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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 4436–4443
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Direct measurement of bend-induced mode deformation in large-mode-area fibers

R. C. G. Smith, A. M. Sarangan, Z. Jiang, and J. R. Marciante  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 4436-4443 (2012)
http://dx.doi.org/10.1364/OE.20.004436


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Abstract

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 single- and few-moded-fibers. However, much more significant mode deformation has been predicted for large-mode-area fibers than for single- or few-moded-fibers. In this paper, the spatial profiles of modes deformed by bending in large-mode-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 OSA

1. Introduction

Large-mode-area fibers for high-power fiber lasers allow for higher continuous-wave and peak powers than conventional single-mode fiber lasers. Their larger fundamental modes offer reduced intensity for the same optical power, thereby limiting nonlinear effects. Large-mode area fibers, however, are nominally multimode. Coiling the fiber is commonly used to reduce the presence of higher-order modes [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]

], but can lead to distortion of the spatial mode profile [2

2. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]

], 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. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]

]
neff2(x,y,z)=n2(x,y)(1+2xR(z))
(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

2. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]

]. Mode deformation has been measured in both single-mode [3

3. Z. W. Bao, M. Miyagi, and S. Kawakami, “Measurements of field deformations caused by bends in a single-mode optical fiber,” Appl. Opt. 22(23), 3678–3680 (1983). [CrossRef] [PubMed]

] and few-moded [4

4. I. Verrier and J. P. Goure, “Effects of bending on multimode step-index fibers,” Opt. Lett. 15(1), 15–17 (1990). [CrossRef] [PubMed]

] 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-mode-area fibers has been numerically investigated for the case of step-index fibers [5

5. 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]

] and photonic crystal fibers [6

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). [CrossRef]

], which show significantly greater mode deformation than predicted for and observed in single-mode fibers. The most important impact of mode deformation in high-power 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 large-mode-area fibers. The reduction in the effective area has been indirectly measured via the enhancement of nonlinear effects [7

7. J. W. Nicholson, J. M. Fini, A. D. Yablon, P. S. Westbrook, K. Feder, and C. Headley, “Demonstration of bend-induced nonlinearities in large-mode area fibers,” Opt. Lett. 32, 2562–2564 (2007).

]; 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 large-mode-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 step-index fiber [8

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).

]. The geometry of a helix gives the core of a helical fiber an effective bend radius of [9

9. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66, 1025–1031 (1976).

]
RB=Qsin2θQtan2θ=P24π2Q,
(2)
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.

2. Numerical model

3. 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 few-moded 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 helical-core 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
Fig. 2 Measured intensity profile at the cleaved end of a helical-core fiber.
. 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 LP01 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
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.
.

4. Comparison to perturbative models

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
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.
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

Table 2. Beam offset, beam full-width at half maximum (FWHM), and mode overlap of Garth’s perturbative model, Marcuse’s equivalent index model, and Bao’s experimental data

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, 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
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.
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

Table 3. Beam offset, second-moment beam width, and mode overlap of Garth’s perturbative model, Marcuse’s equivalent index model, and measured data

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, 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.

5. 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 large-mode-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 large-mode-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.

Acknowledgments

This work was supported in part by the Defense Advanced Research Projects Agency (DARPA)/Microsystems Technology Office under Grant N66001-09-1-2056, the U.S. Department of Energy (DOE), Office of Inertial Confinement Fusion under Co-operative Agreement DE-FC52-08NA28302, the University of Rochester, and by the New York State Energy Research and Development Authority. The support of DARPA and DOE do not constitute an endorsement by DARPA or the DOE of the views expressed in this paper.

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). [CrossRef] [PubMed]

2.

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]

3.

Z. W. Bao, M. Miyagi, and S. Kawakami, “Measurements of field deformations caused by bends in a single-mode optical fiber,” Appl. Opt. 22(23), 3678–3680 (1983). [CrossRef] [PubMed]

4.

I. Verrier and J. P. Goure, “Effects of bending on multimode step-index fibers,” Opt. Lett. 15(1), 15–17 (1990). [CrossRef] [PubMed]

5.

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]

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). [CrossRef]

7.

J. W. Nicholson, J. M. Fini, A. D. Yablon, P. S. Westbrook, K. Feder, and C. Headley, “Demonstration of bend-induced 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). [CrossRef] [PubMed]

13.

S. J. Garth, “Modes on a bent optical waveguide,” in IEE Proc. J. Optoelectron. 134, 221–229 (1987).

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(140.3510) Lasers and laser optics : Lasers, fiber
(230.7370) Optical devices : Waveguides
(230.2285) Optical devices : Fiber devices and optical amplifiers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: October 14, 2011
Revised Manuscript: December 2, 2011
Manuscript Accepted: December 4, 2011
Published: February 8, 2012

Citation
R. C. G. Smith, A. M. Sarangan, Z. Jiang, and J. R. Marciante, "Direct measurement of bend-induced mode deformation in large-mode-area fibers," Opt. Express 20, 4436-4443 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4436


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References

  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]
  2. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am.66(4), 311–320 (1976). [CrossRef]
  3. Z. W. Bao, M. Miyagi, and S. Kawakami, “Measurements of field deformations caused by bends in a single-mode optical fiber,” Appl. Opt.22(23), 3678–3680 (1983). [CrossRef] [PubMed]
  4. I. Verrier and J. P. Goure, “Effects of bending on multimode step-index fibers,” Opt. Lett.15(1), 15–17 (1990). [CrossRef] [PubMed]
  5. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express14(1), 69–81 (2006). [CrossRef] [PubMed]
  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). [CrossRef]
  7. J. W. Nicholson, J. M. Fini, A. D. Yablon, P. S. Westbrook, K. Feder, and C. Headley, “Demonstration of bend-induced 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. Express15(23), 15402–15409 (2007). [CrossRef] [PubMed]
  13. S. J. Garth, “Modes on a bent optical waveguide,” in IEE Proc. J. Optoelectron.134, 221–229 (1987).

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