Write a short note on He-Ne laser.

A 3 MW laser beam which has a diameter of 1\text{ cm} is foccussed by a lens of focal length 5\text{ cm}. The wavelength of laser is 10,000\text{ \AA}. Calculate the intensity of the focal plane of the lens.

What is a quarter plate? A phase retardation plate of quartz has a thickness 0.1436\text{ mm}. Calculate the wavelength in visible region for which this plate will act as a quarter for plate. The refraction indices of quartz for ordinary and extraordinary rays are 1.5443 and 1.5533 respectively.

What is spatial coherence ? Considering Young's two slit experiment, prove that the distance between the slits must be sufficiently less than (\lambda/\theta) for obtaining fringes of good contrast. \lambda is the wavelength of light used and \theta is the angle subtended by the source at the slits.

Write a short note on Phase and group velocities in dispersive media.

A uniform rope of mass M and length L is hanging from a rigid support. A transverse wave travels along the rope. Calculate the speed of the wave at a distance x from the free end the time taken by the wave to travel the whole length.

The amplitude of damped oscillator of frequency 300\text{ Hz} reduces one-tenth of its initial amplitude after 3000 oscillations. Calculate the damping constant and the time in which its energy will reduce to one-tenth of its initial energy.

Write the equation of motion for an oscillator driven by a simple harmonically varying force. Obtain the condition for maximum energy transfer to the oscillator.

The potential energy for the interaction between two gas molecules can be represented by the function. \phi(r) = -\phi_0 \left[ 2 \left(\frac{r_0}{r}\right)^6 - \left(\frac{r_0}{r}\right)^{12} \right] Where \phi_0 and r_0 are constant. Prove that for small oscillations the frequency is given by \omega = \left( \frac{144 \phi_0}{m r_0^2} \right)^{1/2} Where m is mass of each molecule.

Write a short note on Fourier transformation.

What is meant by temporal and spatial coherence ? Show that the coherence length L = \lambda Q, where \lambda is the mean wavelength and Q represents the purity of a spectral line.

What is a quarter wave plate? Explain its use in the production and detection of circularly polarized light. For calcite for \lambda = 5472 \text{\AA}, \mu_O = 1.658 and \mu_E = 1.486. If the minimum thickness of a plate that can be cut from calcite is 30 \mu\text{ m}, what should be the minimum thickness for preparing a quarter wave plate?

Obtain an expression for the ratio of the probabilities of stimulated and spontaneous emissions. What do you infer from this relation? How is population inversion interpreted thermodynamically?

Describe the working principle of a three level solid state laser giving the transitions involved in the laser action.

Show that the interference fringes obtained in Young's two slit experiment are hyperbolic in shape. Under what conditions these are expected to appear straight?

A fine slit is illuminate by monochromatic light of wave-length 6000 \text{\AA}. A thin wire is placed parallel to the slit and the diffraction pattern is observed on a screen at a distance of 1 m from the wire. In the shadow of the wire equidistant fringes of thickness 1.5 mm are observed. How do you explain this observation ? Calculate he diameter of the wire also.

The angular separation between two distant stars is 1 arc-second. If the effective wavelength of light is 5500 \text{\AA}; what should be the diameter of the objective of a telescope so that the stars are just resolved?

Write a short note on Holography.

In a biprism experiment the fringe-width with lightly of wavelength \lambda = 5900\text{ \AA} is 0.43\text{ mm}. On introducing a mica sheet in the path of one of the interfering rays the central fringe shifts by 1.89\text{ mm}. If refractive index of mica is 1.59, calculate the thickness of the sheet.

Obtain the intensity pattern due to Fraunhofer diffraction at two parallel slits. Each slit has a width ‘a’ and the separation between the slits is d. How many interference fringes will appear in the central diffraction maximum, it d = 4a?