# oblique incidence

when a uniform plane wave  is incident obliquely (making an angle $\theta&space;_{i}$ other than $90^{o}$) to the boundary between the two media then it is known as oblique incidence.

Now consider the situation that is more general case  that is the oblique incidence.

In this case the EM wave (incident wave) not strikes normally the boundary. i.e,  the incident wave is not propagating  along any standard axes (like x,y and z).

Therefore EM wave is moving in a random direction then the general form is    $\overrightarrow{E}&space;=&space;E_{o}&space;\cos&space;(\omega&space;t-\overrightarrow{k}.\overrightarrow{r})$

it is also in the form $\overrightarrow{E}&space;=&space;E_{o}&space;\cos&space;(\overrightarrow{k}.\overrightarrow{r}-\omega&space;t)$.

then $\overrightarrow{k}&space;=&space;k_{x}\overrightarrow{a_{x}}+k_{y}\overrightarrow{a_{y}}+k_{z}\overrightarrow{a_{z}}$ is called the wave number vector (or) the propagation vector.

and $\overrightarrow{r}&space;=&space;x\overrightarrow{a_{x}}+y\overrightarrow{a_{y}}+z\overrightarrow{a_{z}}$ is called the position vector (from origin to any point on the plane of incidence) , then the magnitude of $\overrightarrow{k}$ is related to $\omega$ according to the dispersion.

$k^{2}&space;=&space;k_{x}^{2}+k_{y}^{2}+k_{z}^{2}&space;=&space;\omega&space;^{2}\mu&space;\epsilon$

$\overrightarrow{k}&space;X&space;\overrightarrow{E}&space;=&space;\omega&space;\mu&space;\overrightarrow{H}$.

$\overrightarrow{k}&space;X&space;\overrightarrow{H}&space;=&space;-\omega&space;\epsilon&space;\overrightarrow{E}$.

$\overrightarrow{k}&space;.&space;\overrightarrow{H}&space;=0$.

$\overrightarrow{k}&space;.&space;\overrightarrow{E}&space;=0$.

i.    $\overrightarrow{E},\overrightarrow{H}$ and $\overrightarrow{k}$ are mutually orthogonal.

ii.  $\overrightarrow{E}$  and  $\overrightarrow{H}$ lie on the plane $\overrightarrow{k}&space;.&space;\overrightarrow{r}&space;=&space;k_{x}x+k_{y}y+k_{z}z=constant$.

then the $\overrightarrow{H}$ field corresponding to $\overrightarrow{E}$ field is $\overrightarrow{H}&space;=&space;\frac{1}{\omega&space;\mu&space;}&space;(\overrightarrow{k}&space;X&space;\overrightarrow{E})&space;=&space;\frac{\overrightarrow{a_{k}}&space;X&space;\overrightarrow{E}}{\eta&space;}$.

Now choose oblique incidence of a uniform plane wave at a plane boundary.

the plane defined by the propagation vector $\overrightarrow{k}$ and a unit normal vector $\overrightarrow{a_{n}}$ to the boundary is called the plane of incidence.

the angle $\theta&space;_{i}$ between $\overrightarrow{k}$ and $\overrightarrow{a_{n}}$ is the angle of incidence.

both the incident and reflected waves are in medium 1 while the transmitted wave is in medium 2 .

Now,

$\overrightarrow{E_{i}}&space;=E_{i}\cos&space;(k_{ix}x+k_{iy}y+k_{iz}z-\omega&space;_{i}t)$

$\overrightarrow{E_{r}}&space;=E_{i}\cos&space;(k_{rx}x+k_{ry}y+k_{rz}z-\omega&space;_{r}t)$

$\overrightarrow{E_{t}}&space;=E_{t}\cos&space;(k_{tx}x+k_{ty}y+k_{tz}z-\omega&space;_{t}t)$.

the wave propagates

1.     $\omega&space;_{i}=\omega&space;_{r}=\omega&space;_{t}=\omega$.
2.     $k_{ix}&space;=&space;k_{rx}=k_{tx}=k_{x}$.
3.    $k_{iy}&space;=&space;k_{ry}=k_{ty}=k_{y}$.

(1) indicates that all waves are propagating with same frequency. (2) and (3) shows that the tangential components of propagation vectors be continuous.

$k_{i}&space;\sin&space;\theta&space;_{i}=k_{r}&space;\sin&space;\theta&space;_{r}$  implies  $k_{i}&space;=&space;k_{r}&space;=\beta&space;_{1}&space;=\omega&space;\sqrt{\mu&space;_{1}\epsilon&space;_{1}}$    since $\theta&space;_{r}=\theta&space;_{i}$.

$k_{i}&space;\sin&space;\theta&space;_{i}=k_{t}&space;\sin&space;\theta&space;_{t}$   implies $k_{t}&space;=\beta&space;_{2}&space;=\omega&space;\sqrt{\mu&space;_{2}\epsilon&space;_{2}}$.

$\frac{\sin&space;\theta&space;_{t}}{sin&space;\theta&space;_{i}}=\sqrt{\frac{\mu&space;_{1}\epsilon&space;_{1}}{\mu&space;_{2}\epsilon&space;_{2}}}$

now velocity $u=\frac{\omega&space;}{k}$.

then from Snell’s law       $r_{1}\sin&space;\theta&space;_{i}&space;=&space;r_{2}\sin&space;\theta&space;_{t}$,  where $r_{1}$ and $r_{2}$  are the refractive indices of the two media.

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