Abstract

Methods to produce optimal designs for multi-channel fiber Bragg gratings (FBGs) with identical or close to identical channel-to-channel spectral characteristics are discussed. The proposed approach consists of three distinct steps. The first two steps (preliminary semi-analytic minimization and subsequent fine-tuning) do not depend on the grating design details, but on the number of channels only and can be readily applied to similar problems in other fields, e.g., in radio-physics and coding theory. The third step (spectral characteristic quality improvement) is FBG field specific. A comparison with other known optimization methods shows that the proposed approach yields generally superior results for small to moderate number of channels (N<60).

© 2003 Optical Society of America

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References

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  1. A. Othonos and K. Kalli, Fiber Bragg Gratings (Boston, Artech House, 1999).
  2. H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, �??Quasicontinuous Wavelength Tuning in Super-Structure-Grating (SSG) DBR Lasers,�?? IEEE J. Quantum Electron. 32, 433-441 (1996).
    [CrossRef]
  3. A. V. Buryak and D. Yu. Stepanov, �??Novel multi-channel grating devices,�?? in proceedings of Bragg Gratings, Photosensitivity, and Poling in GlassWaveguides, vol. 60 of Top series, BThB3 (Washington DC, Optical Society of America, 2001).
  4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, �??Optimization of Refractive Index Sampling for Multichannel Fiber Bragg Gratings,�?? IEEE J. Quantum Electron. 39, 91-98 (2003).
    [CrossRef]
  5. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, �??Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,�?? IEEE Photon. Tech. Lett. 14, 1309-1311 (2002).
    [CrossRef]
  6. M. Ibsen, A. Fu, H. Geiger, and R. I. Laming, �??All-fibre 4 x 10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,�?? Electron. Lett. 35, 982-983 (1999).
    [CrossRef]
  7. Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, �??Multi-channel fiber Bragg gratings for dispersion and slope compensaion,�?? in OSA Technical Digest of Optical Fiber Communication Conference, ThAA5, 581-582 (Washington DC, Optical Society of America, 2002).
  8. S. W. Løvseth and D. Yu. Stepanov, �??Analysis of multiple wavelength DFB fiber lasers,�?? IEEE J. Quantum Electron. 37, 770-780 (2001).
    [CrossRef]
  9. S. Narahashi, K. Kumagai, and T. Nojima, �??Minimising peak to average power ratio of multitone signals using steepest descent method,�?? Electron. Lett. 31, 1552-1554 (1995).
    [CrossRef]
  10. M. Friese, �??Multitone signals with low crest factor,�?? IEEE Trans. Commun. 45, 1338-1344 (1997).
    [CrossRef]
  11. A. Othonos, X. Lee, and R. M. Measures, �??Superimposed multiple Bragg gratings,�?? Electron. Lett. 30, 1972-1974 (1994).
    [CrossRef]
  12. G. Sarlet, G. Morthier, R. Baets, D. J. Robbins, and D. C. J. Reid, �??Optimization of multiple exposure gratings for widely tunable laser,�?? IEEE Photon. Techn. Lett. 11, 21-23 (1999).
    [CrossRef]
  13. V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, �??Theory, design, and performance of extended tuning range semiconductor lasers with sampled grating,�?? IEEE J. Quantum Electron. 29, 1824-1834 (1993).
    [CrossRef]
  14. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, �??Long periodic superstructure Bragg gratings in optical fibres,�?? Electron. Lett. 30, 1620- 1622 (1994).
    [CrossRef]
  15. W. H. Loh, F. Q. Zhou, and J. J. Pan, �??Sampled Fiber Grating Based-Dispersion Slope Compensator,�?? IEEE Photon. Techn. Lett. 11, 1280-1282 (1999).
    [CrossRef]
  16. H. Ishii, Y. Tohmori, Y. Yoshikuni, T. Tamamura, and Y. Kondo, �??Multiple-Phase-Shift Super Structure Grating DBR Lasers for Broad Wavelength Tuning,�?? IEEE Photon. Techn. Lett. 5, 613-615 (1993).
    [CrossRef]
  17. Y. Nasu and S. Yamashita, �??Multiple phase-shift superstructure fibre Bragg gratings for DWDM systems,�?? Electron. Lett. 37, 1471-1472 (2001).
    [CrossRef]
  18. R.W. Gerchberg andW. O. Saxton, �??A practical algorithm for the determination of phase from image and diffraction plane pictures,�?? Optik 35, 237-246 (1972).
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge England, Cambridge Univ. Press, 1992).
  20. D. R. Gimlin and C. R. Patisaul, �?? On minimizing the Peak-to-Average Power Ration for the Sum of N Sinusoids,�?? IEEE Trans. Commun. 41, 631-635 (1993).
    [CrossRef]
  21. S. Narahashi and T. Nojima, �??New phasing scheme of N-multiple carriers for reducing peak-to-average power ratio,�?? Electron. Lett. 30, 1382-1383 (1994).
    [CrossRef]
  22. J. Schoukens, Y. Rolain, and P. Guillaume, �??Design of Narrowband, High-Resolution Multisines,�?? IEEE Trans. Instrument. Measur. 45, 750-753 (1996).
    [CrossRef]
  23. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (New York, Wiley & Sons, 1999).
  24. J. Skaar, L. Wang, and T. Erdogan, �??On the Synthesis of Fiber Bragg Gratings by Layer Peeling,�?? IEEE J. Quantum Electron. 37, 165-173 (2001).
    [CrossRef]
  25. L. B¨omer and M. Antweiler, �??Polyphase Barker sequences,�?? Electron. Lett. 25, 1577-1579 (1989); M. Friese and H. Zottmann, �??Polyphase Barker sequences up to length 31,�?? Electron. Lett. 30, 1930-1931 (1994); M. Friese, �??Polyphase Barker sequences up to length 36,�?? IEEE Trans. Inform. Theory 42, 1248-1250 (1996); A. R. Brenner, �??Polyphase Barker sequences up to length 45 with small alphabets,�?? Electron. Lett. 34, 1576-1577 (1998).
  26. E. Van der Ouderaa, J. Schoukens, and J. Renneboog, �??Peak Factor Minimization using a Time-Frequency Domain Swapping Algorithm,�?? IEEE Trans. Instr. Measur. 37, 145-147 (1988).
    [CrossRef]
  27. Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, �??Superposition of chirped fibre Bragg gratings for thirdorder dispersion compensation over 32 WDM channels,�?? Electron. Lett. 38, 1572-1573 (2002).
    [CrossRef]
  28. A. V. Buryak, G. Edvell, A. Graf, K. Y. Kolossovski, and D. Yu. Stepanov, �??Recent progress and novel directions in multi-channel FBG dispersion compensation,�?? in OSA Technical Digest of Conference on Lasers and Electro- Optics (Washington DC, Optical Society of America, 2003).

Electron. Lett.

M. Ibsen, A. Fu, H. Geiger, and R. I. Laming, �??All-fibre 4 x 10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,�?? Electron. Lett. 35, 982-983 (1999).
[CrossRef]

S. Narahashi, K. Kumagai, and T. Nojima, �??Minimising peak to average power ratio of multitone signals using steepest descent method,�?? Electron. Lett. 31, 1552-1554 (1995).
[CrossRef]

A. Othonos, X. Lee, and R. M. Measures, �??Superimposed multiple Bragg gratings,�?? Electron. Lett. 30, 1972-1974 (1994).
[CrossRef]

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, �??Long periodic superstructure Bragg gratings in optical fibres,�?? Electron. Lett. 30, 1620- 1622 (1994).
[CrossRef]

Y. Nasu and S. Yamashita, �??Multiple phase-shift superstructure fibre Bragg gratings for DWDM systems,�?? Electron. Lett. 37, 1471-1472 (2001).
[CrossRef]

S. Narahashi and T. Nojima, �??New phasing scheme of N-multiple carriers for reducing peak-to-average power ratio,�?? Electron. Lett. 30, 1382-1383 (1994).
[CrossRef]

L. B¨omer and M. Antweiler, �??Polyphase Barker sequences,�?? Electron. Lett. 25, 1577-1579 (1989); M. Friese and H. Zottmann, �??Polyphase Barker sequences up to length 31,�?? Electron. Lett. 30, 1930-1931 (1994); M. Friese, �??Polyphase Barker sequences up to length 36,�?? IEEE Trans. Inform. Theory 42, 1248-1250 (1996); A. R. Brenner, �??Polyphase Barker sequences up to length 45 with small alphabets,�?? Electron. Lett. 34, 1576-1577 (1998).

Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, �??Superposition of chirped fibre Bragg gratings for thirdorder dispersion compensation over 32 WDM channels,�?? Electron. Lett. 38, 1572-1573 (2002).
[CrossRef]

IEEE J. Quantum Electron.

V. Jayaraman, Z.-M. Chuang, and L. A. Coldren, �??Theory, design, and performance of extended tuning range semiconductor lasers with sampled grating,�?? IEEE J. Quantum Electron. 29, 1824-1834 (1993).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, �??On the Synthesis of Fiber Bragg Gratings by Layer Peeling,�?? IEEE J. Quantum Electron. 37, 165-173 (2001).
[CrossRef]

S. W. Løvseth and D. Yu. Stepanov, �??Analysis of multiple wavelength DFB fiber lasers,�?? IEEE J. Quantum Electron. 37, 770-780 (2001).
[CrossRef]

H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y. Yoshikuni, �??Quasicontinuous Wavelength Tuning in Super-Structure-Grating (SSG) DBR Lasers,�?? IEEE J. Quantum Electron. 32, 433-441 (1996).
[CrossRef]

A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, �??Optimization of Refractive Index Sampling for Multichannel Fiber Bragg Gratings,�?? IEEE J. Quantum Electron. 39, 91-98 (2003).
[CrossRef]

IEEE Photon. Tech. Lett.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, �??Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,�?? IEEE Photon. Tech. Lett. 14, 1309-1311 (2002).
[CrossRef]

IEEE Photon. Techn. Lett.

W. H. Loh, F. Q. Zhou, and J. J. Pan, �??Sampled Fiber Grating Based-Dispersion Slope Compensator,�?? IEEE Photon. Techn. Lett. 11, 1280-1282 (1999).
[CrossRef]

H. Ishii, Y. Tohmori, Y. Yoshikuni, T. Tamamura, and Y. Kondo, �??Multiple-Phase-Shift Super Structure Grating DBR Lasers for Broad Wavelength Tuning,�?? IEEE Photon. Techn. Lett. 5, 613-615 (1993).
[CrossRef]

IEEE Photon.Techn. Lett.

G. Sarlet, G. Morthier, R. Baets, D. J. Robbins, and D. C. J. Reid, �??Optimization of multiple exposure gratings for widely tunable laser,�?? IEEE Photon. Techn. Lett. 11, 21-23 (1999).
[CrossRef]

IEEE Trans. Commun.

M. Friese, �??Multitone signals with low crest factor,�?? IEEE Trans. Commun. 45, 1338-1344 (1997).
[CrossRef]

D. R. Gimlin and C. R. Patisaul, �?? On minimizing the Peak-to-Average Power Ration for the Sum of N Sinusoids,�?? IEEE Trans. Commun. 41, 631-635 (1993).
[CrossRef]

IEEE Trans. Instr. Measur.

E. Van der Ouderaa, J. Schoukens, and J. Renneboog, �??Peak Factor Minimization using a Time-Frequency Domain Swapping Algorithm,�?? IEEE Trans. Instr. Measur. 37, 145-147 (1988).
[CrossRef]

IEEE Trans. Instrument. Measur.

J. Schoukens, Y. Rolain, and P. Guillaume, �??Design of Narrowband, High-Resolution Multisines,�?? IEEE Trans. Instrument. Measur. 45, 750-753 (1996).
[CrossRef]

Optik

R.W. Gerchberg andW. O. Saxton, �??A practical algorithm for the determination of phase from image and diffraction plane pictures,�?? Optik 35, 237-246 (1972).

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge England, Cambridge Univ. Press, 1992).

A. V. Buryak and D. Yu. Stepanov, �??Novel multi-channel grating devices,�?? in proceedings of Bragg Gratings, Photosensitivity, and Poling in GlassWaveguides, vol. 60 of Top series, BThB3 (Washington DC, Optical Society of America, 2001).

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

Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, �??Multi-channel fiber Bragg gratings for dispersion and slope compensaion,�?? in OSA Technical Digest of Optical Fiber Communication Conference, ThAA5, 581-582 (Washington DC, Optical Society of America, 2002).

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (New York, Wiley & Sons, 1999).

A. V. Buryak, G. Edvell, A. Graf, K. Y. Kolossovski, and D. Yu. Stepanov, �??Recent progress and novel directions in multi-channel FBG dispersion compensation,�?? in OSA Technical Digest of Conference on Lasers and Electro- Optics (Washington DC, Optical Society of America, 2003).

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Figures (3)

Fig. 1.
Fig. 1.

Normalized peak index change as a result of the first two steps in the three-step optimization process. Solid curve shows the analytic estimate (Eq. (7)).

Fig. 2.
Fig. 2.

An illustration of three-stage optimization of a 9-channel dispersion compensator design. Non-trivially modulated phase profile and group delay characteristics are not shown. (a), (b) the amplitude profile and the transmission spectrum obtained after the first step of optimization; (c), (d) the same as (a), (b) but after the second step; (e), (f) result of the third step. The final result is presented in more detail in Fig. 3.

Fig. 3.
Fig. 3.

Details of the 9-channel dispersion compensator design shown in Figs. 2(e,f). (a) amplitude and phase profiles; (b) enlarged (a); (c) central part of the reflection spectrum; (d) group delay. We note that, all fast oscillations of κ(z) profile in the vicinity of the main peak (z≈2.6) were completely eliminated after 30 iterations, though some unimportant weak modulation of the profile in the region z≈3.8 still presents.

Equations (15)

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E b z + i δ E b q ( z ) E f = 0 ,
E f z i δ E f q * ( z ) E b = 0 ,
[ E b ( 0 ) E f ( 0 ) ] = [ 1 t * r 1 t r 2 t 1 t ] [ E b ( L ) E f ( L ) ] ,
q ( z ) = l = 1 N κ ( z ) e i θ ( z ) e i [ ( 2 l N 1 ) Δ k z 2 + ϕ l ] = κ ( z ) e i θ ( z ) S ( z ) ,
arg { S ( z ) } = arctan [ l = 1 N sin ( [ 2 l N 1 ] Δ k z 2 + ϕ l ) l = 1 N cos ( [ 2 l N 1 ] Δ k z 2 + ϕ l ) ] .
S ( z ) = N ( 1 + 2 N Re p = 1 N 1 C p e i p Δ k z ) 1 2 ,
C P = l = 1 N p m l + p m l * , p = 1 , 2 , , N 1 ,
Δ n N = max z S ( z ) Δ n 1 = N + 2 Δ n 1 ,
Δ ( ϕ ) = [ s ( z ) s ( z ) ] 2 = 1 s ( z ) 2 ,
s ( z ) = 1 1 4 N 2 p = 1 N 1 C p 2 + O ( x 3 ) .
Δ n env ( av ) = N ( 1 1 4 N 2 ) Δ n 1 .
F = 1 2 N + O ( 1 N 2 ) .
Δ ( ϕ ) = 1 2 N p = 1 N 1 C p 2 + O ( x 3 x 2 ) ,
Δ ( ϕ ) ϕ l = Im { p = 1 l 1 C p m l p m l * + p = 1 N l C p * m l + p m l * } 2 N 2 Δ ( ϕ ) + O ( x 3 x 2 ) ,
1 2 q ( z 2 ) = + r ( δ ) exp ( i δ z ) d δ .

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