## Explicit solution for Raman fiber laser using Lambert W function

Optics Express, Vol. 15, Issue 8, pp. 4671-4676 (2007)

http://dx.doi.org/10.1364/OE.15.004671

Acrobat PDF (139 KB)

### Abstract

In this paper, an approximate explicit solution for the first-order Raman fiber laser is obtained by using Lambert W function. Good agreement between the explicit solution and numerical simulation is demonstrated. Furthermore, the optimal design of Raman fiber laser is discussed using the proposed solution. The optimal values of fiber length, reflectivity of output fiber Bragg grating and power transfer efficiency are obtained under different pump power. There exists a certain tolerance of the optimal parameters, in which the output power decreases only slightly. The optimal fiber length and reflectivity of output FBG decrease with increasing pump power.

© 2007 Optical Society of America

## 1. Introduction

_{2}O

_{5}-doped silica fibers(PDF), high power Yb-doped dual-cladding fiber lasers (Yb-DCFL) and high-reflectivity fiber Bragg gratings(FBG) [1

1. E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, “CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. **33**, 1542–1544 (1997). [CrossRef]

2. N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun **176**, 219–222 (2000). [CrossRef]

3. M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Elect. **36**, 1117–1122 (2000). [CrossRef]

4. S. Cierullies, H. Renner, and E. Brinkmeyer, “Numerical optimization of multi-wavelength and cascaded Raman fiber lasers,” Opt. Commun. **217**, 233–238 (2003). [CrossRef]

5. I. A. Bufetov and E. M. Dianov, “A simple analytic model of a cw multicascade fibre Raman laser,” Quantum Elect. **30**, 873–877 (2000). [CrossRef]

9. J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, and Y. P. Wang, “Exact analytical solution for Raman fiber laser,” IEEE Photon. Technol. Lett. **18**, 1097–1099 (2006). [CrossRef]

10. C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, “Analytic modeling of the P-doped cascaded Raman fiber lasers,” Opt. Fiber Technol. **13**, 22–26 (2007). [CrossRef]

## 2. Theoretical analysis

1. E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, “CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. **33**, 1542–1544 (1997). [CrossRef]

11. S. D. Jackson and P. H. Muir, “Theory and numerical simulation of nth-order cascaded Raman fiber lasers,” J. Opt. Soc. Am. B **18**, 1297–1306 (2001). [CrossRef]

*i*represent pump(

*i*=0) and Stokes(

*i*=1) waves. The superscripts ± denote forward(+) and backward(-) propagation beams.

*λ*is the wavelengths of pump and Stokes radiations and

_{i}*α*

_{i}is the loss coefficient of Raman fiber at

*λ*.

_{i}*g*refers to the Raman gain efficiency (in W

^{-1}m

^{-1}). At

*z*= 0 and

*z*=

*L*, Eqs. (1a) – (1b ) meet such boundary conditions as

*P*'

_{in}=10

^{-0.1δF-0.1δs}

*P*=

_{in}*η*

_{in}*P*,

_{in}*R*

^{L}_{0}=10

^{-0.2δF-0.4δs},

*R*

_{0}

*R*

^{0}

_{1}=10

^{-0.2δs}

*R*

_{2},

*R*

^{L}_{1}=10

^{-0.2δs}

*R*

_{1}.

*R*

_{0},

*R*

_{1}and

*R*

_{2}are the reflectivity of FBG0, FBG1 and FBG2, respectively. Here, splicing losses of all splicing points and insert losses of all FBGs are assumed to be

*δ*

_{s}and

*δ*

_{F}(in dB), respectively.

*u*= ln(

_{i}*P*

^{+}

_{i}/

*P*

^{-}

_{i})/2

*c*= √P

_{i}^{+}

_{i}/

*P*

^{-}

_{i}is a constant for the coordinates

*z*[12

12. J. AuYeung and A. Yariv, “Theory of cw Raman oscillation in optical fibers,” J. Opt. Soc. Am. B **69**, 803–807(1979). [CrossRef]

*u*are

_{i}*c*as the geometric mean powers and

_{i}*u*(

_{i}*z*) as the gain factors for pump and Stokes radiations. Thus, Eq. (3a) – Eq. (3b) represent the evolvement of the gain factors along the Raman fiber and the geometric mean powers √

*c*are undetermined constants. All boundary conditions are known except

_{i}*u*

_{0}(0). The steady-state conditions for laser oscillation can be obtained by integrating (3a)–(3b) from

*z*=0 to

*z*=

*L*

*i*= 0) and Stokes wave (

*i*= 1)

*δ*

_{0}=

*α*

_{0}

*L*-ln(

*R*

^{L}_{0})/2and

*δ*

_{1}=

*α*

_{1}

*L*-ln(

*R*

^{0}

_{1}

*R*

^{L}_{1})/2 are the single-pass loss factors for pump and Stokes radiation owing to loss of fiber and transmitted loss of FBGs, respectively. We can define the algebraic average powers for pump and Stokes radiations as

*P*̄ = √

*c*. From Eq. (5b), one can find that

_{i}l^{eff}_{i}*P*̄

_{0}will be clamped to the value of

*δ*

_{1}(2

*gL*) when pump power is larger than threshold pump power.

*c*

_{1}= 0.

*P*>

_{in}*P*, we can still assume pump beam attenuate linearly along the fiber with a larger attenuate constant than

_{th}*α*

_{0}. The approximation, which proved to be valid in next section, leads to

*W*

_{0}is the main branch of Lambert W function [13

13. E. M. Wright, “Solution of the Equation *ze ^{z}*=

*a*,” Bull. Am. Math. Soc.

**65**, 89–93(1959). [CrossRef]

*R*

^{L}_{0})/2 ≈ 0 has been assumed in Eq. (9) for

*R*

^{L}_{0}≈ 1 usually.

*l*

^{eff}_{1}by substituting (9) and (5a) into (6) and express √

*c*

_{1}explicitly as follows

*T*=10

_{i}^{-0.2δF-0.2δs}(1-

*R*) is the output transmissivity while considering lumped loss.

_{i}*l*

^{eff}_{1}is similar to (8),

*u*

_{1}(

*z*) is not linearly dependent on

*z*. We can derive

*u*

_{1}(

*z*) by integrating Eq. (3b) from 0 to

*z*

*P*'

_{in}≫

*δ*

_{1}(2

*gL*)(i.e.

*P*'

_{in}≫

*P*̄

_{0}), then

*P*

^{-}

_{0}(0)≪

*P*'

_{in}. Thus one can compare

*P*'

_{in}in with

*P*̄

_{0}to determine whether pump power is depleted or not. In pump-depleted approximation (i.e.,

*P*

^{-}

_{0}(0) ≪

*P*'

_{in}), √

*c*

_{1}and

*p*

^{out}_{1}are dependent linearly on input pump power and the slope efficiency can be obtained from (12)

*P*

^{+}

_{0}(

*L*) ≪

*P*'

_{in}the condition

*P*' ≫

*δ*

_{1}(

*gL*) (i.e.

*P*'

_{in}≫ 2

*P*̄

_{0}) is met, namely pump power is depleted only by single-pass propagation. Thus FBG0 with high reflectivity at pump wavelength is unnecessary under this condition.

## 3. Comparison with the numerical simulation

*g*=1.28 × 10

^{-3}W

^{-1}m

^{-1},

*λ*

_{0}=1.06μm,

*λ*

_{1}=1.24μm,

*α*

^{0}=1.8dB/km,

*α*

_{1}=1.16dB/km. The splicing loss

*δ*

_{s}is 0.02dB and insert loss of FBG

*δ*

_{F}is 0.1dB.

14. J. C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett. **15**, 1698–1700(2003). [CrossRef]

*L*=250m,

*R*

_{1}=30%. The discrepancy between analytical results and numerical simulation is less than 1.2% up to

*P*=20W. When

_{in}*P*>5W, the slope efficiency equals to 70%, which is in excellent agreement with the value calculated from Eq. (16). Figure 2(b) shows the power distributions of the pump and Stokes radiations at Raman fiber when

_{in}*P*=5W. As shown in Fig. 2, the explicit analytical solution agrees well with numerical simulation.

_{in}## 4. Design optimization

*P*

^{out}_{1}/∂

*L*= 0, one can readily deduce the following result

*R*

_{1}when

*P*=5W. From this figure, one can find that the transfer efficiency is maximized (about 64.4%) when

_{in}*L*=230m and

*R*

_{1}=29%. In this figure, the numerical optimal results are also plotted as a comparison with analytical results. They agree well with each other. Figure 3(b) shows the contour diagram of power transfer efficiency versus

*R*

_{1}and

*L*. From the figure, one can also obtain the same optimal parameter values. Additionally, one can find that there exists a certain tolerance of the optimal parameters, in which the output power decreases only slightly. For example, the power transfer efficiency decreases less than 1.5% from the maximum value when the values of

*R*

_{1}and

*L*are selected in the range of contour line 63%.

*P*

^{out}_{1}/∂

*R*

^{L}_{1}=0 . If ∂

*P*

^{out}_{1}/∂

*L*= 0 and ∂

*P*

^{out}_{1}/∂

*R*

^{L}_{1}=0 are met simultaneously, one can obtain the optimal fiber length and reflectivity of output FBG under certain input power. Figure 4 shows the optimal optical fiber length, reflectivity of output FBG and power transfer efficiency as a function of input pump power. As shown in this figure, the optimal fiber length and reflectivity of output FBG decrease with increasing pump power.

## 5. Conclusion

*P*̄

_{0}is clamped to

*δ*

_{1}(2

*gL*) which is determined only by cavity parameters when

*P*≥

_{in}*P*. The pump depletion approximation, under which laser output power increase with increasing pump power, is valid when

_{th}*P*'

_{in}≫

*P*̄

_{0}. Furthermore, if

*P*'

_{in}≫ 2

*P*̄

_{0}, double-pass pumping scheme will be unnecessary.

*L*=230m and

*R*

_{1}=29% with maximum power transfer efficiency 64.4% when

*P*=5W. There exists a certain tolerance of the optimal parameters, in which the output power decreases only slightly. The optimal fiber length and reflectivity of output FBG decrease with increasing pump power.

_{in}## Acknowledgments

## References and links

1. | E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, “CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. |

2. | N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun |

3. | M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Elect. |

4. | S. Cierullies, H. Renner, and E. Brinkmeyer, “Numerical optimization of multi-wavelength and cascaded Raman fiber lasers,” Opt. Commun. |

5. | I. A. Bufetov and E. M. Dianov, “A simple analytic model of a cw multicascade fibre Raman laser,” Quantum Elect. |

6. | S. A. Babin, D. V. Churkin, and E. V. Podivilov, “Intensity interactions in cascades of a two-stage Raman fiber laser,” Opt. Commun. |

7. | B. Burgoyne, N. Godbout, and S. Lacroix, “Theoretical analysis of nth-order cascaded continuous-wave Raman fiber lasers. I. Model and resolution,” J. Opt. Soc. Am. B |

8. | B. Burgoyne, N. Godbout, and S. Lacroix, “Theoretical analysis of nth-order cascaded continuous-wave Raman fiber lasers. II. Optimization and design rules,” J. Opt. Soc. Am. B |

9. | J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, and Y. P. Wang, “Exact analytical solution for Raman fiber laser,” IEEE Photon. Technol. Lett. |

10. | C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, “Analytic modeling of the P-doped cascaded Raman fiber lasers,” Opt. Fiber Technol. |

11. | S. D. Jackson and P. H. Muir, “Theory and numerical simulation of nth-order cascaded Raman fiber lasers,” J. Opt. Soc. Am. B |

12. | J. AuYeung and A. Yariv, “Theory of cw Raman oscillation in optical fibers,” J. Opt. Soc. Am. B |

13. | E. M. Wright, “Solution of the Equation a,” Bull. Am. Math. Soc. 65, 89–93(1959). [CrossRef] |

14. | J. C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(140.3510) Lasers and laser optics : Lasers, fiber

(140.3550) Lasers and laser optics : Lasers, Raman

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: November 14, 2006

Revised Manuscript: March 13, 2007

Manuscript Accepted: March 14, 2007

Published: April 3, 2007

**Citation**

Chaohong Huang, Zhiping Cai, Chenchun Ye, and Huiying Xu, "Explicit solution for Raman fiber laser using Lambert W function," Opt. Express **15**, 4671-4676 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4671

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### References

- E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997). [CrossRef]
- N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000). [CrossRef]
- M. Rini, I. Cristiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded CW Raman fiber lasers," IEEE J. Quantum Elect. 36, 1117-1122 (2000). [CrossRef]
- S. Cierullies, H. Renner, and E. Brinkmeyer, "Numerical optimization of multi-wavelength and cascaded Raman fiber lasers," Opt. Commun. 217, 233-238 (2003). [CrossRef]
- I. A. Bufetov and E. M. Dianov, "A simple analytic model of a cw multicascade fibre Raman laser," Quantum Elect. 30,873-877 (2000). [CrossRef]
- S. A. Babin, D. V. Churkin, and E. V. Podivilov, "Intensity interactions in cascades of a two-stage Raman fiber laser," Opt. Commun. 226, 329-335 (2003). [CrossRef]
- B. Burgoyne, N. Godbout, and S. Lacroix, "Theoretical analysis of nth-order cascaded continuous-wave Raman fiber lasers. I. Model and resolution," J. Opt. Soc. Am. B 22, 764-771 (2005). [CrossRef]
- B. Burgoyne, N. Godbout, and S. Lacroix, "Theoretical analysis of nth-order cascaded continuous-wave Raman fiber lasers. II. Optimization and design rules," J. Opt. Soc. Am. B 22, 772-776 (2005). [CrossRef]
- J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006). [CrossRef]
- C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007). [CrossRef]
- S. D. Jackson and P. H. Muir, "Theory and numerical simulation of nth-order cascaded Raman fiber lasers," J. Opt. Soc. Am. B 18, 1297-1306 (2001). [CrossRef]
- J. AuYeung and A. Yariv, "Theory of cw Raman oscillation in optical fibers," J. Opt. Soc. Am. B 69, 803-807(1979). [CrossRef]
- E. M. Wright, "Solution of the Equation zez=a," Bull. Am. Math. Soc. 65, 89-93(1959). [CrossRef]
- J. C. Bouteiller, "Spectral modeling of Raman fiber lasers," IEEE Photon. Technol. Lett. 15, 1698-1700(2003). [CrossRef]

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