## Arbitrary waveguide connector based on embedded optical transformation |

Optics Express, Vol. 18, Issue 16, pp. 17273-17279 (2010)

http://dx.doi.org/10.1364/OE.18.017273

Acrobat PDF (851 KB)

### Abstract

Arbitrary connector for waveguides of different cross sections is proposed and designed theoretically based on the embedded optical transformation theory. First, the general expressions of constitutive tensors of the metamaterials filled in the connector are derived. Second, there are some full-wave simulations that validate the constitutive tensors derived. The results show that the connector with metamaterials inclusions with designed constitutive parameters can fulfill the reflectionless transmission of electromagnetic waves between waveguides of different cross sections. Finally, connectors of several forms are investigated parametrically, and two sets of constitutive tensors that can be physically achieved by existing metamaterials are gotten. It is believed that this study provides a feasible way to fulfill the efficient transmission of electromagnetic waves between waveguides of different cross sections.

© 2010 OSA

## 1. Introduction

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

2. 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]

3. H. S. Chen, B. I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. **99**(6), 063903 (2007). [CrossRef] [PubMed]

4. B. L. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B **76**(12), 121101 (2007). [CrossRef]

5. W. Yan, M. Yan, and M. Qiu, “Non-magnetic simplified cylindrical cloak with suppressed zeroth order scattering,” Appl. Phys. Lett. **93**(2), 021909 (2008). [CrossRef]

6. H. Ma, S. B. Qu, Z. Xu, and J. F. Wang, “The open cloak,” Appl. Phys. Lett. **94**(10), 103501 (2009). [CrossRef]

7. C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express **16**(23), 19366–19374 (2008). [CrossRef]

8. C. W. Qiu, L. Hu, X. F. Xu, and Y. J. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **79**(4), 047602 (2009). [CrossRef] [PubMed]

10. B. Ivsic, Z. Sipus, and S. Hrabar, “Analysis of Uniaxial Multilayer Cylinders Used for Invisible Cloak Realization,” IEEE Trans. Antenn. Propag. **57**(5), 1521–1527 (2009). [CrossRef]

11. F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. **103**(7), 073901 (2009). [CrossRef] [PubMed]

12. J. S. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901 (2008). [CrossRef] [PubMed]

14. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**(5912), 366–369 (2009). [CrossRef] [PubMed]

15. Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D Appl. Phys. **42**(3), 035408 (2009). [CrossRef]

16. X. D. Luo, T. Yang, Y. W. Gu, H. Y. Chen, and H. R. Ma, “Conceal an entrance by means of superscatterer,” Appl. Phys. Lett. **94**(22), 223513 (2009). [CrossRef]

17. T. R. Zhai, Y. Zhou, J. Zhou, and D. H. Liu, “Polarization controller based on embedded optical transformation,” Opt. Express **17**(20), 17206–17213 (2009). [CrossRef] [PubMed]

18. 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]

19. D. H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express **17**(10), 7807–7817 (2009). [CrossRef] [PubMed]

20. W. X. Jiang, T. J. Cui, X. Y. Zhou, X. M. Yang, and Q. Cheng, “Arbitrary bending of electromagnetic waves using realizable inhomogeneous and anisotropic materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(6), 066607 (2008). [CrossRef]

25. 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]

26. P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express **18**(2), 767–772 (2010). [CrossRef] [PubMed]

## 2. Derivation of the connector’s constitutive tensors based on embedded optical transformation

*y*

_{1}(

*x*), and the curve that connects A and B’ is defined as

*y*

_{2}(

*x*). The functions of the two curves can be selected arbitrarily, so long as they can satisfy the numerical values at the points of A, B’ and C, D’, respectively. Hence the Jacobian transformation matrix can be gotten based on Eq. (1):which represents the derivative of the transformed coordinates with respect to the original coordinates. Using the property that Maxwell’s equations are form invariant in the original space and transformed space, the permittivity and permeability tensors of the medium in the transformed space can be expressed as: where

## 3. Full-wave simulations and parametric investigations

_{10}mode of 2GHz can be transmitted. The simulation domain is just like Fig. 1(a), and the port 1 is selected to illuminate the incident wave. Here it should be noticed that the simulations are done in the transformed space, but the constitutive parameters in Eq. (5a)–(5c) are expressed as the function of

*x*and

*y*, which are variables of the original space. Here the variables of the original space should be replaced by the variables of transformed space, which can be gotten through Eq. (1):

*ε*

_{xy}owes a part of negative values, no matter what kinds of boundary function

*y*

_{2}(

*x*’) and

*y*

_{1}(

*x*’) are selected. This also can be explained through Eq. (5b). When the connector is symmetrical, the non-diagonal component of the constitutive tensor can be simplified as:

*y*

_{2}(

*x*’) must be monotone decreasing at some numerical range of

*x*’, this results in the negative value of

*ε*

_{xy}. For the unsymmetrical structures, the function

*y*

_{2}(

*x*’) and

*y*

_{1}(

*x*’) can be selected properly, and all the components of the constitutive tensor can be kept positive, as shown in Fig. 3(j)–3(o). Although one or two components maybe less than 1, these two sets of constitutive tensors are easy to be physically achieved by existing 2D metamaterials.

*y*

_{2}(

*x*’) and

*y*

_{1}(

*x*’) are selected as lines with the same slope of

*k*

_{0}. The simplified constitutive tensor can be expressed as:

*k*

_{0}≥1.

## 4. Conclusion

## Acknowledgement

## References and links

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

2. | 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 |

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

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

5. | W. Yan, M. Yan, and M. Qiu, “Non-magnetic simplified cylindrical cloak with suppressed zeroth order scattering,” Appl. Phys. Lett. |

6. | H. Ma, S. B. Qu, Z. Xu, and J. F. Wang, “The open cloak,” Appl. Phys. Lett. |

7. | C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express |

8. | C. W. Qiu, L. Hu, X. F. Xu, and Y. J. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

9. | B. I. Popa and S. A. Cummer, “Cloaking with optimized homogeneous anisotropic layers,” Phys. Rev. A |

10. | B. Ivsic, Z. Sipus, and S. Hrabar, “Analysis of Uniaxial Multilayer Cylinders Used for Invisible Cloak Realization,” IEEE Trans. Antenn. Propag. |

11. | F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. |

12. | J. S. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. |

13. | X. F. Xu, Y. J. Feng, Y. Hao, J. M. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett. |

14. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science |

15. | Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D Appl. Phys. |

16. | X. D. Luo, T. Yang, Y. W. Gu, H. Y. Chen, and H. R. Ma, “Conceal an entrance by means of superscatterer,” Appl. Phys. Lett. |

17. | T. R. Zhai, Y. Zhou, J. Zhou, and D. H. Liu, “Polarization controller based on embedded optical transformation,” Opt. Express |

18. | 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. |

19. | D. H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express |

20. | W. X. Jiang, T. J. Cui, X. Y. Zhou, X. M. Yang, and Q. Cheng, “Arbitrary bending of electromagnetic waves using realizable inhomogeneous and anisotropic materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

21. | Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. |

22. | W. Q. Ding, D. H. Tang, Y. Liu, L. X. Chen, and X. D. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. |

23. | Y. G. Ma, N. Wang, and C. K. Ong, “Application of inverse, strict conformal transformation to design waveguide devices,” J. Opt. Soc. Am. A |

24. | X. F. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express |

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

26. | P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 15, 2010

Revised Manuscript: July 19, 2010

Manuscript Accepted: July 19, 2010

Published: July 29, 2010

**Citation**

Kuang Zhang, Qun Wu, Fan-Yi Meng, and Le-Wei Li, "Arbitrary waveguide connector based on embedded optical transformation," Opt. Express **18**, 17273-17279 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17273

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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]
- 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]
- H. S. Chen, B. I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 (2007). [CrossRef] [PubMed]
- B. L. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76(12), 121101 (2007). [CrossRef]
- W. Yan, M. Yan, and M. Qiu, “Non-magnetic simplified cylindrical cloak with suppressed zeroth order scattering,” Appl. Phys. Lett. 93(2), 021909 (2008). [CrossRef]
- H. Ma, S. B. Qu, Z. Xu, and J. F. Wang, “The open cloak,” Appl. Phys. Lett. 94(10), 103501 (2009). [CrossRef]
- C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express 16(23), 19366–19374 (2008). [CrossRef]
- C. W. Qiu, L. Hu, X. F. Xu, and Y. J. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 047602 (2009). [CrossRef] [PubMed]
- B. I. Popa and S. A. Cummer, “Cloaking with optimized homogeneous anisotropic layers,” Phys. Rev. A 79(2), 023806 (2009). [CrossRef]
- B. Ivsic, Z. Sipus, and S. Hrabar, “Analysis of Uniaxial Multilayer Cylinders Used for Invisible Cloak Realization,” IEEE Trans. Antenn. Propag. 57(5), 1521–1527 (2009). [CrossRef]
- F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. 103(7), 073901 (2009). [CrossRef] [PubMed]
- J. S. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
- X. F. Xu, Y. J. Feng, Y. Hao, J. M. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett. 95(18), 184102 (2009). [CrossRef]
- R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
- Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D Appl. Phys. 42(3), 035408 (2009). [CrossRef]
- X. D. Luo, T. Yang, Y. W. Gu, H. Y. Chen, and H. R. Ma, “Conceal an entrance by means of superscatterer,” Appl. Phys. Lett. 94(22), 223513 (2009). [CrossRef]
- T. R. Zhai, Y. Zhou, J. Zhou, and D. H. Liu, “Polarization controller based on embedded optical transformation,” Opt. Express 17(20), 17206–17213 (2009). [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]
- D. H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express 17(10), 7807–7817 (2009). [CrossRef] [PubMed]
- W. X. Jiang, T. J. Cui, X. Y. Zhou, X. M. Yang, and Q. Cheng, “Arbitrary bending of electromagnetic waves using realizable inhomogeneous and anisotropic materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066607 (2008). [CrossRef]
- Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009). [CrossRef]
- W. Q. Ding, D. H. Tang, Y. Liu, L. X. Chen, and X. D. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96(4), 041102 (2010). [CrossRef]
- Y. G. Ma, N. Wang, and C. K. Ong, “Application of inverse, strict conformal transformation to design waveguide devices,” J. Opt. Soc. Am. A 27(5), 968–972 (2010). [CrossRef]
- X. F. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express 18(10), 10168–10176 (2010). [CrossRef] [PubMed]
- 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]
- P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express 18(2), 767–772 (2010). [CrossRef] [PubMed]

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