Complex Reflection and Transmission Coefficient in oblique incidence

[baby_1]

Homework Statement

Complex Reflection and Transmission Coefficient in oblique incidence

Relevant Equations

Snell equations

Hello,
Something has made me confused after studying the Snell equations these days. Regarding the Balanis Advanced engineering electromagnetic( the pages have been attached), and based on that the reflection and transmission coefficient can be complex I need to rewrite the (5-23a) again:
$$Cos(\theta _{i})e^{-j\beta _{1}xSin(\theta _{i})}+\Gamma Cos(\theta _{r})e^{-j\beta _{1}xSin(\theta _{r})}=TCos(\theta _{t})e^{-j\beta _{2}xSin(\theta _{t})}$$
Due to $\Gamma=a+jb$ and $T=c+jd$
$$Cos(\theta _{i})e^{-j\beta _{1}xSin(\theta _{i})}+\sqrt{a^2+b^2} Cos(\theta _{r})e^{-j\beta _{1}xSin(\theta _{r})+jtan^{-1}(\frac{b}{a})}=\sqrt{c^2+d^2}Cos(\theta _{t})e^{-j\beta _{2}xSin(\theta _{t})+jtan^{-1}(\frac{d}{c})}$$
so it is not easy to say that just (5-24b) equation (Snell's law of refraction) should be correct because of the phase of reflection and transmission coefficients. They can change the equality. How do we count on the Snell equation when these parameters are complex ( as Balanis mentioned that)

 

[NateD]

?Thank you. A:The phase of the reflection and transmission coefficients is not important for Snell's law of refraction. The only thing of importance is the ratio of the sines of the incident and refractive angles. The phase is essentially a constant, and does not affect the value of the ratio.