## Semi-empirical multi-port lattice model for long-period fiber grating analysis under arbitrary temperature distributions

Optics Express, Vol. 16, Issue 2, pp. 598-606 (2008)

http://dx.doi.org/10.1364/OE.16.000598

Acrobat PDF (273 KB)

### Abstract

We propose a semi-empirical model for the complete analysis (spectrum, bandwidth, and wavelength/phase shifts) of a temperature-tuned long-period fiber grating (LPFG) filter. By applying the multi-port lattice model to LPFGs, while deriving and utilizing the empirically determined temperature-dependence of core-to-cladding intermodal dispersions, we achieve a precise, practical means of spectrum analysis. Excellent agreement of the model with the experimental results was obtained over wide spectral ranges.

© 2008 Optical Society of America

## 1. Introduction

1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band rejection filters,” J. Lightwave Technol. **14**, 58–65 (1996). [CrossRef]

5. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. **21**, 336–338 (1996). [CrossRef] [PubMed]

5. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. **21**, 336–338 (1996). [CrossRef] [PubMed]

6. Y. Liu, J. A. Willims, L. Zhang, and I. Bennion, “Phase shifted and cascaded long period fiber gratings,” Opt. Commun. **164**, 27–31 (1999). [CrossRef]

7. M. Harurnoto, M. Shigehara, and H. Suganurna, “Gain-flattening filter using long-period fiber gratings,” J. Lightwave Technol. **20**, 1027–1033 (2002). [CrossRef]

8. X. Shu, T. Allsop, B. Gwandu, and L. Zhang, “High-temperature sensitivity of long-period gratings in B—Ge codoped fiber,” Photon. Technol. Lett. **13**, 818–820 (2001). [CrossRef]

9. J. K. Bae, J. Bae, S. H. Kim, N. Park, and S. B. Lee, “Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters,” Photon. Technol. Lett. **17**, 1226–1228 (2005). [CrossRef]

3. V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. **25**, 203–205 (2000). [CrossRef]

## 2. Formulation of the principle

*T*) and coupling coefficient κ(λ,

*T*) of the cladding mode, we first focus our attention on κ. Employing the notation in [3

3. V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. **25**, 203–205 (2000). [CrossRef]

11. J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. **12**, 797–810 (2004). [CrossRef]

11. J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. **12**, 797–810 (2004). [CrossRef]

12. T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A **14**, 1760–1773 (1997). [CrossRef]

*C*(λ,

*T*) is the overlap integral between the core and cladding mode filed over the fiber cross section[10

10. J. Bae, J. Chun, and S. B. Lee, “Synthesis of long-period fiber gratings with the inverted Erbium gain spectrum using the multiport lattice filter model,” J. Lightwave Technol. **22**, 1976–1986 (2004). [CrossRef]

12. T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A **14**, 1760–1773 (1997). [CrossRef]

*T*), without any loss of generality we now employ the expression in [3

3. V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. **25**, 203–205 (2000). [CrossRef]

_{0}(λ,

*T*

_{0}), and the differential dispersion function ΔΦ(λ,

*T*-

*T*

_{0}) - as follows,

^{-6}/°C) of silica-based single mode fiber [13

13. K. Shima, K. Himeno, T. Sakai, S. Okude, A. Wada, and R. Yamauchi, “A novel temperature-insensitive long-period fiber grating using a boron-codoped-germanosilicate-core fiber,” Proc. Optical Fiber Communication Conf. Dallas, TX, 347–348, (1997). [CrossRef]

*T*)=β

^{(0)}-β

^{(p)}, we note that ΔΦ(λ,

*T*-

*T*

_{0}) can be alternatively expressed in terms of the effective index of the core and cladding modes.

*b*[14],

*V*[14],

*n*

^{(0)}

*=*

_{eff}*b*(

*n*-

_{co}*n*)+

_{cl}*n*and then Δ

_{cl}*n*

^{(0)}

*=Δ[*

_{eff}*b*(

*n*-

_{co}*n*)+

_{cl}*n*], where

_{cl}*a*is the core radius. Now, rewriting ΔΦ(λ, T-T

_{0}) in Eq. (3) in terms of

*b*, and its differential, Δ

*b*, we get,

*V*and also by using (4) for the calculation of Δ

*V*and Δ

*b*;

*V*=2.405 λ

_{cutoff.T0}/λ and using Δ

*n*≅Δ

_{cl}*n*

^{(p)}

*(as all the cladding modes ~ up to 20*

_{eff}^{th}- are confined almost entirely within the cladding material), we derive the interim expression of ΔΦ(λ,

*T*-

*T*

_{0}), which can be used in Eq. (2); expressed as the product of the wavelength dependent part and temperature dependent component,

_{cutoff,T0}is the cutoff wavelength of the fiber at fixed temperature.

*n*

_{co}-

*n*

_{cl}) as a function of temperature. For this purpose, we first obtain Φ(λ,

*T*) and its differential, ΔΦ(λ,

*T*), from the measurement, and then use the fitting Eq. (9) to get Δ(

*n*

_{co}-

*n*

_{cl}).

## 3. Determination of parameters

*T*) values obtained from the spectrum measurement on the different LPFGs (with Λ=625, 540, 496, 440, 400 and 344um for the 1

^{st}to 6

^{th}cladding modes and Λ=344, 348, 352, 356, and 360um for the 7th cladding mode.

*T*was varied between 26 and 140°C. The Φ(λ,

*T*) values were calculated from the measured peak wavelengths and grating period, Λ, at each λ and

*T*[3

**25**, 203–205 (2000). [CrossRef]

*T*-

*T*

_{0}) values derived from Fig. 1 for the 1

^{st}~7

^{th}cladding modes (wavelengths between 1000nm to 1650nm). It is worth mentioning that the measured ΔΦ(λ,

*T*-

*T*

_{0}) and Δ(

*n*

_{co}-

*n*

_{cl}) values for the different cladding modes overlap in a perfect manner, consistent with the result based on the full-model analysis [8

8. X. Shu, T. Allsop, B. Gwandu, and L. Zhang, “High-temperature sensitivity of long-period gratings in B—Ge codoped fiber,” Photon. Technol. Lett. **13**, 818–820 (2001). [CrossRef]

*T*-

*T*

_{0}) for each and every cladding mode. Especially, by taking advantage of the linear relationship observed in Fig. 2(b) between Δ(

*n*

_{co}-

*n*

_{cl}) and the temperature variation, one can rewrite the Eq. (9), at least within the tuning range of interest (25~140°C), as follows;

_{cutoff, T0}at fixed temperature, and a fitting constant α

_{T}, which can be determined from simple measurements of the uniform LPFG spectrum at several different temperatures (focusing on only one of the cladding modes).

## 4. Application of the semi-empirical LPFG analysis model

9. J. K. Bae, J. Bae, S. H. Kim, N. Park, and S. B. Lee, “Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters,” Photon. Technol. Lett. **17**, 1226–1228 (2005). [CrossRef]

_{cutoff,0}=1153nm. For this specific fiber, the measured value of α

_{T}was 1.66×10

^{-6}/°C.

10. J. Bae, J. Chun, and S. B. Lee, “Synthesis of long-period fiber gratings with the inverted Erbium gain spectrum using the multiport lattice filter model,” J. Lightwave Technol. **22**, 1976–1986 (2004). [CrossRef]

*k*uniform LPFG sections, we now build the transfer matrix

*Q*in (11) from the product of the

*M*(

_{k}*T*)’s, viz. the transfer matrix of the uniform LPFGs at temperature

_{k}*T*, with its elements calculated from (1) and (7), as follows :

_{k}*E*

^{(0)}(

*L*) and

*E*

^{(p)}(

*L*) are the resulting electric fields in the core and cladding, respectively.

### 4.1 Example 1

_{1}=348µm and Λ

_{2}=360µm for LPFG1 and LPFG2, respectively, as shown in Fig. 4), but utilizing the same cladding mode (7

^{th}). For the prediction of the LPFG spectral shape, given the inherent limitations in the conventional, lightweight approaches such as the 2×2 matrix model or higher order fitting method, one can employ the full modal analysis [12

12. T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A **14**, 1760–1773 (1997). [CrossRef]

*T*

_{0}) obtained in the previous section was used. With the wavelength dependent curvature of the intermodal dispersion function of the 7

^{th}cladding mode Φ

^{(7)}(Fig. 5(b), data from Fig. 1), the intermodal dispersion comes to have a different slope for the different grating periods, Λ1 and Λ2, resulting in different spectral shapes at λ1

_{res}and λ2

_{res}. Utilizing Eqs. (1) and (10) to get κ and δ, respectively, and then using (11) to get the final spectrum, the analytically obtained spectrum peak wavelength λ

_{res}and FWHM (full width half maximum) bandwidth Δλ

_{FWHM}were 1506nm and 46nm for LPFG1, and 1628.4nm and 25nm for LPFG2, respectively, exactly overlapping with the experimentally determined values [Fig. 5(a)].

### 4.2 Example 2

_{1}=Λ

_{3}=356µm, Λ

_{2}=433µm). For the analysis, the dispersion function Φ(λ,

*T*

_{0}) and its temperature differential ΔΦ(

*T*-

*T*

_{0}) obtained in the previous section were used. Since the core mode of LPFG1/LPFG3 mainly couples with the 7

^{th}cladding mode, and the core mode of LPFG2 couples with the 6

^{th}cladding mode, all the other cladding modes were ignored in the calculation within the wavelength range of interest.

^{th}cladding mode to LPFG3’s 7

^{th}cladding mode was observed, exhibiting interference fringes within the resonance band of the composite grating (with the free spectral range of the comb FSR

_{comb}=12nm). Figure 7(c) also shows the spectrum of the LPFG at a temperature of 120°C. Not only a shift in the resonance wavelength, but also changes in FSR

_{comb}of the 7

^{th}cladding mode (12nm to 9nm) were predicted from the model and were confirmed by experiment. It should be mentioned that the changes in FSR

_{comb}are mainly due to the temperature induced vertical shift in the dispersion curve Φ

^{(7)}(for LPFGs 1 and 3), and corresponding changes in the intermodal dispersion slope at the resonance frequency. It is also worth noting that even with almost identical ΔΦ(or Δ(

*n*

_{co}-

*n*

_{cl})) values for the 6

^{th}and 7

^{th}cladding modes, the resonance wavelength shift of the 7

^{th}cladding mode is much larger than that of the 6

^{th}cladding mode, because of the steeper intermodal dispersion slope of the former.

_{res}of LPFG2’s 6

^{th}cladding mode was observed which, it is important to note, was accompanied by spectral changes of the 7

^{th}cladding mode [Fig. 7(e)].

^{th}cladding modes between LPFG1 and LPFG3 which, although minor, led to changes in the resonance peaks and spectral profiles. It is important to note that the above example clearly demonstrates the successful treatment of multiple cladding modes and the couplings in-between them using our simplistic, semi-empirical format. Accurate analysis on the temperature dependencies of LPFGs were also achieved (which has not been practically feasible with the full-modal analysis or high-order fitting method) for our approach, using external parameter sets measured with minimal efforts.

### 4.3 Example 3

^{th}:dash, 7

^{th}:dot).

## 6. Conclusion

## References and links

1. | A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band rejection filters,” J. Lightwave Technol. |

2. | B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, and R. I. Laming, “High-performance optical fiber polarizers based on long-period grating in birefringent optical fibers,” Photon. Technol. Lett. |

3. | V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. |

4. | B. J. Eggleton, R. E. Slusher, J. B. Judkins, J. B. Stark, and A. M. Vengsarkar, “All-optical switching in long period fiber gratings,” Opt. Lett. |

5. | A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. |

6. | Y. Liu, J. A. Willims, L. Zhang, and I. Bennion, “Phase shifted and cascaded long period fiber gratings,” Opt. Commun. |

7. | M. Harurnoto, M. Shigehara, and H. Suganurna, “Gain-flattening filter using long-period fiber gratings,” J. Lightwave Technol. |

8. | X. Shu, T. Allsop, B. Gwandu, and L. Zhang, “High-temperature sensitivity of long-period gratings in B—Ge codoped fiber,” Photon. Technol. Lett. |

9. | J. K. Bae, J. Bae, S. H. Kim, N. Park, and S. B. Lee, “Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters,” Photon. Technol. Lett. |

10. | J. Bae, J. Chun, and S. B. Lee, “Synthesis of long-period fiber gratings with the inverted Erbium gain spectrum using the multiport lattice filter model,” J. Lightwave Technol. |

11. | J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. |

12. | T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A |

13. | K. Shima, K. Himeno, T. Sakai, S. Okude, A. Wada, and R. Yamauchi, “A novel temperature-insensitive long-period fiber grating using a boron-codoped-germanosilicate-core fiber,” Proc. Optical Fiber Communication Conf. Dallas, TX, 347–348, (1997). [CrossRef] |

14. | G. P. Agrawal, |

**OCIS Codes**

(060.2330) Fiber optics and optical communications : Fiber optics communications

(350.2770) Other areas of optics : Gratings

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 14, 2007

Revised Manuscript: December 28, 2007

Manuscript Accepted: January 3, 2008

Published: January 8, 2008

**Citation**

Jun Kye Bae, Namkyoo Park, Jinho Bae, Dongyeon Koh, Sang Hyuck Kim, and Sang Bae Lee, "Semi-empirical multi-port lattice model for long-period
fiber grating analysis under arbitrary
temperature distributions," Opt. Express **16**, 598-606 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-598

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### References

- A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band rejection filters," J. Lightwave Technol. 14, 58-65 (1996). [CrossRef]
- B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, and R. I. Laming, "High-performance optical fiber polarizers based on long-period grating in birefringent optical fibers," Photon. Technol. Lett. 9, 1370-1372 (1997). [CrossRef]
- V. Grubsky and J. Feinberg, "Long-period fiber gratings with variable coupling for real-time sensing applications," Opt. Lett. 25, 203-205 (2000). [CrossRef]
- B. J. Eggleton, R. E. Slusher, J. B. Judkins, J. B. Stark and A. M. Vengsarkar, "All-optical switching in long period fiber gratings," Opt. Lett. 22, 883-885 (1997). [CrossRef] [PubMed]
- A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano and, C. R. Davidson, "Long-period fiber-grating-based gain equalizers," Opt. Lett. 21, 336-338 (1996). [CrossRef] [PubMed]
- Y. Liu, J. A. Willims, L. Zhang, and I. Bennion, "Phase shifted and cascaded long period fiber gratings," Opt. Commun. 164, 27-31 (1999). [CrossRef]
- M. Harurnoto, M. Shigehara, and H. Suganurna, "Gain-flattening filter using long-period fiber gratings," J. Lightwave Technol. 20, 1027 - 1033 (2002). [CrossRef]
- X. Shu, T. Allsop, B. Gwandu and L. Zhang, "High-temperature sensitivity of long-period gratings in B-Ge codoped fiber," Photon. Technol. Lett. 13, 818-820 (2001). [CrossRef]
- J. K. Bae, J. Bae, S. H. Kim, N. Park and S. B. Lee, "Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters," Photon. Technol. Lett. 17, 1226-1228 (2005). [CrossRef]
- J. Bae, J. Chun, and S. B. Lee, "Synthesis of long-period fiber gratings with the inverted Erbium gain spectrum using the multiport lattice filter model," J. Lightwave Technol. 22, 1976-1986 (2004). [CrossRef]
- J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, "Analysis for long period fiber gratings using thermal kernel function," Opt. Express 12, 797-810 (2004). [CrossRef]
- T. Erdogan, "Cladding-mode resonances in short- and long- period fiber grating filters," J. Opt. Soc. Am. A 14, 1760 - 1773 (1997). [CrossRef]
- K. Shima, K. Himeno, T. Sakai, S. Okude, A. Wada, and R. Yamauchi, "A novel temperature-insensitive long-period fiber grating using a boron-codoped-germanosilicate-core fiber," Proc. Optical Fiber Communication Conf. Dallas, TX, 347-348, (1997). [CrossRef]
- G. P. Agrawal, Fiber-Optic Communications Systems (New York: Wiley, 1997), Chap. 2.

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