Abstract

We provide additional details on the bend perturbation calculation necessitated by labelling errors in Fig. 3(a) of Optica 2, 267 (2015) [CrossRef]  , and provide one citation missing from the original manuscript.

© 2017 Optical Society of America

In the original manuscript, we had labeled the y-axis of Fig. 3(a) as “ap,” which was defined in (3) with a radial dependence [1]. This implied radial dependence is a typographical error. To derive the correct axis label, we begin with the two degenerate OAM states of a given |L|:

ΨL+=σ^+F|L|(r)eiLϕ
ΨL=σ^F|L|(r)eiLϕ.

Coupling between these two states requires polarization coupling from σ^+ to σ^. As this is precisely the effect conventional polarization controllers (polcons) enact in single mode fibers (SMF), we assume that this mode coupling term is of normalized order 1 [2]. Thus, coupling between ΨL+ and ΨL by a polcon with scalar perturbation component, Δn2, as given by (3) in the original manuscript, leads directly to (4). To evaluate (4), we use the model for bend perturbations introduced by Blake et al. [3], where the scalar effect of a bend is given by:

Δn2(r,ϕ)=ei2πnλ(1χ)rθcosϕ.

We take the wavelength to be 1550 nm, n is the index of refraction of Silica at 1550 nm, and χ=0.22 is a factor to account for stress-induced changes to the refractive index of the fiber when bent [3]. θ is the bend angle, assumed small. (E2) can be transformed using the Jacobi–Anger expansion, and directly inserted into (4) in the original manuscript. Using (5) in the original manuscript, we obtain:

k|2L|=ΨL+|Δn2|ΨL=F|L|(r)|J2|L|(2πnλ(1χ)rθ)|F|L|(r).

We assume that the bend of the polcon lies along a circle of radius 2.8 cm, as is the case for our polcon experimentally, and define the local bend as being an arc along the circumference of said circle such that the center of the fiber is displaced by the radius of the fiber by the end of the bend. The angle which defines this sector of a circle is geometrically equivalent to the bend angle. Using a fiber radius of 125 μm, this bend angle is calculated to be 0.095 radians, which we round to 0.1 radians as an order of magnitude approximation. The unperturbed mode fields, F|L|(r), are calculated from a measured fiber refractive index profile, and (E3) is thus evaluated numerically to populate Fig. 3(a) in the original manuscript. The correct figure label should be kp as in (E3), with the integration over the cross section of the fiber having been carried out. Additionally, the x-axis in Fig. 3(a) in the original manuscript should have referred to p=2L, rather than p=ΔL/2 as it is labeled.

 figure: Fig. E1.

Fig. E1. Coupling coefficient, in dB scale, due to bend perturbations in an air core fiber according to (E3), for multiple bend angles, θ. For each angle, the coefficients rapidly decrease as the Fourier series order increases, in agreement with the intuition that angular momentum should be conserved.

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In the original paper, we have written “The polcon may be understood as transfer of OAM from the bend perturbation to the field itself [25].” Reference [3] ([26] in main text) should also be cited here. Where [26] is currently cited in the main text, [2] (from this erratum) should be cited instead.

REFERENCES

1. P. Gregg, P. Kristensen, and S. Ramachandran, "Conservation of orbital angular momentum in air-core optical fibers," Optica 2, 267–270 (2015). [CrossRef]  

2. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, "Bending-induced birefringence in single-mode fibers," Opt. Lett. 5, 273–275 (1980). [CrossRef]  

3. J. N. Blake, H. E. Engan, H. J. Shaw, and B. Y. Kim, "Analysis of intermodal coupling in a two-mode fiber with periodic microbends," Opt. Lett. 12, 281–283 (1987). [CrossRef]  

References

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  1. P. Gregg, P. Kristensen, and S. Ramachandran, "Conservation of orbital angular momentum in air-core optical fibers," Optica 2, 267–270 (2015).
    [Crossref]
  2. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, "Bending-induced birefringence in single-mode fibers," Opt. Lett. 5, 273–275 (1980).
    [Crossref]
  3. J. N. Blake, H. E. Engan, H. J. Shaw, and B. Y. Kim, "Analysis of intermodal coupling in a two-mode fiber with periodic microbends," Opt. Lett. 12, 281–283 (1987).
    [Crossref]

2015 (1)

1987 (1)

1980 (1)

Blake, J. N.

Eickhoff, W.

Engan, H. E.

Gregg, P.

Kim, B. Y.

Kristensen, P.

Ramachandran, S.

Rashleigh, S. C.

Shaw, H. J.

Ulrich, R.

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Figures (1)

Fig. E1.
Fig. E1. Coupling coefficient, in dB scale, due to bend perturbations in an air core fiber according to (E3), for multiple bend angles, θ. For each angle, the coefficients rapidly decrease as the Fourier series order increases, in agreement with the intuition that angular momentum should be conserved.

Equations (4)

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ΨL+=σ^+F|L|(r)eiLϕ
ΨL=σ^F|L|(r)eiLϕ.
Δn2(r,ϕ)=ei2πnλ(1χ)rθcosϕ.
k|2L|=ΨL+|Δn2|ΨL=F|L|(r)|J2|L|(2πnλ(1χ)rθ)|F|L|(r).

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