## Power transfer between neighboring planar waveguides |

Optics Express, Vol. 20, Issue 3, pp. 3152-3157 (2012)

http://dx.doi.org/10.1364/OE.20.003152

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### Abstract

The ability to control light over very small distances is a problem of fundamental importance for a vast range of applications in communications, nanophotonics, and quantum information technologies. For this purpose, several methods have been proposed and demonstrated to confine and guide light, for example in dielectric and surface plasmon polariton (SPP) waveguides. Here, we study the interaction between different kinds of planar waveguides, which produces dramatic changes in the dispersion relation of the waveguide pair and even leads to mode suppression at small separations. This interaction also produces a transfer of power between the waveguides, which depends on the gap and propagation distances, thus providing a mechanism for optical signal transfer. We analytically study the properties of this interaction and the power transfer in different structures of interest including plasmonic and particle-array waveguides, for which we propose an experimental realization of these ideas.

© 2012 OSA

12. E. Verhagen, M. Spasenovic, A. Polman, and L. Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. **102**, 203904 (2009). [CrossRef] [PubMed]

*iκ*is imaginary outside the waveguide, thus producing evanescent fields that cannot propagate away from it. These evanescent tails spread out of the guiding structure with a characteristic 1/

_{z}*e*–field-amplitude-decay penetration distance

*L*= 1/

_{z}*κ*[13].

_{z}*k*

_{||}parallel to the plane of the structure. Particle array waveguides (a) have a confining resonance within the First Brillouin Zone (FBZ). This resonance defines a dispersion relation with vanishing group velocity when the parallel wave vector approaches

*π*/

*a*and degenerate TE and TM modes when the particles are spherical [6

6. X. M. Bendana and F. J. Garcia de Abajo, “Confined collective excitations of self-standing and supported planar periodic particle arrays,” Opt. Express **17**, 18826–18835 (2009). [CrossRef]

*i*for

*σ*-polarized light and

*d*is the separation distance. This model can be easily generalized to an arbitrary number of waveguides. The results of this model are shown in Fig. 1(d–f) for different combinations of waveguides. The effect of interaction is qualitatively the same in all of these cavities: for large distances, the modes are at the position of the original non-interacting resonances; however, as the separation is decreased, mode repulsion takes place down to a critical distance below which one of the modes is pushed beyond the light line of the surrounding medium, thus being broadened and effectively non contributing to propagating optical signal due to coupling to radiation; in contrast, the remaining mode is still pushed towards larger parallel wave vectors that increase its degree of confinement. This repulsive interaction is a common effect in many optical and plasmonic hybridized systems when similar modes are placed at interacting distances [16

16. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. **4**, 899–903 (2004). [CrossRef]

*k*

_{||}mode in Fig. 1(f) (the symmetric mode) is well defined down to

*k*

_{||}= 0. Nevertheless, there is a critical distance

*d*below which the mode is suppressed, which occurs at

*k*

_{||}= 0. This point describes non-propagating modes of vanishing group velocity.

*d*, the two modes accumulate different phase as they propagate along the structure. Therefore, the power density moving in each part (i.e., each individual waveguide) of the interacting structure (

*P*) varies along the propagation direction when both modes are excited, thus producing power transfer back and forth between both waveguides. For simplicity, we focus on symmetric geometries (e.g., the MIM structure or a double layer of equal-size particles), in which the field amplitudes

_{i}*c*(

_{i}*x*) in waveguide

*i*= 1, 2 at the position

*x*along the propagation direction satisfies the equations where

*c*and

_{s}*c*are the complex amplitudes of the symmetric and antisymmetric modes.

_{a}*d*varies very smoothly along the propagation direction. In such a system, the interaction (and therefore, also the wave vector of the modes) varies adiabatically along the propagation direction

*x*. Within the adiabatic regime, we can neglect losses and propagation constant shifts if the penetration distance of the modes outside the waveguide is much smaller than the bending radius. Treating as a perturbation all terms in the Maxwell equations that differ from the straight waveguide, one can show that radiative losses in a curved waveguide of large bending radius

*R*scale as (

*L*)

_{z}/R^{2}, where

*L*is the penetration distance into the surrounding medium (i.e., the transversal extension of the mode). Small radiative losses ∼ 3% per optical cycle have been already predicted for small rings of radius

_{z}*R*twice smaller than the wavelength and

*L*∼ 0.01 [17

_{z}/R17. A. Manjavacas and F. J. Garcia de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. **9**, 1285–1289 (2009). [CrossRef]

*L*

*≈ 2.65*

_{z}*μ*m ≪

*R*≥ 100

*μ*m, leading to (

*L*)

_{z}/R^{2}∼ 7 × 10

^{−4}) renders radiative losses negligible compared to ohmic losses. This produces a power transfer that depends on the nature of the waveguides, as well as on the details of the variation of the separation distance with

*x*. Starting with a mode prepared in guide 1 at large distances between the waveguides at position

*x*

_{0}, the net power in each guide at a subsequent position

*x*reduces to [18

18. 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**, 310–312 (2009). [CrossRef] [PubMed]

19. A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. **90**, 211101 (2007). [CrossRef]

*k*

_{||}(

*d*) is the wave vector difference between the two modes, which is calculated from the Fabry-Perot condition (Eq. (1)), rendering

*k*

_{||}for each mode as a function of separation

*d*(see Fig. 1). In the derivation of these equations, we consider the incident wave to be equally split between in-phase symmetric and antisymmetric modes at

*x*

_{0}(this places the weight of the incident wave only in waveguide 1 according to Eqs. (2)), we follow the adiabatic evolution of the modes (i.e., they pick up a phase as they propagate), and we recombine the resulting amplitudes using Eqs. (2) to obtain the power density

*P*∝ |

_{i}*c*|

_{i}^{2}.

*R*≈ 200

*μ*m for a

*d*= 3

*μ*m gap. If we continue increasing the radius, total transfer occurs at intermediate propagation distances, from where the power would be transferred back to the first waveguide. The corresponding oscillations of the power back and forth between both curved waveguides along the propagation direction in the interaction arc is clearly observed for a large value of the radius in Fig. 3(c).

*d*≈ 3

*μ*m for

*R*= 200

*μ*m), the power is completely transferred to the second waveguide. Smaller gap values produce several cycles of power transfer and, when the gap distance is smaller than a critical value, the suppression of one of the hybrid modes leads to a featureless distribution of power on both sides of the system. In Fig. 3(a), we show that the power changes rapidly between both waveguides when we increase the bending radius

*R*or decrease the gap distance

*d*. When the constitutive guiding structure is surrounded by a dielectric material, such as in Fig. 3(b), the modes become leaky below the light line and the gap distance has larger critical values, and therefore the changes in the output signal are slower and occur for larger values of the radius.

## Acknowledgments

## References and links

1. | S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater “Plasmonics—a route to nanoscale optical devices,” Adv. Mater. |

2. | D. Marcuse |

3. | H. Raether, |

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

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

6. | X. M. Bendana and F. J. Garcia de Abajo, “Confined collective excitations of self-standing and supported planar periodic particle arrays,” Opt. Express |

7. | T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today |

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

9. | M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B |

10. | M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. |

11. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

12. | E. Verhagen, M. Spasenovic, A. Polman, and L. Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. |

13. | J. D. Jackson, |

14. | E. D. Palik |

15. | M. Born and E. Wolf, |

16. | P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. |

17. | A. Manjavacas and F. J. Garcia de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. |

18. | 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. |

19. | A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.7370) Optical devices : Waveguides

(230.7390) Optical devices : Waveguides, planar

(230.7400) Optical devices : Waveguides, slab

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 30, 2011

Revised Manuscript: January 6, 2012

Manuscript Accepted: January 10, 2012

Published: January 26, 2012

**Citation**

X. M. Bendaña and F. J. García de Abajo, "Power transfer between neighboring planar waveguides," Opt. Express **20**, 3152-3157 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-3152

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### References

- S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater “Plasmonics—a route to nanoscale optical devices,” Adv. Mater.13, 1501 (2001). [CrossRef]
- D. MarcuseTheory of Dielectric Optical Waveguides (Academic, 1974).
- H. Raether, Surface Plasmons (Springer-Verlag, 1988).
- P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B61, 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. B63, 125417 (2001). [CrossRef]
- X. M. Bendana and F. J. Garcia de Abajo, “Confined collective excitations of self-standing and supported planar periodic particle arrays,” Opt. Express17, 18826–18835 (2009). [CrossRef]
- T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today61, 44 (2008). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics”, Nature (London)424, 824–830 (2003). [CrossRef]
- M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B62, 16356–16359 (2000). [CrossRef]
- M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett.23, 1331–1333 (1998). [CrossRef]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B60, 5751 (1999). [CrossRef]
- E. Verhagen, M. Spasenovic, A. Polman, and L. Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett.102, 203904 (2009). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
- E. D. PalikHandbook of Optical Constants and Solids (Academic, 1985).
- M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).
- P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett.4, 899–903 (2004). [CrossRef]
- A. Manjavacas and F. J. Garcia de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett.9, 1285–1289 (2009). [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, 310–312 (2009). [CrossRef] [PubMed]
- A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett.90, 211101 (2007). [CrossRef]

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