Abstract

Metal-helix based metamaterials have been introduced as compact and broadband circular polarizers. However, the end of the metal wire together with the helix center defines an axis in space, which unavoidably breaks the rotational symmetry at the metamaterial surface. This introduces linear birefringence. Symmetry can be recovered by considering an integer number, e.g. N = 4, of intertwined helices arranged to a square array. We show that the operation principles are fundamentally different though. Metamaterial circular polarizers based on N = 4 helices, unlike single helices, inherently require absorption of the constituent metal. Otherwise, the combination of a four-fold rotational axis and time-inversion symmetry strictly forbids circular-polarizer action. Our symmetry analysis is confirmed by extensive numerical calculations comparing results for perfect electric conductors with those for a free-electron Drude metal with finite damping.

© 2012 OSA

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References

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  1. 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,” Science325(5947), 1513–1515 (2009).
    [CrossRef] [PubMed]
  2. J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: A numerical parameter study,” Opt. Express18(2), 1059–1069 (2010).
    [CrossRef] [PubMed]
  3. C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett.105(24), 247401 (2010).
    [CrossRef] [PubMed]
  4. M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bichiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
    [CrossRef]
  5. A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
    [CrossRef] [PubMed]
  6. J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
    [CrossRef]
  7. M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett.32(17), 2547–2549 (2007).
    [CrossRef] [PubMed]
  8. M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett.35(10), 1593–1595 (2010).
    [CrossRef] [PubMed]
  9. Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun3, 870 (2012).
    [CrossRef] [PubMed]
  10. Z. Y. Yang, M. Zhao, P. X. Lu, and Y. F. Lu, “Ultrabroadband optical circular polarizers consisting of double-helical nanowire structures,” Opt. Lett.35(15), 2588–2590 (2010).
    [CrossRef] [PubMed]
  11. Z. Yang, M. Zhao, and P. Lu, “Improving the signal-to-noise ratio for circular polarizers consisting of helical metamaterials,” Opt. Express19(5), 4255–4260 (2011).
    [CrossRef] [PubMed]
  12. J. D. Kraus and R. Marhefka, Antennas: For All Applications, 3rd ed. (McGraw-Hill, 2003).
  13. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys.67(5), 717–754 (2004).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).
  15. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).
  16. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).
  17. C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron.16(2), 367–375 (2010).
    [CrossRef]
  18. J. Fischer and M. Wegener, “Three-dimensional optical laser lithography beyond the diffraction limit,” Laser Photon. Rev. doi: 10.1002/lpor.201100046.
    [CrossRef]

2012

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun3, 870 (2012).
[CrossRef] [PubMed]

2011

A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
[CrossRef] [PubMed]

Z. Yang, M. Zhao, and P. Lu, “Improving the signal-to-noise ratio for circular polarizers consisting of helical metamaterials,” Opt. Express19(5), 4255–4260 (2011).
[CrossRef] [PubMed]

2010

2009

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bichiral photonic crystals,” Adv. Mater. 21, 4680–4682 (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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

2007

2004

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys.67(5), 717–754 (2004).
[CrossRef]

Alù, A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun3, 870 (2012).
[CrossRef] [PubMed]

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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Belkin, M. A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun3, 870 (2012).
[CrossRef] [PubMed]

Braun, P. V.

A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
[CrossRef] [PubMed]

Burger, S.

Chan, C. T.

C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett.105(24), 247401 (2010).
[CrossRef] [PubMed]

Decker, M.

M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett.35(10), 1593–1595 (2010).
[CrossRef] [PubMed]

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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Frölich, A.

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

Gansel, J. K.

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: A numerical parameter study,” Opt. Express18(2), 1059–1069 (2010).
[CrossRef] [PubMed]

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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Giessen, H.

A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
[CrossRef] [PubMed]

Gissibl, T.

A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
[CrossRef] [PubMed]

Kaschke, J.

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

Klotzbücher, T.

A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
[CrossRef] [PubMed]

Kriegler, C. E.

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron.16(2), 367–375 (2010).
[CrossRef]

Latzel, M.

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

Li, H.

C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett.105(24), 247401 (2010).
[CrossRef] [PubMed]

Linden, S.

M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett.35(10), 1593–1595 (2010).
[CrossRef] [PubMed]

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: A numerical parameter study,” Opt. Express18(2), 1059–1069 (2010).
[CrossRef] [PubMed]

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron.16(2), 367–375 (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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Lu, P.

Lu, P. X.

Lu, Y. F.

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys.67(5), 717–754 (2004).
[CrossRef]

Radke, A.

A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
[CrossRef] [PubMed]

Rill, M. S.

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron.16(2), 367–375 (2010).
[CrossRef]

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bichiral photonic crystals,” Adv. Mater. 21, 4680–4682 (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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Soukoulis, C. M.

Thiel, M.

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bichiral photonic crystals,” Adv. Mater. 21, 4680–4682 (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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett.32(17), 2547–2549 (2007).
[CrossRef] [PubMed]

von Freymann, G.

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bichiral photonic crystals,” Adv. Mater. 21, 4680–4682 (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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett.32(17), 2547–2549 (2007).
[CrossRef] [PubMed]

Wegener, M.

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett.35(10), 1593–1595 (2010).
[CrossRef] [PubMed]

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: A numerical parameter study,” Opt. Express18(2), 1059–1069 (2010).
[CrossRef] [PubMed]

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron.16(2), 367–375 (2010).
[CrossRef]

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bichiral photonic crystals,” Adv. Mater. 21, 4680–4682 (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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett.32(17), 2547–2549 (2007).
[CrossRef] [PubMed]

Wei, Z.

C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett.105(24), 247401 (2010).
[CrossRef] [PubMed]

Wu, C.

C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett.105(24), 247401 (2010).
[CrossRef] [PubMed]

Yang, Z.

Yang, Z. Y.

Yu, X.

C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett.105(24), 247401 (2010).
[CrossRef] [PubMed]

Zhao, M.

Zhao, R.

Zhao, Y.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun3, 870 (2012).
[CrossRef] [PubMed]

Adv. Mater.

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bichiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[CrossRef]

Adv. Mater. (Deerfield Beach Fla.)

A. Radke, T. Gissibl, T. Klotzbücher, P. V. Braun, and H. Giessen, “Three-dimensional bichiral plasmonic crystals fabricated by direct laser writing and electroless silver plating,” Adv. Mater. (Deerfield Beach Fla.)23(27), 3018–3021 (2011).
[CrossRef] [PubMed]

Appl. Phys. Lett.

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett.100(10), 101109 (2012).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Electron.16(2), 367–375 (2010).
[CrossRef]

Nat Commun

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun3, 870 (2012).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

C. Wu, H. Li, Z. Wei, X. Yu, and C. T. Chan, “Theory and experimental realization of negative refraction in a metallic helix array,” Phys. Rev. Lett.105(24), 247401 (2010).
[CrossRef] [PubMed]

Rep. Prog. Phys.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys.67(5), 717–754 (2004).
[CrossRef]

Science

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,” Science325(5947), 1513–1515 (2009).
[CrossRef] [PubMed]

Other

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999).

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

J. D. Kraus and R. Marhefka, Antennas: For All Applications, 3rd ed. (McGraw-Hill, 2003).

J. Fischer and M. Wegener, “Three-dimensional optical laser lithography beyond the diffraction limit,” Laser Photon. Rev. doi: 10.1002/lpor.201100046.
[CrossRef]

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Figures (3)

Fig.
       1
Fig. 1

Illustration of the unit cells of helix-metamaterial geometries considered in this work. In each case, top view (left) and oblique view (right) are depicted. (a) Single (N = 1) left-handed metal helices with axial period a arranged to a square array with lateral lattice constant a. The metal wire diameter is d=0.1×a, the helix diameter is D=0.6×a. Note that the end of the metal wire together with the center axis of the helices defines an axis in space. This leads to a one-fold rotational axis. (b) Similarly arranged N = 4 intertwined helices recover a four-fold rotational axis compatible with the square-array symmetry. (c) Arrangement of single helices like in (a) effectively recovering four-fold rotational symmetry of the overall structure by laterally displacing the four helices from (b) within one new unit cell with lattice constant 2a.

Fig. 2
Fig. 2

Calculated normal-incidence intensity transmittance (solid), reflectance (solid), and conversion spectra (dashed). Incident left-handed circular polarization (LCP) is shown in red, incident right-handed circular polarization (RCP) in blue. In cases where no blue curve is visible, the blue curve is identical to the red one to within the curve linewidth. The left-handed N = 1 (left column, i.e., (a) and (c)) and N = 4 (right column, i.e., (b) and (d)) structures are defined in Fig. 1(a) and Fig. 1(b), respectively. The insets at the top repeat the top view onto a unit cell. For the top row of the overall 2×2 matrix (i.e., (a) and (b)), the metal is treated as a lossless perfect electric conductor (PEC), for the bottom row (i.e., (c) and (d)) as a free-electron Drude model with finite damping/losses (gold parameters).

Fig. 3
Fig. 3

As Fig. 2(d), but for more than one helix pitch along the helix axis as indicated. (a) same as Fig. 2(d) with 1 pitch, (b) 2 pitches, (c) 3 pitches, and (d) 6 pitches. Note that all conversions (dashed) are strictly zero. In reflection, no blue solid curves are visible because they are identical to the red ones to within the curve linewidth. The equal gray areas in (a)-(d) highlight the frequency interval for which broadband circular-polarizer action is observed for several axial pitches.

Equations (13)

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

E r = r E i =( r xx r xy r yx r yy ) E i
r =( r xx r xy r xy r xx )
S = 1 2 ( 1 i 1 i )
r circ = S   r   S 1 =( r xx + r xy 0 0 r xx r xy )=( r RCP LCP 0 0 r LCP RCP ).
E t,LCP = t LCP LCP   E i,LCP    and    E r,RCP = t RCP LCP   E i,LCP .
0= t RCP RCP r RCP LCP * +r ' RCP LCP t LCP LCP *
1= r LCP RCP r RCP LCP * +t ' LCP LCP t LCP LCP * .
1=r ' RCP LCP r ' RCP LCP * +t ' LCP LCP t ' LCP LCP * .
| t LCP LCP | 2 | t RCP RCP | 2 | r RCP LCP | 2 = | 1 r LCP RCP r RCP LCP * | 2 .
| t RCP RCP | 2 | t LCP LCP | 2 | r LCP RCP | 2 = | 1 r RCP LCP r LCP RCP * | 2 .
| t LCP LCP | 2 | t RCP RCP | 2 | r RCP LCP | 2 = | t RCP RCP | 2 | t LCP LCP | 2 | r LCP RCP | 2 .
| t LCP LCP |=| t RCP RCP |
| r RCP LCP |=| r LCP RCP |.

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