volume | PIER Journals

Abstract

An alternative, field-based formulation of transformation optics is proposed. Field transformations are expressed in the language of boundary conditions for the electromagnetic fields facilitated through the introduction of generalized potential functions. It is shown that the field-based approach is equivalent to the conventional coordinate-transformation approach but is preferable when looking for specific field distribution. A set of example devices such as invisibility cloaks, concentrators, rotators, and transformation optics lenses capable of creating light beams with predetermined field distribution (e.g., Gaussian and sinusoidal) is studied to validate the effectiveness of the field-based formulation. As for the boundary conditions for the cloaked region the absence of the normal component of the Poynting vector is justified. In the frames of the field-based approach the physical reasons behind infinite components (singularities) of the material parameters of transformation optics devices are straightforwardly revealed.

1. Dolin, L., "About the possibility of three-dimensional electromagnetic systems with inhomogeneous anisotropic filling," Izvestiya Vuzov: Radiophysics, Vol. 4, 964-967, 1961. Google Scholar

2. Ward, A. J. and J. B. Pendry, "Refraction and geometry in Maxwell's equations," J. Mod. Opt., Vol. 43, 773-793, 1996.
doi:10.1080/09500349608232782 Google Scholar

3. Pendry, J. B., D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science, Vol. 312, 1780-1783, 2006.
doi:10.1126/science.1125907 Google Scholar

4. Leonhardt, U., "Optical conformal mapping," Science, Vol. 312, 1777-1780, 2006.
doi:10.1126/science.1126493 Google Scholar

5. Schurig, D., 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, Vol. 314, 977-980, 2006.
doi:10.1126/science.1133628 Google Scholar

6. Leonhardt, U. and T. G. Philbin, "Transformation optics and the geometry of light," Prog. Opt., Vol. 53, 69-152, 2009.
doi:10.1016/S0079-6638(08)00202-3 Google Scholar

7. Leonhardt, U. and T. G. Philbin, Geometry and Light: The Science of Invisibility, Dover, Mineola, 2010.

8. Cummer, S. A., B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E, Vol. 74, 036621, 2006.
doi:10.1103/PhysRevE.74.036621 Google Scholar

9. Schurig, D., J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express, Vol. 14, 9794-9804, 2006.
doi:10.1364/OE.14.009794 Google Scholar

10. Gabrielli, L. H., J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructure cloak operating at optical frequencies," Nat. Photonics, Vol. 3, 461-463, 2009.
doi:10.1038/nphoton.2009.117 Google Scholar

11. Qiu, C.-W., L. Hu, B. Zhang, B.-I. Wu, S. G. Johnson, and J. D. Joannopoulos, "Spherical cloaking using nonlinear transformations for improved segmentation into concentric isotropic coatings," Opt. Express, Vol. 17, 13467-13478, 2009.
doi:10.1364/OE.17.013467 Google Scholar

12. Kwon, D. and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett., Vol. 92, 013505, 2008.
doi:10.1063/1.2830698 Google Scholar

13. Yan, M., W. Yan, and M. Qiu, "Invisibility cloaking by coordinate transformation," Prog. Opt., Vol. 52, 261-304, 2009.
doi:10.1016/S0079-6638(08)00006-1 Google Scholar

14. Cai, W., U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics, Vol. 1, 224-227, 2007.
doi:10.1038/nphoton.2007.28 Google Scholar

15. Kildishev, A. V., W. Cai, U. K. Chettiar, and V. M. Shalaev, "Transformation optics: Approaching broadband electromagnetic cloaking," New J. Phys., Vol. 10, 115029, 2008.
doi:10.1088/1367-2630/10/11/115029 Google Scholar

16. Kildishev, A. V. and V. M. Shalaev, "Engineering space for light via transformation optics," Opt. Lett., Vol. 33, 43-45, 2008.
doi:10.1364/OL.33.000043 Google Scholar

17. Nicolet, A., F. Zolla, and S. Guenneau, "Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section," Opt. Lett., Vol. 33, 1584-1586, 2008.
doi:10.1364/OL.33.001584 Google Scholar

18. Greanleaf, A., Y. Kurilev, M. Lassas, and G. Uhlmann, "Invisibility and inverse problems," Bulletin of the Americam Mathematical Society, Vol. 46, 55-97, 2009.
doi:10.1090/S0273-0979-08-01232-9 Google Scholar

19. Luo, Y., H. Chen, J. Zhang, L. Ran, and J. A. Kong, "Design and analytical fullwave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations," Phys. Rev. B, Vol. 77, 125127, 2008.
doi:10.1103/PhysRevB.77.125127 Google Scholar

20. Zharova, N. A., I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, "Ideal and nonideal invisibility cloaks," Opt. Express, Vol. 16, 21369-21374, 2008.
doi:10.1364/OE.16.021369 Google Scholar

21. Valentine, J., J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater., Vol. 8, 568-571, 2009.
doi:10.1038/nmat2461 Google Scholar

22. Urzhumov, Y. A. and D. R. Smith, "Transformation optics with photonic band gap media," Phys. Rev. Lett., Vol. 105, 163901, 2010.
doi:10.1103/PhysRevLett.105.163901 Google Scholar

23. Han, T., C. Qiu, and X. Tang, "An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation," J. Opt., Vol. 12, 095103, 2010.
doi:10.1088/2040-8978/12/9/095103 Google Scholar

24. Tuniz, A., B. T. Kuhlmey, P. Y. Chen, and S. C. Fleming, "Weaving the invisible thread: Design of an optically invisible metamaterial fibre," Opt. Express, Vol. 18, 18095-18105, 2010.
doi:10.1364/OE.18.018095 Google Scholar

25. Rahm, M., D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, "Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations," Photon. Nanostruct.: Fundam. Applic., Vol. 6, 87, 2008.
doi:10.1016/j.photonics.2007.07.013 Google Scholar

26. Chen, H. and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett., Vol. 90, 241105, 2007.
doi:10.1063/1.2748302 Google Scholar

27. Lai, Y., H. Chen, Z.-Q. Zhang, and C. T. Chan, "Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell," Phys. Rev. Lett., Vol. 102, 093901, 2009.
doi:10.1103/PhysRevLett.102.093901 Google Scholar

28. Lai, Y., J. Ng, H. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, "Illusion optics: The optical transformation of an object into another object," Phys. Rev. Lett., Vol. 102, 253902, 2009.
doi:10.1103/PhysRevLett.102.253902 Google Scholar

29. Li, C., X. Meng, X. Liu, F. Li, G. Fang, H. Chen, and C. T. Chan, "Experimental realization of a circuit-based broadband illusionoptics analogue," Phys. Rev. Lett., Vol. 105, 233906, 2010.
doi:10.1103/PhysRevLett.105.233906 Google Scholar

30. Schultheiss, V. H., S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, "Optics in curved space," Phys. Rev. Lett., Vol. 105, 143901, 2010.
doi:10.1103/PhysRevLett.105.143901 Google Scholar

31. Luo, Y., L.-X. He, Y. Wang, H. L. W. Chan, and S.-Z. Zhu, "Changing the scattering of sheltered targets," Phys. Rev. A, Vol. 83, 043809, 2011.
doi:10.1103/PhysRevA.83.043809 Google Scholar

32. Han, T., C.-W. Qiu, and X. Tang, "Distributed external cloak without embedded antiobjects," Opt. Lett., Vol. 35, 2642-2644, 2010.
doi:10.1364/OL.35.002642 Google Scholar

33. Jiang, W. X., T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, "Cylindrical-to-plane-wave conversion via embedded optical transformation," Appl. Phys. Lett., Vol. 92, 261903, 2008.
doi:10.1063/1.2953447 Google Scholar

34. Ward, A. J. and J. B. Pendry, "Calculating photonic Green's functions using a nonorthogonal finite-difference time-domain method," Phys. Rev. B, Vol. 58, 7252-7259, 1998.
doi:10.1103/PhysRevB.58.7252 Google Scholar

35. Shyroki, D. M., "Note on transformation to general curvilinear coordinates for Maxwell's curl equations," , 2003, Preprint, arXiv:physics/0307029v2 .
doi:10.1109/LMWC.2006.884768 Google Scholar

36. Shyroki, D. M., "Squeezing of open boundaries by Maxwell-consistent real coordinate transformation," IEEE Microwave and Wireless Components Letters, Vol. 16, 576-578, 2006.
doi:10.1109/TMTT.2007.897841 Google Scholar

37. Shyroki, D. M., "Efficient Cartesian-grid-based modeling of rotationally symmetric bodies," IEEE Transactions on Microwave Theory and Techniques, Vol. 55, 1132-1138, 2007.
doi:10.1109/TMTT.2007.914637 Google Scholar

38. Shyroki, D. M., "Exact equivalent straight waveguide model for bent and twisted waveguides," IEEE Transactions on Microwave Theory and Techniques, Vol. 56, 414-419, 2008.
doi:10.1088/1367-2630/10/11/115033 Google Scholar

39. Smolyaninov, I. I., "Transformational optics of plasmonic metamaterials," New J. Phys., Vol. 10, 115033, 2008.
doi:10.1021/nl100800c Google Scholar

40. Huidobro, P. A., M. L. Nesterov, L. Martin-Moreno, and F. J. Garca-Vidal, "Transformation optics for plasmonics," Nano Lett., Vol. 10, 1985-1890, 2010.
doi: --- Either ISSN or Journal title must be supplied. Google Scholar

41. Liu, Y., T. Zentgraf, G. Bartal, and X. Zhang, "Transformational plasmonics," Nano Lett., Vol. 10, 1991-1997, 2010.
doi:10.1364/OE.18.012027 Google Scholar

42. Kadic, M., S. Guenneau, and S. Enoch, "Transformational plasmonics: Cloak, concentrator and rotator," Opt. Express, Vol. 18, 12027-12032, 2010.
doi:10.1103/PhysRevLett.105.233901 Google Scholar

43. Aubry, A., D. Y. Lei, S. A. Maier, and J. B. Pendry, "Interaction between plasmonic nanoparticles revisited with transformation optics," Phys. Rev. Lett., Vol. 105, 233901, 2010.
doi:10.1088/1367-2630/9/3/045 Google Scholar

44. Cummer, S. A. and D. Schurig, "One path to acoustic cloaking," New J. Phys., Vol. 9, 45, 2007.
doi:10.1088/0022-3727/43/11/113001 Google Scholar

45. Chen, H. and C. T. Chan, "Acoustic cloaking and transformation acoustics," J. Phys. D, Vol. 43, 113001, 2010.
doi:10.1103/PhysRevLett.101.134501 Google Scholar

46. Farhat, M., S. Enoch, S. Guenneau, and A. B. Movchan, "Broadband cylindrical acoustic cloak for linear surface waves in a fluid," Phys. Rev. Lett., Vol. 101, 134501, 2008.
doi:10.1103/PhysRevLett.106.253901 Google Scholar

47. Popa, B.-I., L. Zigoneanu, and S. A. Cummer, "Experimental acoustic ground cloak in air," Phys. Rev. Lett., Vol. 106, 253901, 2011.
doi:10.1063/1.3068749 Google Scholar

48. Alitalo, P., F. Bongard, J.-F. Zurcher, J. Mosig, and S. Tretyakov, "Expermental verification of broadband cloaking using a volumetric cloak composed of periodically stacked cylindrical transmission-line networks," Appl. Phys. Lett., Vol. 94, 014103, 2009.
doi:10.1103/PhysRevLett.103.103905 Google Scholar

49. Tretyakov, S., P. Alitalo, O. Luukkonen, and C. Simovski, "Broadband electromagnetic cloaking of long cylindrical objects," Phys. Rev. Lett., Vol. 103, 103905, 2009.
doi:10.1126/science.1166949 Google Scholar

50. Liu, R., C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband ground-plane cloak," Science, Vol. 323, 366-369, 2009. Google Scholar

51. Ma, H. F. and T. J. Cui, "Three-dimensional broadband ground-plane cloak made of metamaterials," Nat. Commun., Vol. 1, 21, 2010.
doi:10.1038/ncomms1176 Google Scholar

52. Chen, X., Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, "Macroscopic invisibility cloaking of visible light," Nat. Commun., Vol. 2, 176, 2011.
doi:10.1103/PhysRevLett.106.033901 Google Scholar

53. Zhang, B., Y. Luo, X. Liu, and G. Barbastathis, "Macroscopic invisible cloak for visible light," Phys. Rev. Lett., Vol. 106, 033901, 2011.
doi:10.1103/PhysRevLett.102.213901 Google Scholar

54. Smolyaninov, I. I., V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, "Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking," Phys. Rev. Lett., Vol. 102, 213901, 2009.
doi:10.1126/science.1186351 Google Scholar

55. Ergin, T., N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, "Three-dimensional invisibility cloak at optical wavelengths," Science, Vol. 328, 337, 2010.
doi:10.1364/OL.36.002059 Google Scholar

56. Fischer, J., T. Ergin, and M. Wegener, "Three-dimensional polarization-independent visible-frequency carpet invisibility cloak," Opt. Lett., Vol. 36, 2059-2061, 2011.
doi:10.1088/1464-4258/11/1/015104 Google Scholar

57. Collins, P. and J. McGuirk, "A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks," J. Opt. A: Pure Appl., Vol. 11, 015104, 2009.
doi:10.1063/1.3026532 Google Scholar

58. Jiang, W. X., T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, "Invisibility cloak without singularity," Appl. Phys. Lett., Vol. 93, 194102, 2008.
doi:10.1063/1.3168652 Google Scholar

59. Hu, J., X. Zhou, and G. Hu, "Nonsingular two dimensional cloak of arbitrary shape," Appl. Phys. Lett., Vol. 95, 011107, 2009.
doi:10.1103/PhysRevLett.99.233901 Google Scholar

60. Yan, M., Z. Chao, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett., Vol. 99, 233901, 2007.
doi:10.1103/PhysRevLett.101.203901 Google Scholar

61. Li, J. and J. B. Pendry, "Hiding under the carpet: A new strategy for cloaking," Phys. Rev. Lett., Vol. 101, 203901, 2008.
doi:10.1364/JOSAB.28.000922 Google Scholar

62. Huang, L., D. Zhou, J. Wang, Z. Li, X. Chen, and W. Lu, "Generalized transformation for nonmagnetic invisibility cloak with minimized scattering," J. Opt. Soc. Am. B, Vol. 28, 922-928, 2011.
doi:10.1364/OE.18.013038 Google Scholar

63. Han, T. and C.-W. Qiu, "Isotropic nonmagnetic flat cloaks degenerated from homogeneous anisotropic trapeziform cloaks," Opt. Express, Vol. 18, 13038-13043, 2010.
doi:10.1103/PhysRevLett.99.063903 Google Scholar

64. Chen, Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett., Vol. 99, 063903, 2007.
doi:10.1088/1367-2630/11/11/113001 Google Scholar

65. Novitsky, A., C.-W. Qiu, and S. Zouhdi, "Transformation-based spherical cloaks designed by an implicit transformation-independent method: Theory and optimization," New J. Phys., Vol. 11, 113001, 2009.
doi:10.1103/PhysRevE.80.016604 Google Scholar

66. Qiu, C.-W., A. Novitsky, H. Ma, and S. Qu, "Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloaks," Phys. Rev. E, Vol. 80, 016604, 2009.
doi:10.1103/PhysRevE.72.016623 Google Scholar

67. Alu, A. and N. Engheta, "Achiving transparency with plasmonic and metamaterial coatings," Phys. Rev. E, Vol. 72, 016623, 2005.
doi:10.1088/1367-2630/14/1/013054 Google Scholar

68. Rainwater, D., A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Al, "Experimental verification of three-dimensional plasmonic cloaking in free-space," New J. Phys., Vol. 14, 013054, 2012.
doi:10.1103/PhysRevLett.102.233901 Google Scholar

69. Alu, A. and N. Engheta, "Cloaking a sensor," Phys. Rev. Lett., Vol. 102, 233901, 2009.
doi:10.1088/1367-2630/10/11/115028 Google Scholar

70. Tretyakov, S. A., I. S. Nefedov, and P. Alitalo, "Generalized field-transforming metamaterials," New J. Phys., Vol. 10, 115028, 2008.
doi:10.1088/1367-2630/10/11/115022 Google Scholar

71. Yaghjian, A. D. and S. Maci, "Alternative derivation of electromagnetic cloaks and concentrators," New J. Phys., Vol. 10, 115022, 2008.
doi:10.1088/2040-8978/13/3/035104 Google Scholar

72. Novitsky, A. V., "Inverse problem in transformation optics," J. Opt., Vol. 13, 035104, 2011.
doi:10.1103/PhysRevLett.58.1499 Google Scholar

73. Durnin, J., J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett., Vol. 58, 1499-1504, 1987.
doi:10.1103/PhysRevLett.99.213901 Google Scholar

74. Siviloglou, G. A., J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett., Vol. 99, 213901, 2007.
doi:10.1088/2040-8978/13/2/024003 Google Scholar

75. McCall, M. W., A. Favaro, P. Kinsler, and A. Boardman, "A spacetime cloak, or a history editor," J. Opt., Vol. 13, 024003, 2011.
doi:10.1088/2040-8978/13/2/024007 Google Scholar

76. Cummer, S. A. and R. T. Thompson, "Frequency conversion by exploiting time in transformation optics," J. Opt., Vol. 13, 024007, 2011.
doi:10.1038/nature10695 Google Scholar

77. Fridman, M., A. Farsi, Y. Okawachi, and A. L. Gaeta, "Demonstration of temporal cloaking," Nature, Vol. 481, 62-65, 2012.
doi:10.1364/JOSAB.28.001082 Google Scholar

78. , , , http://www.comsol.com/.
doi:10.1088/2040-8978/13/7/075103

79. Zang, Zang and C. Jiang, "A rotatable and amplifying optical transformation device," J. Opt. Soc. Am. B, Vol. 28, 1082-1087, 2011. Google Scholar

80. Perczel, J., C. Garcia-Meca, and U. Leonhardt, "Partial transmutation of singularities in optical instruments," J. Opt., Vol. 13, 075103, 2011. Google Scholar

81. Fedorov, F. I., Optics of Anisotropic Media, Izdatelstvo AN BSSR, Minsk, 1958.

82. Fedorov, F. I., Theory of Gyrotropy, Nauka i Tehnika, Minsk, 1976.

83. Fedorov, F. I. and V. V. Filippov, Reflection and Transmission of Light by Transparent Crystals, Nauka i Tehnika, Minsk, 1976.

84. Fedorov, F. I., Theory of Elastic Waves in Crystals, Plenum Press, New York, 1968.

85. Fedorov, F. I., Lorentz Group, Nauka, Moscow, 1979.
doi:10.1103/PhysRevD.5.787

86. Fedorov, F. I., "To the theory of total reflection," Doklady Akademii Nauk SSSR, Vol. 105, 465-469, 1955.
doi:10.1103/PhysRevLett.93.083901 Google Scholar

87. Imbert, C., "Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam," Phys. Rev. D, Vol. 5, 787-796, 1972. Google Scholar

88. Onoda, M., S. Murakami, and N. Nagaosa, "Hall effect of light," Phys. Rev. Lett., Vol. 93, 083901, 2004. Google Scholar

89. Serdyukov, A. N., I. V. Semchenko, S. A. Tretyakov, and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, 2001.
doi: --- Either ISSN or Journal title must be supplied.

90. Post, E. J., Formal Structure of Electromagnetics, Noth-Holland Publishing Company, Amsterdam, 1962.
doi: --- Either ISSN/ISBN or Series/Volume title must be supplied.

Dr. Andrey V. Novitsky

Dr. Sergei V. Zhukovsky

Dr. Leonid M. Barkovsky

Prof. Andrei V. Lavrinenko