Abstract

We placed active magnetic metamaterials on metallic surface to implement a tunable reflector with excellent agile performance. By incorporating active elements into the unit cells of the magnetic metamaterial, this active magnetic metamaterial can be tuned to switch function of the reflector among a perfect absorber, a perfect reflector and a gain reflector. This brings about DC control lines to electrically tune the active magnetic metamaterial with positive loss, zero loss and even negative loss. The design, analytical and numerical simulation methods, and experimental results of the tunable reflector are presented.

© 2014 Optical Society of America

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2013 (4)

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

H. Wang, L. Wang, “Perfect selective metamaterial solar absorber,” Opt. Express 21(S6), A1078–A1093 (2013).
[CrossRef]

L. Sun, X. Yang, J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103, 201109 (2013).

W. Xu, S. Sonkusale, “Microwave diode switchable metamaterial reflector/absorber,” Appl. Phys. Lett. 103, 031902 (2013).

2012 (5)

2010 (1)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

2009 (2)

2008 (3)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

B. Kanté, A. de Lustrac, J. M. Lourtioz, S. N. Burokur, “Infrared cloaking based on the electric response of split ring resonators,” Opt. Express 16(12), 9191–9198 (2008).
[CrossRef] [PubMed]

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

2007 (1)

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).

2006 (2)

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

2005 (1)

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

2004 (2)

A. Tennant, B. Chambers, “Adaptive radar absorbing structure with PIN diode controlled active frequency selective surface,” Smart Mater. Struct. 13(1), 122–125 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

2003 (1)

S.A. Ramakrishna, J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).

1999 (1)

B. Chambers, “A smart radar absorber,” Smart Mater. Struct. 8(1), 64–72 (1999).
[CrossRef]

Akin, T.

Akozbek, N.

N. Mattiucci, R. Trimm, G. D’Aguanno, N. Akozbek, M.J. Bloemer, “Tunable, narrow-band, all-metallic microwave absorber,” Appl. Phys. Lett. 101, 141115 (2012).

Alaee, R.

Averitt, R. D.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Bartal, G.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Bloemer, M.J.

N. Mattiucci, R. Trimm, G. D’Aguanno, N. Akozbek, M.J. Bloemer, “Tunable, narrow-band, all-metallic microwave absorber,” Appl. Phys. Lett. 101, 141115 (2012).

Burokur, S. N.

Chambers, B.

A. Tennant, B. Chambers, “Adaptive radar absorbing structure with PIN diode controlled active frequency selective surface,” Smart Mater. Struct. 13(1), 122–125 (2004).
[CrossRef]

B. Chambers, “A smart radar absorber,” Smart Mater. Struct. 8(1), 64–72 (1999).
[CrossRef]

Chen, H.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

Chen, H. T.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Chipouline, A.

Cummer, S. A.

D’Aguanno, G.

N. Mattiucci, R. Trimm, G. D’Aguanno, N. Akozbek, M.J. Bloemer, “Tunable, narrow-band, all-metallic microwave absorber,” Appl. Phys. Lett. 101, 141115 (2012).

De Angelis, F.

de Lustrac, A.

Di Fabrizio, E.

Dong, Y.

Y. Dong, T. Itoh, “Metamaterial-based antennas,” Proc. IEEE 100(7), 2271–2285 (2012).
[CrossRef]

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Ekmekci, E.

Fang, N.

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

Farhat, M.

Gao, J.

L. Sun, X. Yang, J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103, 201109 (2013).

Gossard, A. C.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Huang, Y.

Itoh, T.

Y. Dong, T. Itoh, “Metamaterial-based antennas,” Proc. IEEE 100(7), 2271–2285 (2012).
[CrossRef]

Kanté, B.

Kildishev, A. V.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Landy, N. I.

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

Lederer, F.

Lee, H.

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

Liu, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Lourtioz, J. M.

Mattiucci, N.

N. Mattiucci, R. Trimm, G. D’Aguanno, N. Akozbek, M.J. Bloemer, “Tunable, narrow-band, all-metallic microwave absorber,” Appl. Phys. Lett. 101, 141115 (2012).

Mock, J. J.

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

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Nikolaenko, A. E.

Padilla, W. J.

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

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Papasimakis, N.

Pendry, J. B.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Pendry, J.B.

S.A. Ramakrishna, J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).

Popa, B. I.

Premaratne, M.

Ramakrishna, S.A.

S.A. Ramakrishna, J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).

Rockstuhl, C.

Rukhlenko, I. D.

Sajuyigbe, S.

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

Schurig, D.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Shen, L.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

Smith, D. R.

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

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Sonkusale, S.

W. Xu, S. Sonkusale, “Microwave diode switchable metamaterial reflector/absorber,” Appl. Phys. Lett. 103, 031902 (2013).

Stacy, A. M.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

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

Sun, L.

L. Sun, X. Yang, J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103, 201109 (2013).

Taylor, A. J.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Tennant, A.

A. Tennant, B. Chambers, “Adaptive radar absorbing structure with PIN diode controlled active frequency selective surface,” Smart Mater. Struct. 13(1), 122–125 (2004).
[CrossRef]

Topalli, K.

Trimm, R.

N. Mattiucci, R. Trimm, G. D’Aguanno, N. Akozbek, M.J. Bloemer, “Tunable, narrow-band, all-metallic microwave absorber,” Appl. Phys. Lett. 101, 141115 (2012).

Turhan-Sayan, G.

Wang, H.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

H. Wang, L. Wang, “Perfect selective metamaterial solar absorber,” Opt. Express 21(S6), A1078–A1093 (2013).
[CrossRef]

Wang, L.

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Wen, G.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Xu, W.

W. Xu, S. Sonkusale, “Microwave diode switchable metamaterial reflector/absorber,” Appl. Phys. Lett. 103, 031902 (2013).

Yang, X.

L. Sun, X. Yang, J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103, 201109 (2013).

Yao, J.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Yuan, H. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Yuan, Y.

Zhang, B.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

Zhang, X.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

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

Zheludev, N. I.

Zheludev, N.I.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

Zheng, B.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

Zhu, W.

Zide, J. M. O.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett (1)

L. Sun, X. Yang, J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103, 201109 (2013).

Appl. Phys. Lett. (2)

N. Mattiucci, R. Trimm, G. D’Aguanno, N. Akozbek, M.J. Bloemer, “Tunable, narrow-band, all-metallic microwave absorber,” Appl. Phys. Lett. 101, 141115 (2012).

W. Xu, S. Sonkusale, “Microwave diode switchable metamaterial reflector/absorber,” Appl. Phys. Lett. 103, 031902 (2013).

Nat. Commun (1)

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N.I. Zheludev, B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nat. Commun. 4, 2652 (2013).

Nature (2)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Opt. Express (7)

Phys. Rev. B (1)

S.A. Ramakrishna, J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).

Phys. Rev. Lett (1)

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

Y. Dong, T. Itoh, “Metamaterial-based antennas,” Proc. IEEE 100(7), 2271–2285 (2012).
[CrossRef]

Science (4)

D. R. Smith, J. B. Pendry, M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

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

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Smart Mater. Struct. (2)

A. Tennant, B. Chambers, “Adaptive radar absorbing structure with PIN diode controlled active frequency selective surface,” Smart Mater. Struct. 13(1), 122–125 (2004).
[CrossRef]

B. Chambers, “A smart radar absorber,” Smart Mater. Struct. 8(1), 64–72 (1999).
[CrossRef]

Other (3)

S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice, (Wiley, 1966).

D. M. Pozar, Microwave Engineering (Wiley, 2012), p.174.

F. Capolino, Theory and Phenomena of Metamaterials (CRC Press LLC, 2009), Vol. 1.

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Figures (12)

Fig. 1
Fig. 1

Illustration of reflectivity reduction or amplification from metallic surfaces with material slab.

Fig. 2
Fig. 2

(a) Tunable reflector with active magnetic metamaterial on metallic surface; (b) A unit cell of the active magnetic metamaterial under consideration.

Fig. 3
Fig. 3

One periodic unit cell of the active magnetic metamaterials. (r = 20.5, a = 1,θ = 50°, l1 = 5 and g = 0.5; units in mm)

Fig. 4
Fig. 4

The fabricated tunable reflector with six periodic unit cells.

Fig. 5
Fig. 5

Measurement setup for tunable reflector.

Fig. 6
Fig. 6

Tunable reflector with six units measured by using a TEM cell test fixture.

Fig. 7
Fig. 7

Measured reflectivity/absorbance of the tunable reflector as perfect absorber when the power supplies are off.

Fig. 8
Fig. 8

Measured reflectivity results of active magnetic metamaterials, when V1 = 0V, 3V, 5V, 6V, 9V (i.e. phase shift≈0°, 45°, 90°, 135°, 180°) and V2 = 2V, 3V, 6V, 12V (i.e. attenuation≈13dB, 10dB, 5dB, 2dB).

Fig. 9
Fig. 9

Measured reflectivity that varies from −20dB to + 15dB.

Fig. 10
Fig. 10

Equivalent circuit of the active magnetic metamaterial cell.

Fig. 11
Fig. 11

S-matrix network of simulation model of the tunable reflector with n unit cells of active magnetic metamaterials on metallic surface.

Fig. 12
Fig. 12

Numerically simulated reflectivity of the tunable reflector.

Equations (6)

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i s = V s,inc V sd jωX+ Z in ' , i d = i d1 + i d2 = G eqv ( V s,inc V sd ) jωX + V d,inc V ds jωX+ Z out ' ,
a mm = S( i s + i d ) μ 0 H inc ( ω 0 2 ω 2 1 ) 1 ,
μ r = χ m 1= μ 0 a mm V c 1= S( i s + i d ) H inc V c ( ω 0 2 ω 2 1 ) 1 1,
μ r '' = μ 0 S 2 V c Z 0 ω 2 X 2 + Z 0 2 ( 2ω A r ω 2 cos(θ)+ A r Z 0 2X sin(θ) ) ( ω 0 2 ω 2 1 ) 1
[ b 1 b 2 b 2 n + 1 ] = ( S 11 S 12 S 1 , 2 n + 1 S 21 S 21 S 2 , 2 n + 1 S 2 n + 1 , 1 S 2 n + 1 , 2 S 2 n + 1 , 2 n + 1 ) [ a 1 a 2 a 2 n + 1 ]
[ a 1 a 2 a 2 n ] = [ b 1 ' b 2 ' b 2 n ' ] = ( S 11 ' S 12 ' S 1 , 2 n ' S 21 ' S 22 ' S 2 , 2 n ' S 2 n , 1 ' S 2 n , 2 ' S 2 n , 2 n ' ) [ a 1 ' a 2 ' a 2 n ' ] = [ S ] 2 n + 1 , 2 n + 1 [ b 1 b 2 b 2 n ]

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