## Abstract

We report on the propagation characteristics of a plasmonic waveguide structure based on two coupled rectangular wedges. Dispersion, propagation loss, and field distributions are investigated by three-dimensional finite-difference time-domain method. The considered structure supports only one supermode over 30THz bandwidth, and the calculated propagation loss at *λ*=1.55μm is 0.0257dB/μm, which is lower than the existing report by 1.7 times while keeping comparable field localizations. The all-planar structure in conjunction with the linearly dispersive characteristic over a wide operational bandwidth signifies its great potential for optical signal transporting in nanophotonic circuits.

© 2007 Optical Society of America

## 1. Introduction

Photonic components are superior to electronic ones in terms of the large operational bandwidth, fast switching speed, and low cross-talk, enabling all-optical signal processing and system on chip (SOC) applications. With the growing demand of miniaturized photonic devices and high density optical integrated circuits (OICs), plasmonic waveguides (PWs) have re-grab people’s attention for the great capability of overcoming diffraction limit [1–5] by transporting lightwave in form of surface waves along the interface between metal and dielectrics. In contrast to conventional optical waveguides operated based on total internal reflection (TIR), PWs first turn the incident light into surface plasmon polaritons (SPPs) [6] and later transport SPPs along the metal-dielectric interface due to boundary condition constraints. As a consequence, the electromagnetic field maintains its maximum at the interface while evanescently decays into both-side surroundings. In order to compromise the enhanced field localization and the propagation length, a variety of PW structures such as rectangular metallic stripes [7], metallic rods [8], metallic nanochains [9], metallic gaps [10,11], and trapezium-like nanowedges [12] etc have been proposed and widely investigated. Up to now, only channel plasmon polaritons (CPPs) [13–15] in metallic V-grooves have been demonstrated to satisfy simultaneously low propagation loss and strong localization [16]. In addition, theoretical simulation predicts that CPPs can be guided through sharp bends with nearly zero propagation loss [17] and high tolerance to structural imperfections [16], pushing a step further toward to the construction of compact nanophotonic circuits.

However, from the fabrication point of view, it is not easy to fabricate sharp V-shaped grooves by using standard semiconductor lithography techniques. To reduce the run-to-run variations and to increase the reliability of the manufacturing process, an all-planar structure with relaxed fabrication tolerance is favored. Here, we analyze a structure based on two coupled rectangular wedges as shown in Fig. 1. Unlike the so-called slot waveguides [18], the two coupled metallic films were deposited on a common metallic slab which may effectively suppress the modes located at the two bottom corners. The idea is analogous to the difference between a dielectric rib waveguide and a stripe waveguide where the lateral confinement of the former waveguide is much less than that of the latter one and single mode operation with an oversized core is thereby feasible for the rib waveguide [19]. This property allows one to tailor the structural parameters of the analyzed PW and make it at all time single-moded. As a result, a supermode consisting of two coupled wedge-plasmon-polaritons [20] [21] can be guided along the upper corners of the trench. In the following sections, the propagation characteristics including the modal propagation constant, attenuation coefficient and dispersion coefficient are addressed.

## 2. Simulation methodology

A three-dimensional finite-difference time-domain (3D FDTD) algorithm is developed for analyzing the propagation properties of the PWs. The development of the 3D-FDTD code is based on the standard Yee cell [22]. For modeling the interaction of light with metal, the recursive algorithm [23] is included into the FDTD framework in order to update the calculated electromagnetic fields. Specifically, the metal is modeled via the classical Drude’s model [24] in which the time dependent relative permittivity is parameterized as *ε*(*t*) = *ε*
_{∞}{1 + *ω*
^{2}
_{P}[1 - exp(-*v _{c}t*)] /

*v*}, where

_{c}*ε*

_{∞}is the background relative permittivity of the metal,

*ω*is the angular plasma frequency, and

_{P}*v*is the collision frequency. The mesh layout, boundaries and excitation conditions for the calculation are depicted in Fig. 2(a) and 2(b). As shown in Fig. 2, finite computational domain is realized by imposing 10-cell perfectly matched layer (PML) [25] to absorb the outgoing electromagnetic waves without producing significant reflections back into the simulation domain. The cell size and the time step used to discretize the space domain and time domain are 30nm×30nm×30nm (~

_{c}*λ*/50) and 0.05fs, respectively. This mesh layout results in a calculation volume of 36μm×2.4μm×2.7μm, corresponding to 1200×80×90 cells. An electric dipole oriented in the y-direction is placed at the central part of the coupled waveguides and a continuous sinusoidal wave

*J*= sin(

_{y}*ωt*) , where

*ω*is the angular frequency, is assigned to the dipole as the temporal current signature for the excitation. To obtain a steady-state result, 4000 time steps is used to allow the field evolved to a constant distribution in regardless of the location of the excitation.

## 3. Results and discussion

The typical field distributions for the coupled plasmonic waveguide excited at wavelength of 1.55 μm are shown in Fig. 3. The electromagnetic mode located at the two top corners has odd parity with respect to the symmetry plane, corresponding to a symmetric supermode. The associated effective index and attenuation coefficient for the supermode is calculated using the fast Fourier transformation (FFT) method in conjunction with curve fitting techniques. Three waveguides with various structural parameters are investigated in order to find the optimum trade-off between the field localization and the propagation length. As shown in Fig. 4, it is found that the modal effective index of coupled plasmonic waveguides at wavelength of 1.55 μm increases with the height of the gap, and asymptotically approaches to a saturation value. On the other hand, the propagation loss decreases with the increasing of the gap height. In general, the higher the mode index, the stronger the field localized. Therefore, the analyzed waveguide structure holds a good compromise between the field localization and the propagation length is clearly in evidence. Compare with the results obtained by the effective index method (EIM) [26], the modal effective indices in our study are larger, indicating better field confinement. This discrepancy is due to the nature of the modeling approach, where the EIM ignores the possible coupling effect and is unable to accurately determine the mode shape, in particular, close to the mode cutoff condition [21]. As the rib height increases to 1080 nm and the spacing between the two rib waveguides is set to 600 nm, we obtain the lowest propagation loss of 0.0257dB/μm and the associated lateral field localization at the full-width at half-maximum valued at 0.84μm. Compare with the CPPs propagating in metallic V-gooves, the field localization is comparable but the propagation distance is almost doubled.

The dispersion characteristics are further analyzed by scanning the carrier frequency of the excitation from 180THz to 230THz, corresponding to the wavelength ranging from 1.30μm to 1.67μm. As shown in Fig. 5, the coupled rib PWs shows a linear dispersion relation with only slightly dependence on the structural parameters. This result indicates a constant dispersion coefficient over a broad frequency range of ~50THz. The correspondent propagation loss is also calculated which shows a flat spectrum over the same frequency range, as shown in Fig. 6. These peculiar properties may allow the realization of a wide-band operation, dispersion compensator free, nano-scaled lightwave circuits.

## 4. Conclusions

We systematically analyze the propagation characteristics of the supermode supported by two coupled semi-infinite rib plasmonic waveguides using 3D-FDTD method. Fast Fourier transformation (FFT) is used for the determination of the propagation constant and the attenuation coefficient. It is found that only one supermode can exist regardless of the height of the gap between the two coupled rib plasmonic waveguides. The lowest propagation loss found is 0.0257 dB/μm, which is lower than that of a CPP in V-groove by 1.7 times, and the associated lateral field confinement is comparable to that of the CPPs. This result indicates that we have successfully compromised the field localization and the propagation loss at the same time. In addition, the investigated waveguide has a constant dispersion coefficient with a flat attenuation spectrum (< 0.05 dB/μm) over 30THz. Moreover, the structure is easier to fabricate due to the all-planar structural features and it is expected that these properties can be useful for lightwave transporting in a nano-photonic circuit.

Note that during the preparation of this manuscript, a paper that treats a similar problem using effective-index method was published [26].

## Acknowledgments

This work was sponsored by the National Science Council, Taiwan (R.O.C.). The author would like to thank for the grant support under contract number 95-2221-E-008-123-MY3.

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