Abstract

An indefinite permittivity medium (IPM) has been fabricated and optically characterized in mid-infrared spectral range (10.7 µm−11.3 µm). Phase and amplitude transmission measurements reveal two remarkable properties of IPMs: (i) transmission of sub-diffraction waves (as short as λ/4) can exceed those of diffraction-limited ones, and (ii) sub-diffraction waves can propagate with negative refractive index. We describe a novel double-detector optical technique relying on the interference between sub-diffraction and diffraction-limited waves for accurate measurement of the transmission amplitude and phase of the former.

© 2010 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
    [CrossRef] [PubMed]
  2. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [CrossRef] [PubMed]
  3. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
    [CrossRef]
  4. I. V. Lindell, S. A. Tretyakov, K. A. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
    [CrossRef]
  5. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37(4), 259–263 (2003).
    [CrossRef]
  6. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
    [CrossRef] [PubMed]
  7. A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
    [CrossRef]
  8. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73(11), 113110 (2006).
    [CrossRef]
  9. X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75(4), 045103 (2007).
    [CrossRef]
  10. M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
    [CrossRef] [PubMed]
  11. D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21(5), 1032–1043 (2004).
    [CrossRef]
  12. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
    [CrossRef]
  13. D. R. Smith and D. Schurig, “Spatial filtering using media with indefinite permittivity and permeability tensor,” Appl. Phys. Lett. 82(14), 2215–2217 (2003).
    [CrossRef]
  14. T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
    [CrossRef] [PubMed]
  15. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [CrossRef] [PubMed]
  16. D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. 84(22), 4403–4405 (2004).
    [CrossRef]
  17. P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
    [CrossRef]
  18. Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” arXiv:0910.3981v2 (2009).
  19. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  20. D. Korobkin, Ya. Urzhumov, and G. Shvets, “Enhanced near-field resolution in mid-infrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468–478 (2006).
    [CrossRef]
  21. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near filed,” J. Mod. Opt. 50, 1419–1430 (2003).
  22. E. A. Ray, M. J. Hampton, and R. Lopez, “Simple demonstration of visible evanescent-wave enhancement with far-field detection,” Opt. Lett. 34(13), 2048–2050 (2009).
    [CrossRef] [PubMed]
  23. Y. Xiong, Zh. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
    [CrossRef] [PubMed]
  24. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
    [CrossRef] [PubMed]
  25. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
    [CrossRef] [PubMed]
  26. D. E. Aspnes, “Bounds to average internal fields in two-component composites,” Phys. Rev. Lett. 48(23), 1629–1632 (1982).
    [CrossRef]
  27. R. A. Depine, M. E. Inchaussandague, and A. Lakhtakia, “Classification of dispersion equations for homogeneous, dielectric-magnetic, uniaxial materials,” J. Opt. Soc. Am. A 23(4), 949–955 (2006).
    [CrossRef]
  28. M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
    [CrossRef]
  29. E. D. Palik, ed., Handbook of Optical Constants of Solids. (Academic Press, Orlando, Fla., 1985).
  30. W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
    [CrossRef]
  31. N. Gedik and J. Orenstein, “Absolute phase measurement in heterodyne detection of transient gratings,” Opt. Lett. 29(18), 2109–2111 (2004).
    [CrossRef] [PubMed]

2010 (1)

2009 (4)

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
[CrossRef] [PubMed]

E. A. Ray, M. J. Hampton, and R. Lopez, “Simple demonstration of visible evanescent-wave enhancement with far-field detection,” Opt. Lett. 34(13), 2048–2050 (2009).
[CrossRef] [PubMed]

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

2007 (4)

Y. Xiong, Zh. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75(4), 045103 (2007).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

2006 (6)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73(11), 113110 (2006).
[CrossRef]

T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[CrossRef] [PubMed]

D. Korobkin, Ya. Urzhumov, and G. Shvets, “Enhanced near-field resolution in mid-infrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468–478 (2006).
[CrossRef]

R. A. Depine, M. E. Inchaussandague, and A. Lakhtakia, “Classification of dispersion equations for homogeneous, dielectric-magnetic, uniaxial materials,” J. Opt. Soc. Am. A 23(4), 949–955 (2006).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

2005 (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

2004 (4)

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. 84(22), 4403–4405 (2004).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21(5), 1032–1043 (2004).
[CrossRef]

N. Gedik and J. Orenstein, “Absolute phase measurement in heterodyne detection of transient gratings,” Opt. Lett. 29(18), 2109–2111 (2004).
[CrossRef] [PubMed]

2003 (5)

M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
[CrossRef]

D. R. Smith and D. Schurig, “Spatial filtering using media with indefinite permittivity and permeability tensor,” Appl. Phys. Lett. 82(14), 2215–2217 (2003).
[CrossRef]

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37(4), 259–263 (2003).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
[CrossRef] [PubMed]

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near filed,” J. Mod. Opt. 50, 1419–1430 (2003).

2001 (1)

I. V. Lindell, S. A. Tretyakov, K. A. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
[CrossRef]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1982 (1)

D. E. Aspnes, “Bounds to average internal fields in two-component composites,” Phys. Rev. Lett. 48(23), 1629–1632 (1982).
[CrossRef]

1959 (1)

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[CrossRef]

Alekseyev, L.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Alekseyev, L. V.

Aspnes, D. E.

D. E. Aspnes, “Bounds to average internal fields in two-component composites,” Phys. Rev. Lett. 48(23), 1629–1632 (1982).
[CrossRef]

Barnakov, Y. A.

Belov, P. A.

P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73(11), 113110 (2006).
[CrossRef]

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37(4), 259–263 (2003).
[CrossRef]

Blaikie, R. J.

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. 84(22), 4403–4405 (2004).
[CrossRef]

Bonner, C. E.

Braun, P. X.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

Chassagne, T.

M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
[CrossRef]

Cheng, L.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

Choa, F.-S.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

Dazord, J.

M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
[CrossRef]

Depine, R. A.

Dignam, M. M.

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

Dryden, D.

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Ferro, G.

M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
[CrossRef]

Franz, K. J.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Gedik, N.

Gmachl, C.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Hampton, M. J.

Hao, Y.

P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73(11), 113110 (2006).
[CrossRef]

He, S.

X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75(4), 045103 (2007).
[CrossRef]

Hernandez, M. J.

M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
[CrossRef]

Hillenbrand, R.

T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[CrossRef] [PubMed]

Hoffman, A. J.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Howard, S. S.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Hughes, S.

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

Ilvonen, S.

I. V. Lindell, S. A. Tretyakov, K. A. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
[CrossRef]

Inchaussandague, M. E.

Jacob, Z.

Jin, Y.

X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75(4), 045103 (2007).
[CrossRef]

Kleinman, D.

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[CrossRef]

Kolinko, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21(5), 1032–1043 (2004).
[CrossRef]

Korobkin, D.

T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[CrossRef] [PubMed]

D. Korobkin, Ya. Urzhumov, and G. Shvets, “Enhanced near-field resolution in mid-infrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468–478 (2006).
[CrossRef]

Lakhtakia, A.

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Li, H.

Li, X.

X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75(4), 045103 (2007).
[CrossRef]

Lindell, I. V.

I. V. Lindell, S. A. Tretyakov, K. A. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
[CrossRef]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Liu, Zh.

Y. Xiong, Zh. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Lopez, R.

Mayy, M.

Melville, D. O. S.

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. 84(22), 4403–4405 (2004).
[CrossRef]

Mock, J. J.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[CrossRef]

Monteil, Y.

M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
[CrossRef]

Narimanov, E.

Narimanov, E. E.

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[CrossRef] [PubMed]

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Nataraj, G.

Nikoskinen, K. A.

I. V. Lindell, S. A. Tretyakov, K. A. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
[CrossRef]

Noginov, M. A.

Orenstein, J.

Patterson, M.

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

Pendry, J. B.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near filed,” J. Mod. Opt. 50, 1419–1430 (2003).

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Podolskiy, V. A.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
[CrossRef] [PubMed]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near filed,” J. Mod. Opt. 50, 1419–1430 (2003).

Ray, E. A.

Reza, A.

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

Rye, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[CrossRef]

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

Schurig, D.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21(5), 1032–1043 (2004).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, “Spatial filtering using media with indefinite permittivity and permeability tensor,” Appl. Phys. Lett. 82(14), 2215–2217 (2003).
[CrossRef]

Shvets, G.

D. Korobkin, Ya. Urzhumov, and G. Shvets, “Enhanced near-field resolution in mid-infrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468–478 (2006).
[CrossRef]

T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[CrossRef] [PubMed]

Sivco, D. L.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21(5), 1032–1043 (2004).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, “Spatial filtering using media with indefinite permittivity and permeability tensor,” Appl. Phys. Lett. 82(14), 2215–2217 (2003).
[CrossRef]

Spitzer, W. G.

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[CrossRef]

Sridhar, A.

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

Stewart, W. J.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near filed,” J. Mod. Opt. 50, 1419–1430 (2003).

Sun, C.

Y. Xiong, Zh. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Taubner, T.

T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[CrossRef] [PubMed]

Thongrattanasiri, S.

Tretyakov, S. A.

I. V. Lindell, S. A. Tretyakov, K. A. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
[CrossRef]

Urzhumov, Ya.

T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[CrossRef] [PubMed]

D. Korobkin, Ya. Urzhumov, and G. Shvets, “Enhanced near-field resolution in mid-infrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468–478 (2006).
[CrossRef]

Van Vlack, C.

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

Walsh, D.

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[CrossRef]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near filed,” J. Mod. Opt. 50, 1419–1430 (2003).

Wolf, C. R.

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. 84(22), 4403–4405 (2004).
[CrossRef]

Xiong, Y.

Y. Xiong, Zh. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Yao, P.

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Y. Xiong, Zh. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Zhu, G.

Appl. Phys. Lett. (3)

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[CrossRef]

D. R. Smith and D. Schurig, “Spatial filtering using media with indefinite permittivity and permeability tensor,” Appl. Phys. Lett. 82(14), 2215–2217 (2003).
[CrossRef]

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. 84(22), 4403–4405 (2004).
[CrossRef]

J. Appl. Phys. (1)

A. J. Hoffman, A. Sridhar, P. X. Braun, L. Alekseyev, S. S. Howard, K. J. Franz, L. Cheng, F.-S. Choa, D. L. Sivco, V. A. Podolskiy, E. E. Narimanov, and C. Gmachl, “Midinfrared semiconductor optical metamaterials,” J. Appl. Phys. 105(12), 122411 (2009).
[CrossRef]

J. Cryst. Growth (1)

M. J. Hernandez, G. Ferro, T. Chassagne, J. Dazord, and Y. Monteil, “Study of surface defects on 3C–SiC films grown on Si(111) by CVD,” J. Cryst. Growth 253(1-4), 95–101 (2003).
[CrossRef]

J. Mod. Opt. (1)

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near filed,” J. Mod. Opt. 50, 1419–1430 (2003).

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

J. Opt. Soc. Am. B (2)

Microw. Opt. Technol. Lett. (2)

I. V. Lindell, S. A. Tretyakov, K. A. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
[CrossRef]

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37(4), 259–263 (2003).
[CrossRef]

Nano Lett. (1)

Y. Xiong, Zh. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Nat. Mater. (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. (1)

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[CrossRef]

Phys. Rev. B (4)

P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73(11), 113110 (2006).
[CrossRef]

X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75(4), 045103 (2007).
[CrossRef]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

P. Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80(19), 195106 (2009).
[CrossRef]

Phys. Rev. Lett. (3)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
[CrossRef] [PubMed]

D. E. Aspnes, “Bounds to average internal fields in two-component composites,” Phys. Rev. Lett. 48(23), 1629–1632 (1982).
[CrossRef]

Science (3)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

T. Taubner, D. Korobkin, Ya. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Other (2)

Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” arXiv:0910.3981v2 (2009).

E. D. Palik, ed., Handbook of Optical Constants of Solids. (Academic Press, Orlando, Fla., 1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

(a) Parallel and transverse components of the dielectric permittivities of effective anisotropic medium. (b) Real (solid lines) and imaginary (dashed lines) components of k m for m = 0,1,2 Fourier orders. A 0° incident angle is assumed, so k || m = 2πm / d 1, and k m are calculated from Eq. (1). The inset shows the excitation of the Fourier orders by the object grating.

Fig. 2
Fig. 2

Directions of Poynting flux (red color) and wave-vector (black color) for 0th (solid line) and 1st (dashed line) Fourier orders at λ=10.7 µm (left) and 11.3 µm using an angle of incidence of 30°. The 0th order is evanescent and not shown for λ=11.3 µm.

Fig. 3
Fig. 3

(a) Schematic of the IPM structure. (b) SEM image of the IPM structure cross-section. (c) Moiré pattern of both gratings in transmitted light. (d) SEM picture of the object grating.

Fig. 4
Fig. 4

Experimental set-up for intensity measurements: the incident laser beam scatters at the object grating and launches diffraction-limited (0th order) and sub-diffraction (higher order) electromagnetic waves into the IPM. These modes are scattered into the far field towards the detector by the diagnostic grating.

Fig. 5
Fig. 5

Experimental demonstration of the anti-cutoff [13] phenomenon. (a) Normalized intensities of Fourier harmonics vs wavelength. (b) Ratio of 1st to 0th and 2nd to 0th Fourier harmonics vs wavelength.

Fig. 6
Fig. 6

Experimental set-up for phase measurements.

Fig. 7
Fig. 7

Signals from Detector 1 (red) and Detector 2 (black) at (a) λ=10.753 µm and (b) λ=11.107 µm. Note the sign change of the phase shift between the detector signals ϕD 1ϕD 2. Beam profile measured near Detector 1 for λ=11.171 µm: (c) single (delayed) arm only, (d) interference minimum, and (e) interference maximum. The grey rectangle in the center represents the size of the MCT detector relative to the beam.

Fig. 8
Fig. 8

Phase shift between 0th and 1st Fourier harmonics: experiment (red dots), effective medium theory (green curve) and COMSOL simulations (blue curve).

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

ε ¯ ¯ = ε e z e z + ε | | ( e x e x + e y e y ) ,
ε | | 0 ,
1 / ε 0.
Re ( ε | | ( λ s ) ) = 0 ,
Re ( ε 1 ( λ s ) ) 0.
k | | m = 2 π λ sin ( θ i ) + 2 π d 1 m .
ε | | = ( ε S i C + ε S i O 2 ) / 2 ,
ε = 2 ε S i C ε S i O 2 / ( ε S i C + ε S i O 2 )
Re ( ε S i C ( λ s ) ) = Re ( ε S i O 2 ( λ s ) ) .
k | | 2 ε + k 2 ε | | = ω 2 c 2 ,
k | | ( m , n ) = 2 π λ sin ( θ i ) + 2 π d 1 m 2 π d 2 n ,
sin θ m n = sin θ i + λ ( m d 1 n d 2 ) ,
S = Re [ k ε | | ] | H 0 | 2 2 , and S | | = Re [ k | | ε ] | H 0 | 2 2 ,
φ m = Re ( k m )     l ,
ε S i C ( ω ) = ε ω 2 ω L O 2 i Γ ω ω 2 ω T O 2 i Γ ω ,
E D 1 = E D 1 0 e i φ 0 + i φ d i ω t + E D 1 1 e i φ 1 i ω t ,
I D 1 = | E D 1 | 2 = | E D 1 0 | 2 + | E D 1 1 | 2 + 2 | E D 1 0 E D 1 1 * | cos ( φ d + φ 0 φ 1 ) ,
I D 1 = I d e l , 1 0 + I n o n d e l , 1 1 + 2 I d e l , 1 0 I n o n d e l , 1 1 cos ( φ d + φ 0 φ 1 ) .
E D 2 = E D 2 0 e i φ 0 i ω t + E D 2 1 e i φ 1 + φ d i ω t .
I D 2 = | E D 2 | 2 = | E D 2 0 | 2 + | E D 2 1 | 2 + 2 | E D 2 0 E D 2 1 * | cos ( φ d φ 0 + φ 1 ) ,
I D 2 = I n o n d e l , 2 0 + I d e l , 2 1 + 2 I n o n d e l , 2 0 I d e l , 2 1 cos ( φ d φ 0 + φ 1 ) .
K D 1 = 2 I d e l , 1 0 I n o n d e l , 1 1 I d e l , 1 0 + I n o n d e l , 1 1 , and K D 2 = 2 I n o n d e l , 2 0 I d e l , 2 1 I n o n d e l , 2 0 + I d e l , 2 1

Metrics