## On long-range plasmonic modes in metallic gaps

Optics Express, Vol. 15, Issue 21, pp. 13669-13674 (2007)

http://dx.doi.org/10.1364/OE.15.013669

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### Abstract

Satuby and Orenstein [Opt. Express **15**, 4247–4252 (2007)] reported the discovery and numerical and experimental investigation of long-range surface plasmon-polariton eigenmodes guided by wide (6 to 12 μm) rectangular gaps in 400 nm thick gold films using excitation of vacuum wavelength λ_{
vac
} = 1.55 μm. In this paper, we carry out a detailed numerical analysis of the two different types of plasmonic modes in these structures. We show that no long-range eigenmodes exists for these gap plasmon waveguides, and that the reported “modes” are likely to be beams of bulk waves and surface plasmons, rather than guided modes of the considered structures.

© 2007 Optical Society of America

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

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

4. D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface”, Opt. Lett. **29**, 1069–1071 (2004). [CrossRef] [PubMed]

11. Y. Satuby and M. Orenstein, “Surface-Plasmon-Polariton modes in deep metallic trenches- measurement and analysis,” Opt. Express **15**, 4247–4252 (2007). [CrossRef] [PubMed]

11. Y. Satuby and M. Orenstein, “Surface-Plasmon-Polariton modes in deep metallic trenches- measurement and analysis,” Opt. Express **15**, 4247–4252 (2007). [CrossRef] [PubMed]

11. Y. Satuby and M. Orenstein, “Surface-Plasmon-Polariton modes in deep metallic trenches- measurement and analysis,” Opt. Express **15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

_{Au}= - 96.90 +

*i*10.97); vacuum wavelength λ

_{vac}= 1.55 μm; and the gold film (and the gap) is surrounded by (filled with) a uniform dielectric with the permittivity ε

_{d}= 2.25. The numerical results presented here are from the commercial finite-element frequency-domain solver of Maxwell’s equations (COMSOL). These results have also been verified by two previously developed and tested in-house numerical algorithms: a 3D finite-difference time-domain (FDTD) solver [1

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

10. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. **87**, 061106 (2005). [CrossRef]

12. D. F. P. Pile, “Compact-2D FDTD for waveguides including materials with negative dielectric permittivity, magnetic permittivity and refractive index,” Appl. Phys. B **81**, 607–613 (2005). [CrossRef]

**87**, 261114 (2005). [CrossRef]

8. D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. **100**, 013101 (2006). [CrossRef]

10. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. **87**, 061106 (2005). [CrossRef]

12. D. F. P. Pile, “Compact-2D FDTD for waveguides including materials with negative dielectric permittivity, magnetic permittivity and refractive index,” Appl. Phys. B **81**, 607–613 (2005). [CrossRef]

*n*) ≡ Re[

_{eff}*k*λ

_{vac}/(2π)] for the two existing plasmonic eigenmodes in the considered structure as functions of the gap width

*w*. Here,

*k*is the wave number of the guided plasmonic mode in the gap. The fundamental mode is formed by the coupled wedge plasmons whose electric field distribution [1

**87**, 261114 (2005). [CrossRef]

*s*)

_{f}a_{g}_{cw}to denote this fundamental plasmonic mode to reflect the corresponding symmetry of the charge distribution across the film (index

*f*) and across the gap (index

*g*); the indices

*cw*indicate that this mode is formed by coupled wedge plasmons. If the gap width is smaller than the penetration depth of the plasmon field, the symmetry of the fundamental plasmonic mode ((

*s*)

_{f}a_{g}_{cw}) results in a strong attractive interaction of the opposite electric charges across the gap, leading to the strongest mode localization and largest (compared to any other mode in the considered structure) effective refractive index (i.e. the smallest phase velocity) [1

**87**, 261114 (2005). [CrossRef]

*n*(and mode localization) increases with decreasing gap width and/or film thickness (Fig. 2(a) and see also [1

_{eff}**87**, 261114 (2005). [CrossRef]

*s*)

_{f}a_{g}_{cw}mode does not have a minimum cut-off film thickness or gap width. This mode was also considered in [2

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

*s*)

_{f}s_{g}_{cw}eigenmodes with the

*s*symmetry of the charge distribution (Fig. 2(d)) can only exist at gap widths

_{f}s_{g}*w*≥ 0.6 μm. If

*w*is reduced below 0.6 μm, the repulsive interaction between the same charges across the gap makes

*n*for the (

_{eff}*s*)

_{f}s_{g}_{cw}mode smaller than that for the surface plasmons on the top and bottom interfaces of the gold film (dotted line in Fig. 2(a)). The (

*s*)

_{f}s_{g}_{cw}mode becomes non-eigen, i.e., leaking into the non-localized surface plasmons. Increasing the gap width results in reducing the repulsive interaction of charges across the gap, thus increasing the effective refractive index for the (

*s*)

_{f}s_{g}_{cw}mode (Fig. 2(a)). If the gap width is significantly larger than the penetration depth of the electromagnetic field into the dielectric, the effect of symmetry of the charge distribution across the gap becomes negligible (see the field distribution for the (

*s*)

_{f}a_{g}_{cw}mode in Fig. 3(a) for the gap width

*w*= 6 μm which is the smallest gap width considered in [11

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

*s*)

_{f}a_{g}_{cw}and (

*s*)

_{f}s_{g}_{cw}modes tend to the same value (Fig. 2(a)). Physically, this value is equal to the effective index for the plasmon mode guided by the two 90° coupled wedges formed by a rectangular cross-section of the metal film, i.e., when the gap width is equal to infinity (Fig. 3(b)). The effective mode index in this case is

*n*≈ 1.553 +

_{eff}*i*0.0054, the real part of which is given by the dashed line in Fig. 2(a).

*w*= 6 μm in Fig. 3(a)), the differences between the distributions of electric field magnitude in the (

*s*)

_{f}a_{g}_{cw}and (

*s*)

_{f}s_{g}_{cw}modes are negligible, and Fig. 3(a) may equally be used to represent either of these modes. Increasing the gap width only results in further separation of the wedge plasmons coupled across the gap, and further reduction of the field in the middle of the gap.

*s*)

_{f}a_{g}_{cw}and (

*s*)

_{f}s_{g}_{cw}modes in the considered structure are presented in Fig. 2(b) as functions of gap width. As can be seen from Fig. 2(b), the maximal propagation distance corresponds to the (

*s*)

_{f}s_{g}_{cw}mode and is ≈ 26 μm. Clearly, this is not a long-range mode. For comparison, the propagation distance for a non-localized surface plasmon on a smooth gold-dielectric interface (corresponding to λ

_{vac}= 155, ε

_{Au}= -96.90 +

*i*10.97, and ε

_{d}= 2.25) is ≈ 61 μm.

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

*a*)

_{f}a_{g}_{cw}(circles), (

*s*)

_{f}a_{g}_{gp1}(crosses), and (

*a*)

_{f}a_{g}_{gp1}(squares) plasmonic modes. The indices

*gp*for the last two modes indicate that these modes are formed by gap plasmons (Fig. 1(c)) rather than coupled wedge plasmons (Figs. 1(b,d)); the additional index 1 is used to denote the order of these modes, because higher order

*gp*-type plasmonic modes (e.g., the (

*s*)

_{f}a_{g}_{gp2}mode) can in principle exist in the considered gap plasmon waveguides (but at significantly smaller gap widths). Note also that the

*gp*-modes always correspond to antisymmetric charge distribution across the gap. (Symmetric

*gp*-plasmons always leak into surface plasmons because the repulsive interaction of the same charges across the gap always results in reducing their wave vector below that of surface plasmons.) Nevertheless, to preserve the consistency of notation, we will formally include the symbol

*a*in brackets when denoting these modes.

_{g}*a*)

_{f}a_{g}_{cw}, (

*s*)

_{f}a_{g}_{gp1}, and (

*a*)

_{f}a_{g}_{gp1}plasmonic modes in the considered structures is limited to very small gap widths. For example, the respective upper cut-off gap widths for these modes are ≈ 36 nm, ≈ 8.7 nm and ≈ 4.5 nm (Fig. 3(a)). These upper cut-off widths are considerably smaller than the gap widths of 6 – 12 μm considered in [11

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

*gp*-modes at gap widths of 6 – 12 μm. Therefore, the coupled plasmons propagating on the sides of the gap (trench) cannot be the source for the main field lobe of the trench mode, as was incorrectly (in our opinion) suggested in the third paragraph of section 2 in paper [11

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

*n*= 1.496 + 0.000164

_{eff}*i*(for the trench of 6 μm width) and

*n*= 1.49768 + 0.0000513

_{eff}*i*(for the trench of 10 μm width) [11

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

*n*= 1.50) and effective refractive index for surface plasmons at the top and bottom interfaces of the metal film (

_{d}*n*= 1.518). This means that the suggested “long-range plasmonic modes” cannot be structural eigenmodes, because such modes would be leaking into both the surrounding dielectric medium and surface plasmons at the film interfaces.

_{SP}**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

**15**, 4247–4252 (2007). [CrossRef] [PubMed]

## References and links

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

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

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

4. | D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface”, Opt. Lett. |

5. | D. K. Gramotnev and D. F. P. Pile. “Single-mode sub-wavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface”, Appl. Phys. Lett. |

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

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

8. | D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. |

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

10. | D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. |

11. | Y. Satuby and M. Orenstein, “Surface-Plasmon-Polariton modes in deep metallic trenches- measurement and analysis,” Opt. Express |

12. | D. F. P. Pile, “Compact-2D FDTD for waveguides including materials with negative dielectric permittivity, magnetic permittivity and refractive index,” Appl. Phys. B |

13. | Y. Satuby and M. Orenstein, “Surface plasmon poloariton waveguiding: From multimode stripe to a slot geometry,” Appl. Phys. Lett. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 25, 2007

Revised Manuscript: September 4, 2007

Manuscript Accepted: September 4, 2007

Published: October 4, 2007

**Citation**

David F. P. Pile, Dmitri K. Gramotnev, Rupert F. Oulton, and Xiang Zhang, "On long-range plasmonic modes in metallic gaps," Opt. Express **15**, 13669-13674 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13669

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### References

- D. F. P. Pile, T. Ogawa, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005). [CrossRef]
- G. Veronis and S. Fan, "Guided subwavelength plasmonic mode support 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 and D. K. Gramotnev, "Channel plasmon-polariton in a triangular groove on a metal surface," Opt. Lett. 29, 1069-1071 (2004). [CrossRef] [PubMed]
- D. K. Gramotnev and D. F. P. Pile, "Single-mode sub-wavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface," Appl. Phys. Lett. 85, 6323-6325 (2004). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005). [CrossRef] [PubMed]
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon sub-wavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006). [CrossRef] [PubMed]
- D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 013101 (2006). [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 (2000). [CrossRef]
- D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005). [CrossRef]
- Y. Satuby and M. Orenstein, "Surface-Plasmon-Polariton modes in deep metallic trenches- measurement and analysis," Opt. Express 15, 4247-4252 (2007). [CrossRef] [PubMed]
- D. F. P. Pile, "Compact-2D FDTD for waveguides including materials with negative dielectric permittivity, magnetic permittivity and refractive index," Appl. Phys. B 81, 607-613 (2005). [CrossRef]
- Y. Satuby and M. Orenstein, "Surface plasmon poloariton waveguiding: From multimode stripe to a slot geometry," Appl. Phys. Lett. 90, 251104 (2007). [CrossRef]

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