Abstract

We present a manufacturing method based on the dynamic use of phase plates to photowrite Bragg gratings. This process allows for control of the local value of the index modulation envelope in the grating. The application to apodized fiber Bragg gratings is discussed.

© 2002 Optical Society of America

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References

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  1. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  2. C. Martinez, P. Ferdinand, “Phase-shifted fibre Bragg grating photowriting using a UV phase plate in a modified Lloyd mirror configuration,” Electron. Lett. 34, 1687–1688 (1998).
    [CrossRef]
  3. C. Martinez, P. Ferdinand, “Analysis of phase-shifted fiber Bragg gratings written with phase plates,” Appl. Opt. 38, 3223–3228 (1999).
    [CrossRef]
  4. J. Singh, M. Zippin, “Apodized fiber Bragg gratings for DWDM applications using uniform phase mask,” in Proceedings of the Twenty-Fourth European Conference on Optical Communication (Lerko Print S. A., Madrid, Spain, 1998), Vol. 1, pp. 189–190.
  5. B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
    [CrossRef]
  6. J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
    [CrossRef]
  7. J. J. Pan, Y. Shi, “Steep skirt fibre Bragg grating fabrication using a new apodised phase mask,” Electron. Lett. 33, 1895–1896 (1997).
    [CrossRef]
  8. P.-Y. Cortes, F. Ouellette, S. LaRochelle, “Intrinsic apodisation of Bragg gratings written using UV-pulse interferometry,” Electron. Lett. 34, 396–397 (1998).
    [CrossRef]
  9. R. Kashyap, A. Swanton, D. Armes, “Simple technique for apodising chirped and unchirped fibre Bragg gratings,” Electron. Lett. 32, 1226–1228 (1996).
    [CrossRef]
  10. M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
    [CrossRef]
  11. P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2.
    [CrossRef]
  12. M. Yamada, K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  13. N. Roussel, S. Magne, C. Martinez, P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. Mater. Devices Syst. 5, 119–132 (1999).
    [CrossRef]
  14. P. Krug, R. Stolte, R. Ulrich, “Measurement of index modulation along an optical fiber Bragg grating,” Opt. Lett. 20, 1767–1769 (1995).
    [CrossRef] [PubMed]
  15. D. K. W. Lam, B. K. Garside, “Characterization of single-mode optical fiber filters,” Appl. Opt. 20, 440–445 (1981).
    [CrossRef] [PubMed]
  16. H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
    [CrossRef]

2000 (1)

H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
[CrossRef]

1999 (2)

N. Roussel, S. Magne, C. Martinez, P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. Mater. Devices Syst. 5, 119–132 (1999).
[CrossRef]

C. Martinez, P. Ferdinand, “Analysis of phase-shifted fiber Bragg gratings written with phase plates,” Appl. Opt. 38, 3223–3228 (1999).
[CrossRef]

1998 (2)

C. Martinez, P. Ferdinand, “Phase-shifted fibre Bragg grating photowriting using a UV phase plate in a modified Lloyd mirror configuration,” Electron. Lett. 34, 1687–1688 (1998).
[CrossRef]

P.-Y. Cortes, F. Ouellette, S. LaRochelle, “Intrinsic apodisation of Bragg gratings written using UV-pulse interferometry,” Electron. Lett. 34, 396–397 (1998).
[CrossRef]

1997 (1)

J. J. Pan, Y. Shi, “Steep skirt fibre Bragg grating fabrication using a new apodised phase mask,” Electron. Lett. 33, 1895–1896 (1997).
[CrossRef]

1996 (2)

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

R. Kashyap, A. Swanton, D. Armes, “Simple technique for apodising chirped and unchirped fibre Bragg gratings,” Electron. Lett. 32, 1226–1228 (1996).
[CrossRef]

1995 (3)

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

P. Krug, R. Stolte, R. Ulrich, “Measurement of index modulation along an optical fiber Bragg grating,” Opt. Lett. 20, 1767–1769 (1995).
[CrossRef] [PubMed]

1989 (1)

1987 (1)

1981 (1)

Albert, J.

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

Armes, D.

R. Kashyap, A. Swanton, D. Armes, “Simple technique for apodising chirped and unchirped fibre Bragg gratings,” Electron. Lett. 32, 1226–1228 (1996).
[CrossRef]

Barcelos, S.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
[CrossRef]

Bilodeau, F.

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

Chan, Y.

H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
[CrossRef]

Cole, M. J.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
[CrossRef]

Cortes, P.-Y.

P.-Y. Cortes, F. Ouellette, S. LaRochelle, “Intrinsic apodisation of Bragg gratings written using UV-pulse interferometry,” Electron. Lett. 34, 396–397 (1998).
[CrossRef]

Erickson, L. E.

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

Ferdinand, P.

N. Roussel, S. Magne, C. Martinez, P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. Mater. Devices Syst. 5, 119–132 (1999).
[CrossRef]

C. Martinez, P. Ferdinand, “Analysis of phase-shifted fiber Bragg gratings written with phase plates,” Appl. Opt. 38, 3223–3228 (1999).
[CrossRef]

C. Martinez, P. Ferdinand, “Phase-shifted fibre Bragg grating photowriting using a UV phase plate in a modified Lloyd mirror configuration,” Electron. Lett. 34, 1687–1688 (1998).
[CrossRef]

Garside, B. K.

Glenn, W. H.

Hill, K. O.

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2.
[CrossRef]

Jiang, H.

H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
[CrossRef]

Johnson, D. C.

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

Kashyap, R.

R. Kashyap, A. Swanton, D. Armes, “Simple technique for apodising chirped and unchirped fibre Bragg gratings,” Electron. Lett. 32, 1226–1228 (1996).
[CrossRef]

Krug, P.

Lam, D. K. W.

Lam, Y.

H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
[CrossRef]

Laming, R. I.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
[CrossRef]

LaRochelle, S.

P.-Y. Cortes, F. Ouellette, S. LaRochelle, “Intrinsic apodisation of Bragg gratings written using UV-pulse interferometry,” Electron. Lett. 34, 396–397 (1998).
[CrossRef]

Loh, W. H.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
[CrossRef]

Magne, S.

N. Roussel, S. Magne, C. Martinez, P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. Mater. Devices Syst. 5, 119–132 (1999).
[CrossRef]

Malo, B.

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

Martinez, C.

N. Roussel, S. Magne, C. Martinez, P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. Mater. Devices Syst. 5, 119–132 (1999).
[CrossRef]

C. Martinez, P. Ferdinand, “Analysis of phase-shifted fiber Bragg gratings written with phase plates,” Appl. Opt. 38, 3223–3228 (1999).
[CrossRef]

C. Martinez, P. Ferdinand, “Phase-shifted fibre Bragg grating photowriting using a UV phase plate in a modified Lloyd mirror configuration,” Electron. Lett. 34, 1687–1688 (1998).
[CrossRef]

Meltz, G.

Morey, W. W.

Ouellette, F.

P.-Y. Cortes, F. Ouellette, S. LaRochelle, “Intrinsic apodisation of Bragg gratings written using UV-pulse interferometry,” Electron. Lett. 34, 396–397 (1998).
[CrossRef]

Pan, J. J.

J. J. Pan, Y. Shi, “Steep skirt fibre Bragg grating fabrication using a new apodised phase mask,” Electron. Lett. 33, 1895–1896 (1997).
[CrossRef]

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2.
[CrossRef]

Roussel, N.

N. Roussel, S. Magne, C. Martinez, P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. Mater. Devices Syst. 5, 119–132 (1999).
[CrossRef]

Sakuda, K.

Shi, Y.

J. J. Pan, Y. Shi, “Steep skirt fibre Bragg grating fabrication using a new apodised phase mask,” Electron. Lett. 33, 1895–1896 (1997).
[CrossRef]

Singh, J.

J. Singh, M. Zippin, “Apodized fiber Bragg gratings for DWDM applications using uniform phase mask,” in Proceedings of the Twenty-Fourth European Conference on Optical Communication (Lerko Print S. A., Madrid, Spain, 1998), Vol. 1, pp. 189–190.

Stolte, R.

Swanton, A.

R. Kashyap, A. Swanton, D. Armes, “Simple technique for apodising chirped and unchirped fibre Bragg gratings,” Electron. Lett. 32, 1226–1228 (1996).
[CrossRef]

Thériault, S.

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

Ulrich, R.

Yamada, M.

Yuan, X.-C.

H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
[CrossRef]

Zervas, M. N.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
[CrossRef]

Zhou, Y.

H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
[CrossRef]

Zippin, M.

J. Singh, M. Zippin, “Apodized fiber Bragg gratings for DWDM applications using uniform phase mask,” in Proceedings of the Twenty-Fourth European Conference on Optical Communication (Lerko Print S. A., Madrid, Spain, 1998), Vol. 1, pp. 189–190.

Appl. Opt. (3)

Electron. Lett. (7)

C. Martinez, P. Ferdinand, “Phase-shifted fibre Bragg grating photowriting using a UV phase plate in a modified Lloyd mirror configuration,” Electron. Lett. 34, 1687–1688 (1998).
[CrossRef]

B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprimed using a phase mask,” Electron. Lett. 31, 223–225 (1995).
[CrossRef]

J. Albert, K. O. Hill, B. Malo, S. Thériault, F. Bilodeau, D. C. Johnson, L. E. Erickson, “Apodisation of the spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222–223 (1996).
[CrossRef]

J. J. Pan, Y. Shi, “Steep skirt fibre Bragg grating fabrication using a new apodised phase mask,” Electron. Lett. 33, 1895–1896 (1997).
[CrossRef]

P.-Y. Cortes, F. Ouellette, S. LaRochelle, “Intrinsic apodisation of Bragg gratings written using UV-pulse interferometry,” Electron. Lett. 34, 396–397 (1998).
[CrossRef]

R. Kashyap, A. Swanton, D. Armes, “Simple technique for apodising chirped and unchirped fibre Bragg gratings,” Electron. Lett. 32, 1226–1228 (1996).
[CrossRef]

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995).
[CrossRef]

Opt. Commun. (1)

H. Jiang, X.-C. Yuan, Y. Zhou, Y. Chan, Y. Lam, “Single-step fabrication of diffraction gratings on hybrid sol-gel glass using holographic interference lithography,” Opt. Commun. 185, 19–24 (2000).
[CrossRef]

Opt. Fiber Technol. Mater. Devices Syst. (1)

N. Roussel, S. Magne, C. Martinez, P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. Mater. Devices Syst. 5, 119–132 (1999).
[CrossRef]

Opt. Lett. (2)

Other (2)

J. Singh, M. Zippin, “Apodized fiber Bragg gratings for DWDM applications using uniform phase mask,” in Proceedings of the Twenty-Fourth European Conference on Optical Communication (Lerko Print S. A., Madrid, Spain, 1998), Vol. 1, pp. 189–190.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic view of the setup used for apodization with the phase plate process.

Fig. 2
Fig. 2

Results of spectral modeling for the different apodization functions summarized in Table 1. The spectra are given in decibels, depending on the spectral shift toward Bragg wavelength Δλ = λ - λ B . The theoretical spectrum of a uniform grating is also presented. All the spectra verify that R = 97% and ΔλFWHM = 430 pm.

Fig. 3
Fig. 3

Simulation of the index modulation variation with time along the grating length during an erasure experiment. The bottom half of the plots relates to growing the grating with respect to a first exposure (t < T/2). For time >T/2 the plots represent the erasure: (a) ∂Δn/∂tI and (b) ∂Δn/∂tI 2.

Fig. 4
Fig. 4

Model that represents grating growth: (a) exposure with an UV pattern (symbolized by a succession of dark and bright fringes) leads to a growth in the refractive index that creates a square grating and (b) if the pattern is π phase shifted another growth of the refractive index leads to the erasure of the grating.

Fig. 5
Fig. 5

Normalized movement of one phase plate is given for super Gaussian apodization. The inset is a simulation of the growth of the grating first-order index modulation that results from this movement.

Fig. 6
Fig. 6

Kinetics of inscription (t ≤ 300 s and t > 600 s, thin lines) and erasure (300 s < t ≤ 600 s, thick lines) of a 4-mm-long Bragg grating.

Fig. 7
Fig. 7

Evolution of the refractive-index modulation of a 4-mm-long Bragg grating during erasure.

Fig. 8
Fig. 8

Comparison between the experimental (solid curve) and theoretical envelopes of fiber Bragg gratings achieved with the phase plate process. The first curve represents a linear movement of the phase plate (T = 20 min and L = 4 mm). The second and third curves represent Gaussian (T = 15 min, L = 4 mm, p = 2) and super Gaussian (T = 15 min, L = 4 mm, p = 1.5, G = 2) envelope functions.

Fig. 9
Fig. 9

Comparison between the theoretical super Gaussian apodization function (p = 1.5, G = 2, L = 6 mm, squares) and the measure of the normalized index modulation envelope of an apodized Bragg grating manufactured with the phase plate movement described in Table 2 (T = 20 min, solid curve).

Fig. 10
Fig. 10

Comparison between the experimental (solid curve) and the theoretical (squares) spectra of a super Gaussian apodized fiber Bragg grating (p = 1.5, G = 2, L = 6 mm, T = 20 min for experiment and Δn 0 = 3.6 × 10-4 for theory). The spectral response of a 4-mm-long uniform grating is also given (dotted curve). Inset: spectral reflection of the super-Gaussian apodized fiber Bragg grating in percent.

Tables (2)

Tables Icon

Table 1 Parameters Used to Simulate the Apodized Grating Spectral Responses

Tables Icon

Table 2 Movement of Both Phase Plates for Gaussian Apodization

Equations (37)

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

Δnz=Δnav+Δnmod Tzcos2πΛ z;
Tz=exp-z-L/22GL/2p2G.
I0z=Iav+Imod cos2πΛ z, 0zL,
Iπz=Iav+Imod cos2πΛ z+π, 0zL.
I0z+Iπz=2Iav.
Δnn0, I, t=0tΔntn0, I, udu.
Δnz, T/2=0T/2Δntnc, I0z, udu.
Δnz, T=Δnz, T/2+0T/2Δntnc+Δnz, T/2, Iπz, udu,
gz=Δnz×Λ/2z,
Λ/2z=k δz-k Λ2,
Gν  Δn˜ν * 2/Λν  k Δn˜ν+k 2Λ.
G1Λ  k Δn˜1Λ2k+1.
G1Λ  Δn˜1Λ.
Δnzmin, t=0,
Δnzmax, t=0tΔntnc, Imod, udu.
Δnzmin, t=0t-T/2Δntnc, Imod, udu,
Δnzmax, t=0T/2Δntnc, Imod, udu.
Δnzmin, T=Δnzmax, T.
Δnmodt=0tΔntudu,
Δnavt= 120tΔntudu.
Δnmodt=0T/2Δntudu-0t-T/2Δntudu,
Δnavt= 120T/2Δntudu+ 120t-T/2Δntudu.
tπz=zπ-1t.
Δnmodz=0T-tπzΔntudu-0tπzΔntudu,
Δnavz= 120T-tπzΔntudu+ 120tπzΔntudu.
Δn0t=0tΔntudu.
Δnmodz=Δn0T-tπz-Δn0tπz,
Δnavz=Δn0T-tπz+Δn0tπz2.
Tz=Δn0T-tπz-Δn0tπzΔn0T.
Δn0t=η0t.
tπz=T/21-Tz,
Δnavz=η0T/2.
0tπzT/21-T0.
For 0  t  T21 - T0, zπt=T-1T-2tT.
For T21 - T0  t  T, zπt=T-1T0.
zπt= L2+ L2p2GlnTT-2t
zπt= L2- L2p2GlnTT-2t

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