## Analytical solution for wave propagation through a graded index interface between a right-handed and a left-handed material

Optics Express, Vol. 17, Issue 8, pp. 6747-6752 (2009)

http://dx.doi.org/10.1364/OE.17.006747

Acrobat PDF (137 KB)

### Abstract

We have investigated the transmission and reflection properties of structures incorporating left-handed materials with graded index of refraction. We present an exact analytical solution to Helmholtz’ equation for a graded index profile changing according to a hyperbolic tangent function along the propagation direction. We derive expressions for the field intensity along the graded index structure, and we show excellent agreement between the analytical solution and the corresponding results obtained by accurate numerical simulations. Our model straightforwardly allows for arbitrary spectral dispersion.

© 2009 Optical Society of America

## 1. Introduction

1. C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: New Frontiers in Optics,” Adv. Mater. **18**, 1941–1952 (2006). [CrossRef]

2. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of *ε* and *μ*,” Sov. Phys. Usp **10**, 509–514 (1968). [CrossRef]

3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys.: Cond. Matter **10**, 4785–4788 (1998). [CrossRef]

4. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. **47**, 2075–2084 (1999). [CrossRef]

5. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

1. C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: New Frontiers in Optics,” Adv. Mater. **18**, 1941–1952 (2006). [CrossRef]

6. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency,” Phys. Rev. Lett. **102**, 053901 (2009). [CrossRef] [PubMed]

7. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

8. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science **308**, 534–537 (2005). [CrossRef] [PubMed]

9. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

10. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature **450**, 397–401 (2007). [CrossRef] [PubMed]

11. N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials,” Science **317**, 1698–1702 (2007). [CrossRef] [PubMed]

12. P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. **92**, 203111 (2008). [CrossRef]

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

14. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

## 2. Field equations

*iωt*) dependency. Furthermore, we assume that the effective medium approximation can be made and that the materials are isotropic, so that their optical properties can be described by the effective dielectric permittivity and the effective magnetic permeability. For most metamaterials, the effective medium assumption is valid, because their constituents elements are on the subwavelength level. The geometry of the problem is illustrated in Fig. 1. The electric field is directed along the

*y*-axis,

**E**(

**r**) =

*E*(

*x*)

**e**

_{y}, whereas the magnetic field is directed along the

*z*-axis,

**H**(

**r**) =

**H**(

*x*)

**e**

_{z}. The propagation direction of the wave is along the

*x*-axis. Since the fields depend only on the

*x*-coordinate, we have

*ε*=

*ε*(

*ω*,

*x*) and

*μ*=

*μ*,(

*ω*,

*x*) are the frequency-dependent dielectric permittivity and magnetic permeability, respectively.

*E*(

*x*) or

*H*(

*x*):

*ε*(

*x*) and

*μ*(

*x*) may be completely arbitrary, even on space scales faster than the wavelength of the radiation, on the condition of course that the effective medium approximation remains valid.

*F*(

*x*) and

*G*(

*x*) instead of the functions

*E*(

*x*) and

*H*(

*x*) using the following transformations:

*F*(

*x*) and

*G*(

*x*):

*ε*(

*x*) and

*μ*(

*x*), Eqs. (6)–(7) are generally reduced to the hypergeometric equation, allowing for analytical solution in terms of suitable hypergeometric functions.

## 3. Analytical solutions of the field equations

*ρ*is a positive real parameter describing the steepness of the transition from the right-handed material at the left-hand side of the plane

*x*= 0 to the left-handed material at the right-hand side of the plane

*x*= 0. There is no restriction on the functions

*μ*(

_{eff}*ω*) and

*ε*(

_{eff}*ω*) (except of course for such restrictions as the Kramers-Kronig relationships), so that our method allows for arbitrary spectral dispersion. The reader should note that the impedance

*E*

_{0}and

*H*

_{0}are constant amplitudes, and

*ρ*. Let us now choose the solution with the minus sign in the exponent of the expression (12), i.e.,

*E*(

*x*) in the limits

*x*± ∓∞,

**k**

_{-∞}= +

*κ*

**e**

_{x}in the right-handed material far from the interface (

*x*→ ∞). This is a wave that propagates in the +

*x*direction, i.e., a wave propagating to the right. On the other hand, for

*x*→ +∞, the wave has wavevector

**k**

_{+∞}= -

*κ*

**e**

_{x}; this represents a wave of which the phase fronts propagate in the -

*x*direction. However, since we have a left-handed material for

*x*> 0, the energy flux (Poynting’s vector) is still propagating from left to right. This is perfectly consistent with the fact that there is no reflection on this structure. This is also apparent from the fact that ∣ cosh(

*ρx*)

^{-iκ/ρ}∣ = 1, so that ∣

*E*(

*x*) ∣ is constant throughout the structure.

## 4. Comparison with numerical results

*λ*

_{0}= 1 μm and

*ε*

_{eff}(

*λ*

_{0}) = μ

_{eff}(

*λ*

_{0}) = 1. Figures 3(a)–(b) and Figs. 3(c)–(d) are for different transition steepness. We see that there is excellent agreement between the analytical and numerical results.

## 5. Conclusion

## Acknowledgments

## References and links

1. | C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: New Frontiers in Optics,” Adv. Mater. |

2. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

3. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys.: Cond. Matter |

4. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. |

5. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

6. | P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency,” Phys. Rev. Lett. |

7. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

8. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science |

9. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

10. | K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature |

11. | N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials,” Science |

12. | P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. |

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

14. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

15. | S. A. Ramakrishna and J. B. Pendry, “Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry,” Phys. Rev. B |

16. | A. O. Pinchuk and G. C. Schatz, “Metamaterials with gradient negative index of refraction,” J. Opt. Soc. Am. A |

17. | C. G. Parazzoli, B. E. C. Koltenbah, R. B. Greegor, T. A. Lam, and M. H. Tanielian, “Eikonal equation for a general anisotropic or chiral medium: application to a negative-graded index-of-refraction lens with an anisotropic material,” J. Opt. Soc. Am. B |

18. | N. Dalarsson, M. Maksimovic, and Z. Jaksic, “A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile,” Science of Sintering |

19. | D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “A gradient index metamaterial,” Phys. Rev. E |

20. | N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. |

21. | P. Yeh, |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 27, 2009

Revised Manuscript: April 6, 2009

Manuscript Accepted: April 6, 2009

Published: April 8, 2009

**Citation**

Mariana Dalarsson and Philippe Tassin, "Analytical solution for wave propagation through a graded index interface between a right-handed and a left-handed material," Opt. Express **17**, 6747-6752 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6747

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### References

- C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-Index Materials: New Frontiers in Optics," Adv. Mater. 18, 1941-1952 (2006). [CrossRef]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and," Sov. Phys. Usp 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in ThinWire Structures," J. Phys.: Cond. Matter 10, 4785-4788 (1998). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075- 2084 (1999). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009). [CrossRef] [PubMed]
- J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005). [CrossRef] [PubMed]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Optical Hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-8256 (2006). [CrossRef] [PubMed]
- K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007). [CrossRef] [PubMed]
- N. Engheta, "Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials," Science 317, 1698-1702 (2007). [CrossRef] [PubMed]
- P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008). [CrossRef]
- U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- S. A. Ramakrishna and J. B. Pendry, "Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry," Phys. Rev. B 69, 115115 (2004). [CrossRef]
- A. O. Pinchuk and G. C. Schatz, "Metamaterials with gradient negative index of refraction," J. Opt. Soc. Am. A 24, A39-A44 (2007). [CrossRef]
- C. G. Parazzoli, B. E. C. Koltenbah, R. B. Greegor, T. A. Lam, and M. H. Tanielian, "Eikonal equation for a general anisotropic or chiral medium: application to a negative-graded index-of-refraction lens with an anisotropic material," J. Opt. Soc. Am. B 23, 439-450 (2006). [CrossRef]
- N. Dalarsson, M. Maksimovic, and Z. Jaksic, "A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile," Science of Sintering 39, 185-191 (2007). [CrossRef]
- D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005). [CrossRef]
- N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, "Metamaterials: electromagnetic enhancement at zero-index transition," Opt. Lett. 33, 2350-2352 (2008). [CrossRef] [PubMed]
- P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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