Abstract

We present a planar waveguide model and a mechanism based on standing wave resonances to interpret the unity absorptions of ultrathin planar metamaterial absorbers. The analytical model predicts that the available absorption peaks of the absorber are corresponding to the fundamental mode and only its odd harmonic modes of the standing wave. The model is in good agreement with numerical simulation and can explain the main features observed in typical ultrathin planar metamaterial absorbers. Based on this model, ultrathin planar metamaterial absorbers with multi-band absorptions at desired frequencies can be easily designed.

© 2012 OSA

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2012

2011

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, “Ultrathin multiband gigahertz metamaterial absorbers,” J. Appl. Phys.110(1), 014909 (2011).
[CrossRef]

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, “Dual-band perfect absorption and field enhancement by interaction between localized and propagating surface plasmons in optical metamaterials,” J. Opt.13(7), 075005 (2011).
[CrossRef]

Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5(6), 4641–4647 (2011).
[CrossRef] [PubMed]

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B83(16), 165107 (2011).
[CrossRef]

Y. Ma, Q. Chen, J. Grant, S. C. Saha, A. Khalid, and D. R. S. Cumming, “A terahertz polarization insensitive dual band metamaterial absorber,” Opt. Lett.36(6), 945–947 (2011).
[CrossRef] [PubMed]

X. Shen, T. J. Cui, J. Zhao, H. F. Ma, W. X. Jiang, and H. Li, “Palarization-independent wide-angle triple-band metamaterial absorber,” Opt. Express19(10), 9401–9407 (2011).
[CrossRef] [PubMed]

Y. Jin, S. Xiao, N. A. Mortensen, and S. L. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express19(12), 11114–11119 (2011).
[CrossRef] [PubMed]

B. Zhang, Y. Zhao, Q. Hao, B. Kiraly, I.-C. Khoo, S. Chen, and T. J. Huang, “Polarization-independent dual-band infrared perfect absorber based on a metal-dielectric-metal elliptical nanodisk array,” Opt. Express19(16), 15221–15228 (2011).
[CrossRef] [PubMed]

M. B. Pu, C. G. Hu, M. Wang, C. Huang, Z. Y. Zhao, C. T. Wang, Q. Feng, and X. G. Luo, “Design principles for infrared wide-angle perfect absorber based on plasmonic structure,” Opt. Express19(18), 17413–17420 (2011).
[CrossRef] [PubMed]

J. Grant, Y. Ma, S. Saha, A. Khalid, and D. R. S. Cumming, “Polarization insensitive, broadband terahertz metamaterial absorber,” Opt. Lett.36(17), 3476–3478 (2011).
[CrossRef] [PubMed]

J. Sun, L. Liu, G. Dong, and J. Zhou, “An extremely broad band metamaterial absorber based on destructive interference,” Opt. Express19(22), 21155–21162 (2011).
[CrossRef] [PubMed]

2010

D. Y. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers,” Phys. Rev. B82(20), 205117 (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. B27(3), 498–504 (2010).
[CrossRef]

X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett.104(20), 207403 (2010).
[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 (2010).
[CrossRef] [PubMed]

2009

Q.-Y. Wen, H.-W. Zhang, Y.-S. Xie, Q.-H. Yang, and Y.-L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization,” Appl. Phys. Lett.95(24), 241111 (2009).
[CrossRef]

2008

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

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 (2010).
[CrossRef] [PubMed]

D. Y. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers,” Phys. Rev. B82(20), 205117 (2010).
[CrossRef]

Cai, G.

P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, “Dual-band perfect absorption and field enhancement by interaction between localized and propagating surface plasmons in optical metamaterials,” J. Opt.13(7), 075005 (2011).
[CrossRef]

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 (2010).
[CrossRef] [PubMed]

Chen, H.-T.

H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express20(7), 7165–7172 (2012).
[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 (2010).
[CrossRef] [PubMed]

Chen, Q.

Chen, S.

Cheng, Q.

H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, “Ultrathin multiband gigahertz metamaterial absorbers,” J. Appl. Phys.110(1), 014909 (2011).
[CrossRef]

Cui, T. J.

H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, “Ultrathin multiband gigahertz metamaterial absorbers,” J. Appl. Phys.110(1), 014909 (2011).
[CrossRef]

X. Shen, T. J. Cui, J. Zhao, H. F. Ma, W. X. Jiang, and H. Li, “Palarization-independent wide-angle triple-band metamaterial absorber,” Opt. Express19(10), 9401–9407 (2011).
[CrossRef] [PubMed]

Cumming, D. R. S.

Ding, P.

P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, “Dual-band perfect absorption and field enhancement by interaction between localized and propagating surface plasmons in optical metamaterials,” J. Opt.13(7), 075005 (2011).
[CrossRef]

Dong, G.

Fan, C.

P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, “Dual-band perfect absorption and field enhancement by interaction between localized and propagating surface plasmons in optical metamaterials,” J. Opt.13(7), 075005 (2011).
[CrossRef]

Feng, Q.

Grant, J.

Hao, J.

J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B83(16), 165107 (2011).
[CrossRef]

Hao, Q.

He, S.

He, S. L.

Hu, C. G.

Hu, W.

P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, “Dual-band perfect absorption and field enhancement by interaction between localized and propagating surface plasmons in optical metamaterials,” J. Opt.13(7), 075005 (2011).
[CrossRef]

Huang, C.

Huang, T. J.

Jiang, W. X.

Jiang, Z. H.

Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5(6), 4641–4647 (2011).
[CrossRef] [PubMed]

Jin, Y.

Khalid, A.

Khoo, I.-C.

Kiraly, B.

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]

Li, H.

X. Shen, T. J. Cui, J. Zhao, H. F. Ma, W. X. Jiang, and H. Li, “Palarization-independent wide-angle triple-band metamaterial absorber,” Opt. Express19(10), 9401–9407 (2011).
[CrossRef] [PubMed]

H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, “Ultrathin multiband gigahertz metamaterial absorbers,” J. Appl. Phys.110(1), 014909 (2011).
[CrossRef]

Li, L.

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

Liang, C.

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

Liang, E.

P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, “Dual-band perfect absorption and field enhancement by interaction between localized and propagating surface plasmons in optical metamaterials,” J. Opt.13(7), 075005 (2011).
[CrossRef]

Liu, L.

Liu, X.

X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett.104(20), 207403 (2010).
[CrossRef] [PubMed]

Liu, Y.-L.

Q.-Y. Wen, H.-W. Zhang, Y.-S. Xie, Q.-H. Yang, and Y.-L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization,” Appl. Phys. Lett.95(24), 241111 (2009).
[CrossRef]

Liu, Z.

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

Luo, X. G.

Ma, H. F.

Ma, Y.

Mayer, T. S.

Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5(6), 4641–4647 (2011).
[CrossRef] [PubMed]

Mock, J. J.

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]

Mortensen, N. A.

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 (2010).
[CrossRef] [PubMed]

D. Y. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers,” Phys. Rev. B82(20), 205117 (2010).
[CrossRef]

Padilla, W. J.

X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett.104(20), 207403 (2010).
[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]

Pu, M. B.

Qiu, M.

J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B83(16), 165107 (2011).
[CrossRef]

Saha, S.

Saha, S. C.

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]

Shchegolkov, D. Y.

D. Y. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers,” Phys. Rev. B82(20), 205117 (2010).
[CrossRef]

Shen, X.

Shen, X. P.

H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, “Ultrathin multiband gigahertz metamaterial absorbers,” J. Appl. Phys.110(1), 014909 (2011).
[CrossRef]

Simakov, E. I.

D. Y. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers,” Phys. Rev. B82(20), 205117 (2010).
[CrossRef]

Smith, D. R.

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]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

Starr, A. F.

X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett.104(20), 207403 (2010).
[CrossRef] [PubMed]

Starr, T.

X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett.104(20), 207403 (2010).
[CrossRef] [PubMed]

Sun, J.

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 (2010).
[CrossRef] [PubMed]

Toor, F.

Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5(6), 4641–4647 (2011).
[CrossRef] [PubMed]

Wang, C. T.

Wang, M.

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

Wen, Q.-Y.

Q.-Y. Wen, H.-W. Zhang, Y.-S. Xie, Q.-H. Yang, and Y.-L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization,” Appl. Phys. Lett.95(24), 241111 (2009).
[CrossRef]

Werner, D. H.

Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5(6), 4641–4647 (2011).
[CrossRef] [PubMed]

Xiao, S.

Xie, Y.-S.

Q.-Y. Wen, H.-W. Zhang, Y.-S. Xie, Q.-H. Yang, and Y.-L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization,” Appl. Phys. Lett.95(24), 241111 (2009).
[CrossRef]

Xue, Q.

P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, “Dual-band perfect absorption and field enhancement by interaction between localized and propagating surface plasmons in optical metamaterials,” J. Opt.13(7), 075005 (2011).
[CrossRef]

Yang, Q.-H.

Q.-Y. Wen, H.-W. Zhang, Y.-S. Xie, Q.-H. Yang, and Y.-L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization,” Appl. Phys. Lett.95(24), 241111 (2009).
[CrossRef]

Yang, Y.

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

L. Li, Y. Yang, and C. Liang, “A wide-angle polarization-insensitive ultra-thin metamaterial absorber with three resonant modes,” J. Appl. Phys.110(6), 063702 (2011).
[CrossRef]

Ye, Y. Q.

Yuan, L. H.

H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, “Ultrathin multiband gigahertz metamaterial absorbers,” J. Appl. Phys.110(1), 014909 (2011).
[CrossRef]

Yun, S.

Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5(6), 4641–4647 (2011).
[CrossRef] [PubMed]

Zhang, B.

Zhang, H.-W.

Q.-Y. Wen, H.-W. Zhang, Y.-S. Xie, Q.-H. Yang, and Y.-L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization,” Appl. Phys. Lett.95(24), 241111 (2009).
[CrossRef]

Zhang, X.

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

Zhao, J.

Zhao, Y.

Zhao, Z. Y.

Zhou, B.

H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, “Ultrathin multiband gigahertz metamaterial absorbers,” J. Appl. Phys.110(1), 014909 (2011).
[CrossRef]

Zhou, J.

J. Sun, L. Liu, G. Dong, and J. Zhou, “An extremely broad band metamaterial absorber based on destructive interference,” Opt. Express19(22), 21155–21162 (2011).
[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 (2010).
[CrossRef] [PubMed]

Zhou, L.

J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B83(16), 165107 (2011).
[CrossRef]

ACS Nano

Z. H. Jiang, S. Yun, F. Toor, D. H. Werner, and T. S. Mayer, “Conformal dual-band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5(6), 4641–4647 (2011).
[CrossRef] [PubMed]

Appl. Phys. Lett.

Q.-Y. Wen, H.-W. Zhang, Y.-S. Xie, Q.-H. Yang, and Y.-L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization,” Appl. Phys. Lett.95(24), 241111 (2009).
[CrossRef]

J. Appl. Phys.

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Figures (4)

Fig. 1
Fig. 1

Illustration of standing wave resonance model of a typical ultrathin planar metamaterial absorber. (a) Structure of the waveguide and the incident EM wave; (b) Zoomed vectors and their components of the diffractive EM waves from adjacent gaps; (c) Normalized electric field of the fundamental mode and its odd harmonic modes at x = 0 and z = - d/2 in the case of strong diffraction effect calculated according to Eq. (3).

Fig. 2
Fig. 2

Diagram of the ultrathin THz planar metamaterial absorber based on standing wave resonances and the simulation results in the case of incident wave with TM polarization. (a) Front view at the top and side view at the bottom of the structure; (b) Simulated absorption spectra induced by incident EM wave with TM (black line) and TE Polarization (red line). The dashed line represents the frequencies of the standing wave modes calculated according to Eq. (6). (c) Simulated absorption vs. the thickness d of the dielectric spacer layer of the absorber; (d) Electric field component Ey at x = 0 corresponding to different absorption peaks at different phase when the strength of Ey is strongest; (e) Electric field component Ez at x = 0 corresponding to different absorption peaks at different phase when the strength of Ez is strongest; (f) The power loss density distribution at x = 0.

Fig. 3
Fig. 3

Distributions of electric field component Ez at z = −1.7µm corresponding to eight absorption peaks shown in Fig. 2(b) in the case of incident EM wave with TM polarization. The inset shows the front view of the unit cell.

Fig. 4
Fig. 4

Ultrathin planar metamaterial absorber with a cross metallic top layer. (a) The front view of the unit cell (top) and the side view of the unit cell (bottom); (b) Simulation result of the absorption spectrum of an infrared absorber. The dashed line represents the fundamental mode calculated with Eq. (6). (c) Electric field component Ez at x = 0 µm; (d) Electric field component Ez at z = - 0.065 µm. (e) Simulation results of the 1st and 3rd modes of the cross-waveguide style absorber working at the infrared and THz domain vs. the results calculated with Eq. (6) at different dimension L of the cross. (f) Simulation results of the optimized thicknesses d1 and d3 corresponding to the 1st and the 3rd modes at different dimension of the cross, respectively. The straight lines are the linear fits.

Equations (8)

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

E z + = E z0 exp[i( k y y ω 0 t+ ϕ 0 +π)]
E z = E z0 exp[i( k y y ω 0 t+ ϕ 0 )]
E zsw = E z + + E z =2Esinθexp[ μ( L 2 | y | ) ]sin( 2πnsinθ λ 0 )sin( 2πc λ 0 t+ ϕ 0 )
f sw =(2j1) c 2nLsinθ ,j=1,2,3,...
λ sw = 2nLsinθ (2j1) ,j=1,2,3,...
f sw (2j1) c 2nL ,j=1,2,3,...
λ sw 2nL (2j1) ,j=1,2,3,...
L λ 1 2n( λ 1 ) = c 2n( f 1 ) f 1

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