## Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions |

Optics Express, Vol. 19, Issue 14, pp. 13020-13030 (2011)

http://dx.doi.org/10.1364/OE.19.013020

Acrobat PDF (1626 KB)

### Abstract

We propose a novel optical transformation to design homogeneous isotropic bends connecting multiple waveguides of different cross sections which can ideally tunnel the wave along any directions through multiple waveguides. First, the general expressions of homogeneous and anisotropic parameters in the bend region are derived. Second, the anisotropic material can be replaced by only two kinds of isotropic materials and they can be easily arranged in planarly stratified configuration. Finally, an arbitrary bender with homogeneous and isotropic materials is constructed, which can bend electromagnetic wave to any desired directions. To achieve the utmost aim, an advanced method is proposed to design nonmagnetic, isotropic and homogeneous bends that can bend waves along arbitrary directions. More importantly, all of the proposed bender has compact shape due to all flat boundaries, while the wave can still be perfectly tunneled without mode distortion. Numerical results validate these functionalities, which make the bend much easier in fabrication and application.

© 2011 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. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901 (2008). [CrossRef] [PubMed]

3. Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. **102**(9), 093901 (2009). [CrossRef] [PubMed]

4. H. Y. Chen, B. Hou, S. Y. Chen, X. Y. Ao, W. J. 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]

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,” Photon. Nanostruct. Fundam. Appl. **6**(1), 87–95 (2008). [CrossRef]

6. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

7. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B **77**(3), 035122 (2008). [CrossRef]

8. 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]

9. 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] [PubMed]

12. B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**(8), 085103 (2009). [CrossRef]

9. 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] [PubMed]

12. B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**(8), 085103 (2009). [CrossRef]

13. X. J. Wu, Z. F. Lin, H. Y. Chen, and C. T. Chan, “Transformation optical design of a bending waveguide by use of isotropic materials,” Appl. Opt. **48**(31), G101–G105 (2009). [CrossRef] [PubMed]

14. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. **105**(10), 104913 (2009). [CrossRef]

15. 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]

9. 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] [PubMed]

13. X. J. Wu, Z. F. Lin, H. Y. Chen, and C. T. Chan, “Transformation optical design of a bending waveguide by use of isotropic materials,” Appl. Opt. **48**(31), G101–G105 (2009). [CrossRef] [PubMed]

14. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. **105**(10), 104913 (2009). [CrossRef]

15. 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]

16. 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]

17. K. Zhang, Q. Wu, F. Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express **18**(16), 17273–17279 (2010). [CrossRef] [PubMed]

**78**(6), 066607 (2008). [CrossRef] [PubMed]

13. X. J. Wu, Z. F. Lin, H. Y. Chen, and C. T. Chan, “Transformation optical design of a bending waveguide by use of isotropic materials,” Appl. Opt. **48**(31), G101–G105 (2009). [CrossRef] [PubMed]

*N*has to be large enough to ensure the validity of the effective medium theory [18

18. 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]

**78**(6), 066607 (2008). [CrossRef] [PubMed]

**48**(31), G101–G105 (2009). [CrossRef] [PubMed]

*N*(which always requires only 2 types of isotropic materials because of the manipulated homogeneity).

**78**(6), 066607 (2008). [CrossRef] [PubMed]

14. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. **105**(10), 104913 (2009). [CrossRef]

12. B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**(8), 085103 (2009). [CrossRef]

**48**(31), G101–G105 (2009). [CrossRef] [PubMed]

16. 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]

17. K. Zhang, Q. Wu, F. Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express **18**(16), 17273–17279 (2010). [CrossRef] [PubMed]

15. 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]

## 2. Theoretical analysis

**78**(6), 066607 (2008). [CrossRef] [PubMed]

**96**(4), 041102 (2010). [CrossRef]

*BCD*and

*ABD*in virtual space (

*x*,

*y*,

*z*) are transformed into

*BC'D'*(region I) and

*ABD'*(region II) in real space (

*x'*,

*y'*,

*z'*), respectively. Then EM waves incident from port 1 are able to fluently flow out from port 2 without any reflection. Here the coordinates of points

*A, B, C, D*,

*C', D'*are constants, and can be expressed as

*AB*and

*AD*where

*BCD*to Region I (denoted by the triangle

*BC'D'*) can be expressedwhereAccording to the principles of transformation optics [1

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

*ABD'*is transformed from the triangle

*ABD*and the transformation equations can be expressed aswhereThe parameters of region II thus becomewhere

18. 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]

19. A. Novitsky, C. W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” N. J. Phys. **11**(11), 113001 (2009). [CrossRef]

*x*-axis, as shown in Fig. 1(b).

**48**(31), G101–G105 (2009). [CrossRef] [PubMed]

*N*types of isotropic materials with non-parallel interfaces (

*N*refers to the initial number of discretized anisotropic layers in Ref [9

**78**(6), 066607 (2008). [CrossRef] [PubMed]

## 3. Simulation results and discussion

*A*,

*B*,

*C*,

*D*as (0, −0.05), (0, 0.05), (0.05, 0.05), (0.05, −0.05), respectively.

*a*= 10 cm, and the cutoff frequency is 1.5 GHz. The coordinates of

*C'*,

*D'*are (0.05, −0.1) and (0.15, −0.1) respectively, and the simulation frequency is 2 GHz under transverse magnetic (TM) polarization. Figure 2 shows the magnetic field distribution of the equal vertical bender using homogeneous materials with and without anisotropy. Figure 2(a) corresponds to the bender without transformation material, and Fig. 2(b) corresponds to the bender with transformation material. When the bend region is empty, the phase distortions caused by reflections in the bend region is clearly observed (Fig. 2(a)). If the designed transformation material is placed in the bend, it can be seen that the EM waves are tunneled via the bend region without any reflections (Fig. 2(b)). Based on the effective medium theory, the bend could be realized through an alternating layered isotropic structure. Here we choose

*M*= 10 for both Region I and Region II, and the effective isotropic parameters can be found as

*z*direction. The effective isotropic parameters of the extruded bend can be found as

_{10}is incident. Obviously, perfect performance can be achieved in the extruded bend.

*a*= 10 cm and

*b*= 2 cm, and their cutoff frequency are 1.5 GHz and 7.5 GHz, respectively. The coordinates of

*C'*,

*D'*are (0.05, −0.1) and (0.07, −0.1) respectively, and the simulation frequency is 8 GHz under TM polarization. Figures 3(a) and 3(b) correspond to the bend with and without transformation materials, respectively. Clearly, if no transformation material is placed in the bend, we can observe that there is nearly no transmission from the wide waveguide to the narrow one (Fig. 3(a)). Nevertheless, based on the proposed method, two homogeneous mediums can be designed and embedded into respective regions of the bender as in Fig. 3(b). Then EM waves can be guided form one waveguide to the other without any reflection. Such bend can also be realized by a layered structure, and the effective isotropic parameters can be found

*D'*as (0.05, −0.1) and different

*C'*. Figure 4(a) corresponds to the bend connecting two equal waveguides with bending angle

*C'*as

*C'*as

*C'*as

*C'*as

*A, B, C, D, D'*are (0, −0.05), (0.1, 0.05), (0.2, 0.05), (0.2, −0.05), (0, −0.15), respectively. We set

*C'*as (0, −0.25) for equal bends and as (0, −0.17) for non-equal bends. Figures 5(a) and 5(b) demonstrate the normalized electric fields distribution of the equal bends with and without transformation materials. Clearly, there is nearly no fields transmitted to port 2 without transformation materials in Fig. 5(a), and perfect transmission can be observed in Fig. 5(b) because of the homogeneous transformation materials in bend region. Figures 5(c) and 5(d) show the normalized electric fields distribution of the non-equal bends with and without transformation materials. Comparing the severely phase distortions in Fig. 5(c) and the perfect performance in Fig. 5(d), the design method proposed in this paper is verified again. It is proved that the proposed bends could tunnel waves between two different waveguides at arbitrary directions.

*ABC*and

*BCD*into

*ABC'*and

*BC'D'*, respectively. To verify the advanced design scheme, a collar-shaped waveguide system is constructed and the simulation frequency is 8 GHz under TE polarization. Figures 6(a) and 6(c) correspond to the normalized electric field distribution and average power flow of the system without transformation materials. Clearly, the fields are completely reflected and almost no power could reach port 2. On the contrary, based on the proposed method, when homogeneous mediums are designed and embedded into the two bending regions in Figs. 6(b) and 6(d), highly efficient transmission and well shaped mode pattern are obtained. We also calculate the average power outflow at port 2 in Fig. 6(f). Obviously, the advanced design scheme is valid and perfect transmission through multiple compact bends can still be achieved.

## 4. Advanced design of a nonmagnetic, isotropic and homogeneous bend

*ABD*and

*BCD*into

*ABD'*(region II) and

*BC'D'*(region I), respectively. The constitutive parameters can be obtained according to Eq. (2) and Eq. (4). Since

*AB*=

*BC*=

*BC'*, if we select

*BD'*=

*BC*/2, the constitutive parameters of region I and II are nonmagnetic due to the area of transformed region keeps unchanged, namely

*a*= 10 cm. The frequency is 2 GHz under TM polarization, and the wave is incident from port 1. Figure 7(b) corresponds to the nonmagnetic bender without transformation media (empty bend), in which strong reflection and severe mode distortion will be present. Figure 7(c) shows the magnetic field distribution of the nonmagnetic bender with transformation media of

*θ*approaches

*n*degree bend is composed of two

*n*/2 degree bends. Here we design a

**78**(6), 066607 (2008). [CrossRef] [PubMed]

**96**(4), 041102 (2010). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

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

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

3. | Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. |

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

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,” Photon. Nanostruct. Fundam. Appl. |

6. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

7. | M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B |

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

9. | 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. |

10. | J. T. 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 bend waveguides,” J. Appl. Phys. |

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

12. | B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B |

13. | X. J. Wu, Z. F. Lin, H. Y. Chen, and C. T. Chan, “Transformation optical design of a bending waveguide by use of isotropic materials,” Appl. Opt. |

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

15. | 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. |

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

17. | K. Zhang, Q. Wu, F. Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express |

18. | 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. |

19. | A. Novitsky, C. W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” N. J. Phys. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 9, 2010

Revised Manuscript: February 26, 2011

Manuscript Accepted: April 14, 2011

Published: June 22, 2011

**Citation**

Tiancheng Han, Cheng-Wei Qiu, Jian-Wen Dong, Xiaohong Tang, and Said Zouhdi, "Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions," Opt. Express **19**, 13020-13030 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13020

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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]
- J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
- Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]
- H. Y. Chen, B. Hou, S. Y. Chen, X. Y. Ao, W. J. 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]
- 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,” Photon. Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (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]
- 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] [PubMed]
- J. T. 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 bend waveguides,” 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]
- B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 085103 (2009). [CrossRef]
- X. J. Wu, Z. F. Lin, H. Y. Chen, and C. T. Chan, “Transformation optical design of a bending waveguide by use of isotropic materials,” Appl. Opt. 48(31), G101–G105 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- K. Zhang, Q. Wu, F. Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express 18(16), 17273–17279 (2010). [CrossRef] [PubMed]
- 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]
- A. Novitsky, C. W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” N. J. Phys. 11(11), 113001 (2009). [CrossRef]

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