Abstract

An analytical solution to the fundamental TM0 mode on the model system of a dielectric-semiconductor-insulator-metal four layered planar structure is obtained. A transition from the ‘hybrid plasmonic’ mode to ‘plasmonic only’ mode is ratified by the change from sinusoidal to exponential wave functions in the semiconductor layer as the propagation constant of the TM0 mode exceeds that of the light in bulk form of the semiconductor. A variational method based on the mode hybridization picture is proposed to approximate the dispersion relation of the fundamental TM0 mode. It is demonstrated that the variational method can well produce the dispersion relation of the TM0 mode in the ‘hybrid plasmonic’ region but deviate significantly in ‘plasmonic only’ region if the trial wave function based on mode hybridization is used, which suggests that the mode hybridization idea should only be applied to the ‘hybrid plasmonic’ region.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]

2018 (2)

N. Liu, C. Silien, G. Sun, and B. Corbett, “Low loss photonic nanocavity via dark magnetic dipole resonant mode near metal,” Sci. Rep. 8(1), 17054 (2018).
[Crossref]

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

2016 (1)

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

2014 (1)

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

2013 (1)

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “Theoretical Analysis of Hybrid Plasmonic Waveguide,” IEEE J. Sel. Top. Quantum Electron. 19(3), 4602008 (2013).
[Crossref]

2012 (1)

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

2011 (1)

R.-m. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref]

2010 (1)

2009 (1)

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(7264), 629–632 (2009).
[Crossref]

2008 (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

2003 (1)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

2002 (1)

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299(2-3), 309–312 (2002).
[Crossref]

1995 (1)

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Adams, M. J.

M. J. Adams, Section 2.24, Introduction to optical waveguides (Wiley & Sons Ltd., 1981).

Aitchison, J. S.

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “Theoretical Analysis of Hybrid Plasmonic Waveguide,” IEEE J. Sel. Top. Quantum Electron. 19(3), 4602008 (2013).
[Crossref]

Alam, M. Z.

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “Theoretical Analysis of Hybrid Plasmonic Waveguide,” IEEE J. Sel. Top. Quantum Electron. 19(3), 4602008 (2013).
[Crossref]

Avrutsky, I.

Bartal, G.

R.-m. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref]

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(7264), 629–632 (2009).
[Crossref]

Buchwald, W.

Chang, W.-H.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Chen, H.-Y.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Chen, L.-J.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Cheng, D. K.

D. K. Cheng, Field and wave electromagnetics (Addison – Wesley Publishing Company, Inc., 1983).

Corbett, B.

N. Liu, C. Silien, G. Sun, and B. Corbett, “Low loss photonic nanocavity via dark magnetic dipole resonant mode near metal,” Sci. Rep. 8(1), 17054 (2018).
[Crossref]

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Dabidian, N.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Dai, L.

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(7264), 629–632 (2009).
[Crossref]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Gity, F.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Gladden, C.

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(7264), 629–632 (2009).
[Crossref]

Gocalinska, A.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Gottschalch, V.

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Gwo, S.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Halas, N. J.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

Hammer, M.

Alyona Ivanova, Remco Stoffer, and M. Hammer, “A variational mode solver for optical waveguides based on quasi-analytical vectorial slab mode expansion,” arXiv:1307.1315v2 (2013).

Herzinger, C. M.

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Ivanova, Alyona

Alyona Ivanova, Remco Stoffer, and M. Hammer, “A variational mode solver for optical waveguides based on quasi-analytical vectorial slab mode expansion,” arXiv:1307.1315v2 (2013).

Jackson, J. D.

J. D. Jackson, Classical electrodynamics (John Wiley & Sons, Inc, 1999).

Justice, J.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Kim, J. S.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Li, B.-H.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Li, G.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Li, Q.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

Li, Y.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Li, Z. P.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

Lieber, C. M.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Liu, N.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

N. Liu, C. Silien, G. Sun, and B. Corbett, “Low loss photonic nanocavity via dark magnetic dipole resonant mode near metal,” Sci. Rep. 8(1), 17054 (2018).
[Crossref]

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Liu, X.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Lu, M.-Y.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Lu, Y. J.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Ma, R.-m.

R.-m. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref]

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(7264), 629–632 (2009).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

McCarthy, B.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Mojahedi, M.

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “Theoretical Analysis of Hybrid Plasmonic Waveguide,” IEEE J. Sel. Top. Quantum Electron. 19(3), 4602008 (2013).
[Crossref]

Molenaar, J.

E. van Groesen and J. Molenaar, Continuum modeling in the physical sciences (Society for Industrial and Applied Mathematics, 2007).

Nordlander, P.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

Oulton, R. F.

R.-m. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref]

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(7264), 629–632 (2009).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Pan, D.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

Pelucchi, E.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Pemble, M.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Povey, I.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

Qian, F.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Qiu, X. G.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

Ruppin, R.

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299(2-3), 309–312 (2002).
[Crossref]

Sanders, C. E.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Schubert, M.

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Shih, C.-K.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Shvets, G.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Silien, C.

N. Liu, C. Silien, G. Sun, and B. Corbett, “Low loss photonic nanocavity via dark magnetic dipole resonant mode near metal,” Sci. Rep. 8(1), 17054 (2018).
[Crossref]

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Snyder, P. G.

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Soref, R.

Sorger, V. J.

R.-m. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref]

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(7264), 629–632 (2009).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Stoffer, Remco

Alyona Ivanova, Remco Stoffer, and M. Hammer, “A variational mode solver for optical waveguides based on quasi-analytical vectorial slab mode expansion,” arXiv:1307.1315v2 (2013).

Sum, T. C.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Sun, G.

N. Liu, C. Silien, G. Sun, and B. Corbett, “Low loss photonic nanocavity via dark magnetic dipole resonant mode near metal,” Sci. Rep. 8(1), 17054 (2018).
[Crossref]

van Groesen, E.

E. van Groesen and J. Molenaar, Continuum modeling in the physical sciences (Society for Industrial and Applied Mathematics, 2007).

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Wang, C.-Y.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Wang, W. H.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

Wei, H.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Woollam, J. A.

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Wu, C. H.

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

Xiong, Q.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Xu, H.

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Xu, H. X.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

Yao, H.

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Zentgraf, T.

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(7264), 629–632 (2009).
[Crossref]

Zhang, Q.

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Zhang, S. P.

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

Zhang, X.

R.-m. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref]

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(7264), 629–632 (2009).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Chem. Rev. (1)

H. Wei, D. Pan, S. P. Zhang, Z. P. Li, Q. Li, N. Liu, W. H. Wang, and H. X. Xu, “Plasmon Waveguiding in Nanowires,” Chem. Rev. 118(6), 2882–2926 (2018).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “Theoretical Analysis of Hybrid Plasmonic Waveguide,” IEEE J. Sel. Top. Quantum Electron. 19(3), 4602008 (2013).
[Crossref]

J. Appl. Phys. (1)

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, and J. A. Woollam, “Optical constants of GaxIn1-xP lattice matched to GaAs,” J. Appl. Phys. 77(7), 3416–3419 (1995).
[Crossref]

Nano Lett. (1)

N. Liu, A. Gocalinska, J. Justice, F. Gity, I. Povey, B. McCarthy, M. Pemble, E. Pelucchi, H. Wei, C. Silien, H. Xu, and B. Corbett, “Lithographically Defined, Room Temperature Low Threshold Subwavelength Red-Emitting Hybrid Plasmonic Lasers,” Nano Lett. 16(12), 7822–7828 (2016).
[Crossref]

Nat. Commun. (1)

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref]

Nat. Mater. (1)

R.-m. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref]

Nat. Photonics (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Nature (1)

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(7264), 629–632 (2009).
[Crossref]

Opt. Express (1)

Phys. Lett. A (1)

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299(2-3), 309–312 (2002).
[Crossref]

Sci. Rep. (1)

N. Liu, C. Silien, G. Sun, and B. Corbett, “Low loss photonic nanocavity via dark magnetic dipole resonant mode near metal,” Sci. Rep. 8(1), 17054 (2018).
[Crossref]

Science (2)

Y. J. Lu, J. S. Kim, H.-Y. Chen, C. H. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. G. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, S. Gwo, and C.-K. Shih, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

Other (8)

E. van Groesen and J. Molenaar, Continuum modeling in the physical sciences (Society for Industrial and Applied Mathematics, 2007).

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Alyona Ivanova, Remco Stoffer, and M. Hammer, “A variational mode solver for optical waveguides based on quasi-analytical vectorial slab mode expansion,” arXiv:1307.1315v2 (2013).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

J. D. Jackson, Classical electrodynamics (John Wiley & Sons, Inc, 1999).

D. K. Cheng, Field and wave electromagnetics (Addison – Wesley Publishing Company, Inc., 1983).

M. J. Adams, Section 2.24, Introduction to optical waveguides (Wiley & Sons Ltd., 1981).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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Figures (5)

Fig. 1.
Fig. 1. (a) Diagram of the dielectric-semiconductor-insulator-metal 4 layered planar structure. (b) Dispersion curves of TM0 mode obtained from Eq. (4) (black) and Eq. (6) (blue) and from COMSOL simulation package (dashed green) respectively with permittivity ɛd = ɛi = 2.9, ɛm = 1-ωp22 with ωp = 1.4 × 1016 rad·Hz, 2a = 110 nm and h = 6 nm. The intersection point between the light line (dashed red) and the dispersion curve corresponds to the transition frequency ωT. (c) The ratio of propagation constant β of TM0 mode with respect to the propagation constant of light in bulk semiconductor as a function of β. Here ${n_c} = \sqrt {{\varepsilon _c}} $ and ${k_0} = \frac{\omega }{c}$.
Fig. 2.
Fig. 2. Hybrid plasmonic to plasmonic only transition frequency ${\omega _T}$ (solid curves) and corresponding propagation constant β (dashed curves) as a function of (a) relative permittivity of core semiconductor ${\varepsilon _c}$ with three different$\; {\varepsilon _d}$ values at 2a = 110 nm and h = 6 nm, (b) relative permittivity of thin gap material ${\varepsilon _d}$ with three different ${\varepsilon _c}$ values at 2a = 110 nm and h = 6 nm, (c) core semiconductor thickness 2a with three different $\; {\varepsilon _d}$ values at ${\varepsilon _c} = 13$ and h = 6 nm, and (d) gap thickness h with three different ${\varepsilon _c}$ values at ${\varepsilon _d}$ = 2.9 and 2a = 110 nm. In this figure, ɛm = 1-ωp22 with ωp = 1.4 × 1016 rad·Hz.
Fig. 3.
Fig. 3. (a) Diagrams showing the Ex and Ez components of fundamental TM0 mode at the metal-insulator interface (red), in the top three layers (blue) and the superposition of the two modes (pink) in ‘hybrid plasmonic’ region. (b) Dispersion curves of TM0 mode at the metal-dielectric interface (red), that in the dielectric-semiconductor-dielectric 3 layers (blue), that obtained through the variational method assuming a hybrid plasmonic mode with lossless materials (pink) and those obtained from Eq. (4) (dashed black) and Eq. (6) (dashed green). The geometric and material parameters used here are the same as used in Fig. 1. The grey dashed line indicates the transition point obtained from Fig. 1. (c) Change of photonic to plasmonic component ratio in the hybrid plasmonic mode as a function of β with dispersion curve shown in (b). A = sinφ, B = cosφ.
Fig. 4.
Fig. 4. (a) Dispersion curves of TM0 mode at the Ag-insulator interface (red), that in the insulator-GaInP-insulator 3 layers (blue), that obtained through the variational method assuming a hybrid plasmonic mode with lossy materials (pink) and that obtained from COMSOL simulation package (dashed green). The geometric parameters used here are the same as used in Fig. 1 and Fig. 3. The permittivities of Ag and GaInP are obtained from Ref. [17] and [14] respectively. The right axis indicates the ratio of Real(β) of TM0 mode obtained from the variational method with respect to Real(β) of light in bulk GaInP. (b) Imag(β) of the four modes described in (a). The inset shows the angle φ associated with the change of photonic to plasmonic component ratio in the hybrid plasmonic mode. Angle $\phi \; $for the optimized trial wave function occurs at $\phi = 0.$
Fig. 5.
Fig. 5. (a) Diagrams showing the Ey field of fundamental TE0 mode in the top three layers (green solid), its image mode (green dash) and the superposition of the two modes (cerulean blue). (b) Dispersion curves of TE0 mode calculated from Eq. (8) using wave functions (Eq. 11) detailed in section 4 (cerulean blue), that obtained from COMSOL simulation package (black dash). The dispersion curve of TE0 mode in insulator-semiconductor-insulator 3 layers is given for comparison (green). The geometric and material parameters used here are the same as used in Fig. 1.

Equations (22)

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{ ω ε 0 ε ( z ) E + i C H = β R H ω μ 0 μ H i C E = β R E
{ H y ( z ) = B d e i β x e k d ( z a ) H y ( z ) = A e i β x cos ( k c z θ ) H y ( z ) = B i 1 e i β x e k i ( z + a ) + B i 2 e i β x e k i ( z + a ) H y ( z ) = C e i β x e k m ( z + a + h ) for z > a for | z | < a for ( a + h ) < z < a for z < ( a + h )
{ E x ( z ) = i B d k d ω ε 0 ε d e i β x e k d ( z a ) E x ( z ) = i A k c ω ε 0 ε c e i β x sin ( k c z θ ) E x ( z ) = i B i 1 k i ω ε 0 ε i e i β x e k i ( z + a ) + i B i 2 k i ω ε 0 ε i e i β x e k i ( z + a ) E x ( z ) = i C k m ω ε 0 ε m e i β x e k m ( z + a + h ) for z > a for | z | < a for ( a + h ) < z < a for z < ( a + h )
s i n ( 2 k c a ) ε d k c c o s ( 2 k c a ) ε c k d s i n ( 2 k c a ) ε c k d + c o s ( 2 k c a ) ε d k c = ε c k d ε d k c ( ε m k d + ε d k m ) e 2 k d h ε m k d + ε d k m ( ε m k d + ε d k m ) e 2 k d h + ε m k d ε d k m
{ H y ( z ) = A 1 e i β x e κ c ( z + a ) + A 2 e i β x e κ c ( z + a ) for | z | < a E x ( z ) = i A 1 κ c ω ε 0 ε c e i β x e κ c ( z + a ) + i A 2 κ c ω ε 0 ε c e i β x e κ c ( z + a ) for | z | < a
( ε c k d + ε d κ c ) e 4 κ c a + ε c k d ε d κ c ( ε c k d + ε d κ c ) e 4 κ c a ε c k d + ε d κ c = ε c k d ε d κ c ( ε m k d + ε d k m ) e 2 k d h ε m k d + ε d k m ( ε m k d + ε d k m ) e 2 k d h + ε m k d ε d k m
ω T = ε d c 2 a ε c ε c ε d ε m ε c ε d ( 1 + e 2 ω T ε c ε d h / c ) ε d ε c ε m ( 1 e 2 ω T ε c ε d h / c ) ε m ε c ε d ( 1 e 2 ω T ε c ε d h / c ) ε d ε c ε m ( 1 + e 2 ω T ε c ε d h / c )
ω = β ( E , R H H , R E ) i ( E , C H H , C E ) ε 0 E , ε E + μ 0 H , μ H
β ( E , R H H , R E ) i ( E , C H H , C E ) = | A | 2 ω t ( ε 0 E t , ε t E t + μ 0 H t , H t ) + | B | 2 ω b ( ε 0 E b , ε b E b + μ 0 H b , H b ) + A B ω b ( ε 0 E t , ε b E b + μ 0 H t , H b ) + B A ω t ( ε 0 E b , ε t E t + μ 0 H b , H t )
ε 0 E , ε E + μ 0 H , μ H = | A | 2 ( ε 0 E t , ε E t + μ 0 H t , H t ) + | B | 2 ( ε 0 E b , ε E b + μ 0 H b , H b ) + A B ( ε 0 E t , ε E b + μ 0 H t , H b ) + B A ( ε 0 E b , ε E t + μ 0 H b , H t )
{ E y ( z ) = A cos k c a ( 1 e k d ( 2 a + 2 h ) ) e i β x e k d ( z a ) for z > a E y ( z ) = A e i β x cos k c z A cos k c a e i β x e k d ( z + a + 2 h ) for | z | < a E y ( z ) = A cos k c a e i β x e k d ( z + a ) A cos k c a e 2 k d h e i β x e k d ( z + a ) for  ( a + h ) < z < a E y ( z ) = 0 for z < ( a + h )
{ E y ( z ) = A cos ( k c a θ ) e i β x e k d ( z a ) for z > a E y ( z ) = A e i β x cos ( k c z θ ) for | z | < a E y ( z ) = B b 1 e i β x e k d ( z + a ) + B b 2 e i β x e k d ( z + a ) for  ( a + h ) < z < a E y ( z ) = C e i β x e k m ( z + a + h ) for z < ( a + h )
{ H t y ( z ) = A t cos ( k c a ) e i β x e k d t ( z a ) E t x ( z ) = i A t cos ( k c a ) k d t ω t ε 0 ε d e i β x e k d t ( z a ) E t z ( z ) = A t cos ( k c a ) β ω t ε 0 ε d e i β x e k d t ( z a )  for  z > a
{ H t y ( z ) = A t cos ( k c a ) e i β x e k d t ( z + a ) E t x ( z ) = i A t cos ( k c a ) k d t ω t ε 0 ε d e i β x e k d t ( z + a ) E t z ( z ) = A t cos ( k c a ) β ω t ε 0 ε d e i β x e k d t ( z + a ) for z < a
{ H t y ( z ) = A t cos ( k c z ) e i β x E t x ( z ) = i A t sin ( k c z ) k c ω t ε 0 ε c e i β x E t z ( z ) = A t cos ( k c z ) β ω t ε 0 ε c e i β x for | z | < a
( β 2 ω t 2 ε d c 2 ) ε c 2 = ( ω t 2 ε c c 2 β 2 ) ε d 2 ta n 2 [ ( ω t 2 ε c c 2 β 2 ) a ]
{ H b y ( z ) = A b e i β x e k d b ( z + a + h ) E b x ( z ) = i A b k d b ω b ε 0 ε d e i β x e k d b ( z + a + h ) E b z ( z ) = A b β ω b ε 0 ε d e i β x e k d b ( z + a + h ) for z > ( a + h )
{ H b y ( z ) = A b e i β x e k m ( z + a + h ) E b x ( z ) = i A b k m ω b ε 0 ε m e i β x e k m ( z + a + h ) E b z ( z ) = A b β ω b ε 0 ε m e i β x e k m ( z + a + h ) for z < ( a + h )
{ H y ( z ) = s i n φ A t cos ( k c a ) e i β x e k d t ( z a ) + c o s φ A b e i β x e k d b ( z + a + h ) E x ( z ) = s i n φ A t i cos ( k c a ) k d t ω t ε 0 ε d e i β x e k d t ( z a ) + c o s φ A b i k d b ω b ε 0 ε d e i β x e k d b ( z + a + h ) E z ( z ) = s i n φ A t cos ( k c a ) β ω t ε 0 ε d e i β x e k d t ( z a ) c o s φ A b β ω b ε 0 ε d e i β x e k d b ( z + a + h ) for z > a
{ H y ( z ) = s i n φ A t cos ( k c z ) e i β x + c o s φ A b e i β x e k d b ( z + a + h ) E x ( z ) = s i n φ A t i sin ( k c z ) k c ω t ε 0 ε c e i β x + c o s φ A b i k d b ω b ε 0 ε d e i β x e k d b ( z + a + h ) E z ( z ) = s i n φ A t cos ( k c z ) β ω t ε 0 ε c e i β x c o s φ A b β ω b ε 0 ε d e i β x e k d b ( z + a + h ) for | z | < a
{ H y ( z ) = s i n φ A t cos ( k c a ) e i β x e k d t ( z + a ) + c o s φ A b e i β x e k d b ( z + a + h ) E x ( z ) = s i n φ A t i cos ( k c a ) k d t ω t ε 0 ε d e i β x e k d t ( z + a ) + c o s φ A b i k d b ω b ε 0 ε d e i β x e k d b ( z + a + h ) E z ( z ) = s i n φ A t cos ( k c a ) β ω t ε 0 ε d e i β x e k d t ( z + a ) c o s φ A b β ω b ε 0 ε d e i β x e k d b ( z + a + h ) for ( a + h ) < z < a
{ H y ( z ) = s i n φ A t cos ( k c a ) e i β x e k d t ( z + a ) + c o s φ A b e i β x e k m ( z + a + h ) E x ( z ) = s i n φ A t i cos ( k c a ) k d t ω t ε 0 ε d e i β x e k d t ( z + a ) c o s φ A b i k m ω b ε 0 ε m e i β x e k m ( z + a + h ) E z ( z ) = s i n φ A t cos ( k c a ) β ω t ε 0 ε d e i β x e k d t ( z + a ) c o s φ A b β ω b ε 0 ε m e i β x e k m ( z + a + h ) for  z < ( a + h )