- #1

baby_1

- 159

- 15

- 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:

[tex]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})}[/tex]

Due to [itex]\Gamma=a+jb[/itex] and [itex]T=c+jd[/itex]

[tex]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})}[/tex]

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)

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:

[tex]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})}[/tex]

Due to [itex]\Gamma=a+jb[/itex] and [itex]T=c+jd[/itex]

[tex]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})}[/tex]

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)