1. Introduction

Surface plasmons are evanescent electromagnetic waves highly localized on metal-dielectric interfaces. They can, in suitable configurations, be directly excited by incident light. Combined with advances in nanofabrication and simulation methods, this has led to a field of true nanoscale optics, called plasmonics [1–3]. Recent research, especially in active plasmonics, has shown great promise in leading to future applications. Bergman and Stockman proposed a SPASER [4], we proposed a plasmonic band gap laser [5], and Seidel et al. [6] observed stimulated amplification of surface plasmons in Kretschmann configuration with a fluorescent dye as a gain material. More recently Noginov et al. [7] observed enhanced scattering from a silver aggregate surrounded by optically pumped dye molecules. Recently, they also observed stimulated emission of surface plasmons using the Kretschmann configuration with a dye-doped polymer film [8].

Our proposed plasmonic band gap laser consists of a two-dimensionally corrugated thin silver film — a plasmonic crystal — coated with a fluorescent laser dye layer [5]. In this laser the dye, pumped by a short pulse laser, excites surface plasmon waves by near-field coupling while simultaneously providing stimulated amplification of the standing surface plasmon waves at the plasmonic band gap edge. A part of the energy of the plasmons is finally coupled out as a free-space laser beam, emitted perpendicular to the surface.

The laser uses a long-range surface plasmon (LRSP) mode [9] to minimize the absorption loss. However, the excited fluorescent molecules have several possible decay routes other than LRSP modes, such as spontaneous radiation into free-space, excitation of electron-hole pairs in the metal, and excitation of short-range surface plasmon (SRSP) modes. To obtain lasing, the stimulated emission rate must be higher than the sum of the decay rates to these undesired routes. The lower the undesirable decay rates, the lower the pumping threshold. In other words, a high energy transfer rate to the LRSP mode is required for lasing.

The theory of the energy transfer from a fluorescent molecule, treated as an oscillating dipole, to a metal surface was established by Chance et al. [10]. Later, Weber and Eagen [11] employed this theory to calculate the coupling efficiency from a fluorescent molecule to a surface plasmon mode on a silver-dielectric interface. They pointed out that the efficiency reached 93% in optimal conditions.

Recently, Winter et al. [12] calculated the coupling efficiency from a fluorescent molecule to surface plasmons on metallic thin films supporting both LRSP and SRSP modes. The authors found out that the energy transfer rate to SRSP modes is always higher than to LRSP modes. Furthermore, they also noticed that reducing the film thickness in order to minimize absorption losses also decreases the energy transfer rate to LRSP modes, while the coupling to the SRSP modes correspondingly increases.

In this paper we report the calculation result of the coupling efficiency of fluorescent energy from a fluorescent molecule to LRSP modes in plasmonic crystals featuring band gaps. Based on the results, we introduce a way to increase the energy coupling efficiency to LRSP modes without a simultaneous detrimental increase of coupling to SRSP modes.

2. Suppression of short-range surface plasmons

 

Fig. 1. Geometry of the calculated model. There is no variation in the y-direction. The ambient is assumed to be Alq₃ doped with DCM with a dielectric constant of ε ₁ = 2.89. The fill factor of the grating is 75%.

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We consider metallic thin films embedded in a dielectric medium. In this situation, the film supports both LRSP and SRSP modes, provided that it is thin enough. However, once a periodic corrugation is introduced onto the interfaces of the films, the situation changes.

First, we examine the dispersion relation of the surface plasmon modes supported by one-dimensionally corrugated metallic thin films. The material of choice is silver and both surfaces are periodically corrugated. Even in the case of one-dimensional plasmonic crystals, various structures can be considered. Here we investigate only symmetrically corrugated films with a rectangular profile having a fill factor (ratio of ridge width to the grating pitch) of 0.75 as shown in Fig. 1, because this structure exhibits the widest plasmonic band gap and nearly zero radiation losses at the upper edge of the plasmonic band gap, which are required for laser applications [13]. The pitch, the maximum thickness, and the minimum thickness of the silver film are denoted by Λ, d ₁, and d ₂, respectively.

The dispersion relation is represented in ω−k space by the loci of the maxima of reflectance, which is equivalent to the zeroth order diffraction efficiency. The diffraction efficiency of the corrugated silver film was calculated with rigorous coupled wave analysis (RCWA) [14–16] for a TM-polarized incident plane wave. Figure 2 shows the calculated zeroth order efficiency for various minimum film thickness d ₂ values. In the figure the in-plane wave vector k_x is normalized by the grating vector K = 2π/Λ. In the calculations we fixed the pitch of the grating and the maximum thickness of the silver film to Λ = 360 nm and d ₁ = 40 nm, respectively. We used the refractive indices of silver reported by Johnson and Christy [17]. The dielectric constant of the surrounding medium is assumed to be ε ₁ =2.89, which corresponds to a possible gain medium, Alq₃ doped with DCM [5]. On the right hand side of the light line the incident wave is evanescent. In this situation the surface plasmon resonance enhances the reflection field, so that the reflectance may exceed unity.

 

Fig. 2. (a)-(g): Efficiency of the reflected zeroth order diffraction by the corrugated thin silver films with different minimum film thicknesses d ₂ and (h): the loci of the reflection maxima shown in (g) to clarify the existing modes.

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In the case of d ₂ = 40 nm, i.e. no corrugation, there are two modes, the LRSP (high energy) mode and the SRSP (low energy) mode. For d ₂ = 35, 30, and 20 nm, clear plasmonic band gaps can be observed in both LRSP and SRSP modes. As the minimum thickness d ₂ decreases, the band gap increases because the gap width is roughly proportional to the square of the amplitude of the corrugation [18]. Simultaneously, the energy difference between the LRSP mode and the SRSP mode increases because it depends on the average thickness of the metallic film. In this d ₂ region, at the energies of the band gap edges for the LRSP mode, the density of modes increases, as evidenced by the flat dispersion curves. As a result, the energy coupling efficiency from fluorescent molecules to the LRSP mode increases. However, even at the LRSP band edge, the coupling efficiency to the LRSP mode is still lower than that to the SRSP mode due to the broader resonance of the SRSP.

In the cases of d ₂ = 10, 5, and 0 nm, at the upper LRSP band-edge, no energy couples from fluorescent molecules to the SRSP modes because there is no SRSP mode at this energy for these thicknesses d ₂. As a result, most of the energy radiated by the molecule is expected to couples to the LRSP mode. In the cases of d ₂ = 5 and 0 nm a new mode appears, overlapping the lower LRSP mode. However, this mode has no effect on the energy coupling at the LRSP band-gap-edge we are interested in. We will discuss this mode later in Section 4.

Usually, the scattering of surface plasmons into free-space propagating modes increases as the grating amplitude increases. However, at the upper LRSP band-edge the plasmon mode can not scatter into free space because of the opposite parity of the surface plasmon and free-space radiation [13]. Only diffraction into other surface plasmon modes takes place. Thus, in this case the corrugation does not cause any radiation losses.

3. Fraction of power coupled to the LRSP modes

Next, we calculate the energy coupling efficiency from excited fluorescent molecules to the LRSP modes. A molecule, treated as an oscillating dipole, is placed just above the center of one of ridge parts of the corrugation as shown in Fig. 1. The distance of the dipole from the silver surface is denoted by h. For simplicity we assume there is no variation in y direction — the dipoles form a continuous array along a line parallel to the grating. When the dipole vector is parallel to y axis, this situation is identical to the oscillating current in an infinitely long conducting wire, which does not emit photons. Thus, here we consider only the case where the dipole vector is in the xz plane. The angle between the dipole vector and the z axis is denoted by θ.

We calculated the energy dissipation of oscillating dipoles with the method of Ford and Weber [19] in a two-dimensional configuration. If the plasmonic crystal underlays the fluorescent molecules as shown in Fig. 1, not only the specularly reflected field but also the diffracted field contributes to the energy dissipation. The energy dissipation from the oscillating dipole is given by ℘ = (ω/ε ₁ )Im(μ*·E), where ε ₁ is the dielectric constant of the ambient, μ is the dipole moment of the excited molecule, E is the electric field emitted from the dipole including all diffraction orders due to the plasmonic crystal, and * denotes a complex conjugate. The dissipation of energy due to the plasmonic crystal per unit length along the y axis is given by

where k_xm = k _x0+mK and k_zm = [ε ₁(ω/c)² − k_xm ²]^1/2 are the in-plane component and the normal component of the wave vector for the m-th diffraction order, respectively. The in-plane wave vector k _x0 corresponds to the radiation from the dipole incident to the interface. η_m is the m-th order diffraction efficiency for a TM-polarized incidence. μ_x = |μ| sinθ, μ_y = |μ| cosθ and μ is the dipole moment per unit length along the y axis.

Figure 3 shows the energy dissipation as a function of the in-plane wave vector for minimum metal thicknesses d ₂ = 0, 30, and 40 nm. The calculation was done at energies 1.975 eV and 1.952 eV for d ₂ = 0 and 30 nm, respectively. These values correspond to the higher band-edge energy of the LRSP mode in each structure. The dissipation for d ₂ = 40 nm was calculated at the energy of 1.936 eV. The assumed distance of the dipole from the surface and orientation of the dipole were h = 100 nm and θ = 45°, respectively. There are several peaks; peaks at k_x = mK are related to LRSP modes, while the other peaks are related to SRSP modes. However, all the peaks do not necessarily represent energy dissipation by surface plasmons.

To clarify the coupling process, let us take a closer look at which peaks really correspond to energy dissipation by LRSPs. The horizontal axis of Fig. 3 can be considered a wave vector not only for the incidence from the dipole but also for the diffracted waves by the plasmonic crystal. This is simply due to reciprocity. Here, we regard this wave vector k_x as that of the diffracted waves.

 

Fig. 3. Energy dissipation from the oscillating dipole as a function of the wave vector parallel to the metal surface for various values of the minimum thickness d ₂ of the metal film. The dipole height is assumed to be h = 100 nm.

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Let us focus on the peak at k_x/K = 2. It is caused by the following process. Initially, both the incident and the diffracted radiation field with k_x/K = 1 excite the LRSP mode, which has a long lifetime. Then the LRSP mode is diffracted by the corrugation into waves with k_x/K = 2. Finally, the energy at k_x/K = 2 is dissipated by the excitation of electron hole pairs, that is, lossy surface waves. In this process the surface plasmons only play a role of an energy reservoir. All the other peaks at k_x/K = m, excluding m = 1, are caused by lossy surface waves. Therefore, we can conclude that only the peak at k_x/K = 1 represents the dissipation by the LRSP mode, while the peak appearing at a slightly larger wave vector represents the dissipation by the SRSP mode. The other peaks with wave vectors k_x/K < 1 represent dissipation by free space radiation and the remaining peaks with k_x/K >~ 1.2 represent that by lossy surface waves.

The energy coupling efficiencies to the LRSP and SRSP modes are given by the integration of the dissipated energy in the wave vector region of each corresponding peak. We examined the coupling efficiency of the dipole energy to the LRSP at the frequency of the upper plasmonic band-gap-edge, ~ 1.95 eV. Figure 4 shows the efficiency of the energy coupling to the LRSP mode compared to the total dissipated energy as a function of the distance between the dipole and the top surface of the silver film for various corrugation amplitudes.

When the interfaces of the film are flat, i.e. d ₂ = 40 nm, the energy coupling efficiency to the LRSP mode is quite small, no more than 23%, because most energy couples to the SRSP mode, as pointed out by Winter et al. [12]. However, once a corrugation is introduced into the interfaces, the LRSP coupling efficiency increases dramatically. Even with a very small corrugation amplitude of 2.5 nm, i.e. d ₂ = 35 nm, the maximum efficiency reaches 40%. This is mainly caused by the increase of the mode density of the LRSP mode at the band-gap-edge, which is inversely proportional to the slope of the dispersion curve.

The coupling efficiency to the LRSP mode almost monotonically increases as the corrugation amplitude increases. For the structure with d ₂ = 0 nm the maximum energy coupling efficiency exceeds 55%. For d ₂ = 10, 5, and 0 nm, although there is no SRSP mode at the calculated frequency, the increase in the coupling efficiency is not as high as expected. The reason for this is a increased dissipation by lossy surface waves. This can be seen in Fig. 3, where the energy dissipation for d ₂ = 0 nm is larger than that for d ₂ = 30 nm at around k_x/K = 1.5, a wave vector that corresponds to lossy surface waves.

 

Fig. 4. Coupling efficiency of the energy of dipole to the LRSP as a function of the distance between the dipole and the top surface of the silver film for various of minimum film thickness d ₂.

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4. The new mode

Here we discuss the newly appearing mode in the structures with d ₂ = 5 and 0 nm, as shown in Fig. 2. The dispersion curve of the mode seems to follow the lower LRSP mode mode quite closely both in position and shape, although it crosses the LRSP mode in the case of d ₂ = 0 nm due to its larger curvature. In order to investigate the characteristics of this mode, we calculated its field distribution supported by the structure with d ₂ = 0 nm.

Figure 5(a) shows the resulting magnetic field, normalized by the amplitude of the incident field. Figure 5(b) depicts the corresponding charge distribution deduced from the field distribution. For comparison, the charge distributions of the SRSP modes at the upper band gap edge and the lower band gap edge, for example, at points B and C shown in Fig. 2, are shown in Figs. 5(c) and (d), respectively. The charge distribution of the new mode is different from the SRSP mode at the lower band gap edge but similar to that at the higher band gap edge. Thus, this mode may have different properties from the ordinary SRSP modes. There have been many papers that researched surface plasmon modes in metallic slit arrays [20–24]. However, they have not studied such ultra thin metallic films as we investigated, where surface plasmons in both interfaces directly couple with each other.

To rule out the possibility that the new mode could be an artifact of the simulation method, we checked the results with Comsol Multiphysics, a commercial software package based on the finite-element method. The results confirmed that the new mode is indeed an eigenmode of the structures with d ₂ = 0 and 5 nm at k_x/K = 1, and that position of the mode in energy is very close to the lower LRSP branch.

One may attribute the mode to Rayleigh anomalies, because the dispersion curve for the new mode approaches the light lines for diffraction orders while deviating from k_x/K = 1. Steele et al. [25] have reported that dispersion relation of Rayleigh anomalies have band gaps in a thin metallic film with a narrow slit array. However, the reported gap width is much narrower than that obtained in our calculations. Further investigation is needed to evaluate the exact properties and significance of this mode.

 

Fig. 5. (a): Calculated magnetic field distribution of the alternative mode supported by the structure with d ₂ = 0 nm at point A shown in Fig. 2 and (b): the corresponding charge distribution. (c) and (d): Charge distributions of the SRSP modes at the upper and lower band gap edge, for example, points B and C shown in Fig. 2, respectively.

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5. Conclusion

In conclusion, we introduced metallic thin films with two periodically corrugated interfaces that do not support SRSP modes while still supporting LRSP modes, if the the grooves are deep enough. In these structures the energy coupling efficiency from excited molecules to LRSP modes increases as the depth of the grooves increases. When the structure is a metallic slit array, the maximum coupling efficiency reaches 55%. This structure can be used to suppress undesirable decay routes and reduce the lasing threshold of our proposed plasmonic band gap laser.