## Anomalous refractive effects in honeycomb lattice photonic crystals formed by holographic lithography |

Optics Express, Vol. 18, Issue 16, pp. 16302-16308 (2010)

http://dx.doi.org/10.1364/OE.18.016302

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### Abstract

We have investigated for the first time the anomalous refractive effects of a photonic crystal (PhC) formed by holographic lithography (HL) with triangular rods arranged in a honeycomb lattice in air. Possibilities of left-handed negative refraction and superlens are discussed for the case of TM2 band with the index contrast *n* = 3.4:1. In contrast to the conventional honeycomb PhC made of regular rods in air, the HL PhCs show left-handed negative refraction over a wider and higher frequency range with high transmissivity (>90%), and the effective indices quite close to −1 for a wide range of incident angles with a larger all-angle left-handed negative refraction (AALNR) frequency range (Δ*ω*/*ω* ≈14.8%). Calculations and FDTD simulations demonstrate the high-performance negative refraction properties can happen in the holographic structures for a wide filling ratio and can be modulated by changing the filling ratio easily.

© 2010 OSA

## 1. Introduction

1. V. G. Veselago, “Electrodynamics of Substances with Simultaneously Negative Values of Sigma and Mu,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

3. S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. **68**(2), 449–521 (2005). [CrossRef]

4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**(5514), 77–79 (2001). [CrossRef] [PubMed]

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

*ε*<0), such media are realized by metamaterials composed of metal/dielectric composites. However, the absorption loss in the metal limits potential optical applications. In contrast to left-handed materials, photonic crystals made of synthetic periodic dielectric materials can exhibit an extraordinarily high, nonlinear dispersion such as negative refraction and self-focusing properties that are solely determined by the characteristics of their band structures and equal frequency contours (EFC) [5

5. K. Ren, Z. Y. Li, X. B. Ren, S. Feng, B. Y. Cheng, and D. Z. Zhang, “Three-dimensional light focusing in inverse opal photonic crystals,” Phys. Rev. B **75**(11), 115108 (2007). [CrossRef]

7. G. Sun, A. S. Jugessur, and A. G. Kirk, “Imaging properties of dielectric photonic crystal slabs for large object distances,” Opt. Express **14**(15), 6755–6765 (2006). [CrossRef] [PubMed]

14. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**(25), 3152–3155 (1990). [CrossRef] [PubMed]

15. S. D. Gedney, “An Anisotropic PML Absorbing Media for FDTD Simulation of Fields in Lossy Dispersive Media,” Electromagnetics **16**(4), 399–415 (1996). [CrossRef]

## 2. Structures and calculations

16. X. L. Yang, L. Z. Cai, and Q. Liu, “Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,” Opt. Express **11**(9), 1050–1055 (2003). [CrossRef] [PubMed]

*I*

_{t}, 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 [17

17. A. J. Turberfield, M. Campbell, D. N. Sharp, M. T. Harrison, and R. G. Denning, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature **404**(6773), 53–56 (2000). [CrossRef] [PubMed]

*ε*= 11.56 (i.e.

*n*= 3.4) to analyze the dispersion characteristics of the holographic PhCs. With the increase of intensity threshold

*I*

_{t}the pattern of inverse structure changes from a noncontinuous to a continuous one as shown in Fig. 1 , which consists of honeycomb lattice with trigonal rods arranged in air. When

*I*

_{t}= 2.0, corresponding to the filling ratio of

*f*= 25.6%, the rods just begin to be connected and then the veins become thicker with

*I*

_{t}increasing.

*I*

_{t}= 2.0 in the four lowest bands is plotted in Fig. 2 , in which the inset gives the corresponding irreducible Brillouin zone and three high-symmetry lattice points. The light line in a vacuum (

*ω*=

*ck*) separates the shadow region (

*ω*>

*ck*) and the shadowless region (

*ω*<

*ck*). As a rule, the modes in shadow region are oscillatory in air, the photonic bands above the light cone can be probed experimentally with polarized-angular dependent reflectivity measurements [18

18. S. Inoue and Y. Aoyagi, “Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding,” Phys. Rev. B **69**(20), 205109 (2004). [CrossRef]

*ωa/*2πc (where c is the light velocity and

*a*is the lattice constant), exists between the second and third bands. The frequencies are normalized as

*ωa/*2πc (or

*a/*λ). The symmetrical point Г is the Brillouin center and the corresponding frequencies of the second and third bands decrease with the operational reference point shifting from Г to the vicinity points M or K, which indicate that Left-handed refractive effect is possibly in the PhC around the point Г [10

10. R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “All-angle left-handed negative refraction in Kagomé and honeycomb lattice photonic crystals,” Phys. Rev. B **73**(16), 165310 (2006). [CrossRef]

9. C. Y. Luo, S. G. Johnson, J. D. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B **65**(20), 201104 (2002). [CrossRef]

*ω*<0.348, the air EFCs are surrounded by the corresponding one in the PhC, and

*ω*≤ 0.5 × 2πc/

*a*

_{s}(where

*a*

_{s}is the ГK interface period, here

*a*

_{s}=

*a*). The foregoing analyses reveal that the necessary conditions for AALNR [9

9. C. Y. Luo, S. G. Johnson, J. D. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B **65**(20), 201104 (2002). [CrossRef]

*ω*/

*ω*≈14.8%) in the HL honeycomb PhC, which is higher and wider than the result from 0.2 to 0.218 (Δ

*ω*/

*ω*≈8%) for the honeycomb lattice PhC composed of regular circle columns [10

10. R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “All-angle left-handed negative refraction in Kagomé and honeycomb lattice photonic crystals,” Phys. Rev. B **73**(16), 165310 (2006). [CrossRef]

*v**[19]. The propagation direction of light beam in any medium is given by the energy velocity vector. According to definition the group velocity vector*

_{gr}

*v**= ∇*

_{gr}

_{k}*ω*is always oriented perpendicular to the EFC in the direction along which the frequency is increasing. For the second band of the PhC, the

*v**is pointed inward from the EFC, and the phase velocity vector*

_{gr}

*v**and*

_{ph}

*v**can be antiparallel like in Veselago’s metamaterials [1*

_{gr}1. V. G. Veselago, “Electrodynamics of Substances with Simultaneously Negative Values of Sigma and Mu,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

*n*

_{eff}can be obtained. The band structure predicts the behavior of the fundamental wavevector, which can be used to calculate the effective index. We define an effective index as the ratio of the (local) fundamental wavenumber and the wavenumber in free space. Figure 4 shows the effective index

*n*

_{eff}in a wide frequency region from 0.30 to 0.39 augmenting with the increasing frequency, where the blue solid line denotes

*n*

_{eff}= −1. The frequency of

*ω*= 0.348, corresponding to the effective index

*n*

_{eff}= −1, should be the optimal frequency for a 2D photonic-crystal-based superlens. In Fig. 3, the normal is along ΓM direction, the blue circle represents the EFC of

*ω*= 0.348 in air, the dashed line means the conservation of the parallel components of wave vectors, and the blue, green and red arrows denote the directions of incident wave vector

*K**, refractive wave vector*

_{i}

*K**and group velocity*

_{r}

*V*_{gr}, respectively, and the blue circle is of comparable size with the corresponding PhC EFC, where

*K**makes an angle of 30° with the normal ΓM and*

_{i}

*K*_{r}·

*V*_{gr}< 0 with

*V*_{gr}pointing to the negative direction with the same angle to the normal, which demonstrates the effective index are quite close to −1.

*f*and the effective index

*n*

_{eff}. Figure 5 gives the trend of frequency range varying with filling ratio for anomalous refractive effect, where the red solid line represents the optimal frequencies for PhC superlens with

*n*

_{eff}= −1. When the filling ratio varying from 4% to 48%, the frequency scope decreases from 0.143 to the minimum of 0.086 at

*f*= 15% (or

*I*

_{t}= 1.85) and then increases to 0.115. By numerical calculations we know that the minimum

*n*

_{eff}varies from −1.18 to −2.38 and the optimal frequency with

*n*

_{eff}= −1 (the red solid line) falls from 0.506 to 0.303 with the filling ratio increasing, which imply it is easier to acquire negative refraction or PhC superlens in the high frequency range by decreasing the filling ratio of HL honeycomb PhC.

## 3. Numerical simulations

*f*= 0.256. The high index contrast honeycomb structure can achieve good optical confinement, but it also induces serious impedance mismatch that leads to strong reflection and scattering at the interface between the input free space and the surface of PhC slab. To conquer this obstacle, we dispose the surface of the PhC slab with the trigonal dielectric flange (cut 0.4

*a*) as shown in Fig. 6(a) to effectively reduce the reflection and scattering losses [20

20. T. Matsumoto, K. S. Eom, and T. Baba, “Focusing of light by negative refraction in a photonic crystal slab superlens on silicon-on-insulator substrate,” Opt. Lett. **31**(18), 2786–2788 (2006). [CrossRef] [PubMed]

21. S. S. Xiao, M. Qiu, Z. C. Ruan, and S. L. He, “Influence of the surface termination to the point imaging by a photonic crystals slab with negative refraction,” Appl. Phys. Lett. **85**(19), 4269–4271 (2004). [CrossRef]

*ω*= 0.348, which is incident upon the PhC slab with angles of

*θ*= 30° and 60° to the normal of the interface. The simulated wave patterns are shown in Fig. 6(b) and 6(c) respectively. In both cases the incident beams are refracted in the opposite directions of the reflected beams, namely, the effective refractive index of this PhC is

*n*

_{eff}= −1, and the negative refraction in this holographic PhC is an absolutely left-handed behavior with

*K*_{r}·

*V*_{gr}<0, which is in good accordance with the calculation results in section II.

*ω*= 0.348 is located on the upper side of the PhC slab. In Fig. 7(a) , the distance from the point source to the upper interface (i.e. the object distance) is

*d*

_{o1}= 8.0

*a*, and the image approximately locates at the edge of the lower interface (i.e. the image distance

*d*

_{i1}= 0). In Fig. 7(b), the object distance is

*d*

_{o2}= 3.5

*a*and the relevant image distance becomes

*d*

_{i2}≈4.6

*a*. It is obvious that the sum of

*d*

_{o}and

*d*

_{i}is mostly a constant for this PhC slab, which satisfies the Snell’s law for a flat lens with

*n*

_{eff}= −1. Moreover, it is worth emphasizing that in Fig. 7(b) there is an internal focus inside the PhC slab, which is a clear evidence of LHMs following the rules of geometric optics. The similar simulations have been made for other PhC slabs with different filling ratios to prove that superlens properties can be acquired in the HL honeycomb PhC slabs easily. For the fixed source and PhC slab, the focus distance can be modulated by changing the frequency of incident light freely. If the same point source is located in front of the PhC slab with the interface normal to ГK direction, the field pattern is shown in Fig. 7(c). It is clear that the wave is difficult to propagate through the PhC slab perpendicularly due to the symmetry mismatch, yet, some beams leak out with ± 60° to the normal, which can be explained for the excited Bloch wave is nearly in the ΓM direction.

*ω*

_{0}= 0.424

*ωa/*2πc incidents normally upon the PhC slab with

*f*= 10.1% (or

*I*

_{t}= 1.75). The transmission spectrum is measured at the back of the PhC slab. Figure 8 shows the transmission (red line) and reflection (blue line) spectra, the black arrow indicates the frequency with

*n*

_{eff}= −1. In accord with the predicted results by the frequency scope of Fig. 5, the Gaussian wave over a wide range of frequencies from 0.38 to 0.46 can propagate through the HL honeycomb PhC slab and the other part is forbidden by the side PBGs. Particularly, the Gaussian wave with frequency below the light cone can propagate through the PhC slabs with a high transmission (>90%), which is not affected by undesired diffraction. For the part of waves with the frequency above light cone, the reflection and diffraction losses lead to the decrease of transmissivity. Obviously, comparing with the conventional honeycomb lattices made of regular rods in air, the left-handed negative refraction with high transmissivity (>90%) can be more easily acquired in a wider and higher frequency scope in the HL honeycomb PhCs.

## 4. Conclusions

*n*= 3.4:1. We studied the relation between the filling ratio

*f*and the effective refractive index

*n*

_{eff}, and demonstrate that the left-handed negative refraction exists over a wider and higher range of frequency with high transmissivity (>90%) compared with the conventional honeycomb PhC made of regular rods in air, and the frequency range can be upgraded by decreasing the filling ratio. Moreover, it is found that the effective index is quite close to −1 for a wide range of incident angles with a larger AALNR frequency range (Δ

*ω*/

*ω*≈14.8%). Typical left-handed behaviors such as negative refraction, flat superlens are simulated by FDTD method, which accord well with the results of our numerical calculations. We believe that these results are important and useful for understanding and designing the anomalous optical behavior of PhC applications fabricated by HL.

## Acknowledgement

## References and links

1. | V. G. Veselago, “Electrodynamics of Substances with Simultaneously Negative Values of Sigma and Mu,” Sov. Phys. Usp. |

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

3. | S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. |

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

5. | K. Ren, Z. Y. Li, X. B. Ren, S. Feng, B. Y. Cheng, and D. Z. Zhang, “Three-dimensional light focusing in inverse opal photonic crystals,” Phys. Rev. B |

6. | P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. |

7. | G. Sun, A. S. Jugessur, and A. G. Kirk, “Imaging properties of dielectric photonic crystal slabs for large object distances,” Opt. Express |

8. | 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 |

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

10. | R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “All-angle left-handed negative refraction in Kagomé and honeycomb lattice photonic crystals,” Phys. Rev. B |

11. | T. Asatsuma and T. Baba, “Aberration reduction and unique light focusing in a photonic crystal negative refractive lens,” Opt. Express |

12. | L. Gan, Y. Z. Liu, J. Y. Li, Z. B. Zhang, D. Z. Zhang, and Z. Y. Li, “Ray trace visualization of negative refraction of light in two-dimensional air-bridged silicon photonic crystal slabs at 1.55 microm,” Opt. Express |

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

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

15. | S. D. Gedney, “An Anisotropic PML Absorbing Media for FDTD Simulation of Fields in Lossy Dispersive Media,” Electromagnetics |

16. | X. L. Yang, L. Z. Cai, and Q. Liu, “Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,” Opt. Express |

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

18. | S. Inoue and Y. Aoyagi, “Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding,” Phys. Rev. B |

19. | K. Sakoda, Optical Properties of Photonic Crystals, Springer Series in Optical Sciences 80 (Springer-Verlag, Berlin, 2001). |

20. | T. Matsumoto, K. S. Eom, and T. Baba, “Focusing of light by negative refraction in a photonic crystal slab superlens on silicon-on-insulator substrate,” Opt. Lett. |

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

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 26, 2010

Revised Manuscript: June 1, 2010

Manuscript Accepted: June 28, 2010

Published: July 19, 2010

**Citation**

G. Y. Dong, X. L. Yang, and L. Z. Cai, "Anomalous refractive effects in honeycomb lattice photonic crystals formed by holographic lithography," Opt. Express **18**, 16302-16308 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16302

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### References

- V. G. Veselago, “Electrodynamics of Substances with Simultaneously Negative Values of Sigma and Mu,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68(2), 449–521 (2005). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
- K. Ren, Z. Y. Li, X. B. Ren, S. Feng, B. Y. Cheng, and D. Z. Zhang, “Three-dimensional light focusing in inverse opal photonic crystals,” Phys. Rev. B 75(11), 115108 (2007). [CrossRef]
- P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006). [CrossRef] [PubMed]
- G. Sun, A. S. Jugessur, and A. G. Kirk, “Imaging properties of dielectric photonic crystal slabs for large object distances,” Opt. Express 14(15), 6755–6765 (2006). [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(8), 1708–1711 (2006). [CrossRef]
- C. Y. Luo, S. G. Johnson, J. D. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002). [CrossRef]
- R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “All-angle left-handed negative refraction in Kagomé and honeycomb lattice photonic crystals,” Phys. Rev. B 73(16), 165310 (2006). [CrossRef]
- T. Asatsuma and T. Baba, “Aberration reduction and unique light focusing in a photonic crystal negative refractive lens,” Opt. Express 16(12), 8711–8719 (2008). [CrossRef] [PubMed]
- L. Gan, Y. Z. Liu, J. Y. Li, Z. B. Zhang, D. Z. Zhang, and Z. Y. Li, “Ray trace visualization of negative refraction of light in two-dimensional air-bridged silicon photonic crystal slabs at 1.55 microm,” Opt. Express 17(12), 9962–9970 (2009). [CrossRef] [PubMed]
- X. Y. Ao and S. L. He, “Three-dimensional photonic crystal of negative refraction achieved by interference lithography,” Opt. Lett. 29(21), 2542–2544 (2004). [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(25), 3152–3155 (1990). [CrossRef] [PubMed]
- S. D. Gedney, “An Anisotropic PML Absorbing Media for FDTD Simulation of Fields in Lossy Dispersive Media,” Electromagnetics 16(4), 399–415 (1996). [CrossRef]
- X. L. Yang, L. Z. Cai, and Q. Liu, “Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,” Opt. Express 11(9), 1050–1055 (2003). [CrossRef] [PubMed]
- A. J. Turberfield, M. Campbell, D. N. Sharp, M. T. Harrison, and R. G. Denning, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). [CrossRef] [PubMed]
- S. Inoue and Y. Aoyagi, “Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding,” Phys. Rev. B 69(20), 205109 (2004). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals, Springer Series in Optical Sciences 80 (Springer-Verlag, Berlin, 2001).
- T. Matsumoto, K. S. Eom, and T. Baba, “Focusing of light by negative refraction in a photonic crystal slab superlens on silicon-on-insulator substrate,” Opt. Lett. 31(18), 2786–2788 (2006). [CrossRef] [PubMed]
- S. S. Xiao, M. Qiu, Z. C. Ruan, and S. L. He, “Influence of the surface termination to the point imaging by a photonic crystals slab with negative refraction,” Appl. Phys. Lett. 85(19), 4269–4271 (2004). [CrossRef]

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