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. 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]
4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
5. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]
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]
10. A. O. Pinchuk and G. C. Schatz, "Metamaterials with gradient negative index of refraction," J. Opt. Soc. Am. A 24, A39-A44 (2007). [CrossRef]
11. B. K. Juluri, S. Chin, S. Lin, T. R. Walker, L. Jensen, and T. J. Huang, "Propagation of designer surface plasmons in structured conductor surfaces with parabolic gradient index," Opt. Express 17, 2997-3006 (2009). [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]
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]
14. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (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. 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]
22. F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (2006). [CrossRef]
23. H.-T. Chien and C.-C. Chen, "Focusing of electromagnetic waves by periodic arrays of air holes with gradually varying radii," Opt. Express 14, 10759-10764 (2006). [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]
27. Y. A. Kravtsov, and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
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]
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]
31. P. Lalanne, "Effective medium theory applied to photonic crystals composed of cubic or square cylinders," Appl. Opt. 35, 5369-5380 (1996). [CrossRef] [PubMed]
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]
33. L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Electr. Eng. 94, 65-68 (1947).
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. 34, 145-159 (2002).
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]
36. A. Sihvola, Electromagnetic mixing formulas and applications (The Institution of Electrical Engineers, London, United Kingdom, 1999). [CrossRef]
37. W. G. Egan and D. E. Aspnes, "Finite-wavelength effects in composite media," Phys. Rev. B 26, 5313-5320 (1982). [CrossRef]
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]
40. R. K. Lüneburg, The mathematical theory of optics (University of California Press, Los Angeles, CA, 1944).
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 Lüneburg lens," Microwave Opt. Technol. Lett. 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]
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]
47. N. I. Landy and W. J. Padilla, "Guiding light with conformal transformations," Opt. Express 17, 14872-14879 (2009). [CrossRef] [PubMed]
48. Z. L. Mei and T. J. Cui, "Arbitrary bending of electromagnetic waves using isotropic materials," J. Appl. Phys. 105, 104913 (2009). [CrossRef]

Appl. Opt.

• P. Lalanne, "Effective medium theory applied to photonic crystals composed of cubic or square cylinders," Appl. Opt. 35, 5369-5380 (1996). [CrossRef] [PubMed]

Appl. Phys. Lett.

• P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999). [CrossRef]
• Q. Cheng, H. F. Ma, and T. J. Cui, "Broadband planar Luneburg lens based on complementary metamaterials," Appl. Phys. Lett. 95, 181901 (2009). [CrossRef]

J. Appl. Phys.

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

J. Lightwave Technol.

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

J. Opt. Soc. Am. A

• A. O. Pinchuk and G. C. Schatz, "Metamaterials with gradient negative index of refraction," J. Opt. Soc. Am. A 24, A39-A44 (2007). [CrossRef]
• 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]

J. Phys. D Appl. Phys.

• 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]

Microwave Opt. Technol. Lett.

• G. Zouganelis and D. Budimir, "Effective dielectric constant and design of sliced Lüneburg lens," Microwave Opt. Technol. Lett. 49, 2332-2337 (2007). [CrossRef]

Nano Lett.

• 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]

Nat. Mater.

• J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 569-571 (2009). [CrossRef]

Nat. Photonics

• L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructures cloak operating at optical frequencies," Nat. Photonics 3, 461-463 (2009). [CrossRef]
• W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]

Opt. Commun.

• R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000). [CrossRef]

Opt. Express

• N. I. Landy and W. J. Padilla, "Guiding light with conformal transformations," Opt. Express 17, 14872-14879 (2009). [CrossRef] [PubMed]
• 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]
• 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]
• B. K. Juluri, S. Chin, S. Lin, T. R. Walker, L. Jensen, and T. J. Huang, "Propagation of designer surface plasmons in structured conductor surfaces with parabolic gradient index," Opt. Express 17, 2997-3006 (2009). [CrossRef] [PubMed]
• H.-T. Chien and C.-C. Chen, "Focusing of electromagnetic waves by periodic arrays of air holes with gradually varying radii," Opt. Express 14, 10759-10764 (2006). [CrossRef] [PubMed]
• H. Kurt and D. S. Citrin, "Graded index photonic crystals," Opt. Express 15, 1240-1253 (2007). [CrossRef] [PubMed]

Opt. Lett.

• E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 30, 2278-2280 (2005). [CrossRef] [PubMed]
• 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]

Opt. Quantum Electron.

• 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. 34, 145-159 (2002).

Phys. Rev.

• Y. Jiao, S. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method," Phys. Rev. 70, 036612 (2004). [CrossRef]

Phys. Rev. B

• A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998). [CrossRef]
• W. G. Egan and D. E. Aspnes, "Finite-wavelength effects in composite media," Phys. Rev. B 26, 5313-5320 (1982). [CrossRef]
• W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989). [CrossRef]
• W. Smigaj, and B. Gralak, "Validity of the effective-medium approximation of photonic crystals," Phys. Rev. B 77, 235445 (2008). [CrossRef]
• 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]
• F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (2006). [CrossRef]
• B. Vasić, G. Isić, R. Gajić, and K. Hingerl, "Coordinate transformation based design of confined metamaterial structures," Phys. Rev. B 79, 085103 (2009). [CrossRef]
• 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]

Phys. Rev. E

• P. A. Belov and C. R. Simovski, "Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers," Phys. Rev. E 72, 026615 (2005). [CrossRef]
• 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]
• D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005). [CrossRef]

Phys. Rev. Lett.

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

Proc. Inst. Electr. Eng.

• L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Electr. Eng. 94, 65-68 (1947).

Science

• J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
• D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
• J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
• 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]
• U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]

Other

• R. K. Lüneburg, The mathematical theory of optics (University of California Press, Los Angeles, CA, 1944).
• A. Sihvola, Electromagnetic mixing formulas and applications (The Institution of Electrical Engineers, London, United Kingdom, 1999). [CrossRef]
• Y. A. Kravtsov, and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).

Click here to see a list of articles that cite this paper

Related Journal Articles

• Effects of material composition on the superlens frequency of photonic crystals (JOSAB)
• Backward nonlinear surface Tamm states in left-handed metamaterials (OE)
• Controlling thermal radiation by photonic quantum well structure with zero-averaged-refractive-index gap (JOSAB)
• Matched coordinates and adaptive spatial resolution in the Fourier modal method (OE)
• Analytical theory of wave propagation through stacked fishnet metamaterials (OE)

Related Conference Papers

• Near-field observations of self-collimation in photonic crystal superlattices
• Experimental evidence for the mirage effect and giant dispersive phenomena in graded photonic crystals
• Analytical engineering of index maps for the design of all-dielectric metametarials
• Proposal for analytical design of all-dielectric metametarials
• Transformed Cladding Waveguides: Confining Light with All-Dielectric Metamaterials