## Analytic models of spectral responses of fiber-grating-based interferometers on FMC theory |

Optics Express, Vol. 20, Issue 4, pp. 4009-4017 (2012)

http://dx.doi.org/10.1364/OE.20.004009

Acrobat PDF (865 KB)

### Abstract

In this paper the analytic models (AMs) of the spectral responses of fiber-grating-based interferometers are derived from the Fourier mode coupling (FMC) theory proposed recently. The interferometers include Fabry-Perot cavity, Mach-Zehnder and Michelson interferometers, which are constructed by uniform fiber Bragg gratings and long-period fiber gratings, and also by Gaussian-apodized ones. The calculated spectra based on the analytic models are achieved, and compared with the measured cases and those on the transfer matrix (TM) method. The calculations and comparisons have confirmed that the AM-based spectrum is in excellent agreement with the TM-based one and the measured case, of which the efficiency is improved up to ~2990 times that of the TM method for non-uniform-grating-based in-fiber interferometers.

© 2012 OSA

## 1. Introduction

1. Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, “Widely tunable long-period waveguide grating couplers,” Opt. Express **14**(26), 12644–12654 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12644. [CrossRef] [PubMed]

2. S. K. Liaw, L. Dou, and A. S. Xu, “Fiber-bragg-grating-based dispersion-compensated and gain-flattened raman fiber Amplifier,” Opt. Express **15**(19), 12356–12361 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12356. [CrossRef] [PubMed]

3. K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. **10**(7), 1006–1008 (1998). [CrossRef]

*vice versa*. Therefore two cascaded LPFGs function as a Mach-Zehnder (MZ) interferometer [4

4. X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,” Opt. Lett. **23**(7), 509–510 (1998), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-23-7-509. [CrossRef] [PubMed]

5. P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. **15**(8), 1576–1580 (2004). [CrossRef]

6. G. D. Marshall, R. J. Williams, N. Jovanovic, M. J. Steel, and M. J. Withford, “Point-by-point written fiber-Bragg gratings and their application in complex grating designs,” Opt. Express **18**(19), 19844–19859 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-19-19844. [CrossRef] [PubMed]

7. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**(8), 1277–1294 (1997). [CrossRef]

8. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. **26**(16), 3474–3478 (1987). [CrossRef] [PubMed]

9. L. A. Weller-Brophy and D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A **2**(6), 863–871 (1985). [CrossRef]

10. J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. **93**, 385–401 (2009). [CrossRef]

12. E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. **37**(8), 755–787 (2005). [CrossRef]

13. H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron. **34**(5-6), 481–492 (2002). [CrossRef]

14. E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. **15**(8), 1295–1302 (1997). [CrossRef]

15. A. Bouzid and M. A. G. Abushagur, “Scattering analysis of slanted fiber gratings,” Appl. Opt. **36**(3), 558–562 (1997). [CrossRef] [PubMed]

16. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **48**(6), 4758–4767 (1993). [CrossRef] [PubMed]

19. X. K. Zeng, “Application of Fourier mode coupling theory to real-time analyses of nonuniform Bragg gratings,” IEEE Photon. Technol. Lett. **23**(13), 854–856 (2011). [CrossRef]

20. X. K. Zeng and K. Liang, “Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period,” Opt. Express **19**(23), 22797–22808 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22797. [CrossRef] [PubMed]

## 2. Analytic models

### 2.1 FMC theory for grating-based interferometers

*z*is the wave propagation direction,

*Δn*(

*z*) is the profile of the index perturbation in an interferometer,

*m*and

*s*are the orders of propagating mode and coupled mode, respectively,

*B*(

_{m}*z*) and

*B*(

_{s}*z*) are amplitude coefficients,

*β*and

_{m}*β*are propagation constants; sign “±” is replaced with “+” for counter-directional coupling, and with “-” for co-directional coupling;

_{s}*k*is the coupling coefficient between the propagating and coupled modes, defined aswhere

_{s}*ε*

_{0}is the permittivity of vacuum,

*ω*is angular frequency,

*n*

_{0}is the refractive index of fiber core,

*r*and

*ϕ*are the axes of cylindrical coordinate system,

*A*is cross-section area,

*A∞*is whole cross-section area,

*E*and

_{m}*E*are the electric fields of the propagating and coupled modes, respectively.

_{s}*z*is located at the center of any interferometer, the index perturbation

*Δn*(

*z*) of the interferometer consisting of two identical fiber gratings, can be modeled aswhere

*L*is the length of each fiber grating,

*P*is the interval between the two cascaded gratings, Δ

*n*

_{0}(

*z*) is the perturbation distribution of one fiber grating shifted mathematically to the central position. In Michelson interferometer,

*P*equals 2 times the space between the LPFG and the terminal reflector.

*Δn*

_{0}(

*z*) in one fiber grating. We define an equation aswhere

*ν*is independent variable as defined previously,

_{s}*γ*(

_{s}*ν*) and

_{s}*η*(

_{s}*ν*) represent the real and imaginary, respectively, components due to the Fourier transform. Equation (7) can be implemented by discrete Fourier transform suitable for all types of index perturbations, also by analytic solution if there exists closed-form Fourier integration suitable for some perturbations. By substituting Eqs. (5), (6) and (7) into Eq. (4), we can solve the integral Eq. (4) for

_{s}*B*(

_{m}*z*) and

*B*(

_{s}*z*), and then obtain the reflectivity

*R*= |

*B*

_{s}(

*z*

_{0})|

^{2}/|

*B*

_{m}(

*z*

_{0})|

^{2}and the

*bar*-transmission

*T*= |

*B*

_{m}(

*z*

_{1})|

^{2}/|

*B*

_{m}(

*z*

_{0})|

^{2}of FBG- and LPFG-based interferometers, respectively, given by where

*ν*= (

_{B}*n*+

_{m}*δ*

_{n0}+

*n*)/

_{s}*λ*,

*ν*= (

_{L}*n*+

_{m}*δ*

_{n0}-

*n*)/

_{s}*λ*. Equations (8) and (9) are the general solutions for the spectra responses of the aforementioned interferometers which may be constructed by uniform or non-uniform gratings. Once the perturbation Δ

*n*

_{0}(

*z*) of one fiber grating is determined, the spectral responses of the corresponding interferometer can be modeled and calculated on Eqs. (7), (8) and (9), which provide analytic solutions for the interferometers formed by uniform-gratings and some NU-gratings. The followings are some examples of the analytic models.

### 2.2 For uniform-grating-based interferometers

*n*

_{0u}(

*z*) of one uniform grating can be written to bewhere

*Λ*and

*δ*

_{n}are the period and amplitude, respectively, of the perturbation in one fiber grating. In Eq. (10), the difference between FBG and LPFG is only the value of period

*Λ*. By substituting Eq. (10) into Eq. (7), we can derive the closed-form of the Fourier transform result, and then get

*γ*= 0 and

_{su}*η*aswhere sinc(x) = sin(x)/x. Equations (8) and (9), together with Eq. (11), let us get the analytic solutions for the reflectivity

_{su}*R*of the uniform-FBG-based FP cavity, and for the bar-transmission

_{Fu}*T*of the uniform-LPFG-based MZ or Michelson interferometer, governed by Eqs. (12) and (13), respectively where

_{Mu}*σ*=

_{Β}*v*1/Λ and

_{B}−*σ*=

_{L}*v*1/Λ are the detuning parameters of counter- and co-directional couplings, respectively.

_{L}−### 2.3 For GA-grating-based interferometers

*n*

_{0G}(

*z*) of one GA-grating can be modeled aswhere

*α*is GA-coefficient,

*Λ*and

*δ*

_{n}are the nominal period and amplitude, respectively, of the perturbation. By incorporating Eq. (14) with Eqs. (7), (8) and (9), we can derive the closed-form of the Fourier transform of the perturbation Δ

*n*

_{0G}(

*z*), and then obtain the analytic solutions for the reflectivity

*R*of the GA-FBG-based FP cavity, and for the bar-transmission

_{FG}*T*of the GA-LPFG-based MZ or Michelson interferometer, described by Eqs. (15) and (16), respectively

_{MG}## 3. Simulations and comparisons

### 3.1 Uniform-FBG-based FP cavity

*n*= 1.4635 of core mode LP

_{m}_{01}. The two FBGs are identical, and cascaded to form a FP cavity, which are described by the following parameters: period

*Λ*= 0.532

*μm*, grating length

*L*= 1000.16

*μm*(integer times the period), perturbation amplitude

*δ*= 4 × 10

_{n}^{−4}, normalized coupling coefficient

*k*= 2531πΝ/s and interval

_{s}*P*= 2.5

*mm*. The transmission

*T*of the FP cavity can be measured easily by an optical spectrum analyzer, where

_{Fu}*T*= 1-

_{Fu}*R*. Figures 2(a) and 2(b) plot the calculated reflectivities and transmissions, respectively, of the FP cavity, according to Eq. (12) (solid lines) and the TM method (dotted lines). Figure 2(c) shows the measured transmission of a FBG-based FP cavity with the similar parameters abovementioned. Figures 2(a), 2(b) and 2(c) indicate that the Eq. (12)-based spectrum is very close to the TM-based one and the measured case, and that the spectral response of uniform-FBG-based FP cavity is with some sidelobes.

_{Fu}### 3.2 Uniform-LPFG-based MZ and Michelson interferometers

*n*= 1.46533 and

_{m}*n*= 1.462 of core mode and cladding mode, respectively. The two LPFGs are also identical, and cascaded to construct a MZ interferometer with the following parameters:

_{s}*Λ*= 400

*μm*,

*L*= 60

*mm*,

*P*= 550

*mm*,

*δ*= 1.8 × 10

_{n}^{−4}and

*k*= 69.8πΝ/s. Figure 3(a) illustrates the calculated transmissions of the MZ interferometer, depending on Eq. (13) (solid lines) and the TM method (dotted lines). Figure 3(b) exhibits the measured transmission of a MZ interferometer with the similar parameters abovementioned. Figures 3(a) and 3(b) show that the Eq. (13)-based spectrum is in good agreement with the TM-based one, and is also very close to the measured case. A Michelson interferometer can be formed by a terminal reflector and one LPFG with the same parameters abovementioned. If the value of the interval between the LPFG and the terminal reflector without attenuation is equal to 275

_{s}*mm*, the spectral response of the Michelson interferometer is the same as the MZ interferometer.

### 3.3 GA-grating-based interferometers

*n*= 1.49 of core mode LP

_{m}_{01}. The FP cavity is described by the following parameters: GA-coefficient

*α*= 20,

*Λ*= 0.52

*μm*,

*L*= 3

*mm*,

*P*= 4

*mm*,

*δ*= 3.5 × 10

_{n}^{−4}and

*k*= 2531πΝ/s. Figure 4(a) plots the calculated reflectivities of the GA-FBG-based FP cavity, according to Eq. (15) (solid lines) and the TM method (dotted lines). In the TM method, each GA-FBG is divided into 150 piecewise-uniform segments.

_{s}*n*= 1.4654 and

_{m}*n*= 1.4619 of core mode and cladding mode, respectively, which results in a MZ interferometer. The MZ interferometer is described by the parameters:

_{s}*α*= 20,

*Λ*= 400

*μm*,

*L*= 40

*mm*,

*P*= 150

*mm*,

*δ*= 4 × 10

_{n}^{−4}and

*k*= 115.7πΝ/s. Figure 4(b) demonstrates the calculated transmissions of the GA-LPFG-based MZ or Michelson interferometers, by using Eq. (16) (solid lines) and the TM method (dotted lines). In the TM method, each GA-LPFG is also divided into 150 piecewise-uniform segments. Figures 4(a) and 4(b) indicate that the calculated spectra based on Eqs. (15) and (16) are in excellent agreements with those on the TM method, for GA-grating-based interferometers.

_{s}### 3.4 Efficiency of analytic models

## 4. Conclusion

## Acknowledgments

## References and links

1. | Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, “Widely tunable long-period waveguide grating couplers,” Opt. Express |

2. | S. K. Liaw, L. Dou, and A. S. Xu, “Fiber-bragg-grating-based dispersion-compensated and gain-flattened raman fiber Amplifier,” Opt. Express |

3. | K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. |

4. | X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,” Opt. Lett. |

5. | P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. |

6. | G. D. Marshall, R. J. Williams, N. Jovanovic, M. J. Steel, and M. J. Withford, “Point-by-point written fiber-Bragg gratings and their application in complex grating designs,” Opt. Express |

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

8. | M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. |

9. | L. A. Weller-Brophy and D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A |

10. | J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res. |

11. | H. Kogelnik, “Filter response of nonuniform almost-periodic structure,” Bell Syst. Tech. J. |

12. | E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron. |

13. | H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron. |

14. | E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. |

15. | A. Bouzid and M. A. G. Abushagur, “Scattering analysis of slanted fiber gratings,” Appl. Opt. |

16. | L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

17. | X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for fiber Bragg gratings,” Acta Phys. Sin. |

18. | X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. |

19. | X. K. Zeng, “Application of Fourier mode coupling theory to real-time analyses of nonuniform Bragg gratings,” IEEE Photon. Technol. Lett. |

20. | X. K. Zeng and K. Liang, “Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period,” Opt. Express |

21. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(050.2770) Diffraction and gratings : Gratings

(230.0230) Optical devices : Optical devices

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: December 2, 2011

Revised Manuscript: January 14, 2012

Manuscript Accepted: January 16, 2012

Published: February 2, 2012

**Citation**

Xiangkai Zeng, Lai Wei, Yingjun Pan, Shengping Liu, and Xiaohui Shi, "Analytic models of spectral responses of fiber-grating-based interferometers on FMC theory," Opt. Express **20**, 4009-4017 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4009

Sort: Year | Journal | Reset

### References

- Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, “Widely tunable long-period waveguide grating couplers,” Opt. Express14(26), 12644–12654 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-26-12644 . [CrossRef] [PubMed]
- S. K. Liaw, L. Dou, and A. S. Xu, “Fiber-bragg-grating-based dispersion-compensated and gain-flattened raman fiber Amplifier,” Opt. Express15(19), 12356–12361 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12356 . [CrossRef] [PubMed]
- K. P. Koo, M. LeBlanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett.10(7), 1006–1008 (1998). [CrossRef]
- X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,” Opt. Lett.23(7), 509–510 (1998), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-23-7-509 . [CrossRef] [PubMed]
- P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol.15(8), 1576–1580 (2004). [CrossRef]
- G. D. Marshall, R. J. Williams, N. Jovanovic, M. J. Steel, and M. J. Withford, “Point-by-point written fiber-Bragg gratings and their application in complex grating designs,” Opt. Express18(19), 19844–19859 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-19-19844 . [CrossRef] [PubMed]
- T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15(8), 1277–1294 (1997). [CrossRef]
- M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt.26(16), 3474–3478 (1987). [CrossRef] [PubMed]
- L. A. Weller-Brophy and D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A2(6), 863–871 (1985). [CrossRef]
- J. J. Liau, N. H. Sun, S. C. Lin, R. Y. Ro, J. S. Chiang, C. L. Pan, and H. W. Chang, “A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory,” Prog. Electromagn. Res.93, 385–401 (2009). [CrossRef]
- H. Kogelnik, “Filter response of nonuniform almost-periodic structure,” Bell Syst. Tech. J.55, 109–126 (1976).
- E. Mazzetto, C. G. Someda, J. A. Acebron, and R. Spigler, “The fractional Fourier transform in the analysis and synthesis of fiber Bragg gratings,” Opt. Quantum Electron.37(8), 755–787 (2005). [CrossRef]
- H. V. Baghdasaryan and T. M. Knyazyan, “Modeling of linearly chirped fiber Bragg gratings by the method of single expression,” Opt. Quantum Electron.34(5-6), 481–492 (2002). [CrossRef]
- E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol.15(8), 1295–1302 (1997). [CrossRef]
- A. Bouzid and M. A. G. Abushagur, “Scattering analysis of slanted fiber gratings,” Appl. Opt.36(3), 558–562 (1997). [CrossRef] [PubMed]
- L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics48(6), 4758–4767 (1993). [CrossRef] [PubMed]
- X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for fiber Bragg gratings,” Acta Phys. Sin.59, 8597–8606 (2010).
- X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin.59, 8607–8614 (2010).
- X. K. Zeng, “Application of Fourier mode coupling theory to real-time analyses of nonuniform Bragg gratings,” IEEE Photon. Technol. Lett.23(13), 854–856 (2011). [CrossRef]
- X. K. Zeng and K. Liang, “Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period,” Opt. Express19(23), 22797–22808 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22797 . [CrossRef] [PubMed]
- J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127(6), 1918–1939 (1962). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.