Abstract

The effect of chirality on the modal dispersion characteristics of optical fibers is investigated. Using a step-index fiber as a numerical example, we show that adding chirality to the core changes the modal normalized propagation constants b, group delay, and waveguide dispersion drastically. Some potential applications are briefly discussed.

© 1994 Optical Society of America

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References

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  1. D. Marcuse, “Selected topics in the theory of telecommunication fibers,” in Optical Fiber Telecommunication II, S. E. Miller, ed. (Academic, New York, 1988).
  2. E. G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988), Chaps. 4 and 5.
    [CrossRef]
  3. G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991), Chaps. 2 and 3.
  4. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 8.
  5. S. E. Miller, A. G. Chynoweth, Optical Telecommunications (Academic, New York, 1979), p. 103.
  6. B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” IEEE J. Light-wave Technol. LT-4, 967–979 (1986).
    [CrossRef]
  7. D. Marcuse, C. Lin, “Low dispersion single-mode fiber transmission-the question of practical versus theoretical maximum transmission bandwidth,” IEEE J. Quantum Electron. QE-17, 869–878 (1981).
    [CrossRef]
  8. M. A Saifi, S. J. Jang, L. B. Cohen, J. Stone, “Triangular-profile single-mode fiber,” Opt. Lett. 7, 43–45 (1982).
    [CrossRef] [PubMed]
  9. W. L. Mammel, L. G. Cohen, “Numerical prediction of fiber transmission characteristics from arbitrary refractive-index profiles,” Appl. Opt. 21, 699–703 (1982).
    [CrossRef] [PubMed]
  10. D. L. Jaggard, N. Engheta, “Chirality in electromagnetics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.
  11. A. Lakhtakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristic of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. 28, 90–94 (1986).
    [CrossRef]
  12. A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Vol. 335 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1989).
  13. N. Engheta, A. R. Mickelson, “Transition radiation caused by a chiral plate,” IEEE Trans. Antennas Propag. AP-30, 1213–1216 (1982).
    [CrossRef]
  14. N. Engheta, M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 15, 639–647 (1990).
    [CrossRef]
  15. M. W. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 289–301 (1990).
    [CrossRef]
  16. M. Chien, Y. Kim, H. Grebel, “Mode conversion in optically active and isotropic waveguides,” Opt. Lett. 14, 826–828 (1989).
    [CrossRef] [PubMed]
  17. C. Eftimiu, L. W. Pearson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
    [CrossRef]
  18. N. Engheta, P. Pelet, “Surface waves in chiral layers,” Opt. Lett. 16, 723–725 (1991).
    [CrossRef] [PubMed]
  19. M. S. Kluskens, E. H. Newman, “Microstrip lines on a chiral substrate,” IEEE Trans. Microwave Theory Tech. 39, 1889–1891 (1991).
    [CrossRef]
  20. P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
    [CrossRef]
  21. P. Pelet, N. Engheta, “Coupled-mode theory for chirowaveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
    [CrossRef]
  22. J. M. Sveden, “Propagation analysis of chirowaveguides using the finite-element method,” IEEE Trans. Microwave Theory Tech. 38, 1488–1496 (1990).
    [CrossRef]
  23. M. S. Kluskens, E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448–1455 (1990).
    [CrossRef]
  24. M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
    [CrossRef]
  25. M. H. Uman, V. V. Varadan, V. K. Varadan, “Rotation and dichroism associated with microwave propagation neffin chiral composite samples,” Radio Sci. 26, 1327–1334 (1991).
    [CrossRef]
  26. V. V. Varadan, Y. Ma, V. K. Varadan, “Effects of chiral microstructure on em wave propagation in discrete random media,” Radio Sci. 24, 785–792 (1989).
    [CrossRef]
  27. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.
  28. R. A. Waldron, Theory of Guided Electromagnetic Waves (Van Nostrand, London, 1969), Chap. 7.

1991 (4)

N. Engheta, P. Pelet, “Surface waves in chiral layers,” Opt. Lett. 16, 723–725 (1991).
[CrossRef] [PubMed]

M. S. Kluskens, E. H. Newman, “Microstrip lines on a chiral substrate,” IEEE Trans. Microwave Theory Tech. 39, 1889–1891 (1991).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

M. H. Uman, V. V. Varadan, V. K. Varadan, “Rotation and dichroism associated with microwave propagation neffin chiral composite samples,” Radio Sci. 26, 1327–1334 (1991).
[CrossRef]

1990 (6)

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

P. Pelet, N. Engheta, “Coupled-mode theory for chirowaveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
[CrossRef]

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

M. S. Kluskens, E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448–1455 (1990).
[CrossRef]

N. Engheta, M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 15, 639–647 (1990).
[CrossRef]

M. W. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 289–301 (1990).
[CrossRef]

1989 (3)

M. Chien, Y. Kim, H. Grebel, “Mode conversion in optically active and isotropic waveguides,” Opt. Lett. 14, 826–828 (1989).
[CrossRef] [PubMed]

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

V. V. Varadan, Y. Ma, V. K. Varadan, “Effects of chiral microstructure on em wave propagation in discrete random media,” Radio Sci. 24, 785–792 (1989).
[CrossRef]

1986 (2)

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristic of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. 28, 90–94 (1986).
[CrossRef]

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” IEEE J. Light-wave Technol. LT-4, 967–979 (1986).
[CrossRef]

1982 (3)

1981 (1)

D. Marcuse, C. Lin, “Low dispersion single-mode fiber transmission-the question of practical versus theoretical maximum transmission bandwidth,” IEEE J. Quantum Electron. QE-17, 869–878 (1981).
[CrossRef]

Ainslie, B. J.

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” IEEE J. Light-wave Technol. LT-4, 967–979 (1986).
[CrossRef]

Chien, M.

Chynoweth, A. G.

S. E. Miller, A. G. Chynoweth, Optical Telecommunications (Academic, New York, 1979), p. 103.

Cohen, L. B.

Cohen, L. G.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

Day, C. R.

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” IEEE J. Light-wave Technol. LT-4, 967–979 (1986).
[CrossRef]

Eftimiu, C.

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

Engheta, N.

N. Engheta, P. Pelet, “Surface waves in chiral layers,” Opt. Lett. 16, 723–725 (1991).
[CrossRef] [PubMed]

M. W. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 289–301 (1990).
[CrossRef]

N. Engheta, M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 15, 639–647 (1990).
[CrossRef]

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

P. Pelet, N. Engheta, “Coupled-mode theory for chirowaveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
[CrossRef]

N. Engheta, A. R. Mickelson, “Transition radiation caused by a chiral plate,” IEEE Trans. Antennas Propag. AP-30, 1213–1216 (1982).
[CrossRef]

D. L. Jaggard, N. Engheta, “Chirality in electromagnetics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.

Grebel, H.

Jaggard, D. L.

D. L. Jaggard, N. Engheta, “Chirality in electromagnetics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.

Jang, S. J.

Keiser, G.

G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991), Chaps. 2 and 3.

Kim, Y.

Kluskens, M. S.

M. S. Kluskens, E. H. Newman, “Microstrip lines on a chiral substrate,” IEEE Trans. Microwave Theory Tech. 39, 1889–1891 (1991).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448–1455 (1990).
[CrossRef]

Kowarz, M. W.

N. Engheta, M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 15, 639–647 (1990).
[CrossRef]

M. W. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 289–301 (1990).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristic of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. 28, 90–94 (1986).
[CrossRef]

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

Lin, C.

D. Marcuse, C. Lin, “Low dispersion single-mode fiber transmission-the question of practical versus theoretical maximum transmission bandwidth,” IEEE J. Quantum Electron. QE-17, 869–878 (1981).
[CrossRef]

Ma, Y.

V. V. Varadan, Y. Ma, V. K. Varadan, “Effects of chiral microstructure on em wave propagation in discrete random media,” Radio Sci. 24, 785–792 (1989).
[CrossRef]

Mammel, W. L.

Marcuse, D.

D. Marcuse, C. Lin, “Low dispersion single-mode fiber transmission-the question of practical versus theoretical maximum transmission bandwidth,” IEEE J. Quantum Electron. QE-17, 869–878 (1981).
[CrossRef]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 8.

D. Marcuse, “Selected topics in the theory of telecommunication fibers,” in Optical Fiber Telecommunication II, S. E. Miller, ed. (Academic, New York, 1988).

Mickelson, A. R.

N. Engheta, A. R. Mickelson, “Transition radiation caused by a chiral plate,” IEEE Trans. Antennas Propag. AP-30, 1213–1216 (1982).
[CrossRef]

Miller, S. E.

S. E. Miller, A. G. Chynoweth, Optical Telecommunications (Academic, New York, 1979), p. 103.

Neumann, E. G.

E. G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988), Chaps. 4 and 5.
[CrossRef]

Newman, E. H.

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Microstrip lines on a chiral substrate,” IEEE Trans. Microwave Theory Tech. 39, 1889–1891 (1991).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448–1455 (1990).
[CrossRef]

Pearson, L. W.

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

Pelet, P.

N. Engheta, P. Pelet, “Surface waves in chiral layers,” Opt. Lett. 16, 723–725 (1991).
[CrossRef] [PubMed]

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

P. Pelet, N. Engheta, “Coupled-mode theory for chirowaveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
[CrossRef]

Saifi, M. A

Stone, J.

Sveden, J. M.

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

Uman, M. H.

M. H. Uman, V. V. Varadan, V. K. Varadan, “Rotation and dichroism associated with microwave propagation neffin chiral composite samples,” Radio Sci. 26, 1327–1334 (1991).
[CrossRef]

Varadan, V. K.

M. H. Uman, V. V. Varadan, V. K. Varadan, “Rotation and dichroism associated with microwave propagation neffin chiral composite samples,” Radio Sci. 26, 1327–1334 (1991).
[CrossRef]

V. V. Varadan, Y. Ma, V. K. Varadan, “Effects of chiral microstructure on em wave propagation in discrete random media,” Radio Sci. 24, 785–792 (1989).
[CrossRef]

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristic of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. 28, 90–94 (1986).
[CrossRef]

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

Varadan, V. V.

M. H. Uman, V. V. Varadan, V. K. Varadan, “Rotation and dichroism associated with microwave propagation neffin chiral composite samples,” Radio Sci. 26, 1327–1334 (1991).
[CrossRef]

V. V. Varadan, Y. Ma, V. K. Varadan, “Effects of chiral microstructure on em wave propagation in discrete random media,” Radio Sci. 24, 785–792 (1989).
[CrossRef]

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristic of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. 28, 90–94 (1986).
[CrossRef]

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

Waldron, R. A.

R. A. Waldron, Theory of Guided Electromagnetic Waves (Van Nostrand, London, 1969), Chap. 7.

Appl. Opt. (1)

IEEE J. Light-wave Technol. (1)

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” IEEE J. Light-wave Technol. LT-4, 967–979 (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Marcuse, C. Lin, “Low dispersion single-mode fiber transmission-the question of practical versus theoretical maximum transmission bandwidth,” IEEE J. Quantum Electron. QE-17, 869–878 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

N. Engheta, A. R. Mickelson, “Transition radiation caused by a chiral plate,” IEEE Trans. Antennas Propag. AP-30, 1213–1216 (1982).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448–1455 (1990).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “A parametric study of microwave reflection characteristic of a planar achiral–chiral interface,” IEEE Trans. Electromagn. Compat. 28, 90–94 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

M. S. Kluskens, E. H. Newman, “Microstrip lines on a chiral substrate,” IEEE Trans. Microwave Theory Tech. 39, 1889–1891 (1991).
[CrossRef]

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

J. Appl. Phys. (2)

P. Pelet, N. Engheta, “Coupled-mode theory for chirowaveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
[CrossRef]

N. Engheta, M. W. Kowarz, “Antenna radiation in the presence of a chiral sphere,” J. Appl. Phys. 15, 639–647 (1990).
[CrossRef]

Opt. Lett. (4)

Radio Sci. (3)

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

M. H. Uman, V. V. Varadan, V. K. Varadan, “Rotation and dichroism associated with microwave propagation neffin chiral composite samples,” Radio Sci. 26, 1327–1334 (1991).
[CrossRef]

V. V. Varadan, Y. Ma, V. K. Varadan, “Effects of chiral microstructure on em wave propagation in discrete random media,” Radio Sci. 24, 785–792 (1989).
[CrossRef]

Other (9)

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

R. A. Waldron, Theory of Guided Electromagnetic Waves (Van Nostrand, London, 1969), Chap. 7.

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

D. L. Jaggard, N. Engheta, “Chirality in electromagnetics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.

D. Marcuse, “Selected topics in the theory of telecommunication fibers,” in Optical Fiber Telecommunication II, S. E. Miller, ed. (Academic, New York, 1988).

E. G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988), Chaps. 4 and 5.
[CrossRef]

G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991), Chaps. 2 and 3.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 8.

S. E. Miller, A. G. Chynoweth, Optical Telecommunications (Academic, New York, 1979), p. 103.

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Figures (12)

Fig. 1
Fig. 1

Configuration of a chiral step-index optical fiber.

Fig. 2
Fig. 2

Normalized phase constant β/k0 for the HEn1 mode of a chiral step-index optical fiber, with ξc = 0.001 and r = 1.1. Our results (solid curves) agree well with the numerical results of Ref. 22 (dotted curves).

Fig. 3
Fig. 3

Normalized phase constant β/k0 for a chiral step-index optical fiber of different ξc with r = 2.56, for the HE11 mode. Dotted curve, ξc = 0; solid curve, ξc = 10−5; dashed–dotted curve, ξc = 10−4; dashed curve, ξc = 10−3.

Fig. 4
Fig. 4

Normalized phase constant β/k0 for the HE11 mode of a chiral step-index optical fiber with different r. ξc = 0.001. Dotted curve, r = 1.1; solid curve, r = 1.5; dotted–dashed curve, r = 2.0; dashed curve, r = 2.56.

Fig. 5
Fig. 5

Normalized phase constants β/k0 of the HE11 and H12 modes versus the normalized parameter k0a. Parameters: n1 = n2 + Δn, n2 = 1.46, Δn = 0.012.

Fig. 6
Fig. 6

Normalized phase constants β/k0 of the HE11, HE−1.1, and HE01 modes versus the normalized parameter k0a for different ξc. Parameters: n1 = n2 + Δn, n2 = 1.46, Δn = 0.012.

Fig. 7
Fig. 7

Effective group refractive index Neff of the fundamental HE11 mode versus vacuum wavelength λ. Parameters: n1 = n2 + Δn, n2 = 1.46, Δn = 0.012.

Fig. 8
Fig. 8

Waveguide dispersion D of the fundamental HE11 mode versus vacuum wavelength λ. Parameters: n1 = n2 + Δn, n2 = 1.46, Δn = 0.012.

Fig. 9
Fig. 9

Functional relations between the normalized parameter b and the normalized frequency parameter V for the fundamental HE11 mode and higher-order modes, with chirality ξc = 0.001. The curves with symbols represent HE1m (m = 1–3), and the unmarked curves represent HE0m (m = 1–3).

Fig. 10
Fig. 10

Functional relations between the normalized parameter b and the normalized frequency parameter V for the fundamental HE11 mode and higher-order modes, with chirality ξc = 0.001. The curves with symbols represent HE1m (m = 1–3), and the unmarked curves represent HE−1,m (m = 1–3).

Fig. 11
Fig. 11

Normalized parameter b versus the normalized frequency parameter V for different chiralities: (a) HE11 mode, (b) HE−1.1 mode. The values of ξc are given next to their corresponding curves.

Fig. 12
Fig. 12

Normalized parameter b versus the normalized frequency parameter V for different chiralities for the fundamental HE11 mode and the first two higher-order modes, HE12 and HE13. (a) Curves with symbols, ξc = 0; unmarked curves, ξc = 0.0004. (b) Curves with symbols, ξc = 0.0001; unmarked curves, ξc = 0.0006.

Equations (18)

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× × ( E H ) - 2 ω μ ξ c × ( E H ) - k 2 ( E H ) = 0 ,
t 2 E z + c E z + j d H z = 0 ,
t 2 H z + f H z - j g E z = 0 ,
c = f = ( k + 2 + k - 2 ) / 2 - β 2 ,             d = 2 ω 2 μ 2 ξ c , g = 2 ω 2 μ 2 ξ c / η c 2 k ± = ± ω μ ξ c + k 2 + ( ω μ ξ c ) 2 ,             η c 2 = η / ( 1 + η 2 ξ c 2 ) , η = μ / .
[ E r ; E θ ; H r ; H θ ] T = [ a i j ] i , j = 1 - 4 [ E z r ; 1 r E z θ ; H z r ; 1 r H z θ ] T , a 11 = a 22 = a 33 = a 44 = - j β ( k + 2 + k - 2 2 - β 2 ) / Δ 2 , a 12 = a 21 = a 34 = - a 43 = - ω μ ξ c ( k 2 + β 2 ) / Δ 2 , a 13 = a 24 = 2 ω 2 μ 2 β ξ c / Δ 2 , a 14 = - a 23 = - j ω μ ( k 2 - β 2 ) / Δ 2 , a 31 = a 42 = - 2 ω 2 μ 2 β ξ c / η c 2 Δ 2 , a 32 = - a 41 = j ω μ ( k 2 - β 2 ) / η c 2 Δ 2 , Δ 2 = ( k + 2 - β 2 ) ( k - 2 - β 2 ) .
( t 2 + s 1 2 ) ( t 2 + s 2 2 ) ( E z H z ) = 0 ,
( t 2 + s 1 2 ) ( E z H z ) = 0 ,             ( t 2 + s 2 2 ) ( E z H z ) = 0 ,
E z = [ A 1 J n ( s 1 r ) + A 2 J n ( s 2 r ) ] exp ( - j n θ ) ,
H z = [ τ 1 A 1 J n ( s 1 r ) + τ 2 A 2 J n ( s 2 r ) ] exp ( - j n θ ) ,
τ 1 = - j g / ( s 1 2 - f ) ,             τ 2 = - j g / ( s 2 2 - f ) ;
E z = A 3 K n ( p r ) exp ( - j n θ ) ,             H z = A 4 K n ( p r ) exp ( - j n θ ) ,
| J n ( s 1 a ) J n ( s 2 a ) - K n 0 τ 1 J n ( s 1 a ) τ 2 J n ( s 2 a ) 0 - K n P 1 P 2 - n β K n p 2 a j ω μ 0 K n p Q 1 Q 2 - j ω 0 K n p - n β K n p 2 a | = 0 ,
P i = a 21 s i J n ( s i a ) + a 22 ( - j n / a ) J n ( s i a ) + a 23 τ i s i J n ( s i a ) + ( - j n / a ) τ i a 24 J n ( s i a ) ; Q i = a 41 s i J n ( s i a ) + a 42 ( - j n / a ) J n ( s i a ) + a 43 τ i s i J n ( s i a ) + ( - j n / a ) τ i a 44 J n ( s i a ) ,             i = 1 , 2 ; K n = K n ( p a ) ;             K n = K n ( p a ) .
τ = L d β d ω = - L λ 2 2 π c d β d λ , D = 1 L d τ d λ = - 1 2 π c ( 2 λ d β d λ + λ 2 d 2 β d λ 2 ) ,
τ = L c ( β ¯ - λ d β ¯ d λ ) = L c N eff ,             D = - 1 c λ d 2 β ¯ d λ 2 ,
Δ β = β 1 - β 2 = [ β 1 ( ω ) ω - β 2 ( ω ) ω ] ω = ω 0 ( ω - ω 0 ) ,
| J n ( s 1 a ) 0 - K n 0 0 J n ( s 2 a ) 0 - K n - n β h 2 a J n ( s 1 a ) j ω μ 0 h J n ( s 2 a ) - n β p 2 a K n j ω μ 0 p K n - j ω h J n ( s 1 a ) - n β h 2 a J n ( s 2 a ) j ω 0 p K n - n β p 2 a K n | = 0 ,
[ r J n ( u 1 ) u 1 J n ( u 1 ) + K n ( u 2 ) u 2 K n ( u 2 ) ] [ J n ( u 1 ) u 1 J n ( u 1 ) + K n ( u 2 ) u 2 K n ( u 2 ) ] = [ n β ( u 2 2 + u 1 2 ) k 0 u 2 2 u 2 2 ] 2 ,

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