Thin transparent films produce colours when a white light falls on them or they act as antireflecting coatings. Find the conditions for these characteristics.

(i) Give the equation of motions for a particle executing simple harmonic motion for undamped, damped and forced vibrations, and explain the terms.

(ii) Give a plot between the displacement and time for each case of the above and interpret.

(iii) What are the conditions for critical damping and resonance?

(i) When a monochromatic light passes through a single slit, dark and bright fringes are observed on a screen kept far away from the slit. Draw an amplitude and intensity pattern of the light on the screen.

(ii) Write an expression for intensity distribution and give the conditions for bright and dark fringes.

(iii) If the slit width is 0\cdot 3\text{ mm} and the distance between the screen and the slit is about 30\text{ cm}, then what is the angle of diffraction for the first minimum? (\lambda = 5 \times 10^{-5}\text{ cm})

(i) Explain the reason for pulse broadening due to intermodal and material dispersion. Deduce the relation of pulse broadening for intermodal dispersion in optical fiber. (ii) A step index fiber in air has a numerical aperture of 0\cdot 16, a core refractive index of 1\cdot 45 and a core diameter of 60~\mu\text{m}. Determine the normalized frequency for the fiber when light at a wavelength of 0\cdot 8~\mu\text{m} is transmitted. Also estimate the number of guided modes propagating in the fiber.

A parallel beam of light of wavelength 5890~\text{\AA} is incident at an angle of 30^\circ on a plane transmission grating with 15000 lines/inch. Find the highest order of spectrum that can be observed.

Discuss the properties of Cornu spiral. Show that the spiral can be used to obtain the intensity distribution in the Fresnel's diffraction pattern due to a straight edge.

(i) In a Michelson's interferometer, 100 fringes cross the field of view when the movable mirror is displaced through 2\cdot 894 \times 10^{-3}~\text{cm}. Calculate the wavelength of the monochromatic source of light. (ii) A shift of 200 fringes is observed when the movable mirror of a Fabry-Pérot interferometer is shifted by 0\cdot 0298~\text{mm}. Calculate the wavelength of the incident radiation.

Although the principle of operation of a basic LASER is based upon two energy levels, why does one need a 3-level or a 4-level scheme to achieve satisfactory lasing ? Explain your answer with special reference to a Ruby-laser.

Discuss absorption loss in an optical fibre comparing and contrasting the intrinsic and extrinsic absorption mechanisms.

State and explain Fermat's principle of extremum path and use the same to deduce the laws of reflection and refraction of light.

(i) What is Holography ? (ii) Show with simple diagrams, how a hologram is written and read using a laser. (iii) Mention some important applications of holography.

(i) Using the concept of spontaneous and stimulated emission of radiation, obtain the relation between Einstein's A and B coefficients. (ii) What is the physical significance of Einstein's A coefficient ? (iii) Justify why lasing action is much more difficult at X-ray frequency than in case of infrared frequency spectrum.

The vibrations of a string fixed at both ends is represented by the equation y = 2\sin\frac{\pi x}{3}\cos 50\pi t\ (m) If the above stationary wave is produced due to the superposition of two component waves of the same frequency, velocity and amplitude travelling in opposite directions, find (i) the exact equations of the displacements associated with the vibrations of the component waves and (ii) the distance between two consecutive nodes of the stationary wave.

Prove that the group velocity V_g of electromagnetic waves in a dispersive medium is given by V_g = \frac{c}{n + \omega\ dn/d\omega} where c is the velocity of light in vacuum and n is the refractive index of the medium for the angular frequency \omega of the waves.

State Huygen's principle. Using this principle and suitable diagram, show that this principle could lead to Snell's law of refraction.

A diffraction grating has 5000\text{ lines per cm}. For illumination at normal incidence, determine the dispersive power of the grating in the second order spectrum in the range of wavelengths around 500\text{ n.m.}

Discuss the principles of Fresnel's half period zone and explain how these are used in the construction of a zone plate. Show how the zone plate has several foci.

Plane polarized light of wavelength 6000\text{ \AA} is incident on a thin quartz plate cut with faces parallel to the optic axis. Given, \mu_0 = 1.544 and \mu_e = 1.553, calculate the following quantities : (i) The ratio of the intensities of the ordinary and the extraordinary light if the planes of vibration of two incident lights make an angle of 30^\circ with the optic axis. (ii) The minimum thickness of the plate for which ordinary and the extraordinary waves will combine to produce plane polarized light.

Obtain the equation of motion of a simple pendulum using Lagrangian formalism. Hence, obtain expression for the angular frequency of a simple harmonic oscillator.

Explain, with neat diagrams, what you mean by monochromatic spherical aberration with reference to paraxial focus, marginal focus and circle of least confusion.