Closed Form Analytic Expressions for the Evanescent and Traveling Components of the Electromagnetic Green Function and for Defocused Hemispherical Focusing of Electromagnetic Waves

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents closed-form analytic expressions for electromagnetic field components, with important applications in high-NA optical systems. This is a technically sound and theoretically important paper. It will make a valuable contribution to the field. I recommend acceptance after minor revisions addressing the following points.

1. The author presents in Eq. 1 the electric field components for traveling electric and magnetic dipole waves in the focal region, expressed via angular spectrum representation (Debye integral). The field described by Eq. 1 appears to lack the inherent singularity expected at the origin for dipole radiation. Is Eq. 1 derived under the assumption that the field is reconstructed by time-reversed propagation of the far-field radiation pattern of a dipole at the focus? In other words, does Eq. 1 represent the field formed by back-propagating the dipole’s far-field distribution through the focusing system, thereby naturally filtering out the singular self-field term?
2. The connection between the hemispherical focusing integrals (Eqs. 1-2, alpha = pi/2) and the complete spherical solution (Eq. 13, alpha = pi). Whether Eq. 13 was obtained through direct extension of Eqs. 1-2 integrals to alpha = pi with analytical solution, or independent spherical wave expansion methods
3. Weyl expansion. Is the Weyl expansion mathematically equivalent to an angular spectrum decomposition of Green function for a dipole? Does the Weyl expansion reduce to the Whittaker expansion if only traveling wave components in the Weyl expansion are retained?
4. 12. What is the difference between the notations q and w? They appear to be the same. After Eq. 12, the author states that “but w is useful to describe forward propagating”. However, in standard wave propagation theory, the forward propagation direction should be determined by the sign of the longitudinal wave vector component (kz), not the radial coordinate w.
5. Equation 18. I would like to understand the mathematical origin of the sign(u) function in the Green's function expression. Does this term emerge from solving the partial differential equation for the Green's function, where separate solution forms must be considered for the z > 0 and z < 0 regions?

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, the author, based on the works of Bertilone [4] and Arnoldus [6,7], presented (without derivation) more symmetric expressions for the evanescent (21) and traveling (27) wave for an electric dipole, written in terms of the Lommel functions U0 and U1. The calculation of the integrals (11) in terms of the Lommel functions is based on the reference integral 2.12.10.3 from [39]. The work can be published after the author takes into account the comments.

Comments

1. It can be clarified that adding the real function of the plane wave spectrum A(θ) to the integrals (2) will not change the form of the expressions for intensity (3), field (5), and flux (line 58).
2. Expression (3) is written in the plane of the focus, where the integrals (2) are real, so the sign of the real part is superfluous.
3. It is necessary to indicate for which case the expression (5) is given: for left or right circular polarization.
4. For radial polarization (7) it is said that such a field is formed only by the longitudinal electric field in the initial plane. It is necessary to indicate by which field in the initial plane the azimuthal polarization in (9) is formed.
5. For equation (13) it is necessary to explain what complete spherical ED focusing is. And how it differs from the 4Pi hemispherical case?

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf