Question 373: Ray Optics

Question

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here, z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices n
1
and n
2
(> n
1
), as shown in the figure. A ray of light is incident with angle $	heta$
i
from medium 1 and emerges in medium 2 with refraction angle $	heta$
f
with a lateral displacement $l$.
Which of the following statement(s) is (are) true?
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l0d6yw0w/ecbd5642-e23b-4ef8-b8fd-2eccfa594c99/0f6c82f0-9c27-11ec-823a-13aee278b549/file-1l0d6yw0x.png?format=png)

Accepted Answer

From Snell's law,
$
\begin{aligned}
n_1 \sin 	heta_i=n\left(z_1ight) \sin 	heta_{z_1}=n\left(z_2ight) \sin 	heta_{z_2} & =\ldots \ldots \\
\ldots & =n(d) \sin 	heta_d=n_2 \sin 	heta_f
\end{aligned}
$
The lateral displacement of a ray by a parallel slab depends on the angle of incidence $	heta_i$, refractive index $n_1$, slab thickness $d$ and the refractive index of the slab $n$ (it is independent of $n_2$ because $l$ is measured before the start of medium 2, there is no effect of $n_2$ on $l$ ). Consider the parallel slab shown in the figure.
By Snell's law, $n_1 \sin i_1=n \sin r_1$ and $n \sin i_2=$ $n_2 \sin r_2$. By geometry, $i_2=r_1$ and the lateral displacement is given by
$
l=d \frac{\sin r_1}{\cos r_1}=\frac{d n_1 \sin i_1}{\sqrt{n^2-n_1^2 \sin ^2 i_1}} .
$
Hence, $l$ is independent of $n_2$ but it is dependent of $n(z)$.
![](https://app-content.cdn.examgoal.net/fly/@width/image/1li47pnd2/f5b23999-bf12-4756-bf4e-ee7e7bf9dd90/ce8e4260-fb92-11ed-b649-11dfbee6d258/file-1li47pnd3.png?format=png)

Answer

$l$ is dependent on n(z)

Answer

n
1
sin$	heta$
i
= (n
2
$-$ n
1
) sin$	heta$
f

Answer

n
1
sin$	heta$
i
= n
2
sin$	heta$
f

Answer

$l$ is independent of n
2

Ray Optics

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