## Nano plasmon polariton modes of a wedge cross section metal waveguide

Optics Express, Vol. 14, Issue 19, pp. 8779-8784 (2006)

http://dx.doi.org/10.1364/OE.14.008779

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### Abstract

Optical plasmon-polariton modes confined in both transverse dimensions to significantly less than a wavelength are exhibited in open waveguides structured as sharp metal wedges. The analysis reveals two distinctive modes corresponding to a localized mode on the wedge point and surface mode propagation on the abruptly bent interface. These predictions are accompanied by unique field distributions and dispersion characteristics.

© 2006 Optical Society of America

## 1. Introduction

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

2. P. Berini, “Plasmon-polariton modes guided by a metal film of a finite width bounded by different dielectrics,” Opt. Express **7**, 329 (2000). [CrossRef] [PubMed]

3. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” J. Nonlinear Physics and Materials **11**, 65 (2002). [CrossRef]

4. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a Dielectric Wedge,” Phys. Rev. B **6**, 3810 (1972). [CrossRef]

5. A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B **14**, 5526 (1976). [CrossRef]

6. A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: Analytical solution,” Phys. Rev. B **32**, 6045 (1985). [CrossRef]

4. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a Dielectric Wedge,” Phys. Rev. B **6**, 3810 (1972). [CrossRef]

7. 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]

## 2. Analysis of wedge plasmonic modes

*K*. Satisfying the boundary conditions at each interface point is impossible for a single Modified Bessel function at each medium, thus the solution is a series of K-functions.

_{v}(qr)exp(ivθ) or ~K

_{iv}(qr)exp(-vθ).

*q*is the radial momentum, and v denoted the K-function order. Taking the order to be imaginary is more suitable since it implies azimuthal hyperbolic dependence, rather than harmonic, which better describes the azimuthal decay of the plasmonic solutions away from the interface. Moreover, the {K

_{iv}(qr)}

_{v}is a complete set (in contrary to {K

_{v}(qr)}

_{v}) with basis functions being square integrable and having vanishing power at infinity (decaying faster than r

^{-1}).

_{v}K

_{iv}(qr)exp(-vθ), the expansion coefficients, as a function of the order, fv, define the v-spectrum of the solution. The general v-spectrum is continuous on the indefinite interval, [0,∞). However the compactness of this expansion allows for a sufficient representation on a limited interval, [0,v

_{max}]. The criteria for sufficiently high v

_{max}were the fulfillment of the boundary conditions to 1% accuracy. As the order of a basis function is increased, the azimuthal decay is enhanced while the radial one is decreased. The confinement of the plasmonic fields around the wedge tip is translated to effectively bounding the v-spectrum. This could be inferred also from the Kontorovich-Lebedev transform [11

11. G. Z. Forristall and J. D. Ingram, “Evaluation of distributions useful in Kontorovich-Lebedev transform theory,” SIAM J. Math. Anal. **3**, 561 (1972). [CrossRef]

_{max}may cause errors mostly at the interfaces. Thus, constructing and testing the solution, according to the boundary conditions, on the interface assures of the solution accuracy on the entire domain.

## 3. Modal characteristics

_{z}and H

_{z}), four independent boundary conditions apply for each r>0 on the interface θ=0:

*f*

_{i}(i=1…4) coefficients are frequency and

*β*dependent (

*β*- the modal propagation constant), given explicitly in the Appendix. From Eq. (1) it is apparent that only hybrid modes are possible. Out of the four symmetries we study in this text the solutions having the E(even)-H(odd) symmetry, exited with symmetrical electrical field.

^{2}-

_{D,M})

^{0.5}, k

_{0}is the free-space wave number, and ε is the relative dielectric constant. For H

_{z}o a,b,cosh are replaced by c,d,sinh in Eq. (2), v which denotes the K-function imaginary order, is real and continuous values. Since each of the basic field functions in both sides of the interface has a different r-dependence, the boundary conditions for all r values dictate a superposition of wave solutions. This is a major complication absent in the electrostatic analysis, for which one assumes substantial

*β*-values resulting with similar basis functions on both sides of interface (i.e. q

_{M}=q

_{D}).

_{iv}(q

_{D}r)}, using orthogonality of K

_{iv}(qr)/r, and characteristics of the K-function integral, yields 4N algebraic equations:

_{P}=137nm. The same calculation scheme may support the case of complex permittivity, namely complex

*ε*

_{M}and

*β*.

*β*, accompanied by tighter localization around the wedge point. The E

_{z}field component at pt. (I) is depicted in Fig. 2, and drops off symmetrically for all transverse directions – into the metal and dielectric. At the same effective index value, the lower dispersion branch exhibits different field characteristics, as demonstrated at Fig. 3. Comparison of the E

_{z}component reveals that a main lobe is still localized on the wedge point but secondary lobes appear as well on the metal interfaces. Decreasing the modal frequency, the number of side lobes as well as the respective modal cross section increase. The distribution of the pointing vector directed along z (S

_{z}) at dispersion points I and II are depicted in Figs 4(a) and 4(b), accordingly. Especially at point II, although the power propagates predominantly in the dielectric, more than 90% of the mode’s intensity and power (S

_{z}) are guided within a cross section of 0.01% of free-space wavelength - well under the “diffraction limit”. In Fig. 5 it is apparent that also for the lower branch – the mode becomes more localized towards the wedge point as β is increased.

_{spp}decreases. The cutoff frequency should increase with the wedge angle, as the lower dispersion branch may diverge less from the single surface SPP curve, due to reduced coupling between the modes near the wedge point. A cutoff angle is thus expected at each given wavelength, above which a solution does not exist. This resembles the cutoff angle that was observed in some specific FDTD simulations for different symmetry modes (E

^{e}H

^{e}) [7

7. 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]

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

7. 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]

## 4. Conclusion

## Appendix

_{2}F

_{1}denotes the Hypergeometric function.

## Acknowledgment

## References and Links

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

2. | P. Berini, “Plasmon-polariton modes guided by a metal film of a finite width bounded by different dielectrics,” Opt. Express |

3. | V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” J. Nonlinear Physics and Materials |

4. | L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a Dielectric Wedge,” Phys. Rev. B |

5. | A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B |

6. | A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: Analytical solution,” Phys. Rev. B |

7. | 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. |

8. | I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B |

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

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

11. | G. Z. Forristall and J. D. Ingram, “Evaluation of distributions useful in Kontorovich-Lebedev transform theory,” SIAM J. Math. Anal. |

12. | B. Prade and J. Y. Vinet, “Guided optical waves in fibers with negative dielectric constant,” J. Light. Tech. |

13. | H. Reather, |

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

**OCIS Codes**

(200.4650) Optics in computing : Optical interconnects

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: June 20, 2006

Revised Manuscript: August 21, 2006

Manuscript Accepted: August 28, 2006

Published: September 18, 2006

**Citation**

Eyal Feigenbaum and Meir Orenstein, "Nano plasmon polariton modes of a wedge cross section metal waveguide," Opt. Express **14**, 8779-8784 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8779

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824 (2003). [CrossRef] [PubMed]
- P. Berini, "Plasmon-polariton modes guided by a metal film of a finite width bounded by different dielectrics," Opt. Express 7, 329 (2000). [CrossRef] [PubMed]
- V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes in metal nanowires and left-handed materials," J. Nonlinear Phys. Mater. 11, 65 (2002). [CrossRef]
- L. Dobrzynski, and A. A. Maradudin, "Electrostatic edge modes in a Dielectric Wedge," Phys. Rev. B 6, 3810 (1972). [CrossRef]
- A. Eguiluz, and A. A. Maradudin, "Electrostatic edge modes along a parabolic wedge," Phys. Rev. B 14, 5526 (1976). [CrossRef]
- A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, "Electrostatic edge modes of a hyperbolic dielectric wedge: Analytical solution," Phys. Rev. B 32, 6045 (1985). [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]
- I. V. Novikov, and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002). [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]
- D. F. P. Pile, and D. K. Gramotnev, "Channel plasmon-polariton in a triangular groove on a metal surface," Opt. Lett. 29, 1069 (2004). [CrossRef] [PubMed]
- G. Z. Forristall and J. D. Ingram, "Evaluation of distributions useful in Kontorovich-Lebedev transform theory," SIAM J. Math. Anal. 3, 561 (1972). [CrossRef]
- B. Prade and J. Y. Vinet, "Guided optical waves in fibers with negative dielectric constant," J. Lightwave Technol. 12, 6 (1994). [CrossRef]
- H. Reather, Surface plasmon (Springer, Berlin, 1988).
- M. P. Nezhad, K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polariton on planar metallic waveguides," Opt. Express 12, 4072 (2004). [CrossRef] [PubMed]

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