## General conformal transformation method based on Schwarz-Christoffel approach |

Optics Express, Vol. 19, Issue 16, pp. 15119-15126 (2011)

http://dx.doi.org/10.1364/OE.19.015119

Acrobat PDF (1104 KB)

### Abstract

A general conformal transformation method (CTM) is proposed to construct the conformal mapping between two irregular geometries. In order to find the material parameters corresponding to the conformal transformation between two irregular geometries, two polygons are utilized to approximate the two irregular geometries, and an intermediate geometry is used to connect the mapping relations between the two polygons. Based on these manipulations, the approximate material parameters for TE and TM waves are finally obtained by calculating the Schwarz-Christoffel (SC) mappings. To demonstrate the validity of the method, a phase modulator and a plane focal surface Luneburg lens are designed and simulated by the finite element method. The results show that the conformal transformation can be expanded to the cases that the transformed objects are with irregular geometries.

© 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]

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 **314**(5801), 977–980 (2006). [CrossRef] [PubMed]

4. 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]

6. G. Yuan, X. Dong, Q. Deng, H. Gao, C. Liu, Y. Lu, and C. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express **18**(6), 6327–6332 (2010). [CrossRef] [PubMed]

2. U. Leonhardt, “Optical conformal mapping,” Science **312**(5781), 1777–1780 (2006). [CrossRef] [PubMed]

7. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express **17**(17), 14872–14879 (2009). [CrossRef] [PubMed]

11. C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. **97**(4), 044101 (2010). [CrossRef]

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

7. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express **17**(17), 14872–14879 (2009). [CrossRef] [PubMed]

11. C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. **97**(4), 044101 (2010). [CrossRef]

8. 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] [PubMed]

10. M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A **81**(3), 033837 (2010). [CrossRef]

11. C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. **97**(4), 044101 (2010). [CrossRef]

## 2. The Conformal Transformation Method

*Q*

_{1},

*Q*

_{2}are the two irregular geometries in the complex plane.

*P*

_{1},

*P*

_{2}are the two polygons used to approximate

*Q*

_{1},

*Q*

_{2}. In transformation optics terms,

*Q*

_{1},

*P*

_{1}are in the virtual space, and

*Q*

_{2},

*P*

_{2}are in the physical space [12

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

*R*is the intermediate geometry (here it is a rectangle) used to connect

*P*

_{1},

*P*

_{2}.

*Z, ζ, ω*denote their coordinates, and

*A, B, C, D*mark the four vertices of them. By employing SC mapping [13

13. T. A. Driscoll, “A MATLAB toolbox for Schwarz-Christoffel mapping,” ACM Trans. Math. Softw. **22**(2), 168–186 (1996). [CrossRef]

*z*=

*f*

_{1}(

*ζ*),

*ω*=

*f*

_{2}(

*ζ*) which map the intermediate geometry

*R*onto the two polygons

*P*

_{1},

*P*

_{2}, respectively, can be calculated out. Use the same method, their inverse mapping

*ζ*=

*g*

_{1}(

*z*),

*ζ*=

*g*

_{2}(

*ω*) can also be achieved.

*P*

_{1}onto

*P*

_{2}expressed by

*f*

_{1},

*f*

_{2}is deduced based on the relations given in Ref. [8

8. 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] [PubMed]

9. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express **18**(1), 244–252 (2010). [CrossRef] [PubMed]

*ε*and the permeability

*μ*) corresponding to the conformal mapping

*ω*=

*f*(

*z*) for TE and TM waves can be given byWhere

*u*,

*v*represent the real and imaginary part of the complex function

*ω*=

*f*(

*z*), respectively, namely

*ω*=

*f*(

*z*) ≡

*u*(

*x*,

*y*) +

*i v*(

*x*,

*y*), and the virtual space is assumed to be vacuum. Because

*ω*=

*f*(

*z*) is an analytic function, its derivative can be written as

*f*′(

*z*) =

*∂*+

_{x}u*i ∂*[15]. Then Eq. (1) becomeswhere |∙| represents the norm of a complex function. Since

_{x}v*ω*=

*f*(

*z*) can be expressed in the form of composite function:

*ω*=

*f*

_{2}[

*g*

_{1}(

*z*)], where

*ζ*=

*g*

_{1}(

*z*) is the inverse function of

*z*=

*f*

_{1}(

*ζ*). Thus using of differential chain rule yields

*ζ = g*

_{1}(

*z*) is also an analytic function, thus

*g*

_{1}′(

*z*) = 1 /

*f*

_{1}′(

*ζ*). Then Eq. (3) can be rewritten as

*f*′(

*z*) =

*f*

_{2}′(

*ζ*) /

*f*

_{1}′(

*ζ*). Consequently, the approximate material parameters for TE and TM waves expressed by Eq. (2) finally becomes

*P*

_{1}onto

*P*

_{2}rather than from

*Q*

_{1}onto

*Q*

_{2}. However, because

*P*

_{1},

*P*

_{2}are approximations of

*Q*

_{1},

*Q*

_{2}, thus the material parameters given by Eq. (4) can be used to approximate the ones corresponding to the mapping from

*Q*

_{1}onto

*Q*

_{2}. With this idea, the procedure of the CTM can be described as following:

- Step 1: Use two polygons
*P*_{1},*P*_{2}to approximate the two irregular geometries*Q*_{1},*Q*_{2}. - Step 2: Use an intermediate geometry
*R*to connect the two polygons*P*_{1},*P*_{2}. For instance, a rectangle or a disk can be utilized as the intermediate geometry. - Step 3: By employing SC mapping (the SC mapping can be computed by, for example, the toolbox in Ref. [13]), calculate the conformal mapping
13. T. A. Driscoll, “A MATLAB toolbox for Schwarz-Christoffel mapping,” ACM Trans. Math. Softw.

**22**(2), 168–186 (1996). [CrossRef]*z*=*f*_{1}(*ζ*) between*R*and*P*_{1}, and*ω*=*f*_{2}(*ζ*) between*R*and*P*_{2}. - Step 4: Compute the approximate material parameters corresponding to the transformation from
*Q*_{1}onto*Q*_{2}by Eq. (4).

*P*

_{1},

*P*

_{2}should be equal to ensure that a conformal mapping exists between them. If a disk is used as the intermediate geometry, conformal centers in

*P*

_{1},

*P*

_{2}should be defined. In the step 4, the enough side numbers of the two polygons

*P*

_{1},

*P*

_{2}should be selected to ensure an accurate calculation of the material parameters.

## 3. Design and Simulation for Two Types of Transformation Devices

### 3.1 Phase Modulator

*Q*

_{1}is the geometry before transformation, and in the physical space,

*Q*

_{2}is the phase modulator after transformation. In this case, both

*Q*

_{1}and

*Q*

_{2}can be considered as irregular. Because

*Q*

_{1}is not a disk as it appears, but a topological quadrilateral [15]. Hence, whether its boundary

*ABCD*is circular or irregular will not cause any difference in essence to the CTM procedure.

*O*of

*Q*

_{1}, then the phase of the excited electromagnetic (EM) field on the boundary

*ABCD*of

*Q*

_{1}will be equal, namely

*ABCD*is an equal phase surface. And then, if

*Q*

_{1}is transformed onto

*Q*

_{2}, the boundary

*ABCD*of

*Q*

_{2}will also be the equal phase surface for the point source on the center of

*Q*

_{2}. Because the

*AB*,

*CD*sides of

*Q*

_{2}are arcs whose centers are

*F*

_{1},

*F*

_{2}, respectively. Therefore, as exhibited in Fig. 2,

*F*

_{1},

*F*

_{2}will become two focal points. To verify this idea, the proposed CTM is employed to calculate the material parameters corresponding to the conformal mapping from

*Q*

_{1}onto

*Q*

_{2}(Note that the material parameters cannot be calculated by the solving Laplace’s equation method proposed in Ref [8

8. 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] [PubMed]

*Q*

_{1}nor

*Q*

_{2}is a rectangle). In this case, the procedure of the CTM can be specified as following:

- Step 1: A regular 74-gon
*P*_{1}is selected to approximate*Q*_{1}, and a polygon*P*_{2}is selected to approximate*Q*_{2}with the boundaries*AB*,*CD*which are composed of 20 line segments respectively to approximate the arcs*AB*,*CD.*The conformal modulus of*P*_{1}and*P*_{2}both are 2.45 to ensure that a conformal mapping exists between them. - Step 2: A rectangle
*R*with conformal modulus equals to 2.45 is used to connect*P*_{1}and*P*_{2}. (In mathematics, one method to calculate conformal modulus of a topological quadrilateral is to map it onto a rectangle by SC mapping, and the ratio of height to width of the rectangle equals to the conformal modulus of the topological quadrilateral, and of course, it also equals to the conformal modulus of the rectangle.) - Step 3: The conformal mapping
*f*_{1},*f*_{2}are calculated numerically by employing the SC mapping program. - Fig. 3The simulation results of the phase modulator. (a) The refractive index distribution in the phase modulator. (b) The z component of the electric field when a point source is on the center of the phase modulator.

*F*

_{1},

*F*

_{2}. This result agrees well with our design expectations.

### 3.2 Plane Focal Surface 2D Luneburg Lens

16. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. **6**(3), 151–155 (2011), doi:. [CrossRef] [PubMed]

17. A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express **19**(6), 5156–5162 (2011). [CrossRef] [PubMed]

*R*is the radius of the lens and

*r*is the distance from a point on the lens to the lens center. The focusing property of the Luneburg lens is that every point on the cylindrical surface is the focal point for the incident plane waves on the opposite side of the lens.

*Q*

_{1}is the 2D Luneburg lens before transformation. If

*Q*

_{1}is mapped onto

*Q*

_{2}, then accordingly, the cylindrical focal surface

*AB*of

*Q*

_{1}(marked in green) will be mapped to the plane focal surface

*AB*of

*Q*

_{2}. Consequently, in the physical space,

*Q*

_{2}becomes a PFS Luneburg lens. To this end, the CTM is also utilized to find the approximate material parameters (The solving Laplace’s equation method is invalid in this case either, since neither of

*Q*

_{1},

*Q*

_{2}is a rectangle). In this case, the CTM procedure can be specified as following:

- Step 1: A polygon
*P*_{2}is used to approximate*Q*_{2}. The*MN*boundary of*P*_{2}which is composed of 60 line segments is used to approximate the*MN*arc of*Q*_{2}. And the conformal center of*P*_{2}is (u, v) = (0, 0). - Step 2: The conformal mapping
*f*_{2}are calculated numerically by employing SC mapping program. - Fig. 5Simulation results of the PFS Luneburg lens. (a) is the refractive index distribution in the PFS Luneburg lens. (b)(c)(d) are the Z component of the electric field when the point source is located at (u, v) = (−0.5, 0), (−0.5, 0.1), (−0.5, 0.2), respectively. (e)(f) are the Z component of the electric field when plane waves incident on the PSF Luneburg lens.

*Q*

_{1}itself is a disk which can be used as the intermediate geometry, thus no other intermediate geometry is used. For the same reason, no polygon is used to approximate

*Q*

_{1}. Consequently, Eq. (2) is used instead of Eq. (4) to compute the approximate material parameters in the step 3. Additionally, due to the property of SC mapping, if a disk is used as the intermediate geometry, a conformal center should be defined as in the step 1.

*u*axis incidents on the PSF Luneburg lens, and then focuses on the point (−0.5, 0). Similarly, Fig. 5(f) shows that when the angle between the propagation direction of the plane wave and

*u*axis is 30 degrees, the wave focuses on another point whose location is (−0.5, 0.14). Therefore, the PSF Luneburg lens can also be used as a receiving antenna. Note that in Fig. 5(e), 5(f), because the impedance on the

*MN*boundary of the PSF Luneburg lens is not perfectly matched with the impedance of vacuum, thus a small part of energy of the incident wave is reflected.

## 4. Discussion of the Numerical Errors

*P*

_{2}(in Fig. 1). In order to achieve accurate material parameters, the polygons should be set with enough sides so that the width of the distorted region is much smaller than the operating wavelength of the EM wave.

*MN*arc of

*Q*

_{2}is approximated by 15 line segments, 60 line segments, and 90 line segments, respectively. Then, the widths of the distorted regions of the material parameters as shown in Fig. 6 are about 0.2cm, 0.02cm, and 0.01cm, respectively. Since the operating wavelength of the PFS Luneburg lens in our simulation is 0.2cm which is already equivalent to the width of the distorted region in the 15 line segments approximation, thus the 15 line segments approximation is unacceptable. In the 60 line segments and 90 line segments approximations, the widths of the distorted regions are ten percent and five percent of the operating wavelength, respectively. In these cases, the macroscopic EM fields will not sensitive to the detail material parameters distributions in the distorted regions, but mainly depend on the average material parameters in the regions. As a result, these approximations are acceptable. The simulation results in Fig. 5 based on the 60 line segments approximation suggest that the approximation is already accurate enough for the 0.2 cm operating wavelength. Based on the analysis, we believe that the enough polygons sides should be taken to ensure the width of the distorted region being less than ten percent of the operating wavelength for achieving accurate transformation.

## 5. Conclusion

## Acknowledgments

## References and links

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

2. | U. Leonhardt, “Optical conformal mapping,” 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. | 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. |

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

6. | G. Yuan, X. Dong, Q. Deng, H. Gao, C. Liu, Y. Lu, and C. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express |

7. | N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express |

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

9. | J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express |

10. | M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A |

11. | C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. |

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

13. | T. A. Driscoll, “A MATLAB toolbox for Schwarz-Christoffel mapping,” ACM Trans. Math. Softw. |

14. | T. A. Driscoll and L. N. Trefethen, |

15. | P. Henrici, |

16. | T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. |

17. | A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 10, 2011

Revised Manuscript: June 24, 2011

Manuscript Accepted: June 26, 2011

Published: July 21, 2011

**Citation**

Linlong Tang, Jinchan Yin, Guishan Yuan, Jinglei Du, Hongtao Gao, Xiaochun Dong, Yueguang Lu, and Chunlei Du, "General conformal transformation method based on Schwarz-Christoffel approach," Opt. Express **19**, 15119-15126 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15119

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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]
- U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (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 (2008). [CrossRef] [PubMed]
- M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008). [CrossRef]
- G. Yuan, X. Dong, Q. Deng, H. Gao, C. Liu, Y. Lu, and C. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express 18(6), 6327–6332 (2010). [CrossRef] [PubMed]
- N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]
- 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] [PubMed]
- J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express 18(1), 244–252 (2010). [CrossRef] [PubMed]
- M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010). [CrossRef]
- C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. 97(4), 044101 (2010). [CrossRef]
- J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
- T. A. Driscoll, “A MATLAB toolbox for Schwarz-Christoffel mapping,” ACM Trans. Math. Softw. 22(2), 168–186 (1996). [CrossRef]
- T. A. Driscoll and L. N. Trefethen, Schwartz-Christoffel Mapping (Cambridge University Press, 2002).
- P. Henrici, Applied and Computational Complex Analysis, Volume 3: Discrete Fourier Analysis, Cauchy Integrals, Construction of Conformal Maps, Univalent Functions (Wiley, 1986).
- T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:. [CrossRef] [PubMed]
- A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19(6), 5156–5162 (2011). [CrossRef] [PubMed]

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