Abstract

In this paper, we propose a new and versatile mechanism for electrical tuning of planar metamaterials: strong coupling of metamaterial resonances to engineered intersubband transitions that can be tuned through the application of an electrical bias. We present the general formalism that allows calculating the permittivity tensor for intersubband transitions in generic semiconductor heterostructures and we study numerically the specific case of coupling and tuning metamaterials in the thermal infrared through coupling to biased GaAs semiconductor quantum wells. This tuning mechanism can be scaled from the visible to the far infrared by the proper choice of metamaterials and semiconductor heterostructures.

© 2012 OSA

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  6. X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
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    [CrossRef]
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  32. V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, “Quantum theory of quantum-well polaritons in semiconductor microcavities,” Phys. Rev. B Condens. Matter 49(13), 8774–8779 (1994).
    [CrossRef] [PubMed]
  33. E. R. Brown, S. J. Eglash, and K. A. McIntosh, “Observation of normal-incidence intersubband absorption in n-type Al0.09Ga0.91Sb quantum wells,” Phys. Rev. B Condens. Matter 46(11), 7244–7247 (1992).
    [CrossRef] [PubMed]
  34. Q. Du, J. Alperin, and W. I. Wang, “Infrared electroabsorption modulation in AlSb/InAs/AlGaSb/GaSb/AlSb stepped quantum-wells grown by molecular-beam epitaxy,” Appl. Phys. Lett. 67(15), 2218–2219 (1995).
    [CrossRef]
  35. L. A. Samoska, B. Brar, and H. Kroemer, “Strong far-infrared intersubband absorption under normal incidence in heavily n-type doped nonalloy GaSb-AlSb superlattices,” Appl. Phys. Lett. 62(20), 2539–2541 (1993).
    [CrossRef]
  36. Y. Zhang, N. Baruch, and W. I. Wang, “Normal incidence infrared photodetectors using intersubband transitions in GaSb l-valley quantum-wells,” Appl. Phys. Lett. 63(8), 1068–1070 (1993).
    [CrossRef]

2012 (1)

Y. Todorov and C. Sirtori, “Intersubband polaritons in the electrical dipole gauge,” Phys. Rev. B 85(4), 045304 (2012).
[CrossRef]

2011 (3)

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

D. Dietze, A. Benz, G. Strasser, K. Unterrainer, and J. Darmo, “Terahertz meta-atoms coupled to a quantum well intersubband transition,” Opt. Express 19(14), 13700–13706 (2011).
[CrossRef] [PubMed]

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

2010 (2)

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

2009 (3)

2007 (1)

C. M. Soukoulis, T. Koschny, J. Zhou, M. Kafesaki, and E. N. Economou, “Magnetic response of split ring resonators at terahertz frequencies,” Phys. Status Solidi, B Basic Res. 244(4), 1181–1187 (2007).
[CrossRef]

2006 (1)

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

2004 (1)

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

1996 (1)

1995 (1)

Q. Du, J. Alperin, and W. I. Wang, “Infrared electroabsorption modulation in AlSb/InAs/AlGaSb/GaSb/AlSb stepped quantum-wells grown by molecular-beam epitaxy,” Appl. Phys. Lett. 67(15), 2218–2219 (1995).
[CrossRef]

1994 (1)

V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, “Quantum theory of quantum-well polaritons in semiconductor microcavities,” Phys. Rev. B Condens. Matter 49(13), 8774–8779 (1994).
[CrossRef] [PubMed]

1993 (4)

L. A. Samoska, B. Brar, and H. Kroemer, “Strong far-infrared intersubband absorption under normal incidence in heavily n-type doped nonalloy GaSb-AlSb superlattices,” Appl. Phys. Lett. 62(20), 2539–2541 (1993).
[CrossRef]

Y. Zhang, N. Baruch, and W. I. Wang, “Normal incidence infrared photodetectors using intersubband transitions in GaSb l-valley quantum-wells,” Appl. Phys. Lett. 63(8), 1068–1070 (1993).
[CrossRef]

L. Wendler and E. Kandler, “Intrasubband and intersubband plasmon-polaritons in semiconductor quantum-wells,” Phys. Status Solidi, B Basic Res. 177(1), 9–67 (1993).
[CrossRef]

P. H. Tsao, “Derivation and implications of the symmetry property of the permittivity tensor,” Am. J. Phys. 61(9), 823–825 (1993).
[CrossRef]

1992 (3)

C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and Y. A. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60(2), 151 (1992).
[CrossRef]

F. G. Pikus, “Excitons in quantum wells with a two dimensional electron gas,” Sov. Phys. Semicond. 26, 2633 (1992).

E. R. Brown, S. J. Eglash, and K. A. McIntosh, “Observation of normal-incidence intersubband absorption in n-type Al0.09Ga0.91Sb quantum wells,” Phys. Rev. B Condens. Matter 46(11), 7244–7247 (1992).
[CrossRef] [PubMed]

1991 (1)

E. Rosencher and P. Bois, “Model system for optical nonlinearities: asymmetric quantum wells,” Phys. Rev. B Condens. Matter 44(20), 11315–11327 (1991).
[CrossRef] [PubMed]

1990 (1)

E. R. Brown and S. J. Eglash, “Calculation of the intersubband absorption strength in ellipsoidal-valley quantum wells,” Phys. Rev. B Condens. Matter 41(11), 7559–7568 (1990).
[CrossRef] [PubMed]

1989 (1)

E. Rosencher, P. Bois, J. Nagle, and S. Delattre, “Second harmonic generation by intersubband transitions in compositionally asymmetrical MQWs,” Electron. Lett. 25(16), 1063 (1989).
[CrossRef]

1967 (1)

F. Stern and W. E. Howard, “Properties of semiconductor surface inversion layers in the electric quantum limit,” Phys. Rev. 163(3), 816–835 (1967).
[CrossRef]

Alperin, J.

Q. Du, J. Alperin, and W. I. Wang, “Infrared electroabsorption modulation in AlSb/InAs/AlGaSb/GaSb/AlSb stepped quantum-wells grown by molecular-beam epitaxy,” Appl. Phys. Lett. 67(15), 2218–2219 (1995).
[CrossRef]

Atwater, H. A.

Averitt, R. D.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009).
[CrossRef]

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Aydin, K.

Azad, A. K.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009).
[CrossRef]

Baruch, N.

Y. Zhang, N. Baruch, and W. I. Wang, “Normal incidence infrared photodetectors using intersubband transitions in GaSb l-valley quantum-wells,” Appl. Phys. Lett. 63(8), 1068–1070 (1993).
[CrossRef]

Benz, A.

Bois, P.

E. Rosencher and P. Bois, “Model system for optical nonlinearities: asymmetric quantum wells,” Phys. Rev. B Condens. Matter 44(20), 11315–11327 (1991).
[CrossRef] [PubMed]

E. Rosencher, P. Bois, J. Nagle, and S. Delattre, “Second harmonic generation by intersubband transitions in compositionally asymmetrical MQWs,” Electron. Lett. 25(16), 1063 (1989).
[CrossRef]

Boreman, G. D.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

Boyd, E. M.

Brar, B.

L. A. Samoska, B. Brar, and H. Kroemer, “Strong far-infrared intersubband absorption under normal incidence in heavily n-type doped nonalloy GaSb-AlSb superlattices,” Appl. Phys. Lett. 62(20), 2539–2541 (1993).
[CrossRef]

Brener, I.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Brown, E. R.

E. R. Brown, S. J. Eglash, and K. A. McIntosh, “Observation of normal-incidence intersubband absorption in n-type Al0.09Ga0.91Sb quantum wells,” Phys. Rev. B Condens. Matter 46(11), 7244–7247 (1992).
[CrossRef] [PubMed]

E. R. Brown and S. J. Eglash, “Calculation of the intersubband absorption strength in ellipsoidal-valley quantum wells,” Phys. Rev. B Condens. Matter 41(11), 7559–7568 (1990).
[CrossRef] [PubMed]

Capasso, F.

C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and Y. A. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60(2), 151 (1992).
[CrossRef]

Chen, H. T.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Chen, H.-T.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009).
[CrossRef]

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Cho, Y. A.

C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and Y. A. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60(2), 151 (1992).
[CrossRef]

Cich, M. J.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009).
[CrossRef]

Coffey, K. R.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

Darmo, J.

Delattre, S.

E. Rosencher, P. Bois, J. Nagle, and S. Delattre, “Second harmonic generation by intersubband transitions in compositionally asymmetrical MQWs,” Electron. Lett. 25(16), 1063 (1989).
[CrossRef]

Dicken, M. J.

Dietze, D.

Du, Q.

Q. Du, J. Alperin, and W. I. Wang, “Infrared electroabsorption modulation in AlSb/InAs/AlGaSb/GaSb/AlSb stepped quantum-wells grown by molecular-beam epitaxy,” Appl. Phys. Lett. 67(15), 2218–2219 (1995).
[CrossRef]

Economou, E. N.

C. M. Soukoulis, T. Koschny, J. Zhou, M. Kafesaki, and E. N. Economou, “Magnetic response of split ring resonators at terahertz frequencies,” Phys. Status Solidi, B Basic Res. 244(4), 1181–1187 (2007).
[CrossRef]

Eglash, S. J.

E. R. Brown, S. J. Eglash, and K. A. McIntosh, “Observation of normal-incidence intersubband absorption in n-type Al0.09Ga0.91Sb quantum wells,” Phys. Rev. B Condens. Matter 46(11), 7244–7247 (1992).
[CrossRef] [PubMed]

E. R. Brown and S. J. Eglash, “Calculation of the intersubband absorption strength in ellipsoidal-valley quantum wells,” Phys. Rev. B Condens. Matter 41(11), 7559–7568 (1990).
[CrossRef] [PubMed]

Enkrich, C.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Fedotov, V. A.

Gabbay, A.

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

Gin, A.

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Ginn, J. C.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

Goodhue, W.

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Goodhue, W. D.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Gossard, A. C.

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Hoffman, A. J.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Howard, W. E.

F. Stern and W. E. Howard, “Properties of semiconductor surface inversion layers in the electric quantum limit,” Phys. Rev. 163(3), 816–835 (1967).
[CrossRef]

Hradil, Z.

V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, “Quantum theory of quantum-well polaritons in semiconductor microcavities,” Phys. Rev. B Condens. Matter 49(13), 8774–8779 (1994).
[CrossRef] [PubMed]

Hutchinson, A. L.

C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and Y. A. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60(2), 151 (1992).
[CrossRef]

Ioffe, A. F.

Ivchenko, E. L.

Kafesaki, M.

C. M. Soukoulis, T. Koschny, J. Zhou, M. Kafesaki, and E. N. Economou, “Magnetic response of split ring resonators at terahertz frequencies,” Phys. Status Solidi, B Basic Res. 244(4), 1181–1187 (2007).
[CrossRef]

Kaliteevski, M. A.

Kandler, E.

L. Wendler and E. Kandler, “Intrasubband and intersubband plasmon-polaritons in semiconductor quantum-wells,” Phys. Status Solidi, B Basic Res. 177(1), 9–67 (1993).
[CrossRef]

Kavokin, A. V.

Koschny, T.

C. M. Soukoulis, T. Koschny, J. Zhou, M. Kafesaki, and E. N. Economou, “Magnetic response of split ring resonators at terahertz frequencies,” Phys. Status Solidi, B Basic Res. 244(4), 1181–1187 (2007).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Kroemer, H.

L. A. Samoska, B. Brar, and H. Kroemer, “Strong far-infrared intersubband absorption under normal incidence in heavily n-type doped nonalloy GaSb-AlSb superlattices,” Appl. Phys. Lett. 62(20), 2539–2541 (1993).
[CrossRef]

Kuo, P.

Langston, W.

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Li, J.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Linden, S.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Ma, J.

McIntosh, K. A.

E. R. Brown, S. J. Eglash, and K. A. McIntosh, “Observation of normal-incidence intersubband absorption in n-type Al0.09Ga0.91Sb quantum wells,” Phys. Rev. B Condens. Matter 46(11), 7244–7247 (1992).
[CrossRef] [PubMed]

Miao, X.

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Nagle, J.

E. Rosencher, P. Bois, J. Nagle, and S. Delattre, “Second harmonic generation by intersubband transitions in compositionally asymmetrical MQWs,” Electron. Lett. 25(16), 1063 (1989).
[CrossRef]

Nesvizhskii, A. I.

O'hara, J. F.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Padilla, W. J.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009).
[CrossRef]

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Passmore, B.

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Peralta, X. G.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Peters, D. W.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

Pikus, F. G.

F. G. Pikus, “Excitons in quantum wells with a two dimensional electron gas,” Sov. Phys. Semicond. 26, 2633 (1992).

Plum, E.

Pryce, I. M.

Quattropani, A.

V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, “Quantum theory of quantum-well polaritons in semiconductor microcavities,” Phys. Rev. B Condens. Matter 49(13), 8774–8779 (1994).
[CrossRef] [PubMed]

Reno, J.

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Rosencher, E.

E. Rosencher and P. Bois, “Model system for optical nonlinearities: asymmetric quantum wells,” Phys. Rev. B Condens. Matter 44(20), 11315–11327 (1991).
[CrossRef] [PubMed]

E. Rosencher, P. Bois, J. Nagle, and S. Delattre, “Second harmonic generation by intersubband transitions in compositionally asymmetrical MQWs,” Electron. Lett. 25(16), 1063 (1989).
[CrossRef]

Samoska, L. A.

L. A. Samoska, B. Brar, and H. Kroemer, “Strong far-infrared intersubband absorption under normal incidence in heavily n-type doped nonalloy GaSb-AlSb superlattices,” Appl. Phys. Lett. 62(20), 2539–2541 (1993).
[CrossRef]

Savona, V.

V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, “Quantum theory of quantum-well polaritons in semiconductor microcavities,” Phys. Rev. B Condens. Matter 49(13), 8774–8779 (1994).
[CrossRef] [PubMed]

Schwendimann, P.

V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, “Quantum theory of quantum-well polaritons in semiconductor microcavities,” Phys. Rev. B Condens. Matter 49(13), 8774–8779 (1994).
[CrossRef] [PubMed]

Shaner, E.

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Shelton, D. J.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

Sinclair, M. B.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

Sirtori, C.

Y. Todorov and C. Sirtori, “Intersubband polaritons in the electrical dipole gauge,” Phys. Rev. B 85(4), 045304 (2012).
[CrossRef]

C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and Y. A. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60(2), 151 (1992).
[CrossRef]

Sivco, D. L.

C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and Y. A. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60(2), 151 (1992).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis, T. Koschny, J. Zhou, M. Kafesaki, and E. N. Economou, “Magnetic response of split ring resonators at terahertz frequencies,” Phys. Status Solidi, B Basic Res. 244(4), 1181–1187 (2007).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Stern, F.

F. Stern and W. E. Howard, “Properties of semiconductor surface inversion layers in the electric quantum limit,” Phys. Rev. 163(3), 816–835 (1967).
[CrossRef]

Strasser, G.

Sweatlock, L. A.

Taylor, A. J.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009).
[CrossRef]

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Todorov, Y.

Y. Todorov and C. Sirtori, “Intersubband polaritons in the electrical dipole gauge,” Phys. Rev. B 85(4), 045304 (2012).
[CrossRef]

Tsai, D. P.

Tsao, P. H.

P. H. Tsao, “Derivation and implications of the symmetry property of the permittivity tensor,” Am. J. Phys. 61(9), 823–825 (1993).
[CrossRef]

Unterrainer, K.

Vangala, S.

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

Walavalkar, S.

Waldman, J.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Wang, W. I.

Q. Du, J. Alperin, and W. I. Wang, “Infrared electroabsorption modulation in AlSb/InAs/AlGaSb/GaSb/AlSb stepped quantum-wells grown by molecular-beam epitaxy,” Appl. Phys. Lett. 67(15), 2218–2219 (1995).
[CrossRef]

Y. Zhang, N. Baruch, and W. I. Wang, “Normal incidence infrared photodetectors using intersubband transitions in GaSb l-valley quantum-wells,” Appl. Phys. Lett. 63(8), 1068–1070 (1993).
[CrossRef]

Wanke, M. C.

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Wegener, M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Wendler, L.

L. Wendler and E. Kandler, “Intrasubband and intersubband plasmon-polaritons in semiconductor quantum-wells,” Phys. Status Solidi, B Basic Res. 177(1), 9–67 (1993).
[CrossRef]

Wendt, J. R.

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

Wright, J. B.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Young, E. W.

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Zhang, Y.

Y. Zhang, N. Baruch, and W. I. Wang, “Normal incidence infrared photodetectors using intersubband transitions in GaSb l-valley quantum-wells,” Appl. Phys. Lett. 63(8), 1068–1070 (1993).
[CrossRef]

Zheludev, N. I.

Zhou, J.

C. M. Soukoulis, T. Koschny, J. Zhou, M. Kafesaki, and E. N. Economou, “Magnetic response of split ring resonators at terahertz frequencies,” Phys. Status Solidi, B Basic Res. 244(4), 1181–1187 (2007).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Zide, J. M.

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Am. J. Phys. (1)

P. H. Tsao, “Derivation and implications of the symmetry property of the permittivity tensor,” Am. J. Phys. 61(9), 823–825 (1993).
[CrossRef]

Appl. Phys. Lett. (6)

Q. Du, J. Alperin, and W. I. Wang, “Infrared electroabsorption modulation in AlSb/InAs/AlGaSb/GaSb/AlSb stepped quantum-wells grown by molecular-beam epitaxy,” Appl. Phys. Lett. 67(15), 2218–2219 (1995).
[CrossRef]

L. A. Samoska, B. Brar, and H. Kroemer, “Strong far-infrared intersubband absorption under normal incidence in heavily n-type doped nonalloy GaSb-AlSb superlattices,” Appl. Phys. Lett. 62(20), 2539–2541 (1993).
[CrossRef]

Y. Zhang, N. Baruch, and W. I. Wang, “Normal incidence infrared photodetectors using intersubband transitions in GaSb l-valley quantum-wells,” Appl. Phys. Lett. 63(8), 1068–1070 (1993).
[CrossRef]

X. Miao, B. Passmore, A. Gin, W. Langston, S. Vangala, W. Goodhue, E. Shaner, and I. Brener, “Doping tunable resonance: toward electrically tunable mid-infrared metamaterials,” Appl. Phys. Lett. 96(10), 101111 (2010).
[CrossRef]

A. Gabbay, J. Reno, J. R. Wendt, A. Gin, M. C. Wanke, M. B. Sinclair, E. Shaner, and I. Brener, “Interaction between metamaterial resonators and intersubband transitions in semiconductor quantum wells,” Appl. Phys. Lett. 98(20), 203103 (2011).
[CrossRef]

C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and Y. A. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60(2), 151 (1992).
[CrossRef]

Electron. Lett. (1)

E. Rosencher, P. Bois, J. Nagle, and S. Delattre, “Second harmonic generation by intersubband transitions in compositionally asymmetrical MQWs,” Electron. Lett. 25(16), 1063 (1989).
[CrossRef]

J. Opt. Soc. Am. B (1)

Metamaterials (Amst.) (1)

X. G. Peralta, I. Brener, W. J. Padilla, E. W. Young, A. J. Hoffman, M. J. Cich, R. D. Averitt, M. C. Wanke, J. B. Wright, H. T. Chen, J. F. O'hara, A. J. Taylor, J. Waldman, W. D. Goodhue, J. Li, and J. Reno, “External modulators for terahertz quantum cascade lasers based on electrically-driven active metamaterials,” Metamaterials (Amst.) 4(2-3), 83–88 (2010).
[CrossRef]

Nano Lett. (1)

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11(5), 2104–2108 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3(3), 148–151 (2009).
[CrossRef]

Nature (1)

H.-T. Chen, W. J. Padilla, J. M. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Rev. (1)

F. Stern and W. E. Howard, “Properties of semiconductor surface inversion layers in the electric quantum limit,” Phys. Rev. 163(3), 816–835 (1967).
[CrossRef]

Phys. Rev. B (1)

Y. Todorov and C. Sirtori, “Intersubband polaritons in the electrical dipole gauge,” Phys. Rev. B 85(4), 045304 (2012).
[CrossRef]

Phys. Rev. B Condens. Matter (4)

V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, “Quantum theory of quantum-well polaritons in semiconductor microcavities,” Phys. Rev. B Condens. Matter 49(13), 8774–8779 (1994).
[CrossRef] [PubMed]

E. R. Brown, S. J. Eglash, and K. A. McIntosh, “Observation of normal-incidence intersubband absorption in n-type Al0.09Ga0.91Sb quantum wells,” Phys. Rev. B Condens. Matter 46(11), 7244–7247 (1992).
[CrossRef] [PubMed]

E. R. Brown and S. J. Eglash, “Calculation of the intersubband absorption strength in ellipsoidal-valley quantum wells,” Phys. Rev. B Condens. Matter 41(11), 7559–7568 (1990).
[CrossRef] [PubMed]

E. Rosencher and P. Bois, “Model system for optical nonlinearities: asymmetric quantum wells,” Phys. Rev. B Condens. Matter 44(20), 11315–11327 (1991).
[CrossRef] [PubMed]

Phys. Status Solidi, B Basic Res. (2)

C. M. Soukoulis, T. Koschny, J. Zhou, M. Kafesaki, and E. N. Economou, “Magnetic response of split ring resonators at terahertz frequencies,” Phys. Status Solidi, B Basic Res. 244(4), 1181–1187 (2007).
[CrossRef]

L. Wendler and E. Kandler, “Intrasubband and intersubband plasmon-polaritons in semiconductor quantum-wells,” Phys. Status Solidi, B Basic Res. 177(1), 9–67 (1993).
[CrossRef]

Science (1)

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Sov. Phys. Semicond. (1)

F. G. Pikus, “Excitons in quantum wells with a two dimensional electron gas,” Sov. Phys. Semicond. 26, 2633 (1992).

Other (10)

J. Faist and C. Sirtori, Intersubband Transitions in Quantum Wells: Physics and Device Applications I, Semiconductors and Semimetals (Academic Press, 2000), Vol. 62.

M. Helm, IntersubbandTransitions in Quantum Wells:Physics and Device Applications, Semiconductors and Semimetals (Academic Press, 2000), Vol. 62.

G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, 1 ed. (Wiley-Interscience, 1991).

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2009).

T. Cui, D. Smith, and R. Liu, Metamaterials: Theory, Design, and Applications (Springer, 2009).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific Publishing Company, 2004).

R. W. Boyd, Nonlinear Optics (Academic Press, 1992), chap. 3.

The issue of the symmetry of the permittivity tensor when dealing with transparent or passive media as been studied in numerous textbooks and publications (for example see [25]). For simplicity, we choose to display a non-symmetric tensor with the understanding that for computations we use only one diagonal half.

http://www.lumerical.com , “Fdtd solutions.”

C. Weisbuch and B. Vinter, Quantum Semiconductor Structures—Fundamentals and Applications (Academic Press, 1991).

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Figures (7)

Fig. 1
Fig. 1

(a) A lower bandgap material sandwiched between two higher bandgap materials. The solid lines are the minimum (maximum) of the conduction (valence) band. (b) A schematic diagram of the inplane dispersion of the confined electronic conduction band levels

Fig. 2
Fig. 2

(a) Schematic diagram of the split-ring resonator (SRR) used with the dimension definitions. (b) Calculated transmission spectrum of the SRR in (a) on top of a GaAs substrate designed to resonate at 33 THz, for both linear polarizations. The arrow indicates the frequency at which plots (c-h) were calculated. (c-e) The enhanced field amplitude (relative to the incident field E0, of the three components Ex, Ey and Ez, respectively. The plots correspond to a plane located 15nm below the interface. (f-h) The dependence of the field components on z (the growth direction). (f-h) were drawn along a single line passing through the points denoted in (c-e) by a + sign.

Fig. 3
Fig. 3

(a) The potential (solid black lines) of a two asymmetric coupled GaAs QWs with Al0.5Ga0.5As barriers. (b) The potential under a bias of −75 kV/cm. (c) The dependence of three lowest ISTs on the applied bias.

Fig. 4
Fig. 4

A schematic diagram of the layer sequence used for the FDTD calculation. The first (lowest) layer is a GaAs substrate. The following layers are a repetition of a unit cell, which is composed of a 10nm layer (associated with the calculated ISTs susceptibility) sandwiched between two 20nm Al0.5Ga0.5As layers. The upper layer is a 25nm cap GaAs layer. The top most structure is a gold SRR.

Fig. 5
Fig. 5

(a) The calculated susceptibility of the IST (2→1) for an applied bias of 23.1 kV/cm used for the FDTD calculation. (b) The calculated transmission spectra for several applied biases. The red curve shows the resonant condition where the IST energy coincides with the SRR resonance. (c) The calculated transmission spectra at resonance (red curve in (b)) of both polarizations (solid curves). The dashed curve is the calculated transmission specra for the same structure where the QW layers were replaced by GaAs layers. (d) The energy minima of each transmission spectrum (Ey polarization) as a function of the applied bias. Anti-crossing behavior is clearly observed, indicating strong coupling between the MM resonator and the IST.

Fig. 6
Fig. 6

(a) The degree of amplitude modulation, defined in Eq. (1.14), for several applied biases. The spectra are offset from each other in the y-axis direction and therefore the labels refer only to the first spectrum. (b) The modulation degree at a certain frequency (32.7 THz) as function of the applied biases. The data corresponds to the dashed red line in (a).

Fig. 7
Fig. 7

(a) calculated transmission spectra when a different number of QWs were used in the FDTD simulation. (b) The spectra in (a) were fitted by two Lorentzians. The peak frequencies of the Lorentzians are shown by the black circles and indicate the level splitting. The solid blue curve is a plot of Eq. (1.15) when using A = 1 and α avg ΔZ=0.1 .

Equations (23)

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H I = e 2 ( A w p ^ + p ^ w A )
H I =e( w zx A x +w zy A y +w zz A z ) p ^ z
H I =e( w zx w zz E x + w zy w zz E y +E z ) z ^
( H I ) nm = d nm 1 w zz w E, d nm =e(0,0, z nm )
P z = 1 0 h M,n,m,k d mn w ( H I ) nm ( f m,k f n,k ) e iωt ω ω mn + mn /2
P z = Ne M C ε 0 M,m j w zj w zz z m1 ( H I ) 1m ( ω ω m1 + m1 /2 )
χ zj ( ω )= Ne 2 M C ε 0 M,m v=x,y,z ( w zj w zv w zz 2 ) | z m1 | 2 ( ω ω m1 + m1 /2 )
ε( ω )= ε b ( ω )+( 0 0 0 0 0 0 χ zx ( ω ) χ zy ( ω ) χ zz ( ω ) )
ε GaAs QW ( ω )=( ε GaAs b ( ω ) 0 0 0 ε GaAs b ( ω ) 0 0 0 ε GaAs b ( ω ) )+( 0 0 0 0 0 0 0 0 1 ) χ 0 ( ω )
χ 0 ( ω )= Ne 2 ε 0 | z 21 | 2 ( ω ω 21 + 21 /2 )
ε GaSb QW(L<4nm) =( ε GaSb b ( ω ) 0 0 0 ε GaSb b ( ω ) 0 0 0 ε GaSb b ( ω ) )+( 0 0 0 0 0 0 0.2 0.2 1 ) χ 0 ( ω )
( ν MM MM Ω/2 Ω * /2 ν IST IST )
ν ± = ν MM + ν IST +i( γ MM + γ IST ) 2 ± 1 4 Ω 2 + [ ν MM ν IST +i( γ MM γ IST ) ] 2
M( ν )= T QW (ν) T ref (ν) T ref (ν)
Ω=A 1 e avg ΔZN QW 1 e 2 α avg ΔZ
E z ( z )= E 0 e αz
d ( μ ) = i=1 N QW C i ( μ ) d i
C i ( 1 ) = C 0 e αZ i , C 0 = [ i N QW e 2αZ i ] 1 2
i C i ( μ )* C 0 e αZ i = δ μ,1
d ( μ ) E= i=1 N QW C i ( μ ) d i E 0 e αZ i = dE 0 i=1 N QW C i ( μ ) e αZ i = dE 0 C 0 δ μ,1
C 0 2 =( i=1 N QW | e αz 1 | 2 )= e 2 αz 1 i=0 N QW 1 e 2αΔZi = e 2 αz 1 1 e 2 αΔZN QW 1 e 2αΔZ
N eff = e αz 1 1 e 2 αΔZN QW 1 e 2αΔZ
ν + ν =A 1 e 2 α avg ΔZN QW 1 e 2 α avg ΔZ

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