Wave Optics

$I = {I_0}{\cos ^2}\theta $

Intensity of polarized light $ = {{{I_0}} \over 2}$

$ \Rightarrow $ Intensity of untransmitted light $ = {I_0} - {{{I_0}} \over 2} = {{{I_0}} \over 2}$

$I = {I_0}{\left( {{{\sin \theta } \over \theta }} \right)^2}$

and $\theta = {\pi \over \lambda }\left( {{{ay} \over D}} \right)$

For principal maximum y = 0

$ \therefore $ $\theta $ = 0

When $\theta = 0$, the intensity formula becomes :

$I = I_0\left(\frac{\sin \theta}{\theta}\right)^2 = I_0\left(\frac{\sin 0}{0}\right)^2$

Here, we have an indeterminate form $\frac{0}{0}$. However, by taking the limit of the expression as $\theta$ approaches 0, we find that :

$\lim\limits_{\theta \to 0} \left(\frac{\sin \theta}{\theta}\right) = 1$

Therefore, the intensity at the principal maximum remains the same :

$I = I_0\left(\frac{\sin \theta}{\theta}\right)^2 = I_0(1)^2 = I_0$

Hence intensity will remain same.

${y \over D} \ge 1.22{\lambda \over d}$

$ \Rightarrow D \le {{yd} \over {\left( {1.22} \right)\lambda }}$

$ = {{{{10}^{ - 3}} \times 3 \times {{10}^{ - 3}}} \over {\left( {1.22} \right) \times 5 \times {{10}^{ - 7}}}}$

$ = {{30} \over {61}} \approx 5m$

$\therefore$ ${D_{\max }} = 5m$

The angle of incidence for total polarization is given by

$\tan \theta = n \Rightarrow \theta = {\tan ^{ - 1}}n$

Where $n$ is the refractive index of the glass.

For constructive interference $d\,\sin \theta = n\lambda $

Given $d = 2\lambda \Rightarrow \sin \theta = {n \over 2}$

$n = 0,1, - 1,2, - 2$ hence five maxima are possible.

To observe the phenomenon of interference, you need two sources that emit radiation of the same frequency and have a definite phase relationship. This is because interference is a result of the superposition of waves, which requires the waves to be coherent. Coherence is achieved when the waves have a constant phase difference, which implies that they are of the same frequency and have a definite phase relationship (a phase relationship that does not change with time).

So, the correct answer is :

Option D : of the same frequency and having a definite phase relationship.