208
J. Eur. Opt. Society-Rapid Publ. 21, 21( 2025)
Figure
2. Electric fields( in red) at the origin of the coordinate axes. We see( although shown apart) 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 wavevectors are also shown in blue.
Since the phases of r jj; t jj
X
; t jj
Z are not 0 ° or 180 °, the complex amplitudes for the reflected and transmitted waves are neither in phase nor opposite with respect to the incident one. In fact, the three complex quantities r jj; t jj
X
; t jj
Z have been expressed with phase angles in degrees in the argument of the complex exponential, clearly illustrating the differences relative to the incident wave.
Figure 2 shows a snapshot of the relevant vectors at the origin of the coordinate system at the instant when the incident electric field(~ E i ð0Þ) reaches its maximum. The coordinate axes are the same as for Figure 1, but now only the interface is depicted( as a horizontal line), with its normal direction represented by a dotted line for clarity. The directions of the incident, reflected, and transmitted waves are shown in blue. While these vectors are plotted slightly apart for visual clarity, it should be noted that all electric field vectors( displayed in red) correspond to a common point: the origin of the coordinates system. According to equation( 17), the amplitude of the reflected electric field equals that of the incident field. However, since the reflected field is not in phase with the incident field at its maximum, the length of the reflected E-vector appears shorter. The fields E~ i ð0Þ; E~ r ð0Þ; E~ t ð0Þ; E~ i ð0Þþ E~ r ð0Þ are shown. It can be observed that the tangential( horizontal) component of the electric field( the X-coordinate of the E~ t ð0Þ; E~ i ð0Þþ E~ r ð0Þ vectors) remains continuous across the interface between the two media, while the normal component( their Z-coordinate) exhibits discontinuity.
It is worth noting that, unlike the case of a transparent interface below the critical angle, the incident and reflected electric fields here are neither in phase nor in opposite phase. Consequently, Figure 2 provides valuable insight into the not-intuitive behavior of the fields at the interface.
Figure 3 illustrates the same electric fields as shown in Figure 2, along with the electric fields at a moment 1 / 18 of a time period later. Comparing the two figures reveals that the incident and reflected electric fields of the homogeneous transverse waves in medium“ i” are both smaller, leading to a total electric field that rotates clockwise in the X-Z plane. Similarly, in medium“ t”, the electric field increases in magnitude, with its vector rotating also clockwise in Figure 3. For illustration, the dots forming ellipses represent the trajectories of the tips of the total electric field vectors in media“ i” and“ t” over a full period of the wave. As in Figure 2, we observe that the tangential components of the electric fields in medium“ i” and medium“ t” remain continuous across the interface, with their horizontal projections aligning perfectly. In contrast, the normal components exhibit discontinuity.
Despite this detailed depiction, critical information about the transmitted wave remains missing – specifically, its decay rate. Using equation( 55) from Section 1.5.4 of [ 2 ], the real exponential term governing the in-depth decay of the electric field is given by:
exp
� xz v 2 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sin 2 h i
� 1 ¼ exp �2p zk2 sin 2 h i � 1: ð19Þ n 2 n 2
Since v 2 and k 2 represent the speed and wavelength in the second medium, the final expression in( 19)( as indicated in [ 2 ]) suggests that“ the amplitude decreases very rapidly