Waveguide couplers are important building blocks for photonics circuitry. Conventional photonic (and microwave) waveguides support modes which propagate above certain waveguide width [1]. The structural cutoff of the waveguide under this width may yield a total back reflection (in closed waveguide – e.g. metallic) or reflection + radiation loss (in open waveguide – e.g. dielectric) of an incoming wave propagating in a narrowing taper when approaching the cutoff width value (Fig. 1(a)). In a plasmonic gap waveguide [2,3] i.e. – a dielectrically loaded gap between two real metal plates, the lowest (TM₀) mode is a slow wave (surface plasmon polariton) at all frequencies. A plasmonic wave can never exhibit a structural cutoff in any closed system even for a vanishing size (above the boundaries of the macroscopic model, namely > few nms). Therefore a taper made in such a waveguide does not exhibit a cutoff, which may be utilized for bridging between the micro to the nano world [4] and focus light effectively [5] (Fig. 1(b)). This taper configuration was first suggested as an adiabatic taper for a metal slab waveguide, [5] and later for a gap structure including non-adiabatic plasmonic gap couplers [4,6].

The plasmonic gap may support also backward waves (negative index modes) [7] and related negative refraction phenomena were recently demonstrated in simulations [8] and experimentally [9]. Here we show that the unique characteristics of such waves result in extraordinary structural cutoff, where the waveguide supports the mode only below a certain waveguide’s width. This yields a taper design which is inverted, as illustrated in Fig. 1(c).

 

Fig. 1. Waveguide tapers schematics: (a) photonic or microwave taper may exhibit a cutoff when its lateral dimension is reduced; (b) plasmonic taper supporting slow-waves and light does not exhibit a cutoff; (c) plasmonic taper supporting backward waves exhibit a cutoff when its lateral dimension is enhanced.

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The dielectric function of noble metals (e.g., Au, Ag) at the optical frequency range has a negative real part and the imaginary part related to conductivity, increases with the frequency [10]. When two such metal bulks are separated by a dielectric layer, the resulting gap waveguide exhibits the (lossless text-book [11]) dispersion curves of Fig. 2(a) (depicted for various gap widths, d, where the frequency, ω, is normalized by the plasma frequency – ω_p (plasma wavelength for Au is λ_p=137nm), and the effective index is n_eff=β/k₀, where β and k₀ are the propagation constant and the free space momentum. The TM₀ mode, which has symmetrical H-field, is a slow (plasmonic) mode (below the light line) for any gap width. However, the TM₁ mode (anti-symmetrical H-field) flips between plasmonic (slow) mode to non-plasmonic (fast) mode as the frequency is changed – namely it crosses the light line. In the fast mode regime a cutoff frequency is exhibited when n_eff=0. For large gap width the TM₁ dispersion curve is positive, however at gap width < ~50nm (for Au and a dielectric gap with refractive index =3.5) the slop of the dispersion is (seemingly) turning negative. The negative slop may be erroneously interpreted as negative group velocity, however when metal conductivity is rigorously included and the causal solution is selected (the solution that decays away from the source) – it is obtained that the negative slop dispersion branch is misplaced and the actual solution exhibits positive group velocity but negative phase velocity (negative index modes) (as can be seen in Fig. 2(b)).

Thus the actual solution of the dispersion relations for TM₁ mode of plasmonic gap is depicted in Fig. 2(b) for gap width of 30nm, where the metal permittivity is taken from measured results [10]. At each frequency the effective index of the three modes with the lowest attenuation (for antisymmetric excitation: TM₁ TM₃ and TM₅) in the causal half plane are depicted. Only the negative index branch at the high frequency range (circled by red dashed line in the figure) is a propagating mode, while the other part of the dispersion curve exhibits rapid attenuation. A proper propagating mode is exhibiting Figure Of Merit (FOM= Re{n_eff}/Im{n_eff}) larger than 1, while modes with FOM < 1 are related to as evanescent mode and are over-damped.

 

Fig. 2. (a) Lossless dispersion relations of TM₀ and TM₁ modes in “plasmonic gap” for different gap widths- d, metal permittivity according to loss-less Drude model (λ_p=137nm). (b) Actual dispersion Including loss) of the first 3 anti-symmetric H-field modes for d=30nm. Metal permittivity is taken from measurements [10]. ℰ_D=3.5².

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The same rigorous solution of TM₁ mode at various gap widths results in the dispersion curves of Fig 3(a), (only real effective index for proper propagating solutions – FOM>1 is depicted). For all parameters – the dispersion slope is positive but the index is sometimes negative (Media 1 is showing the negative phase propagation as calculated by FDTD simulation. Details will be discussed later). At the negative index regime – and for a given frequency (that is higher than the ω_spp frequency) these backward waves exhibit a structural cutoff, however it occurs as the gap width is enhanced. In the specific case of λ=622nm (ω=0.22ω_p) the geometrical dispersion curves (of the anti-symmetrical TM modes) are depicted in Fig. 3(b), showing that a cutoff is obtained as the gap size is enhanced above ~45nm. This is typical to negative index modes in any waveguide, e.g. in [12]

 

Fig. 3. (a) Dispersion relations of TM₁ modes in “plasmonic gap” for different gap widths, λ_p=137nm (b) Geometrical dispersion of the first 3 anti-symmetric modes: negative index mode reaching a cutoff (blue) and two evanescent modes (TM3 in red & TM5 in green), λ=622nm (ω=0.22ω_p), Au-Dielectric-Au layers. (Metal permittivity is according to measurements [10], ℰ_D=3.5²) (Media 1).

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The latter suggests that in this frequency regime, a narrow taper with expanding gap width will result eventually in back reflection of the incoming wave, which is counterintuitive compared to ordinary photonic tapers. The inverted taper was analyzed by applying transmission matrix formalism, as illustrated in Fig. 4(a). The taper is assumed to be comprised of linearly varying width plasmonic gap sections stacked along the propagation axis. The dispersion curves show that the only propagating one is the TM₁, while the other modes belong to the evanescent family (over-damped with FOM<l). Therefore the transfer matrix calculations can be based solely on the TM₁ mode. The resulting distribution of the Pointing vector along propagation (S_z) in a specific inverted taper is depicted in Fig. 4(b). At the entrance to the taper (z=0) the gap width is small enough and the TM₁ mode is propagating. A source located at z<0 is launching power to the positive z direction, and being a backward wave the phase propagates backwards into the source. Within the dielectric medium the power flows (necessarily) in the direction of the phase (back into the source), but in the metal the power flows away from the source – which is clearly seen in the Figure. The backwards wave (negative index) characteristic is stemming directly from the fact that more power is propagating in the metal than in the dielectric and therefore most of the power flows anti parallel to the phase. The taper is broadening along its 100nm length to 50nm width at the exit which is above the structural cutoff. Therefore the wave is totally reflected from this taper.

 

Fig. 4. (a) Calculation schematics: modeling the taper as a chain of thin slices, using transfer matrix and propagation in each slice, (b) Power (Pointing vector) distribution along propagation, Sz, for a taper going from 20nm to 50nm gap width. λ=622nm, ℰ_D=3.5², ℰ_M=-86.1-i8.16.

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These modes are expected above the asymptotic ω_spp frequency, and since this high frequency range is close to the plasma frequency, the metal conductivity causes a substantial attenuation. However, the FOM here is about 3-4 – much higher than any other negative index waves in this frequency regime.

In order to better visualize the interesting propagation effects we present in Fig. 5 snap shots of full wave simulations at a longer wavelength (1.5μm) and in order to be in the ‘exotic’ negative index regime we artificially reduced ω_spp by using a fictitious dielectric with n=12 (ω_spp can be actually reduced by diluting the metal [13]). For this scenario the calculated cutoff width is 30nm. Exciting the taper in its narrower port, by an anti-symmetrical field, results in backward waves. Due to the slow-light (low group velocity) nature of these waves, the slow energy propagation is apparent in the simulations – light pulse which would have propagated 15μm in free space, was advanced only by 0.2μm for the same propagation time (central snapshot). As the net light power is propagating away from the source of excitation the phase is evidently moving towards the source (better visualized in the movie, Media 2). As the light reaches the cutoff width at the wider gap region of the taper, its power is back reflected – validating the inverted taper concept.

 

Fig. 5. Negative index waves in an inverted taper: snapshots of H-field distribution (FDTD simulation). λ=1.5μm, ℰ_D=12², din=22nm, dout=32nm, Length=0.5μm. (Media 2)

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In conclusion, the lossless dispersion curves for the TM₁ modes in thin plasmonic gap revealed negative slopes that are attributed by causality to backward waves. These waves propagate only for thin enough gaps and therefore a back reflection from a tapered gap waveguide is exhibited when the taper is broadening – the inverse phenomenon to that of microwave of photonic tapers. This idea may be extended to other structures where the dispersion slope of negative index modes is increased with reduction in the waveguide’s dimension – such as in the case of TM₁ mode of metal slab [2] or NIM layered structure [12,14].