## General properties of two-dimensional conformal transformations in electrostatics |

Optics Express, Vol. 19, Issue 21, pp. 20035-20047 (2011)

http://dx.doi.org/10.1364/OE.19.020035

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### Abstract

Electrostatic properties of two-dimensional nanosystems can be completely described by their non-trivial geometry modes. In this paper we prove that these modes as well as the corresponding eigenvalues are invariant under any conformal transformation. This invariance suggests a new way to study electrostatic conformal transformations, while also providing an in-depth interpretation of the behavior exhibited by singular plasmonic nanoparticles.

© 2011 OSA

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

7. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. **10**, 115023 (2008). [CrossRef]

7. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. **10**, 115023 (2008). [CrossRef]

8. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. **9**, 387–396 (2010). [CrossRef] [PubMed]

9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977–980 (2006). [CrossRef] [PubMed]

11. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science **328**, 337–339 (2010). [CrossRef] [PubMed]

12. M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. **13**, 024003 (2011). [CrossRef]

13. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. **95**, 041106 (2009). [CrossRef]

14. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. **100**, 063903 (2008). [CrossRef] [PubMed]

15. D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express **16**, 18731–18738 (2008). [CrossRef]

17. J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter **14**, 8463–8479 (2002). [CrossRef]

21. B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media **17**, 581–614 (2007). [CrossRef]

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B **82**, 125430 (2010). [CrossRef]

27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. **12**, 093030 (2010). [CrossRef]

*i.e.*, broadband response and divergent electric fields around the singularities.

28. D. J. Bergman and D. Stroud, in Solid State Physics , H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. **46**, pp. 148–270. [CrossRef]

29. G. W. Milton, *The Theory of Composites* (Cambridge University Press, 2002). [CrossRef]

*ɛ*

_{1}(

*ω*) and

*ɛ*

_{2}(

*ω*) respectively, can be described by a set of non-trivial eigenmodes

*φ*of the following generalized eigenproblem where

_{n}*s*represents the corresponding eigenvalues (see Appendix A). The function

_{n}*θ*(

**r**) characterizes the geometry of the composite:

*θ*(

**r**) = 1 when

**r**is inside the medium with dielectric constant

*ɛ*

_{1}and equals 0 elsewhere. To simplify the following discussion,

*ɛ*

_{2}(

*ω*) is assumed to be 1. Since this equation depends exclusively on the geometry, but not on the material composition, the resultant eigenmodes are therefore referred to as geometry modes [28

28. D. J. Bergman and D. Stroud, in Solid State Physics , H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. **46**, pp. 148–270. [CrossRef]

*s*of a non-trivial mode must be real and limited to the range 0 <

_{n}*s*< 1, and the normalized eigenmodes

_{n}*φ*can be used to expand an arbitrary function for points

_{n}**r**that are inside the

*ɛ*

_{1}material [30

30. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. **90**, 027402 (2003). [CrossRef] [PubMed]

33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. **12**, 4947–4960 (1979). [CrossRef]

*any*conformal coordinate transformation. To prove this fact, we rewrite its left-hand side as where the Einstein summation convention is employed, and the metric tensor

*g*≡

_{ij}**e**

*·*

_{i}**e**

*=*

_{j}*gδ*because of the conformal transformation. Similarly the right-hand side can be reformulated as Evidently the transformed equation is identical to the previous one if we interpret the new equation as being in a right-handed Cartesian system and keep

_{ij}*θ*and

*φ*unchanged.

_{n}*φ*and their corresponding eigenvalue

_{n}*s*immediately suggest a new way to interpret and study the electrostatic response of the transformed structure. This new approach depends exclusively on the geometry, as described by the function

_{n}*θ*(

**r**), and does not need any information regarding the material parameters

*ɛ*(

*ω*) or the external potential. For instance, we can expand the total potential

*φ*in terms of the associated eigenmodes, where

_{t}*k*=

*ω*/

*c*,

*s*(

*ω*) = 1/(1 –

*ɛ*

_{1}) being the Bergman’s spectral parameter and

*φ*

_{0}being the external potential. Notice that we do not include the radiation damping here. Consequently, once these eigenmodes of the original geometry are known, we can obtain the potential

*φ*of any transformed or derivative geometry by simply calculating the expansion coefficients (

_{t}*φ*|

_{n}*φ*

_{0}). In other words, transforming geometry is equivalent to transforming the external source, or more precisely, changing the expansion coefficient of each eigenmode. Furthermore, the time-averaged power

*P*

_{a}absorbed by the structure can be written as (See Appendix B) where

*I*= Σ

_{e}*|*

_{n}*β*(

_{n}*φ*|

_{n}*φ*

_{0})|

^{2}represents the integration of the electric field intensity |

*E*|

^{2}inside the particle. In a similar way, the extinction may be expressed as When we transform the external field

*φ*

_{0}accordingly,

*i.e.*,

*φ*

_{0}(

*x, y*) =

*φ*′

_{0}(

*u, v*) with

*u*and

*v*being the new coordinates, all the energy quantities,

*P*

_{a},

*I*

_{e}and

*P*

_{ex}, are evidently invariant. On the other hand, if the external potential is fixed with respect to the

*w*coordinate where

*w*=

*u*+

*iv*, it follows that these energy quantities are proportional to

*a*

^{2}when the conformal mapping has a form

*a*×

*w*(

*z*) with

*z*=

*x*+

*iy*and

*a*is real. In other words, the larger the particle area, the stronger the absorption and extinction.

*surface modes*of the particle [34

34. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, 1998). [CrossRef]

35. Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. **35**, 1431–1433 (2010). [CrossRef] [PubMed]

*n*th mode is strictly

*s*(

*ω*) =

*s*, or equivalently

_{n}*ɛ*

_{1}= 1 – 1/

*s*, when we do not include the radiation loss. Notice that the corresponding permittivity

_{n}*ɛ*

_{1}of the particle should be real and negative since 0 <

*s*< 1 [34

_{n}34. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, 1998). [CrossRef]

36. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. **97**, 206806 (2006). [CrossRef] [PubMed]

*n*th

*surface plasmonic resonance*(SPR) is given by

*s*(

*ω*–

_{n}*iγ*) =

_{n}*s*with

_{n}*ω*being real resonant frequency and

_{n}*γ*being the relaxation rate [28

_{n}28. D. J. Bergman and D. Stroud, in Solid State Physics , H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. **46**, pp. 148–270. [CrossRef]

37. D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. **91**, 253902 (2003). [CrossRef]

*s*are conserved under any conformal mapping, the transformed structure will have the same SPRs as the original structure at the identical resonant frequencies. For instance, since a metallic cylinder can be transformed from a metal-dielectric interface by

_{n}*w*=

*e*, its SPR hence can be determined by the nonretarded surface-plasmon condition of the metal-dielectric interface [38

^{z}38. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**, 131–314 (2005). [CrossRef]

*w*=

*w*(

*z*). It is further assumed that the external potential

*φ*

_{0}(

*u, v*) =

*p*+

_{u}u*p*, which corresponds to a uniform electric field −

_{v}v*p*

_{u}**e**

*–*

_{u}*p*

_{v}**e**

*. The expansion coefficients in this case can be written as where In addition,*

_{v}*ρ*

_{1}= 1 and

*ρ*

_{2}= ∓

*e*

^{–|k|(d1+d2)}for positive

*k*, and they should be interchanged when

*k*< 0. Here we use the Cauchy-Riemann equations of conformal mappings as well as

*w*=

*e*, where the radii of the two cylinders are

^{z}*r*

_{1}=

*e*

^{d1}and

*r*

_{2}=

*e*

^{d2}respectively [39

39. P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. **94**, 231111 (2009). [CrossRef]

*p*is assumed to be zero because of the structural symmetry. Evidently, only eigenmodes with

_{v}*k*= ±1 are excited. We further use these coefficients to calculate the induced potential

*φ*. It is found that

_{i}*σφ*/[

_{i}*φ*

_{0}(1 –

*ɛ*

_{1})], with

34. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, 1998). [CrossRef]

*ɛ*

_{1}) will excite a surface mode as long as its expansion coefficient is not zero. Consequently the transformed system will present broadband response in principle. One example is the crescent studied in [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B **82**, 125430 (2010). [CrossRef]

*w*= 1/

*z*and

*d*

_{1}> 0 (Figure 1(c)). The expansion coefficients are found to be (See Appendix D) which can be further used to achieve identical electric fields as those presented in Ref. [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B **82**, 125430 (2010). [CrossRef]

*φ*

_{k,}_{±}|

*φ*

_{0})| depends exclusively on the amplitude of the external electric field. The energy quantities such as the absorption hence do not depend on the direction of the incident field [22

**82**, 125430 (2010). [CrossRef]

*φ*

_{k,}_{±}|

*φ*

_{0})|

^{2}on

*k*achieved by setting

*p*=

_{u}*p*= 1,

_{v}*d*

_{1}=

*d*and

*d*

_{1}= 2

*d*with

*d*being an arbitral real quantity. As mentioned, each eigenmode of the crescent is excited, consequently |(

*φ*

_{k,}_{±}|

*φ*

_{0})|

^{2}is always positive. Furthermore, in this case these expansion coefficients are strongly localized in the regime 0.3 ≤ |

*k*|

*d*≤ 1.6 with a maximum around |

*k*|

*d*= 0.72.

27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. **12**, 093030 (2010). [CrossRef]

*d*

_{1}< 0 and

*d*

_{2}> 0, using the conformal mapping given by

*w*= 1/

*z*(Figure 1(d)). The corresponding expansion coefficients are calculated as (See Appendix D) Again all the eigenmodes are excited by the external source. Note that when

*d*

_{1}= −

*d*

_{2}the cylinders have the same radius. The kissing cylinders then possess both

*u*and

*v*mirror symmetry. Hence, the even mode

*φ*

_{k,}_{+}or the odd mode

*φ*

_{k,}_{−}can be excited by a

*u*- or

*v*- polarized electric field respectively. An example dependence of |(

*φ*

_{k,}_{±}|

*φ*

_{0})|

^{2}on

*k*is plotted in Figure 2, with

*p*=

_{u}*p*= 1 as well as

_{v}*d*

_{1}= −

*d*and

*d*

_{1}= 2

*d*. Similar to the curve associated with the crescent, the eigenmodes with 0.2 ≤ |

*k*|

*d*≤ 1.6 contribute significantly to the absorption cross section, which guarantees that the kissing cylinders will exhibit broadband behavior.

**82**, 125430 (2010). [CrossRef]

27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. **12**, 093030 (2010). [CrossRef]

*ω*, the resonant frequency of the

_{n}*n*-th surface mode. The total potential of Eq. (5) can therefore be approximated as

*β*(

_{n}*φ*|

_{n}*φ*

_{0})

*φ*. Since only

_{n}*φ*is position dependent, the amplitude of the corresponding electric field is proportional to |∇

_{n}*| = |*

_{w}φ_{n}*dz/dw*||∇

*|. Taking the crescent as an example, |*

_{z}φ_{n}*dz/dw*| =

*x*

^{2}+

*y*

^{2}since

*w*= 1/

*z*. Consequently, when we map the point (

*a,*∞) in

*xy*coordinates to the singular point of the crescent (the origin of the

*uv*coordinates), the corresponding electric field tends to infinity. On the other hand, |

*dz/dw*| = 1/

*e*for the two coaxial cylinders we studied above, and the resultant electric field for this case only depends on

^{x}*r*and does not exhibit singular behavior. Note that |

*dz/dw*|

^{2}is actually the stretching factor for area when we transform the

*xy*coordinates to the

*uv*coordinates. The field enhancements because of |

*dz/dw*| hence are purely induced by the coordinate transformations. Furthermore, we want to point out that, although the electric field at a few specific points can be infinite,

*I*, the integration of the electric field intensity over the system, is finite and invariant under any conformal transformation.

_{e}*i.e.*, broadband response and divergent electric fields in the neighborhood of the singularities.

## A. Properties of Eq. (1)

33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. **12**, 4947–4960 (1979). [CrossRef]

*G*(

**r**

*,*

**r**′) which satisfies the following Laplace’s equation Notice that both

*φ*and

_{n}*G*should satisfy the homogeneous Dirichlet-Neumann boundary conditions. If a scalar product of two functions further is defined as the operator

*Ĝ*is then Hermitian. To prove this fact, we can show that

*Ĝ*is Hermitian, such that its eigenvalues

*s*are all real. Moreover, which suggests that the eigenvalues lie between 0 and 1. As pointed out in Reference [33

_{n}33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. **12**, 4947–4960 (1979). [CrossRef]

## B. Absorption and extinction

*φ*are a complete orthogonal set [33

_{n}**12**, 4947–4960 (1979). [CrossRef]

*ɛ*

_{1}material, can be expanded in terms of these modes Using the fact that where

*φ*being the induced potential, we achieve the following The total potential hence can be given as Moreover, the corresponding electric field is given by

_{i}**E**= −∇

*φ*. Here the radiation loss is not included.

## C. Geometry modes of one-dimensional finite slab

*d*

_{1},

*d*

_{2}]. The possible solution has a form

*ae*

^{|k|x}

*e*for

^{iky}*x*<

*d*

_{1},

*ce*

^{–|k|x}

*e*for

^{iky}*x*>

*d*

_{2}, and

*b*

_{1}

*e*

^{|k|x}

*e*+

^{iky}*b*

_{2}

*e*

^{−|k|x}

*e*in the middle. The boundary conditions at

^{iky}*x*=

*d*

_{1}and

*x*=

*d*

_{2}yield where

*m*=

*e*

^{|k|d1}and

*n*=

*e*

^{|k|d2}. Two sets of eigenmodes can be obtained. The eigenvalues of the odd modes are given by and the associated eigenmodes are

## D. Expansion coefficients of different structures

**Two coaxial cylinders:**The conformal mapping used here is

*w*=

*e*. To obtain the expansion coefficients, we use Eqs. (10) and (11). It is found that as well as where use has been made of the fact that Because of the structural symmetry we assume that

^{z}*p*= 0, and the resultant expansion coefficients are Evidently, only eigenmodes with

_{v}*k*= ±1 are excited. Making use of these coefficients, the induced potential

*φ*= Σ

_{i}*(*

_{n}*β*– 1)(

_{n}*φ*|

_{n}*φ*

_{0})

*φ*can be expressed as with

_{n}*r*

_{1}=

*e*

^{d1},

*r*

_{2}=

*e*

^{d2}, and Finally we achieve, by using the fact that

*φ*

_{0}=

*p*=

_{u}u*p*cos

_{u}e^{x}*y*=

*p*cos

_{u}r*θ*, the following result:

**Crescent:**It is known that a slab, with

*d*

_{1}> 0, can be transformed to a crescent by a conformal mapping of

*w*= 1/

*z*. Consequently

*g*=

*dw/dz*= −1/

*z*

^{2}. By using Eqs. (10) and (11), it is found that Since

*x*≥

*d*

_{1}

*>*0, the second integration has a pole at

*y*=

*ix*in the upper half complex plane. Consequently for negative

*k*, we have Similar procedures can be applied to calculate

*k*, which yield because of the fact that Here the integration has a pole at

*y*= −

*ix*in the lower half complex plane. Hence the resultant expansion coefficients are given by Notice that for same

*k*, the even mode and the odd mode result in identical expansion coefficients.

**Kissing cylinder dimer:**A kissing dimer can be transformed from a slab with

*d*

_{1}< 0 and

*d*

_{2}> 0, by using a conformal mapping of

*w*= 1/

*z*.

*F*, To evaluate the second integration, we notice that its integrand has a pole at

_{k}*y*=

*ix*, which is located in the upper half plane for positive

*x*while in the lower half plane for negative

*x*. By carefully choosing the integral contour, it is found that for negative

*k*, and for positive

*k*. The corresponding expansion coefficients are then given by

## E. Electric field of the crescent

*z*=

_{w}*dz/dw*. By representing the geometrical mode

*φ*

_{k,}_{±}as we finally arrive at

*ϑ*(

*x*) = 1 for positive

*x*while

*ϑ*(

*x*) = 0 when

*x*< 0. Furthermore, using Eq. (8),

*β*

_{k,}_{±}can be reformulated as where

*t*=

*d*

_{2}

*– d*

_{1}being the slab thickness, and

*eγ*= (

*ɛ*

_{1}

*–*1)/(

*ɛ*

_{1}+ 1). Moreover, since

*k*is continuous, the electric field, taking the

*u*component as an example, may be expressed as

*ɛ*

_{1}) < −1 as well as the Im(

*ɛ*

_{1}) is positive and close to zero. Consequently

*e*is located in the first quadrant of the complex plane. Further assuming that only the poles of the integrand, given by

^{γ}*k*= ±

*γ*/

*t*, contribute to the integration, we finally obtain for positive

*y*, and for negative

*y*. The electric field in the

*x*<

*d*

_{1}region, where

*a*

_{−}= (1+

*e*

^{−γ})

*e*

^{−2γd1/t}and

*b*

_{−}= 0, can then be expressed as By using the same notation as Ref. [22

**82**, 125430 (2010). [CrossRef]

*g*= 1, the above equation is found to be identical to Eq. (28) of Ref. [22

**82**, 125430 (2010). [CrossRef]

## Acknowledgments

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31. | M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. |

32. | M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B |

33. | D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. |

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

35. | Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. |

36. | F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. |

37. | D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. |

38. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. |

39. | P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 27, 2011

Revised Manuscript: July 13, 2011

Manuscript Accepted: September 17, 2011

Published: September 29, 2011

**Citation**

Yong Zeng, Jinjie Liu, and Douglas H. Werner, "General properties of two-dimensional conformal transformations in electrostatics," Opt. Express **19**, 20035-20047 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20035

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