## Abstract

A new class of optical fiber, the SHARC fiber, is analyzed in a high-power fiber amplifier geometry using the gain-filtering properties of confined-gain dopants. The high-aspect-ratio (~30:1) rectangular core allows mode-area scaling well beyond 10,000 μm^{2}, which is critical to high-pulse-energy or narrow-linewidth high-power fiber amplifiers. While SHARC fibers offer modally dependent edge loss at the wide “semi-guiding” edge of the waveguide, the inclusion of gain filtering adds further modal discrimination arising from the variation of the spatial overlap of the gain with the various modes. Both methods are geometric in form, such that the combination provides nearly unlimited scalability in mode area. Simulations show that for kW-class fiber amplifiers, only the fundamental mode experiences net gain (15 dB), resulting in outstanding beam quality. Further, misalignment of the seed beam due to offset, magnification, and tilt are shown to result in a small (few percent) efficiency penalty while maintaining kW-level output with 99% of the power in the fundamental mode for all cases.

© 2012 OSA

## 1. Introduction

High-power and high-energy fiber lasers and amplifiers have greatly benefitted from large-mode-area (LMA) fibers [1]. Since nonlinear effects typically depend on intensity, scaling to larger mode areas means that higher cw and peak powers can be carried in the fiber with the same intensities as in singe-mode fibers. Since LMA fibers are nominally multimode in nature, some sort of mode filtering is required in order to deliver an output beam having the desired beam quality that is provided exclusively by the fundamental (lowest order) mode.

Numerous LMA fiber architectures have been developed to provide the requisite mode filtering, such as low-NA fibers with bend loss [2], chirally-coupled core fiber [3], higher-order-mode (HOM) fiber [4], photonic crystal rod-type fibers [5], leakage-channel fibers [6], and gain filtering via confined-gain dopants [7]. Although each fiber architecture has demonstrated scaling beyond the conventional single-mode-fiber limit, each also has drawbacks that prohibit indefinite scaling, even if only due to practical packaging considerations.

A semi-guiding high-aspect-ratio-core (SHARC) fiber was recently introduced [8] as a new fiber architecture that combines the beneficial aspects of solid-state slab lasers and fiber lasers. This new class of fiber, which incorporates a rectangular core, was modeled extensively to elucidate its mode-filtering mechanism, to demonstrate its potential as a very-large mode area (VLMA) fiber, and to understand the limitations of its operation. Passive SHARC fibers were shown to provide a modal discrimination mechanism in the wide core dimension that does not rely on coiling the fiber, a relative modal discrimination that is independent of the core area, a record-breaking mode-area scaling beyond 10,000 μm^{2}, and the ability to form a compact coiled package without impacting either the propagation loss, the mode content, or the spatial overlap of the fundamental mode with the gain profile. However, all previous effort was aimed at understanding the passive SHARC fiber.

In this paper, we explore the benefits of the SHARC fiber in high-power fiber amplifiers. The specific architecture adopted for this case is the inclusion of gain filtering through the spatial distribution of gain dopants. Specifically, we show that gain-filtered SHARC fiber can provide kW-level amplification with loose tolerances for seed beam injection and >99% fundamental-mode content at the amplifier output under all conditions.

This paper is organized as follows. In Section 2, this new class of active semi-guiding high-aspect-ratio-core (SHARC) fibers is reviewed along with the gain-filtering benefits of tailored dopant profiles. Detailed analytic modeling results, which elucidate the principles and fundamental operation of the active fiber and serve as a guide in the fiber design, are described in Section 3. In Section 4, intense numerical simulations confirm the analytical predictions and further demonstrate the strength of the gain-filtered SHARC concept, including strict evaluation of alignment tolerances. A discussion of practical implications of this new class of active fibers is presented in Section 5, with concluding remarks following in Section 6.

## 2. Semi-guiding high-aspect-ratio fiber and gain filtering

The semi-guiding high-aspect-ratio core (SHARC) fiber concept schematically shown in Fig. 1
is a radical departure from conventional fiber designs. It embodies a high-aspect-ratio rectangular core that enables scaling to a very large area, up to 30,000 μm^{2} or more, while retaining a thin, mechanically flexible narrow dimension. Despite the large core area, a SHARC fiber can still be coiled in the fast-axis direction, depicted in Fig. 1. This enables SHARC fibers to form a compact package comparable to that of state-of-the-art LMA fibers (which have significantly smaller mode area ~400 μm^{2}). Despite this common packaging strategy, coiled SHARC fibers do not suffer from the significant performance challenges of conventional LMA fibers as the core area is increased. In particular, as the core size of conventional LMA fibers increases, the mode area becomes severely reduced due to bend-induced mode deformation [9, 10]. In contrast, the SHARC fiber increases the core area by expanding only the slow-axis dimension, while maintaining a constant thickness in the fast-axis direction in which the coiling occurs. Although mode deformation exists, in principle, in the fast-axis dimension of the SHARC fiber, the small fast-axis waveguide dimension (~15 µm) prohibits any significant mode deformation similar to what is observed in small-core (~20 µm) LMA fibers.

As denoted by the key descriptor “semi-guiding,” this fiber design specifies conventional index-based guiding via total internal reflection (TIR) in only one transverse dimension (the “fast axis,” as labeled in Fig. 1). Specifically, and as Fig. 1 indicates, index-based TIR guiding only occurs along the two large surfaces of the rectangular core where they are in contact with the fast-axis cladding layers. The slow-axis core edges are designed not to support TIR; in fact, the index step at the core-edge boundary is kept very small (Δn ~100 ppm or less), such that all modes suffer propagation losses as they radiate out beyond the slow-axis core edges. Although the fast-axis direction can be designed to be single-mode, slow-axis mode control is achieved in the passive fiber by exploiting the natural process of “loss filtering” [8]. This process arises from the fact that all slow-axis modes suffer radiation loss into the cladding through the “open” core edges, with the lowest-order mode having significantly less loss than any other mode. Using this approach, the mode-dependent loss can effectively discriminate in favor of the lowest-order mode, analogous to the desirable mode filtering in other LMA fibers.

Perhaps most importantly, the SHARC-fiber core-area scaling necessary to access higher power levels is achieved by increasing only the slow-axis direction, and this proceeds at a practically fixed core-to-cladding ratio. This fact allows power to be scaled with a constant effective pump absorption coefficient and therefore a constant total fiber length. The SHARC fiber architecture also scales output power at a constant pump-etendue per output watt, thereby ensuring the possibility of generating higher output power levels without having to invent new pump-diode packages with increasingly higher brightness. As a quantitative example, carrying 3-kW of single-frequency optical power will require core dimensions of 20 μm × 1.5 mm, for a total core area of 30,000 μm^{2}, which is equivalent to a circular core having a diameter of ~200 μm. In this example, stimulated Brillouin scattering (SBS) suppression occurs by virtue of the large core area and low intensity, which lead to an SBS threshold power in excess of 3 kW even for a kHz-range laser bandwidth. Hence, in order to deliver multi-kW-level optical powers, SHARC fibers do not require additional SBS suppression techniques such as multi-GHz signal modulation [11, 12], with its associated system complexity, or acoustic waveguide management [13–15].

The inclusion of gain into the SHARC fiber provides another opportunity for mode control, as depicted in Fig. 1. Gain filtering, a process that provides modal discrimination via gain instead of loss, has recently been investigated as a mode-control method in high-power fiber amplifiers [7, 16, 17]. Nominally, this is achieved by spatially tailoring the gain dopant profile, such as the step profile depicted in Fig. 1.

The top row of Fig. 2 depicts the impact of spatial gain saturation in conventional optical fiber amplifiers. Nominally seeded by the fundamental mode (left), the amplifier gain (red) is saturated as the optical power grows, leaving a spatial hole “burnt” into the gain profile at the center of the fiber where the fundamental mode intensity is the highest (center image). Higher order modes (HOMs), most of which have intensity nulls as the center of the fiber, can extract the gain at the edges of the fiber, resulting in higher net gain for the HOMs and degraded beam quality at the output of the fiber amplifier. By confining the gain to the central portion of the waveguide while maintaining the same refractive index profile (Fig. 2., bottom row), the gain extraction by the fundamental mode saturates nearly all of the gain, leaving no gain at the waveguide edges for the HOMs to exploit.

The maximum benefit of gain filtering is obtained by designing the transverse ytterbium dopant profile to optimize the overlap of the gain with the fundamental mode while minimizing the gain-overlap of all other modes, performing a global optimization at all levels of saturation. It has been recently shown that gain filtering in fiber amplifiers can lead to better beam quality than the injected seed beam [7, 17].

Gain filtering possesses two unique features that set it apart from all other mode-control techniques: lossless filtering and geometrical overlap. The first factor is unique amongst all other mode-filtering techniques that, while providing loss to higher-order modes, also provide loss for the fundamental mode. This makes gain filtering the highest efficiency mode-filtering method available. Second, mode filtering relies on the geometric overlap of the modes with the gain profile. Rather than relying on the difference between modal indices that necessarily decreases with increasing core area, gain filtering is indefinitely scalable, since the mode profiles essentially do not change with increasing core area.

One anticipated drawback to gain filtering in round fibers is the aforementioned mode deformation, where the mode becomes compressed towards the outside edge of the bend for large core diameters. Although the reduced mode size is detrimental to most LMA fiber applications, for very large cores (~100 µm) the displacement of the mode towards the edge of the waveguide can reduce the effectiveness of gain filtering by altering the overlap of the deformed modes with the centralized gain region [7].

The SHARC fiber offers a unique advantage that can exploit gain filtering without this packaging limitation. The SHARC fiber is coiled in the fast-axis direction, but the gain filtering is applied in the slow-axis direction, as depicted in Fig. 1. Therefore, no slow-axis mode offset will be incurred, and the mode overlap with the gain will remain unchanged regardless of core area or coiling diameter. Consequently, the integration of gain filtering into the SHARC fiber yields the ideal architecture for significant core-area scaling of high-power fiber amplifiers to 30,000 µm^{2} and beyond.

## 3. SHARC fiber amplifier analytic calculations

As was discussed in detail in [8], the SHARC fiber geometry lends itself nicely to separation of variables such that the fast- and slow-axis physics can be handled nearly independently of each other. This makes direct analytical modeling possible, from which the primary physics can be obtained. The validity of this assumption and its benefits were confirmed with rigorous three-dimensional beam propagation modeling (BPM) simulations [8].

The first analysis pertains to the impact of gain filtering on the SHARC modes. Simplistically, the net gain experienced by a particular mode is mathematically given by the spatial overlap of the mode with the available gain. In practical terms, the available gain is the nominal gain as saturated by all of the modes in the fiber. For the case of interest, the fundamental mode will carry most of the power in the fiber. As such, the differential gain g_{k} experienced by a given mode *k* as it propagates along the z-direction is given by the equation

_{ss}is small-signal gain profile, I

_{sat}is the saturation intensity, Φ

_{k}is the profile of the

*k*

^{th}mode (power-normalized to unity), and I

_{0}(x,z) is the intensity of the fundamental mode. Note that this term carries both power and spatial dependence such that the gain will be

*locally*saturated in the transverse dimension.

Using Eq. (1), the differential gain of each mode can be calculated as a function of saturation level. Figure 3 shows the results of these calculations at various saturation levels for flat-top and Gaussian gain profiles of varying distribution width. In each case, the gains are normalized to that of the fundamental mode, which is therefore represented by a black dashed line at unity. The other (higher-order) modes are labeled in the upper left figure.

For the cases of no saturation (I_{0}/I_{sat} = 0, bottom figures), the plots simply represent the overlap of the modes with the gain dopant profile. As would be expected in this case, the fundamental mode experiences the highest gain when the waveguide is doped to match the mode profile. However, it has been previously shown that saturation can drastically change this picture, since the gain becomes locally saturated where the mode has the highest intensity (and therefore extracts the highest gain locally) [7, 18]. Similar to the case for conventional (round) optical fibers [7], the optimum operation condition for mode discrimination changes with local gain saturation. All saturated (I/I_{sat} > 0) plots in Fig. 3 show that the fundamental mode has the *smallest* gain of all calculated modes when the normalized gain width is unity. However, similar to the case of round fibers, by confining the gain to the central portion of the waveguide where the fundamental mode has the highest intensity, the saturation happens more uniformly, without leaving residual gain near the edge of the waveguide for higher-order modes to exploit. This is clearly observable in all plots of Fig. 3. For the flat-top case (left-hand side), the optimal gain width is ~45% of the waveguide width, allowing the fundamental mode 1.4x higher differential gain than any other mode in the fiber.

In contrast to the case of conventional round fibers where the flat-top profile provides better mode discrimination than Gaussian gain profiles [7], the Gaussian gain profile provides slightly better performance than flat-top gain profiles in SHARC fibers. At a relative gain width of 0.35, the fundamental mode has 1.6x higher gain than any other mode in the fiber. Keep in mind that these are *differential* gain values that simplistically lead to exponential gain. As such, the modal discrimination will be much larger than 60%, as will be calculated shortly.

Although the Gaussian gain profile yields higher modal discrimination, the remainder of this paper will explore the optimized flat-top gain width (45% of the waveguide width). Such flat-top confined gain fibers have already been fabricated in round fiber geometries [16, 17].

To analytically calculate the modal discrimination in a realistic SHARC fiber amplifier using gain filtering, it is necessary to integrate the gain along the fiber length; the saturation condition in the amplifier changes as a function of propagation distance because the signal experiences gain as it propagates. The analysis therefore needs to proceed in several steps. First the local nominal gain of the amplifier is calculated as a function of propagation distance into the fiber. From this, the gain experienced by each mode is calculated according to Eq. (1). Finally, the integrated gain of each mode along the length of the whole amplifier is combined with the SHARC fiber’s edge loss to calculate the net modal gain in the fiber amplifier.

Optical power evolution in conventional laser media proceeds according to

where P_{k}is the power in the

*k*-th mode, g

_{k}is given by Eq. (1), and α

_{k}is the modal loss. For rare-earth ionic systems and resonant (in-band) pumping conditions, the laser kinetics behave as a quasi-three-level system. For such a type of system, the saturation intensity for signal wavelength is affected by saturation of the pump transition, and the

*effective*signal saturation intensity becomes dependent on the pump saturation level [19]. Since the pump intensity I

^{pump}(z) necessarily varies along the fiber due to absorption by the ytterbium ions, the effective signal saturation also becomes z-dependent, taking the following formHere I

_{sat}= hν

_{s}/σ

_{s}τ and I

^{p}

_{sat}= hν

_{p}/σ

_{p}τ are the nominal saturation intensities for signal and pump defined by the upper manifold lifetime τ, and the transition cross sections σ

_{s}and σ

_{p}at corresponding signal and pump wavelengths λ

_{s}= c/ν

_{s}and λ

_{p}= c/ν

_{p}, respectively.

In the quasi-three-level kinetics model, the small signal gain factor g_{ss} depends on the doping density and the signal wavelength λ_{s}, but is also nominally dependent on the pump saturation level I^{pump}/I^{p}_{sat}. However, this dependence approaches a constant value for typical conditions of strong pump saturation (the case relevant to high-power Yb:fiber lasers) and can therefore be ignored, allowing g_{ss} to be constant along the length of the fiber.

To first order, bi-directional pumping leads to nearly uniform pump distribution along the length of the fiber. Consequently, Eq. (3) can be taken as constant along the fiber by assuming proper values for the pump intensity and its saturation intensity. In this approximation, Eq. (2) can be solved for the fundamental mode assuming an amplifier with a fixed gain, and g_{ss} as the constant fitting parameter. Typical fiber amplifier performance (gain = 30, efficiency = 80%) and a 1-kW nominal output power lead to the starting values listed in Table 1
. The geometrical cross-sections are taken from the core and cladding of the fabricated SHARC fiber described in [8].

Using the values listed in Table 1 with the emission and absorption cross-sections calculated for ytterbium-doped silica glass fiber [20], a small signal gain of 12.9 dB/m is obtained by numerically solving Eq. (2). Note that this value is the *modal* gain. Given the optimized gain width of 45% of the waveguide width, the *material* gain required to yield the 30-gain amplifier (assuming fundamental-mode operation) is 18.7 dB. Note that this value is typical for a ytterbium-doping level of ~0.2 wt%, a level much lower than is conventionally required for high-power dual clad fiber lasers due to the very high (2.5%) core/cladding area ratio inherent in the SHARC fiber geometry.

The final piece to analytically modeling a SHARC fiber amplifier is the inclusion of the edge loss with the relative modal gains calculated via Eq. (1). As described in the introduction and in more detail in [8], the modal loss due to the non-TIR waveguide edges is given by

_{0}is the effective (1-D) index in the slow-axis core, and n

_{1}is the complex index of the slow-axis cladding that includes the diffractive loss. Combining Eq. (4) with Eqs. (2) and (3) allows prediction of the net modal gains integrated through the SHARC fiber amplifier as a function of the index-step in the slow-axis waveguide. The results, shown in Fig. 4 , clearly demonstrate the high modal discrimination expected from SHARC fiber amplifiers employing gain filtering. Not only does the gain-filtered SHARC fiber exhibit modal discrimination in excess of 10 dB over most of the range, but there is a very large practical fabrication range [21] of index step (~400 ppm) for which

*only*the fundamental mode experiences net gain, indicated by the shaded region in Fig. 4. This combination of loss and gain filtering therefore enables single-transverse-mode operation for very multimode [N

_{modes}≈(w/λ)NA ≈50 slow axis modes] fiber amplifier.

Figure 4 shows that the modal discrimination in the SHARC fiber amplifier will be in the range of 12-15 dB. This is 3-4x larger than the modal discrimination of the passive SHARC fiber alone (see Fig. 5 of [8]), implying that the second mode is being eliminated primarily via gain filtering. Since the modal discrimination provided by gain filtering depends on the amplifier gain and not its length, the slow-axis core dimension of the SHARC fiber amplifier can therefore be expanded nearly indefinitely without increasing the fiber length and therefore compromising nonlinear thresholds. Figure 4 demonstrates the potential of the SHARC fiber amplifiers in providing extremely large mode areas without compromising modal discrimination, leading to excellent output beam quality at kW-level output powers.

## 4. SHARC fiber amplifier numerical simulations

In order to confirm the behavior of the SHARC fiber amplifier predicted by the analytic modeling, full numerical simulations of the active SHARC fiber were performed using the Beam Propagation Method (BPM) [22]. The primary advantage of this method is its ability to model spatially dependent structures, such as the refractive index and gain profiles, with an arbitrary launch field. The output of the model describes the complete spatial profile of the optical field without having *a priori* knowledge of the modes of the fiber. In this method, the paraxial wave equation is used to model the signal beam along the fiber axis, z, as

_{0}= 2π/λ and n

_{co}is the refractive index of the waveguide core. On the right hand side of Eq. (5), the first term represents diffraction, where ${\nabla}_{T}^{2}$ is the transverse Laplacian, and the second term accounts for the cross-sectional refractive index profile, n(x,y). The last term incorporates the spatial dependence of the gain doping profile through g

_{ss}(x,y) and accounts for spatially localized saturable gain in three dimensions.

Equation (5) was solved with a finite-difference scheme (FD-BPM) [23] using the Alternating Direction Implicit (ADI) method [24]. The need to model long propagation lengths required that transparent boundary conditions be implemented [25]. The geometry modeled in the simulations corresponds to the geometry of Fig. 1, with the specific numerical values listed in Table 2
. Note that the value of I_{sat} used in the simulations is the conventionally calculated value modified according to Eq. (3) and the related discussion and parameters used in Section 3. In these simulations, a slow-axis index step of zero was selected for generality; as was discussed in [8], this situation does not represent the highest model discrimination, but does imply a simple fabrication geometry.

The first simulation was designed to demonstrate the effectiveness of modal discrimination by incorporating gain filtering into the SHARC fiber. In this simulation, a kW-level amplifier (gain = 30) was modeled using the parameters in Table 2, with the input power being evenly distributed along the slow axis of the input (flat-top profile). Such a profile will excite all even modes of the SHARC waveguide with nearly equal amplitude. However, as shown in Fig. 5, all of the higher-order modes that appear early on in the propagation (most prominently before 0.4 m) are filtered out via edge loss and confined gain, leaving the amplified output in the fundamental mode only. Such a simulation clearly demonstrates the effectiveness of incorporating gain filtering in the SHARC fiber for amplification with high modal discrimination.

To illustrate this logic of this approach, Fig. 6 shows the calculated 2D modes of the SHARC waveguide in relative scale to the waveguide and ytterbium-doping structures. The fundamental mode (k = 0) is nearly entirely contained in the gain region, while the overlap of the second mode with the gain region is less than 35%. It is precisely this concept that leads to exceptional mode filtering by using confined gain regions. However, the geometrical overlap of the mode with the ytterbium-doped region is not sufficient to fully describe the problem since local transverse saturation of the gain changes each mode’s ability to extract gain, as discussed in Section 3.

The next step in assessing the modal properties of the SHARC fiber amplifier using gain filtering was to launch each mode independently into the amplifier and calculate its net gain. The results of these simulations are shown in Fig. 7 , along with the analytic modeling results presented in Section 3. The simulations agree with the analytic calculations in that the combined gain- and edge-loss-filtering yield almost 15 dB of gain margin for the fundamental mode over the second mode, and much higher margin over all other modes. In this regime, the second mode experiences zero net gain, as indicated in the figure. Further, the agreement between the BPM simulations and the analytic model is exceptional, particularly for the modes of interest (the first two modes, which will couple the most strongly). The discrepancy between the models can be accounted for by the numerical precision of the mode calculations.

Although Fig. 5 demonstrates the power of gain filtering in the SHARC fiber, its practical aspects result from alignment of the seed beam into the SHARC fiber. One of the most difficult tasks facing the use of LMA fibers in general is launching the seed beam into the fundamental mode with high efficiency. This is particularly difficult in practical applications where the physical environments are not as stable as in typical laboratory settings. Therefore, understanding the alignment tolerances of the gain-filtered SHARC fiber is of critical importance for practical applications.

In the next series of simulations, a Gaussian beam is launched into fiber amplifier with three different conditions: offset in the slow-axis direction, improper slow-axis magnification, and angular offset (tilt) in the slow axis direction. In each case, the modal content and output power (amplifier efficiency) are used to calculate the effects of improper spatial alignment of the seed beam into the gain-filtered SHARC fiber amplifier.

Figure 8(a) shows the results of simulations when a Gaussian beam matching the fundamental mode is injected into the fiber with an offset in the slow-axis direction. The first observation is that although the higher-order-mode (HOM) content increases with increasing misalignment, the fraction of output power in the fundamental mode remains above 99% even for a 120-µm offset, more than 25% of the waveguide width. This is particularly striking considering the poor launch efficiency (~40% into higher-order modes). The second observation is that there is a price to misaligning the input beam: reduced amplifier efficiency. The misaligned seed beam excites many modes in the SHARC fiber, with larger offsets translating to lower powers being launched into the fundamental mode. Since the higher-order modes experience net loss (as per Fig. 7), the saturation level in the amplifier is dictated by the power in the fundamental mode, translating to reduced amplifier saturation and less power in the output. However, for reasonable alignment tolerances (50 µm = 11% of the waveguide width) the penalty to the amplifier efficiency is less than 2%.

As a demonstration of the filtering phenomenon, Fig. 8(b) shows the slow-axis field amplitude propagating through the amplifier when seeded with a 120-µm offset. The offset of the launched seed beam can be observed at the left side of the figure. Similar to the case shown in Fig. 5 for the flat-top launch, the strong mode filtering inherent in this configuration leads to amplifier output in the fundamental mode.

Figure 9(a) shows the results of similar simulations when a Gaussian beam with improper slow-axis beam width is injected on-axis into the fiber. Again, the combination of gain filtering and SHARC fiber edge loss leads to very high mode discrimination at the amplifier output, greater than 99.4% in this case. Due to the symmetric launch conditions, a large fraction of power is always launched into the fundamental mode, as indicated by the upper horizontal axis in Fig. 9(a). Correspondingly, the amplifier efficiency reduction is much smaller, less than 4% over the entire simulated range. Note that this range ( ± 35% mode mismatch) implies rather poor alignment conditions.

Figure 9(b) shows the shows the slow-axis field amplitude propagating through the amplifier when the Gaussian seed beam is de-magnified in the slow-axis dimension to 0.65 times the width that would best match the fundamental mode. The smaller beam is readily observable at the left-hand side of the simulation domain. Although this beam is coupled into many SHARC fiber modes, their observed beating rapidly diminishes as the higher-order modes are filtered out during propagation through the amplifier.

Finally, Fig. 10(a) shows the results of simulations when a Gaussian beam matched to the fundamental mode is injected on-axis into the fiber with wavefront tilt in the slow-axis direction. The wavefront tilt is normalized to the full-width diffraction-limited divergence angle of the fundamental SHARC mode as a basis for intuitive experimental understanding, since this relative angle translates throughout the optical injection system via the Lagrange invariant [26]. Figure 10(a) once again demonstrates the remarkable modal filtering properties of the gain-filtered SHARC fiber amplifier. Regardless of the amount of angular misalignment, the fundamental mode carries over 99.2% of the output power. The penalty in amplifier efficiency is also modest over the entire range modeled, and less than 2% for reasonable (0.8x diffraction limit) alignment tolerances.

Figure 10(b) shows the slow-axis field amplitude propagating through the amplifier when the Gaussian seed beam is tilted to 1.5 times the diffraction limit. The off-axis propagation direction of the beam is clearly evident early in the propagation (left-hand side of plot), but eventually is corrected as only the fundamental mode experiences significant gain during propagation through the amplifier.

Figures 8-10 demonstrate that the gain-filtered SHARC fiber amplifier is extremely tolerant to misalignment in the injection of the seed laser beam. In fact, this architecture shows a “beam clean-up” type of behavior: regardless of the injection HOM content, even as large as 40%, the output HOM content it always less than 1%.

It should be noted that the simple model used in Section 3 to derive the small-signal gain parameter was for an amplifier with a gain of 30 without including the effects of edge loss or confined gain. With perfect launch conditions, the results of the BPM simulations including edge loss and confined gain results in a ~4% higher net gain, implying higher amplifier efficiency. Although the edge loss in SHARC fibers will contribute to reducing the amplifier efficiency, it is a distributed loss. The saturable nature of the gain enables a reduced signal to extract higher gain, resulting in a smaller efficiency reduction than one might expect. Moreover, the confined gain used here for gain filtering also leads to higher efficiency since the gain is confined to the high-intensity portion of the mode, which can extract the gain very efficiently [16, 27]. The net result is that SHARC amplifier efficiency is slightly higher than what one might expect from a conventional LMA fiber amplifier.

## 5. Discussion and practical implications of the SHARC fiber amplifier

As implied in Fig. 1, the SHARC fiber has numerous thermal advantages over conventional round fiber for use in high-power amplifiers. First, since the core is much larger than conventional LMA fibers, the pump light is absorbed over a much larger area resulting in lower heat-source density. Second, the slab-like geometry allows for easier heat extraction than for cylindrical fibers, both in terms of heat-transfer surface area of the core into the cladding as well as the flat surface of the cladding wrapped around a cooled mandrill (shown in Fig. 1). This argument is similar to the advantages of slab vs. rod geometries in high-power solid-state laser systems. Finally, the large core area of the SHARC fiber significantly increases the SBS threshold, allowing the use of longer fibers and enabling lower heat load per unit length.

It is important to realize that, unlike large-core photonic crystal fibers, a SHARC fiber amplifier enables very large core areas while maintaining the all-glass monolithic architecture that is one of the principal attractions of existing fiber lasers. As an example, it is reasonable to expect that the front-end of a SHARC fiber amplifier chain will have one or more LMA fiber preamplifiers. In this case, one would like to have an all-glass coupler that can be spliced between the final LMA fiber preamplifier and the final-stage SHARC fiber power amplifier to (a) shape the circular mode of the LMA preamplifier into a high-aspect-ratio (~30:1 or greater) ellipse, and (b) collimate the launched signal as it enters the SHARC fiber amplifier. These two functions can be performed simultaneously, without any free-space optics, using the signal coupler schematically shown in Fig. 11 . This coupler comprises a short length of passive SHARC-like fiber that exploits index-based guiding to maintain a constant fast-axis dimension all the way from the LMA fiber output to the SHARC fiber amplifier input. In the slow-axis dimension, the coupler functions as a one-dimensional quarter-pitch planar GRIN lens by employing a slow-axis index gradient. Such a scheme allows the signal beam to expand and match the optimum input size for the SHARC amplifier input, providing a nominally collimated beam in the slow-axis dimension. Simulations show that coupler lengths in the range 10-20 mm will suffice for this type of application, depending on the precise dimensions of the LMA and SHARC amplifier fibers.

The other important function to achieve without free-space optics is pump coupling into the SHARC fiber amplifier. This can be achieved with a planar analog of the ubiquitous pump combiners that are used with conventional circular-core dual-clad fibers. The pump coupler concept for SHARC fibers is schematically shown in Fig. 12 . This coupler is spliced between two sections of active fiber (only short sections of which are illustrated) and is designed such that it allows pump radiation to enter the active fiber without imposing any appreciable loss on the signal beam, which also passes through the pump coupler. In order to accomplish this, the pump radiation is configured to enter and propagate at an angle relative to the SHARC fiber axis, thus allowing the pump radiation to efficiently enter the active fiber from the edge of the fiber and propagate along the plane of the semi-guiding core. The semi-guiding core of the pump coupler also efficiently carries the signal from one end to the other. Specifically, since the signal-beam fast-axis dimension may only be ~20 µm, the guiding in that direction is absolutely essential for efficient propagation of the beam from one active fiber through the pump coupler into the next active fiber. However, the slow-axis signal-beam dimension may be approximately ~1 mm or more. Since the pump-coupler length may be only ~5 to 10 mm, very little diffraction occurs in the wide direction, so no guiding is required by the pump coupler in that dimension. Figure 12 shows a pump coupler designed for bi-directional pumping, but this concept can also be adapted to applications in which both pump fibers inject pump power in the same direction.

In considering this pumping approach, it is useful to appreciate how large the acceptance etendue is for pumping a SHARC fiber. Consider a 20 μm x 1 mm core contained within a 200 μm x 1.2 mm pump cladding, and an outer cladding providing an NA of 0.45 for the outer boundaries of the pump cladding. This geometry presents a full-angle beam-parameter product of 1120 mm-mrad in the wide dimension, which is roughly equivalent to a linear array of > 10 pump fibers having a 200-μm core diameter and 0.22 NA. A SHARC fiber amplifier therefore easily accommodates efficient launch of a large number of state-of-the-art fiber-coupled pump diode packages.

## 6. Conclusions

A new class of optical fiber, the SHARC fiber, was analyzed in a high-power fiber amplifier geometry using the gain-filtering properties of confined gain dopants. The high-aspect-ratio (~30:1) rectangular core allows mode-area scaling well beyond 10,000 μm^{2}, which is critical to high-pulse-energy or narrow-linewidth high-power fiber amplifiers. While SHARC fibers offer modally dependent edge loss at the wide “semi-guiding” edge of the waveguide, the inclusion of gain filtering adds further modal discrimination via distribution of the spatial overlap of the gain with the various modes. Both methods are geometric in form, such that the combination provides nearly unlimited scalability in mode area. Simulations showed that for kW-class fiber amplifiers, only the fundamental mode experiences net gain (15 dB), resulting in outstanding beam quality. Further, misalignment of the seed beam due to offset, magnification, and tilt were shown to result in small (few percent) efficiency penalty and kW-level output with 99% of the power in the fundamental mode for all cases.

## Acknowledgment

The authors wish to thank Robert Byren, Roberta Gotfried, and John Zolper for the research grant that enabled the collaborative relationship between the University of Rochester and Raytheon.

## References and links

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