The refractive indices of material of wavelength 5090\text{ \AA}, 5340\text{ \AA} and 5890\text{ \AA} are equal to 1.64, 1.640 and 1.630 respectively. Estimate the phase group velocities of light near \lambda = 5340\text{ \AA}.

An interference pattern is obtained by using two coherent sources of light, and the intensity variation is observed to be \pm 10\% of the average intensity. Determine the relative intensities of the interfering sources.

Define coherent length. A helium-neon laser emits radiation at wavelength \lambda = 632.8\text{ nm} with \Delta\lambda = 2\text{ pm}. Calculate the coherence wavelength.

Give an account of the origin of optical activity in eqartz crystal. A wafer of crystalline eqartz of thickness. 2.945 \times 10^{-5}\text{ m} is used to change a beam of linearly polarised light (\lambda = 589\text{ nm}) into circularly polarised light. Find the difference in refractive index for the two waves in the crystal, assuming this to be minimum thickness that will produce the effect.

Explain the general principle of laser action. What do you mean by population inversion ? Discuss the transitions involved in the ruby laser. A pulsed laser is rate at 10\text{mw}. It generates 3\text{ ns} wide pulses at frequency 500\text{ Hz}. Compute the instaneous power in the pulse.

A vibrating source is moving in a medium with speed V_s which is greater than the speed V of propagation of the wave in the medium. Apply Huygens principle to show that a conical wave front of half angle \sin^{-1}\left(\frac{V}{V_s}\right) is generated.

A ruby laser produces a beam of light of wavelength 0.3\text{ \AA} with a circular cross-section 1\text{ cm} in diameter. Calculate the diameter of this beam at a distance of 1000\text{ kilometers}.

Give an outline of Fresnel's explanation of optical rotation. How does optical rotation due to a material vary with \lambda? For an optically active material the difference between the refractive indices for right-handed and left-handed vibrations (\mu_R - \mu_L) for \lambda = 4500\text{ \AA} is 12 \times 10^{-5}. Estimate the optical rotation caused by 1\text{mm} thick plate in light of 1 = 4500\text{ \AA}. [Assume (m_R - m_L) as independent of \mu.]

Show that the particle velocity in case of a plane progressive wave is given by \frac{\partial y}{\partial t} = -V \frac{\partial y}{\partial x} (where V is the wave velocity). Hence obtain the differential equation of wave motion.

Give a mathematical analysis of forced vibration and hence explain the phenomenon of amplitude resonance.

A particle executing simple harmonic motion, has an acceleration (\pi^2/3)\text{ cm/sec}^2 when the displacement is 1\text{ cm}. Determine the period.

A diffraction grating with 3 \times 10^4 lines is used in the second order in the range of wavelength 6000\text{\AA}. Find the smallest \Delta\lambda it can resolve.

Distinguish between Fresnel and Fraunhofer classes of diffraction of light. Discuss the theory of a plane grating and hence find an expression for the angular dispersion of a plane grating.

State Rayleigh Criterison for limit of resolution. Show that \frac{I_{\text{middle}}}{I_{\text{max}}} = \frac{8}{\pi^2}

Define geometrical resolving power of an optical instrument. Derive the expression for the resolving power of a microscope and comment on the relation between resolving power and magnifying power, if any exists.

A soap film of refractive index 1.33 is illuminated with light of different wavelength at an angle of 45^\circ. There is complete destructive interference for \lambda = 5890\text{\AA}. Find thickness of the film.

Write a note on He-Ne laser.

Deduce the possible thickness of a quarter wave plate of quartz which is to be used for sodium light of wavelength 5890 \text{\AA}, (\mu_o = 1.558, \mu_e = 1.486.)

Write a note on Purity of spectral lines and coherence length.

Deduce an expression for the diffraction waves at single slit and discuss how it is related with a Fourier transform.