Abstract

We report for the first time on rigorous numerical simulations of a helical-core fiber by using a full vectorial method based on the transformation optics formalism. We modeled the dependence of circular birefringence of the fundamental mode on the helix pitch and analyzed the effect of a birefringence increase caused by the mode displacement induced by a core twist. Furthermore, we analyzed the complex field evolution versus the helix pitch in the first order modes, including polarization and intensity distribution. Finally, we show that the use of the rigorous vectorial method allows to better predict the confinement loss of the guided modes compared to approximate methods based on equivalent in-plane bending models.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Rigorous simulations of coupling between core and cladding modes in a double-helix fiber

Maciej Napiorkowski and Waclaw Urbanczyk
Opt. Lett. 40(14) 3324-3327 (2015)

Direct measurement of bend-induced mode deformation in large-mode-area fibers

R. C. G. Smith, A. M. Sarangan, Z. Jiang, and J. R. Marciante
Opt. Express 20(4) 4436-4443 (2012)

References

  • View by:
  • |
  • |
  • |

  1. V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
    [Crossref]
  2. R. Birch, “Fabrication and characterization of circularly birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
    [Crossref]
  3. J. P. Koplow, D. A. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000).
    [Crossref] [PubMed]
  4. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006).
    [Crossref] [PubMed]
  5. X. Ma, C. Zhu, I.-N. Hu, A. Kaplan, and A. Galvanauskas, “Angular-momentum coupled optical waves in chirally-coupled-core fibers,” Opt. Express 19(27), 26515–26528 (2011).
    [Crossref] [PubMed]
  6. X. Ma, C. Zhu, I.-N. Hu, A. Kaplan, and A. Galvanauskas, “Single-mode chirally-coupled-core fibers with larger than 50µm diameter cores,” Opt. Express 22(8), 9206–9219 (2014).
    [Crossref] [PubMed]
  7. M. P. Varnham, R. D. Birch, and D. N. Payne, “Helical-core circularly birefringent fibres,” in Proc.IOOC-ECOC (1985.)
  8. J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quantum Electron. 16(5), 455–461 (1984).
    [Crossref]
  9. J. Qian, “Coupled-mode theory for helical fibers,” IEE Proc-J. 135, 178–182 (1988).
  10. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976).
    [Crossref]
  11. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66(10), 1025–1031 (1976).
    [Crossref]
  12. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
    [Crossref]
  13. K. Kaufman, R. Terras, and R. Mathis, “Curvature loss in multimode optical fibers,” J. Opt. Soc. Am. 71(12), 1513–1518 (1981).
    [Crossref]
  14. Z. Jiang and J. R. Marciante, “Mode-area scaling of helical-core, dual-clad fiber lasers and amplifiers using an improved bend-loss model,” J. Opt. Soc. Am. B 23(10), 2051–2058 (2006).
    [Crossref]
  15. R. C. G. Smith, A. M. Sarangan, Z. Jiang, and J. R. Marciante, “Direct measurement of bend-induced mode deformation in large-mode-area fibers,” Opt. Express 20(4), 4436–4443 (2012).
    [Crossref] [PubMed]
  16. M. Li, X. Chen, A. Liu, S. Gray, J. Wang, D. T. Walton, and L. A. Zenteno, “Limit of effective area for single-mode operation in step-index large mode area laser fibers,” J. Lightwave Technol. 27(15), 3010–3016 (2009).
    [Crossref]
  17. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976).
    [Crossref]
  18. A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
    [Crossref]
  19. A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
    [Crossref]
  20. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
    [Crossref] [PubMed]
  21. A. Nicolet, A. B. Movchan, S. Guenneau, and F. Zolla, “Asymptotic modelling of weakly twisted electrostatic problems,” C. R. Mécanique 334, 91–97 (2006).
  22. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
    [Crossref] [PubMed]
  23. T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. S. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30(11), 2921–2927 (2013).
    [Crossref]

2014 (1)

2013 (1)

2012 (2)

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

R. C. G. Smith, A. M. Sarangan, Z. Jiang, and J. R. Marciante, “Direct measurement of bend-induced mode deformation in large-mode-area fibers,” Opt. Express 20(4), 4436–4443 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (1)

2007 (2)

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

2006 (5)

2004 (1)

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

2000 (1)

1987 (1)

R. Birch, “Fabrication and characterization of circularly birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

1984 (1)

J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quantum Electron. 16(5), 455–461 (1984).
[Crossref]

1981 (1)

1976 (3)

Agha, Y. O.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

Barnett, S. M.

Biancalana, F.

T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. S. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30(11), 2921–2927 (2013).
[Crossref]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Birch, R.

R. Birch, “Fabrication and characterization of circularly birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

Birch, R. D.

M. P. Varnham, R. D. Birch, and D. N. Payne, “Helical-core circularly birefringent fibres,” in Proc.IOOC-ECOC (1985.)

Chamorovsky, Y. K.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Chen, X.

Clarkson, W. A.

Cole, J. H.

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

Conti, C.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Cooper, L. J.

Galvanauskas, A.

Goldberg, L.

Gray, S.

Gubin, V. P.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Guenneau, S.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, A. B. Movchan, S. Guenneau, and F. Zolla, “Asymptotic modelling of weakly twisted electrostatic problems,” C. R. Mécanique 334, 91–97 (2006).

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

Hu, I.-N.

Isaev, V. A.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Jiang, Z.

Kang, M. S.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Kaplan, A.

Kaufman, K.

Kliner, D. A.

Koplow, J. P.

Lee, H. W.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Li, M.

Liu, A.

Ma, X.

Marciante, J. R.

Marcuse, D.

Mathis, R.

Morshnev, S. K.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Movchan, A. B.

A. Nicolet, A. B. Movchan, S. Guenneau, and F. Zolla, “Asymptotic modelling of weakly twisted electrostatic problems,” C. R. Mécanique 334, 91–97 (2006).

Nicolet, A.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, A. B. Movchan, S. Guenneau, and F. Zolla, “Asymptotic modelling of weakly twisted electrostatic problems,” C. R. Mécanique 334, 91–97 (2006).

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

Oussov, A. I.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Payne, D. N.

M. P. Varnham, R. D. Birch, and D. N. Payne, “Helical-core circularly birefringent fibres,” in Proc.IOOC-ECOC (1985.)

Pendry, J. B.

Ross, J. N.

J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quantum Electron. 16(5), 455–461 (1984).
[Crossref]

Russell, P. S. J.

T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. S. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30(11), 2921–2927 (2013).
[Crossref]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Sahu, J. K.

Sarangan, A. M.

Sazonov, A. I.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Schermer, R. T.

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

Schurig, D.

Smith, D. R.

Smith, R. C. G.

Starostin, N. I.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Terras, R.

Varnham, M. P.

M. P. Varnham, R. D. Birch, and D. N. Payne, “Helical-core circularly birefringent fibres,” in Proc.IOOC-ECOC (1985.)

Walton, D. T.

Wang, J.

Wang, P.

Weiss, T.

T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. S. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30(11), 2921–2927 (2013).
[Crossref]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Wong, G. K. L.

T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. S. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30(11), 2921–2927 (2013).
[Crossref]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Xi, X. M.

Zenteno, L. A.

Zhu, C.

Zolla, F.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, A. B. Movchan, S. Guenneau, and F. Zolla, “Asymptotic modelling of weakly twisted electrostatic problems,” C. R. Mécanique 334, 91–97 (2006).

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

C. R. Mécanique (1)

A. Nicolet, A. B. Movchan, S. Guenneau, and F. Zolla, “Asymptotic modelling of weakly twisted electrostatic problems,” C. R. Mécanique 334, 91–97 (2006).

Electron. Lett. (1)

R. Birch, “Fabrication and characterization of circularly birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

Eur. Phys. J. Appl. Phys. (1)

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

IEEE J. Quantum Electron. (1)

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (4)

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

Opt. Express (4)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quantum Electron. 16(5), 455–461 (1984).
[Crossref]

Quantum Electron. (1)

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovsky, and A. I. Oussov, “Use of spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Science (1)

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Wave Random Complex (1)

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

Other (2)

J. Qian, “Coupled-mode theory for helical fibers,” IEE Proc-J. 135, 178–182 (1988).

M. P. Varnham, R. D. Birch, and D. N. Payne, “Helical-core circularly birefringent fibres,” in Proc.IOOC-ECOC (1985.)

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 (12)

Fig. 1
Fig. 1 Cross-section of the analyzed helical-core fiber. Arrow shows the twist direction.
Fig. 2
Fig. 2 Effective indices n’eff of the fundamental modes calculated versus the twist rate 1/Λ for the helical-core fiber with core diameters d = 60 µm (blue), 40 µm (red), 20 µm (green) and core offset Q = 100 µm, λ = 1053 nm. Fundamental modes have either left-handed circular (solid line) or right-handed circular (dotted line) polarization.
Fig. 3
Fig. 3 Comparison between circular birefringence in Cartesian coordinates calculated numerically using transformation optics formalism (color lines) and obtained from Eq. (16) (black line) as a function of the twist rate 1/Λ. Simulation parameters: Q = 100 µm, λ = 1053 nm.
Fig. 4
Fig. 4 Calculated effective mode offset Qeff versus the twist rate 1/Λ related to the displacement of the fundamental mode in a helical-core fiber. Insets show intensity profiles calculated for 1/Λ = 0.2 mm−1. Simulation parameters: d1 = 60 µm (blue), d2 = 40 µm (red), d3 = 20 µm (green), Q = 100 µm and λ = 1053 nm.
Fig. 5
Fig. 5 Comparison of circular birefringence calculated versus 1/Λ for λ = 1053 nm (solid lines) and λ = 500 nm (dotted lines) (a) and versus the wavelength for 1/Λ = 0.15 mm−1 (b). Simulation parameters d1 = 60 µm (blue), d2 = 40 µm (red), Q = 100 µm; black lines represent birefringence claculated using Eq. (15).
Fig. 6
Fig. 6 Effective refractive indices n’eff calculated in the helicoidal coordinate system versus the twist rate 1/Λ for four first-order modes of a helical-core fiber (a) and the difference between respective n’eff and the average value n’av of effective indices of four modes (b). Simulation parameters: d = 20 µm, Q = 100 µm, λ = 1053 nm.
Fig. 7
Fig. 7 Difference between neff of individual modes and the average value nav of effective indices of four modes calculated in the Cartesian coordinate system versus the twist rate 1/Λ. In graph (a) we present the simulation results for the full analyzed range, while in graph (b) only for a weakly twisted fiber. Simulation parameters: d = 20 µm, Q = 100 µm, λ = 1053 nm.
Fig. 8
Fig. 8 Evolution of the intensity profiles of the first-order modes in a helical-core fiber (d = 20 µm, Q = 100 µm) versus twist rate 1/Λ. Wavelength λ = 1053 nm.
Fig. 9
Fig. 9 Difference in effective indices neff for the modes pairs HE21’-/TM01’ + (red) and HE21’+/TE01’- (blue) and circular birefringence of the fundamental modes calculated versus the twist rate 1/Λ. Simulation parameters d = 20 µm, Q = 100 µm, λ = 1053 nm.
Fig. 10
Fig. 10 Surface distribution of the ellipticity angle in TM01’ + mode for 1/Λ = 0.08 mm−1 (a) and ellipticity angle in the modes TE01’- (blue) and TM01’ + (red) calculated at the extreme points 1 (ξ1 = 100 µm, ξ2 = 10 µm - solid line) and 2 (ξ1 = 110 µm, ξ2 = 0 - dashed line) versus 1/Λ (b). Simulation parameters: d = 20 µm, Q = 100 µm, λ = 1053 nm.
Fig. 11
Fig. 11 Comparison of the loss calculated with rigorous (red lines) and approximate (black lines) methods [14] as a function of the helix pitch Λ. Solid lines indicate the fundamental modes, while dashed lines the first-order modes with lower loss. Simulation parameters: d = 60 µm, Q1 = 100 µm, Q2 = 300 µm, λ = 1053 nm.
Fig. 12
Fig. 12 Confinement loss characteristics calculated for the fundamental mode (solid line) and the first order modes of lower losses (modes pair HE21’+/TE01’- dashed line) versus the helix pitch Λ determining the single-mode operation range. Simulation parameters: d1 = 60 µm (blue), d2 = 40 µm (red), d3 = 20 µm (green), Q = 100 µm, λ = 1053 nm.

Equations (17)

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

x = ξ 1 cos ( A ξ 3 ) + ξ 2 sin ( A ξ 3 ) y = - ξ 1 sin ( A ξ 3 ) + ξ 2 cos ( A ξ 3 ) z = ξ 3 ,
| A | = 2 π Λ 1 ,
ε i ' j ' = | det ( J i ' i ) | 1 J i ' i J j ' j ε i j ,
μ i ' j ' = | det ( J i ' i ) | 1 J i ' i J j ' j μ i j ,
J i ' i = x i ' x i .
ε ' = ( 1 + A 2 ξ 2 2 A 2 ξ 1 ξ 2 A ξ 2 A 2 ξ 1 ξ 2 1 + A 2 ξ 1 2 A ξ 1 A ξ 2 A ξ 1 1 ) ε ,
μ ' = ( 1 + A 2 ξ 2 2 A 2 ξ 1 ξ 2 A ξ 2 A 2 ξ 1 ξ 2 1 + A 2 ξ 1 2 A ξ 1 A ξ 2 A ξ 1 1 ) μ .
E ( r , θ , z , t ) = E ( r , θ ) exp ( j β z ) exp ( j ω t ) ,
E r ( r ) sin ( m ( θ A z ) ) = j E r ( r ) 2 [ exp ( j m ( θ A z ) ) exp ( j m ( θ A z ) ) ] ,
E θ ( r ) cos ( m ( θ A z ) ) = E θ ( r ) 2 [ exp ( j m ( θ A z ) ) + exp ( j m ( θ A z ) ) ] ,
1 2 [ j E r ( r ) E θ ( r ) ] exp ( j m θ ) exp ( j m A z ) = E R H C ( r , θ ) exp ( j m A z )
1 2 [ j E r ( r ) E θ ( r ) ] exp ( j m θ ) exp ( j m A z ) = E L H C ( r , θ ) exp ( j m A z ) .
E R H C ( r , θ , z , t ) = E R H C ( r , θ ) exp ( j m A z ) exp ( j β z ) exp ( j ω t ) = = E R H C ( r , θ ) exp ( j ( β + m A ) z ) exp ( j ω t ) = = E R H C ( r , θ ) exp ( j β R H C z ) exp ( j ω t ) ,
E L H C ( r , θ , z , t ) = E L H C ( r , θ ) exp ( j m A z ) exp ( j β z ) exp ( j ω t ) = = E L H C ( r , θ ) exp ( j ( β m A ) z ) exp ( j ω t ) = = E L H C ( r , θ ) exp ( j β L H C z ) exp ( j ω t ) .
n ' e f f = ( β ± m A ) λ 2 π = n e f f ± m λ Λ .
B = 2 λ ( S Λ ) Λ S = 2 λ ( ( 2 π Q ) 2 + Λ 2 Λ ) Λ ( 2 π Q ) 2 + Λ 2 ,
Q e f f = ( 2 B s Λ 2 B s λ Λ ) 2 Λ 2 4 π 2 .

Metrics