## Far-field analysis of axially symmetric three-dimensional directional cloaks |

Optics Express, Vol. 21, Issue 8, pp. 9397-9406 (2013)

http://dx.doi.org/10.1364/OE.21.009397

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### Abstract

Axisymmetric radiating and scattering structures whose rotational invariance is broken by non-axisymmetric excitations present an important class of problems in electromagnetics. For such problems, a cylindrical wave decomposition formalism can be used to efficiently obtain numerical solutions to the full-wave frequency-domain problem. Often, the far-field, or Fraunhofer region is of particular interest in scattering cross-section and radiation pattern calculations; yet, it is usually impractical to compute full-wave solutions for this region. Here, we propose a generalization of the Stratton-Chu far-field integral adapted for *2.5D formalism*. The integration over a closed, axially symmetric surface is analytically reduced to a line integral on a meridional plane. We benchmark this computational technique by comparing it with analytical Mie solutions for a plasmonic nanoparticle, and apply it to the design of a three-dimensional polarization-insensitive cloak.

© 2013 OSA

## 1. Introduction

1. G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express **20**, 4176–4188 (2012) [CrossRef] [PubMed] .

2. J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters **9**, 887–891 (2009) [CrossRef] [PubMed] .

*ρz*-plane.

*N*×

_{ρ}*N*×

_{z}*N*sized problem, where the indices indicate the number of mesh points in each of the dimensions, it is instead only necessary to solve

_{ϕ}*m*two-dimensional problems of size

_{max}*N*×

_{ρ}*N*. The reduction in dimensionality typically results in extremely large computational savings in terms of space and processing time.

_{z}3. A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag. **47**, 1260–1266 (1999) [CrossRef] .

4. R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag. **41**, 975–981 (1993) [CrossRef] .

*et al.*designed a directional, three-dimensional isotropic-medium cloak by revolving a two-dimensional conformal cloak [5

5. Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys. **111**, 053105 (2012) [CrossRef] .

6. C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science **337**, 1072–1074 (2012) [CrossRef] [PubMed] .

## 2. Quasi-two-dimensional modeling

*2.5D modeling*. In the 2.5D formulation, all fields are decomposed in terms of the azimuthal mode number

*m*∈ ℤ. The decomposition proceeds by expanding the fields in cylindrical harmonics with respect to the variable

*ϕ*: If the geometry is

*ϕ*-independent (i.e. it is rotationally symmetric), each cylindrical harmonic propagates independently. This can be easily demonstrated by direct substitution of Eqs. (1) into Maxwell’s equations and considering that

**D**

^{(m)}=

*ε*(

*ρ*,

*z*)

**E**

^{(m)}and

**B**

^{(m)}=

*μ*(

*ρ*,

*z*)

**H**

^{(m)}. For each azimuthal number

*m*, an

*mth*wave equation can be written as: where the operator ∇

^{(m)}× is obtained from the curl operator by substituting derivatives with respect to

*ϕ*with a factor

*im*in cylindrical coordinates. The set of Eqs. (2) can be solved on the two-dimensional cross-section of the simulation domain using a standard scattered-field formulation for any arbitrary excitation. For many wave forms of interest, the cylindrical harmonic expansion converges rapidly, and therefore it can be truncated at a relatively small

*m*=

*m*. An incident plane wave typically serves as a convenient field excitation. A plane wave can be decomposed in cylindrical harmonics as follows. Consider a wave polarized such that the magnetic field is transverse to the

_{max}*z*-axis (TM

*polarization), i.e.*

_{z}*H*= 0 as shown in Fig. 1(a) and the electric field is: In the previous expression

_{z}*θ*is the angle between the wave vector and the

_{i}*z*-axis. Taking advantage of the periodicity with respect of

*ϕ*, it is possible to expand the

*z*-component of

**E**as Fourier series [7

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

*E*. However this would introduce differing factors of sin

_{x}*ϕ*and cos

*ϕ*in both components

*E*and

_{ϕ}*E*, violating our requirement that the azimuthal variations must be the same for all fields. It is more convenient to derive all remaining cylindrical-coordinate components directly from

_{ρ}*E*. Using Maxwell’s curl equations we obtain [7

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

*z*-axis by reintroducing the

*ϕ*-dependent factor

*e*in the sums of Eqs. (1).

^{imϕ}### 2.1. Far-field calculation

**E**

*along a given direction*

_{far}**r̂**can be expressed as an integral of the near-fields

**E**and

**H**over an arbitrary surface

*S*enclosing the scattering object using the well-known Stratton-Chu formula [8

8. J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. **56**, 99–107 (1939) [CrossRef] .

**n̂**is the unit normal to

*S*,

*k*is the wave number and

*ϕ*′ ∈ [0, 2

*π*]. The result of this integration is: where

*ds*is the line element on the surface

*S*corresponding to

*ϕ*′ = constant. The linear kernels

*ξ*=

*kρ*′

*sinθ*, that is

*J*=

_{m}*J*(

_{m}*kρ*′

*sinθ*). Details of the above derivation are included in the Appendix A. What is important here is that by explicitly assuming the azimuthal dependence, it is possible to analytically integrate along the revolved direction

*ϕ*. Thus, the surface integration is reduced to a much faster line integration along the boundary of the revolved cross-section domain. This expression is valid for any arbitrary structures that possess axial symmetry.

### 2.2. Validation

*z*-direction toward negative values, impinging on a dielectric sphere and polarized such that the electric field lies along the

*x*-direction (equivalently, on the plane

*ϕ*= 0), as shown in Fig. 1(a). From the optical theorem, the extinction cross-section

*σ*is found from the far-field amplitude in the forward direction

_{ext}*σ*=

_{abs}*W*/

_{abs}*I*

_{0}, where

*W*can be easily calculated from the integral: performed over the volume Ω occupied by the particle. Writing the fields using similar expressions as in (1), gives: where we denoted the revolved cross-sectional area as Ω′. We note that non-zero contributions exists only for

_{abs}*n*=

*m*, for which the integration over the azimuthal variable gives a factor of 2

*π*. The previous integral finally reduces to: The volume integration is thus reduced to an integration across the revolved cross-section area only. The total scattering cross-section is then obtained as

*σ*=

_{sca}*σ*−

_{ext}*σ*. Figure 1(b) shows the comparison of absorption, extinction, and scattering efficiencies calculated using the standard Mie theory and those obtained by 2.5D simulations using Eq. (8) for the gold nanosphere of radius

_{abs}*R*= 30 nm. Notice that the two curves coincide exactly. Figure 1(c) shows the scattering differential cross-section in the plane of polarization at the electric field (

*E*-plane) and the magnetic field (

*H*-plane) respectively, for the wavelength corresponding to the localized surface plasmon resonance. The scattering pattern reproduces the characteristic

*donut*shape of the dipolar radiation.

## 3. Unidirectional 3D cloak

*et al.*[10

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

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

11. U. Leonhardt, “Optical conformal mapping,” Science **312**, 1777–1780 (2006) [CrossRef] [PubMed] .

13. N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials **12**, 25–28 (2013) [CrossRef] .

5. Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys. **111**, 053105 (2012) [CrossRef] .

14. Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. **13**, 024002 (2011) [CrossRef] .

15. Y. Urzhumov and D. R. Smith, “Low-loss directional cloaks without superluminal velocity or magnetic,” Opt. Lett. **37**, 4471 (2012) [CrossRef] [PubMed] .

*z*and filled by homogeneous and anisotropic materials. The double-cone shaped cloaked object of height 2

*a*and width 2

*b*is coated by a perfectly conducting infinitely thin film. The surrounding cloaking device has a cylindrical boundary of height 2

*a*and width 2

*B*= 2

*a*. We refer to these sectors as regions I, I′, and II, as shown in Fig. 2(a). Using standard transformation optics techniques [10

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

*χ*=

*B*/(

*B*−

*b*) and

*γ*=

*b/a*, we obtain that the non-zero material properties are given by: Region I′ is filled with a mirror imaged material from region I, and

*μ̿*=

*ε̿*. A full-wave simulation performed using the 2.5D technique shows that the near-field scattering from the cloaked object appears strongly reduced (see Fig. 2(b)). However, the apparent smoothness or lack of scattering of the near-fields does not always indicate the far-field performance, especially when phase accumulation errors are present. Performing the far-field integration described above, we are able to determine exactly the expected reduction in scattering cross-section in the far-field. If we define the geometrical cross-sectional area as

*A*=

*πb*

^{2}, the total scattering cross-sections are

*σ*= 2.080

_{sca}*A*and

*σ*= 0.077

_{sca}*A*for the uncloaked and cloaked geometries respectively. The cloaking device reduces the total cross-section to less than 4% of the original scatterer. In Fig. 3 we show the differential scattering cross-section of the uncloaked object compared to the case where the object is surrounded by the cloaking device. The cloaking reduces the differential scattering at least of one order of magnitude in all directions with respect to uncloaked object.

## 4. Conclusion

## A. Appendix: Far-field formula

**V**=

**n̂**×

**E**or

**V**=

**n̂**×

**H**. In the 2.5D formulation all the fields are expressed in the orthonormal basis

*ρ̂*,

*ϕ̂*,

**ẑ**〉 and can be decomposed in terms of the azimuthal mode number

*m*as in Eq. (1). Substituting an analogous expression for

**V**gives: where

**M**and vectors

**a**and

**b**,

**Ma**×

**Mb**=

*det*(

**M**)[

**M**

^{−1}]

^{T}**a**×

**b**. For the orthonormal properties of the matrix

*det*(

^{−1}]

*=*

^{T}**r′**·

**r̂**and

*ϕ*coordinate explicit. Before proceeding in our calculation it is useful to notice that since all fields have the same azimuthal dependence,

*e*, we can safely assume that also the far-field will show the same behavior. We can write then: The calculation of the far-field can be reduced to determining the far-field coefficients

^{imϕ}*ϕ*= 0. With this in mind and substituting the Eqs. (21) and (22) into Eq. (19) we find: where we have assumed

**r̂**∈ {

*ϕ*= 0},

*ξ*=

*kρ*′ sin

*θ*and: In order to solve the integrals of Eq. (24), consider the following integral: where

*J*is the

_{m}*m*-th order Bessel function of the first kind. Using Eq. (26) we can easily perform the integrations of Eq. (24) over

*ϕ*∈ [0, 2

*π*]. Considering that the surface element can be written as

*dS*′ =

*ρ*′

*dϕ*′

*ds*′ gives: where all Bessel’s functions are evaluated in

*ξ*=

*kρ*′ sin

*θ*, that is

*J*=

_{m}*J*(

_{m}*kρ*′ sin

*θ*). The remaining integration is performed along the boundary

*s*of the domain cross-section. The quantity in the brackets becomes then:

*J*

_{m−1}+

*J*

_{m+1}= 2

*mJ*/

_{m}*ξ*and

*J*

_{m−1}−

*J*

_{m+1}= 2

*J*′

*.*

_{m}**r̂**,

*ϕ̂*,

*θ̂*〉 of spherical coordinates reads then: where: Performing the products in Eq. (29) and using Eqs, (27) and (28) finally gives the following expression for the far-field coefficients: where the matrices

**E**

*(*

_{far}*ϕ*,

*θ*) can be found using the expression (23). This expression is valid for any arbitrary structures that possess axial symmetry.

## Acknowledgments

## References and links

1. | G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express |

2. | J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters |

3. | A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag. |

4. | R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag. |

5. | Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys. |

6. | C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science |

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

8. | J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. |

9. | H. C. van de Hulst, |

10. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

11. | U. Leonhardt, “Optical conformal mapping,” Science |

12. | Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. |

13. | N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials |

14. | Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. |

15. | Y. Urzhumov and D. R. Smith, “Low-loss directional cloaks without superluminal velocity or magnetic,” Opt. Lett. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(290.5850) Scattering : Scattering, particles

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Scattering

**History**

Original Manuscript: February 25, 2013

Revised Manuscript: April 2, 2013

Manuscript Accepted: April 2, 2013

Published: April 9, 2013

**Citation**

Cristian Ciracì, Yaroslav Urzhumov, and David R. Smith, "Far-field analysis of axially symmetric three-dimensional directional cloaks," Opt. Express **21**, 9397-9406 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9397

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### References

- G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express20, 4176–4188 (2012). [CrossRef] [PubMed]
- J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters9, 887–891 (2009). [CrossRef] [PubMed]
- A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag.47, 1260–1266 (1999). [CrossRef]
- R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag.41, 975–981 (1993). [CrossRef]
- Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys.111, 053105 (2012). [CrossRef]
- C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012). [CrossRef] [PubMed]
- J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955). [CrossRef]
- J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56, 99–107 (1939). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons Ltd, New York, 1957).
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Optical conformal mapping,” Science312, 1777–1780 (2006). [CrossRef] [PubMed]
- Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009). [CrossRef]
- N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials12, 25–28 (2013). [CrossRef]
- Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011). [CrossRef]
- Y. Urzhumov and D. R. Smith, “Low-loss directional cloaks without superluminal velocity or magnetic,” Opt. Lett.37, 4471 (2012). [CrossRef] [PubMed]

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