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Cancellation of non-conservative scattering forces in optical traps by counter-propagating beams: erratum

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Abstract

In a previous Letter [Opt. Lett. 40, 1900 (2015) [CrossRef]  ], we asserted that two counter-propagating beams must be polarized with opposite handedness to cancel scattering forces. A more careful calculation shows that this is not the case. The correct condition is achieved by beams with the same handedness, as derived in this Erratum.

© 2016 Optical Society of America

In a previous Letter [1], we asserted in the paragraph following Eq. (16) that “if beam 1 is collimated and everywhere circularly polarized, then beam 2 must also be circularly polarized, but with the opposite handedness.” A more careful application of Eq. (16) in [1] shows that this is not the case.

Circular polarization across paraxial input beam 1 can be described by E⃗1i(θ,ϕ)=f(θ,ϕ)eiψ(x^±iy^), where f(θ,ϕ) is a real-valued scalar function that determines the amplitude across the beam, and ψ is a real-valued scalar. Applying Eq. (16) in [1] gives the solution for beam 2:

E⃗2i(θ,ϕ)=v⃗*E⃗1i*(θ,ϕ+π)
=ei(γ+ψ)f(θ,ϕ+π)(x^±iy^),
where v⃗eiγ(x^y^z^), represents an element-wise multiplication, * indicates the complex conjugate, and γ takes a value between 0 and 2π. Clearly, the solution for beam 2 is polarized identically to beam 1.

Thus, if beam 1 is collimated and everywhere circularly polarized, then beam 2 must also be circularly polarized with the same handedness. This also implies that Fig. 2(b) in [1] is incorrect: the two circular arrows should be identical, as shown in Fig. 1. These errors affect neither the derivations of any equations nor the main conclusions given in [1].

 figure: Fig. 1.

Fig. 1. Corrected version of Fig. 2(b) from [1] showing a circularly polarized beam pair where the intensity varies across each beam. The arrows indicate the direction or handedness of the polarization. The correction is to make both arrows identical. The color bar indicates the normalized intensity of the beams.

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REFERENCE

1. S. Divitt, L. Rondin, and L. Novotny, Opt. Lett. 40, 1900 (2015). [CrossRef]  

References

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  1. S. Divitt, L. Rondin, and L. Novotny, Opt. Lett. 40, 1900 (2015).
    [Crossref]

2015 (1)

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Figures (1)

Fig. 1.
Fig. 1. Corrected version of Fig. 2(b) from [1] showing a circularly polarized beam pair where the intensity varies across each beam. The arrows indicate the direction or handedness of the polarization. The correction is to make both arrows identical. The color bar indicates the normalized intensity of the beams.

Equations (2)

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E⃗ 2 i ( θ , ϕ ) = v⃗ * E⃗ 1 i * ( θ , ϕ + π )
= e i ( γ + ψ ) f ( θ , ϕ + π ) ( x ^ ± i y ^ ) ,

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