## Design of multichannel DWDM fiber Bragg grating filters by Lagrange multiplier constrained optimization

Optics Express, Vol. 14, Issue 23, pp. 11002-11011 (2006)

http://dx.doi.org/10.1364/OE.14.011002

Acrobat PDF (139 KB)

### Abstract

We present the synthesis of multi-channel fiber Bragg grating (MCFBG) filters for dense wavelength-division-multiplexing (DWDM) application by using a simple optimization approach based on a Lagrange multiplier optimization (LMO) method. We demonstrate for the first time that the LMO method can be used to constrain various parameters of the designed MCFBG filters for practical application demands and fabrication requirements. The designed filters have a number of merits, i.e., flat-top and low dispersion spectral response as well as single stage. Above all, the maximum amplitude of the index modulation profiles of the designed MCFBGs can be substantially reduced under the applied constrained condition. The simulation results demonstrate that the LMO algorithm can provide a potential alternative for complex fiber grating filter design problems.

© 2006 Optical Society of America

## 1. Introduction

1. T. Erdogan, “Fiber grating spectra,” J. of Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

2. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sincsampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. **10**, 842–844 (1998). [CrossRef]

3. X.-F. Chen, Y. Luo, C.-C. Fan, and S.-Z. Xie, “Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system,” IEEE Photon. Technol. Lett. **12**, 1013–1015 (2000). [CrossRef]

4. W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled fiber grating based dispersions lope compensation,” IEEE Photon. Technol. Lett. **11**, 1280–1282 (1999). [CrossRef]

2. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sincsampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. **10**, 842–844 (1998). [CrossRef]

*N*times higher [7

7. A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. **39**, 91–98 (2003). [CrossRef]

8. K. Y. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Y. Stepanov, “Three-step design optimization for multi-channel fiber Bragg gratings,” Opt. Express **11**, 1029–1038, (2003). [CrossRef] [PubMed]

9. S. Baskar, P. N. Suganthan, N. Q. Ngo, A. Alphones, and R. T. Zheng, “Design of triangular FBG filter for sensor applications using covariance matrix adapted evolution algorithm,” Opt. Commun. **260**, 716–722 (2006). [CrossRef]

13. H. Li and Y. Sheng, “Direct design of multichannel fiber Bragg grating with discrete layer-peeling algorithm,” IEEE Photon. Technol. Lett. **15**, 1252–1254 (2003). [CrossRef]

14. Q. Wu, P. L. Chu, and H. P. Chan, “General design approach to Multichannel Fiber Bragg Grating,” J. of Lightwave Technol. **24**, 1571–1580 (2006). [CrossRef]

15. N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A **53**, 1879–1885 (1996). [CrossRef] [PubMed]

## 2. LMO algorithm for FBG design

1. T. Erdogan, “Fiber grating spectra,” J. of Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

*R*and

*S*are the forward- and backward mode amplitudes,

_{c}is the designed coupling coefficient distribution function with Δn(z) being the envelope function of the grating index modulation, λ

_{c}is the central wavelength and ε the overlapping factor. In this study the κ(z) function will be assumed to be real for the ease of practical fabrication. The main idea of the LMO method is to define an objective functional that needs to be minimized, such as,

*r*(

*λ*)=|

*S*(0)/

*R*(0)|

^{2}is the calculated reflection spectrum, L is the total length of the grating, and β is a positive number acting as a weighting parameter for the constraint control. In the defined cost functional, Eq. (2), the spatially coupling coefficient κ(z) is used to shape an output reflection spectrum

*r*(

*λ*) of a given reflection spectrum

*r*(

_{d}*λ*) and to minimize both the reflection spectra difference and the norm of the coupling coefficient profiles simultaneously.

*R*=

*R*+

_{R}*iR*,

_{I}*S*=

*S*+

_{R}*iS*,

_{I}*μ*=

_{R}*μ*+

_{R,R}*iμ*and

_{R,I}*μ*=

_{S}*μ*+

_{S,R}*iμ*.

_{S,I}*J*, a variational method for Eq (2) is used with respect to the forward- and backward- modes R and S through the Lagrange multipliers

*μ*and

_{R}*μ*. The resulting equations of motion for the Lagrange multipliers are

_{S}*=*

_{r}*r*(

*λ*)-

*r*(

_{d}*λ*) is the discrepancy between the output and target reflection spectrum. Then, the cost functional

*J*is varied with respect to the coupling coefficient function κ(z)

*κ*(

_{ini}*z*) and let

*κ*(

_{old}*z*)=

*κ*(

_{ini}*z*).

*μ*(0) and

_{R}*μ*(0) by using Eq. (4). Then, the propagations of the Lagrange-multiplier functions

_{S}*μ*(

_{R}*z*) and

*μ*(

_{S}*z*) from

*z*=0 to

*z*=

*L*can be obtained by solving the Eq. (3).

## 3. Design results and discussion

*r*

_{0}is the maximum reflectivity, λ

_{c}is the central wavelength, Δ

_{CS}is channel spacing, and Δ

*λ*is the bandwidth for each channel. The total number of the calculated spectral points is set to be 200 and the central wavelength is set to be 1.55×10

^{-3}mm (1550nm). The units of λ and

*L*are

*mm*, and κ(z) is

*mm*

^{-1}. In the LMO algorithm for synthesizing MCFBGs, α is an ad hoc constant and β is a weighting parameter which is zero for unconstrained conditions and nonzero for the constrained coupling coefficient design. In this study case, we find that the best value of α is around 5×10

^{4}, which can achieve an optimal and smoother convergence. The constrain on the value of the coupling constant can be more enforced with the sacrifice of the reflectivity spectrum quality by increasing the values of the weighting parameter β. We choose a value of β=10

^{-7}for the comparison with the unconstrained situation β=0 in this designed case.

_{CS}=50GHz and the Δ

*λ*is 0.16nm for each channel corresponding to a bandwidth of full width at half maximum about 0.32nm and bandwidth 0.35nm in -50dB. The simulation results are shown in Fig. 1. The designed reflection spectrum meets excellently with the target spectrum as in Fig. 1(a), with close to 30dB isolation outside the channels and with low dispersion inside the channels (deviation <±100 ps/nm in 75% of stopband and the maximum value is about ±350ps/nm within whole channels). A detailed dispersion profile in one channel of the designed two-channel FBG is shown in Fig. 1(c). The apodization profile of the index modulation for this two-channel FBG filter is shown in Fig. 1(d), with the maximum index modulation only slightly larger than the single-channel one (initial Gaussian apodization). In this work, we only consider the spectral reflectivity amplitude optimization, just to demonstrate the present optimization method. This is why we don’t have a flat group delay (dispersionless) response here.

*λ*=0.088nm corresponding to a bandwidth of full width at half maximum 0.16nm and bandwidth 0.2nm in -50dB is synthesized. The simulation results appear in Fig. 2. Again, the designed reflection spectrum meets very well with the target spectrum in Fig. 2(a) with more than 30dB isolation outside the channels. In Fig. 2(c), the dispersion profile in one channel of the 8-channel FBG is shown. The apodization profile of the index modulation for this eight-channel FBG filter is shown in Fig. 2(d), with the maximum index modulation 1.6 times higher than the single-channel one (initial Gaussian apodization). It should be noted that the simulation is finished after hundreds of iterations. The convergence of the LMO method for MCFBG syntheses is efficient and monochromatic. Unlike other phase sampling approaches, no additional Monte-Carlo based optimization algorithm is used here. The typical evolution curves of the calculated average error (total error divided by the number of spectral points) for the cases of N=2, 4, 8 channel numbers are shown in Fig. 3. The reason why the initial error increases with the channel number is simply because we use the same Gaussian apodization function as the initial guess for all the design cases. When the channel number is increased, the initial error is increased due to the larger mismatch. However, the important thing here is that the convergence behavior (or trend) for different channel numbers is basically the same as can be seen in Fig. 3, despite the different magnitude of the initial errors. That is, the convergence quality actually does not degrade due to the increase of the channel numbers. This proves the suitability of the present method for designing complicated multichannel FBG filters.

^{-7}is used to control the maximum index modulation in the apodization profile. In Fig. 4, it can be seen that the maximum index modulation of a MCFBG could be significantly decreased to the same magnitude as the single-channel FBG case by slightly sacrificing the channel reflectivity.

## 4. Conclusion

## Acknowledgments

## References and links

1. | T. Erdogan, “Fiber grating spectra,” J. of Lightwave Technol. |

2. | M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sincsampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. |

3. | X.-F. Chen, Y. Luo, C.-C. Fan, and S.-Z. Xie, “Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system,” IEEE Photon. Technol. Lett. |

4. | W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled fiber grating based dispersions lope compensation,” IEEE Photon. Technol. Lett. |

5. | H. Lee and G. P. Agrawal, “Purely phase-sampled fiber Bragg gratings for broad-band dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. |

6. | Q Wu, C. Yu, K. Wang, X. Wang, Z. Yu, H. P. Chan, and P. L. Chu, “New sampling-based design of simultaneous compensation of both dispersion and dispersion slope for multichannel fiber Bragg gratings,” IEEE Photon. Technol. Lett. |

7. | A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. |

8. | K. Y. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Y. Stepanov, “Three-step design optimization for multi-channel fiber Bragg gratings,” Opt. Express |

9. | S. Baskar, P. N. Suganthan, N. Q. Ngo, A. Alphones, and R. T. Zheng, “Design of triangular FBG filter for sensor applications using covariance matrix adapted evolution algorithm,” Opt. Commun. |

10. | S. Baskar, R. T. Zheng, A. Alphones, N. Q. Ngo, and P. N. Suganthan, “Particle swarm optimization for the design of low-dispersion Fiber Bragg Gratings,” IEEE Photon. Technol. Lett. |

11. | R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. |

12. | J. Skaar, L. Wang, and T. Erdogen, “On the synthesis of fiber Bragg grating by layer peeling,” IEEE J. Quantum Electron. |

13. | H. Li and Y. Sheng, “Direct design of multichannel fiber Bragg grating with discrete layer-peeling algorithm,” IEEE Photon. Technol. Lett. |

14. | Q. Wu, P. L. Chu, and H. P. Chan, “General design approach to Multichannel Fiber Bragg Grating,” J. of Lightwave Technol. |

15. | N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A |

16. | N. Wang and H. Rabitz, “Optimal control of population transfer in an optical dense medium,” J. Chem. Phys. |

17. | R. Buffa, “Optimal control of population transfer through the continuum,” Opt. Commun. |

18. | F. I. Lewis, |

19. | S. A. Rice and M. Zhao, |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

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

(230.1480) Optical devices : Bragg reflectors

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 28, 2006

Revised Manuscript: October 23, 2006

Manuscript Accepted: October 26, 2006

Published: November 13, 2006

**Citation**

Cheng-Ling Lee, Ray-Kuang Lee, and Yee-Mou Kao, "Design of multichannel DWDM fiber Bragg grating filters by Lagrange multiplier constrained optimization," Opt. Express **14**, 11002-11011 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11002

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### References

- T. Erdogan, "Fiber grating spectra," J. of Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]
- M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sincsampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photon. Technol. Lett. 10, 842-844 (1998). [CrossRef]
- X.-F. Chen, Y. Luo, C.-C. Fan, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000). [CrossRef]
- W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based dispersions lope compensation," IEEE Photon. Technol. Lett. 11, 1280-1282 (1999). [CrossRef]
- H. Lee and G. P. Agrawal, "Purely phase-sampled fiber Bragg gratings for broad-band dispersion and dispersion slope compensation," IEEE Photon. Technol. Lett. 15, 1091-1093 (2003). [CrossRef]
- Q. Wu, C. Yu, K. Wang, X. Wang, Z. Yu, H. P. Chan, and P. L. Chu, "New sampling-based design of simultaneous compensation of both dispersion and dispersion slope for multichannel fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 381-383 (2005). [CrossRef]
- A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003). [CrossRef]
- K. Y. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Y. Stepanov, "Three-step design optimization for multi-channel fiber Bragg gratings," Opt. Express 11, 1029-1038, (2003). [CrossRef] [PubMed]
- S. Baskar, P. N. Suganthan, N. Q. Ngo, A. Alphones, and R. T. Zheng, "Design of triangular FBG filter for sensor applications using covariance matrix adapted evolution algorithm," Opt. Commun. 260,716-722 (2006). [CrossRef]
- S. Baskar, R. T. Zheng, A. Alphones, N. Q. Ngo, and P. N. Suganthan, "Particle swarm optimization for the design of low-dispersion Fiber Bragg Gratings," IEEE Photon. Technol. Lett. 17, 615-617 (2005). [CrossRef]
- R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 29,1824-1834 (1993).
- J. Skaar, L. Wang, and T. Erdogen, "On the synthesis of fiber Bragg grating by layer peeling," IEEE J. Quantum Electron. 37, 165-173 (2001). [CrossRef]
- H. Li and Y. Sheng, "Direct design of multichannel fiber Bragg grating with discrete layer-peeling algorithm," IEEE Photon. Technol. Lett. 15, 1252-1254 (2003). [CrossRef]
- Q. Wu, P. L. Chu, and H. P. Chan, "General design approach to Multichannel Fiber Bragg Grating," J. of Lightwave Technol. 24, 1571-1580 (2006). [CrossRef]
- N. Wang and H. Rabitz, "Optimal control of pulse amplification without inversion," Phys. Rev. A 53, 1879-1885 (1996). [CrossRef] [PubMed]
- N. Wang and H. Rabitz, "Optimal control of population transfer in an optical dense medium," J. Chem. Phys. 104, 1173-1178 (1996). [CrossRef]
- R. Buffa, "Optimal control of population transfer through the continuum," Opt. Commun. 153, 240-244 (1998). [CrossRef]
- F. I. Lewis, Optimal Control, (Wiley, New York, 1986).
- S. A. Rice and M. Zhao, Optimal Control of Molecular Dynamics, (Wiley, New York, (2000).

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