The dispersion relation for deep water waves is given by \omega^2 = gk + ak^3 where g and a are constants. Obtain expressions for phase velocity and group velocity in terms of the wavelength \lambda. \omega and k represent the angular frequency and wave number, respectively.
A plano-convex lens of radius 1\cdot 0\text{ m} is placed on an optically flat glass plate. The space between the lens and the glass plate is filled with a liquid. The diameter of the 5^{\text{th}} ring changes to 3\cdot 0 \times 10^{-3}\text{ m}. Calculate the refractive index of the liquid when the ring is bright. The wavelength of light used is 589\text{ nm}.
Explain the lasing action of \text{He}-\text{Ne} laser using energy level diagram and describe the operation of \text{He}-\text{Ne} laser experimental set-up.
Write down the equation of motion of a weakly damped harmonic oscillator driven by a harmonic force. Obtain an expression for the maximum amplitude of oscillation under steady-state conditions.
Explain why information carrying capacity of an optical fibre can be enhanced by reducing the pulse dispersion. How does one minimize pulse dispersion using a graded index optical fibre?
An oscillator of mass 0\cdot 01\text{ kg} draws maximum power at a frequency of 96\text{ Hz} with half power points at 93\text{ Hz} and 99\text{ Hz}. If the amplitude of driving force is 3\cdot 88\text{ N}, calculate (i) quality factor, (ii) the damping factors and amplitude of the oscillator at resonance.
The displacement associated with a three-dimensional plane wave is given by \Psi(x,y,z,t) = a\cos\left[\frac{\sqrt{3}}{2}kx + \frac{1}{2}ky - \omega t\right]. Calculate the angles made by the propagating wave with the x, y and z-axes.
A parallel beam of light from a He–Ne laser (\lambda=630 nm) is made to fall on a narrow slit of width 0.2\times10^{-3} m. The Fraunhofer diffraction pattern is observed on a screen placed in the focal plane of a convex lens of focal length 0.3 m. Calculate the distance between the
(i) first two minima and
(ii) first two maxima on the screen.
A block of mass m is attached to one end of a combination of two massless springs connected in series. The other end of the combination is fixed to a rigid support. The spring constants of the two springs are k_1 and k_2, respectively. The block is free to move on a frictionless horizontal surface. If the block is pulled a little and then released, calculate the frequency of oscillation of the block.
Explain the physical significance of resolving power of a grating with relevant mathematical expression.
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?
Show that a travelling wave on the string, clamped on both the ends, undergoes a phase change of \pi. Hence obtain the time-independent form of the wave equation representing a standing wave on the string.
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.
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.
A pulse of \lambda_0=600\,\mathrm{nm} and \Delta\lambda=10\,\mathrm{nm} propagates through a fibre which has a material dispersion coefficient of 50\,\mathrm{ps} per km per nm at 600\,\mathrm{nm}. Calculate the pulse broadening in traversing a 10\,\mathrm{km} length of the fibre. If the pulse width at the input of the fibre is 12\,\mathrm{ns}, what will be the pulse width at the output of the fibre?
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} ?
Explain the Fraunhofer diffraction at a single slit and obtain the condition for minima. Derive the expression for resolving power of a grating.
What are Einstein's A and B coefficients ? Establish a relation between them.
