1. Introduction

The designer of a high power fiber amplifier suitable for use in coherent beam combination faces an interesting task of balancing numerous design parameters to simultaneously achieve kW power, high beam quality, and strong suppression of stimulated Brillouin scattering (SBS). As one example, using shorter fibers with larger mode areas helps suppress SBS, but it probably lowers the threshold for modal instabilities. Similarly, broadening the bandwidth of the signal light by phase modulation suppresses SBS, but if the resulting linewidth is more than a few GHz it makes beam combining more difficult. These are only two of several conflicting requirements that must be properly balanced.

The topic of this paper is how to maintain high purity of the fundamental mode while still suppressing SBS in a kW amplifier. In an earlier paper [1] we showed that a moving refractive index grating created by the optical field of two interfering modes with a small frequency offset between them can lead to exponential growth of an initially weak, higher order mode, resulting in severe mode degradation when the signal power exceeds a sharp threshold. The question we consider here is how, if that instability can be avoided, mode competition might affect modal purity, and how much weight should be given to this effect in designing an amplifier.

The term mode competition refers to the differing gains of various transverse modes in the presence of saturation of the population inversion, often referred to as transverse hole burning. If two competing modes have parallel polarizations, interference between the modal fields will be shown to be an important consideration. If they have orthogonal polarizations, only their irradiances matter. Both cases must be considered because either polarization maintaining (PM) or non-PM fiber may be favored by other considerations. For example, the use of a non-PM fiber should raise the modal instability threshold mentioned earlier. The fundamental and higher order modes sample different portions of the core, so small birefringence variations across the core cause their polarizations to evolve at different rates. This means the net influence of the irradiance grating caused by interference between the modes is reduced, as is the mode coupling strength [1]. Non-PM fiber is also attractive for its ease of fabrication. On the other hand, a PM fiber offers as much as a factor of two improvement in the SBS threshold [2, 3].

It is widely known that spatial hole burning by a strong fundamental mode leaves a population inversion that has a higher overlap with high order modes than with the fundamental. This has been shown to lead to higher gain of the higher order modes, and thus loss of modal purity [4–10]. It is also a demonstrated, but perhaps under appreciated, fact that spatial hole burning can suppress the growth of higher order modes, thus improving the quality of the output beam [11–13]. We present here a study of how these effects might influence modal purity of an amplifier designed for use in coherent beam combining. Whether hole burning leads to enhancement or suppression of higher order modes depends on whether or not the light in higher order modes is coherent with the light in the fundamental mode. Incoherence leads to enhancement, coherence to suppression. In a non-PM fiber each effect occurs at various points along the fiber with the expectation of intermediate values for the gain enhancement or suppression.

Field interference can occur only if the light in the two modes is sufficiently spectrally/temporally coherent. Coherence between two modes is strong if they have the same spectrum and if the group velocity induced temporal walk off between them is much less than the coherence time of the light. Walk off times are of order 1 ps/m in large mode area fibers typical of high power amplifiers [14], so for spectral widths of a few GHz, coherence remains high over the full length of a 10 m long amplifier.

Numerous modeling papers have addressed the growth of higher order modes in large mode area, Yb-doped fiber amplifiers in the incoherent case [4–10]. They usually assume that the launched signal light is predominantly in the fundamental mode, but with a specified small fraction accidentally injected or scattered into some of the higher order modes. The most commonly employed models compute the local upper state populations based on the irradiances of the pump and certain populated signal modes, neglecting interference. The upper state population distribution is then used to compute the power gain for each signal mode, along with the pump loss. The gains of the higher order modes are often found to be greater than that of the fundamental mode. Such models have been applied to straight fibers with step index cores and uniform doping across the core [4], where it is found that the output power fraction in the higher order modes is increased by gain saturation. This is sometimes cited as an important cause of beam degradation in high power amplifiers. Similar conclusions are drawn for bent fiber, with even higher relative gains in the higher order modes due to the bend-induced displacement and compression of the fundamental mode [5]. To counteract this effect, doping of the active ion can be confined to the center of the core so it overlaps best with the fundamental mode [6–10, 15]. However, it is noted that confined doping is less effective in bent than in straight fibers [5].

A few studies have included intermodal coherence [11–13]. They showed that, rather than experiencing enhanced gain, a weak higher order mode in the presence of a strong fundamental mode experiences depressed gain relative to the fundamental mode. Andermahr and Fallnich showed in experimental measurements [11] and in numerical modeling using a beam propagation method [12] that parallel polarized higher order mode light was gain depressed while orthogonally polarized light was gain enhanced. In their model they assumed the step index fiber was straight, with full core doping. Their model did not include pump depletion. Similarly, using analytical methods to study planar waveguides, Vysotsky and Napartovich [13] found that the inclusion of intermodal interference suppressed the gain of higher order modes in the presence of a strong fundamental.

In this paper we present modeling results for a test high power fiber amplifier, illustrating the important difference in behavior of the two polarizations in higher order modes, for both straight and bent fibers, with and without confined doping. In one set of model runs we assume there is no mixing of the two polarizations, as should be true of high quality PM fiber. In another set of runs for non-PM fibers we assume the fundamental and higher order mode polarizations evolve at different rates. We simulate this by periodically swapping the fields of the two polarizations in the higher order modes, with an arbitrarily chosen period of 40 mm.

2. The test fiber amplifier

For our study we chose the amplifier parameters listed in Table 1. This design does not represent any actual design, but it is representative of high power fiber amplifiers, while also allowing efficient modeling. We usually inject 30 W in the fundamental mode (LP₀₁) and 0.3 W total in select higher order modes. When the doping diameter is reduced from 30 μm to study the effects of confined doping, we increase the doping density by the ratio of doped areas to keep the pump absorption constant without increasing the fiber length. Longer fibers would lead to longer modeling run times. We model both a straight fiber and one bent to a 50 mm radius, studying the influence of doping confinement in both.

Table 1. Parameters of Test Amplifier

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3. Our models

3.1. Modes

We use a matrix eigen value method to compute the modes [16]. The basis functions comprise sine expansions with twenty terms in x and y directions. The three modes we will use are illustrated in Fig. 1. For short we call the LP₀₁ mode m1, the LP_{11 h} mode polarized parallel to m1 is called m2_||, its orthogonally polarized counterpart is m2_⊥, and mode LP₀₂ is m6.

 

Fig. 1 Mode irradiance profiles in straight fiber (top) and fiber bent to 50 mm radius (bottom). We call them m1 (left), m2 (center), and m6 (right).

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3.2. Irradiance based model

This model is appropriate for a single higher order mode polarized orthogonal to the fundamental mode. The irradiance profile is used in the steady state expression for the upper state population fraction,

The upper state population computed from Eq. (1) is used to compute the change in pump power using

3.3. Field based models

To include modal interference we use a split-step beam propagation method (BPM) to model the evolution of the modes. The first half of the split step is propagation of the total signal field over one dz step using FFT methods. The second half adds the phase due to the refractive index profile of the fiber, including bending, over the dz step. The second half also includes computation of the signal gain. To compute the gain the steady state upper level population is computed using Eq. (1), where I_s is computed from the square of the full optical field rather than from the sum of the irradiances of the individual modes. From n_u the gain/loss of the signal field is computed and added to the propagating field. We compute the modal content of the signal field at regular intervals along the fiber by computing the overlap integral of the propagating field with the modal fields of interest. The pump wave is propagated in the same way as in the previous model. This model runs in approximately 100 minutes.

3.4. Modeling notes

To allay doubts whether the first modeling method correctly models orthogonally polarized modes, we validated it by modeling the amplifier using the BPM model with the two orthogonally polarized modes propagated separately so there is no interference between them. The upper state population was computed from the pump irradiance and sum of the irradiances of the two oppositely polarized signal fields. We found negligible difference between this and the irradiance based model. We use the same two-field BPM model to simulate a non-PM fiber by periodically swapping the two orthogonally polarized higher order mode fields. First one field and then the other interferes with the fundamental mode. We do not claim this is necessarily a realistic treatment. It treats the limiting case of a large difference in the birefringence of the fundamental and higher order mode, which is useful given our lack of detailed knowledge of actual polarization evolution.

We verified convergence of the models by doubling the grid resolution in each of the dimensions, x, y, (from 1.2×1.2 μm) and z (from 2 μm). We also verified that mixtures of modes propagate as expected if the doping concentration is set to zero, or if the pump power is set to zero.

We also showed that doubling the doping density while keeping the amplifier length constant was nearly equivalent to keeping the doping constant and doubling the length. This is used below to facilitate modeling of restricted area doping.

In our earlier paper [1] we argued and demonstrated that stationary refractive index gratings caused by spatial hole burning, either through the Kramers-Kronig effect or through nonuniform heating, cannot cause power transfer between modes. Indeed, when we included such refractive index changes in our BPM models, we found no significant difference from the results without them. Therefore refractive index changes associated with modal interference are not included in the modeling results presented here.

4. Modeled amplifier performance

In the following subsections we present model results for straight and bent fiber with different doping diameters, from full core to one third of the core. We believe the trends are quite clear from the limited number of examples we can present here.

For later comparisons we present in Fig. 2 the baseline performance of co- and counter-pumped, straight, fully doped, amplifiers with the parameters of Table 1. The power in all higher order modes is zero.

 

Fig. 2 Baseline model results for co-pumped amplifier (top) and counter-pumped amplifier (bottom) using the parameters in Table 1. Higher order modal input powers are set to zero.

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4.1. Full core doping

Our first example clearly illustrates the difference between the behavior of higher order modes of parallel and orthogonal polarizations. Mode competition suppresses the growth of the || polarized modes relative to the strong fundamental mode, but enhances the growth of the ⊥ polarized modes. Fig. 3 shows plots of the power in modes m2_|| and m2_⊥ relative to the m1 power, for a co-pumped amplifier (top) and for a counter-pumped amplifier (bottom). The power in m2 is initially 1% of the power in m1. The irradiance based model is used to compute m2_⊥, while the field based BPM model is used for m2_|| and for non-PM fiber. Because our model for non-PM fiber switches the polarization many times at regular intervals, the light in either polarization state at any position has propagated nearly equal distances in each polarization state. This implies the non-PM curve should approximate the geometric mean of the parallel and orthogonal curves, and we verified this.

 

Fig. 3 Power in m2_|| (solid) or m2_⊥ (dashed) or the sum of m2_|| and m2_⊥ (chain) divided by power in m1 for a co-pumped amplifier (top) and a counter-pumped amplifier (bottom). The fiber is straight and the full core is doped. For the non-PM fiber the m2 power is launched half in each polarization, and the fields for the two polarizations of m2 are swapped every 20 mm. Other properties are listed in Table 1 and Fig. 1, and the pump and m1 powers are nearly identical to those in Fig. 2.

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In the co-pumped case most of the separation in power between m2_|| and m2_⊥ occurs before z = 2 m. For greater distances the pump is weak and the signal strong (see upper plot in Fig. 2) so the upper state population is strongly suppressed across the full core, and m2 with either polarization experiences approximately the same gain as m1. In the counter-pumped case the separation between m2_⊥ and m2_|| continues to grow over the full length of the fiber because the signal and pump powers are comparable the whole way, causing a relatively constant, moderate gain saturation.

At this point a physical explanation for the difference in gains of the two polarizations is called for. The cause of gain enhancement of m2_⊥ is fairly obvious. The strong m1 depletes the gain in the center of the core where it is strongest. Mode m2 is weak near the center but strong farther from the core center where depletion is weaker. Consequently m2_⊥ is less sensitive to the gain saturation than m1, so it has higher gain than m1. The suppression of m2_|| must be related to its interference with m1. When the phase between m1 and m2_|| is π/2, there is no interference, and the situation is exactly the same as for m2_⊥, so m2_|| has higher gain than mode one. However, when the phase between the modes is zero, the irradiance profile is asymmetric in the plane of the lobes of m2. On one side the fields of m1 and m2_|| interfere constructively, on the other destructively. Gain is higher on the destructive side due to a less depleted upper state population there, so that side of the field grows more there than the opposite. The result is a symmetrization of the field across the core. Although it might not be readily apparent, modal decomposition shows that this is equivalent to the suppression of m2 relative to m1. So, depending on the relative phases of the two modes, m2 is either enhanced or suppressed. In all the cases we have modeled the suppression is stronger than the enhancement, with the net result being mode suppression. This phase sensitivity is apparent in the magnified plot of the m2 power divided by the m1 power shown in Fig. 4. As expected, the oscillation period is half the mode beat length, with growth and suppression behaving exactly as we just described.

 

Fig. 4 Ratio of power in m2_|| to power in m1 for $P_s^1\hspace{0.17em}=\hspace{0.17em}300\hspace{0.17em}\text{W}$ and $P_s^2\hspace{0.17em}=\hspace{0.17em}3\hspace{0.17em}\text{W}$. The beat length is 3.63 mm for these two modes. Depending on the phase between the two modes, m2_|| growth relative to m1 is either positive or negative, but the net change in this ratio over a full cycle is negative.

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4.2. Confined doping

One widely discussed method of improving modal purity is to confine the doping to a central portion of the core [5, 6, 8–10, 15]. This improves the overlap of the gain with the fundamental mode relative to the higher order modes. We test this using our models, increasing doping concentration as necessary to maintain constant pump absorption.

Fig. 5 summarizes our results for m2 (top) and m6 (bottom) in unbent fiber. The gain G_m is defined as the output power of mode m divided by its input power. The doping diameter is varied from 10 μm up to the full core diameter of 30 μm. As expected, the gain of m2_⊥ can be reduced below that of the fundamental by reducing the doping diameter below 24 μm. Similarly, the gain of m6_⊥ can be reduced below the fundamental by reducing the doping diameter below 28 μm. One important point is that, according to these plots, confined doping influences the || polarized modes in nearly the same way as the ⊥ modes. The curves for a non-PM fiber in all four graphs can be estimated by taking the geometric mean of the parallel and orthogonal curves.

 

Fig. 5 A comparison of the relative gains vs. doping diameter for co-pumping. Other parameters are listed in Table 1. The gain G_m is the output power divided by the input power in the m ^th mode. The upper plot is the gain of modes m2_⊥ and m2_|| relative to m1; the lower is the gain of modes m6_⊥ and m6_|| relative to m1. For non-PM fiber the gain of mode 2 is reasonably approximated by the geometric mean of the two curves shown.

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Fig. 6 summarized our results for a similar calculation using fiber bent to a 50 mm radius. Bending increases the gains of m2 and m6 of both polarizations relative to m1. The ratio of powers for the two polarizations of m2 and m6 is approximately the same as in unbent fiber, and the effect of confined doping is also similar.

 

Fig. 6 A comparison of the relative gain vs. doping diameter for fiber bent to a radius of 50 mm in co-pumped amplifier (see Fig. 1 for bent fiber mode profiles). Other parameters are listed in Table 1. The gain G_m is the output power divided by the input power in the m ^th mode. The upper plot is the gain of modes m2_⊥ and m2_|| relative to m1; the lower is the gain of modes m6_⊥ and m6_|| relative to m1. For non-PM fiber the gain of mode 2 is reasonably approximated by the geometric mean of the two curves shown.

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5. Discussion

In the results presented here we have used 1% of the total power in the higher order mode. However, our modeling shows that the degree of suppression is quite constant for higher power ratios, up to 25% power in the higher order mode. Further, if several higher order modes are populated at the 1% level we find each interacts with the fundamental mode nearly independently, and each is suppressed as though the other modes were absent.

We note that the fundamental mode is not unique in creating a differential gain between the two polarizations of a much weaker mode. It appears that any strong mode can produce this effect on any weak mode.

6. Conclusions

In coherent beam combining a pure fundamental (LP₀₁) mode is desired from each fiber amplifier. The second mode (LP₁₁) probably poses the biggest problem because it adds variable uncorrected tilts to the beam which degrades efficiency of the combination process. This mode is also easily populated at launch by slight tilts and displacements of the amplifier fiber relative to the input beam. Only the light in LP ₁₁ polarized parallel to the fundamental at the output end is of real concern because the orthogonally polarized light, while it wastes a certain amount of pump power, does not interfere with beam combination.

As we have demonstrated, in a PM fiber the parallel polarized light of the non-dominant mode is substantially suppressed by gain saturation. Differential gain between the fundamental and higher order modes due to mode competition does not worsen the beam quality as is sometimes claimed, but rather helps purify the output beam. In non-PM fiber the polarizations of the fundamental and higher order mode do not evolve together, so at the fiber output the power in a higher order mode is on average nearly equal in the two polarizations, and its power is intermediate between the suppressed and enhanced levels. Here again, however, mode competition does not degrade mode purity. Only in tightly bent fiber does the mode purity degrade due to mode competition.

If greater suppression of higher order modes is required than is available from the mode competition effect, confined doping can be used. We have shown that this suppression adds with that provided by competition. However, confined doping may be incompatible with tight bending.

The overall conclusion must be that the effects of mode competition on beam quality from high power fiber amplifiers are modest, and except for tightly bent fiber are unlikely to strongly impact the beam quality. However, it is noteworthy that to the extent it does have an influence, it is usually positive, contradicting the notion of degradation due to mode competition. In any case, we have demonstrated that the effects can be modeled realistically in reasonable run times, so further exploration through modeling is straightforward.

Finally we note the well known method of bending the fiber to smaller radii than we used can be an effective way to suppress higher order modes. The differential bend loss between the fundamental and higher order modes may be sufficient to suppress higher order modes without excess loss of the fundamental [17]. Done properly, this can have a much stronger influence on beam quality than does mode competition.