I am trying to solve a problem in which light is normally incident on a material of refractive index n which is coated with an anti-reflective coating of refractive index $n^{\frac 1 2}$ and thickness equal to $\frac 1 4 \lambda$ ($\lambda$ being the wavelength). I need to show that under these conditions there is no reflected wave leaving the surface of the anti-reflective coating. I am taking the wave to be travelling along the z-axis, with the coating and material on the xy-plane.
The boundary conditions for interfaces between the materials are: $$\epsilon_1 E^{\perp}_1= \epsilon_2 E^{\perp}_2$$ $$B^{\perp}_1= B^{\perp}_2$$ $$\vec E^{\parallel}_1= \vec E^{\parallel}_2$$ $$\frac 1 {\mu} \vec B^{\parallel}_1=\frac 1 {\mu} \vec B^{\parallel}_2$$
In order to do this I believe I need to show that the wave reflected from the anti-reflective coating interferes destructively with one transmitted through the coating after having been reflected from the inner material.
So I would start with expressions for the electric and magnetic fields incident on the coating:
$\tilde {\vec E_I} (z,t) = \tilde E_{0_I} e^{i(k_1z- \omega t)} \hat x$
$\tilde {\vec B_I} (z,t) = \frac 1 v \tilde E_{0_I} e^{i(k_1z- \omega t)} \hat y$
This would then generate expressions for reflected and transmitted waves in the coating (to keep it shorter I'll only show the electric field related expressions):
$\tilde {\vec E_{R_1}} (z,t) = \tilde E_{0_{R_1}} e^{i(-k_1z- \omega t)} \hat x$
$\tilde {\vec E_{T_1}} (z,t) = \tilde E_{0_{T_1}} e^{i(k_2z- \omega t)} \hat x$
I can then apply the boundary conditions at the interface between air and the coating to relate the expressions: $$ \tilde {\vec E_{R_1}} + \tilde {\vec E_{T_1}} = \tilde {\vec E_I}$$
$\tilde {\vec E_{T_1}}$ is then incident on the inner material of refractive index n, again reflecting and transmitting:
$\tilde {\vec E_{R_2}} (z,t) = \tilde E_{0_{R_2}} e^{i(-k_2z- \omega t)} \hat x$
$\tilde {\vec E_{T_2}} (z,t) = \tilde E_{0_{T_2}} e^{i(k_3z- \omega t)} \hat x$
Boundary conditions can again be applied to relate expressions at the interface between the coating and the inner material: $$\tilde {\vec E_{R_2}} +\tilde {\vec E_{T_2}} =\tilde {\vec E_{T_1}}$$
Now is where I start to struggle a bit. Once the wave reflects from the inner material I suppose I need to add another reflection and transmission between the coating and air media like so: $$\tilde {\vec E_{R_2}} +\tilde {\vec E_{R_3}} =\tilde {\vec E_{T_3}}$$ Where $\tilde {\vec E_{T_3}}$ is the wave leaving the surface of the anti-reflective coating. At this point it seems like I have more variables than I can solve for with my equations, and thats not even including the magnetic field analysis. I think I must be overcomplicating the problem or missing some key simplification so I'd appreciate it if someone could give me some advice. Thank you!
