Abstract

We demonstrate that a combination of multipole and Bloch methods is well suited for calculating the modes of air core photonic crystal fibers. This includes determining the reflective properties of the cladding, which is a prerequisite for the modal calculations. We demonstrate that in the presence of absorption, the modal losses can be substantially smaller than in the corresponding bulk medium.

© 2001 Optical Society of America

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References

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  1. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic bandgap guidance in optical fibres,” Science 282, 1476–1478 (1998).
    [Crossref] [PubMed]
  2. J. Broeng, S.E. Barkou, T. Sondergaard, and A. Bjarklev, “Analysis of air-guiding photonic band gap fibres,” Opt. Lett. 21, 1547–1549(2000).
  3. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fibre Technol. 6, 181–191 (2000).
    [Crossref]
  4. A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
    [Crossref]
  5. Tanya M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennet, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50–56 (2000).
    [Crossref]
  6. B.J. Eggleton, P.S. Westbrook, C.A. White, C. Kerbage, R.S. Windeler, and G.L. Burdge, “Claading-modd-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084–1100 (2000).
    [Crossref]
  7. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers,” submitted J. Opt. Soc. Am. B.
  8. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, “Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 1: formulation,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
    [Crossref]
  9. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, “Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 2: properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
    [Crossref]
  10. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am A 17, 1012 (2000)
    [Crossref]
  11. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
    [Crossref]
  12. T.P White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. 26, 488–490 (2001).
    [Crossref]
  13. R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and Bao Ke-Da, “Green’s functions, lattice sums, and Rayleigh’s Identity for a dynamic scattering problem”, ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)
  14. N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135 (1995)
    [Crossref]
  15. R.C. McPhedran, N.A. Nicorovici, L.C. Botten, and K.A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808 (2000).
    [Crossref]
  16. G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical diffraction,” in preparation for Phys. Rev. E.
  17. F. W. J. Olver, “Bessel Functions of Integer Order,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1972), pp. 355–433.

2001 (2)

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
[Crossref]

T.P White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. 26, 488–490 (2001).
[Crossref]

2000 (9)

R.C. McPhedran, N.A. Nicorovici, L.C. Botten, and K.A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808 (2000).
[Crossref]

J. Broeng, S.E. Barkou, T. Sondergaard, and A. Bjarklev, “Analysis of air-guiding photonic band gap fibres,” Opt. Lett. 21, 1547–1549(2000).

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fibre Technol. 6, 181–191 (2000).
[Crossref]

A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
[Crossref]

Tanya M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennet, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50–56 (2000).
[Crossref]

B.J. Eggleton, P.S. Westbrook, C.A. White, C. Kerbage, R.S. Windeler, and G.L. Burdge, “Claading-modd-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084–1100 (2000).
[Crossref]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, “Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 1: formulation,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[Crossref]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, “Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 2: properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[Crossref]

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am A 17, 1012 (2000)
[Crossref]

1998 (1)

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic bandgap guidance in optical fibres,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

1995 (1)

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135 (1995)
[Crossref]

Andrés, M.V.

A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
[Crossref]

Andrés, P.

A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
[Crossref]

Asatryan, A. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
[Crossref]

Asatryan, A.A.

Barkou, S.E.

Bennet, P.J.

Birks, T.A.

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic bandgap guidance in optical fibres,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Bjarklev, A.

Botten, L. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
[Crossref]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135 (1995)
[Crossref]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical diffraction,” in preparation for Phys. Rev. E.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and Bao Ke-Da, “Green’s functions, lattice sums, and Rayleigh’s Identity for a dynamic scattering problem”, ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)

Botten, L.C.

Brechet, F.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fibre Technol. 6, 181–191 (2000).
[Crossref]

Broderick, N.G.R.

Broeng, J.

J. Broeng, S.E. Barkou, T. Sondergaard, and A. Bjarklev, “Analysis of air-guiding photonic band gap fibres,” Opt. Lett. 21, 1547–1549(2000).

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic bandgap guidance in optical fibres,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Burdge, G.L.

de Sterke, C. M.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
[Crossref]

de Sterke, C.M.

T.P White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. 26, 488–490 (2001).
[Crossref]

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers,” submitted J. Opt. Soc. Am. B.

Eggleton, B.J.

Enoch, S.

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am A 17, 1012 (2000)
[Crossref]

Ferrando, A.

A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
[Crossref]

Gralak, B.

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am A 17, 1012 (2000)
[Crossref]

Grubits, K.A.

R.C. McPhedran, N.A. Nicorovici, L.C. Botten, and K.A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808 (2000).
[Crossref]

Ke-Da, Bao

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and Bao Ke-Da, “Green’s functions, lattice sums, and Rayleigh’s Identity for a dynamic scattering problem”, ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)

Kerbage, C.

Knight, J.C.

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic bandgap guidance in optical fibres,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Kuhlmey, B.T.

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers,” submitted J. Opt. Soc. Am. B.

Marcou, J.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fibre Technol. 6, 181–191 (2000).
[Crossref]

Martijn de Sterke, C.

Maystre, D.

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers,” submitted J. Opt. Soc. Am. B.

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
[Crossref]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135 (1995)
[Crossref]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical diffraction,” in preparation for Phys. Rev. E.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and Bao Ke-Da, “Green’s functions, lattice sums, and Rayleigh’s Identity for a dynamic scattering problem”, ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)

McPhedran, R.C.

Miret, J.J.

A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
[Crossref]

Monro, Tanya M.

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
[Crossref]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135 (1995)
[Crossref]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and Bao Ke-Da, “Green’s functions, lattice sums, and Rayleigh’s Identity for a dynamic scattering problem”, ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical diffraction,” in preparation for Phys. Rev. E.

Nicorovici, N.A.

Olver, F. W. J.

F. W. J. Olver, “Bessel Functions of Integer Order,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1972), pp. 355–433.

Pagnoux, D.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fibre Technol. 6, 181–191 (2000).
[Crossref]

Renversez, G.

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers,” submitted J. Opt. Soc. Am. B.

Richardson, D.J.

Robinson, P.A.

Roy, P.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fibre Technol. 6, 181–191 (2000).
[Crossref]

Russell, P.St.J.

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic bandgap guidance in optical fibres,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Silvestre, E.

A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
[Crossref]

Smith, G. H.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical diffraction,” in preparation for Phys. Rev. E.

Sondergaard, T.

Steel, M.J.

Tayeb, G.

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am A 17, 1012 (2000)
[Crossref]

Westbrook, P.S.

White, C.A.

White, T.P

White, T.P.

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers,” submitted J. Opt. Soc. Am. B.

Windeler, R.S.

J. Lightwave Technol. (2)

J. Math. Phys. (1)

R.C. McPhedran, N.A. Nicorovici, L.C. Botten, and K.A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808 (2000).
[Crossref]

J. Opt. Soc. Am A (1)

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am A 17, 1012 (2000)
[Crossref]

J. Opt. Soc. Am. A (2)

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

A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, “Vector description of higher order modes in photonic crystal fibres,” J. Opt. Soc. Am. B 17, 1333–1340 (2000)
[Crossref]

Opt. Fibre Technol. (1)

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fibre Technol. 6, 181–191 (2000).
[Crossref]

Opt. Lett. (2)

Phys. Rev. E (2)

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, and C. M. de Sterke, “Photonic band calculations using scattering matrices”, Phys. Rev. E 64, 046603 (2001)
[Crossref]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135 (1995)
[Crossref]

Science (1)

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic bandgap guidance in optical fibres,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Other (4)

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers,” submitted J. Opt. Soc. Am. B.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical diffraction,” in preparation for Phys. Rev. E.

F. W. J. Olver, “Bessel Functions of Integer Order,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1972), pp. 355–433.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and Bao Ke-Da, “Green’s functions, lattice sums, and Rayleigh’s Identity for a dynamic scattering problem”, ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)

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Figures (4)

Fig. 1.
Fig. 1.

Geometry of the unit cell (defined by the fundamental translation vectors e1 and e2. Phase origins P 1 and P 2 and representations of the incoming (δ ±) and outgoing (f ±) plane wave trains is depicted.

Fig. 2.
Fig. 2.

Dispersion diagram for a hexagonal microstructured optical fibre overlaid with the light line (magenta curve) and the modal dispersion curve (red curve). The fibre data are: hole diameter d=4.026 µm, hole spacing Λ=5.7816 µm, central hole diameter dc =13.1 µm.

Fig. 3.
Fig. 3.

Wavelength variation of the absorption reduction factor for the fibre of Fig. 2.

Fig. 4.
Fig. 4.

Longitudinal components of the electric field, the magnetic field and the Poynting vector at the wavelength, λ=3.428µm, where the MOF of Fig. 2 has maximum reduction in material absorption. At this wavelength, the matrix index is ne =1.39+i0.0003.

Equations (18)

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

E z = m [ A m E l J m ( k e r c l ) + B m E l H m ( 1 ) ( k e r c l ) ] e im arg ( r c l ) ,
E z = l = 1 N c m B m E l H m ( 1 ) ( k e r c l ) e im arg ( r c l ) + m A m E 0 J m ( k e r ) e im θ ,
A E l = j l 𝓗 l j B E j + 𝓙 l 0 A E 0 , with 𝓗 lj = [ 𝓗 nm lj ] , 𝓙 l 0 = [ 𝓙 nm l 0 ] ,
𝓗 nm lj = H n m ( 1 ) ( k e c lj e i ( n m ) arg ( c lj ) , 𝓙 l 0 = J m n ( k e c l ) e i ( m n ) ) arg ( c l ) ,
B E 0 = l = 1 N c 𝓙 0 l B E l , where 𝓙 0 l = [ 𝓙 nm 0 l ] , 𝓙 nm 0 l = J n m ( k e c l ) e i ( n m ) arg ( c l ) ,
B El = R EE , l A E l + R EH , l A H l , B H l = R HE , l A E l + R HH , l A H l .
R ˜ l = ( R EE , l R EH , l R HE , l R HH , l ) , 𝓡 = diag ( R ˜ l )
𝓐 = 𝓗 𝓑 + 𝓙 B 0 A ˜ 0 , B ˜ 0 = 𝓙 0 B 𝓑 , with 𝓙 B 0 = [ 𝓙 l 0 ] and 𝓙 0 B = [ 𝓙 0 l ] .
𝓜 𝓑 ( I 𝓡 𝓢 ) 𝓑 = 0 , where 𝓢 = 𝓗 ˜ + 𝓙 ˜ B 0 R ˜ 0 𝓙 ˜ 0 B .
M ˜ B ˜ ( I R ˜ S ˜ ) B ˜ = 0 , with S ˜ A = ( S A 0 0 S A ) .
S n A j l H n 0 lj e i k 0 · ( c j c l ) = j 0 H n ( 1 ) ( k c j ) e in arg ( c j ) e i k 0 · c j ,
E t = s ξ s 1 2 ( δ s E e i χ s y + f s E + e i χ s y ) R s E + ξ s 1 2 ( δ s H e i χ s y + f s H + e i χ s y ) R s H
y ̂ × K t = s ξ s 1 2 ( δ s E e i χ s y f s E + e i χ s y ) R s E + ξ s 1 2 ( δ s H e i χ s y f s H + e i χ s y ) R s H
F W ( 0 ) Δ = [ I + 2 ω ξ ˜ T K ˜ ( I ˜ R ˜ S ˜ G ) 1 R ˜ J ˜ ξ ˜ ] Δ ,
W ( 0 ) = ( T 1 ( 0 ) R 1 ( 0 ) R 2 ( 0 ) T 2 ( 0 ) ) ,
W F = 0 where W = [ T μ I R R T μ 1 I ] ,
𝓘 = 1 2 ( I I I I ) , we form W = 𝓘 W 𝓘 T = ( T ' + R c I is I is I T R c I ) ,
𝓋 i 1 T g i = 1 2 c g i where 𝓋 i = I + ( T R ) ( T ± R ) , ( i = 1 , 2 ) .

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