## Derivation of Raman threshold formulas for CW double-clad fiber amplifiers

Optics Express, Vol. 17, Issue 10, pp. 8476-8490 (2009)

http://dx.doi.org/10.1364/OE.17.008476

Acrobat PDF (239 KB)

### Abstract

We show that the classic Raman threshold formula is unsuitable to accurately predict the onset of Raman scattering in high-power CW double-clad fiber amplifiers. Consequently new analytical formulas for the Raman threshold are obtained and their accuracy is tested. Using these new formulas, the dependence of the Raman threshold on various parameters is studied.

© 2009 Optical Society of America

## 1. Introduction

1. J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron . **13**, 537–545 (2007). [CrossRef]

3. S. W. Allison, G. T. Gillies, D. W. Magnuson, and T. S. Pagano, “Pulsed laser damage to optical fibers,” Appl. Opt . **24**, pp. 3140–3145, (1985). [CrossRef] [PubMed]

5. J. Kim, P. Dupriez, C. Codemard, J. Nilsson, and J. K. Sahu, “Suppression of stimulated Raman scattering in a high power Yb-doped fiber amplifier using a W-type core with fundamental mode cut-off,” Opt. Express **14**, 5103–5113 (2006). [CrossRef] [PubMed]

6. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express **14**, 2715–2720 (2006) [CrossRef] [PubMed]

*maximum possible power*was very difficult to determine beforehand and its estimation was a matter of experience and/or computer simulations. This is because, contrary to the case of passive fibers, no analytic formula was available to calculate the Raman threshold in active fibers.

7. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt . **11**, 2489–2494 (1972). [CrossRef] [PubMed]

## 2. Classic Raman Threshold formula applied to high-power CW fiber amplifiers

7. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt . **11**, 2489–2494 (1972). [CrossRef] [PubMed]

*A*is the effective mode area of the fiber,

_{eff}*g*is the Raman gain coefficient, and

_{R}*L*= (1-

_{eff}*e*

^{-αpL})/

*α*is the effective fiber length, with

_{p}*α*standing for propagation losses.

_{p}*α*with the fiber gain rate in the effective length term. However, there are many reasons that suggest that this simple approach doesn’t work. On the one hand, the formula above, when applied to fiber amplifiers, assumes a perfect exponential signal growth. This would restrict the usability of Eq. (1) to unsaturated contra-directionally pumped amplifiers. On the other hand, the classic formula assumes that the Raman scattering undergoes the same loss/gain (due to the active fiber) as the signal to be amplified. This is clearly not true in a conventional fiber amplifier since the wavelength shift of the Raman scattering ensures that the fiber cross-sections are very different at the signal and Raman wavelengths (see [8

_{p}8. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-Doped Fiber Amplifiers,” IEEE J. Quantum Electron . **33**, pp. 1049–1056 (1997). [CrossRef]

9. Y. Wang, “Stimulated Raman scattering in high-power double-clad fiber lasers and power amplifiers,” Opt. Eng . **44**, pp. 114202-1–114202-12 (2005). [CrossRef]

## 3. Derivation of the Raman threshold formulas for high-power CW fiber amplifiers

*co-directional*or counter-directional with the signal power, or

*bidirectional*. In each of these configurations the evolution of the signal power inside of the active fiber is very different, which ultimately leads to very different Raman thresholds [9

9. Y. Wang, “Stimulated Raman scattering in high-power double-clad fiber lasers and power amplifiers,” Opt. Eng . **44**, pp. 114202-1–114202-12 (2005). [CrossRef]

*roughly estimate*the value of the Raman threshold in this situation is to use a total pump power equal to the average value of the Raman threshold in the co- and counter-propagating directions, distributed in such a way that the pump in each direction is not higher than its corresponding Raman threshold.

*β*is the factor that selects the level of the Raman threshold with respect to the output signal power (e.g. 0.1 for a Raman output power that is 10% of the signal output power), and

*γ*stands for the integral on

_{R}*α*.

_{R}*P*and

_{signal}(L), L_{eff}, γ_{R}*P*. As commented before, the expressions for these parameters are dependent on the pump configuration of the amplifier. Therefore, from this point on, the co- and counter-propagating pump cases will be studied separately.

_{R o}### 3.1. Co-directional pumping configuration

11. Y. Wang, C. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun . **242**, 487–502 (2004). [CrossRef]

*I*and

_{p}*I*are the pump and signal intensities respectively where the + or − superscripts represent the propagation direction.

_{Signal}*N*is the doping-ion concentration,

*N*is the population density of the excited state and

_{2}*N*is the population density of the ground state.

_{1}*σ*and

_{aj}*σ*are the absorption and emission cross-sections of the doping-ions respectively at the wavelength

_{ej}*λ*(with

_{j}*j*=

*p*or

*signal*, to denote pump or laser signal).

*τ*is the spontaneous emission lifetime,

*A*is the doped area of the fibre,

*A*is the effective modal area of the laser signal and

_{eff}*A*is the effective area of the pump signal.

_{eff p}*Γ*and

_{p}*Γ*are the overlapping factors between the doped area and the pump or the laser radiation respectively.

_{signal}*σ*and

_{p}*σ*are the attenuation factors of the fibre at the pump and laser signal wavelengths.

_{signal}*h*is the Planck constant, and

*c*the speed of light in vacuum.

*I*≫

_{signal}*I*all along the fiber. This approximation is reasonable for double-clad fibers since, in them, the pump core area is typically tens of times larger than the signal core area. Thus, in this case, neglecting the propagation losses at the pump wavelength and taking into account that

_{pump}*P*=

_{i}*I**

_{i}*A*, the solutions for the co-propagating case are:

_{eff i}*C*and

*ζ*are given by the following expressions:

*α*≪

_{signal}*ζ*), it can be shown that the average value of the signal and pump powers along the fiber are given by:

*γ*and

_{R}*P*. In order to calculate the value of

_{R o}*γ*, that represents the fiber gain/loss at the Raman Stokes wavelength, the complete differential equation that governs the growth of the Raman Stokes signal should be used (i.e. an expanded version of Eq. (2)):

_{R}*α′*represents the propagation loss coefficient of the fiber at the Raman Stokes wavelength,

_{R}*Γ*is the overlapping factor between the doped area and the mode at the Raman Stokes wavelength, and

_{R}*σ*and

_{aR}*σ*are the absorption and emission cross-sections at the Raman Stokes wavelength respectively. This last equation can be analytically solved assuming that the Raman Stokes is small enough as not to have a big influence on the inversion profile of the fiber. Thus, the output power of the Raman scattering is given by:

_{eR}*γ*corresponds to the exponent of the second exponential term.

_{R}*γ*represents the Raman Stokes wavelength and

_{R}*∆λ*is the ASE bandwidth at the Raman Stokes wavelength (normally coinciding with the Raman gain bandwidth, i.e. ~5nm after [11

_{ASE}11. Y. Wang, C. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun . **242**, 487–502 (2004). [CrossRef]

*P*is given by [7

_{R spon}7. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt . **11**, 2489–2494 (1972). [CrossRef] [PubMed]

*B*is the effective bandwidth of the Raman gain (~1 THz). The problem at this point is that Eq. (14) has no analytical solution because both

_{eff}*N*

_{2}and

*P*vary along the length of the fiber. Therefore, in order to obtain the desired analytical solution an approximation is required. In this case

_{signal}*N*

_{2}and

*P*will be approximated by their average values and they will be considered constant along the fiber. Thus, the solution of Eq. (14) becomes:

_{signal}*P*given by Eq. (10) and

_{signal ave}*N*

_{2}ave being calculated from Eq. (5) by substituting

*P*and

_{signal}*P*by

_{p}*P*and

_{signal ave}*P*respectively. In Eq. (16) the term multiplying the exponential represents

_{p ave}*P*. Therefore, at this point all the terms required by Eq. (4) to calculate the Raman threshold have already been calculated. Thus, putting all the terms together we arrive at the following expression for the Raman threshold in the co-propagating pump configuration:

_{R o}**11**, 2489–2494 (1972). [CrossRef] [PubMed]

### 3.2. Counter-directional pumping configuration

*L*is smaller in the counter-propagating pump configuration, which means that the Raman scattering and any other nonlinear effect will have a higher threshold (as also reported in [9

_{eff}9. Y. Wang, “Stimulated Raman scattering in high-power double-clad fiber lasers and power amplifiers,” Opt. Eng . **44**, pp. 114202-1–114202-12 (2005). [CrossRef]

*γ*and

_{R}*P*are still given by Eq. (12) and Eq. (16) respectively, with the only exception that now:

_{R o}## 4. Approximations of the Raman threshold formulas

### 4.1. Approximations for the co-directional pumping configuration

*γ*which makes it difficult to approximate. If a non-iterative formula is required, then

_{R}*γ*has to be approximated by a constant value

_{R}*. This can be done by considering that*γ ¯

_{R}*γ*is a function which starts at a certain value for low pump powers and then tends asymptotically to another one for larger pumps. We have found that the average of these two extreme values of

_{R}*γ*works well when trying to approximate this function by a constant. Thus,

_{R}*f*(

*x*)=

*xBe*

^{-Ax+C}. It can be shown that, in the range of parameters of interest, this function can be well approximated by a Boltzmann function with the following parameters:

*P*, the following approximation for the Raman threshold in the co-propagating pump case is obtained:

_{p}**11**, 2489–2494 (1972). [CrossRef] [PubMed]

*γ*is not too high, i.e. the formula below should not be used when operating a system near the lower wavelength range of the gain bandwidth and with high doping concentrations. Additionally, as will be seen in the following sections, this approximation provides a very conservative value of the Raman threshold. This is because, given the nonlinear dependence of the Raman scattering on the signal intensity and fiber length, the propagation of a high constant power along the whole fiber length represents a worse case than a small initial signal that grows along the fiber (and which average power equals that of the constant signal) because the latter will only generate significant Raman in a much shorter section of the fiber. On the other hand, in this approach the parameter

_{R}*β*that controls the percentage of output signal power transferred to the Raman Stokes at the threshold is lost. In this sense, the formula is only able to provide a pump power value at which it is still safe to operate the system without having any detrimental effect created by the Raman scattering. In spite of the limitations listed above, we feel that this formula, given its simplicity, can still be interesting to a group of people that only want to assess whether their system will be operating free of Raman. Therefore, using Eq. (1) and Eq. (10) this simple approximation takes the form:

### 4.2. Approximations for the counter-directional pumping configuration

*P*≫

_{p o}*P*, is:

_{signal o}## 5. Simulations and discussion

### 5.1. Accuracy of the Raman threshold formulas

^{26}ions/m

^{3}doping concentration and 1.1*10

^{-13}m/W Raman gain coefficient. The pump wavelength is 976nm and the signal wavelength 1030nm. The input signal power to the amplifier is set to 50 Watt. In order to carry out the simulations a model based on that described in [11

11. Y. Wang, C. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun . **242**, 487–502 (2004). [CrossRef]

*β*=0.1 (i.e ten percent of the output power contained in the Raman Stokes component). Under these circumstances the formulas predict a Raman threshold for the pump of 756 Watt in the co-directional pumping case, and of 1843 Watt for counter-directional pump. The results of the simulations using exactly the pump powers predicted by the equations are given in Fig. 1. As can be seen at first sight, the results are

*visually*accurate with the Raman power growing up to the point of being ~10% of the signal power (with the consequent depletion on the signal power). The numerical accuracy of the Raman threshold obtained with Eq. (17) and Eq. (22) can be evaluated by taking into account that the simulations presented in Fig. 1 predict an output Raman level of 56.2 Watt and 194.1 Watt respectively. These output Raman powers represent a ~8.7% and a ~13.5% of their respective output signal powers.

*β*=0.01). Besides, in this case the signal wavelength was set to 1064 nm. The last column of the table represents the Raman output power normalized to the signal output power and expressed as a percentage.

**44**, pp. 114202-1–114202-12 (2005). [CrossRef]

*I*≫

_{signal}*I*) and, therefore, they are also subject to limitations. Thus, these formulas loose their accuracy when predicting the Raman threshold for fibers with relatively large core areas and relatively small cladding diameters (for example a fiber with 20 μm core and 125 μm cladding diameters). However, amplifiers based on these fibers normally exhibit Raman thresholds that are well beyond the fiber damage threshold (because of the large core area and their short lengths) and, therefore, they can usually be considered Raman-free.

_{pump}**44**, pp. 114202-1–114202-12 (2005). [CrossRef]

### 5.2. Influence of fiber length and doping concentration on the Raman threshold

^{25}ions/m

^{3}) on, the dependence of the Raman threshold on this parameter is greatly reduced (for both co- and counter-propagating pumping configurations). This suggests that, in general it will be more advantageous to work with higher doping concentrations and shorter fiber lengths.

*N*=1*10

^{26}is 3257 Watt, and for the same fiber length but

*N*=1.6*10

^{26}the threshold is ~3900Watt). Moreover, this increase in the Raman threshold comes with no penalty of the amplification efficiency, thus allowing the extraction of higher output signal powers from the fiber amplifier. This behaviour might seem counterintuitive at first, but it can easily be explained by the evolution of the signal power along the amplifier. In the example given above, with the higher doping concentration the fiber is much longer than the absorption length (white line). This implies that the pump power remaining at the signal input end is so small that it is not able to generate the necessary inversion required to amplify the signal. Therefore, in the first few meters of the amplifier the signal propagates almost unamplified and it is only in the last half of the fiber that it undergoes strong amplification. On the contrary, with the lower doping concentration the length of the fiber is similar to the absorption length, which implies that the signal will be amplifier from the very beginning of the fiber. The result is that the effective length of the amplifier with the lower doping concentration is longer than that of the amplifier with the higher doping level. Of course this behaviour can only be seen on the counter-propagating configuration, since with a co-propagating pump the signal always undergoes the strongest amplification in the first meters of the fiber. Furthermore, if the signal wavelength changes to, say 1030nm, this behaviour is also not observed (due to the stronger influence of the ASE).

*N*, which automatically gives a higher Raman threshold to the fiber amplifier (see white lines in Fig. 2). However, what this behaviour does highlight is the fact that the Raman threshold in fiber amplifiers is an extremely complex parameter that depends on the interplay of many different factors.

## 6. Conclusions

## Acknowledgments

## References and links

1. | J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron . |

2. | D. Gapontsev, IPG Photonics, “6kW CW Single Mode Ytterbium Fiber Laser in All-Fiber Format,” in “Solid State and Diode Laser Technology Review” (Albuquerque, 2008) |

3. | S. W. Allison, G. T. Gillies, D. W. Magnuson, and T. S. Pagano, “Pulsed laser damage to optical fibers,” Appl. Opt . |

4. | G. P. Agrawal, |

5. | J. Kim, P. Dupriez, C. Codemard, J. Nilsson, and J. K. Sahu, “Suppression of stimulated Raman scattering in a high power Yb-doped fiber amplifier using a W-type core with fundamental mode cut-off,” Opt. Express |

6. | J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express |

7. | R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt . |

8. | R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-Doped Fiber Amplifiers,” IEEE J. Quantum Electron . |

9. | Y. Wang, “Stimulated Raman scattering in high-power double-clad fiber lasers and power amplifiers,” Opt. Eng . |

10. | R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron . |

11. | Y. Wang, C. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun . |

12. | F. Röser, D. N. Schimpf, J. Rothhardt, T. Eidam, J. Limpert, A. Tünnermann, and F. Salin, “Gain limitations and consequences for short length fiber amplifiers,” in OSA Topical Meeting on Advanced Solid-State Photonics (ASSP, 2008), paper WB22. |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(190.5650) Nonlinear optics : Raman effect

(190.5890) Nonlinear optics : Scattering, stimulated

(060.3510) Fiber optics and optical communications : Lasers, fiber

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 25, 2009

Revised Manuscript: April 3, 2009

Manuscript Accepted: April 4, 2009

Published: May 5, 2009

**Citation**

Cesar Jauregui, Jens Limpert, and Andreas Tünnermann, "Derivation of Raman treshold formulas for CW double-clad fiber amplifiers," Opt. Express **17**, 8476-8490 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8476

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### References

- J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, "The rising power of fiber lasers and amplifiers," IEEE J. Sel. Top. Quantum Electron. 13, 537-545 (2007). [CrossRef]
- D. Gapontsev, IPG Photonics, "6kW CW Single Mode Ytterbium Fiber Laser in All-Fiber Format," in "Solid State and Diode Laser Technology Review" (Albuquerque, 2008)
- S. W. Allison, G. T. Gillies, D. W. Magnuson, and T. S. Pagano, "Pulsed laser damage to optical fibers," Appl. Opt. 24, 3140-3145 (1985). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, NY, 1995).
- J. Kim, P. Dupriez, C. Codemard, J. Nilsson, and J. K. Sahu, "Suppression of stimulated Raman scattering in a high power Yb-doped fiber amplifier using a W-type core with fundamental mode cut-off," Opt. Express 14, 5103-5113 (2006). [CrossRef] [PubMed]
- J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, "Extended single-mode photonic crystal fiber lasers," Opt. Express 14, 2715-2720 (2006) [CrossRef] [PubMed]
- R. G. Smith, "Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering," Appl. Opt. 11, 2489-2494 (1972). [CrossRef] [PubMed]
- R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, "Ytterbium-Doped Fiber Amplifiers," IEEE J. Quantum Electron. 33, 1049-1056 (1997). [CrossRef]
- Y. Wang, "Stimulated Raman scattering in high-power double-clad fiber lasers and power amplifiers," Opt. Eng. 44, 114202-1 - 114202-12 (2005). [CrossRef]
- R. H. Stolen, "Polarization effects in fiber Raman and Brillouin lasers," IEEE J. Quantum Electron. QE-15, 1157-1160 (1979). [CrossRef]
- Y. Wang, C. Xu, and H. Po, "Analysis of Raman and thermal effects in kilowatt fiber lasers," Opt. Commun. 242, 487-502 (2004). [CrossRef]
- F. Röser, D. N. Schimpf, J. Rothhardt, T. Eidam, J. Limpert, A. Tünnermann, and F. Salin, "Gain limitations and consequences for short length fiber amplifiers," in OSA Topical Meeting on Advanced Solid-State Photonics (ASSP, 2008), paper WB22.

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