Refraction of Light Through a Glass Slab (Lateral shift or displacement)
Consider a rectangular glass slab of refractive index $n$ and thickness $t$.
Let a ray of light AO be incident at O such that $i$ be the incident angle.After refraction in the glass, the ray of light bends towards the normal NN' and goes along OB such that $r$ is the angle of refraction
According to the law of refraction,
$$n_g = \frac{\sin i}{\sin r} \quad \dots (i)$$
At point B, the ray of light again suffers a refraction and emerges out of the glass slab into the air along BC. For refraction at B, $r$ is the angle of incidence and $i$ is the angle of refraction also known as emergent angle.
Again, according to the law of refraction,
${ }^{g}n_{a} = \frac{\sin r_1}{\sin i'} \quad \dots (ii)$
We know that,
${ }^{a}n_{g} = \frac{1}{{ }^{g}n_{a}}$
Using eqns. (i) and (ii), we get,
$ \frac{\sin i}{\sin r} = \frac{1}{\frac{\sin r_1}{\sin i'}} = \frac{\sin i'}{\sin r_1}$
$\frac{\sin i}{\sin r} = \frac{\sin i'}{\sin r_1}$
or
$i = i'$
This shows that a ray of light incident obliquely on the parallel sided glass slab emerges out parallel to the incident ray.
However, the incident ray is laterally displaced.
Lateral Shift or Displacement :
The perpendicular distance between the direction of incident ray and the emergent ray is known as the lateral shift.
To calculate the lateral shift produced by the glass slab, draw BL perpendicular to the initial path OD of the incident ray.
The perpendicular distance BL = $d$ is the lateral shift.
From right angled triangle $\Delta$ BOL,
$\sin (i - r_{1}) = \frac{BL}{OB} = \frac{d}{OB}$
$d = OB \sin (i - r_{1}) \quad \dots (i)$
From $\Delta BON'$,
$\cos r = \frac{ON'}{OB} = \frac{t}{OB}$
$OB = \frac{t}{\cos r} \quad \dots (ii)$
Substituting the value of eqn. (ii) in eqn. (i), we get,
$d = \frac{t \sin (i - r_{1})}{\cos r} \quad \dots (iii)$
Note : Lateral shift will be maximum if $\cos r_{1} = 1 ( r_{1} = 0)$ and $\sin (i - r_{1}) = 1(i - r_{1} = 90^\circ$).$
$\therefore$ Maximum lateral shift = thickness of slab ($t$).