The optical properties of a medium are governed by the relative permittivity `(epsi_(r))` and relative permeability `(mu_(r))`. The refractive index is defined as `sqrt(mu_(r)epsi_(r))=n`. For ordinary material, `epsi_(r) gt 0 ` and `mu_(r) gt 0` and the positive sign is taken for the squre root.
In 1964, a Russian scientist V. Veselago postualted the existance of material with `epsi_(r) lt 0` and `mu_(r) lt 0` . Since, then such metamaterial have been produced in the laboratories and their optical properties studied. For such materials `n= - sqrt(mu_(r) epsi_(r))`. As light enters a medium of such refractive index the phases travel away from the direction of propagation.
(i) According to the description above show that if rays of light enter such a medium from air ( refractive index =1) at an angle theta in 2nd quadrant, then the refracted beam is in the 3rd quadrant.
(ii) Prove that Snell's law holds for such a medium.

Explanation

Let us assume that the given postulate is true, then two parallel ray would proceed as shown in the figure below.

(i) Let AB represent the incident wavefront and DE represent the refracted wavefront. All points on a wavelength must be in same phase and in turn, must have the same optical path length.
Thus, `-sqrt(epsi_(r)mu_(r)) AE=BC-sqrt(epsi_(r)mu_(r))CD`
`BC=sqrt(epsi_(r) mu_(r))(CD-AE)`
`BC gt 0, CD gt AE`
As showing that the postualte is reasonable. If however, the light proceeded in the same it does for ordinary material ( viz. in the fourth quadrat , Fig. 2)
Then, `-sqrt(epsi_(r)mu_(r)) AE=BC-sqrt(epsi_(r)mu_(r))CD`
`BC=sqrt(epsi_(r) mu_(r))(CD-AE)`
If `BC gt 0`, then `CD gt AE`
which is obvious from Fig. (i).
Hence, the postulate reasonable.
However, if the light proceeded in the sense it does for ordinary material, ( going from 2nd quadrant to 4 the quadrant ) as shown in Fig. (i)., then proceeding as above,
`-sqrt(epsi_(r)mu_(r)) AE=BC-sqrt(epsi_(r)mu_(r))CD`
or `BC=sqrt(epsi_(r) mu_(r))(CD-AE)`
As `AE gt CD`, therefore `BC lt 0` which is not possible. Hence, the given postulate is correct.
(ii) From Fig. (i)
`BC = AC sin theta_(i) `
and `CD-AE=AC sin theta_(r)`
As `BC=sqrt(mu_(r) epsi_(r))" " [CD-AE=BC]`
`:. AC sin theta_(i)=sqrt(epsi_(r) mu_(r)) AC sin theta_(r)`
or `(sin theta_(i))/(sin theta_(r))=sqrt(epsi_(r)mu_(r)=n`
Which proves Snell's law.