1. Thermal mode instability background

The inexorable demand for higher-power lasers inevitably runs into roadblocks as one physical limitation or another restricts progress. In the development of high-power, single-mode, ytterbium-doped fiber lasers, the demand for optical power density beyond the onset of undesirable nonlinearities of conventional fiber has forced the use of large mode-area (LMA) fibers. The basic incarnation of these optical fibers remains a step-index profile, but with increased core diameter from single-mode (6 µm) to a 25-µm core diameter [1–4]. Since this size of core diameter moves the fiber outside of the single-mode operating regime, the numerical aperture of LMA fibers are lowered to around 0.06 to reduce the number of supported modes [5,6]. At low powers, this strategy allows the higher-order modes to be filtered via bend-loss [4,5] and has resulted in multi-kW fiber laser systems with near diffraction-limited beam quality.

Some applications, such as coherent combination of multiple fiber amplifiers, require narrowband amplification to very high power levels. Local thermal profiles that occur due to quantum defect between absorption and emission cause local refractive-index changes. This, combined with the coherent nature of the light in the fiber, results in dynamic coupling of light from the fundamental mode to higher-order modes at rates that would cause unacceptable losses via discrete spatial filtering due to ∼100% of power transferring between modes. The onset of this instability has a threshold-like behavior that separates regimes of stable operation from regimes of dynamic mode instability [7–9].

Numerous methods have been proposed and demonstrated [10–18] to combat this phenomenon. These efforts have largely been extrinsic, such as [11–13,15,17]. Extrinsic methods are largely system-level and can be implemented on commercial-class fibers. Intrinsic methods aim to reduce the amplifier susceptibility to thermal mode instability (TMI) by modifying the fiber itself. One such example is [18]. A significant advantage can be gained by using intrinsic methods since they can work in conjunction with extrinsic methods.

In this work, a more in-depth understanding of TMI physics and the factors that influence its threshold have been developed. These have been applied to develop a new class of optical fibers based on refractive index and ytterbium doping profiling that leads to increase of the TMI threshold by a factor of two. The work is divided into three main parts: Section 2 focuses on an expanded computation model based on Naderi et al. [19], Section 3 reveals new insight into the understanding of TMI physics, and Section 4 describes a new class of optical fibers designed to take advantage of the new physical understanding. Discussions and concluding remarks are given in Section 5.

2. Expanded TMI model

A number of approaches have been taken to model this phenomenon, ranging from grating-based phenomenological models to beam-propagation methods [19–26]. Like Naderi et al. [19], the model presented here simulates LMA fiber amplifier behavior by modeling the modally/spatially and temporally resolved optical, thermo-optical, and thermal behavior of high-power ytterbium-doped fiber-amplifiers (YDFAs). The spatio-temporal model contains full 3D spatial resolution of the thermal and population inversion, while reducing the spatial dependence of the complex optical field to a discrete set of modes with complex amplitude. This quantization offers a significant advantage in terms of computation time, but does not quite capture the entire set of physics. Specifically, the modes are fixed through the simulation, with only perturbative coupling between the modes to modify their behavior (power and phase) in order to facilitate rapid computation. For the case of high-power fiber amplifiers, this represents a minor flaw in using a modally decomposed model since the same thermal properties that lead to coupling between the modes also leads to modification of the transverse mode profiles and, more importantly, their eigenvalues (represented by their propagation coefficients or effective indices). Thus, the varying thermal characteristic along the length of the fiber means that the “natural mode” of the fiber (i.e., the ideal waveguide mode solution) can also change both along the length of the fiber and in time as the thermal characteristic dynamically changes.

To address this limitation while retaining all of the benefits of modal decomposition modeling, each mode profile and its associated eigenvalue are allowed to vary along the fiber as the average core temperature along the length of the fiber. Specifically, the modes of the fiber are calculated as a function of temperature, which manifests in the well-known quadratic graded-index profile for cylindrical symmetries [27]. The sets of modes (one set of unique transverse modes at each temperature) are solved using a finite-difference method mode-solver. In this way, any given mode (e.g., LP₀₁, LP₁₁, etc.) has both a spatial profile and a propagation coefficient that vary (discretely) as a function of core temperature. The difference between the propagation constants of the first two fiber modes (LP₀₁ and LP₁₁) is important to the TMI physics in particular. Despite the potential for small changes to make large impacts in chaotic systems, the variations of the modes with temperature have only a minor impact on the TMI threshold for conventional step-index fibers. However, some designs investigated in this work rely on small index changes in the fiber, so the thermal dependencies will be maintained to ensure accuracy.

As derived directly from Maxwell’s equations [23], the evolution of the complex optical (electric) field in the fiber amplifier is given by

In the expanded model presented in this work, $\varphi $_i, and β_i are temperature dependent (generated from an interpolated look up table based on the finite-difference method mode solutions), and are thus implicitly axially and time dependent as well. The modes and their eigenvalues are calculated independently via finite-difference mode solver using the heat generated in the fiber as described by Brown and Hoffman [27]. The refractive index perturbation (δn) is the difference from the initial heat equation determined by Brown and that of the numerical solution to the heat equation given in Eq. (4). Longitudinal heat flow is neglected due to the scale of the longitudinal thermal features, as determined by the difference in mode propagation constants, which is on the order of centimeters, while the transverse dimensions of interest are tens of microns. The resulting 2D heat equation is given by

While the signal intensity is spatially resolved, the pump intensity is considered to be well mixed spatially and is thus described by a uniform intensity across the core as

Equations (1)-(3) and (6)-(8) are resolved rapidly (ns-scale) in comparison to the evolution of the thermal profile (µs-scale). Therefore, their solutions are considered as nearly instantaneous and as such are solved in between temporal steps of the thermal model. Adiabaticity is assumed in implementing the varying mode profiles and propagation coefficients both down the length of the fiber and in time. Provided the thermal profile changes occur over sufficiently long (optical) lengths, any given mode of the fiber will adiabatically change along the length of the fiber with minimal coupling between modes (when referring solely due to the axial temperature changes, and not including transverse effects). The resulting computational algorithm is as follows: once the spatial temperature profile is calculated at a given point in the fiber, the resulting average core temperature is used to interpolate (a) the profile of each mode at that point in the fiber (using the previously solved mode set) and (b) their propagation coefficients β_i. Akin to the adiabatic physics in the actual (physical) fiber amplifier, adiabatic conversion with propagation is assumed such that the power does not transfer between modes; rather the mode’s spatial profile and effective index change as the mode propagates down the fiber without transfer of power to any other modes (neglecting transverse thermal effects). In this way, the evolution of the mode profiles and the beating between the modes are taken into account in a natural and physical way, and the remaining (transverse) thermal perturbations can be truly perturbative.

In numerically implementing the model, an analytical solution to Brown and Hoffman [27] was used to derive an initial thermal solution for the fiber, which was used to develop an approximately thermally resolved mode set that allows mode profiles and propagation constants to change as a function of temperature in the fiber. A 4^th order Runge-Kutta solver was used for the propagation of the modally resolved electromagnetic wave and its interaction with the ytterbium-doped gain medium. An alternating direction implicit (ADI) method was used to solve a time-dependent heat equation with two resolution zones: high resolution near the core and low resolution near the outer edge of the fiber. For generality, a square mesh was used rather than polar mesh as in Naderi et al. [19]. We expect the square mesh to have negligible impact on the run speed and allow greater control of the index and gain-dopant description of fibers away from the central axis. Note, this thermal simulation is solved assuming no symmetry conditions.

The set of common parameters used in the simulations is given in Table 1, as taken from Refs. [23,27–28]. As with Naderi et al. [19], two resolutions of thermal meshes were used to capture the physics of the core while retaining the ability to compute the whole system in a reasonable timeframe. The high-resolution mesh is made sufficiently small in resolution and sufficiently large in extent to represent the physics occurring in LMA fiber cores up to 50 µm in diameter. The low-resolution mesh is included to accurately model a thermal system that is referenced to an environment outside that of the 230-µm diameter fiber without catastrophically increasing the computational time. To avoid a strictly abrupt transition at the boundary between these two meshes, the low-resolution grid is interpolated for matching to the high-resolution grid. In addition, having a cylindrical boundary on a Cartesian grid was compensated by modifying the boundary element to have the same volume and surface as a cylinder. Both of the above methodologies were checked against Brown and Hoffman [27] and found to be in agreement. Axial optical and thermal resolutions differ because the Runge-Kutta method requires half-step information in the refractive index. Each simulation begins with instantaneous turn on of pump and signal and an approximation of the thermal profile provided by Brown and Hoffman [27]. This produces an initial condition of symmetry, which must be settled out during the full simulation that accounts for the physical asymmetries. The simulation is allowed to settle for 20-40 ms, after which simulations below TMI threshold stabilize and above TMI settle into their final dynamic operating characteristics.

Table 1. TMI simulation parameters

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3. TMI in LMA fiber and critical thermal processes

The nature of TMI is insidious in that even though beam quality can be retained at low power, LMA fibers driven to higher powers exhibit thermally induced mode instability [8] that is characterized by a significant fraction of optical power being exchanged between modes of the fiber at kHz rates. To date, some methods have been demonstrated to reduce the impact of TMI via treatments such as active mode control, gain saturation management, pump/signal control, and fiber chemistry and geometry [10–19]. All of these would lead to higher optical intensities, which would require the exploration of fiber amplifiers with larger core areas. Therefore, it is prudent to understand what happens to the TMI threshold when the core area is increased for conventional step-index fiber.

Though the model describes the coupling between all supported fiber modes, coupling between the fundamental and the next highest order mode is the strongest and thus will determine the TMI threshold in standard (step-index) fiber types [23]. One interesting aspect allowed by the rate equation model simulations is the ability to observe the transition between stable and unstable operation. For the kW-class amplifiers considered here, 35 W of signal power was seeded into the fundamental (LP₀₁) mode, with 1.75 W (5% of the fundamental-mode power) launched into the higher-order (LP₁₁) mode. The pump was co-propagated with the signal launch, as is standard for kW-class LMA-fiber amplifiers.

The model was applied to fibers with core diameters varying from 20 to 50 µm, each with 230 µm cladding diameter. The doping density of the core was adjusted between 2.5 and 12 × 10²⁵ m⁻³ to maintain the same fiber length and output power for a given pump power as the core diameter is adjusted. Simulations were run with 25 W increments in pump power, and the TMI threshold was determined as the power when dynamic power exchange between modes was observed to be in excess of 10% of the fundamental mode power.

Figure 1 shows the resulting TMI threshold for core diameters ranging from 20 µm to 50 µm for forward a forward-pumped amplifier. The error bars indicate the 25 W pump power increment. Figure 1 shows a monotonically decreasing TMI threshold with increasing core diameter. At first glance, this seems counter-intuitive for many reasons. First, the heat is generated over a larger area, so the heat density [Q in Eq. (4)] is lower. Second, the heat generation is physically closer to the fiber edge where the heat is removed. While thermal lensing has been modeled to have larger impact on larger core diameters [27,28], the impact on TMI physics is substantially different.

 

Fig. 1. TMI threshold of co-pumped, step-index LMA fiber amplifier as a function of fiber core diameter.

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TMI occurs due to the time lag between the thermal profile and the rate of phase shift of the optical profile. Therefore, the distance that the thermal profile must shift is inherently tied to the mode profiles. Consider an LMA fiber with a 25-µm, 0.06-NA core that carries 95% of the power in the fundamental (LP₀₁) mode, with the remaining 5% in one of the LP₁₁ modes. These fields, along with the resultant coherently combined optical intensity profile, are shown in Fig. 2. Note that the peak of the combined intensity, which extracts the most gain and causes the most heat deposition via the quantum defect, is offset from the center of the fiber. As the thermal profile builds, the phase is shifted between the LP₀₁ and LP₁₁ modes, until the optical peak is shifted to the other side of the fiber center. While the local generation rate of this profile can shift as quickly as the modal interference pattern shifts, the relaxation of a thermal pattern into a symmetric equilibrium that is required for a dissipation of the generated grating is based on the geometry of the pattern and the thermal properties of the fiber. If we assume the thermal conductivity of fibers of interest are similar to an inconsequential degree, then it is the maximum separation of the peak intensity from the peak of the fundamental mode is a useful measure for how long a thermal pattern will lag in its dissipation. This maximum thermal inertial peak shift (TIPS) of the heat density is therefore determined by the off-axis coherently-combined intensity profile.

 

Fig. 2. Optical profiles of a 25 µm 0.06-NA core fiber: LP₀₁ mode field carrying 95% of the power (green), LP₁₁ mode field carrying 5% of the power (blue), combined intensity profile (red), and the fiber core refractive index profile (purple). Orange arrows indicate TIPS.

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In a comparable LMA fiber with 50-µm core diameter, the TIPS would be nearly twice as long, indicating that the thermal lag would be nearly twice the duration as that of the 25-µm LMA fiber. While this conceptually explains the decreasing TMI trend with increasing core diameter as shown in Fig. 1, the physics is more complicated. The larger fiber will have both a small overall heat density and smaller localized gain extraction. As such, the result is that the TMI threshold is not simply a 50% reduction for a double-sized core, but a smaller reduction as shown in Fig. 1.

To validate the physics of this hypothesis, simulations were run for fiber amplifiers using LMA fibers having core diameters of 20 µm and 50 µm. Specifically, the simulations were run just above the respective TMI threshold for each fiber. The dynamics of the modal power transfer are captured in Fig. 3, which also shows the conceptual TIPS for the two fibers. The modal power exchange dynamics are clearly much slower in the 50 µm LMA fiber than in the 20 µm LMA fiber, indicating that the TIPS is critical in the TMI process. These new physics are critical to not only properly understanding the physical phenomenon, but also in the design of both extrinsic and intrinsic TMI-mitigation strategies.

 

Fig. 3. Depiction of TIPS (top) and power in the LP₀₁ mode as a function of time (bottom) for (a) 20 µm LMA fiber, and (b) 50 µm LMA fiber.

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4. Engineered fibers for mitigating TMI

While the TIPS was identified as critical to the TMI process in Section 3, the equations shown in Section 2 reveal a key to mitigating TMI using new fiber designs. Specifically, Eqs. (2) and (3) involve the spatial overlap of the two modes of interest (the fundamental LP₀₁ and the LP₁₁) with a third quantity: the refractive index perturbation and the population inversion, respectively. Let us first consider the refractive index perturbation in Eq. (2). Notionally, the refractive index is perturbed everywhere in the fiber (via dn/dT) since the entire fiber is hotter than room temperature under pumped conditions. However, the thermal transport physics revealed in Section 3 indicate that the perturbation in the entire fiber is not critical to the TMI physics.

To gain some understanding, simulations were performed in a confined-gain fiber [29,30]. This type of fiber was selected for simulation specifically because the region of gain doping is different from the refractive index step that defines the core. In these simulations, two sets of data were extracted, one each for below and above the TMI threshold. In each of these data sets, the temperature profile along the length of the fiber was extracted at four locations across the transverse dimension of the fiber, as depicted in Fig. 4(a).

 

Fig. 4. Temperature variation along fiber length below (b) and above (c) TMI threshold at various transverse locations in the fiber as indicated in the depiction of the confined-gain LMA fiber (a). Colored curves in (b) and (c) are offset for clarity.

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Below the TMI threshold [Fig. 4(b)], the temperature variation is identical at every point across the transverse dimension of the fiber. The variations are sinusoidal, indicating that the perturbation is not yet sufficiently strong to induce TMI. Most importantly, they are the same magnitude everywhere in the fiber core. Above the TMI threshold [Fig. 4(c)], a significantly different picture emerges. First, the deviation, while still periodic, becomes decidedly non-sinusoidal. More importantly, the variations are of significantly larger magnitude in the gain region of the fiber core compared to the passive region of the fiber core. This means that although Eq. (2) states that the overlap of the modes with the refractive index perturbation as a whole should be reduced, the most significant impact would be to reduce the spatial overlap of the modes with the region of heat generation. This new understanding is in agreement with the concept of the TIPS, in that the heat generation is the key driving force of TMI, not the overall thermal distribution in the fiber.

Equation (3) shows that the spatial overlap of the modes should be minimized with respect to the population inversion. Note that while this is strictly not viable to reduce all the way to zero, there is freedom in gain design that can actually improve amplifier efficiency while reducing the size of the gain region, e.g. using confined gain [31].

Taken together, these insights on Eqs. (2) and (3) lead to minimizing the spatial overlap between the two modes of interest and the gain region (where the heat is generated). Merging these concepts with the insights in Section 3 leads to a set of design rules that can be used to mitigate TMI intrinsically via fiber design:

4.1 Confined-gain fiber

Confined gain, where the gain does not extend all the way to the edge of the fiber core as depicted in Fig. 4(a), has been recognized as one of the key methods for mode control in VLMA fibers. Not only has it been well modeled [30,32], but it has been experimentally validated including powers over 800 W [29,33–35]. The key to “gain filtering” is not to introduce loss into the amplifier to filter out the unwanted modes, but rather to deny gain to higher-order modes (HOMs) under all circumstances (i.e. levels of gain saturation).

Although the impact of confined gain on the TMI threshold was originally studied in the AFRL publication [14], the radial confinement was limited to 80% of the core diameter. While the results still showed an increase in the TMI, the full potential of confined gain was not realized in those simulations. Theoretical studies [30,32] report that 50-60% of the core diameter represents the best modal suppression. Since filtering out HOMs necessarily results in a reduction of TMI tendencies (via increased threshold), it is prudent to optimize the effectiveness of gain filtering. In addition, confining the gain to the central portion of the core conforms to Design Rule A: reducing the spatial overlap between the gain and the optical modes. While the gain overlap with the fundamental mode is not significantly decreased, the overlap of the gain with the LP₁₁ mode is.

In our simulations, we used a 50% doping diameter, meaning the ytterbium doping extends from the center of the fiber 50% of the way to the edge of the core. This reduction in dopant volume creates a lower total number of dopants. In order to maintain similar amplification and thermal behavior, the doping density was increased to produce the same pump utilization in the length of fiber. Note that the room temperature modes of a confined-gain fiber are identical to those of a conventional LMA fiber (i.e. the step-index core shared by both fibers determines the mode shape). With the same cladding diameter, this represents the same approximate laser kinetics as a 30 µm LMA fiber core. The results of the simulations showed a substantial benefit in the TMI threshold over conventional LMA fibers. It is notable that the TMI threshold for the 50 µm confined-gain fiber is 30% higher than the TMI threshold for a conventional 25 µm LMA fiber. In Section 5, these results are captured and graphically compared to the other results in Fig. 9.

4.2 Trefoil fiber

Design Rule B requires that the fundamental mode and the higher-order modes have lower spatial overlap. In order to separate the fundamental mode from the higher order modes (HOMs), a three coupled-core fiber design was conceived of that would concentrate a fundamental (super)mode (LP₀₁) amongst the cores that significantly confines the optical field to the center of the fiber, as depicted in Fig. 5(left). Note that in each core, the optical field is concentrated at the edge of each core near the center of the fiber.

 

Fig. 5. Lowest-order modes of a Trefoil fiber. The green circles annotate the core locations, while the red ovals are guides to the eye for marking the fundamental mode power.

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The triangular symmetry generates an assortment of modes of similar order to that of the LP₁₁, meaning they have two distinct lobes with opposite phase. The two unique LP₁₁ modes (with different eigenvalues, not including rotational or polarization variants) are shown in Fig. 5 (center, right). In the LP_11,a mode, one of the lobes looks simply like the fundamental mode of a single core. The other lobe, however, is concentrated across the other two cores but offset away the center of the fiber. Comparing the LP₀₁ and LP_11,a modes indicates that the overlap is only relatively strong in the left-most core. The LP_11,b mode has a substantially different shape, with optical field largely absent in one core, and the lobes primarily concentrated in the other two cores. Comparing this with the LP₀₁ mode shows a significantly lower overlap with the LP₀₁, even with respect to the LP_11,a mode.

Since the LP_11,a mode has higher overlap with the LP₀₁ mode, it was selected for use in the simulations to represent a worst-case scenario. The optical intensity of the LP₀₁ mode concentrated to one side of each core suggests that confining the gain to half the core would be advantageous both in terms of gain filtering as well as Design Rule A (reducing overlap between gain and modes). In these simulations, the gain in this fiber was confined to the halves of the cores closest to the center, assuming each core could be constructed from two half-rods. The resulting TMI threshold from this fiber increased substantially, yielding a 37% improvement. In Section 5, these results are captured and graphically compared to the other results in Fig. 9.

4.3 Cladded linear index gradient (CLING) fiber

Optical waveguides are mathematically equivalent to quantum mechanical systems, where the refractive index profile is the equivalent of the energy potential, optical modes are the equivalent of the probability distribution, and effective indices are the equivalent of the eigen energies. Equations (2) and (3) are in fact exact mathematical prescriptions for the coupling between two quantum mechanical eigenstates based on a perturbation.

An optical waveguide is well represented by a quantum mechanical potential viewed upside down. The eigenvalues (effective indices) are positioned in the “well” at an appropriate “energy level” (location in refractive index space). This concept is shown in Fig. 6(a). Note that the modes in this case are approximately the same size and thus have high spatial (intensity) overlap. Recognizing that the mode size is largely dictated by the edges of the “well” (refractive index boundary), the modes can be made different sized by altering the spatial shape of the refractive index profile. For example, if the “well” were tapered (i.e., a graded-index), the second mode would be larger than the fundamental mode. This concept is depicted in Fig. 6(b), and conceptually satisfies Design Rule B.

 

Fig. 6. Comparison of optical modes and eigen levels between step-index and graded-index fibers. (a) Modes of a conventional step-index fiber are nearly the same size. (b) In contrast, modes of a graded-index fiber are not the same size, with higher-order modes being increasingly larger.

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Regarding the graded index, a linear index grade is more tolerant than a conventional quadratic-type graded index. In the linear case, a change in the effective index makes a corresponding change in the mode size. By contrast in the quadratic case, a small change in the effective index can make a drastic change in the mode size for the fundamental mode in particular. In contrast to the cartoon shown in Fig. 6(b), (a) continual grade in the fiber is undesirable from two perspectives: fabrication and optical. The core fabrication (e.g., CVD) will be radially smaller and less expensive if the index grade does not continue all the way down to the cladding index value. Having an index step to close off the index grade also helps confine the tails of the modes optically and provides a way to apply additional numerical aperture to the design (such that the NA is not exceedingly low) without extending the gradient area.

Therefore, due to both physics and tolerances, the optimized design is a cladded linear index graded fiber, or CLING fiber. The specific design used a conventional step-index fiber with a 0.06 NA that was modified by bumping the center up and the edges down by the same amount. The total core diameter must be expanded reasonably beyond the fundamental mode size in order to allow sufficient expansion of the LP₁₁ mode.

Figure 7 shows both the LP₀₁ and LP₁₁ modes for two fibers having the same effective area for the fundamental mode: the CLING fiber (blue and orange, respectively) and an LMA fiber (purple and green, respectively). Note that although the fundamental modes are approximately the same size and shape, the LP₁₁ mode of the CLING fiber is substantially (almost 40%) wider. Radially, this means that the 2D overlap is reduced by nearly a factor of two, in principle significantly raising the TMI threshold.

 

Fig. 7. Power normalized modes of a CLING fiber (red) compared to modes of a conventional 50-µm step-index fiber (blue) of the same average NA (0.06). The edge index step of the CLING fiber is 1.4511 while the graded portion spans 0.0004 in index to a peak of 1.4515.

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In addition to spatially separating the modes as per Design Rule B, the CLING fiber also satisfies Design Rule C. Consider the case shown in Fig. 2 of a 25 µm LMA fiber carrying 95% of the power in the fundamental mode and 5% in the LP₁₁ mode. A similar analysis was made of a CLING fiber that has the same fundamental mode area as the 25 µm LMA fiber. Under the same conditions as the LMA fiber, the CLING fiber provides a significantly smaller TIPS, as shown in Fig. 8. This occurs for two reasons. First, the fundamental mode of the CLING fiber is more centrally peaked than the Gaussian-like LMA fiber, which is more rounded. This means that the peak of the coherently added fields will be harder to move away from the fiber center in the CLING fiber than in the LMA fiber. Second, the LP₁₁ mode is much broader and flatter in the CLING than in the LMA fiber. This means that it will be harder for the LP₁₁ mode to pull the peak of the coherently added fields away from the center of the fiber.

 

Fig. 8. Optical profiles of a 25 µm LMA fiber and CLING fiber with the same fundamental mode area: LP₀₁ mode field carrying 95% of the power (green), LP₁₁ mode field carrying 5% of the power (blue), combined intensity profile (red), and the fiber core refractive index of the 25 µm LMA fiber profile (purple, dashed for comparison only for CLING data). Orange arrows indicate TIPS.

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The CLING fiber, following both Design Rules B and C, shows a significant increase in TMI threshold in comparison to the conventional step-index LMA fiber type. With a TMI threshold of 725W the CLING fiber shows a 52% increase in the TMI laser threshold over a conventional LMA fiber of the same fundamental mode size (50 µm effective core diameter). Similarly, CLING fiber designed to match the fundamental mode of a 25 µm core LMA fiber resulted in a pump threshold of 1175W, a 68% improvement over the conventional step index fiber. In Section 5, these results are captured and graphically compared to the other results in Fig. 9.

 

Fig. 9. TMI threshold of high-power fiber amplifiers as a function of effective core diameter using various active fiber types: conventional LMA (red), Confined gain (green), Trefoil (blue), and CLING (purple). All configurations are co-pumped and of the same fiber length. Note: the term “effective core diameter” means having the same mode area as an LMA fiber of that core diameter.

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5. Discussion and summary

The TMI threshold results for conventional LMA fiber, from Section 3, and the three novel fiber types from of Section 4 are compiled together and shown in Fig. 9. Each of the new fiber types has a significantly improved TMI threshold over conventional LMA fiber types. The new fiber types have higher TMI threshold at 50 µm effective core diameter (having the same mode area as the fundamental mode of and LMA fiber with a 50 µm core) than LMA fiber at much smaller core diameters. This effectively means that the new fiber types can be used to inhibit nonlinearities without sacrificing TMI threshold. Notably, the TMI threshold for the confined-gain fiber at 50 µm effective diameter is much higher than the threshold of the industry-standard 20 µm LMA fiber.

In the simulations presented, none of the LMA or new fiber types used bend loss for mode filtering. As such, the TMI thresholds may be higher in all cases. The amount of bend loss for the new fiber types and the impact of bend-induced mode deformation remains as a separate study.

Figure 1 revealed that scaling to larger core diameters for high-power applications is simply not viable since the TMI threshold decreases for increasing core diameter. However, the three new fiber classes presented all offer ways to simultaneously increase the core diameter and the TMI threshold.

It must be noted that these new fiber types are not mutually exclusive. For example, confined gain can be used with the CLING fiber to allow it to follow all three Design Rules A, B, and C. This functionality is shown in Fig. 10. Note that the spatial overlap between the gain doping region and the LP₁₁ mode is substantially reduced compared to conventional LMA fiber. A substantial advantage may be leveraged by combining confined gain with the CLING fiber. In addition, there is likely a whole host of new fiber designs that can be developed using the design rules laid out in Section 4.

 

Fig. 10. LP₀₁ (left) and LP₁₁ (right) modes of LMA fiber with conventional gain doping (top) and CLING fiber with confined-gain doping (bottom).

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Finally, these intrinsic TMI-mitigation fiber designs are fully compatible with extrinsic TMI mitigation techniques. Although the physical mechanisms behind the intrinsic and extrinsic TMI mitigation strategies are quite different, the benefits may not be simply additive. Meaning, two technique that each yields a factor of 2 increase may not yield a factor of 4 increase when used in combination.

In summary, an improved spatio-temporal fiber amplifier model for simulating thermal mode instability (TMI) in high-power fiber amplifiers was developed. The model was applied to reveal new physics regarding the thermal physics that is critical to the TMI process, which are not the glass volume or the cooling method, but rather the transit path length of the quantum-defect-defined thermal peak in the fiber amplifier. The new physics and model analysis were applied to create a set of design rules to guide the development of new fiber types specifically for intrinsically mitigating TMI. These rules and the improved model were applied to three new fiber concepts for mitigating TMI in high-power fiber amplifiers. All three fiber types were shown to substantially increase the TMI threshold, up to a factor of 2 in some cases. In addition, all three new fiber classes offer ways to simultaneously increase the core diameter and the TMI threshold.