Vol. 139
Latest Volume
All Volumes
PIER 185 [2026] PIER 184 [2025] PIER 183 [2025] PIER 182 [2025] PIER 181 [2024] PIER 180 [2024] PIER 179 [2024] PIER 178 [2023] PIER 177 [2023] PIER 176 [2023] PIER 175 [2022] PIER 174 [2022] PIER 173 [2022] PIER 172 [2021] PIER 171 [2021] PIER 170 [2021] PIER 169 [2020] PIER 168 [2020] PIER 167 [2020] PIER 166 [2019] PIER 165 [2019] PIER 164 [2019] PIER 163 [2018] PIER 162 [2018] PIER 161 [2018] PIER 160 [2017] PIER 159 [2017] PIER 158 [2017] PIER 157 [2016] PIER 156 [2016] PIER 155 [2016] PIER 154 [2015] PIER 153 [2015] PIER 152 [2015] PIER 151 [2015] PIER 150 [2015] PIER 149 [2014] PIER 148 [2014] PIER 147 [2014] PIER 146 [2014] PIER 145 [2014] PIER 144 [2014] PIER 143 [2013] PIER 142 [2013] PIER 141 [2013] PIER 140 [2013] PIER 139 [2013] PIER 138 [2013] PIER 137 [2013] PIER 136 [2013] PIER 135 [2013] PIER 134 [2013] PIER 133 [2013] PIER 132 [2012] PIER 131 [2012] PIER 130 [2012] PIER 129 [2012] PIER 128 [2012] PIER 127 [2012] PIER 126 [2012] PIER 125 [2012] PIER 124 [2012] PIER 123 [2012] PIER 122 [2012] PIER 121 [2011] PIER 120 [2011] PIER 119 [2011] PIER 118 [2011] PIER 117 [2011] PIER 116 [2011] PIER 115 [2011] PIER 114 [2011] PIER 113 [2011] PIER 112 [2011] PIER 111 [2011] PIER 110 [2010] PIER 109 [2010] PIER 108 [2010] PIER 107 [2010] PIER 106 [2010] PIER 105 [2010] PIER 104 [2010] PIER 103 [2010] PIER 102 [2010] PIER 101 [2010] PIER 100 [2010] PIER 99 [2009] PIER 98 [2009] PIER 97 [2009] PIER 96 [2009] PIER 95 [2009] PIER 94 [2009] PIER 93 [2009] PIER 92 [2009] PIER 91 [2009] PIER 90 [2009] PIER 89 [2009] PIER 88 [2008] PIER 87 [2008] PIER 86 [2008] PIER 85 [2008] PIER 84 [2008] PIER 83 [2008] PIER 82 [2008] PIER 81 [2008] PIER 80 [2008] PIER 79 [2008] PIER 78 [2008] PIER 77 [2007] PIER 76 [2007] PIER 75 [2007] PIER 74 [2007] PIER 73 [2007] PIER 72 [2007] PIER 71 [2007] PIER 70 [2007] PIER 69 [2007] PIER 68 [2007] PIER 67 [2007] PIER 66 [2006] PIER 65 [2006] PIER 64 [2006] PIER 63 [2006] PIER 62 [2006] PIER 61 [2006] PIER 60 [2006] PIER 59 [2006] PIER 58 [2006] PIER 57 [2006] PIER 56 [2006] PIER 55 [2005] PIER 54 [2005] PIER 53 [2005] PIER 52 [2005] PIER 51 [2005] PIER 50 [2005] PIER 49 [2004] PIER 48 [2004] PIER 47 [2004] PIER 46 [2004] PIER 45 [2004] PIER 44 [2004] PIER 43 [2003] PIER 42 [2003] PIER 41 [2003] PIER 40 [2003] PIER 39 [2003] PIER 38 [2002] PIER 37 [2002] PIER 36 [2002] PIER 35 [2002] PIER 34 [2001] PIER 33 [2001] PIER 32 [2001] PIER 31 [2001] PIER 30 [2001] PIER 29 [2000] PIER 28 [2000] PIER 27 [2000] PIER 26 [2000] PIER 25 [2000] PIER 24 [1999] PIER 23 [1999] PIER 22 [1999] PIER 21 [1999] PIER 20 [1998] PIER 19 [1998] PIER 18 [1998] PIER 17 [1997] PIER 16 [1997] PIER 15 [1997] PIER 14 [1996] PIER 13 [1996] PIER 12 [1996] PIER 11 [1995] PIER 10 [1995] PIER 09 [1994] PIER 08 [1994] PIER 07 [1993] PIER 06 [1992] PIER 05 [1991] PIER 04 [1991] PIER 03 [1990] PIER 02 [1990] PIER 01 [1989]
2013-04-17
3D Printed Lattices with Spatially Variant Self-Collimation
By
Progress In Electromagnetics Research, Vol. 139, 1-14, 2013
Abstract
In this work, results are given for controlling waves arbitrarily inside a new type of spatially variant lattice. To demonstrate the concept, an unguided beam was made to flow around a 90° bend without diffracting or scattering. Control of the field was achieved by spatially varying the orientation of the unit cells throughout a self-collimating photonic crystal, but in a manner that almost completely eliminated deformations to the size and shape of the unit cells. The device was all-dielectric, monolithic, and made from an ordinary dielectric with low relative permittivity (εr = 2.45). It was manufactured by fused deposition modeling, a form of 3D printing, and its performance confirmed experimentally at around 15 GHz.
Citation
Raymond C. Rumpf, Javier Pazos, Cesar R. Garcia, Luis Ochoa, and Ryan Wicker, "3D Printed Lattices with Spatially Variant Self-Collimation," Progress In Electromagnetics Research, Vol. 139, 1-14, 2013.
doi:10.2528/PIER13030507
References

1. Capolino, F., Theory and Phenomena of Metamaterials, 1st Edition, CRC Press, 2009.
doi:10.1201/9781420054262

2. Shelby, R., D. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science, Vol. 292, 77-79, 2001.
doi:10.1126/science.1058847        Google Scholar

3. Smith, D. R., J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science, Vol. 305, 788-792, 2004.
doi:10.1126/science.1096796        Google Scholar

4. Smith, D. R. and N. Kroll, "Negative refractive index in left-handed materials," Physical Review Letters, Vol. 85, 2933-2936, 2000.
doi:10.1103/PhysRevLett.85.2933        Google Scholar

5. Gralak, B., S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," JOSA A, Vol. 17, 1012-1020, 2000.
doi:10.1364/JOSAA.17.001012        Google Scholar

6. Wu, L., M. Mazilu, and T. F. Krauss, "Beam steering in planar-photonic crystals: From superprism to supercollimator," Journal of Lightwave Technology, Vol. 21, 561, 2003.
doi:10.1109/JLT.2003.808773        Google Scholar

7. Notomi, M., "Negative refraction in photonic crystals," Optical and Quantum Electronics, Vol. 34, 133-143, 2002.
doi:10.1023/A:1013300825612        Google Scholar

8. Baba, T. and M. Nakamura, "Photonic crystal light deflection devices using the superprism effect," IEEE Journal of Quantum Electronics, Vol. 38, 909-914, 2002.
doi:10.1109/JQE.2002.1017606        Google Scholar

9. Enoch, S., G. Tayeb, and B. Gralak, "The richness of the dispersion relation of electromagnetic bandgap materials," IEEE Transactions on Antennas and Propagation, Vol. 51, 2659-2666, 2003.
doi:10.1109/TAP.2003.817549        Google Scholar

10. Baba, T., "Slow light in photonic crystals," Nature Photonics, Vol. 2, 465-473, 2008.
doi:10.1038/nphoton.2008.146        Google Scholar

11. Kosaka, H., T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals: Toward microscale lightwave circuits," Journal of Lightwave Technology, Vol. 17, 2032, 1999.
doi:10.1109/50.802991        Google Scholar

12. Yablonovitch, E., "Inhibited spontaneous emission in solid-state physics and electronics," Physical Review Letters, Vol. 58, 2059-2062, 1987.
doi:10.1103/PhysRevLett.58.2059        Google Scholar

13. Yablonovitch, E., "Photonic crystals," Journal of Modern Optics, Vol. 41, 173-194, 1994.
doi:10.1080/09500349414550261        Google Scholar

14. John, S., "Strong localization of photons in certain disordered dielectric superlattices," Physical Review Letters, Vol. 58, 2486-2489, 1987.
doi:10.1103/PhysRevLett.58.2486        Google Scholar

15. Noda, S., A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature, Vol. 407, 608-610, 2000.
doi:10.1038/35036532        Google Scholar

16. Grann, E. B., M. Moharam, and D. A. Pommet, "Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings," JOSA A, Vol. 11, 2695-2703, 1994.
doi:10.1364/JOSAA.11.002695        Google Scholar

17. Lindell, I., S. Tretyakov, K. Nikoskinen, and S. Ilvonen, "BW media - Media with negative parameters, capable of supporting backward waves," Microwave and Optical Technology Letters, Vol. 31, 129-133, 2001.
doi:10.1002/mop.1378        Google Scholar

18. Smith, D. and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Physical Review Letters, Vol. 90, 77405, 2003.
doi:10.1103/PhysRevLett.90.077405        Google Scholar

19. Smith, D. R., P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, "Design and measurement of anisotropic metamaterials that exhibit negative refraction," IEICE Transactions on Electronics, Vol. E87-C, 359-370, 2004.        Google Scholar

20. Smith, D. R., D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, "Partial focusing of radiation by a slab of indefinite media," Applied Physics Letters, Vol. 84, 2244, 2004.
doi:10.1063/1.1690471        Google Scholar

21. Wood, B., J. Pendry, and D. Tsai, "Directed subwavelength imaging using a layered metal-dielectric system," Physical Review B, Vol. 74, 115116, 2006.
doi:10.1103/PhysRevB.74.115116        Google Scholar

22. Hao, J., Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. Chan, and L. Zhou, "Manipulating electromagnetic wave polarizations by anisotropic metamaterials," Physical Review Letters, Vol. 99, 63908, 2007.
doi:10.1103/PhysRevLett.99.063908        Google Scholar

23. Hoffman, A. J., L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, "Negative refraction in semiconductor metamaterials," Nature Materials, Vol. 6, 946-950, 2007.
doi:10.1038/nmat2033        Google Scholar

24. Elser, J. and V. A. Podolskiy, "Scattering-free plasmonic optics with anisotropic metamaterials," Physical Review Letters, Vol. 100, 66402, 2008.
doi:10.1103/PhysRevLett.100.066402        Google Scholar

25. Yao, J., Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science, Vol. 321, 930-930, 2008.
doi:10.1126/science.1157566        Google Scholar

26. Fang, A., T. Koschny, and C. M. Soukoulis, "Optical anisotropic metamaterials: Negative refraction and focusing," Physical Review B, Vol. 79, 245127, 2009.
doi:10.1103/PhysRevB.79.245127        Google Scholar

27. Garcia, C. R., J. Correa, D. Espalin, J. H. Barton, R. C. Rumpf, R. Wicker, and V. Gonzalez, "3D printing of anisotropic metamaterials," Progress In Electromagnetics Research Letters, Vol. 34, 75-82, 2012.        Google Scholar

28. Ponizovskaya, E., M. Nieto-Vesperinas, and N. Garcia, "Losses for microwave transmission in metamaterials for producing left-handed materials: The strip wires," Applied Physics Letters, Vol. 81, 4470-4472, 2002.
doi:10.1063/1.1527982        Google Scholar

29. Varadan, V. and L. Ji, "Accounting for power `loss' in metamaterials," Metamaterials 2008, Pamplona, Spain, 2008.        Google Scholar

30. Fang, A., T. Koschny, M. Wegener, and C. Soukoulis, "Self-consistent calculation of metamaterials with gain," Physical Review B, Vol. 79, 241104, 2009.
doi:10.1103/PhysRevB.79.241104        Google Scholar

31. Varadan, V. V. and J. Liming, "Does a negative refractive index always result in negative refraction? - Effect of loss," IEEE MTT-S International Microwave Symposium Digest, MTT' 09, 61-64, 2009.        Google Scholar

32. Khurgin, J. B. and G. Sun, "Scaling of losses with size and wavelength in nanoplasmonics and metamaterials," Applied Physics Letters, Vol. 99, 211106, 2011.
doi:10.1063/1.3664105        Google Scholar

33. De Damborenea, J., "Surface modification of metals by high power lasers," Surface and Coatings Technology, Vol. 100, 377-382, 1998.
doi:10.1016/S0257-8972(97)00652-X        Google Scholar

34. Batanov, G., N. Berezhetskaya, I. Kossyi, A. Magunov, and V. Silakov, "Interaction of high-power microwave beams with metal-dielectric media," The European Physical Journal Applied Physics, Vol. 26, 11-16, 2004.
doi:10.1051/epjap:2004016        Google Scholar

35. Petelin, M. and A. Fix, "Comparison of metals in their steadiness to pulse-periodic microwave heating fatigue," IEEE International Vacuum Electronics Conference, 163-164, 2009.        Google Scholar

36. Bilik, V. and J. Bezek, "High power limits of waveguide stub tuners," J. Microw. Power, Vol. 44, 178-186, 2010.        Google Scholar

37. Anzel, I., "High temperature oxidation of metals and alloys,", 325-336, Association of Metallurgical Engineers of Serbia, 2007.        Google Scholar

38. Kosaka, H., T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Applied Physics Letters, Vol. 74, 1212, 1999.
doi:10.1063/1.123502        Google Scholar

39. Witzens, J., M. Loncar, and A. Scherer, "Self-collimation in planar photonic crystals," IEEE Journal of Selected Topics in Quantum Electronics, Vol. 8, 1246-1257, 2002.
doi:10.1109/JSTQE.2002.806693        Google Scholar

40. Iliew, R., C. Etrich, U. Peschel, F. Lederer, M. Augustin, H. J. Fuchs, D. Schelle, E. B. Kley, S. Nolte, and A. Tunnermann, "Diffractionless propagation of light in a low-index photonic-crystal film," Applied Physics Letters, Vol. 85, 5854-5856, 2004.
doi:10.1063/1.1830675        Google Scholar

41. Feng, S., Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, "Focusing properties of a rectangular-rod photonic-crystal slab," Journal of Applied Physics, Vol. 98, 063102, 2005.
doi:10.1063/1.2058190        Google Scholar

42. Iliew, R., C. Etrich, and F. Lederer, "Self-collimation of light in three-dimensional photonic crystals," Optics Express, Vol. 13, 7076-7085, 2005.
doi:10.1364/OPEX.13.007076        Google Scholar

43. Shin, J. and S. Fan, "Conditions for self-collimation in three-dimensional photonic crystals," Optics Letters, Vol. 30, 2397-2399, 2005.
doi:10.1364/OL.30.002397        Google Scholar

44. Lu, Z., S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, "Experimental demonstration of self-collimation inside a three-dimensional photonic crystal," Physical Review Letters, Vol. 96, 173902, 2006.
doi:10.1103/PhysRevLett.96.173902        Google Scholar

45. Kwon, D. H. and D. H. Werner, "Transformation optical designs for wave collimators, flat lenses and right-angle bends," New Journal of Physics, Vol. 10, 115023, 2008.
doi:10.1088/1367-2630/10/11/115023        Google Scholar

46. Mekis, A., J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Physical Review Letters, Vol. 77, 3787-3790, 1996.
doi:10.1103/PhysRevLett.77.3787        Google Scholar

47. Roberts, D., M. Rahm, J. Pendry, and D. Smith, "Transformation-optical design of sharp waveguide bends and corners," Applied Physics Letters, Vol. 93, 251111, 2008.
doi:10.1063/1.3055604        Google Scholar

48. Gabrielli, L. H. and M. Lipson, "Integrated Luneburg lens via ultra-strong index gradient on silicon," Optics Express, Vol. 19, 20122-20127, 2011.
doi:10.1364/OE.19.020122        Google Scholar

49. Spadoti, D. H., L. H. Gabrielli, C. B. Poitras, and M. Lipson, "Focusing light in a curved-space," Optics Express, Vol. 18, 3181-3186, 2010.
doi:10.1364/OE.18.003181        Google Scholar

50. Vasic, B., G. Isic, R. Gajic, and K. Hingerl, "Controlling electromagnetic fields with graded photonic crystals in metamaterial regime," Optics Express, Vol. 18, 20321-20333, 2010.
doi:10.1364/OE.18.020321        Google Scholar

51. Akmansoy, E., E. Centeno, K. Vynck, D. Cassagne, and J. M. Lourtioz, "Graded photonic crystals curve the flow of light: An experimental demonstration by the mirage effect," Applied Physics Letters, Vol. 92, 133501, 2008.
doi:10.1063/1.2901684        Google Scholar

52. Cassan, E., K. V. Do, C. Caer, D. Marris-Morini, and L. Vivien, "Short-wavelength light propagation in graded photonic crystals," Journal of Lightwave Technology, Vol. 29, 1937-1943, 2011.
doi:10.1109/JLT.2011.2151175        Google Scholar

53. Centeno, E. and D. Cassagne, "Graded photonic crystals," Optics Letters, Vol. 30, 2278-2280, 2005.
doi:10.1364/OL.30.002278        Google Scholar

54. Do, K. V., X. Le Roux, D. Marris-Morini, L. Vivien, and E. Cassan, "Experimental demonstration of light bending at optical frequencies using a non-homogenizable graded photonic crystal," Optics Express, Vol. 20, 4776-4783, 2012.
doi:10.1364/OE.20.004776        Google Scholar

55. Li, Y. Y., M. Y. Li, P. F. Gu, Z. R. Zheng, and X. Liu, "Graded wavelike two-dimensional photonic crystal made of thin films," Applied Optics, Vol. 47, C70-C74, 2008.
doi:10.1364/AO.47.000C70        Google Scholar

56. Rumpf, R. C. and J. Pazos, "Synthesis of spatially variant lattices," Optics Express, Vol. 20, 15263-15274, 2012.
doi:10.1364/OE.20.015263        Google Scholar

57. Hussein, M. I., "Reduced Bloch mode expansion for periodic media band structure calculations," Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, Vol. 465, 2825-2848, 2009.
doi:10.1098/rspa.2008.0471        Google Scholar

58. Johnson, S. G. and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Optics Express, Vol. 8, 173-190, 2001.
doi:10.1364/OE.8.000173        Google Scholar

59. Guo, S. and S. Albin, "Simple plane wave implementation for photonic crystal calculations," Optics Express, Vol. 11, 167-175, 2003.
doi:10.1364/OE.11.000167        Google Scholar

60. Rumpf, R. C., "Simple implementation of arbitrarily shaped total-field/scattered-field regions in finite-difference frequency-domain," Progress In Electromagnetics Research, Vol. 36, 221-248, 2012.
doi:10.2528/PIERB11092006        Google Scholar

61. Gibson, I., D. W. Rosen, and B. Stucker, Additive Manufacturing Technologies: Rapid Prototyping to Direct Digital Manufacturing, 459, Springer, London, New York, 2010.

62. Wohlers, T., "Wohlers report 2012: Additive manufacturing and 3D printing state of the industry,", Annual Worldwide Progress Report, Wohlers Associates, Fort Collins, 2012.        Google Scholar

63. Palmer, J., B. Jokiel, C. Nordquist, B. Kast, C. Atwood, E. Grant, F. Livingston, F. Medina, and R. Wicker, "Miniature RF components enabled by mesoscale rapid manufacturing,", 2005.        Google Scholar

64. Palmer, J., B. Jokiel, C. Nordquist, B. Kast, C. Atwood, E. Grant, F. Livingston, F. Medina, and R. Wicker, "Mesoscale RF relay enabled by integrated rapid manufacturing," Rapid Prototyping Journal, Vol. 12, 148-155, 2006.
doi:10.1108/13552540610670726        Google Scholar

65. Choi, J. W., E. MacDonald, and R. Wicker, "Multi-material microstereolithography," The International Journal of Advanced Manufacturing Technology, Vol. 49, 543-551, 2010.
doi:10.1007/s00170-009-2434-8        Google Scholar

66. Choi, J. W., H. C. Kim, and R. Wicker, "Multi-material stereolithography," Journal of Materials Processing Technology, Vol. 211, 318-328, 2011.
doi:10.1016/j.jmatprotec.2010.10.003        Google Scholar

67. Choi, J. W., F. Medina, C. Kim, D. Espalin, D. Rodriguez, B. Stucker, and R. Wicker, "Development of a mobile fused deposition modeling system with enhanced manufacturing flexibility," Journal of Materials Processing Technology, Vol. 211, 424-432, 2011.
doi:10.1016/j.jmatprotec.2010.10.019        Google Scholar

68. Lopes, A. J., E. MacDonald, and R. B. Wicker, "Integrating stereolithography and direct print technologies for 3D structural electronics fabrication," Rapid Prototyping Journal, Vol. 18, 129-143, 2012.
doi:10.1108/13552541211212113        Google Scholar

69. Wicker, R. B. and E. W. MacDonald, "Multi-material, multi-technology stereolithography," Virtual and Physical Prototyping, Vol. 7, 181-194, 2012.
doi:10.1080/17452759.2012.721119        Google Scholar

70. Botten, L., T. White, C. M. de Sterke, and R. McPhedran, "Wide-angle coupling into rod-type photonic crystals with ultralow reflectance," Physical Review E, Vol. 74, 026603, 2006.
doi:10.1103/PhysRevE.74.026603        Google Scholar

71. Sigaj, W. and B. Gralak, "Semianalytical design of antireflection gratings for photonic crystals," Physical Review B, Vol. 85, 035114, 2012.
doi:10.1103/PhysRevB.85.035114        Google Scholar