## Nonlinear couplers with tapered plasmonic waveguides |

Optics Express, Vol. 20, Issue 9, pp. 9403-9408 (2012)

http://dx.doi.org/10.1364/OE.20.009403

Acrobat PDF (810 KB)

### Abstract

We suggest and demonstrate numerically that, by employing tapered waveguides in the geometry of a directional coupler, we can enhance dramatically the performance for optical switching of nonlinear plasmonic couplers operating at the nanoscale, overcoming the detrimental losses but preserving the subwavelength confinement. We demonstrate that, by an appropriate choice of the taper angle of the coupled metal-dielectric slot waveguides, we can compensate for the amplitude decrease and enhance the sharpness of the response for the switching operation.

© 2012 OSA

## 1. Introduction

*plasmons*[3

3. D. K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics **4**, 83–91 (2010). [CrossRef]

4. N. C. Panoiu and R. M. Osgood, “Subwavelength nonlinear plasmonic nanowire,” Nano Lett. **4**, 2427–2430 (2004). [CrossRef]

5. W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. **6**, 1027–1030 (2006). [CrossRef]

6. J. A. H. van Nieuwstadt, M. Sandke, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. **97**, 146102 (2006). [CrossRef] [PubMed]

7. R. Yang, M. A. G. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express **16**, 20142–20148 (2008). [CrossRef] [PubMed]

8. D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B **93**, 99–106 (2008). [CrossRef]

10. T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Design and characterization of dielectric-loaded plasmonic directional couplers,” J. Lightwave Technol. **27**, 5521–5528 (2009). [CrossRef]

11. A. Degiron, S. Y. Cho, T. Tyler, N. M. Jokerst, and D. R. Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. **11**, 015002 (2009). [CrossRef]

14. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. **12**, 187–189 (1987). [CrossRef] [PubMed]

15. A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express **16**, 21209–21214 (2008). [CrossRef] [PubMed]

16. J. R. Salgueiro and Yu. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. **97**, 081106 (2010). [CrossRef]

17. A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. **105**, 116804 (2010). [CrossRef] [PubMed]

19. S. I. Bozhevolnyi and K. V. Nerkararyan, “Adiabatic nanofocusing of channel plasmon polaritons,” Opt. Lett. **35**, 541–543 (2010). [CrossRef] [PubMed]

16. J. R. Salgueiro and Yu. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. **97**, 081106 (2010). [CrossRef]

## 2. Model and approach

*z*-axis (which will be considered the propagation direction) and lie both external borders forming an angle

*α*so that the waveguide width decreases linearly with distance as shown in Fig. 1. Considering the TM mode described by the electric and magnetic fields

**E**=

*E*

_{x}**x**+

*E*

_{z}**z**and

**H**=

*H*

_{y}**y**, we can write the permittivity of the medium as follows, where

*ε*is a complex number with negative real part and an imaginary part which accounts for the loss in the metallic claddings and

_{m}*ε*the linear permittivity of the dielectric cores. The Kerr coefficient

_{d}*γ*does not appear in the above equation since we have rescaled the fields

**E**and

**H**by the factor

*γ*

^{−1}

^{/}^{2}. Additionally we will consider spatial variables rescaled by the vacuum wavenumber

*k*

_{0}=

*ω/c*, being

*ω*the frequency and

*c*the speed of light in vacuum.

17. A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. **105**, 116804 (2010). [CrossRef] [PubMed]

18. A. R. Davoyan, I. V. Shadrivov, Yu. S. Kivshar, and D. K. Gramotnev, “Optimal tapers for compensating losses in plasmonic waveguides,” Phys. Status Solidi RRL **4**, 277–279 (2010). [CrossRef]

*α*∼ 1 deg.) the effect of power concentration due to the taper fully compensated the power loss in the metal claddings. This model can be in principle also applied to the present directional coupler considering mode amplitudes at each of the waveguides

*A*

_{1,2}(

*z*) and writing the coupled equations: where

*σ*, Γ,

*N*and

*K*are parameters calculated applying a perturbation theory to the linear mode in order to describe losses and nonlinear effects, and considering an adiabatic approach to describe the effect of the width change of the waveguide with the propagation distance. In that way,

*σ*describes power carried by the guides and so

*∂*accounts for the effect of the tapering shape, Γ describes de effect of losses,

_{z}σ*N*the nonlinearity and

*K*is the coupling coefficient describing the power transference between both waveguides [20

20. E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. **22**, 988–993 (1986). [CrossRef]

16. J. R. Salgueiro and Yu. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. **97**, 081106 (2010). [CrossRef]

21. S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propagat. **45**, 392–400 (1997). [CrossRef]

**J**=

*∂*

_{t}**P**which is introduced in the Maxwell’s equations,

*∂*

_{t}**D**= ∇ ×

**H**−

**J**, being

**D**the electric displacement, and modeled by solving an additional differential equation,

*ω*is the plasma frequency and

_{p}*η*is the electron collision frequency describing the power losses. Both parameters can be calculated from Drude’s model

*ε*) = −8.25 and Im(

*ε*) = 0.3 deduced from the optical constants of silver at

*λ*≈ 480 nm [22

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

*ω*= 1, we obtain

*ω*= 3.043 and

_{p}*η*= 0.031.

23. R. M. Joseph and T. Allen, “Spatial soliton deflection mechanism indicated by FD-TD Maxwell’s equations modelling,” IEEE Photon. Technol. Lett. **6**, 1251–1254 (1994). [CrossRef]

**D**=

*ε*(|

**E**|

^{2})

**E**taking the nonlinear permittivity as in Eq. (1). For the simulations we took

*ε*= 2.25 (pure silica) though the nonlinear model is general for any Kerr material just rescaling back the fields by the factor

_{d}*γ*

^{1/2},

*γ*is the Kerr coefficient, after the simulation. At each time step, after obtaining the magnetic component and then the two components for the displacement vector,

*D*and

_{x}*D*, the calculation of the electric field components requires the solution of a nonlinear cubic equation at each point of the mesh. This is efficiently done by a single Newton step, starting from the value of the electric components at the previous time step.

_{z}## 3. Results and discussions

*α*= 1.9 deg. We deduced that for angles slightly lower than

*α*= 2.0 degrees loss is compensated by the focusing effect of the taper waveguides. We should remark that irradiance is a more convenient parameter than power for this study as it represents the optical density inside the waveguide core, that can result enhanced by a progressive decrease of the core width. The total power, however, will always decrease as an effect of the optical loss in the metallic layers. In fact Fig. 2(c) shows how total power in the second waveguide decreases when the taper angle increases. This is due to the fact that when the waveguide core width is smaller a larger fraction of the modal field lies inside the metallic cladding and relative loss increase. Nevertheless the irradiance inside the core increases when the angle increases [Fig. 2(d)] because the power concentration inside the core partially compensates loss.

*α*= 1.9 deg. The three images correspond to linear, intermediate and nonlinear regimes reached when input power is increasedly larger. Though loss still exist, the irradiance inside the core is enhanced as was shown in Fig. 2(d) and this is the reason why switching results more effective. This is clearly seen in the switching curve shown in Fig. 4 where the relative irradiance for a beat length is plotted against the input irradiance. The case for non tapered waveguides (

*α*= 0.0) was already studied in Ref. [16

**97**, 081106 (2010). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | M. I. Brongersma and P. G. Krik, Eds. |

2. | S. Maier, |

3. | D. K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics |

4. | N. C. Panoiu and R. M. Osgood, “Subwavelength nonlinear plasmonic nanowire,” Nano Lett. |

5. | W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. |

6. | J. A. H. van Nieuwstadt, M. Sandke, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. |

7. | R. Yang, M. A. G. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express |

8. | D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B |

9. | Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. |

10. | T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Design and characterization of dielectric-loaded plasmonic directional couplers,” J. Lightwave Technol. |

11. | A. Degiron, S. Y. Cho, T. Tyler, N. M. Jokerst, and D. R. Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. |

12. | V. M. Agranovich, V. S. Babichenko, and V. Y. Chernyak, “Nonlinear surface polaritons,” Sov. Phys. JETP |

13. | G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. |

14. | D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. |

15. | A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express |

16. | J. R. Salgueiro and Yu. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. |

17. | A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. |

18. | A. R. Davoyan, I. V. Shadrivov, Yu. S. Kivshar, and D. K. Gramotnev, “Optimal tapers for compensating losses in plasmonic waveguides,” Phys. Status Solidi RRL |

19. | S. I. Bozhevolnyi and K. V. Nerkararyan, “Adiabatic nanofocusing of channel plasmon polaritons,” Opt. Lett. |

20. | E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. |

21. | S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propagat. |

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

23. | R. M. Joseph and T. Allen, “Spatial soliton deflection mechanism indicated by FD-TD Maxwell’s equations modelling,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(130.4815) Integrated optics : Optical switching devices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 7, 2012

Revised Manuscript: March 21, 2012

Manuscript Accepted: April 2, 2012

Published: April 9, 2012

**Citation**

José R. Salgueiro and Yuri S. Kivshar, "Nonlinear couplers with tapered plasmonic waveguides," Opt. Express **20**, 9403-9408 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-9403

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### References

- M. I. Brongersma and P. G. Krik, Eds. Surface Plasmon Nanophotonics (Springer, 2007), p. 268
- S. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), p. 219
- D. K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010). [CrossRef]
- N. C. Panoiu and R. M. Osgood, “Subwavelength nonlinear plasmonic nanowire,” Nano Lett.4, 2427–2430 (2004). [CrossRef]
- W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6, 1027–1030 (2006). [CrossRef]
- J. A. H. van Nieuwstadt, M. Sandke, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett.97, 146102 (2006). [CrossRef] [PubMed]
- R. Yang, M. A. G. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express16, 20142–20148 (2008). [CrossRef] [PubMed]
- D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B93, 99–106 (2008). [CrossRef]
- Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett.34, 810–812 (2009). [CrossRef]
- T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Design and characterization of dielectric-loaded plasmonic directional couplers,” J. Lightwave Technol.27, 5521–5528 (2009). [CrossRef]
- A. Degiron, S. Y. Cho, T. Tyler, N. M. Jokerst, and D. R. Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys.11, 015002 (2009). [CrossRef]
- V. M. Agranovich, V. S. Babichenko, and V. Y. Chernyak, “Nonlinear surface polaritons,” Sov. Phys. JETP32, 512–515 (1980).
- G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys.58, 2453–2459 (1985). [CrossRef]
- D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett.12, 187–189 (1987). [CrossRef] [PubMed]
- A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express16, 21209–21214 (2008). [CrossRef] [PubMed]
- J. R. Salgueiro and Yu. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett.97, 081106 (2010). [CrossRef]
- A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett.105, 116804 (2010). [CrossRef] [PubMed]
- A. R. Davoyan, I. V. Shadrivov, Yu. S. Kivshar, and D. K. Gramotnev, “Optimal tapers for compensating losses in plasmonic waveguides,” Phys. Status Solidi RRL4, 277–279 (2010). [CrossRef]
- S. I. Bozhevolnyi and K. V. Nerkararyan, “Adiabatic nanofocusing of channel plasmon polaritons,” Opt. Lett.35, 541–543 (2010). [CrossRef] [PubMed]
- E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron.22, 988–993 (1986). [CrossRef]
- S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propagat.45, 392–400 (1997). [CrossRef]
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
- R. M. Joseph and T. Allen, “Spatial soliton deflection mechanism indicated by FD-TD Maxwell’s equations modelling,” IEEE Photon. Technol. Lett.6, 1251–1254 (1994). [CrossRef]

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