Can D_1 and D_2 lines of sodium light (\lambda_{D_1}=5890\mathring{\mathrm{A}}\ \text{and}\ \lambda_{D_2}=5896\mathring{\mathrm{A}}) be resolved in second-order spectrum if the number of lines in the given grating is 450? Explain.

Obtain an expression for the resolving power of a grating explaining the Rayleigh's criterion of resolution.

The refractive indices of core and cladding in a step index optical fiber are 1.52 and 1.48 respectively. The diameter of the core is 30\,\mu\mathrm{m}. If the operating wavelength is 1.3\,\mu\mathrm{m}, calculate the V parameter and the maximum number of modes supported by the fiber.

In Michelson interferometer, 100 fringes cross the field of view when the movable mirror is displaced through 0.029\,\text{mm}. Calculate the wavelength of the light source used.

Explain the principle of (i) induced absorption, (ii) spontaneous emission and (iii) stimulated emission. Show that the ratio of Einstein's coefficients is given by \frac{A}{B}=\frac{8\pi h\nu^{3}}{c^{3}}

What is the role of an optical resonator in a laser? Why does one prefer curved mirrors instead of plane mirrors in designing an optical resonator?

Find out the phase and group velocities of a radio wave of frequency \omega = \sqrt{2}\omega_p in the ionosphere (as a dielectric medium) of refractive index n = \sqrt{1 - \frac{\omega_p^2}{\omega^2}}. Here, \omega_p is the ionospheric plasma frequency.

A convex lens of focal length 20\,\mathrm{cm} is placed after a slit of width 0.5\,\mathrm{mm}. If a plane wave of wavelength 5000\,\mathring{\mathrm{A}} falls normally on the slit, calculate the separation between the second minima on either side of the central maximum.

Using matrix method, find out the equivalent focal length for a combination of two thin lenses of focal lengths f_1 and f_2 separated by a distance a.

Using the concept of Einstein’s A and B coefficients for a two-level atomic system under thermal equilibrium, determine the ratio of the number of atoms per unit volume in the two levels experiencing spontaneous and stimulated emission. How does the principle of population inversion lead to the gain mechanism in the active medium of the laser?

Find out the angle between the reflected and refracted rays when a parallel beam of light is incident on a dielectric surface at an angle equal to the Brewster's angle. Explain how do you use this concept to produce linearly polarized light.

How does holography differ from conventional photography? What are the requirements for the formation and reading of a hologram?

In a Young double slit experiment, the first bright maximum is displaced by y=2\,\text{cm} from the central maximum. If the spacing between slits and distance from the screen are 0.1\,\text{mm} and 1\,\text{m} respectively, find the wavelength of light.

Explain the working principle of a 3-level laser with a specific example. Comment on why the third level is needed.

What is the physical significance of Einstein's A-coefficient? Explain why it is more difficult to achieve Lasing action at X-ray wavelength than at infra-red wavelength.

Show that the group velocity is equal to particle velocity. Also prove that the group velocity of the photons is equal to c, the velocity of light.

For a multimode step index optical fibre, the core refractive index is 1.5 and fractional index difference is 0.001. Calculate the pulse broadening for 1\,\mathrm{km} length of the fibre. Over a length of 2\,\mathrm{km} of the fibre, calculate the minimum pulse separation that can be transmitted without overlap.

In a certain engine, a piston undergoes vertical SHM with an amplitude of 10\,\mathrm{cm}. A washer rests on the top of the piston. As the motor is slowly speeded up, at what frequency will the washer no longer stay in contact with the piston?

Considering a plane transmission diffraction grating, where d is the distance between two consecutive ruled lines, m as the order number and \theta as the angle of diffraction for normal incidence, calculate the angular dispersion \frac{d\theta}{d\lambda} for an incident light of wavelength \lambda.

Explain why information carrying capacity of an optical fibre can be enhanced by reducing the pulse dispersion. How does one minimize pulse dispersion using a graded index optical fibre?