward normal component at the ith interface in the jth medium side. Note that E_n^s(S_1)=(ε_c/ε_s)E_n^c(S_1) and E_n^s(S_2)=(ε_d/ε_s)E_n^d(S_2).
The  ratio of integrated field intensities in the shell and core regions takes the form


    L=∫ dV_s |E|^2/∫ dV_c |E|^2
    =
    ε_d/ε_s∫ dS_2Φ^* E_n^d/∫ dS_1Φ^* E_n^c - ε_c/ε_s,


where the potentials Φ are continuous at the interfaces. For nanostructures whose shape permits separation of variables, the potential  can be written as Φ (r)=R (ξ)Σ(η,ζ), where ξ is the radial (normal) coordinate and the pair (η,ζ) parametrizes the surface. With surface area element dS=h_ηh_ζ dη dζ and normal derivative ∇_n=h_ξ^-1(∂/∂ξ), where h_i are the scale factors (i=ξ,η,ζ), the fraction of integrals   takes the form


    I=∫ dS_2Φ^* E_n^d/∫ dS_1Φ^* E_n^c
    =
       R_d (ξ_2)R_d' (ξ_2)/R_c (ξ_1)R_c' (ξ_1)
    ×   ∫∫ dη_2 dζ_2 (h_η_2h_ζ_2/h_ξ_2)  |Σ |^2/∫∫ dη_1 dζ_1 (h_η_1h_ζ_1/h_ξ_1)  |Σ |^2.


Below we outline evaluation of L for core-shell NP  described by two confocal prolate spheroids with semi-axises  a   and  b. The two shell surfaces corresponds to ξ_1=a /f and  ξ_2=sa /f, where f =√(a^2-b ^2) is half distance between the foci, and s>1 characterizes the shell thickness. Evaluation of  angular integrals yields


    I
    = R_d (ξ_2)R_d' (ξ_2)/R_c (ξ_1)R_c' (ξ_1) ξ_2^2-1/ξ_1^2-1.


For longitudinal dipole mode (l=1, m=0), we have R_c = P_1(ξ) for ξ<ξ_1, R_s = AP_1(ξ)+BQ_1(ξ) for ξ_2<ξ<ξ_2, and R_d = CQ_1(ξ) for ξ>ξ_2, yielding


    L=C^2 ε_d/ε_s Q_1(ξ_2)Q'_1(ξ_2)/P_1(ξ_1)P'_1(ξ_1) ξ_2^2-1/ξ_1^2-1 - ε_c/ε_s,

where C and ε_c(ω_pl) are determined from the continuity of R_i and ε_iR'_i across the interfaces.








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