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 $n_1\left(\ne n_2\right)$. 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 $\theta$ from the line joining them, so that $\theta$ 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?

• A. The phase difference between the two rays is independent of $d$.
• B. The two rays interfere constructively at the detector.
• C. The phase difference between the two rays depends on $n_1$ but is independent of $n_2$.
• D. The phase difference between the two rays vanishes only for certain values of $d$ and the angle of incidence of the beam, with $\theta$ being the corresponding angle of refraction.

Answer 1

Answer: (A), (B)

Optical path difference $\to$
$\Delta x=n_1(d\tan \theta )\sin \alpha -n_2(d\tan \theta )\sin \theta$
$=\left(n_1\sin \alpha -n_2\sin \theta \right)d\tan \theta$
$=0$
$\Rightarrow \Delta \varphi =0$