Abstract

In this work, it is shown how the shapes of surface plasmon dispersion curves can be engineered by manipulating the distribution of the electromagnetic fields in multilayer structures, which themselves are controlled by the free electron density in metal-like materials, such as doped semiconductors in the THz spectral range. By having a nonuniform free electron density profile, reduced relative to that in typical bulk metals, the electromagnetic fields of surface plasmons are distributed in different metallic materials that have different complex dielectric permittivities. As the in-plane component of surface plasmon’s wave-vector increases, they become more confined to a particular layer of the multilayer structure and have energies that are predictable by considering the permittivity of the layer in which the fields are most concentrated. Unusual and arbitrary shapes of surface plasmon dispersion curves can be designed, including stair steps and dovetails shapes.

© 2013 Optical Society of America

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2013 (1)

2012 (1)

2010 (2)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

2009 (1)

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt.56, 1908–1915 (2009).
[CrossRef]

2008 (2)

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

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[CrossRef]

2007 (1)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature450, 397–401 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (4)

A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett.86, 141102 (2005).
[CrossRef]

K. Li, X. Li, M. Stockman, and D. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71, 1–5 (2005).
[CrossRef]

A. Karalis, E. Lidorikis, M. Ibanescu, J. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett.95, 1–4 (2005).
[CrossRef]

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

2004 (1)

2003 (2)

W. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–831 (2003).
[CrossRef] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003).
[CrossRef] [PubMed]

2002 (2)

P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B At. Mol. Opt. Phys.35, R31 (2002).
[CrossRef]

F. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B66, 155412 (2002).
[CrossRef]

1995 (3)

1981 (1)

E. Litwin-Staszewska, W. Szymanska, and P. Piotrzkowski, “The electron mobility and Thermoelectric Power in InSb at Atmospheric and Hydrostatic Pressures,” Phys. Status Solidi B106, 551–559 (1981).
[CrossRef]

Azad, A. K.

A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett.86, 141102 (2005).
[CrossRef]

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Barnes, W.

W. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–831 (2003).
[CrossRef] [PubMed]

Bergman, D.

K. Li, X. Li, M. Stockman, and D. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71, 1–5 (2005).
[CrossRef]

Bigelow, M.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003).
[CrossRef] [PubMed]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature450, 397–401 (2007).
[CrossRef] [PubMed]

Boltasseva, A.

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

Boyd, R.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Boyd, R. W.

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt.56, 1908–1915 (2009).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003).
[CrossRef] [PubMed]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A21, 2442–2446 (2004).
[CrossRef]

Bykov, D. A.

Cai, G.

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Catrysse, P. B.

Chegel, V.

V. Chegel, Y. Demidenko, V. Lozovski, and A. Tsykhonya, “Influence of the shape of the particles covering the metal surface on the dispersion relations of surface plasmons,” Surf. Sci.602, 1540–1546 (2008).
[CrossRef]

Cheng, Y.

Cotter, N. P. K.

Demidenko, Y.

V. Chegel, Y. Demidenko, V. Lozovski, and A. Tsykhonya, “Influence of the shape of the particles covering the metal surface on the dispersion relations of surface plasmons,” Surf. Sci.602, 1540–1546 (2008).
[CrossRef]

Dereux, A.

W. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–831 (2003).
[CrossRef] [PubMed]

Ding, P.

Doskolovich, L. L.

Ebbesen, T.

W. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–831 (2003).
[CrossRef] [PubMed]

Emani, N.

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

Fan, C.

Gaeta, A.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

García-Vidal, F.

F. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B66, 155412 (2002).
[CrossRef]

Gauthier, D.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

He, J.

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature450, 397–401 (2007).
[CrossRef] [PubMed]

Hill, S.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

Hu, W.

Ibanescu, M.

A. Karalis, E. Lidorikis, M. Ibanescu, J. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett.95, 1–4 (2005).
[CrossRef]

Ishii, S.

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

Joannopoulos, J.

A. Karalis, E. Lidorikis, M. Ibanescu, J. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett.95, 1–4 (2005).
[CrossRef]

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Karalis, A.

A. Karalis, E. Lidorikis, M. Ibanescu, J. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett.95, 1–4 (2005).
[CrossRef]

König, F.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

Kuklewicz, C.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

Leonhardt, U.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003).
[CrossRef] [PubMed]

Li, K.

K. Li, X. Li, M. Stockman, and D. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71, 1–5 (2005).
[CrossRef]

Li, X.

K. Li, X. Li, M. Stockman, and D. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71, 1–5 (2005).
[CrossRef]

Liang, E.

Lidorikis, E.

A. Karalis, E. Lidorikis, M. Ibanescu, J. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett.95, 1–4 (2005).
[CrossRef]

Litwin-Staszewska, E.

E. Litwin-Staszewska, W. Szymanska, and P. Piotrzkowski, “The electron mobility and Thermoelectric Power in InSb at Atmospheric and Hydrostatic Pressures,” Phys. Status Solidi B106, 551–559 (1981).
[CrossRef]

Lozovski, V.

V. Chegel, Y. Demidenko, V. Lozovski, and A. Tsykhonya, “Influence of the shape of the particles covering the metal surface on the dispersion relations of surface plasmons,” Surf. Sci.602, 1540–1546 (2008).
[CrossRef]

Maier, S.

S. Maier, Plasmonics - Fundamentals and Applications, Vol. 76 of Chemistry and Materials Science (Springer, 2010).

Martín-Moreno, L.

F. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B66, 155412 (2002).
[CrossRef]

McNab, S. J.

Milonni, P. W.

P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B At. Mol. Opt. Phys.35, R31 (2002).
[CrossRef]

Moharam, M. G.

Naik, G.

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

Nevière, M.

Okawachi, Y.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Philbin, T. G.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

Piotrzkowski, P.

E. Litwin-Staszewska, W. Szymanska, and P. Piotrzkowski, “The electron mobility and Thermoelectric Power in InSb at Atmospheric and Hydrostatic Pressures,” Phys. Status Solidi B106, 551–559 (1981).
[CrossRef]

Pommet, D. A.

Popov, E.

Preist, T. W.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Vol. 11 of Springer Tracts in Modern Physics (Springer-Verlag, 1988).

Reinisch, R.

Robertson, S.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

Sambles, J. R.

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Schweinsberg, A.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Selker, M. D.

Shalaev, V.

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

Sharping, J.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Soljacic, M.

A. Karalis, E. Lidorikis, M. Ibanescu, J. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett.95, 1–4 (2005).
[CrossRef]

Stockman, M.

K. Li, X. Li, M. Stockman, and D. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71, 1–5 (2005).
[CrossRef]

Szymanska, W.

E. Litwin-Staszewska, W. Szymanska, and P. Piotrzkowski, “The electron mobility and Thermoelectric Power in InSb at Atmospheric and Hydrostatic Pressures,” Phys. Status Solidi B106, 551–559 (1981).
[CrossRef]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature450, 397–401 (2007).
[CrossRef] [PubMed]

Tsykhonya, A.

V. Chegel, Y. Demidenko, V. Lozovski, and A. Tsykhonya, “Influence of the shape of the particles covering the metal surface on the dispersion relations of surface plasmons,” Surf. Sci.602, 1540–1546 (2008).
[CrossRef]

Vlasov, Y. A.

Wang, J.

West, P.

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Xue, Q.

Yeh, P.

P. Yeh, Optical Waves in Layered Media, Vol. 95 of Wiley Series in Pure and Applied Optics(Wiley, 1988).

Zhang, W.

A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett.86, 141102 (2005).
[CrossRef]

Zhao, Y.

A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett.86, 141102 (2005).
[CrossRef]

Zhu, Z.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Zia, R.

Appl. Phys. Lett. (1)

A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett.86, 141102 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt.56, 1908–1915 (2009).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. B At. Mol. Opt. Phys. (1)

P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B At. Mol. Opt. Phys.35, R31 (2002).
[CrossRef]

Laser Photonics Rev. (1)

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010).
[CrossRef]

Nat. Mater. (1)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9, 193–204 (2010).
[CrossRef] [PubMed]

Nature (2)

W. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–831 (2003).
[CrossRef] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature450, 397–401 (2007).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (2)

F. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B66, 155412 (2002).
[CrossRef]

K. Li, X. Li, M. Stockman, and D. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71, 1–5 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

A. Karalis, E. Lidorikis, M. Ibanescu, J. Joannopoulos, and M. Soljačić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett.95, 1–4 (2005).
[CrossRef]

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 1–4 (2005).
[CrossRef]

Phys. Status Solidi B (1)

E. Litwin-Staszewska, W. Szymanska, and P. Piotrzkowski, “The electron mobility and Thermoelectric Power in InSb at Atmospheric and Hydrostatic Pressures,” Phys. Status Solidi B106, 551–559 (1981).
[CrossRef]

Science (2)

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008).
[CrossRef] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003).
[CrossRef] [PubMed]

Surf. Sci. (1)

V. Chegel, Y. Demidenko, V. Lozovski, and A. Tsykhonya, “Influence of the shape of the particles covering the metal surface on the dispersion relations of surface plasmons,” Surf. Sci.602, 1540–1546 (2008).
[CrossRef]

Other (3)

P. Yeh, Optical Waves in Layered Media, Vol. 95 of Wiley Series in Pure and Applied Optics(Wiley, 1988).

S. Maier, Plasmonics - Fundamentals and Applications, Vol. 76 of Chemistry and Materials Science (Springer, 2010).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Vol. 11 of Springer Tracts in Modern Physics (Springer-Verlag, 1988).

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Figures (6)

Fig. 1
Fig. 1

The structure with two underlying dielectric layers, namely the thin film and the semi-infinite substrate composed of vacuum is shown in (a). The amplitude |R| of the scattered TM polarized plane wave, assuming a unit amplitude TM polarized beam incident from above is shown in (b). The peaks in |R|2 indicate the locations in ωk space of the even and odd SP modes. It should be noted that for evanescent SP modes, |R|2 is not bound by unity as in the reflection coefficient for propagating plane waves.

Fig. 2
Fig. 2

The multilayer InSb structure studied in this work is shown in (a). The structure is designed to produce multiple stair steps in the DC with progressively lower frequency SP modes as kx increases. The values of ε′m and −ε′m/ε″m for the three different semiconductor layers is shown in (b). At ν = νstep,4 (given by Eq. (5)) Layers 2 and 3 act as dielectrics and Layer 4 acts as a metal (see points marked by ”Δ”) with a value of −ε′m/ε″m = 1.78 (see point ”c”); at ν = νstep,3 Layer 2 acts as a dielectric and Layer 3 and 4 act as metals (see points marked by ”+”) with a value of −ε′m/ε″m = 1.41 for Layer 3 (see point ”b”); at ν = νstep,2 Layer 2, 3 and 4 are all metallic in behavior (see points marked by ”x”) with a value of −ε′m/ε″m = 1.25 for Layer 2 (see point ”a”).

Fig. 3
Fig. 3

The reflectance (i.e., |R|2), assuming a unit amplitude incident beam is shown in (a). The DC of the SPs has a stair step shape. The magnetic fields for points o, + and Δ are shown in Figs. 4 and 5. The value of the logarithm of |S−1| of the structure shown in (b). Self-sustaining resonant modes, such as SPs, occur when |S−1| = 0, and the regions of local minima of |S−1| for real frequency and kx imply the presence of resonating poles in S.

Fig. 4
Fig. 4

The value of Re(Hz) for ν = 1.02 THz and kx = 2.5 μm−1 (Point ”o” in Fig. 3(a)). It is seen that the field of this SP is spread throughout the semiconductor, and with Layer 4 occupying the most volume within the semiconductor, this layer largely determines the frequency of this SP mode.

Fig. 5
Fig. 5

The value of Re(Hz) for ν = 0.91 THz and kx = 18 μm−1 (Point ”+” in Fig. 3(a)) is shown in (a). The field for this higher kx mode extends into Layer 4 to a lesser degree than shown the SP mode of Fig. 4 and is concentrated in Layer 3. Thus Layer 3 largely determines the frequencies of these SPs. The value of Re(Hz) for ν = 0.75 THz and kx = 50 μm−1 (Point ”Δ” in Fig. 3(a)) is shown in (b). The very large value of kx causes the field in the semiconductor to be concentrated in the top-most layer, i.e., thus this layer largely determines the frequencies of these SPs.

Fig. 6
Fig. 6

The simulated reflection coefficient using a RCWA method for a dovetail dispersion. The structure consist of a top and bottom layer with the identical an doping concentration of 4.3 × 1015cm−1 and thicknesses of 15 nm and 1μm respectively. the middle layer has a lower doping of 2.5 × 1015cm−1 and thickness of 100 nm. As can be seen, the SP modes transitions from a high energy to a lower energy, and then reappears at the high energy for larger kx.

Equations (5)

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k sp = k o ( ε m ε d ε m + ε d ) 1 / 2
ε m = ε m + i ε m = ε ω p 2 ω ( ω + i Γ )
( B t A b ) = S ( A t B b ) = ( S 11 S 12 S 21 S 22 ) ( A t B b )
( A t B b ) = S 1 ( B t A b )
ν step , i = ω step 2 π = ω p , i 2 Γ 2 ( ε + ε d ) / 4 2 π ε + ε d

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