Question 12: Ray Optics

Question

A monochromatic light wave is incident normally on a glass slab of thickness $d$, as shown in the figure. The refractive index of the slab increases linearly from $n_1$ to $n_2$ over the height $h$. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?

Accepted Answer

At point A on one end of the slab, the wavefront takes a time $$ t $$ to reach the other end. The time $$ t $$ can be expressed as the ratio of the distance traveled, $$ d $$, to the speed of the wave in the medium, which is $$ \frac{c}{n_1} $$ (where $$ c $$ is the speed of light in vacuum and $$ n_1 $$ is the refractive index of the medium at A). Therefore, the time taken $$ t $$ is given by $$ t = \frac{d}{\frac{c}{n_1}} = \frac{n_1 d}{c} $$.
Simultaneously, consider the wavefront at point B, located in a different medium with a refractive index $$ n_2 $$. In the same time $$ t $$, the wavefront at B travels a distance $$ s_1 $$. The distance $$ s_1 $$ can be calculated as the product of the speed of light in the medium at B, $$ \frac{c}{n_2} $$, and the time $$ t $$, yielding $$ s_1 = \frac{c}{n_2} 	imes \frac{n_1 d}{c} = \frac{d n_1}{n_2} $$.
Once the wavefront at B reaches the end of the slab, an additional time $$ t_1 $$ elapses. The time $$ t_1 $$ can be determined by considering the remaining distance the wavefront must travel in the slab, which is $$ d - s_1 $$, and dividing it by the speed of light in the medium at B. This gives us $$ t_1 = \left(\frac{d - s_1}{c}ight) n_2 = \left(d - \frac{d n_1}{n_2}ight) \frac{n_2}{c} = \frac{d(n_2 - n_1)}{c} $$.
During this time $$ t_1 $$, the wavefront at A will travel a distance $$ s_2 $$. The distance $$ s_2 $$ is simply the speed of light in vacuum multiplied by the time $$ t_1 $$, leading to $$ s_2 = c t_1 = d(n_2 - n_1) $$.
Finally, the angle $$ 	heta $$ can be determined using the tangent function, where $$ 	an 	heta $$ is the ratio of the difference in distances traveled by the wavefronts in the two media to the height $$ h $$ of the slab.
Therefore, $$ 	an 	heta = \frac{(n_2 - n_1) d}{h} $$.

Answer

It will deflect up by an angle $	an ^{-1}\left[\frac{\left(n_2^2-n_1^2ight) d}{2 h}ight]$.

Answer

It will deflect up by an angle $	an ^{-1}\left[\frac{\left(n_2-n_1ight) d}{h}ight]$.

Answer

It will not deflect.

Answer

The deflection angle depends only on $\left(n_2-n_1\right)$ and not on the individual values of $n_1$ and $n_2$.

Ray Optics

A monochromatic light wave is incident normally on a glass slab of thickness d, as shown in the figure. The refractive index of the slab increases linearly from n_1 to n_2 over the height h. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?

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