J. Eur. Opt. Society-Rapid Publ. 21, 21( 2025) 213
For finding the Z component in medium“ t” we have to use the algebraic formula for the complex square root in( A. 9) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � k z t
¼ ðk 0 ðn t þ ij t ÞÞ 2 � k x 2 � � i
¼ Re k z t þ iIm k z t rffiffiffiffiffiffiffiffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffi!
( A þ B B � A A ¼
¼ k 0 þ i; q
n 2 t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � j2 t � n2 i sin2 ðhÞ 2 2 B ¼ A 2 þ ð2n t j t Þ 2: ðA: 15Þ
Finally, the wavelength k t is obtained from Reð ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
� k t Þ ¼ k x 2 � i þ Reðk z t Þ
2
¼ 2p k t ðA: 16Þ and the propagation angle in medium“ t” is! u ¼ tan �1 k x �i
Re k z: ðA: 17Þ t
Since the value k t in medium“ t”( defined as the distance for the periodicity of the phase) is found from( A. 16), this wavelength changes with the incidence angle and it is not a fixed quantity depending only on the refractive index of medium“ t”, likefor transparent materials.
The procedure for computing k z t given by( A. 15) is very convenient since is general and automatically gives the complete answer: if Imðk z t Þ¼0( since Imð ~ k t Þ has only Z component as explained) it means that Imð ~ k t Þ0 and the transmitted wave is homogeneous. Besides, the value of angle u is also obtained from( A. 17), including evanescent waves as particular case.
Thus, for all practical isotropic interfaces from a transparent medium“ i” to any material in medium“ t”, withtheaboveprocedure summarized in equations( A. 13)–( A. 17) we will know all the wavevectors for the waves, besides the incident one ð ~ k i Þ. For the transmitted wave Reð ~ k t Þ defines the wavelength and propagation direction and Imð ~ k t Þ represents the decay of the transmitted inhomogeneous wave.