Ultra-short plasmonic splitters and waveguide cross-over based on coupled surface plasmon slot waveguides

Composite metal-dielectric-metal (MDM) surface plasmon polariton (SPP) structures are first proposed to realize the ultra-short optical splitters with simplified designs. The operation mechanism is based on the contra-directional coupling achieved in composite plasmonic slot waveguides. In certain cases, the switching function can also be realized. It is further shown that based on the same physical mechanism multidielectric-core composite MDM structures could serve as a novel plasmonic waveguide crossover component with low cross talk and high throughput. ©2011 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (200.4660) Optical logic; (240.6680) Surface plasmons; (250.5300) Photonic integrated circuits. References and links 1. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006). 2. E. 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Introduction
Today's semiconductor industry has successfully scaled the dimensions of electronic devices down to the order of ~50 nm.Unfortunately, purely electronic circuits operating above ~10 GHz seriously suffer from heat generation and signal delay [1,2].Photonics offers an effective solution to this problem by implementing communications and processing systems based on photonic circuits, which however are dictated by diffraction limit (the typical dimension is on the order of ~1 μm) [2].This implies that electronic and optical devices usually show complementary characteristics in terms of sizes and speed.Against this background surface plasmon polaritons (SPPs) [3] have been suggested to act as novel digital data carriers in information processing, because plasmonics offers both the compactness of electronics and the speed of optics [4].Over the past years, various plasmonic waveguide structures such as metallic stripes [5], metal-dielectric-metal (MDM) and dielectric-metaldielectric (DMD) structures [6][7][8], dielectric-loaded SPP waveguides [9], V-shaped metal grooves [10], and Λ-shaped metal wedges [11] have been prototyped as interconnects for information transport.
Recently, MDM plasmon waveguides have been attracting a great deal of attention partially due to its strong modal confinement [6][7][8]12].On the other hand, MDM structures exhibit a negative refractive index for the guided plasmon, and indeed two-dimensional negative refraction has been observed in these plasmonic waveguides [13,14].In their works, the backward modes in MDM waveguides are found to be responsible for negative refraction at optical frequencies [13,14].Several research works have involved waveguides with negative index components and backward waves, and interesting properties of guidance were reported [15][16][17][18][19].Moreover, the interaction between positive index and negative index waveguides was theoretically investigated, and the phenomenon of contra-directional coupling was demonstrated [20,21].Compared to co-directional plasmonic couplers [22,23] in which the coupling lengths typically excess several micrometers, couplers that operate under the contra-directional coupling condition exhibit much shorter coupling lengths due to a rapid coupling rate [24].Although the coupling of backward and forward waves could possibly play an important role in nanophotonics, so far few efforts [24,25] have been made to the design of integrated optical devices and components based on such contra-directional coupling mechanism.
In this paper, we will show that contra-directional coupling achieved in plasmonic MDM slot waveguides can be used to realize novel devices with simplified designs.Firstly, the guiding SPP modes including the field-symmetric (forward) and anti-symmetric (backward) modes in MDM structures are numerically investigated.The contra-directional coupling between two adjacent MDM waveguides with different dielectric core but operating in the same wavelength range is also discussed.Then, we propose a composite MDM structure composed of a silver-GaP-silver structure sandwiched between two silver-Si 3 N 4 -silver SPP waveguides to realize the ultra-short plasmonic splitters at visible frequencies.In certain cases, the switching function can also be achieved.Furthermore, it is shown that based on the same physical mechinism these multi-dielectric-core composite structures could serve as a novel plasmonic waveguide crossover component with low cross talk and high throughput.

Dispersion relation and coupling characteristics of MDM structures
Throughout this paper, we use a commercial FEM software package (COMSOL Multiphysics) to numerically evaluate the performances of the proposed waveguide structures.In the calculations, silicon nitride (refractive index: n = 2.03) and gallium phoshpide (refractive index: n = 3.59) are chosen as dielectric gaps, and the metal is assumed to be silver with its dielectric constants described by a Drude model: where ω p = 1.29 × 10 16 rad/s is the plasma frequency of bulk silver and τ is the relaxation time.We begin our discussion by first considering ideal structures in which silver is lossless (τ 1 = 0).Note that the use of perfect metal allows us to acquire meaningful physical insights in higher optical frequencies, e.g., roughly above 600 THz [26].Later in this paper, the effect of realistic metal losses will be discussed.In a MDM structure the SPPs at each metal-dielectric interface may couple with each other to form field symmetric and anti-symmetric modes depending on the excitation frequency and surface plasmon frequency [27].Thus, it provides us a possibility to excite these two modes at the same frequency in two different MDM structures.Figure 1(a) illustrates the dispersion properties of planar Ag-Si 3 N 4 -Ag and Ag-GaP-Ag waveguides in which both the dielectric cores have the same thickness t = 20 nm.In the calculations, the complex propagation constant γ = β + iα with α and β being the attenuation constant and phase constant, respectively, is obtained from the mode analysis solver of COMSOL.It is well known that for SPPs propagating on a planar metal/dielectric interface, surface plasmon resonance frequency ω Ag-dielectric is given by ω Ag-dielectric = ω p /(1 + ε d ) 1/2 , where ε d is the relative permittivity of the dielectric layer adjacent to the semi-infinite metal.Obviously, the surface plasmon resonance frequency for a silver-Si 3 N 4 interface (ω Ag-Si3N4 = 0.447ω p ) is larger than that for a silver-GaP interface (ω Ag-GaP = 0.268ω p ).As shown in Fig. 1(a), below the silver-GaP surface plasmon resonance frequency both the Ag-Si 3 N 4 -Ag and Ag-GaP-Ag structures support a field symmetric mode (forward mode) with a positively sloped dispersion curve.Between the silver-GaP and silver-Si 3 N 4 surface plasmon resonance frequencies, the Ag-Si 3 N 4 -Ag waveguide still supports a field symmetric mode, whereas the Ag-GaP-Ag structure now supports an anti-symmetric field mode (backward mode).In this case, the dispersion curve for the Ag-GaP-Ag waveguide exhibits a negative slope, meaning that the energy and phase fronts propagate in opposite directions.To visualize the characteristics of these two eigen modes, the magnetic field y-component (H y ) distributions are calculated at the frequency ω Ag-GaP < ω = 0.31ω p < ω Ag-Si3N4 and are shown in Figs.1(c) and 1(d) for the planar Ag-Si 3 N 4 -Ag and Ag-GaP-Ag SPP waveguides, respectively.It is clearly seen that the former waveguide has a symmetric magnetic field distribution, whereas an anti-symmetry field distribution is observed for the latter structure.Furthermore, the two dispersion curves are observed to cross at a frequency ω = 0.3ω p (corresponding wavelength: ~490 nm), which makes a strong coupling between the symmetric and anti-symmetric modes possible.Now let us consider a composite two-dielectric-core MDM waveguide formed by stacking the individual Ag-Si 3 N 4 -Ag and Ag-GaP-Ag structures in a neighborhood, in which the thickness of Ag layer between Si 3 N 4 and GaP is set to be 20 nm to ensure an efficient coupling.There are two main regions shown in the dispersion curve of the composite MDM structure: in the frequency range 0.28ω p < ω < 0.33ω p (around the crossing point ω = 0.3ω p of two individual MDM dispersion curves) the forward mode in the Ag-Si 3 N 4 -Ag waveguide does couple with the backward mode in the Ag-GaP-Ag structure [Fig.1(a), red curves with symbols], and away from this frequency range the composite MDM structure dispersion curve tends to overlap with the individual MDM structure dispersion curves [Fig.1(a), red curves without symbols], which implies that there exists only weak or negligible coupling between two individual MDM structures.The resultant superimposed eigen mode, e.g., at a frequency of ω = 0.3ω p , formed in this two-dielectric-core composite MDM structure with its H y and electric field x-component distributions are shown in Fig. 1(e).It is noted that the superimposed eigen mode is evanescent in nature as a result of contra-directional coupling.Indeed, unlike a zero attenuation constant in the individual MDM structures (α = 0, due to the use of perfect silver there is no absorption), significant attenuation is observed for the composite MDM structure in the frequency range 0.28ω p < ω < 0.33ω p [Fig.1(b)], which in turn confirms the occurrence of contra-directional mode coupling between Ag-Si 3 N 4 -Ag and Ag-GaP-Ag waveguides.It is worth noting that in the contra-directional and strong coupling regime two modes exist simultaneously [24].One mode carries power forward in the GaPcore and backward in the Si 3 N 4 -core MDM waveguide [solid line with solid circles, Fig. 1(b)], while the opposite is true for the other mode [solid line with open triangles, Fig. 1(b)].Which mode actually dominates will depend on which dielectric-core waveguide is excited.Since in the following discussions the signal is assumed to be excited from the Si 3 N 4 -core waveguide, we only consider the mode that carries power forward in the Si 3 N 4 -core and backward in the GaP-core waveguide.For revealing the guidance characteristic of the composite two-dielectric-core SPP waveguide, a 90° bend is introduced into the Ag-Si 3 N 4 -Ag waveguide.To decrease the bending loss all the sharp corners are rounded with a 30 nm-radius curvature [6].Geometrical parameters are displayed in the schematic [Fig.2(a)].In our simulations, a symmetric eigen mode at a frequency of ω = 0.3ω p is excited, and propagates along the left input port of the Si 3 N 4 core.The corresponding magnetic field (H y ) distribution in the composite structure is shown in Fig. 2(b).It is clearly seen that after its propagation through the bend portion, this symmetric mode is converted into a super-imposed evanescent eigen mode supported by the composite waveguide.Due to the contra-directional coupling effect, a field anti-symmetric propagating mode is then induced in the Ag-GaP-Ag waveguide, which propagates backward in the Ag-GaP-Ag waveguide.As a consequence, a symmetric mode guided forward in the Ag-Si 3 N 4 -Ag waveguide is transferred into an anti-symmetric mode propagating backward in the upper portion of the Ag-GaP-Ag structure.

Design and analysis of plasmonic splitters
As discussed above, a composite Ag-Si 3 N 4 -Ag-GaP-Ag waveguide exhibits the large attenuation constant in the contra-directional coupling regime (corresponding to a frequency range 0.28ω p < ω < 0.33ω p ), which indicates that the energy transfers from the input Si 3 N 4core waveguide to the coupled GaP-core waveguide at an exponential rate.Therefore, high efficient coupling can be achieved within a very short coupling length.It is such short contradirectional coupling length that paves a possible way to realize ultra-small devices with simplified designs.
For the accomplishment of plasmonic splitters, we propose a composite three-dielectriccore structure with mirror symmetry by sandwiching a silver-GaP-silver waveguide between two silver-Si 3 N 4 -silver waveguides.Figure 3(a) shows the schematic of the proposed splitter structure, in which two 90° bend Si 3 N 4 -core ports (port A and B) are used to input signals, and output signals are monitored from two bottom ports (port C1 and C2) of the Si 3 N 4 -core waveguides, and as well from the upper port (port D) of the GaP-core waveguide.Strictly speaking, 100% coupling can only be achieved when the composite MDM waveguide is semiinfinitely long [24].By adopting the definition of the surface plasmon propagation length [3], the contra-directional coupling length (L C ) is defined here as a distance along the direction of propagation where the input power exponentially decay to 1/e, i.e., L C = 1/α, which means that ~63% power has been transferred from input Si 3 N 4 -core waveguide to the GaP-core waveguide.As shown in Fig. 3(b), the contra-directional coupling length for a threedielectric-core waveguide in the frequency range 0.28ω p < ω < 0.33ω p is on the order of ~40 nm.Thus, in the design a 210 nm (~5L C ) long three-dielectric-core waveguide has already ensured a sufficiently high coupling level.In principle, the proposed splitter structure is a plasmonic interferometer based on contra-directional coupling.Inspired by optical switches reported in [28] and [29] where a phase difference between two optical signals has been initialized to implement the switching functions, we will also introduce the phase shift into our proposed plasmonic splitters.Actually, such a phase shift in plasmonic devices has been experimentally achieved by using all-optical, electro-optical, thermo-optical or field effects [30][31][32][33], which could be incorporated in the future plasmonic chips.As a first example, the external signals (ω = 0.30ω p ) directed to the inputs A and B have equal amplitude and only the phase of that directed to input B is varied.The output power (normalized to the total input power) of port C1, port C2, and port D are plotted against the phase shift between input A and input B in Fig. 3(c).It is seen that the outputs of port C1 and C2 are equal, i.e., C1:C2 = 1.More importantly, the splitting ratio between ports D and C1 (C2) is found to depend on the phase shift introduced in input B.
To explain the physical origin of the above phenomena, a general case that a combination of simultaneous input A = a and B = 1  a with a relative phase shift Δφ is considered.Here, a and (1  a) represent the normalized input power from ports A and B, respectively.Note that the three-dielectric-core waveguide supports two eigen modes: symmetric and anti-symmetric modes.In that case, both magnetic fields 0 exp


     (from port B) can be considered as a superposition of these two eigen modes [34], and thus can be expressed as: Note that in Eqs. ( 2) and ( 3) the first term (symmetric part) contributes to the output power of port C1 (C2), while the second term (anti-symmetric part) is responsible for the output power of port D. By combining Eqs. ( 2) and ( 3) with their original forms, the parameters of A s , A a , Δφ 1 , and Δφ 2 are immediately given in Eqs. ( 4) and ( 5): When a is equal to 1/2, i.e., the inputs A and B have equal amplitude, the normalized output power of port C1(C2) and port D can be described as: (2 ) sin ( / 2), where the coefficients k C and k D are multiplied due to the existence of bend loss.It is found that Eqs. ( 6) and ( 7) with k C /2 = 0.395 and k D = 0.833 can best fit the normalized output power curves shown in Fig. 3(c).Thus, the splitting ratio between ports D and C1 (or C2) is found to show phase shift dependence in a manner of D/C 1 = 2.11tan 2 (Δφ/2), which implies that the splitting ratio could be adjusted continuously by controlling the phase difference.In addition, as seen from the magnetic field distribution for a typical splitter with Δφ = 0.4π [Fig.3(d)] the symmetric-field modes output from ports C1 and C2 always have the same phase.Particularly, since the coupling length (L C ) can almost be regarded as a constant value in the whole contra-directional coupling regime, the behavior of the proposed splitter is expected to be not degraded when the operation frequency varied within the range of 0.28ω p < ω < 0.33ω p , which has been confirmed in our simulations for another two frequencies ω = 0.28ω p and 0.33ω p (data not shown here).This is quite different from the previously reported plasmonic co-directional couplers, where the maximum power transfer occurs periodically along the direction of propagation, and the coupling length itself is dependent on the operation wavelength due to the materials dispersion [23].In that case, even when the phase difference is introduced into a co-directional coupler with a constant length, similar results can only be achieved for a specified wavelength, i.e., variation of operation wavelength will significantly degrade the behavior of co-directional couplers.Now we shall consider another case that the original power-ratio between input A and input B is varied while the phase shift between them is fixed.Here the phase shift is restricted to a range from 0 to π.In Fig. 4(a), the normalized output power is plotted as functions of the normalized input power for three typical phase shifts.The output power of port C1 and C2 is always identical, regardless of the input power difference between port A and B. For Δφ = 0.2π, the output power of port C1/C2 (port D) first increases and then decreases (first decreases and then increases) with increasing the input power of port A (correspondingly, with a decreasing input power of port B since the total input power is kept constant).An inverse tendency of the output power is observed for the case of Δφ = 0.8π.Again, from the magnetic field distribution for input A = 20%, input B = 80% and Δφ = 0.2π, it is observed that the symmetric-field modes output from ports C1 and C2 are always in phase [Fig.4(b)].Additionally, a special case is found for the phase shift Δφ = 0.5π, in which the output power of ports C1 (C2) and D does not change as the power-ratio is varied.The reason is simply because in Eq. ( 4) when cos(Δφ) = 0 contributions from symmetric and anti-symmetric parts of magnetic fields are equal (A s = A a ).In fact, with assistance of two-beam interference the switching functions are readily achieved in the present plasmonic splitters.The mechanism is based on the constructive or destructive interference between two backward modes in the GaP core waveguides coupled respectively from the identical left and right Si 3 N 4 core waveguides if an appropriate phase shift is introduced.When the external optical signal (ω = 0.3ω p ) is simultaneously input from ports A and B, a destructive interference of two backward modes is observed in the GaP-core waveguide [Fig.5(a)].In this case, two input eigen modes seems to propagate along the Si 3 N 4 -core waveguides without bearing the contra-directional coupling.It is found that ~40% power (normalized to the total input power) output from port C1 and as well from port C2 ("on" state).Meanwhile, no output power is observed in port D ("off" state).If an additional π phase shift is introduced into one of two inputs, a constructive interference of two backward modes now occurs in the center GaP-core waveguide [Fig.5(b)], which results in that more than ~80% power output from port D ("on" state), but no output power is obtained from Ports C1 and C2 ("off" state).This means that opposite switching functions are now realized on the port D and ports C1/C2.In the above cases, perfect metal (without loss) is used to acquire meaningful physical insights.However, in the practice the realistic metal loss has to be taken into account.For this purpose, we will discuss the effect of metal loss by assuming the relaxation time τ = 1.25 × 10 14 s in the Drude model described in Eq. (1). Figure 6(a) shows attenuation constant for a three-dielectric-core waveguide in the presence of metal loss.By comparing Fig. 6(a) with Fig. 3(b), it is immediately seen that significant attenuation occurs out of the contradirectional coupling regime, which is purely caused by metal loss.However, the attenuation constants within the contra-directional coupling regime are very close to the ones extracted from the perfect metal model, which means that the contra-directional coupling effect still dominates over the metal loss in the frequency range 0.28ω p < ω < 0.33ω p .A possible way to decrease the whole metal loss is to truncate the size of plasmonic devices as minimum as possible.Since all the operation of the proposed devices is based on the contra-directional coupling, to ensure a sufficient high coupling level the length of three-dielectric-core waveguide should be on the order of several coupling length.As shown in Fig. 6(b), we only keep the area indicated by the dashed-line square box (140 nm × 140 nm).In this case, the remained three-dielectric-core waveguide is 70 nm long (~2L C ).
After reducing the device size, we reevaluate the performance of plasmonic devices here.As shown in the inset of Fig. 6(b), the magnetic field (H y ) distribution (ω = 0.3ω p , input A = 50%, input B = 50% and Δφ = π) is almost the same as that for perfect metal model shown in Fig. 5(b).According to our simulations, when the metal loss is taken into account the level of output power from both ports C1/C2 and D significantly drops.For example, the output power from port D has dropped from ~80% (lossless model) to ~40% (lossy model) for the case of ω = 0.3ω p , input A = 50%, input B = 50% and Δφ = π.At the same time, we shall point out that the phase-shift dependent tendency of the output power does not change.For practical purposes, we suggest that by introducing gain materials into the proposed structure to enlarge such a narrow difference and compensate the unavoidable metal loss [35][36][37].

Design and analysis of waveguide cross-over components
The ability to intersect waveguides is very important in constructing high density integrated optical circuits, owing to the desire for complex systems involving multiple waveguides [38].In order to decrease the crosstalk and insertion loss, different mechanisms have been proposed and investigated intensively [39].In this section, we will demonstrate that based on the contra-directional coupling composite multi-dielectric-core MDM waveguides can also be used to realize waveguide crossover components with high throughput.Moreover, dispersion curves for symmetric and anti-symmetric modes of dual-GaP-core waveguide are found to cross the dispersion of forward mode in single-Si 3 N 4 -core waveguide at the frequencies of ω = 0.33ω p and 0.29ω p , respectively.As a result, two contra-directional coupling effects are expected to occur around these two crossing points.It is important to note that the frequency range for strong contra-directional coupling between symmetric mode of dual-GaP-core waveguide and forward mode of single-Si 3 N 4 -core waveguide is from 0.325ω p to 0.345ω p , and for another case is from 0.28ω p to 0.3ω p .This allows our proposed structure to operate at a frequency within a range of 0.28ω p < ω < 0.3ω p .
As a first case, the external signal (ω = 0.29ω p ) is directed to port A of single-Si 3 N 4 -core waveguide and excite a symmetric mode.As shown in Fig. 8(a), the input power first transfers to the dual-GaP-core waveguide in a very short distance due to the strong contra-directional coupling between forward mode of single-Si 3 N 4 -core waveguide and backward antisymmetric mode of dual-GaP-core waveguide.After that, the power in dual-GaP-core waveguide immediately couples to another single-Si 3 N 4 -core waveguide based on the same mechanism.In this way, ~90% power (normalized to the input power) are output from port B, i.e., it reaches ~90% horizontal throughput (~10% power reflect back due to the bending loss).When a symmetric mode (ω = 0.29ω p ) of dual-GaP-core waveguide is excited on port C, it is seen from the magnetic field (H y ) distribution [Fig.8(b)] that the input signal propagates directly along the vertical dual-GaP-core waveguide without suffering any cross talk because there is almost no coupling between symmetric mode of dual-GaP-core waveguide and forward mode of single-Si 3 N 4 -core waveguide.In this case it reaches nearly 100% (normalized to the input power) vertical throughput (output from port D).Actually, the symmetric mode dispersion branch of a dual-GaP-core waveguide [black solid line in Fig. 7(b)] is originated from that of GaP-Ag-GaP (DMD) waveguide [40].Thus, it provides us a possible way to transfer the symmetric mode of GaP-Ag-GaP waveguide to the symmetric mode of dual-GaP-core waveguide, and vice versa.Figure 8(c) schematically shows this transition structure, where a 12-nm-thick silver layer surrounded by GaP medium acts as an input DMD waveguide.As seen from Fig. 8(d), after the symmetric mode is excited on this GaP-Ag-GaP waveguide, it indeed propagates and transfers to the desired symmetric mode of dual-GaP-core waveguide in an efficient manner (~90% efficiency).

Conclusion
In conclusion, the guiding SPP modes including the symmetric and anti-symmetric field modes in MDM structures are numerically investigated.We show that the contra-directional coupling between adjacent MDM waveguides with different dielectric cores can be achieved with high coupling efficiency.Based on the contra-directional coupling, we theoretically demonstrate that a composite three-dielectric-core MDM structure with mirror symmetry can be used to realize ultra-short plasmonic splitters with simplified designs.In certain cases, the switching functions can also be achieved in the proposed plasmonic splitters with assistance of phase shift.Furthermore, it is shown that based on the same physical mechanism composite multi-dielectric-core MDM structures could serve as a novel plasmonic waveguide crossover component with low cross talk and high throughput.

Fig. 1 .
Fig. 1. (Color online) (a) Phase constants of planar Ag-Si3N4-Ag (dark yellow solid line), Ag-GaP-Ag (blue solid line), and Ag-Si3N4-Ag-GaP-Ag structures (co-directional and weak coupling regime: red dashed line; contra-directional and strong coupling regime: red solid line with symbols).The thicknesses of dielectric cores (both Si3N4 and GaP) and Ag layer between Si3N4 and GaP are 20 nm.(b) Attenuation constant of the composite Ag-Si3N4-Ag-GaP-Ag structure.Red dashed line represents the co-directional and weak coupling regime.Red solid line with solid circles represents the mode that carries power forward in the GaP-core waveguide.Red solid line with open triangles represents the mode carries power forward in the Si3N4-core waveguide.(c)-(d) Magnetic field y-component (Hy) distributions of the eigen modes at a frequency of ω = 0.31ωp in the single dielectric-core waveguides.(e) Hy distribution (red solid line) and electric field x-component (Ex) distributions (the mode carries power forward in the GaP-core waveguide: dark-yellow dashed line; the mode carries power forward in the Si3N4-core waveguide: blue dashed line) for a composite dual-dielectric-core waveguide at a frequency of ω = 0.3ωp.

Fig. 2 .
Fig. 2. (Color online) (a) The schematic of a composite two-dielectric-core waveguide together with structure sizes.(b) The magnetic field (Hy) distribution in this structure when an eigen mode at a frequency of ω = 0.3ωp is launched from the left input port.

Fig. 3 .
Fig. 3. (Color online) (a) The schematic of our proposed plasmonic splitter containing a composite three-dielectric-core structure with mirror symmetry.(b) The attenuation constant (red line with open triangles) and coupling length (blue line with solid circles) of a threedielectric-core composite structure as a function of frequency without metal loss.Note that only the mode carries power forward in the Si3N4-core waveguide is considered here.(c) The normalized output power of ports C1 (C2) and D as a function of phase shift between input A and B. The normalized input power of port A and B are 50%.(d) The magnetic field (Hy) distribution for the phase shift Δφ = 0.4π.

Fig. 4 .
Fig. 4. (Color online) (a) The output power as functions of the normalized input power for three typical phase shifts: Δφ = 0.2π, 0.5π and 0.8π.(b) The magnetic field (Hy) distribution for the case of 20% input power in port A, 80% power input in port B, and Δφ = 0.2π.

Fig. 5 .
Fig. 5. (Color online) The magnetic field distributions for the case that the normalized input power of port A and B are 50%, but with different phase shift: (a) Δφ = 0 and (b) Δφ = π.

Fig. 6 .
Fig. 6. (Color online) (a) The attenuation constant of the proposed three-dielectric-core composite structure with metal loss as a function of frequency.Only the mode carries power forward in the Si3N4-core waveguide is shown here.(b) In the lossy case, only the area indicated by the dashed-line square box (140 nm × 140 nm) is kept to decrease the whole metal loss.The inset is the magnetic field (Hy) distribution for the constructive interference.

Fig. 7 .
Fig. 7. (Color online) (a) A schematic of the waveguide cross-over component composed of composite multi-dielectric-core waveguides.The dielectric cores (both Si3N4 and GaP) and Ag layer between Si3N4 and GaP are 20 nm thick.(b) Phase constants of planar Ag-Si3N4-Ag (blue solid line), Ag-GaP-Ag-GaP-Ag (symmetric field mode: black solid line; anti-symmetric field mode: dark-yellow solid line), and Ag-GaP-Ag-GaP-Ag-Si3N4-Ag structures (co-directional and weak coupling range: red solid lines without symbols; contra-directional and strong coupling range: red solid lines with circles).

Figure 7 (
Figure 7(a) shows the schematic of our proposed waveguide cross-over component.Two Ag-Si 3 N 4 -Ag waveguides (Si 3 N 4 core: 20 nm) are placed on each sides of a composite Ag-GaP-Ag-GaP-Ag structure (GaP core: 20 nm, middle Ag layer thickness: 20 nm).The thickness of Ag layer between Si 3 N 4 and GaP is also 20 nm.The dispersion properties of these two MDM waveguides are calculated within a frequency range ω Ag-GaP < ω < ω Ag-Si3N4 and shown in Fig. 7(b).As already discussed in section 2, the single-Si 3 N 4 -core waveguide (Ag-Si 3 N 4 -Ag structure) only exists a forward mode in this frequency range [blue solid line in Fig. 7(b)].For the composite dual-GaP-core waveguide (Ag-GaP-Ag-GaP-Ag structure), both symmetric field [black solid line in Fig. 7(b)] and anti-symmetric field [dark-yellow solid line in Fig. 7(b)] modes are present in the same frequency range, and both are backward modes.Moreover, dispersion curves for symmetric and anti-symmetric modes of dual-GaP-core waveguide are found to cross the dispersion of forward mode in single-Si 3 N 4 -core waveguide at the frequencies of ω = 0.33ω p and 0.29ω p , respectively.As a result, two contra-directional coupling effects are expected to occur around these two crossing points.It is important to note that the frequency range for strong contra-directional coupling between symmetric mode of dual-GaP-core waveguide and forward mode of single-Si 3 N 4 -core waveguide is from 0.325ω p to 0.345ω p , and for another case is from 0.28ω p to 0.3ω p .This allows our proposed structure to operate at a frequency within a range of 0.28ω p < ω < 0.3ω p .As a first case, the external signal (ω = 0.29ω p ) is directed to port A of single-Si 3 N 4 -core waveguide and excite a symmetric mode.As shown in Fig.8(a), the input power first transfers to the dual-GaP-core waveguide in a very short distance due to the strong contra-directional coupling between forward mode of single-Si 3 N 4 -core waveguide and backward antisymmetric mode of dual-GaP-core waveguide.After that, the power in dual-GaP-core waveguide immediately couples to another single-Si 3 N 4 -core waveguide based on the same mechanism.In this way, ~90% power (normalized to the input power) are output from port B, i.e., it reaches ~90% horizontal throughput (~10% power reflect back due to the bending loss).When a symmetric mode (ω = 0.29ω p ) of dual-GaP-core waveguide is excited on port C, it is seen from the magnetic field (H y ) distribution [Fig.8(b)] that the input signal propagates directly along the vertical dual-GaP-core waveguide without suffering any cross talk because there is almost no coupling between symmetric mode of dual-GaP-core waveguide and forward mode of single-Si 3 N 4 -core waveguide.In this case it reaches nearly 100% (normalized to the input power) vertical throughput (output from port D).Actually, the symmetric mode dispersion branch of a dual-GaP-core waveguide [black solid line in Fig.7(b)] is originated from that of GaP-Ag-GaP (DMD) waveguide[40].Thus, it provides us a possible way to transfer the symmetric mode of GaP-Ag-GaP waveguide to the symmetric mode of dual-GaP-core waveguide, and vice versa.Figure 8(c) schematically shows this transition structure, where a 12-nm-thick silver layer surrounded by GaP medium acts as an input DMD waveguide.As seen from Fig. 8(d), after the symmetric mode is excited on this

Fig. 8 .
Fig. 8. (Color online) (a) The magnetic field (Hy) distribution for the case that the input signal is excited in port A. The output power is along the horizontal path and output from port B. (b) The magnetic field (Hy) distribution for the case that the input signal is excited in port C. The output power is along the vertical path and output from port D. (c) A schematic of the transition structure used to transfer the symmetric mode of GaP-Ag-GaP waveguide to the symmetric mode of dual-GaP-core waveguide.(d) The magnetic field (Hy) distribution for the transition process.