## Layered superlensing in two-dimensional photonic crystals

Optics Express, Vol. 14, Issue 23, pp. 11178-11183 (2006)

http://dx.doi.org/10.1364/OE.14.011178

Acrobat PDF (442 KB)

### Abstract

We demonstrate layered superlensing in two-dimensional photonic crystals structured by both square and triangular lattices. In virtue of equifrequency contour analysis and FDTD calculation, both near field and far field imaging are displayed. Layered superlensing consisting of only triangular lattice photonic crystal is also studied and it exhibits more flexibility than the single layer counterpart. That is, the objective distance can be changed freely while keeping the image distance constant and vice versa. Hence, farther field imaging is achieved.

© 2006 Optical Society of America

## 1. Introduction

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

2. R. W. Ziolkowski, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E **64**, 056625 (2001). [CrossRef]

3. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media- media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. **31**, 129–133 (2001). [CrossRef]

4. 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**, 4184–4187 (2000). [CrossRef] [PubMed]

5. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**, 4773–4776 (1996). [CrossRef] [PubMed]

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

7. L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Cheng, T. M. Grzegorczyk, and J. A. Kong, “Experimental study on several left-handed metamaterials,” PIER **51**, 249–279 (2005). [CrossRef]

8. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B **62**, 10696–10705 (2000). [CrossRef]

10. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a Negative Refractive Index,” Phys. Rev. Lett. **90**, 107402 (2003). [CrossRef] [PubMed]

9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B **62**, 10696–10705 (2000). [CrossRef]

13. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B **65**, 201104 (2002). [CrossRef]

13. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B **65**, 201104 (2002). [CrossRef]

14. X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two dimensional photonic crystal,” Opt. Express **12**, 2919–2924 (2004). [CrossRef] [PubMed]

15. Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B **68**, 245110 (2003). [CrossRef]

21. C. Li, J. M. Holt, and A. L. Efros, “Far-field imaging by the Veselago lens made of a photonic crystal,” J. Opt. Soc. Am. B **23**, 490–497 (2006). [CrossRef]

22. C. Y. Li, J. M. Holt, and A. L. Efros, “Imaging by the Veselago lens based upon a two-dimensional photonic crystal with a triangular lattice,” J. Opt. Soc. Am. B **23**, 963–968 (2006). [CrossRef]

23. C. Shen, K. Michielsen, and H. De Raedt, “Image transfer by cascaded stack of photonic crystal and air layers,” Opt. Express **14**, 879–886 (2006). [CrossRef] [PubMed]

## 2. Numerical method

25. M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Optics Commun. **80**, 3–4 (1991). [CrossRef]

26. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

28. S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propagat. **44**, 1630–1639 (1996). [CrossRef]

*a*. Time and frequency are then expressed in units of

*a*/

*c*and

*c*/

*a*, respectively, where

*c*denotes the velocity of light in vacuum.

*f*=

*ωa*/2

*πc*is the dimensionless frequency. Square meshes with mesh size

*δ*=0.02

*a*and time step Δ

*t*=0.95

*δ*/√2

*c*were employed. More than 10000 time steps were run to reach the steady state.

## 3. Layered superlens structured by triangular lattice PC

*r*=0.4

*a*(

*a*is the lattice constant) embedded in dielectric matrix with permittivity 12.96. Its photonic bands (TM modes) are illustrated in Fig. 1(a) showing negative refraction of the first case at frequency 0.306 (normalized by

*ωa*/2

*πc*) with effective refractive index

*n*=-1 [9

9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B **62**, 10696–10705 (2000). [CrossRef]

*a*/4-0.2)

*a*.

*D*+

_{o}*D*=

_{i}*T*holds [14

14. X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two dimensional photonic crystal,” Opt. Express **12**, 2919–2924 (2004). [CrossRef] [PubMed]

*D*is the distance from the object to one side of the PC slab (we called it objective distance hereafter),

_{o}*D*is the distance from the other side of the PC slab to the image (we called it image distance hereafter), and

_{i}*T*is the thickness of the slab. If the objective distance

*D*is greater than the thickness of the PC slab

_{o}*T*, the image won’t appear.

*T*) is to draw the object nearer by a distance of 2*

_{o}*T*and a virtual image is formed, hence we call the first lens objective lens. The image lens sees the virtual image and a real one is formed, hence we call the second lens image lens. So, even the objective distance is larger than the thickness of objective lens a real image is still formed which farther free the object from near field imaging. Likewise, even the image distance is greater than the thickness of the image lens a image is still formed as pictured in Fig. 2(a).

_{o}23. C. Shen, K. Michielsen, and H. De Raedt, “Image transfer by cascaded stack of photonic crystal and air layers,” Opt. Express **14**, 879–886 (2006). [CrossRef] [PubMed]

## 4. Layered superlens structured by both square and triangular lattice PC

*r*=

*a*(

_{tri}*a*is the lattice constant) embedded in dielectric matrix with permittivity 12.96 while the square lattice photonic crystal consists of a periodic array of infinitely long, cylindrical dielectric rods with radius

_{tri}*r*=0.3

*a*(

_{squa}*a*is the lattice constant) and permittivity 14 embedded in air. Their photonic bands (TM modes) are shown in Fig. 1, respectively. the triangular lattice PC show negative refractive of the first case at frequency 0.306 (normalized by

_{squa}*ωa*/2

_{tri}*πc*) with effective refraction index

*n*=-1 [9

**62**, 10696–10705 (2000). [CrossRef]

*ωa*/2

*squa**πc*) of negative refraction of the second case discussed above [13

13. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B **65**, 201104 (2002). [CrossRef]

*a*, pegging 0.306*

_{tri}*ωa*/2

_{tri}*πc*to 0.192*

*ωa*/2

_{squa}*πc*. Specifically, the triangular lattice constant

*a*is 0.306/0.192 that of the square lattice constant

_{tri}*a*.

_{squa}**65**, 201104 (2002). [CrossRef]

*n*form -1 for different angles of incidence. Anyway, superlensing effect of square-triangular layered PC was observed in the transversal direction and the field propagation map are shown in Fig. 4(a) and Fig. 4(b). Similarly, the triangularsquare lattice PC is also studied and the field propagation maps are displayed in Fig. 4(c) and Fig. 4(d).

## 5. Discussions and conclusions

## Acknowledgments

## References and links

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

2. | R. W. Ziolkowski, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E |

3. | I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media- media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. |

4. | 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. |

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

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

7. | L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Cheng, T. M. Grzegorczyk, and J. A. Kong, “Experimental study on several left-handed metamaterials,” PIER |

8. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

9. | M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B |

10. | S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a Negative Refractive Index,” Phys. Rev. Lett. |

11. | S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B |

12. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature (London) |

13. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B |

14. | X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two dimensional photonic crystal,” Opt. Express |

15. | Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B |

16. | S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction,” Appl. Phys. Lett. |

17. | X. Zhang, “Image resolution depending on slab thickness and object distance in a two-dimensional photoniccrystal-based superlens,” Phys. Rev. B |

18. | X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B |

19. | A. Martinez and J. Marti, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B |

20. | A. Martinez and J. Marti, “Analysis of wave focusing inside a negative-index photonic-crystal slab,” Opt. Express |

21. | C. Li, J. M. Holt, and A. L. Efros, “Far-field imaging by the Veselago lens made of a photonic crystal,” J. Opt. Soc. Am. B |

22. | C. Y. Li, J. M. Holt, and A. L. Efros, “Imaging by the Veselago lens based upon a two-dimensional photonic crystal with a triangular lattice,” J. Opt. Soc. Am. B |

23. | C. Shen, K. Michielsen, and H. De Raedt, “Image transfer by cascaded stack of photonic crystal and air layers,” Opt. Express |

24. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

25. | M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Optics Commun. |

26. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

27. | A. Taflove and S. C. Hagness, |

28. | S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propagat. |

**OCIS Codes**

(110.2960) Imaging systems : Image analysis

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: August 23, 2006

Revised Manuscript: October 21, 2006

Manuscript Accepted: October 22, 2006

Published: November 13, 2006

**Citation**

Haifei Zhang, Linfang Shen, Lixin Ran, Yu Yuan, and Jin Au Kong, "Layered superlensing in two-dimensional photonic crystals," Opt. Express **14**, 11178-11183 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11178

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### References

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of and," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- R. W. Ziolkowski, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001). [CrossRef]
- I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media- media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001). [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, 4184-4187 (2000). [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, 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. Microwave Theory Technol. 47, 2075-2084 (1999). [CrossRef]
- L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Cheng, T. M. Grzegorczyk, and J. A. Kong, "Experimental study on several left-handed metamaterials," PIER 51, 249-279 (2005). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000). [CrossRef]
- S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a Negative Refractive Index," Phys. Rev. Lett. 90, 107402 (2003). [CrossRef] [PubMed]
- S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003). [CrossRef]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature (London) 423, 604-605 (2003). [CrossRef] [PubMed]
- C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002). [CrossRef]
- X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two dimensional photonic crystal," Opt. Express 12,2919-2924 (2004). [CrossRef] [PubMed]
- Z.-Y. Li and L.-L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003). [CrossRef]
- S. Xiao, M. Qiu, Z. Ruan, and S. He, "Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction," Appl. Phys. Lett. 85, 4269-4271 (2004). [CrossRef]
- X. Zhang, "Image resolution depending on slab thickness and object distance in a two-dimensional photoniccrystal-based superlens," Phys. Rev. B 70, 195110 (2004). [CrossRef]
- X. Wang and K. Kempa, "Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs," Phys. Rev. B 71, 085101 (2005). [CrossRef]
- A. Martinez and J. Marti, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005). [CrossRef]
- A. Martinez and J. Marti, "Analysis of wave focusing inside a negative-index photonic-crystal slab," Opt. Express 13, 2858-2868 (2005). [CrossRef] [PubMed]
- C. Li, J. M. Holt, and A. L. Efros, "Far-field imaging by the Veselago lens made of a photonic crystal," J. Opt. Soc. Am. B 23, 490-497 (2006). [CrossRef]
- C. Y. Li, J. M. Holt, and A. L. Efros, "Imaging by the Veselago lens based upon a two-dimensional photonic crystal with a triangular lattice," J. Opt. Soc. Am. B 23, 963-968 (2006). [CrossRef]
- C. Shen, K. Michielsen, and H. De Raedt, "Image transfer by cascaded stack of photonic crystal and air layers," Opt. Express 14, 879-886 (2006). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- M. Plihal, A. Shambrook, and A. A. Maradudin, "Two-dimensional photonic band structures," Optics Commun. 80, 3-4 (1991). [CrossRef]
- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics—the Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2000).
- S. D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices," IEEE Trans. Antennas Propagat. 44, 1630-1639 (1996). [CrossRef]

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