## Numerical simulations of long-range plasmons

Optics Express, Vol. 14, Issue 4, pp. 1611-1625 (2006)

http://dx.doi.org/10.1364/OE.14.001611

Acrobat PDF (375 KB)

### Abstract

We present simulations of plasmonic transmission lines consisting of planar metal strips embedded in isotropic dielectric media, with a particular emphasis on the long-range surface plasmon polariton (SPP) modes that can be supported in such structures. Our computational method is based on analyzing the eigenfrequencies corresponding to the wave equation subject to a mixture of periodic, electric and magnetic boundary conditions. We demonstrate the accuracy of our approach through comparisons with previously reported simulations based on the semi-analytical method-of-lines. We apply our method to study a variety of aspects of long-range SPPs, including tradeoffs between mode confinement and propagation distance, the modeling of bent waveguides and the effect of disorder and periodicity on the long-ranging modes.

© 2006 Optical Society of America

## 1. Introduction

1. J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J.-P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B **60**, 9061–9068 (1999). [CrossRef]

18. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) **424**, 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**, 10484–10503 (2000). [CrossRef]

5. 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]

6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B **63**, 125417 (2001). [CrossRef]

*long-range*SPPs [20

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

21. J.J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B **33**, 5286–5201 (1986). [CrossRef]

15. S.J. Al-Bader, “Optical Transmission on Metallic Wires - Fundamental Modes,” IEEE J. Quantum Electron. **40**, 325–329 (2004). [CrossRef]

16. Rashid Zia, Anu Chandran, and Mark L. Brongersma, “Dielectric waveguide model for guided surface polaritons,” Opt. Lett. **30**, 1473–1475 (2005). [CrossRef] [PubMed]

7. R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width,” Opt. Lett. **52**, 844–846 (2000). [CrossRef]

8. R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express **13**, 977–984 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-977. [CrossRef] [PubMed]

14. A. Boltasseva, S.I. Bozhevolnyi, T. Søndergaard, T. Nikolajsen, and K. Leosson, “Compact Z-add-drop wavelength filters for long-range surface plasmon polaritons” Opt. Express **13**, 4237–4243 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-11-4237. [CrossRef] [PubMed]

12. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. **85**, 5833–5835 (2004). [CrossRef]

22. J.P. Kottmann, O.J.F. Martin, D.R. Smith, and S. Schultz, “Spectral response of plasmon resonant nanoparticles with a non-regular shape,” Opt. Express **6**, 213–219 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=oe-6-11-213. [CrossRef] [PubMed]

23. J.P. Kottmann, O.J.F. Martin, D.R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. **341**, 1–6 (2001). [CrossRef]

## 2. Numerical method

### 2.1. Infinitely long and straight waveguides

*f*(

*z*+

*d*) =

*exp*(

*i*φ)

*f*(

*z*), where

*z*is the space coordinate parallel to the propagation direction and

*d*is the distance between the two planes of periodicity. Once φ is fixed, then an eigenvalue problem is specified by the materials within the unit cell and the boundary conditions. At every value of φ, an infinite number of eigenmodes can, in principle, be obtained, each with a different complex frequency.

*k*

_{zr}) is obtained by repeating the simulations for different values of

*ε*

_{Ag}; then

*k*

_{zi}, the imaginary part of the propagation constant parallel to the surface, can be calculated at any frequency according to the relation

*k*

_{zi}= 2πv” /v

_{g}, where v” is the imaginary part of the frequency and v

_{g}is the group velocity i.e. the derivative of v(

*k*

_{zr}). It should be noted that the use of periodic boundary conditions always folds the dispersion of the SPP modes when

*k*

_{zr}reaches the edge of the Brillouin zone π/

*d*. For non-periodic systems, we work with a sufficiently small value of

*d*(typically between 10 nm and 100 nm) so that the zone-folding occurs at frequencies beyond the spectral range we consider. It is usually desirable to minimize

*d*since it reduces the size of the computational domain and thereby improves the calculation process. However,

*d*should be non-zero even for truly 2-D problems (such as solving for the modes of the smooth rectangular waveguide of Fig. 1(a)) because we use an eigensolver for 3-D geometries.

### 2.2. Bent waveguides

_{0}, the mean radius of curvature (defined as the distance between the origin of the cylindrical coordinates and the middle of the strip), and ρ

_{s}the position of the outer surface of the unit cell; we maintain the previous notations for the width and thickness of the strip. The periodic boundary conditions applied at the edges of the unit cell are now that a field component satisfies

*f*(

*θ*

_{0}+

*θ*

_{1}) =

*exp*(

*i*φ)

*f*(

*θ*

_{0}), where

*θ*

_{0}and (

*θ*

_{0}+

*θ*

_{1}) are the position of the two planes of periodicity in cylindrical coordinates, and φ is the phase delay (see Fig. 1(b)). Note that when ρ

_{0}»

*w*and

*θ*

_{1}« 1, the previous equality can be approximated by

*f*(

*θ*

_{0}+

*θ*

_{1}) ≃

*exp*(

*i*

*θ*

_{1}ρ

_{0}

*k*

_{zr})

*f*(

*θ*

_{0}), which provides a qualitative relationship between φ and the in-plane component of the SPP wave vector

*k*

_{zr}.

25. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. **66**, 216–220 (1976). [CrossRef]

_{s}. We accordingly surround this surface with a Perfect Matching Layer (PML)–a fictitious anisotropic material that fully absorbs the electromagnetic fields impinging upon it. PMLs essentially push the boundary infinitely far away from the structure [26

26. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

27. R. Mittra and U. Pekel, “A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave Guid. Wave Lett. **5**, 84–86 (1995). [CrossRef]

## 3. Testing the simulation method

5. 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]

*ε*

_{d}= 4. In this case, the field distribution of the modes must be either symmetric or antisymmetric with respect to the two symmetry planes perpendicular to the propagation direction (here we refer to the symmetry of the electric field component parallel to the

*y*-direction, see Fig. 1(a)). For each of the four different field symmetries that are thus possible, a series of a fundamental and several higher-order modes can be excited along the strip in much the same way as electromagnetic modes supported by hollow waveguides. Following these considerations, each family can be generated separately by placing the proper combination of electric and/or magnetic walls halfway through the structure. Consequently only one quarter of the structure needs to be simulated, which significantly reduces the calculation time. For the eigensolver method used here–as with nearly all finite-difference or finite-element methods–the simulation of a rectangular strip is not straightforward because the sharp 90 degree corners of the structure generate strong field singularities that cannot be solved properly without using an extremely fine mesh with a large number of elements. In the following, we avoid this problem and shorten the calculation time by slightly rounding the corners of the strips. As we will see, this modification does not alter the results significantly.

_{vac}= 633 nm for strips of width

*w*= 1

*μ*m; the radius of the rounded corners is

*r*= 5 nm. We have labeled the modes according to the nomenclature proposed by Berini [5

**61**, 10484–10503 (2000). [CrossRef]

*E*

_{y}, the transverse electric field component of highest amplitude, is symmetric or antisymmetric with respect to the horizontal and vertical plane of symmetry, respectively. The subscript

*b*means that the modes are bounded to the surface and the superscript indicates the number of maxima in the spatial distribution of

*E*

_{y}along the largest dimension.

*E*

_{y}corresponding to the four fundamental modes are shown in Figs. 4 and 5 for the thicknesses

*t*= 200 nm and

*t*= 40 nm. In order to test the accuracy of our simulation technique, we solved the fields for the whole structure, without the use of perfect electric or magnetic walls along the symmetry planes of the strip. Although the different field distributions are fully consistent with the above discussion, the eigensolver finds that some modes are truly degenerate which is in contradiction to the conclusions of Berini who found all modes to be nondegenerate [5

**61**, 10484–10503 (2000). [CrossRef]

*t*= 200 nm. However, they represent in this case the symmetric solution with respect to the horizontal symmetry axis. In addition, the degeneracy is lifted as the thickness of the strip decreases. This is due to the fact that the fields now surround the whole structure, thus allowing the left and right side of the structure to couple.

**61**, 10484–10503 (2000). [CrossRef]

28. 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]

## 4. Further results and discussion

6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B **63**, 125417 (2001). [CrossRef]

*w*= 1 μm,

*t*= 20 nm, λ

_{vac}= 633 nm) for gradually higher asymmetric environments. It can be seen that the mode becomes strongly asymmetric when increasing the mismatch between the refractive indexes of the substrate (

*n*

_{sub}= 1.52) and the upper cladding (

*n*

_{up}). The electromagnetic field dramatically expands into the upper cladding so that the absorption losses into the metal are minimized–the propagation length increasing from 65 microns when

*n*

_{up}= 1.53 to 0.9 mm when

*n*

_{up}= 1.56. And yet, the optical field remains firmly bounded to the interface between the strip and the dielectric substrate; in fact it becomes even better confined as the asymmetry increases.

6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures,” Phys. Rev. B **63**, 125417 (2001). [CrossRef]

*t*= 40 nm,

*w*= 1

*μ*m,

*ε*

_{d}= 4) with various radii of curvature ρ

_{0}. As shown in Fig. 9(a) for ρ

_{0}= 15

*μ*m and ρ

_{0}= ∞, the mode dispersion relation remains largely independent of the radius of curvature even for very sharp bends. To gain further insight into this behavior, we map the electric field of the mode for different values of ρ

_{0}. Figure 9(b) summarizes these simulations performed at the free-space wavelength λ

_{vac}=633 nm. As ρ

^{0}decreases, the field evolves from a symmetric pattern surrounding the strip to a localized and highly asymmetric distribution around the outer corner. In other words, the bending improves the field confinement and hence allows the mode to propagate without suffering significant curvature losses. It is however likely that experimental transmission lines operating in similar conditions would lose more power than what the present results suggest. Real bends are often composed of combinations of straight and curved segments, in which case transition losses between the connecting segments can become important, as suggested by the significant mismatch between the field patterns of Fig. 9(b). It would be thus interesting to analyze in detail the coupling between structures of different curvatures. This would require simulating the junction and a significant length of strip, since the geometry of the problem does not allow the efficient use of periodic boundary conditions. Unfortunately the size of the computational domain required would be too large to perform tractable simulations using the eigensolver as outlined. We note that the metal thickness and consequently the field confinement of the structures considered in this section are too large for the

8. R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express **13**, 977–984 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-977. [CrossRef] [PubMed]

*P*is kept constant while the effect of the modulation depth is explored. The structure consists of a Ag strip of rectangular cross-section with a square wave modulation of the top and bottom surface (Fig. 11(a)). Strictly speaking, such a surface profile contains an infinite number of spatial Fourier components which each may influence the SPP modes. However, it is known that the lowest harmonic 2π/

*P*dominates the scattering of the SPP modes so that the role of the higher Fourier components can be ignored for sake of clarity [30

30. W.L. Barnes, T.W. Preist, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B **54**, 6227–6244 (1996). [CrossRef]

*ε*

_{d}= 4. The length of the unit cell in the direction of propagation is set to one period of the surface modulation. Thus, the phase advance between the two planes of periodicity is such that the corresponding propagation constant (

*k*

_{zr}) always lies within the first Brillouin zone.

*k*

_{zr}reaches the edge of the first Brillouin zone π/

*P*. The mode propagation at frequencies below or above the gap is significantly affected by the surface modulation depth because the scattering losses typically increase with the modulation height. Figure 11(c) illustrates this point by plotting the imaginary part of the frequency as a function of the wave vector for the different branches of Fig. 11(b).

34. A. Lai, C. Caloz, and T. Itoh, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Magazine **5**, 34–50 (2004). [CrossRef]

35. R. Islam, F. Elek, and G.V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directiona power flow,” Electron Lett. **40**, 315–317 (2004). [CrossRef]

36. V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of e and *μ*,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

37. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

38. M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A: Pure Appl. Opt. **7**, S12–S22 (2005). [CrossRef]

39. S. O’Brien and J.B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites” J. Phys. Condens. Matter **14**, 4035–4044 (2002). [CrossRef]

40. K.C. Huang, M.L. Povinelli, and J.D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. **85**543–545 (2004). [CrossRef]

*dω*/

*dk*and

*ω*/

*k*, respectively, have opposite sign, meaning that the energy flux points toward the opposite direction of the wave front propagation. Those waves can thus be considered as if having a negative propagation constant or as if traveling in a medium with a negative refractive index. This is exactly the case for the upper branch modes of Fig. 11(b) so they can exhibit many of the interesting and unusual properties initially proposed for bulk metamaterials.

## 5. Conclusion

## Acknowledgments

## References and links

1. | J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J.-P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B |

2. | J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J.-P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B |

3. | B. Lamprecht, J.R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, and F.R. Aussenegg, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. |

4. | J.-C. Weeber, M.U. González, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips” Appl. Phys. Lett. |

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

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

7. | R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width,” Opt. Lett. |

8. | R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express |

9. | S. Jette-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express |

10. | P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmon-polariton waveguides” J. Appl. Phys. |

11. | T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. |

12. | T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. |

13. | A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M.S. Larsen, and S.I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. |

14. | A. Boltasseva, S.I. Bozhevolnyi, T. Søndergaard, T. Nikolajsen, and K. Leosson, “Compact Z-add-drop wavelength filters for long-range surface plasmon polaritons” Opt. Express |

15. | S.J. Al-Bader, “Optical Transmission on Metallic Wires - Fundamental Modes,” IEEE J. Quantum Electron. |

16. | Rashid Zia, Anu Chandran, and Mark L. Brongersma, “Dielectric waveguide model for guided surface polaritons,” Opt. Lett. |

17. | H. Raether, |

18. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) |

19. | D.M. Pozar, |

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

21. | J.J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B |

22. | J.P. Kottmann, O.J.F. Martin, D.R. Smith, and S. Schultz, “Spectral response of plasmon resonant nanoparticles with a non-regular shape,” Opt. Express |

23. | J.P. Kottmann, O.J.F. Martin, D.R. Smith, and S. Schultz, “Dramatic localized electromagnetic enhancement in plasmon resonant nanowires,” Chem. Phys. Lett. |

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

25. | D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. |

26. | J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

27. | R. Mittra and U. Pekel, “A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave Guid. Wave Lett. |

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

29. | G. A. Farias and A. A. Maradudin, “ Effect of surface roughness on the attenuation of surface polaritons on metal films,” Phys. Rev. B |

30. | W.L. Barnes, T.W. Preist, S.C. Kitson, and J.R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B |

31. | W.L. Barnes, S.C. Kitson, T.W. Preist, and J.R. Sambles, “Photonic surfaces for surface-plasmon polaritons,” J. Opt. Soc. Am. A |

32. | P.E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, “Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers,” Appl. Phys. Lett. |

33. | S.A. Maier, M.D. Friedman, P.E. Barclay, and O. Painter, “Experimental demonstration of fiber-accessible metal nanoparticle plasmon waveguides for planar energy guiding and sensing,” Appl. Phys. Lett. |

34. | A. Lai, C. Caloz, and T. Itoh, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Magazine |

35. | R. Islam, F. Elek, and G.V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directiona power flow,” Electron Lett. |

36. | V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of e and |

37. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

38. | M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A: Pure Appl. Opt. |

39. | S. O’Brien and J.B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites” J. Phys. Condens. Matter |

40. | K.C. Huang, M.L. Povinelli, and J.D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. |

41. | C. Kittel, |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: December 16, 2005

Revised Manuscript: January 31, 2006

Manuscript Accepted: February 1, 2006

Published: February 20, 2006

**Citation**

Aloyse Degiron and David Smith, "Numerical simulations of long-range plasmons," Opt. Express **14**, 1611-1625 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1611

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### References

- J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J.-P. Goudonnet, "Plasmon polaritons of metallic nanowires for controlling submicron propagation of light," Phys. Rev. B 60, 9061-9068 (1999). [CrossRef]
- J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J.-P. Goudonnet, "Near-field observation of surface plasmon polariton propagation on thin metal stripes," Phys. Rev. B 64045411 (2001). [CrossRef]
- B. Lamprecht, J.R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, and F.R. Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79, 51-53 (2001). [CrossRef]
- J.-C. Weeber, M.U. González, A.-L. Baudrion, and A. Dereux, "Surface plasmon routing along right angle bent metal strips" Appl. Phys. Lett. 87, 221101, (2005). [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]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric structures," Phys. Rev. B 63, 125417 (2001). [CrossRef]
- R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width," Opt. Lett. 52, 844-846 (2000). [CrossRef]
- R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-977. [CrossRef] [PubMed]
- S. Jette-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides," Opt. Express 13, 4674-4682 (2005)http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4674. [CrossRef] [PubMed]
- P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmonpolariton waveguides" J. Appl. Phys. 98, 043109 (2005). [CrossRef]
- T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, "Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths," Appl. Phys. Lett. 82, 668-670 (2003). [CrossRef]
- T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, "Surface plasmon polariton based modulators and switches operating at telecom wavelengths," Appl. Phys. Lett. 85, 5833-5835 (2004). [CrossRef]
- A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M.S. Larsen, and S.I. Bozhevolnyi, "Integrated optical components utilizing long-range surface plasmon polaritons," J. Lightwave Technol. 23, 413-422 (2005). [CrossRef]
- A. Boltasseva, S.I. Bozhevolnyi, T. Søndergaard, T. Nikolajsen, and K. Leosson, "Compact Z-add-drop wavelength filters for long-range surface plasmon polaritons" Opt. Express 13, 4237-4243 (2005)http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-11-4237. [CrossRef] [PubMed]
- S.J. Al-Bader, "Optical Transmission on Metallic Wires - Fundamental Modes," IEEE J. Quantum Electron. 40, 325-329 (2004). [CrossRef]
- Rashid Zia, Anu Chandran, and Mark L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005). [CrossRef] [PubMed]
- H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 424, 824-830 (2003). [CrossRef] [PubMed]
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