## Controlling electromagnetic fields with graded photonic crystals in metamaterial regime |

Optics Express, Vol. 18, Issue 19, pp. 20321-20333 (2010)

http://dx.doi.org/10.1364/OE.18.020321

Acrobat PDF (1283 KB)

### Abstract

Engineering of a refractive index profile is a powerful method for controlling electromagnetic fields. In this paper, we investigate possible realization of isotropic gradient refractive index media at optical frequencies using two-dimensional graded photonic crystals. They consist of dielectric rods with spatially varying radii and can be homogenized in broad frequency range within the lowest band. Here they operate in metamaterial regime, that is, the graded photonic crystals are described with spatially varying effective refractive index so they can be regarded as low-loss and broadband graded dielectric metamaterials. Homogenization of graded photonic crystals is done with Maxwell-Garnett effective medium theory. Based on this theory, the analytical formulas are given for calculations of the rods radii which makes the implementation straightforward. The frequency range where homogenization is valid and where graded photonic crystal based devices work properly is discussed in detail. Numerical simulations of the graded photonic crystal based Luneburg lens and electromagnetic beam bend show that the homogenization based on Maxwell-Garnett theory gives very good results for implementation of devices intended to steer and focus electromagnetic fields.

© 2010 Optical Society of America

## 1. Introduction

*a*is much smaller then the wavelength

*λ*so they can be regarded as media with effective parameters. In most of reported realizations, the ratio Ω =

*a*/

*λ*was not negligible but the effective parameters were still well defined. This transitional regime between the effective medium and the photonic crystal regime is denoted as the metamaterial regime [1

1. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E **71**, 036617 (2005). [CrossRef]

2. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E **71**, 036609 (2005). [CrossRef]

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

6. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science **323**, 366–369 (2009). [CrossRef] [PubMed]

7. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. **102**, 183903 (2009). [CrossRef] [PubMed]

8. Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express **17**, 18354–18363 (2009). [CrossRef] [PubMed]

9. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science **305**, 847–848 (2004). [CrossRef] [PubMed]

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

16. Z. L. Mei, J. Bai, and T. J. Cui, “Gradient index metamaterials realized by drilling hole arrays,” J. Phys. D Appl. Phys. **43**, 055404 (2010). [CrossRef]

13. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. **8**, 569–571 (2009). [CrossRef]

15. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express **17**, 12922–12928 (2009). [CrossRef] [PubMed]

16. Z. L. Mei, J. Bai, and T. J. Cui, “Gradient index metamaterials realized by drilling hole arrays,” J. Phys. D Appl. Phys. **43**, 055404 (2010). [CrossRef]

17. P. S. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. **17**, 1982 (1999). [CrossRef]

18. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E **70**, 036612 (2004). [CrossRef]

19. E. Centeno and D. Cassagne, “Graded photonic crystals,” Opt. Lett. **30**, 2278–2280 (2005). [CrossRef] [PubMed]

20. E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B **73**, 235119 (2006). [CrossRef]

21. H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express **15**, 1240–1253 (2007). [CrossRef] [PubMed]

24. S. Astilean, P. Lalanne, P. Chavel, E. Cambril, and H. Launois, “High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm,” Opt. Lett. **23**, 552–554 (1998). [CrossRef]

25. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogeneous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. **98**, 243901 (2007). [CrossRef] [PubMed]

26. U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, “Implementation of a graded-index medium by use of subwavelength structures with graded fill factor,” J. Opt. Soc. Am. A **22**, 724–733 (2005). [CrossRef]

## 2. Graded photonic crystals in metamaterial regime

*n*(

*x*,

*y*) in

*xy*-plane is shown in Fig. 1(a). In order to realize this profile, it is firstly approximated with discrete one as shown in Fig. 1(b). Each

*ij*-cell is a square of side

*a*, the coordinates of its central point are

*x*and

_{i}*y*and the cell refractive index,

_{j}*n*, is equal to

_{ij}*n*(

*x*,

_{i}*y*). The ray path through the GRIN medium is governed by equations of Hamiltonian optics [27]. Hamiltonian for the

_{j}*ij*-cell is a plane wave dispersion

*c*is speed of light in vacuum and

*k*is modulus of wave vector in

*xy*-plane.

*xy*-plane is shown in Fig. 1(c). Dielectric rods of only one material are used while their radii are spatially varying.

*ε*is the permittivity of dielectric rods and

_{rod}*ε*is the permittivity of the host medium. The field propagation can be treated by equations of Hamiltonian optics [17

_{host}17. P. S. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. **17**, 1982 (1999). [CrossRef]

18. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E **70**, 036612 (2004). [CrossRef]

28. W. Smigaj and B. Gralak, “Validity of the effective-medium approximation of photonic crystals,” Phys. Rev. B **77**, 235445 (2008). [CrossRef]

29. P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E **72**, 026615 (2005). [CrossRef]

*n*(

^{α}_{eff}*k*))

_{ij}being the frequency dependent effective refractive index obtained from dispersion curves of a PC whose unit cell is the

*ij*-cell whereas

*α*stands for transverse magnetic (TM, magnetic field is in

*xy*-plane) and transverse electric mode (TE, electric field is in

*xy*-plane). Equations (1) and (2) have to be equivalent in order that the GPC from Fig. 1(c) would implement GRIN medium from Fig. 1(b), which gives the condition to determine a radius of rod,

*r*, of the

_{ij}*ij*-cell:

*a*and wavelength

*λ*, Ω =

*a*/

*λ*, (introduced at the beginning of the paper) will be hereafter denoted as normalized frequency. In this paper we studied GPCs with SiO

_{2}(

*ε*= 4.5) and Si (

*ε*= 11.8) rods whose EFCs can be approximated with circles locally up to even Ω

_{max}≈ 0.44 in the lowest band. This means that the GPCs can be homogenized [29

29. P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E **72**, 026615 (2005). [CrossRef]

_{max}which is a decisive reason for broadband work of GPC based devices. In Fig. 2 are shown calculated values

*n*(

^{α}_{eff}*k*) (blue lines) of a PC for five different values

*r*/

*a*. The dispersion curves of the PC were calculated using COMSOL Multiphysics by considering single unit cell of the PC with periodic boundary conditions. The eigenvalue problem was then solved numerically while the effective refractive index was calculated from Eq. (2). Due to spatial dispersion, the effective refractive index starts to differ from the value in the long-wavelength limit

*n*. But for the normalized frequencies Ω ≲ 0.25, we can adopt the approximation

^{α}_{eff}*n*(

^{α}_{eff}*k*) ≈

*n*, so Eq. (3) reads

^{α}_{eff}30. S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Effective dielectric constant of periodic composite structures,” Phys. Rev. B **48**, 14936–14943 (1993). [CrossRef]

*ε*(

_{κ}*κ*=

*x*,

*y*,

*z*). In a general case, the PCs are biaxial, but in the considered case of circular cilynders in a square lattice, they are uniaxial,

*ε*=

_{x}*ε*=

_{y}*ε*, that is, they are isotropic in

_{plane}*xy*-plane [32

32. P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic Crystal Optics and Homogenization of 2D Periodic Composites,” Phys. Rev. Lett. **82**, 719–722 (1999). [CrossRef]

30. S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Effective dielectric constant of periodic composite structures,” Phys. Rev. B **48**, 14936–14943 (1993). [CrossRef]

35. A. Kirchner, K. Busch, and C. M. Soukoulis, “Transport properties of random arrays of dielectric cylinders,” Phys. Rev. B **57**, 277–288 (1998). [CrossRef]

*ε*of uniaxial PC is given as [36

_{plane}36. A. Sihvola, *Electromagnetic mixing formulas and applications*, (The Institution of Electrical Engineers, London, United Kingdom, 1999). [CrossRef]

*ε*is obtained using the exact formula, that is, as the volume average of

_{z}*ε*and

_{rod}*ε*

_{host}*f*stands for the filling fraction of rods,

*f*=

*r*

^{2}

*π*/

*a*

^{2}and

*L*= 1/2 is the depolarizing factor of rods in the long-wavelength limit for

_{plane}*xy*-plane. The effective refractive indices based on EMT theory are

*r*/

*a*< 0.4. This is in accordance with the results reported in Ref. [34]. For TE mode, the electric field is normal to the rods in PCs and the main assumption in the derivation of MG theory is that a rod is placed in the local field of homogeneously polarized matter [36

36. A. Sihvola, *Electromagnetic mixing formulas and applications*, (The Institution of Electrical Engineers, London, United Kingdom, 1999). [CrossRef]

*r*/

*a*> 0.4 even in the long-wavelength limit. In the case of TM mode, the electric field is parallel to the rods so the PC is homogenously polarized even for large filling fractions and field averaging give exact value in the long-wavelength limit.

*n*can be expressed using EMT, the condition from Eq. (4) now reads (

^{α}_{eff}*n*)

^{α}_{EMT}_{ij}=

*n*. Radii of rods in the GPC from Fig. 1 (c) can be determined by putting this condition in Eqs. (5) and (6) with spatially dependent filling fraction,

_{ij}*f*=

_{ij}*r*

^{2}

_{ij}

*π*/

*a*

^{2}. Finally, in the case of TE polarization, the radius

*r*is obtained from Eq. (5) as

_{ij}*r*/

*a*< 0.4.

*n*(

^{α}_{eff}*k*) ≈

*n*for Ω ≲ 0.25 which is consistent with the result reported in Ref. [37

^{α}_{EMT}37. W. G. Egan and D. E. Aspnes, “Finite-wavelength effects in composite media,” Phys. Rev. B **26**, 5313–5320 (1982). [CrossRef]

*n*(

^{α}_{eff}*k*) and

*n*is more pronounced. By calculating

^{α}_{EMT}*n*using extended MG theories [38

^{α}_{EMT}38. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B **39**, 9852–9858 (1989). [CrossRef]

39. R. Ruppin, “Evaluation of extended Maxwell-Garnett theories,” Opt. Commun. **182**, 273–279 (2000). [CrossRef]

*n*(

^{α}_{eff}*k*) in Fig. 2, not only its value in the long-wavelength limit but the effective index

*n*would be then Ω dependent. On the other hand, Eqs. (7) and (8), enable frequency independent design applicable in broad frequency range.

^{α}_{EMT}## 3. Device implementation by graded photonic crystals in metamaterial regime

*n*, to maximal value

_{min}*n*, common characteristic for all cells in GPCs which implement devices is free-space wavelength

_{max}*λ*

_{0}, while wavelength

*λ*=

*λ*

_{0}/

*n*and the normalized frequencies Ω =

*a*/

*λ*vary from cell to cell. Therefore, the excitation frequency in simulations will be expressed in terms of Ω

_{0}=

*a*/

*λ*

_{0}, while Ω will lie between

*n*Ω

_{min}_{0}and

*n*Ω

_{max}_{0}.

### 3.1. 2D Luneburg lens

41. P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic crystals as optical components,” Appl. Phys. Lett. **75**, 2725–2727 (1999). [CrossRef]

42. G. Zouganelis and D. Budimir, “Effective dielectric constant and design of sliced Luneberg lens,” Microwave Opt. Technol. Let. **49**, 2332–2337 (2007). [CrossRef]

43. Q. Cheng, H. F. Ma, and T. J. Cui, “Broadband planar Luneburg lens based on complementary metamaterials,” Appl. Phys. Lett. **95**, 181901 (2009). [CrossRef]

*R*is the lens radius and

*ρ*is modulus of radius vector of the lens internal points. In order to implement this permittivity profile (

*n*= 1,

_{min}*n*= 1.41), SiO

_{max}_{2}rods (

*ε*= 4.5) were used whereas the host medium was vacuum. Radii of rods were calculated using Eq. (7). The simulation results for the original lens with refractive index profile given by Eq. (9) are shown in Fig. 3(a) whereas the results for the GPC lens are given in Fig. 3(b). Excitation frequency in the simulated case was Ω

_{0}= 0.14 while Ω ∈ (0.14,0.2). As can be seen the GPC lens works as well as the original one, the plane wave from the left side is focused onto the opposite, right side, while the field within the lens behaves as a plane wave locally which indicates that the effective medium approximation is valid. To give exact criterion for the description of the lens performances and to suggest possible way for experimental testing, we calculated reflection in front of the GPC lens as a function of excitation frequency Ω

_{0}, Fig. 4, while the reflection from the original lens is given for comparison. As can be seen, for Ω

_{0}= 0.14, the reflections from both lenses are negligible meaning that the incident field does not resolve periodic structures of the GPC lens perceiving it as a locally homogeneous medium such as the original lens.

_{0}= 0.08, Ω ∈ (0.08,0.11), are shown in Fig. 5(a). As can be seen, the lens works fine while the reflection is low, Fig. 4. However, the focus became wider, it was moved to the lens interior and field intensities in the focus were decreased. These lower performances are because the original lens was designed in the limit of geometrical optics. Nevertheless, simulations showed that the lens worked well even for Ω

_{0}= 0.067, far beyond the geometrical optics when free-space wavelength was equal to the lens radius. It should be emphasized that existence of the lower frequency limit is not consequence of the implementation with GPCs.

_{0}= 0.3, Ω ∈ (0.3,0.42), where the reflection is still low and at the same level as for the original lens, Fig. 4. As can be seen from Fig. 5(b), the GPC lens works very fine for this frequency, focal point is narrower, field intensities are greater, so focusing is even better since the work at higher frequencies is closer to the limit of geometrical optics. But for Ω ≳ 0.25, the difference between the effective index based on MG theory and the effective index calculated from dispersion curves of PC results in slight changes of refractive index profile within the GPC lens so the focus point is moved toward the lens interior. Therefore, for the frequencies Ω ≳ 0.25, implementation of the Luneburg lens with GPCs is still possible since it is not strongly sensitive to slight changes of refractive index distribution but for the precise control, the effective index should be calculated from dispersion curves of PCs.

_{0}above 0.3 leads to strong reflection from the GPC lens, Fig. 4. Since the reflection from the original lens is still negligible, the increased reflection means that the incident field does not perceive the GPC lens as a locally homogeneous medium anymore. This can be also observed as a standing wave pattern in a front of and within the central part of the GPC lens, Fig. 5(c), as a result of Bragg reflections from the central cells. The effective refractive index is the largest in the center of the lens,

*n*=

*n*, implying the lowest value of

_{max}*λ*so this is the place where Ω reaches the Bragg reflection condition firstly. With further increasing of Ω

_{0}, the Bragg reflection condition becomes fulfilled in parts of the GPC lens with lower value of effective refractive index, so Bragg reflections start to appear from the side cells in the GPC lens as well, Fig. 5(d). Therefore, based on calculation of reflection from the GPC lens, we conclude that Ω

_{0}≈ 0.3 is the maximal excitation frequency and Ω ≈ 0.42 is the upper frequency limit for proper work of GPC lens.

_{max}≈ 0.44, the maximal normalized frequency for the homogenization and work in metamaterial regime for PCs. This means that Bragg reflections firstly appear when Ω starts to overcome Ω

_{max}within the part of a GPC based device with the highest effective refractive index. For the lens, this is the central part. Therefore, for the proper work of the device, the excitation frequency have to satisfy the approximative condition Ω

_{0}≲ Ω

_{max}/

*n*. The advantage of this formula is because Ω

_{max}_{max}can be simply obtained by calculating EFC for unit cell which implements the highest effective refractive index.

### 3.2. Electromagnetic beam bend

44. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**, 085103 (2009). [CrossRef]

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

46. S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. **8**, 4243–4247 (2008). [CrossRef]

*w*=

*u*+

*iv*, is transformed to annular segment in polar coordinates of

*z*-complex plane,

*z*=

*x*+

*iy*, as shown in Fig. 6. The constant

*C*can be used for adjusting dielectric profile within the bend. If the square domain is empty that is, if refractive index within square domain is

*n*= 1, refractive index profile within the bend is [45

_{w}45. U. Leonhardt, “Optical Conformal Mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

18. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E **70**, 036612 (2004). [CrossRef]

16. Z. L. Mei, J. Bai, and T. J. Cui, “Gradient index metamaterials realized by drilling hole arrays,” J. Phys. D Appl. Phys. **43**, 055404 (2010). [CrossRef]

*n*= 1.25 to

_{min}*n*= 2.86. This is a significantly larger variation of the refractive index compared to the Luneburg lens so a denser distribution of rods was required in order to obtain good behaviour. The bend was implemented with Si (

_{max}*ε*= 11.8) rods in vacuum background, Eq. (8) was used to determine radii of rods while

*C*= 1. The simulation results for the original and GPC based bend are shown in Fig. 7(a) and 7(b), respectively, with excitation frequency Ω

_{0}= 0.11 while Ω∈ (0.137,0.32). As can be seen, the field distribution in the GPC bend matches the field in the original one and the field perceives the GPC bend as an effective medium. In order to exactly compare performances of GPC bend and the original one, we calculated transmissions through both of them which could be also appropriate for experimental testing. The calculation results are shown in Fig. 8 and for Ω

_{0}= 0.11 both bends have the same transmission.

47. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express **17**, 14872–14879 (2009). [CrossRef] [PubMed]

8. Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express **17**, 18354–18363 (2009). [CrossRef] [PubMed]

**43**, 055404 (2010). [CrossRef]

48. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. **105**, 104913 (2009). [CrossRef]

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

_{0}= 0.068.

_{0}= 0.15 (Ω is between 0.187 and 0.43). The simulation results for this excitation frequency are shown in Fig. 9(b). The bending effect works well, but there is a little phase delay (about half a wavelength) in the GPC bend compared to the original one (simulation not shown here). This indicates a slight difference between original distribution of refractive index and the one implemented with GPC as a result of applied homogenization based on EMT. The bending effect showed robustness to this difference but in cases where refractive index have to be precisely controlled, it should be calculated from PC dispersion curves.

_{0}≈ 0.15 transmission for the GPC bend decreases abruptly while it stays at the same level for the original bend, Fig. 8. This means that the incoming field does not perceive the GPC bend as an effective medium anymore. Bragg reflections start to appear from the cells along the inner bend edge because the refractive index is the highest and the wavelength is the lowest here and this is the place where Bragg condition is satisfied firstly. Since the bending effect does not work anymore, Ω

_{0}≈ 0.15 is maximal allowed excitation frequency while Ω ≈ 0.43 is the upper frequency limit for proper work of GPC bend. As in the case of the Luneburg lens, the upper frequency limit Ω ≈ 0.43 is very close to Ω

_{max}. Therefore, achieving of that limit can be again interpreted as a result of increasing of Ω above Ω

_{max}for the cells with the highest effective refractive index in GPC. As a result, the maximal allowed excitation frequency can be again determined using the approximative formula Ω

_{0}≲ Ω

_{max}/

*n*.

_{max}## 4. Conclusion

_{max}, the maximal normalized frequency for the homogenization and work in metamaterial regime for GPCs. As a result, the maximal excitation frequency for proper work of the GPC devices can be determined using approximative formula Ω

_{0}≲ Ω

_{max}/

*n*. Although the original devices with GRIN medium were designed in the limit of geometrical optics, simulations showed that the GPC based bend and lens worked very well outside the limit of geometrical optics.

_{max}## Acknowledgments

## References and links

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32. | P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic Crystal Optics and Homogenization of 2D Periodic Composites,” Phys. Rev. Lett. |

33. | L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Elec. Eng. |

34. | M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, “Design and optimization of 2D photonic crystal waveguides based on silicon,” Opt. Quantum Electron. |

35. | A. Kirchner, K. Busch, and C. M. Soukoulis, “Transport properties of random arrays of dielectric cylinders,” Phys. Rev. B |

36. | A. Sihvola, |

37. | W. G. Egan and D. E. Aspnes, “Finite-wavelength effects in composite media,” Phys. Rev. B |

38. | W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B |

39. | R. Ruppin, “Evaluation of extended Maxwell-Garnett theories,” Opt. Commun. |

40. | R. K. Lüneburg, |

41. | P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic crystals as optical components,” Appl. Phys. Lett. |

42. | G. Zouganelis and D. Budimir, “Effective dielectric constant and design of sliced Luneberg lens,” Microwave Opt. Technol. Let. |

43. | Q. Cheng, H. F. Ma, and T. J. Cui, “Broadband planar Luneburg lens based on complementary metamaterials,” Appl. Phys. Lett. |

44. | B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B |

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

46. | S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartal, and X. Zhang, “Ray optics at a deep-subwavelength scale: a transformation optics approach,” Nano Lett. |

47. | N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express |

48. | Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. |

**OCIS Codes**

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 30, 2010

Revised Manuscript: August 13, 2010

Manuscript Accepted: August 23, 2010

Published: September 9, 2010

**Citation**

Borislav Vasic, Goran Isic, Rados Gajic, and Kurt Hingerl, "Controlling electromagnetic fields with graded photonic crystals in metamaterial regime," Opt. Express **18**, 20321-20333 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20321

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