## Abstract

Amplification of the single-interface and long-range surface plasmon-polariton modes is studied in planar metallic structures incorporating gain media formed by Rhodamine 6G dye molecules in solution. We employ a theoretical model that accounts for the nonuniformity of the gain medium close to the metal surface due to position-dependent dipole lifetime and pump irradiance. The results of this model are used as a baseline for a comparative study against two simplified models: one neglects the position-dependent dipole lifetime while the other assumes a uniform gain medium. The discrepancies between the models are explained in terms of the mode overlap with the gain distribution near the metal. For the cases under analysis, the simplified models estimate the required pump irradiance with deviation factors that vary from 1.45 at the lossless conditions to 8 for gains near saturation. The relevance of describing properly the amount of gain interacting with the SPP mode and the role played by the dipole quantum efficiency are discussed.

© 2009 Optical Society of America

## 1. Introduction

Surface plasmon-polaritons (SPPs) are electromagnetic waves coupled to surface plasma oscillations that propagate along the interface between a dielectric and a metal with a negative real part of permittivity [1]. A variety of plasmonic structures have been analyzed exhibiting interesting properties [2], but the intrinsic propagation loss of SPP modes introduces limitations for several potential applications. This problem has created a strong interest in understanding the physics of SPP amplification [3]–[7] as well as exploring experimentally its feasibility [8]–[13]. The rapid progress in active plasmonics has provided exciting results; for instance, stimulated amplification of SPP modes has been observed [9, 10, 11], and very recently, lasing in sub-wavelength plasmonic waveguides was claimed [12].

Optical dipoles (dye molecules, rare-earth ions, atoms, or semiconductor quantum dots) are of particular interest because they can be integrated with dielectric hosts and conveniently used in the construction of active SPP waveguides. Indeed, much research has been done on the complex radiative behaviour of dipoles in close proximity to metallic surfaces [14, 15]. However, few investigations have addressed its consequences on SPP amplification [16, 17, 18]. Close to the metallic surface the pump radiation is generally not uniform and the increased density of modes and coupling to electrons in the metal quench the dipole’s lifetime. These two factors play an important role in shaping the gain distribution interacting with a mode during the amplification process. It is the purpose of this paper to address these phenomena.

This paper builds on our previous theoretical work that describes SPP amplification in planar structures accounting for the nonuniformity of optically pumped dipolar gain media [18]. Here, we describe the model in greater detail and employ it to study the amplification of the long-range SPP (LRSPP) and the single-interface SPP modes in two planar structures incorporating gain media formed by Rhodamine 6G (R6G) dye molecules in solution. The analysis considers two dye concentrations taking into account the concentration-dependent quantum efficiency of R6G. We present a comparative study between the proposed model and two simplified models that do not account entirely for the nonuniform nature of the gain near the metal surface. We discuss the relevance of including such effects by analyzing the gain-mode overlap described by the different gain models and express the importance of the dipole quantum efficiency in determining the amount of gain interacting with the SPP mode.

## 2. Geometries Investigated

The analysis consider the structures shown in Figs. 1(a) and (b). Structure 1 consists of a 20 nm-thick silver film extending infinitely over the (*x,z*)-plane. The bottom cladding is formed by 2 *µ*m of CYTOP, a lossless dielectric, which in turn sits on a semi-infinite silicon substrate; the top surface is covered by R6G dye molecules in a mixture of ethanol and methanol. A semi-infinite CYTOP superstrate lies on top of the dye solution, holding it to within a 1-*µ*m-thick layer. For structure 2 the silver region extends infinitely below the dye region.

In both structures, dye molecules are excited at broadside by a monochromatic pump signal of wavelength *λ*
_{p}=532 nm (frequency-doubled Nd:YAG), which is near the peak absorption of R6G. SPP mode amplification is studied at the peak emission wavelength of the dye, *λ*
_{e}=560 nm. We assume that the real permittivity of the dye solution is matched to CY-TOP at *λ*
_{e}, making structure 1 symmetric. This can be achieved using a mixture of approximately 57% methanol and 43% ethanol as the dye solvent [22]. The analysis considers two dye concentrations, *C _{a}*=5 mM and

*C*=40 mM, which correspond to dye volume densities of 3.01×10

_{b}^{18}cm

^{-3}and 24.08×10

^{18}cm

^{-3}, respectively. The concentration-dependent properties of R6G are approximated either in pure methanol or pure ethanol from information available in the literature [23]–[26]. Tables 1 and 2 in the appendix summarize respectively the material relative permittivities and photophysical parameters of R6G at the two wavelengths of interest.

Structure 1 supports LRSPP and short-range SPP modes [19]. However, the gain analysis of the latter will be omitted because of its large propagation losses. Structure 2 supports the single-interface SPP mode. The proposed structures and pump arrangement are physically realizable and could serve directly to validate the results reported herein.

## 3. Gain model

The energy levels and transition rates of R6G are depicted schematically in Fig. 1(c). It shows the three most relevant electronic states and associated vibrational and rotational overlapping energy levels (represented by the shaded areas) of the singlet manifold. The parameters in the figure are as follows: *S _{i}* denote the ground (

*i*=0), first excited (

*i*=1), and second excited (

*i*=2) energy states, and Ni indicates their respective molecular populations;

*σ*

^{0}

_{p}and

*σ*

^{0}

_{a}(

*σ*

^{1}

_{p}and

*σ*

^{1}

_{a}) are the absorption cross-sections of state

*S*

_{0}(

*S*

_{1}) at

*λ*and

_{p}*λ*, respectively;

_{e}*τ*

_{21}is the lifetime (decay time) from state

*S*

_{2}to

*S*

_{1}, which is approximately the lifetime of state

*S*

_{2}[24];

*σ*and

_{e}*τ*are, respectively, the emission cross-section at

*λ*and lifetime of the

_{e}*S*

_{1}state;

*I*is the pump irradiance at

_{p}*λ*and

_{p}*I*is the stimulated emission irradiance at

_{e}*λ*. We neglect transitions to the triplet manifold assuming that the pump signal duration is short compared to the reciprocal inter-system crossing rate of 2.8

_{e}*µs*[24]. Yet, we also assume that the pump signal duration is long compared to

*τ*, such that steady state conditions are reached. These assumptions are reasonable for R6G molecules pumped by typical ~10 ns-pulse frequency-doubled Nd:YAG lasers.

The small-signal gain coefficient of the dye at λe is given by [27]

where *σ**_{e}=*σ _{e}*-

*σ*

^{1}

_{a}is the effective emission cross-section of the S1 state. To obtain N1 at any point in the dye region one must solve the rate equations locally, accounting for the position dependent lifetime and pump radiation. Thus, we replace the constants

*τ*and

*I*with their respective position dependent expressions, which will be obtained in the following subsections.

_{p}Solving the rate equations locally under steady state conditions and using the small signal approximation, one obtains

where *N* is the total molecular population, *ω _{p}* is the pump angular frequency, and

*ħ*is the reduced Plank constant. In writing Eq. (2) we have taken

*N*

_{2}=0 assuming that

*τ*

^{-1}

_{21}is much faster than the rate at which

*S*

_{2}is populated,

*I*

_{p}σ^{1}

_{p}≪

*ħw*/

_{p}*τ*

_{21}. This holds for

*I*≪6×10

_{p}^{9}W=cm

^{2}, which also holds throughout the paper. Substituting Eq. (2a) into Eq. (1) one obtains the local small-signal gain coefficient

The present analysis assumes an infinitesimally small amplifier length, in which ASE does not play a role. In a real device, the effect of ASE could be significant when the amplifier’s length and dye concentration are large enough. ASE renders g inhomogeneous along the propagation axis, peaking at the center of the amplifier and tailing off towards the facets. Thus, one should expect the present model to be accurate for modeling short-length SPP amplifiers.

#### 3.1. Position dependent pump irradiance

We model the pump signal as a linearly polarized plane wave of wavelength *λ _{p}* and irradiance

*I*

^{0}

_{p}that illuminates the structure from the top with propagation direction normal to the metal plane. Its absorption trough the dye is described by Eq. (2b) and satisfies the following approximations:

where we have used the fact that *σ*
^{1}
_{p}<*σ*
^{0}
_{p} for R6G, as it is typically the case for most dyes. The behavior of *I _{p}*(

*y*) is non-linear and monotonic between these limiting cases. A rigorous non-linear treatment is out of the scope of this paper; instead we henceforth employ the linear approximation described by the top form in Eq. (4), which is also valid for an optically thin dye region regardless of the value of

*I*. Under this approximation, the relative permittivity of the dye at

_{p}*λ*can be expressed as

_{p}where *ε*′_{r,p} is the relative permittivity of the dye solvent. Using this expression we compute the pump electric-field distribution throughout the structure, *E _{p}*(

*y*), using a transfer matrix formalism [28]. Then, the irradiance distribution is obtained as

where *η*(*y*) is the characteristic impedance of the medium at the location y.

The normalized pump irradiance distribution, *I _{p}*(

*y*)/

*I*

^{0}

_{p}, in Structure 1 and 2 is shown in Figs. 2(a) and (b), respectively. Each figure show results for the two dye concentrations. The pump distribution above the metal is similar in both structures, following a standing-wave pattern due to field reflections throughout the structure. The peak values in both structures reach similar values because the lower-cladding thickness in structure 1 was chosen to reduce resonant coupling into the slab mode and increase the peak irradiance in the dye region.

The dashed curves in Fig. 2(b) illustrate the case where the bottom form in Eq. (4) is satisfied. Thus, the difference between dashed and solid curves represents the worst case error due to the linear approximation. For concentration *C _{a}* the dye layer is optically thin and the error is only 8%. This is not the case for concentration

*C*, where the error reaches 95%. Similar errors apply to Structure 1. In general, the absorption is overestimated when the adopted linear approximation is not valid. In such a case,

_{b}*N*

_{1}(y) in Eq. (2b) and consequently

*g*(

*y*) in Eq. (3) take lower-bounded values set by the low-

*I*linear absorption.

_{p}#### 3.2. Position dependent dipole lifetime

The position-dependent lifetime, *τ*(*y*), in Eq. (3) is obtained following a classical treatment that considers molecules as dipoles being driven by the reflection of its own radiation at the metal surface [14, 15]. Within this framework, the normalized decay rate of a dipole located at a distance *y* from the metal surface and with dipole-moment oriented along *µ* can be written as

where *γ _{nr}*=1-

*ϕ*is the normalized non-radiative decay rate and

*ϕ*is the dipole’s quantum efficiency;

*u*=

*k*

_{‖}/

*k*is the component of the dipole’s field wave vector parallel to the metal plane (

*k*

_{‖}) normalized to the magnitude of the dipole’s far-field wave vector (

*k*); and

*P*is the dipole’s power dissipation density normalized to the power dissipation far from the metal. The components of

_{µ}*P*with dipole-moment orientation perpendicular (⊥) and parallel (‖) to the metal plane are

_{µ}$${P}_{\parallel}(u,y)=\mathrm{Re}\genfrac{}{}{0.1ex}{}{3}{4}\genfrac{}{}{0.1ex}{}{u}{\sqrt{1-{u}^{2}}}\left\{\left(1+{r}_{s}\mathrm{exp}\left(i2{k}_{\parallel}y\right)\right)+\left(1-{u}^{2}\right)\left(1-{r}_{p}\mathrm{exp}\left(i2{k}_{\parallel}y\right)\right)\right\},$$

where *r*
_{S,P} are the Fresnel reflection coefficients of the multilayer structure for waves polarized perpendicular (S) and parallel (P) to the metal plane.

For concentration *C _{a}*, the rotational reorientation time of R6G in a similar solvent,

*τ*, is much smaller than the lifetime,

_{or}*τ*(see Table 2), leading to an isotropic excited dipole population. On the contrary,

*τ*is comparable to

_{or}*τ*for concentration

*C*; nonetheless Förster-type energy transfer between dipoles renders the excited dipole population isotropic [23]. Hence, for our analysis we consider the isotropic normalized power dissipation density obtained by averaging over all possible dipole-moment orientations

_{b}The results of Eq. (9) at *λ _{e}* for several metal-dipole separations are shown in Fig. 3(a) and (b) for structures 1 and 2, respectively. The features present over different intervals of

*u*are associated with the particular decay channel labeled on the figure. These decay channels include coupling to radiative modes (RAD), where the dipole emits a photon; coupling to SRSPP, LRSPP, and single-interface SPP, where the dipole emits a SPP; and coupling to electron-hole (EH) pairs in the metal, where the dipole’s energy is transferred directly to the metal in a dipole-dipole interaction. The latter, characterized by the broad peaks present at large

*u*values and short dipole-metal separations, is strongly suppressed by electron screening when

*u*reaches the upper limit of the EH excitation continuum [14]. For silver at

*λ*this limit is

_{e}*u*=2.2

*k*(

_{F}/k*k*being the Fermi wave vector). We account approximately for this effect by limiting the u-domain to values below 2.2

_{F}*k*as indicated by the vertical dashed lines in Fig. 3.

_{F}/kThe lifetime of an isotropically oriented dipole as a function of its distance from the metal surface is given by

where *γ*̂(*y*) is the normalized decay rate of an isotropically oriented dipole obtained from Eq. (7) using Eq. (9) as the integrand. The results of Eq. (10) normalized to *τ* are shown in Fig. 4 for both structures and for both dye concentrations. For concentration *C _{a}*,

*ϕ*=0.9 and Eq. (7) is dominated by the second term, which comprises the different decay channels discussed above. In this case, dipole relaxation through the multiple decay channels causes the characteristic strong quenching near the metal surface and the oscillatory behaviour in

*τ*(

*y*). For concentration

*C*, the quantum efficiency drops to

_{b}*ϕ*=0.08 due to Förster-type energy transfer between dipoles [23] and Eq. (7) is dominated by the non-radiative term. Thus, for concentration

*C*the lifetime is practically position independent, except for very short dipole-metal separations where the EH, SRSPP, and SPP decay channels still have significant effects.

_{b}The above mentioned effects have been confirmed experimentally for two dyes with different intrinsic quantum efficiency [29]. Yet, there are no experimental reports addressing the case where the quantum efficiency is altered due to a Förster type energy transfer between dipoles.

## 4. Numerical analysis

The gain medium is characterized by its inhomogeneous complex permittivity at *λ _{e}*,

where *ε*′_{r,e} is the relative permittivity of the dye solvent. The SPP modes of the structures are obtained using the transfer matrix method [30]. In this calculation, *ε _{r,e}*(

*y*) is divided into 15,000 layers of equal thickness, keeping its value constant within each layer. We take the SPP mode propagation along the z-axis with mode fields having a complex phase of the form exp[

*z*(-

*iβ*+

*α*)]. Thus, a positive (negative) value of

*α*indicates amplification (attenuation) as the mode propagates in the positive

*z*-direction. The

*mode power gain coefficient*is given by

In what follows we compare the results obtained using three different models for *g*(*y*). Our baseline model is described in Section 3 and will be referred to as model *A*; model *B* accounts for *I _{p}*(

*y*) but assumes a uniform lifetime of the from

*τ*(

*y*)=

*τ*; and model

*C*assumes a uniform pump distribution of the form

*I*(

_{p}*y*)=

*I*

^{0}

*and the same uniform lifetime as model*

_{p}*B*. In all our calculations we have used the material relative permittivities and concentration-dependent photophysical parameters of R6G listed in the appendix.

#### 4.1. Gain-mode overlap

The mode-gain overlap is depicted in Figs. 5(a,b) and (c,d) for the LRSPP mode of structure 1 and single-interface SPP mode of structure 2, respectively. They show the mode’s normalized |*E _{y}*|

^{2}distribution and the dye’s gain coefficient distributions obtained with models

*A, B*, and

*C*(labeled

*g*, and

_{A}, g_{B}*g*) for both dye concentrations and two pump irradiances:

_{C}*I*=300 kW/cm

_{p,l}^{2}and

*I*=10 MW/cm

_{p,h}^{2}for concentration

*C*, and

_{a}*I*=3 MW/cm

_{p,l}^{2}and

*I*=30MW/cm

_{p,h}^{2}for concentration

*C*. The magenta dashed lines indicate the gain saturation limit,

_{b}*g*=

_{s}*Nσ**

_{em}. The mode distributions obtained with the different gain models and pump irradiances on each figure are very similar and lay on top of each other. The embedded movies show the evolution of

*g*(

*y*) and |

*E*|

_{y}^{2}over the range 0<

*I*

^{0}

_{p}<30 MW/cm

^{2}.

In calculating *g*(*y*) we have included the effects of gain reduction due to the formation of weakly-fluorescent dimers at high concentrations. The dimer molar fraction, *d*, is obtained from [23] as 0:08 and 0 for concentration *C _{b}* and

*C*, respectively. The gain coefficients are obtained from Eq. (3) by letting

_{a}*N*→

*N*(1-

*d*). The large

*g*(

*y*) values obtained have not been demonstrated experimentally with R6G; however, they should be achievable in principle since the two dye concentrations are well bellow the solubility limit (~0.6 M in methanol) of R6G.

In general, the values of *g _{A}, g_{B}*, and

*g*obtained for Structure 1 are similar to those obtained for structure 2. This is expected as we note from Figs. 2 and 4 that

_{C}*I*(

_{p}*y*) and

*τ*(

*y*) are similar for both structures. For concentration

*C*and pump irradiance

_{a}*I*the gain coefficients are far from the saturation limit;

_{p},l*g*and

_{A}*g*follow closely the pump profile given by Eq. (6) whereas

_{B}*g*is well below their peaks because model C does not account for pump reflection. However,

_{C}*g*is significantly suppressed for

_{A}*y*<150 nm, which reduces the gain-mode overlap. This is consequence of the lifetime quenching discussed in Section 3.2. On the other hand, for

*I*, the three gain models approach the saturation limit. The high pump irradiance overcomes to some extent the lifetime quenching reducing the discrepancy between

_{p,h}*g*and

_{A}*g*to within a smaller region,

_{B}*y*<80 nm. The results for concentration

*C*show two main differences. First, saturation effects are not as evident because of the large dye concentration; and second, the low quantum efficiency reduces the lifetime quenching near the metal surface causing

_{b}*g*and

_{A}*g*to differ within a reduced range,

_{B}*y*<90 nm, that depends weakly on the pump irradiance.

The gain-mode overlap can be very different in both structures because of the different mode penetration depths into the gain medium. For the LRSPP mode, |*E _{y}*|

^{2}drops a factor of

*e*

^{-2}at

*y*=400 nm whereas for the single-interface SPP mode it occurs at y=155 nm. Thus, in the latter case the gain nonuniformities close to the metal surface have a stronger impact in the gain-mode overlap.

#### 4.2. Mode power gain versus pump irradiance

In this subsection we study the properties of *G* as a function of *I*
^{0}
_{p} using the three gain models previously described. Figs. 6(a,b) and (c,d) show *G*(*I*
^{0}
_{p}) computed with the three models (labeled *G _{A}, G_{B}*, and

*G*) for the LRSPP and single-interface SPP modes, respectively. The results are computed for the range 0<

_{C}*I*

^{0}

_{p}<30 MW/cm

^{2}. The dashed magenta lines indicate the saturation limit,

*G*=

_{s}*G*(

*I*

^{0}

_{p}→∞), and the quantity

*D*=

*G*-

_{s}*G*(0) given on each figure represents the amplification range. Observe that for cases with the same dye concentration,

*D*is approximately 2:45 times larger for the single-interface SPP mode. The factor of 2 is expected because only half of the LRSPP mode interacts with the gain medium. The remainder is attributed to the larger bulk sensitivity of the single-interface SPP mode [31].

The lossless condition calculated with model *A* is indicated by the black dashed lines on the figures. For the LRSPP mode, this condition is achieved with reasonably low pump irradiances; ie., *I*
^{0}
_{p}=175 kW/cm^{2} and *I*
^{0}
_{p}=430 kW/cm^{2} for concentrations *C _{a}* and

*C*, respectively. The value for concentration

_{b}*C*is larger because of the larger dye absorption at

_{b}*λ*, which attenuates the amplified light, and because of the larger dye absorption at

_{e}*λ*, which reduces the pump irradiance in the dye region as shown in Fig 2. On the other hand, the lossless condition for single-interface SPP mode can be achieved only with concentration

_{p}*C*when

_{b}*I*

^{0}

_{p}=3.455 MW/cm

^{2}. Note that this value could be overestimated by up to a factor of two because it does not satisfy the linear absorption approximation as stated by the top form of Eq. (4), which was adopted in the calculations. On the contrary, the

*I*

^{0}

_{p}values required for lossless propagation of the LRSPP mode do satisfy this approximation.

The pump irradiances required for lossless propagation in both structures are attainable using commercial lasers; however, the maximum *I*
^{0}
_{p} value that can be sustained is determined by the metal damage threshold. Laser-induced metal damage develops over a number of phases that aggravate with increasing irradiance [32]. The first signs of damage appear in the form of microscopic defects and deformations on the surface; as the irradiance increases the surface melts and eventually evaporates leading to ablation. The damage threshold of a plasmonic structure must be defined in terms of the metal quality needed to support a particular SPP mode and could be lower than the ablation threshold, which has been estimated as ~160 MW/cm^{2} for bulk silver exposed during 20 minutes to pulsed (0.8 J/cm^{2}, 5 ns duration, 0.16 Hz repetition rate) 532 nm radiation [33]. For thin films, factors such as loss of adhesion due to thermal stress [34] and lower heat dissipation capacity could further reduce this damage threshold.

From the results in Fig. 6(a–d) we note that model *A* generally requires higher *I*
^{0}
_{p} values to achieve the same *G* values estimated by the other two models because it considers factors that reduce the dye’s gain over the mode region. To quantify these deviations we define the factor

where *x* stands for either model *B* or C and *I _{p,A}* (

*I*) is the pump irradiance required to obtain a particular value of

_{p,x}*G*using model

*A*(model

*x*).

*f*is plotted for both concentrations in Figs. 7(a,b) and (c,d) for the LRSPP and single-interface SPP modes, respectively.

Consider first the factors associated with model *B*, which result from neglecting the lifetime position dependence. From the fB curves for concentration Ca and structure 1 (structure 2) we note that model A estimates pump irradiances that are factors of 1.3 to 2.5 (1.5 to 6.8) larger than those of model B. The larger values obtained for structure 2 occur because the mode-gain overlap of the single-interface SPP mode is more sensitive to the changes in *g*(*y*) that result from neglecting lifetime quenching included in *τ* (*y*). Yet, for concentration *C _{b}* and for both structures,

*f*is close to unity for most values of

_{B}*G*. This accentuates that model

*B*can be a good approximation to model A when the dipole quantum efficiency is low because in this case

*τ*(

*y*)≈

*τ*.

Consider now the factors associated with model *C*, which result from neglecting both the lifetime and pump position dependence. Interestingly, for concentration *C _{a}*, there exist a region over which

*f*≈1 (also shown in Figs. 6(a) and (c) as similar

_{C}*G*and

_{A}*G*curves in the region

_{C}*I*

^{0}

_{p}<200 kW/cm

^{2}). This is merely coincidental and cannot be generalized for an arbitrary structure. Indeed, this is supported by the results for concentration

*C*and structure 1 (structure 2), where

_{b}*f*ranges from 1.45 to 2.55 (1.27 to 2.5). In general,

_{C}*G*reaches saturation with lower

_{C}*I*

^{0}

_{p}values than

*G*because model

_{A}*C*assumes a perfect gain-mode overlap and neglects the pump absorption through the dye. This results in large

*f*values at large

_{C}*G*values. A clear example is given in Fig. 6(c), where the

*I*

^{0}

_{p}deviation (D) covers almost one decade; this yields

*f*=8 as shown in Fig. 7(c).

_{C}The values of *f _{B}* and

*f*at the lossless propagation condition are indicated on Figs. 7(a), (b), and (d). Some of them represent deviations of ~50% in the pump irradiance estimated by model

_{C}*A*. These values are rather significant, especially if the pump irradiance is close to the damage threshold of the metal. Also, from the discussion above, we note that model

*A*provides large corrections to the other two models around the gain saturation region. Thus, the use of model

*A*would be particularly important for weaker gain media, where the lossless condition could be achieved close to gain saturation.

## 5. Summary

In summary, we study the amplification of the LRSPP and single-interface SPP modes in planar metallic structures incorporating gain media formed by Rhodamine 6G dye molecules in solution. We employ a theoretical model that accounts for the nonuniform gain distribution in close proximity to the metal and its overlap with the SPP mode. The analysis considers two dye concentrations, 5 mM and 40 mM, accounting for the concentration-dependent quantum efficiency of R6G. Within the limitations of the model, the results suggest that lossless propagation is possible in the visible for both SPP modes using the proposed structures and pump arrangement. For the LRSPP mode, the lossless propagation condition was obtained using reasonable pump irradiances; ie., 175 kW/cm^{2} and 430 kW/cm^{2} for 5 mM and 40 mM dye concentration, respectively. For the single-interface SPP mode, lossless propagation was obtained only with the 40 mM dye concentration using an intense pump irradiance of 3.45 MW/cm^{2}.

The proposed model is taken as baseline to perform a comparative study against two simplified models: one neglects the lifetime position dependence, while the other assumes a uniform gain medium. The discrepancies between the baseline and the simplified models are explained in terms of gain-mode overlap. For the cases under analysis, the simplified models estimate the required pump irradiance with deviation factors that vary from 1.45 at the lossless conditions to 8 for gains near saturation. The relevance of describing properly the amount of gain interacting with the SPP mode and the role played by the dipole quantum efficiency are discussed. In particular, the results show that the discrepancies due to ignoring the lifetime position dependence are not considerable when the dye’s quantum efficiency is low.

## 6. Appendix: Relative permittivities and R6G parameters

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