## Squeezing and expanding light without reflections via transformation optics |

Optics Express, Vol. 19, Issue 4, pp. 3562-3575 (2011)

http://dx.doi.org/10.1364/OE.19.003562

Acrobat PDF (1452 KB)

### Abstract

We study the reflection properties of squeezing devices based on transformation optics. An analytical expression for the angle-dependent reflection coefficient of a generic three-dimensional squeezer is derived. In contrast with previous studies, we find that there exist several conditions that guarantee no reflections so it is possible to build transformation-optics-based reflectionless squeezers. Moreover, it is shown that the design of antireflective coatings for the non-reflectionless case can be reduced to matching the impedance between two dielectrics. We illustrate the potential of these devices by proposing two applications in which a reflectionless squeezer is the key element: an ultra-short perfect coupler for high-index nanophotonic waveguides and a completely flat reflectionless hyperlens. We also apply our theory to the coupling of two metallic waveguides with different cross-section. Finally, we show how the studied devices can be implemented with non-magnetic isotropic materials by using a quasi-conformal mapping technique.

© 2011 OSA

## 1. Introduction

## 2. Theory

*E*field intensities in the input and output media are consistent with the conservation of total power flow. It is also worth mentioning that the squeezer provides a compressed version of the fields inside it. This compression is transferred to the outside world near the squeezer. However, once the electromagnetic wave has exited the squeezer, it is subject to the diffraction laws of the output medium. Thus, the Gaussian beam exiting the squeezer will diverge as it propagates. This is mainly observed in Fig. 2(b), as this divergence is faster in air than in the medium with n = 4.

## 3. Applications

10. D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. **38**(7), 949–955 (2002). [CrossRef]

11. G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. **92**(13), 131101 (2008). [CrossRef]

2. L. Vivien, S. Laval, E. Cassan, X. Le Roux, and D. Pascal, “2-D taper for low-loss coupling between polarization-insensitive microwaveguides and single-mode optical fibers,” J. Lightwave Technol. **21**(10), 2429–2433 (2003). [CrossRef]

12. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. **11**(1), 232–240 (2005). [CrossRef]

*λ*= 1.5μm, and a waveguide with sub-wavelength width

*w*= 1 μm and refractive index

*n*= 4, as in the example of Fig. 2 (the problem would be very similar if we used silicon, since

*n*= 3.45 in this band).

_{Si}*z*due to an inefficient mode matching. To extend the application to larger compression factors, antireflective coatings are necessary. In a 3D problem, compression in both transversal directions is demanded. Since in this application we have normal incidence, a squeezer fulfilling condition 3 with the proper antireflective coating can be employed. As for its size, the squeezer can be as short as desired. Nonetheless, the necessary constitutive parameters become extreme as we reduce

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

14. B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**(8), 85103 (2009). [CrossRef]

15. V. M. Shalaev, “Physics. Transforming light,” Science **322**(5900), 384–386 (2008). [CrossRef] [PubMed]

16. Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. **94**(20), 203108 (2009). [CrossRef]

15. V. M. Shalaev, “Physics. Transforming light,” Science **322**(5900), 384–386 (2008). [CrossRef] [PubMed]

17. A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. **32**(23), 3432–3434 (2007). [CrossRef] [PubMed]

18. D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. **10**(11), 115039 (2008). [CrossRef]

*z*direction is made, its width must be extremely small. The above theoretical results, enables us to design an expander that implements a completely flat reflectionless hyperlens. Following a similar approach to that of [17

17. A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. **32**(23), 3432–3434 (2007). [CrossRef] [PubMed]

*z*direction as was shown above), while expanding the fields in

*x*direction at the same time. This transformation can be described by

19. 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]

20. X. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express **18**(10), 10168–10176 (2010). [CrossRef] [PubMed]

*a*, to another waveguide W2 of transverse size

*a*/2, where we only excite the first TE mode at the left end of waveguide W1. In Fig. 6 , we show the norm of the electric field for different solutions to this problem. In Fig. 6(a), no coupler is used and we just linearly change the metallic boundary of the waveguide. The strong modulation appearing in waveguide W1 and the transition waveguide indicates that high reflections are taking place at the boundary between the transition waveguide and waveguide W2.

*a*, to a transverse size of

*a*/2 (

19. 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]

20. X. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express **18**(10), 10168–10176 (2010). [CrossRef] [PubMed]

*S*

_{11}as a function of the refractive index of the filling medium of waveguide W2, for the case where the coupler is used [as in Figs. 6(b) and 6(c)]. A pronounced minimum is clearly seen very close to

## 4. Practical implementation

19. 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]

20. X. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express **18**(10), 10168–10176 (2010). [CrossRef] [PubMed]

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

21. 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]

## 5. Conclusion

## Appendix A. Derivation of the reflection and transmission coefficients

*k*

_{y}= 0. In addition, the problem is simplified due to the fact that both the auxiliary layer and the outer medium are characterized by diagonal constitutive parameters. Given these simplifications and considering that

## Acknowledgements

## References and links

1. | R. Yang, M. A. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express |

2. | L. Vivien, S. Laval, E. Cassan, X. Le Roux, and D. Pascal, “2-D taper for low-loss coupling between polarization-insensitive microwaveguides and single-mode optical fibers,” J. Lightwave Technol. |

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

4. | U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. |

5. | 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. |

6. | M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express |

7. | W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” |

8. | T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. |

9. | E. Hecht, Optics, (Addison Wesley, 4th edition, 2001). |

10. | D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. |

11. | G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. |

12. | T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. |

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

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

15. | V. M. Shalaev, “Physics. Transforming light,” Science |

16. | Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. |

17. | A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. |

18. | D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. |

19. | P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express |

20. | X. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express |

21. | 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 |

**OCIS Codes**

(220.3630) Optical design and fabrication : Lenses

(230.0230) Optical devices : Optical devices

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 26, 2010

Revised Manuscript: December 30, 2010

Manuscript Accepted: January 3, 2011

Published: February 9, 2011

**Citation**

C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, "Squeezing and expanding light without reflections via transformation optics," Opt. Express **19**, 3562-3575 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3562

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### References

- R. Yang, M. A. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express 16(24), 20142–20148 (2008). [CrossRef] [PubMed]
- L. Vivien, S. Laval, E. Cassan, X. Le Roux, and D. Pascal, “2-D taper for low-loss coupling between polarization-insensitive microwaveguides and single-mode optical fibers,” J. Lightwave Technol. 21(10), 2429–2433 (2003). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006). [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]
- 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]
- W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231v1 (2008).
- T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005). [CrossRef]
- E. Hecht, Optics, (Addison Wesley, 4th edition, 2001).
- D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002). [CrossRef]
- G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008). [CrossRef]
- T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005). [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]
- B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009). [CrossRef]
- V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]
- Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. 94(20), 203108 (2009). [CrossRef]
- A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32(23), 3432–3434 (2007). [CrossRef] [PubMed]
- D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. 10(11), 115039 (2008). [CrossRef]
- 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]
- X. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express 18(10), 10168–10176 (2010). [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. Express 18(6), 6089–6096 (2010). [CrossRef] [PubMed]

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