Question 54: Waves Optics

Question

A double slit setup is shown in the figure. One of the slits is in medium 2 of refractive index $n_{2}$. The other slit is at the interface of this medium with another medium 1 of refractive index $n_{1}\left(
eq n_{2}ight)$. The line joining the slits is perpendicular to the interface and the distance between the slits is $d$. The slit widths are much smaller than $d$. A monochromatic parallel beam of light is incident on the slits from medium 1. A detector is placed in medium 2 at a large distance from the slits, and at an angle $	heta$ from the line joining them, so that $	heta$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.

Which of the following statement(s) is(are) correct?

Accepted Answer

$
	an 	heta  =\frac{\mathrm{AB}}{d}$
And, $
\mathrm{AB}  =d 	an 	heta $ ........ (i)
$
\mathrm{BC}  =\mathrm{AB} \sin \alpha $
$
 =d 	an 	heta \sin \alpha $
$
\mathrm{AD}  =\mathrm{AB} \sin 	heta
$
Path difference in vacuum
$$
\begin{aligned}
\Delta x & =n_1 \mathrm{BC}-n_2 \mathrm{AD} \\
\Delta x & =n_1(\mathrm{AB}) \sin \alpha-n_2(\mathrm{AB} \sin 	heta)
\end{aligned}
$$
Hence, $\mathrm{AB}\left(n_1 \sin \alpha-n_2 \sin 	hetaight)=0$

Answer

The phase difference between the two rays is independent of $d$.

Answer

The two rays interfere constructively at the detector.

Answer

The phase difference between the two rays depends on $n_{1}$ but is independent of $n_{2}$.

Answer

The phase difference between the two rays vanishes only for certain values of $d$ and the angle of incidence of the beam, with $	heta$ being the corresponding angle of refraction.

Waves Optics

Choose the correct answer: