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Figure
4. Electric fields( in red) at the origin of the coordinate axes, as in Figure 2. We see( shown separately) the vectors E~ i ð0Þ; E~ r ð0Þ; E~ t ð0Þ; E~ i ð0Þþ E~ r ð0Þ corresponding to the exact time where the incident(~ E i ð0Þ) field is maximum. The red dots show the tips of the electric fields for a full time period T. In contrast with Figure 3, now the ellipse in medium“ i” is described counterclockwise, while the ellipse in medium“ t” rotates clockwise.
The phases in( 21)–( 22) are neither 0 ° nor 180 °. Asbefore, the detailed calculations for the three complex quantities r jj; t jj
X
; t jj
Z have been expressed with phase angles written in degrees in the argument of the complex exponentials.
Figure 4 depicts a snapshot at the instant when the incident electric field reaches its maximum. Figure 4 illustrates the incident and reflected electric fields, along with the resulting electric field in medium“ i” and the transmitted electric field. As before in Figure 3, the dots correspond to the tips of the electric field vectors over a full time period.
We could demonstrate now that the ellipse in medium“ i” is described rotating counterclockwise, while the ellipse in medium“ t” is turned clockwise. As expected from basic theory, the tangential component of the electric field remains continuous across the interface, whereas the normal component does not. It is important to note that in this case, where the second medium is absorbing, the wave in medium“ t” is inhomogeneous and not purely transverse [ 11, 12 ]. The direction of phase propagation in medium
“ t” is indicated and labeled as Reð ~ k t Þ. The decay rate would correspond to Imð ~ k t Þ, a vector oriented in the upward normal direction. To fully characterize the inhomogeneous transmitted wave, three key parameters are required:
( i) the direction of phase propagation,( ii) the wavelength, and( iii) the rate of decay of the wave( as we define by( 19)). The theoretical framework for these calculations is provided in [ 10 ], while the numerical procedures are elaborated upon in Appendix.
The computed results, assuming k i = 587.0 nm, are the following: h t ¼ 54:96; k t ¼ 747:7nm; ðfrom ðA: 17ÞÞ; ð23Þ ðfrom ðA: 16ÞÞ; ð24Þ
exp �0:714 z
; ðfrom ðA: 15ÞÞ: ð25Þ k t
4 Summary and conclusions
We have shown how Fresnel formulas can be applied to a general isotropic interface( transparent incident medium and absorbing second medium) by employing our equations( 8)–( 12), which conveniently represent the fields within a unified coordinate system.