1. Introduction

A single metal/dielectric interface between two semi-infinite media is probably the simplest optical waveguide. Light propagates along the interface in the form of surface waves called surface plasmon polaritons (SPP), as a result of the interaction between photons and the free electrons of the metal [1]. The huge development of plasmonic (or SPP) waveguides during the last decade originates in their unique property to confine light at a single interface [2]. It has been the key for the realization of efficient bio-sensing devices [3, 4] and quantum cascade lasers in the far infrared and THz ranges [5, 6]. In practice, simple plasmonic waveguides are made of thin metal stripes of width in the micrometer range [7, 8, 9, 10, 11]. They allow light confinement at the sub-wavelength scale in direction perpendicular to the metal surface and at the wavelength scale in the transverse direction. Many recent researches were devoted to the achievement of two-dimensional light confinement perpendicular to the propagation direction. A wide variety of plasmonic waveguides have been proposed: metal nanowires [12], metal nanoparticles chains [13, 14], plasmonic bandgap structures [15], subwavelength metal grooves [16], and nanoscale gaps in metal structures [17]. Based on these concepts, subwave-length passive and active optical components have also been proposed [18, 19, 20]. However, the integration of plasmonic waveguides in photonic circuits and devices suffers from two major limitations [21]: the metal absorption (non-radiative losses), and the control of the free-space (radiative) coupling of surface plasmon polaritons.

It is well known that the radiative and non radiative damping of surface waves are modified by a periodical pattern [24]. The periodical structure play the role of a crystal for surface waves, and allows plasmonic band engineering [25, 26]. In such plasmonic crystals, the control of radiative and non radiative damping is related to band gap opening between coupled surface modes having low group velocities.

In contrast to the plasmonic crystals described above, thin nanostructured plasmonic waveguides deposited on a substrate allow to control light absorption and light emission (free-space coupling) of surface plasmon polaritons without modification of their group velocities [27]. This phenomenon is related to the existence of two types of surface waves propagating either at the air/metal interface or at the metal/substrate interface.

In this article, we analyze the physical origin of this phenomenon. Our study is focused on plasmonic waveguides made of thin gold films drilled with very narrow slits and deposited on a GaAs substrate. Previous studies of similar structures were devoted to the coupling between surface waves and free-space propagating light [28, 29]. Here, the interactions between air/gold and gold/GaAs surface waves are investigated experimentally and theoretically. We show that the excitation of SPP leads to Fano-type resonances with specific light localization into the slits of the metal film. This light localization is a key for the enhancement or inhibition of the free-space coupling of surface waves. It results in two propagation regimes: a low-loss propagation regime, and a radiative propagation regime. Both are evidenced by high-resolution angle-resolved transmission measurements and rigorous electromagnetic calculations. We show that the modulation of the transmission intensity along the plasmonic dispersion band is a signature of the modulation of radiative losses of surface waves. This analysis is supported by mode calculations that allow to distinguish the radiative and non radiative losses of SPP modes. It is found that non-radiative losses due to metal absorption are negligible in case of radiative propagation regime. On the contrary, low-loss propagation regime results in very efficient inhibition of free-space coupling despite numerous radiative channels due to diffracted waves.

 

Fig. 1. (a) Schematic view of the nanostructured metallic waveguide made of a gold grating with subwavelength slits (period d, slit width a, thickness t) deposited on a GaAs substrate. Surface plasmon polaritons (SPP) propagating in x-direction on air/metal and metal/substrate interfaces are schematically represented in red. (b) Scanning electron microscope image of the sample. (c) Dispersion curves of SPP propagating along flat semi-infinite air/metal (dark) and GaAs/metal (grey) interfaces. (d) Schematic diagram of SPP dispersion curves for air/metal and metal/substrate interfaces with slightly perturbating periodical patterns.

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2. Dispersion diagram of a thin 1D nanostructured plasmonic waveguide

2.1. Experimental and numerical methods

Thin gold metallic gratings have been fabricated on GaAs substrates, with nanoscale slit width a = 160 nm, period d = 2.9 μm, and metal thickness t = 40 nm (Figs. 1(a) and 1(b)). They have been prepared by electron-beam lithography using a double lift-off technique described in reference [26]. The area of typical grating is 2 × 2 mm². A scanning electron microscope image of the grating is shown in Fig. 1(b).

Angle-resolved transmission measurements have been optimized for high angle resolution [26]. The light was linearly polarized and focused on the sample using an elliptical mirror. The incident spot diameter was 1.7 mm, and the angular resolution was set to Δθ = ±0.3°. Zero-order transmission measurements have been carried out from θ = 0° to θ = 45° in 0.1° increments, by means of a Fourier transform spectrometer (Bruker Equinox 55) and an InSb detector in the 1 μm - 5.4 μm wavelength range. In the following, we plot the absolute transmission intensity through the whole sample (metallic grating on GaAs substrate). The spectral resolution was set to 10 cm^-1 in order to eliminate the Fabry-Perot resonances that occur in the 350 μm-thick GaAs substrate.

Electromagnetic calculations have been carried out, and are compared to experimental results. They are based on the modal expansion in the grating region, and on a plane wave decomposition in both air and substrate regions [30]. The permittivity of gold is modelized by a Drude model: ε_m = 1 - ω ² _p/(ω ² + iωγ). We choose ω_p = 1.29 × 10¹⁶ s^-1 and γ = 1.14 × 10¹⁴ s^-1 as a good approximation for the permittivity of gold in the 1 μm - 5.4 μm wavelength range [31]. The permittivity of GaAs is taken from reference [31]. With these data, the comparison between numerical and experimental results have shown a slight spectral shift of gold/GaAs surface plasmon resonances. It is attributed to a slight modification of the optical index of the substrate at the gold/GaAs interface. It has been taken into account by replacing the refractive index of the substrate n _GaAs by an effective index n _eff = 0.985 × n _GaAs [27].

2.2. Evidence of air/gold and gold/GaAs plasmonic bands

Figure 2 shows the measured (a) and calculated (b) TM-polarized zero-order transmission intensity as a function of the wavenumber σ = 1/λ and the in-plane wavevector k_x = k ₀ sin θ, where k ₀ = 2πσ. An excellent agreement is found between experimental and numerical results. Both transmission diagrams exhibit two series of bands with two different slopes. They clearly reveal the dispersion curves of SPP propagating on both the air/gold interface and the gold/GaAs interface.

The two series of dispersion curves can be described as follows. At the resonance, transmission is enhanced due to the coupling between SPP and diffracted waves. This phenomenon occurs when k ^(p) _x ≈ ±k_spp, where k ^(p) _x = k ₀ sin θ + p2π/d is the in-plane wavevector of the p-order diffracted wave (p = 0, ± 1,…), and $k_{\mathrm{spp}}=k_0/\sqrt{\frac{1}{{\epsilon }_d}+\frac{1}{{\epsilon }_m}}$ is the wavevector of SPP propagating on a flat semi-infinite metal-dielectric interface (see the dispersion relation in Fig. 1(c)), ε_d and ε_m are the permittivities of the dielectric and metallic media, respectively. Since gold is a high conductivity metal, the slope of SPP bands is approximately proportional to 1/n where n is the optical index of the dielectric medium (n = 1 in air and n ≃ 3.4 in the GaAs substrate). The resulting plasmonic band structure is schematically represented in Fig. 1(d) for air/metal and substrate/metal periodically modulated surfaces. In the following, A_p (respect. S_p) plasmonic bands refer to SPP modes coupled to the p-order diffracted wave in the air (respect. in the substrate).

It is remarkable to observe numerous plasmonic bands in Fig 2(a). In particular, metal/substrate S_p plasmonic bands are distinguishable up to p = ±8 orders in experiments. The metal absorption increases with σ, and induces a broadening of SPP resonances, which is clearly seen in Fig. 2.

2.3. Plasmonic band gap

The coupling between two counter-propagating surface waves of the same interface gives rise to plasmonic band gaps at the center and at the boundaries of the first Brillouin zone (k_x = 0 and k_x = ±π/d) [24]. As an example, the plasmonic band gap due to the coupling between A₊₁ and A_-2 modes is shown in insets of Figs. 2(a) and 2(b). In this case, the tiny slits (a/d = 5.5%) induces an extremely narrow gap (δσ/σ ≃ 1.6% according to experimental data). In this example, the high-energy coupled-mode is radiative (bright dot), whereas the low-energy mode is non-radiative. It is worth noting that the plasmonic band gap between S_p bands does not appear in transmission spectra since the resonance widths are much greater than the expected band gaps.

2.4. Fano resonances

Plasmonic resonances shown in Fig. 2 are characterized as Fano resonances, featuring both a minimum and a maximum in transmission spectra (respectively dark and bright bands in transmission diagrams) [32]. Theoretical investigations of transmission metallic gratings have shown that surface resonances can be associated to poles and zeros of an isolated interface [33]. Here both interfaces are uncoupled in the most part of the dispersion diagram since the upper and lower interfaces of the grating are different in nature. Then, in the spectral proximity of surface resonances, the transmission intensity through the whole structure can be approximated by:

 

Fig. 2. Transmission intensity measurements (left (a)) and calculations (right (b)) as a function of the wavenumber σ = 1/λ and the in-plane wavevector k_x, in TM polarization through a gold grating on a GaAs substrate (d = 2.9 μm, a = 160 nm, t = 40 nm). Insets: detailed view of the intersection between A₊₁ and A_-2 dispersion curves at the boudaries of the first Brillouin zone. A detailed view of the white dashed-dotted frames are shown in Figs. 4(a) and 4(b).

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where ω_p and ω ₀ are complex-value frequencies of poles and zeros, respectively, and β is a normalization factor.

Figure 3 shows a transmission spectrum around a Fano resonance of air/metal surface mode for an incident angle of θ = 16°. The same description can be made for metal/substrate resonances. The zero and pole of surface modes are related to the minimum and maximum of the transmission intensity, respectively. They can be interpretated as the result of interference between two radiative channels: direct transmission and resonant transmission through the excitation of surface waves [34, 35].

The maps of the magnetic field intensity illustrate the light-grating interactions around Fano resonances (see A and B in Fig. 3). In case (A) of transmission minimum, the magnetic field H_y (and the tangential electric field component E_x) is nearly zero around the slits as the result of destructive interference between the incident light and the SPP wave. It drastically reduces the electromagnetic coupling between the upper and lower side of the grating through the apertures. On the contrary, in case (B) of transmission maximum, the magnetic field intensity is increased into the slits due to constructive interference between the incident light and the SPP wave, enabling enhanced radiative coupling.

The interaction between A_p and S_p Fano resonances is studied in the following section. A striking feature of transmission diagrams is the absence of band gap phenomenon at the intersections between air/metal (A_-2) and substrate/metal (S_p) plasmonic bands. Nevertheless we shall show that the coupling between both surface modes induces a modification of radiative and non-radiative losses and that these modulations originate from the peculiar field distribution at the Fano resonances.

 

Fig. 3. Left: calculated transmission spectrum around a Fano resonance of air/metal surface mode (incident angle θ = 16°). Right: map of the magnetic field intensity at the minimum transmission wavelength (A) and at the maximum transmission wavelength (B). Red regions show high intensity regions. Dark rectangles show the metallic grating.

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3. Enhancement and inhibition of the radiative coupling of surface waves

In this section, we focus our study on the A_-2 SPP mode schematically represented in red in Fig. 1(d), and we analyze the coupling between surface waves propagating on the two different interfaces.

3.1. Evidence of two (radiative/low-loss) propagation regimes

Detailled views of Figs. 2(a) and 2(b) are shown in Figs. 4(a) and 4(b). Transmission diagrams plotted in Figs. 4(a) and 4(b) exhibit intensity modulations along the bright band of the A_-2 dispersion curve. Both maxima and minima are located at the intersections between the A_-2 mode and the gold/GaAs (S₊₅, S_-6, S₊₆) SPP modes. They can be explained as follows.

Light transmission enhancement is evidenced at the intersections between the A_-2 bright band and one of the S_p bright bands (red arrows in Fig. 4). This is a signature of the enhancement of the radiative coupling of SPP modes to free-space plane waves. Both air/metal and substrate/metal SPP waves are involved in the resonance and coupled together. Light transmission inhibition occurs at the crossing of the A_-2 bright band and Sp dark bands (blue arrows in Fig. 4). In this case, the air/metal SPP wave is nearly uncoupled to the substrate side of the grating. The decrease of transmission intensity reveals the diminution of the radiative losses of SPP resonances [36, 37].

We emphasize that the inhibition of the radiative coupling of air/metal surface modes is due to the zero of the Fano resonance of metal/substrate surface modes (dark bands). As explained above, the zero of Fano resonances is associated to a nearly zero electromagnetic field intensity in the apertures, see Fig. 3 (A). The narrow slits represent the diffracting feature of the structure, so that the stationary minimum of the field in the vicinity of the slits induces an inhibition of the field emitted by the grating.

Hence, light transmission modulation evidences two different regimes for the propagation of surface waves: radiative propagation regime and low loss propagation regime. They are schematically depicted in Fig. 4 (red and blue frames). It must be noticed that the interaction between A_p and S_p modes has a strong effect on the enhancement and inhibition of light transmission despite the absence of coupled mode and bandgap opening. Moreover, this coupling exists with surface waves propagating in the same or opposite directions, their different group velocities remaining unperturbated.

 

Fig. 4. Detailed view of transmission intensity measurements (left (a)) and calculations (right (b)) as a function of the wavenumber σ and the in-plane wavevector k_x. The intensity modulation along the A_-2 SPP band (diagonal) reveals two different propagation regimes: radiative modes induced by a coupling between air/metal and substrate/metal surface waves (red frames), and low-loss modes involving only air/metal surface waves (blue frames).

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3.2. Propagation length

This modification of radiative losses can be studied by the help of a modal analysis as described below. Firstly, we have extracted the transmission maxima T(σ) along the A _-2 bright band of Figs. 4(a) and 4(b). The inverse transmission 1/T(σ) is reported in Figs. 5(a) (measurements) and 5(b) (calculations). Secondly, we have calculated the complex frequency (ω′ - i/2τ) of the A_-2 electromagnetic mode [30], where τ is the resonance lifetime. The propagation length Λ is deduced from Λ = ν_gτ with ν_g ≃ c. The results are plotted in Figs. 5(a) and 5(b) (red curve). In accordance with the previous description, the modification of radiative losses induces identical spectral modulations for 1/T and Λ. These results confirm that the measurements of 1/T along the A_-2 bright band are a signature of the modulation of radiative losses of surface waves.

3.3. Radiative and non-radiative losses

To further analyze non-monotonous variations of the propagation length, we have distinguished radiative and non-radiative losses of SPP modes. Radiative losses are calculated by neglecting the metal absorption [30] (the imaginary part of ν_m is set to 0). They take into account all the (transmitted and reflected) propagating diffracted waves. Figure 6 shows a comparison of the resulting radiative propagation length Λ_r (green) to the propagation length Λ of the real structure (red). It is found that both curves merge in the case of radiative propagation regime (minima of Λ): non-radiative losses due to metal absorption are negligible, and the radiative efficiency Λ/Λ_r is close to 1. On the other hand, the difference between the red and green curves for longer propagation lengths evidences the strong inhibition of radiative emission of SPP waves in case of low loss propagation regimes (Λ/Λ_r = 0.25 at σ = 0.544 μm^-1).

3.4. Inhibition of free-space coupling

The inhibition of free-space coupling is further illustrated in 3Fig. 7. The propagation length (A) of the A_-2 surface modes is compared to the propagation length of surface modes of (coutinuous) metal films without slit, with different thicknesses. We consider the leaky mode propagating on the air-side of the metal film. In the case of non-radiative propagation regimes (maxima of the red curve), it is found that the propagation losses are very close to the continuous film with the same thickness (t = 40 nm) despite numerous radiative channels due to diffracted waves propagating in the substrate and in the air. For instance, the propagation length of the A_-2 mode reaches A = 165 μm at σ = 0.544 μm (λ = 1.84 μm) which corresponds to 86 % of the value of the same structure without slit.

 

Fig. 5. The inverse transmission intensities 1/T(σ) along the A_-2 SPP bright band are extracted from Figs. 4(a) and 4(b). They are reported in (a) (measurements, black points) and (b) (calculations, red points) and compared to the theoretical propagation length ? (red solid lines).

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Fig. 6. Calculated propagation length of SPP on nanostructured plasmonic waveguides with a lossy metal (red solid line) and a lossless metal (green solid line). Dark arrows show radiative (R) and non-radiative low-loss (NR) propagation regimes.

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Fig. 7. Comparison between the propagation length of surface modes on nanostructured plasmonic waveguides (red line), non-structured (continuous) metal films with the same thickness (t = 40 nm, dark line) and with smaller thicknesses (t = 25 nm and t = 37 nm).

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In case of continuous metal films, radiative losses of air/metal surface plasmons can also occur via evanescent coupling through the metal film into plane waves propagating in the substrate. Obviously, the thinner the metal layer, the greater the radiative losses. Figure 7 shows that the modulations of losses of the A_-2 mode follow the losses of metal films with thicknesses in the range t = 25 – 37 nm.

4. Conclusion

In conclusion, we have shown that thin metallic gratings with very narrow slits deposited on a dielectric substrate support two different propagation regimes, leading to a modulation of the propagation length. This property originates in the coupling between Fano resonances of air/metal and substrate/metal surface modes. The radiative propagation regime shows negligible absorption losses, whereas low loss propagation regime allows propagation lengths above 150 μm in the 1.5 – 2 μm wavelength range. The theoretical analysis of radiative and non-radiative losses show that free-space coupling of SPP modes can be inhibited efficiently in case of non-radiative propagation regimes.

It has to be emphasized that thin plasmonic waveguides enable to control radiative and non radiative losses without modification of the group velocity of surface modes. This is very different from classical plasmonic crystals, in which radiative losses can be modified via coupled modes having low group velocities. These properties open a route toward the design of nanostructured plasmonic waveguides with finely tuned radiative losses. Propagation lengths above 100 μm makes feasible the integration of thin nanostructured plasmonic waveguides in photonic circuits [18, 19, 20]. The ability to control the trade-off between propagation length and light emission is also of high importance in surface-emitting devices like quantum cascade lasers [5, 6].