The motion of a damped mechanical oscillator is represented by m\ddot{x}+\alpha\dot{x}+\beta x=0 where m, \alpha and \beta are constants. The oscillator is critically damped. The system is given an impulse at x=0 and t=0, resulting in an initial velocity v. After how much time the system experiences maximum displacement?
Use matrix method to obtain an expression for the focal length of a coaxial combination of two thin lenses having focal lengths f_1 and f_2 separated by distance d.
Derive an expression for intermodal dispersion for a multimodal step-index fibre.
A step-index silica fiber consists of core of refractive index 1\cdot 450 and diameter 90\ \mu\text{m}. If the numerical aperture of the fiber is 0\cdot 16, calculate : (i) the acceptance angle of the fiber, (ii) refractive index of the cladding, and (iii) if the fiber is immersed in water of refractive index 1\cdot 33, how does the acceptance angle change ?
An optical waveguide has core and cladding of refractive indices 1\cdot 46 and 1\cdot 45 respectively. (i) Calculate the maximum number of modes with wavelength \lambda = 900\text{ nm} that can travel if it is a planar waveguide of core thickness 10\ \mu\text{m}. (ii) What is the number of modes if it is a cylindrical fiber of diameter 10\ \mu\text{m} ?
What are Einstein's A and B coefficients ? Establish a relation between them.
Calculate the minimum thickness of a quartz plate which would behave as a quarter-wave plate for wavelength of light, \lambda=6000\,\mathring{\mathrm{A}}. The refractive indices for ordinary and extraordinary rays are \mu_o=1.544 and \mu_e=1.553.
In the steady state forced vibration of the damped harmonic oscillator, show that the amplitude of the driven system is maximum when p = \sqrt{\omega^2 - 2b^2} and further the value of the maximum amplitude is \frac{f}{2b \sqrt{\omega^2 - b^2}}, where the symbols have their usual significance. (Here you start from the steady state solution)
A plano-convex lens of crown glass is connected to a concavo-convex lens of flint glass so that the convex surface of the first and the concave surface of the second fit exactly. If the combination is achromatic and the combined focal length is -50\text{ cm}, determine the radii of curvature of the faces of the flint glass lens. The dispersive power of the crown glass is one-half of that of the flint glass and the refractive indices for yellow light are respectively 1\cdot 500 and 1\cdot 625 respectively.
What is Fermat's principle ? Derive the laws of reflection and refraction with the help of this principle, when the light is incident on a curved surface.
Show with proper mathematical analysis that the ratio of Einstein's A and B coefficients depends upon the energy separation between the two energy levels participating in the optical transitions. What is the physical significance of A coefficient? \text{(5 marks)} Justify the statement, ``It is very difficult to develop an X-ray laser''. \text{(5 marks)}
Bring out the essential differences between the physical principles of spontaneous and stimulated emission of radiation. Why is it difficult to get efficient lasing action in case of an ideal two-level material system? Can you propose a scheme to enhance efficiency? Discuss.
A plane wave has the following expression for its electric field: \vec{E}=\hat{x}E_{0x}\cos(\omega t-kz+\alpha)+\hat{y}E_{0y}\cos(\omega t-kz+\beta) If the phase difference is defined as \delta=\beta-\alpha, under what conditions do we achieve elliptic polarization? What are the conditions for circular polarization?
The Fraunhofer single-slit diffraction intensity is given by I=I_0\frac{\sin^2 x}{x^2} where x=\frac{\pi d y}{\lambda l}, with l as the distance from slit to source, d the slit width, y the detector distance and \lambda the wavelength. What is the value of cumulative intensity \int_{-\infty}^{\infty} I(y)\,dy?
When a thin film of a transparent material is put behind one of the slits in Young's double-slit interference experiment, the zero-order fringe moves to the position previously occupied by the fourth-order bright fringe. The index of refraction of the film is n=1.2 and the wavelength of light, \lambda=5000\,\mathring{\mathrm{A}}. Determine the thickness of the film.
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?
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.
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.
A light spring of relaxed length 'a_0' is suspended from a point. It carries a mass 'm' at its lower free end, which stretches it through a distance l. Show that the vertical oscillation of the system is simple harmonic in nature and has time period T = 2\pi \sqrt{l / g}, where g is the acceleration due to gravity.
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?
