Ultra-broadband interconnection between two SPP nanostrips by a photorefractive soliton waveguide

Hamed Tari; Alessandro Bile; Arif Nabizada; Eugenio Fazio

doi:10.1364/OE.489886

1. Introduction

Surface plasmon polariton waves (SPP) have had a great development in recent years, both in the field of sensors [1] and for applications in the development of new optical devices with very high integration, i.e. in dimensions much smaller than the light wavelength [2]. In fact, SPP waves can be confined within very small area at the interface between a metal and an insulating layer, beyond the diffraction limit of the conventional optical waveguides and provide extra miniaturization of the interconnections in the chip [3].

However, optical propagation in the metal layer imposes an intrinsic cost to the propagation length due to the very high absorption that light undergoes in a conductor. Furthermore, the obtainable integrated devices are mainly linear in nature, since it is difficult if not impossible to excite optical nonlinearities in metals. Consequently, the limited propagation and the absence of nonlinearity of SPP waves strongly reduces their applicability, especially in the visible wavelength range. To solve these problems, several technological solutions have been proposed concerning the waveguide architecture such as the realization of wedge plasmons polariton [4], channel plasmon polariton [5] or the use of cylindrical geometries [6].

These structures are partially able to improve the propagation distance while maintaining a good optical multiplexing capability thanks to the linearity of the refractive response. However, still none of the proposed solutions is capable of either transmitting signals over long distances or producing active addressing.

Recently, an innovative method of interconnection between metallic waveguides sup-porting SPP has been proposed using photorefractive soliton channels [7]. A soliton waveguide is a system that exploits the variation of the refractive index induced by a soliton beam to confine and propagate light signals [8]. By exploiting the photorefractive nonlinearity, for example, of lithium niobate substrates [9], waveguides can be made with a refractive index contrast as high as 10⁻³−10⁻⁴, even 10–20 mm long and above all with very low losses propagation (0.04 and 0.07 dB/cm [10]) much lower than any waveguide made with traditional techniques. This is possible because such structures are only self-written by the light which “chooses” the best refractive index profile for diffraction-free propagation.

Furthermore, being self-written, their characteristics depend on the specifics of the light that generated them and also that passes through them: in this way, soliton waveguides can “learn” information and memorize it [11]. This memory effect of photorefractive materials provides exceptional characteristics to these channels written in them, paving the way for a wide range of applications including, for example, the realization of neural networks based on graphs of intersecting soliton channels, whose junctions can open or close in consequence of the information to be processed [12–15].

In this paper we analyze the interconnection of two metallic strips acting as waveguides for SPP signals with self-written solitonic channels. Light propagating in the form of SPP undergoes diffraction if the metallic guiding strip is abruptly interrupted. Using photorefractive substrates, this diffracted light can confine itself in a soliton channel, whose propagation characteristics can be suitably controlled and modulated.

2. SPP propagation

2.1 Materials’ geometry and method

The analyzed system geometry is shown in Fig. 1.

Fig. 1. Sample scheme (out of scale).

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A metallic nanostrip of Silver, as thick as 250 nm, is placed at the top of a thin layer of Indium-Tin-Oxide (ITO) and it acts as a waveguide for SPP waves propagating at the lower interface between the metal and ITO. The ITO layers (top and bottom) have a dual role: they act as a dielectric for light (in the VIS-NIR spectral range, down to 1.3 µm wavelength, with a positive dielectric constant unlike silver [16]) and simultaneously form the plates of a plane capacitor to apply a bias static electric field necessary for the soliton formation [17] (as described in paragraph 3). The top ITO layer, as thick as 500 nm, isolates the SPP interface from the photorefractive substrate: thus, the SPP propagation is not at all affected by the photorefractive nonlinearity. Its thickness was designed to have a reduced divergence of the diffracted light at the end of the metal strip, as described in the next paragraph 2.2. The Ag nanostrip was 20 µm long, abruptly stopped at the end. Thus, the SPP wave propagates until this metallic strip end and then diffracts in the surrounding media. The second conductive stack, present at the bottom of the substrate, is constituted by a 10 nm thick layer of ITO followed by 40 nm of silver. In this case a very thin layer of ITO was chosen to ensure both a very efficient tunnelling of the light that must reach the bottom metallic layer to recouple the radiation as SPP wave and an efficient static bias application. On the bottom of all, a further semi-infinite insulating sub-layer allows the formation of the second SPP wave along the lower metal strip.

The SPP propagation at the silver-ITO interface is described by the following SPP-wavevector components [18] (see appendix for details):

(1)$$\left\{ {\begin{array}{l} {{k_x} = {k_0}\sqrt {\frac{{\varepsilon {^{\prime}_M}.{\varepsilon_{ITO}}}}{{\varepsilon {^{\prime}_M} + {\varepsilon_{ITO}}}}} \cdot \left[ {1 + i\frac{{\varepsilon {^{\prime\prime}_M}}}{{2\varepsilon^{\prime}_M{^2}}} \cdot \frac{{\varepsilon {^{\prime}_M}.{\varepsilon_{ITO}}}}{{\varepsilon {^{\prime}_M} + {\varepsilon_{ITO}}}}} \right]}\\ {{k_{z,metal}} = i\; {k_0}\frac{{\varepsilon {^{\prime}_M}}}{{\sqrt {\varepsilon {^{\prime}_M} + {\varepsilon_{ITO}}} }}\; }\\ {{k_{z,ITO}} ={-} i\; {k_0}\frac{{{\varepsilon_{ITO}}}}{{\sqrt {\varepsilon {^{\prime}_M} + {\varepsilon_{ITO}}} }}\; } \end{array}} \right.$$

where ${k_0}$ is the light wavevector in the vacuum, $\varepsilon {^{\prime}_M}$ and $\varepsilon {^{\prime\prime} _M}$ are the real and imaginary part of the complex permittivity of the metal and ${\varepsilon _{ITO}}$ the permittivity of the ITO dielectric layer (we have neglected its imaginary component as we are considering it in its insulating regime).

By appropriately sizing the metal nanostrip (larger than 200 nm), we forced the light to propagate exclusively at the lower metal-dielectric interface, avoiding coupling with the upper one.

After SPP propagation along with the Ag/ITO interface, the SPP wave will be diffracted from the end of the strip, partly in the photorefractive medium and partly in the upper insulator.

The whole system is covered by a 2-µm thick film of SiO₂ as insulating cladding.

2.2 SPP propagation and diffraction

In Fig. 2 we report the simulations of the SPP wave (at 532 nm) propagation along the metallic nanostrip and its diffraction when the nanostrip in interrupted. In Fig. 2(A)–2(B) we have reported the SPP diffraction from a metallic nanostrip completely buried inside and homogeneous medium, i.e., without the presence of any media interface. As you can see, the metallic discontinuity induces a large diffraction that spreads all around. The presence of an interface between the cladding and the substrate (Fig. 2(C)) forces the diffracted light to propagate more inside the substrate than the cladding. That is because the metallic edge points towards the substrate and the diffracted light cannot return in the cladding due to an internal total reflection at the interface. If now a thin film of ITO is located below the metallic strip (2D), the light diffraction inside the cladding is spatially more confined due to a Fabry-Perot resonance filtering that reduces angular dispersion.

Fig. 2. Surface-Plasmon-Polariton wave propagation and diffraction. A) Magnification of B) showing the SPP propagation along the metallic nanostrip and its diffraction when the strip is interrupted. C) SPP diffraction directly inside the photorefractive substrate, without the ITO layer. D) SPP diffraction in presence of an ITO layer as thick as 500 nm. E) Divergence angle of the diffracted light as function of the ITO layer thickness. F) Angle of the highest diffracted light intensity.

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Therefore, the ITO layer plays a fundamental role on the collimation of the diffracted beam, even in the linear regime, i.e. without the action of nonlinearity. In fact, as it can be seen in Fig. 2(E), by varying the thickness of the ITO from 250 nm up to 500 nm a strong reduction of the angular dispersion of the diffracted light is observed at the end of the metallic strip. This drop down is generated by the spatial filtering that the ITO layer exerts as a Fabry Perot resonator (Fig. 2(E)). Taking advantage of Fig. 2(E), it was decided to use a 500 nm ITO layer in order to have an oriented and collimated diffraction as much as possible, in order to favor the subsequent soliton formation. The Fabry-Perot resonance influences the absolute angle of diffraction too, as shown in Fig. 2(F). However, this is a minor effect (of the order of few degrees) and does not affect the total dynamics so much.

3. Photorefractive screening soliton generation

3.1 Photorefractive screening soliton: model and materials

We have solved the complete set of equations describing the photorefractive nonlinearity and the screening soliton formation [19]. Light photons (F) can be absorbed by donors (${N_D}$) of a photorefractive crystal inducing electron transitions (${N_e}$) in the conduction band. Thus, the ionized donor population $N_D^ + $ rate equation is:

(2)$$\frac{{dN_D^ + }}{{dt}} = \sigma F({{N_D} - N_D^ + } )- \gamma {N_e}N_D^ + $$

where σ is the absorption cross section and γ the recombination probability. Consequently, a local charge population is generated, formed by excited donors, electrons and intrinsic acceptors:

(3)$$\rho = q({N_D^ +{+} {N_A} - {N_e}} ),$$

where q is the modulus of the electron charge, and according to the Navier–Stokes equation, can migrate in the form of electric current inside the photorefractive crystal because of diffusion/self-generated photovoltaic field/conduction induced by an externally-applied static electric bias:

(4)$$\vec{J} = \; \mu kT\vec{\nabla }{N_e} + q\mu {N_e}\vec{E} + \eta \sigma F({{N_d} + N_D^ + } )\hat{c}$$

where µ is the electron mobility, k the Boltzmann constant, and η the photovoltaic efficiency. The $\hat{c}$ direction corresponds to the fast crystallographic axis, i.e. the optical axis of the photorefractive crystal, along which the photorefractive nonlinearity is strongest. In Eq. (4) the $\vec{E}$ term describes the whole local electric field, combination of the applied bias and the photogenerated screening one:

(5)$$\vec{\nabla } \cdot {\vec{D}_{SC}} = \rho ,$$

thus

(6)$$\vec{E} = {\vec{E}_{bias}} + \frac{{{{\vec{D}}_{SC}}}}{{{\varepsilon _{PR}}}}.$$

Such field is the one responsible for the nonlinear refractive index due to the electro-optical effect:

(7)$$\left\{ {\begin{array}{l} {{n_x} = {n_y} = {n_0}}\\ {{n_z} = {n_e} - \frac{1}{2}n_e^3{r_{33}}E.\frac{1}{{1 + \frac{I}{{{I_{sat}}}}}}} \end{array}} \right..$$

Here ${r_{33}}$ is the electro-optical coefficient of the photorefractive material, ${n_0}$ and ${n_e}$ identify the ordinary and extraordinary refractive indices respectively for uniaxial crystals, and ${I_{sat}}$ the luminous intensity that produces a saturation of the material's absorption.

This is the classic model of photorefractivity defined for example by the Zozulya-Anderson theory [20].

For the solution of Eqs. (2)-(7), we have considered a Strontium-Barium-Niobate (SBN) crystal doped with Cerium as photorefractive substrate. Such material exhibits large electro-optic coefficient, a rapid dielectric relaxation with almost no residual electrical memory [21]. Thus, it is one of the best candidates for the soliton-confinement purposes.

In tab. 1 we report the numerical data used.

Table 1. Material parameters used for the numerical experiments [22]

View Table

3.2 Soliton formation

In the experiment we injected 2 µW power of light 532 nm as SPP wave propagating along the lower interface of the upper metallic strip. An electric bias as high as 5 kV/cm was also applied across the photorefractive material (using the two Ag-ITO stacks as plates of a plane capacitor) to generate the screening soliton beams.

In Fig. 3 the formation of the soliton channel is shown as a function of time: the images of the spatial field of charge are shown in detail at the top, the variation of the refractive index due to the electro-optical effect at the center and the intensity profiles at the bottom. The simulations are reported up to 75 s after the light signal is turned on, which means from the beginning of the SPP wave diffraction to the end of the metal nanostrip.

Fig. 3. Formation of a screening soliton from the SPP diffraction.

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At the beginning (t = 0) there are no photo-excited charges and, consequently, the refractive index variation is homogeneous and caused by the applied electric bias only. The light has a linear diffraction governed by both the abrupt interruption of the metallic SPP waveguide and by the Fabry-Perot resonance of the ITO layer underlying the silver strip, as described in the previous paragraph. As time passes, the absorption of light sets in motion the electric charges which induce a screening electric field, cause of a local modification of the refractive index. This modulation of the refractive index is responsible for focusing the light which, in turn, increases its local intensity too. This is virtuous feedback which slowly leads the system to self-confine along a channel of photoexcited charges which corresponds to a channel of varied refractive index (induced solitonic waveguide).

The presence of the lower metallic plate speeds up this evolution: in fact, this metal acts as a reservoir of free charges which, due to the light and the electric bias, can be transferred to the photorefractive crystal, accelerating the variation of the refractive index and, consequently, the confinement of the light.

3.3 Solitonic waveguiding

The modified refractive index channel induced by the photo-excited charges acts as a waveguide not only for the writing light but for other wavelengths too [23]. In particular, this waveguide can be also exploited by other wavelengths capable of propagating in the form of SPP but which cannot induce nonlinearities by themselves in the photorefractive material due to very low or no intrinsic absorption. This is for example the case of the radiation at 1064 nm, used here as a carried signal (Fig. 4).

Fig. 4. Intensity of the waveguided light at 1064 nm within a soliton channel created at 532 nm.

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This radiation can still propagate efficiently as an SPP wave along the silver nanostrip, obviously with a different wave vector than that of the 532 nm radiation due to both the different wavelength and the dispersion of the refractive indices. This difference does not alter the propagation but may originate a different diffraction pattern at the end of the nanostrip which could affect the coupling inside the photoinduced waveguide. However, the spatial filtering exerted by the ITO nanolayer underlying the silver selects the wave vectors that can be transmitted, reducing the angular dispersion and favoring the coupling within the soliton channel.

As you can see from Fig. 4, the light coupled inside the channel oscillates transversely (swing effect [24]), varying its size (the color change along propagation does not depend on the power losses but on the difference in section that the beam assumes).

We have calculated a coupling efficiency from SPP to the soliton waveguide of the order 85–90% (nominally 88 ± 5%) and propagation losses within channel of about 15%. About 70% of the power launched as SPP engraves the lower metallic layer.

4. Recoupling of the soliton channel into the second metallic nanostrip

The light carried inside the soliton waveguide can be recoupled into the underlying metallic nanostrip still in the form of SPP wave. The necessary condition for the light to be recoupled is the satisfaction of the dispersion condition of the wave vectors in Kretschmann configuration [25], as shown in Fig. 5(a). By appropriately choosing the material of the insulating sub-layer (Fig. 1), the dispersion curve of the light and of the plasma wave can be tuned so as to ensure their overlap for an extended region in the space k/k0 (fig.5a).

Fig. 5. Waveguiding and SPP recoupling at different wavelengths. A) The dispersion curves have been matched along an extended area in order to ensure an almost-constant recoupling B) over a large range of wavelengths. C) Signals at different wavelengths propagating within the solitonic waveguide and recoupled as SPP wave at the bottom metallic nanostrip.

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In the specific case under examination, i.e. with a photorefractive substrate identified by an SBN crystal, the dispersion matching in fig.5a has been obtained by choosing the Norland Optical Adhesive 170 (NOA-170) as insulating sub-layer, which provides the proper refractive index mismatch.

NOA-170 is a yellow liquid adhesive that cures to a clear film when exposed to ultraviolet and/or visible light, providing a cost-effective and practical solution to the problem. Such material has a refractive index ranging from 1.71 to 1.73, which ensures a good recoupling of light as SPP for a very broadband of wavelengths (fig.5b). It is worth mentioning that the coupling efficiency calculated here for the interconnection is defined as the ratio between the IR signal power recoupled as SPP and the power transported into the solitonic channel. As you can see, the SPP recoupling gets an almost flat efficiency: between 40 and 42% in the range 1070–1190 nm (bandwidth more than 100 nm), and in any case higher than 35 in the range 1050–1250 nm (bandwidth of about 200 nm). In Fig. 5(C) we have represented the soliton waveguiding and the recoupling for different wavelengths, covering the most important telecom bands.

The overall propagation losses of the solitonic interconnection have been evaluated by comparing the SPP power arriving at the diffraction point (end of the upper waveguide) and the power recoupled in the form of SPP wave at the lower metallic strip, and they reach a value of about 6–7 dB. We estimate that these losses are mainly due to the couplings of light entering and leaving the soliton waveguide channel since it is known [10] that these waveguides have very low propagation losses. In our case we have estimated propagation losses of the order of 10⁻⁴−10⁻⁵dB/µm or less, which are significantly lower than those reported in literature, which range from 0.14 dB/µm to 0.95 dB/µm [26,27], for devices without solitonic interconnections.

5. Conclusions

The generation of a solitonic channel is certainly an excellent tool for interconnecting distant metallic waveguides together, thus lengthening the propagation of the SPP waves and allowing the realization of complex circuits. The use of a photorefractive material as a means to realize the interconnections offers a further important advantage: to provide the plasmon signals with a usable nonlinearity for active systems, able to perform intelligent signal processing. Specific circuit geometries based on soliton waveguides have already demonstrated [12] that they can learn information, memorize it and use it for fully optical recognition. The present study allows the transfer of optical knowledge in the nonlinear field and in signal processing to the plasmonic framework.

Appendix: SPP wavevector derivation

Let’s consider the propagation of a surface wave at the interface between a metal (medium 1) and an insulating (medium 2). Neglecting the absorption of medium 2, we consider the dielectric constant ɛ₁ complex and ɛ₂ real:

(8)$${\tilde{\varepsilon }_1} = \varepsilon {^{\prime}_1} - i\varepsilon {^{\prime\prime} _1}$$

(9)$${\tilde{\varepsilon }_2} = \varepsilon {^{\prime}_2}$$

Since an SPP wave is an evanescent wave in both media constituting the interface, we expect the propagative components (k_x) of the wave vectors to be complex, to account for ohmic absorption losses in the metal, while the transversal ones (k_z) are purely imaginary:

(10)$${\tilde{k}_x} = k{^{\prime}_x} - ik{^{\prime\prime} _x}$$

(11)$${\tilde{k}_z} ={-} ik{^{\prime\prime} _z}$$

with the conditions that in each medium:

(12)$$\tilde{k}_x^2 + \tilde{k}_{z1}^2 = {\tilde{\varepsilon }_1}k_0^2$$

(13)$$\tilde{k}_x^2 + \tilde{k}_{z2}^2 = {\tilde{\varepsilon }_2}k_0^2.$$

where k₀ is the wavevector in the vacuum. Please note that in both media ${\tilde{k}_x}$ gets the same value because the two evanescent wave components must propagate at the same speed independently on where they are. The Gauss's theorem ensures that, in the absence of free electric charges, the dielectric displacement vector has zero divergence in either medium. From this we derive the following relations:

(14)$${\tilde{k}_x}{E_{1x}} + {\tilde{k}_{z1}}{E_{1z}} = 0$$

(15)$${\tilde{k}_x}{E_{2x}} + {\tilde{k}_{z2}}{E_{2z}} = 0$$

where the components of the electric fields must satisfy the following conditions of continuity at the interface:

(16)$${E_{1x}} = {E_{2x}}$$

(17)$${\tilde{\varepsilon }_1}{E_{1z}} = {\tilde{\varepsilon }_2}{E_{2z}}.$$

Using Eqs. (16) into (15) and subtracting (14) and (15) we get:

(18)$${\tilde{k}_{z1}}{E_{1z}} = {\tilde{k}_{z2}}{E_{2z}}$$

that, together with Eq. (17), gives rise to:

(19)$$\frac{{{{\tilde{k}}_{z1}}}}{{{{\tilde{\varepsilon }}_1}}} = \frac{{{{\tilde{k}}_{z2}}}}{{{{\tilde{\varepsilon }}_2}}}.$$

Subtracting (12) and (13) it is possible to obtain a second relation connecting together ${\tilde{k}_{z1}}$ and ${\tilde{k}_{z2}}$:

(20)$$\tilde{k}_{z1}^2 - \tilde{k}_{z2}^2 = ({{{\tilde{\varepsilon }}_1} - {{\tilde{\varepsilon }}_2}} )k_0^2.$$

From Eqs. (19) and (20) we obtain the expression for the z-components of the wavevectors:

(21)$$\tilde{k}_{z1}^2 = k_0^2\; \frac{{\tilde{\varepsilon }_1^2}}{{{{\tilde{\varepsilon }}_1} - {{\tilde{\varepsilon }}_2}}}$$

(22)$$\tilde{k}_{z2}^2 = k_0^2\; \frac{{\tilde{\varepsilon }_2^2}}{{{{\tilde{\varepsilon }}_1} - {{\tilde{\varepsilon }}_2}}}.$$

Still using Eqs. (12) and (13) it is possible to determine the x-component of the wavevectors:

(23)$$\tilde{k}_x^2 = k_0^2\; \frac{{{{\tilde{\varepsilon }}_1}{{\tilde{\varepsilon }}_2}}}{{{{\tilde{\varepsilon }}_1} - {{\tilde{\varepsilon }}_2}}}\; .$$

Now we impose the conditions that

(24)$$\tilde{k}_x^2 \in \,\mathrm{{\mathbb C}}$$

(25)$$\tilde{k}_{z1}^2,\tilde{k}_{z2}^2 \in \,\mathrm{{\mathbb R}}.$$

Under the assumption that for a metal

(26)$$|{\varepsilon {^{\prime}_1}} |\gg |{\varepsilon {^{\prime\prime}_1}} |$$

we obtain the final expressions:

(27)$${\tilde{k}_x} = {k_0}\sqrt {\frac{{\varepsilon {\mathrm{^{\prime}}_1}.\varepsilon {^{\prime}_2}}}{{\varepsilon {\mathrm{^{\prime}}_1} + \varepsilon {^{\prime}_2}}}} \cdot \left[ {1 + i\frac{{\varepsilon {^{\prime\prime}_1}}}{{2\varepsilon \mathrm{^{\prime}}_1^2}} \cdot \frac{{\varepsilon {\mathrm{^{\prime}}_1}.\varepsilon {^{\prime}_2}}}{{\varepsilon {\mathrm{^{\prime}}_1} + \varepsilon {^{\prime}_2}}}} \right]$$

(28)$$|{{k_{z1}}} |= {k_0}\frac{{\varepsilon {\mathrm{^{\prime}}_1}}}{{\sqrt {\varepsilon {\mathrm{^{\prime}}_1} + \varepsilon {^{\prime}_2}} }}$$

(29)$$|{{k_{z2}}} |= {k_0}\frac{{\varepsilon {^{\prime}_2}}}{{\sqrt {\varepsilon {\mathrm{^{\prime}}_1} + \varepsilon {^{\prime}_2}} }}$$

Funding

Sapienza Università di Roma (AteneoPiccoli2022#000300_22-FazioAMDG, AteneoSeedPNRR2021i#000300_21-Fazio, AvvioRicerca2022-AR2221814D17193B-Bile).

Disclosures

The authors declare no conflicts of interest

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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SBN:Ce	Ordinary refractive index (n_o)	2.359 @532 nm ; 2.250 @1064nm
	Extraordinary refractive index (n_e)	2.283 @532 nm ; 2.225 @1064nm
	Photorefractive coefficient (r₃₃)	1340 pm/V
	Absorption cross-section (σ)	2.553·10⁻²³ m² @532 nm ; 0 @1064nm
	Charge mobility (µ)	2.27·10⁻⁶ m²/Vs
	Total donor density (N_D)	4.7·10²⁴m⁻³
	Total acceptor density (N_A)	1.7·10²²m⁻³
	Recombination rate (γ)	1.0·10⁻¹⁶ m³/s

ITO.	DC-conductivity	1.3 ·10⁶ s/m

Ultra-broadband interconnection between two SPP nanostrips by a photorefractive soliton waveguide

Abstract

1. Introduction

2. SPP propagation

2.1 Materials’ geometry and method

2.2 SPP propagation and diffraction

3. Photorefractive screening soliton generation

3.1 Photorefractive screening soliton: model and materials

3.2 Soliton formation

3.3 Solitonic waveguiding

4. Recoupling of the soliton channel into the second metallic nanostrip

5. Conclusions

Appendix: SPP wavevector derivation

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (29)

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