Abstract

A method for measuring modal birefringence and polarization mode dispersion in highly birefringent fibers is presented. It employs a white-light interference phenomenon arising between polarization modes as a result of mode coupling induced by a pointlike lateral force applied in approximately one half the length of the tested fiber. This permits the use of a Wollaston prism without a delay line as an analyzing interferometer. Results of measurements of modal birefringence and polarization mode dispersion at λ0 = 826 nm are reported for four commercially available fibers: the York Bow-Tie, the Fujikura Panda, the Andrew E-type, and the Andrew D-type.

© 1993 Optical Society of America

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References

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  1. W. Ecikhoff, O. Krumpholz, “Determination of ellipticity of monomode glass fibers from measurements of scattered light intensity,” Electron. Lett. 12, 405–406 (1976).
    [CrossRef]
  2. A. Simon, R. Ulrich, “Evolution of polarization along a single mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
    [CrossRef]
  3. R. Ulrich, A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
    [CrossRef] [PubMed]
  4. K. Takada, J. Noda, R. Ulrich, “Precision measurement of modal birefringence of highly birefringent fibers by periodic lateral force,” Appl. Opt. 24, 4387–4391 (1985).
    [CrossRef] [PubMed]
  5. R. Calvani, R. Caponi, F. Cisterino, G. Coppa, “Fiber birefringence measurements with an external stress method and heterodyne polarization detection,” J. Lightwave Technol. LT-5, 1176–1182 (1987).
    [CrossRef]
  6. R. Calvani, R. Caponi, F. Cisterino, “Real-time heterodyne fiber polarimetry with narrow- and broad-band sources,” J. Lightwave Technol. LT-4, 877–883 (1986).
    [CrossRef]
  7. K. Kikuchi, T. Okoshi, “Wavelength-sweeping technique for measuring the beat length of linearly birefringent optical fibers,” Opt. Lett. 8, 122–123 (1983).
    [CrossRef] [PubMed]
  8. M. Nakazawa, T. Horiguchi, M. Tukoda, N. Uchida, “Polarization beat-length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–514 (1981).
    [CrossRef]
  9. T. Okoshi, S. Ryu, K. Emura, “Measurements of polarization parameters of single-mode optical fibers,” J. Opt. Commun. 2, 134–141 (1981).
  10. S. Huang, Z. Lin, “Measuring the birefringence of single-mode fibers with short beat length or nonuniformity: a new method,” Appl. Opt. 24, 2355–2361 (1985).
    [CrossRef] [PubMed]
  11. Y. Sasaki, N. Shibata, J. Noda, “Splicing of single-polarization fibers by an optical short pulse method,” Electron. Lett. 18, 997–998 (1982).
    [CrossRef]
  12. S. Rashleigh, R. Ulrich, “Polarization mode dispersion in single-mode fibers,” Opt. Lett. 3, 60–62 (1978).
    [CrossRef] [PubMed]
  13. W. K. Burns, R. P. Moeller, “Measurement of polarization mode dispersion in high-birefringence fibers,” Opt. Lett. 8, 195–197 (1983).
    [CrossRef] [PubMed]
  14. M. Monerie, P. Lamonir, L. Jeunhomme, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 16, 907–908 (1981).
    [CrossRef]
  15. K. Mochizuki, Y. Namihira, H. Wakabayashi, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 17, 153–154 (1981).
    [CrossRef]
  16. N. Shibata, M. Tateda, S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron. QE-18, 53–58 (1982).
    [CrossRef]
  17. J. P. von der Weid, L. Thèvanez, J. P. Pellaux, “Interferometric measurements of chromatic and polarization mode dispersion in highly birefringent single-mode fibers,” Electron. Lett. 23, 151–152 (1987).
    [CrossRef]
  18. L. Thèvanez, V. de Coulon, J. P. von der Weid, “Polarization mode interferometry in birefringent single-mode fibers,” Opt. Lett. 12, 619–621 (1987).
    [CrossRef]
  19. L. Thèvanez, J. P. Pellaux, N. Gisin, J. P. von der Weid, “Birefringence measurements in fibers without polarizer,” J. Lightwave Technol. LT-7, 1207–1213 (1986).
  20. J. Vobian, “Chromatic and polarization mode dispersion measurements of single-mode fibers with a Mach–Zehnder interferometer between 1200 and 1700 nm,” J. Opt. Commun. 11, 29–36 (1990).
    [CrossRef]
  21. K. Okomato, T. Hosaka, “Polarization-dependent chromatic dispersion in birefringent optical fibers,” Opt. Lett. 12, 290–292 (1987).
    [CrossRef]
  22. S. C. Rashleigh, “Measurements of fiber birefringence by wavelength scanning: effect of dispersion,” Opt. Lett. 8, 336–339 (1983).
    [CrossRef] [PubMed]
  23. B. Dyott, J. R. Cozens, D. G. Morris, “Preservation of polarization in optical-fiber waveguides with elliptical cores,” Electron. Lett. 15, 380–382 (1979).
    [CrossRef]
  24. N. Chinone, R. Ulrich, “Elasto-optic polarization measurement in optical fiber,” Opt. Lett. 6, 16–18 (1981).
    [CrossRef] [PubMed]
  25. R. B. Sosman, “Electrical and optical properties of silica,” in International Critical Tables of Numerical Data, Physics, Chemistry and Technology, E. W. Washburn, ed. (McGraw-Hill, New York, 1929), vol. 6, pp. 341–344.
  26. J. H. Shields, J. W. Ellis, “Dispersion of birefringence of quartz in the near infrared,” J. Opt. Soc. Am. 46, 263–265 (1956).
    [CrossRef]

1990

J. Vobian, “Chromatic and polarization mode dispersion measurements of single-mode fibers with a Mach–Zehnder interferometer between 1200 and 1700 nm,” J. Opt. Commun. 11, 29–36 (1990).
[CrossRef]

1987

K. Okomato, T. Hosaka, “Polarization-dependent chromatic dispersion in birefringent optical fibers,” Opt. Lett. 12, 290–292 (1987).
[CrossRef]

R. Calvani, R. Caponi, F. Cisterino, G. Coppa, “Fiber birefringence measurements with an external stress method and heterodyne polarization detection,” J. Lightwave Technol. LT-5, 1176–1182 (1987).
[CrossRef]

J. P. von der Weid, L. Thèvanez, J. P. Pellaux, “Interferometric measurements of chromatic and polarization mode dispersion in highly birefringent single-mode fibers,” Electron. Lett. 23, 151–152 (1987).
[CrossRef]

L. Thèvanez, V. de Coulon, J. P. von der Weid, “Polarization mode interferometry in birefringent single-mode fibers,” Opt. Lett. 12, 619–621 (1987).
[CrossRef]

1986

L. Thèvanez, J. P. Pellaux, N. Gisin, J. P. von der Weid, “Birefringence measurements in fibers without polarizer,” J. Lightwave Technol. LT-7, 1207–1213 (1986).

R. Calvani, R. Caponi, F. Cisterino, “Real-time heterodyne fiber polarimetry with narrow- and broad-band sources,” J. Lightwave Technol. LT-4, 877–883 (1986).
[CrossRef]

1985

1983

1982

Y. Sasaki, N. Shibata, J. Noda, “Splicing of single-polarization fibers by an optical short pulse method,” Electron. Lett. 18, 997–998 (1982).
[CrossRef]

N. Shibata, M. Tateda, S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron. QE-18, 53–58 (1982).
[CrossRef]

1981

N. Chinone, R. Ulrich, “Elasto-optic polarization measurement in optical fiber,” Opt. Lett. 6, 16–18 (1981).
[CrossRef] [PubMed]

M. Monerie, P. Lamonir, L. Jeunhomme, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 16, 907–908 (1981).
[CrossRef]

K. Mochizuki, Y. Namihira, H. Wakabayashi, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 17, 153–154 (1981).
[CrossRef]

M. Nakazawa, T. Horiguchi, M. Tukoda, N. Uchida, “Polarization beat-length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–514 (1981).
[CrossRef]

T. Okoshi, S. Ryu, K. Emura, “Measurements of polarization parameters of single-mode optical fibers,” J. Opt. Commun. 2, 134–141 (1981).

1979

R. Ulrich, A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
[CrossRef] [PubMed]

B. Dyott, J. R. Cozens, D. G. Morris, “Preservation of polarization in optical-fiber waveguides with elliptical cores,” Electron. Lett. 15, 380–382 (1979).
[CrossRef]

1978

1977

A. Simon, R. Ulrich, “Evolution of polarization along a single mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
[CrossRef]

1976

W. Ecikhoff, O. Krumpholz, “Determination of ellipticity of monomode glass fibers from measurements of scattered light intensity,” Electron. Lett. 12, 405–406 (1976).
[CrossRef]

1956

Burns, W. K.

Calvani, R.

R. Calvani, R. Caponi, F. Cisterino, G. Coppa, “Fiber birefringence measurements with an external stress method and heterodyne polarization detection,” J. Lightwave Technol. LT-5, 1176–1182 (1987).
[CrossRef]

R. Calvani, R. Caponi, F. Cisterino, “Real-time heterodyne fiber polarimetry with narrow- and broad-band sources,” J. Lightwave Technol. LT-4, 877–883 (1986).
[CrossRef]

Caponi, R.

R. Calvani, R. Caponi, F. Cisterino, G. Coppa, “Fiber birefringence measurements with an external stress method and heterodyne polarization detection,” J. Lightwave Technol. LT-5, 1176–1182 (1987).
[CrossRef]

R. Calvani, R. Caponi, F. Cisterino, “Real-time heterodyne fiber polarimetry with narrow- and broad-band sources,” J. Lightwave Technol. LT-4, 877–883 (1986).
[CrossRef]

Chinone, N.

Cisterino, F.

R. Calvani, R. Caponi, F. Cisterino, G. Coppa, “Fiber birefringence measurements with an external stress method and heterodyne polarization detection,” J. Lightwave Technol. LT-5, 1176–1182 (1987).
[CrossRef]

R. Calvani, R. Caponi, F. Cisterino, “Real-time heterodyne fiber polarimetry with narrow- and broad-band sources,” J. Lightwave Technol. LT-4, 877–883 (1986).
[CrossRef]

Coppa, G.

R. Calvani, R. Caponi, F. Cisterino, G. Coppa, “Fiber birefringence measurements with an external stress method and heterodyne polarization detection,” J. Lightwave Technol. LT-5, 1176–1182 (1987).
[CrossRef]

Cozens, J. R.

B. Dyott, J. R. Cozens, D. G. Morris, “Preservation of polarization in optical-fiber waveguides with elliptical cores,” Electron. Lett. 15, 380–382 (1979).
[CrossRef]

de Coulon, V.

Dyott, B.

B. Dyott, J. R. Cozens, D. G. Morris, “Preservation of polarization in optical-fiber waveguides with elliptical cores,” Electron. Lett. 15, 380–382 (1979).
[CrossRef]

Ecikhoff, W.

W. Ecikhoff, O. Krumpholz, “Determination of ellipticity of monomode glass fibers from measurements of scattered light intensity,” Electron. Lett. 12, 405–406 (1976).
[CrossRef]

Ellis, J. W.

Emura, K.

T. Okoshi, S. Ryu, K. Emura, “Measurements of polarization parameters of single-mode optical fibers,” J. Opt. Commun. 2, 134–141 (1981).

Gisin, N.

L. Thèvanez, J. P. Pellaux, N. Gisin, J. P. von der Weid, “Birefringence measurements in fibers without polarizer,” J. Lightwave Technol. LT-7, 1207–1213 (1986).

Horiguchi, T.

M. Nakazawa, T. Horiguchi, M. Tukoda, N. Uchida, “Polarization beat-length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–514 (1981).
[CrossRef]

Hosaka, T.

Huang, S.

Jeunhomme, L.

M. Monerie, P. Lamonir, L. Jeunhomme, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 16, 907–908 (1981).
[CrossRef]

Kikuchi, K.

Krumpholz, O.

W. Ecikhoff, O. Krumpholz, “Determination of ellipticity of monomode glass fibers from measurements of scattered light intensity,” Electron. Lett. 12, 405–406 (1976).
[CrossRef]

Lamonir, P.

M. Monerie, P. Lamonir, L. Jeunhomme, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 16, 907–908 (1981).
[CrossRef]

Lin, Z.

Mochizuki, K.

K. Mochizuki, Y. Namihira, H. Wakabayashi, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 17, 153–154 (1981).
[CrossRef]

Moeller, R. P.

Monerie, M.

M. Monerie, P. Lamonir, L. Jeunhomme, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 16, 907–908 (1981).
[CrossRef]

Morris, D. G.

B. Dyott, J. R. Cozens, D. G. Morris, “Preservation of polarization in optical-fiber waveguides with elliptical cores,” Electron. Lett. 15, 380–382 (1979).
[CrossRef]

Nakazawa, M.

M. Nakazawa, T. Horiguchi, M. Tukoda, N. Uchida, “Polarization beat-length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–514 (1981).
[CrossRef]

Namihira, Y.

K. Mochizuki, Y. Namihira, H. Wakabayashi, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 17, 153–154 (1981).
[CrossRef]

Noda, J.

K. Takada, J. Noda, R. Ulrich, “Precision measurement of modal birefringence of highly birefringent fibers by periodic lateral force,” Appl. Opt. 24, 4387–4391 (1985).
[CrossRef] [PubMed]

Y. Sasaki, N. Shibata, J. Noda, “Splicing of single-polarization fibers by an optical short pulse method,” Electron. Lett. 18, 997–998 (1982).
[CrossRef]

Okomato, K.

Okoshi, T.

K. Kikuchi, T. Okoshi, “Wavelength-sweeping technique for measuring the beat length of linearly birefringent optical fibers,” Opt. Lett. 8, 122–123 (1983).
[CrossRef] [PubMed]

T. Okoshi, S. Ryu, K. Emura, “Measurements of polarization parameters of single-mode optical fibers,” J. Opt. Commun. 2, 134–141 (1981).

Pellaux, J. P.

J. P. von der Weid, L. Thèvanez, J. P. Pellaux, “Interferometric measurements of chromatic and polarization mode dispersion in highly birefringent single-mode fibers,” Electron. Lett. 23, 151–152 (1987).
[CrossRef]

L. Thèvanez, J. P. Pellaux, N. Gisin, J. P. von der Weid, “Birefringence measurements in fibers without polarizer,” J. Lightwave Technol. LT-7, 1207–1213 (1986).

Rashleigh, S.

Rashleigh, S. C.

Ryu, S.

T. Okoshi, S. Ryu, K. Emura, “Measurements of polarization parameters of single-mode optical fibers,” J. Opt. Commun. 2, 134–141 (1981).

Sasaki, Y.

Y. Sasaki, N. Shibata, J. Noda, “Splicing of single-polarization fibers by an optical short pulse method,” Electron. Lett. 18, 997–998 (1982).
[CrossRef]

Seikai, S.

N. Shibata, M. Tateda, S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron. QE-18, 53–58 (1982).
[CrossRef]

Shibata, N.

N. Shibata, M. Tateda, S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron. QE-18, 53–58 (1982).
[CrossRef]

Y. Sasaki, N. Shibata, J. Noda, “Splicing of single-polarization fibers by an optical short pulse method,” Electron. Lett. 18, 997–998 (1982).
[CrossRef]

Shields, J. H.

Simon, A.

R. Ulrich, A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
[CrossRef] [PubMed]

A. Simon, R. Ulrich, “Evolution of polarization along a single mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
[CrossRef]

Sosman, R. B.

R. B. Sosman, “Electrical and optical properties of silica,” in International Critical Tables of Numerical Data, Physics, Chemistry and Technology, E. W. Washburn, ed. (McGraw-Hill, New York, 1929), vol. 6, pp. 341–344.

Takada, K.

Tateda, M.

N. Shibata, M. Tateda, S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron. QE-18, 53–58 (1982).
[CrossRef]

Thèvanez, L.

J. P. von der Weid, L. Thèvanez, J. P. Pellaux, “Interferometric measurements of chromatic and polarization mode dispersion in highly birefringent single-mode fibers,” Electron. Lett. 23, 151–152 (1987).
[CrossRef]

L. Thèvanez, V. de Coulon, J. P. von der Weid, “Polarization mode interferometry in birefringent single-mode fibers,” Opt. Lett. 12, 619–621 (1987).
[CrossRef]

L. Thèvanez, J. P. Pellaux, N. Gisin, J. P. von der Weid, “Birefringence measurements in fibers without polarizer,” J. Lightwave Technol. LT-7, 1207–1213 (1986).

Tukoda, M.

M. Nakazawa, T. Horiguchi, M. Tukoda, N. Uchida, “Polarization beat-length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–514 (1981).
[CrossRef]

Uchida, N.

M. Nakazawa, T. Horiguchi, M. Tukoda, N. Uchida, “Polarization beat-length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–514 (1981).
[CrossRef]

Ulrich, R.

Vobian, J.

J. Vobian, “Chromatic and polarization mode dispersion measurements of single-mode fibers with a Mach–Zehnder interferometer between 1200 and 1700 nm,” J. Opt. Commun. 11, 29–36 (1990).
[CrossRef]

von der Weid, J. P.

L. Thèvanez, V. de Coulon, J. P. von der Weid, “Polarization mode interferometry in birefringent single-mode fibers,” Opt. Lett. 12, 619–621 (1987).
[CrossRef]

J. P. von der Weid, L. Thèvanez, J. P. Pellaux, “Interferometric measurements of chromatic and polarization mode dispersion in highly birefringent single-mode fibers,” Electron. Lett. 23, 151–152 (1987).
[CrossRef]

L. Thèvanez, J. P. Pellaux, N. Gisin, J. P. von der Weid, “Birefringence measurements in fibers without polarizer,” J. Lightwave Technol. LT-7, 1207–1213 (1986).

Wakabayashi, H.

K. Mochizuki, Y. Namihira, H. Wakabayashi, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 17, 153–154 (1981).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

A. Simon, R. Ulrich, “Evolution of polarization along a single mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
[CrossRef]

Electron. Lett.

W. Ecikhoff, O. Krumpholz, “Determination of ellipticity of monomode glass fibers from measurements of scattered light intensity,” Electron. Lett. 12, 405–406 (1976).
[CrossRef]

M. Nakazawa, T. Horiguchi, M. Tukoda, N. Uchida, “Polarization beat-length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–514 (1981).
[CrossRef]

Y. Sasaki, N. Shibata, J. Noda, “Splicing of single-polarization fibers by an optical short pulse method,” Electron. Lett. 18, 997–998 (1982).
[CrossRef]

M. Monerie, P. Lamonir, L. Jeunhomme, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 16, 907–908 (1981).
[CrossRef]

K. Mochizuki, Y. Namihira, H. Wakabayashi, “Polarization mode dispersion measurements in long single-mode fibers,” Electron. Lett. 17, 153–154 (1981).
[CrossRef]

J. P. von der Weid, L. Thèvanez, J. P. Pellaux, “Interferometric measurements of chromatic and polarization mode dispersion in highly birefringent single-mode fibers,” Electron. Lett. 23, 151–152 (1987).
[CrossRef]

B. Dyott, J. R. Cozens, D. G. Morris, “Preservation of polarization in optical-fiber waveguides with elliptical cores,” Electron. Lett. 15, 380–382 (1979).
[CrossRef]

IEEE J. Quantum Electron.

N. Shibata, M. Tateda, S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron. QE-18, 53–58 (1982).
[CrossRef]

J. Lightwave Technol.

R. Calvani, R. Caponi, F. Cisterino, G. Coppa, “Fiber birefringence measurements with an external stress method and heterodyne polarization detection,” J. Lightwave Technol. LT-5, 1176–1182 (1987).
[CrossRef]

R. Calvani, R. Caponi, F. Cisterino, “Real-time heterodyne fiber polarimetry with narrow- and broad-band sources,” J. Lightwave Technol. LT-4, 877–883 (1986).
[CrossRef]

L. Thèvanez, J. P. Pellaux, N. Gisin, J. P. von der Weid, “Birefringence measurements in fibers without polarizer,” J. Lightwave Technol. LT-7, 1207–1213 (1986).

J. Opt. Commun.

J. Vobian, “Chromatic and polarization mode dispersion measurements of single-mode fibers with a Mach–Zehnder interferometer between 1200 and 1700 nm,” J. Opt. Commun. 11, 29–36 (1990).
[CrossRef]

T. Okoshi, S. Ryu, K. Emura, “Measurements of polarization parameters of single-mode optical fibers,” J. Opt. Commun. 2, 134–141 (1981).

J. Opt. Soc. Am.

Opt. Lett.

Other

R. B. Sosman, “Electrical and optical properties of silica,” in International Critical Tables of Numerical Data, Physics, Chemistry and Technology, E. W. Washburn, ed. (McGraw-Hill, New York, 1929), vol. 6, pp. 341–344.

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Figures (6)

Fig. 1
Fig. 1

Diagram of the interferometric system for measuring PMD and modal birefringence in HB fibers.

Fig. 2
Fig. 2

Spectral power density of the Sharp laser diode operating in a broadband regime.

Fig. 3
Fig. 3

Successive steps in the signal processing method: (a) white-light interference pattern, (b) nonmodulated component of intensity, (c) normalized higher harmonics, and (d) product of the visibility and coherence function.

Fig. 4
Fig. 4

Shift of the coherence function δMc expressed in the number of interference fringes versus the displacement δL of the coupling point for the Andrew and York fibers.

Fig. 5
Fig. 5

Repeatability test for measurements of modal birefringence in the Andrew and York fibers.

Fig. 6
Fig. 6

Shift of the interference fringes δMf versus the displacement δL of the coupling point for the Andrew and York fibers.

Tables (1)

Tables Icon

Table 1 Results of the Measurement of Phase (ΔnHB) and Group (ΔNHB) Birefringences, Beat Length(LB), and PMD (τ) for Four Commercially Available Fibers at λ0 = 826 nm

Equations (33)

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

τ = 1 c ( Δ n HB - λ d Δ n HB d λ ) ,
E 0 ( ω ) = [ 1 1 ] E 0 ( ω ) ,
T 1 = [ exp i Δ β HB L 1 0 0 1 ] ,
T 2 = [ exp i Δ β HB L 2 0 0 1 ] ,
T C = [ C x x C x y C y x C y y ] ,
T WP = [ exp i δ WP ( y ) 0 0 1 ] ,
T A = 1 2 [ 1 1 1 1 ] ,
δ WP ( y ) = ( y - y 0 ) Δ β WP tan α ,
I s ( ω ) = E x ( ω ) 2 + E y ( ω ) 2 ,
E = T A T WP T 2 T C T 1 E 0 .
I s ( ω ) = E 0 ( ω ) 2 ( C x x 2 + C y x 2 + C x y 2 + C y y 2 + 2 Re [ ( C x x C y x * + C x y * C y y ) exp i L 1 Δ β HB ( ω ) ] + 2 Re { ( C x x C y x * + C x y * C y y ) × exp i [ L 2 Δ β HB ( ω ) + ( y - y 0 ) Δ β WP ( ω ) tan α ] } + 2 Re { C x x C y y * exp i [ ( L 1 + L 2 ) × Δ β HB ( ω ) + ( y - y 0 ) Δ β WP ( ω ) tan α ] } + 2 Re { C y x C x y * exp i [ ( L 2 - L 1 ) × Δ β HB ( ω ) + ( y - y 0 ) Δ β WP ( ω ) tan α ] } ) ,
I = 0 I s ( ω ) d ω ,
Δ β WP ( ω ) = Δ β WP ( ω 0 ) + d Δ β WP d ω ( ω - ω 0 ) ,
Δ β HB ( ω ) = Δ β HB ( ω 0 ) + d Δ β HB d ω ( ω - ω 0 ) ,
τ HB = d Δ β HB d ω = 1 c ( Δ n HB - λ 0 d Δ n HB d λ ) = Δ N HB c ,
d Δ β WP d ω = 1 c ( Δ n WP - λ 0 d Δ n WP d λ ) = Δ N WP c ,
I ( y ) = I 0 ( y ) ( C x x 2 + C y x 2 + C x y 2 + C y y 2 + 2 Re [ ( C x x C y x * + C x y * C y y ) exp i ω 0 c L 1 Δ n HB ] × γ ( Δ ω c L 1 Δ N HB ) + 2 Re { ( C x x C y x * + C x y * C y y ) × exp i ω 0 c [ L 2 Δ n HB + ( y - y 0 ) Δ n WP tan α ] } × γ { Δ ω c [ L 2 Δ N HB + ( y - y 0 ) Δ N WP tan α ] } + 2 Re { C x x C y y * exp i ω 0 c [ ( L 1 + L 2 ) Δ n HB + ( y - y 0 ) Δ n WP tan α ] } γ { Δ ω c [ ( L 1 + L 2 ) × Δ N HB + ( y - y 0 ) Δ N WP tan α ] } + 2 Re { C y x C x y * exp i ω 0 c [ ( L 2 - L 1 ) Δ n HB + ( y - y 0 ) Δ n WP tan α ] } γ { Δ ω c [ ( L 2 - L 1 ) × Δ N HB + ( y - y 0 ) Δ N WP tan α ] } ) ,
C x x = cos 2 β exp i δ c + sin 2 β , C x y = cos β sin β ( exp i δ c - 1 ) , C x y = cos β sin β ( exp i δ c - 1 ) , C y y = sin 2 β exp i δ c + cos 2 β ,
I ( y ) = I 0 ( y ) ( 1 + V cos { ω 0 c [ Δ n HB ( L 2 - L 1 ) + Δ n WP ( y - y 0 ) tan α ] } γ { Δ ω c [ Δ N HB ( L 2 - L 1 ) + Δ N WP ( y - y 0 ) tan α ] } ) ,
V = cos 2 β sin 2 β exp i δ c - 1 2 C x x 2 + C x y 2 + C y x 2 + C y y 2 .
L 1 = L 1 - δ L ,             L 2 = L 2 + δ L ,
Δ N HB ( L 2 - L 1 ) + Δ N WP ( y - y 0 ) tan α = 0
Δ N HB ( L 2 - L 1 + 2 δ L ) + Δ N WP ( y - y 0 - δ y c ) tan α = 0
δ y c = Δ N HB Δ N WP tan α ( 2 δ L ) .
δ y f = Δ n HB Δ n WP tan α ( 2 δ L ) .
Δ y = λ 0 Δ n WP tan α .
δ M c = δ y c Δ y = Δ N HB Δ n WP λ 0 Δ N WP ( 2 δ L ) ,
δ M f = δ y f Δ y = Δ n HB λ 0 ( 2 δ L ) .
I 0 ( y ) = 1 2 Δ y - Δ y Δ y I ( y ) d y = 1 2 Δ y - Δ y Δ y I 0 ( y ) d y - 1 2 Δ y V - Δ y Δ y I 0 ( y ) γ { Δ ω c [ Δ N HB ( L 2 - L 1 ) + Δ N WP ( y - y 0 ) tan α ] } cos { ω 0 c [ Δ n HB ( L 2 - L 1 ) + Δ n WP ( y - y 0 ) tan α ] } d y .
V γ ( y ) = 1 2 Δ y - Δ y Δ y | I ( y ) I 0 ( y ) - 1 | d y ,
Δ N HB = λ 0 Δ N WP Δ n WP δ M c 2 δ L .
Δ n HB = λ 0 ( δ M f 2 δ L ) ,
L B = 2 δ L δ M f .

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