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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3258–3271
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Impact of modal interference on the beam quality of high-power fiber amplifiers

Cesar Jauregui, Tino Eidam, Jens Limpert, and Andreas Tünnermann  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3258-3271 (2011)
http://dx.doi.org/10.1364/OE.19.003258


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Abstract

Recent work on high-power fiber amplifiers report on a degradation of the output beam quality or even on the appearance of mode instabilities. By combining the transversally resolved rate equations with a 3D Beam propagation method we have managed to create a model able to provide an explanation of what we believe is at the root of this effect. Even though this beam quality degradation is conventionally linked to transversal hole burning, our simulations show that this alone cannot explain the effect in very large mode area fibers. According to the model presented in this paper, the most likely cause for the beam quality degradation is an inversion-induced grating created by the interplay between modal interference along the fiber and transversal hole burning.

© 2011 OSA

1. Introduction

Since the introduction of the double-clad design, the power of fiber laser systems has grown exponentially [1

1. A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010). [CrossRef] [PubMed]

]. Both CW and pulsed lasers have profited from this renaissance of fiber lasers, with output powers in excess of 6kWatts already demonstrated in CW regime [2

2. D. Gapontsev and I. P. G. Photonics, “6kW CW single mode ytterbium fiber laser in all-fiber format,” in Solid State and Diode Laser Technology Review (Albuquerque, 2008).

] and approaching the kilowatt level for fs-pulse fiber lasers [3

3. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010). [CrossRef] [PubMed]

]. With this rapid evolution of the output powers it is natural to start asking the question of which the ultimate power limit of fiber systems is [4

4. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. Barty, “Ultimate power limits of optical fibers, ” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMO6, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OMO6.

]. Whichever the studies on this topic, they all agree that this limit will be the result of some non-linear effect or combination of them. Thus, whereas there is no discussion that the ultimate limit will be given by self-focusing (leading to the physical destruction of the fiber), before this limit is reached there are a plethora of non-linear effects that can seriously compromise the performance of the laser system to the point of rendering it useless for practical purposes. Among these non-linear effects there can be found Self-Phase Modulation (SPM), Raman scattering and Four-Wave Mixing [5

5. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).

].

Apart from presenting the model, in this work it has been used to analyze the impact of the already mentioned inversion grating on the higher order mode content of the output beam of high-power fiber amplifiers. The simulations show that the existence of this inversion-induced grating can lead to substantial amounts of energy transfer from the fundamental mode to the HOM. Additionally, the dependence of this energy transfer on different parameters such as fiber length and pump core diameter is studied.

The paper is arranged as follows: in section 2 the simulation model is presented and its advantages and limitations are discussed. In section 3 the physical origin for the growth of HOM content in the output beam of a high-power fiber amplifier is presented and the dependences of this effect on some parameters are investigated with the model. Finally, some conclusions are drawn.

2. Simulation model

As said before, the model presented herein is the result of a combination between a 3D BPM and the TR-rate equations. Therefore, this model will inherit the virtues and limitations of those two simulation methods. On the one hand, it means that this new model is a full three-dimensional simulation of an active fiber, and in that sense one that offers a deep insight into the impact of mode propagation on the active processes taking place in the fiber. However, on the other hand this means that this new model is necessarily computationally intensive. In order to reduce the computation time, the model used in this work is a steady-state one, and therefore all the information on the dynamics of the effects is lost. In spite of this, the model presented herein offers a very useful approach to understand the impact of mode interference on the performance of high-power fiber amplifiers. Additionally, since the BPM is a one-way propagation model, this new model is only able to simulate single pass fiber amplifiers. However, the big advantage of BPM, and therefore of this new model, is that it is not based on fiber modes but on electric field beams instead, which implies that the beam is not decomposed into individual modes during propagation. Therefore, this model can also simulate effects generated by leaky or even radiation modes.

Since the model is a combination between BPM and the TR-rate equations, in the following, these two simulation methods will be presented separately, and finally the way to combine them will be explained.

2.1. Beam propagation method

In this section the basic theory of BPM is reviewed. The notation and general description of the method will follow those presented in [13

13. C. Xu and W. Huang, “Finite-difference beam propagation method for guide-wave optics,” Prog. Electromagn. Res. 11, 1–49 (1995) (PIER).

].

The starting point is the vectorial Helmholtz wave equation for a linear and isotropic medium:
2Es+n(x,y,z)2k2Es=(Es)
(1)
where Esstands for the electric field vector of the signal, n(x,y,z) is the three-dimensional refractive index of the material, and k is the wavenumber. Considering that the complex index of refraction n(x,y,z) typically varies slowly in the direction of propagation (z direction in this case), the vectorial Helmholtz equation can be rewritten as a function of the transverse electric field components Es,t:

2Es,t+n2k2Es,t=t[tEs,t1n2t(n2Es,t)]
(2)

Now, assuming a one-way propagation of the light, i.e. that the light propagates only in the +z direction, the electric field of the signal can be separated into a slowly varying envelope As,t and a fast oscillating phase term:
Es,t(x,y,z)=As,t(x,y,z)ejnokz
(3)
where no is a reference refractive index close to the actual effective index of the beam in the fiber (i.e. it should be chosen so that the envelope varies slowly in the propagation direction). Introducing Eq. (3) into Eq. (2) it is possible to obtain the so-called one-way wave equation:
z(j2nokz)As,t(x,y,z)=PAs,t(x,y,z)
(4)
where the operator P is given by:

PAs,t=t2As,t(x,y,z)+(n2no2)k2As,t(x,y,z)t[tAs,t(x,y,z)1n2t(n2As,t(x,y,z))]
(5)

By discretizing the operator P in Eq. (7), it is possible to obtain a system of linear equations which can be expressed in matrix form as:
A[As,t]l+1=B[As,t]l
(8)
where [As,t]l+1 and [As,t]l are the electric field vectors at positions (l+1)Δz and lΔz, respectively. Additionally, A and B are non-symmetric complex band matrixes. These matrixes can be efficiently inverted using the BiCG-STAB method [16

16. H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992). [CrossRef]

].

In order to solve the linear system of Eqs. (8), we have used the transparent boundary conditions as described in [17

17. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=ol-16-9-624. [CrossRef] [PubMed]

]. Besides, it must be said that in our simulations the scalar approximation (i.e. just one polarization) was used, since all the fibers under analysis are weakly guiding or/and polarization maintaining. However, it is important to highlight that the model, as presented in this section, is general and supports the semi-vectorial or full-vectorial implementations as well.

2.2. Transversally-resolved steady state rate equations

In this section the transversally resolved steady state rate equations are briefly presented. The notation used in this section closely follows that from [14

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

]. In obtaining the TR-rate equations several assumptions were made: (1) two-level systems are considered where the excited state absorption is neglected, (2) the pump light is assumed to be homogeneously distributed across the fiber cross-section, (3) polarization effects are ignored, and (4) monochromatic signals are considered throughout the calculations. Thus, taking into account these assumptions, the transversally resolved rate equations are:
N2(x,y,z)N1(x,y,z)=[Pp+(z)+Pp(z)]σapΓp(x,y)hυp+Ps+(z)σasΓs(x,y)hυs[Pp+(z)+Pp(z)]σepΓp(x,y)hυp+1τ+Ps+(z)σesΓs(x,y)hυsdPp±(z)dz={x1x2y1y2[σepN2(x,y,z)σapN1(x,y,z)]Γp(x,y)dxdy}Pp±(z)αpPp±(z)dPs+(z)dz={x1x2y1y2[σesN2(x,y,z)σasN1(x,y,z)]Γs(x,y)dxdy}Ps+(z)αsPs+(z)
(9)
where N1(x,y,z) and N2(x,y,z) are the population densities of the lower and upper lasing levels at the position (x,y,z). Additionally, Pp(z) and Ps(z) are the pump and signal powers along the propagation direction z, respectively. The signs + and – on the powers represent the propagation direction (either +z or –z). On the other hand, σap and σas are the absorption cross-sections at the pump and signal wavelengths, respectively. Similarly, σep and σes are the emission cross-sections at the pump and signal wavelengths, respectively. Besides, h is the Planck constant, τ is the lifetime in the excited state, υp and υs are the pump and signal frequencies, respectively. In addition, αp and αs are the attenuation coefficients of the pump and signal due to their propagation through the fiber, respectively. It is also worth noting that the integration limits x1, x2, y1, y2 are chosen to sweep the complete core area. Finally, Γp(x,y) and Γs(x,y) are the power filling distributions of pump and signal, which can be expressed as follows:
Γp(x,y)=1AcladandΓs(x,y)=ψ(x,y)ψ(x,y)dxdy
(10)
where Aclad is the area of the pump core, and ψ(x,y) is the transversal intensity distribution of the signal beam.

It is important to note at this point that the equations presented in Eqs. (9) and (10) differ from those in [14

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

]. This is because the rate equations have been modified to make them compatible with the BPM. As a consequence, for example, the signal power Ps can only propagate in the forward direction and, therefore, it is only represented by a + sign in Eq. (9). Additionally, only one signal beam is considered (instead of one per fiber mode as in [14

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

]). The reason is that in the present model there is no need to decompose the beam into the fiber modes because BPM uses the complete electric field. Thus, the power filling distributions of Eq. (10) will be calculated with the actual beam shape obtained from the BPM propagation. Among other advantages (such as being a more exact simulation of what actually takes place in the fiber), this strategy allows taking mode interference into account.

In order to be able to program Eqs. (9) and (10) in a computer, they have to be discretized. However, the beam obtained by BPM does not have to be necessarily radial-symmetric (as a result of mode interference), which implies that in this model it is not possible to divide the fiber in concentric rings as done in [14

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

]. Therefore, the discretization has to be done in a rectangular grid with grid steps Δx and Δy in the x- and y-directions respectively. Thus, the discrete transversally resolved steady state rate equations are:
N2(m,k)(z)N1(m,k)(z)=[Pp+(z)+Pp(z)]σapΓp(m,k)hυpA(m,k)+Ps+(z)σasΓs(m,k)hυsA(m,k)[Pp+(z)+Pp(z)]σepΓp(m,k)hυpA(m,k)+1τ+Ps+(z)σesΓs(m,k)hυsA(m,k)dPp±(z)dz=mk[σepN2(m,k)(z)σapN1(m,k)(z)]Γp(m,k)Pp±(z)αpPp±(z)dPs+(z)dz=mk[σesN2(m,k)(z)σasN1(m,k)(z)]Γs(m,k)Ps+(z)αsPs+(z)
(11)
with:
A(m,k)=ΔxΔyΓp(m,k)=A(m,k)AcladΓs(m,k)=ψ(mΔx,kΔy)mkψ(mΔx,kΔy)N(m,k)=N1(m,k)+N2(m,k)
(12)
where N(m,k) represents the total ion concentration at the transversal point (mΔx, kΔy). We have programmed these equations in a computer and solved them using the Runge-Kutta methods.

2.2. Active BPM model

n(m,k)=n(mΔx,kΔy)=ΔnN2(m,k)N(m,k)+jλ4π(σesN2(m,k)σasN1(m,k))
(13)

Note that in this approach it is considered that the index of refraction and the gain of the fiber do not undergo substantial changes within the fiber section. Additionally, it might be argued that, if the index change is predominantly due to temperature, it will be reversed with respect to that considered in Eq. (13) (i.e. the areas with lower inversion, that is, with higher depletion, have a higher temperature and therefore a higher refractive index). This does not change the fact that the inversion profile generates an index change that mimics it (albeit maybe inversed) and has some time persistence. Thus, in the following, for illustration purposes, the index change given in Eq. (13) will be assumed. Deviations of the actual index profile from that considered in Eq. (13) will result in different mode coupling efficiencies, but its effect will still be (qualitatively) that described in this paper: an energy transfer between different modes.

Using this complex refractive index, the BPM can be used to obtain the beam intensity distribution at the new RK iteration point. This new beam distribution, calculated as described above, implicitly takes into account the differential gain observed by the different fiber modes due to THB. This new transversal intensity distribution of the beam ψ(x,y) obtained for each iteration after the propagation process is used to calculate the new power filling factors Γp(m,k)and Γs(m,k). Then, using these new power filling factors, the system of Eqs. (11) can be solved. At this point it is worth noting that, when solving the TR-rate equations in this fashion, it is implicitly assumed that the beam does not change too much from one iteration to the next. This approximation is sufficiently good, at least for LMA fibers.

3. Simulation results: Impact of modal interference on the beam quality of high-power fiber lasers

Besides, straight low NA very large core step-index fibers have the additional computational advantage that, since they have to be kept straight, they have a limited length of ~2m at most (for practical reasons), which further reduces the computation time. This way, in the simulations presented in the following, the computation took in average 1 hour per meter length in a conventional desktop computer.

The fibers are simulated in the following have the following parameters: 1.45 core index, 0.017 NA, 80 μm core diameter, 62 μm diameter of the doped core region with an active ion concentration of 3.25·1025 ion/m3. The fiber cross-sections employed are those measured in [18

18. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]

]. These parameters are typical for state-of-the-art very large mode area fibers [6

6. J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express 13(4), 1055–1058 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-4-1055. [CrossRef] [PubMed]

]. Note that these fibers do exhibit a certain preferential gain for the LP01 since the doped region is smaller than the whole core area. This tends to favor the amplification of the fundamental mode against the amplification of the higher order modes [14

14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

,19

19. T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B 19(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539. [CrossRef]

]. In spite of this, it is shown in the following that the HOMs can grow faster than the fundamental mode (by means of an inversion grating that facilitates an energy transfer in the HOM direction), which is a counterintuitive result.

3.1. Physical origin of the beam quality degradation

Figure 2
Fig. 2 (a) Evolution of the beam intensity along a 1m long active fiber with an initial excitation of 95% LP01 and 5% LP11 modes (only fiber core shown). (b) Corresponding inversion profile showing the areas with non-depleted inversion (only fiber core shown).
shows the simulation results corresponding to a 1m long fiber with the characteristics given above and 280 μm pump core diameter. In these simulations the input signal power is 30 W at 1064 nm, and the pump power is 300 W at 976 nm. Even though the model is full 3D, Fig. 2 shows only a cut of the results in the x-z plane. At the input of the fiber 95% of the energy was coupled in the fundamental LP01 mode and 5% in the LP11 mode (with the right orientation to shift the center of gravity of the beam in the x-z plane). As seen in Fig. 2(a), the evolution of the beam intensity along a fiber, when considering mode interference, creates periodic changes of the beam (in this example seen as a periodic shift of the center of gravity of the beam) (see e.g. Fig. 1). This gives rise to periodic core areas where the inversion (here defined as N2/N) has not been efficiently depleted (see Fig. 2(b)) which, in turn, via effects such as the resonantly induced index change of doped fibers discussed in [9

9. M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. 3(1), 44–64 (1997). [CrossRef]

11

11. A. A. Fotiadi, O. L. Antipov and P. Megret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” Frontiers in Guided Wave Optics and Optoelectronics, 209–234 (2010).

] and/or the temperature dependence of the refractive index [12

12. L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11(9), 1435–1446 (1993). [CrossRef]

] (since core areas with different inversion levels will exhibit different temperatures), result in core regions with a locally higher refractive index. The reader should note that in order to obtain these local index changes it is mandatory to consider the interplay between local THB and mode interference. These periodic index variations, provided that the mode interference is stable (which will be discussed next), create a long period grating. This grating, having been generated by mode beating, has in turn exactly the right period to transfer energy between the two interfering modes (LP01 and LP11), so that at the end of the fiber the HOM content can grow substantially. This alone can reduce the beam quality of the laser output and, additionally, it is our belief that it may trigger the mode instabilities reported elsewhere [3

3. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010). [CrossRef] [PubMed]

].

3.2. Beam quality degradation: simulation results

In the following some simulation results are presented, and the basic dependence of the inversion-induced grating on various parameters (such as fiber length or pump core diameter) is analyzed. In all the cases the excitation of the modes at the beginning of the fiber is distributed as follows: 95% of the energy is coupled into the LP01 and 5% in the LP11.

ci(z)=A^s,t(x,y,z)ϕ^*i(x,y)dxdy
(14)

Note that in Eq. (14) the ^ symbol indicates that the electric fields have been normalized to have a power of 1 W.

As no inversion-grating has been considered in this first simulation, the relative mode content shown in Fig. 3(b) reveals a progressive increase of the LP01 content thanks to the effect of the preferential gain. Please note that even though this simulation does not consider the effect of the inversion grating it still includes THB. Thus, these results show that THB alone cannot explain the beam quality degradation observed in some experiments (at least when some amount of preferential gain is included in the fiber design).

A closer look at Fig. 4 provides some clues to understand the effect of the inversion-induced grating. On the one hand, as any fiber grating, its coupling efficiency depends on the number of periods that it comprises. Thus, since the period length is only determined by the transverse opto-geometrical characteristics of the fiber, and not by its length, a shorter fiber should substantially reduce the amount of energy transfer. On the other hand, it can be seen in Fig. 4 that the strong THB at the end of the fiber “erases” the grating (by depleting the inversion) and, therefore, in this fiber section the efficiency of the energy transfer should be reduced. This is confirmed by Fig. 5, where it can be seen that the energy transfer rate decreases towards the end of the fiber. Actually, the highest energy transfer rate in this figure can be found in the central region of the fiber, where correspondingly in Fig. 4 the highest inversion grating contrast is to be found (see left plots). Thus, the amplification characteristics, i.e. the degree of saturation, of the amplifier play additionally an important role.

Even though not shown in here, our simulations also show that the effect of this inversion-induced grating tends to be in general smaller at 1030 nm. This can be understood taking into account that for the fiber lengths considered in here there is more gain, and therefore, potentially more THB at 1030 nm than at 1064 nm.

4. Conclusions

Acknowledgments

The authors acknowledge financial support from the Thüringen Ministry of Education, Science and Culture (TMBWK) through the “Modenfeldstabilisierung in Hochleistungsfaserlaser und -verstärkersystemen” – MOFA project. The authors also want to acknowledge the European Research Council for financial support under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement n° [240460] “PECS”.

References and links

1.

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010). [CrossRef] [PubMed]

2.

D. Gapontsev and I. P. G. Photonics, “6kW CW single mode ytterbium fiber laser in all-fiber format,” in Solid State and Diode Laser Technology Review (Albuquerque, 2008).

3.

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010). [CrossRef] [PubMed]

4.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. Barty, “Ultimate power limits of optical fibers, ” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMO6, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OMO6.

5.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).

6.

J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express 13(4), 1055–1058 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-4-1055. [CrossRef] [PubMed]

7.

C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-10-8476. [CrossRef] [PubMed]

8.

N. Andermahr and C. Fallnich, “Optically induced long-period fiber gratings for guided mode conversion in few-mode fibers,” Opt. Express 18(5), 4411–4416 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-5-4411. [CrossRef] [PubMed]

9.

M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. 3(1), 44–64 (1997). [CrossRef]

10.

J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M. J. F. Digonnet, “Experimental and theoretical analysis of the resonant nonlinearity in Ytterbium-doped fiber,” J. Lightwave Technol. 16(5), 798–806 (1998). [CrossRef]

11.

A. A. Fotiadi, O. L. Antipov and P. Megret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” Frontiers in Guided Wave Optics and Optoelectronics, 209–234 (2010).

12.

L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11(9), 1435–1446 (1993). [CrossRef]

13.

C. Xu and W. Huang, “Finite-difference beam propagation method for guide-wave optics,” Prog. Electromagn. Res. 11, 1–49 (1995) (PIER).

14.

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]

15.

N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express 16(24), 20038–20046 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-24-20038. [CrossRef] [PubMed]

16.

H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992). [CrossRef]

17.

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=ol-16-9-624. [CrossRef] [PubMed]

18.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]

19.

T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B 19(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539. [CrossRef]

20.

F. Wijnands, H. J. W. M. Hoekstra, G. J. M. Krijnen, and R. M. de Ridder, “Modal fields calculation using the finite difference beam propagation method,” J. Lightwave Technol. 12(12), 2066–2072 (1994). [CrossRef]

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.2400) Fiber optics and optical communications : Fiber properties

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: October 14, 2010
Revised Manuscript: December 24, 2010
Manuscript Accepted: January 3, 2011
Published: February 3, 2011

Virtual Issues
February 18, 2011 Spotlight on Optics

Citation
Cesar Jauregui, Tino Eidam, Jens Limpert, and Andreas Tünnermann, "Impact of modal interference on the beam quality of high-power fiber amplifiers," Opt. Express 19, 3258-3271 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3258


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References

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  10. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M. J. F. Digonnet, “Experimental and theoretical analysis of the resonant nonlinearity in Ytterbium-doped fiber,” J. Lightwave Technol. 16(5), 798–806 (1998). [CrossRef]
  11. A. A. Fotiadi, O. L. Antipov and P. Megret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” Frontiers in Guided Wave Optics and Optoelectronics, 209–234 (2010).
  12. L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11(9), 1435–1446 (1993). [CrossRef]
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  14. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef] [PubMed]
  15. N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express 16(24), 20038–20046 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-24-20038 . [CrossRef] [PubMed]
  16. H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992). [CrossRef]
  17. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991), http://www.opticsinfobase.org/abstract.cfm?URI=ol-16-9-624 . [CrossRef] [PubMed]
  18. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]
  19. T. Bhutta, J. I. Mackenzie, D. P. Shepherd, and R. J. Beach, “Spatial dopant profiles for transverse-mode selection in multimode waveguides,” J. Opt. Soc. Am. B 19(7), 1539–1543 (2002), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-19-7-1539 . [CrossRef]
  20. F. Wijnands, H. J. W. M. Hoekstra, G. J. M. Krijnen, and R. M. de Ridder, “Modal fields calculation using the finite difference beam propagation method,” J. Lightwave Technol. 12(12), 2066–2072 (1994). [CrossRef]

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