## Negative refractions in two-dimensional photonic crystals formed by holographic lithography

Optics Express, Vol. 15, Issue 13, pp. 8003-8009 (2007)

http://dx.doi.org/10.1364/OE.15.008003

Acrobat PDF (526 KB)

### Abstract

We present a study on negative refractions in the four lowest bands of two-dimensional (2D) square lattices formed by holographic lithography (HL) and compare these features with those of a lattice of the same kind but with regular dielectric columns. The plane wave calculations and FDTD simulations have shown that in some bands or for some interfaces the negative refraction can only happen in holographic structures, and generally the rightness of holographic structures and regular structures of the same kind may be different.

© 2007 Optical Society of America

## 1. Introduction

2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

*μ*<0) interconnected with a set of metallic rods (

*ε*<0). Photonic crystals (PhCs) made of periodically modulated dielectric or metallic materials can also show effective negative refraction [6

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

*ε*or

*μ*of a PhC is impossible, many anomalous effects such as negative refraction and “super prism” [7

7. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B **58**, R10096–10099 (1998). [CrossRef]

8. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature **423**, 604–605 (2003). [CrossRef] [PubMed]

## 2. Structures

16. L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, “Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap” J. Opt. Soc. Am. B **23**, 1708–1711 (2006). [CrossRef]

*c*and a threshold intensity

*I*

_{t}, which is a specific value the region with light intensity below it can be removed and the region above it will remain due to photopolymerization for negative photoresist, we may wash away the region of

*I*<

*I*

_{t}to get a normal structure. By filling this structure with a material of high dielectric constant and then removing the template, an inverse structure can be obtained [19

19. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature **404**, 53–56 (2000). [CrossRef] [PubMed]

*ε*=11.4 for gallium arsenide as an example and choose

*c*= 0.16 and

*I*

_{t}= 2.72 which lead to the largest band gap as we analyzed before [16

16. L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, “Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap” J. Opt. Soc. Am. B **23**, 1708–1711 (2006). [CrossRef]

*I*

_{t}= 2.72 respectively, and (c) is a regular structure having the same filling ratio (FR) as (a) but circular columns with

*r*= 0.364

*a*, here

*r*is the radius of column and

*a*the lattice constant.

## 3. Wave vector diagrams and FDTD results

*f*=

*ωa*/2π

*c*is used, where

*c*is the velocity of light in vacuum.

*K*_{i}, refractive wave vector

*K*_{f}and group velocity

*V*_{g}, respectively; and the relative frequency in air is

*f*= 0.28 for (a) and (c), and

*f*= 0.24 for (b). From Figs. 2(a) and 2(b) we can see that in the outside region of the vicinity of M point EFCs are convex having an inward-pointing group velocity

*V*_{g}, but the black dashed line intersects two points at the EFC of photonic lattices, the upper one has

*V*_{g}pointing towards the source, while the bottom one has

*V*_{g}pointing away from the source. This means that only the bottom point contributes to a propagating beam or a transmitted beam. The refractive wave vector

*K*_{f}pointing to the intersect point is on the same side of the normal as

*K*_{i}is, thus right-handed (RH) negative refractions happen with

*K*_{f}∙

*V*_{g}> 0 and

*V*_{g}pointing to the negative direction [13

13. R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express **13**, 8596–8605 (2005). [CrossRef] [PubMed]

*V*_{g}and positive refractions. In order to get a comprehensive comparison, we have also investigated the regular structures with

*r*/

*a*changing form 0.1 ~ 0.5, since the columns will overlap and not be circles if

*r*/

*a*> 0.5. The frequencies of EFCs reduce with increasing column radius, and we find that negative refractions only happen in one situation of

*r*/

*a*= 0.5 when the columns are just joined together with each other. This means that negative refractions cannot occur in regular structures with separated circular dielectric columns for the first band of TE modes. To demonstrate these phenomena more clearly, we give the FDTD simulated wave patterns of TE 1 in Figs. 3(a), 3(b) and 3(c) for the cases of Figs. 2(a), 2(b) and 2(c), respectively. From Fig. 3 we can see that

*V*_{g}points to the negative direction in (a) and (b) and positive in (c), exactly coincident with the analysis of Fig. 2. The

*V*_{g}in holographic normal structure (b) is the largest, since the corresponding EFCs have the densest distribution and the largest gradient. The EFC plots are sparser for the inverse structure, and sparsest for regular lattices.

*K*_{f}∙

*V*_{g}< 0 with

*V*_{g}pointing to the negative direction. The corresponding FDTD simulations of refractions for these two structures are given in GIF image format with increasing incident angles from 5° to 60° in Figs. 5(a) and 5(b), where the incident frequency is taken as 0.35 and 0.32 respectively. From these images we can see clearly that in almost the whole angular range negative refraction exists in the two structures. On the contrary, for lattices with regular circular columns in Fig. 4(c), there is only one convex EFC of

*f*= 0.36 around the origin, thus negative refraction happens only for the waves incident across ΓM interface; and different from the former two lattices, the negative refraction here is similar as in Figs. 2 (a) and 2(b) with

*K*_{f}∙

*V*_{g}> 0 and

*V*_{g}is negative. The FDTD simulations given in Fig. 6 are the wave propagation patterns of TE waves at an incidence of 20° on the ΓM interface, (a), (b) and (c) are for the three structures respectively. The consistency of pointing direction of

*V*_{g}in these figures demonstrated our analysis again.

*r*/

*a*is large enough the EFCs of regular structures become similar to those of holographic structures, and negative refraction may happen in these situations similarly as in HL structures, but the column radius is restricted within a narrow range.

*r*/

*a*= 0.5. And for TE2 mode, holographic structures can yield left-handed negative refractions when EMWs are incident either on ΓM or ΓX interface, for EFCs around Γ point are convex and have inward gradients; on the other hand, the regular lattice can only achieve right-handed negative refractions for ΓM interface incident when

*r*/

*a*is not large enough. The most similar properties for these three structures occur in the third band, where LH negative refraction happens in all lattices for a small wave band as the shapes of their EFCs are quite alike. On the contrary, the propagation properties in holographic and regular structures with a small ratio of

*r*/

*a*in band 4 are considerably different. The overlap of bands 3 and 4 in holographic structures leads to dense EFCs and negative refraction in band 4 lacked in regular structure. The negative refraction may appear in regular structures only for a large enough

*r*/

*a*. In table 1 we give all frequency bands that could intricate negative refractions for each structure, from it one can see that negative refraction seems more likely to happen in holographic structures, at least they cover a larger wave band.

## 5. Conclusions

## Acknowledgment

## References and links

1. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Uspekhi Fiz. Nauk |

2. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

3. | 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 |

4. | J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. |

5. | L. Chen, S. L. He, and L. F. Shen, “Finite-size effects of a left-handed material Slab on the image quality,” Phys. Rev. Lett. |

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

7. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B |

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

9. | R. Gajic, F. Kuchar, R. Meisels, J. Radovanovic, K. Hingerl, J. Zarbakhsh, J. Stampfl, and A. Woesz, “Physical and materials aspects of photonic crystals for microwaves and millimetre waves,” Zeitschrift fuer Metallkunde |

10. | K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, “Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens,” Phys. Rev. B. |

11. | P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and lefthanded electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. |

12. | S. Foteinopoulou1 and C. M. Soukoulis “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B. |

13. | R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express |

14. | E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. |

15. | X. Y. Ao and S. L. He, “Three-dimensional photonic crystal of negative refraction achieved by interference lithography,” Opt. Lett. |

16. | L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, “Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap” J. Opt. Soc. Am. B |

17. | L. Z. Cai, C. S. Feng, M. Z. He, and X. L. Yang, “Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps” Opt. Express |

18. | K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. |

19. | M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature |

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 13, 2007

Revised Manuscript: May 23, 2007

Manuscript Accepted: May 23, 2007

Published: June 12, 2007

**Citation**

X. X. Shen, X. L. Yang, L. Z. Cai, G. Y. Dong, X. F. Meng, X. F. Xu, and Y. R. Wang, "Negative refractions in two-dimensional photonic crystals formed by holographic lithography," Opt. Express **15**, 8003-8009 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8003

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### References

- V. G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Usp. Fiz. Nauk 92, 517-526 (1967).
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans Microwave Theory Tech 47, 2075-2084 (1999). [CrossRef]
- J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901-083904 (2003). [CrossRef] [PubMed]
- L. Chen, S. L. He, L. F. Shen, "Finite-size effects of a left-handed material Slab on the image quality," Phys. Rev. Lett. 92, 107404-1-4 (2004). [CrossRef] [PubMed]
- M. Notomi, "Theory of light propagation in strong modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B. 62, 10696-10705 (2000). [CrossRef]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-10099 (1998). [CrossRef]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003). [CrossRef] [PubMed]
- R. Gajic, F. Kuchar, R. Meisels, J. Radovanovic, K. Hingerl, J. Zarbakhsh, J. Stampfl, and A. Woesz, "Physical and materials aspects of photonic crystals for microwaves and millimetre waves," Z. Metallkd. 95, 618-623 (2004).
- K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B. 70, 205125-1-5 (2004). [CrossRef]
- P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and lefthanded electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401-1-4 (2004). [CrossRef] [PubMed]
- S. Foteinopoulou1 and C. M. Soukoulis "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B. 72, 165112-1-20 (2005). [CrossRef]
- R. Gajiæ, R. Meisels, F. Kuchar, and K. Hingerl, "Refraction and rightness in photonic crystals," Opt. Express 13, 8596-8605 (2005). [CrossRef] [PubMed]
- E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case employing nonspherical atoms," Phys. Rev. Lett. 67, 2295-2298 (1991). [CrossRef] [PubMed]
- X. Y. Ao and S. L. He, "Three-dimensional photonic crystal of negative refraction achieved by interference lithography," Opt. Lett. 29, 2542-2544 (2004). [CrossRef] [PubMed]
- L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, "Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap" J. Opt. Soc. Am. B 23, 1708-1711 (2006). [CrossRef]
- L. Z. Cai, C. S. Feng, M. Z. He, and X. L. Yang, "Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps" Opt. Express 13, 4325-4330 (2005). [CrossRef] [PubMed]
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
- M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000). [CrossRef] [PubMed]

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