Energy Backflow in Tightly Focused Fractional Order Vector Vortex Beams with Binary Topological Charges
Round 1
Reviewer 1 Report
This work reported a longitudinal energy evolution on the focal plane of vector modes by using the Richards-Wolf diffraction integral. They use fractional order vector vortex beams, which are subjected to the tight focusing through a larger numerical aperture. The results show that the backflow energy exist only when the binary topological charges n and m satisfy n+m=2 or n-m=-2. More intriguingly, the longitudinal energy component on the focus plane can be controlled in a flexible way by means of the binary topological charges. Overall speaking, the results look interesting. I think this work can deserve publication in the Journal: Photonics. In addition, I suggest that authors consider the following comment points:
1). The section of the abstract should be tidier and more concise. Especially the last paragraph of the abstract needs to be rewritten. This paragraph mainly tells readers what this work mainly does and should have a clear difference from the abstract.
2). The authors should extend a bit along the potential application of this work.
3). Either” vector vortex beams” or “vortex vector beams” should be in the same typing form in the whole text.
4). Sect. 2 “Materials and Methods” could be revised as “Theory and methods”.
5). For the integers of m and n, Eq. (2) should be substituted into Eq. (1) and simplified by using Euler’s formula and Bessel identity to obtain the electric field expression with Bessel function expressed by a single integral, which is conducive to analysis and improving the speed of numerical calculation.
6). For the non-integers of m and n, is it directly to simulate numerically by Eq. (1)? Alternatively, the fractional phase terms are first expanded into a superposition of integer order using Fourier series [M. V. Berry, J. Opt. A: Pure Appl. Opt. 6, 259 (2004)], and then one resorts to numerical simulation.
7). A brief explanation of the main findings is needed. For an example, Fig. 5 shows the backflow energy of vortex vector beams with different values of m+n and m-n. The reasons for the results in this case need to be analyzed.
8). The quality of this manuscript should be improved. For instance, Line 40, “The states of polarization and …” should be revised as “The SoPs and …”; Line 44, “In 2000, Novitsky…” should be revised as “In 2007, A. V. Novitsky and D. V. Novitsky…”; and Line 49, “Kotlyar et. al. found that that …” should be revised as “Kotlyar et. al. found that …”.
Author Response
Dear Professor,
Thank you very much for your kind and patience. The suggestions and comments are highly valuable and helpful for improving the manuscript. We have carefully revised the manuscript in accordance with the comments and suggestions. Below are the details of our reply to all the comments.
On behalf of all authors and sincerely yours,
Yan Wu
Reply to Reviewer 1
This work reported a longitudinal energy evolution on the focal plane of vector modes by using the Richards-Wolf diffraction integral. They use fractional order vector vortex beams, which are subjected to the tight focusing through a larger numerical aperture. The results show that the backflow energy exist only when the binary topological charges n and m satisfy n+m=2 or n-m=-2. More intriguingly, the longitudinal energy component on the focus plane can be controlled in a flexible way by means of the binary topological charges. Overall speaking, the results look interesting. I think this work can deserve publication in the Journal: Photonics. In addition, I suggest that authors consider the following comment points:
Comment 1:
“1) The section of the abstract should be tidier and more concise. Especially the last paragraph of the manuscript needs to be rewritten. This paragraph mainly tells readers what this work mainly does and should have a clear difference from the abstract.”
Response 1:
We would like to thank the reviewer for the comment and suggestion. The verbose part of the abstract was cut out and the last paragraph of the manuscript was rewritten.
Abstract: Using the Richards-Wolf diffraction integral, the longitudinal energy evolution on the focal plane of the fractional order vector vortex (FOVV) beams were studied. These beams possess a vortex topological charge n and a polarization topological charge m, and are subjected to tight focusing through a larger numerical aperture. Our investigation reveals the existence of backflow energy when the binary topological charges n and m satisfy the conditions of n+m=2 or n-m=-2. The component circularly polarized vortex beams of (i.e. the minus second-order vortex right circularly polarized beam) and (i.e. the second-order vortex left circularly polarized beam), play significant roles in the generation of reverse energy flux at the focal region. For FOVV beams with binary topological charges n and m, whose sum and difference are integers, the longitudinal energy on the focal plane exhibits axial symmetry. If the sum or the difference of the topological charges n and m is not an integer, the axisymmetric longitudinal energy on the focal plane is disrupted.
Conclusions: For the tightly focused FOVV beams with binary topological charges, the reverse energy flow appears near the focal region when the fractional topological charge m and the vortex topological charge n satisfy the condition of n+m=2 or n−m=−2. Our research reveals that the minus second-order vortex right circularly polarized beam, described by , and the second-order vortex left circularly polarized beam described by , play significant roles in generating the reverse energy flow at the focal region. This finding provides a comprehensive explanation for the observed maximum negative energy flux density of the tightly focused incident vortex-free second-order (n=0, m=2) radial polarization laser field [4]. Moreover, if the algebraic sum and the difference of the two topological charges m and n of a FOVV beam are integers, the longitudinal energy on the focal plane exhibits axis symmetry. Adjusting the appropriate values of n and m, one could obtain the laser beams with a customized longitudinal energy distribution at the focal region.
Comment 2:
“2) The authors should extend a bit along the potential application of this work.”
Response 2:
The discussion section containing the potential application is added, as follows
Discussion
Different from the homogeneous polarized laser fields with uniform SOP, the vector vortex beams exhibit diverse SOP characteristics at different spatial positions, accompanied by spiral wavefronts and phase singularities. Because of the vector nature, the SOP even the direction of the electric field will be rearranged after being focused by a high NA, which leads to the variation of the direction of the energy flow of laser field, even appearing the reverse energy flow. Based on the vortex nature each photon of the vortex light field carries a specific amount of orbital angular momentum, which serves as both an information carrier and a means of particles manipulating. For FOVV beams, whose different focusing characteristics are mainly caused by the transverse electric vector distributions on the cross-section of the incident laser fields. By taking a FOVV beam as a superposition of two circularly polarized vortex beams, the focusing properties are demonstrated directly by the focusing cases of the component vortex circularly polarized beams. The tightly focusing property of the FOVV beam broadens the means of microparticle manipulation. By matching the binary topological charges of the required FOVV beams, one can anticipate even design the energy flow distribution on the focal plane for better manipulation of microscopic particles.
Comment 3:
“3) Either” vector vortex beams” or “vortex vector beams” should be in the same typing form in the whole text.”
Response 3:
- The term is uniformly written as “vector vortex beams” in the whole revised manuscript.
Comment 4:
“4) Sect. 2 “Materials and Methods” could be revised as “Theory and methods.”
Response 4:
Sect. 2 is revised as “Theory and methods”.
Comment 5:
“5) For the integers of m and n, Eq. (2) should be substituted into Eq. (1) and simplified by using Euler’s formula and Bessel identity to obtain the electric field expression with Bessel function expressed by a single integral, which is conducive to analysis and improving the speed of numerical calculation.”
Response 5:
Thank you for the comment. Generally, the integration in Eq. (1) should be simplified as a single integral for the integers of m and n. however, for the fractional m and n in this work, Eq. (1) cannot be simplified as a single integral since there is not analytical expression for azimuthal integration with the fractional m and n. therefore, we have to carry out the double integral in this work.
Comment 6:
“6) For the non-integers of m and n, is it directly to simulate numerically by Eq. (1)? Alternatively, the fractional phase terms are first expanded into a superposition of integer order using Fourier series [M. V. Berry, J. Opt. A: Pure Appl. Opt. 6, 259 (2004)], and then one resorts to numerical simulation.”
Response 6:
In this work, the integral with the non-integers of m and n is directly numerically simulated by Eq. (1). we thank the reviewer provide another approach to achieve the integral by using Fourier series.
Comment 7:
“7) A brief explanation of the main findings is needed. For an example, Fig. 5 shows the backflow energy of vortex vector beams with different values of m+n and m-n. The reasons for the results in this case need to be analyzed.”
Response 7:
Fig. 5 shows the focusing properties of the vortex circularized beams with various integer vortex orders. The main reason for the different focusing performance is the different electric vector distributions on the cross-section of the laser beams. We add the explanations as follows:
Similarly, by comparing figures 5(b) with 5(c), it is obviously that the cases of the vortex left circularly polarized laser fields with integer vortex order and the vortex right circularly polarized laser fields with the corresponding opposite negative integer vortex order are identical too. Further calculations revealed that for an arbitrary integer s the distributions of the electric vector of the pair components on the cross-section of the laser beams of and are exactly identical, so after being tightly focused the property of the energy flux near the focal region are just the same.
Comment 8:
“8) The quality of this manuscript should be improved. For instance, Line 40, “The states of polarization and …” should be revised as “The SoPs and …”; Line 44, “In 2000, Novitsky…” should be revised as “In 2007, A. V. Novitsky and D. V. Novitsky…”; and Line 49, “Kotlyar et. al. found that that …” should be revised as “Kotlyar et. al. found that …”
Response 8:
- The spelling problems mentioned have been all corrected, and the whole manuscript has been examined and revised.
Reviewer 2 Report
The study of tightly focused vector light field is a hot topic at present. When vector beams are focused by a high-numerical-aperture objective, the backward longitudinal energy could be obtained near the focal plane, which is very useful in manipulating particles of the order of several nanometers to several hundred microns. The authors investigated the longitudinal energy evolution on the focal plane of the fractional order vector vortex (FOVV) beams, by utilizing the Richards-Wolf diffraction integral. They demonstrated the existence of backflow energy when the binary topological charges n and m satisfy the conditions of n+m=2 or n−m=−2. The author did a good manuscript preparing and the simulation results were interesting. In view of the observation of energy backflow under certain conditions, the manuscript can be accepted for publication after minor revision. However, there are some misunderstandings, and the authors should address the following comments:
1) The main issue with the current manuscript is the insufficient description of the scientific significance of the work. To enhance the readability of the manuscript, some descriptions on the importance of backflow energy are needed in the Introduction section.
2) The vortex beams possess extensive potential applications, and the authors are encouraged to cite and compare together, such as optical communication (Optics Communications, vol. 452, pp. 116-123, 2019; Nature Photonics, vol. 15, pp. 901-907, 2021. ), detection and imaging (IEEE Sensors Journal, 2023, 23(4), 4078-4084; Measurement, 2022, 189, 110600.).
3) There is also observable energy backflow in Figure 1(a) and Figure 2(a), which does not satisfy the conditions n+m=2 and n−m=−2. Please explain on this.
4) The authors should check the existing statement, for example, in Lines 44, 53 and 71, the ‘numerical aperture’ should be ‘NA’.
5) A few words about the possible applications in the future are recommended.
None
Author Response
Dear Professor,
We are grateful for your suggestions and comments that are highly valuable and helpful for improving the manuscript. We have carefully revised the manuscript in accordance with the comments and suggestions. Below are the details of our reply to all the comments. The manuscript has also been revised accordingly. Thank you very much for your kind and patience.
On behalf of all authors and sincerely yours,
Yan Wu
Reply
The study of tightly focused vector light field is a hot topic at present. When vector beams are focused by a high-numerical-aperture objective, the backward longitudinal energy could be obtained near the focal plane, which is very useful in manipulating particles of the order of several nanometers to several hundred microns. The authors investigated the longitudinal energy evolution on the focal plane of the fractional order vector vortex (FOVV) beams, by utilizing the Richards-Wolf diffraction integral. They demonstrated the existence of backflow energy when the binary topological charges n and m satisfy the conditions of n+m=2 or n−m=−2. The author did a good manuscript preparing and the simulation results were interesting. In view of the observation of energy backflow under certain conditions, the manuscript can be accepted for publication after minor revision. However, there are some misunderstandings, and the authors should address the following comments:
Comment 1:
“1) The main issue with the current manuscript is the insufficient description of the scientific significance of the work. To enhance the readability of the manuscript, some descriptions on the importance of backflow energy are needed in the Introduction section.”
Response 1:
We add the lines 44-52 to explain the importance of the research. As follows, “Focusing the vector vortex beams is widely used both in optical data storage [11, 12], optical communication [13-16], detection and imaging [17,18], and optical micromanipulation [19]. Because the focusing characteristics are closely rely on the polarization of the light, the vector and the vortex natures of the focused beam is of vital importance. By utilizing a spatial light modulator (SLM) in conjunction with a common path interferometric arrangement, factional vector vortex beams with arbitrary topological charges could be generated [20]. However, the characteristics of the longitudinal energy flow within tightly focused factional order vector vortex (FOVV) beams remain unclear. ”
Comment 2:
“2) The vortex beams possess extensive potential applications, and the authors are encouraged to cite and compare together, such as optical communication (Optics Communications, vol. 452, pp. 116-123, 2019; Nature Photonics, vol. 15, pp. 901-907, 2021. ), detection and imaging (IEEE Sensors Journal, 2023, 23(4), 4078-4084; Measurement, 2022, 189, 110600.).”
Response 2:
We added the referred papers as Refs. [15,16,18] in the revised manuscript for better clarification.
[15]Xiaozhou Cui, Xiaoli Yin, Huan Chang, Huanyu Liao, Xiaozheng Chen, Xiangjun Xin, Yongjun Wang. Experimental study of machine-learning-based orbital angular momentum shift keying decoders in optical underwater channels. Optics Communications 2019, 452, 116-123.
[16] Xu Ouyang, Yi Xu, Mincong Xian, Ziwei Feng, Linwei Zhu, Yaoyu Cao, Sheng Lan, Bai-Ou Guan, Cheng-Wei Qiu, Min Gu & Xiangping Li. Synthetic helical dichroism for six-dimensional optical orbital angular momentum multiplexing. Nature Photonics 2021, 15, 901-907.
[18] Yunlai Wang, Yanzhe Wang, and Zhongyi Guo. OAM radar based fast super-resolution imaging. Measurement 2022, 189, 110600.
Comment 3:
“3) There is also observable energy backflow in Figure 1(a) and Figure 2(a), which does not satisfy the conditions n+m=2 and n−m=−2. Please explain on this.”
Response 3:
We add the explanation in lines 140-145 in paragraph 4th of the section “3. Results ”.
“Reviewing the Figures 1(a3) and 1(c3), one can find that if the sum of the binary topological charges deviates from 2, or the difference of them deviate from -2, the axisymmetric longitudinal energy on the focal plane is disrupted, and as the deviation value increasing the reverse energy becomes smaller (as shown in Fig. 1(a3)) even none (as shown in Fig. 1(c3)).”
Comment 4:
“4) The authors should check the existing statement, for example, in Lines 44, 53 and 71, the ‘numerical aperture’ should be ‘NA’.”
Response 4:
After the declaration the term is unified to “NA” completely.
Comment 5:
“5) A few words about the possible applications in the future are recommended.”
Response 5:
The section of “Discussion” is added including the possible applications. As follows
Discussion
Different from the homogeneous polarized laser fields with uniform SOP, the vector vortex beams exhibit diverse SOP characteristics at different spatial positions, accompanied by spiral wavefronts and phase singularities. Because of the vector nature, the SOP even the direction of the electric field will be rearranged after being focused by a high NA, which leads to the variation of the direction of the energy flow of laser field, even appearing the reverse energy flow. Based on the vortex nature each photon of the vortex light field carries a specific amount of orbital angular momentum, which serves as both an information carrier and a means of particles manipulating. For FOVV beams, whose different focusing characteristics are mainly caused by the transverse electric vector distributions on the cross-section of the incident laser fields. By taking a FOVV beam as a superposition of two circularly polarized vortex beams, the focusing properties are demonstrated directly by the focusing cases of the component vortex circularly polarized beams. The tightly focusing property of the FOVV beam broadens the means of microparticle manipulation. By matching the binary topological charges of the required FOVV beams, one can anticipate even design the energy flow distribution on the focal plane for better manipulation of microscopic particles.
Round 2
Reviewer 1 Report
The authors fully responded to the comments. However, the quality of the revised manuscript should be improved. For example, Line 32, “…focused the negative energy…” should be revised as “… focused, the negative energy…”; Line 215, “For FOVV beams, whose different focusing characteristics are mainly…” should be revised as “Different focusing characteristics of FOVV beams are mainly…”.
Author Response
We are very grateful to the reviewers for their kindness and patience to improving the manuscript. We have revised the manuscript carefully in accordance with the suggestions. The amendments made in the revised version are highlighted with yellow background in the revised manuscript.
