1. Introduction

Active fibers have proven wide applicability in high-power laser sources in order to offer a good beam quality. Continuous wave fiber lasers and amplifiers with a single transverse mode and a single polarization have been demonstrated with output powers of several 100W [1, 2].

Nevertheless, limitations of power scalability arise from the high power density within the core, which can cause undesirable nonlinear effects. An increase of the core size in large-mode-area (LMA) fibers can substantially reduce the power density, but in general also allows the propagation of higher-order modes, which reduce the beam quality and the polarization extinction ratio [3]. Several fiber designs that offer a large mode area on the one hand and suppress higher-order modes on the other hand are current subjects of research: Some promising approaches are chirally-coupled core structures [4], leakage channel fibers [5], LMA photonic crystal fibers [6] and gain-guided fibers [7]. However, also with these designs the maximum core size that ensures pure single mode propagation is limited. Therefore, active fibers of highest output powers are operated in the transition regime between single-mode and multi-mode propagation. In this so-called few-mode regime the examination of transverse mode behavior is of importance to understand how a few higher-order modes influence the beam quality and the polarization characteristics. An adequate modeling is needed to predict how modal perturbations are amplified and how they can be suppressed. Moreover, the use of higher-order modes for fiber lasers and amplifiers has been proposed [8, 9], wherefore an appropriate modeling of mode interaction is also essential.

The mode content can be influenced by passive and active coupling processes: We denote mode changes that result from local variations of the refractive index [10, 11] as passive mode coupling. For example, mechanical stress caused by bending of the fiber locally changes the refractive index and the birefringence, which affects the modal power distribution and the polarization. Bending the fiber can excite guided higher-order modes, but it can also be used to suppress unintended higher-order modes by coupling them into unguided (leaky) modes. Additionally, local gain variations can cause different amplifications of individual modes, what we will call active mode coupling. Also active mode coupling influences the modal power distribution and the polarization.

In this paper we concentrate ourselves on active mode coupling, which we model by solving the Fresnel wave equation. Our approach is based on the beam propagation method (BPM) of [12] with the addition of a saturable gain factor. In recent experiments we observed that transverse modes experience a polarization dependent amplification in a LMA fiber amplifier [13]. We showed that different transverse modes prefer to propagate within different polarization states at high gains. The main aim of this paper is to verify these observations with numerical simulations and to offer a better understanding of the underlying physical effects by visualizing the field distribution inside the fiber.

While earlier approaches [14, 15, 16], which modeled effects of transverse gain variation in LMA fibers, relied on a set of rate equations for each mode, the BPM calculates the electric field distribution and, therefore, is less restrictive concerning spatial amplification. In Section 2 we describe our numerical approach and discuss advantages and disadvantages compared to using the rate equations. The numerical results of a two dimensional (2D) and a three dimensional (3D) simulation are presented and compared with experimental results in Section 3.

2. Modeling the electric field propagation in an active fiber

2.1. Fresnel wave equation with saturable gain

The propagation of the electric field E⃗ can be described by the Helmholtz equation

where the spatial coordinates are r⃗=(x, z) in the 2D-simulation and r⃗=(r,ϕ, z) in the 3D-simulation. The symmetry axis of the fiber equals the propagation direction and is chosen as z-axis. The wave number k(r⃗) depends on the local refractive index n(r⃗) and α(r⃗), the gain (α>0) or loss (α<0) coefficient, respectively.

where $k_0=\frac{2\pi }{\lambda }$ is the vacuum wave number and λ the vacuum wavelength. By taking local saturation into account the gain coefficient α becomes a function of the local intensity I(r⃗), the small-signal-gain α _ss(r⃗) and the saturation intensity I _sat:

The Fresnel approximation [12] is used to get rid of the fast oscillations in z-direction. By assuming $\overrightarrow{E}\left(\overrightarrow{r}\right)=\overrightarrow{A}\left(\overrightarrow{r}\right)e^{-i{k}_{\mathrm{av}}·z}$, where k_av is the average wave number, and by using the slowly-varying-envelope-approximation $\frac{{\partial }^2\overrightarrow{A}}{\partial z^2}=0$ Eq. (1) can be written as:

where ${\overrightarrow{\nabla }}_t^2=\frac{{\partial }^2}{\partial x^2}$ in two dimensions and ${\overrightarrow{\nabla }}_t^2=\frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{{\partial }^2}{\partial {\phi }^2}$ in three dimensions.

2.2. Modal power content and modal polarization state

After the calculation of the electric field A⃗(r⃗) the local intensity I(r⃗)=(cε₀n/2)·|A(r⃗)|² can be derived, where c is the velocity of light, ε₀ the vacuum permittivity and n the refractive index. P(z) is the power in the slice at position z and is calculated by P(z)=∫I(r⃗t)dr⃗_t, where r⃗t=x in two dimensions and r⃗_t=(r,ϕ) in three dimensions.

As there is only a discrete number of transverse modes, which are able to propagate within the fiber, it is helpful to decompose the electric field A⃗(r⃗) into the corresponding basis of transverse modes. The complex overlap vector c⃗_i(z) represents the content within the ith mode E_i(r⃗_t) at the position z:

Then the modal power is |c⃗i|². A⃗(r⃗) and c⃗_i(z) are considered to have two orthogonal polarization components, which are each orthogonal to z, and that we denote p- and s- polarization:

All information about the modal polarization state is contained in the normalized overlap vector c⃗i/|c⃗_i|². Therewith the modal polarization state can be illustrated e.g. on the Poincaré sphere. In addition, it can also be helpful to subdivide the modal power into the polarization directions, then |c_i,p|² and |c_i,s|² denote the power in the ith mode in p- and s-polarization, respectively.

2.3. Comparison to the rate equation approach

Earlier approaches modeling modal gain in active fibers extended the classic rate equations by defining photon numbers for each mode [14, 15, 16]. As typically also the photon numbers of the pump light are included, these approaches can adequately simulate pump absorption, reabsorption of the signal and spontaneous emission. Modal gain is included by defining gain coefficients α_i for each mode:

where I_i is the intensity distribution of the ith mode. The number of variables is limited by the number of modes, which simplifies the numerical solution. However, as only the modal intensity distributions are used, effects of modal interference and thus the influence of modal polarizations are not adequately modeled.

Applying the beam propagation method is less restrictive concerning transverse effects, as the amplification of the complete vectorial electric field is calculated. Thereby, modal interference and modal polarization are naturally included. On the other hand, this method does not include pump effects and spontaneous emission so far. In our simulations we consider a constant small-signal-gain along the z-direction, what is an acceptable approximation for the amplifier of [13] and comparable setups. Moreover, we neglect amplified spontaneous emission (ASE), as the power of ASE was much smaller than the signal power in [13] and therefore had a negligible influence on the mode coupling. For experiments with a significant amount of ASE its prediction stays an issue and will be subject of further simulations.

2.4. Algorithm and parameters

The propagation of the electric field is calculated by solving Eq. (4) with a Runge-Kutta method of fourth order. At each z-step the intensity I(r⃗_t) and thereby the local gain coefficient α(r⃗_t) are calculated. The parameters were chosen with regard to the LMA amplifier of [13] and are listed in table 1. The fiber had a step index profile and the refractive indices of the core and the cladding correspond to a numerical aperture of NA=0.06. The amplifier was seeded with an incident power P _in=100mW, which was equal to the saturation power P _sat=100mW. The saturation intensity I _sat is the ratio of the saturation power and the core area.

Table 1. Modeling parameters used for the 2D- and the 3D-simulations.

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The 2D-simulations showed that the mode behavior (i.e. the behavior of the modal power and the modal polarization) does not depend on the fiber length L, but on the total gain G(z)=P(z)/P(z=0). This allows to simulate a shorter fiber length without losing comparability with the experimental data of [13], what is advantageous with regard to the calculation time. We carefully checked that a shorter length L of the fiber combined with a higher small-signal gain does not influence the mode behavior, if G(z=L) is constant and if L is larger than the mode beating period. This mode beating results from different propagation constants of the transverse modes (βLP₀₁=9.1721·10⁶m^-1 and βLP₁₁=9.1701·10⁶m^-1). Therefore, we set the fiber length to L=0.1m and chose the small-signal gain to achieve a total gain of G(z=L)≈25, which corresponds to the total gain of the experiments in [13].

3. Modeling results

The Fresnel equation (Eq. (4)) has been solved numerically in two and three dimensions. Subsequently, the modal powers and the modal polarizations have been derived from the calculated electric field. First, the fundamental and the first higher-order mode have been propagated. Both modes show the same dependence on y, which means that each cut in x-direction results in an intensity profile of the same shape. Therefore the results of the simulation (concerning modal powers and polarizations) were the same in two and three dimensions. Due to the much shorter computation time the 2D-simulation has been used in this case. Second, the 3D-simulation has been applied to model the propagation of three differently polarized modes. As the third mode shows a different dependence on y, the electric field, which consists of the superposition of the three modes, has no fixed symmetry axis in contrast to the 2d-case, so that the 3D-simulation has to be used.

3.1. 2D-modeling: Mode behavior of two transverse modes

 

Fig. 1. Analytically calculated electric field of 2D transverse modes: (a) fundamental mode, (b) first higher-order mode.

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Figure 1 shows the electric fields of the analytically calculated 2D modes E₀(x) and E₁(x) [10], which correspond to the fiber modes LP ₀₁ and LP ₁₁, respectively. These modes have been used to define the initial electric field:

The fundamental mode contained 70% of the power and was chosen to be purely p-polarized. The higher-order mode contained 30% of the power, which was equally distributed to p- and s-polarization. This choice allows to examine the different evolution of the p- and s-polarized parts of the higher-order mode and the modal powers are similar to the experimental ones of [13].

Figure 2(a) shows the amplifier gain, which increases almost linearly along the propagation distance, as the amplifier is saturated. Figure 2(b) illustrates the intensity distribution within the fiber. The power in each slice has been normalized in order to visualize the change of the transverse intensity distribution independently of the total power. The interference between the modes E ₀ and E ₁ changes over z due to their different propagation constants. This causes a mode beating, which looks like a zigzag beam propagation. At locations of lower intensity the gain is less saturated resulting in a transverse spatial hole burning (TSHB). It can be recognized in Fig. 2(a) that the interference contrast and thereby the TSHB decreases in z-direction.

 

Fig. 2. Propagation of two modes: (a) gain (shown in linear scale) over propagation distance, (b) intensity distribution within the fiber (normalized for each slice), (c) relative modal powers in modes for p- and s- polarizations, (d) modal polarizations illustrated on the Poincaré sphere (the circles correspond to the initial conditions).

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Taking a look to the relative modal powers in p- and s-polarization (Fig. 2(c)) helps to understand the reasons for this observation. The power in the fundamental mode stays mainly p-polarized during the propagation, only a small increase of the s-polarized part is observed. The higher-order mode has equal powers in both polarizations at the beginning of the fiber. The s-polarized part of the higher-order mode is preferentially amplified related to the TSHB. On the Poincaré sphere (Fig. 2(d)) it can be also observed that the polarization states become more and more different from each other with increasing gain. The modes move towards opposite sides of the sphere. This is the reason for the reduction of the TSHB, as differently polarized electric fields cannot interfere and thus the interference contrast of E ₀ and E ₁ decreases.

The difference between both polarization states can be quantified by the spherical distance, which is the shortest path along the surface of the Poincaré sphere. Figure 3 shows the spherical distances derived from (a) the numerical simulations and (b) from measurements (see [13] for experimental details). An increase of the spherical distance with the gain is observed in the simulation as well as in the experiment. Moreover, a strong increase of the spherical distance at low gain and a saturation at high gain occur equally in simulations and experiment. In contrast to the simulation results the measured spherical distance nearly reaches the maximum distance of π and becomes faster saturated at high gain. One explanation for this deviation is that in the simulation only the active mode coupling has been taken into account, while in the experiment additionally passive mode coupling and an effective birefringence exist for each mode [17].

 

Fig. 3. The spherical distance as a measure for the difference of the modal polarizations: (a) simulation data, (b) experimental data from [13]. (The gain is shown in linear scale.)

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3.2. 3D-modeling: Mode behavior of three transverse modes

In order to model the propagation of the superposition of three differently polarized modes LP ₀₁, LP ₁₁ and LP _1′1 the 3D-simulation has been used. Compared to the 2D-modeling, the 3D-simulation needs a much longer calculation time (about a factor of 1000), which is a result of the third dimension and smaller z-steps (see table 1) that are necessary to maintain algorithm stability. Figure 4 shows the intensity profiles: (a–c) of the individual modes, (d) of the superposition of these modes used as initial condition, and (e) after the amplification in the fiber amplifier. The intensity profiles at the beginning and at the end of the fiber change locally due to mode beating. The images (d) and (e) represent examples, which illustrate that the intensity is distributed more homogeneously at the end of the fiber. This more uniform distribution corresponds to a reduced TSHB.

 

Fig. 4. The intensity distribution: (a–c) of the modes LP ₀₁, LP ₁₁ and LP _1′1, (d) of the superposition of the modes used as initial condition, and (e) after amplification in the fiber.

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More details are recognized from the relative modal powers (Fig. 5(a)) and the modal polarization states (Fig. 5(b)), which characterize the mode behavior. The relative modal power in LP ₀₁ decreases, while the relative modal powers in the higher-order modes LP ₁₁ and LP ₁′1 increase, as they can stronger profit from the unsaturated gain. Moreover, it can be observed, that the power of LP ₁′₁ increases somewhat stronger than the power of LP ₁₁. The LP ₁′₁ mode is stronger amplified, because the spherical distance between LP ₁′₁ and LP ₀₁ is larger than that between LP ₁₁ and LP ₀₁. In the experiments, which we presented in [13], the decrease of the relative power in LP ₀₁ and the different amplification of LP ₁₁ and LP ₁′1 have not been observed. We suppose that passive mode coupling and mode dependent bending losses had a stronger influence on the modal powers, so that the effects of active mode coupling were not recognizable.

Focusing on the polarization states, active mode coupling shows a stronger influence and a better agreement with the experiments is found. Figure 5(b) shows the evolution of polarization states on the Poincaré sphere. (The initial polarization states of the modes are marked by the circles.) Similar to the propagation of only two modes it can be recognized that the modal polarization states move away from each other with increasing gain.

In order to compare the simulated polarization behavior with experiments we derived the spherical distances for each combination of the three modes and calculated the sum of all distances (Fig. 6(a)). Figure 6(b) shows the corresponding curves that were derived from experimental data (see [13] for measurement details). While all spherical distances (d ₁, d ₂ and d ₃) increase with the gain and saturate at high gain values in the simulation, one spherical distance (d ₁) slightly decreases in the presented measurements. Moreover, the spherical distances d ₂ and d ₃ in Fig. 6(b) approach and touch each other, what is not observed in the simulation. Actually, in a series of measurements the spherical distances showed a strong dependence on the experimental conditions like the fiber position or room temperature, so that Fig. 6(b) represents an exemplary measurement. However, the sum of the spherical distances (d ₄) tended towards 2π in all measurements, what agrees well with the behavior in the simulation. 2π is the maximal value for the sum of three spherical distances that is reached, if all polarization states are on a great circle of the sphere. This behavior of modes to take different polarization states at high gain, can be attributed to the active mode coupling.

 

Fig. 5. Propagation of three modes: (a) modal powers, (b) modal polarization states illustrated on the Poincaré sphere (the circles correspond to the initial conditions). (The gain is shown in linear scale.)

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Fig. 6. For the case that three modes propagate we calculated the spherical distances between: LP ₀₁ and LP ₁₁ (d ₁), LP ₀₁ and LP ₁′1 (d ₂), LP ₁₁ and LP ₁′₁ (d ₃), and the sum of all distances (d ₄=d ₁+d ₂+d ₃). Figure (a) shows simulation data and (b) experimental data (see [13] for measurement details). (The gain is shown in linear scale.)

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Summarizing, also for the case of three propagating modes the active mode coupling has a strong effect on the modal polarization states. Although in experiments additionally the passive mode coupling influences modal powers and polarization states, the preference to take different polarization states at high gain is in good agreement with the simulation.

4. Conclusion and outlook

We have modeled transverse mode interaction in a LMA fiber amplifier using an extended beam propagation method and have simulated for the first time, to the best of our knowledge, polarization depended mode amplification. In good agreement with experimental results it was observed that the spherical distance between transverse modes increases with the gain, i.e. the transverse modes prefer to take different polarizations at high gain.

As the local refractive index and the local gain are used for the calculation of the electric field distribution, our approach can easily be extended to model arbitrary gain and refractive index profiles, e.g. a ring doping or refractive index changes by bending the fiber. Moreover, this approach can potentially be extended to LMA fiber lasers.