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. 2018 May 9;8:7431. doi: 10.1038/s41598-018-25501-w

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© The Author(s) 2018

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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Two possible schemes to estimate the absorption coefficient α. In the single-mode case (a) and (b) there is no correlation between probe and reference beam, i.e. 〈ΔN_pΔN_R〉 = 0 while in the two-mode case (c) and (d) 〈ΔN_pΔN_R〉 ≠ 0. Different possibilities of input states and absorption estimators for both the schemes are discussed in the text.