## Spectral Talbot effect in sampled fiber Bragg gratings with super-periodic structures

Optics Express, Vol. 15, Issue 14, pp. 8812-8817 (2007)

http://dx.doi.org/10.1364/OE.15.008812

Acrobat PDF (212 KB)

### Abstract

Due to the equivalence between an integral multiple of 2π and zero in the phase space, a general configuration of sampled fiber Bragg gratings (SFBGs) with super-periodic structures has been introduced and investigated. These super-periodic structures can be used to implement spectral Talbot effect with a large degree of freedom, as long as any one of the three parameters involved in quadratic phase profile is an even number. Then the phase profiles of such SFBGs are analyzed in detail. Although their phase increments are constant or non-constant periodic functions in different cases, theoretical analysis and simulations show that the obtained filtering characteristics are the same. In contrast to uniform SFBGs with identical sampling period, multiplied filtering channels and similar group-delay characteristic are achieved for these SFBGs with super-periodic structures.

© 2007 Optical Society of America

## 1. Introduction

1. 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. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. **21**, 2074–2083 (2003). [CrossRef]

2. X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, “Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating,” IEEE Photon. Technol. Lett. **12**, 1501–1503 (2000). [CrossRef]

4. C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. **29**, 1590–1592 (2004). [CrossRef] [PubMed]

13. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. **7**, 728–744 (2001). [CrossRef]

4. C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. **29**, 1590–1592 (2004). [CrossRef] [PubMed]

7. 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]

12. Y. T. Dai, X. F. Chen, J. Sun, and S. Z. Xie, “Wideband multichannel dispersion compensation based on a strongly chirped sampled Bragg grating and phase shifts”, Opt. Lett. **31**, 311–313 (2006). [CrossRef] [PubMed]

8. X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. **18**, 1371–1373 (2006). [CrossRef]

9. Y. Nasu and S. Yamashita, “Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique,” J. Lightwave Technol. **23**, 1808–1817 (2005). [CrossRef]

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express **14**, 4270–4279 (2006). [CrossRef] [PubMed]

## 2. General phase condition for spectral Talbot effect

*n*(

*z*) by

*n*is the “dc” index modulation (the fringe visibility is specified as 1),

*u*(

*z*) and

*θ*(

*z*) represent the sampling function and the phase profile,

*β*

_{0}=2

*π*/Λ

_{0}is the central spatial frequency, and Λ

_{0}is the central grating period. As for Talbot effect, the phase profile

*θ*(

*z*) is always the decisive factor. Aided by Taylor expansion formula,

*θ*(

*z*) can be written as

*N*is an integer determined by

_{k}*k*,

*m*(a positive integer) is the multiplying factor of Talbot effect, and

*P*is the sampling period. If

*m*= 1, Eq. (3) corresponds to integer Talbot effect; if

*m*= 2, 3, 4, ⋯, it stands for fractional effect. Then we return to the phase profile, Eq. (2). There are so many terms on its right side that it is hard to derive each

*θ*if we treat all terms as an entity. For simplicity, by means of treating each term as an independent part to satisfy Eq. (3) respectively,

_{r}*θ*can be easily obtained as:

_{r}*N*is an integer determined by

_{k,r}*k*and

*r*As a coefficient of Taylor series, each

*θ*should be a constant along z-axis. Namely

_{r}*θ*is required to be independent of the variation of

_{r}*k*. To eliminate the influence of

*k*, we replace

*N*with a fixed number

_{k,r}*N*

_{k,r}^{́}. Consequently,

*θ*can be expressed by Eq. (4).

_{r}## 3. Super-periodic structures for implementing spectral Talbot effect

### 3.1 Phase transition points for super-periodic structures

*π*(2

*Nπ*) is equivalent to zero, where

*N*is an integer. With this equivalence, the value of Eq. (1) keeps unchanged when we replace

*θ*(

*z*) = 2

*Nπ*with

*θ*(

*z*) = 0 . Assuming that these transition points are located at sampling points

*t*is an integer. Such new super-periodic structures are reproductions of the original structure around

*z*= 0 . In this way, such a SFBG comprises identical, repeating super-periodic structures, the so-called SFBG with super-periodic structures.

*s*×

*m*)

*P*, where

*s*is a positive integer. When only quadratic phase profile is considerer, Eq. (5) can be rewritten as

*s*

^{2}

*mN*

_{k,2}

^{́}) must be an even number to satisfy Eq. (6). As long as any parameter in the set (

*s*,

*m*,

*N*

_{k,2}

^{́}) is an even number, we can find suitable transition points. Note that Eq. (6) provides a general configuration for super-periodic structures. As reported in Ref [8

8. X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. **18**, 1371–1373 (2006). [CrossRef]

*N*

_{k,2}

^{́}has been taken into consideration. Here, besides

*N*

_{k,2}

^{́},

*m*can be adjusted properly to obtain transition points. What is more, a new parameter

*s*is introduced to get transition points for the first time, which allows a larger degree of freedom than before to obtain Talbot effect. When

*s*> 1, the corresponding phase increment is greatly different from those investigated previously [6–9

6. J. Azaña, C. Wang, and L. R. Chen, “Spectral self-imaging phenomena in sampled Bragg gratings,” J. Opt. Soc. Am. B **22**, 1829–1841 (2005). [CrossRef]

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express **14**, 4270–4279 (2006). [CrossRef] [PubMed]

*θ*(

*k*) is not a linear function any more, but a periodic function. The simplest example is that “

*θ*(

*k*) is equal to a constant, and this constant may be zero or

*mN*

_{k,2}

^{́}

^{π}according to Eq. (7) derived through the combination of Eqs. (2), (3) and (4).

*θ*(

*k*) can be a non-constant periodic function. There are probably a variety of such periodic functions and we will show a schematic diagram about such functions of certain case later in this work.

### 3.2 Concrete super-periodic structures and simulations

*s*= 1 is offered, either

*m*or

*N*

_{k,2}

^{́}is required to be an even number, so that spectral Talbot effect can be realized. In this case, “

*θ*(

*k*) is zero or

*mN*

_{k,2}

^{́}

*π*, and detailed phase profiles are presented in Fig. 1. Fig. 1(a) illustrates the quadratic phase profile of conventional chirped-SFBG. For SFBGs with super-periodic structures, the phase profile for “

*θ*(

*k*) = 0 is shown in Fig. 1(b), while the phase profile for “

*θ*(

*k*) =

*mN*

_{k,2}

^{́}

*π*is illustrated in Fig. 1(c). We can see that each period of phase profiles originates from the section around

*z*= 0 . It is convenient to derive a corresponding chirp effect Λ(

*z*) for Fig. 1(b), which is indicated in Fig. 1(d). However, it is not easy to get a resulting chirp effect for Fig. 1(c), and now we just introduce such a phase profile in theory.

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

*n*= 5×10

^{-4}, Λ

_{0}= 521.89nm, the effective index

*n*=1.485, the duty cycle

_{eff}*p*= 0.08, and the total number of sampling periods

*T*= 24. The sampling period is set as

*P*= 1.0 mm, implying a channel spacing of 0.8 nm for uniform SFBGs. If

*m*is an even number (

*m*= 2), the resulting filtering characteristics are shown in Fig. 2. Obviously, compared with uniform SFBGs, the number of channels over a given wavelength region has increased to

*m*times, while group-delay characteristic is nearly identical in shape. Here, we skip analogous simulations about the parity of

*N*

_{k,2}

^{́}(such as

*N*

_{k,2}

^{́}= 2) as they have been discussed in Ref [8

8. X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. **18**, 1371–1373 (2006). [CrossRef]

*s*> 1. If

*s*= 3, 5, 7, 9⋯, one between

*m*and

*N*

_{k,2}

^{́}is required to be an even number, which is identical to the case

*s*= 1. More importantly, if

*s*= 2, 4, 6, 8⋯, the spectral Talbot phenomenon can always be realized, regardless of the parity of

*m*or

*N*

_{k,2}

^{́}.

*s*×

*m*)

*P*, not (

*mP*). According to Fig. 3(a), we are capable of deriving a periodic function “

*θ*(

*k*) shown in Fig. 3(b), which might be the most remarkable difference between the two cases (

*s*= 1 and

*s*> 1).

*m*times and the group-delay characteristic is nearly identical with that of uniform SFBG. In addition, such derived parameter sets can be utilized to design temporal, periodic Talbot filters [14

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express **14**, 4270–4279 (2006). [CrossRef] [PubMed]

### 3.3 Discussions

**14**, 4270–4279 (2006). [CrossRef] [PubMed]

2. X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, “Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating,” IEEE Photon. Technol. Lett. **12**, 1501–1503 (2000). [CrossRef]

4. C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. **29**, 1590–1592 (2004). [CrossRef] [PubMed]

*s*,

*m*, N

_{k,2}

^{́}), rather than only

*m*, can be adjusted properly. As far as we know, this obtained degree of freedom is the largest one to implement spectra Talbot effect. Furthermore, proposed SFBGs allow a larger tolerance on chirp coefficient because super-periodic structures can bring with a relatively lower deviation on phase profile [8

**18**, 1371–1373 (2006). [CrossRef]

## 4. Conclusion

*s*,

*m*, N

_{k,2}

^{́}) is an even number, a series of equivalent transition points can be determined, which corresponds to new super-periodic structures. On the basis of this principle, we succeed in achieving the spectral Talbot phenomena with a large degree of freedom. In the meantime, the phase profile and phase condition (phase increment) with different parameters are analyzed. The phase increment is zero or non-zero constant if

*s*= 1, while the increment is a non-constant periodic function if

*s*> 1.

## Acknowledgments

## References and links

1. | 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. |

2. | X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, “Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating,” IEEE Photon. Technol. Lett. |

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

4. | C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. |

5. | L. R. Chen and J. Azaña, “Spectral Talbot phenomena in sampled arbitrarily chirped Bragg gratings,” Opt. Commun. |

6. | J. Azaña, C. Wang, and L. R. Chen, “Spectral self-imaging phenomena in sampled Bragg gratings,” J. Opt. Soc. Am. B |

7. | 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. |

8. | X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. |

9. | Y. Nasu and S. Yamashita, “Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique,” J. Lightwave Technol. |

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

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

12. | Y. T. Dai, X. F. Chen, J. Sun, and S. Z. Xie, “Wideband multichannel dispersion compensation based on a strongly chirped sampled Bragg grating and phase shifts”, Opt. Lett. |

13. | J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. |

14. | J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express |

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

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

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

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 30, 2007

Revised Manuscript: June 8, 2007

Manuscript Accepted: June 8, 2007

Published: June 28, 2007

**Citation**

Xi-Hua Zou, Wei Pan, Bin Luo, Meng-Yao Wang, and Wei-Li Zhang, "Spectral Talbot effect in sampled fiber Bragg gratings with super-periodic structures," Opt. Express **15**, 8812-8817 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8812

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### References

- 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]
- X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, "Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 1501-1503 (2000). [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]
- C. Wang, J. Azaña, and L. R. Chen, "Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures," Opt. Lett. 29, 1590-1592 (2004). [CrossRef] [PubMed]
- L. R. Chen and J. Azaña, "Spectral Talbot phenomena in sampled arbitrarily chirped Bragg gratings," Opt. Commun. 250, 302-308 (2005). [CrossRef]
- J. Azaña, C. Wang, and L. R. Chen, "Spectral self-imaging phenomena in sampled Bragg gratings," J. Opt. Soc. Am. B 22, 1829-1841 (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]
- X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, "Periodically chirped sampled fiber Bragg gratings for multichannel comb filter," IEEE Photon. Technol. Lett. 18, 1371-1373 (2006). [CrossRef]
- Y. Nasu and S. Yamashita, "Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique," J. Lightwave Technol. 23, 1808-1817 (2005). [CrossRef]
- H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation," J. Lightwave Technol. 21, 2074-2083 (2003). [CrossRef]
- F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995). [CrossRef]
- Y. T. Dai, X. F. Chen, J. Sun, and S. Z. Xie, "Wideband multichannel dispersion compensation based on a strongly chirped sampled Bragg grating and phase shifts," Opt. Lett. 31, 311-313 (2006). [CrossRef] [PubMed]
- J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001). [CrossRef]
- J. Azaña and S. Gupta, "Complete family of periodic Talbot filter for pulse repetition rate multiplication," Opt. Express 14, 4270-4279 (2006). [CrossRef] [PubMed]
- T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]

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