Theory of Diffraction by Holes of Arbitrary Sizes

Round 1

Reviewer 1 Report (New Reviewer)

Comments and Suggestions for Authors

Reviewer: In Manuscript electronics-3632902, the author developed an electromagnetic theory that describes the coupling between resonant cavities and RF waveguides, and further developed Bethe's small aperture polarization method. In addition, an accurate analytical expression for the reflection coefficient as a physical parameter function of the cavity waveguide system was proposed, which is applicable to any geometric shape, material, and frequency. The manuscript is well-written, with clear figures, good organization, a professional experimental study, and reasonable analysis. Therefore, I believe this manuscript is suitable for acceptance and publication in Electronics.

There are a few minor issues that need to be addressed:

1. When abbreviations are first given in the manuscript, the full name should be provided, such as RF. The author needs to check the manuscript for further verification.
2. Are all the formulas provided by the author in the manuscript self-proposed, or are they variations of other corresponding formulas? If not proposed by the author, the source should be provided.

Author Response

Comment 1: In Manuscript electronics-3632902, the author developed an electromagnetic theory that describes the coupling between resonant cavities and RF waveguides, and further developed Bethe's small aperture polarization method. In addition, an accurate analytical expression for the reflection coefficient as a physical parameter function of the cavity waveguide system was proposed, which is applicable to any geometric shape, material, and frequency. The manuscript is well-written, with clear figures, good organization, a professional experimental study, and reasonable analysis. Therefore, I believe this manuscript is suitable for acceptance and publication in Electronics.

There are a few minor issues that need to be addressed:

(1) When abbreviations are first given in the manuscript, the full name should be provided, such as RF. The author needs to check the manuscript for further verification.

Risponse 1: We thank the referee for their positive evaluation and constructive comments.

In response to the minor issue raised, we have carefully reviewed the manuscript and ensured that all abbreviations are defined upon their first appearance in the text.

We appreciate the referee's valuable feedback and support for the publication of our work.

Comment 2:    Are all the formulas provided by the author in the manuscript self-proposed, or are they variations of other corresponding formulas? If not proposed by the author, the source should be provided.}

Risponse 2:  Regarding the question about the formulas: we confirm that all the key analytical expressions and formulas presented in the manuscript are original and self-proposed by the author, unless otherwise explicitly stated. For the cases where existing theories or methods were extended or referenced (such as Bethe's small aperture polarization method), appropriate citations have been included in the manuscript to acknowledge the original sources.

We have reviewed the manuscript to ensure that all necessary references are properly cited.

We appreciate the referee's helpful suggestions, which have contributed to improving the clarity and completeness of the paper.  

Reviewer 2 Report (New Reviewer)

Comments and Suggestions for Authors

This paper presents an analytical framework for evaluating the reflection coefficient in high-gradient RF cavities coupled to waveguides via small apertures. By extending Bethe’s diffraction theory and incorporating Collin’s modifications, the authors develop a general method applicable to arbitrary geometries and frequencies.

The derivation of the reflection and transmission coefficients is thorough, and the resulting expressions are generalized to a wide class of coupling configurations.

Line 229. It is not clear from the text which equation is analogous to the reflection coefficient using circuit theory. Authors should compare with equation 44.

A major drawback in this work is the absence of quantitative verification. Comparation of eq 44 and 48 is purely qualitative. Authors should provide quantitative calculations for some particular case in order to validate their model. This is essential for this manuscript and should be added. While the theory is general, a numerical validation against full-wave simulation (even in an appendix) would further strengthen the claims.

Author Response

Comment 1:  This paper presents an analytical framework for evaluating the reflection coefficient in high-gradient RF cavities coupled to waveguides via small apertures. By extending Bethe's diffraction theory and incorporating Collin?s modifications, the authors develop a general method applicable to arbitrary geometries and frequencies.

The derivation of the reflection and transmission coefficients is thorough, and the resulting expressions are generalized to a wide class of coupling configurations.

Line 229. It is not clear from the text which equation is analogous to the reflection coefficient using circuit theory. Authors should compare with equation 44.

Risponse 1: To clarify the analogy, Eq. (47) is presented as the equivalent expression for the reflection coefficient derived using circuit theory for the $TE_{10}$ mode, as originally given by Collin [2]}. This formulation is directly compared to Eq. (44), which represents the reflection coefficient for the $TM_{010}$ mode in our analysis. Both equations share a similar structure, featuring a complex rational expression with terms proportional to the electric or magnetic polarizability and a normalization constant related to the power flow of the respective modes ($P_{01}$ for $TM_{010}$ and $\beta_{10}$ for $TE_{10}$). Additionally, each contains a second-order dependence on the ratio of wavenumbers, $\frac{k}{k_c}$ or $\frac{k_0}{k_{101}}$, and reflects the geometric influence of the resonant cavity (cylindrical versus rectangular). This comparison is provided to illustrate the consistency of our approach with classical circuit-based methods and to highlight the physical parallels between the two modal systems.

Comment 2:  A major drawback in this work is the absence of quantitative verification. Comparation of eq 44 and 48 is purely qualitative. Authors should provide quantitative calculations for some particular case in order to validate their model. This is essential for this manuscript and should be added. While the theory is general, a numerical validation against full-wave simulation (even in an appendix) would further strengthen the claims.

Risponse 2:  We respectfully acknowledge the referee's comment. However, we would like to emphasize that our work is fundamentally analytical in nature. The purpose of this study is to present a self-consistent analytical model rooted in electromagnetic theory, using polarizability and mode matching techniques to derive the reflection coefficient for the $TM_{01}$ mode.\\

The comparison with the classical result obtained by Collin, which we present in Eq. (48), serves to highlight the structural and physical analogies between our model and well-established analytical results. This comparison is meaningful within the analytical framework, as both models describe similar physical phenomena using comparable parameters (e.g., polarizabilities, mode normalization, and geometrical dependencies). Our goal is not to replicate numerical results, but to show how analytical tools can offer deep insight into the physical mechanisms of wave propagation and scattering in resonant structures.

It is important to note that not all analytical works must be validated by numerical simulations. Analytical models are often developed to provide conceptual clarity, closed-form solutions, and scaling laws that are sometimes obscured in purely numerical approaches. While numerical simulations are valuable for validating complex geometries or optimizing designs, our intention here is to provide a mathematically transparent derivation that can serve as a reference point for further theoretical or computational investigations. 

 

Reviewer 3 Report (New Reviewer)

Comments and Suggestions for Authors

This article presents the electromagnetic theory describing the coupling between a resonant cavity and an RF waveguide by means of an aperture. This is based on a technique described in a 1944 paper by H. A. Bethe entitled “Theory of diffraction by small holes”. In that paper, Bethe described a solution for the diffraction of electromagnetic radiation by a hole that is smaller than the wavelength using a method based on fictitious magnetic charges and currents in the hole that satisfy the boundary conditions. The submitted paper, entitled “Theory of diffraction by holes of arbitrary sizes” develops the work of Bethe, Collin, and others to derive equations for the scattering parameters and reflection coefficient for a TM01 mode.

 

This paper presents some important results that could be applied to aid waveguide design for accelerators and may be of interest to some readers. Unfortunately, however, the lack of clarity in the presentation impedes understanding and detracts from the work. In addition, the validation of this theory is somewhat limited, and there is no discussion of the limits of applicability or limitations of the theory presented here. While these issues unfortunately mean that the paper is not suitable for publication in its current form, I believe it could be made suitable for publication after careful revision. Detailed comments provided below.

 

 

General comments

1. There is a lack of motivation for this work. While a very brief statement on the motivation is provided in the abstract, there is none whatsoever provided in the introduction. It is also not clear what the novelty of this work compared to previous work is. It is essential to provide motivation and context, as well as specific use cases, to justify the importance and novelty of this work.

2. It is not clear what the range of applicability of this work is. While the title suggests that this theory should be applicable to holes of arbitrary sizes, the text seems to suggest that this is only applicable to small holes comparable to the wavelength of radiation. The range of applicability and limitations of this theory require discussion.

3. Simulation results should be provided to validate the theoretical framework presented here.

4. In general, there is a lack of care and clarity in the presentation. Terms in equations are not defined clearly or are not defined at all, and the text often does not make it clear how equations are derived. Care should be taken to make this paper as clear and understandable as possible to readers. Regarding symbols in equations, it may be helpful to readers to include a list of all symbols used throughout the paper. In addition, vectors should be consistently bold, even when referenced in the text, to prevent confusion. Furthermore, key results should be highlighted, and the meaning and implications of equations described.

5. When referencing long papers and books, it would be helpful to the reader to provide more specific indications of where the equations or material being referenced can be found within those texts.

 

Specific comments

1. Line 12: The abstract mentions RF cavities operating in the GV/m range. Can the authors point to examples of cavities reliably operating in this range?

2. Lines 13-14: How compact are these “compact structures”?

3. Line 26: The first sentence claims that “the reflection coefficient plays a fundamental role”. The reflection coefficient ought to be defined and its fundamental role described.

4. Line 34: Only a partial author list of the cited papers is provided here. Is there a reason other authors have been omitted?

5. Lines 41-42: What exactly is meant by a dielectric iris supported by a perfectly conducting screen. Is the dielectric a window? What is the importance of a dielectric iris? Does the theory only apply in this case?

6. Line 44: What is the importance of a circular wall? Does the theory only apply in this case?

7. Line 50: The polarizability ought to be defined. It seems the polarizability and polarization coefficient are used interchangeably. It would be better to be consistent.

8. Fig. 1: This image is low quality, poorly typeset, and difficult to read. It is not clear from the caption and text below which way E_sB and H_sB should be propagating. The caption mentions an “incoming wave from the right”, yet the arrow is pointing to the right. The caption is also largely a repeat of the text below, which is not particularly helpful. H_sF and H_sB are referenced in the caption but not in the figure.

9. Line 60: “Scattering parameters” should be defined.

10. Line 61: What exactly is this statement saying? Is this theory only valid for holes that are smaller than the wavelength, or for holes that are comparable to the wavelength, or for arbitrary-sized holes?

11. Line 66: It is not clear what “a resonant cavity can lose in accuracy” means. Is this referring to measurement inaccuracies?

12. An outline of the paper, including what is presented in which section, should be provided at the end of the introduction.

13. Line 80: Why is E_BF not shown in Fig. 1?

14. Eq. (3): E, H, J_0, r, \beta_n are not defined.

15. Lines 96-101: This is not expressed clearly.

16. Eq. (5) and (6): r, theta, z, and corresponding quantities with subscripts, are not defined.

17. Eq. (7): epsilon is not defined (unless epsilon_0 is meant).

18. Line 114: j has been used since Eq. (3) and should be defined earlier.

19. Eq. (11) to (14): \Sigma ought to be larger or brackets used to make it clear what is being summed.

20. Eq. (15): This equation should be explained, since it is not the standard formulation of the Lorentz reciprocity theorem.

21. Eq. (16): Why have E_1 and H_1 become E- and H-? Why has n become z? These things needed explaining.

22. Eq. (19): How did \zeta_01- enter this equation?

23. Eq. (20): Are these cross products?

24. Eq. (21): Should \beta_n be \beta, as defined on line 127.

25. Line 133: How does P_{01} come about from Eq. (20)?

26. Line 134: It would be helpful to provide equation numbers when referencing equations.

27. Line 136: \Xi is undefined. Does \zeta have the same meaning as defined on line 93?

28. Line 136: h_n does not appear in Eq. (22).

29. Line 136: \omega_e is not defined.

30. Eq. (23): Both \beta and \beta_n appear in this equation. What is the difference?

31. Eq. (24): It is probably not necessary to include both Eq. (23) and Eq. (24) since the simplification is trivial and can be shown in one step. There ought to be some discussion, though, of these equationn – what does it tell us?

32. Line 145: Why is P_{01}, which was first defined on line 133, now defined differently?

33. Eq. (25): What is the difference between e_n and e_z defined in line 136?

34. Eq. (26): This equation ought to be explained.

35. Line 147: n should be defined earlier.

36. Lines 148-149: \eta_{01} is presumably the impedance of free space. What are the units, and why is this stated to be “exactly defined”?

37. Eq. (27) and (28): These equations are stated without justification. How do they come about?

38. Eq. (29): Why is the integral over z and not S_1? Why has e_{t01} become e_{01}? What is b? There appears to be a sign error introduced when evaluating \beta.

39. Line 153: Is this equation explicitly stated in [1]? It would be helpful to provide a more specific reference. Moreover, in line 88, \alpha is defined as the attenuation constant per unit length. Should this be \alpha_e?

40. Lines 156-158: A clearer and more extensive explanation is needed here.

41. Eq. (31): P_z and E_z^0 are undefined.

42. Line 103: Is it the exponent that is being discussed here? Is the second term not proportional to a^5?

43. Eq. (34): The derivation of this equation requires an explanation. It should also be explained how this reduces to -2/3a^3 in Bethe’s theory (line 153).

44. Line 177: Should mention the aperture is small compared to wavelength.

45. Eq. (35): Typesetting is awry.

46. Eq. (36): This equation is introduced without any explanation. Is it derived from Eq. (24)?

47. Eq. (37): This is just a restatement of Eq. (34).

48. Eq. (38) and (39): \Gamma should be defined.

49. Line 202: “the equation above” would be better written as Eq. (36).

50. Eq. (41): It should be explained how the Heaviside step function enters this equation. There appears to be a sign error in the definition of f(r).

51. Eq. (42): The relation between z and n should be explained. Why are these equations not written as vectors like Eq. (5) and (6)?

52. Line 216: Should expand on or provide a reference for “self consistent solution method”.

53. Eq. (40) and Eq. (44) appear to be the key results in this paper, though they are accompanied by very little explanation or discussion.

54. Eq. (45): This is just a repetition of Eq. (30).

55. Eq. (46): This is just a repetition of Eq. (34).

56. Lines 220-222: This is just a repetition of lines 196-199.

57. Eq. (47): Very limited explanation of this equation.

58. Eq. (48): Is this an expression for the reflection coefficient? It would be better to write this as an equation. Why is the TM01 mode compared to a TE10 mode?

59. Eq. (49): b and Q are undefined.

60. Eq. (50): Why has t e_{t01} in Eq. (23) become h_^{t01} here?

61. Eq. (51): How does t enter this equation? Where does the denominator r come in?

Comments on the Quality of English Language

There are several minor English errors.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 4 Report (New Reviewer)

Comments and Suggestions for Authors

1- Define "RF" in abstract.

2- Do not use "we" in your writing.  Too many uses which is not recommended in technical paper writing.

3-The theoretical analysis presented for diffraction through holes of arbitrary sizes is promising, especially its application to TM01 mode coupling through small apertures. However, the manuscript would be significantly modified by including some validation for the analytical method.

Author Response

Comment 1:  Define "RF" in abstract.

Risponse 1: We have addressed this comment by defining "RF" as radio frequency in the abstract. 

 

Comment 2:  Do not use "we" in your writing.  Too many uses which is not recommended in technical paper writing.

Risponse 2:  We have removed the use of "we" in most instances throughout the manuscript to align with the conventions of formal technical writing.

Comment 3:  The theoretical analysis presented for diffraction through holes of arbitrary sizes is promising, especially its application to TM01 mode coupling through small apertures. However, the manuscript would be significantly modified by including some validation for the analytical method.

Risponse 3: We thank the referee for recognizing the promise of our theoretical analysis, especially in relation to the TM$_{01}$ mode coupling through small apertures. While we appreciate the suggestion to include a validation of the analytical method, we would like to emphasize that this manuscript is focused on presenting a self-consistent analytical model rooted in electromagnetic theory. The comparison made with the classical results from Collin is intended to show structural and physical consistency, not to serve as a numerical benchmark. Our approach provides conceptual clarity and insight into the underlying physics, which is often obscured in numerical treatments. Although numerical validation can be useful, particularly in design or optimization contexts, it is not a necessary requirement for purely theoretical studies. The model's agreement with established analytical results and its rigorous derivation provide confidence in its validity. We agree that future work could expand upon this foundation with numerical simulations, but we believe the current scope-focused on theoretical development?remains coherent and valuable without such additions.\\

However, we would like to emphasize that our work is fundamentally analytical in nature. The purpose of this study is to present a self-consistent analytical model rooted in electromagnetic theory, using polarizability and mode matching techniques to derive the reflection coefficient for the $TM_{01}$ mode.

The comparison with the classical result obtained by Collin, which we present in Eq. (48), serves to highlight the structural and physical analogies between our model and well-established analytical results. This comparison is meaningful within the analytical framework, as both models describe similar physical phenomena using comparable parameters (e.g., polarizabilities, mode normalization, and geometrical dependencies). Our goal is not to replicate numerical results, but to show how analytical tools can offer deep insight into the physical mechanisms of wave propagation and scattering in resonant structures.

 

It is important to note that not all analytical works must be validated by numerical simulations. Analytical models are often developed to provide conceptual clarity, closed-form solutions, and scaling laws that are sometimes obscured in purely numerical approaches. While numerical simulations are valuable for validating complex geometries or optimizing designs, our intention here is to provide a mathematically transparent derivation that can serve as a reference point for further theoretical or computational investigations.

Round 2

Reviewer 2 Report (New Reviewer)

Comments and Suggestions for Authors

I agree with authors answers to the comments. Paper is revised and its quality is improved. I can recommend publication of the manuscript in present form. 

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript presents theoretical derivation of the reflection coefficient for a waveguide-cavity coupling network by modifying Bethe's Theory. The paper is clearly written. The work is of somewhat fundamental interest but lack of details and necessary validation. 

To the referee's opinion, it is necessary for the authors to point out the essential advancement(s) / modification(s) of the developed theory compared to the works in the past. Although some additional information can be found in the supplementary thesis, the authors should clarify, within the scope of this research paper, how this work brings forward the frontier of existing theories developed for power-coupling network, in terms of benefits and theoretical limitations. Furthermore, the modified theory should be properly justified by explicitly comparing with other theories and / or numerical simulations for solving a coupling problem. Throughout the paper draft, the referee could not find the details about such comparisons and corresponding statements. 

Some specific comments and questions are in the following:

1. L25, "concept" to "Concept" 

2. L25, what does it mean by "high accelerating periodic structure"?

3. Resolution of Fig. 1 should be improved, avoiding black dots and shadows; Symbol subscripts should be clearly seen.

4. Some symbols (e.g. E_{R}, E_{T}) are shown in Fig.1 but not yet defined at this point before first-use. Please correct.

5. L66: the ending bracket is missing.

6. L74-76: the authors claim to perform a comparison of the derived reflection coefficient with Ref.[11], however, it is not clear where this comparison is presented throughout the paper draft; Although Collin's work is simply mentioned in between L221 and L225, it is not clear how well the comparison of reflection coefficient can be and what is the main difference if there is any.

7. Formula (3) and (4), the symbol "kc" is not yet defined at this point before first-use.

8. L98-99: "an" deleted in "...induce an electric and magnetic currents" 

9. L110, "Eqs.(3), 4" to "Eqs.(3), (4)"

10.Please define all the symbols used in formula (9) and (10) before first-use. Please introduce changes wherever necessary throughout the paper.

11.L126, "Eq." in "Eq.(15)" not italic

12.L164,L196,L219: "phase delay delay", repetition and redundancy

13.L186-188, rephrase: "...we will calculate...the aperture located between waveguide and the cavity, and will find equations for...".

14.L210, "simplify" to "simplifies"

15.L214, what does it mean by "Self Consistent Solution Method"? If it is elaborated in a publication (thesis) elsewhere, one should point it out because it is not clear what kind of treatment is introduced.

16.L221-L225, what is the result of this comparison? There is no clear statement given. Does the derived formulism by the authors "agree" with Collin's work as cited here? If yes, one should present more details and give a clear statement. If not, possible discrepancy  source(s) should be given.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In my opinion, this paper is a good example of scientific graphomany that presents a lot of mathematical excersizes with zero practical applicability or value. In particular, the authors start with the motivation to use this "theory" for particle accelerators, but do not consider neither a beam loading, nor stored energy effects, nor mupltiple coupled cells with random errors - nothing that is relevant to particle accelerators. For a resonator, coupled to a waveguide, we have a very simple formula of the reflections, based on the loaded Q-factor: S11=(1-+beta)/(1+beta), where beta=Q0/Qext. Where Q0 and Qext can be obtained from simulations, measurements or analytical formulas (see, for example, Lapostolle book). Similar (very simple) formulas exist for beam-loaded cavities and transient processes and are very well known. On countrary, it is not clear how can the monstrous and extremely compicated formulas, derived by the authors, be practically used for accelerator development or measurement.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Despite a heavy copy-and-paste material from some textbook (Pozar?) or the introduction from Ph.D. Thesis, the Author’s response doesn’t address my main criticism: the irrelevance of the presented material to the real-life applications and particularly, to accelerating structures, which the Author mentions in the introduction. This is extremely misleading and must be re-written completely.

First of all, I don’t see how can the presented model be used to calculate accelerating cavities. The 2D model with a slit and some rectangular resonator has nothing to do with the actual accelerating structures. Maybe the authors should present some paratactical example of how their model can be used to solve some practical problem.

Second, there are many publications from 70-90s like SLAC reports, Lapostolle book and most importantly, N. Sobenin, B. Zverev book from CRC press, where they present the analytical solution for a realistic 3D accelerating structure, coupled to the rectangular waveguide. It’s not clear to me how does your model advance the state of the art for RF cavities theory, comparing to already published (and well-known) models.

That said, the authors must either provide a clear explanation of how their model can be used for accelerating cavities calculation, and include an real-life example, or rewrite the introduction completely to eliminate the mention of accelerator and provide a better motivation for their work.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf