## Abstract

A model based on propagation-rate equations with consideration of transverse gain distribution is built up to describe the transverse mode competition in strongly pumped multimode fiber lasers and amplifiers. An approximate practical numerical algorithm by multilayer method is presented. Based on the model and the numerical algorithm, the behaviors of multitransverse mode competition are demonstrated and individual transverse modes power distributions of output are simulated numerically for both fiber lasers and amplifiers under various conditions.

© 2007 Optical Society of America

## 1. Introduction

Rare-earth-doped fiber lasers and amplifiers reveal a number of benefits in terms of
outstanding thermo-optical properties, high conversion efficiency, excellent beam
quality and good compactness compared with the traditional solid-state laser
systems. In the recent years, rapid advances in fiber and laser diode technologies
have led to kilowatt CW fiber lasers with output beams of single transverse mode
quality [1,2] and multi-mJ Q-switched fiber lasers with
diffraction-limited beam quality (M^{2} = 1.1) [3].

However, a further power/energy scaling of fiber lasers or amplifiers is mainly limited by the detrimental effects of various nonlinear processes in fibers such as stimulated Brillouin scattering (SBS), stimulated Raman scattering (SRS) and self-phase modulation (SPM). In general, these nonlinear effects mentioned above scale with the fiber length and are inversely proportional to the mode-field-area of the fiber core. As a result, an effective solution to this problem is to adopt large mode area (LMA) fiber with high doping concentration. But increasing the core size of a fiber eventually will allow it to support more spatial modes which will degrade the beam quality of the laser output. Usually, there are two ways to achieve large mode area fiber while still maintaining in the single mode regime. The most common way is to use multimode fiber with the aid of mode selection technique. Several research groups have suppressed the excitation or propagation of high order transverse modes in multimode fibers by suitably designing the fiber index, tailoring the dopant profiles, coiling the fiber, introducing special cavity configurations, tapering the fiber ends, or carefully adjusting the launch conditions of a seed beam [4]. An alternative way is to use photonic crystal fiber (PCF). The most attractive advantage of PCFs is their endlessly single-mode property combined with unlimited large effective areas in principle, but the bending sensitivity and propagation losses will increase with the enlargement of the fiber core [5].

In this paper, we do pay attention to the former way. In multimode fiber lasers or amplifiers, each potential transverse mode is possible to be excited and propagate because transverse spatial hole burning and mode coupling effects. When we concern some special transverse mode (especially fundamental mode), the mode selection techniques mentioned above will be applied. In any case, the transverse mode competition exists inside the fiber and affects the intensity distribution of output. However, few literatures discuss this behavior in the fiber lasers and amplifiers. References [6,7] have analyzed the strongly pumped fiber amplifiers and lasers in detail, where a single mode fiber is assumed and transverse gain is not considered. References [8–10] have investigated transverse mode discrimination in multimode fibers by spatially controlling the radial dopant distribution, where relative gain of transverse modes in fibers was calculated for mode selection, but the model for describing the behaviors of transverse mode competition and the output power distribution of each mode were not discussed. The goal of this paper is to introduce a model for describing the transverse mode competition in strongly pumped fiber lasers and amplifiers with or without the various mode selection techniques. The model is based on a set of rate equations, in which the interaction between multitransverse modes and active dopant in both transverse and longitudinal directions was considered. An approximate practical numerical algorithm with multilayer method is presented, which makes the equations easy to be solved. Based on the model and the numerical algorithm, the behaviors of transverse mode competition are demonstrated and individual transverse mode power distributions of output are simulated for both fiber lasers and amplifiers under various conditions.

## 2. Theoretical Model

When analyzing the transverse mode competition in the multimode fiber lasers, it is necessary to present the gain distribution in the transverse direction of the fiber. Each transverse mode has individual field intensity distribution, which interacts with the population inversion and shares it. In the competition by sharing the population inversion, the transverse mode which obtains enough gains will be excited and propagate.

The model of fiber lasers that has been developed uses the typical linear cavity as described schematically in Fig. 1. The case of amplifiers is similar except to remove two reflectors.

In this model , the following assumptions are made: (1) two-level systems are considered where the excited state absorption (ESA) is neglected; (2) for a strongly pumped multimode fiber, the pump light is assumed to be homogeneous over the cross section of the fiber cladding and the amplified spontaneous emission (ASE) are ignored; (3) the polarization effects and interactions between neighboring ions (e.g. clustering) are ignored; (4) only one single longitudinal mode is assumed in the fiber.

#### 2.1 Rate Equations

When considering the interaction between the active dopant and each potential transverse mode, the population inversion distribution is not constant in the transverse direction any more. Based on the rate equations presented in Ref. [7] and the assumptions above, the model we developed here to describe the transverse mode competition in multimode fiber lasers and amplifiers can be expressed by the following space-dependent and time-independent steady-state rate equations:

$$\phantom{\rule{2.8em}{0ex}}-\sum _{j}{d}_{\mathrm{ij}}\left[{P}_{\mathrm{si}}^{\pm}\left(z\right)-{P}_{\mathrm{sj}}^{\pm}\left(z\right)\right]$$

where *h* is Planck constant; τ is the spontaneous
lifetime of the upper lasing level; *ν _{p}*
and

*ν*are pump and signal frequencies;

_{s}*a*is the radius of the fiber core;

*N*

_{1}(

*r*,

*φ*,

*z*) and

*N*

_{2}(

*r*,

*φ*,

*z*) are the population densities of the lower and upper lasing levels at the position (

*r*,

*φ*,

*z*) (

*N*(

*r*,

*φ*,

*z*) =

*N*

_{1}(

*r*,

*φ*,

*z*)+

*N*

_{2}(

*r*,

*φ*,

*z*) is the doping concentration distribution which is constant along the fiber axis and has a radial symmetry), respectively; ${P}_{p}^{+}\left(z\right)$ and ${P}_{p}^{-}\left(z\right)$ are the pump powers in the forward and backward directions, respectively; ${P}_{si}^{+}\left(z\right)$ and ${P}_{si}^{+}\left(z\right)$ are the signal powers of the

*i*transverse mode in the forward and backward directions, respectively;

^{th}*σ*(

_{ap}*σ*) and

_{ep}*σ*(

_{as}*σ*) are the pump absorption(emission) and signal absorption(emission) cross-sections, respectively; a and asi are the pump and signal of the

_{es}*i*mode loss factors;

^{th}*d*is the power coupling coefficient between the

_{ij}*i*mode and the

_{th}*j*mode; Γ

^{th}*(*

_{p}*r*,

*φ*) and Γ

*(*

_{si}*r*,

*φ*) are the pump and signal of the

*i*mode power filling distributions which can be expressed as follows

_{th}where
*Ψ _{i}*(

*r*,

*φ*) denotes the distribution function of the

*i*mode intensity;

^{th}*A*and

_{core}*A*are the areas of core and inner cladding respectively. Eq. (1) expresses the ground-state and excited-state population densities

_{clad}*N*

_{1}(

*r*,

*φ*,

*z*) and

*N*

_{2}(

*r*,

*φ*,

*z*) that varies with the forward and the backward directions of the signal powers of the

*i*mode ${P}_{si}^{\pm}\left(z\right)$ as well as the pump powers ${P}_{p}^{\pm}\left(z\right)$ at the same location. Equations (2) and (3) describe the pump and signal (each individual transverse mode) power evolution along the fiber length. In our model, the cross-coupling between modes is also considered which will result in mode conversion and also affect the output intensity distribution [11].

^{th}#### 2.2 Multilayered method for numerical simulations

The theoretical model for describing the mode competition in multimode fibers is
given by Eqs. (1)–(5) completely. But commonly, it is difficult to solve these
complicated equations due to the integrations in Eqs. (2) and (3). Here we present an alternative approximate approach. It
is to divide the cross section of the fiber core into a finite number of
distinctive layers (as shown in Fig. 2), each of which is thin enough so that the dopant
concentration and the population inversion can be deemed as a constant. This
division is reasonable for the index and dopant profiles have circular
symmetries in most cases. Obviously, the accuracy of this method depends on the
number of divided layers *M*. In our simulations, we found that
there is a good balance between the accuracy and the computing time when
*M* = 100 . By applying this method, the Eqs. (1)–(5) can be replaced by the following set of simplified
equations:

where ${A}_{k}=\pi \left({r}_{k}^{2}-{r}_{k-1}^{2}\right)$
(*k* = 1,2
⋯,*M*;*r*
_{0} = 0) is the
area of the *k ^{th}* layer; Γ

*and Γ*

_{pk}*are the pump and signal power filling factors in the*

_{sik}*k*layer. In a step index multimode fiber, the Γ

^{th}*can be given by*

_{sik}where *V* and *U* are, respectively, the normalized
frequency and transverse wave numbers. For *N* number of modes
and *M* number of layers, the total number of coupled equations is
*M* + 2*N* + 2. It
should be pointed out that, for *LP _{mn}* mode, those
modes where

*m*≠ 0 must be calculated twice in Eq (8) for having two helical polarities ($L{P}_{mn}^{c}$ and $L{P}_{mn}^{s}$ ).

#### 2.3 Boundary and initial conditions

To solve the coupled differential Eqs. (6)–(8), additional boundary or initial conditions are needed. For fiber lasers, the boundary conditions are given by

where *L* is the fiber Length; *R*
_{1} and
*R*
_{2} are the reflectivities of the reflectors at
*z*=0 and *z*=*L*,
respectively. At the beginning of the simulation, an initial guess for each
transverse mode is given, and Eqs. (6)–(8) are iterated until the output power
${P}_{si}^{out}=\left(1-{R}_{2}\right){P}_{si}^{+}\left(L\right)$
convergences. For fiber amplifiers, usually, both
the pump and signal propagate in only one direction (the corresponding items in
Eqs. (6)–(8) can be omitted). In the case of a copropagating signal,
giving the initial pump and signal power at *z* = 0, the
evolution of the signal powers can be calculated straightforward from the input
end to the output end by using the forth-order Runge-Kutta method. A
counterpropagating signal is a little more complicated. In which case, an
estimate for the output signal power from the pump end to the other end,
refining the estimate in an iterative procedure until the given signal input
power is reproduced by the calculation at this end.

## 3. Numerical simulation results and discussion

Based on the discussion presented above, the theoretical model and the corresponding
numerical algorithm were built up. In this section, we simulated the behaviors of
transverse mode competition in Yb-doped multimode fiber amplifiers and lasers
respectively under various conditions, such as various dopant distributions, pump
powers and discriminative loss factors for different transverse mode. All of the
mode selection techniques mentioned at the beginning of this paper can be included
in these cases. In the following simulations, the traditional step index multimode
fiber was taken into consideration, and other kinds of fibers can be treated
similarly. In addition, we considered the mode competition in a perfectly flawless
multimode fiber without any other external disturbance, so the power coupling
coefficient *d _{ij}* in Eq. (8) is neglected. The parameters employed in the numerical
simulations are listed in Tab. 1.

First, by using this model, we simulated the power propagation of each mode in fiber
amplifiers with various dopant distributions which was demonstrated in Ref. [9]. The simulated results are shown in Fig. 3, where Γ is the doping confinement factor
defined as the ratio of the maximum gain radius to the core radius and ρ
denotes the normalized radius. In order to describe the competition of every
potential mode during propagation, 50W pump power and 100mW initial copropagated
signal power for each mode are given. The fiber lengths have been calculated until
the amplified signal power is close to the maximum possible in each case. For flat
doping fibers (Fig. 3(a)-(c)), when Γ = 1(in most common cases), the first
high order mode LP_{11} has the maximum output. The output power of
LP_{01}, LP_{11}, LP_{21} and LP_{02} are 8.9W,
16.2W, 12.4W and 4.4W respectively. The percentage of the fundamental mode
(LP_{01}) output in total rises from 21.3% (Γ = 1) to 54.5%
(Γ = 0.75) and 78.5% (Γ = 0.5). From Fig. 3(c), we can see that the output of LP_{02} has
exceeded LP_{11} and LP_{21} when Γ = 0.5. All of these
indicate that small Γ favors the modes which have more power concentrated
in the center of the core. However, Γ could not be too small for ensuring
more gain to the LP_{01} than LP_{02}. Figure 3(d) shows the parabolic doping fibers can also favor
the LP_{01} which raise the percentage of the LP_{01} to 48.5%. The
dependence of the power faction of LP_{01} mode on doping confinement factor
in the flat doping fibers is shown in Fig. 4. We calculated four modes (LP_{01},
LP_{11} LP_{21} and LP_{02}) in the fibers with different
core diameters (28, 50 and 100*μ*m ) at 100W pump power.
It signifies that the optimized doping confinement factor is around
0.5∼0.6. It also indicates that it is possible to achieve single
transverse mode output even when the core diameter is up to
100*μ*m by further optimizing the doping profile and
highest doping concentration.

Figure 5 illustrates the power propagation of each guided
mode in fiber amplifiers with discriminative loss factors which are realized by
coiling the fiber. The bending-loss factors in Fig. 5 are derived from the analysis of Marcuse in Ref. [13]. It can be seen that when bending radius is 7.5
*cm*, there are two modes (LP_{01} and LP_{11})
propagating in the fiber with a certain length. The output power of LP_{11}
decreases slowly with the increase of the fiber length and the maximum
LP_{01} output power is 17W. And when bending radius is reduced to 6
*cm*, only LP_{01} mode remained in the fiber for the
strong suppression of the high order modes. In this case, the output power of
LP_{01} is 37W which is much more than the former, because the power of
LP_{11} drops quickly and LP_{01} monopolizes the gain in the
fiber.

Figure 6 illustrates the power distribution of each mode
varied with the pump powers in flat-doping fiber lasers with Γ = 1. The
fiber with 10m length was pumped equally at double ends. The right sub-figure in Fig. 6 is the details of the left one at low pump power. It
can be seen that not all of potential modes could be excited at one time in the
fiber lasers. When the pump power is low, only LP_{01} mode is
preferentially excited due to its lower threshold. The high order modes
LP_{11} and LP_{21} are excited successively with the increase of
pump power. Then, the outputs of all excited modes increase linearly above their
respective threshold pump powers. The LP_{11} has maximum output and the
percentages of these excited modes are 30% (for LP_{01}), 45% (for
LP_{11}) and 25% (for LP_{21}) respectively at high pump power.

Figure 7 shows the transverse gain distributions
(*N*
_{2}/*N*) at the output end of the
fiber with different pump powers. It indicates that the gain varies along the radial
direction because of transverse mode competition in the fibers. When the pump is low
and only LP_{01} is excited, the gain is lowest in the center and increases
with the radial position which is opposite to the intensity distribution of
LP_{01} mode, arising from the effect of gain saturation. With the
increase of pump power, the gain curve drops and the valley point on it shifts away
from the center for the high order modes (LP_{11} and LP_{21})
account for more and more fraction of the output.

We have also simulated the output power distributions in the fiber lasers with
various dopant profiles and discriminative loss factors which are shown in Fig. 8. It can be seen that, as similar as the case of fiber
amplifiers, decreasing the doping confinement factor Γ or parabolic
doping or coiling the fibers with appropriate bending radius can favor the
LP_{01} mode and even achieve single mode output in this fiber with low
*NA*. Differently, the mode competition in the fiber lasers is
more serious because of the laser cavity configurations.

The simulations above present the transverse mode competition in the multimode fibers
without applying the cross-couple between the modes, because it is hard to determine
the mode coupling matrix *d _{ij}*, although we have
considered this effect in our model. The mode coupling in multimode fibers is a very
complex problem which is caused by many uncertain factors, mainly including the
internal flaw generated during the fabrication procedure and other external
disturbance such as bending, twisting, and etc. In order to exhibit our model has
the ability to process this case, here, we give a simple example to reveal the mode
coupling effect between the fundamental mode LP

_{01}and the closest higher order mode LP

_{11}. Figure 9 shows the power fraction of LP

_{01}mode varies the fiber length in a fiber laser with different mode coupling coefficients between these two modes. The fiber core diameter is 18

*μ*m and inner cladding diameter is 200

*μ*m and other simulation parameters are as same as those in Tab. 1. The mode coupling coefficient From Fig. 9, it can be seen that the mode coupling has important effect on the output intensity distribution, especially in a long fiber. When the fiber length is in the range of several meters, the LP

_{01}mode fraction drops about 20 percent for d=0.01/m (the value is from Ref. 11) and about 5 percent for d=0.001/m. So if more accurate output intensity distribution is needed, further study on mode coupling effect should be taken into account.

## 4. Conclusion

In this paper, a model for describing the transverse mode competition in strongly pumped fiber lasers and amplifiers with or without the mode selection techniques was built up. The model is based on rate equations, in which the interaction between multitransverse modes and active dopant in both transverse and longitudinal directions was considered. In the model, the mode cross-coupling effect was also considered. A practical numerical algorithm with multilayer method was also presented which makes the equations easy to be solved.

Based on the model and the numerical algorithm, the behaviors of multitransverse mode competition were demonstrated and individual transverse modes power distributions of output were simulated numerically in both fiber lasers and amplifiers with various dopant distributions, pump powers and loss factors. The cross-coupling effect between only two modes was also discussed. In the simulations, the traditional step index multimode fiber was taken into consideration, but this model is also applicable for other kinds of multimode fibers (e.g. PCFs). This model could be helpful for designing large mode area fibers, especially for designing such fibers by means of optimizing index and dopant profiles of the fiber, and also could be useful to evaluate the beam quality of multimode fiber lasers.

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