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. 2017 Jun 30;7:4485. doi: 10.1038/s41598-017-04490-2

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© The Author(s) 2017

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Optical forces acting on the particle, when it is at different locations: (a) at the trapping position F_{trans x} and F_{axial x} balance in the xy plan; (b) after the trapping point F_{trans x} exceeds F_{axial x} moving the particle towards the stable point; (c) before the trapping point, both F_{trans x} and F_{axial x} contribute positively to drive the particle to the stable position.