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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 4671–4676
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Explicit solution for Raman fiber laser using Lambert W function

Chaohong Huang, Zhiping Cai, Chenchun Ye, and Huiying Xu  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 4671-4676 (2007)
http://dx.doi.org/10.1364/OE.15.004671


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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

Raman fiber lasers (RFLs) are widely studied as the efficient all-fiber wavelength converter and attractive laser sources in the wavelength range of 1.1-1.9μm. RFLs find many applications in optical communications and sensors for its flexibility to design and all-fiber configuration. Recently significant progresses in RFLs have been made due to the use of ultralow-loss P2O5-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

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]

]. Many numerical methods[3

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

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]

] have been developed to optimize the RFLs. However, the numerical methods are not only time-consuming, but also unstable if the initial guessed values are not properly chosen. Several references [5–8

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]

] have devoted to optimize the Raman fiber laser by analytic approaches. Recently Ref. [9

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]

] presented the exact analytical solution for the single-pass pumping RFL with the assistance of Lambert W function, but no explicit expression was provided. In this paper, we obtain an explicit analytical solution for double-pass pumping RFL under a linear-attenuation approximation for pump propagation. The approximation has been applied to the second-order RFL and proved to be valid [10

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]

]. The proposed explicit solution provides us a clear physical understanding to the optimal design of the laser.

2. Theoretical analysis

The schematic diagram of double-pass pumping Raman fiber laser is shown in Fig. 1. A pair of fiber Bragg grating reflectors, i.e. FBG1 and FBG2, form the Fabry-Perot resonant cavity at the first Stokes wavelength. An additional reflector (i.e. FBG0) with high reflectivity at the pump wavelength yields a double-pass pumping scheme. The reflectivity of FBG0 and FBG2 is larger than 99% at pump and Stokes wavelengths, respectively. FBG1 with relative low reflectivity at the Stokes wavelength can couple the Stokes lights out of the cavity.

Fig. 1. Schematic diagram of a double-pass pumping Raman fiber laser

The forward- and backward-propagated pump and Stokes powers in Raman gain fiber meet the following well-known differential equations [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]

, 11

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]

]

±1P0±dP0±dz=α0gλ1λ0(P1++P1)
(1a)
±1P1±dP1±dz=α1+g(P0++P0)
(1b)

where the subscripts i represent pump(i=0) and Stokes(i=1) waves. The superscripts ± denote forward(+) and backward(-) propagation beams. λi is the wavelengths of pump and Stokes radiations and α i is the loss coefficient of Raman fiber at λi. g refers to the Raman gain efficiency (in W-1m-1). At z = 0 and z = L, Eqs. (1a) – (1b ) meet such boundary conditions as

P0+(0)=Pin´,P0(L)=R0LP0+(L)
P0+(0)=R10P1(0),P1(L)=R1LP1+(L)
(2)

where P'in =10-0.1δF-0.1δs Pin =ηin Pin,RL 0 =10-0.2δF-0.4δs,R 0 R 0 1 =10-0.2δs R 2,RL 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.

Equations (1a)-(1b) can be rewritten as follows by introducing variables ui = ln(P + i/P - i)/2

du0dz=α02gλ1λ0c1cosh(u1)
(3a)
du1dz=α1+2gc0cosh(u0)
(3b)

where √ci = √P+ 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]

] . The boundary conditions for ui are

{u0(0)=ln(Ṕinc0),u0(L)=ln(R0L)2u1(0)=ln(R0L)2,u1(L)=−ln(R1L)2
(4)

We can define √ci as the geometric mean powers and ui (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 √ci are undetermined constants. All boundary conditions are known except u 0(0). The steady-state conditions for laser oscillation can be obtained by integrating (3a)–(3b) from z=0 to z=L

2gλ1λ0c1Ll1eff=lnP´inc0δ0
(5a)
2gc0Ll0eff=δ1
(5b)

where lieff=1L0Lcosh(ui)dz is defined as the normalized effective fiber length for pump (i = 0) and Stokes wave (i = 1)δ 0 = α 0 L-ln(RL 0)/2and δ 1 =α 1 L-ln(R 0 1 RL 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 P̄i=12L0L(Pi++Pi)dz.. It is obvious that P̄ = √cileffi . From Eq. (5b), one can find that P̄0 will be clamped to the value of δ 1(2gL) when pump power is larger than threshold pump power.

We can obtain the following equation by eliminating z from Eq. (3a) – Eq. (3b) and integrating

c1=λ0λ1.α1[u0(L)u0(0)]2gc0{sinh[u0(L)]sinh[u0(0)]}α0[u1(L)u1(0)]2g{sinh[u1(L)]sinh[u1(0)]}
(6)

The equation states that the number of input pump photons is equal to the total number of output pump and Stokes photons from the RFL plus photons dissipated in Raman fiber.

The expression of threshold pump power can be derived from Eqs. (3a)–(3b) or Eqs. (5a)–(5b) while considering √c 1 = 0.

Pth=1ηinα0δ1g(1eα0L)(1+R0Leα0L)
(7)

l0eff={sinh[u0(L)]sinh[u0(0)]}[u0(L)u0(0)]
(8)

Using Eq. (5b) and Eq. (8), we can express u 0(0) or √c 0 explicitly as

u0(0)=lnPin´c0=Pin´gLδ1+12W0[2Pin´gLδ1exp(2Pin´gLδ1)]
(9)

where W 0 is the main branch of Lambert W function [13

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

]. For the sake of simplification, ln(RL 0)/2 ≈ 0 has been assumed in Eq. (9) for RL 0 ≈ 1 usually.

We can obtain the expression of leff 1 by substituting (9) and (5a) into (6) and express √c 1 explicitly as follows

l1eff={sinh[u1(L)]sinh[u1(0)]}[u1(L)u1(0)]
(10)
c1=λ0λ1u0(0)δ02gLl1eff
(11)

The output power can be written as

Piout=Ticieui(L)(i=0,1)
(12)

where Ti =10-0.2δF-0.2δs (1-Ri) is the output transmissivity while considering lumped loss.

Although the expression of leff 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

u1(z)=α1z+u1(0)+2gLc0{sinh[u0(z)]sinh[u0(0)]}[u0(L)u0(0)]
(13)

Using Eq. (9), we can calculate the residual pump power at z=0 and z=L

P0(0)=Pin´exp[2u0(0)]
(14)
P0+(L)=Pin´exp[u0(0)]R0L
(15)

If P'inδ 1(2gL)(i.e. P'inP̄0), then P - 0(0)≪ P'in. Thus one can compare P'in in with P̄0 to determine whether pump power is depleted or not. In pump-depleted approximation (i.e., P - 0(0) ≪ P'in), √c 1 and pout 1 are dependent linearly on input pump power and the slope efficiency can be obtained from (12)

ηsλ0λ1T1R1L12δ11l1eff
(16)

Furthermore, one can find P + 0(L) ≪ P'in the condition P' ≫ δ 1(gL) (i.e. P'in ≫ 2P̄0) 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

To verify the explicit solution, we use a phosphosilicate fiber Raman laser as an example. In comparison the numerical simulation for Eqs. (3a)–(3b) is also carried out using MATLAB BVP solver. The typical parameters of phosphosilicate fiber fabricated by Fiber Optic Research Center of Russia are selected for calculation: g=1.28 × 10-3W-1m-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.

Any spectral broadening for pump and Stokes waves are not considered in the classic model depicted by Eq. (3a) – Eq. (3b). However, the RFL always suffer from the spectral broadening effect. While considering this effect, the model should been modified. The simplest way is to introduce the effective reflectivity of FBG [14

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

]. The effective reflectivity of FBG, which can be estimated by measuring the reflective spectrum of FBG and the output spectra of pump and Stokes waves, is always lower than the nominal reflectivity at Bragg wavelength. For convenience, the effective reflectivity of FBG0 and FBG2 is assumed to be 95%.

Figure 2(a) shows the output pump and Stokes power as a function of input pump power with L=250m, R 1 =30%. The discrepancy between analytical results and numerical simulation is less than 1.2% up to Pin=20W. When Pin>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 Pin=5W. As shown in Fig. 2, the explicit analytical solution agrees well with numerical simulation.

Fig. 2. The comparison between analytic and numerical solution (L=250m, R1=30%). (squares: numerical; lines: analytic). (a) Output power of pump and Stokes radiations versus input pump power; (b) Power distributions of pump and Stokes radiations in Raman fiber when Pin=5W

4. Design optimization

It is well-known that there exits a set of optimal values of fiber length and reflectivity of output FBG which lead to the maximum power transfer efficiency under certain pump power. One can look for the optimal parameters by numerical method. However, the method is time-consuming and unstable. The most interesting subject will be to find the optimal parameters by an analytical method. In this section, we discuss the procedure looking for the optimal parameters using the analytical results in section 2.

Fig. 3. Design optimization of RFL when Pin=5W (squares: numerical; lines: analytical). (a) Optimal fiber length and power transfer efficiency versus R 1; (b) Power transfer efficiency versus R 1 and L

Let ∂Pout 1/∂L = 0, one can readily deduce the following result

L=12ln1R10R1Lln2Pin´gα1(2Pin´gα1α1ln2Pin´gα1)
(17)

This equation shows the optimal fiber length under certain pump power and reflectivity of output FBG. Figure 3(a) shows the optimal fiber length and power transfer efficiency as a function of R 1 when Pin=5W. From this figure, one can find that the transfer efficiency is maximized (about 64.4%) when 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%.

Fig. 4. The optimal fiber length, reflectivity of output FBG and power transfer efficiency vesus input pump power

Similarly, one can obtain the optimal reflectivity of output FBG under certain pump power and fiber length if let ∂Pout 1/∂RL 1=0 . If ∂Pout 1/∂L = 0 and ∂Pout 1/∂RL 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

An explicit analytic solution for the double-pass pumping Raman fiber laser is obtained under the linear-propagation approximation for pump beam. The explicit solution shows good agreement with numerical simulation. The algebraic average power P̄0 is clamped to δ 1(2gL) which is determined only by cavity parameters when PinPth. The pump depletion approximation, under which laser output power increase with increasing pump power, is valid when P'inP̄0. Furthermore, if P'in ≫ 2P̄0, double-pass pumping scheme will be unnecessary.

The optimal design of the 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. For example, the results for phosphosilicate fiber laser show that the optimal cavity parameters are L=230m and R 1=29% with maximum power transfer efficiency 64.4% when Pin=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.

Acknowledgments

The work is partially supported by the Key Project of Fujian Province of China (Grant No.2002F011, 2004HZ01-1-3), Fujian Natural Sciences Project (Grant No.A0310004) and the Innovation Fund of Xiamen University (Grant No.XDKJCX20041003).

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. 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]

6.

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]

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 22, 764–771 (2005). [CrossRef]

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 22, 772–776 (2005). [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]

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]

12.

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

13.

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

14.

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

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

  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]
  6. 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]
  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 22, 764-771 (2005). [CrossRef]
  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 22, 772-776 (2005). [CrossRef]
  9. 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]
  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]
  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]
  12. J. AuYeung and A. Yariv, "Theory of cw Raman oscillation in optical fibers," J. Opt. Soc. Am. B 69, 803-807(1979). [CrossRef]
  13. E. M. Wright, "Solution of the Equation zez=a," Bull. Am. Math. Soc. 65, 89-93(1959). [CrossRef]
  14. J. C. Bouteiller, "Spectral modeling of Raman fiber lasers," IEEE Photon. Technol. Lett. 15, 1698-1700(2003). [CrossRef]

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