## Abstract

A new class of optical fiber is presented that departs from the circular-core symmetry common to conventional fibers. By using a high-aspect-ratio (~30:1) rectangular core, the mode area can be significantly expanded well beyond 10,000 μm^{2}. Moreover, by also specifying a very small refractive-index step at the narrow core edges, the core becomes “semi-guiding,” i.e. it guides in the narrow dimension and is effectively un-guiding in the wide mm-scale dimension. The mode dependence of the resulting Fresnel leakage loss in the wide dimension strongly favors the fundamental mode, promoting single-mode operation. Since the modal loss ratios are independent of mode area, this core structure offers nearly unlimited scalability. The implications of using such a fiber in fiber laser and amplifier systems are also discussed.

© 2011 OSA

## 1. Introduction

Over the last decade, large-mode-area (LMA) fibers have made an enormous impact on high-power and high-energy fiber lasers and amplifiers. Scaling to larger mode areas means that higher powers can be carried in the fiber with the same intensities as in single-mode fibers. Since nonlinear effects typically depend on intensity, LMA fibers have paved the way to higher achievable cw and peak powers by simply raising the threshold powers of the nonlinear effects to levels above the operational powers.

Beyond the conventional single-mode limit, where the V-number of the fiber core exceeds 2.405, LMA fibers become capable of supporting multiple modes. These modes can become coupled and exchange power either if the fiber is disturbed, such as occurs upon coiling the fiber or packaging it into cables, or in the presence of inhomogeneities due to glass fabrication, drawing, heating, or coatings. Generation of higher order modes via such mode scrambling leads to a multimode beam at the output. Therefore, 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.

Several LMA fiber architectures have been developed to accomplish this mode filtering. The workhorse of reliable (commercial) mode-area scaling has been to utilize bend loss [1], which provides larger loss for higher-order modes than for the fundamental mode [2]. Unfortunately, this method has limited scalability since the modal loss depends on the propagation coefficients of the various modes; as the fiber diameter increases, the modal propagation coefficients become more densely packed as a function of mode indices, resulting in a reduced ability to discriminate between the fundamental mode and its nearest neighbors. If some degree of propagation loss can be tolerated (for example, in an active system such as a fiber amplifier), then the method can be scaled to larger core diameters [3].

Other methods can be used to filter out unwanted modes. Chirally coupled core fibers have scaled to 30-μm diameters by coupling high-order azimuthal modes to a helical secondary core coiled around the central LMA core [4]. However, fabrication tolerances will likely prevent this from becoming a commercializable process, similar to the problems with helical-core fibers [5]. Higher-order-mode (HOM) fiber has no distributed filtration mechanism, but relies instead on the relative absence of mode coupling to maintain mode purity [6], although long-period gratings act as a lumped mode filter at the end of the fiber.

Photonic crystal fibers have achieved single-mode operation for core diameters beyond 60 μm [7]. Since the claddings are comprised of pure silica with a pattern of air holes, the NA of photonic crystal fibers can be made almost arbitrarily small, thus allowing the single-mode condition (V < 2.405) to be met for such large core diameters. However, this low NA makes the spatial profile highly susceptible to changes in ambient environmental conditions and slight bends. Although both of these challenges have been overcome by embedding the fiber within a ~2 mm fused-silica rod [8], these fibers face packaging challenges and average-power limits that do not apply to much thinner conventional LMA fibers. Most detrimentally, such rod-like fibers must be kept straight and cannot therefore be formed into a compact coil. This will clearly limit how compact the fiber package can be, and moreover it will ultimately establish average-power limits, since at some point the fiber lengths required to dissipate the thermal load will exceed the maximum linear package dimension that can be tolerated in any specific applications.

Leakage-channel fibers offer another path to core-size scaling while maintaining some capability to coil the fiber [9]. In this approach, the cladding surrounding the core is not continuous, but it has sizeable gaps that are periodically located around the core circumference. These gaps allow higher-order modes to preferentially “leak” out of the core, with the core-dependent loss rates being optimized by specifying the size and spacing of the gaps. In principle, this approach can achieve single-mode operation with core diameters ~100 μm or more while maintaining the ability to be coiled. However, calculations show that a 100-μm core would have a rather large minimum bend radius of 1.25 m to maintain propagation losses below 1 dB/m [9]. Once again, as core-size scaling is shown to be possible while maintaining good beam quality, the power-scaling benefit comes with the disadvantage of a significantly larger package size.

In this paper, we present a new class of large-mode-area optical fiber that provides: (i) a modal discrimination mechanism that does not rely on coiling the fiber; (ii) relative modal discrimination that is independent of the core area; (iii) compact coiling without impacting either the propagation loss, the mode content, or the spatial overlap of the fundamental mode with the gain profile; and (iv) record-breaking mode-area scaling beyond 10,000 μm^{2}.

This paper is organized as follows. In Section 2, this new class of semi-guiding high-aspect-ratio-core (SHARC) fibers is described. Detailed analytic modeling results, which elucidate the principles and fundamental operation of the fiber, are described in Section 3. In Section 4, intense numerical simulations confirm the analytical predictions and further demonstrate the strength of the SHARC concept. Section 5 presents results proving that SHARC fibers can be formed into a compact coil, similar to LMA fibers, with no excess loss. Applications of this new class of fibers are discussed in Section 6, with concluding remarks following in Section 7.

## 2. Semi-Guiding High-Aspect-Ratio Fiber

The semi-guiding high-aspect-ratio core (SHARC) fiber concept schematically shown in Fig. 1
departs from conventional fiber designs in several basic features. First, it embodies a high-aspect-ratio rectangular core that enables scaling to a very large area, up to 30,000 μm^{2} or more (conventional LMA fibers have core areas ~400 μm^{2}), while retaining a thin, mechanically flexible narrow dimension. Second, the aspect ratio of the core will typically range from 30:1 to 100:1 or more, depending on the output power requirement; such aspect ratios are significantly greater than those of currently available rectangular-core fibers [10].

Third, as is denoted by the key descriptor “semi-guiding,” this 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 core edges of the slow axis. This will be discussed in more detail in Sections 3 and 4.

Power-handling scalability in the SHARC fiber is based on the following strategy. First, the narrow (fast-axis) cladding dimension will be small, ~0.5 mm or less, so that the fiber can be coiled. The fast-axis core will have a typical LMA NA (~0.06) and sufficiently thin core to allow single- or few-mode operation in the fast-axis direction. Second, the total core area required to accommodate the desired optical power will be provided by scaling the wide (slow-axis) dimension linearly with power, such that the power is carried at a constant intensity. Hence, power-limiting processes such as optical damage and stimulated scattering are held constant as the power is increased.

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. 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], with its associated fabrication complexity.

In addition to enabling significant power scaling and despite the highly multimode core, this unique fiber design is intrinsically capable of robust single-mode operation in both transverse dimensions. Although the fast-axis direction can be designed to be single-mode, slow-axis mode control is achieved by exploiting the natural process of “loss filtering,” which will be described in detail in Section 3. This process arises from the fact that all slow-axis modes suffer radiation loss into the cladding through the “open” core edges, where the index step is designed to be very small but not necessarily zero. As is schematically depicted in Fig. 2 , the radiation loss is highly mode-dependent (increasing approximately as the second power of the slow-axis mode number), 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.

## 3. SHARC Fiber Analytic Calculations

From a modeling perspective, the SHARC geometry lends itself nicely to separation of variables, meaning 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. Consider factorizing the problem in two transverse dimensions, the slow and fast axes, which we will mathematically denote as the x and y directions, respectively. The rectangular shape of the waveguide simplifies the problem to the well-known one-dimensional solution for the modes of a planar waveguide, and the complete rectangular waveguide mode is a product of the two orthogonal planar waveguide modes.

Mathematically, the SHARC modes of transverse orders *s* and *f* can then be represented by M_{s,f}(x,y,z,t) = E_{s,f}(x,y)exp(−iωt + iβ_{s,f}z), where the transverse mode pattern can be factorized as E_{s,f}(x,y) ≅ X_{s}(x)Y_{f}(y). The parameter β_{s,f} = k_{0}(n_{co} − δn_{f} − δn_{s}) is the propagation coefficient of the mode, where k_{0} = (2π/λ) is the wavenumber, and δn_{s} and δn_{f} quantify how the effective refractive indexes in the slow- and fast-axis directions, respectively, differ from the core and fast axis cladding material indexes n_{co} and n_{cl} that are shown in Fig. 1. Specifically, the planar (1-D) waveguide mode Y_{f}(y) formed by the fast-axis claddings has an effective index n_{eff} = n_{co} − δn_{f}, and n_{eff} lies in the range n_{cl} < n_{eff} < n_{co}. The slow-axis correction δn_{s} is defined in a manner analogous to δn_{f}; it is nominally small compared to δn_{f}, and it is induced by the planar waveguide mode solution X_{s}(x) along the wide slow-axis dimension. The accuracy of this simplified approach was validated by comparing the analytic modes to a numerical calculation of the modes using a finite difference method for a representative SHARC core geometry. A normalized overlap integral of the analytic and numerically calculated modes yielded a value larger than 99.99%, which fully justifies the mode factorization approach presented in this section.

The analytical portion of this paper is focused on the slow-axis factor of the mode solution, X_{s}(x)exp(−ik_{0}δn_{s}z), with the goal of investigating the mode characteristics of weakly confined modes that have essentially no total internal reflection (TIR) at the edges of a very wide multimode planar stripe. For the narrow planar waveguide along the fast axis, which has TIR index-guiding walls, the mode structure, Y_{f}(y)exp(−ik_{0}δn_{f}z), is well known [16]. To simplify these analytical calculations, the fast-axis planar waveguide is assumed to be single-mode, although the basic propagation properties and the resulting mode discrimination also arise in the case of a multi-mode fast-axis waveguide, as is discussed below in connection with Section 4. As such, the modes of the two-dimensional SHARC structure are spatially differentiated by the slow-axis mode profiles alone, and thus only need a single mode order, the parameter *s*, for mathematical representation.

Before analyzing the slow-axis modes, it is important first to understand the impact of the fast-axis waveguide in the slow-axis dimension. Using standard planar waveguide theory [16], one can calculate the effective index of the fast-axis mode. True to any conventional waveguide, its exact value will lie between the refractive indices of the core and the cladding, n_{cl} < n_{eff} < n_{co}. Since the slow-axis cladding index n_{scl} is nominally the same as, or very close to, the core index, the effective fast-axis mode index n_{eff} is generally less than that of the slow-axis mode, n_{eff} < n_{scl} . This result has two consequences. First, the difference between n_{eff} and n_{co} will allow Fresnel reflections at the fast-axis claddings that largely contain the mode power in the fast-axis dimension. Second, the slow-axis dimension will never exhibit TIR guiding, and the modes will leak out of the core region. This leakage is precisely the mechanism that leads to mode filtering that discriminates against the higher-order modes. Specifically, due to the absence of TIR guiding in the slow-axis direction, all modes of the SHARC fiber experience loss, and as such are termed *leaky modes* [17]. Such modes have a fixed, predetermined spatial profile, but their propagation coefficient is a complex number.

If the refractive index of the core material n_{co} is equal to the slow-axis cladding index n_{scl}, the slow axis interface between the regions appears to be “optically open,” as if the core modes can freely propagate into the slow-axis claddings. However, partial reflections at the borders cannot be eliminated. Recall that the light-propagation environment is very different on opposite sides of the slow-axis edge: in the fast axis direction, light is TIR guided in the slow-axis core, but freely diffracting in the slow-axis cladding. The propagation characteristics inside and outside the core are distinct even when n_{co} = n_{scl}. As a consequence, the effective index of the fast-axis mode is lower than that of the slow-axis cladding and therefore will result in an effective Fresnel reflection. One may be inclined to think that the slow-axis interface can be truly eliminated by matching the slow-axis cladding index to the effective index of the fast-axis mode. However, the fast-axis mode is represented by the combination of a core index and cladding index, as the oscillatory and evanescent portions of the mode reside in each respective region. As such, it is impossible to match the slow-axis cladding to both core and fast-axis cladding simultaneously. Therefore, there is always a Fresnel reflection at the interface, as will be evident in later simulations.

Light radiated from the leaky modes through the interface from the core to the slow-axis claddings is lost in the fiber cladding and does not return to the core. This process therefore serves as an optical loss mechanism for all of the core modes. This simple fact helps to support the following assumption, the validity of which will be justified by later results: radiative loss into the slow-axis claddings can be modeled phenomenologically in our 1-D analysis by adding physical absorption into the cladding material. In practice, the cladding can be almost absolutely lossless, but the presence of the radiative leakage out of the core makes the cladding function as an effective absorber.

Mathematically, absorption in a medium implies an imaginary part of the refractive index. The slow-axis effective planar waveguide can thus be represented, as in Fig. 3
, by the refractive index n_{0} inside the core and by the complex refractive index n_{1} in the surrounding cladding:

_{0}is, in fact, n

_{eff}, the effective index of the fast-axis planar waveguide mode. The cladding index n

_{1}has a real part, n

_{scl}, given by the material index of the slow-axis claddings, and an imaginary part, κ, phenomenologically added to emulate radiation leaking from the core, as depicted in Fig. 3. There is no

*a priori*information with which to quantify the effective loss parameter κ, so it is kept as a variable of the problem to be determined by matching either to experimental data, if available, or to results of a more accurate model. However, strong radiative losses would represent an impractical waveguide, so the additional condition κ << 1 is imposed.

Classical analytical results for step-index planar waveguides, as presented in [16] for example, are used for the modes X_{s}(x)exp(iβ_{s}z) and their propagation constants β_{s} = k(n_{0} − δn_{s}). The modes are known to be harmonic functions inside the core, |x|< w/2, giving either cos(q_{s}x) for even orders (s = 0, 2, 4…) or sin(q_{s}x) for odd orders (s = 1, 3, 5…), with exponential tails, exp[−p_{s}(|x|−w/2)], outside the core, for |x|> w/2. The parameters q_{s} and p_{s} are nominally found through the boundary conditions at the interface in conjunction with standard, simple, geometric relations applying to the propagation coefficient β_{s}. However, in the case shown in Fig. 3, the propagation coefficient will be complex since the cladding includes an effective loss. The mode loss is given by the imaginary part of the propagation coefficient, α_{s} = 2Im(β_{s}). The propagation coefficient is calculated from the standard equation

_{s}w/2 is an offset, in units of the argument q

_{s}x for harmonic dependence, between the border, x = w/2, and the closest zero point, X

_{s}(x) = 0, of the mode of order

*s*. The parameter ξ is found as a solution of the transcendental equation

_{0}and n

_{1}, the functional form is fully applicable for the complex value for n

_{1}from Eq. (1) as well. In this case, the parameters ξ and β

_{s}also become complex-valued.

For the present application, the corresponding waveguide in the slow-axis dimension is very wide and highly multimode. Its corresponding V-number is therefore very large, V = (πh/λ)(n_{0}
^{2} – n_{1}
^{2})^{1/2} >> 1. For waveguides having large V-numbers, the mode amplitudes are nearly zero at the core interface, x = ± w/2. The corresponding profiles for the three lowest-order modes are shown in Fig. 4
.

In terms of the dispersion relation Eq. (3), the vanishing of the mode amplitudes near the waveguide boundary mathematically implies that ξ_{s} << 1. Applying this condition, Eq. (3) can be solved explicitly using power decompositions. Taking only the lowest order terms in the small parameter 1/V, the modal loss rate becomes

_{0}and n

_{1}given by Eq. (1).

Dependences for the modal loss rates as a function of the real material index step between the cladding and core, n_{scl} − n_{co}, are plotted in Fig. 5
for the three lowest modes of a 450-μm wide channel and a wavelength of 1.06 μm. The value n_{eff} = 1.450803 was calculated separately for λ = 1.06 μm light via a 1-D finite difference mode solver for a waveguide thickness h = 15 μm using a n_{co} = 1.45 and n_{cl} = 1.449, a typical index step for an LMA-type fiber. The value κ = 10^{−4} was used for this plot, corresponding to an effective bulk absorption coefficient α_{scl} ≅ 6 cm^{−1} in the cladding. This value of κ was chosen to match the curves calculated via beam propagation simulations (detailed in the Section 4).

Equation (4) and the corresponding plots in Fig. 5 represent the key analytic results for understanding the slow-axis mode behavior for the SHARC fiber. First, these results demonstrate that loss rates for all of the modes are sharply peaked for a small index step over the effective index, n_{scl} − n_{eff} ≅ 0. The fact that this peak does not occur when the core index equals the slow cladding index, as one might expect, is a result of propagation physics, which is captured by the separation of variables applicable to this rectangular geometry.

Second, consistent with Eq. (4), the results also show that the ratios of the mode-loss rates (i.e. the relative spacings of the loss-rate curves) are independent of the specific value of the slow-axis cladding index step. As a consequence, the mode-loss peaks define a useful index-step range of about a few hundred parts per million, outside of which the absolute loss rates become too low to be practical for mode filtering; this range represents a reasonable fabrication tolerance for designing the refractive index of the core and slow-axis cladding materials.

Third, the loss rate for the fundamental mode is much lower than that of all the higher-order modes. Mathematically, Eq. (4) says that the loss rate of the modes scales as (1 + s)^{2}. The implication is that the loss of the fundamental mode is four times lower than that of the second mode (s = 1), and nine times lower than the third mode (s = 2). Such strong discrimination properties inherent to the SHARC fiber suggest that nearly single-mode behavior will be obtained through a highly multimode waveguide of sufficient length L, if α_{1}L > 1. If one also applies the condition α_{0}L = α_{1}L/4 << 1, the transmission efficiency of the lowest mode through the waveguide is still sufficiently high for practical applications.

More importantly, Eq. (4) shows that the loss ratio for different modes is independent of the core width, implying that the modal discrimination properties do not change with arbitrarily increasing core size. Equation (4) provides further scaling rules for the absolute mode losses, namely that the loss rates scale with the core width as 1/w^{3}, and with wavelength as λ^{2}. Such dependences are typical for many lossy waveguides [18]. This means that for any width w and wavelength λ, one can determine an appropriate fiber length such that α_{1}L > 1 but α_{0}L << 1, which supports single-mode low-loss propagation through the fiber.

The physical reason behind the scaling factors is as follows. For grazing incidence at the waveguide-edge interface, the Fresnel transmission (loss) due to a single reflection decreases in proportion to the angle θ, which is defined as the angle of slant incidence (complimentary to the commonly used angle of incidence). The number of bounces the mode of order *s* experiences per unit length of propagation is approximately θ_{s}/w. The loss is therefore proportional to the product of the two, yielding a loss that scales in proportion to θ_{s}
^{2}/w. Since a planar waveguide mode of order *s* has spatial features on the order of w/(1 + s), it therefore diverges approximately as θ_{s} ≈(1 + s)λ/w. Using this relation yields a loss rate that scales in proportion to θ_{s}
^{2}/w = (1 + s)^{2}λ^{2}/w^{3}, which is readily observable in Eq. (4). The final factor in Eq. (4) is usually defined by the nature of the waveguide interface, differing by the nature of the channel loss (e.g. metal, anti-guiding dielectric, etc.); the last term in Eq. (4) quantifies this factor for SHARC waveguides.

## 4. SHARC Fiber Numerical Simulations

In order to confirm the behavior of the SHARC fiber predicted by the analytic modeling, full numerical simulations of the structure were performed using the Beam Propagation Method (BPM) [19]. The primary advantage of this method is its ability to model spatially dependent structures, such as the refractive index profile, 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

Equation (5) was solved with a finite-difference scheme (FD-BPM) [20] using the Alternating Direction Implicit (ADI) method [21]. The desire to model long propagation lengths required that transparent boundary conditions be implemented [22]. The two geometries modeled in the simulations correspond to the geometry of Fig. 1, with the specific numerical values listed in Table 1 .Initial simulations were performed for a 160-μm x 15-μm core high-aspect-ratio fiber under two conditions: (a) the conventional fully guiding core, where the slow- and fast-axis cladding had the same value (0.001 less than the core refractive index), and (b) the semi-guiding core, where the slow-axis refractive index is the same as that of the core. In both cases, a flat-top intensity profile (uniform across the core in both transverse dimensions) was launched into the 5-cm long fiber.

The top movie in Fig. 6 (Media 1, Media 2) shows the transverse (x,y) intensity profile as the light propagates down the fiber (into the plane of the figure) with the propagation distance in μm shown at the top of the figure. The dimensions of the figure are the same as the simulation window (300 μm × 46 μm) and are thus not shown to relative scale. The intensity profile shown exhibits typical behavior in a conventional fully guiding multimode waveguide, including the re-imaging properties of multi-mode interference structures (i.e. the Talbot effect) [23] which occurs at around 35 mm, in agreement with calculations using the known analytic formula [24] for the 160-μm waveguide width. In contrast, the bottom movie in Fig. 6 (Media 2, Media 1) reveals the mode filtering properties of the SHARC fiber. Near the waveguide entrance, both fibers exhibit similar mode-beating effects. However, as the light propagates down the waveguide, most of the higher-order modes, represented by high spatial frequencies (small spatial features), are filtered out of the SHARC fiber after about 10 mm of propagation length. Beyond that distance, essentially single-mode slow-axis propagation is retained as the resulting mode continues to weaken due to the leakage out of the core.

Since Fig. 6 (Media 1, Media 2) shows that the waveguides are single-mode in the fast-axis direction, these numerical simulations confirm the validity of using the effective index approximation in separating the fast- and slow-axis dimensions for analysis in Section 3. Nonetheless, full three-dimensional BPM is utilized throughout the simulations in this paper.

Subsequent simulations were performed for a much wider high-aspect-ratio fiber, with core dimensions of 450-μm x 15-μm. Figure 7 shows the intensity profile in the slow-axis dimension (x) propagating along the fiber axis (z, from left to right in the figure) at the middle of the core (y = 0) when a beam propagates down the fiber axis of the fully guiding core (top) and the semi-guiding core (bottom). Figure 7(a) clearly shows the presence of all of the modes that were launched into the fully guiding fiber as well as the Talbot re-imaging effect, occurring at around 29 cm, in agreement with the standard analytic calculation. In contrast, Fig. 7(b) shows the depletion of high-spatial-frequency components of the optical field within the first 8 cm of propagation. Interestingly, the SHARC fiber also shows an image at the Talbot plane, but the image is formed without the high-spatial-frequency components, and, as such, looks rather smooth.

As discussed in Section 3 and shown in Fig. 1, the slow-axis cladding may not have precisely the same index as the core. Both for design purposes and to understand the impact of fabrication tolerances, it is desirable to know the modal loss properties as a function of the slow-axis cladding refractive index. By launching specific modes into the SHARC fiber, the loss of each mode can be calculated after BPM propagation through a specific length of fiber. Figure 8 shows the propagation loss of the first three modes in the SHARC fiber as a function of the index step of the slow-axis cladding relative to that of the core. With the exceptions described later, Fig. 8 reliably reproduces the trends and results shown in Fig. 5, most importantly the significantly increasing loss with increasing mode order.

At an index step of − 80 ppm, the modal loss shows a peak that might be valuable for designing short fibers with high modal discrimination (for example, as might be used in fiber amplifiers). Other applications may require less loss of the fundamental mode, but similar discrimination. That the ratio of the modal losses does not change with index step implies that for a specific modal discrimination (i.e. desired output beam quality), the total net propagation loss of the fundamental mode is fixed. This feature was predicted by Eq. (4). Variation of the index step n_{co} − n_{scl} represents an extra degree of freedom that can be exercised in design optimization, and it allows one to match potentially contradictory requirements for fiber length and available core width to satisfy the low-loss single-mode propagation condition: α_{1}L > 1 but α_{0}L << 1.

At a slow-axis cladding index step of − 250 ppm, the fiber enters the completely guiding regime, as noted in Fig. 8 by the lack of loss for any modes in the shaded region. While a one-dimensional or symmetric two-dimensional waveguide would enter the guiding regime at any step less than zero, the impact of the fast-axis in this spatially separable geometry requires a lower index step than the effective index of the fast-axis mode (somewhere between 0 and − 1000 ppm for this simulation) in order to enter the fully guiding regime. The analytic calculation shown in Fig. 5 does not exhibit this feature, because for that calculation the loss is artificially introduced as absorption within the slow-axis cladding. Therefore, when the index step is such that the light would enter the guiding regime in the analytic model, the cladding still exhibits loss since the evanescent tail extends into the absorbing claddings. In reality, this absorption artifact is not physically present, as indicated by the BPM simulations that properly account for the loss and guiding properly via diffraction and index profile.

Another difference between the approximate analytical-model plots of Fig. 5 and the accurate simulations of Fig. 8 is the shape of the curves as the index step approaches the guiding regime. This difference is attributed to two factors: approach to the TIR regime, and an additional loss channel, neither of which is captured by the simplified analytical model of factorized mode structures. The additional leakage channel can be understood as follows. Although the slow-axis cladding index n_{scl} can be slightly less than the core index n_{co}, making that interface guiding, the slow-axis cladding index can still be greater than the signal-cladding index n_{cl}, making that small interface anti-guiding. Hence, optical power residing within the fast-axis claddings near this interface can be drawn out into the slow-axis cladding, as schematically indicated by the red arrows in Fig. 9
. This is precisely what happens to the evanescent tail of the fast-axis mode in the vicinity of the core edges.

Keeping in mind that effective loss filtering is possible even with fabrication tolerances on the refractive indices as large as 10^{−4} (100 ppm), the above discussion for Fig. 8 has several implications. First, the benefits of the SHARC structure are inherent in the geometry and can be exploited over a range of index values, rather than being limited to a singular point-design that would be nearly impossible to realize. Second, the slow-axis cladding can be made of the same exact material as the core, which makes fabrication of the total structure simpler than requiring a specific slow-axis cladding index. Finally, realistic fabrication tolerances will also allow the design objectives to be met if, in fact, a different specific slow-axis refractive index is desired.

## 5. Bend Loss in SHARC Fibers

As mentioned previously, one of the key benefits of this geometry is the separation of fast-axis and slow-axis dimensions. In particular, this benefit allows for compact packaging, since the fiber can be coiled in the fast-axis dimension without any substantial detriments. To underscore this statement, we performed simulations for a 1-m length of SHARC fiber in a coiled configuration. The launch end of the fiber was held straight for 5 cm. The fiber was then adiabatically transitioned over a 5-cm length to the final coil diameter. The opposite procedure was implemented on the output end of the fiber. In this way, the realistic packaging configuration depicted in the inset of Fig. 10 was modeled.

In these BPM simulations, the equivalent index method [25] was used to simulate the effects of bending. In this method, the refractive index profile is modified as

_{b}is the bend radius that changes with propagation distance as previously described.

As the bend radius decreases, the tail of the fast-axis mode in the direction of the bend extends increasingly further into the cladding. This extended tail contains two contributions: the evanescent tail of the mode, which extends deeper into the cladding due to bend-induced mode deformation, and propagating light radiating out of the waveguide, which is the expected bend loss. In performing these simulations, it was necessary to extend the spatial extent of the fast-axis computational window to prevent the mode tails from carrying any significant power at the computational boundary. If the computational boundary had not been sufficiently far from the fiber boundary, the power in the evanescent tail that reached the computational boundary would still have been finite. Since the model implicitly assumes any power reaching a computational boundary continues to propagate evanescently (and thus without loss) beyond the boundary, the latter case would have been incapable of accurately quantifying the loss, and the simulation would not have properly accounted for the bent nature of the waveguide represented by Eq. (6).

Figure 10 shows the excess loss due to bending as a function of the bending radius in the fast-axis dimension. The excess loss remains negligible until the bend radius reaches about 15 cm, at which point it starts to increase almost exponentially with decreasing bend radius. The bend radius at which the loss starts to increase is somewhat larger than that of conventional LMA fibers if one takes the core diameter equal to SHARC core thickness, d = h. There are two factors responsible for this behavior. First, as the fiber is bent, mode deformation tends to compress the mode to the outer edge of the waveguide. Although this effect has negligible impact on the effective mode area for a narrow fast-axis waveguide, the evanescent tail at the outer bend edge of the waveguide becomes larger, which allows the leakage channel described by Fig. 9 to be enhanced. Second, the bend loss in rectangular-core fibers is, in general, more significant than that of circular-core fibers. Light leaks out of bent waveguides due to violation of the TIR condition. In a circular-core fiber, light leaks out of the very edge of the core when bent, much like liquid leaking out of a tipped cup. In a rectangular-core fiber, however, light leaks out along of the entire edge of the bent core, which leads to higher loss.

It is important to note, however, that from an applications perspective, the bend loss of 0.5 dB that occurs at a reasonable packaging radius of 7 cm is quite acceptable in a high-gain fiber amplifier system. Further, the loss due to coiling can be reduced without substantially sacrificing mode area by simply designing the core appropriately; a thinner fast-axis core with higher NA will reduce the impact of coiling without changing the modal properties of the SHARC fiber.

Finally, and perhaps most significantly, the beam propagation simulations also reveal that the mode content in the fast-axis direction remains nearly constant regardless of bend radius, with higher-order mode content at the output of the fiber varying less than 0.1% under all coiling conditions, including an uncoiled fiber. Once again this finding emphasizes the benefits of the high-aspect-ratio core in the separation of fast and slow axes.

## 6. Discussion and Implications of the SHARC Fiber

When considering a glass fiber whose shape significantly deviates from cylindrical, one may anticipate a host of fabrication difficulties, including (a) manufacturing the large fiber structures, (b) fabricating the perform with high (~30:1) aspect ratio, and (c) maintaining the rectangular cladding shape shown in Fig. 1 while drawing the fiber. Figure 11 shows the cross-section of a cleaved high-aspect-ratio core fiber fabricated at OFS Laboratories [26]. The resultant fiber has core dimensions of 14 μm × 390 μm. Although this particular fiber core is guided in both transverse dimensions, the resulting 28:1 core aspect ratio indicates that fabricating a high-aspect-ratio core can be readily accomplished using modern fabrication techniques. In fact, several high-aspect-ratio-core fibers have been fabricated at OFS Laboratories with even larger core sizes, up to 32 μm × 600 μm, and the more recent fibers have much flatter core edges that are also perpendicular to the wide core surfaces.

The SHARC fiber offers an important feature not common to any other very-large-core-area fibers: mechanical flexibility. Despite the large core area, a SHARC fiber can still be coiled in the fast-axis direction, as is shown schematically in Fig. 12 . 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).

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. Specifically, LMA fibers having larger core areas require successively larger bend radii to avoid excessive bend losses. But as the core size increases the mode area becomes severely reduced by bend-induced mode deformation [27,28]. 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. If the fiber is single-mode in the fast-axis direction, fast-axis mode-area reduction is negligible. If the fiber is designed to support a few fast-axis modes, the mode-area reduction is comparable to, but actually somewhat less than the reduction for common LMA fibers. In this few-mode case, the fast-axis coiling eliminates the second or higher-order modes, as is currently done in conventional (20 to 25-μm diameter) LMA fibers.

SHARC fibers can be readily adapted for use in fiber lasers and amplifiers by including ytterbium or other rare-earth dopants into the core. Such doped fibers have already been fabricated at OFS Laboratories. The SHARC fiber architecture offers substantial benefits in such applications. First, the SHARC architecture promises an inherent increase in SBS threshold due to its very large mode area and resulting low intensity. For example, an order-of-magnitude increase in SBS threshold relative to that of a 25-μm LMA fiber is achievable by simply designing a SHARC fiber with a 25 μm × 250 μm core. In fact, the fibers that have already been fabricated by OFS would exhibit an increase in the SBS threshold by a factor of almost 40. This feature makes the SHARC fiber a very attractive candidate for high-power narrowband fiber amplifiers.

Second, the rectangular cross-section of SHARC fibers provides the same thermal management benefits that slab lasers routinely achieve relative to solid-state rod lasers. Hence, this new class of fibers, when used in a laser configuration, combines the advantageous features of conventional fiber lasers and slab lasers, thereby extending the available performance envelope beyond what is possible with either of those existing technologies independently.

Third, 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 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.

Finally, bending an active LMA fiber usually displaces and compresses the mode radially away from the longitudinal axis of the active core area, reducing the overlap integral of the mode with the active core, and, hence, decreasing the gain. This effect becomes especially problematic for gain-filtered fibers [29]. For a SHARC fiber, fast-axis bending does not affect the slow axis mode distribution, so the gain for the fundamental mode and, hence, the lasing efficiency are not impacted much by the bending.

## 6. Conclusions

We have described the concept, analysis, and simulations of semi-guiding high-aspect-ratio-core (SHARC) fibers. This new class of fiber breaks the near-circular core symmetry common to traditional fibers, offering a host of advantages over common circular fibers. By using a high-aspect-ratio (~30:1) rectangular core, the mode area can be significantly expanded well beyond 10,000 μm^{2} while the natural loss filtering promotes single-mode operation in the slow-axis direction. Further, the SHARC fiber design produces Fresnel losses where the modal-loss ratios are constant with mode area, and this implies nearly unlimited scalability. We also outlined a few of the benefits that accrue from using the SHARC fiber architecture in fiber laser and amplifier systems.

## Acknowledgments

The authors are grateful to Robin Reeder and Brian Boland for useful discussions of mode loss mechanisms in a SHARC fiber. The authors also greatly appreciate the pioneering work done by Dennis Trevor and David DiGiovanni of OFS Laboratories in developing processes to fabricate high-aspect-ratio rectangular-core fibers.

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