Direct measurement of bend-induced mode deformation in large-mode-area fibers

In 1976 Marcuse developed an equivalent index model to predict the effects of bending in waveguides, and predicted deformation of the spatial modes in bent optical fibers. Perturbative approaches have been previously applied and tested to predict the behavior of singleand fewmoded-fibers. However, much more significant mode deformation has been predicted for large-mode-area fibers than for singleor few-moded-fibers. In this paper, the spatial profiles of modes deformed by bending in largemode-area fibers are measured for the first time. A finite difference method employing the equivalent index model is used to calculate the modes of the helical fiber, which show an offset that is twice as large as that predicted for single-mode fiber, and mode compression that is five times greater. These calculated results are compared to the experimental data, yielding significantly better agreement than previous perturbative approaches. ©2012 Optical Society of America OCIS codes: (060.2270) Fiber characterization; (140.3510) Lasers, fiber; (230.2285) Fiber devices and optical amplifiers; (230.7370) Waveguides. References and links 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). 2. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). 3. Z. W. Bao, M. Miyagi, and S. Kawakami, “Measurements of field deformations caused by bends in a singlemode optical fiber,” Appl. Opt. 22(23), 3678–3680 (1983). 4. I. Verrier and J. P. Goure, “Effects of bending on multimode step-index fibers,” Opt. Lett. 15(1), 15–17 (1990). 5. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006). 6. J. C. Baggett, T. M. Monro, K. Furusawa, V. Finazzi, and D. J. Richardson, “Understanding bending losses in holey optical fibers,” Opt. Commun. 227(4-6), 317–335 (2003). 7. J. W. Nicholson, J. M. Fini, A. D. Yablon, P. S. Westbrook, K. Feder, and C. Headley, “Demonstration of bendinduced nonlinearities in large-mode area fibers,” Opt. Lett. 32, 2562–2564 (2007). 8. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31, 226–228 (2006). 9. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66, 1025–1031 (1976). 10. W. P. Huang, ed., Method for Modeling and Simulation of Guided-wave Optoelectronic Devices (EMW Publishing, 1995.) 11. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations and the Schrödinger Equation (Wiley, 2001). 12. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15(23), 15402–15409 (2007). 13. S. J. Garth, “Modes on a bent optical waveguide,” in IEE Proc. J. Optoelectron. 134, 221–229 (1987).

reduce the presence of higher-order modes [1], but can lead to distortion of the spatial mode profile [2], which is commonly referred to as mode deformation.
In 1976, Marcuse showed that by a transformation of coordinates, a coiled optical fiber can be mathematically treated as a straight fiber with an equivalent index distribution given by [2] 2 2 2 ( , , ) ( , ) 1 ( ) where n(x,y) is the material index of the waveguide structure, x is the dimension in the direction of the bend, and R is the radius of the fiber coil.The alteration in the equivalent index given in Eq. ( 1) results in the mode being compressed toward the edge of the core opposite the bend direction, reducing the effective area of the mode [2].Mode deformation has been measured in both single-mode [3] and few-moded [4] fibers.Perturbative models have been accurately applied to describe the measured deformation in those fibers, but those perturbative models are less effective in describing the mode deformation of large-mode-area fibers, as will be described in Section 4.
In the context of high-power fiber lasers, the effect of mode deformation on large-modearea fibers has been numerically investigated for the case of step-index fibers [5] and photonic crystal fibers [6], which show significantly greater mode deformation than predicted for and observed in single-mode fibers.The most important impact of mode deformation in highpower fiber lasers is the predicted mode compression, which reduces the effective area of the mode thereby decreasing the nonlinear thresholds that large-mode-area fibers are meant to increase.
However, mode deformation has never been directly experimentally measured in largemode-area fibers.The reduction in the effective area has been indirectly measured via the enhancement of nonlinear effects [7]; both stimulated Raman scattering and self-phase modulation increased as the fiber coiling diameter was reduced, indicating that the bending resulted in reduced mode area as predicted by theoretical mode calculations using Marcuse's equivalent index model.
In this paper, we present the first direct measurements of mode deformation in largemode-area fibers, and compare the results to mode predictions using both perturbative analytic approaches and direct numerical calculations.The measurements were performed using large-mode-area helical-core fibers.Helical-core fibers have been investigated recently to implement bend loss without the long-term stresses induced in a coiled conventional stepindex fiber [8].The geometry of a helix gives the core of a helical fiber an effective bend radius of [9]  , sin tan 4 where Q is the offset of the core and θ is the trajectory angle of the helix defined by tan(θ) = 2πQ/P, where P is the pitch of the helix.Helical-core fibers are useful for directly measuring mode deformation because the bend is integrated into the straight fiber, and therefore, continues to the cleaved end of the fiber without any need for a physical fiber coil.Moreover, a helical-core fiber does not have intrinsic stress, unlike a conventional coiled fiber.Therefore, the measured effects of bending in a helical-core fiber can be directly compared with Marcuse's effective index model, which does not include any dependence on mechanically induced stress.

Numerical model
The modes of the equivalently bent fiber were calculated using the finite difference method [10,11].The parameters used to solve for the fiber mode are similar to the helical-core fiber used in the experiment: 39-μm core diameter, 0.09 core NA and a bend radius of 12.9 mm.The exact values of the core and NA were selected to find the modes that yielded the best agreement with the experimental measurement.Figure 1  LP ) calculated for the bent fiber.The modes were identified by the number of intensity lobes and location of phase changes in the electric field from positive to negative.Note that the degeneracy of the two LP 11 modes is broken due to bending, resulting in the two orientations having different propagation constants.For the same reason, only one of the LP 21 modes is a bound waveguide mode.Figure 1 reveals that all of the modes tend to be compressed toward the outside of the fiber core in the plane of the bend, similar to previous calculations of modes in bent fibers.

Measurement
The experiment was performed using a ~10m length of helical-core fiber with a 40 ± 1-μm diameter and a nominally 0.10 NA (V = 12 at 1053 nm).The core is offset from the center of the fiber by 100 ± 3 μm and the helix pitch is 7.1 ± 0.1 mm, which results in an effective bend radius of 12.9 ± 0.7 mm.1053nm light with a linewidth of <70kHz was launched from a single-mode fiber via a microscope objective into the helical fiber, one end of which had been tapered to achieve an effective bend radius of ~4.3m, thus approximating a straight albeit fewmoded fiber (V = 3.7 at 1053 nm).In this way, the round mode of the single-mode fiber could be mode matched to the fundamental mode of the effectively-straight tapered helical fiber.After launch, this mode should then adiabatically transform through the taper to the fundamental mode of the effectively bent helical-core fiber.On the other end of the helicalcore fiber, the flat-cleaved fiber output was collimated with another microscope objective and imaged onto a CCD camera with a magnification of ~100.The intensity recorded by the camera was then Fourier filtered to remove etaloning effects and noise.The resulting intensity profile is displayed in Fig. 2. Mode deformation can be seen in the distorted and curved appearance of the intensity profile.The image in Fig. 2 looks very similar to the deformed LP 01 mode shown in Fig. 1(a), which has been rotated and scaled to match the measured data via optimized mode-overlap calculation and is displayed in Fig. 3.
Since perfect free-space launch into the fundamental mode of a multimode fiber is not possible, some amount of light will end up in the higher-order modes shown in Figs.1(b)-(d).Therefore, a more accurate match to the experimental data can be obtained by coherently Fig. 2. Measured intensity profile at the cleaved end of a helical-core fiber.Fig. 3. Model prediction of the LP01 mode of the bent fiber.The white line marks the edge of the fiber core and the white dot marks the center of the core.including these higher order modes and optimizing their relative phases and amplitudes.Scaling the total power to match the experimental measurement, the optimization is performed by minimizing the normalized root-mean-square difference (nrmsd) between the measured data and the model predictions as described by the equation ( ) ) min( ) where i represents the data points in x-y space.The modal power distribution and relative phase differences between the modes that resulted in the optimal overlap are shown in Table 1.The intensity nrmsd between the experimental and theoretical data is 1.7% for the composite model using all four modes, and the resulting intensity pattern is displayed in Fig. 4. Compared to the fundamental mode (shown in Fig. 3), the composite beam profile appears more oblong, its tails better matching the measured data.It is interesting to note that although the LP 01 mode carries only 75% of the total power, the match with the experimental data (indicated in Fig. 2) shows only ~2% rms error.This is yet further confirmation of an earlier conclusion [12] that many beam quality measurements (such as M 2 ) do not by themselves yield sufficient information to reveal the modal content of the beam exiting the fiber.LP modes of the fiber.The white line marks the edge of the fiber core and the white dot marks the center of the core.

Comparison to perturbative models
The perturbative models for mode deformation were created by Bao [3] and Garth [13], the latter of which was used by Verrier [4] to explain their experimental findings.The model by Garth is very similar to that used by Bao and is more publicly accessible, so it is used in this work for comparison to Marcuse's model.Both Marcuse's and Garth's models start with the wave equation, with the origin of the coordinate system located at the center of the fiber's bend.Both models then impose a mathematical transformation to obtain an equation for the wave equation in the local coordinates of the fiber.At that point, Marcuse's model makes an approximation, by allowing the core radius be small compared to the bend radius, which results in the aforementioned effective index given by Eq. ( 1).Marcuse's approximate equation must then be solved numerically.Garth's model, on the other hand, does not make that approximation, instead defining a perturbative solution to the equation with a series of terms using existing LP modal solutions whose successive amplitudes scale with increasing orders of core radius over bend radius.This perturbative model is explicitly solved for only two terms, effectively making the same approximation as Marcuse (small core radius compared to bend radius) albeit at a different point in the derivation.The importance of this issue will be discussed in Section 5.
In applying these two models, a comparison is first made between the fundamental mode profiles of a single-mode fiber using Garth's perturbative model, Marcuse's equivalent index model using a finite difference solver, and Bao's measured data.Figure 5 shows line-outs of the intensity profiles along the axis of the bend for a fiber with a 4.8-µm diameter, 0.095-NA core that is coiled at a 4.5-mm bend radius.All the curves clearly indicate mode deformation (shifting and compression in the direction of the bend).Garth's model strikingly reproduces Bao's measurements, while the mode deformation predicted by Marcuse's model overestimates the shift and predicts a larger tail in the cladding.The line-outs were used to calculate the beam offset from the center of the fiber core, the beam width via full-width half-maximum (FWHM), and the mode field overlap.The results of these calculations, shown in Table 2, indicate that Garth's perturbative model is more effective at predicting the mode deformation of a single-mode fiber than Marcuse's equivalent index method, although both yield numerically acceptable solutions.These two models were then applied to the large-mode-area helical-core fiber measured in Section 3. Figure 6 shows line-outs of the intensity profiles along the axis of the bend for a fiber using Garth's perturbative model, Marcuse's equivalent index model using a finite difference solver, and the experimentally measured data presented in Section 3. The figure clearly indicates that perturbative models are no longer applicable and that numerical solutions must be implemented using Marcuse's equivalent index method.As before, the line outs of the large-mode-area fiber were used to calculate the beam offset from the center of the fiber core, the beam width as calculated by the second moment method, and the mode field overlap.The results of these calculations, shown in Table 3, confirm that Garth's perturbative model is no longer applicable, and that Marcuse's equivalent index model, while underestimating the mode tails on both sides of the main peak, offers a realistic match to experimentally measured data.

Discussion and conclusions
The calculations and measurements presented in this paper highlight the need for multiple models when predicting the effects of bending on the modes of optical fibers.The force driving this need is not entirely transparent, however.The models of both Marcuse and Garth make approximations of the core radius being small compared to the bend radius.Indeed, both the single-mode and LMA cases presented in Section 4 conform to this approximation, with the ratio of core radius to bend radius being of the same magnitude, namely 0.5x10 −3 and 1.5x10 −3 respectively.Single-mode fibers are well confined, and therefore experience small perturbations when bent at reasonable radii.As such, their behavior is readily predictable using perturbative approaches.Large-mode-area fibers, on the other hand, have larger cores and usually lower numerical aperture, allowing multiple confined modes and leaky modes.Although the fundamental mode of such a fiber is highly confined to the core, it is not tightly controlled by the core-cladding interface.As such, the modes in large-mode-area fibers are easily distorted, especially when coiled sufficiently to strip higher-order modes.This difference is clearly noticeable when comparing the beam compression and offset between single-mode and largemode-area fibers.The beam in the large-mode-area fiber is significantly more compressed (5x) and much further away from the center of the core (2x) than the beam in the single-mode fiber.
Understanding this in terms of the models, the place in the derivation where the small core-radius/bend-radius approximation is imposed becomes critical.Whereas Garth's model includes only a single perturbative mode term to the precise bend, Marcuse's effectively includes all mode terms via numerical solution of the perturbative bend.This conclusion obviates the possibility of using a perturbative model for bent LMA fibers, instead requiring the use of direct numerical solution via Marcuse's effective index model.
In conclusion, the deformation of modes in large-mode-area optical fibers predicted by Marcuse's equivalent index model was directly measured for the first time.Although both Marcuse's model and the predicted mode deformation for large-mode-area fibers have been used extensively (particularly in the field of high-power fiber lasers), direct observation of mode deformation in these fibers has proved elusive.In this work, helical-core fibers were used to enable direct experimental observation of the deformed spatial mode of a bent largemode-area fiber.Further, comparison of the experimental measurements to modes numerically calculated using the equivalent index model resulted in a normalized rms intensity difference of 1.7%.In addition, a numerical model was necessary for the application of this model, since simpler, perturbative models were demonstrated to be not applicable to large-mode-area fibers, due to the larger effect of the bending than accounted for with the perturbative models.Such data directly validates for the first time both Marcuse's equivalent index model and the prediction of mode deformation in bent large-mode-area optical waveguides.

Fig. 1 .
Fig. 1.Calculated spatial profiles of the first four fiber modes.The dashed white lines mark the edges of the fiber core and the white dots mark the centers of the cores.The fibers are bent toward the bottom of the figure.

Fig. 5 .
Fig. 5. Intensity line-outs along the direction of the bend for a 4.5-µm 0.095-NA single-mode fiber bent at a 4.5-mm radius: prediction using Garth's model (red line); finite difference calculation using Marcuse's equivalent index model (blue line); and data measured by Bao et al. [3] (black dots).The gray shaded areas indicate the cladding cladding region, while white indicates the core.

Fig. 6 .
Fig. 6.Intensity line-outs along the direction of the bend for the large-mode-area helical-core fiber used in the experiment: prediction using Garth's model (red line); finite difference calculation using Marcuse's equivalent index model (blue line); and measured data (black dots) from Fig. 2. The gray shaded areas indicate the cladding region, while white indicates the core.