Abstract

We derive a differential equation that relates the Mueller matrices of an optical system at adjacent frequencies in the presence of polarization mode dispersion and polarization-dependent loss (PDL). We then demonstrate that a solution of this equation based on the Magnus expansion yields a description of the Mueller matrix in orders of the principal state vector that coincides with previously reported results for systems without PDL.

© 2006 Optical Society of America

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  1. J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. USA 97, 4541-4550 (2000).
    [CrossRef] [PubMed]
  2. H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
    [CrossRef]
  3. B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Quantum Electron. 6, 317-329 (2000).
    [CrossRef]
  4. W. Magnus, "On the exponential solution of differential equations for a linear operator," Commun. Pure Appl. Math. 7, 649-673 (1954).
    [CrossRef]
  5. J. Oteo and J. Ros, "From time ordered products to Magnus expansion," J. Math. Phys. 41, 3268-3277 (2000).
    [CrossRef]
  6. D. Yevick, M. Chanachowicz, M. Reimer, M. O'Sullivan, W. Huang, and T. Lu, "Chebyshev and Taylor approximations of polarization mode dispersion for improved compensation bandwidth," J. Opt. Soc. Am. A 22, 1662-1667 (2005).
    [CrossRef]
  7. D. Yevick, T. Lu, W. Huang, and W. Bardyszewski, "Operator expansions for polarization mode dispersion analysis and compensation," J. Opt. Soc. Am. A 23, 445-460 (2006).
    [CrossRef]
  8. N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. QE-22, 2131-2140 (1986).
    [CrossRef]
  9. E. Collett, Polarized Light in Fiber Optics (PolaWave Group, 2003), Chap. 14, pp. 379-431.
  10. E. W. Weisstein, "Matrix direct product" (Wolfram Research, 1999), http://mathworld.wolfram.com/MatrixDirectProduct.html.
  11. A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," 2004, http://arXiv.org/abs/math-ph/0412061.
  12. B. Huttner and N. Gisin, "Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses," Opt. Lett. 22, 504-506 (1997).
    [CrossRef] [PubMed]
  13. W. E. Baylis, "Special relativity with 2×2 matrices," Am. J. Phys. 48, 918-925 (1980).
    [CrossRef]
  14. A. Eyal and M. Tur, "A modified Poincare sphere technique for the determination of polarization-mode dispersion in the presence of differential gain/loss," in Optical Fiber Communication Conference (OFC) Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), p. 340.
  15. Y. Li and A. Yariv, "Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses," J. Opt. Soc. Am. B 17, 1821-1827 (2000).
    [CrossRef]
  16. M. Glasner and D. Yevick, "Generalized propagation techniques for longitudinally varying refractive index distributions," Math. Comput. Modell. 16, 177-182 (1992).
    [CrossRef]
  17. M. Glasner, D. Yevick, and B. Hermansson, "Generalized propagation formulas of arbitrarily high order," J. Chem. Phys. 95, 8266-8272 (1991).
    [CrossRef]
  18. R. Barakat, "Exponential versions of the Jones and Mueller-Jones polarization matrices," J. Opt. Soc. Am. A 13, 158-163 (1996).
    [CrossRef]
  19. D. Gottlieb, "Skew symmetric bundle maps on space-time," Contemp. Math. 220, 117-141 (1998).
    [CrossRef]
  20. T. Kudou, M. Iguchi, M. Masuda, and T. Ozeki, "Theoretical basis of polarization mode dispersion equalization up to the second order," J. Lightwave Technol. 18, 614-617 (2000).
    [CrossRef]
  21. A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarization mode dispersion," Electron. Lett. 41, 1658-1659 (1999).
    [CrossRef]
  22. M. Glasner, D. Yevick, and B. Hermansson, "High-order generalized propagation techniques," J. Opt. Soc. Am. B 8, 413-415 (1991).
    [CrossRef]

2006 (1)

D. Yevick, T. Lu, W. Huang, and W. Bardyszewski, "Operator expansions for polarization mode dispersion analysis and compensation," J. Opt. Soc. Am. A 23, 445-460 (2006).
[CrossRef]

2005 (1)

2003 (1)

2000 (5)

Y. Li and A. Yariv, "Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses," J. Opt. Soc. Am. B 17, 1821-1827 (2000).
[CrossRef]

T. Kudou, M. Iguchi, M. Masuda, and T. Ozeki, "Theoretical basis of polarization mode dispersion equalization up to the second order," J. Lightwave Technol. 18, 614-617 (2000).
[CrossRef]

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

J. Oteo and J. Ros, "From time ordered products to Magnus expansion," J. Math. Phys. 41, 3268-3277 (2000).
[CrossRef]

J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. USA 97, 4541-4550 (2000).
[CrossRef] [PubMed]

1999 (1)

A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarization mode dispersion," Electron. Lett. 41, 1658-1659 (1999).
[CrossRef]

1998 (1)

D. Gottlieb, "Skew symmetric bundle maps on space-time," Contemp. Math. 220, 117-141 (1998).
[CrossRef]

1997 (1)

1996 (1)

1992 (1)

M. Glasner and D. Yevick, "Generalized propagation techniques for longitudinally varying refractive index distributions," Math. Comput. Modell. 16, 177-182 (1992).
[CrossRef]

1991 (2)

M. Glasner, D. Yevick, and B. Hermansson, "Generalized propagation formulas of arbitrarily high order," J. Chem. Phys. 95, 8266-8272 (1991).
[CrossRef]

M. Glasner, D. Yevick, and B. Hermansson, "High-order generalized propagation techniques," J. Opt. Soc. Am. B 8, 413-415 (1991).
[CrossRef]

1986 (1)

N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. QE-22, 2131-2140 (1986).
[CrossRef]

1980 (1)

W. E. Baylis, "Special relativity with 2×2 matrices," Am. J. Phys. 48, 918-925 (1980).
[CrossRef]

1954 (1)

W. Magnus, "On the exponential solution of differential equations for a linear operator," Commun. Pure Appl. Math. 7, 649-673 (1954).
[CrossRef]

Aiello, A.

A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," 2004, http://arXiv.org/abs/math-ph/0412061.

Barakat, R.

Bardyszewski, W.

D. Yevick, T. Lu, W. Huang, and W. Bardyszewski, "Operator expansions for polarization mode dispersion analysis and compensation," J. Opt. Soc. Am. A 23, 445-460 (2006).
[CrossRef]

Baylis, W. E.

W. E. Baylis, "Special relativity with 2×2 matrices," Am. J. Phys. 48, 918-925 (1980).
[CrossRef]

Chanachowicz, M.

Collett, E.

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, 2003), Chap. 14, pp. 379-431.

Eyal, A.

A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarization mode dispersion," Electron. Lett. 41, 1658-1659 (1999).
[CrossRef]

A. Eyal and M. Tur, "A modified Poincare sphere technique for the determination of polarization-mode dispersion in the presence of differential gain/loss," in Optical Fiber Communication Conference (OFC) Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), p. 340.

Frigo, N.

N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. QE-22, 2131-2140 (1986).
[CrossRef]

Geiser, C.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Gisin, N.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

B. Huttner and N. Gisin, "Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses," Opt. Lett. 22, 504-506 (1997).
[CrossRef] [PubMed]

Glasner, M.

M. Glasner and D. Yevick, "Generalized propagation techniques for longitudinally varying refractive index distributions," Math. Comput. Modell. 16, 177-182 (1992).
[CrossRef]

M. Glasner, D. Yevick, and B. Hermansson, "High-order generalized propagation techniques," J. Opt. Soc. Am. B 8, 413-415 (1991).
[CrossRef]

M. Glasner, D. Yevick, and B. Hermansson, "Generalized propagation formulas of arbitrarily high order," J. Chem. Phys. 95, 8266-8272 (1991).
[CrossRef]

Gordon, J.

J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. USA 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Gordon, J. P.

Gottlieb, D.

D. Gottlieb, "Skew symmetric bundle maps on space-time," Contemp. Math. 220, 117-141 (1998).
[CrossRef]

Hermansson, B.

M. Glasner, D. Yevick, and B. Hermansson, "Generalized propagation formulas of arbitrarily high order," J. Chem. Phys. 95, 8266-8272 (1991).
[CrossRef]

M. Glasner, D. Yevick, and B. Hermansson, "High-order generalized propagation techniques," J. Opt. Soc. Am. B 8, 413-415 (1991).
[CrossRef]

Huang, W.

D. Yevick, T. Lu, W. Huang, and W. Bardyszewski, "Operator expansions for polarization mode dispersion analysis and compensation," J. Opt. Soc. Am. A 23, 445-460 (2006).
[CrossRef]

D. Yevick, M. Chanachowicz, M. Reimer, M. O'Sullivan, W. Huang, and T. Lu, "Chebyshev and Taylor approximations of polarization mode dispersion for improved compensation bandwidth," J. Opt. Soc. Am. A 22, 1662-1667 (2005).
[CrossRef]

Huttner, B.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

B. Huttner and N. Gisin, "Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses," Opt. Lett. 22, 504-506 (1997).
[CrossRef] [PubMed]

Iguchi, M.

Kogelnik, H.

H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
[CrossRef]

J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. USA 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Kudou, T.

Li, Y.

Lu, T.

D. Yevick, T. Lu, W. Huang, and W. Bardyszewski, "Operator expansions for polarization mode dispersion analysis and compensation," J. Opt. Soc. Am. A 23, 445-460 (2006).
[CrossRef]

D. Yevick, M. Chanachowicz, M. Reimer, M. O'Sullivan, W. Huang, and T. Lu, "Chebyshev and Taylor approximations of polarization mode dispersion for improved compensation bandwidth," J. Opt. Soc. Am. A 22, 1662-1667 (2005).
[CrossRef]

Magnus, W.

W. Magnus, "On the exponential solution of differential equations for a linear operator," Commun. Pure Appl. Math. 7, 649-673 (1954).
[CrossRef]

Marshall, W. K.

A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarization mode dispersion," Electron. Lett. 41, 1658-1659 (1999).
[CrossRef]

Masuda, M.

Nelson, L. E.

O'Sullivan, M.

Oteo, J.

J. Oteo and J. Ros, "From time ordered products to Magnus expansion," J. Math. Phys. 41, 3268-3277 (2000).
[CrossRef]

Ozeki, T.

Reimer, M.

Ros, J.

J. Oteo and J. Ros, "From time ordered products to Magnus expansion," J. Math. Phys. 41, 3268-3277 (2000).
[CrossRef]

Tur, M.

A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarization mode dispersion," Electron. Lett. 41, 1658-1659 (1999).
[CrossRef]

A. Eyal and M. Tur, "A modified Poincare sphere technique for the determination of polarization-mode dispersion in the presence of differential gain/loss," in Optical Fiber Communication Conference (OFC) Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), p. 340.

Weisstein, E. W.

E. W. Weisstein, "Matrix direct product" (Wolfram Research, 1999), http://mathworld.wolfram.com/MatrixDirectProduct.html.

Woerdman, J. P.

A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," 2004, http://arXiv.org/abs/math-ph/0412061.

Yariv, A.

Y. Li and A. Yariv, "Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses," J. Opt. Soc. Am. B 17, 1821-1827 (2000).
[CrossRef]

A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarization mode dispersion," Electron. Lett. 41, 1658-1659 (1999).
[CrossRef]

Yevick, D.

D. Yevick, T. Lu, W. Huang, and W. Bardyszewski, "Operator expansions for polarization mode dispersion analysis and compensation," J. Opt. Soc. Am. A 23, 445-460 (2006).
[CrossRef]

D. Yevick, M. Chanachowicz, M. Reimer, M. O'Sullivan, W. Huang, and T. Lu, "Chebyshev and Taylor approximations of polarization mode dispersion for improved compensation bandwidth," J. Opt. Soc. Am. A 22, 1662-1667 (2005).
[CrossRef]

M. Glasner and D. Yevick, "Generalized propagation techniques for longitudinally varying refractive index distributions," Math. Comput. Modell. 16, 177-182 (1992).
[CrossRef]

M. Glasner, D. Yevick, and B. Hermansson, "Generalized propagation formulas of arbitrarily high order," J. Chem. Phys. 95, 8266-8272 (1991).
[CrossRef]

M. Glasner, D. Yevick, and B. Hermansson, "High-order generalized propagation techniques," J. Opt. Soc. Am. B 8, 413-415 (1991).
[CrossRef]

Am. J. Phys. (1)

W. E. Baylis, "Special relativity with 2×2 matrices," Am. J. Phys. 48, 918-925 (1980).
[CrossRef]

Commun. Pure Appl. Math. (1)

W. Magnus, "On the exponential solution of differential equations for a linear operator," Commun. Pure Appl. Math. 7, 649-673 (1954).
[CrossRef]

Contemp. Math. (1)

D. Gottlieb, "Skew symmetric bundle maps on space-time," Contemp. Math. 220, 117-141 (1998).
[CrossRef]

Electron. Lett. (1)

A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarization mode dispersion," Electron. Lett. 41, 1658-1659 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. QE-22, 2131-2140 (1986).
[CrossRef]

J. Chem. Phys. (1)

M. Glasner, D. Yevick, and B. Hermansson, "Generalized propagation formulas of arbitrarily high order," J. Chem. Phys. 95, 8266-8272 (1991).
[CrossRef]

J. Lightwave Technol. (2)

J. Math. Phys. (1)

J. Oteo and J. Ros, "From time ordered products to Magnus expansion," J. Math. Phys. 41, 3268-3277 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

Math. Comput. Modell. (1)

M. Glasner and D. Yevick, "Generalized propagation techniques for longitudinally varying refractive index distributions," Math. Comput. Modell. 16, 177-182 (1992).
[CrossRef]

Opt. Lett. (1)

Proc. Natl. Acad. Sci. USA (1)

J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. USA 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Other (4)

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, 2003), Chap. 14, pp. 379-431.

E. W. Weisstein, "Matrix direct product" (Wolfram Research, 1999), http://mathworld.wolfram.com/MatrixDirectProduct.html.

A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," 2004, http://arXiv.org/abs/math-ph/0412061.

A. Eyal and M. Tur, "A modified Poincare sphere technique for the determination of polarization-mode dispersion in the presence of differential gain/loss," in Optical Fiber Communication Conference (OFC) Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), p. 340.

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Equations (63)

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d M d ω M 1 = [ 0 Λ T Λ Ω × ] ,
M ( ω ) = exp [ 0 a T a b × ] M ( ω 0 ) ,
a = Λ 0 Δ ω + Λ 1 Δ ω 2 2 ! + ( Λ 2 1 2 Ω 0 × Λ 1 1 2 Λ 0 × Ω 1 ) Δ ω 3 3 !
b = Ω 0 Δ ω + Ω 1 Δ ω 2 2 ! + ( Ω 2 1 2 Ω 0 × Ω 1 1 2 Λ 1 × Λ 0 ) Δ ω 3 3 !
T ( ω ) = e i 2 ( b + i a ) σ T ( ω 0 ) ,
σ ̃ = ( σ 0 , σ ) = ( σ 0 , σ 1 , σ 2 , σ 3 ) ,
σ 0 = [ 1 0 0 1 ] , σ 1 = [ 1 0 0 1 ] ,
σ 2 = [ 0 1 1 0 ] , σ 3 = [ 0 i i 0 ] ,
M = A ( T T * ) A ,
A = 1 2 [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ]
F G = [ f 00 f 01 f 10 f 11 ] [ g 00 g 01 g 10 g 11 ] = [ f 00 G f 01 G f 10 G f 11 G ] = [ f 00 g 00 f 00 g 01 f 01 g 00 f 01 g 01 f 00 g 10 f 00 g 11 f 01 g 10 f 01 g 11 f 10 g 00 f 10 g 01 f 11 g 00 f 11 g 01 f 10 g 10 f 10 g 11 f 11 g 10 f 11 g 11 ] .
M i j = 1 2 Tr ( σ i T σ j T ) = 1 2 [ σ i T ] m n [ σ j T ] n m = 1 2 [ σ i ] m p T p n [ σ j ] n q T q m = 1 2 [ σ i * ] p m T p n T m q * [ σ j ] n q ,
M i j = 1 2 [ σ ̃ i * ] 2 p + m ( T T * ) 2 p + m , 2 n + q [ σ ̃ j ] 2 n + q ,
σ ̃ 0 = [ 1 0 0 1 ] , σ ̃ 1 = [ 1 0 0 1 ] , σ ̃ 2 = [ 0 1 1 0 ] , σ ̃ 3 = [ 0 i i 0 ] .
T ω T 1 = i 2 W ( ω ) σ ,
T ( ω ) = exp [ i 2 ( β + i α ) σ ] T 0 = [ cos ( λ 2 ) σ 0 i sin ( λ 2 ) ( w ̂ σ ) ] T 0 .
T ω T 1 = i 2 [ λ ω w ̂ + sin ( λ ) w ̂ ω + 2 sin 2 ( λ 2 ) w ̂ × w ̂ ω ] σ ,
M ω M 1 = A [ ( T ω T 1 ) ( T * T * 1 ) + ( T T 1 ) ( T ω * T * 1 ) ] A = i 2 A [ ( Ω σ ) σ 0 σ 0 ( Ω σ ) * ] A + 1 2 A [ ( Λ σ ) σ 0 + σ 0 ( Λ σ ) * ] A .
A ( a σ ) σ 0 A = [ 0 a T a i a × ] ,
A σ 0 ( a σ ) * A = [ 0 a T a i a × ] ,
M ω M 1 = [ 0 Λ T Λ Ω × ] = [ 0 Λ x Λ y Λ z Λ x 0 Ω z Ω y Λ y Ω z 0 Ω x Λ z Ω y Ω x 0 ] .
t ω = [ t ω t ̂ t ω + t t ̂ ω ] = [ 0 Λ T Λ Ω × ] [ t t ] = [ t Λ t ̂ Λ t + t Ω × t ̂ ] ,
t ̂ ω = Λ t ̂ ( Λ t ̂ ) + Ω × t ̂ = t ̂ × ( Λ × t ̂ ) + Ω × t ̂ ,
d M d ω = H ( ω ) M ,
M ( ω ) = T ω exp [ ω 0 ω H ( ω 1 ) d ω 1 ] M 0 .
M ( ω ) = exp [ i = 1 B i ( ω ) ] M 0 ,
B 1 ( ω ) = ω 0 ω d ω 1 H ( ω 1 ) ,
B 2 ( ω ) = 1 2 ω 0 ω d ω 1 ω 0 ω 1 d ω 2 [ H ( ω 1 ) , H ( ω 2 ) ] ,
B 3 ( ω ) = 1 6 ω 0 ω d ω 1 ω 0 ω 1 d ω 2 ω 0 ω 2 d ω 3 ( { [ H ( ω 1 ) , [ H ( ω 2 ) , H ( ω 3 ) ] } + { [ H ( ω 1 ) , H ( ω 2 ) ] , H ( ω 3 ) ] } ) ,
B n ( ω ) = j = 1 n 1 b j j ! ω 0 ω d ω 1 S n ( j ) ( ω 1 ) , n 2 ,
S n ( j ) ( ω ) = m = 1 n j [ B m ( ω ) , S n m ( j 1 ) ( ω ) ] , 2 j n 1 ,
S n ( 1 ) ( ω ) = [ B n 1 ( ω ) , H ( ω ) ] ,
S n ( n 1 ) ( ω ) = { B 1 ( n 1 ) ( ω ) , H ( ω ) } .
H ( ω ) = [ 0 Λ T Λ Ω × ] = h ( Λ ) + c ( Ω ) ,
h ( Λ ) = [ 0 Λ T Λ 0 ] , c ( Λ ) = [ 0 0 0 Ω × ] .
B 1 = c ( Ω 0 Δ ω + Ω 1 Δ ω 2 2 ! + Ω 2 Δ ω 3 3 ! ) + h ( Λ 0 Δ ω + Λ 1 Δ ω 2 2 ! + Λ 2 Δ ω 3 3 ! ) + O ( Δ ω 4 ) .
B 2 = 1 2 0 Δ ω d ω 1 0 ω 1 d ω 2 [ c ( Ω 0 + Ω 1 ω 1 ) + h ( Λ 0 + Λ 1 ω 1 ) , c ( Ω 0 + Ω 1 ω 2 ) + h ( Λ 0 + Λ 1 ω 2 ) ] ,
[ c ( u ) , h ( v ) ] = h ( u × v ) ,
[ c ( u ) , c ( v ) ] = c ( u × v ) ,
[ h ( u ) , h ( v ) ] = c ( v × u )
B 2 = 1 2 Δ ω 3 3 ! { c ( Ω 0 × Ω 1 + Λ 1 × Λ 0 ) + h ( Ω 0 × Λ 1 + Λ 0 × Ω 1 ) } + O ( Δ ω 4 ) .
M ( ω ) = exp [ 0 a T a b × ] M 0 ,
a = Λ 0 Δ ω + Λ 1 Δ ω 2 2 ! + ( Λ 2 1 2 Ω 0 × Λ 1 1 2 Λ 0 × Ω 1 ) Δ ω 3 3 ! + O ( Δ ω 4 ) ,
b = Ω 0 Δ ω + Ω 1 Δ ω 2 2 ! + ( Ω 2 1 2 Ω 0 × Ω 1 1 2 Λ 1 × Λ 0 ) Δ ω 3 3 ! + O ( Δ ω 4 ) ,
T ( ω ) = exp [ i 2 ( b + i a ) σ ] T 0 ,
N c = [ 0 ( a + i b ) T a + i b i ( a + i b ) × ] ,
M ( ω ) = exp [ 0 a T a b × ] M 0 = ( F c F c * ) M 0 ,
F c exp ( N c 2 ) = cosh η 2 I ( 4 ) + 1 η sinh η 2 N c ,
exp [ 0 0 0 b × ] = [ 1 0 0 cos ( b ) I ( 3 ) + sin ( b ) ( b ̂ × ) + ( 1 cos ( b ) ) b ̂ b ̂ T ] ,
exp [ 0 a T a 0 ] = [ cosh ( a ) sinh ( a ) a ̂ T sinh ( a ) a ̂ ( cosh ( a ) 1 ) a ̂ a ̂ T + I ( 3 ) ] .
M ( ω ) = [ 1 0 0 exp [ ( Ω 0 Δ ω + Ω 1 Δ ω 2 2 ! + ( Ω 2 1 2 Ω 0 × Ω 1 ) Δ ω 3 3 ! ) × ] ] M 0
Ω = ψ ω n ̂ + sin ( ψ ) n ̂ ω + [ 1 cos ( ψ ) ] n ̂ × n ̂ ω ,
Ω 0 = ψ 1 n ̂ 0 = d ( ψ n ̂ ) d ω ω 0 ,
Ω 1 = ψ 2 n ̂ 0 + 2 ψ 1 n ̂ 1 = d 2 ( ψ n ̂ ) d ω 2 ω 0 ,
Ω 2 = ψ 3 n ̂ 0 + 3 ψ 2 n ̂ 1 + 3 ψ 1 n ̂ 2 + ψ 1 2 n ̂ 0 × n ̂ 1 = d 3 ( ψ n ̂ ) d ω 3 ω 0 + ψ 1 2 n ̂ 0 × n ̂ 1 ,
d 3 ( ψ n ̂ ) d ω 3 ω 0 = Ω 2 1 2 Ω 0 × Ω 1
R = exp [ ( Ω 0 Δ ω + Ω 1 Δ ω 2 2 ! + ( Ω 2 1 2 Ω 0 × Ω 1 ) Δ ω 3 3 ! ) × ] ,
e F e G = exp ( F + G + 1 2 [ F , G ] + 1 12 [ F , [ F , G ] ] + 1 12 [ G , [ G , F ] ] + ) ,
M comp = exp ( 1 2 N ( 3 ) Δ ω 2 ) exp ( 1 2 N ( 2 ) Δ ω 2 ) exp ( 1 2 N ( 1 ) Δ ω ) exp ( N ( 0 ) Δ ω ) exp ( 1 2 N ( 1 ) Δ ω ) ,
N ( 0 ) = [ 0 0 0 Ω 0 × ] , N ( 1 ) = [ 0 Λ 0 T Λ 0 0 ] ,
N ( 2 ) = [ 0 0 0 Ω 1 × ] , N ( 3 ) = [ 0 Λ 1 T Λ 1 0 ] ,
M ( ω ) = exp [ H 0 Δ ω + H 1 Δ ω 2 2 ! + ( H 2 1 2 [ H 0 , H 1 ] ) Δ ω 3 3 ! + O ( Δ ω 4 ) ] M 0 .
M comp = exp ( 1 6 H 2 Δ ω 3 ) exp ( 2 3 H 0 Δ ω ) exp ( 1 2 H 1 Δ ω 2 ) exp ( 1 3 H 0 Δ ω ) .

Metrics