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We report on the modal competition mediated angular dispersion by heterogeneously coupled plasmonic waveguides. By varying the wavelength of the excitation, the surface waves propagate alongside the upper and lower interfaces can be manipulated in coupled, decoupled, and cutoff regimes. Depending on the coupling states, the output beam can be steered between +15° and −17° for wavelength from λ = 695nm to λ = 675nm. The maximum achieved angular dispersion can be as large as 2.1°/nm. This finding may revolutionize current design concept of spectrometers, providing a significant way to scale down the form factor further into nano-size.
©2010 Optical Society of America
1. Introduction
Surface plasmon waves (SPWs) can propagate along the interface between a metal and dielectric providing nano-scale waveguiding over conventional dielectric waveguides. Bring two interfaces in close proximity, electromagnetic (EM) energy can be squeezed within the gap (or slot) region, leading to further enhancement of modal confinement. This unique property allows a number of important achievements, such as negative refraction [1,2], actively control the beaming of EM radiations [3], and nearly lossless turning around extremely sharp bends [4]. Most research so far concentrate on homo metal/dielectric/metal (MDM) structures [5, 6], where standard characteristic equations (CEs) are solved to give the dispersion relations [7]. However for heterogeneously coupled MDM structures [8, 9], the individual SPW may travel alongside the upper and the lower interfaces at different phase velocities. In this case, CEs under quasi-static conditions can hardly reflect the propagation dynamics within the walk-off distance. The propagation of such quasi-static modes is accompanied by accumulated phase difference, yielding progressively increased wavefront tilt. This in turn determines the angle of propagation for the output beams. This unique property may present in any existing heterogeneously coupled plasmonic waveguides, however so far, the off-plane optical beaming has never been discovered. To date, field beaming effect is widely achieved by plane waves propagating through a single nano-slit surrounded by corrugated grooves [10–12]. Off-plane mode conversion achieved by a single interface plasmonic waveguide incorporating a blazed grating structure was also reported [13]. Here we report the giant angular dispersion effect using heterogeneously coupled MDM structures. By varying the excitation wavelength, the coupling state can be manipulated in coupled, decoupled and cutoff regimes. A figure of merit (FOM) was defined to classify the three regimes and a physical picture based on modal competition well explains our results.
2. Propagation characteristics
The schematic diagram of the structure in this study is depicted in Fig. 1 . The gap between the two semi-infinite metal layers is d=100nm with the refractive index nd=2 and a total length of 200nm. The two metals can be made by Ni and Ni-Ti alloy and the plasma frequencies can be tailored to be ωp=7.4267×1015 Hz and ωp=6.80000×1015 Hz, respectively [14,15]. The electric permittivity for the metals used here was described by Drude model ε(ω)=1-ωp 2/ω(ω+iγc), where the collision frequency are set to γc=4.1689×1014 Hz.
To clarify the propagation characteristics, we first consider lossless conditions. From the CE (1), the propagation constant β and the corresponding modal profile can be calculated.
Where κ is the transverse wavevector and γi=(β2-μ0εik0 2)1/2, i=1,2 are the attenuation coefficients. On the other hand, we treat the same question as two coupled single interface SPW. From the coupled mode equations (CMEs) (2a) and (2b) [16], the modal profiles can be obtained as well. Where a1 and a2 are the field amplitudes of the upper and lower SPWs, respectively. β1 and β2 denote the unperturbed propagation constants. κij, i,j=1,2 here represents the coupling coefficient and β0 is defined as.To explain the calculated results, we define a figure of merit, where FOM≡2κij/Δβ. For FOM>>1, the upper and lower SPWs are relatively strong coupled, forming a supermode propagating at the same phase velocity. The results obtained by CE and CME are in coincident as shown in Fig. 2(a-b) . On the contrary, for FOM<<1, the coupling is very weak and the CE gives solutions of the fundamental mode and the first higher order mode. The two modes are very close to the individual single interface SPW, as shown in Fig. 2(c-d). As a comparison, the result obtained by CME was a hybrid mode forming from the superposition of the modes obtained by CE, as shown in Fig. 2(e-f).
Fig. 2 The comparison of the modal profiles obtained by CE and CMEs. (a-b) for FOM>>1 and (c-f) for FOM<<1.
Next we took loss into account and compared the results with 2D FDTD simulations. In order to simulate an infinite space, fifteen-layer convolution perfectly matched layer (CPML) is allocated on each boundary to absorb the outgoing electromagnetic waves. With CPML, an ignorable reflectivity as small as -59dB was achieved. The cell size and time step were set to 1nm×1nm and 1.66×10-18 sec, respectively. To transform the dispersive permittivity of the metals in time domain, the recursive convolution method was applied. An infinitely long magnetic current source My oriented in the y-direction is placed at the center of waveguide and a continuous sinusoidal wave My=sin(2πc0t/λ0) with the wavelength λ0=650~850 nm is assigned to My as the temporal signature of the excitation. A total time steps of 20000 and an entire simulation area of 2×0.45 μm2 are used in the calculation which effectively leads to a steady-state result. The calculated dispersion relations are shown in Fig. 3(a) . For wavelengths λ≥695nm, only the fundamental mode will be excited and the dispersion relations obtained by FDTD and CE are coincident. With the decrease of the excitation wavelength, the first higher order mode is allowed and the dispersion relations deviated. At the wavelength of 685nm, a hybrid quasi-mode was formed, as predicted by CME previously. Due to the decrease of the coupling strength, the two surface waves are gradually decoupled. As the wavelength is further decreased to λ≤675nm, the optical field in the waveguide splits into two individual surface modes, which no longer coupled with each other. Although the two modes have similar spatial distribution and in principle can both be excited by a symmetric magnetic current source, the first higher order mode (SPW along the upper interface) has higher excitation efficiency for its relatively low loss and better phase matching with the excitation. At the wavelength of 600 nm, the lower surface wave reached cutoff and the resulting modal profile retrogrades to the single interface SPW. The modal evolution was schematically shown in Fig. 3(b).
Fig. 3 (a) The dispersion relations obtained by CE and FDTD methods. (b) Modal evolution with the excitation wavelength.
3. Giant angular dispersion
Based on the obtained dispersion relation and the corresponding picture of modal evolution, a useful application concerning giant angular dispersion can be achieved upon the modal hybridization process. With the wavelength varying from 695nm to 675nm, the dominant mode evolves from the fundamental (lower SPW) to the first higher order mode (upper SPW). By solving the characteristic equation, we found that the transverse wavevectors for the fundamental mode and the first higher order mode are kx0= +0.8795k0 and kx1= −2.0165k0, respectively. As a result, the output beam is capable to be steered with a wide angular distribution depending on the competition of the two modes.
To verify the predicted off-plane angular dispersion, FDTD simulations are carried out. The parameter settings were the same as previously described except the cell size and the time step are doubled. Figure 4 shows the deflection angle as a function of the wavelength. For wavelength from λ=695nm to λ=675nm, the output beam is angularly deflected between +15° and −17°. The deflection angle is linearly related to the wavelength from λ=690nm to λ=680nm, corresponding to a maximum angular dispersion as large as 2.1°/nm. The transverse magnetic field intensities are shown in Fig. 5 and the corresponding time averaged Poynting vectors are shown in Fig. 6 . It is clear in evidence that the electromagnetic energy swirling around the upper corner decreases with the increasing of the wavelength while that around the lower corner behaves the opposite way. The heterogeneously coupled plasmonic structure, as opposed to the homogeneously coupled cases, provides an opportunity to manipulate the competition of the energy vortex in nano scale which gives significant angular dispersion effect.
Fig. 4 Deflection angle as a function of the excitation wavelength. The inset shows the dominant mode and the direction of the output beam.
4. Conclusions
In conclusion, we unraveled the off-plane giant angular dispersion which is yet undiscovered in existing heterogeneously coupled plasmonic waveguides. By varying the excitation wavelength, a wide degree of modal hybridization can be achieved. Illustrations of modal evolution from the lower SPW to the upper SPW between λ=695nm and λ=675nm were detailed. This enables an off-plane optical beaming from +15° to −17°. The maximum achievable angular dispersion can be as large as 2.1°/nm. Compared to the superprism effect in photonic crystals [17], the present research shows comparable performance but much smaller and simple structures, rendering it an alternative to miniaturized spectrometers.
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 NSC 98-2112-M-008-002-MY2.
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