1. Introduction

It is of great interest to guide light at deep subwavelength scales in optoelectronics, partly because it may enable ultra-density integration of optoelectronic circuits. Conventional dielectric waveguides cannot restrict the spatial localization of optical energy beyond the λ ₀/2n limit [1, 2, 3, 4], where λ ₀ is the free space photon wavelength and n is the refractive index of the waveguide. Surface plasmon polaritons (SPPs) waveguides, which utilize the fact that light can be confined at metal-dielectric interface, have shown the potential to guide and manipulate light at deep subwavelength scales [5, 6, 7, 8, 9, 10, 11, 12, 13]. The prospect of integration has motivated significantly recent activities in exploring plasmonic waveguide structures. In constructing highly dense integration of optoelectronic circuits, the ability to intersect waveguides is crucial owing to the desire for complex systems involving multiple waveguides. Usually, waveguide crossings require low intersection loss, low crosstalk, and compact dimensions. Several designs have been proposed for low-loss, low-crosstalk crossing of silicon-on-insulator nanophotonic waveguides [14, 15, 16]. However, to our knowledge, there are few studies about the waveguide crossings for SPP waveguides. Note that previously much attention has been focused on realizing the surface plasmon waveguide with long propagation length [17, 18, 19]. In this paper, we analyze intersection loss of nanoplasmonic waveguides and design compact intersection with no crosstalk. Except for the dispersion of the SPP waveguide, the calculations offered below are performed by the finite-element method (FEM) in frequency domain [20]. The FEM has the advantage of defining the material interfaces accurately using nonuniform triangular meshes and additionally uses an adaptive mesh resolution. The FEM is therefore expected to have a good convergence in numerical calculations [19].

2. Dispersion of surface plasmon polariton waveguides

Consider a subwavelength metal-dielectric-metal (MDM) constructed two-dimensional (2D) plasmonic waveguide. The complex propagation constant β = β_R+jβ_I of surface plasmon polaritons can be obtained exactly by solving the dispersion equation [22]

where $k\left(\beta \right)\left(=\sqrt{{\beta }^2-k_0^2{\epsilon }_1}\right)$and $p\left(\beta \right)\left(=\sqrt{{\beta }^2-k_0^2{\epsilon }_m}\right)$are the wave numbers of SPPs in dielectric and metal, respectively, ε ₁ and ε_m are the dielectric constants of the medium in guide region and metals, respectively, and w is the width of the waveguide. For such a 2D plasmonic waveguide, the fundamental transverse magnetic mode (TM ₀) always exists even when the width is close to zero, while other high-order modes have a cutoff width. To satisfy single-mode condition [22], the width of the plasmonic waveguide should be smaller than λ ₀ arctan ($\frac{\left(\sqrt{\frac{-\mathrm{Re}\left({\epsilon }_m\right)}{{\epsilon }_1}}\right)}{\left(\pi \sqrt{{\epsilon }_1}\right)}$), where arctan is the mathematical arctangent function. For instance, the maximum width of the single-mode condition for the silver-air-silver SPP waveguide is about 720 nm for λ ₀ = 1.55µm. Figure 1 shows the dependence of β/k ₀ of the fundamental SPP mode in the 2D silver-air-silver waveguide on the width w of the waveguide and working wavelength λ ₀ of light in free space, where k ₀ = 2π/λ ₀. From Fig. 1, one can see that the effective refractive index (n _eff=β_R/k ₀) of SPP TM₀ mode is always larger than that of the dielectric, i. e., $n_{\mathrm{eff}}>\sqrt{{\epsilon }_1}.$The loss originating from the intrinsic loss of the metal increases when shrinking the width of the plasmonic waveguide. In this paper, the waveguide width is chosen sufficiently narrow to ensure single-mode propagation in the SPP waveguide.

 

Fig. 1. Dependence of complex propagation constants of SPPs in a 2D silver-air-silver plasmonic waveguide on the width (w) of the waveguide and the working wavelength λ ₀.

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3. Direct crossing for two perpendicular plasmonic waveguides

When considering the waveguide crossing, the confinement in the direction perpendicular to the wave propagation is lost near the crossing region, thus causing diffraction of the light. The diffraction strongly depends on the size and the index-contrast η of the waveguide. The loss for the waveguide crossing of low-index-contrast waveguides is negligible, while the mode diffracts dramatically for the nano-size of high-index-contrast waveguides [14]. Previous studies show that high-index-contrast systems, such as silicon-on-insulator nanowires with η = n _silicon/n _silica = 2.34, have a large intersection loss for the direct waveguide crossing [16]. The silver-air-silver plasmonic waveguide studied here has a size ranging from 20 nm to 500 nm, having a quite large index-contrast of η = n _silver/n _air = 9.3 for the working wavelength 1.55 µm. One would expect that the SPP mode will be significantly diffracted when passing through a nano-scale waveguide crossing. To clearly illustrate the proposed principle, we first analyze the behavior of intersection loss for the direct waveguide crossing of nanoplasmonic waveguides, as shown in the inset of Fig. 2(a). Consider an optical beam incident from port a. Obviously, the beam diffracts when encountering the crossing region. The relative power transmissions are shown in Fig. 2. For simplicity, here we assume that the metal is lossless, i.e, neglecting the imaginary part Im(ε_m) of the metal dielectric permittivity. Figure 2(a) shows the relative power transmissions as a function of the width of the plasmonic silver-air-silver waveguide for the working wavelength 1.55 µm. The squares and circles represent the forward transmittance (a-b) and crosstalk (a-d), respectively. The forward transmittance gradually drops down when the width of the plasmonic waveguide decreases from 500 nm to 50 nm, and the crosstalk slightly increases when decreasing the waveguide’s width. For the case of w = 200 nm, the forward throughput is about -5.89 dB (25.73%) and the crosstalk is about - 6.03 dB (24.96%). From Fig. 2(a), one concludes that this kind of direct waveguide crossing has low transmission and high crosstalk. It is also interesting to note that, when encountering a nano-scale intersection, the throughput is around -6.00 dB (25%), almost the same as the crosstalk. The transmittance spectra, for the case of w = 100 nm, are shown in Fig. 2(b). For the wavelength of our interests, the throughput maintains the same value of -6.00 dB (25%), and similarly for the crosstalk. This result can be explained well with the use of the characteristic impedance concept and transmission line theory [21]. The inset of Fig. 2(b) shows the profile of a steady-state magnetic field at the working wavelength 1.55 µm, which illustrates that the throughput is comparable to the crosstalk. In all calculations mentioned in this paper, the frequency-dependent dielectric function of the silver is described by the lossy Drude model ε(ω)=ε _∞-(ε ₀-ε _∞)ω ² _p/(ω ²+2 iωv_c), where ε _∞/ε ₀ is the relative permittivity at infinite/zero frequency, ω_p is the plasma frequency, and v_c is the collision frequency. We choose ε _∞ = 4.017, ε ₀ = 4.688, ω_p = 1.419 × 10¹⁶ rad/s and v_c = 1.117 × 10¹⁴ rad/s for the Drude model, which fits the experimental data [23] quite well.

 

Fig. 2. (a) The forward transmittance (squares) and crosstalk (circles) for the standard direct crossing as a function of the width (w) of the silver-air-silver plasmonic waveguide for λ₀ = 1.55 µm. The inset shows its corresponding structure. (b) Spectra for the forward transmittance and the crosstalk when w = 100 nm. The inset shows the profile of a steady-state magnetic field at the wavelength 1.55 µm.

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4. Resonant-tunnelling assisted transmittance for subwavelength plasmonic slot waveguides

 

Fig. 3. (a) Transmission spectra of the device [shown in the inset] with different side length of the cavity when w = 100 nm. The solid and dashed lines represent the results for L_x = 700 nm and 660 nm, respectively. (b) Transmission T and reflection R of the device (L_x = 700 nm, L_y = 1000 nm) as a function of the wavelength for w = 50 nm and 100 nm, respectively. The solid and dashed lines represent the results from the FEM method, and the open squares and solid circles are obtained from the coupled-mode theory. The inset shows the profile of a steady-state magnetic field at the resonant frequency for w = 100 nm, which illustrates the complete transmission on resonance.

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To decrease the crosstalk and enhance the forward transmittance of the waveguide crossing, we aim to reduce the diffraction in the crossing region. In order to suppress effectively the diffraction, one generally employs the intersection region significantly exceeding the wavelength [15], which unfortunately results in a low packing density of circuits. In this paper, we utilize the effect of resonant tunnelling through a cavity to enhance the forward transmittance. We consider a system [inset of Fig. 3(a)] consisting of a subwavelength silver-air-silver plasmonic waveguide coupled to a rectangular cavity, which supports a resonant mode of frequency ω ₀. To preserve structural symmetry, the center of the cavity is placed at the center of the plasmonic waveguide and the side lengths of the rectangular cavity are denoted by L_x and L_y. For such a system, the transmission can be described by the resonant tunnelling effect, and one can take advantage of coupled-mode theory [24] to evaluate power transmission T and reflection R on resonance. The expressions are given by

where 1/τ₀ is the decay rate due to the internal loss in the cavity and 1/τ _e is the decay rate of the field in the cavity due to the power escape through the waveguide. From the above equations, one can see the direct relation between the transmittance/reflection and the ratio τ₀/τ_e on resonance. If there is no internal loss in the cavity (1/τ₀→0), the incident wave is completely transmitted on resonance and the spectral width of the resonance is determined by the coupling strength (1/τ_e) between the waveguide and the cavity. Assuming that the metal is lossless, the transmission T of the device with different side length of the cavity are shown in Fig. 3(a), where the width of the plasmonic waveguide w is fixed as 100 nm. From Fig. 3(a), we observe that the resonant frequency of the cavity strongly depends on the side length L _x of the cavity, while it slightly shifts when varying L_y. We also find that, for a fixed value of L_x, the quality factor Q _total = 1/(1/Q _coupling+1/Q _intrinsic) of the resonant system increases when enlarging L_y. Q _total is around 5 when L_x = 700 nm, L_y = 600 nm, and increases to 10 for L_x = 700 nm, L_y = 1000 nm. Figure 3(b) shows the transmission T and reflection R of the device (L_x = 700 nm, L_y = 1000 nm) as a function of the wavelength for w = 50 nm and 100 nm, respectively. In Fig. 3(b) the solid and dashed lines represent the results from the FEM method, and the open squares and solid circles are obtained from the coupled-mode theory. Results from the coupled-mode theory fit very well with those from the FEM method. Since the metal is assumed to be lossless, there is no internal loss in the cavity and there is, therefore, complete transmission on resonance, as seen in Figs. 3(a) and 3(b). In this coupling system, the coupling strength can be tuned by the waveguide’s width. Decreased the width results in a weaker coupling and, therefore, higher quality factor and narrower spectral width of the resonance. For the case of w = 100 nm, Q _total is about 10 and becomes 15 when w = 50 nm. For the lossless case, because of infinitive Q _intrinsic, Q _total is solely determined by Q _coupling, i.e., Q _total=Q _coupling. From Fig. 3(b), we also observe that the resonant frequency of the cavity slightly shifts when w is varied. The inset of Fig. 3(b) shows the profile of a steady-state magnetic field at the resonant frequency for w = 100 nm, which illustrates the complete transmission on resonance. From the distribution of the excited mode in the rectangular cavity, one can explain why the resonant frequency of the cavity is strongly dependent on L_x, while almost independent on L_y, as mentioned above.

Next, let us consider coupling of four branches of perpendicular intersection for two plasmonic waveguides in terms of a resonant cavity at the center, as shown in Fig. 4(c). When the excited resonant mode can be prevented from decaying into the transverse ports, the crosstalk can be prohibited and the system reduces to the resonant tunnelling phenomenon through a cavity. To achieve it, there are general criteria for perpendicular intersection of two waveguides, as mentioned in detail in Ref. [25]. To achieve it, the following conditions should be satisfied: (1) the waveguide should be single-mode with a mirror symmetry plane through its axis and perpendicular to the other one; (2) the cavity must be symmetric with respect to the mirror planes of both waveguides and support resonant modes with different symmetry with respect to waveguides mirror plane. When these requirements are satisfied, due to its orthogonality to the mode in the other waveguide, each resonant state can couple solely to the mode in just one waveguide, thus the crosstalk will be eliminated. From the mode profile [inset of Fig. 3(b)], one can easily see that the excited resonant mode supported by the rectangular cavity is even with respect to one waveguide’s mirror plane and odd with respect to the other. In terms of the general criteria mentioned above, the crosstalk for the perpendicular intersection [Fig. 4(c)] will be prohibited when introducing a rectangular cavity in the intersection. In order to preserve the rotational symmetry of the device, here we analyze the device with the square cavity instead of the rectangular one. Note that the square cavity supports two degenerate modes with different symmetry with respect to waveguides mirror plane. One also notes that for the 2D case the bulk material (silver) can prevent any radiation losses. Following these idea, we design quite a simple crossing intersection with a square resonant cavity, as shown in Fig. 4(c), where the width of the waveguide is 20 nm. Figures 4(a) and 4(b) show the throughput spectra and crosstalk spectra of the device with direct crossing (dashed lines) and cavity-based crossing (solid lines), respectively. Compared with the result for the direct crossing, the throughput for the cavity-based crossing in Fig. 4(a) is really enhanced due to the resonant-tunnelling effect and reaches the unity on resonance. The crosstalk (the solid line) for cavity-assisted crossing, shown in Fig. 4(b), is almost prohibited relative to unmodified crossings. The crosstalk is close to zero in the whole frequency range of interest, which can be naturally understood by the general criteria mentioned above. Figure 4(d) shows the steady-state magnetic field at the resonant frequency for w=20 nm, which illustrates how the energy is fully transmitted forward through the crossing section. We also note that the size of the intersection is quite compact, which is vital for high-density integration.

 

Fig. 4. (a)–(b) Transmission spectra of the device with direct crossing (dashed lines) and cavity-based crossing (solid lines). (c) Intersection of two-dimensional plasmonic waveguides with a square resonant cavity. (d) Profile of a steady-state magnetic field at the resonant wavelength 1.55 µm for w = 20 nm and L = 700 nm.

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All calculations mentioned above are performed when ignoring the material loss. However, it should be emphasized that the metal, silver, is always a lossy material especially in the visible and infrared frequency region. This kind of loss usually limits optical performance of plasmonic devices. For the proposed device studied here, the loss will result in propagation loss of the plasmonic waveguide mode and low quality factor of the unloaded resonant plasmonic cavity. However, when we take the loss into account, the symmetry properties of cavity and waveguide modes will remain unchanged. Thus, the general criteria for eliminating the crosstalk of the perpendicular intersection are still valid, i. e. the crosstalk can be prohibited even when taking material loss into account. From Eqs. (2)–(3), one sees that the spectrum will in general not reach the unity on resonance. This is because the factor 1/τ_o significantly decreases when taking the loss into account and becomes comparable to 1/τ_e. Here, we recalculate the transmission spectra for the perpendicular intersection [Fig. 4(c)] without ignoring the material loss. The source and detector are connected to the coupling region through plasmonic waveguide segments of length 160 nm. Results for the device [Fig. 4(c), L=680 nm, w=20 nm] are shown in Fig. 5. The solid and dashed lines represent the results for ignoring the material loss and considering the loss, respectively. When we take material loss into account, the throughput shown in Fig. 5(a) is only about -5.47 dB on resonance, which originates from the propagation loss of the waveguide mode and the loss from the plasmonic cavity. One can observe from Fig. 1(b) that the propagation loss is quite significant for w = 20 nm. At the telecommunication windows, around 1.5 µm, the SPP propagation length can be greater than the total length of the circuit and reach values close to 1 mm. The propagation loss can thus most likely be reduced for each plasmon device. Apart from the propagation loss, the intersection loss for the device is about -2.7 dB on resonance, which is limited only by the value of Q _intrinsic/Q _coupling. For the lossless case, Q _total is around 20 and becomes 15 when taking the loss into account. Q _intrinsic of the unloaded cavity for the lossless case is much larger than Q _coupling, thus we can obtain Q _coupling=Q _total = 20. Due to the material loss, Q _intrinsic strongly decreases. Assuming that the coupling strength are the same for both cases, one can obtain that Q _intrinsic becomes 60 for the lossy case. In order to improve the transmittance property, what we can do is to increase the value of Q _intrinsic/Q _coupling. Recently, people have used a gain material to compensate for the effect of material loss, thus improving the optical performance of loss-limited plasmonic devices [26, 27]. From Fig. 5(b), one can also observe that the crosstalk is prohibited even when material loss is present. The simulation results fit quite well with the expectations from the analysis mentioned above.

 

Fig. 5. (a) Throughput of the device [Fig. 4(c)] when ignoring the material loss (solid line) and considering the loss (dashed line). (b) Crosstalk of the device when ignoring the material loss and considering the loss.

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5. Summary

In this paper, we have analyzed the intersection loss for two perpendicular plasmonic waveguides. For the direct crossing, when encountering a nano-scale intersection, the throughput is around 25%, almost the same as the crosstalk. Using a general recipe for elimination of crosstalk, we design simple cavity-based structures to enhance the throughput and eliminate the crosstalk. The size of the intersection is compact, which is vital for high-density integration. Numerical results are calculated by FEM in frequency domain, which agree well with those from the coupled-mode theory. Without considering the material loss, the throughput reaches the unity on resonance and the crosstalk is suppressed. Results can be explained in terms of symmetry considerations and resonant tunnelling effects. We find that the effect also exists in the presence of material losses because of the preserved symmetry properties. Apart from the propagation loss, the intersection loss for the proposed device is about -2.7 dB on resonance. Our results may open a way to construct nanoscale crossings for high-density nanoplasmonic integration circuits.