## Design of electromagnetic refractor and phase transformer using coordinate transformation theory

Optics Express, Vol. 16, Issue 10, pp. 6815-6821 (2008)

http://dx.doi.org/10.1364/OE.16.006815

Acrobat PDF (338 KB)

### Abstract

We designed an electromagnetic refractor and a phase transformer using form-invariant coordinate transformation of Maxwell’s equations. The propagation direction of electromagnetic energy in these devices can be modulated as desired. Unlike the conventional dielectric refractor, electromagnetic fields at our refraction boundary do not conform to the Snell’s law in isotropic materials and the impedance at this boundary is matched which makes the reflection extremely low; and the transformation of the wave front from cylindrical to plane can be realized in the phase transformer with a slab structure. Two dimensional finite-element simulations were performed to confirm the theoretical results.

© 2008 Optical Society of America

## 1. Introduction

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

2. 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) [CrossRef] [PubMed]

8. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant coordinate Transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

9. Huanyang. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. **90**, 241105 (2007). [CrossRef]

5. 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]

7. W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photo. **1**, 224–227 (2007). [CrossRef]

## 2. Principle

### 2.1 Refractor

2. 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) [CrossRef] [PubMed]

*x*denotes

^{i}*x*,

*y*,

*z*in original coordinate system,

*x*′ corresponds to

^{i}*x*′,

*y*′,

*z*′ in new coordinate system.

*μ*=

^{i′j′}*ε*and

^{i′j′}*μ*=

*ε*. The original isotropic material changes to be anisotropy.

### 2.2 Phase transformer

*r*

_{2}so that the phase at the output boundary would be the same; width W must be larger than half of the height H for stability.

## 3. Simulation and discussion

### 3.1 Refractor

*θ*to be 45° for example. Four different incident conditions are presented in Fig. 3 performed by the finite-element method. A transverse-electric (TE) plane-wave is used, and the wavelength is specified to be 365nm, which is in the ultraviolet frequency range. The black line demonstrates the direction of the power flow. It is noted that the direction of Poynting vector deviates from the wave vector

*k*in the metamaterial, in other words, the power flow direction is not normal to the wave front in this anisotropic structure.

*θ*

_{1}and refraction angle

*θ*

_{2}can be expressed as follows:

*Δθ*is the designed deflection angle.

*n*=1.6,

_{1}*n*=1.4, and the same

_{2}*n*with

_{2}*n*=1 for the second case (blue line). The third case (red line) is our designed refractor.

_{1}*x*axis. The area for both refractors is 6µm wide and 3µm high. The distribution of Ez is showed in Fig. 5(a); the corresponding time-averaged energy density is presented in Fig. 5(b). Electromagnetic energy propagate straightly in region I and IV but turn left significantly in region II and turn right in region III because of the anisotropic material.

### 3.2 Phase transformer

*x*axis is used for the calculation. The area is 5µm wide and 3.5µm high enclosed by PML. Figure 6(a) demonstrates the distribution of

*z*component of the electric field with the phase transformer in region II and Fig. 6(b) represents the distribution of Ez in free space. Figure 6(c) is the z component of electric field at

*x*=4µm derived from Fig. 6(a) and Fig. 6(b), where the distribution of Ez is more uniform after the phase transformer than in free space.

*y*axis after the phase transformer in Fig. 5(a). The scattering at the boundary between region II and III which is part of our current research cause the unsmooth of Ez in region II and III.

## 5. Conclusion

## Acknowledgments

## References and links

1. | A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. |

2. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

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

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

5. | 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 |

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

7. | W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photo. |

8. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant coordinate Transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007). |

9. | Huanyang. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 4, 2008

Revised Manuscript: April 10, 2008

Manuscript Accepted: April 10, 2008

Published: April 28, 2008

**Citation**

Lan Lin, Wei Wang, Jianhua Cui, Chunlei Du, and Xiangang Luo, "Design of electromagnetic refractor and phase transformer using coordinate transformation theory," Opt. Express **16**, 6815-6821 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6815

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### References

- A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell�??s equation," J. Mod. Opt. 43, 773-793 (1996). [CrossRef]
- 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). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, "Full-wave simulations of electromagnetic cloaking structures," Phy. Rev. E 74,036621 (2006). [CrossRef]
- 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]
- F. Zolla, A. Nicolet, and J. B. Pendry, "Electromagnetic fields analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photo. 1, 224-227 (2007). [CrossRef]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, " Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant coordinate Transformations of Maxwell's Equations," http://www.arxiv.org:physics/0706.2452v1 (2007).
- H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]

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