Performance of Orbital Angular Momentum Communication for a Non-Uniformly Correlated High-Order Bessel–Gaussian Beam in a Turbulent Atmosphere

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

* Each term in the equations needs to be clearly specified. For example: in equations 3 and 4, I'm confused as to the units of v and a (they don't seem consistent between equations).

* In lines 110ff, is the turbulence applied as a single phase screen before Fourier diffraction? Or, is the phase applied at multiple points along the beam path, as turbulence realistically occurs? If it's a single phase screen, please acknowledge/discuss the limitations of this approach to weak turbulence conditions.

* Section 3 presents interesting trends, but explaining the physical mechanisms behind them would significantly strengthen the analysis. For example, the increase in detection probability with wavelength could be explained by slower phase accumulation in longer wavelengths. Similarly, the SNR degradation trends need a physical explanation. Connecting observed trends to plausible mechanisms through reasoning would enhance the reader's understanding and confidence in the results.

* Why is SNR used in Figure 5 (and BER in Figure 6) but detection probability used in other figures?

* Complementing the analytical results with wave optics simulations would provide valuable validation.

* Equation 12 doesn't appear to to come reference 13.

Comments on the Quality of English Language

The English syntax and grammar needs major revision. The abstract is fine, but every other section needs significant improvement.

Author Response

Q1. Each term in the equations needs to be clearly specified. For example: in equations 3 and 4, I'm confused as to the units of v and a (they don't seem consistent between equations).

Reply: Thanks for the reviewer’s comment and suggestion. Where  is the corresponding vector in Fourier space of a position vector r in the polar coordinate system in Eqs.2 and 3. The parameter a=2/kr_c²  is a positive real constant with a coherence length r_c in Eqs.3 and 5. We have checked the parameters in the equations and given the physical meanings of the parameters. The additional words are highlighted in red.

Q2. In lines 110, is the turbulence applied as a single-phase screen before Fourier diffraction? Or, is the phase applied at multiple points along the beam path, as turbulence realistically occurs? If it's a single-phase screen, please acknowledge/discuss the limitations of this approach to weak turbulence conditions.

Reply: Thanks for the reviewer’s comment and suggestion. In this manuscript, we use the Kolmogorov turbulence phase aberrations to analyze the non-uniformly correlated high-order Bessel-Gaussian beam propagating in the turbulent atmosphere, the turbulence spectrum is used to describe the strength of the turbulent atmosphere, the results are the ensemble average [1]. The phase screen applied to simulate the atmospheric turbulence is well, but it is not suitable for the theoretical analysis to obtain the analytical result.

1. Paterson, C. Atmospheric turbulence and orbital angular momentum of single photons for optical communication. Phys. Rev. Lett. 2005, 94, 153901.

Q3. Section 3 presents interesting trends, but explaining the physical mechanisms behind them would significantly strengthen the analysis. For example, the increase in detection probability with wavelength could be explained by slower phase accumulation in longer wavelengths. Similarly, the SNR degradation trends need a physical explanation. Connecting observed trends to plausible mechanisms through reasoning would enhance the reader's understanding and confidence in the results.

Reply: Thanks for the reviewer’s comment and suggestion. We have revised the manuscript according to the reviewers’ suggestion, the additional words are highlighted in red.

Q4. Why is SNR used in Figure 5 (and BER in Figure 6) but detection probability used in other figures?

Reply: Thanks for the reviewer’s comment. In this study, the detection probability indicates the purity of the signal OAM state, and the high detection probability notes the power of the signal OAM state spreading to the neighbor OAM states is small, then the crosstalk is small. The SNR and BER are the important parameters to measure the signal quality of the non-uniformly correlated high-order Bessel-Gaussian beam used in FSO communications. Therefore, we studied the detection probability, SNR, and BER of the non-uniformly correlated high-order Bessel-Gaussian beam propagation in the turbulent atmosphere.

Q5. Complementing the analytical results with wave optics simulations would provide valuable validation.

Reply: Thanks for the reviewer’s comment and suggestion. We have revised the manuscript according to the reviewers’ suggestion, the additional words under the Eq.17 are highlighted in red.

Q6. Equation 12 doesn't appear to come reference 13.

Reply: Thanks for the reviewer’s comment. I checked the equation 12 and reference 13 carefully, the equation 12 in the manuscript is obtained from the equation 18 in the reference 13.

Comments on the Quality of English Language

Q7. The English syntax and grammar needs major revision. The abstract is fine, but every other section needs significant improvement.

Reply: Thanks for the reviewer’s comment and suggestion. We have invited the professional language polishing company to revise the manuscript before submitting the revised manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors researched the detection probability of the signal OAM state, SNR and BER effected by the wavelength, receiver aperture, beam width and topological charge of the non-uniformly correlated high-order Bessel Gaussian beam applied in OAM communications. The results are interesting. In my opinion that this manuscript could be accepted for publication with some revisions. 

1、  How about the relationship between detection probability and beam width?

2、  The results showed that the performance of OAM communications could be improved by the small topological charges and small beam widths. How about non-uniformly correlated zero-order Bessel Gaussian bam with small beam width? Is it better than high-order?

3、  The letters or formulas in the manuscript have different sizes, such as Eqs. (5) and (6), in line 91 and in line 92. The authors should check them carefully.

4、  Some more thorough discussion about SNR and BER would be beneficial, and the manuscript would be more interesting.

5、  On line 167 page 5, the beam width w0 is 1cm, the receiving aperture diameter D is 10cm to 40cm in Fig.3. Therefore, all of the receiving aperture diameter D are larger than the beam width, the authors should check it carefully.

6、  The author should provide the aperture diameter D on line 168.

7、  There is a spelling mistake in the manuscript on line 193. The word “largen” should be replaced by “larger”. The authors should check and correct it carefully.

Author Response

Q1. How about the relationship between detection probability and beam width?

Reply: Thanks for the reviewer’s comment and suggestion. We have revised the manuscript according to the reviewers’ suggestion. We added figure 5 to describe the relationship between detection probability and beam width. The additional words are highlighted in red.

Q2. The results showed that the performance of OAM communications could be improved by the small topological charges and small beam widths. How about non-uniformly correlated zero-order Bessel Gaussian beam with small beam width? Is it better than high-order?

Reply: Thanks for the reviewer’s comment and suggestion. In this manuscript, we studied the non-uniformly correlated high-order Bessel-Gaussian beam carried the OAM state, and the OAM states of the vortex beams have the mutually orthogonal property, these properties have also been implemented for multiplexing/demultiplexing in FSO communication. The center intensity of the non-uniformly correlated zero-order Bessel-Gaussian beam is not zero, so the crosstalk will be generated by the zero-order Bessel Gaussian beam when these beams were applied in OAM multiplexing FSO communications.

Q3. The letters or formulas in the manuscript have different sizes, such as Eqs. (5) and (6), in line 91 and in line 92. The authors should check them carefully.

Reply: Thanks for the reviewer’s comment and suggestion. We have revised the sizes of the parameters in the revised manuscript.

Q4. Some more thorough discussion about SNR and BER would be beneficial, and the manuscript would be more interesting.

Reply: Thanks for the reviewer’s comment and suggestion. We have revised the manuscript according to the reviewers’ suggestion. We added figure 8 to describe the relationship between SNR, BER, and the strength of the turbulent atmosphere. The additional words are highlighted in red.

Q5. On line 167 page 5, the beam width w0 is 1cm, the receiving aperture diameter D is 10cm to 40cm in Fig.3. Therefore, all of the receiving aperture diameter D are larger than the beam width, the authors should check it carefully.

Reply: Thanks for the reviewer’s comment and suggestion. we have checked their values carefully, and the receiving aperture diameter D is 10cm to 40cm is wrong, the correct values are 10mm to 40mm, we have revised Fig.2 in the revised manuscript. 

Q6. The author should provide the aperture diameter D on line 168.

Reply: Thanks for the reviewer’s comment and suggestion. We have given the value of the aperture diameter D=10mm.

Q7. There is a spelling mistake in the manuscript on line 193. The word “largen” should be replaced by “larger”. The authors should check and correct it carefully.

Reply: Thanks for the reviewer’s comment and suggestion. We have revised it in the revised manuscript. We have invited the professional language polishing company to revise the manuscript before submitting the revised manuscript.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thank you for addressing my original concerns and adding clarification to the paper. I have just a few minor comments.

* I noticed an error in my previous comment regarding equation 15. It should have been equation 15, not equation 12, that I believed could benefit from further citation clarification. I apologize for any confusion this may have caused.

* My previous comment on the phase screens aimed to highlight the complex nature of turbulence. It extends beyond a single phase application at the source, as phase perturbations manifest throughout the propagation path, influencing intensity along the way. If the analysis assumes a single accumulated phase, it would only be valid for weak turbulence conditions. To ensure clarity, I suggest the authors explicitly state the intended turbulence representation and discuss whether the chosen parameters (turbulence strength, propagation distance, etc.) fall within the weak turbulence regime.

* My suggestion for numerical diffraction results remains relevant, particularly given my concern about the chosen parameters potentially exceeding the weak turbulence regime. This additional data could help validate the analytical model and solidify the applicability of the results.

* While I appreciate the focus on SNR and BER as key communication metrics, the use of detection probability for specific parameter studies seems inconsistent. For a more comprehensive understanding of turbulence's impact, I recommend utilizing a consistent set of metrics (e.g., plotting SNR against propagation distance alongside the existing detection probability plots). This would facilitate a clearer picture of the system's behavior under different turbulence conditions.

Comments on the Quality of English Language

The English syntax is much improved in this version of the paper. I have a few minor suggestions:

* Lines 20-22: get rid of the newly added "weak turbulent atmosphere" and instead add "weak" in front of atmospheric turbulence on line 21.
* Line 33: replace "terminated" with "degraded".
* Line 39 should read: "performance of communication..."
* Line 120: there should be a period (.) after the equation.
* Line 127: I think "OAM stats" should be changed to "OAM states", correct?
* It looks to me like lines 219-227 are formatted with a smaller font size than the rest of the body text.
* Line 235: "slower phase accumulation..."
* Line 237: "...higher"
* Line 240: replace "other wavelengths" with "shorter wavelengths".
* Line 264: "...larger spots of the beams influenced more by the..." Also, the word "spots" seems to be in a different font.
* Scattered through: sometimes Fig. is capitalized and sometimes it isn't. Make sure to be consistent throughout.

Author Response

Q1. I noticed an error in my previous comment regarding equation 15. It should have been equation 15, not equation 12, that I believed could benefit from further citation clarification. I apologize for any confusion this may have caused.

Reply: Thanks for the reviewer’s comment and suggestion. In Eq.(15) of the manuscript, I(z) describes the intensity of the signal OAM beam, and the denominator denotes the intensity of both signal OAM beam and crosstalk OAM beams. The crosstalk OAM beams were generated by the power of the signal beam spreading to the neighbor OAM states due to the turbulent atmosphere induced [1]. Then the parameter P denotes the detection probability of the signal OAM beam.

    In Eq.(16) of the manuscript, I(z) is the same physical meaning as in Eq. (15), and the denominator denotes the intensity of the crosstalk OAM beams. Then the parameter SNR denotes the signal-to-noise ratio of the signal OAM beam [2, 3].

    Eqs. (15) and (16) in the manuscript are cited from Eqs. (20) and (32) in the Refs 13 and 18, respectively.

    We have described the physical meaning of the Eqs.(15) and (16) in the revised manuscript, the additional words in the revised manuscript are highlighted in blue.

1. Paterson, C. Atmospheric turbulence and orbital angular momentum of single photons for optical communication. Phys. Rev. Lett. 2005, 94, 153901.

2. Chen, M.; Yu, L.; Zhang, Y. Signal/noise ratio of orbital angular momentum modes for a partially coherent modified Bessel-correlated beam in a biological tissue. J. Opt. Soc. Am. A 2017, 34, 2046-2051.

3. Cheng, M.; Dong, K.; Shi, C.; Mohammed, A. H. T.; Guo, L.; Yi, X.; Wang, P.; Li, J. Enhancing Performance of Air–Ground OAM Communication System Utilizing Vector Vortex Beams in the Atmosphere. Photonics 2023, 10, 41.

 

Q2. My previous comment on the phase screens aimed to highlight the complex nature of turbulence. It extends beyond a single phase application at the source, as phase perturbations manifest throughout the propagation path, influencing intensity along the way. If the analysis assumes a single accumulated phase, it would only be valid for weak turbulence conditions. To ensure clarity, I suggest the authors explicitly state the intended turbulence representation and discuss whether the chosen parameters (turbulence strength, propagation distance, etc.) fall within the weak turbulence regime.

Reply: Thanks for the reviewer’s comment and suggestion. The range of the weak turbulence denotes the value of the Cn^2 from 0 to 10^-13m^-2/3 in Chapters 3 and 15 in the Ref.4. Thus, the parameters what we discussed in the manuscript are all in the case of a weak turbulent atmosphere.

4. Andrews LC, Phillips RL. Laser beam propagation through random media. 2nd ed. Bellingham, Washington, USA: SPIE press; 2005.

 

Q3. My suggestion for numerical diffraction results remains relevant, particularly given my concern about the chosen parameters potentially exceeding the weak turbulence regime. This additional data could help validate the analytical model and solidify the applicability of the results.

Reply: Thanks for the reviewer’s comment and suggestion. In the manuscript, the value of the Cn^2 ranges from 5*10^-15m^-2/3 to 1*10^-14m^-2/3. As mentioned in the Q2, the range is satisfied for the weak turbulent atmosphere regime [4].

4. Andrews LC, Phillips RL. Laser beam propagation through random media. 2nd ed. Bellingham, Washington, USA: SPIE press; 2005.

 

Q4. While I appreciate the focus on SNR and BER as key communication metrics, the use of detection probability for specific parameter studies seems inconsistent. For a more comprehensive understanding of turbulence's impact, I recommend utilizing a consistent set of metrics (e.g., plotting SNR against propagation distance alongside the existing detection probability plots). This would facilitate a clearer picture of the system's behavior under different turbulence conditions.

Reply: Thanks for the reviewer’s comment and suggestion. In general, The SNR and BER are analyzed at the received plane to measure the signal quality for the FSO communications, then the propagation distance effect on the SNR and BER are not discussed in the manuscript.

 

Comments on the Quality of English Language

Q4. The English syntax is much improved in this version of the paper. I have a few minor suggestions:

* Lines 20-22: get rid of the newly added "weak turbulent atmosphere" and instead add "weak" in front of atmospheric turbulence on line 21.
* Line 33: replace "terminated" with "degraded".
* Line 39 should read: "performance of communication..."
* Line 120: there should be a period (.) after the equation.
* Line 127: I think "OAM stats" should be changed to "OAM states", correct?
* It looks to me like lines 219-227 are formatted with a smaller font size than the rest of the body text.
* Line 235: "slower phase accumulation..."
* Line 237: "...higher"
* Line 240: replace "other wavelengths" with "shorter wavelengths".
* Line 264: "...larger spots of the beams influenced more by the..." Also, the word "spots" seems to be in a different font.
* Scattered through: sometimes Fig. is capitalized and sometimes it isn't. Make sure to be consistent throughout.

Reply: Thanks for the reviewer’s comment and suggestion. We have revised the manuscript according to the reviewer’s suggestion, the revised words are highlighted in blue.