## Manipulating the field distribution via optical transformation

Optics Express, Vol. 18, Issue 10, pp. 10168-10176 (2010)

http://dx.doi.org/10.1364/OE.18.010168

Acrobat PDF (1399 KB)

### Abstract

Using the coordinate transformation theory, we propose a way to control the field distribution of electromagnetic wave in a waveguide filling with properly designed transformation media. The results show that the field distribution of the electromagnetic wave can be compressed and amplified in both longitudinal and transverse direction. A realizable layered waveguide based on the discrete optical transformation is also designed to manipulate the field distribution of the electromagnetic wave. Potential applications, i.e., turn a narrow (wide) slit into a large (small) window, are investigated in detail.

© 2010 OSA

## 1. Introduction

1. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **72**(1), 016623 (2005). [CrossRef] [PubMed]

33. Y. Luo, J. Zhang, H. Chen, J. Huangfu, and L. Ran, “High-directivity antenna with small antenna aperture,” Appl. Phys. Lett. **95**(19), 193506 (2009). [CrossRef]

*et al*theoretically proposed an invisibility cloak that can hide the objects inside the cloak from detection, the cloak having electrical permittivity and magnetic permeability that are both spatially varying and anisotropic [2

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

34. H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. **91**(18), 183518 (2007). [CrossRef]

35. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. **6**(1), 87–95 (2008). [CrossRef]

36. 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]

*et al*, have proposed a layered lens antenna in which the local properties of fields in a small region can be amplified to a large region [37

37. W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. **93**(22), 221906 (2008). [CrossRef]

## 2. The general modes

*ABCD*and the physical space is

*ABC’D’*. In the transformation region as shown in Fig. 1, the field distribution of EMW can be amplified or compressed both in longitudinal and transverse direction. The geometric coordinate transformation between the new system (

*x', y', z'*) and the original system

*(x, y, z*) can be expressed aswhere (

*x*,

*y*) is the coordinate in virtual space and (

*x'*,

*y'*) is the coordinate in physical space. From Eq. (1), we can find that the field distribution of EMW will be amplified (compressed) in the transverse direction for

*b*>

*a*(

*b*<

*a*). When

*c*>1(

*c*<1), the field distribution of EMW will be amplified (compressed) in the longitudinal direction.

*n*th homogeneous pieces where each layer is composed of homogenous and uniaxially anisotropic metamaterial. Therefore, the geometric coordinate transformation between the new system (

*x', y', z'*) and the original system

*(x, y, z*) can be expressed aswhere

*k(n) = L(m-0.5)/n*(

## 3. Numerical simulations and discussions

*z*aixs propagates from the bottom of the normal region to the transformation region.

*a, b, c,*and

*L*. When

*a =*0.2

*m, b = L =*1

*m,*and

*c =*2, the field distribution from the normal region (free space with perfect electrical conductor boundary) is amplified both in longitudinal and transverse direction in the transformation region. When

*a =*0.2

*m, b = L*= 1

*m, and c =*1/2, the field distribution is amplified in transverse direction but compressed in longitudinal direction in the transformation region. The filed distribution of the EMW can be compressed in transverse direction and amplified in longitudinal direction when

*a =*0.2

*m, b =*0.3

*m, L =*0.5

*m,*and

*c =*2. When the incident electromagnetic wave transports into the transformation region, the field distribution can also be compressed both in longitudinal and transverse direction for

*a =*0.2

*m, b =*0.3

*m, L =*0.5

*m,*and

*c =*1/2. In a word, the field distribution of the EMW in the transformation region is completely decided by the parameters

*a, b, c.*In other words, the field distribution of EMW can be well manipulated by selecting appropriate parameters

*a, b, c*. In addition, we should note that there is reflection at the exit of the transformation region, which is similar to the case of Ref [37

37. W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. **93**(22), 221906 (2008). [CrossRef]

*a<b*(

*a>b*). Here, it should be noted that the more layers we divide the transformation region, the better result we’ll get. The field distribution is shown in Fig. 3 . The field distribution in the layered transformation region (Fig. 3) has almost no difference to in the continuous transformation region (Fig. 2). Therefore, we can manipulate the field distribution of EMW by choosing an appropriate layered homogeneous anisotropy metamaterials. In this case, there is also reflection phenomenon at the exit of the transformation region due to the mismatch at the output port (not shown), which is similar to the continuous coordinate transformation case.

*a*0.2

_{I}=*m, b*1

_{I}=*m, c*1/2,

_{I}=*L*1

_{I}=*m*, the shape of the field distribution is amplified in transverse direction and compressed in longitudinal direction. When the EMW transports into the “mirror” transformation region II with parameters

*a*1

_{II}=*m, b*0.2

_{II}=*m, c*= 2,

_{II}*L*1

_{II}=*m*, the field distribution is amplified in longitudinal direction and compressed in transverse direction. After coming out from the region II, the Electromagnetic field distribution in the top normal region is the same as in the bottom region. In Fig. 4(b)–4(d), the situation is the same as in Fig. 4(a) except for different system parameters.

*a*0.3

_{1}= a_{2}=*m*,

*L*= 0.4

_{1}= L_{2}*m*and

*b*= 0.1

*m.*For the different parameters of

*c*and

_{1}*c*, the small slit can replace different large windows [the right figure of Fig. 5(c) 5(f)] . When the small slit in Fig. 5(a) without transformation media in it, there is not any electromagnetic field distribution in the bottom of the free space as shown in Fig. 5(a). When the small slit in Fig. 5(b) embedded with layered transformation media and

_{2}*c*= 1, the electromagnetic field distribution in the bottom of the free space as shown in Fig. 5(b) is the same as in Fig. 5(c). Compared with Fig. 5(b) and 5(c), the large window in Fig. 5(c) with

_{1}= c_{2}*a’ =*0.3

*m*and

*b’ =*0.8

*m*[see the right figure in Fig. 5(f)] can be replaced by a small slit of Fig. 5(b). In Fig. 5(d) and 5(e), the electromagnetic field distribution in the bottom of the free space is also identical with 3

*c*= 1. But, the small slit replace with a much bigger window with

_{1}= c_{2}*a” =*0.3

*m b” =*1.6

*m*[see the right figure in Fig. 5(g)]. It should be noted that the big window in Fig. 5(e) is much higher than that in Fig. 5(c). But the small slit in Fig. 5(b) and 5(d) is identical. Second, we design a large slit [the left figure of Fig. 6(f) , 6(g)] composed of layered homogeneous anisotropy metamaterials with parameters

*a*0.1

_{1}= a_{2}=*m*,

*L*= 0.5

_{1}= L_{2}*m*and

*b*= 0.5

*m.*In this case, the large slits can replace a small window. The electric field distribution near the slit without transformation media in it is shown in Fig. 6(a). In Fig. 6(b) and 6(c), it is found that the flied distribution of the line source in the bottom of the free space is identical which demonstrates that the small window can be replaced with a large slit. The case of Fig. 6(d) and 6(e) is similar to Fig. 6(b) and 6(c), but the small window in Fig. 6(e) is much shorter than that in Fig. 6(b). These potential applications demonstrate that our device can gather (expand) information from different angles of light sources like an arbitrarily wide (small) window.

## 4. Conclusion

## Acknowledgements

## References and links

1. | A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

2. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” 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. | U. Leonhardt, “Optical conformal mapping,” Science |

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

6. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterial,” Nat. Photonics |

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

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

9. | B. L. Zhang, H. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. |

10. | Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. |

11. | M. Yan, Z. C. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. |

12. | X. Zhou and G. K. Hu, “Acoustic wave transparency for a multilayered sphere with acoustic metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

13. | H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. |

14. | A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. |

15. | D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloak,” Appl. Phys. Lett. |

16. | D.-H. Kwon and D. H. Werner, “Two-dimensional electromagnetic cloak having a uniform thickness for elliptic cylindrical regions,” Appl. Phys. Lett. |

17. | P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. |

18. | H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A |

19. | S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. |

20. | P. Yao, Z. Liang, and X. Jiang, “Limitations of the electromagnetic cloak with dispersive material,” Appl. Phys. Lett. |

21. | W. X. Jiang, T. J. Cui, Q. Cheng, J. Y. Chin, X. M. Yang, R. Liu, and R. Smith, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett. |

22. | C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express |

23. | X. Zhang, H. Y. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express |

24. | Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and the essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. |

25. | J. Hu, X. M. Zhou, and G. K. Hu, “Nonsingular two dimensional cloak of arbitrary shape,” Appl. Phys. Lett. |

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

27. | C. W. Qiu, L. Hu, B. Zhang, B. I. Wu, S. G. Johnson, and J. D. Joannopoulos, “Spherical cloaking using nonlinear transformations for improved segmentation into concentric isotropic coatings,” Opt. Express |

28. | H. Ma, S. Qu, Z. Xu, and J. Wang, “General method for designing wave shape transformers,” Opt. Express |

29. | D.-H. Kwon, “Virtual circular array using material-embedded linear source distributions,” Appl. Phys. Lett. |

30. | 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. |

31. | Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. |

32. | H. Y. 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. |

33. | Y. Luo, J. Zhang, H. Chen, J. Huangfu, and L. Ran, “High-directivity antenna with small antenna aperture,” Appl. Phys. Lett. |

34. | H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. |

35. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. |

36. | 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. |

37. | W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(160.3918) Materials : Metamaterials

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 8, 2010

Revised Manuscript: April 15, 2010

Manuscript Accepted: April 15, 2010

Published: April 30, 2010

**Citation**

Xiaofei Zang and Chun Jiang, "Manipulating the field distribution via optical transformation," Opt. Express **18**, 10168-10176 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-10168

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### References

- A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]
- 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]
- U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (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]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterial,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
- F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32(9), 1069–1071 (2007). [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, 113903 (2007).
- B. L. Zhang, H. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100(6), 063904 (2008). [CrossRef] [PubMed]
- Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007). [CrossRef] [PubMed]
- M. Yan, Z. C. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. 99(23), 233901 (2007). [CrossRef]
- X. Zhou and G. K. Hu, “Acoustic wave transparency for a multilayered sphere with acoustic metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046606 (2007). [CrossRef] [PubMed]
- H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]
- A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33(14), 1584 (2008). [CrossRef] [PubMed]
- D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloak,” Appl. Phys. Lett. 92(1), 013505 (2008). [CrossRef]
- D.-H. Kwon and D. H. Werner, “Two-dimensional electromagnetic cloak having a uniform thickness for elliptic cylindrical regions,” Appl. Phys. Lett. 92(11), 113502 (2008). [CrossRef]
- P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93(24), 243502 (2008). [CrossRef]
- H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77(1), 013825 (2008). [CrossRef]
- S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008). [CrossRef] [PubMed]
- P. Yao, Z. Liang, and X. Jiang, “Limitations of the electromagnetic cloak with dispersive material,” Appl. Phys. Lett. 92(3), 031111 (2008). [CrossRef]
- W. X. Jiang, T. J. Cui, Q. Cheng, J. Y. Chin, X. M. Yang, R. Liu, and R. Smith, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett. 92(26), 264101 (2008). [CrossRef]
- C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16(17), 13414–13420 (2008). [CrossRef] [PubMed]
- X. Zhang, H. Y. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express 16(16), 11764–11768 (2008). [CrossRef] [PubMed]
- Z. Liang, P. Yao, X. Sun, and X. Jiang, “The physical picture and the essential elements of the dynamical process for dispersive cloaking structures,” Appl. Phys. Lett. 92(13), 131118 (2008). [CrossRef]
- J. Hu, X. M. Zhou, and G. K. Hu, “Nonsingular two dimensional cloak of arbitrary shape,” Appl. Phys. Lett. 95(1), 011107 (2009). [CrossRef]
- X. F. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett. 95(18), 184102 (2009). [CrossRef]
- C. W. Qiu, L. Hu, B. Zhang, B. I. Wu, S. G. Johnson, and J. D. Joannopoulos, “Spherical cloaking using nonlinear transformations for improved segmentation into concentric isotropic coatings,” Opt. Express 17(16), 13467–13478 (2009). [CrossRef] [PubMed]
- H. Ma, S. Qu, Z. Xu, and J. Wang, “General method for designing wave shape transformers,” Opt. Express 16(26), 22072 (2008). [CrossRef] [PubMed]
- D.-H. Kwon, “Virtual circular array using material-embedded linear source distributions,” Appl. Phys. Lett. 95(17), 173503 (2009). [CrossRef]
- 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]
- Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]
- H. Y. 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]
- Y. Luo, J. Zhang, H. Chen, J. Huangfu, and L. Ran, “High-directivity antenna with small antenna aperture,” Appl. Phys. Lett. 95(19), 193506 (2009). [CrossRef]
- H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
- 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]
- W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. 93(22), 221906 (2008). [CrossRef]

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