## Anisotropy minimization via least squares method for transformation optics |

Optics Express, Vol. 22, Issue 15, pp. 18490-18498 (2014)

http://dx.doi.org/10.1364/OE.22.018490

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### Abstract

In this work the least squares method is used to reduce anisotropy in transformation optics technique. To apply the least squares method a power series is added on the coordinate transformation functions. The series coefficients were calculated to reduce the deviations in Cauchy-Riemann equations, which, when satisfied, result in both conformal transformations and isotropic media. We also present a mathematical treatment for the special case of transformation optics to design waveguides. To demonstrate the proposed technique a waveguide with a 30° of bend and with a 50% of increase in its output width was designed. The results show that our technique is simultaneously straightforward to be implement and effective in reducing the anisotropy of the transformation for an extremely low value close to zero.

© 2014 Optical Society of America

## 1. Introduction

1. L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. **3**(1), 1–58 (1953). [CrossRef]

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]

## 2. Theoretical development

### 2.1 Transformation optics

3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. **53**, 69–152 (2009). [CrossRef]

3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. **53**, 69–152 (2009). [CrossRef]

### 2.2 Coordinate transformation

*D*. The function is analytic if, and only if, the Cauchy-Riemann conditions are satisfied. In turn, the coordinate transformation represented by

3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. **53**, 69–152 (2009). [CrossRef]

*x' = x*and

*y' = y*[5].

1. L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. **3**(1), 1–58 (1953). [CrossRef]

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]

1. L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. **3**(1), 1–58 (1953). [CrossRef]

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]

7. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express **18**(6), 6089–6096 (2010). [CrossRef] [PubMed]

**101**(20), 203901 (2008). [CrossRef] [PubMed]

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

9. 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]

7. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express **18**(6), 6089–6096 (2010). [CrossRef] [PubMed]

13. D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express **21**(12), 14223–14243 (2013). [CrossRef] [PubMed]

13. D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express **21**(12), 14223–14243 (2013). [CrossRef] [PubMed]

14. D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express **18**(3), 3181–3186 (2010). [CrossRef] [PubMed]

## 3. Mathematical development

### 3.1 Transformation optics for planar waveguides

*θ*relative to the x'-axis such that:

*φ*, the device output boundary angle:

*θ*is not constant over the guide output facet, the electromagnetic wave fronts exiting the device are not planar. Similarly, if the angle

*φ*is not constant, the waveguide output itself is not flat.

### 3.2 Application of the least squares method

*k*’ sample points in the non-transformed medium as shown in Fig. 1.

*E*, summed over each sampling point:

*E*with respect to the coefficients

*E*.

*maxR*', which must be specified in the project. To this end, the algorithm sample the point with highest anisotropy and excluding (or not) the sampled point with lowest anisotropy. In this procedure, the number of sampled points tends to increase. There are evidences [13

13. D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express **21**(12), 14223–14243 (2013). [CrossRef] [PubMed]

*θ*and

*φ*result in the following expression when anisotropy reduction occurs:

*θ*and

*φ*can be defined.

16. C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. **16**(2), 023030 (2014). [CrossRef]

### 3.3 Transformation example

## 4. Results

*maxR*used was 3.

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

9. 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]

_{ij}and B

_{ij}. This is possible since the

## 5. Conclusion

## Acknowledgments

## References and links

1. | L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. |

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

3. | U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. |

4. | G. E. Shilov, |

5. | W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflection less transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008). |

6. | U. Leonhardt, “Optical conformal mapping,” Science |

7. | Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express |

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

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

10. | N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. |

11. | H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. |

12. | Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. |

13. | D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express |

14. | D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express |

15. | K. Astala, T. Iwaniec, and G. Martin, |

16. | C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. |

17. | D. E. Blair, |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(130.2790) Integrated optics : Guided waves

(230.0230) Optical devices : Optical devices

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 14, 2014

Revised Manuscript: June 21, 2014

Manuscript Accepted: June 24, 2014

Published: July 23, 2014

**Citation**

Mateus A. F. C. Junqueira, Lucas H. Gabrielli, and Danilo H. Spadoti, "Anisotropy minimization via least squares method for transformation optics," Opt. Express **22**, 18490-18498 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18490

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### References

- L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math.3(1), 1–58 (1953). [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]
- U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53, 69–152 (2009). [CrossRef]
- G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).
- W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflection less transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).
- U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express18(6), 6089–6096 (2010). [CrossRef] [PubMed]
- N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express17(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. A27(5), 968–972 (2010). [CrossRef] [PubMed]
- N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010). [CrossRef] [PubMed]
- H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1, 124 (2010).
- Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag.61(12), 5910–5922 (2013). [CrossRef]
- D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express21(12), 14223–14243 (2013). [CrossRef] [PubMed]
- D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010). [CrossRef] [PubMed]
- K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).
- C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys.16(2), 023030 (2014). [CrossRef]
- D. E. Blair, Inversion Theory and Conformal Mapping (American Mathematical Society, 2000).

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