Single-Path Spatial Polarization Modulation for Vector Transmission Matrix Measurement and Polarization Control in Scattering Media

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The research is interesting and important because it significantly expands the possibilities of controlling polarization in complex scattering media, which is important for advanced optical applications, including microscopy, telecommunications, and laser beam shaping. The material can be published with minor changes, mainly of a technical kind.

Although the study is experimental, the manuscript is more like a theoretical work. There is no description of the experimental details of the equipment used in the polarimetric setup. In particular, the type and properties of the spatial light modulator (SLM), which works on reflection (resolution and pixel pitch), as well as the tunable liquid-crystal retarder (LCR) are important.

The full-Stokes polarimetry module (FSPM) is apparently built on the basis of a polarizing camera. However, it is not clear what type of camera is used and which additional optical elements are included. If the manufacturer could be identified, it would clarify the equipment's specifications, especially since it is also computer-controlled.

The authors assume that the beam expander, beam splitter, lens, and objective do not alter the polarization state of the light, though this assumption should be verified. Additionally, it should not be presumed that the liquid-crystal retarder and full-Stokes polarimetry module are ideal; manufacturers typically provide the actual characteristics of such components.

The use of abbreviations is inconsistent, with repeated explanations of the same abbreviations as well as omissions. For example, the abbreviation AoLP is explained only in the caption of Fig. 5, but not in Fig. 1, and then again in Appendix A. This should be addressed to improve clarity and presentation. Additionally, the text appears to have not been thoroughly proofread, as evidenced by the presence of the placeholder <?format?> on line 271.

There are no labels a, b, c in Figs. 3 and 4; therefore, the specific parts being referred to can only be inferred from the figure captions.

The manuscript in its current form is understandable only to a narrow circle of specialists well acquainted with the technique of scattered light polarization analysis. Several terms are used without explanation (for example, singular value, singular-value spectra, normalized spectral density). The mention of a single-frequency laser at 532 nm (line 83) refers to a high-performance laser that emits light at a single, extremely narrowband frequency. It would be helpful to clarify whether such a laser was actually used in the experiments, or if the authors merely cited this property without justifying the need for such a laser.

Overall, the manuscript is quite difficult to read, primarily due to the presence of long sentences filled with specialized terminology. It would be beneficial to improve the structure by avoiding combined section titles such as “Experimental Setup and VTM Measurement” or “Discussion and conclusions”. Additionally, the “Results” section title seems somewhat modest and could be made more descriptive. The arrangement of the keywords could also be optimized for clarity and relevance.

Since there are frequent references to the adapted GS algorithm, it was possible to briefly describe GS algorithm and the essence of its adaptation.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors This paper presents a concise, single-path, and interferometer-free scheme for accurately measuring the vector transmission matrix (VTM) in scattering media and achieving polarization control, addressing the limitations of traditional methods which often involve complex setups or incomplete polarization control.

1. In the introduction, it is recommended to clarify the relationship between the proposed scheme and other polarization-related approaches, potentially by including a diagram to illustrate these connections.
2. In the conclusion, consider discussing the potential extension of this scheme to other frequency bands of electromagnetic waves.
3. The study uses a 1mm PTFE layer as the scattering medium, but key physical parameters (such as particle size distribution and refractive index inhomogeneity) are not specified. Additionally, critical experimental parameters—such as the number of pixels and spatial resolution of the SLM, as well as the detection accuracy of the FSPM (e.g., intensity measurement noise level)—should be supplemented.
4. The convergence criterion and convergence speed of the algorithm should be discussed.
5. A performance comparison with mainstream methods would be valuable to readers and is recommended for inclusion.
6. Enlarged versions of the curve graphs should be provided.
7. Relevant application scenarios and potential uses of the method could be added in the introduction to enhance reader interest.

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

The article demonstrates through theoretical derivation and experimental validation that using a single retarder setting leads to coupling between the blocks of the vector transmission matrix (VTM), whereas employing two linearly independent retarder settings (π/2 and 3π/2) successfully decouples and recovers the full VTM.

1. The optical setup in the study utilized a 1 mm thick PTFE (Teflon) as the scattering medium. It remains to be verified whether this method is applicable to scattering media with dichroism or birefringence properties.
2. The model is derived based on a matrix formulation assuming perfect optical components. In practical applications, inherent imperfections in optical elements, such as deviations in the phase delay of retarders, may introduce errors in the calculation of the vector transmission matrix. Would incorporating error terms into the model further enhance the measurement accuracy?
3. In the focusing experiments using the retrieved VTM (Figure 5), the definition of the enhancement range requires further clarification. It should be explicitly stated whether the mentioned "background" refers to the average speckle intensity without wavefront shaping or the local background intensity in the vicinity of the focus.
4. The method relies on precise calibration of the full-Stokes polarimetry module (FSPM). Any calibration errors would directly propagate into the constraint for the relative phase Δφ, potentially causing the Gerchberg-Saxton (GS) algorithm to converge to an incorrect solution. The manuscript does not discuss the calibration procedure or its accuracy.
5. The diversity of reference pool should be further enriched to have state-of-the-art newly work published referring the relation between the polarization information and light propogation in turbid water, such as polarimetric binocular three-dimensional imaging in turbid water, etc.
6. If the phase pattern distribution in the SLM is a digital content image, what would be the distribution of the transmission matrix?

Author Response

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Reviewer 4 Report

Comments and Suggestions for Authors

This manuscript presents a novel and compact method for measuring the vector transmission matrix (VTM) of a scattering medium using a single-path optical setup. The core of the approach is the combination of a phase-only spatial light modulator (SLM) with a subsequent tunable liquid crystal retarder (LCR) to generate spatially varied input polarization states. Crucially, the authors adapt the Gerchberg-Saxton algorithm to retrieve the full VTM from intensity-only, full-Stokes polarimetric measurements at the output, eliminating the need for complex interferometric detection. They theoretically and experimentally demonstrate that using two linearly independent retardance settings of the LCR is necessary and sufficient to decouple the intrinsic blocks of the VTM and avoid spurious correlations. Finally, they effectively utilize the retrieved VTM to demonstrate deterministic polarization-resolved focusing, generating foci with predefined linear and circular polarization states at a target speckle grain. Following are my suggestions:

1. Principle and Practical Implementation of the FSPM: Please provide a more detailed description of the working principle of the Full-Stokes Polarimetry Module (FSPM). Specifically, clarify whether the four polarimetric projections (I₀, I₄₅, I₉₀, I_circ) are acquired sequentially or simultaneously (e.g., via a pixelated polarization filter array). If a simultaneous, pixelated setup is used, the four measurements for a single object point originate from slightly different spatial locations on the sensor. Please explicitly state how this issue of spatial registration is addressed in the data processing to ensure that the Stokes parameters are calculated for the same physical pixel (or after accurate image alignment).
2. Data Processing Pipeline from Raw Images to Algorithmic Constraints: The manuscript mentions the formulas for calculating S₀ to S₃ from the four intensity images, but these parameters do not explicitly appear in the subsequent Gerchberg-Saxton algorithm description. Please clearly articulate the data processing steps that bridge this gap.
3. Physical Meaning of Eₓ and E_y: For absolute clarity, please explicitly define the physical meaning of the output field components Eₓ and E_y in the main text (for instance, when introducing Equation (1)). It should be stated that these represent the complex amplitudes of the output field projected onto a fixed, orthogonal linear polarization basis (e.g., horizontal and vertical).
4. Computational Time and Robustness to Scattering Parameters: The manuscript would benefit from information regarding the time required to achieve the polarization-resolved focusing shown in Figure 5. Please specify the approximate time cost, distinguishing between the VTM measurement phase and the subsequent wavefront calculation phase. Furthermore, please discuss whether and how this time cost depends on the properties of the scattering medium. From a physical perspective, would a thicker or more strongly scattering medium require a larger number of input modes (k) or more algorithm iterations to achieve a precise VTM measurement, thereby increasing the required time?
5. Exploring the Method's Limits: To better define the operational boundaries and robustness of this promising method, it is recommended to include experiments with scattering media of varying thicknesses or scattering strengths. A key question is how the accuracy of the VTM retrieval and the fidelity of the subsequent polarization control degrade as the medium's thickness increases and the polarization memory effect diminishes. Such a study would significantly strengthen the manuscript by mapping the limits of the technique's applicability.

Author Response

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Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

The manuscript might be interesting for the community, but it lacks a few important details. I would recommend the authors to take into account the following remarks and rewrite the manuscript prior it can be accepted.

In the abstract, the authors are encouraged to elaborate on the research background to provide a more comprehensive context for their study. It is essential to highlight its novelty, particularly in relation to established methodologies. 
In the abstract, the authors are suggested to briefly explain the main characteristics of the proposed algorithm to enhance reader comprehension. The authors are suggested to provide main numerical results obtained to enhance reader comprehension.

In the introduction, it is recommended that the authors clearly articulate their main contributions, as well as provide an overview of the manuscript's organization, ideally in separate paragraphs for better clarity.

There is also a lack of quantitative assessment regarding measurement error or noise. The authors should provide information on potential errors, inaccuracies, or confidence intervals associated with their measurements to enhance the robustness of their findings.

I strongly encourage you to enhance the rigor of your work by incorporating a discussion on measurement repeatability and statistical variability. Specifically, providing standard deviations or confidence intervals derived from repeated measurements would significantly bolster the robustness of your conclusions and demonstrate the stability of your algorithm under realistic operational conditions.

In the conclusion, the authors should explicitly address the characteristics and performance metrics of the proposed scheme to reinforce its significance and implications.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

I have no other questions about this article. Thank you to the authors for their explanations. I hope you will consider citing the following article, which I believe offers valuable insights highly relevant to your work on polarization modulation.

Citation:
[1. G. Wu, S. L. Wang, P. Wang and Y. Li, "A Polarization-Doppler Joint Feature-Based Detection Method for Small Targets in Sea Clutter," in IEEE Transactions on Aerospace and Electronic Systems, vol. 60, no. 6, pp. 8791-8804, Dec. 2024, doi: 10.1109/TAES.2024.3433328.

2. Y. Liu, S. L. Wang, G. Wu, P. Wang and Y. Li, "Robust and Unambiguous Four-Channel Monopulse Two-Target Resolution: A Polarimetric Closed-Form Approach," in IEEE Transactions on Signal Processing, vol. 72, pp. 4222-4236, 2024, doi: 10.1109/TSP.2024.3452239.

3. Y. Liu et al., "Polarization Modulation Spectral Approach for Multichannel Radar Forward-Looking Superresolution Imaging," in IEEE Transactions on Aerospace and Electronic Systems, vol. 61, no. 5, pp. 14764-14781, Oct. 2025, doi: 10.1109/TAES.2025.3586826.]

Reviewer 3 Report

Comments and Suggestions for Authors

The authors properly revised the manuscript according to the comments. I think the currernt version can be considered to be accepted.

Reviewer 4 Report

Comments and Suggestions for Authors

I thank the authors for their thorough and thoughtful responses to my comments. The revised manuscript has been strengthened accordingly, and all of my concerns have been adequately addressed. The paper is now much clearer and more robust. I support its acceptance in its current form.

Reviewer 5 Report

Comments and Suggestions for Authors

Authors have carefully addressed all issues