## Time domain simulation of electromagnetic cloaking structures with TLM method

Optics Express, Vol. 16, Issue 9, pp. 6461-6470 (2008)

http://dx.doi.org/10.1364/OE.16.006461

Acrobat PDF (739 KB)

### Abstract

The increasing interest in invisible cloaks has been prompted in part by the availability of powerful computational resources which permit numerical studies of such a phenomenon. These are usually carried out with commercial software. We report here a full time domain simulation of cloaking structures with the Transmission Line Modeling (TLM) method. We first develop a new condensed TLM node to model metamaterials in two dimensional situations; various results are then presented, with special emphasis on what is not easily achievable using commercial software.

© 2008 Optical Society of America

## 1. Introduction

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

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

*et al.*have proposed a TLM node for modeling metamaterials [9

9. 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. Microw. Theory Tech. **53**, 1496–1505 (2005). [CrossRef]

10. J. P. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM-Part 2: Materials with anisotropic properties,” IEEE Trans. Antennas Propag. **47**, 1535–1542 (1999). [CrossRef]

11. L. de Menezes and W. J. R. Hoefer, “Modeling of general constitutive relationships using SCN TLM,” IEEE Trans. Microw. Theory Tech. **44**, 854–861 (1996). [CrossRef]

*et al.*have proposed an approach that allows substituting the anisotropic medium by a concentric layered structure of alternating homogeneous isotropic materials [12

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

*ε*,

_{θ}*μ*) and radial (

_{θ}*ε*,

_{r}*μ*) components of the electromagnetic (EM) parameters can indeed be considered as the effective permittivity or permeability of a composite made of a series of parallel layers whenever the layers are thin enough compared with the wavelength. This statement is based on the effective medium theory and permits to match the angular components with the upper Wiener bound, while the radial components correspond to the lower Wiener bound [13]. We will choose this technique to carry out the cloaking simulation.

_{r}*et al.*, have recently studied the effect of different coordinate transformations [14]. They pointed out that the performance of the non-ideal cloak may be improved by using a nonlinear transformation instead of the well-established linear one.

## 2. TLM topics

### 2.2 Modeling of metamaterials with TLM

*ε*or

_{r}*μ*coexists with usual materials such as vacuum. The first attempt to model metamaterials was reported by So,

_{r}*et al.*, [9

9. 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. Microw. Theory Tech. **53**, 1496–1505 (2005). [CrossRef]

*ε*or

_{r}*μ*values below unity without requiring a decrease in the time step value.

_{r}17. J. A. Portí, J. A. Morente, A. Salinas, E. A. Navarro, and M. Rodríguez-Sola, “A generalized dynamic symmetrical condensed TLM node for the modeling of time-varying electromagnetic media,” IEEE Trans. Antennas Propag. **54**, 2–11 (2006). [CrossRef]

18. 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]

18. 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]

19. 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]

*Z*

_{0}, or admittance

*Y*

_{0}=1/

*Z*

_{0}of the main node lines are chosen to model the free space permittivity,

*ε*

_{0}, and permeability,

*μ*

_{0}. On the one hand, variations on

*ε*with respect to the vacuum value are allowed by adding capacitance to the node. This can be achieved by connecting an open extra port, i.e., a capacitive stub, with admittance

_{y}*Y*

_{y}Y_{0}. On the other hand, variations on

*μ*and

_{x}*μ*require adding inductance which can be achieved by connecting two short-circuited or inductive stubs, with normalized impedance

_{z}*Z*

_{x}Z_{0}and

*Z*

_{z}Z_{0}, respectively. The usual node finally obtained is shown in Fig. 2(a) and the expressions to determine

*Y*,

_{y}*Z*, and

_{x}*Z*are:

_{z}*x*, Δ

*y*, and Δ

*z*represent the length of the node along the

*x*-,

*y*-, and

*z*-directions respectively.

9. 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. Microw. Theory Tech. **53**, 1496–1505 (2005). [CrossRef]

*C*to the capacitance of lines 1 to 4 allows a reduction of the total

_{eq}*ε*below the vacuum value or even to achieve negative permittivity. Concretely, the equation defining

_{y}*ε*becomes

_{y}*ω*by the series connected capacitive ports, lines 6 and 7, yields

*t*

^{2}

*ω*

^{2}/ 4 compared to Eqs. (1), but connectivity of the lines in Figs. 2 are identical, i.e., the scattering matrix, is unaltered with respect to the classical node. Relative permittivity or permeability less than 1 are now easily reachable for a given frequency and, in particular, the zero value which becomes a natural value with this approach without requiring a zero value of Δ

*t*.

*S*remaining absolutely unchanged. Concretely, if the EM parameters are greater than 1 in a certain region of the space, Eqs. (1) have to be chosen while Eqs. (4) have to be used if the EM relative parameters take values below unity.

## 3. Cloaking structures modeling

*R*

_{1}. The incident wave is a 2 GHz TE

_{y}polarized plane wave with electric field parallel to the axis of the cylinder. The incident wave propagation vector,

*, is*

**k***x*-oriented while the magnetic field is

*z*-oriented.

*R*

_{2}. According to the coordinate transformation, the anisotropic relative permittivity and permeability of the cloaking material in cylindrical coordinates have the following radius dependence:

*ε*,

_{y}*μ*, and

_{r}*μ*.

_{θ}12. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express **15**, 11133 (2007). [CrossRef] [PubMed]

*μ*, and

_{A}*μ*can be obtained from Wiener formula:

_{B}*x*- and

*z*-directions is used for the modeling. The node size is Δ

*x*=Δ

*z*=5.10

^{-4}m and the maximum allowable time step for these lengths, Δ

*t*=1.1785ps, is used. Typically, the boundary conditions used in cloaking studies are chosen to be absorbent along the direction of propagation and Perfectly Magnetic Conducting (PMC) for vertical directions, i.e., along the

*z*-direction in this work, which is equivalent to artificially simulate a periodic structure of cylinders along the

*z*-direction. However, the cloaking we are numerically considering is not ideal, and so a certain scattering is expected at each cylinder. In this manner, if we want to characterize the cloaking performance of the coating for a given cylinder, the existence of neighbor cylinders in the periodic structure would artificially magnify the non-ideality of the results. In other words, we are interested in the study of the capability of a single coating structure to make a certain region invisible, and deviations of this ideal goal are better identified if only one coating element are considered. In this sense, the PMC boundaries are not a satisfactory choice but, on the other hand, they cannot be directly substituted by absorbent boundaries because the plane wave condition corresponding to excitation would be broken. The problem is worked out by employing the Huygens surface technique which consists of dividing the mesh into an inner total field (incident and scattered) region and an outer scattered field region. The source conditions for the plane wave excitation are enforced on the interface separating these regions, which presents two benefits: first, absorbing boundary conditions can be directly applied to the limits of the mesh, since excitation has been removed at these points; and second, an illustrative map of the scattered field is directly obtained at the outer region. Finally, the far field is obtained in all cases using the transformation proposed in [20

20. R. Luebbers, D. Ryan, and J. Beggs, “A two-dimensional time-domain near-zone to Far-zone transformation,” IEEE Trans. Antennas Propag. **40**, 848–851 (1992). [CrossRef]

*R*

_{1}=0.1 m while

*R*

_{2}=0.2 m, so that the thin layer condition is satisfied [12

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

*ε*,

_{y}*μ*, and

_{r}*μ*are calculated at the center of the isotropic layers (i.e., on the interface separating layer A and layer B), the corresponding far field pattern is depicted in Fig. 3(a). For comparison, this field has been normalized by the maximum value of the field when no cloaking structure is present. This reference far field pattern, which is checked to perfectly coincide with the theoretical one [21], can be observed in the same figure. A significant reduction of the scattering is noticed in almost all directions at the notable exception of the forward scattering. It is known that the forward scattering of a PEC cylinder is more intense for TE mode compared to the TM mode case [21], so we are dealing with the worst case polarization. By judiciously adjusting the simulation parameters, we will show below that this scattering reduction can be enhanced.

_{θ}*ε*,

_{y}*μ*, and

_{r}*μ*at the center of the isotropic layers, we now calculate them at the boundary between each layer. However, this process gives an infinite value of

_{θ}*μ*for the first layer (i.e. the layer directly adjacent to the inner PEC cylinder) and, therefore, the electromagnetic parameters will not be calculated exactly at the inner boundary of the layers but at a distance 10

_{θ}^{-4}

*δ*of the later, where

*δ*is the thickness of one layer. The far field pattern is displayed in Fig. 3(a) and, compared to the previous case, showing an even stronger scattering reduction. This reduction now occurs for all the direction and the forward scattering has been in particular reduced for 4.3 dB, while the backward scattering reduces for 15.1 dB. This confirms that the cloaking performance is highly sensitive to the parameters’ value of the inner layer; indeed, it is the most affected layer by this change. The simulated electric field is plotted in Fig. 3(b). It is straightaway clear that, once in the cloaking shell, the wave front is deviated and conducted around the cloaked cylinder, while outside the object under consideration it emerges almost unperturbed. Moreover, the Huygens technique allows visualizing in the same picture the scattered field at the outer almost green square region, with the incident field directly filtered out, that ostensibly takes the form of concentric waves. As it has been discussed in [22

22. B. Zhang, H. Chen, B-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B **76**, 121101 (2007). [CrossRef]

*r*=

*R*

_{1}is enforced by Eqs. (5) to be

*ε*→0,

_{y}*μ*→0, and

_{r}*μ*→

_{θ}*∞*. These extreme values induce a magnetic surface current exactly located on the inner boundary of the cloak once illuminated by the incident wave. Such a current shields the cloaked region and then avoids the field to enter this area. On the other hand, the inner layer

*μ*is not infinite in our simulation, so this surface current does not exist here and, then, scattering occurs on the interface of the PEC cylinder, making the cloaking structure of Fig. 3(b) imperfect.

_{θ}## 4. Effect of using reduced parameters on the quality of the cloak

4. 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]

5. S. A. Cummer, B-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E **74**, 036621 (2006). [CrossRef]

*et al.*, have showed that using reduced parameters experiments more scattering than intended and is more than just nonzero reflectance [23

23. M. Yan, Z. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. **99**, 233901 (2007). [CrossRef]

*μ*=1, while the other one to

_{θ}*ε*=1 (the choice

_{y}*μ*is not allowed since the corresponding

_{r}*ε*would not be spatially uniform), both with parameters calculated at the center of the layers. As expected, the performance of the cloaking is deteriorated; however, it is worth noting that in both cases the forward scattering is now reduced especially for the choice

_{y}*ε*=1 (for 7dB). Finally, it is worth noting from Fig. 4(a) that the greater the reduction in the forward direction, the lesser the reduction in the backward direction, which seems to be associated to energy conservation.

_{y}## 5. Effect of non-linear transformation on the quality of the cloak

*α*that changes the electromagnetic parameters of Eqs. (5a), (5b) and (5c) ([14]. The far field pattern, for

*α*=1 (corresponding to a linear transformation),

*α*=0.8, and

*α*=0.5, is depicted in Fig. 4(b) for the full parameters of Eqs. (5a), (5b) and (5c) calculated at the center of the layers. It can be observed that the choice

*α*=0.8 globally improves the cloaking performance, particularly for angle contained between 90° and 270° in the backward direction. Concerning the

*α*=0.5 transformation, it appears that the result is not convincing although it offers some improvement for some directions.

## 6. Frequency dispersion

*et al.*, have presented experimental results demonstrating that a cloaking structure can be constructed in the microwave regime from a metamaterial consisting of split-ring resonators [4

4. 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]

*λ*,

*σ*/

_{2D}*λ*, for the cloaking device compared to that of the simple conducting cylinder. As expected, 2 GHz is the frequency for which the effectiveness of the cloaking material is optimum. However, the resonant nature of dual L-C networks causes that regions of reduced and magnified radiation alternate on both sides of the 2 GHz design frequency.

## 7. Conclusion

## Acknowledgments

## References and links

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

2. | H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. |

3. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

4. | 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 |

5. | S. A. Cummer, B-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E |

6. | C. Christopoulos, |

7. | 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. |

8. | J. A. Portí and J. A. Morente, “A three-dimensional symmetrical condensed TLM node for acoustics,” J. Sound Vibr. |

9. | 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. Microw. Theory Tech. |

10. | J. P. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM-Part 2: Materials with anisotropic properties,” IEEE Trans. Antennas Propag. |

11. | L. de Menezes and W. J. R. Hoefer, “Modeling of general constitutive relationships using SCN TLM,” IEEE Trans. Microw. Theory Tech. |

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

13. | O. Wiener, “Zur theorie der refraktionskonstanten,” Berichteüber die Verhandlungen der Königlich-Sächsischen Gesellsschaft der Wissenschaften zu Leipzig, 256–277 (1910). |

14. | S. Xi, H. Chen, B-I. Wu, B. Zhang, Y. Luo, J. Huangfu, D. Wang, and J. A. Kong, “Effects of different transformations on the performance of a nonideal cylindrical cloak,” Submitted to PIERS. |

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

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

17. | J. A. Portí, J. A. Morente, A. Salinas, E. A. Navarro, and M. Rodríguez-Sola, “A generalized dynamic symmetrical condensed TLM node for the modeling of time-varying electromagnetic media,” IEEE Trans. Antennas Propag. |

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

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

20. | R. Luebbers, D. Ryan, and J. Beggs, “A two-dimensional time-domain near-zone to Far-zone transformation,” IEEE Trans. Antennas Propag. |

21. | C. Balanis, |

22. | B. Zhang, H. Chen, B-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B |

23. | M. Yan, Z. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. 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: March 13, 2008

Revised Manuscript: April 17, 2008

Manuscript Accepted: April 18, 2008

Published: April 22, 2008

**Citation**

Cedric Blanchard, Jorge A. Portí, Bae-Ian Wu, Juan A. Morente, Alfonso Salinas, and Jin Au Kong, "Time domain simulation of electromagnetic
cloaking structures with TLM method," Opt. Express **16**, 6461-6470 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6461

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007). [CrossRef] [PubMed]
- F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007). [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, 977-980 (2006). [CrossRef] [PubMed]
- S. A. Cummer, B-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006). [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]
- J. A. Portí and J. A. Morente, "A three-dimensional symmetrical condensed TLM node for acoustics," J. Sound Vibr. 241, 207-222 (2001). [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]
- J. P. Paul, C. Christopoulos, and D. W. P. Thomas, "Generalized material models in TLM-Part 2: Materials with anisotropic properties," IEEE Trans. Antennas Propag. 47, 1535-1542 (1999). [CrossRef]
- L. de Menezes and W. J. R. Hoefer, "Modeling of general constitutive relationships using SCN TLM," IEEE Trans. Microwave Theory Tech. 44, 854-861 (1996). [CrossRef]
- Y. Huang, Y. Feng, and T. Jiang, "Electromagnetic cloaking by layered structure of homogeneous isotropic materials," Opt. Express 15, 11133 (2007). [CrossRef] [PubMed]
- O. Wiener, "Zur theorie der refraktionskonstanten," Berichteüber die Verhandlungen der Königlich-Sächsischen Gesellsschaft der Wissenschaften zu Leipzig, 256-277 (1910).
- S. Xi, H. Chen, B-I. Wu, B. Zhang, Y. Luo, J. Huangfu, D. Wang, J. A. Kong, "Effects of different transformations on the performance of a nonideal cylindrical cloak," Submitted to PIERS.
- 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 (2007). [CrossRef]
- P. B. Johns, "A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. 35, 370-377 (1987). [CrossRef]
- J. A. Portí, J. A. Morente, A. Salinas, E. A. Navarro, M. Rodríguez-Sola, "A generalized dynamic symmetrical condensed TLM node for the modeling of time-varying electromagnetic media," IEEE Trans. Antennas Propag. 54, 2-11 (2006). [CrossRef]
- J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, 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]
- R. Luebbers, D. Ryan, and J. Beggs, "A two-dimensional time-domain near-zone to Far-zone transformation," IEEE Trans. Antennas Propag. 40, 848-851 (1992). [CrossRef]
- C. Balanis, Advanced Enginnering Electromagnetics, John Wiley & Sons (1989).
- B. Zhang, H. Chen, B-I. Wu, Y. Luo, L. Ran, J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007). [CrossRef]
- M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233901 (2007). [CrossRef]

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