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

Optics Express, Vol. 19, Issue 3, pp. 2562-2572 (2011)

http://dx.doi.org/10.1364/OE.19.002562

Acrobat PDF (1220 KB)

### Abstract

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 multi-dielectric-core composite MDM structures could serve as a novel plasmonic waveguide crossover component with low cross talk and high throughput.

© 2011 OSA

## 1. Introduction

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). [CrossRef]

2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science **311**(5758), 189–193 (2006). [CrossRef] [PubMed]

*μ*m) [2

2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science **311**(5758), 189–193 (2006). [CrossRef] [PubMed]

4. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

5. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B **61**(15), 10484–10503 (2000). [CrossRef]

6. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express **13**(17), 6645–6650 (2005). [CrossRef] [PubMed]

8. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**(12), 2442–2446 (2004). [CrossRef]

9. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B **75**(24), 245405 (2007). [CrossRef]

10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. **95**(4), 046802 (2005). [CrossRef] [PubMed]

11. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. **100**(2), 023901 (2008). [CrossRef] [PubMed]

6. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express **13**(17), 6645–6650 (2005). [CrossRef] [PubMed]

8. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A **21**(12), 2442–2446 (2004). [CrossRef]

12. J. Pan, K. T. He, Z. Chen, and Z. L. Wang, “Realization of subwavelength guiding utilizing coupled wedge plasmon polaritons in splitted groove waveguides,” Opt. Express **18**(16), 16722–16732 (2010). [CrossRef] [PubMed]

13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. **96**(7), 073907 (2006). [CrossRef] [PubMed]

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13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. **96**(7), 073907 (2006). [CrossRef] [PubMed]

14. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science **316**(5823), 430–432 (2007). [CrossRef] [PubMed]

15. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(5), 057602 (2003). [CrossRef] [PubMed]

19. A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. **4**(1), 043509 (2010). [CrossRef]

20. K. Halterman, J. M. Elson, and P. L. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Express **11**(6), 521–529 (2003). [CrossRef] [PubMed]

21. W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, “Interaction Between Negative and Positive Index Medium Waveguides,” J. Lightwave Technol. **26**(21), 3560–3566 (2008). [CrossRef]

22. A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. **90**(21), 211101 (2007). [CrossRef]

23. Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. **34**(3), 310–312 (2009). [CrossRef] [PubMed]

24. Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express **14**(16), 7279–7290 (2006). [CrossRef] [PubMed]

24. Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express **14**(16), 7279–7290 (2006). [CrossRef] [PubMed]

25. W. Liu, A. A. Sukhorukov, A. E. Miroshnichenko, C. G. Poulton, Z. Y. Xu, D. N. Neshev, and Y. S. Kivshar, “Complete spectral gap in coupled dielectric waveguides embedded into metal,” Appl. Phys. Lett. **97**(2), 021106 (2010). [CrossRef]

_{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.

## 2. Dispersion relation and coupling characteristics of MDM structures

*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*= 1.29 × 10

_{p}^{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

26. K. Y. Kim, Y. K. Cho, H.-S. Tae, and J.-H. Lee, “Light transmission along dispersive plasmonic gap and its subwavelength guidance characteristics,” Opt. Express **14**(1), 320–330 (2006). [CrossRef] [PubMed]

27. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B **73**(3), 035407 (2006). [CrossRef]

_{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*+ i

*α*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*is given by

_{Ag-dielectric}*ω*=

_{Ag-dielectric}*ω*/(1 +

_{p}*ε*)

_{d}^{1/2}, where

*ε*is the relative permittivity of the dielectric layer adjacent to the semi-infinite metal. Obviously, the surface plasmon resonance frequency for a silver-Si

_{d}_{3}N

_{4}interface (

*ω*= 0.447

_{Ag-Si3N4}*ω*) is larger than that for a silver-GaP interface (

_{p}*ω*= 0.268

_{Ag-GaP}*ω*). As shown in Fig. 1(a), below the silver-GaP surface plasmon resonance frequency both the Ag-Si

_{p}_{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*) distributions are calculated at the frequency

_{y}*ω*<

_{Ag-GaP}*ω*= 0.31

*ω*<

_{p}*ω*and are shown in Figs. 1(c) and 1(d) for the planar Ag-Si

_{Ag-Si3N4}_{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*= 0.3

*ω*(corresponding wavelength: ~490 nm), which makes a strong coupling between the symmetric and anti-symmetric modes possible.

_{p}_{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 0.28

*ω*<

_{p}*ω*< 0.33

*ω*(around the crossing point

_{p}*ω*= 0.3

*ω*of two individual MDM dispersion curves) the forward mode in the Ag-Si

_{p}_{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*= 0.3

*ω*, formed in this two-dielectric-core composite MDM structure with its

_{p}*H*and electric field

_{y}*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 0.28

*ω*<

_{p}*ω*< 0.33

*ω*[Fig. 1(b)], which in turn confirms the occurrence of contra-directional mode coupling between Ag-Si

_{p}_{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

24. Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express **14**(16), 7279–7290 (2006). [CrossRef] [PubMed]

_{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.

_{3}N

_{4}-Ag waveguide. To decrease the bending loss all the sharp corners are rounded with a 30 nm-radius curvature [6

6. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express **13**(17), 6645–6650 (2005). [CrossRef] [PubMed]

*ω*= 0.3

*ω*is excited, and propagates along the left input port of the Si

_{p}_{3}N

_{4}core. The corresponding magnetic field (

*H*) 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

_{y}_{3}N

_{4}-Ag waveguide is transferred into an anti-symmetric mode propagating backward in the upper portion of the Ag-GaP-Ag structure.

## 3. Design and analysis of plasmonic splitters

_{3}N

_{4}-Ag-GaP-Ag waveguide exhibits the large attenuation constant in the contra-directional coupling regime (corresponding to a frequency range 0.28

*ω*<

_{p}*ω*< 0.33

*ω*), which indicates that the energy transfers from the input Si

_{p}_{3}N

_{4}-core 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 contra-directional coupling length that paves a possible way to realize ultra-small devices with simplified designs.

_{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 semi-infinitely long [24

**14**(16), 7279–7290 (2006). [CrossRef] [PubMed]

*L*) is defined here as a distance along the direction of propagation where the input power exponentially decay to 1/

_{C}*e*, i.e.,

*L*= 1/

_{C}*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 three-dielectric-core waveguide in the frequency range 0.28

*ω*<

_{p}*ω*< 0.33

*ω*is on the order of ~40 nm. Thus, in the design a 210 nm (~5

_{p}*L*) 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

_{C}28. Y. L. Zhang, Y. Zhang, and B. J. Li, “Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals,” Opt. Express **15**(15), 9287–9292 (2007). [CrossRef] [PubMed]

29. S. Q. Zeng, Y. Zhang, B. J. Li, and E. Y.-B. Pun, “Ultrasmall optical logic gates based on silicon periodic dielectric waveguides,” Photonics Nanostruct. Fundam. Appl. **8**(1), 32–37 (2010). [CrossRef]

30. K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics **3**(1), 55–58 (2009). [CrossRef]

33. J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. **9**(2), 897–902 (2009). [CrossRef] [PubMed]

*ω*= 0.30

*ω*) directed to the inputs

_{p}*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*.

*A*=

*a*and

*B*= 1 −

*a*with a relative phase shift Δ

*φ*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

*A*) and

*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*, Δ

_{a}*φ*

_{1}, and Δ

*φ*

_{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: where the coefficients

*k*and

_{C}*k*are multiplied due to the existence of bend loss. It is found that Eqs. (6) and (7) with

_{D}*k*/2 = 0.395 and

_{C}*k*= 0.833 can best fit the normalized output power curves shown in Fig. 3(c). Thus, the splitting ratio between ports

_{D}*D*and

*C1*(or

*C2*) is found to show phase shift dependence in a manner of

*D/C*= 2.11tan

_{1}^{2}(Δ

*φ*/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 Δ

*φ*= 0.4π [Fig. 3(d)] the symmetric-field modes output from ports

*C1*and

*C2*always have the same phase.

*L*) 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

_{C}*not*degraded when the operation frequency varied within the range of 0.28

*ω*<

_{p}*ω*< 0.33

*ω*, which has been confirmed in our simulations for another two frequencies

_{p}*ω*= 0.28

*ω*and 0.33

_{p}*ω*(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

_{p}23. Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. **34**(3), 310–312 (2009). [CrossRef] [PubMed]

*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 Δ

*φ*= 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 Δ

*φ*= 0.8π. Again, from the magnetic field distribution for input

*A*= 20%, input

*B*= 80% and Δ

*φ*= 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 Δ

*φ*= 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(Δ

*φ*) = 0 contributions from symmetric and anti-symmetric parts of magnetic fields are equal (

*A*=

_{s}*A*).

_{a}_{3}N

_{4}core waveguides if an appropriate phase shift is introduced. When the external optical signal (

*ω*= 0.3

*ω*) is simultaneously input from ports

_{p}*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*.

*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 contra-directional 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 0.28

*ω*<

_{p}*ω*< 0.33

*ω*. 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 (~2

_{p}*L*).

_{C}## 4. Design and analysis of waveguide cross-over components

38. S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. **23**(23), 1855–1857 (1998). [CrossRef]

39. W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Compact and low crosstalk waveguide crossing using impedance matched metamaterial,” Appl. Phys. Lett. **96**(11), 111114 (2010). [CrossRef]

_{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*and shown in Fig. 7(b). As already discussed in section 2, the single-Si

_{Ag-Si3N4}_{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*= 0.33

*ω*and 0.29

_{p}*ω*, 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

_{p}_{3}N

_{4}-core waveguide is from 0.325

*ω*to 0.345

_{p}*ω*, and for another case is from 0.28

_{p}*ω*to 0.3

_{p}*ω*. This allows our proposed structure to operate at a frequency within a range of 0.28

_{p}*ω*<

_{p}*ω*< 0.3

*ω*.

_{p}*ω*= 0.29

*ω*) is directed to port

_{p}*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 anti-symmetric 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*= 0.29

*ω*) of dual-GaP-core waveguide is excited on port

_{p}*C*, it is seen from the magnetic field (

*H*) 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

_{y}*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

40. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**(2), 539–554 (1969). [CrossRef]

## 5. Conclusion

## Acknowledgements

## References and links

1. | R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today |

2. | E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science |

3. | H. Raether, |

4. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

5. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B |

6. | L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express |

7. | K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. |

8. | R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A |

9. | T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B |

10. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. |

11. | E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. |

12. | J. Pan, K. T. He, Z. Chen, and Z. L. Wang, “Realization of subwavelength guiding utilizing coupled wedge plasmon polaritons in splitted groove waveguides,” Opt. Express |

13. | H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. |

14. | H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science |

15. | I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

16. | B. L. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary, transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. |

17. | A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express |

18. | A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express |

19. | A. R. Davoyan, I. V. Shadrivov, S. I. Bozhevolnyi, and Y. S. Kivshar, “Backward and forward modes guided by metal-dielectric-metal plasmonic waveguides,” J. Nanophoton. |

20. | K. Halterman, J. M. Elson, and P. L. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Express |

21. | W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, “Interaction Between Negative and Positive Index Medium Waveguides,” J. Lightwave Technol. |

22. | A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. |

23. | Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. |

24. | Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express |

25. | W. Liu, A. A. Sukhorukov, A. E. Miroshnichenko, C. G. Poulton, Z. Y. Xu, D. N. Neshev, and Y. S. Kivshar, “Complete spectral gap in coupled dielectric waveguides embedded into metal,” Appl. Phys. Lett. |

26. | K. Y. Kim, Y. K. Cho, H.-S. Tae, and J.-H. Lee, “Light transmission along dispersive plasmonic gap and its subwavelength guidance characteristics,” Opt. Express |

27. | J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B |

28. | Y. L. Zhang, Y. Zhang, and B. J. Li, “Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals,” Opt. Express |

29. | S. Q. Zeng, Y. Zhang, B. J. Li, and E. Y.-B. Pun, “Ultrasmall optical logic gates based on silicon periodic dielectric waveguides,” Photonics Nanostruct. Fundam. Appl. |

30. | K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics |

31. | M. J. Dicken, L. A. Sweatlock, D. Pacifici, H. J. Lezec, K. Bhattacharya, and H. A. Atwater, “Electrooptic modulation in thin film barium titanate plasmonic interferometers,” Nano Lett. |

32. | J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup-Hansen, L. Markey, and A. Dereux, “Thermo-optic control of dielectric-loaded plasmonic waveguide components,” Opt. Express |

33. | J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. |

34. | K. Iizuka, |

35. | J. Seidel, S. Grafström, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. |

36. | I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B |

37. | M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express |

38. | S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. |

39. | W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Compact and low crosstalk waveguide crossing using impedance matched metamaterial,” Appl. Phys. Lett. |

40. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(200.4660) Optics in computing : Optical logic

(240.6680) Optics at surfaces : Surface plasmons

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: November 30, 2010

Revised Manuscript: December 26, 2010

Manuscript Accepted: January 5, 2011

Published: January 26, 2011

**Citation**

Yi-Jiao Fang, Zhuo Chen, Ling Chen, Kai-Ting He, Zhen-lv Han, and Zhen-Lin Wang, "Ultra-short plasmonic splitters and waveguide cross-over based on coupled surface plasmon slot waveguides," Opt. Express **19**, 2562-2572 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2562

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### References

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