Abstract

Abstract

In this report, multiple-scale analysis (averaging) is used to derive the generalized Schrödinger equations that govern light-wave propagation in strongly-birefringent, randomly-birefringent and rapidly-spun fibers. The averaging procedures are described in Jones space and Stokes space. Despite the differences between the aforementioned fibers, the Stokes-space procedures associated with them are similar, and involve only quantities whose physical significances are known. Not only does the Stokes-space formalism unify the derivations of the aforementioned Schrödinger equations, it also produces equations directly in Jones-Stokes notation, which facilitates subsequent studies of polarization effects in optical systems.

©2007 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Four-wave mixing in a rapidly-spun fiber

C. J. McKinstrie, H. Kogelnik, and L. Schenato
Opt. Express 14(19) 8516-8534 (2006)

Nonlinear wave propagation in a rapidly-spun fiber

C. J. McKinstrie and H. Kogelnik
Opt. Express 14(18) 8072-8087 (2006)

Analytical solution of polarization mode dispersion for triangular spun fibers

Grégory Bouquet, Louis-Anne de Montmorillon, and Pascale Nouchi
Opt. Lett. 29(18) 2118-2120 (2004)

References

  • View by:
  • |
  • |
  • |

  1. G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).
  2. L. F. Mollenauer, J. P. Gordon, and P. V. MamyshevI. Kaminow and T. Koch (Academic Press, 1997), Chapter 12.
  3. S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4, 384–452 (1998).
    [Crossref]
  4. R. J. Essiambre, G. Raybon, and B. MikkelsenI. Kaminow and T. Li (Academic Press, 2002), Chapter 6.
  5. F. Forghieri, R. W. Tkach, and A. R. ChraplyvyI. Kaminow and T. Koch (Academic Press, 1997), Chapter 8.
  6. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
    [Crossref]
  7. S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Commun. E88C, 194–208 (2005).
  8. M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-performance optical-fiber-nonlinearity-based optical waveform monitoring,” J. Lightwave Technol. 23, 2012–2022 (2005).
    [Crossref]
  9. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
    [Crossref]
  10. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett.  16, 1231–1233 (1991).
    [Crossref] [PubMed]
  11. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon , and N. S. Bergano, “Polarization muliplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
    [Crossref]
  12. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
    [Crossref]
  13. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996).
    [Crossref]
  14. A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization-mode dispersion in spun single-mode fibers,” Appl. Opt. 20, 2962–2968 (1981).
    [Crossref] [PubMed]
  15. C. J. McKinstrie and H. Kogelnik, “Nonlinear wave propagation in a rapidly-spun fiber,” Opt. Express 14, 8072–8087 (2006).
    [Crossref] [PubMed]
  16. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1981).
  17. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000).
    [Crossref] [PubMed]
  18. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  19. K. Rottwitt and A. J. StentzI. Kaminow and T. Li (Academic Press, 2002), Chapter 5.
  20. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
    [Crossref]
  21. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [Crossref] [PubMed]
  22. C. R. Menyuk, M. N. Islam, and J. P. Gordon, “Raman effect in birefringent optical fibers,” Opt. Lett. 16, 566–568 (1991).
    [Crossref] [PubMed]
  23. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev.  137, A801–A818 (1965).
    [Crossref]
  24. A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
    [Crossref]

2006 (1)

2005 (2)

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Commun. E88C, 194–208 (2005).

M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-performance optical-fiber-nonlinearity-based optical waveform monitoring,” J. Lightwave Technol. 23, 2012–2022 (2005).
[Crossref]

2002 (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000).
[Crossref] [PubMed]

1998 (1)

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4, 384–452 (1998).
[Crossref]

1996 (2)

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996).
[Crossref]

1992 (1)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon , and N. S. Bergano, “Polarization muliplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

1991 (2)

1987 (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

1986 (1)

1981 (1)

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

1972 (1)

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[Crossref]

1965 (1)

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev.  137, A801–A818 (1965).
[Crossref]

Andrekson, P. A.

M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-performance optical-fiber-nonlinearity-based optical waveform monitoring,” J. Lightwave Technol. 23, 2012–2022 (2005).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Barlow, A. J.

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon , and N. S. Bergano, “Polarization muliplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

Chen, H. H.

Chraplyvy, A. R.

F. Forghieri, R. W. Tkach, and A. R. ChraplyvyI. Kaminow and T. Koch (Academic Press, 1997), Chapter 8.

Essiambre, R. J.

R. J. Essiambre, G. Raybon, and B. MikkelsenI. Kaminow and T. Li (Academic Press, 2002), Chapter 6.

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon , and N. S. Bergano, “Polarization muliplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

Forghieri, F.

F. Forghieri, R. W. Tkach, and A. R. ChraplyvyI. Kaminow and T. Koch (Academic Press, 1997), Chapter 8.

George, N.

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[Crossref]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000).
[Crossref] [PubMed]

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon , and N. S. Bergano, “Polarization muliplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

C. R. Menyuk, M. N. Islam, and J. P. Gordon, “Raman effect in birefringent optical fibers,” Opt. Lett. 16, 566–568 (1991).
[Crossref] [PubMed]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[Crossref] [PubMed]

L. F. Mollenauer, J. P. Gordon, and P. V. MamyshevI. Kaminow and T. Koch (Academic Press, 1997), Chapter 12.

Hansryd, J.

M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-performance optical-fiber-nonlinearity-based optical waveform monitoring,” J. Lightwave Technol. 23, 2012–2022 (2005).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Hellwarth, R. W.

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[Crossref]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

Islam, M. N.

Kogelnik, H.

C. J. McKinstrie and H. Kogelnik, “Nonlinear wave propagation in a rapidly-spun fiber,” Opt. Express 14, 8072–8087 (2006).
[Crossref] [PubMed]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000).
[Crossref] [PubMed]

Lakoba, T. I.

Li, J.

M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-performance optical-fiber-nonlinearity-based optical waveform monitoring,” J. Lightwave Technol. 23, 2012–2022 (2005).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Maker, P. D.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev.  137, A801–A818 (1965).
[Crossref]

Mamyshev, P. V.

L. F. Mollenauer, J. P. Gordon, and P. V. MamyshevI. Kaminow and T. Koch (Academic Press, 1997), Chapter 12.

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

McKinstrie, C. J.

C. J. McKinstrie and H. Kogelnik, “Nonlinear wave propagation in a rapidly-spun fiber,” Opt. Express 14, 8072–8087 (2006).
[Crossref] [PubMed]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Commun. E88C, 194–208 (2005).

Menyuk, C. R.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett.  16, 1231–1233 (1991).
[Crossref] [PubMed]

C. R. Menyuk, M. N. Islam, and J. P. Gordon, “Raman effect in birefringent optical fibers,” Opt. Lett. 16, 566–568 (1991).
[Crossref] [PubMed]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

Mezentsev, V. K.

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4, 384–452 (1998).
[Crossref]

Mikkelsen, B.

R. J. Essiambre, G. Raybon, and B. MikkelsenI. Kaminow and T. Li (Academic Press, 2002), Chapter 6.

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon , and N. S. Bergano, “Polarization muliplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

L. F. Mollenauer, J. P. Gordon, and P. V. MamyshevI. Kaminow and T. Koch (Academic Press, 1997), Chapter 12.

Nayfeh, A. H.

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1981).

Owyoung, A.

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[Crossref]

Payne, D. N.

Radic, S.

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Commun. E88C, 194–208 (2005).

Ramskov-Hansen, J. J.

Raybon, G.

R. J. Essiambre, G. Raybon, and B. MikkelsenI. Kaminow and T. Li (Academic Press, 2002), Chapter 6.

Rottwitt, K.

K. Rottwitt and A. J. StentzI. Kaminow and T. Li (Academic Press, 2002), Chapter 5.

Shapiro, E. G.

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4, 384–452 (1998).
[Crossref]

Stentz, A. J.

K. Rottwitt and A. J. StentzI. Kaminow and T. Li (Academic Press, 2002), Chapter 5.

Stolen, R. H.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

Sunnerud, H.

Terhune, R. W.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev.  137, A801–A818 (1965).
[Crossref]

Tkach, R. W.

F. Forghieri, R. W. Tkach, and A. R. ChraplyvyI. Kaminow and T. Koch (Academic Press, 1997), Chapter 8.

Turitsyn, S. K.

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4, 384–452 (1998).
[Crossref]

Wai, P. K. A.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett.  16, 1231–1233 (1991).
[Crossref] [PubMed]

Westlund, M.

M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-performance optical-fiber-nonlinearity-based optical waveform monitoring,” J. Lightwave Technol. 23, 2012–2022 (2005).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

IEEE J. Quantum Electron. (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

IEICE Trans. Commun. (1)

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Commun. E88C, 194–208 (2005).

J. Lightwave Technol. (3)

M. Westlund, P. A. Andrekson, H. Sunnerud, J. Hansryd, and J. Li, “High-performance optical-fiber-nonlinearity-based optical waveform monitoring,” J. Lightwave Technol. 23, 2012–2022 (2005).
[Crossref]

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon , and N. S. Bergano, “Polarization muliplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

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

Opt. Express (1)

Opt. Fiber. Technol. (1)

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4, 384–452 (1998).
[Crossref]

Opt. Lett. (3)

Phys. Rev. (1)

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev.  137, A801–A818 (1965).
[Crossref]

Phys. Rev. B (1)

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[Crossref]

Proc. Nat. Acad. Sci. (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Other (6)

K. Rottwitt and A. J. StentzI. Kaminow and T. Li (Academic Press, 2002), Chapter 5.

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1981).

R. J. Essiambre, G. Raybon, and B. MikkelsenI. Kaminow and T. Li (Academic Press, 2002), Chapter 6.

F. Forghieri, R. W. Tkach, and A. R. ChraplyvyI. Kaminow and T. Koch (Academic Press, 1997), Chapter 8.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

L. F. Mollenauer, J. P. Gordon, and P. V. MamyshevI. Kaminow and T. Koch (Academic Press, 1997), Chapter 12.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1.
Fig. 1. (a) The thick red line represents the vector a⃗. (b) The thick red, green and blue lines represent the vectors a⃗, p⃗ and q⃗, respectively. The thin curves represent the trajectories of the tips of these vectors produced by rotation about the 1-axis.

Tables (6)

Tables Icon

Table 1. Operators averaged by rotation about the 1-axis

Tables Icon

Table 2. Parallel products averaged by rotation about the 1-axis

Tables Icon

Table 3. Perpendicular products averaged by rotation about the 1-axis

Tables Icon

Table 4. Parallel products averaged by surface integration.

Tables Icon

Table 5. Perpendicular products averaged by naive surface integration.

Tables Icon

Table 6. Statistically averaged operators

Equations (94)

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

z B = i β 2 tt 2 B 2 + i γ B 2 B ,
z B = i β ( i t ) B + i ( γ s B 2 + γ c B 2 ) B ,
z B = i β ( i t ) B + i ( γ c B 2 + γ s B 2 ) B ,
DA = iLA + iNA ,
( D 0 + ε 2 D 2 ) ( ε A 1 + ε 3 A 3 ) = i ( L 0 + ε 2 L 2 ) ( ε A 1 + ε 3 A 3 ) + i ε 3 N 2 A 1 ,
( D 0 i L 0 ) A 1 = 0 ,
( D 0 i L 0 ) A 3 = D 2 A 1 + i L 2 A 1 + i L 2 A 1 ,
A 1 ( z 0 , z 2 ) = U 0 ( z 0 ) B 1 ( z 2 ) ,
D 0 B 3 = D 2 B 1 + i L ˜ 2 B 1 + i N ˜ 2 B 1 ,
D 2 B 1 = i L ̅ 2 B 1 + i N ̅ 2 B 1 ,
DB = i ε 2 L ˜ 2 B + i ε 2 N ˜ 2 B ,
D 0 B 1 = 0 ,
D 0 B 2 = D 1 B 1 + i L ˜ 1 B 1 ,
D 0 B 3 = D 1 B 2 D 2 B 1 + i L ˜ 1 B 2 + i L ˜ 2 B 1 + i N ˜ 2 B 1 .
B 2 ( z 0 , z 2 ) = i 0 z 0 L ˜ 1 ( z ) dz B 1 ( z 2 ) .
K ˜ 2 ( z 0 ) = i L ˜ 1 ( z 0 ) 0 z 0 L ˜ 1 ( z ) dz .
D 2 B 1 = i K ̅ 2 B 1 + i L ̅ 2 B 1 + i N ̅ 2 B 1 ,
U r = σ 0 cos ( ρ 2 ) ir · σ sin ( ρ 2 ) ,
σ j 2 = σ 0 , σ j σ k = ± i σ l ,
σ ˜ 2 = σ 2 cos ρ σ 3 sin ρ ,
σ ˜ 3 = σ 2 sin ρ + σ 3 cos ρ .
a 2 = b 2 cos ρ b 3 sin ρ ,
a 3 = b 2 sin ρ + b 3 cos ρ .
[ O ˜ b ] 0 = μ b + μ b ,
μ = [ b O ˜ b ] 0 b 0 , μ = [ b O ˜ b ] 0 b 0 .
a σ a = ( 2 xy , x 2 y 2 , i x 2 + i y 2 ) .
[ x , y ] = a 0 1 2 [ cos ( θ 2 ) e i ϕ 2 , sin ( θ 2 ) e i ϕ 2 ] ,
a = ( cos θ , sin θ cos ϕ , sin θ sin ϕ ) .
a σ a = a 0 ( sin θ , cos θ cos ϕ i sin ϕ , cos θ sin ϕ + i cos ϕ ) .
p = a 0 ( sin θ , cos θ cos ϕ , cos θ sin ϕ ) ,
q = a 0 ( 0 , sin ϕ , cos ϕ ) .
μ b + μ b = ( γ 0 σ 0 + γ 1 σ 1 ) b
γ 0 = μ + μ b 1 ρ b , γ 1 = μ b 0 ρ b .
z a = i k b σ 1 a + i [ β 0 ( i t ) σ 0 + β 1 ( i t ) σ 1 ] a + i γ K ( a 0 σ 0 a 3 σ 3 3 ) a ,
z b = i [ β 0 ( i t ) σ 0 + β 1 ( i t ) σ 1 ] b + i ( γ 0 b 0 σ 0 + γ 1 b 1 σ 1 ) b ,
a a 3 σ 3 a = b 0 2 sin 2 θ sin 2 ϕ ,
a a 3 σ 3 a = b 0 2 sin θ sin ϕ ( cos θ sin ϕ + i cos ϕ ) ,
[ a a 3 σ 3 a ] ϕ = b 0 2 sin 2 θ 2 ,
[ a a 3 σ 3 a ] ϕ = b 0 2 sin θ cos θ 2 .
i [ s ] 0 = i γ K ( b 0 σ 0 6 b 1 σ 1 6 ) b ,
z a = i ( k b σ 1 + k s σ 3 ) a + i [ β 0 ( i t ) σ 0 + β 1 ( i t ) σ 1 ] a + i γ K ( a 0 σ 0 a 3 σ 3 3 ) a ,
L ˜ 1 = k b ( σ 1 cos ϕ + σ 2 sin ϕ ) ,
0 z 0 L ˜ 1 ( z ) dz = k b [ σ 1 sin ϕ + σ 2 ( 1 cos ϕ ) ] 2 k s .
K ̅ 2 = ( k b 2 2 k s ) σ 3 .
z b = i ( k b 2 2 k s ) σ 3 b + i β 0 ( i t ) σ 0 b + i γ K ( b 0 σ 0 b 3 σ 3 3 ) b .
z b = i δ 1 σ 1 b + i β 0 ( i t ) σ 0 b + i ( γ 0 b 0 σ 0 + γ 1 b 1 σ 1 ) b
K ̂ 2 = i k b 2 σ 1 0 z 0 U ( z ) σ 1 U ( z ) dz .
K ̂ 2 = i ( k b 2 2 k s ) [ σ 0 sin ϕ + i σ 3 ( cos ϕ 1 ) ] .
z a = i ( δ 1 σ 1 + δ 3 σ 3 ) a + i [ β 0 ( i t ) σ 0 + β 1 ( i t ) σ 1 ] a + i γ K ( a 0 σ 0 a 3 σ 3 3 ) a ,
z b = i β 0 ( i t ) σ 0 b + i γ 0 b 0 σ 0 b ,
α σ 0 a + i 2 f R ( a 0 σ 0 ) a + i g R ( a 0 σ 0 + a 1 σ 1 + a 2 σ 2 ) a ,
f R ( a 0 σ 0 ) = 0 t f ( t t ) a 0 ( t ' ) dt σ 0 ,
g R ( a j σ j ) = 0 t g ( t t ) a j ( t ' ) dt σ j
α σ 0 b + i 2 f R ( b 0 σ 0 ) b + i g R ( b 0 σ 0 ) b
i g R [ b 1 σ 1 + ( b 2 σ 2 + b 3 σ 3 ) 2 ] b
i g R ( b 1 σ 1 + b 2 σ 2 ) b
i g R ( 2 b 0 σ 0 3 ) b
x = ( p + m ) 2 1 2 , y = i ( p m ) 2 1 2 .
x 2 y 2 = p m * + p * m ,
x y * + x * y = i ( p m * p * m ) ,
i ( x y * x * y ) = p 2 m 2 .
[ p , m ] = a 0 1 2 [ cos ( θ 2 ) e i ϕ 2 , sin ( θ 2 ) e i ϕ 2 ] ,
a = a 0 ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) .
2 x y = i ( p 2 m 2 ) ,
x 2 y 2 = p 2 + m 2 ,
i ( x 2 + y 2 ) = i 2 p m .
a σ a = a 0 ( sin ϕ i cos θ cos ϕ , cos ϕ i cos θ sin ϕ , i sin θ ) .
p = a 0 ( cos θ cos ϕ , cos θ sin ϕ , sin θ ) ,
q = a 0 ( sin ϕ , cos ϕ , 0 ) .
x = ( i u + v ) 2 1 2 , y = ( i u v ) 2 1 2 .
x 2 y 2 = i ( u v * u * v ) ,
x y * + x * y = u 2 v 2 ,
i ( x y * x * y ) = u v * + u * v .
[ u , v ] = a 0 1 2 [ cos ( θ 2 ) e i ϕ 2 , sin ( θ 2 ) e i ϕ 2 ] ,
a = a 0 ( sin θ sin ϕ , cos θ , sin θ cos ϕ ) .
2 xy = u 2 + v 2 ,
x 2 y 2 = i 2 uv ,
i ( x 2 + y 2 ) = i ( u 2 v 2 ) .
a σ a = a 0 ( cos ϕ i cos θ sin ϕ , i sin θ , sin ϕ i cos θ cos ϕ ) .
p = a 0 ( cos θ sin ϕ , sin θ , cos θ cos ϕ ) ,
q = a 0 ( cos ϕ , 0 , sin ϕ ) .
[ ω 2 c 2 ( k 2 kk · ) ] E ( ω , k ) = 4 π ω 2 P ( ω , k ) ,
κ ( E · E ) E ,
E = A exp ( i ϕ ) + A * exp ( i ϕ ) ,
P = κ [ 2 ( A * · A ) A + ( A · A ) A * ] .
P K = γ K [ 2 ( A * · A ) A 3 + ( A · A ) A * 3 ] .
F R ( E · E ) E + G R ( E E ) · E ,
E · E = A · A exp ( i 2 ϕ ) + 2 A * · A + A * · A * exp ( i 2 ϕ ) ,
E E = A A exp ( i 2 ϕ ) + A * A + A A * + A * A * exp ( i 2 ϕ ) .
P R = 2 F R ( A * · A ) A + G R ( A * A + A A * ) · A ,
F R ( t ) = t f ( t t ) A * ( t ) · A ( t ) dt ,
G R ( t ) = t g ( t t ) [ A * ( t ) A ( t ) + A ( t ) A * ( t ) ] dt .
[ a i ] 0 = 0 , [ a i a k ] 0 = δ ik b 0 2 3 ,
[ R ij ] 0 = 0 , [ R ij R kl ] 0 = δ ik δ jl 3 .

Metrics