Impact of modal interference on the beam quality of high-power fiber amplifiers

doi:10.1364/OE.19.003258

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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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▸ Article Outline
– Abstract
1. Introduction
2. Simulation model
3. Simulation results: Impact of modal interference on the beam quality of high-power fiber lasers
4. Conclusions
– References and links

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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.

1. Introduction

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

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

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

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

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

The conventional way to circumvent the problems related with non-linear fibers is to use Large-Mode-Area fibers, such as the Rod-Type fiber [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]

]. These fibers offer an enlarged fundamental mode diameter that allows mitigating the detrimental impact of non-linear effects, raising for example the Raman threshold typically to the tens to hundreds of kilowatts range [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]

]. The problem with LMA fibers is that they are inherently multimode. The implications of this fact go beyond a mere technical complication to maintain single mode operation. As some works over the last year report, an unexpected modal phenomenon has been observed: at high powers there is a dramatic loss in beam quality which in some cases comes accompanied by mode instabilities (i.e. rapid fluctuations of the output beam shape between two fiber modes) [3

]. As it happens, this effect is currently one of the most limiting factor in average power scaling of high-power laser systems (far more limiting than non-linear effects), and therefore it is urgent to find a solution for it. Additionally, this phenomenon, unlike non-linear effects, has been observed both in pulsed and CW fiber laser systems.

This effect has been commonly linked to Transversal Hole Burning (THB) but, to the best of our knowledge, no further explanation of its origin has been provided. In this paper we provide an explanation of the physical origin of this effect. In this explanation modal interference between the fundamental mode and a higher order mode (HOM) plays a key role. The interplay of THB and modal interference creates an inversion profile along the active fiber which exhibits periodic areas of high non-depleted inversion (which is very different to the smooth inversion profile obtained by THB alone). This, as shown in this paper, gives rise to an inversion grating with exactly the spatial periodicity required to transfer energy from the fundamental mode to the HOM. Note that this inversion-induced grating has similarities with the optically induced long-period grating reported in [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]

], but it should not be mistaken with it. The grating object of this paper is created by inversion-related index of refraction changes originated by a combination of effects such as the resonantly induced index change of doped fibers [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

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 temperature gradients in the fiber core [12

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

]. Therefore, this index change is not of an instantaneous nature as that reported in [8

], but it has a certain time persistence (of the order of the lifetime of the excited state in the doped fiber).

In order to model the causes and effect of this inversion grating, two things are required: a 3D Beam Propagation Model (BPM) [13

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

], and the transversally resolved rate equations of the active fiber [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]

]. Thus, in this work a model is proposed which is the first successful combination (as far as we know) of a 3D-BPM and the transversally resolved rate equations (referred to TR-rate equations in the following). It is shown that this simulation tool can be used to understand the physical origins of the inversion grating, because it offers an accurate solution of the power growth along the amplifier fiber (including pump depletion effects) together with the inclusion of modal interference and the interaction of the beam with the transversal inversion profile. In contrast, normal BPM models do not take THB into account, and conventional TR-rate equations do not include the effect of mode interference. In [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]

], Andermahr et al. have presented a BPM model that can potentially study the interplay between THB and modal interference. However, since this model considers a constant small-signal gain along the fiber, it does not accurately estimate the real amount of THB in high-power fiber amplifiers and it is, therefore, unable to predict any significant energy transfer between the modes due to the inversion grating.

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

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:

\nabla^{2} {\vec{E}}_{s} + n {(x, y, z)}^{2} k^{2} {\vec{E}}_{s} = \nabla (\nabla \cdot {\vec{E}}_{s})

(1)

where

{\vec{E}}_{s}

stands 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

{\vec{E}}_{s, t}

\nabla^{2} {\vec{E}}_{s, t} + n^{2} k^{2} {\vec{E}}_{s, t} = \nabla_{t} [\nabla_{t} \cdot {\vec{E}}_{s, t} - \frac{1}{n^{2}} \nabla_{t} \cdot (n^{2} {\vec{E}}_{s, 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

{\vec{A}}_{s, t}

and a fast oscillating phase term:

{\vec{E}}_{s, t} (x, y, z) = {\vec{A}}_{s, t} (x, y, z) e^{- j n_{o} k z}

(3)

where n_o 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:

\frac{\partial}{\partial z} (j 2 n_{o} k - \frac{\partial}{\partial z}) {\vec{A}}_{s, t} (x, y, z) = P {\vec{A}}_{s, t} (x, y, z)

(4)

where the operator P is given by:

P {\vec{A}}_{s, t} = \nabla_{t}^{2} {\vec{A}}_{s, t} (x, y, z) + (n^{2} - n_{o}^{2}) k^{2} {\vec{A}}_{s, t} (x, y, z) - \nabla_{t} [\nabla_{t} \cdot {\vec{A}}_{s, t} (x, y, z) - \frac{1}{n^{2}} \nabla_{t} \cdot (n^{2} {\vec{A}}_{s, t} (x, y, z))]

(5)

By considering that the energy in optical fibers remains always quite close to the propagation axis, Eq. (5) can be reduced to its paraxial form by applying the Padé approximation:

j 2 n_{o} k \frac{\partial {\vec{A}}_{s, t} (x, y, z)}{\partial z} = P {\vec{A}}_{s, t} (x, y, z)

(6)

which can be approximated by a weighted finite-differences form:

{\vec{A}}_{s, t} (x, y, z + Δ z) = \frac{2 n_{o} k - j Δ z (1 - α) P}{2 n_{o} k + j Δ z α P} {\vec{A}}_{s, t} (x, y, z)

(7)

where Δz is the longitudinal step size and α is a weighting factor that controls the finite difference scheme. Thus, α=0 corresponds to an explicit scheme, whereas α=1 is implicit. Additionally, α=0.5 represents the well-known Crank-Nicholson scheme. As can be appreciated, Eq. (7) relates the electric field at one longitudinal step with the electric field at the previous step, i.e. is a beam propagation equation.

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 {[{\vec{A}}_{s, t}]}^{l + 1} = B {[{\vec{A}}_{s, t}]}^{l}

(8)

where

{[{\vec{A}}_{s, t}]}^{l + 1}

and

{[{\vec{A}}_{s, 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

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

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

]. 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:

\begin{array}{l} \frac{N_{2} (x, y, z)}{N_{1} (x, y, z)} = \frac{\frac{[P_{p}^{+} (z) + P_{p}^{-} (z)] σ_{a p} Γ_{p} (x, y)}{h υ_{p}} + \frac{P_{s}^{+} (z) σ_{a s} Γ_{s} (x, y)}{h υ_{s}}}{\frac{[P_{p}^{+} (z) + P_{p}^{-} (z)] σ_{e p} Γ_{p} (x, y)}{h υ_{p}} + \frac{1}{τ} + \frac{P_{s}^{+} (z) σ_{e s} Γ_{s} (x, y)}{h υ_{s}}} \\ \frac{d P_{p}^{\pm} (z)}{d z} = {\int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} [σ_{e p} N_{2} (x, y, z) - σ_{a p} N_{1} (x, y, z)] Γ_{p} (x, y) d x d y} P_{p}^{\pm} (z) - α_{p} P_{p}^{\pm} (z) \\ \frac{d P_{s}^{+} (z)}{d z} = {\int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} [σ_{e s} N_{2} (x, y, z) - σ_{a s} N_{1} (x, y, z)] Γ_{s} (x, y) d x d y} P_{s}^{+} (z) - α_{s} P_{s}^{+} (z) \end{array}

(9)

where

N_{1} (x, y, z)

and

N_{2} (x, y, z)

are the population densities of the lower and upper lasing levels at the position (x,y,z). Additionally, P_p(z) and P_s(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, σ_ap and σ_as are the absorption cross-sections at the pump and signal wavelengths, respectively. Similarly, σ_ep and σ_es 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, υ_p and υ_s are the pump and signal frequencies, respectively. In addition, α_p and α_s 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 , y ₁ , y ₂ are chosen to sweep the complete core area. Finally, Γ_p(x,y) and Γ_s(x,y) are the power filling distributions of pump and signal, which can be expressed as follows:

Γ_{p} (x, y) = \frac{1}{A_{c l a d}} and Γ_{s} (x, y) = \frac{ψ (x, y)}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ (x, y) d x d y}

(10)

where A_clad 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

]. This is because the rate equations have been modified to make them compatible with the BPM. As a consequence, for example, the signal power P_s 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

]). 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

]. 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:

\begin{array}{l} \frac{N_{2 (m, k)} (z)}{N_{1 (m, k)} (z)} = \frac{\frac{[P_{p}^{+} (z) + P_{p}^{-} (z)] σ_{a p} Γ_{p (m, k)}}{h υ_{p} A_{(m, k)}} + \frac{P_{s}^{+} (z) σ_{a s} Γ_{s (m, k)}}{h υ_{s} A_{(m, k)}}}{\frac{[P_{p}^{+} (z) + P_{p}^{-} (z)] σ_{e p} Γ_{p (m, k)}}{h υ_{p} A_{(m, k)}} + \frac{1}{τ} + \frac{P_{s}^{+} (z) σ_{e s} Γ_{s (m, k)}}{h υ_{s} A_{(m, k)}}} \\ \frac{d P_{p}^{\pm} (z)}{d z} = \sum_{m} \sum_{k} [σ_{e p} N_{2 (m, k)} (z) - σ_{a p} N_{1 (m, k)} (z)] Γ_{p (m, k)} P_{p}^{\pm} (z) - α_{p} P_{p}^{\pm} (z) \\ \frac{d P_{s}^{+} (z)}{d z} = \sum_{m} \sum_{k} [σ_{e s} N_{2 (m, k)} (z) - σ_{a s} N_{1 (m, k)} (z)] Γ_{s (m, k)} P_{s}^{+} (z) - α_{s} P_{s}^{+} (z) \end{array}

(11)

with:

\begin{array}{l} A_{(m, k)} = Δ x \cdot Δ y \\ Γ_{p (m, k)} = \frac{A_{(m, k)}}{A_{c l a d}} \\ Γ_{s (m, k)} = \frac{ψ (m Δ x, k Δ y)}{\sum_{m} \sum_{k} ψ (m Δ x, k Δ y)} \\ N_{(m, k)} = N_{1 (m, k)} + N_{2 (m, k)} \end{array}

(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

Now that the two main building blocks of the model have been briefly summarized, it is possible to describe the way in which they have been combined to create the new active BPM model. For that, the fact that the Runge Kutta (RK) methods are an iterative solution of the differential equations is used. Thus, the solution of each RK iteration is computed for a different point z along the fiber. Exploiting this, it is possible to use the BPM to propagate the beam through the fiber section that goes from the z-position corresponding to the last iteration to the z-position corresponding to the new one. However, prior to doing it, a new complex index profile n(x,y) has to be calculated for this fiber section. This index profile has to take into account both the fiber gain (in its imaginary part) and the inversion-related index change (that occurs as a result of a combination of effects such as the resonantly induced index change [9

–11

] and temperature gradients in the fiber core [12

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

] arising through a transversally inhomogeneous inversion distribution) which will give rise to the grating (in its real part). In the following it will be assumed that this index change Δn is 2·10⁻⁵ between a fully inverted fiber section (i.e. N₂ =N) and a fiber region with a completely depleted inversion (i.e. N₂ ~0). Furthermore, it is assumed that this index change is linearly dependent on the local inversion level. Thus, the complex index of refraction can be obtained from the transversal inversion distribution at the beginning of the fiber section as:

n_{(m, k)} = n (m Δ x, k Δ y) = Δ n \cdot \frac{N_{2 (m, k)}}{N_{(m, k)}} + j \frac{λ}{4 π} (σ_{e s} N_{2 (m, k)} - σ_{a s} N_{1 (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

Γ_{p (m, k)}

and

Γ_{s (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.

As a summary, in this model the simulation proceeds as follows:

1. Define an input beam (i.e. transversal electric field distribution)
2. Propagate the beam using the BPM algorithm until the z-point corresponding to the next iteration of the RK method is reached.
- Calculate the new power filling factors $Γ_{p (m, k)}$ and $Γ_{s (m, k)}$ for each point in the fiber core.
3. Use these power filling factors to solve the system of Eqs. (11).
4. Determine the new z-point for the next iteration of the RK method and repeat from 2.

This loop is repeated until the end of the fiber is reached.

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

In the following the application of the model to study the problem of beam quality degradation in high-power fiber amplifiers is described [3

]. To the best of our knowledge, even though this degradation has been vaguely attributed to THB, an explanation of its physical origin has not been published so far. Please note that, in any case, an in-depth investigation of this effect is outside of the scope of this paper. What it is intended here is to provide a first explanation and description of what we believe is a likely cause of this phenomenon. However, since this beam degradation typically manifests itself as a modal instability, the exact simulation of this effect is beyond the capabilities of the CW model presented above. This also means that there are additionally (still unidentified) processes at work apart from the one described herein. Nevertheless, what is presented and simulated in this paper is what we suspect to be an underlying mechanism triggering these modal instabilities.

In the following, the study is carried out for a straight low NA very large core step-index fiber. The reasons are two-fold, first of all because this mode instability has been reported to happen in this type of fibers (or better said, in index-guided photonic crystal fibers as in [3

], which in a first approximated can be simulated as step-index fibers) and, secondly, because these fibers are numerically less demanding to simulate than small-core fibers. This is due to the fact that the mode spacing (in effective index) is very small in these very large mode area fibers, which gives rise to mode beating in the cm scale (as opposed to the sub-mm scale of small core fibers), which in extreme cases can even be seen with the naked eye. An example of this is shown in Fig. 1 , where different intensities in up-conversion emission can be observed along a ~15cm long 100μm core diameter Rod-type fiber [6

]. Since the cooperative up-conversion emission is related to the local inversion, Fig. 1 points out to alternating fiber sections with high and low inversions.

Fig. 1 Picture of the cooperative up-conversion as seen in a ~15cm long piece of 100μm core diameter Rod-type fiber (the photo has been processed to enhance the interference contrast).

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

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

]. 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

,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 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

–11

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

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).

At this point the question about the stability of this inversion-induced grating must be discussed further. This is because the grating has been created by a standing wave resulting from modal interference, and this is known to be sensitive to external perturbations. However, there are two factors that speak in favor of the stability of this interference. On the one hand, the grating only needs to be stable (at most) in the timescale of the lifetime of the excited ions (which for Yb-doped fibers is ~0.8 ms), and typically, to get energy transfer, it is enough that it is stable during a few transit times of the light in the fiber (which usually represents a much shorter time than the lifetime of the excited ions). This fact drastically reduces the sensitivity of this grating to the most typical perturbations (i.e. mechanic, thermal). Furthermore, high-power fibers have a relatively large external diameter to simultaneously facilitate the coupling of high-power low-brightness pump radiation and the heat extraction. This confers these fibers an inherent stiffness that make the modal interference even less sensitive to external (mechanic) perturbations. An extreme example of such stiff fibers is the Rod-type fiber [6

], which cannot be bent, and in which we have experimentally observed this modal interference (see Fig. 1).

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.

In the first simulation a 2 m long fiber with 280 μm pump core diameter is studied (in the following it will be referred to as the 80/280 fiber). At first, no inversion grating is considered. Thus, Fig. 3(a) shows the power evolution along this fiber when pumped with 300 W in the co-propagating direction. As can be seen, this particular configuration of fiber parameters provides an efficient amplification of the 30 W seed signal. Figure 3(a) is shown as a typical example of the power evolution profile obtained in the simulations (with or without inversion grating). Therefore, the power evolution profile of the following simulations will not be shown, for it looks alike that presented in Fig. 3(a). More interesting are the results presented in Fig. 3(b). There the evolution of the modal content of the beam along the fiber can be observed. In order to get the modal decomposition of the beam, the fiber modes where calculated (using an imaginary distance BPM algorithm [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]

]), and then, exploiting the orthogonality of the fiber modes, the relative content c_i of the i^th fiber mode (with electric field

{\vec{ϕ}}_{i} (x, y)

) in the beam (with electric field amplitude

{\vec{A}}_{s, t} (x, y, z)

) at position z was calculated using the following complex overlap integral:

Fig. 3 (a) Power evolution in a 2 m long 80/280 fiber seeded with 30 W signal power at 1064 nm and co-propagating pumped with 300 W. (b) Corresponding evolution of the relative modal content when no inversion-grating is considered.

c_{i} (z) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\hat{A}}_{s, t} (x, y, z) \cdot {\hat{ϕ}}^{*}_{i} (x, y) d x d y

(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).

Things, however, look different when taking into account the inversion-induced grating arising from the inversion profile. This profile is presented in Fig. 4 for the 2 m long 80/280 fiber. There the inversion-induced grating is clearly seen in the first half of the fiber. In the second half of the fiber, however, THB is so strong that it starts to “erase” the grating by progressively depleting more and more the inversion at the edges of the doped region. In spite of this, the strong periodic inversion changes (observed as vertical lines in the right plot of Fig. 4) in the first half of the fiber, should account for some amount of energy transfer from the fundamental mode towards the higher order mode. As can be seen in Fig. 5 , there is indeed a progressively higher relative content of the LP₁₁ mode in the beam as it propagates through the fiber. This increase of the LP₁₁ modal content is a direct effect of the energy transfer propitiated by the inversion-grating. The growth of the LP₁₁ mode is particularly relevant in this example, because, as discussed before, the fiber used in here has a preferential gain for the LP₀₁. Thus, all the models not considering the effect of mode interference will predict the faster growth of the fundamental mode and, therefore, the progressive reduction of the relative modal content of the LP₁₁ as the beam propagates through the fiber. In this case, on the contrary, even though the inversion grating is in direct competition with the preferential gain, it is clear that it is able to overpower it.

Fig. 4 Inversion map (in the x-z plane) along the 2 m long 80/280 fiber (only core region shown) (right), and corresponding inversion profiles near the center (left up) and near the edge of the core (left down).

Fig. 5 Evolution of the modal content along the 2 m long 80/280 fiber.

As mentioned before, the inversion-related index change has been assumed in this simulation to be 2 ^. 10⁻⁵ between a core area with full inversion (i.e. N₂ /N=1) and a core area with a completely depleted inversion (i.e. N₂ /N~0). This value is in the upper limit of what we consider that can be realistically achieved in a fiber core. However, we have chosen it for illustration purposes so that its effect becomes apparent in the VLMA fibers simulated herein (chosen because they considerably reduce the computation time, as discussed before). In any case, please note that real fibers will tend to have a NA larger than that used in this paper. This will result in much shorter mode-beating periods which, in turn, implies that lower inversion-related index-changes will be required to obtain similar grating efficiencies to those presented in these simulations. This is because the grating efficiency, i.e. the mode coupling efficiency, is a function of the number of periods and the index change. Thus, a lower index change can be compensated by a higher number of grating periods.

It is also worth noticing that there is, additionally, a growth of the radiation modes (in this case mainly represented by the LP₀₂ mode, which even though not strictly guided in this fiber, it is close to its cut-off), which would be translated in extra losses in the practical case. This happens because there is not a perfect mode matching at the fiber input and some of the energy (~0.5%) is coupled to the radiation modes which, even though lossy, are able to also propagate and interfere with the beam along this short fiber length. Thus, an additional (weak) grating is also created, which favors the energy transfer to the radiation modes. The reason why there is an initial amount of energy coupled to the radiation modes has to do with the fact that the modes of the passive structure have been the ones used for the excitation. However, the active structure has a slightly different transversal index profile (due to the refractive index change generated by the inversion). This means that the modes of the passive and active structure are not identical and, therefore, there is some energy lost to the radiation modes due to this mode mismatch (this is also the reason why there are ripples in the traces of the relative modal content plot).

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.

The latter can be demonstrated by simulating a 2 m long fiber with the same signal and pump characteristics as before; the only difference is that now the pump core is 400 μm in diameter (in the following referred to as 80/400 fiber). This makes the amplifier a little less efficient than before (being the output power now ~270 W instead of the ~295 W from the previous simulation) but it also, critically, reduces significantly the amount of THB, as seen in Fig. 6 . Consequently the inversion grating is stronger than before all over the fiber length (see left plots). Therefore, as expected from the discussion above, the energy transfer between the LP₀₁ and the LP₁₁ modes should increase. This is clearly confirmed by Fig. 7 , where it can be seen that the relative modal content of the LP₁₁ mode at the output of the fiber is ~50%. Additionally, it can also be seen that the coupling to radiation modes also becomes stronger (but being this also an effect of the inversion-grating it comes as no surprise).

Fig. 6 Inversion map (in the x-z plane) along the 2 m long 80/400 fiber (only core region shown) (right), and corresponding inversion profiles near the center (left up) and near the edge of the core (left down).

Fig. 7 Evolution of the modal content along the 2m long 80/400 fiber.

A final simulation was carried out to verify that the number of periods of the inversion induced grating is a parameter that strongly influences the final amount of energy coupling between modes. In this case a 1 m long fiber was simulated. In order to keep the results comparable, the pump core dimensions were reduced to keep the same amount of pump absorption as in the first simulation. Thus, the pump core diameter was chosen to be 200μm (this will be referred to as the 80/200 fiber). As in the previous cases the seed signal power was 30 W at 1064 nm, and the pump power was 300 W at 976 nm. As can be observed in Fig. 8 , the inversion-induced grating is quite similar to that shown in Fig. 4 for the first half of the 2m long 80/280 fiber. However, here the number of grating periods is about half of those shown in Fig. 4. This leads to the result shown in Fig. 9 , in which it can clearly be seen that the energy transfer has been strongly reduced. Thus, the LP₁₁ relative modal content amounts to less than 10% at the output of the fiber. In fact, by closely looking at Fig. 9, it can also be seen how in the first half of the fiber the modal content of the LP₁₁ slightly decreases. This is the effect of the preferential gain. Thus, Fig. 9 shows the competition between preferential gain and inversion-induced grating at work.

Fig. 8 Inversion map (in the x-z plane) along the 1 m long 80/200 fiber (only core region shown) (right), and corresponding inversion profiles near the center (left up) and near the edge of the core (left down).

Fig. 9 Evolution of the modal content along the 1m long 80/200 fiber.

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

In this paper a new model for the numerical simulation of high power fiber amplifiers has been presented. To the best of out knowledge this model is the first successful combination of a full 3D BPM with the transversally-resolved rate equation. Thus, this simulation tool takes into account the effects of transversal hole burning, pump depletion, modal interference and inversion-induced local refractive index changes.

Using this model, it has been shown that the interplay between modal interference and transversal hole burning leaves behind a periodic structure of non-depleted inversion that gives rise to a grating. This grating, having been created by mode beating, has exactly the right period to transfer energy between the two interfering beams. This way, our simulations show that via this grating, it is possible that the relative modal content of the LP₁₁ grows as the beam propagates through the fiber, giving thus rise to a reduced beam quality at the output of the amplifier. The influence of this inversion grating can be so strong that it can even overpower the effect of a preferential gain design. This effect is, however, dependent on the signal wavelength, on the saturation characteristics of the amplifier and on the fiber length. Thus, our simulations indicate that a possible way to minimize the impact of this inversion grating is to use short very large mode area fibers and highly saturated amplifiers.

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

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010). [CrossRef] [PubMed]
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).
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]
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 .
G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).
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]
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]
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]
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]
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]
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).
L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11(9), 1435–1446 (1993). [CrossRef]
C. Xu and W. Huang, “Finite-difference beam propagation method for guide-wave optics,” Prog. Electromagn. Res. 11, 1–49 (1995) (PIER).
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]
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]
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]
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]
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]
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]
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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