Mode transformation in waveguiding plasmonic structures

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Abstract

We study transformation of surface modes in plasmonic waveguides when the loss strength is varying from zero to realistic values. For the examples of plasmonic modes guided by cylindrical nanorod and metal–dielectric–metal slot waveguides, we show that transformation of dispersion curves follows one of two fundamental scenarios: degeneracy removal of evanescent modes and mode reconnection. We demonstrate that different parts of the same mode dispersion in lossy structures can have different origin in the lossless case.

Research highlights

► Mode transformation in plasmonic systems is studied. ► Transformation of dispersion with losses is analysed. ► Two fundamental scenarios: mode reconnection and degeneracy removal are observed. ► Evanescent and propagation mode mixing with losses is.

Introduction

Study of surface plasmon polaritons became a rapidly developing field of nanophotonics that is expected to enhance functionalities of the key optical components for information processing on a photonic chip, providing light confinement at subwavelength scale and overcoming the diffraction limit [1], [2]. Basic plasmonic elements are metal–dielectric waveguides that support surface plasmon polaritons. Analysis of the modes in plasmonic waveguides is one of the most important problem which was addressed in many papers, including the modes in circular [3], [4], [5], [6], [7] and planar [8], [9], [10], [11], [12], [13], [14], [15], [16] waveguides.

To date, the properties of guided modes in plasmonic waveguides are studied in two limiting cases: (i) ideal structures without losses (see, e.g., Refs. [3], [9]), and (ii) structures with real losses in metals, see e.g. Refs. [7], [16]. In conventional approach, which is usually employed in optics for the analysis of dielectric waveguides [17], the modal structure and dispersion are studied first without losses. Then the losses are added by means of a perturbation theory, since energy absorption is relatively small in dielectric materials. As a result, the dispersion of guided waves in dielectric waveguides is only weakly modified due to presence of loss. However, unlike dielectric waveguides, the dispersions of the modes guided by metal–dielectric waveguides in lossless and lossy cases are dramatically different (cf. Fig. 15 of Ref. [11] and Fig. 2 of Ref. [16]). Thus, we would like to find out how the dispersion of the modes in loss-free case transforms as we increase absorption parameter towards realistic values.

This problem is closely related to the studies of evanescent and complex modes in dielectric and plasmonic waveguides [18], [19]. In lossy systems, all modes are complex, so that both propagating and evanescent modes become ‘mixed’. As we demonstrate below, the modes of structures with strong absorption appear from merging of the dispersion curves of propagating and evanescent modes, and different spectral parts of the same mode can have different origin.

The evanescent modes are known to be important in many problems involving dielectric waveguides (see, e.g., Ref. [18] and the references therein). Among many examples one may find in the literature, we mention that evanescent modes are necessary for satisfying boundary conditions in the problem of coupling into slow-light photonic crystal waveguide without transition region [20].

In this paper, we study transformation of surface plasmon polariton modes in metal–dielectric structures when the loss strength varies from zero to realistic values, and point out dramatic difference between the modes of lossless and lossy structures. Using two examples, we describe two fundamental generic scenarios of the mode transformation: degeneracy removal of evanescent modes and mode reconnection. Thus, different parts of the spectrum of the same guided mode in realistic structure can have different physical origin.

Section snippets

Mode transformation

We study two basic plasmonic waveguides: metallic nanorod embedded into a dielectric medium, shown in the inset of Fig. 1(a), and a slot waveguide, where a dielectric layer is sandwiched between metal, shown in the inset of Fig. 1(b). To underline generality of the undergoing physics, the two structures of similar characteristic dimensions and material parameters are investigated: both nanorod diameter and slot width are 40 nm, refractive index of the dielectric is ɛd = 2.5, and for metal we take

Conclusions

We have revealed that complex modes describing propagation of surface plasmon polaritons in lossy waveguides appear as a mixture of the originally propagating and evanescent guided modes. We have analyzed transformation of guided modes for two metal–dielectric structures, namely a cylindrical metallic waveguide and metal–dielectric–metal slot waveguide, varying losses from zero to realistic values. We have shown that transformation of dispersion curves in different plasmonic waveguides follows

Acknowledgement

We acknowledge a support from the Australian Research Council, and useful discussions with D.N. Neshev and A.A. Sukhorukov.

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