Simple Analytical Expression of the Voigt Profile

Round 1

Reviewer 1 Report

This manuscript describes several representations of the Voigt profile and shows some numerical simulations. It is divided into several sections. There is an extensive literature discussing the Voigt profile. It would be helpful to mention some more in the introduction (section 1). 

Section 2.1 involves some algebra, duplicating the effort already described in ref. 11+18, finally arriving at expression 25. It seems like that this result is not used subsequently. So what is the purpose of including it?
Section 2.2 approximates, using a Taylor-type expansion, a previous result in the literature. This result is used further.
Section 3 presents some numerical approximations that are shown in several figures. There are many approximations of the Voigt profile in the literature. Please compare those with yours. What is the accuracy of your method? How do these results help in 'several fields of physics, e.g., atmospheric radiative transfer, neutron reactions, molecular spectroscopy, plasma waves, and astrophysical spectroscopy' as promised in the abstract? This generic statement needs to be made more precise: How can somebody else use your results in their research?

Some specific issues:
Line 55: 'And' -> 'and'
Line 59: 'Where" -> 'where'
Line 60: 'And' -> 'and'
Line 62: 'And' -> 'and'
Line 67: 'Where" -> 'where'
Lines 80-82: "a particular integral ... This integral given in Ref. [19]": Please describe which particular integral was referred to, e.g., which integral in Ref. 20.
Line 87: 'Where" -> 'where'
Line 90: 'Where" -> 'where'
Line 95: 'Where" -> 'where'
Line 110: 'Owing to prove' -> Please rephrase
Line 112: 'old result (Eq. (33))': Why do the authors call this result 'old'? Has it been published previously?
Line 142: 'good agreement' -> please quantity

Author Response

January, 16^th, 2021

Letter to the Journal: Quantum reports

 

Manuscript ID: quantumrep-1503844

Title: Simple analytical expression of the Voigt profile

Journal: Quantum reports

Authors:  S. Chib and A. Belafhal.

 

Dear Editor,

 

We sincerely appreciate the valuable comments and suggestions from the editor and the reviewers. The thorough review helped immensely in the shaping of the manuscript. The suggestions and comments have been closely followed and revisions have been made accordingly. The following are the questions extracted from the reviewers’ comments along with our summarized responses. Please see below, our detailed response to comments.

 

Thank you very much for your interest in our manuscript.

 

Yours sincerely,

 

1. Chib and A. Belafhal.

 

 

 

 

 

 

 

 

 

 

 

 

 

• Responses to Reviewer 1:

 

Dear Reviewer,

     Your feedback and suggestions on our manuscript were greatly appreciated. We have improved our paper as a result of your comments and helpful suggestions. All changes in the revised manuscript are made by the use of the “Track changes” function. Now, we would like to describe the change that we have made in the revised manuscript. The details are given as follows:

 

1. There is an extensive literature discussing the Voigt profile. It would be helpful to mention some more in the introduction (section 1). 

Based on your valuable comment suggestion, we have reported some new references.

 

2. Section 2.1 involves some algebra, duplicating the effort already described in ref. 11+18, finally arriving at expression 25. It seems like that this result is not used subsequently. So what is the purpose of including it?

In Section 2.1, we presented the general form of the Voigt function which is expressed by equation 20. By choosing appropriate parameters, we could find the results obtained by references [11, 18]. Thus we can confirm that the result (Eq. 25) represents a special case of our general finding. Fig. 3 is the simulation result based on Eq. (20) and by setting p=1/4 we find the graphical result which corresponds to Eq. (24). We used this last equation instead of Eq. 25 to have homogeneity in the total of the study.

 

3. There are many approximations of the Voigt profile in the literature. Please compare those with yours.

Thanks for your suggestion, that added value to our manuscript. We have presented a comparison between our approximation and that of Teoderscu et al. with the exact expression presented by Srivastava (see Fig.2 of the revised paper). From the obtained results we conclude that our approximation is simple and adequate with the theory. And as you know the analytical expression of the Voigt function is complex and it needs to program functions with several variables (see Eq. (24)).

 

4. What is the accuracy of your method?

Our approximation has a relative average error of 

5. How do these results help in 'several fields of physics, e.g., atmospheric radiative transfer, neutron reactions, molecular spectroscopy, plasma waves, and astrophysical spectroscopy' as promised in the abstract? This generic statement needs to be made more precise: How can somebody else use your results in their research?

Thanks for your carefulness. For example in the field of Astrophysical spectroscopy, if we want to calculate the opacities of hot stellar gases, one will automatically need the Voigt function because, in spectral line theory, the Voigt function describes the independently superimposed influences of the Doppler effect and collisional damping on the evolution of the emission (or absorption). Collision damping on the evolution of the emission (or absorption) curve through a single isolated spectral line and thus the absorption coefficient of a gas is proportional to the Voigt function. Because of the difficulty of programming the exact expression of the Voigt function presented in the literature we had the idea of providing researchers with a simple and easy-to-handle approximation.

 

6. Some specific issues:
   Line 55: 'And' -> 'and'
   Line 59: 'Where" -> 'where'
   Line 60: 'And' -> 'and'
   Line 62: 'And' -> 'and'
   Line 67: 'Where" -> 'where'
   Line 87: 'Where" -> 'where'
   Line 90: 'Where" -> 'where'
   Line 95: 'Where" -> 'where'
   Line 110: 'Owing to prove' -> Please rephrase
   Thank you for your carefulness, we have taken into account all your remarks in the revised version.

 

7. Lines 80-82: "a particular integral ... This integral given in Ref. [19]": Please describe which particular integral was referred to, e.g., which integral in Ref. 20.

Thanks for your suggestion. We are very sorry that we made a mistake in the order of the last two references, normally:

[26] Q. Huang, S. Coëtmellec, F. Duval, A. Louis, H. Leblond, M. Brunel, “Analytical expressions for diffraction-free beams through an opaque disk”, J. Eur. Opt. Soc. 6 (2011) 11031-11037.

[27] A. Belafhal, S. Chib, F. Khannous, T. Usman, “Evaluation of integral transforms using special functions with applications to biological tissues”, J. Comput. App. Math. 40 (2021) 1-23.

In the following, we will present the details of the requested description.

In section 2.2 we based on the integral expressed by Eq. (26) in the reference [Q. Huang, S. Coëtmellec, F. Duval, A. Louis, H. Leblond, M. Brunel, “Analytical expressions for diffraction-free beams through an opaque disk”, J. Eur. Opt. Soc. 6 (2011) 11031-11037.]

                                                        (1)

Taking

 and  so  

Then  

Finally, the quantity I can be rewritten in the following form

 

8. Line 112: 'old result (Eq. (33))': Why do the authors call this result 'old'? Has it been published previously?

Thanks for your suggestion. We called Eq. 33 by the old result because it represents our first approximation of the Voigt function developed in 2000. Yes, it has already published,

118. Belafhal, “The shape of spectral lines: widths and equivalent widths of the Voigt profile”, Commun. 177 (2000) 111-118.

 

9. Line 142: 'good agreement' -> please quantity

Thanks for your suggestion. We add the percentage of agreement in the paragraph and also we include in the figures the difference between our approximations and the numerical method in order to judge the degree of concordance. 

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors present a detailed analytical and numerical study of the commonly used Voigt profile. They compare their new expression and approximation to numerical solutions, to investigate the validity of their results. This type of work is quite valuable, as finding more computationally efficient ways to evaluate Voigt functions may help in large analysis problems. I found the paper to be well-written and quite clear. The figures are sufficiently clear as well. 

One area where the authors could add value to their paper would be to give a sense for how much faster their approach is compared to the other approaches in literature, and the 'brute-force' numerical evaluations they compare their new results to. Of course, this is hardware and implementation-dependent, but the authors may nevertheless be able to generate an estimate of the computational speed-up with their new methods.

- Table 1, second column: what are the numbers in brackets?

- All figures: it may be beneficial to plot the difference between the curves, rather than the curves themselves. E.g. subtracting the Gauss-Hermite result from the result of eqs 31, 33. This would make it easier to judge the qualitative statements about the degree of agreement.

 

 

 

Author Response

January, 16^th, 2021

Letter to the Journal: Quantum reports

 

Manuscript ID: quantumrep-1503844

Title: Simple analytical expression of the Voigt profile

Journal: Quantum reports

Authors:  S. Chib and A. Belafhal.

 

Dear Editor,

 

We sincerely appreciate the valuable comments and suggestions from the editor and the reviewers. The thorough review helped immensely in the shaping of the manuscript. The suggestions and comments have been closely followed and revisions have been made accordingly. The following are the questions extracted from the reviewers’ comments along with our summarized responses. Please see below, our detailed response to comments.

 

Thank you very much for your interest in our manuscript.

 

Yours sincerely,

 

1. Chib and A. Belafhal.

 

 

 

 

 

 

 

 

 

 

 

 

 

• Responses to Reviewer 2:

 

Dear Reviewer,

     Your feedback and suggestions on our manuscript were greatly appreciated. We have improved our paper as a result of your comments and helpful suggestions. All changes in the revised manuscript are made by the use of the “Track changes” function. Now, we would like to describe the change that we have made in the revised manuscript. The details are given as follows:

 

1. One area where the authors could add value to their paper would be to give a sense for how much faster their approach is compared to the other approaches in literature, and the 'brute-force' numerical evaluations they compare their new results to. Of course, this is hardware and implementation-dependent, but the authors may nevertheless be able to generate an estimate of the computational speed-up with their new methods.

Thanks for your suggestion. In the revised paper, we have added a figure in which we have made a comparison between our approximation and that of Teodorescu et al. with the analytical form of Srivastava and we have specified the simulation time of each approximation as well as the relative error  (see Fig. 2).

 

2. Table 1, second column: what are the numbers in brackets?

The mathematical writing (-1) 4.622436696006 is equivalent to . We add a sentence just after Table 1 in the revised paper.

 

3. All figures: it may be beneficial to plot the difference between the curves, rather than the curves themselves. E.g. subtracting the Gauss-Hermite result from the result of Eqs (31, 33). This would make it easier to judge the qualitative statements about the degree of agreement.

According to your valuable remark, we have inserted figures that simulate the subtraction of        the Gauss-Hermite result from the result of Eqs (31) and (33).

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The new version of the manuscript has improved and the authors have addressed the issues raised by this reviewer. 
This reviewer would like to remind the authors to make their manuscript accessible and interesting to an as-wide-as-possible readership. To help readers of the manuscript, it is suggested to incorporate some replies of the authors to this reviewer's comments into the manuscript:
1. The reply 5 involving astrophysical spectroscopy (as a motivation for dealing with Voigt functions).
2. The reply 7: The derivation of equation 26 (section 2.2.).

Author Response

January, 19^th, 2021

Letter to the Journal: Quantum reports

 

Manuscript ID: quantumrep-1503844

Title: Simple analytical expression of the Voigt profile

Journal: Quantum reports

Authors:  S. Chib and A. Belafhal.

 

• Response to Reviewer 1:

 

Dear Reviewer,

We are greatly appreciated your comments that you made on our revised manuscript. According to your valuable suggestions, in the revised version, we have added sentences in the introduction to explain to the readers the reason why we have developed this study and also the derivation of Equation (26) in Section 2.2.