## Optical 3D cavity modes below the diffraction-limit using slow-wave surface-plasmon-polaritons

Optics Express, Vol. 15, Issue 5, pp. 2607-2612 (2007)

http://dx.doi.org/10.1364/OE.15.002607

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### Abstract

Modal volumes at the nano-scale, much smaller than the “diffraction-limit”, with appreciable quality factors, are calculated for a dielectric cavity embedded in a space between metal plates. The modal field is bounded between the metal interfaces in one dimension and can be reduced in size almost indefinitely in this dimension. But more surprisingly, due to the “plasmonic” slow wave effect, this reduction is accompanied by a similar in-plane modal size reduction. Another interesting result is that higher order cavity modes exhibit lower radiation loss. The scheme is studied with effective index analysis, and validated by FDTD simulations.

© 2007 Optical Society of America

## 1. Introduction

1. K. J. Vahala, “Optical microcavities,” Nature **424**, 839 (2003). [CrossRef] [PubMed]

2. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, “Smallest possible electromagnetic mode volume in a dielectric cavity,” IEE Proc. Optoelectron. **145**, 391 (1998). [CrossRef]

5. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003). [CrossRef] [PubMed]

^{3}– where λ

_{0}is the light wavelength in vacuum and n is the medium index of refraction.

6. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. **82**, 1158 (2003). [CrossRef]

## 2. Plasmonic cavity analysis

11. 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 (1997). [CrossRef] [PubMed]

*k*-vector components must be all real at least in one region of space (usually in the core). Thus the available

*k*values are bounded (from above), according to

*k*+

_{x}^{2}*k*+

_{y}^{2}*k*=

_{z}^{2}*k*(

_{0}^{2}ε*k*the vacuum wave-vector and

_{0}*ε*the dielectric constant) which yields (by a simple Fourier analysis) a minimal spatial bound of (λ/2n)

^{3}– known as the diffraction limit. For a metal cladded structure – the plasmon polariton solution is a slow wave such that in all space regions (including the cavity core) at least one k-vector component is imaginary (

*k*for metal layers located at x=constant). Thus there is no bound to

_{x}*k*(from above),

*k*+

_{y}^{2}*k*-|

_{z}^{2}*k*|=

_{x}^{2}*k*, and consequently there is no limit to the mode size reduction. Moreover, the inplane

_{0}^{2}ε*k*-vector (y,z) is enhanced by increasing the vertical component (x), translated to reduced inplane dimensions by reducing the vertical size - which is opposing to the trend in regular dielectric structures. This unique characteristics, emphasizes that in order to obtain cavities of sub-diffraction modal size in all 3D, the metal interfaces are required only in 1D.

6. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. **82**, 1158 (2003). [CrossRef]

12. B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B **44**, 13556 (1991). [CrossRef]

13. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B **33**5186 (1986). [CrossRef]

6. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. **82**, 1158 (2003). [CrossRef]

**82**, 1158 (2003). [CrossRef]

_{eff}(x,z) = β.(x,z)/k

_{0}) is assigned, leading to a 2D in-plane equivalent structure of a dielectric circular cavity. The plasmon-polariton solution is essentially TM and the magnetic field of the cavity mode is:

_{1}and n

_{2}are effective indices of the cylindrical cavity core and cladding; a is the cylinder radius; and

*J*and

_{m}*H*are Bessel and second kind Hankel functions of order m. The valid cavity modes, characterized by azimuthal and radial numbers {m,

^{(2)}_{m}*l*}, have continuous

*H*and

_{θ}*∂*at r=a. The boundary conditions, for a given in-plane cylinder and for an (m) azimuthal order, yield a set of discrete radial solutions (indexed by

_{r}H_{θ.}*l*) – each having a distinct modal frequency.

*k*-vector, resulting in enhanced vertical radiation losses. For the “plasmonic” cavity, the metal layers disallow vertical radiation however they are a source for material loss. Here the inplane radiation losses of the mode into the dielectric clad, decrease with the order of the mode [Fig. 1(b)]. Mathematically it can be traced to the fact that the solution outside the core (r>a), given in Eq. 1 (Hankel function of the second kind), vanishes rapidly for higher orders. On the other hand, the increased modal order is accompanied by a higher self-frequency, which suffers from larger material loss since it is closer to the plasma frequency. The interplay of these two mechanisms is expected to result in an optimized quality factor for modes of intermediate orders.

## 3. Simulative validation

^{3}is more than a order of magnitude smaller than the “diffraction limit”, and has a Q-factor of 170, calculated by the ratio of total stored energy in a volume encompassing the cavity to the total outgoing power from the volume surfaces. The Q-factor can be enhanced further by employing lower loss metals (e.g. silver) or for increased inter-metal gap.

*l*) is extracted. The balance between the vertical metal losses and in-plane radiation losses retains the {0, 2} and {2,1} modes, both having similar eigen frequencies of about 700nm. Their interference matches the mode distribution and spectrum as obtained in the simulation and the expected axial rotation of the pattern is due to the slight difference in the modal propagation constants. The calculated field confinement [Fig. 3(a)] is slightly smaller compared to the simulation results - effective modal volume of (41nm)

^{3}), which may be due to the influence of the imaginary part of the metal dielectric constant, not included in the effective index analysis.

## 4. Conclusion

## Acknowledgments

## References and Links

1. | K. J. Vahala, “Optical microcavities,” Nature |

2. | R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, “Smallest possible electromagnetic mode volume in a dielectric cavity,” IEE Proc. Optoelectron. |

3. | J. Scheuer, W. M. Green, G. A. DeRose, and A. Yariv, “Lasing from a circular Bragg nanocavity with an ultra small modal volume,” Appl. Phys. Lett. |

4. | M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in Photonic Crystals,” Science |

5. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

6. | K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. |

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

8. | P. Grinberg, E. Feigenbaum, and M. Orenstein, “2D Photonic band gap cavities embedded in a plasmonic gap structure - zero modal volume,” LEOS Annual Meeting, Australia (paper TuZ5) (2005). |

9. | H. T. Miyazaki0 and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett |

10. | J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultra small mode volumes in Dielectric Optical Microcavities,” Phys. Rev. Lett. |

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

12. | B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B |

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

14. | J. A. Kong, |

15. | L. C. Andreani, G. Panzarini, and J. M. Ge′rard, “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 23, 2007

Revised Manuscript: February 18, 2007

Manuscript Accepted: February 19, 2007

Published: March 5, 2007

**Citation**

Eyal Feigenbaum and Meir Orenstein, "Optical 3D cavity modes below the diffraction-limit using slow-wave surface-plasmon-polaritons," Opt. Express **15**, 2607-2612 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2607

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### References

- K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003). [CrossRef] [PubMed]
- R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, "Smallest possible electromagnetic mode volume in a dielectric cavity," IEE Proc.: Optoelectron. 145, 391 (1998). [CrossRef]
- J. Scheuer, W. M. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultra small modal volume," Appl. Phys. Lett. 86, 251101 (2005). [CrossRef]
- M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, "Simultaneous inhibition and redistribution of spontaneous light emission in Photonic Crystals," Science 308, 1296 (2005). [CrossRef] [PubMed]
- Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944 (2003). [CrossRef] [PubMed]
- K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158 (2003). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824 (2003). [CrossRef] [PubMed]
- P. Grinberg, E. Feigenbaum, and M. Orenstein, "2D Photonic band gap cavities embedded in a plasmonic gap structure - zero modal volume," LEOS Annual Meeting, Australia (paper TuZ5) (2005).
- H. T. Miyazaki0 and Y. Kurokawa, "Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity," Phys. Rev. Lett 96, 097401 (2006). [CrossRef]
- J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, "Ultra small mode volumes in Dielectric Optical Microcavities," Phys. Rev. Lett. 95, 143901 (2005). [CrossRef] [PubMed]
- 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 (1997). [CrossRef] [PubMed]
- B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556 (1991). [CrossRef]
- J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 335186 (1986). [CrossRef]
- J. A. Kong, Electromagnetic Waves - Progress In Electromagnetics Research 10, (EMW, Cambridge, 1995).
- L. C. Andreani, G. Panzarini, and J. M. Ge´rard, "Strong-coupling regime for quantum boxes in pillar microcavities: Theory," Phys. Rev. B 60, 13276 (1999). [CrossRef]

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