## Exact polarizability and plasmon resonances of partly buried nanowires |

Optics Express, Vol. 19, Issue 23, pp. 22775-22785 (2011)

http://dx.doi.org/10.1364/OE.19.022775

Acrobat PDF (1703 KB)

### Abstract

The electrostatic polarizability for both vertical and horizontal polarization of two conjoined half-cylinders partly buried in a substrate is derived in an analytical closed-form expression. Using the derived analytical polarizabilities we analyze the localized surface plasmon resonances of three important metal nanowire configurations: (1) a half-cylinder, (2) a half-cylinder on a substrate, and (3) a cylinder partly buried in a substrate. Among other results we show that the substrate plays an important role for spectral location of the plasmon resonances. Our analytical results enable an easy, fast, and exact analysis of many complicated plasmonic nanowire configurations including nanowires on substrates. This is important both for comparison with experimental data, for applications, and as benchmarks for numerical methods.

© 2011 OSA

## 1. Introduction

1. M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. Lond. **147**, 145–181 (1857). [CrossRef]

7. J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. **33**, 189–195 (1955). [CrossRef]

10. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. **5**, S16–S50 (2003). [CrossRef]

13. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. **1**, 641–648 (2007). [CrossRef]

14. P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science **308**, 1607–1609 (2005). [CrossRef] [PubMed]

15. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. **98**, 266802 (2007). [CrossRef] [PubMed]

16. F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) **12**, 2844–2861 (2008). [CrossRef]

17. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. **9**, 205–213 (2010). [CrossRef] [PubMed]

13. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. **1**, 641–648 (2007). [CrossRef]

*ɛ*

_{1}and

*ɛ*

_{2}, respectively. The double half-cylinder is placed in two semi-infinite half-spaces with optical properties given by the dielectric constants

*ɛ*

_{3}and

*ɛ*

_{4}. The simpler case of a half-cylinder [18

18. P. C. Waterman, “Surface fields and the T matrix,” J. Opt. Soc. Am. A **16**, 2968–2977 (1999). [CrossRef]

*homogenous*surrounding has, to some extent, been analyzed before [19

19. A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. **76**, 4827–4835 (1994). [CrossRef]

21. M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. **24**, 1267–1277 (2010). [CrossRef]

22. H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. **102**, 044105 (2007). [CrossRef]

22. H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. **102**, 044105 (2007). [CrossRef]

20. A. Salandrino, A. Alu, and N. Engheta, “Parallel, series, and intermediate interconnects of optical nanocircuit elements. 1. Analytical solution,” J. Opt. Soc. Am. B **24**, 3007–3013 (2007). [CrossRef]

21. M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. **24**, 1267–1277 (2010). [CrossRef]

## 2. Theory

*ɛ*=

_{i}*ɛ*(

_{i}*ω*) is implicitly assumed. We also assume that the cross section of the double half-cylinder is small compared to the wavelength, i.e. we take an electrostatic approach. In a static theory, ∇ ×

**E**(

**r**) = 0 and the electrostatic field can be expressed by means of the electrostatic potential

**E**(

**r**) = −∇

*ϕ*(

**r**). In each domain, the electrostatic potential must fulfill Laplace’s equation ∇

^{2}

*ϕ*(

**r**) = 0 ∀

**r**, with the boundary conditions

*ϕ*=

_{i}*ϕ*and

_{j}*ɛ*· ∇

_{i}n̂*ϕ*=

_{i}*ɛ*· ∇

_{j}n̂*ϕ*on

_{j}*S*, where the subscripts

*i*and

*j*refer to the different domains (1,2,3, and 4) and

*S*to the boundaries.

*u*and

*v*[23, 24], which are connected to rectangular coordinates via

*x*= sinh

*u*/(cosh

*u*− cos

*v*) and

*y*= sin

*v*/(cosh

*u*− cos

*v*). The domains of

*u*and

*v*for the different regions are shown in Fig. 2. First we consider the case where the incident field is polarized along the

*y*axis. Such a field will induce a vertically oriented dipole moment in the double half-cylinder. Thus, we look for solutions that are even functions of

*x*and, hence,

*u*. For a unit amplitude incident field

**E**

_{0}=

*ŷ*the incident potential is given as which transforms into In each of the four regions, the scattered part of the potential is expanded as with

*ϕ*̄

*(*

_{i}*λ*,

*v*) =

*c*(

_{i}*λ*)cosh(

*λv*) +

*s*(

_{i}*λ*)sinh(

*λv*) and the inverse transformation given as

*α*of the partly buried double half-cylinder can be found as (see Appendix) Note that the polarizability in Eq. (1) is the normalized polarizability. To convert into standard units

_{v}*α*should be multiplied by the radius of the cylinder squared. By solving the equation system formed from the boundary conditions of the potential and its normal derivative (see Appendix for details),

_{v}*s*

_{3}(

*λ*) can be calculated, and by performing the integral in Eq. (1) we find the vertical polarizability as where

*A*= (

*ɛ*

_{1}+

*ɛ*

_{2}+

*ɛ*

_{3}+

*ɛ*

_{4})[

*ɛ*

_{1}

*ɛ*

_{2}(

*ɛ*

_{3}+

*ɛ*

_{4}) + (

*ɛ*

_{1}+

*ɛ*

_{2})

*ɛ*

_{3}

*ɛ*

_{4}] and

*B*= (

*ɛ*

_{1}

*ɛ*

_{4}−

*ɛ*

_{2}

*ɛ*

_{3})

^{2}. It should be noted that the result presented by Pitkonen in Ref. [21

21. M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. **24**, 1267–1277 (2010). [CrossRef]

*ɛ*

_{3}=

*ɛ*

_{4}is consistent with Eq. (2). This can be demonstrated using identities for polylogarithms to rewrite Pitkonen’s result in terms of the natural logarithm. From Eq. (2) the resonance condition can be identified as For ordinary dielectric surroundings

*ɛ*

_{3},

*ɛ*

_{4}> 0, the resonance condition can only be fulfilled if the dielectric constant of the partly buried double half-cylinder is negative e.g. if the cylinder is made of a free electron-like metal as silver or gold. In this case, the resonances are dipole surface plasmon modes that arise due to the interaction of the free conduction electrons in the metal cylinder with the time-dependent incident field. The plasmon resonance conditions for a vertically polarized incident field for some special geometries are presented in Table 1. Note that the well known result

*ɛ*= −

*ɛ*for a homogenous cylinder [9] is obtained from our general result.

_{h}**E**

_{0}(

**r**) =

*x̂*we have Such a field will induce a horizontally oriented dipole moment in the cylinder and we therefore look for solutions that are odd functions of

*x*and, hence,

*u*. In each of the four regions, we expand the scattered potential as Similarly to the vertically polarized case, the horizontal polarizability may be found as (see Appendix for details) From the equations formed from the boundary conditions of the potential and its normal derivative,

*c*

_{3}(

*λ*) can be calculated (see Appendix). By performing the integral in Eq. (4), we find the horizontal polarizability as It should be noted that there exists a simple symmetry between

*α*and

_{h}*α*. By replacing

_{v}*ɛ*with 1/

_{i}*ɛ*and changing the sign,

_{i}*α*transforms into

_{v}*α*and vice versa. For a horizontally polarized induced dipole moment the plasmon resonance condition simply states that the sum of all the dielectric constants must be zero Special cases for simpler geometries are given in Table 2. Equations (2), (3), (5), and (6) represent the principal results of this work.

_{h}## 3. Results

*ɛ*of the dielectric constant of the cylinder (or half-cylinder) with a fixed small imaginary part as

_{r}*ɛ*=

*ɛ*+ 0.01

_{r}*i*. As we are interested in the plasmon response of the system we consider negative

*ɛ*. First we consider the geometry of a half-cylinder in a homogenous surrounding with

_{r}*ɛ*= 1 (Fig. 3). The result shows that both the real and the imaginary part of the vertical polarizability display a resonant behavior at

_{h}*ɛ*= −

_{r}*ɛ*/3 = −1/3. Such a resonance is commonly referred to as a dipole surface plasmon resonance [25]. Notice how the imaginary part of the polarizability is always positive as it should be for ordinary lossy materials. This is contrary to the numerical results presented by Pitkonen [21

_{h}**24**, 1267–1277 (2010). [CrossRef]

*ɛ*= −3

_{r}*ɛ*= −3. This plasmon resonance is clearly visible in both real and imaginary parts of the polarizability.

_{h}*ɛ*

_{3}= 1) superstrate and quartz (

*ɛ*

_{2}=

*ɛ*

_{4}=

*ɛ*= 1.5

_{h}^{2}) substrate. For vertical polarization the configuration has a surface plasmon resonance at

*ɛ*= −

_{r}*ɛ*

_{h}ɛ_{3}/(2

*ɛ*

_{3}+

*ɛ*) ≈ −0.53. This resonance is clearly seen in both real and imaginary parts of the polarizability. For the horizontal case the surface plasmon resonance is located at

_{h}*ɛ*= −(2

*ɛ*+

_{h}*ɛ*

_{3}) = − 5.5. Note that when compared to the half-cylinder in air (Fig. 3), the plasmon resonances shift significantly, in particular, for the case of a horizontally polarized driving field. This is an important finding because it underlines the fact that the substrate plays a crucial role when the spectral location of the plasmon resonances of metallic nanostructures are established.

*ɛ*

_{1}=

*ɛ*

_{2}=

*ɛ*) partly buried in a quartz substrate (

*ɛ*

_{4}= 1.52 and

*ɛ*

_{3}= 1). The polarizability is presented in Fig. 5. For the case of vertical polarization this configuration has a dipole plasmon resonance at

*ɛ*= −2

*ɛ*

_{3}

*ɛ*

_{4}/(

*ɛ*

_{3}+

*ɛ*

_{4}) ≈ −1.38. For horizontal polarization the plasmon resonance is located at

*ɛ*= −(

*ɛ*

_{3}+

*ɛ*

_{4})/2 = −1.625. For this geometry the polarizability is more complicated. Three peaks in both the real and the imaginary part of the polarizability can be identified from the figure, resulting in a large polarizability over a wide range of

*ɛ*.

_{r}26. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*ɛ*= 1.5

_{h}^{2}) substrate the polarizability is presented in Fig. 6 (b). Note how the resonance in the horizontal polarizability has red-shifted approximately 50 nm due to the presence of the quartz substrate. The resonance wavelength is now close to 420 nm, which is where the real part of the dielectric constant of silver is approximately −5.5. When compared to the half-cylinder in homogenous space, the vertical resonance of the half-cylinder on quartz does not red-shift significantly. The explanation for the larger red-shift seen in the horizontal polarizability is that a larger part of the field is within the substrate for a horizontal dipole moment than for a vertical one. For the partly buried cylinder the polarizabilities as a function of wavelength are presented in Fig. 6 (c). The result shows that the resonances in the vertical and the horizontal polarizabilities are similar to each other. This might be expected because for both polarizations approximately half of the plasmon field will be within the substrate.

## 4. Conclusion

## 5. Appendix

## 5.1. Vertical polarization

*α*of the partly buried double half-cylinder is obtained by comparing the far-field expression of the scattered potential on the

_{v}*y*axis with the scattered potential from a line dipole

*ϕ*(

**r**) =

**p**

*·*

_{v}*r̂/*(2

*πr*), where

**p**

*=*

_{v}*α*·

_{v}**E**

_{0}is the induced dipole moment with the constant incident electric field

**E**

_{0}= −∇

*ϕ*

_{0}(

**r**). On the

*y*axis

*x*= 0 and, hence,

*u*= 0, thus we find

*y*= sin

*v*/(1 − cos

*v*), which in the far field should be large

*y*≫ 1. This means that

*v*≪ 1. In this limit, we Taylor expand the sine and cosine functions to find

*v*≈ 2/

*y*. Using sinh(

*vλ*) ≈

*vλ*and cosh(

*vλ*) ≈ 1 we find the scattered far-field potential on the

*y*axis as where the last integral just is a constant contribution to the potential. This constant is without physical significance as

**E**(

**r**) = −∇

*ϕ*(

**r**). By comparing with the potential of a vertically oriented line dipole

*ϕ*(0,

*y*) =

*p*/(2

_{v}*πy*) the vertical polarizability is easily found as Similarly,

*s*

_{3}(

*λ*) can be calculated from the equation sets formed from the boundary conditions for the potential and its normal derivative. By introducing the constants

*γ*= cosh(

*λπ*/2) and

*ξ*= sinh(

*λπ*/2) we find from the continuity of the potential across the boundaries

*λπ*) =

*ξ*

^{2}+

*γ*

^{2}and sinh(

*λπ*) = 2

*ξγ*. From the transformed incident potential we find We need four expressions that transform into We also need Given these equations it is straightforward to use the boundary conditions for the normal derivative of the potential to set up

*s*

_{3}(

*λ*) can be found as

## 5.2. Horizontal polarization

*y*axis (

*x*≈ 0) with the scattered potential of a horizontal line dipole, which for

*x*small is given as

*ϕ*(

*x*≈ 0,

*y*) =

*xp*/(2

_{h}*πy*

^{2}). In the far field, where

*y*≫ 1, we again find

*v*= 2/

*y*≪ 1 and for

*x*≈ 0 we also have

*u*≈ 0. With these limits, by using the connection between bipolar and rectangular coordinates and Taylor expanding the sine and cosine functions, we find

*x/y*≈

*u/v*, which yields

*u*≈ 2

*x/y*

^{2}. Now by approximating cosh(

*vλ*) ≈ 1 we find the far-field expression of the scattered potential close to the

*y*axis as and thus the horizontal polarizability For horizontal polarization we choose the incident field as

**E**

_{0}(

**r**) =

*x̂*, which yields Thus, we find and The continuity equations for the potential are the same as for the vertical case, Eq. (7), but the boundary conditions for the normal derivative now lead to

*c*

_{3}(

*λ*) as

*c*

_{3}(

*λ*) and

*s*

_{3}(

*λ*). By changing the sign and substituting

*ɛ*with 1/

_{i}*ɛ*,

_{i}*c*

_{3}(

*λ*) transforms into

*s*

_{3}(

*λ*) and vice versa.

## Acknowledgments

## References and links

1. | M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. Lond. |

2. | J. W. Strutt (Lord Rayleigh), “On the scattering of light by small particles,” Phil. Mag. |

3. | L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr. |

4. | G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. |

5. | C. F. Bohren and D. R. Huffman, |

6. | L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag. |

7. | J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. |

8. | H. C. van de Hulst, Light Scattering by Small Particles (Dover, 2000). |

9. | L. Novotny and B. Hecht, |

10. | A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. |

11. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

12. | W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. |

13. | S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. |

14. | P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science |

15. | L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. |

16. | F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) |

17. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. |

18. | P. C. Waterman, “Surface fields and the T matrix,” J. Opt. Soc. Am. A |

19. | A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. |

20. | A. Salandrino, A. Alu, and N. Engheta, “Parallel, series, and intermediate interconnects of optical nanocircuit elements. 1. Analytical solution,” J. Opt. Soc. Am. B |

21. | M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. |

22. | H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. |

23. | P. M. Morse and H. Feshbach, |

24. | H. E. Lockwood, |

25. | S. A. Maier, |

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

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(230.5750) Optical devices : Resonators

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**Citation**

Jesper Jung and Thomas G. Pedersen, "Exact polarizability and plasmon resonances of partly buried nanowires," Opt. Express **19**, 22775-22785 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22775

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### References

- M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. Lond.147, 145–181 (1857). [CrossRef]
- J. W. Strutt (Lord Rayleigh), “On the scattering of light by small particles,” Phil. Mag.41, 447–454 (1871).
- L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr.6, 1–62 (1890).
- G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys.330, 337–445 (1908). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag.36, 365–376 (1918).
- J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, 2000).
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
- A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt.5, S16–S50 (2003). [CrossRef]
- S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005). [CrossRef]
- W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater.19, 3771–3782 (2007). [CrossRef]
- S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon.1, 641–648 (2007). [CrossRef]
- P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308, 1607–1609 (2005). [CrossRef] [PubMed]
- L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007). [CrossRef] [PubMed]
- F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a)12, 2844–2861 (2008). [CrossRef]
- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010). [CrossRef] [PubMed]
- P. C. Waterman, “Surface fields and the T matrix,” J. Opt. Soc. Am. A16, 2968–2977 (1999). [CrossRef]
- A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys.76, 4827–4835 (1994). [CrossRef]
- A. Salandrino, A. Alu, and N. Engheta, “Parallel, series, and intermediate interconnects of optical nanocircuit elements. 1. Analytical solution,” J. Opt. Soc. Am. B24, 3007–3013 (2007). [CrossRef]
- M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl.24, 1267–1277 (2010). [CrossRef]
- H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys.102, 044105 (2007). [CrossRef]
- P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).
- H. E. Lockwood, A Book of Curves (Cambridge University Press, 1963).
- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]

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