## Abstract

Fiber lasers are a highly regarded solid-state laser concept due to their high efficiency, beam quality, and easy thermal management. Unfortunately, the performance of high-power fiber-laser systems is challenged by the onset of detrimental nonlinear effects. Their impact can be reduced dramatically by employing fibers with larger mode-field areas. Even though this is an efficient way to mitigate nonlinear effects, maintaining effective single-mode operation, and with it high beam quality, becomes increasingly difficult as the core is enlarged. In this paper the demands and challenges for the design of a very-large-mode-area (VLMA) fiber are discussed. The benefits of using higher-order mode delocalization as the working principle of active double-clad VLMA fibers are described. Finally, a new low-symmetry large-pitch fiber, which is expected to improve the performance of state-of-the-art fiber-laser systems by increasing higher-order mode delocalization, is proposed and thoroughly analyzed.

© 2014 Optical Society of America

## 1. INTRODUCTION

The characteristics of a rare-earth-doped fiber are directly responsible for the performance of a fiber-laser system at high average powers and for its ability to maintain diffraction-limited beam quality and high efficiency [1]. One of the main challenges for high-power fiber-laser systems—especially for pulsed ones—is the onset of detrimental nonlinear effects such as stimulated Brillouin scattering, stimulated Raman scattering, and self-phase modulation [2]. Crucially, the impact of all these nonlinear effects scales inversely with the effective mode-field area. Consequently, scaling the mode-field area is the most effective mitigation strategy for nonlinear effects in optical fibers. Additionally, the impact of nonlinear effects can be further mitigated by using shorter fiber lengths. For MW-level-peak-power fiber-laser systems this has led to the development of rod-type fibers with mode-field diameters (MFDs) of more than 50 μm and typical fiber lengths of about 1 m [2,3], most commonly employing a double-cladding structure for the guidance of a high-power low-brightness pump radiation. In practice, these active fibers support the propagation of a few modes, but they can be adjusted to effectively operate with a single mode to enable high beam quality and beam-pointing stability. Unfortunately, operating few-mode fibers in an effective single mode becomes increasingly difficult as the core size is enlarged. Therefore, fiber designs that aim at scaling the MFD of the fundamental mode (FM) to sizes beyond 50 times the signal wavelength [known as very-large-mode-area (VLMA) fibers], while still maintaining nearly diffraction-limited beam quality, have to incorporate mechanisms that enforce effective single-mode operation. Over the last decade many mechanisms have been proposed and incorporated into advanced VLMA fiber designs with the sole purpose of preferentially amplifying one single mode in a fiber. This can be done by introducing losses to the propagation of the higher-order modes (HOMs), by having structures that provide a higher amplification factor to the FM than to the HOM, or by propagating in a single HOM that ensures reduced mode coupling.

The main two exponents of the first strategy (i.e., HOM attenuation) are the so-called chirally coupled core (CCC) fibers [4] and the leakage-channel fibers (LCFs) [5]. The CCC fibers are characterized by having one or more satellite cores wrapped around the main signal core. This arrangement allows for a resonant coupling of the HOMs of the main core into the satellite core, where the radiation is lost due to very high bending losses. On the other hand, the LCFs rely on an open core structure that provides much higher leakage losses for the HOM than for the FM. This way, after the light has propagated a certain length through this fiber, only the FM will survive. In order to generate the leakage losses, the LCFs have to either dispose of a double-clad structure or exploit bending losses [6].

As mentioned above, the second strategy of enforcing effective single-mode operation is by preferentially amplifying the FM of a fiber. One implementation of this idea is implementing doped areas that are smaller than the fiber core so that the overlap of the HOMs with this doped region (and with it the gain) is significantly reduced with respect to the overlap/gain of the FM [7–9]. However, the most recent and successful implementation of the preferential amplification of the FM in double-clad VLMA fiber is based on a concept known as HOM delocalization [10]. In this approach the fibers are explicitly designed to deform HOMs from the core region. This results in a double benefit: on the one hand the HOMs are more difficult to excite with a Gaussian beam (which are the typical beams found in fiber-laser systems), and, on the other hand, the amplification factor of the HOMs is substantially reduced. Even though, as mentioned above, there are other approaches that can be successfully exploited to achieve single-mode operation in VLMA fibers, the concept of HOM delocalization is one of the newest and most promising ones, and, therefore, it will be thoroughly discussed in this paper. There are currently two main representatives of this approach: the distributed mode filtering (DMF) rod [11], which achieves delocalization by resonantly coupling the HOMs outside of the core, and the large-pitch fibers (LPFs) [10], in which the delocalization of the HOMs is assisted by avoided crossing [12]. Recently a new fiber design that also exploits HOM delocalization assisted by avoided crossings has been proposed: multitrench fiber (MTF) [13]. However, to the best of our knowledge, no experimental demonstration of VLMA MTFs has been published so far.

The underlying idea of the last strategy (i.e., HOM propagation) employs the fact that nearest-neighbor mode coupling in multimode fibers is reduced when propagating in a single HOM, typically a ${\mathrm{LP}}_{0\mathrm{m}}$ mode of high order, instead of the FM. At the same time, bend resistance is greatly increased. Thus, in HOM fibers long-period gratings are used to couple light from the FM to the HOM and, after amplification, vice versa [14,15].

Among all the different VLMA fiber designs proposed so far, the LPF concept [10] is the one that currently holds most of the performance records for pulsed VLMA-fiber-laser systems with nearly diffraction-limited beam quality. For example, several hundred watts average power [16], up to 26 mJ pulse energy [17], and 3.8 GW peak power [18] have been demonstrated with these fibers. Moreover, recently a Tm-doped LPF design has been presented that possesses the largest MFD ($\sim 65\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$) published so far for effectively single-mode fibers around 2 μm emission wavelength [19]. These results justify why LPFs have become in recent years a reference in performance for pulsed fiber-laser systems, to the point that they are used as a benchmark to evaluate novel fiber designs.

The short length of high-peak-power fiber amplifiers combined with the high average power levels that are routinely delivered by these fibers can easily result in peak thermal loads of several 10 W/m. Such high thermal loads give rise to thermal effects that lead to a significant shrinking of the MFD in VLMA fibers [20]. More importantly, these high thermal loads ultimately result in the onset of mode instabilities [1,21]. Mode instabilities refer to the threshold-like degradation of the output beam of an active fiber observed once a certain average power threshold (in the range of some 100 W or even kW) has been reached. Mode instabilities are firmly believed to be induced by the interplay of thermal effects and modal interference in the active fiber [22–24]. The strong impact of thermal effects in the performance of VLMA fibers illustrated by the examples cited above implies that the influence of the thermal load in high-power fiber amplifiers has to be included as an integral part of the evaluation of any future high-power fiber design.

In the first part of this paper the challenges for mode-area scaling while maintaining single-mode operation are investigated employing common step-index fibers (SIFs) as an example. The next part focuses on maintaining effective single-mode operation in few-mode fibers. Based on this general discussion, the benefits of HOM delocalization are analyzed. Finally, an improved LPF with reduced symmetry is proposed that is expected to offer better performance than the standard LPF design.

## 2. MODE-FIELD-AREA SCALING

In the following, some general laws of mode-field-area scaling are pointed out. These considerations will be illustrated for a SIF, but they can be easily transferred to other fiber designs and are, therefore, general. Furthermore, the shape of the FM (intensity profile) and its size relative to the core will be used as parameters that have to be maintained during scaling (in the following referred to as normalized mode shape). By doing this, a similar confinement to the core, a similar amplification characteristic, and a similar beam quality at the output of the fiber are ensured independently of the core size. However, when scaling the fiber this way, other mode properties, such as the effective indices or the propagation losses, will necessarily change with the core size.

As is widely known, the core of a SIF has a slightly higher refractive index than the surrounding cladding, which allows for optical guidance based on total internal reflection. The guiding properties of a SIF can be characterized by the $V$ parameter that is derived from the core diameter $2a$, the wavelength $\lambda $, and the numerical aperture NA, according to the following equation:

Fibers with a $V$ parameter less than 2.405 and an infinitely extended cladding are strictly single-mode, i.e., the core guides exclusively a nearly Gaussian FM [25]. All SIFs that share a specific $V$ parameter possess modes with identical normalized mode shape regardless of their wavelength, core size, or core–cladding index step. Therefore, scaling should be performed with a constant $V$ parameter, which implies that the MFD increases linearly with the core diameter. In order to increase the mode area while leaving the $V$ parameter constant, the NA has to be decreased linearly; i.e., scaling the core size by a factor of 2 requires the NA to be halved. Additionally, in the weakly guiding approximation, the core–cladding index step $\mathrm{\Delta}n$ is proportional to ${\mathrm{NA}}^{2}$. Thus, linearly scaling the core diameter implies a quadratically decreasing index step $\mathrm{\Delta}n$ to maintain a given $V$ parameter.

The validity of this scaling law is not limited to the intentionally designed index steps between the core and cladding of a SIF, but it is also applicable to any other index change [25]. In any waveguide, an index change can be induced by various effects, such as bending [6], the thermal load [26,27], the Kerr nonlinearity, or the so-called resonantly induced refractive index changes [28]. All these effects have to be scaled accordingly, as will be highlighted in the next paragraphs. Even though all of these effects can alter the waveguide properties during operation, the first two are the most common, and therefore our following discussion will revolve around them.

The influence of bending was comprehensively studied in [6], and it has already been shown that mode-area scaling requires the bending radius to be scaled proportional to ${\mathrm{NA}}^{-3}$ [29]. This means that doubling the core diameter and increasing the bending radius by a factor of ${2}^{3}$ yields equivalent normalized mode shapes, as this scales the applied index profile quadratically. The cubic behavior of the bending radius highlights the need for straight rod-type structures for MFDs beyond 50 μm [2,10]. By keeping the fiber straight, the rod-type design avoids any bending-induced mode deformations and reduces mode mixing due to microbendings [30].

The thermally induced index changes that occur in active fibers due to quantum-defect heating during amplification can be approximated by a parabolic index gradient inside the doped area and a logarithmic decay outside of this region [26,28]. With constant material parameters and geometries, the thermal load caused by optical amplification in an active fiber is directly proportional to the thermally induced index gradient. In terms of mode-area scaling, this index gradient has to be treated identically to the index step of SIFs; i.e., the thermal load decreases quadratically when scaling the core size. In other words, doubling the core diameter and reducing the thermal load by a factor of 4, leads to identical normalized mode shapes. This dependence has been experimentally verified in [20].

In Fig. 1, these scaling laws are exemplarily illustrated for a SIF with $V=7$ under thermal load and bending. So, as mentioned before, in order to scale the MFD by a factor of 2 (from 50 to 100 μm in the unbent case) while maintaining the same normalized mode shapes, the core diameter has to be doubled (from 67 to 134 μm in this case), and the index step has to be reduced by a factor of 4 to yield the same $V$ parameter. Additionally, the thermal load also has to be reduced by a factor of 4, and, simultaneously, the bending radius has to be increased by a factor of 8. Applying these scaling principles leads to the same mode set, which is proportionally scaled. In consequence, it can be deduced that larger cores become increasingly more sensitive to perturbations.

The quadratically increasing influence of index changes when scaling the MFD is also valid for photonic-crystal fibers (PCFs). In particular, the tolerance to the index matching between the doped core material and the undoped surrounding cladding material of an active PCF becomes quadratically smaller for larger core diameters [31].

Finally, a general dependence of mode-area scaling on the wavelength can also be deduced from Eq. (1). For a given NA, the core diameter can be scaled linearly with the wavelength. For instance, a wavelength of 2 μm as used in Tm-doped fibers allows for a two times larger MFD compared to 1 μm while maintaining the same NA and $V$ parameter.

All scaling properties are summarized in Table 1. For any mode-area-scalable fiber concept it is vital to study the absolute parameters that most severely alter the mode properties. In the case of active VLMA fibers these are the index-matching accuracy (especially for PCFs), bending, and the thermo-optic effect. Furthermore, in general it is sufficient to calculate these effects for one distinct MFD, e.g., 50 μm, and apply the scaling properties for larger variants or different wavelengths.

## 3. EFFECTIVE SINGLE-MODE OPERATION

This section highlights different concepts for effective single-mode operation.

Single-clad SIFs can be strictly single-mode for $V<2.405$. Theoretically, a SIF can be indefinitely scaled if the $V$ parameter is kept constant, but this would require a linear reduction of the NA or a quadratic reduction of the index contrast, which has some technological limitations, in particular for doped cores. In practice, an accuracy of $\pm 0.01$ can be achieved in the NA, which leads to few-mode guidance for MFDs of more than 20 μm at 1 μm wavelength, i.e., in Yb-doped fibers. A more accurate control of the NA can be achieved by index-guiding PCFs [32]. In these fibers averaging over the small low-index inclusions in the cladding leads to an effective NA, which allows for strictly single-mode (single-clad) fibers with up to 33 μm MFD [33]. For larger cores the scaling requires an even better control of the effective NA than what is achievable today, and, consequently, these fibers become few-mode again.

Effective single-mode operation can be enforced in multimode fibers by suppressing the HOMs as they propagate along the waveguide. A very strong suppression mechanism can arise from the differential modal propagation losses induced by an infinite or absorbing cladding (i.e., single-clad fibers) [34], by bending [6], or by resonances [4,35]. All these mechanisms are based on higher propagation losses for the HOMs compared to the FM. By choosing an appropriate fiber length the HOMs can be suppressed by, e.g., 20 dB, and effective single-mode operation is achieved. However, in general the propagation loss difference between different modes collapses for larger core sizes, which significantly reduces the efficiency of this discrimination mechanism as the core diameter is increased [6,31]. Additionally, since in these fibers the modal energy is typically lost at the outer fiber boundary, the propagation losses are strongly decreased if a thermally induced parabolic index profile is created during active operation, since the modes become progressively better confined around the fiber axis. Moreover, the propagation losses of straight double-clad fibers are negligible as the double-clad structure that strongly confines the pump also confines the signal radiation [34]. However, it can be argued that some sort of propagation losses for individual modes may be achieved by mode mixing [30], but their exact value can hardly be calculated. In conclusion, the HOM discrimination due to propagation losses is not scalable and cannot be applied to most high-power fiber designs.

Theoretically, effective single-mode operation can be realized in few-mode fibers only by adjusting the excitation conditions [30]. A fiber amplifier is typically seeded by an oscillator or amplifier delivering a nearly Gaussian beam profile. Therefore, the modal excitation with a Gaussian beam profile is of crucial interest. This Gaussian beam can be size-, position-, and angle-mismatched with respect to the ideal case, which leads to a multidimensional parameter space for the evaluation of the excitation. In a strictly single-mode fiber any mismatch will only decrease the power content in the FM, but in a multimode fiber this power can be coupled to guided HOMs. To illustrate this effect the excitation of the mode set of a SIF with $V$ parameter of 7 is simulated for a size- and position-matched Gaussian beam, which is tilted by an increasing angle (Fig. 2). It can be seen that one distinct HOM, namely the ${\mathrm{LP}}_{11}$, is predominantly excited for small angle mismatch. For larger angles ${\mathrm{LP}}_{02}$ and ${\mathrm{LP}}_{12}$ are excited as well. Modes of higher orders have progressively finer structured intensity profiles and phase patterns, which makes the accidental excitation of these modes by an incident Gaussian beam more and more unlikely. Pure excitation of the FM becomes increasingly difficult as the core size and, with it, the number of modes are scaled [30].

## 4. HOM DELOCALIZATION

As briefly mentioned in Section 1, a promising novel approach for effective single-mode operation is HOM delocalization. This concept influences simultaneously the excitation and amplification of modes inside a few-mode fiber. The LPF was the first representative of a fiber design exploiting HOM delocalization, and it showed a significant increase in the mode-instability threshold [10,16]. In the meantime this operation principle has been transferred to other fiber designs, namely the DMF fiber [11] and MTF [13], both based on resonant effects. These different approaches highlight the fundamental nature of this operation principle.

The principle of operation of HOM delocalization will be discussed in the following. Figure 3 illustrates the mode excitation of a fiber employing HOM delocalization, in this exemplary case and without loss of generality, and a LPF, a structure that is discussed in detail later. As the FM maintains a nearly Gaussian profile, its excitation is very similar for LPF (Fig. 3) and SIF (Fig. 2). However, the HOM excitation is fairly different. While for a SIF the HOM power content (shaded area) is distributed among only a few distinct modes, the strong HOM deformations distribute the same overall HOM power content into many different HOMs. This difference seems to be negligible at first glance, but the combination of excitation and amplification reveals a very strong suppression of HOMs in a fiber amplifier.

The gain of an individual mode in an active fiber is mainly determined by the overlap $\gamma $ of the mode with the doped region,

This parameter can be used to evaluate the quality of HOM suppression by analyzing the differential amplification of each relevant HOM with respect to the FM. In the simplest approximation the amplification of different modes can be calculated by assuming a small-signal gain behavior:

It has to be noted that most high-power VLMA fibers are not operated in small-signal gain, but they operate near or above the saturation intensity instead [36]. However, even in saturated cases, the general conclusions extracted from the small-signal gain case are valid. Typical rod-type fibers possess a small-signal gain greater than 30 dB/m, but their effective gain is lowered to about 20 dB/m due to saturation effects. Consequently, the amplification characteristics of these fibers can be approximated, in the simplest case, by assuming an effective small signal gain of 20 dB (${g}_{0}l=4.6$). In order to convey an idea of the impact of the different modal overlaps with the doped region, some examples can be calculated. Thus, while a mode with 100% overlap would experience an amplification factor of 100, modes with 85%, 40%, and 20% possess amplification factors of 50, 6, and just 3, respectively. Clearly, modes with lower overlap values can be dramatically suppressed at the output of the fiber (with respect to the power content of the FM) by differential gain.

The combination of excitation and amplification is illustrated in Fig. 4. The basis for this graph is the power fraction in each mode (taken from Figs. 2 and 3), which is amplified according to their individual doping overlap [Eqs. (2) and (3)]. The differential amplification of each mode leads to a very strong suppression of individual HOMs. Clearly, not only the power content of each individual mode but also the combined HOM content (shaded area) is dramatically reduced for the LPF employing HOM delocalization compared to a few-mode SIF. It should be noted that this evaluation is only an example to show the general strength of this effect. With higher gain values the suppression of HOMs is even stronger. Furthermore, effects such as thermal load may alter the overlap with the doped region of the modes during active operation, an effect that will be discussed later on.

Considering the excitation and amplification in combination brings an interesting fork in the design process of an active fiber. The question is which HOMs should be further considered in the design process. This is a crucial decision that determines how to optimize a given design. Interestingly the set of relevant HOMs is different if the fiber is going to be used as an oscillator or as an amplifier. In the first case the relevant HOMs are those exhibiting the higher overlap with the doped region, since these will see the highest gain. On the other hand, if the fiber is going to be used as an amplifier, many modes can be excluded due to their negligible “accidental” excitability. For instance, ${\mathrm{LP}}_{31}$-like HOMs can have higher overlaps with the doped region than ${\mathrm{LP}}_{1\mathrm{m}}$-like HOMs [37], but these fine structured modes are difficult to excite with a Gaussian-like beam, and, therefore, they have no impact on the single-mode operation of a fiber amplifier.

Typically the ${\mathrm{LP}}_{1\mathrm{m}}$-like HOMs are the most relevant modes since they are the first to be strongly excited in a fiber with a slight misalignment of the seed beam. Furthermore, these modes usually appear during mode instabilities [1]. Therefore, the suppression of ${\mathrm{LP}}_{1\mathrm{m}}$-like HOMs is currently one of the main goals of advanced high-power VLMA fibers designed to be used primarily as amplifiers.

In conclusion, HOM delocalization can be a very strong mechanism to enforce effective single-mode operation of a fiber amplifier. A further refinement for HOM delocalizing fibers can be thought of: introducing absorptive regions in the pump core of these fibers. In such fibers the delocalized HOMs possess a large overlap with the absorbing material, whereas the FM will only have very limited contact with it. Thus, with proper gain/loss management [38], the HOMs can be suppressed further as they propagate along the fiber.

## 5. LARGE-PITCH FIBER

The LPF concept exploits the enormous design flexibility of PCFs [32,39] to achieve and maximize HOM delocalization. While the modal-sieve concept employing propagation losses can be used for HOM suppression of a single-clad PCF [40], this modal discrimination mechanism cannot be exploited any longer for double-clad fibers. Nevertheless, the low-index structure can still be arranged in such a way that an efficient HOM delocalization can be achieved. The principle of HOM deformation has already been discussed (Figs. 3 and 4), but the LPF structure employs additional features worthwhile to highlight.

The mode set of an exemplary LPF design for increasing thermal load is illustrated in Fig. 5. While for a SIF the ${\mathrm{LP}}_{11}$ typically possesses the highest overlap among the ${\mathrm{LP}}_{1\mathrm{m}}$-modes, a ${\mathrm{LP}}_{13}$-like HOM has the highest overlap for the LPF structure. The exchange of mode orders has a very beneficial effect, when applying thermal load to this structure (i.e., in active operation). As all ${\mathrm{LP}}_{1\mathrm{m}}$-like HOMs share the same symmetry class they are involved in avoided crossings [12,41]. Therefore, the ${\mathrm{LP}}_{13}$-like HOM is deformed and the ${\mathrm{LP}}_{12}$-like HOM takes over its role as the most confined HOM for an increasing thermal load. A further increase of thermal load leads to the final ${\mathrm{LP}}_{11}$-like HOM taking over the role as the HOM with the highest overlap. In consequence, avoided crossings can decrease the HOM overlap for a large range of thermal loads, and, therefore, the effect of HOM delocalization becomes even stronger. The evaluation of HOM delocalization as performed in Fig. 4 (i.e., a “cold” fiber) is a worst-case scenario for a LPF.

Another outstanding property of the LPF design is its scalability. The modal intensity profiles of a LPF behave, under proportional scaling of the structure, like those of an ideal SIF with a constant $V$ parameter. This effect is well known for endlessly single-mode PCFs [40], where the effective $V$ parameter of an approximately infinite PCF becomes independent of the hole-to-hole spacing (pitch) $\mathrm{\Lambda}$ for relative hole diameters $d/\mathrm{\Lambda}>0.1$ and $\mathrm{\Lambda}>10\xb7\lambda $. Therefore, the LPF structure can be scaled proportionally to larger cores while, e.g., the normalized mode shapes are maintained. Additionally, this also implies that the LPF design operates over a very large wavelength region, since increasing the pitch is equivalent to decreasing the wavelength. This scalability is not a general feature of HOM delocalization but a special feature of the nonresonant design approach of the LPF.

## 6. LOW-SYMMETRY LARGE-PITCH FIBER

Despite the strong effective single-mode operation of a LPF, the thermal load will (for a high enough power) end up strongly modifying the waveguide and with it its guiding properties, in particular the amount of HOM delocalization. In this section we propose a new LPF design with increased HOM delocalization, which improves the performance at high thermal loads compared to the standard LPF.

The symmetry of a waveguide structure influences the symmetry of the guided modes [42]. Even though this is true for all modes, the FM is the one being least influenced by symmetry changes of the cladding; instead it will adapt itself to the inner core geometry. In contrast, the ${\mathrm{LP}}_{1\mathrm{m}}$-like HOMs prefer mirror symmetries. Thus, by altering the fiber symmetry, these classes of modes can be further deformed to increase the HOM delocalization [43].

On the way to less symmetric structures spirals are an interesting idea, which has been proposed by our group [44] and adopted by others [37,45]. In theory spiral structures can be asymmetric without any mirror or rotational symmetries, but the HOM delocalization is mainly affected by geometries with a structure size comparable to the transverse wavelength of the modes [39,43]. For some spiral structures it can be seen that the HOMs are swirled out of the core region, which can enhance the HOM delocalization.

A design implementing such a swirling structure, but still based on a hexagonal lattice for standard stack-and-draw technique, is proposed in Fig. 6. This low-symmetry LPF exploits holes of different diameters to increase HOM delocalization.

## 7. METHODOLOGY AND COMPARABILITY

The evaluation of fiber designs is a very difficult task since it involves sweeping a multidimensional parameter field. Furthermore, the simulation of a fiber amplifier can become very challenging and computationally intense as soon as effects such as thermal waveguide changes, mode coupling, and mode instabilities are included. In fact, the development of such simulation tools is a topic of ongoing research in several groups around the world. In many cases the inclusion of these effects requires some assumptions, which have not been understood in detail and/or have yet to be verified experimentally. Herein, we propose a simple, yet powerful, method to estimate the performance of new fiber designs, which can be deduced purely from the fiber design (i.e., there is no need to simulate the laser operation of the fiber). This method is based on the experimental observation showing that a reduced overlap of the HOMs with the doped region is beneficial in terms of the mode-instability threshold and that thermal waveguide changes have a strong impact on the guiding properties of a high-power fiber design. There are many additional parameters that can and should be considered for the design of a fiber amplifier or laser—such as the signal and pump wavelength, doping concentration, pump absorption, and fiber length—but again, this complicates the evaluation process and distracts from the fundamental operating principle of different fiber designs. In fact, none of this is strictly necessary to carry out an estimation of the performance level of a new fiber design. According to our simulations and experimental observations, the evolution of the modal overlap with the doped region with an increasing thermal load seems to be a very important property that allows performing such first estimations of the expected performance of a new high-power fiber design based on HOM delocalization.

For this evaluation the refractive index profile of a given fiber design is arranged and the doped area is defined. Based on the analytic approximations from [26,28] the thermally induced index change can be added and the corresponding mode set is calculated. Finally, Eq. (2) is used to calculate the overlap with the doped region for each mode.

Additionally to the methodology, the comparability of different fiber designs is also a very error-prone task. A proper choice of parameters is necessary to grant comparability among different fiber designs. As has been shown in the first part, the scaling laws require the MFD to be constant. Furthermore, for the design of an amplifier, the evolution of the signal power along the fiber (which ultimately determines the strength of nonlinear effects) should be comparable, by maintaining the doped area, pump-clad area, doping concentration, and FM overlap with the doped region. On the other hand, geometrical parameters such as the core diameter or the hole size do not have to be matched as they bear no meaning for physical effects.

For the following evaluation a MFD of 50 μm, a cladding diameter of 200 μm, a doped region with 50 μm diameter, and a FM overlap with a doped region of about 85% are fixed. These parameters are achieved for a LPF with a pitch of 35 μm and $d/\mathrm{\Lambda}=0.3$. For the low-symmetry LPF the relative hole diameters can be chosen as 0.2 and 0.4 for the smaller and larger holes, respectively. All simulations are performed at a wavelength of 1030 nm, which implies a Yb-doped fiber amplifier. Such systems have reached average output powers of 300 W with fiber lengths of 1 m. In the counterpumped configuration an exponential evolution of the thermal load from some W/m up to about 40 W/m can be expected, which highlights the necessity to evaluate this thermal range.

With the scaling laws given above, this evaluation is equivalent to a fiber with, e.g., a two times larger core diameter and a four times smaller thermal load. More interestingly, this evaluation can be directly transferred to a Tm-based fiber amplifier with a signal around 2 μm and the same MFD but with four times higher thermal load—these values suit the experimental situation very well due to the increased quantum defect of Tm-doped silica pumped around 790 nm.

## 8. PERFORMANCE EVALUATION

Figure 7 depicts the evolution of the modal overlaps with increasing thermal load for the LPF fiber design (blue lines) and the low-symmetry LPF (green lines). As already illustrated in Fig. 3, the general trend of continually increased HOM overlap with increasing thermal load (e.g., as would be seen in a SIF) is interrupted by two avoided crossings. Over a large thermal load range (0–20 W/m) the performance of the LPF is even better than that of the “cold” fiber. Consequently the HOM overlap with the doped region averaged over the whole thermal load range considered herein is 45%. The proposed low-symmetry LPF (Fig. 6, green lines) allows us to enhance the HOM delocalization even further, leading to a mean HOM overlap of just 32%. In combination with the exponential gain evolution and excitation, this design is expected to show an even more robust effective single-mode operation and higher mode-instability threshold.

For the experimental realization of a fiber design the impact of manufacturing tolerances has to be studied as well. For a PCF the index matching between the doped region and the host material is crucial [31]. An index mismatching accuracy of $-2\mathrm{e}-5$ has been achieved experimentally [20]. Therefore, this accuracy has been assumed in the following evaluation, summarized in Fig. 8. A lower core index can deform the FM of the waveguide. Due to the combination of smaller and larger air holes the low-symmetry LPF is more sensitive to an index depression of the doped region than the standard LPF. This can affect the efficiency and beam quality of the low-symmetry fiber design for very low average powers (even though the relative air hole sizes can be fine-tuned to get rid of this effect at low powers). However, in high-power operation the thermal load necessary to overcome these deformations is easily reached and the index depression will hardly affect the performance. At the same time, the HOM suppression is improved as the HOMs are pushed out of the core region. Therefore, the index depression effectively shifts the region with strong HOM suppression to higher thermal loads compared to a perfectly matched fiber (Fig. 7). For the range of thermal loads studied herein the mean HOM overlap is 37% and 26% for the LPF and low-symmetry LPF, respectively. Therefore, the improved HOM delocalization especially for high thermal loads of the low-symmetry LPF is expected to outweigh any possible performance degradation at low powers.

Finally, and as mentioned above, the beam quality is a very important parameter. One might expect a decrease in beam quality for the asymmetric design, but due to the partly increased hole sizes, the FM confinement and beam quality are maintained with respect to the standard LPF. The calculated FM of both designs possesses a theoretical ${\mathrm{M}}^{2}<1.2$.

## 9. CONCLUSION

Fiber-laser systems have shown their benefits regarding high efficiency, diffraction-limited beam quality, and easy thermal management. However, the long interaction length of the tightly confined light with the material of the fiber makes these waveguides very sensitive to nonlinear effects. Thus, the performance evolution of fiber-laser systems has been enabled by the development of fibers that can mitigate nonlinear effects. Over the years it has become clear that the most effective technique for the mitigation of nonlinear effects in an active fiber is scaling of the MFD. However, as highlighted in the paper, there are some physical laws that are obeyed when scaling the size of a fiber, and they result in the fiber becoming more and more sensitive to external perturbations. Additionally, with the current technological limitations fibers with very large MFDs ($>50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$) become few-mode. Therefore, in order to avoid a penalization in the beam quality and/or pointing stability of the radiation emitted by fiber-laser systems, advanced techniques are required to preserve effective single-mode operation even with VLMAs. From the techniques proposed so far so-called HOM delocalization seems to be the most robust and promising. In this technique, the inner structure of the fiber is designed in such a way that the HOMs are pushed away from the core and/or doped region of the fiber. As thoroughly discussed above, this brings a twofold benefit: on the one hand the HOMs have a lower overlap with the doped region and, therefore, undergo a significantly lower amplification factor than the FM; on the other hand, the delocalized HOMs cannot be efficiently excited by an incoming Gaussian beam, which results in the energy being spread over many HOMs instead of being concentrated in just one. The distribution of power in many HOMs, as has been shown in the paper, is advantageous since it results in a strong preferential amplification of the FM. Additionally, HOM delocalization is strongly suspected to be able to increase the mode-instability threshold in fiber-laser systems. HOM delocalization was first exploited in LPF and later incorporated in other designs such as the DMF rod or the MTF. Currently, the LPF design has enabled most of the performance records in ultrafast fiber-laser systems with several hundred watts average power, up to 26 mJ pulse energy, and 3.8 GW peak power. Another outstanding property of the LPF design is its scalability: the LPF structure can be scaled proportionally to larger cores while the normalized mode shapes are maintained.

A further refinement of the LPF design has been presented in this paper, the so-called low-symmetry LPF. The inner structure of this design has a considerably reduced number of symmetries resulting in a stronger delocalization of the HOMs. For the first time, to the best of our knowledge, a comparative analysis of different fiber designs including index depressions of the doped region and a variable thermal load has been presented. From this study it can be expected that a low-symmetry LPF can achieve an improved delocalization of the HOMs over a broader thermal load range than the standard LPF, which should enable more robust effective single-mode operation.

## FUNDING INFORMATION

German Federal Ministry of Education and Research (BMBF); European Research Council (ERC) (617173 ACOPS and240460 PECS); Thuringian Ministry for Economy, Labour and Technology (TMWAT, Project No. 2011 FGR 0103); European Social Fund (ESF).

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