Abstract

We suggest a novel method for significant improvement of fiber Bragg grating (FBG) spectral characteristics based on compensation of repeatable errors during the grating writing process. The method is based on an open loop scheme in which a test grating is written first and then a corrected design is obtained and implemented by writing another (compensated) grating. The results are presented for a FBG dispersion compensator.

© 2002 Optical Society of America

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  1. K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, Appl. Phys. Lett. 32, 647 (1978).
    [CrossRef]
  2. A. Othonos and K. Kalli, Fiber Bragg Gratings (Artech House, Norwood, Mass., 1999).
  3. M. Bass, J. M. Enoch, E. W. Van Stryland, and W. L. Wolfe, eds., Handbook of Optics (McGraw-Hill, New York, 2001), Chap. 13.5.
  4. J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.
  5. Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
    [CrossRef]
  6. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999).
    [CrossRef]
  7. W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, Opt. Lett. 20, 2051 (1995).
    [CrossRef] [PubMed]
  8. J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.
  9. R. Feced and M. N. Zervas, IEEE J. Quantum Electron. 35, 1105 (1999).
    [CrossRef]
  10. L. Poladian, Opt. Lett. 25, 787 (2000).
    [CrossRef]
  11. J. Skaar, L. Wang, and T. Erdogan, IEEE J. Quantum Electron. 37, 165 (2001).
    [CrossRef]
  12. E. Brinkmeyer, Opt. Lett. 20, 810 (1995).
    [CrossRef] [PubMed]
  13. J. Skaar, in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Vol. 61 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper BThB2-1.
  14. T. Komukai, T. Inui, and M. Nakazawa, IEEE Photon. Technol. Lett. 12, 816 (2000).
    [CrossRef]
  15. M. Bohn, W. Rosenkranz, and G. Mohs, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), paper ThGG38, p. 665.
    [CrossRef]

2001 (1)

J. Skaar, L. Wang, and T. Erdogan, IEEE J. Quantum Electron. 37, 165 (2001).
[CrossRef]

2000 (2)

T. Komukai, T. Inui, and M. Nakazawa, IEEE Photon. Technol. Lett. 12, 816 (2000).
[CrossRef]

L. Poladian, Opt. Lett. 25, 787 (2000).
[CrossRef]

1999 (1)

R. Feced and M. N. Zervas, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

1995 (2)

1978 (1)

K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, Appl. Phys. Lett. 32, 647 (1978).
[CrossRef]

Ayliffe, P.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Baker, V.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Barcelos, S.

Beauchesne, G. A.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Bennett, P. J.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Bohn, M.

M. Bohn, W. Rosenkranz, and G. Mohs, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), paper ThGG38, p. 665.
[CrossRef]

Boylan, C.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Brennan, J. F.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Brinkmeyer, E.

Byrd, C. H.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Caldwell, R.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Clements, S. J.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Cole, M. J.

Elder, D. E.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Erdogan, T.

J. Skaar, L. Wang, and T. Erdogan, IEEE J. Quantum Electron. 37, 165 (2001).
[CrossRef]

Feced, R.

R. Feced and M. N. Zervas, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Fells, J. A. J.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Fujii, Y.

K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, Appl. Phys. Lett. 32, 647 (1978).
[CrossRef]

Grubsky, V.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Hadjifotiou, A.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Hernandez, E.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Hill, K. O.

K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, Appl. Phys. Lett. 32, 647 (1978).
[CrossRef]

Inui, T.

T. Komukai, T. Inui, and M. Nakazawa, IEEE Photon. Technol. Lett. 12, 816 (2000).
[CrossRef]

Johnsen, D. C.

K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, Appl. Phys. Lett. 32, 647 (1978).
[CrossRef]

Kalli, K.

A. Othonos and K. Kalli, Fiber Bragg Gratings (Artech House, Norwood, Mass., 1999).

Kanellopoulos, S. E.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Kawasaki, B. S.

K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, Appl. Phys. Lett. 32, 647 (1978).
[CrossRef]

Komukai, T.

T. Komukai, T. Inui, and M. Nakazawa, IEEE Photon. Technol. Lett. 12, 816 (2000).
[CrossRef]

Laming, R. I.

Lee, W. S.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Li, H.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Loh, W. H.

Madsen, C. K.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999).
[CrossRef]

Matthews, M. R.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Mohs, G.

M. Bohn, W. Rosenkranz, and G. Mohs, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), paper ThGG38, p. 665.
[CrossRef]

Motaghian, S. M. R.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Nakazawa, M.

T. Komukai, T. Inui, and M. Nakazawa, IEEE Photon. Technol. Lett. 12, 816 (2000).
[CrossRef]

Othonos, A.

A. Othonos and K. Kalli, Fiber Bragg Gratings (Artech House, Norwood, Mass., 1999).

Pan, Z.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Poladian, L.

Popelek, J.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Priddle, H. F. M.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Rosenkranz, W.

M. Bohn, W. Rosenkranz, and G. Mohs, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), paper ThGG38, p. 665.
[CrossRef]

Rothenberg, J. F.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Sahil, S.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Sinha, P. J.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Skaar, J.

J. Skaar, L. Wang, and T. Erdogan, IEEE J. Quantum Electron. 37, 165 (2001).
[CrossRef]

J. Skaar, in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Vol. 61 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper BThB2-1.

Song, Y. W.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Starodubov, D.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Valenti, J. A.

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

Wakefield, J.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Wang, L.

J. Skaar, L. Wang, and T. Erdogan, IEEE J. Quantum Electron. 37, 165 (2001).
[CrossRef]

Wilcox, R.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Willner, A. E.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

Zervas, M. N.

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999).
[CrossRef]

Appl. Phys. Lett. (1)

K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, Appl. Phys. Lett. 32, 647 (1978).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. Feced and M. N. Zervas, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, IEEE J. Quantum Electron. 37, 165 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. Komukai, T. Inui, and M. Nakazawa, IEEE Photon. Technol. Lett. 12, 816 (2000).
[CrossRef]

Opt. Lett. (3)

Other (8)

J. F. Brennan, E. Hernandez, J. A. Valenti, P. J. Sinha, M. R. Matthews, D. E. Elder, G. A. Beauchesne, and C. H. Byrd, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD12-1.

A. Othonos and K. Kalli, Fiber Bragg Gratings (Artech House, Norwood, Mass., 1999).

M. Bass, J. M. Enoch, E. W. Van Stryland, and W. L. Wolfe, eds., Handbook of Optics (McGraw-Hill, New York, 2001), Chap. 13.5.

J. A. J. Fells, P. J. Bennett, R. Feced, P. Ayliffe, J. Wakefield, H. F. M. Priddle, V. Baker, S. E. Kanellopoulos, C. Boylan, S. Sahil, W. S. Lee, S. J. Clements, and A. Hadjifotiou, in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper PD11-1.

Y. W. Song, S. M. R. Motaghian, D. Starodubov, J. F. Rothenberg, Z. Pan, H. Li, R. Wilcox, J. Popelek, R. Caldwell, V. Grubsky, and A. E. Willner, in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), p. 580.
[CrossRef]

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999).
[CrossRef]

J. Skaar, in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Vol. 61 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper BThB2-1.

M. Bohn, W. Rosenkranz, and G. Mohs, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), paper ThGG38, p. 665.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Examples of the compensation technique: (a) target theoretical spectral characteristics, (b) experimentally measured spectral characteristics (GDR Δτ1τ1exp-τ1th), (c) experimentally measured spectral characteristics of the compensated grating.

Equations (13)

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

LD=1BDΔλ,
ΔτΔλD2=B-1.
Ebz+iδEb-qzEf=0, Efz+iδEf-q*zEb=0,
Eb0Ef0=T11T12T21T22EbLEfL
Tˆ=1t*r1t-r2t1t,
arg tω=Hlntω.
τ2=2Ln0c-2Hln 1-R/ω-τ1,
maxzτiexp-τirecmaxzΔτi,
maxzR1exp-R2expmaxzR1exp-R.
qavexp=q1exp1-ρ+q2expρ,
ρ=1-cosπ0zqthzdz/0Lqthzdz/2.
Δκcorr=κth-κavexp,  Δθcorr=θth-θavexp.
κcomp=κth+Δκcorr,  θcomp=θth+Δθcorr

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