## Resonant-tunnelling-assisted crossing for subwavelength plasmonic slot waveguides

Optics Express, Vol. 16, Issue 19, pp. 14997-15005 (2008)

http://dx.doi.org/10.1364/OE.16.014997

Acrobat PDF (250 KB)

### Abstract

We theoretically investigate properties of crossing for two perpendicular subwavelength plasmonic slot waveguides. In terms of symmetry consideration and resonant-tunnelling effect, we design compact cavity-based crossing structures for nanoplasmonic waveguides. Our results show that the crosstalk is practically eliminated and the throughput reaches the unity on resonance. Simulation results are in agreement with those from coupled-mode theory. Taking the material loss into account, the symmetry properties of the modes are preserved and the crosstalk remains suppressed, while the throughput is naturally lowered. Our results may open a way to construct nanoscale crossings for high-density nanoplasmonic integration circuits.

© 2008 Optical Society of America

## 1. Introduction

*λ*

_{0}/2

*n*limit [1

1. M. Lipson, “Guiding, modulating, and emitting light on silicon - challeges and opportunities,” J. Lightwave Technol. **23**, 4222–4238 (2005). [CrossRef]

2. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. **23**, 401–412 (2005). [CrossRef]

3. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron **11**, 232–240 (2005). [CrossRef]

4. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. **22**, 475–477 (1997). [CrossRef] [PubMed]

*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

6. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. **2**, 229–232 (2003). [CrossRef] [PubMed]

8. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. **30**, 3359–3361 (2005). [CrossRef]

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

10. D. F. P. Pile, T. Ogawa, D. K. Gramotven, Y. Matsuzaki, K. C. Vernon, T. Yamaguchi, K. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionallly localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. **87**, 261114 (2005). [CrossRef]

11. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature **440**, 508–511 (2006). [CrossRef] [PubMed]

12. S. Xiao, L. Liu, and M. Qiu, “Resonator channel drop filters in a plasmon-polaritons metal,” Opt. Express **14**, 2932–2937(2006). [CrossRef] [PubMed]

13. 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**, 2442–2446 (2004). [CrossRef]

14. W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Opt. Lett. **32**, 2801–2803 (2007). [CrossRef] [PubMed]

15. T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. Part 1 **43**, 646–647 (2004). [CrossRef]

16. H. Chen and A. W. Poon, “Low-loss multimode-interference-based crossings for silicon wire waveguides,” IEEE Photon. Technol. Lett. **18**, 2260–2262 (2006). [CrossRef]

17. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. , **47**, 1927–1930 (1981). [CrossRef]

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

19. J. Jung, T. Sondergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polaritons waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B , **76**, 035434 (2007). [CrossRef]

19. J. Jung, T. Sondergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polaritons waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B , **76**, 035434 (2007). [CrossRef]

## 2. Dispersion of surface plasmon polariton waveguides

*β*=

*β*+

_{R}*jβ*of surface plasmon polaritons can be obtained exactly by solving the dispersion equation [22

_{I}22. I. P. Kamonow, W. L. Mammel, and H. P. Weber, “Metal-clad optical waveguides: analytical and experimental study,” Appl. Opt. **13**, 396–405 (1974). [CrossRef]

*ε*

_{1}and

*ε*are the dielectric constants of the medium in guide region and metals, respectively, and

_{m}*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

22. I. P. Kamonow, W. L. Mammel, and H. P. Weber, “Metal-clad optical waveguides: analytical and experimental study,” Appl. Opt. **13**, 396–405 (1974). [CrossRef]

*λ*

_{0}arctan (

*λ*

_{0}= 1.55

*µ*m. 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}= 2

*π*/

*λ*

_{0}. From Fig. 1, one can see that the effective refractive index (

*n*

_{eff}=

*β*/

_{R}*k*

_{0}) of SPP TM

_{0}mode is always larger than that of the dielectric, i. e.,

## 3. Direct crossing for two perpendicular plasmonic waveguides

*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

14. W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Opt. Lett. **32**, 2801–2803 (2007). [CrossRef] [PubMed]

*η*=

*n*

_{silicon}/

*n*

_{silica}= 2.34, have a large intersection loss for the direct waveguide crossing [16

16. H. Chen and A. W. Poon, “Low-loss multimode-interference-based crossings for silicon wire waveguides,” IEEE Photon. Technol. Lett. **18**, 2260–2262 (2006). [CrossRef]

*η*=

*n*

_{silver}/

*n*

_{air}= 9.3 for the working wavelength 1.55

*µ*m. 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*) 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

_{m}*µ*m. 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

21. G. VeronisS. Fan
“Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. **87**, 131102 (2005). [CrossRef]

*µ*m, 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*), where

_{c}*ε*

_{∞}/

*ε*

_{0}is the relative permittivity at infinite/zero frequency,

*ω*is the plasma frequency, and

_{p}*v*is the collision frequency. We choose

_{c}*ε*

_{∞}= 4.017,

*ε*

_{0}= 4.688,

*ω*= 1.419 × 10

_{p}^{16}rad/s and

*v*= 1.117 × 10

_{c}^{14}rad/s for the Drude model, which fits the experimental data [23] quite well.

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

15. T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. Part 1 **43**, 646–647 (2004). [CrossRef]

*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*and

_{x}*L*. 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

_{y}*T*and reflection

*R*on resonance. The expressions are given by

_{0}is the decay rate due to the internal loss in the cavity and 1/τ

*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 τ*

_{e}_{0}/τ

*on resonance. If there is no internal loss in the cavity (1/τ*

_{e}_{0}→0), the incident wave is completely transmitted on resonance and the spectral width of the resonance is determined by the coupling strength (1/τ

*) between the waveguide and the cavity. Assuming that the metal is lossless, the transmission*

_{e}*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*of the cavity, while it slightly shifts when varying

_{x}*L*. We also find that, for a fixed value of

_{y}*L*, the quality factor

_{x}*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*= 700 nm,

_{x}*L*= 600 nm, and increases to 10 for

_{y}*L*= 700 nm,

_{x}*L*= 1000 nm. Figure 3(b) shows the transmission

_{y}*T*and reflection

*R*of the device (

*L*= 700 nm,

_{x}*L*= 1000 nm) as a function of the wavelength for

_{y}*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*, while almost independent on

_{x}*L*, as mentioned above.

_{y}25. S. G. Johnson, C. Manolatou, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. **23**, 1855–1857 (1998). [CrossRef]

*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.

*significantly decreases when taking the loss into account and becomes comparable to 1/τ*

_{o}*. 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),*

_{e}*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

*µ*m, 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

26. S. A. Marier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. **258**, 295–299 (2006). [CrossRef]

27. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express **12**, 4072–4079 (2006). [CrossRef]

## 5. Summary

## Acknowledgments

## References and links

1. | M. Lipson, “Guiding, modulating, and emitting light on silicon - challeges and opportunities,” J. Lightwave Technol. |

2. | W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. |

3. | T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron |

4. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

5. | T. W. Ebbesen, C. Genet, and S. I. Bozhebolnyi, “Surface-plasmon circuitry,” Phys. Today |

6. | S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. |

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

8. | G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. |

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

10. | D. F. P. Pile, T. Ogawa, D. K. Gramotven, Y. Matsuzaki, K. C. Vernon, T. Yamaguchi, K. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionallly localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. |

11. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature |

12. | S. Xiao, L. Liu, and M. Qiu, “Resonator channel drop filters in a plasmon-polaritons metal,” Opt. Express |

13. | 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 |

14. | W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Opt. Lett. |

15. | T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. Part 1 |

16. | H. Chen and A. W. Poon, “Low-loss multimode-interference-based crossings for silicon wire waveguides,” IEEE Photon. Technol. Lett. |

17. | D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. , |

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

19. | J. Jung, T. Sondergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polaritons waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B , |

20. | J. Jin, |

21. | G. VeronisS. Fan
“Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. |

22. | I. P. Kamonow, W. L. Mammel, and H. P. Weber, “Metal-clad optical waveguides: analytical and experimental study,” Appl. Opt. |

23. | E. D. Palik, |

24. | H. A. Haus, |

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

26. | S. A. Marier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. |

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

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: June 25, 2008

Revised Manuscript: September 5, 2008

Manuscript Accepted: September 5, 2008

Published: September 9, 2008

**Citation**

Sanshui Xiao and Niels A. Mortensen, "Resonant-tunnelling-assisted crossing for
subwavelength plasmonic slot
waveguides," Opt. Express **16**, 14997-15005 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14997

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### References

- M. Lipson, "Guiding, modulating, and emitting light on silicon - challeges and opportunities," J. Lightwave Technol. 23, 4222-4238 (2005). [CrossRef]
- W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, "Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology," J. Lightwave Technol. 23, 401-412 (2005). [CrossRef]
- T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron 11, 232-240 (2005). [CrossRef]
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, "Guiding of a one-dimensional optical beam with nanometer diameter," Opt. Lett. 22, 475-477 (1997). [CrossRef] [PubMed]
- T. W. Ebbesen, C. Genet, and S. I. Bozhebolnyi, "Surface-plasmon circuitry," Phys. Today 61, 44-50 (2008).
- S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003). [CrossRef] [PubMed]
- K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Opt. Lett. 82, 1158-1160 (2003).
- G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359-3361 (2005). [CrossRef]
- L. Liu, Z. Han, and S. He, "Novel surface plasmon waveguide for high integration," Opt. Express 13, 6645-6650 (2005). [CrossRef] [PubMed]
- D. F. P. Pile, T. Ogawa, D. K. Gramotven, Y. Matsuzaki, K. C. Vernon, T. Yamaguchi, K. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionallly localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006). [CrossRef] [PubMed]
- S. Xiao, L. Liu, and M. Qiu, "Resonator channel drop filters in a plasmon-polaritons metal," Opt. Express 14, 2932-2937(2006). [CrossRef] [PubMed]
- 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, 2442-2446 (2004). [CrossRef]
- W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, "Low-loss, low-cross-talk crossings for silicon-oninsulator nanophotonic waveguides," Opt. Lett. 32, 2801-2803 (2007). [CrossRef] [PubMed]
- T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, "Low loss intersection of Si photonic wire waveguides," Jpn. J. Appl. Phys. Part 1 43, 646-647 (2004). [CrossRef]
- H. Chen and A.W. Poon, "Low-loss multimode-interference-based crossings for silicon wire waveguides," IEEE Photon. Technol. Lett. 18, 2260-2262 (2006). [CrossRef]
- D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981). [CrossRef]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000). [CrossRef]
- J. Jung, T. Sondergaard, and S. I. Bozhevolnyi, "Theoretical analysis of square surface plasmon-polaritons waveguides for long-range polarization-independent waveguiding," Phys. Rev. B 76, 035434 (2007). [CrossRef]
- J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).
- G. Veronis and S. Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]
- I. P. Kamonow, W. L. Mammel, and H. P. Weber, "Metal-clad optical waveguides: analytical and experimental study," Appl. Opt. 13, 396-405 (1974). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N. J., 1984).
- S. G. Johnson, C. Manolatou, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Elimination of cross talk in waveguide intersections," Opt. Lett. 23, 1855-1857 (1998). [CrossRef]
- S. A. Marier, "Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides," Opt. Commun. 258, 295-299 (2006). [CrossRef]
- M. P. Nezhad. K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides," Opt. Express 12, 4072-4079 (2006). [CrossRef]

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