Abstract

An infrared polarizer is designed with a predicted extremely high extinction ratio exceeding 3 × 1016 and transmittance higher than 89% for one polarization in the wavelength region from 1.6 to 2.3 µm. Moreover, the performance does not start to deteriorate until 60° tilting angle. The wide-angle high transmission is attributed to the excitation of magnetic polaritons and suitable LC circuit models, which could predict the resonance wavelengths quantitatively, are developed to better understand the underlying mechanisms. The proposed structure can be tuned by controlling the geometrical parameters for different potential applications such as polarizers, beamsplitters, filters, and transparent electrodes.

© 2013 OSA

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2013 (1)

B. Zhao, L. P. Wang, Y. Shuai, and Z. M. Zhang, “Thermophotovoltaic emitters based on a two-dimensional grating/thin-film nanostructure,” Int. J. Heat Mass Transfer (submitted to) (2013).

2012 (4)

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

L. P. Wang and Z. M. Zhang, “Wavelength-selective and diffuse emitter enhanced by magnetic polaritons for thermophotovoltaics,” Appl. Phys. Lett.100(6), 063902 (2012).
[CrossRef]

G. Kang, Y. Fang, I. Vartiainen, Q. Tan, and Y. Wang, “Achromatic polarization splitting effect of metallic gratings with sub-50 nm wide slits,” Appl. Phys. Lett.101(21), 211104 (2012).
[CrossRef]

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun.3, 870 (2012).
[CrossRef] [PubMed]

2011 (3)

2010 (3)

2009 (1)

2008 (3)

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Transmission enhancement through nanoscale metallic slit arrays from the visible to mid-infrared,” J. Comput. Theor. Nanosci.5, 201–213 (2008).

Y.-B. Chen, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of submicron metallic slit arrays,” J. Heat Transf.- Trans. ASME 130(8), 082404 (2008).
[CrossRef]

2007 (2)

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Z. Y. Yang and Y. F. Lu, “Broadband nanowire-grid polarizers in ultraviolet-visible-near-infrared regions,” Opt. Express15(15), 9510–9519 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-15-9510 .
[CrossRef] [PubMed]

2006 (2)

2005 (3)

L. Zhou and W. Liu, “Broadband polarizing beam splitter with an embedded metal-wire nanograting,” Opt. Lett.30(12), 1434–1436 (2005).
[CrossRef] [PubMed]

F. Miyamaru and M. Hangyo, “Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B71(16), 165408 (2005).
[CrossRef]

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

2003 (1)

Y. T. Pang, G. W. Meng, Q. Fang, and L. D. Zhang, “Silver nanowire array infrared polarizers,” Nanotechnology14(1), 20–24 (2003).
[CrossRef]

1997 (1)

Ahn, S.-W.

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

Alù, A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun.3, 870 (2012).
[CrossRef] [PubMed]

Belkin, M. A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun.3, 870 (2012).
[CrossRef] [PubMed]

Bower, J. E.

Buonanno, M.

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Carr, D. W.

Chan, H. B.

Chen, J.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Chen, L.

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Chen, Y.-B.

Y.-B. Chen, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of submicron metallic slit arrays,” J. Heat Transf.- Trans. ASME 130(8), 082404 (2008).
[CrossRef]

B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Transmission enhancement through nanoscale metallic slit arrays from the visible to mid-infrared,” J. Comput. Theor. Nanosci.5, 201–213 (2008).

Cheng, C.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Chou, S. Y.

Chua, S. J.

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

Cirelli, R. A.

Collins, R. T.

Cui, Y.

Datla, R. U.

David, C.

Deng, L. Y.

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

Deng, X.

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Ding, J.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Ekinci, Y.

Fan, Y.-X.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Fang, Q.

Y. T. Pang, G. W. Meng, Q. Fang, and L. D. Zhang, “Silver nanowire array infrared polarizers,” Nanotechnology14(1), 20–24 (2003).
[CrossRef]

Fang, Y.

G. Kang, Y. Fang, I. Vartiainen, Q. Tan, and Y. Wang, “Achromatic polarization splitting effect of metallic gratings with sub-50 nm wide slits,” Appl. Phys. Lett.101(21), 211104 (2012).
[CrossRef]

Ferry, E.

Flammer, P. D.

Fu, S.

S. Xie, H. Li, S. Fu, H. Xu, X. Zhou, and Z. Liu, “The extraordinary optical transmission through double-layer gold slit arrays,” Opt. Commun.283(20), 4017–4024 (2010).
[CrossRef]

Furtak, T. E.

Gentile, T. R.

Hangyo, M.

F. Miyamaru and M. Hangyo, “Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B71(16), 165408 (2005).
[CrossRef]

He, S.

Hollingsworth, R. E.

Hu, J.

Huang, X.-R.

Kang, G.

G. Kang, Y. Fang, I. Vartiainen, Q. Tan, and Y. Wang, “Achromatic polarization splitting effect of metallic gratings with sub-50 nm wide slits,” Appl. Phys. Lett.101(21), 211104 (2012).
[CrossRef]

Kim, J.-S.

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

Kim, S. H.

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

Klemens, F.

Lee, B. J.

Y.-B. Chen, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of submicron metallic slit arrays,” J. Heat Transf.- Trans. ASME 130(8), 082404 (2008).
[CrossRef]

B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Transmission enhancement through nanoscale metallic slit arrays from the visible to mid-infrared,” J. Comput. Theor. Nanosci.5, 201–213 (2008).

Lee, K.-D.

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

Lee, S.-H.

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

Li, H.

S. Xie, H. Li, S. Fu, H. Xu, X. Zhou, and Z. Liu, “The extraordinary optical transmission through double-layer gold slit arrays,” Opt. Commun.283(20), 4017–4024 (2010).
[CrossRef]

Li, W.-D.

Liu, D.

Liu, H.

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

Liu, W.

Liu, X.

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Liu, Z.

S. Xie, H. Li, S. Fu, H. Xu, X. Zhou, and Z. Liu, “The extraordinary optical transmission through double-layer gold slit arrays,” Opt. Commun.283(20), 4017–4024 (2010).
[CrossRef]

Lu, Y. F.

Marcet, Z.

Meng, G. W.

Y. T. Pang, G. W. Meng, Q. Fang, and L. D. Zhang, “Silver nanowire array infrared polarizers,” Nanotechnology14(1), 20–24 (2003).
[CrossRef]

Migdall, A. L.

Miner, J.

Miyamaru, F.

F. Miyamaru and M. Hangyo, “Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B71(16), 165408 (2005).
[CrossRef]

Pai, C. S.

Pang, Y. T.

Y. T. Pang, G. W. Meng, Q. Fang, and L. D. Zhang, “Silver nanowire array infrared polarizers,” Nanotechnology14(1), 20–24 (2003).
[CrossRef]

Park, J.-D.

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

Peltzer, J. J.

Peng, R.-W.

Peng, Y.

Ren, F.-F.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Shi, D.-J.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Shuai, Y.

B. Zhao, L. P. Wang, Y. Shuai, and Z. M. Zhang, “Thermophotovoltaic emitters based on a two-dimensional grating/thin-film nanostructure,” Int. J. Heat Mass Transfer (submitted to) (2013).

Sigg, H.

Solak, H. H.

Tai, S.

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Tan, Q.

G. Kang, Y. Fang, I. Vartiainen, Q. Tan, and Y. Wang, “Achromatic polarization splitting effect of metallic gratings with sub-50 nm wide slits,” Appl. Phys. Lett.101(21), 211104 (2012).
[CrossRef]

Tanner, D. B.

Taylor, J. A.

Teng, J. H.

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

Vartiainen, I.

G. Kang, Y. Fang, I. Vartiainen, Q. Tan, and Y. Wang, “Achromatic polarization splitting effect of metallic gratings with sub-50 nm wide slits,” Appl. Phys. Lett.101(21), 211104 (2012).
[CrossRef]

Walters, F.

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Wang, H.-T.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Wang, J. J.

L. Chen, J. J. Wang, F. Walters, X. Deng, M. Buonanno, S. Tai, and X. Liu, “Large flexible nanowire grid visible polarizer made by nanoimprint lithography,” Appl. Phys. Lett.90(6), 063111 (2007).
[CrossRef]

Wang, L. P.

B. Zhao, L. P. Wang, Y. Shuai, and Z. M. Zhang, “Thermophotovoltaic emitters based on a two-dimensional grating/thin-film nanostructure,” Int. J. Heat Mass Transfer (submitted to) (2013).

L. P. Wang and Z. M. Zhang, “Wavelength-selective and diffuse emitter enhanced by magnetic polaritons for thermophotovoltaics,” Appl. Phys. Lett.100(6), 063902 (2012).
[CrossRef]

L. P. Wang and Z. M. Zhang, “Effect of magnetic polaritons on the radiative properties of double-layer nanoslit arrays,” J. Opt. Soc. Am. B27(12), 2595–2604 (2010).
[CrossRef]

Wang, Y.

G. Kang, Y. Fang, I. Vartiainen, Q. Tan, and Y. Wang, “Achromatic polarization splitting effect of metallic gratings with sub-50 nm wide slits,” Appl. Phys. Lett.101(21), 211104 (2012).
[CrossRef]

Woo, K.

Wu, Q. Y.

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

Wu, Q.-Y.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Xie, S.

S. Xie, H. Li, S. Fu, H. Xu, X. Zhou, and Z. Liu, “The extraordinary optical transmission through double-layer gold slit arrays,” Opt. Commun.283(20), 4017–4024 (2010).
[CrossRef]

Xu, H.

S. Xie, H. Li, S. Fu, H. Xu, X. Zhou, and Z. Liu, “The extraordinary optical transmission through double-layer gold slit arrays,” Opt. Commun.283(20), 4017–4024 (2010).
[CrossRef]

Xu, J.

C. Cheng, J. Chen, D.-J. Shi, Q.-Y. Wu, F.-F. Ren, J. Xu, Y.-X. Fan, J. Ding, and H.-T. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B78(7), 075406 (2008).
[CrossRef]

Yang, Z. Y.

Ye, Z.

Yoon, P.-W.

S.-W. Ahn, K.-D. Lee, J.-S. Kim, S. H. Kim, J.-D. Park, S.-H. Lee, and P.-W. Yoon, “Fabrication of a 50 nm half-pitch wire grid polarizer using nanoimprint lithography,” Nanotechnology16(9), 1874–1877 (2005).
[CrossRef]

Zhai, T.

Zhang, L.

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

Zhang, L. D.

Y. T. Pang, G. W. Meng, Q. Fang, and L. D. Zhang, “Silver nanowire array infrared polarizers,” Nanotechnology14(1), 20–24 (2003).
[CrossRef]

Zhang, X. H.

L. Y. Deng, J. H. Teng, L. Zhang, Q. Y. Wu, H. Liu, X. H. Zhang, and S. J. Chua, “Extremely high extinction ratio terahertz broadband polarizer using bilayer subwavelength metal wire-grid structure,” Appl. Phys. Lett.101(1), 011101 (2012).
[CrossRef]

Zhang, Z. M.

B. Zhao, L. P. Wang, Y. Shuai, and Z. M. Zhang, “Thermophotovoltaic emitters based on a two-dimensional grating/thin-film nanostructure,” Int. J. Heat Mass Transfer (submitted to) (2013).

L. P. Wang and Z. M. Zhang, “Wavelength-selective and diffuse emitter enhanced by magnetic polaritons for thermophotovoltaics,” Appl. Phys. Lett.100(6), 063902 (2012).
[CrossRef]

L. P. Wang and Z. M. Zhang, “Effect of magnetic polaritons on the radiative properties of double-layer nanoslit arrays,” J. Opt. Soc. Am. B27(12), 2595–2604 (2010).
[CrossRef]

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B. Zhao, L. P. Wang, Y. Shuai, and Z. M. Zhang, “Thermophotovoltaic emitters based on a two-dimensional grating/thin-film nanostructure,” Int. J. Heat Mass Transfer (submitted to) (2013).

J. Comput. Theor. Nanosci. (1)

B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Transmission enhancement through nanoscale metallic slit arrays from the visible to mid-infrared,” J. Comput. Theor. Nanosci.5, 201–213 (2008).

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[CrossRef]

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S. Xie, H. Li, S. Fu, H. Xu, X. Zhou, and Z. Liu, “The extraordinary optical transmission through double-layer gold slit arrays,” Opt. Commun.283(20), 4017–4024 (2010).
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Figures (3)

Fig. 1
Fig. 1

The proposed nanostructure and its performance as an IR polarizer: (a) Schematic of a period of the double-layer grating; (b) Spectral transmittance for TM waves and the extinction ratio at normal incidence; (c) Contour plot of the transmittance as a function of the wavelength and angle of incidence for TM waves. The parameters used for the calculation are P = 500 nm, tm = 400 nm, ts = 30 nm, and Wg = 150 nm.

Fig. 2
Fig. 2

The enhancement of the magnetic field and current loop when the MPs are excited and the simple LC circuit models: (a,b) Contour plots of the dimensionless field distribution | H y / H 0 | 2 for P1 and P2, respectively. The arrows indicate the directions of the current flow; (c,d) LC models for P1 and P2, respectively, based on the magnetic field and current density distributions.

Fig. 3
Fig. 3

The effect of certain geometric parameters on the normal transmittance for TM waves using the parameters given in Fig. 1 as the base set: (a) Ag grating thickness tm ; (b) Spacer thickness ts ; (c) Period P by keeping W g /P=0.3 ; (d) Grating strip width P W g . Note that the circle and triangle marks indicate the locations of P1 and P2, respectively, according to the base parameters.

Equations (7)

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

L 1 = L e,1 = W g ε 0 ω p 2 δl
L 2 = L 2 = L e,2 + L m,2 = (P2 W g ) 4 ε 0 ω p 2 δl + μ 0 (P2 W g ) t s 8l
L 3 = L e,3 + L m,3 = t m ε 0 ω p 2 δl + μ 0 W g t m 2l
C g1 = c 1 ε 0 t m l W g and C s = c 2 ε 0 ε d (P2 W g )l 4 t s
ω p1 = ( 2 C s + 1 C g1 ) 1 L 1 +2 L 2 +2 L 2 +2 L 3
C g2 = c 3 ε 0 t m l W g + c 4 ε 0 ε d t s l W g
ω p2 = 1 C g2 ( L 1 +2 L 3 )

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