## Tunable wavelength dependent nanoswitches enabled by simple plasmonic core-shell particles |

Optics Express, Vol. 21, Issue 22, pp. 26052-26068 (2013)

http://dx.doi.org/10.1364/OE.21.026052

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### Abstract

In this paper we demonstrate the feasibility of using a plasmonic core-shell particle to function as a wavelength dependent switch for integration into nanoantenna structures. First, a quasistatic analysis is performed and the necessary conditions are derived which allow the particle to operate in either a short- or an open-circuit state. These conditions dictate that the core and the shell permittivity values need to have opposite sign. Consequently, at optical wavelengths where noble metals are modeled as Drude dielectrics, these conditions can be easily realized. As a matter of fact, it is demonstrated that a realistic core-shell particle can exhibit both the short- and open-circuit states, albeit at different wavelengths. Our analysis is extended by examining the same problem beyond the quasistatic limit. For this task we utilize an inhomogeneous spherical transmission line representation of the core-shell particle. The conditions are derived for the particle that yield either an input admittance or impedance equal to zero. It is further demonstrated that these conditions are the short wavelength generalization of their quasistatic counterparts.

© 2013 Optical Society of America

## 1. Introduction

*i.e.*they are simple to fabricate) and increased tunability with robust operation. Towards this end, herein we explore the optical properties of a plasmonic core-shell particle in the context of providing a practical solution for the development of a polarization independent and tunable nanocircuit element that behaves as an effective wavelength dependent switch.

5. A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave. **3**(1), 37–60 (1989). [CrossRef]

*i.e.*the ratio between the inner and outer radius). Specifically, in [6

6. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B **76**(24), 245403 (2007). [CrossRef]

7. A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B **82**(3), 035117 (2010). [CrossRef]

8. N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. **95**(9), 095504 (2005). [CrossRef] [PubMed]

*et al.*in [9

9. H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys. **103**(9), 094112 (2008). [CrossRef]

*et al.*in [10

10. A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys. **11**(1), 013026 (2009). [CrossRef]

11. A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE **99**(10), 1669–1681 (2011). [CrossRef]

_{r}input admittance of a core-shell particle, which essentially corresponds to the TM

_{r}input admittance of a radially inhomogeneous spherical transmission line. A set of conditions is derived that yield a zero input admittance, and a zero input impedance. The main characteristics of these conditions is that, as expected, they are functions of the structure’s wavenumbers as well as the input admittance of the core and the admittance of the fringing field in the shell region, due to scattering from the core. It is further demonstrated that in the limit of an electrically small core-shell particle, the short- and open-circuit conditions for the dynamic case yield the ones obtained from the quasistatic analysis. Throughout this paper the

## 2. Quasi-static analysis

### 4.1 Loseless material

12. A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res. **62**, 317–331 (2006). [CrossRef]

12. A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res. **62**, 317–331 (2006). [CrossRef]

### 4.2 Lossy material

13. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

12. A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res. **62**, 317–331 (2006). [CrossRef]

## 3. Dynamic analysis

14. J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods **10**(5-6), 441–450 (1962). [CrossRef]

### 3.1 Spherical transmission line representation of a core-shell particle

_{r}response, for completeness we include the TE

_{r}expressions as well. The quantities corresponding to the TM

_{r}excitation are denoted by the “

_{r}excitation are denoted by the “

_{r}and impedances for TE

_{r}excitation. The input TM

_{r}admittance and TE

_{r}impedance of the core-shell particle, normalized to

14. J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods **10**(5-6), 441–450 (1962). [CrossRef]

_{r}admittance of this field is given by

_{r}input impedance for mode

_{r}mode contributes insignificantly to the particle’s optical response.

### 3.2 Short- and open-circuit conditions

_{r}admittance and TE

_{r}impedance of the core, normalized to

_{r}admittanceand the following TE

_{r}impedance

_{r}admittance and TE

_{r}impedance of a spherical transmission line (looking towards its origin) with a refractive index

_{r}admittance and TE

_{r}impedance may be rewritten asand

13. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

### 3.3 Discussion

_{r}input admittance of a core-shell particle can be associated to that of a homogeneous sphere whose permittivity properties are described by Eq. (11). In order to validate this statement the following task was performed: a homogeneous sphere was assumed with a radius of 20 nm and a permittivity that can be represented as the equivalent of a parallel connection of RLC circuits [16

16. R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag. **45**(11), 1602–1610 (1997). [CrossRef]

_{r}admittance and the TE

_{r}impedance of the homogeneous sphere given by Eq. (30) and Eq. (31) are equivalent to Eq. (40) and Eq. (41), respectively. In this analysis only mode

*i.e.*from direct application of the CM mixing rule. It can be seen that the effective permittivity obtained by the dynamic analysis exhibits its Lorentz resonance at a frequency shifted with respect to the one predicted by the quasistatic formulation.

_{r}and TE

_{r}polarizations in Fig. 6, along with the corresponding quantities for the core-shell particle. Evidently, the effective homogeneous sphere can mimic very well the TM

_{r}impedance of the core-shell particle, however it is noticed that a minor discrepancy occurs in the TE

_{r}response. The spurious resonance displayed by Z

_{TE}occurs at the same frequency where the real part of the effective permittivity exhibits its Lorentz resonance. In order to get a better understanding of the nature of this artifact we perform a Taylor expansion on Eq. (31), which yields

_{r}impedance is proportional to the properties of the homogeneous sphere. Consequently, when the latter resonates, the impedance will also exhibit a Lorentz-type resonance, as indicated in Fig. 6(c) and 6(d). This artifact is also translated into a spurious peak in the effective particle’s extinction efficiency as can be seen in Fig. 7. However, the magnitude of this peak is extremely minute, which is in accordance with the fact that the values of Z

_{TM}are three orders of magnitude larger than those of Z

_{TE}.

_{TM≡}Z

_{θ}corresponds to the Z

_{TM}input impedance of a homogeneous sphere whose permittivity is given by Eq. (49). On the other hand, Z

_{TE≡}Z

_{φ}is given by the Z

_{TE}input impedance of a homogeneous sphere whose dielectric properties are equal to those of the shell. This last conclusion can be drawn from the observation that for the particular core-shell particle under consideration we have

## 4. Conclusions

## References and links

1. | J. R. Wait, |

2. | W. C. Chew, |

3. | M. Kerker, |

4. | A. Sihvola and L. V. Lindell, “Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers,” J. Electromagnet. Wave. |

5. | A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave. |

6. | J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B |

7. | A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B |

8. | N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. |

9. | H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys. |

10. | A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys. |

11. | A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE |

12. | A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res. |

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

14. | J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods |

15. | A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. |

16. | R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(290.5850) Scattering : Scattering, particles

(260.2065) Physical optics : Effective medium theory

**ToC Category:**

Plasmonics

**History**

Original Manuscript: August 14, 2013

Revised Manuscript: October 2, 2013

Manuscript Accepted: October 3, 2013

Published: October 24, 2013

**Citation**

Anastasios H. Panaretos and Douglas H. Werner, "Tunable wavelength dependent nanoswitches enabled by simple plasmonic core-shell particles," Opt. Express **21**, 26052-26068 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26052

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### References

- J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, 1962).
- W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
- M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
- A. Sihvola and L. V. Lindell, “Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers,” J. Electromagnet. Wave.2(2), 741–756 (1988).
- A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave.3(1), 37–60 (1989). [CrossRef]
- J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007). [CrossRef]
- A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B82(3), 035117 (2010). [CrossRef]
- N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005). [CrossRef] [PubMed]
- H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys.103(9), 094112 (2008). [CrossRef]
- A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys.11(1), 013026 (2009). [CrossRef]
- A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE99(10), 1669–1681 (2011). [CrossRef]
- A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res.62, 317–331 (2006). [CrossRef]
- P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
- J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods10(5-6), 441–450 (1962). [CrossRef]
- A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys.22(10), 1242–1246 (1951). [CrossRef]
- R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag.45(11), 1602–1610 (1997). [CrossRef]

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