An electromagentic wave falls normally on the boundary between two isotropic dielectrics with refractive indies `n_(1)` and `n_(2)`. Making use of the continuity condition for the tangential components, `E` and `H` across the boundary, demonstrate that at the interface teh electric field vector `E`
(a) of the transmitted wave experienced no Phase jump,
(b) of the reflected wave is subjected to the phase jump equal to `pi` refractive from a medium of higher optical density.

Explanation

(a) Suppose the vector `oversetrarr(E),oversetrarr(E'), oversetrarr(E''),` correspond to the incident, reflected and the transmitted wave. Due to the continuity of the tangential component of the electric filed across the interface, it follows that
`E_(tau) + E'_(tau) = E''_(tau).....(1)`
where the subscrip `tau` means tangential.
The energy flux density is `oversetrarr(E) xx oversetrarr(H) = oversetrarr(S)`.
Since `H sqrt(mu mu_(0)) = Esqrt(epsilon epsilon_(0))`
`H = E sqrt((epsilon_(0))/(mu_(0))) sqrt(epsilon//mu) = n sqrt((epsilon_(0))/(mu_(0)))E`
Now `S_(~) nE^(2)` and since the light is incident normally
`n_(1)E_(tau)^(2) = n_(1)E'_(tau)^(2) + nE''_(tau)^(2).....(2)`
or `n_(1) (E_(tau)^(2) - E'_(tau)^(2)) = n_(2)E''_(tau)^(2)`
so `n_(1) (E_(tau) - E'_(tau)) = n_(2)E''_(tau).........(3)`
so `E''_(tau) = (2n_(1))/(n_(1) + n_(2))E_(tau)`
Since `E''_(tau)` and `E_(tau)` have the same sign, there is no phase change involved in this case.
(b) from (1)&(3)
`(n_(2) + n_(1)) E'_(tau) + (n_(2) - n_(1)) E_(tau) = 0`
or `E'_(tau) = (n_(1) - n_(2))/(n_(1) + n_(2)) E_(tau)`.
If `n_(2) gt n_(1)`, then `E'_(tau) & E_(tau)` have opposite signs. Thus the reflected wave has an abrupt change of phase by `pi` if `n_(2) gt n_(1)` i.e on reflection from the intenface between two media when light is incident from the rarrer to denser medium.