J. Eur. Opt. Society-Rapid Publ. 21, 21( 2025) 205
Figure
1. Coordinate axes for the present work. The plane of incidence is the plane XZ. The Y-axis points inside the plane of the paper. The interface is the plane XY. The wavevectors are shown in blue. The vectors corresponding to the electric fields for the“ p” case are shown in red.
are often not covered in detail in most textbooks, which instead refer the reader to the work of [ 2 ] for a more rigorous treatment. In this reference, the approach involves using the incident angle h i with Snell’ s law in its“ complex” form to determine the transmitted angle h t. This value is then substituted into equations( 1)–( 4) for further analysis.
Introducing a complex angle for the refracted wave challenges the previous demonstration based on the geometrical projection of fields associated with homogeneous plane waves. Consequently, the physical explanation of the phenomenon must be supplemented to ensure that the practical interpretation of equations( 1)–( 4) becomes intuitive and clear. The question is analyzed in detail in [ 9 ]. Basically two mathematical procedures for addressing the general isotropic interface are compared there: the so called“ phase and attenuation” vector representation and the“ complex angle” notation. Since these procedures are mathematically complete and well-suited for the problem, we do not intend to repeat them here. The general derivation of the Fresnel coefficients( with specific notation) and a particular numerical example are also developed there. Rather, our goal is to help understanding the use of the complex angle notation since, as the author literally points out in the text of [ 9 ], that method“ severely lacks in terms of physical interpretability”.
We propose a comprehensive procedure for applying equations( 1)–( 4) for the transparent-absorbing isotropic interface at any incidence angle, offering a complete interpretation of the resulting practical outcomes. In addition to determining and explicitly writing the amplitudes and phases of all coexisting electric fields, we will illustrate the physical effects introduced by the phase differences in the fields of the reflected and transmitted waves. For the cases where the transmitted wave becomes inhomogeneous, we will also give its propagation direction and penetration decay rate. Finally, although our analysis is limited to the monochromatic case, we believe this work serves as a valuable resource at the undergraduate and graduate levels, as well as for researchers and professionals.
The structure of the paper is as follows. First, we review the application of the Fresnel formulas to the most general case of an isotropic interface, assuming a transparent incident medium, and explicitly compute the Cartesian components of the electric fields of the waves. We examine the specific case of incidence above the critical angle, illustrating the behavior of the electric fields at the interface. Finally, we address incidence on an absorbing substrate, again highlighting the interplay between the various electric fields. For completeness, detailed explanations of the nature of inhomogeneous and evanescent waves are provided in Appendix.
2 Application of the Fresnel formulas in the general plane isotropic interface
The classical analysis of light incidence on a plane interface results in the well-known Fresnel equations( 1)–( 4). The detailed derivation begins with defining the coordinate axes at the interface. For the“ p” case, our Figure 1 aligns with Figure 1 of [ 10 ], and also corresponds to the configuration