## Surface plasmon mode analysis of nanoscale metallic rectangular waveguide

Optics Express, Vol. 15, Issue 19, pp. 12331-12337 (2007)

http://dx.doi.org/10.1364/OE.15.012331

Acrobat PDF (199 KB)

### Abstract

A detailed study of guided modes in a nanoscale metallic rectangular waveguide is presented by using the effective dielectric constant approach. The guided modes, including both traditional waveguide mode and surface plasmon mode, are investigated for the silver rectangular waveguide. The mode evolution in narrow waveguide is also discussed with the emphasis on the dependence of mode dispersion with waveguide height. Finally, the red-shift of the cutoff wavelength of the fundamental mode is observed when the waveguide height decreases, contrary to the behavior of regular metallic waveguide with PEC boundary. The comprehensive analysis can provide some guideline in the design of subwavelength optical devices based on the dispersion characteristics of metallic rectangular bore.

© 2007 Optical Society of America

## 1. Introduction

1. P. N. Prasad, *Nanophotonics* (Wiley-Interscience, New Jersey, 2004). [CrossRef]

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

3. T. Rindzevicius, Y. Alaverdyan, and B. Sepulveda. “Nanohole plasmons in optically thin gold films,” J. Phys. Chem. C **111**, 1207–1212(2007). [CrossRef]

4. F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B **74**, 205419 (2006). [CrossRef]

5. C. K. Chang, D. Z. Lin, and C. S. Yeh, *et. al.*, “Experimental analysis of surface plasmon behavior in metallic circular slits,” Appl. Phys. Lett. **90**, 061113 (2007). [CrossRef]

6. K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Optical guided dispersions and subwavelength transmissions in dispersive plasmonic circular holes,” Opto-Electron. Rev. **14**, 233–241 (2006). [CrossRef]

7. A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. **239**, 61–66 (2004). [CrossRef]

8. R. Gordon, L. K. S. Kumar, and A. G. Brolo, “Resonant light transmission through a nanohole in a metal film,” IEEE Nanotechnology **5**, 291–294 (2006). [CrossRef]

9. F. M. Kong, K. Li, and B. I. Wu, *et. al.*, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” Prog. Electromagn. Res. **76**, 449–466 (2007) [CrossRef]

10. E. X. Jin and X. Xu, “Finite-difference time-domain studies on optical transmission through planar nano-apertures in a metal Film,” Jpn. J. Appl. Phys. **43**, 407–417(2004). [CrossRef]

11. K. Kawano and T. Kitoh, *Introduction to Optical Waveguide Analysis* (Wiley, Chichester, 2001). [CrossRef]

12. S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express **14**, 9467–9476 (2006). [CrossRef] [PubMed]

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

## 2. Analysis methodology

*ε*

_{r}_{1}, and the relative dielectric constant of the metal at optical frequencies can be approximated by the Drude model

*x*and

*y*directions respectively [Figs. 1(b), 1(c)].

*E*modes (TE-like), in which

^{y}_{mn}*E*and

_{y}*H*are the dominant electromagnetic fields. The wave functions for the

_{x}*x*-direction slab can be written as [15

15. B. I. 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. **93**, 9386 (2003). [CrossRef]

*y*=±

*b*:

*E*

_{z}_{1}=

*E*

_{z}_{2}and

*H*

_{x}_{1}=

*H*

_{x}_{2}, we get

*x*-direction slab

*k*, Eq. (4) can be rewritten as

_{y}*x*-direction slab waveguide can be expressed as

*ε*

_{er}_{1}=

*ε*

_{r}_{1}-(

*k*/

_{y}*k*

_{0})

^{2}. Thus, the metallic waveguide can be regarded as a slab waveguide [Fig.1 (c)] along

*y*-direction with the effective dielectric constant

*ε*

_{er}_{1}in the core and

*ε*

_{r}_{2}(

*ω*) in the cladding and the wave functions can be expressed as

*x*=±

*a*:

*E*

_{y}_{1}=

*E*

_{y}_{2}and

*H*

_{z}_{1}=

*H*

_{z}_{2}, we get

*y*-direction slab:

*ρ*=

_{x}*γ*/

_{x}*k*and

_{x}*γ*

^{2}

*=*

_{x}*k*

^{2}

_{0}(

*ε*

_{er}_{1}-

*ε*

_{r}_{2}(

*ω*))-

*k*

^{2}

*.*

_{x}*k*is imaginary, Eq. (9) can be rewritten as

_{x}*E*modes, there are two types of guided waves in the metallic rectangular waveguide. The first type is the TWG modes which satisfy the guidance conditions Eq. (5) and Eq. (10), and the second type is the SPP modes which satisfy the guidance conditions Eq. (6) and Eq. (10). The TWG modes are denoted as

^{y}_{mn}*E*, while the SPP modes are denoted as

^{y}_{mn}*E*or

^{y}_{me}*E*. The subscripts

^{y}_{mo}*m*and

*n*are integers and they correspond to the number of peaks of each field component in the

*x*and

*y*directions respectively, the subscripts

*e*and

*o*refer to the symmetry of the field distribution (even or odd) in

*x*or

*y*directions.

*E*mode as follows

^{x}_{mn}*γ*

^{2}

*=*

_{x}*k*

^{2}

_{0}(

*ε*

_{r}_{1}-

*ε*

_{r}_{2}(

*ω*))-

*k*

^{2}

*,*

_{x}*γ*

^{2}

*=*

_{y}*k*

^{2}

_{0}(

*ε*

_{r}_{1}-

*ε*

_{r}_{2}(

*ω*))-

*k*

^{2}

*-*

_{x}*k*

^{2}

*,*

_{y}*k*

*=*

_{x}*ik′*, and

_{x}*k*=

_{x}*ik′*(for SPP modes only).

_{x}*k*(or

_{x}*k′*) and

_{x}*k*(or

_{y}*k′*) numerically, and the propagation constant

_{y}*β*can be expressed as

## 3. Results and discussion

*ε*

_{r}_{2}=3.7 and

*ω*=7.1085×10

_{p}^{15}rad/s to model the permittivity of silver. The parameters were calculated by Sönnichsen [16] from experimental data on the reflection and transmission of 25–50 nm thick silver films for wavelengths from 0.188µm to 1.9µm[17

17. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379, (1972). [CrossRef]

*f*=1131THz) is the forbidden area where no guided mode exists. This region is delimited with the refractive index of the silver as solid line. It is shown that the SPP modes usually exist below a critical frequency (

_{p}*f*=1003THz) at which the relative permittivity is negative unity, and the lowest order TWG modes (

_{c}*E*

^{x}_{11},

*E*

^{y}_{11}) exist above the plasmon frequency, but some higher order TWG modes can exist both below and above the plasmon frequency. All the TWG modes locate in fast wave region (

*β*/

*k*

_{0}<1.0), while the SPP modes can migrate from slow wave region to fast wave region as the operating frequency decreases. Although each mode has its own dispersion characteristics, the dispersion curve of some modes can continue smoothly from one mode to another mode, for example: in both cases,

*E*

^{y}_{12}and

*E*

^{y}_{1o},

*E*

^{x}_{31}and

*E*

^{x}_{11}are continuous. Moreover, it is found that some of higher order TWG modes have discontinuous dispersion curves, for example

*E*

^{x}_{21}in 400nm×200nm silver waveguide, one part exists below the plasma frequency and connect with the mode

*E*

^{x}_{o}_{1}, and another part exists above the plasmon frequency and connects with

*E*

^{x}_{41}. Thus, some of TWG modes and SPP modes can coexist at the same frequency with certain waveguide geometries.

*a*is assumed to be 250nm. For comparison, the dispersion curves for the SPP modes

*TM*and

_{e}*TM*in the corresponding slab waveguides with the same heights are also shown in Fig. 3. As the waveguide height decreases, the SPP modes

_{o}*E*

^{x}_{e}_{1}and

*E*

^{x}_{o}_{1}are being suppressed and their cutoff frequencies move up and eventually approach the critical frequency. Also, the slope of curve for the odd SPP mode

*E*

^{y}_{1o}can change from positive to negative when the waveguide height is very small. This means that there exists backward surface wave in the narrow metal waveguide.

*TE*

_{10}mode, and the cut-off wavelength of

*TE*

_{10}is twice the waveguide width. However, the red-shift of cutoff wavelength of fundamental mode

*E*

^{y}_{1e}appears when waveguide aspect ratio decreases. This allows light with longer wavelength to propagate through the rectangular aperture. Moreover, it is found that when the aspect ratio is less than 0.45, it is more suitable for single mode operation.

## 4. Conclusion

*E*

^{x}_{e}_{1}and

*E*

^{x}_{o}_{1}are suppressed and the red-shift of cut off wavelength of the fundamental mode is observed. Furthermore, the slope of curve for the odd SPP mode

*E*

^{y}_{1}

_{o}can change from positive to negative when the waveguide height is very small. This means that there exists backward surface wave in the narrow metal waveguide. Moreover, it is found that when the aspect ratio is less than 0.45, it is more suitable for single mode operation. These results can provide some guideline in the design of nanoscale optical devices based on the dispersion characteristics of metallic rectangular hole.

## References and links

1. | P. N. Prasad, |

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

3. | T. Rindzevicius, Y. Alaverdyan, and B. Sepulveda. “Nanohole plasmons in optically thin gold films,” J. Phys. Chem. C |

4. | F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B |

5. | C. K. Chang, D. Z. Lin, and C. S. Yeh, |

6. | K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Optical guided dispersions and subwavelength transmissions in dispersive plasmonic circular holes,” Opto-Electron. Rev. |

7. | A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. |

8. | R. Gordon, L. K. S. Kumar, and A. G. Brolo, “Resonant light transmission through a nanohole in a metal film,” IEEE Nanotechnology |

9. | F. M. Kong, K. Li, and B. I. Wu, |

10. | E. X. Jin and X. Xu, “Finite-difference time-domain studies on optical transmission through planar nano-apertures in a metal Film,” Jpn. J. Appl. Phys. |

11. | K. Kawano and T. Kitoh, |

12. | S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express |

13. | S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express |

14. | Y. Satuby and M. Orenstein, “Surface-plasmon-polariton modes in deep metallic trenches-measurement and analysis,” Opt. Express |

15. | B. I. 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. |

16. | C. Sönnichsen, “Plasmons in metal nanostructures,” PhD Thesis (Ludwig-Maximilians-Universtät München, München, 2001). |

17. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

(260.2030) Physical optics : Dispersion

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: August 29, 2007

Revised Manuscript: September 7, 2007

Manuscript Accepted: September 8, 2007

Published: September 13, 2007

**Citation**

Fanmin Kong, Bae-Ian Wu, Hongsheng Chen, and Jin Au Kong, "Surface plasmon mode analysis of nanoscale metallic rectangular waveguide," Opt. Express **15**, 12331-12337 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12331

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### References

- P. N. Prasad, Nanophotonics (Wiley-Interscience, New Jersey, 2004). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- T. Rindzevicius, Y. Alaverdyan, and B. Sepulveda. "Nanohole plasmons in optically thin gold films," J. Phys. Chem. C 111, 1207-1212 (2007). [CrossRef]
- F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006). [CrossRef]
- C. K. Chang, D. Z. Lin, C. S. Yeh, et al., "Experimental analysis of surface plasmon behavior in metallic circular slits," Appl. Phys. Lett. 90, 061113 (2007). [CrossRef]
- K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, "Optical guided dispersions and subwavelength transmissions in dispersive plasmonic circular holes," Opto-Electron.Rev. 14, 233-241 (2006). [CrossRef]
- A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004). [CrossRef]
- R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Nanotechnology 5, 291-294 (2006). [CrossRef]
- F. M. Kong, K. Li, B. I. Wu, et al., "Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries," Prog. Electromagn. Res. 76, 449-466 (2007) [CrossRef]
- E. X. Jin and X. Xu, "Finite-difference time-domain studies on optical transmission through planar nano-apertures in a metal Film," Jpn. J. Appl. Phys. 43, 407-417(2004). [CrossRef]
- K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, Chichester, 2001). [CrossRef]
- S. I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467-9476 (2006). [CrossRef] [PubMed]
- S. Collin, F. Pardo, and J. L. Pelouard, "Waveguiding in nanoscale metallic apertures," Opt. Express 15, 4310-4320 (2007). [CrossRef] [PubMed]
- Y. Satuby and M. Orenstein, "Surface-plasmon-polariton modes in deep metallic trenches-measurement and analysis," Opt. Express 15, 4247-4252 (2007). [CrossRef] [PubMed]
- B. I. 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. 93, 9386 (2003). [CrossRef]
- C. Sönnichsen, "Plasmons in metal nanostructures," PhD Thesis (Ludwig-Maximilians-Universtät München, München, 2001).
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]

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