Write down the one-dimensional harmonic oscillator differential equation under damping and its solution for the lightly damped condition, with the meanings of symbols. Determine the dependent energy in the lightly damped condition.
Prove that the group velocity V_g of electromagnetic waves in a dispersive medium with refractive index n(\lambda_0) at wavelength \lambda_0 is given by V_g=\frac{c}{n(\lambda_0)-\lambda_0\dfrac{dn(\lambda_0)}{d\lambda_0}} where c is the free space velocity of light. Find the time taken for the electromagnetic pulse to travel a distance D.
For calcite, the refractive indices of ordinary and extraordinary rays are 1.65836 and 1.48641 at \lambda_0=5893\,\mathring{\mathrm{A}} respectively. A left circularly polarized beam of this wavelength is incident normally on such crystal of thickness 0.005141\,\mathrm{mm} having its optic axis cut parallel to the surface. What will be the state of polarization of the emergent beam?
Explain the physical significance of group velocity from the concept of phase velocity with relevant expressions.
In relation to a plane diffraction grating having 5000 lines per cm and irradiated by light of wavelength 6000\,\mathring{\mathrm{A}}, answer the following:
(i) What is the highest order spectrum which may be observed?
(ii) If the width of opaque space is exactly twice that of the transparent space, which order of spectra will be absent?
A laser beam of 1 micrometer wavelength with 3 megawatts power of beam diameter 10\ \mathrm{mm} is focussed by a lens of focal length 50\ \mathrm{mm}. Evaluate the electric field associated with the light beam at the focal point. (Dielectric permittivity of free space, \varepsilon_0=8\cdot8542\times10^{-12}\ \mathrm{C^2/N-m^2})
What should be the refractive index of cladding of an optical fibre with numerical aperture 0.5 with refractive index of core as 1.5?
Obtain the expression for the primary focal length of Fresnel zone plate.
An unpolarized light beam of intensity 1000\ \mathrm{W/m^2} is incident on an ideal linear polarizer with its transmission axis parallel to vertical direction. Describe an experiment to reduce the intensity of light beam to 500\ \mathrm{W/m^2}.
Two thin symmetrical lenses of two different natures (convex and concave) and of different materials have equal radii of curvature R=15\,\mathrm{cm}. The lenses are put close together and immersed in water \mu_w=4/3. The focal length of the system in water is 30\,\mathrm{cm}. Show that the difference between the refractive indices of two lenses is 1/3.
Show that two convex lenses of the same material kept separated by a distance a, which is equal to the average of two focal lengths, may be used as an achromat, that is, a=\frac{1}{2}(f_1+f_2).
An optical beam of spectral width 7.5\,\mathrm{GHz} at wavelength \lambda = 600\,\mathrm{nm} is incident normally on Fabry-Perot etalon of thickness 100\,\mathrm{mm}. Taking refractive index unity, find the number of axial modes which can be supported by the etalon.
In the propagation of longitudinal waves in a fluid contained in an infinitely long tube of cross-section A, show that \rho=\rho_0\left(1-\frac{\partial\xi}{\partial x}\right) where, \rho_0 = equilibrium density \rho = density of the fluid in the disturbed state \frac{\partial\xi}{\partial x}=\text{volume strain}\quad\left(\left|\frac{\partial\xi}{\partial x}\right|\ll1\right)
Describe Michelson interferometer for evaluation of coherence length of an optical beam. Calculate coherence length of a light beam of wavelength 600\,\mathrm{nm} with spectral width of 0.01\,\mathrm{nm}.
Show that two light beams polarized in perpendicular directions will not interfere.
What are the important properties of a hologram?
Show that the plane of polarisation is rotated through \theta = \frac{\delta}{2} = \frac{\pi d}{\lambda} (\mu_L - \mu_R) in optical rotation where symbols have their usual meanings.
At what temperature are the rates of spontaneous and stimulated emission equal? (Assume \lambda = 500\text{ nm})
A plane-polarised light is incident perpendicularly on a quartz plate cut with faces parallel to optic axis. Find the thickness of the quartz plate which introduces phase difference of 60^\circ between e- and o-rays.
Optical power of 1\text{ mW} is launched into an optical fiber of length 100\text{ m}. If the power emerging from the other end is 0\cdot 3\text{ mW}, calculate the fiber attenuation.
