Comparison between the Propagation Properties of Bessel–Gauss and Generalized Laguerre–Gauss Beams

Round 1

Reviewer 1 Report

In this paper, Sheppard et.al. compared the propagation properties of Bessel-Gauss and generalized Laguerre-Gauss beams, exploring their connections and reviewing asymptotic expressions for generalized Laguerre and Hermite polynomials. It also discusses the practical applications of these types of beams in photonics. While I should admit that I did not go through the math by myself, the presentation makes sense. Overall, I believe that this paper makes a significant contribution to the field of photonics. I recommend that it be accepted for publication after a few minor updates.
I have some broad remarks about the present draft: 1. L29-L35: In this paragraph, the authors touched on applications of LG or HB beams. I believe this article would attract more experimentalists if this section is elaborated further. I think authors should cite some papers that show the real-life use of HG beams and also recent advances in generating such beams. Example: (a) https://doi.org/10.1364/OE.26.003926 , (b) https://doi.org/10.1364/OPEX.12.005448 2. L73: This sentence does not make sense and should be corrected

N/A

Author Response

In this paper, Sheppard et.al. compared the propagation properties of Bessel-Gauss and generalized Laguerre-Gauss beams, exploring their connections and reviewing asymptotic expressions for generalized Laguerre and Hermite polynomials. It also discusses the practical applications of these types of beams in photonics. While I should admit that I did not go through the math by myself, the presentation makes sense. Overall, I believe that this paper makes a significant contribution to the field of photonics. I recommend that it be accepted for publication after a few minor updates.

I have some broad remarks about the present draft: 1. L29-L35: In this paragraph, the authors touched on applications of LG or HB beams. I believe this article would attract more experimentalists if this section is elaborated further. I think authors should cite some papers that show the real-life use of HG beams and also recent advances in generating such beams. Example: (a) https://doi.org/10.1364/OE.26.003926 , (b) https://doi.org/10.1364/OPEX.12.005448 2.

 

We have expanded the list of applications. We have added these two references, and also some more references on applications. But as there are so many applications we do not think it is feasible to refer to all of them.

 

We have also added: ‘The propagation invariance of Bessel beams has been exploited in imaging, trapping and manipulation, nonlinear optics, and laser processing and machining. Apart from using a narrow annular pupil, which is of course very light inefficient, Bessel-like beams can be generated using a deformable mirror, spatial light modulator, micromirror array, binary pupil mask \cite{Wang08}, hologram \cite{Mondal18}, by using an axicon and a lens (or second axicon) to form a bright ring \cite{McLeod,Arimoto,Breen20}, or using a toroidal resonator \cite{Sheppard78}.’

 

L73: This sentence does not make sense and should be corrected

 

We have changed this sentence to make it clearer. ‘We include the case when the azimuthal mode number m is high, so that the radial mode number n cannot be assumed >> m.’

Reviewer 2 Report

My comments

 

[1] The relationships between Hermite-Gauss and Laguerre-Gauss beams were first obtained not in 1993 in Ref.[8], but in 1991 in  DOI:10.1016/0030-4018(91)90534-K.

 

[2] The generalized manifold of hypergeometric beams was considered in 2008 in doi.org/10.1364/JOSAA.25.000262  This should be noted in the text

 

[3] Generally speaking, L guerre-Gauss functions (as well as Bessel functions) allow you to map by unified way any continuous function given at a certain interval. So I don't see any novelty in (4) except to clearly explain the curves in Fig.1.

 

[4] As shown in Ref.[14], generalized LG and HG beams have hidden symmetries inherent in the operator transformations of the Lie group. Do these symmetries remain valid in the approximations made by the authors? I would like the authors to pay closer attention to the deep physical processes that accompany their transformations, and not just get carried away with calculations.

 

[5] Generally speaking, I would like the authors to give more detailed justifications of the need to use their approximations, since modern computer software allows us to do without such approximations for engineering.

Author Response

[1] The relationships between Hermite-Gauss and Laguerre-Gauss beams were first obtained not in 1993 in Ref.[8], but in 1991 in  DOI:10.1016/0030-4018(91)90534-K.

We thank the reviewer for pointing this out. I was aware of this paper, but had forgotten it. It is significant that this paper predated that of Allen. We have included the reference.

[2] The generalized manifold of hypergeometric beams was considered in 2008 in doi.org/10.1364/JOSAA.25.000262  This should be noted in the text

 We thank the reviewer for bringing this to our attention. We have included this reference. We were aware of other papers by the authors, but this one was particularly relevant.

[3] Generally speaking, Laguerre-Gauss functions (as well as Bessel functions) allow you to map by unified way any continuous function given at a certain interval. So I don't see any novelty in (4) except to clearly explain the curves in Fig.1.

We did not mean to claim that it was all that novel. We have changed the following sentence to try to make it clearer what we are trying to say.

‘ It is seen that for any fixed value of m, the Bessel function can be approximated by Laguerre polynomials of different radial orders n, as n does not appear on the left hand side of Eq. 4.’

This is a justification of the plot in Fig.1, which shows how the agreement improves as n is increased.

[4] As shown in Ref.[14], generalized LG and HG beams have hidden symmetries inherent in the operator transformations of the Lie group. Do these symmetries remain valid in the approximations made by the authors? I would like the authors to pay closer attention to the deep physical processes that accompany their transformations, and not just get carried away with calculations.

 We have added more about the symmetries and transformations, including some more references. We feel that the symmetries will not be exactly valid for approximate beams. On the other hand, we think that two beams with close to the same cross-section will diffract in the same way, and this has been born out in our numerical calculations. We have added ‘While these symmetries are expected to break down for approximations to the beams, propagation of a beam and a good approximation will be similar.’

[5] Generally speaking, I would like the authors to give more detailed justifications of the need to use their approximations, since modern computer software allows us to do without such approximations for engineering.

We feel that the similarity between the BG and LG beams is an important physical property. The aim of the paper was to explore this connection in more detail, and compare the several different approximations in the literature. We have added a statement that an advantage of BG is that the relevant scaling is continuously variable, unlike sLG or eLG modes. But gLG allows for better fitting because there is another free parameter. BG beams have been used to fit to experimental output from unstable resonators. Of course, computation is easy nowadays, but analytical studies are import too, in order to bring out patterns of behaviour and improve physical understanding.