## Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings

Optics Express, Vol. 14, Issue 14, pp. 6394-6399 (2006)

http://dx.doi.org/10.1364/OE.14.006394

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### Abstract

In this paper, we describe the properties of Fabry-Perot fiber cavity formed by two fiber Bragg gratings in terms of the grating effective length. We show that the grating effective length is determined by the group delay of the grating, which depends on its diffraction efficiency and physical length. We present a simple analytical formula for calculation of the effective length of the uniform fiber Bragg grating and the frequency separation between consecutive resonances of a Fabry-Perot cavity. Experimental results on the cavity transmission spectra for different values of the gratings’ reflectivity support the presented theory.

© 2006 Optical Society of America

## 1. Introduction

1. J.S. Zyskind, V. Mizrahi, D.J. DiGiovanni, and J.W. Sulhoff, “Short single-frequency erbium-doped fiber laser,” Electron. Lett. **28**, 1385–1387 (1992). [CrossRef]

4. Pavel Polynkin, Alexander Polynkin, Masud Mansuripur, Jerome Moloney, and N. Peyghambarian, “Single-frequency laser oscillator with watts-level output power at 1.5 µm by use of a twisted-mode technique,” Opt. Lett. **30**, 2745–2747 (2005). [CrossRef] [PubMed]

5. J. Canning, N. Groothoff, E. Buckley, T. Ryan, K. Lyytikainen, and J. Digweed, “All-fibre photonic crystal distributed Bragg reflector (PC-DBR) fibre laser,” Opt. Express **11**, 1995–2000 (2003). [CrossRef] [PubMed]

2. Ch. Spiegelberg, J. Geng, Y. Hu, Yu. Kaneda, S. Jiang, and N. Peyghambarian, “Low-noise narrow-linewidth fiber laser at 1550 nm (June 2003),” J. Lightwave Techn. **22**, 57–62 (2004). [CrossRef]

4. Pavel Polynkin, Alexander Polynkin, Masud Mansuripur, Jerome Moloney, and N. Peyghambarian, “Single-frequency laser oscillator with watts-level output power at 1.5 µm by use of a twisted-mode technique,” Opt. Lett. **30**, 2745–2747 (2005). [CrossRef] [PubMed]

6. J.J. Zayhowski, “Limits imposed by spatial hole burning on the single-mode operation of standing-wave laser cavities,” Opt. Lett. **15**, 431–433 (1990). [CrossRef] [PubMed]

7. J. Canning, M. Janos, and M.G. Sceats, “Rayleigh longitudinal profile of optical resonances within waveguide grating structures using sidescattered light,” Opt. Lett. **21**, 609–611 (1996). [CrossRef] [PubMed]

## 2. Effective length of uniform FBGs forming a Fabry-Perot fiber cavity

*R*

_{1,2}are the reflection coefficients for the corresponding gratings,

*L*

_{0}is the length of fiber between FBGs (see Fig.1),

*β*is the propagation constant of the fundamental mode LP

_{01}, and

*φ*

_{1,2}are the phases of the FBGs’ reflection coefficients.

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

*λ*, can be expressed as

*λ*is the free-space wavelength,

*n*

_{g}is the group refractive index for LP

_{01}fiber mode, and the effective lengths of the FBGs,

*L*

_{eff}

_{1,2}, are defined as

*v*

_{g}=

*c*/

*n*

_{g}is the group velocity,

*τ*

_{1,2}=

*∂φ*

_{1,2}/

*∂ω*are the group delays for corresponding FBGs, and

*ω*is the angular optical frequency. Note that for conventional optical fibers

*n*

_{g}≈

*n*

_{eff}, where

*n*

_{eff}is the effective refractive index (the difference between

*n*

_{g}and

*n*

_{eff}is ~1% for fused silica at 1.5 µm), thus

*v*

_{g}in Eq. (3) can be replaced with the fundamental mode velocity

*v*=

*c*/

*n*

_{eff}.

*L*

_{c}, is a sum of the effective lengths of both the FBGs forming the cavity and the distance between them is

*L*

_{c}=

*L*

_{0}+

*L*

_{eff}

_{1}+

*L*

_{eff}

_{2}.

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

*ρ*, is:

*κ*=

*πn*

_{1}/

*λ*is the coupling coefficient (

*n*

_{1}is the grating amplitude and

*λ*is the free-space wavelength),

*ξ*=2

*πn*

_{eff}(

*λ*

^{-1}-

*λ*

_{0}) and

*L*is the grating physical length. The group delay of the light reflected by a FBG, τ, can be found from the reflection coefficient phase:

*φ*is obtained from Eq. (4) as:

*R*=|

*ρ*(

*λ*

_{0})|

^{2}, on the grating amplitude

*n*

_{1}(

*R*=tanh

^{2}(

*πn*

_{1}

*L*/

*λ*

_{0})), the final formula for the grating effective length at the Bragg wavelength is written

*L*

_{eff}/

*L*, on the grating peak reflectivity

*R*for various values of detuning from the peak wavelength

*λ*

_{0}. As it is seen, at relatively low values of the grating diffraction efficiency the grating effective length is near a half of its physical length; at high diffraction efficiencies the effective length approaches zero. Physically, this can be explained by the evident fact that a weak FBG (

*κL*≪1) reflects light power along its length practically homogeneously, but a very intensive FBG (

*κL*≫1) reflects the most of light power from its initial part.

*λ*

_{0}, the effective length is bigger owing to the group delay increase (Fig. 3). The curve for zero-detuning was calculated using Eq. (8), and curves for the detuning values different from zero were calculated using Eqs. (3,5,6). Since in single-frequency DBR lasers the distance between FBGs

*L*

_{0}lies usually in the range of several cm, giving mode-spacing of the order of 10 pm (

*λ*

_{0}±5 pm), Eq. (8) can be used to estimate a minimal FBG effective length.

## 3. Transmittance of a Fabry-Perot cavity formed by two FBGs

*L*

_{1}=

*L*

_{2}=4 cm, and a separation between them,

*L*

_{0}=5 cm. As it is seen the mode spacing grows with an increase of the FBG’s diffraction efficiency owing to a decrease of their effective lengths (see Fig. 2).

12. H. Renner, “Effective-index increase, form birefringence and transition losses in UV-side-illuminated photosensitive fibers,” Opt. Express **11**, 546–560 (2001). [CrossRef]

13. K. Dossou, S. LaRochelle, and M. Fontaine, “Numerical Analysis of the Contribution of the Transverse Asymmetry in the Photo-Induced Index Change Profile to the Birefringence of Optical Fiber,” J. Lightwave Technol. **20**, 1463–1470 (2002). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | J.S. Zyskind, V. Mizrahi, D.J. DiGiovanni, and J.W. Sulhoff, “Short single-frequency erbium-doped fiber laser,” Electron. Lett. |

2. | Ch. Spiegelberg, J. Geng, Y. Hu, Yu. Kaneda, S. Jiang, and N. Peyghambarian, “Low-noise narrow-linewidth fiber laser at 1550 nm (June 2003),” J. Lightwave Techn. |

3. | Yu.O. Barmenkov, A.V. Kir’yanov, J. Mora, J.L. Cruz, and M.V. Andrés, The continuous-wave and giant-pulse operations of a single-frequency Erbium-doped fiber laser. IEEE Photon. Technol. Lett.17, 28–30 (2005). [CrossRef] |

4. | Pavel Polynkin, Alexander Polynkin, Masud Mansuripur, Jerome Moloney, and N. Peyghambarian, “Single-frequency laser oscillator with watts-level output power at 1.5 µm by use of a twisted-mode technique,” Opt. Lett. |

5. | J. Canning, N. Groothoff, E. Buckley, T. Ryan, K. Lyytikainen, and J. Digweed, “All-fibre photonic crystal distributed Bragg reflector (PC-DBR) fibre laser,” Opt. Express |

6. | J.J. Zayhowski, “Limits imposed by spatial hole burning on the single-mode operation of standing-wave laser cavities,” Opt. Lett. |

7. | J. Canning, M. Janos, and M.G. Sceats, “Rayleigh longitudinal profile of optical resonances within waveguide grating structures using sidescattered light,” Opt. Lett. |

8. | Orazio Svelto, |

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

10. | A. Othonos and K. Kalli, |

11. | Raman Kashyap, “ |

12. | H. Renner, “Effective-index increase, form birefringence and transition losses in UV-side-illuminated photosensitive fibers,” Opt. Express |

13. | K. Dossou, S. LaRochelle, and M. Fontaine, “Numerical Analysis of the Contribution of the Transverse Asymmetry in the Photo-Induced Index Change Profile to the Birefringence of Optical Fiber,” J. Lightwave Technol. |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(140.3510) Lasers and laser optics : Lasers, fiber

(230.1480) Optical devices : Bragg reflectors

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: May 22, 2006

Revised Manuscript: June 22, 2006

Manuscript Accepted: June 28, 2006

Published: July 10, 2006

**Citation**

Yuri O. Barmenkov, Dobryna Zalvidea, Salvador Torres-Peiró, Jose L. Cruz, and Miguel V. Andrés, "Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings," Opt. Express **14**, 6394-6399 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-14-6394

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### References

- J.S. Zyskind, V. Mizrahi, D.J. DiGiovanni and J.W. Sulhoff, "Short single-frequency erbium-doped fiber laser," Electron. Lett. 28, 1385-1387 (1992). [CrossRef]
- Ch. Spiegelberg, J. Geng, Y. Hu, Yu. Kaneda, S. Jiang and N. Peyghambarian, "Low-noise narrow-linewidth fiber laser at 1550 nm (June 2003)," J. Lightwave Techn. 22, 57-62 (2004). [CrossRef]
- Yu.O. Barmenkov, A.V. Kir’yanov, J. Mora, J.L. Cruz and M.V. Andrés, The continuous-wave and giant-pulse operations of a single-frequency Erbium-doped fiber laser.IEEE Photon. Technol. Lett. 17, 28-30 (2005). [CrossRef]
- Pavel Polynkin, Alexander Polynkin, Masud Mansuripur, Jerome Moloney and N. Peyghambarian, "Single-frequency laser oscillator with watts-level output power at 1.5 μm by use of a twisted-mode technique," Opt. Lett. 30, 2745-2747 (2005). [CrossRef] [PubMed]
- J. Canning, N. Groothoff, E. Buckley, T. Ryan, K. Lyytikainen and J. Digweed, "All-fibre photonic crystal distributed Bragg reflector (PC-DBR) fibre laser," Opt. Express 11, 1995-2000 (2003). [CrossRef] [PubMed]
- J.J. Zayhowski, "Limits imposed by spatial hole burning on the single-mode operation of standing-wave laser cavities," Opt. Lett. 15, 431-433 (1990). [CrossRef] [PubMed]
- J. Canning, M. Janos and M.G. Sceats, "Rayleigh longitudinal profile of optical resonances within waveguide grating structures using sidescattered light," Opt. Lett. 21, 609-611 (1996). [CrossRef] [PubMed]
- Orazio Svelto, Principles of Lasers, (New York: Plenum Press, 1989), chapter 4.
- T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]
- A. Othonos and K. Kalli, Fiber Bragg gratings: fundamentals and applications in telecommunications and sensing, (Boston: Artech House, 1999), chapter 5.
- Raman Kashyap, "Fiber Bragg Gratings," San Diego: Academic Press, 1999, chapter 4.
- H. Renner, "Effective-index increase, form birefringence and transition losses in UV-side-illuminated photosensitive fibers," Opt. Express 11, 546-560 (2001). [CrossRef]
- K. Dossou, S. LaRochelle and M. Fontaine, "Numerical Analysis of the Contribution of the Transverse Asymmetry in the Photo-Induced Index Change Profile to the Birefringence of Optical Fiber," J. Lightwave Technol. 20, 1463-1470 (2002). [CrossRef]

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