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Achieving very large mode area is a key goal in current research on microstructure and solid fibers for high power amplifiers and lasers. One particular design regime of recent interest has effective area over 1000 square microns and has effectively-single-mode operation ensured by bend losses of the higher-order modes. Simulations show that these fibers are extremely prone to bend-induced distortion and reduction in mode area. The calculated area reduction would significantly impact nonlinear impairments for bend radii relevant to any reasonable spooled package, and can be over 50 percent for bend radii tighter than 10cm. The parabolic-profile design has a natural immunity to bend-induced mode distortion and contraction, and shows superior performance in simulated fair comparisons with other fiber families, including microstructure fibers.
©2006 Optical Society of America
1. Introduction
Fiber amplifiers have been an exciting area of research recently and are poised to displace other sources in many high power applications. The past few years have seen a dramatic increase in output power, and improvements in fiber design promise to enable further leaps forward in the near future [1, 2, 3, 4]. These efforts build on the inherent advantages of fiber as a gain medium, such as the ability of a very long, thin gain medium to effectively dissipate heat.
Several important limitations on total power, such as Raman and Brillouin nonlinear impairments, can be relieved by using fibers with larger mode area. However, several other constraints make this difficult. Even using known techniques for suppression of higher-order modes [5], fibers with effective area greater than a few hundred square microns are difficult to manufacture, handle, and package due to unwanted coupling between modes, and other problems.
There have been many proposals for using microstructure optical fibers (MOFs) to achieve improved large-mode area. Most studies have shown that, while microstructure fibers can have dramatically different performance across broad wavelength ranges [6], fair comparisons at specific wavelengths show modest differences in the tradeoff between effective area and bend sensitivity [7, 8]. One recent result [9, 10] looked at a different MOF regime, with a small number of very large holes. This fiber achieved large effective area, and showed indications of robust operation on a spool of 15 cm diameter. Along with the general push for effective areas above 1000 microns, this result calls for a reexamination of the theoretical bend performance of large-mode-area MOF.
In this paper, we discuss theoretical design of large-mode-area fibers of both solid and microstructure types, emphasizing the need to calculate mode properties (loss, area, etc.) at application-relevant bending radii, and the need to make fair comparisons between fibers of different types, in terms of performance tradeoffs most relevant to applications.
Simulations show that calculated effective area drops significantly for LMA fibers when we simply assume that it is spooled with a realistic bend radius. This is extremely important in the context of an amplifier; since most of the fiber length is bent, the effective area of the bent fiber will determine the onset of nonlinearities. It means that any discussion of straight-fiber areas may be misleading: some LMA designs might seem to offer improved effective area and resistance to nonlinearity according to analysis of their straight-fiber modes, but actually have reduced effective area for any bend radius compatible with relevant packaging. It is important to calculate and compare effective areas for different designs at realistic bend radii. While there have been many studies of bending LMA fibers, bend-induced distortion and area reduction have not previously been studied in detail.
The degradation of fiber modes due to bending suggests the parabolic profile as an interesting candidate for large-mode-area applications. A quadratic index profile has the property that its shape and curvature are preserved under the fiber-bend transformation, which should allow parabolic profiles to maintain higher effective area and better mode shape in the presence of realistic bends. We present encouraging simulation results for examples of parabolic-profile fibers, displaying reduced distortion and contraction of the mode.
Finally we demonstrate how this intuition can be incorporated into a fair comparison of different fiber design families. We present bent-fiber simulations of the recent IMRA fiber [9], and compare this to results for a large number of fibers belonging to simple design families, including the step-index (SIF) and parabolic-profile fibers. This paper illustrates one approach to comparing design strategies for a problem with a complex set of design parameters and and performance metrics. For one reasonable set of assumptions, the analysis indicates superior LMA performance of photonic crystal fibers over step-index fibers, and shows that parabolic-profile fibers perform better than all other fiber types considered here.
2. Bending of large mode area fibers
As we move to very large mode area, many difficulties arise, including macrobend losses, coupling between modes, and sensitivity to nonuniformities in the index profile. Current research addresses these problems by following several different strategies: In some cases a moderately multimode output is tolerated in exchange for extremely large effective areas [11]. Some fibers achieve truly single-mode operation through extremely low effective index contrast of the core, and deal with bend loss mechanically, eliminating bends using a rod-like fiber design [2]. This results in a quasi-solid-state configuration, where only a limited rod length can fit in a reasonable package. This paper assumes the fiber must be spooled and incorporated into a conventional package, and focusses on a strategy of suppressing higher order modes to mitigate bending effects [5, 9]. For this class of fibers, an interesting comparison of different fiber types arises immediately from the need to spool the fiber.
In the current push from core diameters less than 30 microns to those in the 50 micron range, bend-induced distortion of the mode profile increases dramatically, and can no longer be neglected in the assessment of fiber performance. For example, Fig. 1 shows distortion of the fundamental-mode intensity for two fibers: one with moderately large core (30μm diameter, top), and one with very large core (50μm diameter, bottom). The onset of large distortion and mode-area reduction is apparent. The two fibers used in this example have W-shaped index profiles (superficially resembling the letter “W”) as shown in Fig. 2, but the trends are not unique to this profile. We use the name “W3-fiber” to refer to this specific type of W-profile, with outer cladding index n out = n core and R out = 3R core. Both fibers in this example had n core = 1.444 and n core -n clad = 9.6×10-4.
Bending is modeled here using the equivalent index model [12], which accounts for the different path-lengths seen at different transverse positions x as light travels around a bend of radius R bend. Path lengths are adjusted by defining the equivalent index profile n 2 eq,
a modified version of the material index profile n 2, as illustrated in Fig. 3. A number of standard numerical mode-solvers can calculate bent-fiber modes using this model; here, result use a finite-difference mode solver with perfectly matched layer similar to that of [13].
The intuitive equivalent index model explains why the mode is distorted and its area is reduced. Figure 4 shows schematic index plots and fundamental-mode intensity plots for a simple fiber, emphasizing issues relevant to our simulations. The fiber depicted has several higher-order modes (HOMs), and sees a large bend-induced index component relative to the material index contrast. We notice that the bent-fiber has a more asymmetric, and sharply peaked index, leading to a distorted and narrower intensity distribution. Effective index values for the modes are represented by blue or red horizontal lines, and HOM loss (through tunneling to the outer, high-index region) has been depicted by the red arrow and dotted line. This paper focuses on an operating regime where the first group of HOMs have fairly large macrobend losses due to this tunneling, while the fundamental is still low-loss, making the fiber effectively single mode.
More generally, we see that whatever properties we design in the straight fiber, they will not trivially carry over into the equivalent bent fiber index. This presents us with a basic problem is that we do not have direct control over the fiber profile relevant to the actual operating conditions, where the fiber is bent. This can have a large impact, on effective area and other important parameters such as gain overlap, and should be considered in any design optimization.
Fig. 1. Comparison of fundamental mode intensity profiles for two fibers highlights the onset of extreme bend sensitivity as core diameter is pushed beyond 30 microns. The fibers have a W-shaped index profile, with a step-index core (white dashed lines indicate R core and R out). The top two plots show intensities for a fiber with 30-micron core diameter, without and with a bend. The bottom plots are for a 50-micron diameter. Bending to a diameter of 15cm slightly perturbs the mode shape of the smaller-core fiber, but causes a very large displacement, distortion, and contraction in the larger-core fiber.
Fig. 2. This W-shaped index profile was used for the two fibers of Fig. 1. For both fibers, the outer cladding has the same index as the core, and R out = 3R core.
Fig. 4. The equivalent-index model gives us an intuitive picture of bend-induced distortion and area reduction. Bends lead to an index gradient across the core, which tends to push light towards the outside of the bend. The index plot includes the fiber index profile (black) and effective index of the modes (red and blue).
Fig. 5. Effective area is plotted versus bend radius (left), showing a quite significant bend-induced effective area reduction for the larger-core fiber. Circles show bent-fiber simulation results, while the dashed guideline indicates the straight-fiber areas, for comparison. It is also interesting to look at ratio of the areas for the two fibers, A eff,50/A eff,30 (right). This plot shows that the improvement in mode area between the two fibers can become marginal for relevant bend radii (circles), even though the straight-fiber improvement is a factor of 2.3 (black dashed line). Even this mode-area increase is somewhat less than the core-area increase, (50/30)2 ≈ 2.8 (green dashed line).
2.1. Reduced area in solid and microstructure fibers
The dependence of effective area A eff on bend radius R bend is shown in Fig. 5 for the 30 and 50-micron core fibers of the above example (Fig. 1). The standard effective area definition is used here for all fibers
These plots show that modes of the 50-micron fiber have substantially reduced area for any reasonable bend radius. In addition they show that the improvement in effective area obtained by moving from a 30-micron core to a 50-micron core can be marginal if bend radii less than about 10cm are assumed. This is particularly troublesome since the larger 50-micron core in practice results in a number of severe problems, such as mode coupling to HOMs.
We would naturally like to know how bend-induced distortion effects LMA fibers in the literature, for example the recent result by Wong [9]. In fact, using the geometrical parameters quoted for that fiber (hole spacing was 51.1 microns, 4 holes had 46.0 micron diameter, the other two had 39.6 micron diameter), simulations show that bend-induced reduction of effective area is large for this fiber as well. Fig. 6 shows the calculated area versus bend radius, along with the straight fiber value (≈ 1500μm2, in agreement with the published value).
It is then clearly important to quote bent-fiber mode areas in future work in this field. Previously, effective areas quoted in almost all publications have been for straight fibers. This can give misleading impressions about the relative immunity to nonlinearities of different fiber types. This complicates comparisons between fibers, as well as requiring more complicated measurements.
Again considering our intuitive equivalent-index model, the problem seems fairly fundamental. For a given bend radius, using a large core directly implies a large bend-induced index difference across the core. The HOM-suppression strategy generally recommends low contrast (or, in the case of microstructure fibers, low effective index contrast) and so the large bend-induced index gradient will inevitably have some impact on the mode.
Fig. 6. Area plotted versus bend radius (left) for a microstructure fiber [9] shows a large reduction in effective area induced by bends, relative to the straight-fiber value of around 1500 square microns (dashed line). The calculated fiber geometry (right) has two different hole sizes.
Fig. 7. In theory, a pre-corrected fiber could be made to cancel the bend-induced field distortion. This would introduce significant difficulties in fabrication and use of the fiber.
At the same time, problems of distortion, asymmetry, and reduced area are not completely unavoidable. For example, one could theoretically obtain exactly the desired profile with a pre-compensated design. As illustrated in Fig 7, a pre-compensated fiber has a material index profile that is the desired profile minus the expected bend-induced term. If the fiber is bent in exactly the way expected, the equivalent index for the bent fiber will fit the desired profile exactly. While this is an interesting thought experiment, such a fiber may be very difficult to fabricate and to use: One would generally need to produce an asymmetrical index, and control both the diameter and the orientation of the bend. In addition, one will need to consider degradations due to loss and distortion of modes of the unbent fiber. In the next section, we will see that some fibers have natural immunity to bend-induced distortion, without the need for difficult asymmetrical designs prescribed in Fig. 7.
3. Parabolic-profile fibers
The equivalent index model tells us that the effect of a bend is approximately like adding a constant index gradient to the material profile (assuming low contrast). The tighter the bend, the larger the gradient will be. The parabolic profile
Fig. 8. A parabolic-profile fiber would achieve the goal presented in Fig. 7, but without the difficulties associated with an asymmetric fiber design. The parabolic index function is naturally invariant (but translated) under the influence of bending, so that mode fields should be largely free of bend-induced distortion, asymmetry, and contraction. Naturally, the profile beyond the core radius does not have this invariance property.
has a very special property with respect to bends, since it is invariant under addition of a constant gradient. It is then automatically pre-compensated for any bend radius, as long as boundary effects are not too large. This property is illustrated in Fig. 8, which shows both the infinite parabolic profile (red), and the truncated profile (blue) with n(x,y) = n core for r > R core.
We see this mathematically by completing the square,
where Δn = n core -n clad. So the addition of the term Bx is equivalent to a displacement xd,
and addition of a constant index shift C,
Such a transformation should give almost no change in mode size or shape in response to bending, assuming the mode is reasonably well confined to the parabolic region r < R core. Of course, extremely tight bends may displace the mode so much that it is distorted by the boundary. We propose that a truncated parabolic profile fiber will have some immunity to bend-induced reduction of effective area, and may consequently have better resistance to nonlinear impairments. This model also provides simple estimates of bend-induced changes in gain-overlap, since presumably the mode peak is displaced to the index peak at xd.
Simulation results in Fig. 9 confirm these desirable properties of the truncated parabolic profile. Mode field profiles are shown for no bend (middle) and for a 7.5 cm radius bend (right). We see that, at least for these specific examples, the parabolic fiber shows much less distortion than the W3 fibers discussed above. The actual profile simulated (Fig. 10, left) has a finite number of index steps. Finally, the A eff vs. R bend plot in Fig. 10 (right) shows that the parabolic fiber has, as expected, extremely little variation in mode area as fiber is bent, compared to the W3 fibers. We can see that, while the 50 micron W3 fiber has much greater area in the absence of bending, the parabolic fiber has larger area for bend diameters tighter than about 20 cm. Making comparisons between particular fibers of different families will not necessarily give fair results, so we make a more thorough comparison, using many fiber simulations, in the next section. The parabolic fiber considered here has larger diameter (D core = 68 microns) and contrast (Δn = 1.12×10-3) than the fibers of Fig. 1, but the three fibers should be considered comparable, since they have similar fundamental bending loss (at 7.5cm bend radius).
Fig. 9. Simulations confirm the intuitive bend-resistance for a particular example of a fiber with truncated parabolic profile (D core = 68 microns, Δn = .00112). Even for a fairly tight bend, the mode shows essentially no distortion or contraction.
Fig. 10. The truncated parabolic index (left) with finite layers (D core = 68 microns, Δn = .00112) was used for simulations in this section. The simulated effective area (right, black line) is largely independent of the bend radius for the example parabolic fiber. For comparison, the straight parabolic fiber area is shown (black dashed line), and the results for SIF examples of Fig. 5 are repeated (dotted red and blue lines).
4. Comparing design families
To compare fibers with different index-profile shapes, we should not compare the effective area of arbitrary representatives from each family. Rather we should compare the tradeoff between area and other fiber properties, taking into account available degrees of freedom within each family. In this section, we look at several design families, each of which has the same offset index value (n core = 1.444) and two additional degrees of freedom. These are a size parameter (R core or hole spacing Λ) and a contrast parameter (Δ or hole diameter d). We consider the SIF, the W3 and parabolic families shown in Figs. 2 and 10, and the 6-hole uniform photonic crystal fiber (PCF) family (with all hole diameters equal).
By simulating many fibers of each family, we can get a sense of the relevant tradeoffs between mode properties (fundamental loss, effective area, HOM suppression, etc.) as the two design parameters are varied. For simplicity, we focus on three mode properties at a single wavelength (1060nm): fundamental loss, fundamental effective area, and a ratio describing HOM suppression. In practice, it is difficult to summarize HOM-related impairments with a single number, but the HOM suppression ratio used here, defined as the lowest HOM loss divided by the fundamental loss, is one reasonable indicator of whether HOMs problems have been controlled. Where losses are polarization dependent, we use the worst-case polarizations (highest fundamental loss and lowest HOM loss).
For example, Figs. 11 and 12 map out a relevant patch of design space for the truncated parabolic fiber family. Fig. 11 shows the effective area and HOM loss plotted for a range of geometry parameters Δ and R core (also written R c). Many of the trends are intuitive: area increases for larger cores, and decreases slightly as confinement is increased (larger Δ). Losses naturally increase as we move to more strongly-confined fibers, with larger Δ or larger R core. Fundamental bend loss is shown in Fig. 12. As expected, larger contrast gives smaller losses, but interestingly the fundamental loss does not vary monotonically with core size at fixed Δ. All quantities seem to vary smoothly across the design space.
Fig. 11. Plots of effective area and relative HOM suppression as a function of core size and contrast.
In order to distill out an intelligible comparison, we can fix one parameter, and the fundamental loss is a reasonable choice. As illustrated in Fig. 12, we can interpolate a contrast value for each core size, so that all interpolated fibers of all families have the same fundamental bend loss, 0.1dB/meter. All fibers are comparable in this sense, but the detailed comparison depends on the (somewhat arbitrary) assumed values of bend loss and R bend. In Fig. 12, the interpolated contrast values are shown as stars on the dashed line (the loss target).
To allow comparison with the fiber of Fig. 6, we used the estimated fundamental bend loss for that fiber, around 0.1 dB/meter. In fact, this fiber has high-contrast microstructure across a very large area, and it is difficult to obtain very precise loss estimates from simulations. The estimated uncertainty in calculated losses is of order 10 percent for the microstructure fibers (much smaller for solid fibers, due to their smaller material contrast). However, simulations show that small changes in the bend-loss target amount to tiny differences in the final comparison plots shown below.
With one degree of freedom fixed by the bend loss, the performance of each fiber family can be drawn as a single design-tradeoff curve. Figure 13 plots the tradeoff between A eff and HOM suppression for the different families, where best performance is upwards (effectively single mode) and to the right (large mode area). This generalizes previous results on the potential of different fiber types for large-mode-area operation [8]. In particular, the analysis shows that the fiber presented by Wong does indeed seem to extend the mode area beyond what is achievable for the SIF and W3 families. Perhaps more interesting, these results suggest that the parabolic profile fibers offer performance better than all other families considered here, including the microstructure fibers. This gives encouraging support for the intuition presented in Fig. 8. Further study is needed to examine the advantages of this design, especially for area much larger than 600μm2. Convergence analysis shows that the calculated area values have errors at the percent level or better, loss values have errors of up to a few percent for solid fibers, and microstructure fibers may have errors of order ten percent in the plotted loss values and ratios. Efforts are underway to allow more precise estimates for these computationally intensive calculations.
Finally Fig. 14 highlights the potential for inferring misleading results if straight-fiber areas are used instead of bent-fiber areas. By ignoring bend-induced reduction of mode area, this comparison completely overlooks a potentially important advantage of the parabolic designs. The SIF and parabolic designs fall very nearly along the same curve in this picture. This kind of analysis presents an overly optimistic view on the ease of achieving > 1000μm2 effective areas in an effectively single-mode fiber. The difficulty of this goal is clearer once bend-induced distortion is included in the analysis.
Fig. 13. A fair comparison of the different LMA fiber families is distilled from many simulated fibers, interpolating parameters to achieve a common fundamental bend loss of 100dB/km. Good performance is defined by large area (to the right) and HOM suppression (upwards).
Fig. 14. The same type of comparison using unbent fiber effective area values gives a completely different and misleading picture. By ignoring bend-induced effective area reduction, this comparison overstates the advantages of using holes and completely overlooks the advantage of the parabolic design.
5. Conclusions
This paper points out the importance of including bend-induced distortion of fields in any comparison of fibers with very large effective area, and proposes a new type of bend-resistant LMA fiber that is immune to this distortion. Simulations show that distortion induced by bends is important in cases directly relevant to current research in high-power lasers: fibers with core diameters around 50 microns have their effective areas reduced by 50% or more for relatively mild, 15 cm diameter bends. The nonlinearity limit in many applications will be imposed by the long, coiled segment of fiber, even if we assume that a large, undistorted mode is fully restored at the output.
An understanding of bend-induced distortion prompts investigation into designs resistant or pre-compensated for this effect. The parabolic profile stands out because of its ease of fabrication and natural invariance under addition of a bend-induced equivalent index gradient. Simulations confirm that this leads to substantial resistance to bend-induced mode distortion. Results of a large number of simulations for several fiber types have been used to distill fair comparisons of overall fiber performance, including area, bend loss and HOM suppression. For one reasonable set of assumptions, this “fair” comparison supports the superior performance of the parabolic profile relative to other fiber types.
Acknowledgments
I thank David DiGiovanni, Andrew Yablon, and Marc Mermelstein for valuable discussions on large-mode area fiber design. This work was supported in part by the NIST ATP program under award number 70NANB4H3035.
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