## Leaky modes of curved long-range surface plasmon-polariton waveguide

Optics Express, Vol. 14, Issue 26, pp. 13043-13049 (2006)

http://dx.doi.org/10.1364/OE.14.013043

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### Abstract

A three-dimensional method for obtaining the bending losses and field distributions of bent surface plamon-polariton waveguides is presented. The method is based on a so called ‘method of line’, which discretises potential in the direction of the metal-widths, and leads to Airy-equations in the radial direction. From the results obtained. It is confirmed that thiner metal waveguide enable longer-ranging propagation of surface plasmon-polariton mode, but the weakened confinement requires larger bending radii on order to keep radiation loss.

© 2006 Optical Society of America

## 1. Introduction

1. Pierre Berini, “Plasmon-polariton wave guided by thin lossy metal films of finite width:Bounded modes of symmetric structures,” Phys. Rev. B **61**, 484–503 (2000). [CrossRef]

2. Thomas Nikolajsen,
etc., “Polymer-based surface plasmon-polariton stripe waveguides at telecommunication wavelengths,” App. Phys. Lett. **82**, 668–670 (2003). [CrossRef]

## 2. Theory

*r*,

*ϕ*,

*y*). It cosists of a metal film of thickness

*t*and width

*w*has an equivalent complex permittivity

*ε*

_{2}and supported by a semi-infinite homogeneous dielectric substrate of permittivity

*ε*

_{1}and covered by a semi-infinite homogeneous dielectric superstrate of permittivity

*ε*

_{3}. For purpose of analysis, the cross-section of the waveguide is subdivided into three regions I, II, and III. The vector wave equations for the surface plasmon-polariton waveguide are approximated by the scalar Helmholz wave equation, and we only analyze the case of transverse magnetic(TM) modes with E

_{r}, electric fields in the radial direction here since E

_{r}is the main transverse electric field component in most practical structures

*w*/

*t*≫1. In the present article all the fields are assumed to have harmonic time dependance exp(

*jwt*). For TM waves propagating in the

*ϕ*direction, one can write the field component E

_{r}in the following form

*β*is the propagation constant along the mean radius R.

*E*

_{r}(

*r*,

*y*), satisfies the following Helmholtz equation,

*k*

_{0}is the wave number in free space and

*n*(

*y*) is the refractive index. Replacing

*r*=

*R*+

*x*and taking it into consideration that radius

*R*is by far greater than metal thickness

*t*, we can take only the first order of the Taylor expansion of

*D*

_{yy}] of the form of a matrix[6

6. H Diestel, “A method for calculating the guided modes of strip-loaded optical waveguides with arbitrary index profile,” IEEE J. Quantum Electron. **20**, 1288–1292 (1984). [CrossRef]

*ψ*

_{i}} in each area, we obtain from Eq. (5) a set of equations:

*diag*[(

^{2}], where

*k*th line. To decouple the equations we take the sum of the matrices and transform them into the diagonal form by means of the orthogonal transformation:

*T*

_{i}] is the eigenvector and the diagonal matrix [Λ] is the eigenvalue of [

*D*

_{yy}]+

*U*

_{i}}=[

*T*

_{i}]

^{-1}{

*ψ*

_{i}}. We can rewrite Eq. (8) in terms of {

*U*

_{i}} and obtain

7. I. C. Goyal, R. L. Gallawa, and A. K. Ghatak, “Bent planar waveguides and whispering gallery modes: A new method of anaysis,” IEEE J. Lightwave Technol. **8**, 768–774 (1990). [CrossRef]

*Ai*(

*x*goes to -∞(

*Ai*(

*Z*

^{(k)}

_{II}) and

*Bi*(

*x*.

*r*=

*R*,

*r*=

*R*+

*t*leads to the following complex matrix equation:

*β*is complex and its imaginary part is the radiation loss coefficient of the bent surface plasmon-polariton waveguide.

## 3. Results

*w*, wavelength

*λ*, curvature radius

*r*

_{0}and values of indices

*n*

_{1}and

*n*

_{2}, evaluation of Eq. (13) gives us the eigenvalue

*β*of complex number.

*E*

_{r}| of the

1. Pierre Berini, “Plasmon-polariton wave guided by thin lossy metal films of finite width:Bounded modes of symmetric structures,” Phys. Rev. B **61**, 484–503 (2000). [CrossRef]

*r*

_{0}=5mm, respectively. Figure 2(c) shows relative distribution of the mode field along a vertical cut immediately above the metal film for

*r*

_{0}=5mm. As the figures confirm, the bend effects the mode field to shift toward the outer interface. The leaky field is also observed along the outside of the bend.

*r*

_{0}in Fig. 3 As the figures confirm, thick metal waveguide and abrupt bending cause large attenuation. The result also reveals that the propagation losses converge into those of straight waveguides with corresponding thickness as indicated as arrows.

*w*=4

*µm*,

*λ*=1.3

*µm*,

*n*

_{1}=1.535,

*n*

_{2}=0.3859-i7.965)

*w*=4

*µm*,

*λ*=1.3

*µm*,

*n*

_{1}=1.535,

*n*

_{2}=0.3859-i7.965)

## 4. Conclusion

## References and links

1. | Pierre Berini, “Plasmon-polariton wave guided by thin lossy metal films of finite width:Bounded modes of symmetric structures,” Phys. Rev. B |

2. | Thomas Nikolajsen,
etc., “Polymer-based surface plasmon-polariton stripe waveguides at telecommunication wavelengths,” App. Phys. Lett. |

3. | Robert Charbonneau,
etc., “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express |

4. | Robert Charbonneau,
etc., “Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width,” Opt. Lett. |

5. | Pierre Berini, “Curved long-range surface plasmon-polariton waveguides,” Opt. Express |

6. | H Diestel, “A method for calculating the guided modes of strip-loaded optical waveguides with arbitrary index profile,” IEEE J. Quantum Electron. |

7. | I. C. Goyal, R. L. Gallawa, and A. K. Ghatak, “Bent planar waveguides and whispering gallery modes: A new method of anaysis,” IEEE J. Lightwave Technol. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 9, 2006

Revised Manuscript: December 15, 2006

Manuscript Accepted: December 17, 2006

Published: December 22, 2006

**Citation**

Woo-Kyung Kim, Woo-Seok Yang, Hyung-Man Lee, Han-Young Lee, Myung-Hyun Lee, and Woo-Jin Jung, "Leaky modes of curved long-range surface plasmon-polariton waveguide," Opt. Express **14**, 13043-13049 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-13043

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### References

- P. Berini, "Plasmon-polariton wave guided by thin lossy metal films of finite width:bounded modes of symmetric structures," Phys. Rev. B 61, 484-503 (2000). [CrossRef]
- T. Nikolajsen, etc., "Polymer-based surface plasmon-polariton stripe waveguides at telecommunication wavelengths," Appl. Phys. Lett. 82, 668-670 (2003). [CrossRef]
- R. Charbonneau, etc., "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005). [CrossRef] [PubMed]
- R. Charbonneau, etc., "Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000). [CrossRef]
- P. Berini, "Curved long-range surface plasmon-polariton waveguides," Opt. Express 14, 2365-2371 (2006). [CrossRef] [PubMed]
- H. Diestel, "A method for calculating the guided modes of strip-loaded optical waveguides with arbitrary index profile," IEEE J. Quantum Electron. 20, 1288-1292 (1984). [CrossRef]
- I. C. Goyal, R. L. Gallawa, and A. K. Ghatak, "Bent planar waveguides and whispering gallery modes: A new method of anaysis," IEEE J. Lightwave Technol. 8, 768-774 (1990). [CrossRef]

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