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 $n_1\left(\neq n_2\right)$. 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 $\rightarrow$
$ \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 \phi=0$