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used in [ 3 ] to derive equations( 1)–( 4). The derivation proceeds by enforcing the continuity of the tangential components of the electric and magnetic fields at the interface.
A critical condition for the validity of this approach is that all three waves coexisting at the interface( incident, reflected, and transmitted) must be homogeneous. This phenomenon is particularly well illustrated in Figures 4.53 and 4.54 of [ 5 ] which depict external reflection for incidence below and above the Brewster angle, as well as the case of internal reflection.
For instances of internal reflection beyond the critical angle, the classical reference is [ 2 ], Section 1.5.4. Since virtually all authors refer to this source, we provide a summary of the method here. Essentially, Snell’ s law is first applied to determine the transmitted angle h t given the incident angle h i, asfollows:
sin ðh t Þ ¼ n i sinðh i Þ ¼ sinðh iÞ
; n ¼ n t n t n n i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s
¼ 1 � sin2 ðh i Þ ¼i n 2
) cos ðh t Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2 ðh i Þ
� 1: ð5Þ n 2
Finally, cos( h t) is substituted into the previously derived Fresnel equations, followed by a physical interpretation of the resulting waves on an ad hoc basis. Notably, the negative sign is then disregarded, as it pertains to the inhomogeneous wave generated in the second medium.
The case of an absorbing second medium is addressed in [ 2 ], Section 14.2. Again, typically textbooks refer to that approach without further considerations, so we summarize the method here. The analysis in [ 2 ] begins with the derivation of the wave equation for conducting media, which is then compared to the transparent case. Finally, the transmitted wave is introduced by formally substituting the standard refractive index, dielectric constant, and wavevector with their corresponding complex counterparts. Assuming n i 1, the complex refractive index ñ t = n t + ij t must be used in formulas( 1)–( 5). While this approach yields correct results – provided the positive sign is chosen in equation( 5) – it is not entirely satisfactory in didactical terms and does not fully resolve the physical problem. For example, it does not explicitly address the propagation direction of the inhomogeneous wave in the second medium.
In summary, in the references [ 2 – 7 ], the application of the Fresnel formulas to the general case of an isotropic interface can be effectively carried out using equations( 1)–( 5). When all three coexisting waves are homogenous, the separate treatment of“ s” and“ p” polarizations along with the projection of all vectors onto directions parallel and perpendicular to the interface, provides a straightforward and comprehensive approach. This method allows the directions and phases of all amplitudes to be easily determined.
However, when the transmitted wave is evanescent or inhomogeneous, although the separation of“ s” and“ p” cases remains valid, it becomes necessary to fully describe the reflected and transmitted waves, including their phase delays, propagation directions, and wavelengths, and these topics are not fully addressed in references [ 2 – 7 ]. Furthermore, as pointed out in [ 9 ], the use of complex angles in
Snell’ s law is not comprehensible, although its use has passed on in educational purposes over decades. Here, without duplicating the procedures detailed in [ 9 ], we will explain how to use equations( 1)–( 5) and the physical meaning of the subsequent results. Unfortunately, we cannot make a quick comparison between our procedures and the numerical example developed in [ 9 ] for obtaining Figure 3 there, since it corresponds to an interface between two absorbing materials, a configuration that we cannot address. Our developments require a transparent incident medium, while reference [ 9 ] imposes no such restriction.
2.1“ s” case
The s-polarized case is simpler than the p-polarized case because all the electric fields oscillate consistently in the same direction( along the Y-axis in Fig. 1). Since our primary focus is on the behavior of the electric fields at the interface, we will omit the s-polarized case and focus exclusively on the p-polarized case. Anyway, the differences in the magnitude and phase of the r \ and t \ coefficients, as well as those associated with the magnetic fields, play a crucial role in determining the behavior of all Poynting vectors. This aspect, however, falls out of the scope of the present work; for further details, refer to [ 10 ].
2.2“ p” case
In Figure 1, the positive directions of the electric fields for the validity of equations( 1)–( 4) are illustrated in red, while the incident, reflected and transmitted wavevectors are shown in blue. The Y-axis that points inside the XZ plane and the magnetic fields( all of them having the Y direction) are not depicted. For the practical application of Fresnel’ s formulas, the steps are as follows: i) computing sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ii) computing cos ðh t Þ ¼ i r jj Ejj r
¼ n i cos h t � n t cos h i
E jj i n 2 i sin 2 ðh i Þ � 1; ð6Þ n 2 t
; t jj Ejj t ¼ n i cos h t þ n t cos h i
iii) computing
E jj t t jj
X
X cos h t
E jj
¼ t jj; t jj
cos h Z
i
X i
E jj i
E jj t
Z
E jj i
2n i cos h i n i cos h t þ n t cos h i
;
sin h t
¼ t jj: sin h i
Z
To use equation( 8), it is important to note that the two trigonometric quotients are complex quantities. For the reflected beam( see Fig. 1), it is clear that
r jj
X
¼ Ejj rj X ¼ r
E jj i j jj; r jj
Z
¼ Ejj rj Z
¼�r
E jj X i j jj: ð9Þ
Z In summary, E jj
; E jj
; E jj
X Z X
; E jj
Z
; E jj; E jj are the
X i i r r t t
Z ð7Þ
ð8Þ
complex amplitudes of the waves for the incident, reflected, and transmitted electric fields at the interface for the