## Optimization of a continuous phase-only sampling for high channel-count fiber Bragg gratings

Optics Express, Vol. 14, Issue 8, pp. 3152-3160 (2006)

http://dx.doi.org/10.1364/OE.14.003152

Acrobat PDF (316 KB)

### Abstract

A novel continuous phase-only sampling function capable of producing up to 81-channel FBG with excellent channel uniformity and high in-band energy efficiency is presented and optimized by using the simulated annealing algorithm. In order to fabricate this kind of FBGs with a conventional side-writing phase-mask technique, both the diffraction effects and fabrication tolerance of the phase-shifted phase mask have also been addressed. Compared with the numerical results, a 45-channel (spacing of 100 GHz) and an 81-channel (spacing of 50 GHz) phase-only sampled linearly chirped FBG are successfully demonstrated.

© 2006 Optical Society of America

## 1. Introduction

1. F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensation using chirped sampled fibre Bragg gratings,” Electron. Lett. **30**, 899–901 (1995). [CrossRef]

4. M. Durkin, M. Ibsen, M. J. Cole, and R. I. Laming, “1 m long continuously-written fibre Bragg gratings for combined second-and third-order dispersion compensation,” Electron. Lett. **33**, 1891–1893 (1997). [CrossRef]

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

10. 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. Technol. Lett. **14**, 1309–1311 (2002). [CrossRef]

11. 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. **21**, 2074–2083 (2003). [CrossRef]

11. 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. **21**, 2074–2083 (2003). [CrossRef]

17. 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]

18. J. E. Rothenberg, H. Li, J. Popelek, J. Zweiback, and Y. Sheng, “A novel phase-only sampled 45-channel fiber Bragg grating written with a diffraction-compensated phase mask,” Opt. Lett. (to be published). [PubMed]

19. Y. Sheng and L. Sun, “Near-field diffraction of irregular phase gratings with multiple phase-shifts,” Opt. Express **13**, 6111–6116 (2005). [CrossRef] [PubMed]

## 2. Theory of the diffraction compensation phase-mask for side-writing of FBG

*θ*

_{m}are inserted at certain locations as shown in Fig. 1. The diffraction of the incident UV light incident on the mask splits into two at angles of ±

*φ*

_{0}. Their separation in the fiber core at a distance of

*z*from the mask is ∆

*x*= 2

*z*tan

*φ*

_{0}. Base on the fact that the phase in the mask will be split into two two-half amplitude phase-shifts separated with a lateral distance between the +1 and -1 diffracted light [19

19. Y. Sheng and L. Sun, “Near-field diffraction of irregular phase gratings with multiple phase-shifts,” Opt. Express **13**, 6111–6116 (2005). [CrossRef] [PubMed]

*θ*

_{g}(

*x*) in the fiber core is expressed as

*and*θ ˜

_{g}*are Fourier transforms the phase profiles and*θ ˜

_{m}*f*is the spatial frequency of the mask. Assuming that both

*θ*(

_{g}*x*) and

*θ*(

_{m}*x*) are changed periodically with period

*P*, the spatial frequency

*f*is then in the unit of 1/

*P*, and the channel frequency spacing in FBG is given by ∆

*ν*=

*c*/(2

*n*

_{group}

*P*) . Equation (2) shows that the diffraction effect can cause very sever distortion for large magnitude

*f*and made the grating phase vanishes at

*f*= 1/(2∆

*x*).

*, use Eq. (3) to find the spatial spectrum of the mask phase, and then transform it back to find the phase function in mask. Finally, with this diffraction-compensated phase mask, we can exactly imprint the sampling-phase*θ ˜

_{g}*θ*

_{g}(

*x*) in the profile of the FBG rather than the distorted one. Note that this compensated method is not suitable for any binary or multi-levels phase-only sampling functions [11

11. 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. **21**, 2074–2083 (2003). [CrossRef]

*f*= 1/(2∆

*x*), otherwise it would be divided by a small number in Eq. (3) which would lead to inaccurate or unphysical results.

## 3. A continuous phase-only sampling for high channel-count FBG

### 3.1. Phase-only sampling function with diffraction compensation

*n*can be expressed as

*n*

_{1}(

*x*) is the maximum index modulation,

*x*is the position along the grating, Λ is the central pitch of the grating,

*ϕ*

_{g}(

*x*) is the local phase for one channel grating which determines the dispersion of the grating.

*s*(

*x*) stands for a sampling function with period

*P*.

*s*(

*x*) with a period

*P*as

*s*

_{b}(

*x*) is the base sampling function in one period. In general, the base sampling function

*s*

_{b}(

*x*) discussed [10–11

10. 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. Technol. Lett. **14**, 1309–1311 (2002). [CrossRef]

*θ*

_{g}(

*x*) has the general form including more harmonic terms as:

*M*is minimized,

*α*

_{n}and

*β*

_{n}are optimized such that the channel spectrum is flat within the band of interest. Considering the diffraction effect, the pre-compensated phase of the phase mask according to Eq. (3) is then obtained

### 3.2 Optimization of the phase-only sampling function with simulated annealing algorithm

*M*uniform channels could be achieved with

*M*terms in this series since there are two free parameters for each term. In order to obtain optimal magnitudes of the series of

*α*

_{n}

*,β*

_{n}, we exploit the simulated annealing algorithm, where the optimization criteria are the uniformity of the desired channels and the diffraction efficiency

*η*[11

**21**, 2074–2083 (2003). [CrossRef]

*T*.

*S*

_{m}is the

*mth*Fourier coefficient of the base sampling function

*s*

_{b}(

*x*),

*η*(<1) is the target diffraction efficiency for all the in-band channel of 2

*N*+ 1. Depending on the desired channel numbers, a suitable value of

*η*needs to be selected in order to remain the high channel uniformity as well. In our case, we choose it as 0.90. Obviously, the practical diffraction efficiency is equal to

*α*

_{n}

*,β*

_{n}with

*n*= 1,…,

*M*are chosen randomly. The cost function is evaluated by using Eqs. (6), (7), and (10). Then each of

*α*

_{n},

*β*

_{n}is shifted randomly with a uniform density of probability. If the shift of

*α*

_{n}

*,β*

_{n}in the

*k*iteration leads to a decrease of the cost, i.e., ∆

*E*

_{k}=

*E*

_{k}-

*E*

_{k-1}≤ 0, the new

*α*

_{n}

*,β*

_{n}is accepted. Otherwise, if ∆

*E*

_{k}> 0 the new

*α*

_{n}

*,β*

_{n}will still be accepted with a probability

*z*= 10 μm. Considering the refraction happened in the fiber cladding (for cladding index of 1.45) and the radius of cladding, we have ∆

*x*= 25 μm. For convenience, the period of the sampling function is normalized to one. In order to obtain the high diffraction efficiency

*η*, 13, 23, and 41 harmonic terms were selected for 25-, 45-, and 81-channel, respectively. It can be seen that the original sampling design are pretty reasonable. The diffraction efficiencies obtained with the simulated annealing algorithm are 92%, 92% and 84%, respectively. Non-uniformity of the channel intensities are 0.1%, 0.8% and 1.0%, respectively. However, with the phase-split effect due to the mask diffraction, the resulted channels are considerably distorted (as shown in Fig. 2(c), Fig. 3(c), and Fig. 4(c)), which in return means that the compensation to the diffraction effect is necessary once a sampled FBG with high channel-count is written with the side-writing technique.

20. J. Skaar, L Wang, and T. Erdogan, “On the synthesis of fiber Bragg grating by layer peeling,” IEEE J. Quantum Electron. **37**, 165–173 (2001). [CrossRef]

*z*= 10 μm. However, in the practical fabrication, it is strongly desired to know in what extent the deviation of this separation could be allowable. We then investigated the effect of the spacing deviation on the inter-channel uniformity by following equation:

*x*

^{R}is the actual lateral distance between the +1 diffraction and -1 diffraction light at the fiber. ∆

*x*(= 25 μm) is the nominal one for spacing

*z*= 10 μm.

## 4. Experimental results

*θ*

_{m}(

*x*) (as shown in Eq. (8)) and the phase

*ϕ*

_{g}(

*x*) for one seed grating (shown in Eq. (4)) with dispersion of -1360 ps/ns were encoded into the grating, whose local period becomes

*φ*(

*x*) =

*θ*

_{m}(

*x*) +

*ϕ*

_{g}(

*x*) is the local phase. There are two approaches to implement Λ

_{M}(

*x*). Ideally, one can change every period of the grating. In this case, the phase mask must be written with the required accuracy and written continuously without stitching. One can also divide the entire grating into a large number of steps. Each step is considered as a uniform grating with the local average period Λ

_{M}(

*x*). We used the former approach. The phase mask is designed and fabricated with a special “stitch-error-free” lithography tool. The sampled FBG was written with side-writing technique [11

**21**, 2074–2083 (2003). [CrossRef]

18. J. E. Rothenberg, H. Li, J. Popelek, J. Zweiback, and Y. Sheng, “A novel phase-only sampled 45-channel fiber Bragg grating written with a diffraction-compensated phase mask,” Opt. Lett. (to be published). [PubMed]

^{-4}. It can be seen that nearly identical 45 channel with a channel spacing of 100 GHz, useable bandwidth of 0.21 nm, peak-peak group-delay near 20 ps, and chromatic dispersion of about -1374 ps/nm have been obtained. These results agree well with our design data. Noted that this kind of high-quality high channel-count FBG has already been successful used as a dispersion compensator through full C-band for transmission over 640-km SMF fiber [21]. Figure 9 shows the measured results of one typical FBG with 81-channel, where the grating length is 10 cm also and the peak index-modulation required is about 8×10

^{-4}. It can also be seen that nearly identical 81 channel with a channel spacing of 50 GHz, useable bandwidth of 0.11 nm, and chromatic dispersion of about -1400 ps/nm have been obtained. Although the group delay ripple shown in Fig. 9(b) is a little bit large and the inter-channel uniformity is worse than that of 45-channel FBG shown in Fig. 8(b), which is most probably due to the demands of higher index-modulation and more strict precision control for the spacing between the mask and the fiber, this is the first reported for a sampled FBG with up to 81 channels and these results verify that the proposed continuous sampling with pre-diffraction compensation is reasonable.

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | Z. Pan, Y. W. Song, C. Yu, Y. Wang, Q. Yu, J. Popelek, H. Li, Y. Li, and A. E. Willner, “Tunable chromatic dispersion compensation in 40-Gb/s systems using nonlinearly chirped Fiber Bragg Gratings,” J. Lightwave Technol. |

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

4. | M. Durkin, M. Ibsen, M. J. Cole, and R. I. Laming, “1 m long continuously-written fibre Bragg gratings for combined second-and third-order dispersion compensation,” Electron. Lett. |

5. | Y. Painchaud, A. Mailoux, H. Chotard, E. Pelletier, and M. Guy, “Multi-channel fiber Bragg gratings for dispersion and slope compensation,” in |

6. | M. Morin, M. Poulin, A. Mailloux, F. Trepanier, and Y. Painchaud, “Full C-band slope-matched dispersion compensation based on a phase sampled Bragg grating,” in |

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

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

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

10. | 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. Technol. Lett. |

11. | 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. |

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

13. | V. Jayaraman, Z. M. Chuang, and L. A. Coldren, “Theory, design, and performance of extended tuning semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. |

14. | H. Ishii, Y. Tohmori, T. Tamamrua, and Y. Yoshikuni, “Super structure grating (SSG) lasers for broadly tunable DBR lasers,” IEEE Photon. Technol. Lett. |

15. | R. Kashyap, |

16. | L. Poladian, B. Ashton, and W. Ppadden, “Interactive design and fabrication of complex FBGs,” in |

17. | 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. |

18. | J. E. Rothenberg, H. Li, J. Popelek, J. Zweiback, and Y. Sheng, “A novel phase-only sampled 45-channel fiber Bragg grating written with a diffraction-compensated phase mask,” Opt. Lett. (to be published). [PubMed] |

19. | Y. Sheng and L. Sun, “Near-field diffraction of irregular phase gratings with multiple phase-shifts,” Opt. Express |

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

21. | K.-M. Feng, S. Lee, R. Khosravani, S. S. Havstad, and J. E. Rothenberg, “45 ITU-100 channels dispersion compensation using cascaded full C-band sampled FBGs for transmission over 640-Km SMF,” in |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(050.5080) Diffraction and gratings : Phase shift

(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: March 6, 2006

Revised Manuscript: April 6, 2006

Manuscript Accepted: April 7, 2006

Published: April 17, 2006

**Citation**

Hongpu Li, Ming Li, Kazuhiko Ogusu, Yunlong Sheng, and Joshua Rothenberg, "Optimization of a continuous phase-only sampling for high channel–count
fiber Bragg gratings," Opt. Express **14**, 3152-3160 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3152

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### References

- F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fibre Bragg gratings," Electron. Lett. 30, 899-901(1995). [CrossRef]
- Z. Pan, Y. W. Song, C. Yu, Y. Wang, Q. Yu, J. Popelek, H. Li, Y. Li, and A. E. Willner,"Tunable chromatic dispersion compensation in 40-Gb/s systems using nonlinearly chirped Fiber Bragg Gratings," J. Lightwave Technol. 20, 2239-2246 (2002). [CrossRef]
- B. Eggleton, P. A. Krug, L. Poladian, and F. Oullette, "Long periodic superstructure Bragg gratings in optical fibres," Electron. Lett. 30, 1620-1622 (1994). [CrossRef]
- M. Durkin, M. Ibsen, M. J. Cole, and R. I. Laming, "1 m long continuously-written fibre Bragg gratings for combined second-and third-order dispersion compensation," Electron. Lett. 33, 1891-1893 (1997). [CrossRef]
- Y. Painchaud, A. Mailoux, H. Chotard, E. Pelletier, and M. Guy, "Multi-channel fiber Bragg gratings for dispersion and slope compensation," in Proc. Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2002), paper ThAA5.
- M. Morin, M. Poulin, A. Mailloux, F. Trepanier, and Y. Painchaud, "Full C-band slope-matched dispersion compensation based on a phase sampled Bragg grating," in Proc. Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2004), paper WK1.
- M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photon. Technol. Lett. 10, 842-844 (1998). [CrossRef]
- W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based-dispersion slope compensator," IEEE Photon. Technol. Lett. 11, 1280-1282 (1999). [CrossRef]
- A. V. BuryakK. 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]
- 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. Technol. Lett. 14, 1309-1311(2002). [CrossRef]
- H. Li, Y. Sheng, Y. Li, J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high channel counts chromatic dispersion compensation," J. Lightwave Technol. 21, 2074-2083 (2003). [CrossRef]
- H. Lee and G. P. Agrawal, "Purely phase-sampled fiber Bagg gratings for broad-band dispersion and dispersion slope compensation," IEEE Photon. Technol. Lett. 15, 1091-1093 (2003). [CrossRef]
- V. Jayaraman, Z. M. Chuang, and L. A. Coldren, "Theory, design, and performance of extended tuning semiconductor lasers with sampled gratings," IEEE J. Quantum Electron. 29, 1824-1834 (1993). [CrossRef]
- H. Ishii, Y. Tohmori, T. Tamamrua, and Y. Yoshikuni, "Super structure grating (SSG) lasers for broadly tunable DBR lasers," IEEE Photon. Technol. Lett. 4, 393-395 (1993). [CrossRef]
- R. Kashyap, Fiber Bragg Grating (Academic, San Diego, 1999).
- L. Poladian, B. Ashton and W. Ppadden, "Interactive design and fabrication of complex FBGs," in Proc. Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2003), paper WL1.
- 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]
- J. E. Rothenberg, H. Li, J. Popelek, J. Zweiback, and Y. Sheng, "A novel phase-only sampled 45-channel fiber Bragg grating written with a diffraction-compensated phase mask," Opt. Lett. (to be published). [PubMed]
- Y. Sheng and L. Sun, "Near-field diffraction of irregular phase gratings with multiple phase-shifts," Opt. Express 13, 6111-6116 (2005). [CrossRef] [PubMed]
- J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg grating by layer peeling," IEEE J. Quantum Electron. 37,165-173 (2001). [CrossRef]
- K.-M. Feng, S. Lee, R. Khosravani, S. S. Havstad, and J. E. Rothenberg, "45 ITU-100 channels dispersion compensation using cascaded full C-band sampled FBGs for transmission over 640-Km SMF," in Proc. Eur. Conf. Optical Communication (ECOC2003), paper Mo.3.2.5.

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