A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics

doi:10.1364/OE.18.006327

A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics

Guishan Yuan, Xiaochun Dong, Qiling Deng, Hongtao Gao, Chunheng Liu, Yueguang Lu, and Chunlei Du »View Author Affiliations

Optics Express, Vol. 18, Issue 6, pp. 6327-6332 (2010)
http://dx.doi.org/10.1364/OE.18.006327

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▸ Article Outline
– Abstract
1. Introduction
2. Principle
3. Simulation and result
4. Expanded model for the transformer
5. Conclusion
– References and links

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Abstract

A design method was proposed for transforming the scattering cross section’s shape of PEC objects to other arbitrary shapes based on transformation optics. The parameters of the transformer which is a kind of metamaterial being tightly covered to the original PEC object can be determined by the transformation expressions derived in the paper. With the method, the virtual PEC objects with their definite shape of the scattering cross section can be produced visually through the original PEC objects with a designed transformer. The validation was made by comparing the distribution of electromagnetic field of the PEC objects with and without transformer by means of finite-element method. Some examples are also given for demonstrating the effectiveness of the method.

1. Introduction

The concept of transformation optics [1

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

] and its use in designing and realization of a microwave invisible cloak has attracted great interest in the studies of controlling the electromagnetic fields. Many devices with novel properties have been designed in theories and experiments, such as cloak, concentrators, superscatterers, field rotators, phase transformer and so on [3

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(5801), 977–980 (2006). [CrossRef] [PubMed]

–7

L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express 16(10), 6815–6821 (2008). [CrossRef] [PubMed]

]. Although making these devices requires materials with specific permittivity and permeability which cannot be obtained in the nature world, the metamaterials with anisotropic and inhomogeneous behaviors normally, which are formed by different subwavelength structures, can be used for providing the solutions [3

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]

]. As the development of metamaterials, transformed media can be realized not only in microwave but also in optics now [10

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

Among various kinds of present devices, “cloaking” is much attractive because of its novel capability to change the scattering cross section of a perfect electrical conductor (PEC) object. For the “cloak” device, the scattering cross section of the PEC object can be changed to a point, and for the imperfect cloak [11

H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B 76(24), 241104 (2007). [CrossRef]

] it can be changed smaller than the scale of the device. On the contrary, the superscatterer [5

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]

] is corresponding to an amplified scattering cross section of the PEC object. Based on the transformation optics, H.Chen et al. proposed the concept of “reshape” with which the position for scattering cross section of the PEC object can also be shifted for the giving incident electromagnetic(EM) wave although the real PEC object is still in the original position [12

H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008). [CrossRef]

In this paper, we proposed a method for changing the scattering cross section of PEC object to correspond to an object of a different shape based on transformation optics, and the corresponding metamaterial device is called transformer. When the PEC cylinder wears the transformer, it looks as if it was other object instead of the original one.

2. Principle

In the method, the transformation optics is employed for determining the parameters of the transformer. Based on form-invariant coordinate transformations of Maxwell’s equations which has been described in detailed in ref.12 and 13, two domains D and D’, which are called virtual domain and physical domain respectively, can be defined for the virtual PEC object and original PEC object. A given coordinate transformation

x^{i^{'}} = A_{i}^{i^{'}} x^{i}

maps D to D’. If the permittivity and permeability tensors in virtual domain are ε^ij and μ^ij , the relative permittivity and permeability tensors in the physical domain can be calculated by the prescription [13

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]

]

ε^{i^{'} j^{'}} = {[\det (A_{i}^{i^{'}})]}^{- 1} A_{i}^{i^{'}} A_{j}^{j^{'}} ε^{i j}

(1)

μ^{i^{'} j^{'}} = {[\det (A_{i}^{i^{'}})]}^{- 1} A_{i}^{i^{'}} A_{j}^{j^{'}} μ^{i j}

(2)

Where

A_{i}^{i^{'}} = \partial x^{i^{'}} / \partial x^{i}

indicates the Jacobi matrix of transformation and

\det (A_{i}^{i^{'}})

its determinant.

Figure 1 shows the schematic diagram of the design, where a two-dimensional arrangement is supposed for implementing the transformation in the Cylindrical coordinate. In Fig. 1, we assume that the original PEC object is with circular shape and the contour of the destination shape is denoted by the virtual square as an example. In order to transform the scattering cross section corresponding to a circular PEC object to that of a square object, the domain between the outer circle and the virtual square will be transformed to a shell between the inner and outer circle (it is physical domain corresponding to the designed transformer) through a given coordinate transformation. For simplicity, we divide the virtual domain to four equal ones as shown in Fig. 1. The coordinate transformation equations for the example design in Region I can be expressed as

Fig. 1 Schematic diagram of the space transformation for the transformer. The radii of outer and inner circles are R₂ and R₁ , respectively, and the sidelength of the virtual square is 2a. The transformation maps point A to point A’.

{\begin{matrix} r^{'} = \frac{(R_{2} - R_{1}) \cos θ}{R_{2} \cos θ - a} (r - R_{2}) + R_{2} \\ θ^{'} = θ \\ z^{'} = z \end{matrix}

(3)

With above equations, we can obtain the Jacobian transformation matrix in the Cylindrical coordinate to be [12

H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008). [CrossRef]

]

J = (\begin{matrix} \frac{\partial r^{'}}{\partial r} & \frac{\partial r^{'}}{r \partial θ} & 0 \\ \frac{r^{'} \partial θ^{'}}{\partial r} & \frac{r^{'} \partial θ^{'}}{r \partial θ} & 0 \\ 0 & 0 & \frac{\partial z^{'}}{\partial z} \end{matrix}) = (\begin{matrix} a_{11} & a_{12} & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & 1 \end{matrix})

(4)

in which

a_{11} = \frac{(R_{2} - R_{1}) \cos θ}{R_{2} \cos θ - a} a_{12} = \frac{a (R_{2} - R_{1}) \sin θ}{{(R_{2} \cos θ - a)}^{2}} \frac{r - R_{2}}{r} a_{22} = \frac{r^{'}}{r}

(5)

In order to make the corresponding transformation be carried out correctly, the virtual domain should be vacuum in the method, that is,

\bar{ε} = \bar{μ} = δ^{i j}

. We get the constitutive parameter tensors of the medium in the physical domain by the metric invariance of Maxwell’s Eqs. (1) (2):

\bar{ε'} = \bar{μ'} = (\begin{matrix} \frac{a_{11}^{2} + a_{12}^{2}}{a_{11} a_{22}} & \frac{a_{12}}{a_{11}} & 0 \\ \frac{a_{12}}{a_{11}} & \frac{a_{22}}{a_{11}} & 0 \\ 0 & 0 & \frac{1}{a_{11} a_{22}} \end{matrix})

(6)

For simplicity, the TE electromagnetic waves are used in the calculation. Hence only

μ_{r r}

μ_{r θ}

μ_{θ r}

μ_{θ θ}

and

ε_{z z}

are considered.

μ_{r r} = \frac{a_{11}^{2} + a_{12}^{2}}{a_{11} a_{22}} μ_{r θ} = μ_{θ r} = \frac{a_{12}}{a_{11}} μ_{θ θ} = \frac{a_{22}}{a_{11}} ε_{z z} = \frac{1}{a_{11} a_{22}}

(7)

Equation (7) provides the full expressions of permittivity and permeability tensors for the quarter of the shell transformed from Region I. By using rotation operators with rotation angles of π/2, π and 3π/2 around z axis to Eq. (3), we can obtain the transformation formulas for other three quarters of the virtual domain. In the virtual space there is a cylindrical PEC with the shell, whose permittivity and permeability tensors are restricted by Eq. (7), and the shell is the metamaterial transformer. What we expect is that a square prism PEC can be observed instead of the real cylinder PEC object if the transformer is applied to it as clothes since the shape of scattering cross section has changed.

3. Simulation and result

In order to validate our design, we make full-wave simulations based on the finite-element method. For evaluating the transformer’s properties, the near-field distributions are calculated for PEC objects with different shape of scattering cross section and the corresponding patterns of electric field are shown in Fig. 2 . The plane-wave of TE polarization incidents from lower left to upper right corner with frequency 3GHz and unit amplitude, and the inner and outer radii of the shell are R₁ = 0.2m and R₂ = 0.3m, respectively. The side length of the square in virtual space is 2a = 0.2m.

Fig. 2 Snapshot of the total electric field. (a) (b) are the total electric field induced by PEC cylinder with radius R₂ = 0.2m and by PEC square prism with side length 2a = 0.2m, respectively. (c) is the total electric field induced by our designed device transformer.

Figures 2(a) and 2(b) is the snapshot of the total electric field induced by PEC cylinder and PEC square prism, respectively. Figure 2(c) is the snapshot of total electric field induced by our designed device which is composed of a PEC cylinder with the transformer outside. From patterns of electric field in (a) and (c), we can find that when the PEC cylinder wears the transformer, the electric field distribution has been changed significantly. It means that the PEC cylinder was no longer cylindrical to an observer. Comparing the patterns of electric field in (b) and (c) in the region r>R₂ , one can find that they are almost equivalent. If someone observes the device in the region r>R₂ , he will see a PEC square prism instead of a cylindrical one. Obviously, the simulations accord with our design very well and the shape of the scattering cross section has been changed from circular to square in the example.

4. Expanded model for the transformer

In last section, we proposed a transformer that can change the scattering cross section corresponding to a circular PEC object to that of a square one. If other destination contour is set in the virtual domain to replace the virtual square, the corresponding transformer can be designed. Suppose ρ(θ) is the scattering cross section’s boundary of the PEC in virtual space, and the virtual domain is the region between the curve ρ(θ) and the circle with radius of R₂ . The general coordinate transformation equations can be expressed as

{\begin{matrix} r^{'} = \frac{R_{2} - R_{1}}{R_{2} - ρ (θ)} (r - R_{2}) + R_{2} \\ θ^{'} = θ \\ z^{'} = z \end{matrix}

(8)

where ρ(θ) is continuous and piecewise differentiable. This transformation maps the virtual domain to the physical domain which is defined in the region between the two circles with radius R₁ and R₂ . The permittivity and permeability tensors in physical domain can be calculated based on the transformation optics as described in Section 2. If ρ(θ) cannot be differentiable all over the function domain (0≤θ<2π), it is necessary to divide the function domain into several subdomains, in which ρ(θ) is differentiable so that the ρ’(θ) is significative. Thus the relative tensors can be calculated in each subdomain, and tensors of the whole domain are available then. The transformer is just a medium with the calculated tensors of permittivity and permeability. If a PEC cylinder with radius R₁ wears the transformer, its scattering cross section’s boundary would become the curve ρ(θ) instead of its real shape. As the function ρ(θ) is arbitrary, we can change the scattering cross section of a PEC cylinder to various shapes utilizing designed transformers.

According to the shape of the PEC objects, the transformer’s shape can also be designed arbitrarily by changing R₁ , R₂ in Eq. (8) to R₁(θ) R₂(θ) as long as R₁(θ) is less than R₂(θ) all over the function domain (0≤θ<2π).

The further simulations are proposed in Fig. 3 to prove our method. The plane-wave incidents from left to right in these simulations compared with simulations in Fig. 2. It is found by comparing Figs. 3(a) 3(c) and 3(b) 3(d) that the circular scattering cross sections have changed by the transformers to the shape of triangular and cardioid-like respectively in Figs. 3(c) and 3(d). Here in our examples, the curve ρ(θ) is inside the outer circle (ρ(θ)<R₂ ). However, the curve ρ(θ) may be outside the outer circle or has intersections with it, which means ρ(θ) may be larger or smaller than R₂ or even equivalent in one curve. The difference in those three conditions is just related to the refractive-index in transformers. From Eq. (4),

a_{11} = \frac{R_{2} - R_{1}}{R_{2} - ρ (θ)}

a_{22} = \frac{r'}{r}

can be obtained. When ρ(θ) is larger than R₂ , there are a₁₁<0, a₂₂>0. Consequently, μ_rr, μ_θθ, ε_zz are all below zero from Eq. (7) and then the refractive-index will be negative, like the case of superscatterer [5

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]

]. When ρ(θ) is smaller than R₂ , the refractive-index will be positive, just like the case of imperfect cloak [11

H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B 76(24), 241104 (2007). [CrossRef]

], and when ρ(θ) is equivalent to R₂ , that means the intersection will be transformed to a line, and the permittivity and permeability tensors will be in zero and infinite components, just like perfect lens [14

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008). [CrossRef]

Fig. 3 Snapshot of the total electric field. (a) (b) are the total electric field induced by PEC prisms whose cross section are triangular and cardioids-like (the boundary curve ρ(θ) = h(1 + cos(θ)), where h = 0.1m), respectively. (c) and (d) are the total electric field induced by the relative transformers.

5. Conclusion

In this work, we have presented a new transforming device designed by the transformation optics with which we can change the scattering cross section’s shape of a PEC object arbitrarily. With the development of the metamaterials, the designed transformer with anisotropic and inhomogeneous distribution can be pushed closely to the practice and find applications in many aspects like camouflaging some objects from recognition of radar or other detecting systems.

Acknowledgment

This work was supported by Fundamental Research Program of China (No.2006CB302900), High Tech. Program of China (2007AA03Z332) and the Chinese Nature Science Grant (60727006). The authors thank Dr. Haofei Shi and Xingzhan Wei for their kind contributions to the work.

References and links

1.	J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
2.	U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
3.	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(5801), 977–980 (2006). [CrossRef] [PubMed]
4.	M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
5.	T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]
6.	H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]
7.	L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express 16(10), 6815–6821 (2008). [CrossRef] [PubMed]
8.	R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
9.	R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
10.	J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
11.	H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B 76(24), 241104 (2007). [CrossRef]
12.	H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008). [CrossRef]
13.	D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]
14.	M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(230.0230) Optical devices : Optical devices
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Physical Optics

History
Original Manuscript: November 18, 2009
Revised Manuscript: February 27, 2010
Manuscript Accepted: February 28, 2010
Published: March 12, 2010

Citation
Guishan Yuan, Xiaochun Dong, Qiling Deng, Hongtao Gao, Chunheng Liu, Yueguang Lu, and Chunlei Du, "A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics," Opt. Express 18, 6327-6332 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6327

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References

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
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(5801), 977–980 (2006). [CrossRef] [PubMed]
M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]
H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]
L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express 16(10), 6815–6821 (2008). [CrossRef] [PubMed]
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B 76(24), 241104 (2007). [CrossRef]
H. Chen, X. Zhang, X. Luo, H. Ma, and C. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008). [CrossRef]
D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]
M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008). [CrossRef]

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