We propose deep-subwavelength optical waveguides based on metal–dielectric multilayer indefinite metamaterials with ultrahigh effective refractive indices. Waveguide modes with different mode orders are systematically analyzed with numerical simulations based on both metal–dielectric multilayer structures and the effective medium approach. The dependences of waveguide mode indices, propagation lengths, and mode areas on different mode orders, free-space wavelengths, and sizes of waveguide cross sections are studied. Furthermore, waveguide modes are also illustrated with iso-frequency contours in the wave vector space in order to investigate the mechanism of waveguide mode cutoff for high-order modes. The deep-subwavelength optical waveguide with a size smaller than ${\lambda }_0/50$ and a mode area in the order of ${10}^{-4}{\lambda }_0^2$ is realized, and an ultrahigh effective refractive index up to 62.0 is achieved at the telecommunication wavelength. This new type of metamaterial optical waveguide opens up opportunities for various applications in enhanced light–matter interactions.

© 2012 Optical Society of America

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]
2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999). [CrossRef]
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4. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006). [CrossRef]
5. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–075105 (2006). [CrossRef]
6. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007). [CrossRef]
7. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]
8. 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]
9. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007). [CrossRef]
10. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005). [CrossRef]
11. J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009). [CrossRef]
12. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011). [CrossRef]
13. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003). [CrossRef]
14. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16, 15439–15448 (2008). [CrossRef]
15. J. Yao, 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 321, 930 (2008). [CrossRef]
16. Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010). [CrossRef]
17. M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010). [CrossRef]
18. T. Jiang, J. Zhao, and Y. Feng, “Stopping light by an air waveguide with anisotropic metamaterial cladding,” Opt. Express 17, 170–177 (2009). [CrossRef]
19. Y. J. Huang, W. T. Lu, and S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77, 063836 (2008). [CrossRef]
20. W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric field on a slab waveguide made of indefinite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010). [CrossRef]
21. Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012). [CrossRef]
22. J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. 108, 11327–11331 (2011). [CrossRef]
23. X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photon. 6, 450–454 (2012). [CrossRef]
24. A. A. Govyadinov and V. A. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73, 155108 (2006). [CrossRef]
25. M. Yan, L. Thylén, and M. Qiu, “Layered metal-dielectric waveguide: subwavelength guidance, leveraged modulation sensitivity in mode index, and reversed mode ordering,” Opt. Express 19, 3818–3824 (2011). [CrossRef]
26. J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98, 101901 (2011). [CrossRef]
27. G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008). [CrossRef]
28. F.-Y. Meng, Q. Wu, and L.-W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys. A 94, 747–753 (2009). [CrossRef]
29. Z. Shi, G. Piredda, A. C. Liapis, M. A. Nelson, L. Novotny, and R. W. Boyd, “Surface-plasmon polaritons on metal-dielectric nanocomposite films,” Opt. Lett. 34, 3535–3537 (2009). [CrossRef]
30. C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. 35, 610–612 (2010). [CrossRef]
31. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef]
32. T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010). [CrossRef]
33. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef]
34. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011). [CrossRef]
35. 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]
36. C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98, 2963–2971 (1994). [CrossRef]
37. A. Sihvola, Electromagnetic Mixing Formulas and Applications (Institution of Electrical Engineers, 1999).
38. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
39. W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express 18, 5124–5134 (2010). [CrossRef]
40. A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85, 085416 (2012). [CrossRef]
41. J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007). [CrossRef]
42. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007). [CrossRef]
43. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010). [CrossRef]
44. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.
45. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009). [CrossRef]

Appl. Phys. A

• F.-Y. Meng, Q. Wu, and L.-W. Li, “Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic metamaterial,” Appl. Phys. A 94, 747–753 (2009). [CrossRef]

Appl. Phys. B

• Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010). [CrossRef]

Appl. Phys. Lett.

• J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98, 101901 (2011). [CrossRef]
• J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007). [CrossRef]

IEEE Trans. Microwave Theory Tech

• J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999). [CrossRef]

J. Nanomater.

• J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007). [CrossRef]

J. Phys. Chem.

• C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template-synthesized nanoscopic gold Particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98, 2963–2971 (1994). [CrossRef]

Nano Lett.

• X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11, 321–328 (2011). [CrossRef]
• 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]
• Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12, 1443–1447 (2012). [CrossRef]

Nat Photon

• W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat Photon 1, 224–227 (2007). [CrossRef]

Nat. Photon.

• X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photon. 6, 450–454 (2012). [CrossRef]

Nature

• R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef]
• M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011). [CrossRef]
• S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010). [CrossRef]
• M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009). [CrossRef]

Nature Nanotechnol.

• T. M. Babinec, J. M. HausmannBirgit, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nature Nanotechnol. 5, 195–199 (2010). [CrossRef]

Opt. Commun.

• G.-D. Xu, T. Pan, T.-C. Zang, and J. Sun, “Characteristics of guided waves in indefinite-medium waveguides,” Opt. Commun. 281, 2819–2825 (2008). [CrossRef]

Opt. Express

• M. Yan, L. Thylén, and M. Qiu, “Layered metal-dielectric waveguide: subwavelength guidance, leveraged modulation sensitivity in mode index, and reversed mode ordering,” Opt. Express 19, 3818–3824 (2011). [CrossRef]
• Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef]
• Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16, 15439–15448 (2008). [CrossRef]
• T. Jiang, J. Zhao, and Y. Feng, “Stopping light by an air waveguide with anisotropic metamaterial cladding,” Opt. Express 17, 170–177 (2009). [CrossRef]
• Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006). [CrossRef]
• W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express 18, 5124–5134 (2010). [CrossRef]

Opt. Lett.

• M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010). [CrossRef]
• Z. Shi, G. Piredda, A. C. Liapis, M. A. Nelson, L. Novotny, and R. W. Boyd, “Surface-plasmon polaritons on metal-dielectric nanocomposite films,” Opt. Lett. 34, 3535–3537 (2009). [CrossRef]
• C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. 35, 610–612 (2010). [CrossRef]

Phys. Rev. A

• Y. J. Huang, W. T. Lu, and S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77, 063836 (2008). [CrossRef]
• W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric field on a slab waveguide made of indefinite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010). [CrossRef]

Phys. Rev. B

• A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–075105 (2006). [CrossRef]
• A. A. Govyadinov and V. A. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73, 155108 (2006). [CrossRef]
• A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85, 085416 (2012). [CrossRef]
• P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

Phys. Rev. Lett.

• J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005). [CrossRef]
• J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009). [CrossRef]
• D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003). [CrossRef]

Proc. Natl. Acad. Sci.

• J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. 108, 11327–11331 (2011). [CrossRef]

Science

• J. Yao, 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 321, 930 (2008). [CrossRef]
• R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef]
• Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007). [CrossRef]
• J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]
• 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]

Sov. Phys. Usp.

• V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

Other

• A. Sihvola, Electromagnetic Mixing Formulas and Applications (Institution of Electrical Engineers, 1999).
• X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Gain-assisted hyperbolic metamaterials,” in Quantum Electronics and Laser Science Conference (QELS), OSA Technical Digest (Optical Society of America, 2012), paper QTu1G.

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