## Abstract

A finite element steady periodic scalar beam propagation method suited to treating mode instability in amplifiers incorporating fibers with complicated design features is presented and demonstrated on test cases. These incorporate large pitch photonic crystal fibers with confined and down-doped cores in co-pumped and counter-pumped configurations. Seed power fluctuations and slight launch misalignment are proposed and implemented as a noise source to seed the instability. Down doping the confined core by −7.0E-5 was found to increase the instability threshold by 25% in the co-pumped case and 9% in the counter-pumped case. The increase in threshold is correlated with the suppression of higher frequency components of intensity gratings associated with the instability.

## 1. Introduction

High average power fiber-based laser sources have proven to be very effective in a wide range of applications due to their high efficiency, excellent beam quality and large gain bandwidth [1]. Specific characteristics of these sources can be tailored for specific applications. In many cases this process brings the overall design into a parameter space where mode instability limits the average near diffraction-limited power achievable [2]. Despite many years of intense experimental and theoretical research, realized average power gains have been relatively modest. Several approaches have shown benefits including selecting operating wavelengths to minimize quantum defect heating [3], choosing pumping and signal intensities to maximize gain saturation [4,5], choosing core compositions that minimize photodarkening [6], and using directional pumping schemes to even out the heat load along the fiber [7]. However, advances from improved large mode area waveguide designs have been limited. Photonic crystal fibers have shown to provide some waveguide-induced resistance to mode instability [8–13]. One problem hindering advancement in this area has been the lack of availability of time-dependent detailed models of the mode instability process in fibers with complicated waveguide designs. Semi-analytic techniques have been applied to such amplifiers previously [14–16]. The steady periodic beam propagation method has been used successfully to analyze a wide range of characteristics and dependencies of mode instability in fiber amplifiers [17,18]. However, the fast Fourier transform based beam propagation implementation is not well suited to the study of complex waveguide designs due to difficulties in sampling sharp material discontinuities on short length scales within such fibers. The same can be said for beam propagation methods based on the azimuthal harmonic expansion [19]. In contrast, finite element methods excel in handling complex waveguides.

This paper describes and demonstrates a steady periodic mode instability model based on a scalar finite element beam propagation method. Cladding-pumped Ytterbium-doped amplifiers incorporating large pitch photonic crystal fiber designs with confined and depressed-index cores are used as test cases. The simulations assume unavoidable slightly imperfectly-aligned seed launch with realistic seed power fluctuations. The method is described in the next section followed by an analysis and discussion of the dependence of the threshold on waveguide and seed-source parameters. The paper ends with a brief summary of the conclusions drawn.

## 2. Steady periodic finite element beam propagation method

The steady periodic beam propagation method has been described in detail before [17]. The key improvements here are the use of finite element methods to solve the beam propagation and the heat equations. The starting point of this method is the scalar paraxial wave equation that describes the propagation and diffraction of the signal optical field down the axis of the amplifier:

Additionally, co and counter-propagating pump powers${P}_{\pm}$evolve along the length of the amplifier under the assumption that the pump intensity remains constant throughout the inner cladding. The cold fiber waveguide design, laser gain and absorption, the thermo-optic effect, and photodarkening all contribute to the heterogeneous profile of$\epsilon $. To treat mode instability, the time evolution of the quantities in Eq. (1) must be included. This is accomplished by establishing the relevant period of time and then dividing it into $N$discrete time steps small enough to resolve the highest frequency of interest for instability. Periodic boundary conditions are then imposed for the time evolution. This results in a spectrum of discrete frequencies with which the amplifier fluctuates and$N$copies of Eq. (1) each governing the propagation of the field at one time step. At each propagation step and time step, the energy level populations are determined using the steady state laser rate equations. The use of steady state laser rate equations is justified due to the fact that the optical intensities in the doped region are so high that the time dependent populations reach equilibrium fast enough to be considered constant over the time intervals used. This enables the calculation of the heat load due to the quantum defect as well as photodarkening absorption at each point within the active core.

Because the temperature response lags the heat load, the heat load at each time step is necessary to calculate the temperature at any given time step. This is accomplished by solving the heat equation in the frequency domain on each mesh vertex using the finite element method and the temporal discrete Fourier transform (DFT) of the heat load, and transforming the resulting frequency domain temperature profile back to the time domain using the inverse DFT. All material properties of the fiber are constant on each element while the optical, thermal, and population degrees of freedom are defined on the nodes. Also, because photodarkening occurs over a long time scale [20], the time-averaged population inversion is used to determine the absorption. This process produces the update rule for the field based on a space-centered implementation of the Crank-Nicolson method:

*m*indexes the propagation steps and

*j*the time steps. The algebraic expressions for the matrices $M$,$K$, and $E$were derived starting from the appropriate finite element variational form of Eq. (1) as before [19]. The matrix$E$incorporates all contributions to the effective permittivity including the cold waveguide, laser gain and absorption, photodarkening absorption, and thermo-optic distortions. The heat equation in the frequency domain is then given as before [16] by:where the total heat load has contributions from the quantum defect as well as pump and signal photodarkening absorption all within the doped core region of the fiber. This overall approach handles thermally induced waveguide distortions naturally. At each propagation step the optical intensities at each time step are used to generate the heat loading throughout the fiber. The heat Eq. (3) and DFT are then used to determine the temperature at each time and transverse location. The thermally distorted index profile is then used to propagate the intensity which accounts for the resulting distortions of the local fiber modes. This process is then repeated starting with the updated intensity profile. The longitudinal propagation steps are short enough that the thermo-optic feedback loop between waveguide distortions and the intensity profile remains in equilibrium throughout the calculation.

## 3. Large pitch fiber amplifiers with confined down-doped cores

The ultimate purpose of the method described here is to develop fiber designs that suppress instability. Formation of de-stabilizing thermal gratings requires a finite beat length between transverse modes. If it were possible to design a fiber for which the LP_{01}-like and LP_{11}-like modes had the nearly same propagation constant and therefore a very long beat length, it may reduce the effect of these gratings. In this case either the fluctuation frequency would have to decrease or the grating speed would increase as seen by Eq. (4) [18]:

_{11}-like modes that have less overlap with the center of the core while simultaneously serving to preferentially amplify the fundamental mode. It would also partially pre-compensate thermal lensing that occurs under high thermal load [23]. Therefore large pitch fiber amplifiers with confined down-doped cores are an interesting test case.

A base design that is fairly similar to existing ones was chosen [8]. Within a pump cladding with a diameter of 170 µm are two rings of air-holes numbering 18 total on a pitch of 35 µm with a diameter of 10 µm. The confined active down-doped region of the core is hexagonal with a flat-to-flat width of 35 µm. The length of the amplifier was taken to be 1.2 meters and the numerical aperture of the pump cladding was 0.45. A fixed temperature thermal boundary condition was used at the boundary between the pump and outer claddings at a radius of 85 µm. This is justified because only the variations in temperature across the core and not absolute temperatures affect the instability. As long as there exists an enclosing surface sufficiently far away from the center of the core where the temperature is approximately constant, the result is the same. This situation is guaranteed by the form of the thermal diffusion equation that dictates that high azimuthal harmonics in the temperature distribution decay rapidly.

To set the key parameters of the method, the cold modes of the fiber are calculated and then the stimulated thermal Rayleigh scattering (STRS) gain curve for the cold waveguide is calculated using the LP_{01}-like and LP_{11}-like modes and a finite element method [16]. The time step is then chosen so that the maximum frequency component is three times higher than the STRS gain peak. The longitudinal step size is chosen to be no greater than 1/25th of the beat length between these two modes which is sufficient for the method employed. The curvilinear mesh uses edge lengths that are about 1/25th of the core diameter. The Ytterbium emission and absorption cross-sections have been taken from the literature [24]. The photodarkening absorption was determined from the doping concentration (3.5E25 m^{−3}) the area filling factor (0.5) the wavelength conversion coefficient (24.5), and the local inversion fraction as in [25]. The pumping wavelength was 975 nm and the signal wavelength was 1040 nm. All other properties of the fiber are identical to those used in [19]. Figure 1 shows the fiber design cross-section, the fundamental and LP_{11}-like mode profiles, and the STRS gain curves for the range of down-doping values investigated.

## 4. Simulation results and discussion

The model described above was used to simulate these amplifiers. The model was implemented in the C programming language on a distributed memory high performance computing cluster. The Message Passing Interface (MPI) standard was used to arrange the problem on a 2 dimensional logical array of computational cores. The steady periodic algorithm allows all time steps to be calculated simultaneously. Each of 73 time steps employed 9 cores for a total count of 657 cores. Each simulation required less than 2 hours to run. The down doping level ranged from 0 to −7.0E-5 in 1.0E-5 increments. The noise that initiates the instability is assumed to arise from unavoidable small power fluctuations in the seed beam and very slight misalignment of the seed beam with the core. This accurately replicates laboratory conditions under which most amplifiers are tested. This is especially so for all-fiber amplifiers where the seed alignment is fixed by a fusion splicing process that can only align the cores to within a given tolerance. A uniformly distributed random number between 0 and 1 was generated at each time step and mapped linearly onto the range from −5 to 5 µW. This was then added to a seed of 10 W resulting in an RMS fluctuation value of 2.8 µW. This level was chosen after surveying specifications for diode and solid state single-mode seed sources. The launch was taken to be mis-aligned by 0.5 µm in both the horizontal and vertical directions resulting in an overall direction which is not on a symmetry axis of the fiber. This was chosen based on specifications for typical high precision alignment stages.

The data produced by the simulations includes the signal power, pump power, total heat per unit length, the complex field value and temperature at three sampling points within the core (one at the center, one offset in the horizontal direction, and one offset in the vertical direction by approximately a quarter of the mode field diameter) at each propagation step. Additionally the full signal field is stored for a subset of propagation steps for each time step. This stored data enables visualization of the instability dynamics, intensity gratings, thermal index gratings, as well as the determination of the instability threshold. The threshold definition used does not rely on modal decomposition of the output [26]

It is evident that the threshold increases as the index of the doped core is reduced. The threshold trends for the co-pumped and counter-pumped amplifiers are somewhat different. While the counter-pumped threshold initially increases rapidly and then levels off, the co-pumped threshold exhibits a steady rise with down doping. To ascertain what gives rise to this behavior it is instructive to examine the intensity gratings associated with the instability as calculated using the beam propagation method.

Figure 3 shows the evolution of field fluctuation spectrum at one of the off-center sampling points along the length of the amplifier. The random amplitude fluctuations that seeded the amplifier have identical positive and negative frequency spectra at the input end, however, as the signal is amplified along the fiber, the negative frequency components are preferentially amplified giving rise to forward travelling thermal gratings in agreement with the STRS model of instability [27, 28]. The onset of instability is also preceded by frequency mixing as seen by the “filling” in of weakly-seeded spectral components as the signal beam propagates from the seeded end to the output end. The fibers with a more depressed index exhibit delayed onset of this filling in which is consistent with their higher instability thresholds. Close examination of Fig. 3 reveals banded patterns. Physically this means that certain frequency components are suppressed and then arise again as the beam propagates. Notably it appears that the spatial frequency of this behavior is largely independent of the temporal frequency. Also, this spatial frequency is higher in the co-pumped case although the fringes do not appear to be as pronounced as in the counter-pumped case. The fact that the length of these is much longer than the LP_{01}-LP_{11} beat length in the counter-pumped case suggests that they may be associated with interference between different higher order modes which may have a longer beat length between them. Another possibility is interference between different frequency components that would have to travel at different speeds according to Eq. (4) if the beat length was constant.

The plots in Fig. 3 convey a very limited amount of information because they pertain to a single sampling point within the near field amplifier output. More information may be gained by looking at the temporal evolution of the intensity gratings over the last part of the amplifier. Figure 4 shows the intensity at the offset sampling point within the fiber as a function of propagation step which corresponds to distance and time step. While the pattern is somewhat irregular, the gratings appear very clearly revealing interesting behavior. The beat length may be estimated by counting vertical fringes and estimating the number per unit length. It is immediately evident that the beat length of the amplifiers using the down-doped fiber are indeed longer than those using the index-matched core fiber. This lengthening is somewhat reduced in the counter-pumped case due to the higher thermal load (66 W/m compared to 25 W/m in the co-pumped case) and resulting stronger thermal lensing that partly counteracts the down doping of the core. Furthermore, the temporal frequency of the gratings may be estimated using the number of horizontal fringes and the overall time window. Finally, the grating speed may be estimated from the observed beat length and frequencies. Table 1 summarizes these quantities for the four representative amplifier configurations.

It is evident that the gratings arising in these four cases exhibit varied behavior. It is instructive to see which of the characteristics, if any, correlate most closely with the threshold. Typically a larger core size correlates with a stronger STRS coupling coefficient, lower thresholds, and a slower beat frequency [28]. The design approach studied here deliberately seeks to break the relationship between core size and beat frequency by manipulating the beat length while keeping the core size constant. Examining Table 1 reveals that for this particular fiber design higher thresholds correlate with lower beat frequencies. This is not meant to be a general design rule for all amplifiers but rather an illustration of the type of analysis the model presented here permits. The presence of these reduced frequencies is somewhat counterintuitive given the fact that the noise introduced was random in time leading to a random spectrum with a spread-out frequency content and the observed beat frequencies were below the peak STRS gain frequencies shown in Fig. 1. The frequency peak in STRS gain curves arises because the intensity grating moves too fast at frequency shifts larger than the peak serving to wash out the temperature grating while at frequency shifts below the peak the slow movement of the grating lessens the phase shift between the temperature and intensity gratings thus weakening the coupling and therefore the STRS gain. The question then remains why the dominant frequency components in the most down-doped fibers appear to be below the gain peaks shown in Fig. 1. Part of the explanation may be that the strong thermal lensing alters the effective gain curves along the length of the amplifiers. Also, the variance of the direction of the fringes in Fig. 4 indicates the presence of grating components travelling at different speeds. These results suggest that simulating the behavior of amplifiers under seeding conditions incorporating random noise requires a coherent treatment of different STRS frequency components, something that the beam propagation method described here does. Further studies of other different types of fibers may be able to further illuminate the connections between their properties and their instability thresholds.

## 5. Conclusion

A finite element steady periodic scalar beam propagation method suited to treating mode instability in amplifiers incorporating complicated design features was described and demonstrated on test cases of large pitch photonic crystal fiber with confined and down-doped cores in co-pumped and counter-pumped configurations. This method takes a time dependent seed input intensity profile and a constant pump input and determines the time-dependent thermal and optical properties everywhere within the amplifier taking into account the fiber waveguide design, laser gain and absorption, photodarkening, thermal lensing, and moving thermal long period gratings that give rise to instability.

The time dependence of the seed source for the simulations was based on the noise floor of output power variations for plausible seed sources combined with a slight mis-alignment of the seed launch. Although this noise level is virtually imperceptible at the seed end, the STRS process amplifies these fluctuations leading to instability. The simulated thresholds ranged from a low value of 122 Watts to 153 Watts depending on the level of core refractive index reduction. These values are somewhat lower than those reported for some similar amplifiers [6, 26]. This is most likely due to the very strong dependence of the threshold on the doping concentration which affects both quantum defect and photodarkening-induced heating. The value used here may be a little higher than that of the actual fibers used. Also, the experiments may have had better seed launch conditions than those simulated here. Co-pumped amplifiers exhibited stronger instability suppression with down doping of the core than counter-pumped amplifiers. This overall behavior correlated with changes in intensity grating characteristics within the fiber. Specifically, amplifiers with a smaller temporal frequency component within the gratings had higher instability thresholds. It is hoped that this simulation method will contribute to advancing new lines of investigation on how to suppress instability and illuminate new paths of investigation for further power scaling of fiber-based laser sources.

## Acknowledgments

The author would like to acknowledge a grant of computational time from the DoD High Performance Computing Modernization Program, support from the United States Air Force Academy High Performance Computing Research Center, and support from the Air Force Research Laboratory Directed Energy Directorate. The views expressed in this article are those of the author and do not reflect the official policy or position of the US government or the Department of Defense. Distribution A: Approved for public release, distribution unlimited.

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