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
1. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000). [CrossRef] [PubMed]
], which provides larger loss for higher-order modes than for the fundamental mode [2
2. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]
]. 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
3. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]
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
4. C. Liu, G. Chang, N. Litchinitser, A. Galvanauskas, D. Guertin, N. Jabobson, and K. Tankala, “Effectively single-mode chirally-coupled core fiber,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper ME2.
]. However, fabrication tolerances will likely prevent this from becoming a commercializable process, similar to the problems with helical-core fibers [5
5. Z. Jiang and J. R. Marciante, “Loss measurements for optimization of large-mode-area helical-core fibers,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWA3.
]. 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
6. S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006). [CrossRef] [PubMed]
], 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
7. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14(7), 2715–2720 (2006). [CrossRef] [PubMed]
]. 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
8. J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express 13(4), 1055–1058 (2005). [CrossRef] [PubMed]
], 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
9. L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007). [CrossRef]
]. 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
9. L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007). [CrossRef]
]. 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 μm2.
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
Fig. 1 Schematic cross section of a semi-guiding high-aspect-ratio core (SHARC) fiber. Similar to the definition used for semiconductor lasers, the fast and slow axes of the SHARC fiber are indicated in the figure. The rectangular core has thickness h, width w and refractive index nco. The core is surrounded by fast axis claddings of index ncl and slow axis claddings of index nscl.
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 μm2
or more (conventional LMA fibers have core areas ~400 μm2
), 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
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 μm2
, 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
11. G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35(10), 1542–1544 (2010). [CrossRef] [PubMed]
12. C. M. Zeringue, I. Dajani, and G. T. Moore, “Suppression of stimulated Brillouin scattering in optical fibers through phase-modulation: a time dependent model,” Proc. SPIE 7914, 791409, 791409-9 (2011). [CrossRef]
], with its associated system complexity, or acoustic waveguide management [13
13. P. D. Dragic, C.-H. Liu, G. C. Papen, and A. Galvanauskas, “Optical fiber with an acoustic guiding layer for stimulated Brillouin scattering suppression,” CLEO 2006, paper CThZ3.
15. L. Dong, “Limits of stimulated Brillouin scattering suppression in optical fibers with transverse acoustic waveguide designs,” J. Lightwave Technol. 28, 3156 (2010).
], 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
Fig. 2 Two-dimensional (slow axis and propagation axis) depiction of modal loss in a SHARC fiber. Higher-order modes radiate out into the cladding over much shorter propagation distances than do the lower-order modes. In the figure, the modes are spaced out horizontally for clarity.
, 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
can then be represented by Ms,f
(x,y,z,t) = Es,f
(x,y)exp(−iωt + iβs,f
z), where the transverse mode pattern can be factorized as Es,f
(x,y) ≅ Xs
(y). The parameter βs,f
) is the propagation coefficient of the mode, where k0
= (2π/λ) is the wavenumber, and δns
quantify how the effective refractive indexes in the slow- and fast-axis directions, respectively, differ from the core and fast axis cladding material indexes nco
that are shown in Fig. 1
. Specifically, the planar (1-D) waveguide mode Yf
(y) formed by the fast-axis claddings has an effective index neff
, and neff
lies in the range ncl
. The slow-axis correction δns
is defined in a manner analogous to δnf
; it is nominally small compared to δnf
, and it is induced by the planar waveguide mode solution Xs
(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, Xs
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, Yf
z), is well known [16
16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic Publishers, 1983).
]. 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
16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic Publishers, 1983).
], 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, ncl
. Since the slow-axis cladding index nscl
is nominally the same as, or very close to, the core index, the effective fast-axis mode index neff
is generally less than that of the slow-axis mode, neff
. This result has two consequences. First, the difference between neff
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. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1(1), 58–106 (2009). [CrossRef]
]. Such modes have a fixed, predetermined spatial profile, but their propagation coefficient is a complex number.
If the refractive index of the core material nco is equal to the slow-axis cladding index nscl, 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 nco = nscl. 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
Fig. 3 The cross section of the SHARC fiber can be factorized and reduced to two independent step-index planar waveguides for the fast and slow axes. Radiation out of the slow-axis edge of the waveguide (left) can be represented by a complex cladding refractive index in the effective planar waveguide of the slow-axis (right).
, by the refractive index n0
inside the core and by the complex refractive index n1
in the surrounding cladding:
The index n0
is, in fact, neff
, the effective index of the fast-axis planar waveguide mode. The cladding index n1
has a real part, nscl
, 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
16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic Publishers, 1983).
] for example, are used for the modes Xs
z) and their propagation constants βs
). The modes are known to be harmonic functions inside the core, |x|< w/2, giving either cos(qs
x) for even orders (s = 0, 2, 4…) or sin(qs
x) for odd orders (s = 1, 3, 5…), with exponential tails, exp[−ps
(|x|−w/2)], outside the core, for |x|> w/2. The parameters qs
are nominally found through the boundary conditions at the interface in conjunction with standard, simple, geometric relations applying to the propagation coefficient βs
. 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, αs
). The propagation coefficient is calculated from the standard equation
where the parameter ξ = (1 + s)(π/2) – qs
w/2 is an offset, in units of the argument qs
x for harmonic dependence, between the border, x = w/2, and the closest zero point, Xs
(x) = 0, of the mode of order s
. The parameter ξ is found as a solution of the transcendental equation
It is worth mentioning that while Eqs. (2)
have been derived for real values of refractive indexes n0
, the functional form is fully applicable for the complex value for n1
from Eq. (1)
as well. In this case, the parameters ξ and βs
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)(n0
>> 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
Fig. 4 Normalized profiles for the three lowest-order slow-axis modes inside the SHARC core.
In terms of the dispersion relation Eq. (3)
, the vanishing of the mode amplitudes near the waveguide boundary mathematically implies that ξs
<< 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
where both n0
given by Eq. (1)
Dependences for the modal loss rates as a function of the real material index step between the cladding and core, nscl
, are plotted in Fig. 5
Propagation loss for the first three modes (s = 0, 1, and 2) as a function of the slow-axis cladding index step nscl
calculated via Eq. (4)
using w = 450 μm, h = 15 μm, λ = 1.06 μm, neff
= 1.450803, and κ = 10−4
for the three lowest modes of a 450-μm wide channel and a wavelength of 1.06 μm. The value neff
= 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 nco
= 1.45 and ncl
= 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 αscl
≅ 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).
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, nscl
≅ 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 α1
L > 1. If one also applies the condition α0
L = α1
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/w3
, and with wavelength as λ2
. Such dependences are typical for many lossy waveguides [18
18. A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. 12(5), 1015–1025 (1973). [CrossRef] [PubMed]
]. This means that for any width w and wavelength λ, one can determine an appropriate fiber length such that α1
L > 1 but α0
L << 1, which supports single-mode low-loss propagation through the fiber.
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
19. H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood Jr., “Complex propagators for evanescent waves in bidirectional beam propagation method,” J. Lightwave Technol. 18(8), 1155–1160 (2000). [CrossRef]
]. 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
On the right hand side, the first term represents diffraction, where
is the transverse Laplacian, and the second term accounts for the cross-sectional refractive index profile, n(x,y).
was solved with a finite-difference scheme (FD-BPM) [20
20. Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26(8), 1335–1339 (1990). [CrossRef]
] using the Alternating Direction Implicit (ADI) method [21
21. J. Yamauchi, T. Ando, and H. Nakano, “Beam propagation analysis of optical fibres by alternating direction implicit method,” Electron. Lett. 27(18), 1663–1666 (1991). [CrossRef]
]. The desire to model long propagation lengths required that transparent boundary conditions be implemented [22
22. G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28(1), 363–370 (1992). [CrossRef]
]. The two geometries modeled in the simulations correspond to the geometry of Fig. 1
, with the specific numerical values listed in Table 1
Table 1. Computation Parameters used in Simulations
.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.
Subsequent simulations were performed for a much wider high-aspect-ratio fiber, with core dimensions of 450-μm x 15-μm. Figure 7
Fig. 7 Slow-axis intensity profile along the fiber propagation axis when the core is (a) fully guiding, and (b) semi-guiding. A uniform (flat-top) intensity profile is lunched into the core of each fiber at the left side of the figure. The slow-axis edges of the waveguide are shown as horizontal dashed yellow lines. The dimensions of the frames are 800 μm (vertical) × 50 cm (horizontal) and are not shown to relative scale.
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
Fig. 8 Propagation loss for the first three modes as a function of the slow-axis cladding index step. The points represent simulation results, while the curves are guides for the eye. Guiding and semi-guiding regimes are notated in the figure. The fiber core dimensions are w = 450 μm and h = 15 μm, and the wavelength λ = 1.06 μm,
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 nco
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: α1
L > 1 but α0
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 nscl
can be slightly less than the core index nco
, making that interface guiding, the slow-axis cladding index can still be greater than the signal-cladding index ncl
, 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
Fig. 9 Schematic cross section of SHARC fiber showing an additional physical leakage channel captured by BPM simulations.
. 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
Fig. 10 Excess loss due to fast-axis coiling as a function of bend radius for a 1-m long SHARC fiber that has a 5-cm long straight section at the input and output ends of the fiber and 5-cm adiabatic transition regions between the straight sections and the coiled fiber (as shown in the inset). The points represent simulation results, while the curves are guides for the eye. The horizontal axis has been broken into two separate scales to highlight the transition region at small bend radii.
In these BPM simulations, the equivalent index method [25
25. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]
] was used to simulate the effects of bending. In this method, the refractive index profile is modified as
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)
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
Cross-section of a high-aspect-ratio-core fiber fabricated at OFS Laboratories. The core and cladding dimensions are 14 μm × 139 μm and 270 μm × 840 μm, respectively. Photo courtesy of Dennis Trevor and David DiGiovanni of OFS Laboratories [30
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
Fig. 12 Schematic of SHARC fiber in coiled configuration (not to scale). Expansion shows the relative orientation of the fiber core with respect to the coil.
. 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
27. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006). [CrossRef] [PubMed]
28. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008). [CrossRef] [PubMed]
]. 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
29. J. R. Marciante, R. G. Roides, V. V. Shkunov, and D. A. Rockwell, “Near-diffraction-limited operation of step-index large-mode-area fiber lasers via gain filtering,” Opt. Lett. 35(11), 1828–1830 (2010). [CrossRef] [PubMed]
]. 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.