J. Eur. Opt. Society-Rapid Publ. 21, 21( 2025) 209
Figure 3. Electric fields( in red) at the origin of the coordinate axes for two different instants of time. First, for the same time as in Figure 2 and, second, for the time T / 18 later( being T the time period). Thus, we see that both E~ i ð0Þ; E~ r ð0Þ get shorter, making E~ i ð0Þþ E~ r ð0Þ to turn clockwise. Similarly, E~ t ð0Þ gets shorter and turns counterclockwise. The red dots show the tips of the electric fields for a full time period T.
... the effective depth of penetration being... of the order of a wavelength”. However, this statement is somewhat misleading, as when the incidence angle is h i = 55 °( for z n i = 1.5, n t = 1.0, i. e., n = 1.0 / 1.5), the ratio in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 sin the exponent is multiplied by 2p 2 h i
� 1 4:49. However, if the incidence angle were h 2 i = 41.9 °( the critical angle
n
being h C = 41.8 °), we would obtain the following factor multiplying inside the exponential: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2p 2 h i
� 1 0:37: ð20Þ n 2
This result shows that the penetration depth at an incidence of 41.9 ° is more than an order of magnitude greater than at 55 °. In fact, it is evident that the value given by expression( 20) approaches zero as the incidence angle nears 8
><
>: t jj
cos h t
X
¼ t jj ¼ 0:55 exp 2pi cos h i
t jj
sin h t
Z
¼ t jj ¼ 0:55 exp 2pi sin h i
the critical value h i? h C = 41.8 °. Thus, itisnotentirely accurate to say that evanescent waves always decay rapidly, as their decay rate is highly sensitive to the incidence angle.
3.2 Absorbing substrate
As a second example for illustrating the behavior of the fields, consider an interface between two media with refractive indices n i = 1.0, n t = 0.46 + 3.13i( corresponding approximately to brass), and an incidence angle of h i = 40 °. Using equations( 6)–( 9), wefind:
r jj ¼ 0:897 exp 2pi
�134:7 360
See Equation( 22) bottom of this page �60:28 1: 02 exp ½ 2pið0: 33 = 360Þ Š
�59:95 ¼ 0:737 exp 2pi
; 360
0:766
360 �60:28 0: 20 exp ½ 2pið�81: 6 = 360ÞŠ �141:9
¼ 0:175 exp 2pi:
360 0:643
360
; ð21Þ
ð22Þ