Simple Direct Measurement of the Orbital Stokes Parameters in Structured Vortex Beams

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This article proposes a simple method to measure the orbital Stokes parameters of structured vortex beams and to map these states onto the orbital Poincar é sphere. This method only requires the beam intensity pattern to be recorded twice in front of the cylindrical lens and at its bifocal point, making it relatively simple and easy to implement. This work seems interesting. The theory seems to be correct and the experiment is feasible. I recommend its publication in Photonics.

I just have some minor issues:

Why is the method proposed in this article more suitable for certain beams with certain symmetries?

The calculation of the orbital Stokes parameters in the article seems to be quite complex, and it is suggested that the authors discuss their practical application scenarios.

Author Response

We are grateful to the reviewers for their comments, it is really very valuable and allows us to improve our research.

Comments 1: Why is the method proposed in this article more suitable for certain beams with certain symmetries?

Answer: Our approach is based on computer processing of only two intensity pattern shots, which is much simpler and more reliable than other special complex techniques. The restrictions used to simplify the justification of the experiment are not significant, since the theoretical justification uses a universal technique of symplectic transformations. However, additional systematic studies of the reciprocity effect for unstable structured bundles are required, which are beyond the scope of the article. Nevertheless, our preliminary experiments have yielded positive results. We indicated this both in Introduction and Discussion and Conclusions.

Comments 2: The calculation of the orbital Stokes parameters in the article seems to be quite complex, and it is suggested that the authors discuss their practical application scenarios

Answer: Generally speaking, the technique proposed  for specifying the  orbital Stokes parameters is based on computer processing of only two images of the intensity pattern that greatly simplifies the entire measurement process reducing it to good computer software. The step-by-step experimental implementation of the method is described in Section "3.3 Experimental". We have also significantly revised Section "Discussion and Conclusions", where was presented a potential applicability of this method in practice

Reviewer 2 Report

Comments and Suggestions for Authors The manuscript presents a simple and direct method for measuring orbital Stokes parameters in structured vortex beams. By drawing an analogy with polarization Stokes parameters and using second-order intensity moments, the method maps beam states onto the orbital Poincaré sphere. The technique is applied to structured Laguerre-Gaussian and binomial beams, with experimental results supporting the approach. The method is interesting and offers a simplified alternative for measuring OAM in structured beams. It is worth publishing after considering the following revisions:   1. The method relies on the reciprocity effect for symmetric beams. In practice, beams might exhibit slight asymmetries. The authors should discuss how such asymmetries could affect the method's accuracy and applicability. 2. The manuscript lacks a detailed discussion on how experimental noise might influence the results, particularly in the calculation of second-order matrix elements and Stokes parameters. The potential impact of noise and any mitigation strategies should be addressed. 3. While simplifying the process, the method involves multiple steps that might not be suitable for real-time measurements. The authors should discuss its potential limitations in real-time applications.
4. There are some formatting errors, such as on page 5, line 133, where "S3" should be capitalized. A thorough check for such issues is needed before publication.

 

Author Response

We are grateful to the reviewers for their comments, it is really very valuable and allows us to improve our research.

Comments 1: The method relies on the reciprocity effect for symmetric beams. In practice, beams might exhibit slight asymmetries. The authors should discuss how such asymmetries could affect the method's accuracy and applicability.

Answer1: The reciprocity effect considered in Section 3.2 was employed to theoretically substantiate measuring the third orbital Stokes parameter after the astigmatic transformation by a cylindrical lens in Eq. (22). By-turn, the distortions obtained by the structured beam in practice, including the reciprocity effect, are reflected at once in the measurement errors of the orbital Stokes parameters and in mapping onto the Poincare sphere. The estimate and discussion of the measurement error of the orbital Stokes parameters is given in Section 3.3 Experiment.  To clarify the situation, we have made the following clarifications at the lines 257-260.

Comments 2: The manuscript lacks a detailed discussion on how experimental noise might influence the results, particularly in the calculation of second-order matrix elements and Stokes parameters. The potential impact of noise and any mitigation strategies should be addressed.

Answer 2: We have added a paragraph regarding noise reduction in the experimental implementation. See lines 294-302.

Comments 3: While simplifying the process, the method involves multiple steps that might not be suitable for real-time measurements. The authors should discuss its potential limitations in real-time applications.

Answer 3: Indeed, the proposed method includes several steps that may not be optimal for real-time applications, especially in settings that require high data throughput. We have added a long paragraph discussing the limitations of our method in the section “Discussion and Conclusions”. See lines 364.

Comments 4: There are some formatting errors, such as on page 5, line 133, where "S3" should be capitalized. A thorough check for such issues is needed before publication.

Answer: We tried to check the manuscript thoroughly for formatting errors and made the necessary corrections.

Reviewer 3 Report

Comments and Suggestions for Authors

This work introduces a novel technique for measuring orbital Stokes parameters by leveraging an analogy with polarization Stokes parameters and using symplectic methods for the second-order intensity moments matrix. The authors propose a streamlined measurement approach, requiring only two intensity pattern shots, which is efficient and operationally simple. This method is tested using structured Laguerre-Gaussian and binomial beams, demonstrating theoretical and experimental consistency with existing research. The work is well-organized with reliable results, and I recommend it for publication with minor revisions.

1. At the end of the introduction, briefly mention the main methods and key results to give readers a clear impression of this work.

2. Ensure that all symbols used in the equations are clearly defined.

3. In Section 3.3, add more information on how measurement error was estimated.

4. Improve the clarity of Fig. 6 by adding clear labels.

Author Response

We are grateful to the reviewers for their comments, it is really very valuable and allows us to improve our research.

Comments 1:  At the end of the introduction, briefly mention the main methods and key results to give readers a clear impression of this work.

Answer 1: At the end of  Introduction, we have added a paragraph describing the main goal of the paper and the methods we used during its implementation. See lines 56-60.

Comments 2: Ensure that all symbols used in the equations are clearly defined

Answer 2: We tried to check the manuscript thoroughly for formatting errors and made the necessary corrections.

Comments 3: In Section 3.3, add more information on how measurement error was estimated.

Answer 3: We thank the reviewer for valuable comments. We have significantly revised the Discussion and Conclusions section to include information on measurement error. See lines 407–430.

Comments 4: Improve the clarity of Fig. 6 by adding clear labels.

Answer 4: We have tried to improve the clarity of the notations in Figure 6 by clarifying the description of the elements and adding a legend with an explanation.