Abstract

Direct measurement of the ultimate limit of the polarization cross talk of the birefringent fibers caused by anisotropic Rayleigh scattering (ARS) is presented theoretically and experimentally by using polarization optical time-domain reflectometry (POTDR). With POTDR, ARS appears as the bias existing in the backscattered power ratio between the cross-coupled and excited modes. A value twice that of the bias corresponds to the depolarization ratio representing the anisotropy of the fiber core material. The biases measured by birefringent polarization-maintaining fibers are all within 3% of the mean value 0.048.

© 1986 Optical Society of America

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References

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  1. Y. Yamamoto, T. Kimura, “Coherent optical fiber transmission systems,” IEEE J. Quantum Electron. QE-17, 919–935 (1981).
    [CrossRef]
  2. R. Ulrich, M. Johnson, “Fiber-ring interferometer: polarization analysis,” Opt. Lett. 4, 152–154 (1979).
    [CrossRef] [PubMed]
  3. V. Ramaswamy, W. G. French, R. D. Standley, “Polarization characteristics of noncircular core single mode fibers,” Appl. Opt. 17, 3014–3017 (1978).
    [CrossRef] [PubMed]
  4. T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
    [CrossRef]
  5. R. Birch, D. Payne, M. P. Varnham, “Fabrication of polarisation maintaining fibers using gas-phase etching,” Electron. Lett.1036–1037 (1982).
    [CrossRef]
  6. K. Okamoto, Y. Sasaki, N. Shibata, “Mode coupling effects in stress-applied single polarization fibers,” IEEE J. Quantum Electron. QE-18, 1890–1899 (1982).
    [CrossRef]
  7. Y. Sasaki, T. Hosaka, K. Takada, J. Noda, “8 km-long polarisation-maintaining fibres with highly stable polarisation state,” Electron. Lett. 19, 792–794 (1983).
    [CrossRef]
  8. T. Hosaka, Y. Sasaki, K. Okamoto, J. Noda, “Stress-applied polarization-maintaining optical fibers-design and fabrication,” IECE Jpn. J67-C, 741–748 (1984).
  9. E. Brinkmeyer, W. Eickhoff, “Ultimate limit of polarisation holding in single mode fibres,” Electron. Lett. 19, 996–997 (1983).
    [CrossRef]
  10. M. Nakazawa, M. Tokuda, Y. Negishi, “Measurement of polarization mode coupling along a polarization-maintaining optical fiber using a backscattered technique,” Opt. Lett. 8, 546–548 (1983).
    [CrossRef] [PubMed]
  11. K. Takada, K. Okamoto, J. Noda, “Polarization mode coupling with broad-band source in birefringent polarization-maintaining fibers,” J. Opt. Soc. Am. A 2, 753–758 (1985).
    [CrossRef]
  12. D. Marcuse, “Coupled-mode theory for anisotropic optical waveguide,” Bell Syst. Tech. J. 54, 985–995 (1975).
    [CrossRef]
  13. S. C. Rashleigh, W. K. Burns, R. P. Moeller, R. Ulrich, “Polarization holding in birefringent single mode fibers,” Opt. Lett. 7, 40–42 (1982).
    [CrossRef] [PubMed]
  14. K. Bôhm, K. Peterman, E. Weidel, “Performance of Lyot depolarizer with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
    [CrossRef]
  15. M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-polarization operation of highly birefringent bow-tie optical fibres,” Electron. Lett. 19, 246–247 (1983).
    [CrossRef]
  16. M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend behavior of polarisation optical fibres,” Electron. Lett. 19, 679–680 (1983).
    [CrossRef]
  17. J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
    [CrossRef]
  18. H. N. Daglish, “Light scattering in selected optical glasses,” Glass Technol. 11, 30–35 (1970).
  19. E. Brinkmeyer, “Analysis of the backscattering method,” J. Opt. Soc. Am. 70, 1010–1012 (1980).
    [CrossRef]
  20. M. Horiguchi, H. Osanai, “Spectral losses of low-OH-content optical fibres,” Electron. Lett. 12, 310–312 (1976).
    [CrossRef]

1985 (1)

1984 (1)

T. Hosaka, Y. Sasaki, K. Okamoto, J. Noda, “Stress-applied polarization-maintaining optical fibers-design and fabrication,” IECE Jpn. J67-C, 741–748 (1984).

1983 (7)

E. Brinkmeyer, W. Eickhoff, “Ultimate limit of polarisation holding in single mode fibres,” Electron. Lett. 19, 996–997 (1983).
[CrossRef]

M. Nakazawa, M. Tokuda, Y. Negishi, “Measurement of polarization mode coupling along a polarization-maintaining optical fiber using a backscattered technique,” Opt. Lett. 8, 546–548 (1983).
[CrossRef] [PubMed]

Y. Sasaki, T. Hosaka, K. Takada, J. Noda, “8 km-long polarisation-maintaining fibres with highly stable polarisation state,” Electron. Lett. 19, 792–794 (1983).
[CrossRef]

K. Bôhm, K. Peterman, E. Weidel, “Performance of Lyot depolarizer with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-polarization operation of highly birefringent bow-tie optical fibres,” Electron. Lett. 19, 246–247 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend behavior of polarisation optical fibres,” Electron. Lett. 19, 679–680 (1983).
[CrossRef]

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

1982 (3)

R. Birch, D. Payne, M. P. Varnham, “Fabrication of polarisation maintaining fibers using gas-phase etching,” Electron. Lett.1036–1037 (1982).
[CrossRef]

K. Okamoto, Y. Sasaki, N. Shibata, “Mode coupling effects in stress-applied single polarization fibers,” IEEE J. Quantum Electron. QE-18, 1890–1899 (1982).
[CrossRef]

S. C. Rashleigh, W. K. Burns, R. P. Moeller, R. Ulrich, “Polarization holding in birefringent single mode fibers,” Opt. Lett. 7, 40–42 (1982).
[CrossRef] [PubMed]

1981 (2)

Y. Yamamoto, T. Kimura, “Coherent optical fiber transmission systems,” IEEE J. Quantum Electron. QE-17, 919–935 (1981).
[CrossRef]

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

1980 (1)

1979 (1)

1978 (1)

1976 (1)

M. Horiguchi, H. Osanai, “Spectral losses of low-OH-content optical fibres,” Electron. Lett. 12, 310–312 (1976).
[CrossRef]

1975 (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguide,” Bell Syst. Tech. J. 54, 985–995 (1975).
[CrossRef]

1970 (1)

H. N. Daglish, “Light scattering in selected optical glasses,” Glass Technol. 11, 30–35 (1970).

Birch, R.

R. Birch, D. Payne, M. P. Varnham, “Fabrication of polarisation maintaining fibers using gas-phase etching,” Electron. Lett.1036–1037 (1982).
[CrossRef]

Birch, R. D.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-polarization operation of highly birefringent bow-tie optical fibres,” Electron. Lett. 19, 246–247 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend behavior of polarisation optical fibres,” Electron. Lett. 19, 679–680 (1983).
[CrossRef]

Bôhm, K.

K. Bôhm, K. Peterman, E. Weidel, “Performance of Lyot depolarizer with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

Brinkmeyer, E.

E. Brinkmeyer, W. Eickhoff, “Ultimate limit of polarisation holding in single mode fibres,” Electron. Lett. 19, 996–997 (1983).
[CrossRef]

E. Brinkmeyer, “Analysis of the backscattering method,” J. Opt. Soc. Am. 70, 1010–1012 (1980).
[CrossRef]

Burns, W. K.

Daglish, H. N.

H. N. Daglish, “Light scattering in selected optical glasses,” Glass Technol. 11, 30–35 (1970).

Edahiro, T.

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Eickhoff, W.

E. Brinkmeyer, W. Eickhoff, “Ultimate limit of polarisation holding in single mode fibres,” Electron. Lett. 19, 996–997 (1983).
[CrossRef]

French, W. G.

Horiguchi, M.

M. Horiguchi, H. Osanai, “Spectral losses of low-OH-content optical fibres,” Electron. Lett. 12, 310–312 (1976).
[CrossRef]

Hosaka, T.

T. Hosaka, Y. Sasaki, K. Okamoto, J. Noda, “Stress-applied polarization-maintaining optical fibers-design and fabrication,” IECE Jpn. J67-C, 741–748 (1984).

Y. Sasaki, T. Hosaka, K. Takada, J. Noda, “8 km-long polarisation-maintaining fibres with highly stable polarisation state,” Electron. Lett. 19, 792–794 (1983).
[CrossRef]

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Howard, R. E.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

Johnson, M.

Kimura, T.

Y. Yamamoto, T. Kimura, “Coherent optical fiber transmission systems,” IEEE J. Quantum Electron. QE-17, 919–935 (1981).
[CrossRef]

MacChesney, J. B.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

Marcuse, D.

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguide,” Bell Syst. Tech. J. 54, 985–995 (1975).
[CrossRef]

Miya, T.

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Moeller, R. P.

Nakazawa, M.

Negishi, Y.

Noda, J.

K. Takada, K. Okamoto, J. Noda, “Polarization mode coupling with broad-band source in birefringent polarization-maintaining fibers,” J. Opt. Soc. Am. A 2, 753–758 (1985).
[CrossRef]

T. Hosaka, Y. Sasaki, K. Okamoto, J. Noda, “Stress-applied polarization-maintaining optical fibers-design and fabrication,” IECE Jpn. J67-C, 741–748 (1984).

Y. Sasaki, T. Hosaka, K. Takada, J. Noda, “8 km-long polarisation-maintaining fibres with highly stable polarisation state,” Electron. Lett. 19, 792–794 (1983).
[CrossRef]

Okamoto, K.

K. Takada, K. Okamoto, J. Noda, “Polarization mode coupling with broad-band source in birefringent polarization-maintaining fibers,” J. Opt. Soc. Am. A 2, 753–758 (1985).
[CrossRef]

T. Hosaka, Y. Sasaki, K. Okamoto, J. Noda, “Stress-applied polarization-maintaining optical fibers-design and fabrication,” IECE Jpn. J67-C, 741–748 (1984).

K. Okamoto, Y. Sasaki, N. Shibata, “Mode coupling effects in stress-applied single polarization fibers,” IEEE J. Quantum Electron. QE-18, 1890–1899 (1982).
[CrossRef]

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Osanai, H.

M. Horiguchi, H. Osanai, “Spectral losses of low-OH-content optical fibres,” Electron. Lett. 12, 310–312 (1976).
[CrossRef]

Payne, D.

R. Birch, D. Payne, M. P. Varnham, “Fabrication of polarisation maintaining fibers using gas-phase etching,” Electron. Lett.1036–1037 (1982).
[CrossRef]

Payne, D. N.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-polarization operation of highly birefringent bow-tie optical fibres,” Electron. Lett. 19, 246–247 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend behavior of polarisation optical fibres,” Electron. Lett. 19, 679–680 (1983).
[CrossRef]

Peterman, K.

K. Bôhm, K. Peterman, E. Weidel, “Performance of Lyot depolarizer with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

Pleibel, W.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

Ramaswamy, V.

Rashleigh, S. C.

Sasaki, Y.

T. Hosaka, Y. Sasaki, K. Okamoto, J. Noda, “Stress-applied polarization-maintaining optical fibers-design and fabrication,” IECE Jpn. J67-C, 741–748 (1984).

Y. Sasaki, T. Hosaka, K. Takada, J. Noda, “8 km-long polarisation-maintaining fibres with highly stable polarisation state,” Electron. Lett. 19, 792–794 (1983).
[CrossRef]

K. Okamoto, Y. Sasaki, N. Shibata, “Mode coupling effects in stress-applied single polarization fibers,” IEEE J. Quantum Electron. QE-18, 1890–1899 (1982).
[CrossRef]

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Sears, F. M.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

Shibata, N.

K. Okamoto, Y. Sasaki, N. Shibata, “Mode coupling effects in stress-applied single polarization fibers,” IEEE J. Quantum Electron. QE-18, 1890–1899 (1982).
[CrossRef]

Simpson, J. R.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

Standley, R. D.

Stolen, R. H.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

Takada, K.

K. Takada, K. Okamoto, J. Noda, “Polarization mode coupling with broad-band source in birefringent polarization-maintaining fibers,” J. Opt. Soc. Am. A 2, 753–758 (1985).
[CrossRef]

Y. Sasaki, T. Hosaka, K. Takada, J. Noda, “8 km-long polarisation-maintaining fibres with highly stable polarisation state,” Electron. Lett. 19, 792–794 (1983).
[CrossRef]

Tarbox, E. J.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-polarization operation of highly birefringent bow-tie optical fibres,” Electron. Lett. 19, 246–247 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend behavior of polarisation optical fibres,” Electron. Lett. 19, 679–680 (1983).
[CrossRef]

Tokuda, M.

Ulrich, R.

Varnham, M. P.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend behavior of polarisation optical fibres,” Electron. Lett. 19, 679–680 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-polarization operation of highly birefringent bow-tie optical fibres,” Electron. Lett. 19, 246–247 (1983).
[CrossRef]

R. Birch, D. Payne, M. P. Varnham, “Fabrication of polarisation maintaining fibers using gas-phase etching,” Electron. Lett.1036–1037 (1982).
[CrossRef]

Weidel, E.

K. Bôhm, K. Peterman, E. Weidel, “Performance of Lyot depolarizer with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, T. Kimura, “Coherent optical fiber transmission systems,” IEEE J. Quantum Electron. QE-17, 919–935 (1981).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguide,” Bell Syst. Tech. J. 54, 985–995 (1975).
[CrossRef]

Electron. Lett. (7)

E. Brinkmeyer, W. Eickhoff, “Ultimate limit of polarisation holding in single mode fibres,” Electron. Lett. 19, 996–997 (1983).
[CrossRef]

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low loss single polarisation fibres with asymmetric strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

R. Birch, D. Payne, M. P. Varnham, “Fabrication of polarisation maintaining fibers using gas-phase etching,” Electron. Lett.1036–1037 (1982).
[CrossRef]

Y. Sasaki, T. Hosaka, K. Takada, J. Noda, “8 km-long polarisation-maintaining fibres with highly stable polarisation state,” Electron. Lett. 19, 792–794 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-polarization operation of highly birefringent bow-tie optical fibres,” Electron. Lett. 19, 246–247 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend behavior of polarisation optical fibres,” Electron. Lett. 19, 679–680 (1983).
[CrossRef]

M. Horiguchi, H. Osanai, “Spectral losses of low-OH-content optical fibres,” Electron. Lett. 12, 310–312 (1976).
[CrossRef]

Glass Technol. (1)

H. N. Daglish, “Light scattering in selected optical glasses,” Glass Technol. 11, 30–35 (1970).

IECE Jpn. (1)

T. Hosaka, Y. Sasaki, K. Okamoto, J. Noda, “Stress-applied polarization-maintaining optical fibers-design and fabrication,” IECE Jpn. J67-C, 741–748 (1984).

IEEE J. Lightwave Technol. (1)

K. Bôhm, K. Peterman, E. Weidel, “Performance of Lyot depolarizer with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

IEEE J. Quantum Electron. (2)

Y. Yamamoto, T. Kimura, “Coherent optical fiber transmission systems,” IEEE J. Quantum Electron. QE-17, 919–935 (1981).
[CrossRef]

K. Okamoto, Y. Sasaki, N. Shibata, “Mode coupling effects in stress-applied single polarization fibers,” IEEE J. Quantum Electron. QE-18, 1890–1899 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (3)

OSA J. Lightwave Technol. (1)

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “A single-polarization fiber,” OSA J. Lightwave Technol. LT-1, 370–374 (1983).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Propagation of the cross-coupled waves, S z F ( - - - - ) and S z B ( · · · ) produced by the perturbations in the (z ~ z + Δz) fiber range. The large arrow (⇒) shows the light propagation of the HE 11 x mode excited at the input end of the fiber.

Fig. 2
Fig. 2

Light propagation in the POTDR measurement when only the IRS is taken into consideration. The HE 11 x mode is excited at the input end of the fiber. Dashed arrows, ↗ and ↖, show the coupled waves produced in the forward direction by the waveguide fluctuations.

Fig. 3
Fig. 3

Light propagation in the POTDR measurement when the ARS is taken into consideration within the pulse width υgτ.

Fig. 4
Fig. 4

Mode-coupling parameter versus change of propagation constant in the polarization coupling due to the waveguide fluctuation (- - -) and to the ARS (—). β = 476 m−1 and β = 1.4 × 107 m−1 correspond to the polarization coupling for the forward and backward directions, respectively.

Fig. 5
Fig. 5

POTDR measurement system. The analyzer 2 system depicted inside the dashed circle is available to excite one of the principal modes of the test fiber coupled with the polarizer.

Fig. 6
Fig. 6

Polarization cross talk of the transmitted/reflected waves through/at the half-mirror versus the polarization angle of the incident wave into the half-mirror, ζ is the incident angle of the laser beam with respect to the normal line of the half-mirror.

Fig. 7
Fig. 7

Light propagation in the POTDR when not only the He 11 x but also the He 11 y mode is excited at the input end of the fiber owing to the misalignment of the polarizer rotation angle. The vectors [dE(xx), dE(yy)] and [dE(xy), dE(yx)] represent the backscattered waves propagating in the HE*f modes produced by the IRS and ARS, respectively.

Fig. 8
Fig. 8

Backscattered powers, P and P, versus fiber length. The polarization direction of analyzer 1 is (a) perpendicular and (b) parallel to that of the polarizer. (c) The power ratio, P/P, versus fiber length. The mode-coupling parameter of the fiber used in the measurement is h W F = 2.5 × 10 7 m 1.

Fig. 9
Fig. 9

Power ratio, P/P, versus fiber length. The mode-coupling parameter of the fiber is h W F = 2.5 × 10 5 m 1.

Fig. 10
Fig. 10

Measured bias versus the rotation angle of the polarizer. The minimum value of the bias is 0.0470 with the polarizer rotated at an optimum position (θ = 0°) where the power ratio, P/P, versus fiber length is depicted in Fig. 9.

Fig. 11
Fig. 11

Measured bias versus the mode-coupling parameters of the PANDA fibers. The fibers had refractive-index differences of Δ = 0.6% (filled circle) and Δ = 0.26% (open circle).

Fig. 12
Fig. 12

Logarithmic power ratio, 10 log(Pμ/Pν), versus fiber length for single-polarization single-mode fiber when (a) HE 11 x and (b) HE 11 y modes are excited at the input end.

Fig. 13
Fig. 13

Wavelength dependence of polarization cross talk at a 1-km length determined by the ARS compared with those of the PANDA fibers having the mode-coupling parameters, h W F, of (a) 2.5 × 10−5 m−1, (b) 5.0 × 10−6 m−1, (c) 7.9 × 10−7 m−1, and (d) 2.2 × 10−7 m−1.

Fig. 14
Fig. 14

Backscattered powers, P and P, versus fiber length for the conventional single-mode fiber. The polarization direction of the polarizer is (a) perpendicular and (b) parallel to that of the polarizer, (c) The power ratio, P/P, versus fiber length.

Equations (30)

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

E ( y x ) = C F E y F + C B E y B ,
C F = j 0 L Γ y x ( z ) exp { j [ β x z + β y ( L z ) ] } d z
C B = j 0 L Γ y x ( z ) exp [ j ( β x z + β y z ) ] d z ,
| C F | 2 = L ( Γ ˆ ) 2 | G ( β x β y ) | 2
| C B | 2 = L ( Γ ˆ ) 2 | G ( β x + β y ) | 2 ,
| G ( β ) | 2 = + f ( z ) f ( z u ) exp ( j β u ) d u .
h ( β ) = ( Γ ˆ ) 2 2 l / [ 1 + ( β l ) 2 ] .
P / P = ρ + 1 2 sin 2 ( 2 θ ) ,
P ν = P 0 σ υ g τ exp ( 2 α ν z )
P μ = P 0 ρ σ υ g τ exp [ ( α ν + α μ + h W F ) z ] ,
P μ / P ν = ρ exp [ ( α ν α μ h W F ) z ] .
S = 3 Δ ( W 0 / a ) 2 V 2 ,
W 0 / a = 0.65 + 1.619 V 3 / 2 + 2.879 V 6 .
α T = ( 0.248 + 0.76 ) λ 4 ,
h R F = 5.82 × 10 5 Δ ( 0.248 Δ + 0.76 ) ( λ λ c ) 2 ( 0.65 + 1.619 V 3 / 2 + 2.879 V 6 ) 2 .
E x = E cos ( θ ) exp ( j ω 0 t j β x z )
E y = E sin ( θ ) exp ( j ω 0 t j β y z ) ,
d E ( x x ) = j σ Δ z E cos ( θ ) exp ( j ω 0 t 2 j β x z ) ,
d E ( y y ) = j σ Δ z E cos ( θ ) exp ( j ω 0 t 2 j β y z ) ,
d E ( x y ) = j ρ σ Δ z E sin ( θ ) exp ( j ω 0 t j β y z j β x z ) ,
d E ( y x ) = j ρ σ Δ z E cos ( θ ) exp ( j ω 0 t j β x z j β y z ) .
P y = | [ d E ( x x ) + d E ( x y ) ] sin ( θ ) + [ d E ( y y ) + d E ( y x ) ] cos ( θ ) | 2
P x = | [ d E ( x x ) + d E ( x y ) ] cos ( θ ) + [ d E ( y y ) + d E ( y x ) ] sin ( θ ) | 2 .
P y = σ υ g τ | E | 2 [ 1 2 sin 2 ( 2 θ ) + ρ cos 2 ( 2 θ ) ]
P x = σ υ g τ | E | 2 [ cos 4 ( θ ) + sin 4 ( θ ) + ρ sin 2 ( 2 θ ) ] .
P y / P x = ρ + 1 2 sin 2 ( 2 θ ) .
P y B = σ υ g τ P y ( z ) + ρ σ υ g τ P x ( z ) ,
P x B = σ υ g τ P 0 exp ( 2 α x z ) .
d P y B d z = α y P y B + h W F ( P x B P y B ) = ( α y + h W F ) P y B + h W F P x B .
P y B = ρ σ υ g τ P x ( z ) exp [ ( α y + h W F ) z ] = ρ σ υ g τ P 0 exp [ ( α x + α y + h W F ) z ] .

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