## Numerical determination of frequency behavior in cloaking structures based on L-C distributed networks with TLM method

Optics Express, Vol. 16, Issue 13, pp. 9344-9350 (2008)

http://dx.doi.org/10.1364/OE.16.009344

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### Abstract

The increasing interest in metamaterials with negative refractive index has been prompted by a variety of promising optical and microwave applications. Often, the resulting electromagnetic problems to be solve are not analytically derivable; therefore, numerical modeling must be employed and the Transmission Line Modeling (TLM) method constitutes a possible choice. After having greatly simplified the existing TLM techniques for the modeling of metamaterials, we propose in this paper to carry out a frequency study of cloaking structure.

© 2008 Optical Society of America

## 1. Introduction

1. P. B. Johns and R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. Inst. Elec. Eng. **118**, 1203–1208 (1971). [CrossRef]

2. C. Christopoulos, *The Transmission-Line Modeling Method, The Institute of Electrical and Electronic Engineers* (New York and Oxford University Press, Oxford, 1995). [CrossRef]

3. C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, “Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method,” J. Appl. Phys. **102**, 064101 (2007). [CrossRef]

5. C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, “Time domain simulation of electromagnetic cloaking structure with TLM method,” Opt. Express **16**, 6461–6470 (2008). [CrossRef] [PubMed]

*et al.*[9

9. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express **15**, 11133 (2007). [CrossRef] [PubMed]

## 2. TLM modeling of metamaterials

### 2.1 Series node for TM modes

*x*-

*y*plane, the only non-zero field components for a TM mode are

*H*,

_{z}*E*, and

_{x}*E*. This polarization may be simulated by a TLM series node formed with seven transmission lines [10

_{y}10. J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, “On the circuit description of TLM nodes,” Int. J. Electron. **93**, 479–491 (2006). [CrossRef]

*Z*

_{0}=1/

*Y*

_{0}. Line 5, with impedance

*Z*

_{z}*Z*

_{0}, adds extra inductance (shown to be

*L*=

_{z}*Z*

_{z}*Z*

_{0}Δ

*t*/2) to the node, allowing an independent control of the relative permeability

*μ*; while lines 6 and 7, with admittance

_{z}*Y*

_{x}*Y*

_{0}and

*Y*

_{y}*Y*

_{0}respectively, are stubs which add extra capacitance (shown to be

*C*=

_{x}*Y*

_{x}*Y*

_{0}Δ

*t*/2 and

*C*=

_{y}*Y*

_{y}*Y*

_{0}Δ

*t*/2 respectively) to the node, allowing an independent control of the relative permittivity

*ε*and

_{x}*ε*respectively, Δ

_{y}*t*being the TLM time-step.

*H*component, and two parallel circuits, each one constituted by three transmission lines, for the

_{z}*E*and

_{x}*E*components. Each sub-circuit is described by an equation providing

_{y}*Y*,

_{x}*Y*, and

_{y}*Z*:

_{z}*Z*

_{0}is chosen to make

*Z*of Eq. (1) equal to zero in free space (with impedance

_{z}*η*

_{0}) in the case

*Δx*=

*Δy*=

*Δz*. This yields

*Z*

_{0}=

*η*

_{0}/√2.

*is obtained by following the procedure described in [11*

**S**11. J. A. Portí, J. A. Morente, and M. C. Carrión, “Simple derivation of scattering matrix for TLM nodes,” Electron. Lett. **34**, 1763–1764 (1998). [CrossRef]

2. C. Christopoulos, *The Transmission-Line Modeling Method, The Institute of Electrical and Electronic Engineers* (New York and Oxford University Press, Oxford, 1995). [CrossRef]

### 2.2 Metamaterials modeling with a series node

*C*=

*Y*

_{z}*Y*

_{0}Δ

*t*/2, is equivalent to a frequency-dependent negative inductance,

*Y*,

_{x}*Y*, and

_{y}*Z*of Eq. (1) are now given by:

_{z}*ε*and

*μ*below unity and even the zero value, which becomes a natural value with this approach. It is worth noting that the impedance and admittances given by Eq. (1) diverge only by the factor -

*Δt*

^{2}

*ω*

^{2}/4 respectively to those of Eq. (6). The scattering matrix, as well as Eqs. (3-5), is absolutely unaltered with respect to the classical node, which renders the technique very comfortable to use.

### 2.3 Parallel node for TE modes

*x*-

*y*plane, the only non-zero field components for a TE mode are

*E*,

_{z}*H*, and

_{x}*H*. The parallel node is required for such a polarization, and it is obtained as it has been done hereinbefore for the series node. The three circuits describing each component of the EM field are depicted in Fig. 2. Similarly to Eq. (1), the corresponding equations giving

_{y}*Y*,

_{z}*Z*, and

_{x}*Z*are:

_{y}*Z*

_{0}is usually chosen to be

*Z*

_{0}=

*η*

_{0}√2, so that there is no stub if free space is modeled with

*Δx*=

*Δy*=

*Δz*.

### 2.4 Modeling of metamaterials with the 3D Symmetrical Condensed Node

## 3. Numerical results

13. Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and the essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. **92**, 131118 (2008). [CrossRef]

*et al.*[9

9. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express **15**, 11133 (2007). [CrossRef] [PubMed]

5. C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, “Time domain simulation of electromagnetic cloaking structure with TLM method,” Opt. Express **16**, 6461–6470 (2008). [CrossRef] [PubMed]

*R*

_{1}=0.1 m and

*R*

_{2}=0.2 m, respectively, and it is made up of 20 layers. The far field pattern for 2 GHz is depicted in Fig. 3; moreover, by using a simple Fourier Transform, the far field in terms of the frequency for five different angles (from 0° to 180° using 45° steps) is obtained and shown in the same Fig. 3. As expected, the cloaking shell is manifestly efficient for the 2 GHz functional frequency, but it is worth noting that a frequency band appears around this frequency for all directions with the noticeable exception of 0°. This last direction is characterized by a very narrow low radiation region, which shows that it is the most conflictive direction.

*ε*=2.10

^{5}, and relative permeability

*μ*=4.10

^{-5}. It is reasonable to think that these extreme values shield the central region. Nevertheless, the other layers are not perfectly able to steer the radiation around themselves due to both numerical discretization and to the approximation consisting on substituting the theoretical anisotropic material by isotropic layers, which produces the observable forward scattering.

*L*-

*C*network, and are thus expected to perfectly describe it, not only at the design frequency, but also for the whole frequency band.

## 4. Conclusion

*L*-

*C*network, which is well known as an actual technique for implementing the exotic material parameters associated with MM.

## Acknowledgments

## References and links

1. | P. B. Johns and R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission-line matrix,” Proc. Inst. Elec. Eng. |

2. | C. Christopoulos, |

3. | C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, “Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method,” J. Appl. Phys. |

4. | P. P. M. So, H. Du, and W. J. R. Hoefer, “Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks,” IEEE Trans. Microwave Theory Tech. |

5. | C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, “Time domain simulation of electromagnetic cloaking structure with TLM method,” Opt. Express |

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

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

8. | G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically |

9. | Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express |

10. | J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, “On the circuit description of TLM nodes,” Int. J. Electron. |

11. | J. A. Portí, J. A. Morente, and M. C. Carrión, “Simple derivation of scattering matrix for TLM nodes,” Electron. Lett. |

12. | P. B. Johns, “A symmetrical condensed node for the TLM method,” IEEE Trans. Microwave Theory Tech. |

13. | Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and the essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. |

**OCIS Codes**

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(160.3918) Materials : Metamaterials

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 5, 2008

Revised Manuscript: June 4, 2008

Manuscript Accepted: June 6, 2008

Published: June 10, 2008

**Citation**

Cèdric Blanchard, Jorge Portí, Juan-Antonio Morente, Alfonso Salinas, and Bae-Ian Wu, "Numerical determination of frequency behavior
in cloaking structures based on L-C distributed
networks with TLM method," Opt. Express **16**, 9344-9350 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9344

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### References

- P. B. Johns and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix," Proc. Inst. Elec. Eng. 118, 1203-1208 (1971). [CrossRef]
- C. Christopoulos, The Transmission-Line Modeling Method, The Institute of Electrical and Electronic Engineers (New York and Oxford University Press, Oxford, 1995). [CrossRef]
- C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007). [CrossRef]
- P. P. M. So, H. Du, and W. J. R. Hoefer, "Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks," IEEE Trans. Microwave Theory Tech. 53, 1496-1505 (2005). [CrossRef]
- C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, "Time domain simulation of electromagnetic cloaking structure with TLM method," Opt. Express 16, 6461-6470 (2008). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002). [CrossRef]
- Y. Huang, Y. Feng, and T. Jiang, "Electromagnetic cloaking by layered structure of homogeneous isotropic materials," Opt. Express 15, 11133 (2007). [CrossRef] [PubMed]
- J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006). [CrossRef]
- J. A. Portí, J. A. Morente, and M. C. Carrión, "Simple derivation of scattering matrix for TLM nodes," Electron. Lett. 34, 1763-1764 (1998). [CrossRef]
- P. B. Johns, "A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. 35, 370-377 (1987). [CrossRef]
- Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008). [CrossRef]

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