## Waveguide taper engineering using coordinate transformation technology

Optics Express, Vol. 18, Issue 2, pp. 767-772 (2010)

http://dx.doi.org/10.1364/OE.18.000767

Acrobat PDF (238 KB)

### Abstract

Spatial coordinate transformation is a suitable tool for the design of complex electromagnetic structures. In this paper, we define three spatial coordinate transformations which show the possibility of designing a taper between two different waveguides. A parametric study is presented for the three transformations and we propose achievable values of permittivity and permeability that can be obtained with existing metamaterials. The performances of such defined structures are demonstrated by finite element numerical simulations.

© 2010 OSA

## 1. Introduction

*et al*. [1

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]

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

5. 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 Maxwell’s equations,” Photonics and Nanost. Fundam. Appl. **6**(1), 87–95 (2008). [CrossRef]

6. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express **16**(15), 11555–11567 (2008). [CrossRef] [PubMed]

10. D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. **93**(25), 251111 (2008). [CrossRef]

11. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. **102**(18), 183903 (2009). [CrossRef] [PubMed]

12. Z. Wang, Y. Luo, W. Cui, W. Ma, L. Peng, J. Huangfu, H. Chen, and L. Ran, “Controlling the field distribution in waveguides with transformation optics,” Appl. Phys. Lett. **94**(23), 234101 (2009). [CrossRef]

13. F. Kong, B.-I. Wu, J. A. Kong, J. Huangfu, S. Xi, and H. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. **91**(25), 253509 (2007). [CrossRef]

14. P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. **105**(10), 104912 (2009). [CrossRef]

*et al*. [1

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

5. 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 Maxwell’s equations,” Photonics and Nanost. Fundam. Appl. **6**(1), 87–95 (2008). [CrossRef]

14. P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. **105**(10), 104912 (2009). [CrossRef]

## 2. Transformation formulations

*a*and

*b*, and the length of the taper in all three cases is taken to be

*l*. Mathematical expressions defining each formulation of the transformation approaches are given in Fig. 1.

*x*’,

*y*’ and

*z*’ are the coordinates in the transformed (new) space and

*x*,

*y*and

*z*are those in the initial space. As it can be observed from the mathematical expressions, the different formulations depend on the geometric parameters (

*a*,

*b*,

*l*).

*x*and

*y*dependent.where

*x*’,

*y*’) so as to mimic the transformed space. At this stage, we have designed a material with specific desired physical properties. This material can then be described by a permeability and permittivity tensors

*θ*

_{xy}) appears. This nondiagonal term is necessary to guide electromagnetic waves in the

*x-y*plane like it is the case for this taper.

## 3. Simulations and results

_{1}and TE

_{3}) modes of the input waveguide with the E-field directed along the

*z*-axis to verify the conservation of modes through the taper. The waveguides boundaries are assumed as Perfect Electric Conductors (PECs) and matched boundaries conditions are applied to the taper. Verifications are done in the microwave domain for a possible future physical prototyping based on the use of metamaterials [15

15. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

20. A. Sellier, S. N. Burokur, B. Kanté, and A. de Lustrac, “Negative refractive index metamaterials using only metallic cut wires,” Opt. Express **17**(8), 6301–6310 (2009). [CrossRef] [PubMed]

15. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

16. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**(25), 4773–4776 (1996). [CrossRef] [PubMed]

17. J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B **73**(4), 041101 (2006). [CrossRef]

18. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. **88**(4), 041109 (2006). [CrossRef]

19. S. N. Burokur, A. Sellier, B. Kanté, and A. de Lustrac, “Symmetry breaking in metallic cut wire pairs metamaterials for negative refractive index,” Appl. Phys. Lett. **94**(20), 201111 (2009). [CrossRef]

20. A. Sellier, S. N. Burokur, B. Kanté, and A. de Lustrac, “Negative refractive index metamaterials using only metallic cut wires,” Opt. Express **17**(8), 6301–6310 (2009). [CrossRef] [PubMed]

*l*= 5 cm allowing to generate the entire spatial dependence of the material parameters

*θ*

_{xx}(x’),

*θ*

_{zz}(x’),

*θ*

_{xy}(x’,y’) and

*θ*

_{yy}(x’,y’) as shown in Fig. 2 . These distributions are plotted from the expressions given in Table 1. Values of permittivities and permeabilities presented in Fig. 2 account for the control of the electromagnetic field in the taper and the conservation of the propagating modes from waveguide 1 to waveguide 2. Although same spatial distribution profile can be observed for the three different formulations, parameters values are completely different. For linear and parabolic transformations, values of µ

_{yy}are too high to be physically achievable with existing metamaterials. However, it is clear that the exponential transformation leads to values more easily achievable with metamaterials. Moreover, the physical realization of such a metamaterial taper will be facilitated by the slow variation of the material parameters, implying a gradual variation of the geometrical parameters of metamaterial inclusions. We shall note that the components are calculated in the Cartesian system and obey the following dispersion relation in the transverse electric (TE) mode:This equation is obtained from the propagation equation and describes the control of electromagnetic waves in the material. This relation is also important for a future reduction of parameters, which can be done by simplifying the nondiagonal parameter

*θ*

_{xy}to a closed interval near zero when choosing appropriately the length of the taper. For example, by bounding the nondiagonal term of the exponential formulation in Table 1, the condition

_{1}) excitation mode and at 30 GHz for the third (TE

_{3}) excitation mode. Concerning the non tapered junction waveguides, we can observe phase distortions caused by reflections at the junction from the bigger waveguide to the smaller one (Figs. 3(a)-(c)). These distortions become more severe at higher frequencies (30 GHz). However, simulations performed on the tapered waveguides (Figs. 3(d)-(l)) illustrate that electromagnetic waves are properly guided from one waveguide to the other without any impact on the guided mode when the transformed medium is embedded between the two waveguides. The difference in the transformation formulations indicates a change in the path of electromagnetic waves in the tapered section, highlighted by the shaded gray area in Fig. 3. Increasing the frequency improve the transmission between the two waveguides through the tapered section. This phenomenon can be observed when we compare the E-field distributions at 10 GHz and 30 GHz. At 10 GHz, a slight impedance mismatch between the taper output and the small waveguide input can be observed. This phenomenon decreases at higher frequencies, as illustrated for 30 GHz. Figure 3 shows the efficiency of the material parameters defined by coordinate transformation technology.

## 4. Conclusion

## 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. 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 |

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

5. | 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 Maxwell’s equations,” Photonics and Nanost. Fundam. Appl. |

6. | M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express |

7. | M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. |

8. | L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express |

9. | J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguide,” J. Appl. Phys. |

10. | D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. |

11. | H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. |

12. | Z. Wang, Y. Luo, W. Cui, W. Ma, L. Peng, J. Huangfu, H. Chen, and L. Ran, “Controlling the field distribution in waveguides with transformation optics,” Appl. Phys. Lett. |

13. | F. Kong, B.-I. Wu, J. A. Kong, J. Huangfu, S. Xi, and H. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. |

14. | P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. |

15. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. |

16. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

17. | J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B |

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

19. | S. N. Burokur, A. Sellier, B. Kanté, and A. de Lustrac, “Symmetry breaking in metallic cut wire pairs metamaterials for negative refractive index,” Appl. Phys. Lett. |

20. | A. Sellier, S. N. Burokur, B. Kanté, and A. de Lustrac, “Negative refractive index metamaterials using only metallic cut wires,” Opt. Express |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(260.2110) Physical optics : Electromagnetic optics

(260.2065) Physical optics : Effective medium theory

(160.3918) Materials : Metamaterials

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 27, 2009

Revised Manuscript: August 26, 2009

Manuscript Accepted: August 28, 2009

Published: January 6, 2010

**Citation**

Paul-Henri Tichit, Shah Nawaz Burokur, and André de Lustrac, "Waveguide taper engineering using coordinate transformation technology," Opt. Express **18**, 767-772 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-767

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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]
- 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. 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 Maxwell’s equations,” Photonics and Nanost. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
- M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]
- M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008). [CrossRef] [PubMed]
- 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]
- J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguide,” J. Appl. Phys. 104(1), 014502 (2008). [CrossRef]
- D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93(25), 251111 (2008). [CrossRef]
- H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009). [CrossRef] [PubMed]
- Z. Wang, Y. Luo, W. Cui, W. Ma, L. Peng, J. Huangfu, H. Chen, and L. Ran, “Controlling the field distribution in waveguides with transformation optics,” Appl. Phys. Lett. 94(23), 234101 (2009). [CrossRef]
- F. Kong, B.-I. Wu, J. A. Kong, J. Huangfu, S. Xi, and H. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. 91(25), 253509 (2007). [CrossRef]
- P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. 105(10), 104912 (2009). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]
- J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73(4), 041101 (2006). [CrossRef]
- D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]
- S. N. Burokur, A. Sellier, B. Kanté, and A. de Lustrac, “Symmetry breaking in metallic cut wire pairs metamaterials for negative refractive index,” Appl. Phys. Lett. 94(20), 201111 (2009). [CrossRef]
- A. Sellier, S. N. Burokur, B. Kanté, and A. de Lustrac, “Negative refractive index metamaterials using only metallic cut wires,” Opt. Express 17(8), 6301–6310 (2009). [CrossRef] [PubMed]

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