Abstract

We present experimental data on a one-dimensional super-conducting metamaterial that is tunable over a broad frequency band. The basic building block of this magnetic thin-film medium is a single-junction (rf-) superconducting quantum interference device (SQUID). Due to the nonlinear inductance of such an element, its resonance frequency is tunable in situ by applying a dc magnetic field. We demonstrate that this results in tunable effective parameters of our metamaterial consisting of 54 rf-SQUIDs. In order to obtain the effective magnetic permeability μr,eff from the measured data, we employ a technique that uses only the complex transmission coefficient S21.

© 2013 OSA

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Errata

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, "A one-dimensional tunable magnetic metamaterial: erratum," Opt. Express 22, 13041-13042 (2014)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-11-13041

References

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  1. M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett.87, 034102 (2005).
    [CrossRef]
  2. M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
    [CrossRef]
  3. J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
    [CrossRef]
  4. J. Wu, B. Jin, Y. Xue, C. Zhang, H. Dai, L. Zhang, C. Cao, L. Kang, W. Xu, J. Chen, and P. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express19, 12021–12026 (2011).
    [CrossRef] [PubMed]
  5. N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett.90, 163501 (2007).
    [CrossRef]
  6. C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter20, 345220 (2008).
    [CrossRef]
  7. A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun.283, 1633–1639 (2010).
    [CrossRef]
  8. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
    [CrossRef] [PubMed]
  9. P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013).
    [CrossRef]
  10. K. K. Likharev, Dynamics of Josephson Junctions(Gordon and Breach Science, 1991).
  11. M. Tinkham, Introduction to Superconductivity (2nd Edition) (Dover Publications Inc., 2004).
  12. S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
    [CrossRef]
  13. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999).
    [CrossRef]
  14. J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).
  15. D. M. Pozar, Microwave Engineering (2nd Edition) (John Wiley & Sons Inc., 1998) pp. 208–211
  16. J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum.84, 034706 (2013).
    [CrossRef] [PubMed]

2013

P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013).
[CrossRef]

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
[CrossRef]

J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum.84, 034706 (2013).
[CrossRef] [PubMed]

2011

2010

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun.283, 1633–1639 (2010).
[CrossRef]

2008

C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter20, 345220 (2008).
[CrossRef]

2007

N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett.90, 163501 (2007).
[CrossRef]

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

2005

M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett.87, 034102 (2005).
[CrossRef]

2000

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

1999

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999).
[CrossRef]

Anlage, S. M.

J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum.84, 034706 (2013).
[CrossRef] [PubMed]

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett.87, 034102 (2005).
[CrossRef]

Baker-Jarvis, J.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Butz, S.

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
[CrossRef]

P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013).
[CrossRef]

Cao, C.

Cao, W.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Chen, H.

C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter20, 345220 (2008).
[CrossRef]

Chen, H.-T.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Chen, J.

Dai, H.

Du, C.

C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter20, 345220 (2008).
[CrossRef]

Filippenko, L. V.

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
[CrossRef]

Gabitov, I. R.

A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun.283, 1633–1639 (2010).
[CrossRef]

Geyer, R. G.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Grosvenor, C. A.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Gu, J.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Han, J.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

He, M.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999).
[CrossRef]

Holloway, C. L.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Janezic, M. D.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Jin, B.

Johnk, R. T.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Jung, P.

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
[CrossRef]

P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013).
[CrossRef]

Kabos, P.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Kang, L.

Koshelets, V. P.

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
[CrossRef]

Lazarides, N.

N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett.90, 163501 (2007).
[CrossRef]

Li, S.

C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter20, 345220 (2008).
[CrossRef]

Likharev, K. K.

K. K. Likharev, Dynamics of Josephson Junctions(Gordon and Breach Science, 1991).

Maimistov, A. I.

A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun.283, 1633–1639 (2010).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

Orloff, N.

M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett.87, 034102 (2005).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999).
[CrossRef]

Pozar, D. M.

D. M. Pozar, Microwave Engineering (2nd Edition) (John Wiley & Sons Inc., 1998) pp. 208–211

Prozorov, R.

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

Ricci, M. C.

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett.87, 034102 (2005).
[CrossRef]

Riddle, B. F.

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

Shitov, S. V.

P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013).
[CrossRef]

Singh, R.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999).
[CrossRef]

Tian, Z.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Tinkham, M.

M. Tinkham, Introduction to Superconductivity (2nd Edition) (Dover Publications Inc., 2004).

Tsironis, G. P.

N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett.90, 163501 (2007).
[CrossRef]

Ustinov, A. V.

P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013).
[CrossRef]

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
[CrossRef]

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

Wu, J.

Wu, P.

Xing, Q.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Xu, H.

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

Xu, W.

Xue, Y.

Yeh, J.-H.

J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum.84, 034706 (2013).
[CrossRef] [PubMed]

Zhang, C.

Zhang, J. W.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Zhang, L.

Zhang, W.

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

Zhuravel, A. P.

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

Appl. Phys. Lett.

N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett.90, 163501 (2007).
[CrossRef]

P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013).
[CrossRef]

M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett.87, 034102 (2005).
[CrossRef]

J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010).
[CrossRef]

IEEE Trans. Appl. Supercond.

M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999).
[CrossRef]

J. Phys.: Condens. Matter

C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter20, 345220 (2008).
[CrossRef]

Opt. Express

Optics Commun.

A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun.283, 1633–1639 (2010).
[CrossRef]

Phys. Rev. Lett.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum.84, 034706 (2013).
[CrossRef] [PubMed]

Supercond. Sci. Technol.

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013).
[CrossRef]

Other

J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).

D. M. Pozar, Microwave Engineering (2nd Edition) (John Wiley & Sons Inc., 1998) pp. 208–211

K. K. Likharev, Dynamics of Josephson Junctions(Gordon and Breach Science, 1991).

M. Tinkham, Introduction to Superconductivity (2nd Edition) (Dover Publications Inc., 2004).

Supplementary Material (1)

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Figures (6)

Fig. 1
Fig. 1

(a) Sketch of an rf-SQUID. The red cross symbolizes the Josephson junction. (b) Electric equivalent circuit for an rf-SQUID in the small signal approximation. Lgeo is the geometric inductance of the SQUID loop. The red circle indicates the electric circuit model for the junction. R represents the resistance due to a quasiparticle current, Lj is the Josephson inductance and C stands for the capacitance between the superconducting electrodes. (c) Optical micrograph of the rf-SQUID.

Fig. 2
Fig. 2

(a) Optical micrograph of part of the CPW containing a chain of rf-SQUIDs in each gap. (b) Measurement setup including the vector network analyzer (VNA), the bias tees, attenuation and cryogenic amplifier. The green box marks the position of the the part of the waveguide shown in the optical micrograph in (a).

Fig. 3
Fig. 3

Measured transmission magnitude |S21| depending on frequency ν and magnetic flux Φe00.

Fig. 4
Fig. 4

(a) Transmission magnitude (top) and phase (bottom) at an external flux of Φe0 = −0.185Φ0. (b) Real (top) and imaginary (bottom) part of the effective magnetic permeability μr,eff calculated from the transmission data shown in (a). It should be noted that we employ μr,eff = Re(μr,eff) + iIm(μr,eff), opposite to the commonly used definition μ = μ′iμ″.

Fig. 5
Fig. 5

(a) Frequency and flux dependent real part of the effective magnetic permeability Re(μr,eff) calculated from the transmission data shown in Fig. 3. The black dashed line indicates the cut shown in (b). (b) Flux dependence of Re(μr,eff) at a frequency ν = 13.83. Note that only negative flux values are shown.

Fig. 6
Fig. 6

Electrical model of the circuit for one SQUID per unit cell. The unit cells in both pictures are indicated by a dashed box. (a) The CPW is modeled as a transmission line. The SQUIDs couple to the line via a mutual inductance only. (b) Further reduction of the system by including the SQUID influence into the relative, effective permeability μr,eff seen by the line.

Equations (15)

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S = ( Z Z 0 𝟙 ) ( Z + Z 0 𝟙 ) 1
Z = 1 A ˜ 21 ( A ˜ 11 det ( A ˜ ) 1 A ˜ 22 )
A ˜ = A N
A = ( Z L Z C + 1 Z L 1 Z C 1 ) .
l 1 = Z L + 2 Z C Z L 2 + 4 Z L Z C 2 Z C , l 2 = Z L + 2 Z C + Z L 2 + 4 Z L Z C 2 Z C
e 1 = ( 1 , l 2 1 Z L ) T , e 2 = ( 1 , l 1 1 Z L ) T
A = C ( l 1 0 0 l 2 ) C 1
with C = ( e 1 , e 2 ) .
A ˜ = A N = C ( l 1 N 0 0 l 2 N ) C 1 .
0 = ( ( l 2 1 ) l 2 N ( l 1 1 ) l 1 N l 1 l 2 ( l 1 N l 2 N ) Z L l 1 l 2 ( l 1 1 ) l 1 N l 2 ( l 1 1 ) l 1 N ( ( l 1 1 ) l 2 l 1 + 1 ) l 2 N ( l 1 l 2 ) Z L ( l 1 1 ) l 2 N l 1 N l 2 + l 1 N l 1 l 2 ) ( ( S 11 + 1 ) S 22 S 12 S 21 S 11 1 2 S 21 ( ( S 11 + 1 ) S 22 S 12 S 21 + S 11 + 1 ) Z 0 2 S 21 ( S 11 1 ) S 22 S 12 S 21 S 11 + 1 2 S 21 Z 0 ( S 11 1 ) S 22 S 12 S 21 + S 11 1 2 S 21 )
S 21 tot = S 21 in S 21 stl S 21 out ( S 22 in S 12 stl S 21 stl ( S 22 in S 11 stl 1 ) S 22 stl ) S 11 out + S 22 in S 11 stl 1 .
S 21 tot = S 21 in S 21 stl S 21 out .
S 21 tot ( ω , Φ e 0 ) = S 21 in ( ω ) S ˜ 21 stl ( ω ) α ( ω , Φ e 0 ) S 21 out ( ω ) .
S 21 tot , cal ( ω , Φ e 0 ) = S 21 tot ( ω , Φ e 0 ) S 21 tot ( ω , Φ cal ) α ( ω , Φ e 0 )
S 21 stl , reconstructed ( ω , Φ e 0 ) = S 21 tot , cal ( ω , Φ e 0 ) S ˜ 21 stl , stimulated ( ω )

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