## Reflection equalization of the simultaneous dispersion and dispersion-slope compensator based on a phase-only sampled fiber Bragg grating

Optics Express, Vol. 16, Issue 13, pp. 9821-9828 (2008)

http://dx.doi.org/10.1364/OE.16.009821

Acrobat PDF (444 KB)

### Abstract

A novel approach for the reflection equalization of a phase-only sampled fiber Bragg grating (FBG) is presented, where the grating is specially designed as a simultaneous dispersion and dispersion-slope compensator with channels up to 51. The sampling-function used is given with an analytical form with a linearly-chirped sampling period and is optimized by using the simulated annealing algorithm.

© 2008 Optical Society of America

## 1. Introduction

1. U. Peschel, T. Peschel, and F. Lederer, “A compact device for highly efficient dispersion compensation in fiber transmission,” Appl. Phys. Lett. **67**, 2111–2113 (1995). [CrossRef]

3. F. Ouellette, P. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensation using sampled fibre Bragg gratings,” Electron. Lett. **31**, 899–901 (1995). [CrossRef]

4. M. Ibsen, M. Durkin, M. Cole, and R. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. **10**, 842–844 (1998). [CrossRef]

7. A. V. Buryak, K. Kolossovski, and D. Yu. Stepanov, “Optimisation of refractive index sampling for multichannel FBGs,” IEEE J. Quant. Electron. **39**, 91–98 (2003). [CrossRef]

9. Y. T. Dai, X. F. Chen, X. Xu, C. Fan, and S. Z. Xie, “High channel-count comb filter based on chirped sampled fiber Bragg grating and phase shift,” IEEE Photon. Technol. Lett. **17**, 1040–1042 (2005). [CrossRef]

11. H. Li, M. Li, K. Ogusu, Y. Sheng, and J. Rothenberg, “Optimization of a continuous phase-only sampling for high channel-count fiber Bragg gratings,” Opt. Express **14**, 3152–3160 (2006). [CrossRef] [PubMed]

12. J. E. Rothenberg, H. Li, Y. Sheng, J. Popelek, and J. Zweiback, “Phase-only sampled 45 channel fiber Bragg grating written with a diffraction-compensated phase mask,” Opt. Lett. **31**, 1199–1201 (2006). [CrossRef] [PubMed]

13. H. Lee and G. Agrawal, “Purely phase-sampled fiber Bragg gratings for broad-band dispersion and dispersion slope dispersion,” IEEE Photon. Technol. Lett. **15**, 1091–1093 (2003). [CrossRef]

14. H. Lee and G. Agrawal, “Bandwidth equalization of purely phase-sampled fiber Bragg gratings for broadband dispersion and dispersion slope compensation,” Opt. Express **12**, 5595–5602 (2004). [CrossRef] [PubMed]

15. H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, “Advances in the Design and Fabrication of High-Channel-Count Fiber Bragg Gratings,” IEEE J. Lightwave Technol. **25**, 2739–2750 (2007). [CrossRef]

16. Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shifts in phase mask for fiber Bragg grating” IEEE Photon. Technol. Lett. **16**, 1316–1318 (2004). [CrossRef]

14. H. Lee and G. Agrawal, “Bandwidth equalization of purely phase-sampled fiber Bragg gratings for broadband dispersion and dispersion slope compensation,” Opt. Express **12**, 5595–5602 (2004). [CrossRef] [PubMed]

8. Q. Wu, C. Yu, K. Wang, X. Wang, Z. Yu, H. Chan, and P. 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]

## 2. Simultaneous dispersion and dispersion-slope compensation

*Δn*can be expressed as

*Δn*

_{1}(

*z*) is the index-modulation, z is the position along the grating,

*Λ*is the local pitch of a seed grating and it can be expressed as

*Λ*(

*z*)=

*Λ*

_{0}(1-

*C*×

_{g}*z*) for a linearly chirped FBG, where

*Λ*

_{0}is the period at the beginning position of grating and

*C*is the chirp rate of the grating period.

_{g}*s*(

*z*) denotes a sampling function with period of P, in general, it can be expanded in a Fourier series

*S*is the complex-valued Fourier coefficient. To realize the simultaneous dispersion and dispersion-slope compensation, we introduce a chirp in the sampling period the same as the reported in Ref. 5, i.e., we make the local sampling period as

_{m}*P*(

*z*)=

*P*

_{0}(1+

*C*·

_{s}*z*), where

*P*

_{0}is the initial sampling period,

*C*is the linear variation coefficient. For a general case of

_{s}*C*≪1, the sampling function may be expanded and approximately expressed as

_{s}*C*(=

_{eff}*C*-

_{g}*C*·

_{s}*m*Λ

_{0}/

*P*

_{0}) is the equivalent chirp rate of the grating period. It is obviously seen that the chirp in sampling period may be approximately equivalent to the chirp in grating period. Since the dispersion magnitude is inversely proportional to

*C*(which is a linear function of the channel number m), the dispersions in all the channels are no longer identical but changes according to the channel number m, and thus the required dispersion and dispersion slope may be approximately obtained by suitable choosing the value of

_{eff}*C*and

_{g}*C*, respectively. [9

_{s}9. Y. T. Dai, X. F. Chen, X. Xu, C. Fan, and S. Z. Xie, “High channel-count comb filter based on chirped sampled fiber Bragg grating and phase shift,” IEEE Photon. Technol. Lett. **17**, 1040–1042 (2005). [CrossRef]

*Δn*(

_{m}*z*), i.e. the index modulation for the ghost grating m (channel m), is directly proportional to the Fourier coefficient S

_{m}. Therefore, in order to create a FBG with multiple identical channels, in general, one needs to optimize the phase-only sampling function by making all the in-band (for a given channels N) Fourier coefficients S

_{m}identical. [10]

11. H. Li, M. Li, K. Ogusu, Y. Sheng, and J. Rothenberg, “Optimization of a continuous phase-only sampling for high channel-count fiber Bragg gratings,” Opt. Express **14**, 3152–3160 (2006). [CrossRef] [PubMed]

^{2}/km) with length of 110 km. Firstly, we write the phase-only sampling function

*s*(

*z*)with the initial sampling period

*P*

_{0}as

*s*(

*z*)=

*s*(

_{b}*z*)⊗∑

*(*

_{m}δ*z*-

*mP*

_{0}), where

*s*(

_{b}*z*) is the base sampling function in one period which is given as a continuous one with the analytical form:

*s*(

_{b}*z*)=exp[

*iθ*(

_{g}*z*)]. We assume that

*θ*(

_{g}*z*) has the general form including many harmonic terms as:

*M*is minimized, 2

*M*uniform channels could be achieved with

*M*terms in this series since there are two free parameters for each term. By using the simulated annealing algorithm, the parameters

*α*and

_{n}*β*are optimized to make the channel spectrum flat within the band of interest. [10] In our case, we purposely eliminated the spatial frequencies from 17 to 21 to effectively avoid the phase-vanishing effect. [11

_{n}11. H. Li, M. Li, K. Ogusu, Y. Sheng, and J. Rothenberg, “Optimization of a continuous phase-only sampling for high channel-count fiber Bragg gratings,” Opt. Express **14**, 3152–3160 (2006). [CrossRef] [PubMed]

*α*,

_{n}*β*with

_{n}*n*=1⋯16, 22⋯31 which are listed in table 1. Figure 1(a) shows the phase distribution of the optimized sampling function in a normalized period (1mm), which will be transferred into the phase mask with pre-compensation of the diffraction effect. [11

**14**, 3152–3160 (2006). [CrossRef] [PubMed]

^{-5}, and the diffraction efficiency is larger than 80%.

^{2}) at the central wavelength of 1545 nm, parameters for the chirp rates

*C*and

_{sa}*C*are optimally selected as -0.943×10

_{g}^{-4}cm

^{-1}and -1.723×10

^{-5}cm

^{-1}, respectively. Figure 2 shows the calculation results without the reflection equalization. Dispersion about - 1815 ps/nm at wavelength of 1545 nm, dispersion slope of - 6.6 ps/nm

^{2}have been successfully obtained, which are almost the same as what we expect. However, as shown in Fig. 2(b), the reflection between all the inter-channels become no longer identical, i.e., the reflections are linearly decreased with the wavelength decrement.

## 3. Equalization of the Inclined Reflection Spectrum

20. M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. **12**, 498–500(2000) [CrossRef]

*η*denotes the confined coefficient of the electrical field in the fiber core, D (in a unit of ps/nm) denotes the grating dispersion, Δ

*n*is in the unit of 10

^{-4}. From Eq. (6), it can be seen that once the dispersion in each channel are linearly changed, the corresponding channel reflection will not be identical with each other even if all the Fourier coefficients S

_{m}for a given channel number of N are the same. To equalize this distortion appeared in the reflection spectrum, the value of |S

_{m}|

^{2}×|D

_{m}| for each channel should be kept in a constant. Note that the proposal given in Ref. 14

14. H. Lee and G. Agrawal, “Bandwidth equalization of purely phase-sampled fiber Bragg gratings for broadband dispersion and dispersion slope compensation,” Opt. Express **12**, 5595–5602 (2004). [CrossRef] [PubMed]

_{m}is only accurate when the FBG is extremely strong. i.e., the reflection is almost equal to 100%. As a matter of fact, it can be seen from Eq. (4) that any change of the amplitude of S

_{m}will not affect the phase term and thus make no contributions to the dispersion.

*S*|

_{m}^{2}×|

*D*| has been kept in a constant. Firstly, the target Fourier coefficient S

_{m}_{Tm}is determined according to the dispersion spectrum shown in Fig. 2(a). Next, a new continuous phase-only sampling function is proposed and optimized with the simulated annealing algorithm. The cost function is defined as

*α*,

_{n}*β*used for the purpose of the reflection equalization. Figure 3(a) shows the phase distribution of the sampling function. Figure 3(b) shows the obtained channel spectrum in which the sum of the square difference between the target Fourier coefficient |

_{n}*S*|

_{Tm}^{2}and the calculated one |

*S*|

_{Cm}^{2}over all 51 channels is less than 4.3×10

^{-9}. The Fourier coefficient S

_{Cm}of the optimized sampling function is almost identical with the target one S

_{Tm}and the diffraction efficiency is larger than 80%. Figure 4(a) shows the effectively equalized reflection spectrum using the new sampling-function. To clearly show the equalization effect, the reflection spectra with and without reflection equalizations are illustrated in Fig. 4(b). It is seen that the maximum channel-channel reflection difference is decreased from 0.03 to 0.004 (i.e., about 1/10). There also exists some reflection fluctuations among the channels, which are attributed to the differences between the resulted Fourier coefficient and the target ones in the optimization process. Figure 5(a) shows the group delay spectra with and without reflection equalization. Seeing from a randomly selected channel as shown in the inset of Fig. 5(a), we can find that the reflection equalization process make no effect on the group delay spectrum (i.e., the dispersion), which in return means that the Fourier coefficient S

_{m}has no relation with the dispersion slope obtained. Figure 5(b) illustrates the group delay ripples of the central channel with and without reflection equalization which indicates that the maximum ripples are both smaller than 0.2 ps.

## 4. Conclusions

## Acknowledgments

## References and links

1. | U. Peschel, T. Peschel, and F. Lederer, “A compact device for highly efficient dispersion compensation in fiber transmission,” Appl. Phys. Lett. |

2. | A. Isomäki, A. Vainionpää, J. Lyytikäinen, and O. G. Okhotnikov, “Semiconductor mirror for dynamic dispersion compensation,” Appl. Phys. Lett. |

3. | F. Ouellette, P. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensation using sampled fibre Bragg gratings,” Electron. Lett. |

4. | M. Ibsen, M. Durkin, M. Cole, and R. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. |

5. | Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, “Multi-channel fiber Bragg gratings for dispersion and slope compensation,” Optical Fiber Communication Conf. Paper. ThAA5. (2002). |

6. | W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled fiber grating based-dispersion slope compensator,” IEEE Photon. Tech. Lett. |

7. | A. V. Buryak, K. Kolossovski, and D. Yu. Stepanov, “Optimisation of refractive index sampling for multichannel FBGs,” IEEE J. Quant. Electron. |

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

9. | Y. T. Dai, X. F. Chen, X. Xu, C. Fan, and S. Z. Xie, “High channel-count comb filter based on chirped sampled fiber Bragg grating and phase shift,” IEEE Photon. Technol. Lett. |

10. | H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-only sampled fiber Bragg gratings for high channel counts chromatic dispersion compensation,” IEEE J. Lightwave Technol. |

11. | H. Li, M. Li, K. Ogusu, Y. Sheng, and J. Rothenberg, “Optimization of a continuous phase-only sampling for high channel-count fiber Bragg gratings,” Opt. Express |

12. | J. E. Rothenberg, H. Li, Y. Sheng, J. Popelek, and J. Zweiback, “Phase-only sampled 45 channel fiber Bragg grating written with a diffraction-compensated phase mask,” Opt. Lett. |

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

14. | H. Lee and G. Agrawal, “Bandwidth equalization of purely phase-sampled fiber Bragg gratings for broadband dispersion and dispersion slope compensation,” Opt. Express |

15. | H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, “Advances in the Design and Fabrication of High-Channel-Count Fiber Bragg Gratings,” IEEE J. Lightwave Technol. |

16. | Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shifts in phase mask for fiber Bragg grating” IEEE Photon. Technol. Lett. |

17. | M. Morin, M. Poulin, A. Mailloux, F. Trépanier, and Y. Painchaud, “Full C-band slope-matched dispersion compensation based on a phase sampled Bragg grating,” Proceedings of OFC 04, WK1 (2004). |

18. | Y. Painchaud, M Poulin, M. Morin, and M. Guy, “Fiber Bragg grating based dispersion compensator slopematched for LEAF fiber,” Optical Fiber Communication Conf. Paper. OThE2 (2006). |

19. | Y. Painchaud and M. Morin, “Iterative method for the design of arbitrary multi-channel fiber Bragg gratings,” OSA Topical meeting BGPP2007, Paper. BTuB1 (2007). |

20. | M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

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

(230.1480) Optical devices : Bragg reflectors

(260.2030) Physical optics : Dispersion

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: May 20, 2008

Revised Manuscript: June 17, 2008

Manuscript Accepted: June 17, 2008

Published: June 19, 2008

**Citation**

Ming Li and Hongpu Li, "Reflection equalization of the simultaneous dispersion and dispersion-slope compensator based on a phase-only sampled fiber Bragg grating," Opt. Express **16**, 9821-9828 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9821

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### References

- U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995). [CrossRef]
- A. Isomäki, A. Vainionpää, J. Lyytikäinen, and O. G. Okhotnikov, "Semiconductor mirror for dynamic dispersion compensation," Appl. Phys. Lett. 82, 2773-2774 (2003). [CrossRef]
- F. Ouellette, P. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using sampled fibre Bragg gratings," Electron. Lett. 31, 899-901 (1995). [CrossRef]
- M. Ibsen, M. Durkin, M. Cole, and R. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photon. Technol. Lett. 10, 842-844 (1998). [CrossRef]
- Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, "Multi-channel fiber Bragg gratings for dispersion and slope compensation," Optical Fiber Communication Conf. Paper. ThAA5. (2002).
- W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based-dispersion slope compensator," IEEE Photon. Tech. Lett. 11, 1280-1282 (1999). [CrossRef]
- A. V. Buryak, K. Kolossovski, and D. Yu. Stepanov, "Optimisation of refractive index sampling for multi-channel FBGs," IEEE J. Quantum Electron. 39, 91-98 (2003). [CrossRef]
- Q. Wu, C. Yu, K. Wang, X. Wang, Z. Yu, H. Chan, and P. 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]
- Y. T. Dai, X. F. Chen, X. Xu, C. Fan, and S. Z. Xie, "High channel-count comb filter based on chirped sampled fiber Bragg grating and phase shift," IEEE Photon. Technol. Lett. 17, 1040-1042 (2005). [CrossRef]
- H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high channel counts chromatic dispersion compensation," J. Lightwave Technol. 13, 2074-2083 (2003).
- H. Li, M. Li, K. Ogusu, Y. Sheng, and J. Rothenberg, "Optimization of a continuous phase-only sampling for high channel-count fiber Bragg gratings," Opt. Express 14, 3152-3160 (2006). [CrossRef] [PubMed]
- J. E. Rothenberg, H. Li, Y. Sheng, J. Popelek, and J. Zweiback, "Phase-only sampled 45 channel fiber Bragg grating written with a diffraction-compensated phase mask," Opt. Lett. 31, 1199-1201 (2006). [CrossRef] [PubMed]
- H. Lee and G. Agrawal, "Purely phase-sampled fiber Bragg gratings for broad-band dispersion and dispersion slope dispersion," IEEE Photon. Technol. Lett. 15, 1091-1093 (2003). [CrossRef]
- H. Lee and G. Agrawal, "Bandwidth equalization of purely phase-sampled fiber Bragg gratings for broadband dispersion and dispersion slope compensation," Opt. Express 12, 5595-5602 (2004). [CrossRef] [PubMed]
- H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, "Advances in the Design and Fabrication of High-Channel-Count Fiber Bragg Gratings," J. Lightwave Technol. 25, 2739-2750 (2007). [CrossRef]
- Y. Sheng, J. E. Rothenberg, H. Li, Y. Wang, and J. Zweiback, "Split of phase-shifts in phase mask for fiber Bragg grating," IEEE Photon. Technol. Lett. 16, 1316-1318 (2004). [CrossRef]
- M. Morin, M. Poulin, A. Mailloux, F. Trépanier, and Y. Painchaud, "Full C-band slope-matched dispersion compensation based on a phase sampled Bragg grating," Proceedings of OFC 04, WK1 (2004).
- Y. Painchaud, M Poulin, M. Morin, and M. Guy, "Fiber Bragg grating based dispersion compensator slope-matched for LEAF fiber," Optical Fiber Communication Conf. Paper. OThE2 (2006).
- Y. Painchaud and M. Morin, "Iterative method for the design of arbitrary multi-channel fiber Bragg gratings," OSA Topical meeting BGPP2007, Paper. BTuB1 (2007).
- M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, "Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation," IEEE Photon. Technol. Lett. 12, 498-500 (2000). [CrossRef]

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