The Fresnel Approximation and Diffraction of Focused Waves

Round 1

Reviewer 1 Report

This manuscript provides a comprehensive analysis of the diffraction of scalar waves by a screen with a circular aperture, focusing on both collimated beam incidence and focused wave scenarios. The historical background offered enriches the reader's understanding of the topic, and the introduction of various Fresnel approximations, including traditional forms and a generalized version, adds significant value to the existing body of knowledge. The exploration of the focused case, including focal length, Fresnel number, and off-axis behavior, presents a detailed and nuanced view of the phenomena under study. The manuscript is well-organized, logically structured, and addresses a topic of importance in the field of photonics and optical engineering.

Before recommendation for acceptance, some matters should be addressed:

1.     The introduction should review various fields of diffraction of focused waves to broaden the scope of the article such as
i) Engelberg, Jacob, and Uriel Levy. "The advantages of metalenses over diffractive lenses." Nature communications 11.1 (2020): 1991.
ii) Rodrigues Gonçalves, Manuel, et al. "Bright and dark diffractive focusing." Applied Physics B 128.3 (2022): 51.
iii) Weisman, Dror, et al. "Diffractive guiding of waves by a periodic array of slits." Physical Review Letters 127.1 (2021): 014303.

2.     Consider including experimental results such as a discussion on potential experimental setups and the expected outcomes.

3.     In Figure. 5, can you define all the angles in the caption and expand it?

4.     Figures. 9-11, 13, 15, 16, 17 should be enlarged or the fonts inside it should be bigger.

Author Response

The reviewer is thanked for his useful suggestions. The introduction has been extended to include some recent developments:

‘The obvious further work is to extend the treatment to the vectorial case. The topic of diffraction is very diverse, and any paper cannot cover all areas. Other modern research which could be related to the present study includes diffraction by disks, rather than apertures \cite{Aime20}; diffraction of vortex beams \cite{Basistiy93,Khonina14}; and diffractive optics and metalenses \cite{Engelberg20,Goncalves22,Weisman21}. ‘

Concerning experiments, I think that FTDT calculations would be preferable for validation of the results, the optical experiments would be quite difficult to perform. I have extended the discussion section, and added:

‘The focal shift measured experimentally for systems of low NA for acoustic, microwaves and visible light has agreed well with the theoretical predictions \cite{Williams46,Fein49,ONeil49,Lucas82,Farnell58,LiPlatzer83}. Li and Platzer gave results for Fresnel number less than 1 \cite{LiPlatzer83}. To our knowledge, detailed measurements for high NA have not been reported. As for high NA and low Fresnel number, the radius $a$ of the aperture is necessarily quite small, the theoretical treatment could be validated using finite-difference time-domain (TDTD) calculations.’

‘The off-axis behaviour of the predictions of the different approaches for high NA and low Fresnel number could also be compared using FDTD.’

The caption for Figure 5 has been expanded to define the labeling.

For Figs. 9-11, 13, 15, 16, 17 (and some others) the font size of the labelling of the contours has been increased.

Reviewer 2 Report

This is a rather interesting and detailed review on a topic in which the author is a well-known expert.

I just have a few small comments.

1) Although the Historical background given in the manuscript is exhaustive, since the main development of wave theory was carried out many years ago, it would be desirable to add more references to modern research on this topic in the Discussion section [for example, https://doi.org/10.1016/j.ijleo.2005.04.004; https://doi.org/10.3103/S1060992X14020027;
https://doi.org/10.1155/2018/4031793;
https://doi.org/10.1051/0004-6361/201937208; https://doi.org/10.1016/j.ijleo.2023.170950] 

2) There are some typos that need to be corrected:

- page 17, Fig. 6 caption: instead of “as in Eq.62” should be “as in Eq.42”;

- page 27, line 682: instead of “oppsite” should be “opposite”;

- page 28, line 688: instead of “z=2f,” should be “z=0.2f,”;

- page 28, line 691: instead of “asign” should be “assign”;

- page 32, line 770: instead of “eperiments” should be “experiments”;

- page 33, line 780: instead of “approxiate” should be “approximate”;

Author Response

The reviewer is thanked for his useful suggestions.

Some references to some more modern work have been added:

‘The obvious further work is to extend the treatment to the vectorial case. The topic of diffraction is very diverse, and any paper cannot cover all areas. Other modern research which could be related to the present study includes diffraction by disks, rather than apertures \cite{Aime20}; diffraction of vortex beams \cite{Basistiy93,Khonina14}; and diffractive optics and metalenses \cite{Engelberg20,Goncalves22,Weisman21}.’

Teng has been included in the historical review:

‘Teng et al. showed how the amplitude on the axis and along the shadow edge for diffraction of a plane wave by a circular aperture can be calculated analytically using series of Bessel functions, equivalent to Lommel functions \cite{Teng05}.’

The paper by Sandoval is about Fresnel integrals, which can be represented by half-order Lommel functions, but explaining these connections is beyond the scope of the present paper. I couldn’t interpret the final reference mentioned.

The discussion section has been extended.

The typos have been corrected.