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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 1945–1952
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Ring-like solitons in plasmonic fiber waveguides composed of metal-dielectric multilayers

Jie-Yun Yan, Lu Li, and Jinghua Xiao  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 1945-1952 (2012)
http://dx.doi.org/10.1364/OE.20.001945


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Abstract

We design a plasmonic fiber waveguide (PFW) composed of coaxial cylindrical metal-dielectric multilayers in nanoscale, and constitute the corresponding dynamical equations describing the propagation modes in the PFW with the Kerr nonlinearity in the dielectric layers. The physics is connected to the discrete matrix nonlinear Schrödinger equations, from which the highly confined ring-like solitons in scale of subwavelength are found both for the visible lights and the near-infrared lights in the self-defocusing condition. Moreover, when increasing the intensity of the input light the confinement can be further improved due to the cylindrical symmetry of the PFW, which means both the width and the radius of the ring are reduced.

© 2012 OSA

1. Introduction

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

25.. C. Jiang, S. Markutsya, Y. Pikus, and V. V. Tsukruk, “Freely suspended nanocomposite membranes as highly-sensitive sensors,” Nat. Mater. 3, 721–728 (2004). [CrossRef] [PubMed]

27

27.. Y. Fedutik, V. Temnov, U. Woggon, E. Ustinovich, and M. Artemyev, “Exciton-plasmon interaction in a composite metal-insulator-semiconductor nanowire system,” J. Am. Chem. Soc. 129, 14939–14945 (2007). [CrossRef] [PubMed]

] 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

28.. J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Comm. 1, 143 (2010). [CrossRef]

]. 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 ε0εm and ε0εd, 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

29.. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972). [CrossRef]

]; 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 Ez 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.

When the light with the frequency ω propagates in the PFW, the SPPs modes are excited at each interface [30

30.. E. J. Smith, Z. Liu, Y. Mei, and O. G. Schmidt, “Combined surface plasmon and classical waveguiding through metamaterial fiber design,” Nano Lett. 10, 1–5 (2010). [CrossRef]

]. 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 Ez(n), Hϕ(n) and Er(n), 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, nb is the refractive index of the environment, and k0 is the wavenumber in vacuum. The dot-dash line labeled by nbk0 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 βn 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

31.. J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scal localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]

]. The field distributions of the SPPs modes in different SPWs is sensitive to the frequency. For example, the electric field components Ez(n) (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: Er=nan(z)Er(n)eiωt, Ez=nan(z)ε(n)εEz(n)eiωt, and Hϕ=nan(z)Hϕ(n)eiωt, where an 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 In=12SRe(E(n)*×H(n))ezdS here. Thus the equations describing the dynamics of the mode amplitudes an can be determined by the generalized Lorentz reciprocity theorem [32

32.. S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987). [CrossRef]

, 33

33.. S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5, 174–183 (1987). [CrossRef]

]. Finally we found the equations have the form
iddzCA+(BCK)AX|A|2A=0,
(1)
where A is a vector given by the elements an (n = 1, 2, ⋯) and |A|2 represents a diagonal matrix with corresponding elements |an|2. Matrix B is a diagonal matrix with the elements given by the propagation constants βn. C, K and X are matrices, whose elements are expressed respectively as
Cn,n14S(Er(n)Hϕ(n)+Er(n)Hϕ(n))dS,
(2)
Kn,nω4S(ε(n)ε)K˜n,ndS,
(3)
Xn,nSε0n2εd2αnω4K˜n,ndS,
(4)
where K˜n,n=Er(n)Er(n)ε(n)εEz(n)Ez(n), αn=|Er(n)|2+|ε(n)εEz(n)|2.

In the above derivation for Eq. (1), the change of relative dielectric constant εd due to the introduced nonlinearity in the dielectric layers is n2εd2|E|2 with the Kerr coefficient n2 [34

34.. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991). [CrossRef]

]. Also, we have neglected the nonlinear interactions among the SPWs, i.e. items including |an|2an (nn′) 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 |cn,n+1/cn,n|, |cn,n+2/cn,n|, |cn,n+3/cn,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 |cn,n+m/cn,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
iddzA+TA+G|A|2A=0,
(5)
where TC−1BCC−1K and G ≡ −C−1X. These are discrete matrix Schrödinger equations. For the parameters we adopted, we find that matrix T and G can be approximated as a triple diagonal matrix and a diagonal one, respectively. Neglected elements are at least one order of magnitude less than others. Then Eq. (5) are reduced to the nonsymmetric discrete nonlinear Schrödinger equations with variable nonlinear coefficients. In the equations, the matrix T is determined by the interaction between the SPPs mode in each SPW with with its counterpart in the inner nearest SPW and the outer one. Due to the cylindric symmetry of the PFW, T is a nonsymmetric matrix, which leads to the different diffractions towards the center or outside. The matrix X governs the nonlinearity, which will balance the diffraction when solitons forms. The elements |Gn,n| (n = 1,2,⋯), proved to decrease with the n index, give a centripetally increasing nonlinearity. Both these characteristics contribute to the highly confined SPPs modes in the PFW.

The soliton solutions are sought in the form of an(z)=I0uneiρz, where the amplitudes un and ρ are both independent of z. The intensity defined as I = ∑n|an|2 are used in the solutions. We put our emphasis on the unstaggered solitons [17

17.. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007). [CrossRef] [PubMed]

] 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

35.. S. E. Koonin, Computational Physics (Benjamin/Cummings, Menlo Park, 1986).

] is used to find the soliton solutions under the conditions mentioned above.

Figure 2(a) plots the amplitude of the longitudinal field component Ez of the soliton in the PFW with I = 0.05I0. Figure 2(b), with an amplification of one part, presents the distribution of Ez along the radial direction in the case of λ = 700 nm. As Ez 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[βn] 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 Ez of the soliton with the wavelength of λ = 700 nm in the PFW. (b) The distribution of Ez 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.

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.

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.

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.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11004015 and 61078079), National Basic Research Program of China (Grant No. 2010CB923200), and the Doctoral Fund of Ministry of Education of China (Grant No. 20100005120017).

References and links

1..

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and A. Atwater, “Plasmonics: a route to nanoscale optical devices,” Adv. Mater. 13, 1501–1505 (2001). [CrossRef]

2..

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010). [CrossRef]

3..

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express , 1922029–22106 (2011). [CrossRef] [PubMed]

4..

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009). [CrossRef]

5..

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of suface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007). [CrossRef]

6..

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003). [CrossRef]

7..

B. Lee, I. M. Lee, S. Kim, D. H. Oh, and L. Hesselink, “Review on subwavelength confinement of light with plasmonics,” J. Mod. Opt. 57, 1479–1497 (2010). [CrossRef]

8..

A. Husakou and J. Herrmann, “Steplike transmission of light through a metal-dielectric multilayer structure due to an intensity-dependent sign of the effective dielectric constant,” Phys. Rev. Lett. 99, 127402 (2007). [CrossRef] [PubMed]

9..

S. M. Vuković, Z. Jakšić, I. V. Shadrivov, and Y. S. Kivshar, “Plasmonic crystal waveguides,” Appl. Phys. A 103, 615–617 (2011). [CrossRef]

10..

C. W. Lin, K. P. Chen, C. N. Hsiao, S. Lin, and C. K. Lee, “Design and fabrication of an alternating dielectric multi-layer device for surface plasmon resonance sensor,” Sens. Actuators B 113, 169–176 (2006). [CrossRef]

11..

N. N. Akhmediev, “Nonlinear theory of surface polaritons,” Zhurn. Eksp. Teoret. Fiz. 84, 1907–1917 (1983).

12..

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982). [CrossRef]

13..

F. Lederer and D. Mihalache, “An additional kind of nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 59, 151–153 (1986). [CrossRef]

14..

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987). [CrossRef] [PubMed]

15..

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987). [CrossRef] [PubMed]

16..

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229–313 (1989).

17..

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007). [CrossRef] [PubMed]

18..

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104, 106802 (2010). [CrossRef] [PubMed]

19..

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength vortical plasmonic lattice solitons,” Opt. Lett. 36, 1179–1181 (2011). [CrossRef] [PubMed]

20..

Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84, 033855 (2011). [CrossRef]

21..

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear waves in subwavelength waveguide arrays: evanescent bands and the ’phoenix soliton’,” Phys. Rev. Lett. 102, 163902 (2009). [CrossRef] [PubMed]

22..

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402(R) (2007). [CrossRef]

23..

G. Bartal, G. Lerosey, and X. Zhang, “Subwavelength dynamic focusing in plasmonic nanostructures using time reversal,” Phys. Rev. B 79, 201103 (2009). [CrossRef]

24..

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011). [CrossRef]

25..

C. Jiang, S. Markutsya, Y. Pikus, and V. V. Tsukruk, “Freely suspended nanocomposite membranes as highly-sensitive sensors,” Nat. Mater. 3, 721–728 (2004). [CrossRef] [PubMed]

26..

Z. Jakšić, S. M. Vuković, J. Buha, and J. Matovic, “Nanomembrane-based plasmonics,” J. Nanophotonics 5, 051818 (2011). [CrossRef]

27..

Y. Fedutik, V. Temnov, U. Woggon, E. Ustinovich, and M. Artemyev, “Exciton-plasmon interaction in a composite metal-insulator-semiconductor nanowire system,” J. Am. Chem. Soc. 129, 14939–14945 (2007). [CrossRef] [PubMed]

28..

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nat. Comm. 1, 143 (2010). [CrossRef]

29..

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972). [CrossRef]

30..

E. J. Smith, Z. Liu, Y. Mei, and O. G. Schmidt, “Combined surface plasmon and classical waveguiding through metamaterial fiber design,” Nano Lett. 10, 1–5 (2010). [CrossRef]

31..

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scal localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]

32..

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987). [CrossRef]

33..

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5, 174–183 (1987). [CrossRef]

34..

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991). [CrossRef]

35..

S. E. Koonin, Computational Physics (Benjamin/Cummings, Menlo Park, 1986).

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(190.6135) Nonlinear optics : Spatial solitons
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Optics at Surfaces

History
Original Manuscript: November 8, 2011
Revised Manuscript: December 18, 2011
Manuscript Accepted: December 21, 2011
Published: January 13, 2012

Citation
Jie-Yun Yan, Lu Li, and Jinghua Xiao, "Ring-like solitons in plasmonic fiber waveguides composed of metal-dielectric multilayers," Opt. Express 20, 1945-1952 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-1945


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References

  1. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and A. Atwater, “Plasmonics: a route to nanoscale optical devices,” Adv. Mater.13, 1501–1505 (2001). [CrossRef]
  2. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010). [CrossRef]
  3. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express, 1922029–22106 (2011). [CrossRef] [PubMed]
  4. P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon.1, 484–588 (2009). [CrossRef]
  5. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of suface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys.70, 1–87 (2007). [CrossRef]
  6. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003). [CrossRef]
  7. B. Lee, I. M. Lee, S. Kim, D. H. Oh, and L. Hesselink, “Review on subwavelength confinement of light with plasmonics,” J. Mod. Opt.57, 1479–1497 (2010). [CrossRef]
  8. A. Husakou and J. Herrmann, “Steplike transmission of light through a metal-dielectric multilayer structure due to an intensity-dependent sign of the effective dielectric constant,” Phys. Rev. Lett.99, 127402 (2007). [CrossRef] [PubMed]
  9. S. M. Vuković, Z. Jakšić, I. V. Shadrivov, and Y. S. Kivshar, “Plasmonic crystal waveguides,” Appl. Phys. A103, 615–617 (2011). [CrossRef]
  10. C. W. Lin, K. P. Chen, C. N. Hsiao, S. Lin, and C. K. Lee, “Design and fabrication of an alternating dielectric multi-layer device for surface plasmon resonance sensor,” Sens. Actuators B113, 169–176 (2006). [CrossRef]
  11. N. N. Akhmediev, “Nonlinear theory of surface polaritons,” Zhurn. Eksp. Teoret. Fiz.84, 1907–1917 (1983).
  12. V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B47, 167–173 (1982). [CrossRef]
  13. F. Lederer and D. Mihalache, “An additional kind of nonlinear s-polarized surface plasmon polaritons,” Solid State Commun.59, 151–153 (1986). [CrossRef]
  14. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A35, 1159–1164 (1987). [CrossRef] [PubMed]
  15. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett.12, 187–189 (1987). [CrossRef] [PubMed]
  16. D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt.27, 229–313 (1989).
  17. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett.99, 153901 (2007). [CrossRef] [PubMed]
  18. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett.104, 106802 (2010). [CrossRef] [PubMed]
  19. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength vortical plasmonic lattice solitons,” Opt. Lett.36, 1179–1181 (2011). [CrossRef] [PubMed]
  20. Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A84, 033855 (2011). [CrossRef]
  21. O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear waves in subwavelength waveguide arrays: evanescent bands and the ’phoenix soliton’,” Phys. Rev. Lett.102, 163902 (2009). [CrossRef] [PubMed]
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