Abstract

The propagation of light guided in optical fibers is affected in different ways by bending or twisting. Here we treat the polarization properties of twisted six-fold symmetric photonic crystal fibers. Using a coordinate frame that follows the twisting structure, we show that the governing equation for the fiber modes resembles the Pauli equation for electrons in weak magnetic fields. This implies index splitting between left and right circularly polarized modes, which are degenerate in the untwisted fiber. We develop a theoretical model, based on perturbation theory and symmetry properties, to predict the observable circular birefringence (i.e., optical activity) associated with this splitting. Our overall conclusion is that optical activity requires the rotational symmetry to be broken so as to allow coupling between different total angular momentum states.

© 2013 Optical Society of America

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References

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  1. G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
    [CrossRef]
  2. W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
    [CrossRef]
  3. X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
    [CrossRef]
  4. X. Ma, C. H. Liu, G. Chang, and A. Galvanauskas, “Angular-momentum coupled optical waves in chirally coupled core fibers,” Opt. Express 19, 26515–26528 (2011).
    [CrossRef]
  5. C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78, 043828 (2008).
    [CrossRef]
  6. C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
    [CrossRef]
  7. M. Ornigotti, G. Della Valle, D. Gatti, and S. Longhi, “Topological suppression of optical tunneling in a twisted fiber,” Phys. Rev. A 76, 023833 (2007).
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    [CrossRef]
  14. C. Cohen-Tannoudji, Quantum Mechanics (Hermann, 1991).
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    [CrossRef]
  16. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
    [CrossRef]
  17. A. Nicolet, F. Zolla, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel 27, 806–819 (2008).
    [CrossRef]
  18. C. N. Alexeyev and M. A. Yavorsky, “Optical vortices and the higher-order modes of twisted strongly elliptical optical fibers,” J. Opt. A 6, 824–832 (2004).
    [CrossRef]
  19. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2002).
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2013 (1)

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

2012 (1)

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

2011 (2)

X. Ma, C. H. Liu, G. Chang, and A. Galvanauskas, “Angular-momentum coupled optical waves in chirally coupled core fibers,” Opt. Express 19, 26515–26528 (2011).
[CrossRef]

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

2009 (2)

W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
[CrossRef]

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45, 1192–1200 (2009).
[CrossRef]

2008 (2)

A. Nicolet, F. Zolla, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel 27, 806–819 (2008).
[CrossRef]

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78, 043828 (2008).
[CrossRef]

2007 (1)

M. Ornigotti, G. Della Valle, D. Gatti, and S. Longhi, “Topological suppression of optical tunneling in a twisted fiber,” Phys. Rev. A 76, 023833 (2007).
[CrossRef]

2004 (1)

C. N. Alexeyev and M. A. Yavorsky, “Optical vortices and the higher-order modes of twisted strongly elliptical optical fibers,” J. Opt. A 6, 824–832 (2004).
[CrossRef]

2003 (1)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

2002 (1)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2002).
[CrossRef]

2001 (1)

1997 (1)

1986 (1)

1984 (1)

1981 (1)

1979 (1)

1976 (1)

Aghaie, K. Z.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45, 1192–1200 (2009).
[CrossRef]

Alexeyev, C. N.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78, 043828 (2008).
[CrossRef]

C. N. Alexeyev and M. A. Yavorsky, “Optical vortices and the higher-order modes of twisted strongly elliptical optical fibers,” J. Opt. A 6, 824–832 (2004).
[CrossRef]

Barlow, A. J.

Barnett, S. M.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2002).
[CrossRef]

Biancalana, F.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

Birks, T. A.

Botten, L. C.

Chang, G.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, Quantum Mechanics (Hermann, 1991).

Conti, C.

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

Dangui, V.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45, 1192–1200 (2009).
[CrossRef]

de Sterke, C. M.

Della Valle, G.

M. Ornigotti, G. Della Valle, D. Gatti, and S. Longhi, “Topological suppression of optical tunneling in a twisted fiber,” Phys. Rev. A 76, 023833 (2007).
[CrossRef]

Digonnet, M. J. F.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45, 1192–1200 (2009).
[CrossRef]

Fadeyeva, T. A.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

Fan, S.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45, 1192–1200 (2009).
[CrossRef]

Fleming, J. W.

Galvanauskas, A.

Gatti, D.

M. Ornigotti, G. Della Valle, D. Gatti, and S. Longhi, “Topological suppression of optical tunneling in a twisted fiber,” Phys. Rev. A 76, 023833 (2007).
[CrossRef]

Guenneau, S.

A. Nicolet, F. Zolla, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel 27, 806–819 (2008).
[CrossRef]

Kang, M. S.

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

Kino, G. S.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45, 1192–1200 (2009).
[CrossRef]

Knight, J. C.

Lapin, B. P.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

Lee, H. W.

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

Lee, Y. L.

W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
[CrossRef]

Li, L.

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

Liu, C. H.

Longhi, S.

M. Ornigotti, G. Della Valle, D. Gatti, and S. Longhi, “Topological suppression of optical tunneling in a twisted fiber,” Phys. Rev. A 76, 023833 (2007).
[CrossRef]

Love, J.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall1983).

Ma, X.

McPhedran, R. C.

Nicolet, A.

A. Nicolet, F. Zolla, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel 27, 806–819 (2008).
[CrossRef]

Noh, Y. C.

W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
[CrossRef]

Oh, K.

W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
[CrossRef]

Ornigotti, M.

M. Ornigotti, G. Della Valle, D. Gatti, and S. Longhi, “Topological suppression of optical tunneling in a twisted fiber,” Phys. Rev. A 76, 023833 (2007).
[CrossRef]

Padgett, M. J.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

Payne, D. N.

Ramskovhansen, J. J.

Russell, P. S. J.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef]

Sammut, R.

Shin, W.

W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
[CrossRef]

Simon, A.

Snyder, A. W.

Steel, M.

Synder, A. W.

Ulrich, R.

Weiss, T.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

White, T. P.

Wong, G. K.

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

Wong, G. K. L.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

Xi, X. M.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

Yavorsky, M. A.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78, 043828 (2008).
[CrossRef]

C. N. Alexeyev and M. A. Yavorsky, “Optical vortices and the higher-order modes of twisted strongly elliptical optical fibers,” J. Opt. A 6, 824–832 (2004).
[CrossRef]

Yu, B. A.

W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
[CrossRef]

Zheng, X. H.

Zolla, F.

A. Nicolet, F. Zolla, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel 27, 806–819 (2008).
[CrossRef]

Appl. Opt. (4)

Compel (1)

A. Nicolet, F. Zolla, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel 27, 806–819 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45, 1192–1200 (2009).
[CrossRef]

J. Opt. A (2)

C. N. Alexeyev and M. A. Yavorsky, “Optical vortices and the higher-order modes of twisted strongly elliptical optical fibers,” J. Opt. A 6, 824–832 (2004).
[CrossRef]

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

J. Opt. B (1)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282, 3456–3459 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (3)

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78, 043828 (2008).
[CrossRef]

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

M. Ornigotti, G. Della Valle, D. Gatti, and S. Longhi, “Topological suppression of optical tunneling in a twisted fiber,” Phys. Rev. A 76, 023833 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. 110, 143903 (2013).
[CrossRef]

Science (1)

G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Topological excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446 (2012).
[CrossRef]

Other (2)

C. Cohen-Tannoudji, Quantum Mechanics (Hermann, 1991).

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall1983).

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Figures (5)

Fig. 1.
Fig. 1.

Effective index splitting of the fundamental core modes calculated for an ESM PCF in the twisted coordinate frame. The detailed fiber parameters are given in the text. The continuous blue curves denote the numerical calculations for RC polarized modes, the red curves for LC modes, and the dots are the results of perturbation theory. The insets show the normalized z component of the spin angular momentum density. A schematic of the twisted ESM fiber can be seen in the inset on the right-hand side; note that the twist rate is exaggerated for clarity.

Fig. 2.
Fig. 2.

Normalized z component of the Poynting vector distribution for the different total angular momentum contributions of order j+6m in the case of the predominantly LC polarized fundamental core mode (j=1) of an untwisted ESM fiber. Each row represents a different value of m. The columns show (from left to right) the overall intensity for order m and the corresponding LC and RC polarized contributions. Each panel denotes an area of 10μm×10μm centered around the fiber core. Note that more than 98% of the modal power is carried in the LC polarized m=0 contribution.

Fig. 3.
Fig. 3.

Normalized power of different total angular momentum contributions to the predominantly LC polarized core mode of an ESM PCF. The dominant contribution belongs to j=1, which is also the azimuthal order of this mode. Total angular momentum states with order j+6m contribute less significantly with increasing magnitude of the integer values m, whereas all other contributions are basically numerical noise.

Fig. 4.
Fig. 4.

Circular birefringence of the fundamental core modes in an ESM fiber. Red squares denote experimental data, green solid line shows numerical calculations, and black dots depict the perturbation theory. The deviation between perturbation theory and numerical calculations is illustrated by the blue solid line corresponding to the right axis. The agreement between theory, numerical data, and experiment is excellent.

Fig. 5.
Fig. 5.

Effective index neffα of the dominant total angular momentum contributions to the fundamental core modes of a twisted ESM fiber calculated in the laboratory frame, displayed as the deviation from the effective index neff0 of the untwisted fiber. The labeling of the lines is as for the twisted coordinate frame in Fig. 1. Note the dominant α quadratic behavior in contrast to the linear behavior of the effective indices in the twisted coordinate frame. The linear term becomes important only for very small twist rates (inset) and in calculating the circular birefringence of the fiber, when the quadratic terms of LC and RC polarized modes cancel out. The quadratic term has been included in the theory curve by fitting the quadratic order to the numerical data instead of carrying out a higher-order perturbation scheme.

Equations (15)

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

iFz=M0(ρ,ϕ)F.
J=L+S=iϕ+(1001).
exp(iαzJ)M0(ρ,ϕ+αz)exp(iαzJ)=M0(ρ,ϕ).
iGαζ=[M0(ρ,θ)αJ]Gα.
κ0g0=M0g0.
g˜|iγ|g=c16πdAg˜iγg,
κακ0αJ.
g0j|iγJ|g0±j=g0j|exp(iJπ/3)iγJexp(iJπ/3)|g0±j=e±2ijπ/3g0j|iγJ|g0±j=0.
(Gα,LCjGα,RCj)=eiκαjζm[ei(j+6m1)θgα,LCjm(ρ)ei(j+6m+1)θgα,RCjm(ρ)].
(Fα,LCjFα,RCj)=exp(iαzJ)(Gα,LCjGα,RCj)=eiκαjzm[ei(j+6m)(ϕ+αziϕ)gα,LCjm(ρ)ei(j+6m)(ϕ+αz+iϕ)gα,RCjm(ρ)].
βαjm=καj+α(j+6m)=κ0jα(Jj6m)+O(α2).
BC=nRCnLC=α(Jj)λ/π.
M0=ik0(μ00ε)γ3+12k0(ε1T00μ1T)γ2.
γ2=(000i00i00i00i000),γ3=(0010000110000100).
L=c16πdA(HLC*ELCϕ+HRC*ERCϕ+ELC*HLCϕERC*HRCϕ),S=c8πdAIm(HLC*ELC+HRC*ERC),J=L+S.

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