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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 14 — Jul. 4, 2011
  • pp: 13020–13030
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Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions

Tiancheng Han, Cheng-Wei Qiu, Jian-Wen Dong, Xiaohong Tang, and Said Zouhdi  »View Author Affiliations


Optics Express, Vol. 19, Issue 14, pp. 13020-13030 (2011)
http://dx.doi.org/10.1364/OE.19.013020


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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

It is well understood that, if one wants to make use of the isotropic realization (either those method in Refs [9

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

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]

] or our proposed method), The initial discretization layer number 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]

]. Thus, the required types of different isotropic materials will be linearly increased if one adopts the methods in Refs [9

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

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]

], but our method is independent of the initial number N (which always requires only 2 types of isotropic materials because of the manipulated homogeneity).

2. Theoretical analysis

Region II denoted by triangle ABD' is transformed from the triangle ABD and the transformation equations can be expressed as
x=a21x+b21y+c21,y=a22x+b22y+c22,z=z
(3)
where
[a21b21c21]=[xAyA1xByB1xDyD1]1[xAxBxD']    and    [a22b22c22]=[xAyA1xByB1xDyD1]1[yAyByD'].
The parameters of region II thus become
εII=μII=Λ2·Λ2T/det(Λ2)
(4)
where Λ2=[a21,b21,0;a22,b22,0;0,0,1] and det(Λ2)=a21b22a22b21.

It is interesting to note that the constitutive parameters are spatially invariant from Eq. (2) and Eq. (4). This is because no other variables are involved in the material parameters tensors. The homogeneity has been achieved, and thus we focus on achieving isotropy. Because of the symmetry of the tensors εI and εII, we can find a rotation transformation mapping such symmetric tensors into diagonal tensors, from which the effective isotropic media can be derived [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]

,19

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]

]. The permittivity tensor for Region I and Region II in the bend can be expressed as
εi=diag[ζix,ζiy,   ζiz]
(5)
where ζix,y=[εixx+εiyy±(εixxεiyy)2+(2εixy)2]/2, and ζiz=εizz (i=I, or II).

From the rotation transformation, we can derive a relation between rotation angle and original permittivity as
tan(2θi)=2εixy/(εixxεiyy)
(6)
From Eq. (6), the rotation angle can be uniquely determined, which is the same as the angle between the layer and x-axis, as shown in Fig. 1(b).

As shown in Fig. 1(b), we can use the layered structure to fill the bending region and finally an isotropic bend is obtained. Since the ideal anisotropic material is homogeneous in both regions, each region is only composed of 2 types of isotropic materials in a planarly layered pattern: medium-A and medium-B, while Ref [13

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]

] needs 3N types of isotropic materials with non-parallel interfaces (N refers to the initial number of discretized anisotropic layers in Ref [9

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]

]). In contrast, the isotropic materials in our design only need 2 types: medium-A and medium-B.

εiA,   B=ζix±(ζix)2ζixζiy
(7)

3. Simulation results and discussion

In order to demonstrate the feasibility of the waveguide bends proposed in this paper, full-wave simulations are performed to verify the functionality. The waveguides boundaries are assumed to be perfectly conducting (PEC). First, we choose θ0=90 and set the coordinates of A, B, C, D as (0, −0.05), (0, 0.05), (0.05, 0.05), (0.05, −0.05), respectively.

Let us consider a vertical bend connecting two waveguides of equal width with 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
Fig. 2 (Color online). The magnetic field distribution of the designed bend using homogeneous materials for a = 10 cm, with or without anisotropy. (a) a vacuum bend. (b) ideal anisotropic transformation media filled-bend. (c) layered isotropic bend with θI=137 and θII=122 . (d) The average power outflow at port 2. (e) The electric field distribution of the dominant mode TE10 in the extruded bend with 0.04 m thickness along z direction.
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 εIA=12.26, εIB=0.082, μI=1/3, and εIIA=5, εIIB=0.2, μII=1. Figure 2(c) illustrates the magnetic field distribution of the equal-width waveguide bender composed of isotropic alternating layers. In order to quantitatively observe the performance of the bends filled with air, ideal transformation media, and isotropic layered materials, the average power outflow at port 2 is presented in Fig. 2(d). Obviously, the waveguide bend filled with isotropic layered materials behaves as perfectly as the bend filled with ideal transformation media. Hence, the bend can now be implemented using only homogeneous and isotropic materials. We also simulated the extruded bend by extruding the 2D bend along z direction. The effective isotropic parameters of the extruded bend can be found as μIA=12.26, μIB=0.082, εI=1/3, and μIIA=5, μIIB=0.2, εII=1. Figure 2(e) illustrates the electric field distribution of the extruded bend when the dominant mode TE10 is incident. Obviously, perfect performance can be achieved in the extruded bend.

In addition to designing bends with identical cross-sections, it is also possible to design bends connecting two waveguides of different cross-sections, e.g., 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)
Fig. 3 (Color online). The magnetic field distribution of the bend designed using homogeneous materials for a = 10 cm and b = 2 cm, with or without anisotropy. (a) a vacuum bend. (b) ideal anisotropic transformation media filled in the bend. (c-e) Layered isotropic materials filled in the bend with θI=115 and θII=122. (c) M = 10, (d) M = 20, and (e) M = 40. (f) The average power outflow at port 2.
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 εIA=36.5, εIB=0.027, μI=5/3, and εIIA=5, εIIB=0.2, μII=1. Both Region I and Region II are divided into 10 layers in Fig. 3(c), 20 layers in Fig. 3(d), and 40 layers in Fig. 3(e). We also calculate the average power outflow from port 2 when the bend filled with air, ideal transformation media, and isotropic layered materials, as shown in Fig. 3(f). It is obvious that the performance is quite pronounced when the discretization increases. Namely, a homogeneous and isotropic bend connecting two waveguides of different width is derived.

The optical transformation method proposed in this paper is not only used to design vertical waveguide bends (Figs. 2 and 3), it is also applied to design arbitrary bends to bend EM waves to any desired directions. The simulation frequency is 8 GHz under transverse electric (TE) polarization. Figure 4
Fig. 4 (Color online). Normalized electric fields distribution inside the waveguide bends. Two waveguides of identical cross section with a = 10 cm: (a) bending angle 45, (b) bending angle 135. Two waveguides of different cross sections with a = 10 cm and b = 2 cm: (c) bending angle 45, (d) bending angle 135.
shows the normalized electric field with the same D' as (0.05, −0.1) and different C'. Figure 4(a) corresponds to the bend connecting two equal waveguides with bending angle 45 by setting C' as (0.05+0.052,0.1+0.052), Fig. 4(b) corresponds to the bend connecting two equal waveguides with bending angle 135 by setting C' as (0.05+0.052,0.10.052), Fig. 4(c) corresponds to the bend connecting two different waveguides with bending angle 45 by setting C' as (0.05+0.012,0.1+0.012), and Fig. 4(d) corresponds to the bend connecting two different waveguides with bending angle 135 by setting C' as (0.05+0.012,0.10.012). From Fig. 4, it is clear that the incident waves travel through the waveguide bends reflectionlessly and keep their original field patterns in all cases. More importantly, all the bends are constructed with homogeneous materials, which can be also realized with homogeneous and isotropic layered structure.

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

Although homogeneity and isotropy have been achieved in Fig. 2, the bends are still magnetic which translates the fabrication difficulty into magnetism. Here we propose an advanced method to design nonmagnetic, isotropic and homogeneous bend. Figure 7(a)
Fig. 7 (Color online). (a) The schematic illustration of coordinate transformation in the design of advanced nonmagnetic bend by transforming ABD and CBD to ABD’ and C’BD’, respectively. (b) The magnetic field distribution of a vacuum bend with a = 10 cm and θ=60. (c) The magnetic field distribution of the bend in (b) filled with nonmagnetic, homogeneous and anisotropic transformation material. (d) The magnetic field distribution of the bend in (b) filled with layered isotropic dielectric (M = 10) and θ0=150. The embedded figure in (d) illustrates the distribution of constitutive parameters in transformed region.
shows the scheme of nonmagnetic bends by transforming triangles 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 SABD=SABD' and SBCD=SBC'D'.

To validate the advanced method, we consider an equal bend with bending angleθ=60 and 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 εI=εII=[2.33,1.15;1.15,1] and μI=μII=1. Clearly, EM waves completely pass through the bender with homogeneous, non-magnetic and ideal anisotropic transformation media. Figure 7(d) presents the magnetic field distribution inside the homogeneous, non-magnetic and isotropic bend with εA=5.83, εB=0.17 and μ=1. The anisotropy in Fig. 7(c) has been removed by replacing the identical anisotropic material in Region I and II with two isotropic dielectrics. Obviously, the advanced bend is as perfect as the ideal case.

5. Conclusion

In conclusion, we have proposed an advanced bending mechanism that uses exclusively homogeneous, isotropic, nonmagnetic (between two equal waveguides) or magnetic (between two different waveguides) materials. The benders proposed in this paper can connect multiple waveguides with different width and bend EM waves to any desired directions. They are composed of only two blocks of homogeneous materials, and can be easily realized with only two isotropic materials. This design concept may pave a realizable way to the practical applications. The full wave simulation validates the proposed design mechanism and perfect wave tunneling is demonstrated for the advanced bend with the elimination of reflection and mode distortion.

Acknowledgments

This research was supported by National University of Singapore (grant R-263-000-574-1330). It was also supported by France-Singapore collaboration project “CNRS PICS/LIA/GDRI 2009”. J. W. D. is supported by the National Natural Science Foundation of China (NSFC) (10804131) and the Fundamental Research Funds for the Central Universities (2009300003161450). T. C. Han is working towards his PhD in National University of Singapore.

References and links

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]

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. 104(1), 014502 (2008). [CrossRef]

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. 93(25), 251111 (2008). [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]

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]

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]

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

  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]
  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. 104(1), 014502 (2008). [CrossRef]
  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. 93(25), 251111 (2008). [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]
  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]
  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]

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