We present a 2D plasmon waveguide in the form of rows of silver nanorods in hexagonal lattice, that may be used for creating a medium with novel effective electromagnetic properties. Transport of energy due to surface plasmon coupling is investigated with Finite Difference Time Domain method for visible range wavelengths from 400 to 750 nm. For 500 to 750 nm range two-mode nature of the waveguide is shown in simulations. Attenuation factors and group velocities are calculated for transmitted modes.
©2005 Optical Society of America
In the last years there is a great interest in materials like photonic crystals and metamaterials that exhibit novel optical properties because of their subwavelength structure. Shvets  proposed a 2D medium with desired properties that can be composed of a net of waveguides exhibiting those properties. The medium is a negative refractive index material made of narrow channel plasmon waveguides. Recently, a waveguiding periodic arrangement of metal slits was presented  as a method of constructing medium with high refractive index.
We propose a plasmon waveguide based on hexagonal lattice of silver rods that could be a base for 2D net medium with refractive index given by light propagation constant in a single waveguide. Light propagates only in one direction in the waveguide and space isotropy of wavefront propagation in the whole medium is established by different orientations of waveguides. This is an advantage over full photonic crystal structures where isotropy of photonic band is required .
Plasmon waveguides are useful in construction of optical metamaterials because of their small size. Plasmons are coupled oscillations of electromagnetic field and of electrons excited on a surface of a conductor. Surface plasmons concentrate and locally enhance light intensity [4, 5]. This property stimulates investigations on nanoscale plasmonic devices because of their possible applications in near field optics, sensing and data storage [5–12].
Surface plasmons offer a possibility of transporting energy with concentration of light below the diffraction limit [6, 7, 9, 11–13]. A few types of plasmon waveguides were proposed and investigated previously [7–16]. These were attempts to employ surface plasmon propagation along wires, small metal strips [7–10, 12] or grooves in metal . In another approach coupling between plasmons on metal particles and guiding energy by chain of particles was used [14–16].
In our structure, coupling between surface plasmons on neighboring silver rods arranged in hexagonal lattice is employed. Previously, a dielectric structure of nanopillars in square lattice for waveguiding energy was proposed in  and light propagation in channels in metallic photonic crystal was examined in . Resonant coupling of surface plasmons on two metallic nanowires was studied in detail in .
The structure is made of alternating two and three silver nanorods of diameter d=100 nm and lattice constant Λ=200 nm placed in a material with refractive index n=1. It has waveguiding properties in the visible range. We analyze modes and find their propagation constants. The guided modes form four pairs of bands with different field distribution. Propagation of waves is simulated using Finite Difference Time Domain method and estimation of energy losses along the waveguide are made for several wavelengths from the range 400–750 nm.
2. FDTD simulations
We assume that dielectric function of silver is described by Drude model
In the simulations we employ the following parameters ε∞=3.70, ωp=13673 THz and Γ=27.35 THz The parameters were calculated by Sönnichsen  from experimental data on reflection and transmission of 25–50 nm thick silver films obtained for the wavelength range 0.188–1.9 µm by Johnson and Christy .
Real Reε(λ) and imaginary Imε(λ) parts of the dielectric permittivity function of silver calculated according to Drude model are plotted in Fig. 1. In the wide range of wavelengths for which measurements  were made the continuous lines are the best fit curves. In the optical range of our interest, however, we observe a serious discrepancy between the measured values of imaginary part of ε(λ) of silver and the best fit curve. Roughly, the mismatch can be estimated on the level of factor of two. It means that in the visible range absorption losses in silver are underestimated when the Drude model with parameters calculated by Sönnichsen is accepted. Additional absorption losses may result from high surface to volume ratio of the nanowires used to construct the waveguide .
We assume uniformity of the structure and electromagnetic field in the y direction and choose magnetic component of the field parallel to rods (TE mode). Thus, considered field components are Ex, Ez, Hy and Poynting vector length |P|=((Ex Hy)2+(Ez Hy)2)1/2 describes the density of energy.
For simulations we use our own implementation of FDTD method [21, 22]. It uses second order difference equations to compute evolution of electromagnetic field on discrete computational grid. It is able to simulate propagation from finite in space sources like Gaussian or Hermite-Gaussian within volumes limited by periodic Bloch conditions and Uniaxial Perfectly Matched Layers (UPML) as the absorbing boundary conditions. UPML are used to simulate free space surrounding investigated structure. Computations are performed on a PC with 2.8 GHz CPU and 4 GB RAM.
Analysis of modes propagating in the waveguide is made using FDTD method in the structure shown in Fig. 2(a) with Bloch boundary conditions in the z direction, what is equivalent to assumption of propagation constant value k . UPML are used as boundaries in the x direction. Size of the simulation region is 350 nm×2500 nm and space discretization step is Δr=5 nm. At the beginning of computations an initial field distribution consisting only of Hy component is accepted in the whole simulation area to avoid artifacts connected with non-vanishing divergence. This field is periodic in the z direction and Gaussian in the x direction with center shifted from the center of the structure. Then 10000 simulation steps, equal Δt=Δr/2c=8.34×10-18 s each, are made allowing initial field to evolve and the next 10000 steps of simulations are recorded. The summary time spectrum of Hy field component in randomly chosen 2048 points is then analyzed giving information about frequency ω of existing modes with given propagation constant. Then the mode field distribution is obtained using Discrete Fourier Transform of the whole recorded field at a found frequency. Some modes in bands are missing and it can be explained by a lack of proper modes in the initial field. Thorough manipulations with the initial field may lead to solution of that problem.
Figure 2(b) shows a cross section of the waveguide used to study propagation. Size of the simulation region is 5000 nm×2500 nm and space discretization step is Δr=5 nm. The x axis polarized electromagnetic TE wave originates from finite in space Gaussian or Hermite -Gaussian sources positioned at z=0 plane. The sources of different symmetry excite corresponding modes . To avoid numerical switch-on effects amplitude of sources increases from zero to its maximum value during the first three periods. Time discretization step is Δt=Δr/2c=8.34×10-18 s. Simulation area is surrounded by 10 layers thick UPML boundary to cancel possible artificial reflections from the edge. For each wavelength we calculate intensity as an average of Poynting vector lengths over 5 wave periods taken after 10000 time steps.
3.1. Waveguide modes in the visible range
The visible range modes for the first Brillouin Zone (BZ) of the waveguide are shown in Fig. 3(a) as plots interpolated between the calculated values of wavevectors k. The observed dispersion curves form four bands with negative dispersion and another four symmetric bands with respect to the center of BZ with positive dispersion. Each pair of curves represents different structure of the field, see Fig. 3(b, 1–3). Three lowest pairs of branches have energy concentrated within the waveguide. The fourth pair of branches corresponds to a mode with energy concentrated outside the waveguide, see Fig. 3(b, 4). The branch with positive dispersion almost matches the free space dispersion curve for all frequencies and is probably a numerical artifact. The asymmetry of the field in this case is a result of asymmetry of initial field distribution and is connected with the fact that this curve represents two joined beams that propagate in free space on the edges of simulation volume weakly interacting with the structure.
3.2. Light propagation in the waveguide
In propagation simulations we excite single modes with desired symmetry using appropriate sources . In the waveguide propagation of light of wavelength of 600 nm is observed for excitation with both Gaussian and Hermite-Gaussian beams. In the first case, energy is concentrated along the z axis the waveguide. In the second case, due to another shape of the incident beam light propagates mainly between outer nanorods, what can be seen on Figs. 4 and 5. Fields of the output beams preserve Gaussian and Hermite-Gaussian distributions.
In the waveguide energy flows from one set of surface plasmons coupled on two rods to the subsequent set excited on three rods and so on. Difference in intensity distributions results from symmetry of the incident field with respect to the waveguide axis, as it is shown in Figs. 6 and 7.
Figure 8 shows an example of situation where due to off-axis illumination we observe both symmetric and antisymmetric field distributions in the waveguide. We observe serpentine propagation of energy that is interpreted as an effect of symmetric and antisymmetric fields beating and makes a clear evidence that there is a difference in propagation constants between them.
For 600 nm illumination wavelength we calculate the group velocities of symmetric and antisymmetric waves by tracking positions of chosen energy maxima (Fig. 9). We obtain vg=0.84c and vg=0.98c for the symmetric and the antisymmetric fields, correspondingly.
Attenuation of energy in the proposed plasmon waveguide composed of metallic nanorods is high. Figure 10 shows decrease of propagating energy for Gaussian and Hermite-Gaussian illuminating beams calculated for 400, 500, 600 and 700 nm wavelengths.
In Fig. 11 we show attenuation factors calculated for several wavelengths in the range 400–750 nm. The results are obtained from comparison of intensity integrated over the x axis at the input (z=0 nm) and output (z=4000 nm) planes. Both in the input and output planes time averaged Poynting vector lengths are integrated. In the input plane the incident field and that reflected from the input section are taken into account. In the output plane integration includes only the transmitted field. For 500 nm and longer wavelengths the symmetric waves have lower attenuation than the antisymmetric ones. For both fields the rapid increase of attenuation at shorter wavelengths and minimum losses at 550 nm are observed. At 500 nm and 550 nm Gaussian beams we observe interference patterns.
4. Discussion of propagation results
To match results of propagation simulations in the waveguide illuminated with Gaussian and Hermite-Gaussian beams of given time frequency with the mode description we record temporary field distributions along the waveguide length and find its space frequencies in the z direction. The results of that procedure are of low precision, however Fig. 12 shows that we observe modes of the positive dispersion branch from the first BZ as well as modes of the negative dispersion branch of the second BZ. Beating between the modes from the two branches is responsible for the interference pattern in Fig. 10. The difference of propagation constants well corresponds to observed pattern periodicity.
Figure 3(a) and 12 show that propagation of symmetric and antisymmetric field distributions in the waveguide is connected mainly with the first and the second branch of modes, correspondingly. Values of group velocities calculated from propagation of wave maxima are in good agreement with those assessed from positive slopes of dispersion curves. At certain wavelengths attenuation factors shown in Fig. 11 considerably change values. It results from the fact that each mode has its maximum frequency of propagation. Maximum frequencies of mode dispersion curves (branches 1 and 2 in Fig. 3(a)) and frequencies where attenuation changes abruptly are in qualitative agreement.
In the proposed waveguide composed of silver nanorods arranged in hexagonal lattice transfer of energy is due to surface plasmon coupling. In the photonic crystal structure we observe propagation of modes of various field distributions. In the visible range from 500 to 750 nm only two symmetric and antisymmetric modes are guided. We find that theoretical values of attenuation factors change from 3.6 to 14.9 dB/µm and from 10.0 to 17.6 dB/µm for symmetrical and antisymmetrical modes, respectively. For both modes attenuation reaches maximum at 425 nm and minimum at 550 nm. Further analysis of geometrical parameters and properties of the waveguide should lead to lower attenuation. The 2D medium composed of a net of such waveguides may be useful in photonic devices.
This research was sponsored by Polish Ministry of Science and Information Society Technologies grant 3 T08A 081 27. The author participates in the EU 6PR Network of Excellence METAMORPHOSE contract #500 252.
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