## Three-step design optimization for multi-channel fibre Bragg gratings

Optics Express, Vol. 11, Issue 9, pp. 1029-1038 (2003)

http://dx.doi.org/10.1364/OE.11.001029

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### 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

## 1. Introduction

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]

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel 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 4×10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,” Electron. Lett. **35**, 982–983 (1999). [CrossRef]

8. S. W. Lϕvseth 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]

*n*

^{(av)}, accumulates with the number of UV exposures. As a result, the local Bragg wavelengths of the previously written channels become longer, thus requiring corresponding corrections to the period of the interference pattern to be made in advance. Finally, the major drawback of the FBG superimposing is the fact that this average index change grows approximately linearly with the number of UV exposures and, thus, with the number of channels, i.e., Δ

*n*

^{(av)}~

*N*. As it will become clear from the rest of this section, a much better utilization of the fiber photosensitivity is possible using different methods of fabrication of the multi-channel FBGs.

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]

*n*

_{N}that grows linearly with the number of channels

*N*, Δ

*n*

_{N}~

*N*. The efficiency of the comb-sampling approach is low because the fiber is utilized only partially, i.e., there are segments of the fibre without any grating written. If we introduce a fiber utilization figure of merit as

*F*=(Δ

*n*

_{N})

^{2}, where Δ

*N*and

*F*~1/

*N*≪1, i.e., is prohibitively low. Modification of the comb-sampling by writing additional gratings into the unused parts of the fiber [15

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]

*n*

_{N}~

*N*, is a characteristic property of another amplitude-modulation approach, so-called sinc-sampling [6

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

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]

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]

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]

*n*

_{N}~√

*N*(i.e.,

*F*~1); (2) quality of spectral characteristics should not be compromised; and (3) the optimization procedure should not require prohibitively long computer time.

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel 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]

## 2. Formulation of the problem

*E*

_{f}and

*E*

_{b}are the amplitudes of the forward and backward propagating fields, respectively, δ is the normalized frequency detuning from the central Bragg reflection frequency,

*z*is a local distance along FBG,

*q*(

*z*) is a spatial profile of the FBG coupling coefficient, and asterisk denotes complex conjugation. For a reciprocal and lossless (see, e.g., [23] for definitions) FBG of length

*L*we may find complex reflection and transmission coefficients from a transfer matrix, which relates field values at the grating ends

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]

4. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. **39**, 91–98 (2003). [CrossRef]

*N*-channel grating design can be obtained by a dephasing approach, where the slowly varying envelope of a direct summation of

*N*identical gratings, equally spaced in the frequency space, is taken with relative phases ϕ

_{l}for each seeding grating,

*S*(

*z*) is given by

*i*=√-1, Re stands for the real part, and

_{l}:

*m*

_{l}=exp(

*i*ϕ

_{l}). From Eq. (5) it follows that the amplitude modulation of the sampling function is small when autocorrelation of the sequence

*m*

_{l}is low, i.e., when the amplitude of its AACF is small. If |

*C*

_{p}|≤1 for all

*p*=1, 2, …,

*N*-1, the corresponding sequence

*m*

_{l}is called a

*generalized Barker sequence*. Barker sequences have been reported for

*N*up to 45 [25

25. L. Bömer 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).

*N*=2) cannot be optimized as the maximum of the sampling function is always 2 regardless of the values of the dephasingangles ϕ

_{1}, ϕ

_{2}. More generally, one might observe from Eq. (6) that |

*C*

_{N}

_{-1}|=1 for an arbitrary

*N*. This leads to a simple estimate from below for the peak of the sampling function (Eq. (5)),i.e., for the maximum amplitude of index variation (for more details see [10

10. M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. **45**, 1338–1344 (1997). [CrossRef]

*n*

_{1}is the maximum index change for the seeding grating design.

## 3. Optimal design

_{l}, as in Ref. [4

**39**, 91–98 (2003). [CrossRef]

*z*∈[0, 2π/Δ

*k*], which is numerically a burdensome task. A typical number of mesh points in

*z*required to locate the min (max) to an appropriate accuracy is in the order of 10

^{2}×

*N*. Therefore, a single evaluation of the minimizing function only comprises in the order of 10

^{2}×

*N*

^{3}operations. Therefore it is highly desirable to formulate the optimization problem to avoid scanning over continuous variable

*z*. This can be achieved via construction of a functional which comprises integration over

*z*.

### 3.1 Functional approach

*s*(

*z*)≡|

*S*(

*z*)|/√

*N*is its standard deviation over the sampling period,

*z*∈ [0, 2π/Δ

*k*]. Hence, the functional to be minimized can be constructed as follows,

*s*(

*z*)〉≡Δ

*k*/2π

*s*(

*z*)

*dz*. To obtain the last expression in Eq. (8) we used the mean square value of Eq. (5), 〈

*s*(

*z*)

^{2}〉=1. Note that minimization of the standard deviation is equivalent to maximization of the mean value of

*s*(

*z*), i.e., Δ

*s*(

*z*) explicitly, we assume that the second term in Eq. (5),

*x*(

*z*)≡(2/

*N*)Re

*C*

_{p}

*e*

^{ipΔkz}, is much less than 1 for all

*z*∈ [0, 2π/Δ

*k*]. Truncating the Taylor series of

*s*(

*z*) with respect to

*x*(

*z*) at the third term and averaging the result over the period one obtains

_{1}, ϕ

_{2}, …, ϕ

_{N}).

_{l}→ϕ

_{l}+

*a*

_{1}+

*a*

_{2}

*l*, where

*a*

_{1,2}are constants, dimension of the parameter space

*O*(

*N*

^{2}) operations that is a considerable improvement in comparison to the minimax strategies [4

**39**, 91–98 (2003). [CrossRef]

^{6}tries.

### 3.2 Iterative schemes

**39**, 91–98 (2003). [CrossRef]

10. M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. **45**, 1338–1344 (1997). [CrossRef]

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]

*S*(

*z*) into its phase. The peak of the sampling function is reduced significantly at the expense of small side-lobes in its spectrum. The integral size of the side-lobes is proportional to the mean-square deviation of |

*S*(

*z*)| [4

**39**, 91–98 (2003). [CrossRef]

*S*(

*z*) at some level

*S*

_{0}. By subtracting the Fourier transform of the complex error function from the finite spectrum of original

*S*(

*z*) and restoring the amplitude profile of the spectrum to the original form (that includes setting the out-of-band response to zero), one decreases the maximum peak value of

*S*(

*z*). Gradually increasing level

*S*

_{0}, one might significantly reduce the maximum peak of

*S*(

*z*) [10

**45**, 1338–1344 (1997). [CrossRef]

*S*(

*z*) obtained using generalized Barker sequences (see [25

25. L. Bömer 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).

*N*=47, 51 and 53) followed by the clipping (stars). It can be seen that the functional approach on its own yields sufficiently good results. Further optimization based on the clipping algorithm yields the same or better peak optimization than the corresponding two-step optimization of Barker sequences for all 5<

*N*≤45 except

*N*=12, 13, 15, 16 and 40. Functional approach is more simple and computationally more effective then the known algorithms of searching for generalized Barker sequences. Moreover, it yields optimized sampling functions characterized by zero-free profiles that avoids sometimes undesirable phase π-jumps.

### 3.3 Spectral fine-tuning

*n*

_{N}, which, in turn, increases the fiber utilization parameter close to

*F*≈1. In addition, they provide a

*universal*optimization method, i.e., knowing a single

*N*-channel optimal set of dephasing angles allows one to obtain the corresponding design for any given seeding grating by using a simple formula

*q*

_{N}(

*z*)=

*S*

_{N}(

*z*)

*q*

_{1}(

*z*). However, it is important to note that all designs based on

*periodic*sampling are inevitably

*non-ideal*because the neighbouring channels (even well separated ones) may distort the spectra of each other. This is visible in Figs. 2(b,d) as small deviations of the transmission spectra from the square-like shape. For this particular example these deviations are relatively minor. However, for some other grating designs, especially for multi-channel filters with zero in band dispersion, the situation can be worse (see, e.g., [3]). In general, periodic sampling works rather well if the seeding grating has smooth, slowly-varying amplitude and phase, which is usually the case, e.g., for the conventional (i.e., second order only) dispersion compensating devices. In contrast, zero dispersion filters and third order (dispersion slope) dispersion compensators have abrupt jumps in the seeding grating phase and sinc-like seeding grating amplitude dependencies and are much less suited for the use of any pure periodic sampling methods. For such gratings a third optimization step is necessary. It is based on an observation that for weak gratings the first order Born approximation holds:

^{(i)}(

*z*) (

*i*is an iteration number) by

_{1}(

*z*) is the seeding grating amplitude and the constant

*A*

^{(i)}is defined by the normalization condition

*A*

^{(i)}=

^{(i)}}

^{2}

*dz*/

*dz*; (2) using the unchanged multi-channel grating phase and the modified grating amplitude as the input data for the direct scattering solver; (3) modifying the resulting spectrum data in the following way: within the central channel spectral domains the current spectrum data is replaced by pre-iteration data (fixed spectral domain), and outside the central channel spectral domains the current spectrum data is left intact (spectral domain allowed to evolve); (4) finally, solving the inverse scattering problem for the modified spectrum to obtain the optimized multi-channel grating amplitude and phase. That completes one iteration step which can be repeated until fast oscillations in κ

^{(i)}(

*z*) profile are reduced close to an acceptable level.

*z*).

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]

## 4. Conclusion

*N*≲45). Moreover, the method can readily be used to obtain optimization for the number of channels

*N*up to 100 in a reasonable time using conventional personal computers. Finally the described algorithm incorporates a spectrum fine-tuning step, which is specific for FBGs, allowing one to get high quality spectral characteristics with little or no sacrifice of the achieved sampling function minimization level.

*N*-channel FBGs. Thus, for example, broadband FBG devices compensating not only for the average dispersion, but for the dispersion slope as well, may be efficiently designed and implemented. To conclude, the efficient multi-channel optimization methods described in this paper open new exciting opportunities for FBG applications. For example, FBG-based dispersion compensators covering the whole communications’ C-band can replace dispersion compensating fibers as the leading dispersion management tool. The first practically usable experimental implementations of multi-channel FBG-based dispersion compensators with

*N*≫1 have recently been presented [27

27. Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, “Superposition of chirped fibre Bragg gratings for third-order dispersion compensation over 32 WDM channels,” Electron. Lett. **38**, 1572–1573 (2002). [CrossRef]

## Acknowledgment

## References and links

1. | A. Othonos and K. Kalli, |

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. |

3. | A. V. Buryak and D. Yu. Stepanov, “Novel multi-channel grating devices,” in proceedings of Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, 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 Multi-channel Fiber Bragg Gratings,” IEEE J. Quantum Electron. |

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. |

6. | M. Ibsen, A. Fu, H. Geiger, and R. I. Laming, “All-fibre 4×10Gbit/s WDM link with DFB fibre laser transmitters and single sinc-sampled fibre grating dispersion compensator,” Electron. Lett. |

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. Lϕvseth and D. Yu. Stepanov, “Analysis of multiple wavelength DFB fiber lasers,” IEEE J. Quantum Electron. |

9. | S. Narahashi, K. Kumagai, and T. Nojima, “Minimising peak to average power ratio of multitone signals using steepest descent method,” Electron. Lett. |

10. | M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun. |

11. | A. Othonos, X. Lee, and R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. |

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. |

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. |

14. | B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. |

15. | W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled Fiber Grating Based-Dispersion Slope Compensator,” IEEE Photon. Techn. Lett. |

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. |

17. | Y. Nasu and S. Yamashita, “Multiple phase-shift superstructure fibre Bragg gratings for DWDM systems,” Electron. Lett. |

18. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

19. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

20. | D. R. Gimlin and C. R. Patisaul, “On minimizing the Peak-to-Average Power Ration for the Sum of N inusoids,” IEEE Trans. Commun. |

21. | S. Narahashi and T. Nojima, “New phasing scheme of N-multiple carriers for reducing peak-to-average power ratio,” Electron. Lett. |

22. | J. Schoukens, Y. Rolain, and P. Guillaume, “Design of Narrowband, High-Resolution Multisines,” IEEE Trans. Instrument. Measur. |

23. | C. K. Madsen and J. H. Zhao, |

24. | J. Skaar, L. Wang, and T. Erdogan, “On the Synthesis of Fiber Bragg Gratings by Layer Peeling,” IEEE J. Quantum Electron. |

25. | L. Bömer and M. Antweiler, “Polyphase Barker sequences,” Electron. Lett. |

26. | E. Van der Ouderaa, J. Schoukens, and J. Renneboog, “Peak Factor Minimization using a Time-Frequency Domain Swapping Algorithm,” IEEE Trans. Instr. Measur. |

27. | Y. Painchaud, H. Chotard, A. Mailloux, and Y. Vasseur, “Superposition of chirped fibre Bragg gratings for third-order dispersion compensation over 32 WDM channels,” Electron. Lett. |

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). |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(060.2340) Fiber optics and optical communications : Fiber optics components

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 24, 2003

Revised Manuscript: April 22, 2003

Published: May 5, 2003

**Citation**

Kazimir Kolossovski, Rowland Sammut, Alexander Buryak, and Dmitrii Stepanov, "Three-step design optimization for multi-channel fibre Bragg gratings," Opt. Express **11**, 1029-1038 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1029

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### References

- A. Othonos and K. Kalli, Fiber Bragg Gratings (Boston, Artech House, 1999).
- 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 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. 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]
- 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]
- 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]
- 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).
- S. W. Løvseth and D. Yu. Stepanov, �??Analysis of multiple wavelength DFB fiber lasers,�?? IEEE J. Quantum Electron. 37, 770-780 (2001). [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]
- M. Friese, �??Multitone signals with low crest factor,�?? IEEE Trans. Commun. 45, 1338-1344 (1997). [CrossRef]
- A. Othonos, X. Lee, and R. M. Measures, �??Superimposed multiple Bragg gratings,�?? Electron. Lett. 30, 1972-1974 (1994). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- Y. Nasu and S. Yamashita, �??Multiple phase-shift superstructure fibre Bragg gratings for DWDM systems,�?? Electron. Lett. 37, 1471-1472 (2001). [CrossRef]
- 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).
- 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).
- 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]
- 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]
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