Abstract

In this paper, we investigate the modal bifurcation in chiral multilayered fibers through an approach of rigorous modal theory. The mirror symmetry of this theory is presented to provide physical insights into the modal bifurcation. The modes that are double degenerate and mirror images originally in achiral fibers are bifurcated by the chirality in chiral fibers. The modal bifurcation in chiral Bragg fibers is examined as an application of the theory. General guidelines of designing chiral Bragg fibers to possess circular polarization selectivity and wavelength selectivity are proposed from physical considerations and verified by numerical calculations.

© 2013 Optical Society of America

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  1. N. Engheta and D. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Newsletter 30(5), 6–12 (1988).
  2. F. Mariotte, P. Pelet, and N. Engheta, “A review of recent study of guided waves in chiral media,” Prog. Electromagn. Res. 9, 311–350 (1994).
  3. C. Eftimiu and L. Parson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
    [CrossRef]
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    [CrossRef]
  5. K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
    [CrossRef]
  6. R. C. Qiu and I.-T. Lu, “Guided waves in chiral optical fibers,” J. Opt. Soc. Am. A 11, 3212–3219 (1994).
    [CrossRef]
  7. S. Mahmoud, “Guided modes on open chirowaveguides,” IEEE Trans. Microwave Theory Tech. 43, 205–209 (1995).
    [CrossRef]
  8. H. Cory, “Chiral devices-an overview of canonical problems,” J. Electromagn. Waves Appl. 9, 805–829 (1995).
    [CrossRef]
  9. K. M. Flood and D. L. Jaggard, “Single-mode operation in symmetric planar waveguides using isotropic chiral media,” Opt. Lett. 21, 474–476 (1996).
    [CrossRef]
  10. W. N. Herman, “Polarization eccentricity of the transverse field for modes in chiral core planar waveguides,” J. Opt. Soc. Am. A 18, 2806–2818 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. J.-F. Dong, W.-D. Tao, and J. Xu, “Optical power characteristics of guided modes in a double-cladding chiral optical fiber,” Acta Photon. Sin. 36, 1044–1049 (2007).
  14. A. Nair and P. Choudhury, “On the analysis of field patterns in chirofibers,” J. Electromagn. Waves Appl. 21, 2277–2286 (2007).
    [CrossRef]
  15. G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series Vol. 335 (Springer, 1989).
  23. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. A 68, 1196–1201 (1978).
    [CrossRef]
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    [CrossRef]
  28. N. Engheta and P. Pelet, “Modes in chirowaveguides,” Opt. Lett. 14, 593–595 (1989).
    [CrossRef]
  29. C. Brewitt-Taylor, P. Lederer, F. Smith, and S. Haq, “Measurement and prediction of helix-loaded chiral composites,” IEEE Trans. Antennas Propag. 47, 692–700 (1999).
    [CrossRef]
  30. R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
    [CrossRef]

2013 (1)

2012 (1)

J.-F. Dong and J. Li, “Characteristics of guided modes in uniaxial chiral circular waveguides,” Prog. Electromagn. Res. 124, 331–345 (2012).
[CrossRef]

2011 (2)

2009 (1)

G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
[CrossRef]

2007 (2)

J.-F. Dong, W.-D. Tao, and J. Xu, “Optical power characteristics of guided modes in a double-cladding chiral optical fiber,” Acta Photon. Sin. 36, 1044–1049 (2007).

A. Nair and P. Choudhury, “On the analysis of field patterns in chirofibers,” J. Electromagn. Waves Appl. 21, 2277–2286 (2007).
[CrossRef]

2005 (1)

2002 (2)

F. M. Janeiro, C. R. Paiva, and A. L. Topa, “Guidance and leakage properties of chiral optical fibers,” J. Opt. Soc. Am. B 19, 2558–2566 (2002).
[CrossRef]

P. Choudhury and T. Yoshino, “Dependence of optical power confinement on core/cladding chiralities in chirofibers,” Microw. Opt. Technol. Lett. 32, 359–364 (2002).
[CrossRef]

2001 (2)

1999 (1)

C. Brewitt-Taylor, P. Lederer, F. Smith, and S. Haq, “Measurement and prediction of helix-loaded chiral composites,” IEEE Trans. Antennas Propag. 47, 692–700 (1999).
[CrossRef]

1996 (1)

1995 (3)

S. Mahmoud, “Guided modes on open chirowaveguides,” IEEE Trans. Microwave Theory Tech. 43, 205–209 (1995).
[CrossRef]

H. Cory, “Chiral devices-an overview of canonical problems,” J. Electromagn. Waves Appl. 9, 805–829 (1995).
[CrossRef]

R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
[CrossRef]

1994 (2)

R. C. Qiu and I.-T. Lu, “Guided waves in chiral optical fibers,” J. Opt. Soc. Am. A 11, 3212–3219 (1994).
[CrossRef]

F. Mariotte, P. Pelet, and N. Engheta, “A review of recent study of guided waves in chiral media,” Prog. Electromagn. Res. 9, 311–350 (1994).

1993 (1)

K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
[CrossRef]

1992 (1)

1990 (1)

J. Svedin, “Propagation analysis of chirowaveguides using the finite-element method,” IEEE Trans. Microwave Theory Tech. 38, 1488–1496 (1990).
[CrossRef]

1989 (2)

N. Engheta and P. Pelet, “Modes in chirowaveguides,” Opt. Lett. 14, 593–595 (1989).
[CrossRef]

C. Eftimiu and L. Parson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

1988 (1)

N. Engheta and D. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Newsletter 30(5), 6–12 (1988).

1978 (1)

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. A 68, 1196–1201 (1978).
[CrossRef]

1974 (1)

C. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

Argyros, A.

Barbosa, A. M.

Bohren, C.

R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
[CrossRef]

C. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

Brewitt-Taylor, C.

C. Brewitt-Taylor, P. Lederer, F. Smith, and S. Haq, “Measurement and prediction of helix-loaded chiral composites,” IEEE Trans. Antennas Propag. 47, 692–700 (1999).
[CrossRef]

Cao, Y.

Choudhury, P.

A. Nair and P. Choudhury, “On the analysis of field patterns in chirofibers,” J. Electromagn. Waves Appl. 21, 2277–2286 (2007).
[CrossRef]

P. Choudhury and T. Yoshino, “Dependence of optical power confinement on core/cladding chiralities in chirofibers,” Microw. Opt. Technol. Lett. 32, 359–364 (2002).
[CrossRef]

K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
[CrossRef]

Cory, H.

H. Cory, “Chiral devices-an overview of canonical problems,” J. Electromagn. Waves Appl. 9, 805–829 (1995).
[CrossRef]

Docherty, A.

Dong, J.-F.

J.-F. Dong and J. Li, “Characteristics of guided modes in uniaxial chiral circular waveguides,” Prog. Electromagn. Res. 124, 331–345 (2012).
[CrossRef]

J.-F. Dong, W.-D. Tao, and J. Xu, “Optical power characteristics of guided modes in a double-cladding chiral optical fiber,” Acta Photon. Sin. 36, 1044–1049 (2007).

Eftimiu, C.

C. Eftimiu and L. Parson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

Engeness, T.

Engheta, N.

F. Mariotte, P. Pelet, and N. Engheta, “A review of recent study of guided waves in chiral media,” Prog. Electromagn. Res. 9, 311–350 (1994).

N. Engheta and P. Pelet, “Modes in chirowaveguides,” Opt. Lett. 14, 593–595 (1989).
[CrossRef]

N. Engheta and D. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Newsletter 30(5), 6–12 (1988).

Fink, Y.

Flood, K. M.

Genack, A. Z.

G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
[CrossRef]

Goldstein, D.

D. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2010).

Haq, S.

C. Brewitt-Taylor, P. Lederer, F. Smith, and S. Haq, “Measurement and prediction of helix-loaded chiral composites,” IEEE Trans. Antennas Propag. 47, 692–700 (1999).
[CrossRef]

Herman, W. N.

Hunsberger, F.

R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
[CrossRef]

Ibanescu, M.

Jacobs, S.

Jaggard, D.

N. Engheta and D. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Newsletter 30(5), 6–12 (1988).

Jaggard, D. L.

Janeiro, F. M.

Joannopoulos, J.

Johnson, S.

Khastgir, P.

K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
[CrossRef]

Kopp, V. I.

G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series Vol. 335 (Springer, 1989).

Langdon, H.

R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
[CrossRef]

Lederer, P.

C. Brewitt-Taylor, P. Lederer, F. Smith, and S. Haq, “Measurement and prediction of helix-loaded chiral composites,” IEEE Trans. Antennas Propag. 47, 692–700 (1999).
[CrossRef]

Li, J.

Lu, I.-T.

Luebbers, R.

R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
[CrossRef]

Mahmoud, S.

S. Mahmoud, “Guided modes on open chirowaveguides,” IEEE Trans. Microwave Theory Tech. 43, 205–209 (1995).
[CrossRef]

Mariotte, F.

F. Mariotte, P. Pelet, and N. Engheta, “A review of recent study of guided waves in chiral media,” Prog. Electromagn. Res. 9, 311–350 (1994).

Marom, E.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. A 68, 1196–1201 (1978).
[CrossRef]

Misra, V.

K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
[CrossRef]

Nair, A.

A. Nair and P. Choudhury, “On the analysis of field patterns in chirofibers,” J. Electromagn. Waves Appl. 21, 2277–2286 (2007).
[CrossRef]

Neugroschl, D.

G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
[CrossRef]

Ojha, S.

K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
[CrossRef]

Paiva, C. R.

Parson, L.

C. Eftimiu and L. Parson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

Pelet, P.

F. Mariotte, P. Pelet, and N. Engheta, “A review of recent study of guided waves in chiral media,” Prog. Electromagn. Res. 9, 311–350 (1994).

N. Engheta and P. Pelet, “Modes in chirowaveguides,” Opt. Lett. 14, 593–595 (1989).
[CrossRef]

Poladian, L.

Qiu, R. C.

Shvets, G.

G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
[CrossRef]

Singh, K.

K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
[CrossRef]

Skorobogatiy, M.

Smith, F.

C. Brewitt-Taylor, P. Lederer, F. Smith, and S. Haq, “Measurement and prediction of helix-loaded chiral composites,” IEEE Trans. Antennas Propag. 47, 692–700 (1999).
[CrossRef]

Soljacic, M.

Straton, M.

Su, Q.

Svedin, J.

J. Svedin, “Propagation analysis of chirowaveguides using the finite-element method,” IEEE Trans. Microwave Theory Tech. 38, 1488–1496 (1990).
[CrossRef]

Tao, W.-D.

J.-F. Dong, W.-D. Tao, and J. Xu, “Optical power characteristics of guided modes in a double-cladding chiral optical fiber,” Acta Photon. Sin. 36, 1044–1049 (2007).

Topa, A. L.

Trendafilov, S.

G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
[CrossRef]

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series Vol. 335 (Springer, 1989).

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series Vol. 335 (Springer, 1989).

Weisberg, O.

Xu, J.

J.-F. Dong, W.-D. Tao, and J. Xu, “Optical power characteristics of guided modes in a double-cladding chiral optical fiber,” Acta Photon. Sin. 36, 1044–1049 (2007).

Yariv, A.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. A 68, 1196–1201 (1978).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. A 68, 1196–1201 (1978).
[CrossRef]

Yoshikawa, S.

R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
[CrossRef]

Yoshino, T.

P. Choudhury and T. Yoshino, “Dependence of optical power confinement on core/cladding chiralities in chirofibers,” Microw. Opt. Technol. Lett. 32, 359–364 (2002).
[CrossRef]

Zwillinger, D.

D. Zwillinger, Standard Mathematical Tables and Formulae, 31st ed (CRC Press, 2003).

Acta Photon. Sin. (1)

J.-F. Dong, W.-D. Tao, and J. Xu, “Optical power characteristics of guided modes in a double-cladding chiral optical fiber,” Acta Photon. Sin. 36, 1044–1049 (2007).

Chem. Phys. Lett. (1)

C. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

IEEE Antennas Propag. Newsletter (1)

N. Engheta and D. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Newsletter 30(5), 6–12 (1988).

IEEE Trans. Antennas Propag. (2)

C. Brewitt-Taylor, P. Lederer, F. Smith, and S. Haq, “Measurement and prediction of helix-loaded chiral composites,” IEEE Trans. Antennas Propag. 47, 692–700 (1999).
[CrossRef]

R. Luebbers, H. Langdon, F. Hunsberger, C. Bohren, and S. Yoshikawa, “Calculation and measurement of the effective chirality parameter of a composite chiral material over a wide frequency band,” IEEE Trans. Antennas Propag. 43, 123–130 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

S. Mahmoud, “Guided modes on open chirowaveguides,” IEEE Trans. Microwave Theory Tech. 43, 205–209 (1995).
[CrossRef]

J. Svedin, “Propagation analysis of chirowaveguides using the finite-element method,” IEEE Trans. Microwave Theory Tech. 38, 1488–1496 (1990).
[CrossRef]

J. Electromagn. Waves Appl. (2)

A. Nair and P. Choudhury, “On the analysis of field patterns in chirofibers,” J. Electromagn. Waves Appl. 21, 2277–2286 (2007).
[CrossRef]

H. Cory, “Chiral devices-an overview of canonical problems,” J. Electromagn. Waves Appl. 9, 805–829 (1995).
[CrossRef]

J. Opt. A (1)

G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A 11, 074007 (2009).
[CrossRef]

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

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

J. Phys. Soc. Jpn. (1)

K. Singh, P. Choudhury, V. Misra, P. Khastgir, and S. Ojha, “Field cutoffs of three-layer parabolically deformed planar chirowaveguides,” J. Phys. Soc. Jpn. 62, 3778–3782 (1993).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

P. Choudhury and T. Yoshino, “Dependence of optical power confinement on core/cladding chiralities in chirofibers,” Microw. Opt. Technol. Lett. 32, 359–364 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Prog. Electromagn. Res. (2)

F. Mariotte, P. Pelet, and N. Engheta, “A review of recent study of guided waves in chiral media,” Prog. Electromagn. Res. 9, 311–350 (1994).

J.-F. Dong and J. Li, “Characteristics of guided modes in uniaxial chiral circular waveguides,” Prog. Electromagn. Res. 124, 331–345 (2012).
[CrossRef]

Radio Sci. (1)

C. Eftimiu and L. Parson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

Other (3)

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series Vol. 335 (Springer, 1989).

D. Zwillinger, Standard Mathematical Tables and Formulae, 31st ed (CRC Press, 2003).

D. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2010).

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Figures (7)

Fig. 1.
Fig. 1.

Sketch of chiral multilayered fibers with finite layers; rj denotes the radial coordinate of the jth interface numbered from the inside out.

Fig. 2.
Fig. 2.

Comparison of the average energy flux and the polarization state between a mode in some chiral fibers and the corresponding mode of mirror image that exists in the enantiomorphous fiber. S¯ and s are circularly symmetric. Those modal quantities that are not of handedness, including β, S¯z, S¯r and s1, which describe the linear radially or azimuthally polarized component P(s1), are the same. Other quantities that are of handedness including m, which determines the rotational direction of the field pattern at a fixed z as the time elapses, S¯ϕ and s2, which describe the linear radially +π/4 or π/4 polarized component P(s2), and s3, which describes the RCP or LCP component P(s3), are opposite. For one mode, only one of the two polarizations characterized by each Stokes parameter is presented for the sake of clarity of the figure.

Fig. 3.
Fig. 3.

Index profile of Bragg fibers with a chiral core. The two dashed lines indicate the split effective indices of RCP and LCP waves in the bulk core medium. In our calculations, the cladding consists of ten pairs of high/low index layers, with the outermost layer extending to infinity.

Fig. 4.
Fig. 4.

Waveguide dispersion of the fundamental modes. Dispersion curves are not globally flat; wrinkling (inset A) happens in the wavelengths corresponding to the sidelobes of the loss spectrum (Fig. 5). Gray stripes show the regions where the mode with m=1 departs from circular polarization due to mode transitions (inset B).

Fig. 5.
Fig. 5.

Loss of the fundamental modes: (a) δ=0.001, inset shows the Sz (in the core) of a nearly pure RCP mode and (b) δ=0 and δ=0.005, inset shows the Sz (in the core) of a mode with a low purity of circular polarization. The snapshot of Sz is taken at βzωt=0.

Fig. 6.
Fig. 6.

Distributions of s3 and S¯z of the fundamental RCP mode (solid lines, loss 1.0dB/m) and the counterpart of the TE01 mode (dashed lines, loss 2.7dB/m) in the chiral Bragg fiber shown in Fig. 3 with δ=0.001. The vertical dashed lines indicate the interfaces. Two modes are both highly RCP with nearly the same s3 distribution, but there is no bifurcated counterpart for the latter in the same fiber (a LCP mirror image exists in the enantiomorphous fiber with δ=0.001).

Fig. 7.
Fig. 7.

Loss of the fundamental modes. Dispersion of the complex-valued chiral parameter δ is governed by Eq. (7), with δ0=0.005, ω0=1.88×1017Hz and γ=0.0167ω0. A necessary imaginary part is added to nc to eliminate the nonphysical effect induced by the complex δ. Other parameters of the fiber are the same as those of the fiber in Figs. 4 and 5, with δ=0.005. LLCP indicates the loss of LCP wave in the bulk core medium. Gray stripes show the regions where the mode with m=1 departs from circular polarization due to mode transitions.

Equations (13)

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×F±=±k±F±,
F±z=a±Jm(κ±r)+b±Ym(κ±r),F±r=ia±J¯m(κ±,±k±,β,r)+ib±Y¯m(κ±,±k±,β,r),F±ϕ=a±J¯m(κ±,β,±k±,r)b±Y¯m(κ±,β,±k±,r),
B¯m(σ1,σ2,σ3,r)=1σ1[mσ2Bm(σ1r)σ1r+σ3dBm(σ1r)dσ1r],
Ω=[Op+Opq+q],O=MN1(rN1)ΘM21(r1),
Θ=MN11(rN2)MN2(rN2)M41(r3)M3(r3)M31(r2)M2(r2),
M11=Jm(κ+r),M12=Ym(κ+r),M13=Jm(κr),M14=Ym(κr),M21=M11η,M22=M12η,M23=M13η,M24=M14η,M31=J¯m(κ+,β,k+,r),M32=Y¯m(κ+,β,k+,r),M33=J¯m(κ,β,k,r),M34=Y¯m(κ,β,k,r),M41=M31η,M42=M32η,M43=M33η,M44=M34η,
Bm(κ±r)(1)mBm(κr),B¯m(κ±,±k±,β,r)(1)mB¯m(κ,k,β,r),B¯m(κ±,β,±k±,r)(1)mB¯m(κ,β,k,r),
F±zFz,F±rFr,F±ϕFϕ,
(E,H)z(E,H)z,(E,H)r(E,H)r,(E,H)ϕ(E,H)ϕ,
S¯z=R(ErHϕ*EϕHr*)/2,S¯r=R(EϕHz*EzHϕ*)/2,s1=(ErEr*Eϕ*Eϕ)/(ErEr*+Eϕ*Eϕ),
s2=(ErEϕ*+Er*Eϕ)/(ErEr*+Eϕ*Eϕ),s3=i(ErEϕ*Er*Eϕ)/(ErEr*+Eϕ*Eϕ),S¯ϕ=R(EzHr*ErHz*)/2,
(E±,H±)z(E,H)z,(E±,H±)r(E,H)r,(E±,H±)ϕ(E,H)ϕ.
δ=δ0ω02ω22iωγω02,

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