## Calculation of material properties and ray tracing in transformation media

Optics Express, Vol. 14, Issue 21, pp. 9794-9804 (2006)

http://dx.doi.org/10.1364/OE.14.009794

Acrobat PDF (265 KB)

### Abstract

Complex and interesting electromagnetic behavior can be found in spaces with non-flat topology. When considering the properties of an electromagnetic medium under an arbitrary coordinate transformation an alternative interpretation presents itself. The transformed material property tensors may be interpreted as a different set of material properties in a flat, Cartesian space. We describe the calculation of these material properties for coordinate transformations that describe spaces with spherical or cylindrical holes in them. The resulting material properties can then implement invisibility cloaks in flat space. We also describe a method for performing geometric ray tracing in these materials which are both inhomogeneous and anisotropic in their electric permittivity and magnetic permeability.

© 2006 Optical Society of America

## 1. Introduction

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

11. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” (2006). http://xxx.arxiv.org/abs/cond-mat/0607418.

12. T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,” Appl. Phys. Lett. **88**, 081,101 (2006). [CrossRef]

8. 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 (2006). In press. [CrossRef] [PubMed]

## 2. Material properties

*F*

_{αβ}is the tensor of electric field and magnetic induction, and

*G*

^{αβ}is the tensor density of electric displacement and magnetic field, and

*J*

^{β}is the source vector. In component form these tensors are

*x*

^{α})=(

*ct*,

*x*,

*y*,

*z*), and we use the metric signature, +2. All of the information regarding the topology of the space is contained in the constitutive relations

*C*

^{αβµν}is the constitutive tensor representing the properties of the medium, including its permittivity, permeability and bianisotropic properties.

*C*

^{αβµν}is a tensor density of weight +1, so it transforms as [13]

17. A. J. Ward and J. B. Pendry, “Refraction and geometry in maxwell’s equations,” J. Mod. Opt. **43(4)**, 773–793 (1996). [CrossRef]

11. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” (2006). http://xxx.arxiv.org/abs/cond-mat/0607418.

*the same*material properties, but in different spaces. The components in the transformed space are different form those in the original space, due to the topology of the transformation. We will refer to this view as the topological interpretation.

*different*material properties. Both sets of tensor components are interpreted as components in a flat, Cartesian space. The form invariance of Maxwell’s equations insures that both interpretations lead to the same electromagnetic behavior. We will refer to this view as the materials interpretation.

### 2.1. Spherical cloak

*b*, into a spherical shell of inner radius,

*a*, and outer radius,

*b*. Consider a position vector,

**x**. In the original coordinate system (Fig.1(a)) it has components,

*x*, and in the transformed coordinate system (Fig. 1(b)),

^{i}*x*. Of course, its magnitude,

^{i′}*r*, is independent of coordinate system

*g*is the metric of the transformed space. In the materials interpretation, (Fig.1(c)), we consider the components,

_{i′j′}*x*, to be the components of a Cartesian vector, and its magnitude, which we will call

^{i′}*r′*, is found using the appropriate flat space metric

*r*, to a radius,

*r′*, according to the following linear function

*r*≤

*b*, (or equivalently,

*a*≤

*r′*≤

*b*). Outside this domain we assume the identity transformation. (All equations in the remainder of this article apply only to the transformation domain.) We must always limit the transformation to apply only over a finite region of space if we wish to implement it with materials of finite extent. Note that when

*r*=0 then

*r′*=

*a*, so that the origin is mapped out to a finite radius, opening up a whole in space. Note also that when

*r*=

*b*then

*r′*=

*b*, so that space at the outer boundary of the transformation is undistorted and there is no discontinuity with the space outside the transformation domain.

**x**in the original space,

*x*, or the magnitude,

^{i}*r*. We can now drop the primes for aesthetic reasons, and we need not make the distinction between vectors and one-forms as we consider this to be a material specification in flat, Cartesian, three-space, where such distinctions are not necessary. Writing this expression in direct notation

**r**⊗

**r**is the outer product of the position vector with itself, also referred to as a dyad formed from the position vector. We note, for later use, that the determinant can be easily calculated, as above, using an appropriately chosen rotation

### 2.2. Cylindrical cloak

*z*-axis), and one that projects onto the plane normal to the cylinder’s axis.

## 3. Hamiltonian and ray equations

18. Y. A. Kravtsov and Y. I. Orlov, *Geometrical optics of inhomogeneous media* (Springer-Verlag, Berlin, 1990). [CrossRef]

**E**

_{0}and

**H**

_{0}the same units, and

*k*

_{0}=

*ω*/

*c*making

**k**dimensionless. We use constitutive relations with dimensionless tensors

*ε*and

*µ*.

**K**[15],

*ε*and

*µ*are the same symmetric tensor, which we will call

**n**. In this case the dispersion relation has an alternate expression

**n**, or perhaps some other clever way. The latter expression is clearly fourth order in

**k**, but has only two unique solutions. Thus we discover that media with

*ε*=

*µ*is singly refracting [19], unlike for example, uniaxial dielectrics which exhibit an ordinary and extraordinary ray. This can also be seen by noting that free space is singly refracting. A coordinate transformation cannot separate two degenerate ray paths, so the degenerate ray paths of free space will remain so in the transformed coordinate space and thus also in the equivalent media.

**n**), and our Hamiltonian is

*f*(

**x**) is some arbitrary function of position. The equations of motion are [18

18. Y. A. Kravtsov and Y. I. Orlov, *Geometrical optics of inhomogeneous media* (Springer-Verlag, Berlin, 1990). [CrossRef]

*τ*parameterizes the paths. This pair of coupled, first order, ordinary differential equations can be integrated using a standard solver, such as Mathematica’s NDSolve.

## 4. Refraction

**k**

_{1}on one side of the boundary we find

**k**

_{2}on the other side as follows. The transverse component of the wave vector is conserved across the boundary.

**n**is the unit normal to the boundary. This vector equation represents just two equations. The third is obtained by requiring the wave vector to satisfy the plane wave dispersion relation of the mode represented by the Hamiltonian.

**k**

_{2}. Since

*H*is quadratic in

**k**, there will be two solutions, one that carries energy into medium 2, the desired solution, and one that carries energy out. The path of the ray,

*d*

**x**/

*dτ*, indicates the direction of energy flow, so the Hamiltonian can be used to determine which is the desired solution. The desired solution satisfies

**n**is the normal pointing into medium 2. These equations apply equally well to refraction into or out of transformation media. Refracting out into free space is much easier since the Hamiltonian of free space is just,

*H*=

**k**·

**k**-1.

## 5. Cloak Hamiltonians

### 5.1. Spherical cloak

### 5.2. Cylindrical cloak

## 6. Conclusion

## Acknowledgments

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

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

3. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science |

4. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: Negative refraction by photonic crystals,” Nature |

5. | T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz Magnetic Response from Artificial Materials,” Science |

6. | S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science |

7. | D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. |

8. | 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 (2006). In press. [CrossRef] [PubMed] |

9. | A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E |

10. | G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. London A |

11. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” (2006). http://xxx.arxiv.org/abs/cond-mat/0607418. |

12. | T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,” Appl. Phys. Lett. |

13. | E.J. Post, |

14. | J. D. Jackson, |

15. | J. A. Kong, |

16. | D. M. Shyroki, “Exact equivalent-profile formulation for bent optical waveguides,” (2006). Unpublished. |

17. | A. J. Ward and J. B. Pendry, “Refraction and geometry in maxwell’s equations,” J. Mod. Opt. |

18. | Y. A. Kravtsov and Y. I. Orlov, |

19. | H. C. Chen, |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(220.2740) Optical design and fabrication : Geometric optical design

(260.1180) Physical optics : Crystal optics

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: August 18, 2006

Revised Manuscript: September 27, 2006

Manuscript Accepted: September 29, 2006

Published: October 16, 2006

**Citation**

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express **14**, 9794-9804 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-21-9794

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777 (2006). [CrossRef] [PubMed]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788 (2004). [CrossRef] [PubMed]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604 (2003). [CrossRef] [PubMed]
- T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004). [CrossRef] [PubMed]
- S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004). [CrossRef] [PubMed]
- D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88(4), 041,109 (2006).
- 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 (2006). In press. [CrossRef] [PubMed]
- A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005). [CrossRef]
- G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. London A 462, 1364 (2006).
- U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," (2006). http://xxx.arxiv.org/abs/cond-mat/0607418.
- T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett. 88, 081,101 (2006). [CrossRef]
- E.J. Post, Formal structure of electromagnetics (Wiley, New York, 1962).
- J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
- J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley-Interscience, New York, 1990).
- D. M. Shyroki, "Exact equivalent-profile formulation for bent optical waveguides," (2006). Unpublished.
- A. J. Ward and J. B. Pendry, "Refraction and geometry in maxwell’s equations," J. Mod. Opt. 43(4), 773 - 793 (1996). [CrossRef]
- Y. A. Kravtsov and Y. I. Orlov, Geometrical optics of inhomogeneous media (Springer-Verlag, Berlin, 1990). [CrossRef]
- H. C. Chen, Theory of electromagnetic waves: A coordinate free approach, pp. 216-218 (McGraw-Hill, 1985).

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