## Guiding light with conformal transformations

Optics Express, Vol. 17, Issue 17, pp. 14872-14879 (2009)

http://dx.doi.org/10.1364/OE.17.014872

Acrobat PDF (247 KB)

### Abstract

The past decade has seen a revolution in electromagnetics due to the development of metamaterials. These artificial composites have been fashioned to exhibit exotic effects such as a negative index of refraction. However, the full potential of metamaterial devices has only been hinted at. By combining metameterials with transformation optics (TO), researchers have demonstrated an invisibility cloak. Subsequently, quasi-conformal mapping was used to create a device that exhibited a broadband cloaking effect. Here we extend this latter approach to a strictly conformal mapping to create reflection less, inherently isotropic, and broadband photonic devices. Our method combines the novel effects of TO with the practicality of all-dielectric construction. We show that our structures are capable of guiding light in an almost arbitrary fashion over an unprecedented range of frequencies.

© 2009 OSA

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. 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), 063903–063907 (2008). [CrossRef] [PubMed]

4. 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 waveguides,” J. Appl. Phys. **104**(1), 014502 (2008). [CrossRef]

5. 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]

6. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express **15**(22), 14772–14782 (2007). [CrossRef] [PubMed]

7. 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]

10. 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]

11. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**(4), 224–227 (2007). [CrossRef]

12. B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**(8), 085103–085111 (2009). [CrossRef]

13. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901–203904 (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. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, (2009), “Cloaking at Optical Frequencies,” URL http://arxiv.org/abs/0904.3508.

17. U. Leonhardt, “Optical Conformal Mapping,” Science **312**(5781), 1777–1780 (2006). [CrossRef] [PubMed]

*δ*. The quasi-conformal technique will transform each square

*α*of the rectangular cell will be constant for all cells in the transformed space, and is given by,where

*M*is the conformal module of the physical domain and

*m*is that of the virtual domain, and one selects either +1 or −1, whichever yields the larger value for

*α*. The conformal module is the unique value of

*h/w*(Fig. 1) for which the quadrilateral

*Q*, shown in Fig. 1, is conformally equivalent to the rectangular quadrilateral

*R*such that for

*h/w = m*and only for this value there exists a unique conformal mapping that takes the four points {A',B',C',D'} respectively onto the four vertices {A,B,C,D} of

*R*in a counter-clockwise order. For the rectangular virtual domain

*R*, shown in Fig. 1, the conformal module is clearly

*α*presented in Eq. (1) can also be given in terms of the metric tensor

*g*of the transformation,

*ε*is the dielectric constant,

*μ*is the permeability,

*ε*is the background dielectric, and

_{r}*μ*is the background permeability. Typically, both

_{r}*M*and the map itself must be determined numerically [18,19]. We begin by determining the quasiconformal map and

*M*, which allows us to calculate the conformal map.

*m*is unknown, we cannot use (5) to calculate the conformal map. Rather, we must calculate the quasiconformal map by solving the Beltrami equations,where

*f*is constant and invoking the equality of mixed partial derivatives, Eq. (7) may be rewritten asIf

*f*is known, these equations become a set of linear, elliptical partial differential equations that can be discretized and solved

*via*a relaxation method. However, if

*f*is unknown, these equations are nonlinear and

*m*must be calculated iteratively according to

*α*also represents the anisotropy of the material. Therefore, if

*M/m*is not approximately unity, the device cannot be made isotropic, and magnetic coupling, i.e.

20. L. N. Trefethen, “Analysis and design of polygonal resistors by conformal mapping,” Z. Angew. Math. Phys. **35**(5), 692–704 (1984). [CrossRef]

*w*to be the input and output facets of a waveguide, and choose

*w*to be the same in both domains. We are thus free to let

*h*vary in such a way that a strictly conformal map exists. In this fashion the

*w*boundaries can be made reflection less – determined by the metric of the transformation

*g*at the interface [21

21. A previous implementation of an analytic conformal transformation used the exponential function, and therefore did not satisfy this condition. See,S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray Optics at a Deep-Subwavelength Scale: A Transformation Optics Approach,” Nano Lett. **8**(12), 4243–4247 (2008). [CrossRef]

*g*must be expressed in a basis consisting of local coordinates parallel and orthogonal to the port boundaries. If

*g*is equal to the free-space metric, then no reflections will occur on the boundary. Notably, for the conformal map, Eq. (5) becomes exact and we can writeand further this condition is simplified such that only the basis-independent quantity det(

*g*) must be that of free space.

*ε/ε*for a circular beam or waveguide bend, determined computationally by the conformal module of Eq. (1) and specified by Eq. (6). The permittivity has a maximum value of 3.25 and a minimum value of 0.5. The index of refraction (

_{r}*ε*= 30

*ε*at the inside corner.

_{r}13. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901–203904 (2008). [CrossRef] [PubMed]

22. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. **97**(15), 157403 (2006). [CrossRef] [PubMed]

26. A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε -near-zero-filled narrow channels,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(1 Pt 2), 016604–016614 (2008). [CrossRef] [PubMed]

27. M. Rahm, D. Schurig, D. 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. Applic. **6**(1), 87–95 (2008). [CrossRef]

28. A. Degiron and D. R. Smith, “Numerical Simulations of long-range Plasmons,” Opt. Express **14**(4), 1611–1625 (2006). [CrossRef] [PubMed]

29. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. **66**(3), 216–220 (1976). [CrossRef]

*no resonant losses*, since the transformed region is equivalent to a straight waveguide segment. Conformally-mapped regions also present an important simplification to subsequent calculations involving the region, as propagation through the mapped region will be identical to propagation through a straight region of length

*h*in the virtual domain. One may also fabricate miniaturized photonic waveguide components. For example using ‘squeezed’ waveguides, as shown in Fig. 3, we may compress the photonic waveguide, while constructing the bends, splitters, and couplers that constitute the entire photonic circuit.

## 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. | 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. |

4. | 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 waveguides,” J. Appl. Phys. |

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

6. | D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express |

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

8. | 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. |

9. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

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

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

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

13. | J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. 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. | L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, (2009), “Cloaking at Optical Frequencies,” URL http://arxiv.org/abs/0904.3508. |

16. | J. L. J. Valentine, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics”, Nature Materials (2009). |

17. | U. Leonhardt, “Optical Conformal Mapping,” Science |

18. | J. F. Thompson, B. K. Soni, and N. P. Weatherill, eds., Handbook of Grid Generation (CRC Press, Boca Raton, FL, 1994). |

19. | P. Knupp, and S. Steinberg, Fundamentals of Grid Generation (CRC Press, Boca Raton, FL, 1993). |

20. | L. N. Trefethen, “Analysis and design of polygonal resistors by conformal mapping,” Z. Angew. Math. Phys. |

21. | A previous implementation of an analytic conformal transformation used the exponential function, and therefore did not satisfy this condition. See,S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray Optics at a Deep-Subwavelength Scale: A Transformation Optics Approach,” Nano Lett. |

22. | M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. |

23. | R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. |

24. | B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental Verification of Epsilon-Near-Zero Metamaterial Coupling and Energy Squeezing Using a Microwave Waveguide,” Phys. Rev. Lett. |

25. | B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. |

26. | A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε -near-zero-filled narrow channels,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

27. | M. Rahm, D. Schurig, D. 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. Applic. |

28. | A. Degiron and D. R. Smith, “Numerical Simulations of long-range Plasmons,” Opt. Express |

29. | D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 18, 2009

Revised Manuscript: July 27, 2009

Manuscript Accepted: July 29, 2009

Published: August 6, 2009

**Citation**

Nathan I. Landy and Willie J. Padilla, "Guiding light with conformal transformations," Opt. Express **17**, 14872-14879 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14872

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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]
- 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), 063903–063907 (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 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]
- D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15(22), 14772–14782 (2007). [CrossRef] [PubMed]
- 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. 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]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]
- 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]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
- B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 085103–085111 (2009). [CrossRef]
- J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901–203904 (2008). [CrossRef] [PubMed]
- 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]
- L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, (2009), “Cloaking at Optical Frequencies,” URL http://arxiv.org/abs/0904.3508 .
- J. L. J. Valentine, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics”, Nature Materials (2009).
- U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- J. F. Thompson, B. K. Soni, and N. P. Weatherill, eds., Handbook of Grid Generation (CRC Press, Boca Raton, FL, 1994).
- P. Knupp, and S. Steinberg, Fundamentals of Grid Generation (CRC Press, Boca Raton, FL, 1993).
- L. N. Trefethen, “Analysis and design of polygonal resistors by conformal mapping,” Z. Angew. Math. Phys. 35(5), 692–704 (1984). [CrossRef]
- A previous implementation of an analytic conformal transformation used the exponential function, and therefore did not satisfy this condition. See,S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray Optics at a Deep-Subwavelength Scale: A Transformation Optics Approach,” Nano Lett. 8(12), 4243–4247 (2008). [CrossRef]
- M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]
- R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903–023907 (2008). [CrossRef] [PubMed]
- B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental Verification of Epsilon-Near-Zero Metamaterial Coupling and Energy Squeezing Using a Microwave Waveguide,” Phys. Rev. Lett. 100(3), 033903–033907 (2008). [CrossRef] [PubMed]
- B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105(4), 044905 (2009). [CrossRef]
- A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε -near-zero-filled narrow channels,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(1 Pt 2), 016604–016614 (2008). [CrossRef] [PubMed]
- M. Rahm, D. Schurig, D. 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. Applic. 6(1), 87–95 (2008). [CrossRef]
- A. Degiron and D. R. Smith, “Numerical Simulations of long-range Plasmons,” Opt. Express 14(4), 1611–1625 (2006). [CrossRef] [PubMed]
- D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]

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