Abstract

We propose a design of an extremely broad frequency band absorber based on destructive interference mechanism. Metamaterial of multilayered SRRs structure is used to realize a desirable refractive index dispersion spectrum, which can induce a successive anti-reflection in a wide frequency range. The corresponding high absorptance originates from the destructive interference of two reflection waves from the two surfaces of the metamaterial. A strongly absorptive bandwidth of almost 60GHz is demonstrated in the range of 0 to 70GHz numerically. This design provides an effective and feasible way to construct broad band absorber in stealth technology, as well as the enhanced transmittance devices.

© 2011 OSA

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  1. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  2. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
    [CrossRef] [PubMed]
  3. W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).
  4. Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16(9), 6134–6145 (2008).
    [CrossRef] [PubMed]
  5. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
    [CrossRef] [PubMed]
  6. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
    [CrossRef] [PubMed]
  7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  8. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
    [CrossRef] [PubMed]
  9. H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
    [CrossRef] [PubMed]
  10. H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
    [CrossRef]
  11. K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113–083118 (2010).
    [CrossRef]
  12. Y. Cheng, H. Yang, Z. Cheng, and N. Wu, “Perfect metamaterial absorber based on a split-ring-cross resonator,” Appl. Phys., A Mater. Sci. Process. 102(1), 99–103 (2011).
    [CrossRef]
  13. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
    [CrossRef] [PubMed]
  14. C. Wu, Y. Avitzour, and G. Shvets, “Ultra-thin, wide-angle perfect absorber for infrared frequencies,” Proc. SPIE, Proceedings of Metamaterials: Fundamentals and Applications, San Diego, CA, August 10–14 (2008).
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  16. K. B. Alici and E. Ozbay, “Photonic metamaterial absorber designs for infrared solar-cell applications,” Proc. SPIE 7772, 77721B (2010).
    [CrossRef]
  17. Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
    [CrossRef]
  18. B. Wang, Th. Koschny, and C. M. Soukoulis, “Wide-angle and polarization independent chiral metamaterials absorbers,” Phys. Rev. B 80(3), 033108 (2009).
    [CrossRef]
  19. K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113 (2010).
    [CrossRef]
  20. K. B. Alici, A. B. Turhan, C. M. Soukoulis, and E. Ozbay, “Optically thin composite resonant absorber at the near-infrared band: a polarization independent and spectrally broadband configuration,” Opt. Express 19(15), 14260 (2011).
    [CrossRef]
  21. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1980), Chap. 1.
  22. H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
    [CrossRef] [PubMed]
  23. J. Lee and S. Lim, “Bandwidth-enhanced and polarization-insensitive metamaterial absorber using double resonance,” Electron. Lett. 47(1), 8–9 (2011).
    [CrossRef]
  24. H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
    [CrossRef]
  25. S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
    [CrossRef]
  26. Y. Q. Ye, Y. Jin, and S. He, “Omnidirectional polarization-insensitive and broadband thin absorber in the terahertz regime,” J. Opt. Soc. Am. B 27(3), 498–504 (2010).
    [CrossRef]

2011 (4)

C. H. Lin, R. L. Chern, and H. Y. Lin, “Polarization-independent broad-band nearly perfect absorbers in the visible regime,” Opt. Express 19(2), 415–424 (2011).
[CrossRef] [PubMed]

Y. Cheng, H. Yang, Z. Cheng, and N. Wu, “Perfect metamaterial absorber based on a split-ring-cross resonator,” Appl. Phys., A Mater. Sci. Process. 102(1), 99–103 (2011).
[CrossRef]

J. Lee and S. Lim, “Bandwidth-enhanced and polarization-insensitive metamaterial absorber using double resonance,” Electron. Lett. 47(1), 8–9 (2011).
[CrossRef]

K. B. Alici, A. B. Turhan, C. M. Soukoulis, and E. Ozbay, “Optically thin composite resonant absorber at the near-infrared band: a polarization independent and spectrally broadband configuration,” Opt. Express 19(15), 14260 (2011).
[CrossRef]

2010 (8)

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
[CrossRef]

Y. Q. Ye, Y. Jin, and S. He, “Omnidirectional polarization-insensitive and broadband thin absorber in the terahertz regime,” J. Opt. Soc. Am. B 27(3), 498–504 (2010).
[CrossRef]

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113–083118 (2010).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113 (2010).
[CrossRef]

K. B. Alici and E. Ozbay, “Photonic metamaterial absorber designs for infrared solar-cell applications,” Proc. SPIE 7772, 77721B (2010).
[CrossRef]

2009 (4)

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

B. Wang, Th. Koschny, and C. M. Soukoulis, “Wide-angle and polarization independent chiral metamaterials absorbers,” Phys. Rev. B 80(3), 033108 (2009).
[CrossRef]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

2008 (5)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16(9), 6134–6145 (2008).
[CrossRef] [PubMed]

2006 (2)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
[CrossRef] [PubMed]

2000 (1)

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

Alici, K. B.

K. B. Alici, A. B. Turhan, C. M. Soukoulis, and E. Ozbay, “Optically thin composite resonant absorber at the near-infrared band: a polarization independent and spectrally broadband configuration,” Opt. Express 19(15), 14260 (2011).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113 (2010).
[CrossRef]

K. B. Alici and E. Ozbay, “Photonic metamaterial absorber designs for infrared solar-cell applications,” Proc. SPIE 7772, 77721B (2010).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113–083118 (2010).
[CrossRef]

Averitt, R. D.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

Avitzour, Y.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

Azad, A. K.

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

Barrett, J. P.

S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Bilotti, F.

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113–083118 (2010).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113 (2010).
[CrossRef]

Bingham, C. M.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

Chen, F.

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

Chen, H. T.

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

Cheng, Q.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Cheng, Y.

Y. Cheng, H. Yang, Z. Cheng, and N. Wu, “Perfect metamaterial absorber based on a split-ring-cross resonator,” Appl. Phys., A Mater. Sci. Process. 102(1), 99–103 (2011).
[CrossRef]

Cheng, Z.

Y. Cheng, H. Yang, Z. Cheng, and N. Wu, “Perfect metamaterial absorber based on a split-ring-cross resonator,” Appl. Phys., A Mater. Sci. Process. 102(1), 99–103 (2011).
[CrossRef]

Chern, R. L.

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Cui, T. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Cummer, S. A.

S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
[CrossRef] [PubMed]

Fan, K.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

Giessen, H.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Gu, S.

S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
[CrossRef]

Hand, T. H.

S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
[CrossRef]

He, S.

Hentschel, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Jessie, Y.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Jiang, W. X.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Jin, Y.

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kattawar, G. W.

Koschny, Th.

B. Wang, Th. Koschny, and C. M. Soukoulis, “Wide-angle and polarization independent chiral metamaterials absorbers,” Phys. Rev. B 80(3), 033108 (2009).
[CrossRef]

Landy, N. I.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

Lee, J.

J. Lee and S. Lim, “Bandwidth-enhanced and polarization-insensitive metamaterial absorber using double resonance,” Electron. Lett. 47(1), 8–9 (2011).
[CrossRef]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Lim, S.

J. Lee and S. Lim, “Bandwidth-enhanced and polarization-insensitive metamaterial absorber using double resonance,” Electron. Lett. 47(1), 8–9 (2011).
[CrossRef]

Lin, C. H.

Lin, H. Y.

Lin, X. Q.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Liu, N.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Mesch, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

O’Hara, J. F.

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

Ozbay, E.

K. B. Alici, A. B. Turhan, C. M. Soukoulis, and E. Ozbay, “Optically thin composite resonant absorber at the near-infrared band: a polarization independent and spectrally broadband configuration,” Opt. Express 19(15), 14260 (2011).
[CrossRef]

K. B. Alici and E. Ozbay, “Photonic metamaterial absorber designs for infrared solar-cell applications,” Proc. SPIE 7772, 77721B (2010).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113 (2010).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113–083118 (2010).
[CrossRef]

Padilla, W. J.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
[CrossRef] [PubMed]

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

Pilon, D.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

Popa, B.-I.

S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
[CrossRef]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
[CrossRef] [PubMed]

Sajuyigbe, S.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Schurig, D.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Shrekenhamer, D.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

Shvets, G.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Soukoulis, C. M.

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Strikwerda, A. C.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

Tao, H.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

Taylor, A. J.

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

Turhan, A. B.

Urzhumov, Y. A.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Vegni, L.

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113–083118 (2010).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113 (2010).
[CrossRef]

Wang, B.

B. Wang, Th. Koschny, and C. M. Soukoulis, “Wide-angle and polarization independent chiral metamaterials absorbers,” Phys. Rev. B 80(3), 033108 (2009).
[CrossRef]

Weiss, T.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Wu, N.

Y. Cheng, H. Yang, Z. Cheng, and N. Wu, “Perfect metamaterial absorber based on a split-ring-cross resonator,” Appl. Phys., A Mater. Sci. Process. 102(1), 99–103 (2011).
[CrossRef]

Yang, H.

Y. Cheng, H. Yang, Z. Cheng, and N. Wu, “Perfect metamaterial absorber based on a split-ring-cross resonator,” Appl. Phys., A Mater. Sci. Process. 102(1), 99–103 (2011).
[CrossRef]

Yang, P.

Ye, Y. Q.

You, Y.

Yu, G. X.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhai, P. W.

Zhang, X.

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

Zhou, J.

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

Appl. Phys., A Mater. Sci. Process. (1)

Y. Cheng, H. Yang, Z. Cheng, and N. Wu, “Perfect metamaterial absorber based on a split-ring-cross resonator,” Appl. Phys., A Mater. Sci. Process. 102(1), 99–103 (2011).
[CrossRef]

Electron. Lett. (1)

J. Lee and S. Lim, “Bandwidth-enhanced and polarization-insensitive metamaterial absorber using double resonance,” Electron. Lett. 47(1), 8–9 (2011).
[CrossRef]

J. Appl. Phys. (3)

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113–083118 (2010).
[CrossRef]

S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A. Cummer, “A broadband low-reflection metamaterial absorber,” J. Appl. Phys. 108(6), 064913–064918 (2010).
[CrossRef]

K. B. Alici, F. Bilotti, L. Vegni, and E. Ozbay, “Experimental verification of metamaterial based subwavelength microwave absorbers,” J. Appl. Phys. 108(8), 083113 (2010).
[CrossRef]

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

J. Phys. D Appl. Phys. (2)

H. Tao, C. M. Bingham, D. Pilon, K. Fan, A. C. Strikwerda, D. Shrekenhamer, W. J. Padilla, X. Zhang, and R. D. Averitt, “A dual band terahertz metamaterial absorber,” J. Phys. D Appl. Phys. 43(22), 225102 (2010).
[CrossRef]

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and Y. Jessie, “Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D Appl. Phys. 41, 085504–085507 (2008).

Nano Lett. (1)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010).
[CrossRef] [PubMed]

Nat. Mater. (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Opt. Express (4)

Phys. Rev. B (3)

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

B. Wang, Th. Koschny, and C. M. Soukoulis, “Wide-angle and polarization independent chiral metamaterials absorbers,” Phys. Rev. B 80(3), 033108 (2009).
[CrossRef]

H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D. Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J. Padilla, and R. D. Averitt, “Highly flexible wide angle of incident terahertz metamaterial absorber: Design, fabrication, and characterization,” Phys. Rev. B 78(24), 241103 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621–036625 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

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

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901–073904 (2010).
[CrossRef] [PubMed]

Proc. SPIE (1)

K. B. Alici and E. Ozbay, “Photonic metamaterial absorber designs for infrared solar-cell applications,” Proc. SPIE 7772, 77721B (2010).
[CrossRef]

Science (2)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Other (2)

C. Wu, Y. Avitzour, and G. Shvets, “Ultra-thin, wide-angle perfect absorber for infrared frequencies,” Proc. SPIE, Proceedings of Metamaterials: Fundamentals and Applications, San Diego, CA, August 10–14 (2008).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1980), Chap. 1.

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Figures (6)

Fig. 1
Fig. 1

Reflections on different coating system: (a) Lens without coating, the transmittance is not very high because of the reflection on the surface of the lens. (b) AR coating on the lens may improve the transmittance. The destructive interference between reflected waves from the front and back interfaces of the AR coating greatly reduces the reflection and enhances the transmittance of the lens. (c) AR coating eliminates all the reflected waves from the metal by destructive interference, which plays a role of absorber. (d) An ideal refractive index spectrum (top) retrieved by anti-reflection theory, which can realize a successive anti-reflection in whole frequency range in theory. And the S11 (reflectance) of a medium with such refractive index spectrum is the envelope of those dispersed anti-reflection peak, as depicted by the short dash line (bottom).

Fig. 2
Fig. 2

(a) Fabrication of this absorber structure and (b) the typical size of the SRRs and layers’ thickness

Fig. 3
Fig. 3

(a) The simulation model of the metamaterial absorber; (b) S11 of this model is calculated by CST with different thickness of the top layer thickness d4. Peaks resulted from the anti-reflection theory shows redshift with the increase of d4, but there is almost no change on the peaks resulted from the resonance loss, as shown in the inset.

Fig. 4
Fig. 4

(a) Illustration of multiple transmissions and reflections in the enhanced transmittance system. ψ12 implies the phase change brought by transmission from air (1) to AR-film (2). ψ21 implies the phase change brought by transmission from AR-film (2) to air (1). ϕ12 implies the phase change resulted by the reflection at the interfaces between air (1) and AR-film (2). ϕ23 implies the phase change resulted by the reflection at the interfaces between AR-film (2) and medium (3). r12, r23, t12, t21 are amplitudes of Fresnel coefficients and they are real positive numbers. (b) The S parameters of the transmission enhanced system formed by the metamaterial structure. (c) The phase difference of the reflective wave from the two interface of the metamaterial. The vertical dashed lines (green, orange, red and blue) indicate that the phase different is π when the destructive interference occurs.

Fig. 5
Fig. 5

(a) Perfect metamaterial absorber system with a certain size of SRR and favorable thickness of substrate (h1, h2). Moreover, the material (n = 2 + 0.01i) of the substrate is quite low loss. (b) The three SRRs with the typical sizes in Table 1 produce three absorption peaks at frequencies of 2.8GHz, 4.15GHz and 7.15GHz, which is in agreement with the peaks at 2.8GHz, 4.1GHz and 7GHz, respectively, as shown in the inset of Fig. 3.

Fig. 6
Fig. 6

Simulation results of S11 (a) and absorptance (b) at the frequency range of 0 to 70GHz with increased d4. Arrows in (a) show the redshift of the intrinsic anti-reflection peaks with the increase of d4. Bandwidth of absorptance >90% in (b) ranges from 10GHz to 70GHz.

Tables (2)

Tables Icon

Table 1 Typical size of the metamaterial structure

Tables Icon

Table 2 Favorite values of h1, h2 when strong resonant absorption occurs for each SRR

Equations (8)

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

nd= m λ AR 4 ,
nd= mc 4 f AR ,
r ˜ = r 12 exp(i ϕ 12 )+ t 12 r 23 t 21 (iζ) 1 r 21 r 23 exp[i( ϕ 12 + ϕ 23 +2β)] ,
t ˜ = t 12 t 23 exp[i( ψ 12 + ψ 23 +ζ)] 1 r 21 r 23 exp[i( ϕ 12 + ϕ 23 +2β)] ,
R= | r ˜ | 2 = r 12 2 + ( t 12 r 23 t 21 ) 2 +2 r 12 t 12 r 23 t 21 cos( ϕ 12 ζ) 1+ ( r 21 r 23 ) 2 2cos( ϕ 12 + ϕ 23 +2β) r 21 r 23 ,
r 12 t 12 t 21 ,
ϕ 12 ζ=(2N+1)π,
R= | r ˜ | 2 = r 12 2 + ( t 12 t 21 ) 2 2 r 12 t 12 t 21 1+ ( r 21 r 23 ) 2 2cos( ϕ 12 + ϕ 23 +2β) r 21 r 23 .

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