Abstract

We present a method for studying amplification of electromagnetic modes in active, circularly symmetric waveguides with hyperbolic dispersion. Using this method, we obtain a closed-form expression for the modal threshold condition. We find that modal amplification is possible in a region of the radius-wavelength phase-space with small enough radius so that propagation of the mode is permitted while modal energy and phase counter-propagate. At telecommunication frequencies, such a situation is achievable only when the absolute value of the real metal permittivity exceeds that of the active dielectric. We validate our theoretical conclusions with numerical simulations that explain the threshold condition in terms of an energy balance between the longitudinal and radial components of the electric field.

© 2014 Optical Society of America

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  1. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
    [CrossRef] [PubMed]
  2. Z. Jacob, “The classical and quantum optics of hyperbolic metamaterials,” Thesis, Purdue University (2010).
  3. V. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968).
    [CrossRef]
  4. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
    [CrossRef] [PubMed]
  5. V. Podolskiy and E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
    [CrossRef]
  6. C. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
    [CrossRef]
  7. M. Noginov, “Metamaterials with optical gain,” in Tutorials in Metamaterials, M. Noginov and V. Podolskiy, eds. (CRC, 2012), pp. 129–161.
  8. 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(7307), 735–738 (2010).
    [CrossRef] [PubMed]
  9. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
    [CrossRef] [PubMed]
  10. R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
    [CrossRef]
  11. R. Shersby-Harvie, L. Mullett, W. Walkinshaw, J. Bell, and B. Loach, “A theoretical and experimental investigation of anisotropic-dielectric-loaded linear electron accelerators,” Proc. IEE 104B, 273–290 (1957).
  12. V. M. Agranovich and V. E. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985).
    [CrossRef]
  13. C. Chang, “Circular waveguides lined with artificial anisotropic dielectrics,” IEEE Trans. Microw. Theory Tech. 20(8), 517–523 (1972).
    [CrossRef]
  14. W. Yan, M. Wubs, and N. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
    [CrossRef]
  15. C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115(40), 19470–19475 (2011).
    [CrossRef]
  16. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
    [CrossRef]
  17. L. Coldren, S. Corzine, and M. Masanovic, “Gain and current relations,” in Diode lasers and photonic integrated circuits, (Wiley, 2012), pp. 157–246.
  18. J. Smalley, Q. Gu, and Y. Fainman, “Temperature dependence of the spontaneous emission factor in subwavelength semiconductor lasers,” IEEE J. Quantum Electron. 50(3), 175–185 (2014).
    [CrossRef]
  19. G. Naik, V. Shalaev, and A. Boltasseva, “Semiconductors for plasmonics and metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).
  20. C. Riley, T. Kieu, J. Smalley, S. Pan, S. Kim, K. Post, A. Kargar, D. Basov, X. Pan, Y. Fainman, D. Wang, and D. Sirbuly, “Plasmonic tuning of aluminum doped zinc oxide nanostructures by atomic layer deposition,” submitted to Phys. Stat. Sol. (RRL) (2014).
  21. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
    [PubMed]
  22. G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
    [CrossRef] [PubMed]
  23. G. Naik and A. Boltasseva, “A comparitive study of semiconductor-based plasmonic metamaterials,” arXiv, < http://arxiv.org/abs/1108.1531 > (2011).
  24. P. Johnson and R. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [CrossRef]
  25. A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008).
    [CrossRef] [PubMed]
  26. J. S. Smalley, M. W. Puckett, and Y. Fainman, “Invariance of optimal composite waveguide geometries with respect to permittivity of the metal cladding,” Opt. Lett. 38(23), 5161–5164 (2013).
    [CrossRef] [PubMed]
  27. A. Govyadinov and V. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73(15), 155108 (2006).
    [CrossRef]
  28. V. Podolskiy, “Anisotropic and hyperbolic metamaterials,” in Tutorials in Metamaterials, M. Noginov and V. Podolskiy, eds. (CRC, 2012), pp. 163–207.

2014 (1)

J. Smalley, Q. Gu, and Y. Fainman, “Temperature dependence of the spontaneous emission factor in subwavelength semiconductor lasers,” IEEE J. Quantum Electron. 50(3), 175–185 (2014).
[CrossRef]

2013 (3)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[CrossRef] [PubMed]

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

J. S. Smalley, M. W. Puckett, and Y. Fainman, “Invariance of optimal composite waveguide geometries with respect to permittivity of the metal cladding,” Opt. Lett. 38(23), 5161–5164 (2013).
[CrossRef] [PubMed]

2012 (3)

C. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[CrossRef]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[PubMed]

W. Yan, M. Wubs, and N. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[CrossRef]

2011 (2)

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115(40), 19470–19475 (2011).
[CrossRef]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[CrossRef] [PubMed]

2010 (3)

G. Naik, V. Shalaev, and A. Boltasseva, “Semiconductors for plasmonics and metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[CrossRef]

2006 (1)

A. Govyadinov and V. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73(15), 155108 (2006).
[CrossRef]

2005 (1)

V. Podolskiy and E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

1985 (1)

V. M. Agranovich and V. E. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985).
[CrossRef]

1972 (2)

C. Chang, “Circular waveguides lined with artificial anisotropic dielectrics,” IEEE Trans. Microw. Theory Tech. 20(8), 517–523 (1972).
[CrossRef]

P. Johnson and R. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

1968 (1)

V. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich and V. E. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985).
[CrossRef]

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[CrossRef]

Belov, P.

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

Boltasseva, A.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[CrossRef] [PubMed]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[PubMed]

G. Naik, V. Shalaev, and A. Boltasseva, “Semiconductors for plasmonics and metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Chang, C.

C. Chang, “Circular waveguides lined with artificial anisotropic dielectrics,” IEEE Trans. Microw. Theory Tech. 20(8), 517–523 (1972).
[CrossRef]

Chettiar, U. K.

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Christy, R.

P. Johnson and R. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Cortes, C.

C. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[CrossRef]

David, C.

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115(40), 19470–19475 (2011).
[CrossRef]

Drachev, V. P.

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Elser, J.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[CrossRef]

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Fainman, Y.

Fedorov, S.

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

Feng, L.

García de Abajo, F. J.

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115(40), 19470–19475 (2011).
[CrossRef]

Govyadinov, A.

A. Govyadinov and V. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73(15), 155108 (2006).
[CrossRef]

Gu, Q.

J. Smalley, Q. Gu, and Y. Fainman, “Temperature dependence of the spontaneous emission factor in subwavelength semiconductor lasers,” IEEE J. Quantum Electron. 50(3), 175–185 (2014).
[CrossRef]

Han, S.

Ishii, S.

Jacob, Z.

C. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[CrossRef]

Johnson, P.

P. Johnson and R. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Kildishev, A. V.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[PubMed]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[CrossRef] [PubMed]

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Kivshar, Y.

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

Kravtsov, V. E.

V. M. Agranovich and V. E. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985).
[CrossRef]

Lee, S.

Liu, J.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[PubMed]

Liu, Z.

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[CrossRef] [PubMed]

Lomakin, V.

Mizrahi, A.

Molesky, S.

C. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[CrossRef]

Mortensen, N.

W. Yan, M. Wubs, and N. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[CrossRef]

Naik, G.

G. Naik, V. Shalaev, and A. Boltasseva, “Semiconductors for plasmonics and metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).

Naik, G. V.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[CrossRef] [PubMed]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[PubMed]

Narimanov, E.

V. Podolskiy and E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

Newman, W.

C. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[CrossRef]

Nezhad, M. P.

Ni, X.

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[CrossRef] [PubMed]

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Pendry, J. B.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Podolskiy, V.

A. Govyadinov and V. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73(15), 155108 (2006).
[CrossRef]

V. Podolskiy and E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

Podolskiy, V. A.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[CrossRef]

Puckett, M. W.

Rosanov, N.

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

Salakhutdinov, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[CrossRef]

Savelev, R.

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

Shadrivov, I.

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

Shalaev, V.

G. Naik, V. Shalaev, and A. Boltasseva, “Semiconductors for plasmonics and metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).

Shalaev, V. M.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[CrossRef] [PubMed]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[PubMed]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[CrossRef] [PubMed]

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Slutsky, B. A.

Smalley, J.

J. Smalley, Q. Gu, and Y. Fainman, “Temperature dependence of the spontaneous emission factor in subwavelength semiconductor lasers,” IEEE J. Quantum Electron. 50(3), 175–185 (2014).
[CrossRef]

Smalley, J. S.

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Sukhorukov, A.

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

Thoreson, M. D.

Veselago, V.

V. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Wubs, M.

W. Yan, M. Wubs, and N. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[CrossRef]

Xiao, S.

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Yan, W.

W. Yan, M. Wubs, and N. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[CrossRef]

Yuan, H. K.

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Zhang, X.

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[CrossRef] [PubMed]

Adv. Mater. (1)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Smalley, Q. Gu, and Y. Fainman, “Temperature dependence of the spontaneous emission factor in subwavelength semiconductor lasers,” IEEE J. Quantum Electron. 50(3), 175–185 (2014).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

C. Chang, “Circular waveguides lined with artificial anisotropic dielectrics,” IEEE Trans. Microw. Theory Tech. 20(8), 517–523 (1972).
[CrossRef]

J. Opt. (1)

C. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[CrossRef]

J. Phys. Chem. C (1)

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115(40), 19470–19475 (2011).
[CrossRef]

Nat. Mater. (1)

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[CrossRef] [PubMed]

Nature (1)

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(7307), 735–738 (2010).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (5)

A. Govyadinov and V. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73(15), 155108 (2006).
[CrossRef]

P. Johnson and R. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

R. Savelev, I. Shadrivov, P. Belov, N. Rosanov, S. Fedorov, A. Sukhorukov, and Y. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87(11), 115139 (2013).
[CrossRef]

V. Podolskiy and E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

W. Yan, M. Wubs, and N. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[CrossRef]

Phys. Status Solidi (RRL) (1)

G. Naik, V. Shalaev, and A. Boltasseva, “Semiconductors for plasmonics and metamaterials,” Phys. Status Solidi (RRL) 4, 295–297 (2010).

Proc. Natl. Acad. Sci. U.S.A. (1)

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109(23), 8834–8838 (2012).
[PubMed]

Science (1)

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Solid State Commun. (1)

V. M. Agranovich and V. E. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985).
[CrossRef]

Sov. Phys. Usp. (1)

V. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

Other (7)

Z. Jacob, “The classical and quantum optics of hyperbolic metamaterials,” Thesis, Purdue University (2010).

R. Shersby-Harvie, L. Mullett, W. Walkinshaw, J. Bell, and B. Loach, “A theoretical and experimental investigation of anisotropic-dielectric-loaded linear electron accelerators,” Proc. IEE 104B, 273–290 (1957).

M. Noginov, “Metamaterials with optical gain,” in Tutorials in Metamaterials, M. Noginov and V. Podolskiy, eds. (CRC, 2012), pp. 129–161.

L. Coldren, S. Corzine, and M. Masanovic, “Gain and current relations,” in Diode lasers and photonic integrated circuits, (Wiley, 2012), pp. 157–246.

C. Riley, T. Kieu, J. Smalley, S. Pan, S. Kim, K. Post, A. Kargar, D. Basov, X. Pan, Y. Fainman, D. Wang, and D. Sirbuly, “Plasmonic tuning of aluminum doped zinc oxide nanostructures by atomic layer deposition,” submitted to Phys. Stat. Sol. (RRL) (2014).

V. Podolskiy, “Anisotropic and hyperbolic metamaterials,” in Tutorials in Metamaterials, M. Noginov and V. Podolskiy, eds. (CRC, 2012), pp. 163–207.

G. Naik and A. Boltasseva, “A comparitive study of semiconductor-based plasmonic metamaterials,” arXiv, < http://arxiv.org/abs/1108.1531 > (2011).

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Figures (8)

Fig. 1
Fig. 1

Schematic of equivalent waveguides composed of (a) periodic sequence of two distinct materials and (b) bulk anisotropic effective medium. c denotes direction of the optical axis.

Fig. 2
Fig. 2

(a) Real and imaginary effective permittivities, parallel and perpendicular to the optical axis of a periodic sequence of AZO and InGaAsP layers with equal thickness, for InGaAsP carrier concentration of N = 5x1018cm−3. (b) Imaginary permittivities for N = 1x1017cm−3 and N = 1x1019cm−3.

Fig. 3
Fig. 3

(a) Real and imaginary effective permittivities, parallel and perpendicular to the optical axis of a periodic sequence of silver and InGaAsP layers with equal thickness, for InGaAsP carrier concentrations N = 5x1018cm−3. (b) Imaginary permittivities with N = 1x1017cm−3 and N = 1x1019cm−3.

Fig. 4
Fig. 4

(a) Effective indices of TM01 mode at λ0 = 1.55μm as a function of the radius of AZO/InGaAsP composite waveguide with PEC cladding. (b) Contour map of log of propagation length of TM01 mode as a function of guide radius and wavelength for same structure.

Fig. 5
Fig. 5

Threshold and cutoff radius of TM01 mode as a function of wavelength in Ag/InGaAsP waveguide with fill = 0.5 and PEC waveguide, with (a) carrier density of 5e18cm-3, and (b) carrier density as a parameter. In (a) the hatched area corresponds to the region of phase-space satisfying Eq. (4.15). (c) Cutoff and threshold radii for TM01, TM02, and TM03 modes.

Fig. 6
Fig. 6

Evaluation of left-hand side of Eq. (5.3) for TM01 mode at several freespace wavelengths in Ag/InGaAsP waveguide with PEC cladding. Fill = 0.5 and N = 5x1018cm−3.

Fig. 7
Fig. 7

Real and imaginary effective index of TM01 mode at 1.55um in Ag/InGaAsP waveguide with PEC and Ag claddings. Fill = 0.5 and N = 5x1018cm−3.

Fig. 8
Fig. 8

Local and layer-thickness-dependent non-local correction to cutoff and threshold radii of TM01 mode at 1.55um in Ag/InGaAsP waveguide with PEC claddings. Fill = 0.5 and N = 5x1018cm−3.

Equations (44)

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D xy = ε 0 t 1 ε 1 E 1,x + t 2 ε 2 E 2,x t 1 + t 2
D xy = ε ε 0 E x = ε ε 0 E y
ε t 1 ε 1 + t 2 ε 2 t 1 + t 2
E z = ε 0 1 ε 1 1 D 1,z + ε 2 1 D 2,z t 1 + t 2 ,
E z = ( ε 0 ε || ) 1 D z
ε || ( t 1 + t 2 ) ε 1 ε 2 t 1 ε 2 + t 2 ε 1 .
ε (ω,k)= ε (ω,k)+i ε (ω,k)
ε || (ω,k)= ε || (ω,k)+i ε || (ω,k)
ε 1 (ω)= ε 1 (ω)+i ε 1 (ω)
ε 2 (ω)= ε 2 (ω)i ε 2 (ω),
ε 2 (ω)= c ω g(ω) ε 2 .
ε 1 =A+iB
ε 2 =C+iD,
ε = 1 2 ( A+C )
ε = 1 2 ( B+D ),
ε || =2 C( A 2 + B 2 )+A( C 2 + D 2 ) ( A+C ) 2 + ( B+D ) 2
ε || =2 D( A 2 + B 2 )+B( C 2 + D 2 ) ( A+C ) 2 + ( B+D ) 2 .
ε || 2 C( A 2 + B 2 )+A C 2 A 2 + C 2 2| A |C
ε || 2 D( A 2 + B 2 )+B C 2 A 2 + C 2 2| A |C .
ε || 2 C A 2 +A C 2 A 2 + C 2 2| A |C
ε || 2 D A 2 +B C 2 A 2 + C 2 2| A |C .
ε || 2 C A 2 | A | C 2 2 A 2 2| A |C
ε || 2 A 2 ( B+D ) 2 A 2 2| A |C ,
n eff (pq) (ω)= n eff (pq) (ω)+i n eff (pq) (ω) = 1 k 0 ( k z (pq) (ω)+i k z (pq) (ω) ),
n eff (pq) =± ε ( 1 κ (pq) 2 ε || k 0 2 )
n eff (pq) =± ( E+iF )[ 1 κ (pq) 2 ( G+iH ) k 0 2 ] ,
E ε , F ε , G ε || , H ε || .
Case I: | A |<C D<0 E>0, F<0 G<0, H>0
n eff (pq) =+ k 0 2 E( G 2 + H 2 )+ κ (pq) 2 ( E| G |+| F |H ) k 0 2 ( G 2 + H 2 ) ,
n eff (pq) =+ κ (pq) 2 EH| F |[ κ (pq) 2 | G |+ k 0 2 ( G 2 + H 2 ) ] k 0 2 ( G 2 + H 2 )
Case II: | A |>C D<0 E<0, F>0 G>0, H<0 ,
n eff (pq) = κ (pq) 2 ( | E |G+F| H | ) k 0 2 | E |( G 2 + H 2 ) k 0 2 ( G 2 + H 2 ) .
R cut (pq) = Z pq k 0 | E |G+F| H | | E | G 2 +| E | H 2 ,
R cut (pq) Z pq λ 0 2π G = Z pq λ 0 2π ε ||
n eff (pq) = F( G 2 + H 2 ) k 0 2 ( FG| E || H | ) κ (pq) 2 k 0 2 ( G 2 + H 2 ) .
R thres (pq) = Z (pq) k 0 FG| E || H | F( G 2 + H 2 ) ,
R thres (pq) Z (pq) k 0 FG| E || H | F G 2 ,
R thres (pq) < R (pq) < R cut (pq) .
ε || =( t 1 + t 2 ) t 2 D A 2 + t 1 B C 2 ( t 2 A 2 + t 1 C 2 )
U= Ω ( DE )dA = Ω ( ε | E ρ | 2 + ε || | E z | 2 )dA
U || U = Ω | ε || | | E z (pq) | 2 dA Ω | ε | | E ρ (pq) | 2 dA >1
E z (01) = J 0 ( Z (01) ρ/R )exp[ i k z (01) z ]
E ρ (01) =i k z (01) ε || ε E z (01) ρ .
U || U + U clad = Ω | ε || | | E z (pq) | 2 dA Ω | ε | | E ρ (pq) | 2 dA + | ε clad | | E (pq) | 2 dA >1,

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