Prob. $7(20$ pts$)$ A uniform plane wave in the first medium $({}_0,lo{n}_0)$ is incident with an incident angle ${}_{\text{i}\text{n}\text{c}}=45$ onto a second medium $({}_0,2lo{n}_0)$ of a thickness of $5$ cm and further backed by a third medium $({}_0,4lo{n}_0).$ Assume the operating frequency $f=1GHz$ and the electric field is in the hat$(y)$ direction. The problem can be solved by an equivalent transmission line problem, as shown in the figure.

a$.$ Determine the angle ${}_2$ in medium $2,$ and ${}_3$ in medium $3.$

b$.$ This problem can be solved by converting it into a corresponding transmission line problem, as indicated in the figure. What is the corresponding $Z_L$ shown in the figure.

c$.$ Write down the characteristic impedances, $Z_1$ and $Z_2.$ Also, what is the corresponding wavelength in transmission line #$2?$

d$.$ Using Smith Chart to find the input impedance at location $A-A^{\text{'}}.$

e$.$ The reflection coefficient of the problem.

f$.$ Finally, the reflected electric and magnetic fields in the first medium.