1. Introduction and model predictions

Localized surface plasmons (SPs), resonant oscillations of electron density in metallic nanostructures, and propagating surface plasmon polaritons (SPPs), electromagnetic waves coupled to electron density oscillations and propagating along the interface between metal and dielectric, have recently become a hot research topic because of their ability to confine and enhance electromagnetic radiation at the nanometer scale. Localized and propagating surface plasmons and relevant phenomena, which are described by a common term nanoplasmonics, have numerous exiting applications in a variety of sensors [1–3], optoelectronic devices [4–9], and photonic metamaterials [10–12]. However, many of these applications are hindered by one common cause – absorption loss in metal.

The known solution to the loss problem is optical gain added to a dielectric medium. It has been predicted that gain can compensate for absorption loss in propagating [13–15] and localized [16] surface plasmons. Experimentally, six-fold enhancement of localized SPs by optical gain has been demonstrated in Ref. [17]. The compensation of loss in propagating SPP by optical gain, although very small, was first demonstrated in Ref. [18], and the gain sufficient to fully compensate SPP loss in high-quality silver films was achieved in Ref. [19]. In spite of significant progress made in this direction, maintaining required optical gain (of the order of 10³ cm^-1) is a technologically difficult task, which often requires a Q-switched laser [17,19]. This makes the gain solution to the loss problem unpractical for many applications. Ideally, one would want to have a low loss in passive systems, without any gain.

Two metals commonly used in nanoplasmonic and metamaterials applications are silver and gold. Silver has a smaller absorption loss, and gold is a better technological material. Unfortunately, the propagation length of SPP in these metals (in the visible range) is of the order of ten micrometers or shorter.

Can metal-dielectric interfaces be improved to become more suitable for photonic applications? In silver, there is a significant difference between the experimentally measured absorption loss in the visible and ultraviolet ranges of the spectrum [20, 21] and that predicted by the Drude model, which takes into account the contribution of free electrons only [22]. By the first principles calculations [23], we have shown that this difference is due to bound electrons in the layers of silver atoms adjacent to the metal surface. This is evidenced by Fig. 1 comparing the experimental absorption loss with that calculated in bulk silver and silver slabs of different thickness.

 

Fig. 1. Imaginary part of the dielectric constant of silver e” calculated (1) according to the Drude model with plasma frequency ω_p=9.1 eV and loss parameter Γ=0.021 eV; (2) from the first principles for bulk silver; (3–6) for the (1,1,1) surface of silver slabs consisting of 7, 10, 13, and 16 monolayers. Trace 7 - experimental data [20].

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Adsorption of molecules, ions or radicals onto the silver surface can modify surface electronic states and reduce the absorption loss of silver. This effect was demonstrated in Ref. [24], where adsorption of oxygen on Ag(110) removed the effect of surface states and substantially reduced optical response around 1.7 eV (730 nm, 13700 cm^-1).

In this work, we have demonstrated elongation of the SPP propagation length by bringing the silver film in contact with the polymer heavily doped with rhodamine 6G chloride dye.

2. Experimental results and discussion

Experimentally, we have studied SPPs in the attenuated total reflection (ATR) setup shown in Fig 2. Silver films were deposited on the 90 degree glass prism. The thickness of the silver film in different samples varied between 35 and 72 nm. Rhodamine 6G chloride dye (R6G) and polymethyl methacrylate (PMMA) were dissolved in dichloromethane. The solutions were applied to the surface of silver and dried in air, providing a PMMA film doped with R6G. The thickness of the PMMA film was ~10 mm and the concentration of R6G varied between 0 and 100 g/l (2.1x10^-1 M).

 

Fig. 2. Experimental setup: excitation of SPPs in an attenuated total reflection (ATR) geometry.

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In the measurements of the angular dependence of the reflectivity R(θ), the prism was mounted on a motorized goniometer stage. The reflectivity was probed with p polarized 594.1 nm He-Ne laser light. Reflected light was detected by a photomultiplier tube connected to an integrating sphere, which was moved during the scan to follow the walk of the reflected beam. The reflectivity profile R(θ) had a characteristic dip [25] at the angle at which the wave vector of the SPP matched the projection of the photon wave vector to the plane of metal-dielectric interface, Fig. 3. It is described by the known formula

where$r_{ik}^p=\frac{\left(k_{zi}{\epsilon }_k-k_{zk}{\epsilon }_i\right)}{\left(k_{zi}{\epsilon }_k+k_{zk}{\epsilon }_i\right)}$ is the amplitude reflection coefficient for p polarized light at the interface between media i and k (i, k=0,1,2), ε_{i, k} is the dielectric constant, d is the thickness of the metallic film, $k_{zi}=\sqrt{{\epsilon }_i{(\omega ⁄c)}^2-k_x^2}$ is the wave vector in the direction perpendicular to the surface of the metal, k_x=k_phot sinθ ₀ is the wave vector in the direction of the SPP propagation, k_phot is the wave vector of a photon in the glass prism, and q is the angle of incidence; the subscripts 0, 1, and 2 correspond to glass, silver, and PMMA, respectively. Here it is assumed that each of the three media (glass/silver/polymer with dye) is spatially uniform, the boundaries between media are sharply defined, and effects of hybrid states at the interfaces are neglected.

 

Fig. 3. Angular dependence of the reflectivity R(θ) recorded in the setup of Fig. 3, with the concentration of the R6G dye equal to 0 g/l (squares) and 30 g/l (circles). With the addition of dye, the width of the reflectivity profile decreased and the position of its minimum shifted.

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The absorption spectra of PMMA/R6G films featured a characteristic band centered at 530nm, inset of Fig. 4. At high concentration of dye, its long-wavelength wing significantly increased in comparison to that recorded at low concentration of R6G. The imaginary part of the dielectric constant of the PMMA/R6G film ε″ ₂, calculated at λ=594 nm from the absorption data, is plotted as a function of concentration of R6G (in log-log scale) in Fig. 4. At high concentration of dye, the slope of the curve approached 2, which indicated dimerization of dye molecules. Large scatter of the data was due to the known relatively low reproducibility of PMMA films.

We started reflectivity experiments with measuring the angular profile R(θ) in pure glass prism, without any deposited films. By fitting the function R(θ) with Eq. (1) at d=0 and e₂=1, we determined the index of refraction of glass n ₀=(ε₀)^1/2 to be equal to 1.7835 at 594.1 nm, in a very good agreement with the data provided by the manufacturer.

We then repeated the same measurements with the PMMA/R6G films (with different concentrations of the R6G dye) deposited directly onto the prism. The rear surfaces of the polymeric films were intentionally roughened (with very fine sandpaper) to prevent the reflection of light from the polymer/air interface back to the prism. Each reflectivity profile R(θ) was fitted with Eq. (1) at d=0 to determine the dielectric constant of the PMMA/R6G film, Fig. 5.

 

Fig. 4. Imaginary part ε ₂” of the dielectric constant of the PMMA/R6G film as a function of concentration of R6G, N. Diamonds: data calculated from the absorption measurements. Solid line: data points e ₂” fitted with a second order polynomial. Dotted line has the slope equal to h=2. Circles: the values e ₂” used at the fitting of the R(θ) profiles in glass/silver/polymer structures. Inset: Absorption spectrum of PMMA film doped with R6G; trace 1 - N=100 g/l; trace 2 - N=5 g/l.

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Fig. 5. Real part of the dielectric constant ε₁’ of PMMA/R6G as a function of R6G concentration N.

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The reflectivity studies carried out in three-layer structures (glass/silver/polymer) at different concentrations of dye resulted in profiles R(θ) characteristic of SPPs [25], Fig. 3. The most remarkable result of this measurement is that the width W of the dip of the angular profile R(θ) decreased with an addition of R6G to PMMA, up to the concentration of dye N!30 g/l, and then increased again when the concentration of dye was increased further (compare widths W ₁ and W ₂ in Fig. 3). According to Ref. [19], W is inversely proportional to the SPP propagation length L. Correspondingly, the obtained experimental result implies that an addition of R6G dye to PMMA helps to reduce the loss and elongate the propagation length of SPPs.

We then fitted the reflectivity profiles R(θ) using Eq. (1), with the fitting parameters being real and imaginary parts of the dielectric constant of silver, ε₁’ and ε₁”. The real and imaginary parts of the dielectric constants of PMMA/R6G, ε₂’ and ε₂”, were chosen in accord with the measurements presented in Figs. 4 and 5. The determined this way values ε₁’ and ε₁” (at λ=594.1 nm) are plotted against the concentration of R6G in Fig. 6. In the absence of R6G, the value of e1’ coincided within 5% with those published in Refs. [20,21]. The value of ε₁” was smaller than that in Ref. [21] and larger than that in Ref. [20].

Both extracted values ε₁’ and ε₁” were strongly influenced by the presence of R6G dye in the PMMA film, Fig. 6. A particularly strong effect (40% reduction) has been observed in the dependence ε₁”(N) with the increase of the concentration of dye from 0 g/l to 30 g/l.

 

Fig. 6. Obtained from the fitting of R(θ) real (a) and imaginary (b) parts of the dielectric constant of silver, ε₁, as a function of the R6G concentration N. Solid lines – interpolations with second order polynomials.

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This implies that in a framework of a simple model assuming that each of the three media is spatially uniform, the boundaries between media are sharply defined, and the effects of hybrid states at the interfaces are neglected, this reduction of ε₁” would be required to account for the change in the SPP propagation length observed experimentally.

The propagation length L of SPPs can be calculated as [19]

where γ_i is the internal SPP loss, ${\gamma }_i=\frac{\omega }{2c}{\left(\frac{{\epsilon }_1^{\prime }{\epsilon }_2^{\prime }}{{\epsilon }_1^{\prime }+{\epsilon }_2^{\prime }}\right)}^{3⁄2}\left(\frac{{\epsilon }_1^{″}}{{\epsilon }_1^{\prime 2}}+\frac{{\epsilon }_2^{″}}{{\epsilon }_2^{\prime 2}}\right)$, and γ_r is the radiation loss caused by the decoupling of SPPs back to the prism ${\gamma }_r=\left\{\mathrm{Im}\frac{2\omega r_{01}{e}^{i2k_z^0{d}_1}}{c\left({\epsilon }_2-{\epsilon }_1\right)}{\left(\frac{{\epsilon }_2+{\epsilon }_1}{{\epsilon }_2{\epsilon }_1}\right)}^{-3⁄2}\right\}$. Here ω is the oscillation frequency, d ₁ is the thickness of the metallic film, k ⁰ _z is the value of k_z at the resonance angle, and c is the speed of light.

In order to calculate the dependence L(N) for two different thicknesses of the metallic film, d ₁=40 nm and d ₁=80 nm (Fig. 7), we substituted to Eq. (2) functions ε₁(N) and ε2(N) obtained by interpolation of the experimental data in Figs. 4–6 with the second order polynomials. Because the values of the radiative loss γ_r and, correspondingly, the SPP propagation lengths L are different at d ₁=80 nm and d ₁=40 nm, the data sets L(N) (squares and triangles in Fig. 7) were normalized to unity at N=0 for convenience of presentation.

 

Fig. 7. SPP propagation length L as a function of dye concentration N. Solid squares (triangles) – calculations done for real experimental parameters at d ₁=80 nm (40 nm); Open squares (triangles) – calculations done for the hypothetic case of the absence of dye absorption, e ₂ ”=0, at d ₁=80 nm (40 nm). Solid circles – inverse width of the reflectivity profile R(θ). All data sets are normalized to unity at N=0.

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The fact that the two normalized curves L(N) calculated at d ₁=80 nm and d ₁=40 nm (solid squares and triangles) are close to each other suggests that the relative change of the SPP propagation length with the concentration of dye is not strongly dependent on the thickness of the silver film. This justifies the comparison of the calculated values L(N) with the inverse widths W ^-1 of the reflectivity profiles R(θ) (circles in Fig. 7) measured in different samples with the thickness of the silver film varying between 35 and 91 nm and the average thickness being equal to 70 nm, Fig. 7. Not surprisingly (since W ^-1 is proportional to L [19]), these two types of curves closely resemble each other, confirming ~30% elongation of the SPP propagation length with the increase of the dye concentration to N=30 g/l (6.3x10^-2 M). (The slight difference between the experimental and the calculated curves in Fig. 7 is not clearly understood. However, the former one appears to be more accurate since it is based on direct measurements rather than recalculated data.)

The reduction of L with the increase of N above 30 g/l is due to (i) the increase of the absorption losses ε₁” and ε₂” and (ii) the increase of ε₁’, which becomes less negative, Figs. 4,6. Figure 7 demonstrates the strong difference between the values of L calculated under the assumption of ε₂”=0 (no loss in dielectric, open squares and triangles) and at the experimental values of ε ₂” (solid squares and triangles).

We tentatively explain the strong change in ε ₁” by the modification of electronic states in silver layer adjacent to the metal-dielectric interface, including possible formation of the AgCl phase. The preliminary results of ab initio modeling show that the formation of AgCl film can, indeed, affect the absorption loss at the surface. The results of these studies will be reported elsewhere.

3. Summary

To summarize, we have demonstrated that an addition of highly concentrated rhodamine 6G chloride dye to the PMMA film adjacent to the silver film causes 30% elongation of the propagation length of SPP without gain. This opens a new technological dimension to lowloss nanoplasmonics and metamaterials, in which surface phenomena and hybrid electronic states play an enormously important role in determining physical properties of nanocomposites and nanostructures.