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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 5 — Mar. 5, 2007
  • pp: 2607–2612
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Optical 3D cavity modes below the diffraction-limit using slow-wave surface-plasmon-polaritons

Eyal Feigenbaum and Meir Orenstein  »View Author Affiliations


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

2D structures that include metal interfaces are known to relax the 1D confinement limitation. For instance, a mode in a dielectric-gap between two metal interfaces can be reduced in size almost indefinitely by reducing the gap width [6

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]

]. Although calculations show that the mode size may shrink to zero for vanishing interface spacing, we restrict the statement to “almost indefinite confinement” since the macroscopic Maxwell equations may fail at the few nanometers scale. The mode described above is confined only in one dimension and is unlimited in the other two. Exercising the same idea to achieve confinement in two or three dimensions (enclosing the dielectric cavity by metal) fails due to the emergence of a cut-off for the optical mode. Moreover, the incorporation of additional metal interfaces introduces more losses resulting in poor Q-factors.

2. Plasmonic cavity analysis

Hθm,lθr=ejmθ{Jm(k0n1r)raAm,lHm(2)(k0n2r)ra
(1)

Fig. 1. Analysis results: (a) averaged modal size vs. cylinder height at λ0=700nm, according to both our definition of “uncertainty volume” [Eq. (2)] as well as the Purcell effective volume. (b) Outgoing radial power vs. cylinder radius for 20nm gap. nsi=3.5, n0=1, λplasma=137nm.

heff=2σI=2·VAR{I}=2x2I(x)2dxI(x)2dx;Reff2r2I(r)2rdrI(r)2rdr
(2)

In order to facilitate the comparison with other reported cavities, we provide here also the Purcell effective mode volume [Fig. 1(a)] which always yields a significantly lower value.

3. Simulative validation

To validate the predicted cavity performance, we performed FDTD based simulation, incorporating the complex metal dielectric function of the Drude model (which is a good approximation for our wavelengths). A silicon (or GaAs) cylinder 100nm in diameter was inserted into a 20nm gap between gold plates. A broadband pulse was introduced into the cavity from within (Gaussian shaped with x polarization) to allow for multiple resonances excitation. The resulting major cavity mode has a spectral peak of ~700nm, and we used this wavelength to excite the cavity by a CW excitation. The resulting vertical and in-plane distributions of the magnetic field are depicted in Fig. 2.

Fig. 2. FDTD simulations: Gold thickness: 100nm (practically infinite). Excitation: short x polarized pulse with Gaussian y distribution inside the cavity. In-plane and vertical resolutions are 10nm and 1nm respectively. Major spectral peak is at ~700nm. Hθ distribution: (a) vertical plane at z=0 (result obtained from CW excitation at 700nm for higher resolution). Inset: the field profile along y=20nm. (b) in-plane at x=0 (result obtained from the impulse excitation since CW has poorer visualization due to interference with the continuous source).

Fig. 3. Modal calculations by the effective index method for d=20nm: (a) coherent summation of modes {0,2}{2,1} at λ0=700nm, a=50nm (cylinder boundaries in dashed white). (b) Cylinder radius supporting specific modes vs. the wavelength (λ0). Red dashed line denotes the 50nm radius used in FDTD simulations. The inset shows the |Hθ|2 distributions for the different modes for a=50nm (with respective resonance frequencies) (cylinder boundaries in dashed green).

4. Conclusion

Acknowledgments

We would like to acknowledge the Israel Ministry of Science and Technology for a partial support of this research.

References and Links

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]

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. 86, 251101(2005). [CrossRef]

4.

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]

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]

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]

7.

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

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 96, 097401 (2006). [CrossRef]

10.

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]

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]

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 335186 (1986). [CrossRef]

14.

J. A. Kong, Electromagnetic Waves - Progress In Electromagnetics Research 10, (EMW, Cambridge, 1995).

15.

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]

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

  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]
  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. 86, 251101 (2005). [CrossRef]
  4. 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]
  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]
  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]
  7. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824 (2003). [CrossRef] [PubMed]
  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 96, 097401 (2006). [CrossRef]
  10. 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]
  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]
  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 335186 (1986). [CrossRef]
  14. J. A. Kong, Electromagnetic Waves - Progress In Electromagnetics Research 10, (EMW, Cambridge, 1995).
  15. 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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