Abstract

A circular polarization selective microcavity using chiral photonic metamaterials is designed, in which photons of a particular circular polarization dominate in stimulated emissions as a result of cavity resonances. The selection behavior originates from the special chiral reflector, which exhibits two elliptical polarization eigenstates almost identical to the same circular polarization. Theoretical analysis by using Jones matrix is given to explain this interesting phenomenon in detail. A lasing mode with an almost perfect circularly polarized field is present inside this cavity and observable at the output.

© 2013 Optical Society of America

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    [CrossRef]

2012 (1)

2011 (2)

Y. Ye, X. Li, F. Zhuang, and S. W. Chang, “Homogeneous circular polarizers using a bilayered chiral metamaterial,” Appl. Phys. Lett. 99, 031111 (2011).
[CrossRef]

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly-polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef]

2010 (2)

Y. Ye and S. He, “90° polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96, 203501 (2010).
[CrossRef]

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: a numerical parameter study,” Opt. Express 18, 1059–1069 (2010).
[CrossRef]

2009 (4)

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[CrossRef]

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B 79, 035321 (2009).
[CrossRef]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[CrossRef]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[CrossRef]

2007 (1)

2006 (1)

A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. 97, 177401 (2006).
[CrossRef]

2002 (1)

M. Troccoli, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Mid-infrared (λ≈7.4  μm) quantum cascade laser amplifier for high power single-mode emission and improved beam quality,” Appl. Phys. Lett. 80, 4103–4105 (2002).
[CrossRef]

2001 (1)

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

2000 (1)

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

1998 (1)

1993 (1)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

1992 (1)

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

1983 (1)

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Alexander, R. W.

Arakawa, Y.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly-polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef]

Awschalom, D. D.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Bade, K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[CrossRef]

Balandin, A.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Bell, R. J.

Bell, R. R.

Bell, S. E.

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Buhrman, R. A.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Burger, S.

Capasso, F.

M. Troccoli, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Mid-infrared (λ≈7.4  μm) quantum cascade laser amplifier for high power single-mode emission and improved beam quality,” Appl. Phys. Lett. 80, 4103–4105 (2002).
[CrossRef]

Chang, S. W.

S. W. Chang, “Intra-cavity stimulated emissions of photons in almost pure spin states without imposed nonreciprocity,” Opt. Express 20, 2516–2527 (2012).
[CrossRef]

Y. Ye, X. Li, F. Zhuang, and S. W. Chang, “Homogeneous circular polarizers using a bilayered chiral metamaterial,” Appl. Phys. Lett. 99, 031111 (2011).
[CrossRef]

Cho, A. Y.

M. Troccoli, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Mid-infrared (λ≈7.4  μm) quantum cascade laser amplifier for high power single-mode emission and improved beam quality,” Appl. Phys. Lett. 80, 4103–4105 (2002).
[CrossRef]

Chtchelkanova, A. Y.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Daughton, J. M.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Decker, M.

Divincenzo, D.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Dong, J.

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[CrossRef]

Fan, B.

Fedotov, V. A.

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[CrossRef]

A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. 97, 177401 (2006).
[CrossRef]

Gansel, J. K.

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: a numerical parameter study,” Opt. Express 18, 1059–1069 (2010).
[CrossRef]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[CrossRef]

Genack, A. Z.

Gmachl, C.

M. Troccoli, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Mid-infrared (λ≈7.4  μm) quantum cascade laser amplifier for high power single-mode emission and improved beam quality,” Appl. Phys. Lett. 80, 4103–4105 (2002).
[CrossRef]

He, S.

Y. Ye and S. He, “90° polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96, 203501 (2010).
[CrossRef]

Iwamoto, S.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly-polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef]

Jiang, H. W.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Joannopoulos, J. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Klein, M. W.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (EMW, 2005).

Konishi, K.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly-polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef]

Kopp, V. I.

Koschny, T.

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[CrossRef]

Kriegler, C. E.

Kumagai, N.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly-polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef]

Kuwata-Gonokami, M.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly-polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef]

Lederer, F.

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B 79, 035321 (2009).
[CrossRef]

Li, X.

Y. Ye, X. Li, F. Zhuang, and S. W. Chang, “Homogeneous circular polarizers using a bilayered chiral metamaterial,” Appl. Phys. Lett. 99, 031111 (2011).
[CrossRef]

Linden, S.

Long, L. L.

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Menzel, C.

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B 79, 035321 (2009).
[CrossRef]

Mor, T.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Nomura, M.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly-polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011).
[CrossRef]

Ordal, M. A.

Paul, T.

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B 79, 035321 (2009).
[CrossRef]

Pikus, G. E.

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Plum, E.

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[CrossRef]

Rockstuhl, C.

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B 79, 035321 (2009).
[CrossRef]

Rogacheva, A. V.

A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. 97, 177401 (2006).
[CrossRef]

Roukes, M. L.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Roychowdhury, V.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Ruther, M.

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[CrossRef]

Schwanecke, A. S.

A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. 97, 177401 (2006).
[CrossRef]

Shelankov, A. L.

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Sivco, D. L.

M. Troccoli, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Mid-infrared (λ≈7.4  μm) quantum cascade laser amplifier for high power single-mode emission and improved beam quality,” Appl. Phys. Lett. 80, 4103–4105 (2002).
[CrossRef]

Soukoulis, C. M.

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[CrossRef]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[CrossRef]

Thiel, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[CrossRef]

Treger, D. M.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Troccoli, M.

M. Troccoli, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, “Mid-infrared (λ≈7.4  μm) quantum cascade laser amplifier for high power single-mode emission and improved beam quality,” Appl. Phys. Lett. 80, 4103–4105 (2002).
[CrossRef]

Vithana, H. K. M.

von Freymann, G.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[CrossRef]

von Molnr, S.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Vrijen, R.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Wang, K.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Ward, C. A.

Wegener, M.

Wolf, S. A.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294, 1488–1495 (2001).
[CrossRef]

Yablonovitch, E.

R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. Divincenzo, “Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures,” Phys. Rev. A 62, 012306 (2000).
[CrossRef]

Ye, Y.

Y. Ye, X. Li, F. Zhuang, and S. W. Chang, “Homogeneous circular polarizers using a bilayered chiral metamaterial,” Appl. Phys. Lett. 99, 031111 (2011).
[CrossRef]

Y. Ye and S. He, “90° polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96, 203501 (2010).
[CrossRef]

Zheludev, N. I.

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[CrossRef]

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Computer Simulation Technology (CST), http://www.cst.com .

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Figures (4)

Fig. 1.
Fig. 1.

(a) Top view of the two-dimensional square array of gold helices. (b) The single right-handed gold helix in a unit cell. The axes indicate the polarization and the propagation directions of the incident wave. (c) The corresponding spectra for the magnitudes of reflection coefficients. The gray region denotes the stop band of the gold helix array.

Fig. 2.
Fig. 2.

(a) Unit cell of the composite chiral reflector composed of the BR and gold helix array. (b) The photonic band structure of the BR (without gold helix array). (c) The magnitudes of reflection coefficients for the composite chiral reflector. The yellow regions in (b) and (c) denote the photonic bandgap.

Fig. 3.
Fig. 3.

(a) Schematic diagram of the multiple reflections in the microcavity. (b) The magnitude and (c) phase of the reflection coefficients r⃖2,++ and r⃖1. The vertical construction line in (b) indicates the resonant frequency.

Fig. 4.
Fig. 4.

|E|-field pattern of the mode 1 in xz plane. The red arrows represent a snapshot of the electric field directions in different positions. The black arrows denote the rotation directions.

Equations (14)

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rCP=(r++r+r+r),
w1,2=|r+||r+|+|r+|(±r+/r+1),
Λ1,2=r++±r+r+,
|Λ1,2|=|r++±r+r+||r++|+|r+r+|=|r++|+|r+||r+|0.25.
|Λ1,2|=|r++±r+r+||r++||r+r+|=|r++||r+||r+|0.85.
Er=R⃖totalE0=R⃖1E0+T⃖1PR⃖2PT⃗1E0+T⃖1PR⃖2P(R⃗1PR⃖2P)T⃗1E0+T⃖1PR⃖2P(R⃗1PR⃖2P)2T⃗1E0+T⃖1PR⃖2P(R⃗1PR⃖2P)3T⃗1E0+,
R⃖total=R⃖1+T⃖1PR⃖2P(I+M+M2+)T⃗1=R⃖1+T⃖1PR⃖2P(IM)1T⃗1,
R⃖2=(r⃖2,++0r⃖2,+r⃖2,++).
T⃖1=(t⃖100t⃖1)=t⃖1I,R⃗1=r⃗1I,R⃖1=r⃖1I.
R⃖total=(r⃖total,++r⃖total,+r⃖total,+r⃖total,)=(r⃖1+e2iβlr⃖2,++t⃗1t⃖1(1e2iβlr⃖2,++r⃗1)0e2iβlr⃖2,+t⃗1t⃖1(1e2iβlr⃖2,++r⃗1)2r⃖1+e2iβlr⃖2,++t⃗1t⃖1(1e2iβlr⃖2,++r⃗1)).
r⃗1r⃖1t⃗1t⃖1=r⃗1/r⃖1*=r⃖1/r⃗1*=ei(ϕ1+ϕ1),
r⃖total,++=r⃖total,=r⃖1+e2iβlr⃖2,++t⃗1t⃖1e2iβlr⃖2,++r⃗1r⃖1(1e2iβlr⃖2,++r⃗1)=r⃖1e2iβlr⃖2,++(r⃗1r⃖1t⃗1t⃖1)(1e2iβlr⃖2,++r⃗1)=eiϕ1(|r⃖1||r⃖2,++|ei(ϕ1+ϕ2+2βl))(1|r⃖2,++||r⃗1|ei(ϕ1+ϕ2+2βl)),
R⃖total=r⃖total,+(0010).
Mφ=R⃗1PR⃖2Pφ=r⃗1e2iklr⃖2φ=φ{|r⃖2,++||r⃗1|eαl=1ϕ1+ϕ2+2βl=2π.

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