1. Introduction

How to control the propagation of light is a most important subject in optics. Using the technology of the optical fiber waveguides (OFWs) to pilot the light makes a big advance towards all-optical signal processing. As far as the ever-accelerated miniaturization of optical devices is concerned, however, the conventional OFWs seem to be difficult to satisfy the requirement because of the diffraction limitation for the optical components of dielectric photonic materials. Recently, it is shown that the limitation may be overcome in the rapid developing field of plasmonics [1–4], based on the properties of the surface plasmon polaritons (SPPs), a confined mode localized at the interfaces between metal and dielectric materials [5, 6]. Subsequently wide attentions have been paid to miscellaneous nanostructures with metamaterials involved in pursuit of subwavelength confinement of the light [7]. Among these progresses, several researchers have predicted the subwavelength control of the light in various lattices through the formation of solitons when the nonlinearity is considered. It has been found that the planar nonlinear metal-dielectric multilayers (MDMs), a stack of alternating metal and dielectric nanolayers, can effectively manipulate the propagation of light [8–16] and put a subwavelength confinement on it [17–24]. For example, Zhang’s group theoretically found the subwavelength discrete solitons in the nanoscaled periodic structures consisting of MDMs [17]. Ye et al. also predicted the stable fundamental, vortical and multipole plasmonic lattice solitons in arrays of metallic nanowires embedded in a nonlinear medium [18–20].

In this paper we design a plasmonic fiber waveguide (PFW) by rolling the MDMs to cylindrical shape to guide the light. The PFW can control the light in two-dimensional (2D) transverse space when the light propagates along the axial direction in analogy to the OFWs but based on the plasmonics rather than the optics. The nonlinearity in the dielectric layers is introduced to realize the subwavelength confinement. Based on the advanced nano-technologies [25–27] as well as the sophisticated fiber fabrications, the PFW is supposed to be an easily-fabricated structure. Actually spherical hyperlens made of the MDMs have been realized in experiment [28]. Besides, in contrast to the lattices made of planar MDMs, the structure of the PFW has the cylindrical symmetry, which produces the centripetal confinement when solitons form. Our results show that the energy could be highly confined in the cross section with size far small than the operating wavelength. This kind of confinement combines the advantages of fiber’s long distance transportation with minimized transverse space caused by the cooperation of the nonlinearity and the SPPs’ field enhancement effect. This is significant for applications of plasmonics in future communications and large-scale integrations.

2. Theory and solutions

The structure of the PFW is composed of alternating metal and dielectric coaxial cylindrical layers, as shown in Fig. 1(a) and (b), where the coaxial cylindrical metal layers are labeled as n = 1,2,··· along the radial direction. The thicknesses of the metal layers and the dielectric layers are represented by w and d, respectively. Note that the dielectric material at center is a cylinder with the radius of d. The dielectric constants of metal and dielectric materials are ε₀ε_m and ε₀ε_d, respectively. ε₀ is dielectric constant of the vacuum. A set of typical parameters used in the paper are that: the metal is Ag and its dielectric constants are taken from Ref. [29]; the dielectric constant of the dielectric material is 10; the thicknesses are w = 10nm and d = 40nm, respectively. In fact, the following theory is still applicable when the parameters d and w are functions of n.

 

Fig. 1. (Color online) (a) Cross section of the PFW. (b) Schematics of the PFW. Only first few metal layers are drawn for demonstration. (c) Dispersion of SPPs mode in the 1-th SPW. Distributions of E_z in different SPWs in case of λ = 700 nm (d), and λ = 1550 nm (e), respectively. For clarity, only that in SPWs with indexes 1, 2, 6, and 7 are drawn. The mode amplitudes in different SPWs have been normalized by their energy flows in z direction respectively.

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When the light with the frequency ω propagates in the PFW, the SPPs modes are excited at each interface [30]. Thus the PFW can be considered as a plasmonic multiwaveguide system, of which the i-th single plasmonic waveguide (SPW) is constituted by the corresponding i-th coaxial metal layer and the dielectric environment. The interactions among the propagating modes of SPPs in these SPWs would cause the energy diffraction if no other mechanism counteracts it, as shown below. Firstly we study the guiding modes of the SPWs. By exactly solving the Maxwell equations, we can get the dispersion relations and field distributions [E⁽ⁿ⁾(r), H⁽ⁿ⁾(r)] of the SPPs mode in n-th SPW. Here we only consider the fundamental TM mode, with non-vanishing components $E_z^{\left(n\right)}$, $H_{\varphi }^{\left(n\right)}$ and $E_r^{\left(n\right)}$, which are functions of the radial coordinate r. The dispersion relation of 1-th SPW is given in Fig. 1(c), where β is the propagation constant of the SPPs, n_b is the refractive index of the environment, and k₀ is the wavenumber in vacuum. The dot-dash line labeled by n_bk₀ is the dispersion of the light in the background. From Fig. 1(c) one can see that the propagation constant β has a strong dependence on the frequency and the real part of it is always bigger than that of the background, which is the characteristics of the SPPs. Calculations show that β_n have a small variation for a given frequency, especially for the SPWs with high indexes, which is reasonable as the high-index SPWs can be approximated as planar waveguides with the same width of w. The dispersion relation when n is big enough consist with the results in the planar dielectric-metal-dielectric waveguides [31]. The field distributions of the SPPs modes in different SPWs is sensitive to the frequency. For example, the electric field components $E_z^{\left(n\right)}$ (n = 1,2,⋯,6,7,⋯)of the SPWs are plotted in Fig. 1(d) for λ = 700 nm and in Fig. 1(e) for λ = 1550 nm. It is clear that the interaction between adjacent modes in case of λ = 700 nm is small enough to take a nearest-neighbor approximation while it is not in case of λ = 1550 nm.

For the whole PFW, we express the total electric field E(r,z) and magnetic field H(r,z) in the superposition of the modes in all SPWs: $E_r={\sum }_n{a}_n\left(z\right)E_r^{\left(n\right)}{e}^{-i\omega t}$, $E_z={\sum }_n{a}_n\left(z\right)\frac{{\epsilon }^{\left(n\right)}}{\epsilon }{E}_z^{\left(n\right)}{e}^{-i\omega t}$, and $H_{\varphi }={\sum }_n{a}_n\left(z\right)H_{\varphi }^{\left(n\right)}{e}^{-i\omega t}$, where a_n is the mode amplitude of the n-th SPW, ε and ε⁽ⁿ⁾ are the dielectric constant of the PFW and of n-th SPW, respectively. For simplicity all these modes of SPWs have been normalized by their energy flows $I_n=\frac{1}{2}{\int }_S\text{Re}\left({\mathbf{E}}^{\left(n\right)*}\times {\mathbf{H}}^{\left(n\right)}\right)\cdot {\mathbf{e}}_{\mathbf{z}}\mathit{dS}$ here. Thus the equations describing the dynamics of the mode amplitudes a_n can be determined by the generalized Lorentz reciprocity theorem [32, 33]. Finally we found the equations have the form

In the above derivation for Eq. (1), the change of relative dielectric constant ε_d due to the introduced nonlinearity in the dielectric layers is $n_2{\epsilon }_d^2{|E|}^2$ with the Kerr coefficient n₂ [34]. Also, we have neglected the nonlinear interactions among the SPWs, i.e. items including |a_n′|²a_n (n ≠ n′) are neglected. If the distance d is big enough so that the overlapping of adjacent SPWs’ fields can be ignored, the matrix C can also be approximated as a diagonal matrix. In the case shown in Fig. 1(d), the values of |c_n,n+1/c_n,n|, |c_n,n+2/c_n,n|, |c_n,n+3/c_n,n| are <0.09, <0.002, <0.00003, respectively. But for the case of strong interaction shown in Fig. 1(e), the approximation is not acceptable. The values of |c_n,n+m/c_n,n| decrease below 0.01 until m ≥9. Therefore the Eq. (1) is suitable to general cases, which can also be written as a more simplified form

The soliton solutions are sought in the form of $a_n\left(z\right)=\sqrt{I_0}{u}_n{e}^{i\rho z}$, where the amplitudes u_n and ρ are both independent of z. The intensity defined as I = ∑_n|a_n|² are used in the solutions. We put our emphasis on the unstaggered solitons [17] formed in the self-defocusing media, with the Kerr coefficient set as −1 × 10⁻¹⁵m²/V² and I₀ about 5 × 10⁻⁴W. The number of layers is chosen big enough to ensure the vanishing fields at the boundary. The Guass-Seidel method [35] is used to find the soliton solutions under the conditions mentioned above.

Figure 2(a) plots the amplitude of the longitudinal field component E_z of the soliton in the PFW with I = 0.05I₀. Figure 2(b), with an amplification of one part, presents the distribution of E_z along the radial direction in the case of λ = 700 nm. As E_z reflects the intensity of the light, it is concluded from Fig. 2(a) that the energy mainly concentrates inside the region with the radius of about 400 nm. It is noteworthy that the maximal intensity is at the circle with radius about 250 nm other than at the center, which can be seen more clearly from Fig. 2(b). That is why we call it as the ring-like solitons. It is the nonlinearity that confines the energy in the region with the scale of subwavelength, where the effect of field enhancement provided by the SPPs is prominent, which in turn makes the nonlinearity more achievable. The cylindric symmetry enables energy to concentrate towards the axis, which further promotes the confinement of energy. The propagations of solitons can be obtained by solving the evolution of Eq. (5) with the obtained solitons as the initial values. The lossless propagation of the soliton along 40μm in the z direction is drawn in Fig. 2(c). The linear case under which the diffraction dominates is also plotted in Fig. 2(d) for comparison. Moreover, Fig. 2(b) also shows that metal layers hold a large portion of the energy. For this reason in addition to a relatively big value of Im[β_n] for visible lights, the soliton experiences a high loss in the PFW if the loss in the metal is considered, as shown in Fig. 2(e). A gain medium is therefore necessary for the PFW working at the visible lights.

 

Fig. 2. (Color online) (a) The profile of the longitudinal field component E_z of the soliton with the wavelength of λ = 700 nm in the PFW. (b) The distribution of E_z along the radial direction, where the inset is an amplification of one part. (c) Nonlinear and (d) linear propagation over 40 μm distance in the lossless PFW. (e) Propagation of the soliton in the same PFW when the loss in metal is taken into account.

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In the case of λ = 1550 nm, the ring-like solitons are also found in the same PFW. The propagation of the soliton with I = 80I₀ along 40μm in the lossless situation is plotted in Fig. 2(a), while the lossless linear propagation is shown in Fig. 2(b). Compared with the case of λ = 700 nm, the diffraction is outstanding, because of the strong interactions among the SPPs in different SPWs as discussed before. Consequently a relatively strong intensity is needed to form the highly confined solitons. As the portion of the field residing inside the metal layers decreases correspondingly, the propagation distance increases to some extent in the lossy PFW, which is clearly seen by comparing Fig. 3(c) with Fig. 2(e).

 

Fig. 3. (Color online) (a) Propagation of the soliton along 40 μm in the lossless PFW when the wavelength λ is 1550nm. (b) Linear propagation along 4 μm in the same PFW. (c) Propagation of the soliton along 4 μm when the loss is considered.

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For the lattices solitons, increasing both the intensity and the nonlinear coefficient will enhance the nonlinear effect and thus make the energy confined more tightly. In the PFW, this characteristics is particularly obvious. The profiles of the solitons under three values of the intensity I = 20I₀, 100I₀, 180I₀ are drawn in Figs. 4(a)–(c). The energy of the soliton is confined in a ring of decreasing width with the intensity increasing. Furthermore, because of the nonsymmetry of the matrix T, increasing the intensity also makes the energy move centripetally. It is an exciting result that the highly confined soliton in Fig. 4(c) has a transverse size far small than its wavelength. The peaks of these solitons with more values of intensities are plotted in Fig. 4(d). Calculations show that when the intensity reaches a certain value, the peak stops moving. The reason is that the nonsymmetry of the diffraction is trivial for the considerable nonlinearity in this situation. Besides, as the matrixes T and G are determined by the field distributions of all SPWs, the soliton’s shape can be modified by adjusting the radius and width of each metal layer.

 

Fig. 4. (Color online) (a–c) The profiles of the solitons with the intensity of 20I₀, 100I₀, 180I₀ respectively. The values in each figure have been normalized to their maximum. (d) Radius of the soliton’s peak changing with the intensity.

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3. Conclusion

In conclusion, we designed a PFW composed of the MDMs and predicted theoretically the form of highly confined subwavelength solitons in the PFW both for the visible light and the near-infrared light. The equations describing the propagations of the solitons have been obtained in a compact matrix form. This kind of PFWs is expected to take the role of OFWs in future in the plasmonics-based communications and other nano-photonic applications, such as lithography, beam shaping, etc.