Single Scattering Dynamics of Vector Bessel–Gaussian Beams in Winter Haze Conditions

Round 1

Reviewer 1 Report (Previous Reviewer 3)

Comments and Suggestions for Authors

I think the authors have revised the manuscript very clearly, and it can be accepted now.

Author Response

Comments 1: I think the authors have revised the manuscript very clearly, and it can be accepted now.

Response: Thank you very much for your positive feedback and for taking the time to review our manuscript. We are glad to hear that the revisions have clarified the content, and we appreciate your acknowledgment of our efforts.

Reviewer 2 Report (Previous Reviewer 4)

Comments and Suggestions for Authors

The authors' explanations not only do not remove the questions asked earlier, but also raise new ones.

 

Question one.

Consider the propagation of two Gaussian beams with the same power but different widths. Following the authors' reasoning (we quote, with minimal correction):

"The spatial distribution of radiation intensity in Gaussian beams with different radii leads to different interactions with the scatterers. Wider Gaussian beams typically have a larger beam spot size and a more distributed energy profile. This leads to a lower local intensity of radiation at each individual point, which in turn reduces the strength of the interaction between the beam and individual scatterers. ... This reduction in intensity weakens the interaction with scatterers, leading to a lower attenuation rate compared to narrower Gaussian beams, where the energy is more concentrated and hence the interaction is stronger."

Do the authors agree with this conclusion?

The author’s state: "the distribution of radiation intensity affects the scattering cross-section." This means that the scattering cross-section of an individual particle depends on the intensity of the incident radiation, i.e. s=s(I). To simplify the reasoning, we assume that all particles in the medium are identical, and their concentration is N_a. Then, for the scattering coefficient of the medium we obtain a=s(I)N_a= a(I), i.e. the scattering coefficient also depends on the intensity. In this case, the attenuation in the initial section of propagation (within the geometric approximation) will be subject to the equation

dI(z)/dz= - a(I)I(z)

The authors do not provide the form of the function a=a(I), but in any form the decay of the radiation intensity will differ from the exponential decay (Bouguer's law). But this contradicts the authors' earlier statements that the decay occurs in accordance with Bouguer's law.

 

Which statement is correct? "The radiation decays according to Bouguer's law" or "the cross-section depends on the intensity"?

Author Response

Comments 1：Consider the propagation of two Gaussian beams with the same power but different widths. Following the authors' reasoning (we quote, with minimal correction):

"The spatial distribution of radiation intensity in Gaussian beams with different radii leads to different interactions with the scatterers. Wider Gaussian beams typically have a larger beam spot size and a more distributed energy profile. This leads to a lower local intensity of radiation at each individual point, which in turn reduces the strength of the interaction between the beam and individual scatterers. ... This reduction in intensity weakens the interaction with scatterers, leading to a lower attenuation rate compared to narrower Gaussian beams, where the energy is more concentrated and hence the interaction is stronger."

Do the authors agree with this conclusion?

The author’s state: "the distribution of radiation intensity affects the scattering cross-section." This means that the scattering cross-section of an individual particle depends on the intensity of the incident radiation, i.e. s=s(I). To simplify the reasoning, we assume that all particles in the medium are identical, and their concentration is Na. Then, for the scattering coefficient of the medium we obtain a(I)=s(I)Na, i.e. the scattering coefficient also depends on the intensity. In this case, the attenuation in the initial section of propagation (within the geometric approximation) will be subject to the equation

dI(z)/dz= -a(I)I(z)

The authors do not provide the form of the function a=a(I), but in any form the decay of the radiation intensity will differ from the exponential decay (Bouguer's law). But this contradicts the authors' earlier statements that the decay occurs in accordance with Bouguer's law.

Which statement is correct? "The radiation decays according to Bouguer's law" or "the cross-section depends on the intensity"?

Response: Thank you for your thoughtful comments and for pointing out the key issues related to the scattering cross-section and its dependence on intensity, as well as the consistency with Bouguer’s law. We appreciate the opportunity to clarify and refine our explanations.

Firstly, we fully agree with your conclusion that the spatial distribution of radiation intensity in Gaussian beams with different radii leads to varying interactions with scatterers. As seen in the figure, A wider beam, due to its larger spot size, distributes energy over a broader area, resulting in a lower local intensity at each point. This reduced intensity naturally leads to weaker interactions with scatterers, resulting in a lower attenuation rate compared to narrower beams where the energy is more concentrated and the interaction with scatterers is stronger. We believe this conclusion is entirely consistent with our reasoning, which emphasizes the role of spatial intensity distribution in determining scattering behavior.

Figure The Efficiency factors of spherical particle under Gaussian beam irradiation at different beam width w₀ = 3.18, 5.30 and 7.42 μm. (a) Absorption efficiency factor, (b) Extinction efficiency factor, and (c) Scattering efficiency factor.

Regarding your concern about the scattering coefficient a(I), which depends on the intensity I, we recognize the potential discrepancy with traditional exponential decay described by Bouguer’s law. When a(I) varies with intensity, the decay of radiation intensity may indeed differ from the standard exponential form. However, we would like to clarify that the scattering coefficient's dependence on the spatial distribution of radiation intensity is central to our approach, rather than solely the magnitude of the intensity at any given point. This is an important distinction because it emphasizes that the decay is influenced by the spatial variation in intensity—such as the intensity profile across the beam—rather than a uniform attenuation as would be expected from a constant scattering coefficient.

Although the form of the radiation decay in this model may not strictly follow the classical exponential decay of Bouguer's law, we believe that this difference arises from considering the more complex interaction between the beam's intensity distribution and the scatterers in the medium. Our model is not intended to contradict Bouguer's law; rather, it extends the law by incorporating the spatially varying intensity in non-uniform beams, such as vortex beams with different OAM. In this way, we are not suggesting a fundamental contradiction but rather offering a more nuanced understanding of the attenuation process in such scenarios.

We hope this clarifies our approach, and we will ensure that these points are addressed more explicitly in the revised manuscript. Once again, we thank you for your thoughtful feedback, which has helped us refine the manuscript further.

Author Response File: Author Response.docx

Reviewer 3 Report (New Reviewer)

Comments and Suggestions for Authors

In this manuscript, the scattering dynamics of vector Bessel-Gaussian (BG) beams in 10

winter haze conditions are discussed. Moreover, the role of Orbital Angular Momentum (OAM) modes in scattering behavior and the transmission attenuation of BG beams through haze-laden atmospheres are analyzed. In my opinion, the paper is clearly written and the result is simple to find. But some issues need the authors to add and address because of insufficient content, which may helpful for promoting this work to broadly audience. 

1) From the line 243 to 264, i.e. talking about the influence of the OAM modes, the results are not clear. Above all, why did you choose OAM modes 1 and 3? They are both odd modes. What about the even modes? Are their conclusions the same as those of the odd modes? Besides, it shows in Fig.3 that at shorter wavelengths, the RCS is primarily affected by the coating. However, at large wavelengths, the RCS is mainly influenced by the OAM modes. What is the physical essence?

2) In the line 283, it is said”the RCS magnitude is strongly influenced by the coating material and thickness”In my opinion, there is no significant change in the RCS. Also, this part are mainly talking about the effect of polarization. Why was there no comparison of the impact of different polarization states on the RCS?

3) It states ”Higher OAM modes, which are characterized by a broader spatial energy distribution, result in a lower local intensity at the location of individual particles. ” in the line 329 to 331. Where did this conclusion come from? It contradicts the conclusion in line 117. Also, based on the result from Fig 3(d), the lower mode should exhibit a broader spatial energy distribution.

4) In the line 336, it is said “For m = 0 (non-vortex beams), soot particles exhibit the greatest impact on transmittance.” While, the soot particles shows the less impact on transmittance comparing with other particles from the Fig 7(a).

5) What is the definition of particle coordinate system in the line 136? The word “Sault” is wrong in the Fig7.

Author Response

Comments 1：In this manuscript, the scattering dynamics of vector Bessel-Gaussian (BG) beams in winter haze conditions are discussed. Moreover, the role of Orbital Angular Momentum (OAM) modes in scattering behavior and the transmission attenuation of BG beams through haze-laden atmospheres are analyzed. In my opinion, the paper is clearly written and the result is simple to find. But some issues need the authors to add and address because of insufficient content, which may helpful for promoting this work to broadly audience.

Response: Thank you for your valuable feedback on our manuscript. We appreciate your positive remarks regarding the clarity of the writing and the simplicity of the findings. We are grateful for your constructive suggestions and will address the issues you have highlighted to further enhance the manuscript.

Comments 2： 1) From the line 243 to 264, i.e. talking about the influence of the OAM modes, the results are not clear. Above all, why did you choose OAM modes 1 and 3? They are both odd modes. What about the even modes? Are their conclusions the same as those of the odd modes? Besides, it shows in Fig.3 that at shorter wavelengths, the RCS is primarily affected by the coating. However, at large wavelengths, the RCS is mainly influenced by the OAM modes. What is the physical essence?

Response: Thank you for your insightful comments and for pointing out areas that would benefit from further clarification.

Regarding your first comment about the choice of OAM modes 1 and 3 in the manuscript, we recognize that the influence of OAM modes on the RCS can indeed be studied across both odd and even modes. The selection of modes 1 and 3 was made primarily due to their frequent use in previous studies (Shi, C.; Cheng, M.; Guo, L. "Attenuation Characteristics of Bessel-Gaussian Vortex Beam by a Wet Dust Particle," Opt. Commun. 514, 128138, 2022). These modes were chosen to explore the basic scattering behaviors in our study. As highlighted in our previous work (Shi, C.; Guo, L.; Cheng, M.; et al., "Scattering of a High-Order Vector Bessel-Gaussian Beam by a Spherical Marine Aerosol," J. Quant. Spectrosc. Radiat. Transfer 265, 107552, 2021), the general trend observed is that an increase in the OAM mode number typically results in a reduction in the RCS, due to the more distributed energy inherent in higher-order OAM modes.

However, due to space limitations in the manuscript, we opted not to include data for even modes, as their trends are qualitatively similar to those of the odd modes. The primary difference lies in the magnitude of the RCS, but the overall trend remains consistent: a higher OAM mode number generally leads to a lower RCS. To avoid overcrowding the figure (Fig. 3) with too many curves, we focused on displaying the odd modes. We believe this approach ensures clarity while still capturing the essential trends.

In response to your second question regarding the influence of coatings at shorter wavelengths and the dominance of OAM modes at longer wavelengths, we agree that this observation pertains to the underlying physical interactions between the BG beams and the scatterers. At shorter wavelengths, the scattering behavior is primarily governed by the surface properties of the haze particles, such as the coatings (ice or water), since the wavelength is comparable to the particle size. As the wavelength increases, however, the scattering dynamics shift, and the effect of OAM modes becomes more pronounced. This is because the OAM modes influence the spatial distribution of the scattered energy, resulting in a more significant impact on the RCS as the wavelength grows.

We hope this clarification addresses your concerns. We will revise the manuscript to incorporate these explanations and provide additional context where necessary.

Comments 3： 2) In the line 283, it is said "the RCS magnitude is strongly influenced by the coating material and thickness” In my opinion, there is no significant change in the RCS. Also, this part are mainly talking about the effect of polarization. Why was there no comparison of the impact of different polarization states on the RCS?

Response: Thank you for your valuable feedback and insightful comments on our manuscript. We sincerely appreciate your constructive suggestions, which have helped us improve the clarity and focus of our work.

In response to your comment regarding the statement "the RCS magnitude is strongly influenced by the coating material and thickness," we have carefully reconsidered this section and made the necessary revisions. Upon further reflection, we agree that the impact of inner-to-outer diameter ratio on the RCS is not as significant as initially indicated, particularly in light of the strong influence of polarization states on the scattering patterns. We have revised the manuscript to clarify that while the coating material and its thickness do play a role, the observed variations in RCS are primarily due to the polarization state, which redistributes the scattered energy across different angles without significantly altering the overall RCS magnitude.

Additionally, we have expanded the discussion on the role of polarization in scattering. As revised in the manuscript, all RCS distributions maintain symmetry, with scattering intensity varying based on the orientation of the particles. While forward scattering remains concentrated in the 30° and 330° directions, backward scattering shows more complex variations across polarization states. These differences are tied to the polarization characteristics of the incident BG beam. However, despite these intricate angular distributions, the RCS magnitude remains largely unaffected, with polarization states such as radial and azimuthal showing larger overall amplitudes, but not significantly altering the total RCS magnitude.

We believe these revisions provide a clearer understanding of the interplay between coating materials, polarization states, and the resulting scattering patterns. We hope that these changes effectively address your concerns and improve the manuscript.

Comments 4：3) It states ”Higher OAM modes, which are characterized by a broader spatial energy distribution, result in a lower local intensity at the location of individual particles. ” in the line 329 to 331. Where did this conclusion come from? It contradicts the conclusion in line 117. Also, based on the result from Fig 3(d), the lower mode should exhibit a broader spatial energy distribution.

Response: Thank you for your valuable feedback. I would like to further clarify the point raised regarding the statement: "Higher OAM modes, which are characterized by a broader spatial energy distribution, result in a lower local intensity at the location of individual particles" (lines 329-331). This conclusion is based on the inherent properties of BG beams, where an increase in OAM mode leads to a larger hollow size in the beam intensity profile. As a result, the energy is distributed over a larger area, which lowers the local intensity at each point in the beam. This reduced intensity naturally results in weaker interactions with scatterers, leading to lower attenuation compared to beams with smaller spot sizes, where the energy is more concentrated and the interaction with scatterers is stronger.

Regarding the apparent contradiction with the conclusion in line 117, I agree that additional clarification is necessary. In line 117, we refer to the RCS angular distribution in Fig. 3(d). As shown, both higher and lower OAM modes exhibit similar RCS intensities in the forward direction, indicating that the overall scattering strength does not differ significantly. The seeming contradiction arises because, although higher OAM modes have a broader spatial energy distribution, which theoretically reduces local intensity and scattering strength, the RCS remains comparable in the forward direction, as illustrated in Fig. 3(d). This suggests that while higher OAM modes lead to weaker scattering in some regions due to reduced energy intensity, the forward scattering behavior is largely unaffected.

I hope this further explanation clears up the confusion, and I appreciate your understanding.

Comments 5：4) In the line 336, it is said “For m = 0 (non-vortex beams), soot particles exhibit the greatest impact on transmittance.” While, the soot particles shows the less impact on transmittance comparing with other particles from the Fig 7(a).

Response: Thank you for your valuable comment. Upon reviewing the figure and the explanation, we acknowledge that the original statement regarding soot particles needs clarification. The significant impact of soot particles on transmittance is primarily attributed to their refractive index, particularly the imaginary part, which is notably higher than that of other particles. This higher imaginary refractive index leads to greater absorption and a more significant reduction in transmission. The comparison in Figure 7(a) is made based on the attenuation caused by different particles, and the soot particles exhibit the largest impact due to their higher absorption characteristics rather than just scattering. Therefore, we have revised the text to clarify this point by emphasizing the role of the refractive index in influencing transmittance.

Comments 6：5) What is the definition of particle coordinate system in the line 136? The word “Sault” is wrong in the Fig7.

Response: Thank you for your valuable feedback. Regarding the definition of the particle coordinate system in line 136, I apologize for the oversight. I have now included a more detailed description in the revised manuscript. Specifically, I consider a vector BG beam propagating along the z-axis and illuminating a concentric spherical particle. The particle has an inner radius a, an outer radius b, and refractive indices m₁ and m₂. The surrounding medium is assumed to be vacuum. The incident beam is positioned in the coordinate system o′x′y′z′ with its center at o′. For the particle, the corresponding coordinate system is oxyz with its center at o. The coordinates of o′ in the oxyz system are (x₀, y₀, z₀).

Regarding the term "Sault" in Fig. 7, this was a typographical error. It should indeed be "sulfate," and I have corrected it in the revised figure.

Reviewer 4 Report (New Reviewer)

Comments and Suggestions for Authors

The study of the scattering of beams used in optical communications is an important issue. There are several ambiguities in the presented work that need to be corrected.

1.       In the upper atmosphere, ice crystals are formed that are not spherical. Why are these phenomena not taken into account?

2.       In the formulas on page 3 the symbol "m" is used. Then "m" describes the refractive index - I suggest changing the symbols.

3.       In the formulas on page 3 the symbol "n" is used. Then "n" or “m” describes the order of the Bessel function - I suggest changing the symbols.

4.       Table 1. Complex refractive indices are shown. What are these values? Are they obtained from measurements or in some other way? Are the complex refractive indices or the real ones used in subsequent calculations? If they are not taken into account in subsequent calculations, why are they shown?

5.       It is also unclear to me whether calculations for vortex beams are performed only for first-order vortices. In beams in which we have an order of, for example, 4 vortices break up into constellations. Such beams were used in positioning laser beams. Each vortex point of such a beam falls on a different area of ​​the droplet - how does scattering look then? Neglecting this effect is not taking into account experimental realities. What is the point of theoretical calculations for situations other than observed ones?

Author Response

Comments 1：The study of the scattering of beams used in optical communications is an important issue. There are several ambiguities in the presented work that need to be corrected.

Response: Thank you for your constructive feedback. I appreciate your observation regarding the ambiguities in the presented work. I have reviewed the manuscript thoroughly and have made the necessary corrections to clarify these points. Specific areas where ambiguity was noted have been addressed with additional explanations, clearer definitions, and enhanced descriptions of the underlying concepts. I hope these revisions improve the overall clarity and precision of the work.

Comments 2：1.       In the upper atmosphere, ice crystals are formed that are not spherical. Why are these phenomena not taken into account?

Response: Thank you for your insightful comment. Indeed, the formation of ice crystals in the upper atmosphere is a complex process, and it is well acknowledged that these crystals are not perfectly spherical. However, in the context of this study, we focus on a simplified model where the difference between the inner and outer dimensions of the particle is minimal. This allows us to approximate the particle shape as spherical, particularly under the assumption that the outer water layer is thin relative to the overall size. While we recognize that non-spherical shapes could influence the scattering behavior, we intentionally limit this study to a first-order approximation for simplicity and tractability. The primary focus of this work is to investigate the impact of the refractive index, which plays a significant role in the scattering behavior.

We fully agree that future studies should explore non-spherical configurations, especially as they may provide more accurate predictions for practical scenarios. However, incorporating such complexity would significantly increase the intricacy of the results, particularly in the case of structured light, which warrants further investigation in subsequent studies.

 

Comments 3：2.       In the formulas on page 3 the symbol "m" is used. Then "m" describes the refractive index - I suggest changing the symbols.

Response: In response to your comment, I have revised the manuscript to improve clarity. Specifically, I have changed the symbol "m" to m₁ for the refractive index of the inner layer and m₂ for the refractive index of the outer layer. Additionally, the term "OAM mode" has been modified to s to better represent the corresponding variable.

Comments 4：3.       In the formulas on page 3 the symbol "n" is used. Then "n" or “m” describes the order of the Bessel function - I suggest changing the symbols.

Response: We greatly appreciate your thorough review and insightful comments. In the formulas on page 3, the symbols "n" and "m" are specifically used to represent the summation indices. The particle size distribution is described by the function fff. To eliminate any potential ambiguity, we have meticulously reviewed the manuscript to ensure that the meaning of each symbol is clearly defined and used consistently throughout the text.

Comments 5：4.       Table 1. Complex refractive indices are shown. What are these values? Are they obtained from measurements or in some other way? Are the complex refractive indices or the real ones used in subsequent calculations? If they are not taken into account in subsequent calculations, why are they shown?

Response: Thank you for your detailed observations and for raising this important point. The complex refractive indices presented in Table 1 are obtained from the reference: Han, Q., Rossow, W. B., & Lacis, A. A. (1994). Near-global survey of effective droplet radii in liquid water clouds using ISCCP data [J]. Journal of Climate, 7(4), 465–497. These values are widely used and are considered standard in similar studies.

The real and imaginary parts of the refractive indices are utilized in subsequent calculations, as they are essential for accurately describing the scattering and absorption characteristics of the haze particles. Their inclusion in Table 1 serves to provide a comprehensive overview of the optical properties used in the analysis and ensures transparency for readers interested in replicating or extending the study. To address your concern, we have added a clarification in the revised manuscript to highlight the relevance and use of these values in the calculations.

Comments 6：5.       It is also unclear to me whether calculations for vortex beams are performed only for first-order vortices. In beams in which we have an order of, for example, 4 vortices break up into constellations. Such beams were used in positioning laser beams. Each vortex point of such a beam falls on a different area of the droplet - how does scattering look then? Neglecting this effect is not taking into account experimental realities. What is the point of theoretical calculations for situations other than observed ones?

Response: Thank you for your insightful comments. As you rightly pointed out, during long-distance atmospheric propagation, the phase distribution of vortex beams can indeed undergo vortex splitting, especially in the presence of atmospheric turbulence. In our current scattering model, we primarily focus on energy transfer and the overall scattering effects. However, we acknowledge that the phenomenon of vortex splitting is crucial, especially in higher-order vortices such as the case you mentioned, where beams with multiple vortex points (e.g., 4) split into constellations.

Considering this effect is an important step forward, and we plan to explore the vortex splitting behavior in future studies, particularly in the context of atmospheric scattering and turbulence. However, it is worth noting that the current modeling approach may not be directly applicable for this, and more advanced methods, such as particle scattering and turbulent phase screens or other suitable techniques, may be required to accurately capture the dynamics of this phenomenon.

Additionally, we have observed that in particle scattering screen methods, the impact of particle scattering on the phase of light beam is relatively weak compared to the effects induced by atmospheric turbulence. This is a key distinction, and it underscores that the Mie scattering modeling approach, while useful, has its limitations when it comes to fully capturing the complexity of atmospheric scattering dynamics.

We appreciate your suggestion and agree that neglecting such effects would overlook the experimental realities. The theoretical framework we have developed aims to provide a first approximation, but further refinements are certainly necessary to better align with real-world observations.

Reviewer 5 Report (New Reviewer)

Comments and Suggestions for Authors

In this paper, the authors have investigated the scattering properties of vector Bessel-Gaussian beam based on the generalized Lorentz-Mie theory. The evolution of OAM is evaluated, the results may have potential application in FSO. In my opinion, the paper is well written and can be accepted after considering following comments:

1. The references of equations in Section 2 should be cited, if the equations are not derived.

2. When the visible light is considered, what will happen? In the analysis, the infrared light is discussed in Section 3.

3. If the distance increases, what will happen? For example, 2km, 10km.

Author Response

Comments 1：In this paper, the authors have investigated the scattering properties of vector Bessel-Gaussian beam based on the generalized Lorentz-Mie theory. The evolution of OAM is evaluated, the results may have potential application in FSO. In my opinion, the paper is well written and can be accepted after considering following comments:

Response: We sincerely appreciate your thorough review and constructive feedback on our manuscript. Your positive comments about our work and its potential application in free-space optics (FSO) are very encouraging. We are glad to hear that you found the paper well-written and of interest. We have carefully considered your comments and have revised the manuscript accordingly to improve its quality.

Comments 2：1. The references of equations in Section 2 should be cited, if the equations are not derived.

Response: We have carefully reviewed the equations in Section 2. For the key equations that are not derived in detail, we have added appropriate references to their original sources. These citations ensure proper attribution and provide additional context for readers. For the basic equations that are widely used and well-established in the literature, we have not provided specific references, as they are considered foundational knowledge.

Comments 3：2. When the visible light is considered, what will happen? In the analysis, the infrared light is discussed in Section 3.

Response: We appreciate your interest in the extension of our work to visible light. While our analysis in Section 3 primarily focuses on infrared light, the results for visible light can indeed be inferred from our model. It is well known that visible light interacts more strongly with fog and haze particles compared to infrared light, leading to more complex angular distributions of the radar cross-section (RCS) and stronger attenuation due to scattering and absorption.

Furthermore, our model is fully capable of calculating the scattering properties for visible light. However, as infrared light is less affected by atmospheric conditions, we chose to focus on it in this work due to its practical advantages in atmospheric haze conditions.

Comments 4：3. If the distance increases, what will happen? For example, 2km, 10km.

Response: We appreciate your inquiry about the effect of distance on the transmission of the vector BG beam. In a haze or foggy environment, the power attenuation of structured light follows the Beer-Lambert law, which predicts rapid attenuation as the transmission distance increases. For long distances (such as 2 km or 10 km), the attenuation would indeed become more pronounced, resulting in significant power loss.

Furthermore, due to the limitations of the model presented in the paper, the beam waist of the BG beam is assumed to be comparable to the size of the atmospheric particles. This leads to a strong divergence of the beam over long distances, making it less suitable for long-range transmission. We acknowledge that this property limits the practical application of BG beams for extended distances, as the beam experiences significant spread and power loss.

In future work, we plan to address this issue by refining the model to account for beam divergence more effectively and exploring solutions to improve long-distance transmission efficiency.

Round 2

Reviewer 2 Report (Previous Reviewer 4)

Comments and Suggestions for Authors

In their responses to the comments, the authors confirmed their thesis that "the spatial distribution of radiation intensity in Gaussian beams with different radii but the same power leads to different interactions with scatterers." However, this statement is absurd. In addition, it is not clear what the physical nature of such interaction is in this case. On this basis, I cannot support the publication of this manuscript

Author Response

Comments 1: In their responses to the comments, the authors confirmed their thesis that "the spatial distribution of radiation intensity in Gaussian beams with different radii but the same power leads to different interactions with scatterers." However, this statement is absurd. In addition, it is not clear what the physical nature of such interaction is in this case. On this basis, I cannot support the publication of this manuscript

Response: Thank you once again for your thorough and constructive feedback. We sincerely appreciate the time you’ve dedicated to reviewing our manuscript. We understand that you have concerns regarding our explanation of the relationship between the spatial distribution of radiation intensity in Gaussian beams and their interaction with scatterers. We would like to take this opportunity to clarify our reasoning and address your points in detail.

You noted that our statement regarding the spatial distribution of radiation intensity in Gaussian beams with different radii leading to varying interactions with scatterers seems "absurd." We understand that this may be counterintuitive, especially when considering traditional models, and would like to offer a more thorough explanation of our viewpoint.

Our argument is based on the assumption that the scattering process is influenced by the local intensity of the radiation at different points within the beam, which in turn affects the scattering cross-section (ESR) of the individual particles. This is not to suggest that the nature of the scattering fundamentally changes, but rather that the strength of the scattering interaction can vary depending on how the energy is distributed spatially across the beam.

Key Points to Consider:

Beam Intensity Distribution: The Gaussian beam, being a spatially varying distribution of energy, has different intensity values at different positions within the beam profile. In the case of a beam with a larger spot size (wider beam), the intensity at each point is lower compared to a beam with a smaller spot size (narrower beam) that concentrates the same power in a smaller region. This lower intensity at each point reduces the interaction between the beam and the scatterers. This is analogous to how the effect of light on a surface varies depending on how concentrated or diffuse the light is.

Scattering Cross-Section: The scattering cross-section of a particle is not necessarily constant—it can vary depending on the intensity of the incident light, especially when considering nonlinear effects. For example, the cross-section can increase in response to higher intensity radiation. However, in the regime we are discussing (linear scattering), the reduced intensity in the larger Gaussian beam profile leads to a weaker interaction with scatterers compared to a narrower beam with higher intensity at each point.

Absence of Contradiction with Bouguer’s Law: While we understand your concern regarding the apparent contradiction between our explanation and Bouguer's law, we would like to emphasize that our approach is not intended to replace or contradict this law. In the classical single-scattering approximation under Bouguer's law, the attenuation is uniform, as you pointed out. However, when we account for spatially varying intensity in different beam profiles, we introduce a more nuanced model where the attenuation rate can vary due to the local intensity at each point within the beam. This does not invalidate Bouguer’s law but rather suggests that attenuation in spatially varying beams could deviate from the traditional exponential decay in certain contexts.

Physical Nature of the Interaction: To address your concern about the "physical nature" of the interaction, we would like to clarify that our argument is based on the spatial distribution of intensity and how that affects the interaction strength between the radiation and scatterers. We are not suggesting a fundamental alteration to the scattering mechanism itself but pointing out that lower local intensities in wider beams result in weaker interactions with scatterers, and higher local intensities in narrower beams result in stronger interactions. This effect is especially significant when considering scattering coefficients that depend on intensity.

We hope this detailed explanation clarifies our reasoning. We believe that our discussion of the spatial intensity distribution and its impact on scattering interactions is not only physically sound but also extends the understanding of attenuation in systems with varying beam profiles.

Reviewer 3 Report (New Reviewer)

Comments and Suggestions for Authors

In my opinion, the paper is clearly written and well organized after careful revision by the authors. It can be accept in present form.

 

Author Response

Thank you very much for your positive feedback and for recognizing the efforts we have made in revising the manuscript. We sincerely appreciate your time and constructive comments, which have greatly helped improve the clarity and organization of our work.

Reviewer 4 Report (New Reviewer)

Comments and Suggestions for Authors

no comments

Author Response

Thank you very much for your positive feedback and for recognizing the efforts we have made in revising the manuscript. We sincerely appreciate your time and constructive comments, which have greatly helped improve the clarity and organization of our work.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Report on photonics-3375841:

The authors investigated the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions. By employing the generalized Lorenz-Mie theory, they analyze the impact of the transition from water to ice coatings on haze particles and its effect on the scattering properties of Bessel-Gaussian beams. The results reveal that the presence of ice-coated haze particles significantly alters the scattering and extinction efficiencies of Bessel-Gaussian beams, in comparison to water-coated particles. Such vector Bessel-Gaussian results may have potential applications in diffraction-free devices.

These findings highlight the importance of accurate light scattering modeling in the presence of ice-coated particles and underscore the critical influence of particle concentration, humidity, and orbital angular momentum modes in optimizing optical systems for operation in hazy and adverse atmospheric conditions. The manuscript reports a significant advance and offers incremental improvement to existing work in the references. The similar Airy beams [OL 38, 4585 (2013)], and vortex beams [OL 42, 3097 (2017)] in atomic and atomic-like ensemble are studied theoretically and experimentally. The authors should distinguish and clarify three-types diffraction-free: Airy beams, vortex beams, and Bessel-Gaussian beams ? The authors also give a brief discussion on diffraction-free beams issue in the introduction section to highlight the novelty of the current work. The possible experimental realization techniques of single scattering of vector Bessel-Gaussian beams in winter haze conditions should be discussed. In addition, the English writing and figures quality throughout this manuscript might be further revised to meet the standard of Photonics.

The author’s work of the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions is useful. This research provides valuable insights into the interaction between vector Bessel-Gaussian beams and environmental haze, offering a foundation for the advancement of optical communication and environmental monitoring technologies in faze atmospheric conditions. After above improvement, I could recommend publication of the manuscript in Photonics.

Comments on the Quality of English Language

The English writing and figures quality throughout this manuscript might be further revised to meet the standard of Photonics.

Author Response

Comments 1：The authors investigated the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions. By employing the generalized Lorenz-Mie theory, they analyze the impact of the transition from water to ice coatings on haze particles and its effect on the scattering properties of Bessel-Gaussian beams. The results reveal that the presence of ice-coated haze particles significantly alters the scattering and extinction efficiencies of Bessel-Gaussian beams, in comparison to water-coated particles. Such vector Bessel-Gaussian results may have potential applications in diffraction-free devices.

Response: We would like to sincerely thank the reviewer for their careful reading of our manuscript and for providing insightful comments. We are glad that our study on the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions was well-received. Regarding the reviewer's comment on the impact of the transition from water to ice coatings on haze particles and its effect on the scattering properties of BG beams, we appreciate the recognition of this important aspect of our work. As pointed out, the ice coating significantly alters the scattering and extinction efficiencies, and this finding indeed enhances the understanding of the scattering behavior in winter haze conditions. We have now highlighted this effect more clearly in the revised manuscript, especially in the discussion section, where we elaborate further on the implications of these changes for practical applications in diffraction-free devices.

Comments 2： These findings highlight the importance of accurate light scattering modeling in the presence of ice-coated particles and underscore the critical influence of particle concentration, humidity, and orbital angular momentum modes in optimizing optical systems for operation in hazy and adverse atmospheric conditions. The manuscript reports a significant advance and offers incremental improvement to existing work in the references. The similar Airy beams [OL 38, 4585 (2013)], and vortex beams [OL 42, 3097 (2017)] in atomic and atomic-like ensemble are studied theoretically and experimentally. The authors should distinguish and clarify three-types diffraction-free: Airy beams, vortex beams, and Bessel-Gaussian beams ? The authors also give a brief discussion on diffraction-free beams issue in the introduction section to highlight the novelty of the current work. The possible experimental realization techniques of single scattering of vector Bessel-Gaussian beams in winter haze conditions should be discussed. In addition, the English writing and figures quality throughout this manuscript might be further revised to meet the standard of Photonics.

Response: Thank you for your constructive feedback. We would like to clarify a few points and provide additional context regarding the current limitations and future directions of our work. Regarding the experimental aspect, we acknowledge the complexity of real atmospheric conditions, which involve both particle scattering and turbulence effects. However, due to the challenges of conducting experiments in such environments, our study currently focuses on a simplified numerical simulation of single-particle scattering. This simulation serves as a theoretical reference and lays the groundwork for future experimental investigations, which will incorporate the more complex effects of turbulence and particle scattering as the next step in our research. Regarding the diffraction-free beams, we appreciate your comments on distinguishing between Airy beams, vortex beams, and Bessel-Gaussian beams. In our work, we primarily use the Bessel-Gaussian beam as a representative example to explore the behavior of diffraction-free beams in the context of particle scattering. While Airy and vortex beams are also significant, our study focuses on the specific case of Bessel-Gaussian beams, and we will clarify this distinction in the revised manuscript. We will also reference the relevant works you mentioned, including the studies on Airy beams [OL 38, 4585 (2013)] and vortex beams [OL 42, 3097 (2017)], in the updated manuscript. Regarding the experimental realization of single scattering of vector Bessel-Gaussian beams in winter haze conditions, we will discuss the theoretical framework and potential experimental setups for future studies. While the current paper focuses on numerical simulations, we recognize the importance of bridging the gap between theory and experiment, and this will be addressed in the revised introduction. We revised the introduction section to better highlight the novelty of our work and incorporate the suggested improvements, including a clearer explanation of diffraction-free beams and their relevance to the present study. Additionally, we have reviewed and improved the quality of the English writing and figures to ensure they meet the standards of Photonics.

Comments 3：The author’s work of the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions is useful. This research provides valuable insights into the interaction between vector Bessel-Gaussian beams and environmental haze, offering a foundation for the advancement of optical communication and environmental monitoring technologies in faze atmospheric conditions. After above improvement, I could recommend publication of the manuscript in Photonics.

Response: We would like to sincerely thank the reviewer for their careful reading of our manuscript and for providing insightful comments. We are glad that our study on the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions was well-received.

Regarding the reviewer's comment on the impact of the transition from water to ice coatings on haze particles and its effect on the scattering properties of Bessel-Gaussian beams, we appreciate the recognition of this important aspect of our work. As pointed out, the ice coating significantly alters the scattering and extinction efficiencies, and this finding indeed enhances the understanding of the scattering behavior in winter haze conditions. We have now highlighted this effect more clearly in the revised manuscript, especially in the discussion section.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is rather comprehensive description of ice haze in atmosphere that takes into account almost all parameters - particles shape and size, including sell-core particles comprising different phases (ice and water), concentration, as well as optical conditions - wavelength, polarization, different Orbital Angular Momentum (OAM) modes . The scattering dynamics of vector Bessel-Gaussian (BG) beams in winter haze environments is described theoretically in details. 

The illustrations and references are relevant. Winter haze conditions are reviewed in different locations, e.g. "Haze Pollution in Beijing".

Author Response

Comments 1：The author’s work of the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions is useful. This research provides valuable insights into the interaction between vector Bessel-Gaussian beams and environmental haze, offering a foundation for the advancement of optical communication and environmental monitoring technologies in faze atmospheric conditions. After above improvement, I could recommend publication of the manuscript in Photonics.

Response: We would like to sincerely thank the reviewer for their careful reading of our manuscript and for providing insightful comments. We are glad that our study on the single scattering dynamics of vector Bessel-Gaussian beams in winter haze conditions was well-received.

Regarding the reviewer's comment on the impact of the transition from water to ice coatings on haze particles and its effect on the scattering properties of Bessel-Gaussian beams, we appreciate the recognition of this important aspect of our work. As pointed out, the ice coating significantly alters the scattering and extinction efficiencies, and this finding indeed enhances the understanding of the scattering behavior in winter haze conditions. We have now highlighted this effect more clearly in the revised manuscript, especially in the discussion section.

Comments 2：The paper is rather comprehensive description of ice haze in atmosphere that takes into account almost all parameters - particles shape and size, including sell-core particles comprising different phases (ice and water), concentration, as well as optical conditions - wavelength, polarization, different Orbital Angular Momentum (OAM) modes . The scattering dynamics of vector Bessel-Gaussian (BG) beams in winter haze environments is described theoretically in details.

The illustrations and references are relevant. Winter haze conditions are reviewed in different locations, e.g. "Haze Pollution in Beijing".

Response: We sincerely thank the reviewer for their detailed and constructive feedback. We appreciate the recognition of the comprehensiveness of our study, particularly in addressing key parameters such as particle shape, size, concentration, and optical conditions, as well as the theoretical exploration of the scattering dynamics of vector Bessel-Gaussian beams in winter haze environments.

Regarding the mention of "Haze Pollution in Beijing," we would like to clarify that while we referenced winter haze conditions as a general example, we did not incorporate specific atmospheric parameters for any particular location, such as Beijing. The study focuses on a simplified numerical simulation of the scattering dynamics under idealized winter haze conditions, without using actual atmospheric data from specific regions. The example was included to illustrate the potential environmental scenario, rather than to imply the use of real atmospheric parameters. We will update the manuscript to clarify this point and ensure that the reference to geographical locations does not cause any confusion.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors researched the scattering dynamics of vector B-G beams in winter haze environments, and the impact of different OAM modes on the performances has been proved. The results are good enough to be accepted. However, here are some suggestions for them to improve the quality of the paper.

1.  I think some references about the vortex beams and the applications should be updated in the introduction section, and they cited lots of old references. Tailoring spatiotemporal dynamics of plasmonic vortices. Opto-Electron Adv 6, 220133 (2023); Robust measurement of orbital angular momentum of a partially coherent vortex beam under amplitude and phase perturbations. Opto-Electron Sci 3, 240001 (2024). Functionality multiplexing in high-efficiency metasurfaces based on coherent wave interferences. Opto-Electron Adv 7, 240086 (2024). 

2. The length of this article can be proved, for example some of the equations can be deleted.

Author Response

Comments 1：The authors researched the scattering dynamics of vector B-G beams in winter haze environments, and the impact of different OAM modes on the performances has been proved. The results are good enough to be accepted. However, here are some suggestions for them to improve the quality of the paper.

Response: We would like to express our sincere gratitude to the reviewer for their positive evaluation of our work. We are pleased to hear that the reviewer finds the results of our study on the scattering dynamics of vector BG beams in winter haze environments, and the impact of different OAM modes, to be valuable. We also appreciate the reviewer’s constructive suggestions for further improving the quality of the paper. We are committed to addressing these points in our revised manuscript and carefully implement the recommended changes to enhance the overall quality and clarity of the paper.

Comments 2：I think some references about the vortex beams and the applications should be updated in the introduction section, and they cited lots of old references. Tailoring spatiotemporal dynamics of plasmonic vortices. Opto-Electron Adv 6, 220133 (2023); Robust measurement of orbital angular momentum of a partially coherent vortex beam under amplitude and phase perturbations. Opto-Electron Sci 3, 240001 (2024). Functionality multiplexing in high-efficiency metasurfaces based on coherent wave interferences. Opto-Electron Adv 7, 240086 (2024).

Response: We would like to sincerely thank the reviewer for their valuable feedback and for suggesting updates to the references in the introduction. We fully agree that more recent studies on vortex beams and their applications should be included to reflect the latest developments in the field. As per your suggestion, we will update the introduction by incorporating the following references:

1. Tailoring spatiotemporal dynamics of plasmonic vortices, Opto-Electron Adv, 6, 220133 (2023)
2. Robust measurement of orbital angular momentum of a partially coherent vortex beam under amplitude and phase perturbations, Opto-Electron Sci, 3, 240001 (2024)
3. Functionality multiplexing in high-efficiency metasurfaces based on coherent wave interferences, Opto-Electron Adv, 7, 240086 (2024)

In addition, we revised the introduction to better highlight the latest advancements in vortex beam research, specifically in the context of their applications in optical technologies. This also ensure that our manuscript is up-to-date and aligned with recent developments in the field.

Comments 3：The length of this article can be proved, for example some of the equations can be deleted.

We sincerely thank the reviewer for their constructive feedback. We understand the concern regarding the length of the article, and we have carefully reviewed the manuscript to improve its conciseness. In response to the suggestion, we have deleted several equations that were deemed redundant or less critical to the main discussions in the paper. These changes help streamline the content while preserving the key technical details essential for understanding the core findings of our study.

Reviewer 4 Report

Comments and Suggestions for Authors

The paper considers the problem of scattering of Bessel-Gaussian (BG) vector beams under regional conditions of winter haze formed by particles containing water-coating and ice-coating insoluble cores. The problem is solved numerically based on the Lorentz-Mie theory in the single scattering approximation. An estimate of the influence of haze concentration and beam parameters on the scattered field characteristics is given. It is shown that an increase in particle concentration leads to a decrease in transmission with decreasing wavelength and increasing core density. The authors paid special attention to the influence of the orbital angular momentum (OAM) of the BG beam on the transmission level. However, the conclusions drawn raise questions:

1. The authors provide the following conclusion (lines 413-415): “It is observed that higher OAM modes exhibit a reduced interaction with particles due to their broader energy distribution, while lower OAM modes, with energy concentrated in a smaller region, are more susceptible to scattering and attenuation.” It is not clear what kind of interaction reduction the authors are talking about. The problem is solved in the single scattering approximation, for which it is typical that the attenuation of the radiation power obeys the Bouguer law. In this case, the decrease in the beam power along the path will be the same, regardless of the OAM of the radiation, and single scattering does not distort the transverse intensity distribution.

2. The results in Fig. 3 indicate that the scattering indicatrices (RCS) for radiation with the same wavelengths and different OAM values ​​are different. These results require clarification. Each elementary volume of a homogeneous medium has the same indicatrix corresponding to the particle size spectrum (39). The absolute value of RCS for an elementary volume is proportional to the magnitude of the intensity incident on this volume. The integral RCS should be proportional to the integral value of the power of the entire scattering volume. But taking into account the fact stated in the previous remark, the integral values ​​of the power should be equal, provided that the initial values ​​of the power of radiation with different OAM values ​​are the same.

3. The study of the propagation of vector BG beams in the atmosphere increases the relevance of the study. However, this study does not consider the influence of such important characteristics of BG beams as their non-diffractive propagation and self-healing properties. The influence of OAM on the scattering characteristics is studied. However, to determine this influence, Laguerre-Gaussian beams with a simpler structure could be considered, for example. The question arises, how will the choice of the beam type affect the results at the same OAM values?

4. The manuscript contains technical errors. In Figure 4, the letter designations (a), (b), (c), ... do not match the designations in the figure caption.

Author Response

Comments 1：The paper considers the problem of scattering of Bessel-Gaussian (BG) vector beams under regional conditions of winter haze formed by particles containing water-coating and ice-coating insoluble cores. The problem is solved numerically based on the Lorentz-Mie theory in the single scattering approximation. An estimate of the influence of haze concentration and beam parameters on the scattered field characteristics is given. It is shown that an increase in particle concentration leads to a decrease in transmission with decreasing wavelength and increasing core density. The authors paid special attention to the influence of the orbital angular momentum (OAM) of the BG beam on the transmission level. However, the conclusions drawn raise questions:

Response: We would like to sincerely thank the reviewer for their thoughtful feedback and for providing valuable insights into our work. We appreciate the recognition of the study's focus on the scattering dynamics of BG vector beams in winter haze environments with particles containing both water-coating and ice-coating insoluble cores.

Regarding the concerns raised about the conclusions, we acknowledge that further clarification may be needed to ensure the accuracy and robustness of our findings. We will carefully review the manuscript to address the questions raised and provide additional explanations where necessary. Specifically, we will:

1. Clarify the influence of haze concentration and beam parameters: We expanded on how changes in particle concentration affect the transmission, and provide a more detailed discussion of the physical mechanisms involved, especially in relation to varying wavelength and core density.
2. Address the role of OAM: While we highlighted the impact of OAM on transmission, we further elucidated the relationship between OAM and scattering characteristics, and explain why this influence is particularly important in the context of BG beams.

These clarifications are included in the revised manuscript to ensure that the conclusions are both clear and well-supported by the numerical results. We also check the overall consistency of the results to resolve any outstanding questions.

Comments 2：The authors provide the following conclusion (lines 413-415): “It is observed that higher OAM modes exhibit a reduced interaction with particles due to their broader energy distribution, while lower OAM modes, with energy concentrated in a smaller region, are more susceptible to scattering and attenuation.” It is not clear what kind of interaction reduction the authors are talking about. The problem is solved in the single scattering approximation, for which it is typical that the attenuation of the radiation power obeys the Bouguer law. In this case, the decrease in the beam power along the path will be the same, regardless of the OAM of the radiation, and single scattering does not distort the transverse intensity distribution.

Response: We would like to sincerely thank the reviewer for their insightful comment regarding the conclusion about the interaction between higher OAM modes and particles. We agree that the original phrasing in the manuscript was unclear and could lead to misinterpretation. We appreciate the opportunity to refine our explanation.

The reviewer correctly points out that in the single scattering approximation, the attenuation of radiation power follows the Beer-Lambert law, and the transverse intensity distribution of the beam is not distorted. We acknowledge this and would like to clarify that the broader spatial distribution of energy in higher OAM modes affects the localized interaction with particles, which can impact the effective extinction coefficient. Specifically, larger OAM modes result in a larger beam spot size with less concentrated energy, leading to a weaker interaction with individual haze particles. This reduced interaction results in a lower extinction coefficient, meaning that under the Beer-Lambert law, the attenuation is weaker, and the transmittance is higher for higher OAM modes.

To address the reviewer’s concerns, we propose the following revisions:

1. Clarifying the effect of OAM modes on extinction and transmittance: We revised the explanation to highlight that the larger beam size and less concentrated energy of higher OAM modes lead to reduced energy attenuation during interaction with individual particles, which effectively decreases the extinction coefficient and increases the transmittance. This clarification aligns with the physical principles of the single scattering approximation.
2. Revising the conclusion statement: We have rewritten the conclusion to explicitly state that higher OAM modes exhibit weaker attenuation due to their broader energy distribution, while still satisfying the Bouguer law.

We believe this revised explanation addresses the reviewer’s concern by providing a clearer and more accurate description of the relationship between OAM modes and scattering, while ensuring consistency with the single scattering approximation. We will update the manuscript accordingly to reflect these changes in both the conclusion and the relevant sections of the text.

Comments 3：The results in Fig. 3 indicate that the scattering indicatrices (RCS) for radiation with the same wavelengths and different OAM values are different. These results require clarification. Each elementary volume of a homogeneous medium has the same indicatrix corresponding to the particle size spectrum (39). The absolute value of RCS for an elementary volume is proportional to the magnitude of the intensity incident on this volume. The integral RCS should be proportional to the integral value of the power of the entire scattering volume. But taking into account the fact stated in the previous remark, the integral values of the power should be equal, provided that the initial values of the power of radiation with different OAM values are the same.

Response: We would like to thank the reviewer for their insightful comments and for pointing out the need for clarification regarding the results in Figure 3, specifically the differences observed in the RCS for radiation with the same wavelengths and different OAM values. We understand the concern raised regarding the interpretation of the results, especially in light of the fact that each elementary volume of a homogeneous medium should have the same indicatrix corresponding to the particle size spectrum, as stated in the previous remarks. The absolute value of the RCS for an elementary volume is indeed proportional to the intensity of the incident radiation, and the integral RCS should theoretically be proportional to the total power scattered by the entire volume, assuming that the initial powers for different OAM values are the same.

To address this issue, we have reviewed our numerical simulations and the associated calculations in greater detail. The observed differences in the RCS are primarily due to the inherent properties of BG beams, which carry OAM. These properties lead to different intensity distributions and interference patterns in the scattered field, even when the initial power of radiation with different OAM values is the same.

In response to your comment, we have clarified the following points in the revised manuscript:

1. Impact of OAM on intensity distribution: The key difference between the OAM modes lies in the spatial distribution of the beam intensity. While the total power of radiation with different OAM values remains the same, the OAM modes cause different intensity profiles within the scattering volume. This results in variations in the RCS at different scattering angles, as seen in the results presented in Figure 3.
2. Integral RCS and OAM modes: We have elaborated on the fact that the integral of the RCS over the entire scattering volume should remain constant for the same initial power. However, the angular distribution of the scattered intensity (i.e., the RCS indicatrix) depends on the OAM value because of the way the beam's phase front interacts with the particles. These variations in the angular scattering pattern do not affect the total integrated scattered power but do influence the shape and distribution of the RCS.

We hope this explanation helps to clarify the underlying principles and resolves any concerns. Thank you again for your constructive feedback.

Comments 4：The study of the propagation of vector BG beams in the atmosphere increases the relevance of the study. However, this study does not consider the influence of such important characteristics of BG beams as their non-diffractive propagation and self-healing properties. The influence of OAM on the scattering characteristics is studied. However, to determine this influence, Laguerre-Gaussian beams with a simpler structure could be considered, for example. The question arises, how will the choice of the beam type affect the results at the same OAM values?

Response: We would like to sincerely thank the reviewer for their thoughtful and constructive comments. We address the points raised as follows:

Non-diffractive Propagation and Self-healing Properties: We appreciate the reviewer’s suggestion to discuss the self-healing properties of BG beams. In the revised manuscript, we have revisited this aspect in the introduction, noting that while BG beams exhibit self-healing properties when obstructed by a single obstacle, real-life scenarios involving atmospheric turbulence or scattering disturb the entire beam, including all impinging conical waves. This disruption significantly diminishes or eliminates the possibility of self-healing. Given the focus of our study on scattering dynamics in haze conditions, where such disturbances are prevalent, the manuscript does not analyze self-healing in detail but acknowledges it as an important characteristic in controlled environments.

LG Beams: We appreciate the suggestion to explore LG beams, but our study focuses on BG beams due to their unique characteristics, such as non-diffractive properties and better resilience against scattering in atmospheric conditions like winter haze. Additionally, we have previously compared BG and LG beams in atmospheric turbulence, concluding that BG beams are more suitable for short-range transmission environments, while LG beams perform better in long-range transmission due to their ability to maintain OAM integrity. Since our study focuses on short-range scattering in haze environments, BG beams are the most appropriate choice.

Choice of Beam Type (BG vs. LG) and Impact on Results: We will add a brief discussion to clarify the impact of the beam type on results. While both BG and LG beams can carry OAM, BG beams are particularly suited for short-range transmission in scattering environments like haze due to their atmospheric penetration and resilience to distortion. The focus of this study is on BG beams to assess their scattering dynamics in winter haze, rather than on a broad comparison of beam types.

We thank the reviewer for their valuable feedback and incorporate these points in the revised manuscript.

Comments 5：The manuscript contains technical errors. In Figure 4, the letter designations (a), (b), (c), ... do not match the designations in the figure caption.

Response: We would like to thank the reviewer for their careful review of our manuscript. Regarding the concern about the figure designations in Figure 4, we have thoroughly checked both the figure and the corresponding caption. We can confirm that now the letter designations in the figure are correctly aligned with those in the caption:

(a) X-polarized

(b) Y-polarized

(c) Left circularly polarized

(d) Right circularly polarized

(e) Azimuthally polarized

(f) Radially polarized

We appreciate the reviewer’s attention to detail and will address any further concerns to ensure the manuscript meets the required standards.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Now, I recommend publication of Photonics.

Reviewer 4 Report

Comments and Suggestions for Authors

The authors' response to comments 2, 3 is unsatisfactory.

Comments 2. The authors provide the following conclusion (lines 413-415): “It is observed that higher OAM modes exhibit a reduced interaction with particles due to their broader energy distribution, while lower OAM modes, with energy concentrated in a smaller region, are more susceptible to scattering and attenuation.” It is not clear what kind of interaction reduction the authors are talking about. The problem is solved in the single scattering approximation, for which it is typical that the attenuation of the radiation power obeys the Bouguer law. In this case, the decrease in the beam power along the path will be the same, regardless of the OAM of the radiation, and single scattering does not distort the transverse intensity distribution

Response: We would like to sincerely thank the reviewer for their insightful comment regarding the conclusion about the interaction between higher OAM modes and particles. We agree that the original phrasing in the manuscript was unclear and could lead to misinterpretation. We appreciate the opportunity to refine our explanation

The reviewer correctly points out that in the single scattering approximation, the attenuation of radiation power follows the Beer-Lambert law, and the transverse intensity distribution of the beam is not distorted. We acknowledge this and would like to clarify that the broader spatial distribution of energy in higher OAM modes affects the localized interaction with particles, which can impact the effective extinction coefficient. Specifically, larger OAM modes result in a larger beam spot size with less concentrated energy, leading to a weaker interaction with individual haze particles. This reduced interaction results in a lower extinction coefficient, meaning that under the Beer-Lambert law, the attenuation is weaker, and the transmittance is higher for higher OAM modes.

To address the reviewer’s concerns, we propose the following revisions:

1. Clarifying the effect of OAM modes on extinction and transmittance: We revised the explanation to highlight that the larger beam size and less concentrated energy of higher OAM modes lead to reduced energy attenuation during interaction with individual particles, which effectively decreases the extinction coefficient and increases the transmittance. This clarification aligns with the physical principles of the single scattering approximation.
2. Revising the conclusion statement: We have rewritten the conclusion to explicitly state that higher OAM modes exhibit weaker attenuation due to their broader energy distribution, while still satisfying the Bouguer law.

We believe this revised explanation addresses the reviewer’s concern by providing a clearer and more accurate description of the relationship between OAM modes and scattering, while ensuring consistency with the single scattering approximation. We will update the manuscript accordingly to reflect these changes in both the conclusion and the relevant sections of the text.

 

I disagree. The statement that "lower energy concentration of modes with higher OAM leads to a decrease in energy attenuation when interacting with individual particles" is correct if the scattering cross-section of an individual particle (ESR) depends on the intensity of the incident radiation. And this is possible only in the presence of nonlinear effects of the interaction of radiation with matter. But nothing is said about this in the article

Author Response

Please find the attachment.

Author Response File: Author Response.pdf