Explain the refractive index distribution for a graded index optical fiber given as \begin{aligned} n^2(r) &= n_1^2 \left[ 1 - 2\delta \left( \frac{r}{a} \right)^q \right], & r < a \\ &= n_1^2 [1 - 2\delta] = n_2^2, & r > a \end{aligned} with usual meaning of symbols. Sketch and name the above profiles with justification for q = 1, 2 and \infty.
Explain physically why dispersion is diminished in multimode parabolic-index fiber than in step-index fiber.
Deduce an expression for resolving power of a diffraction grating, explaining its meaning. What will be the value of the number of slits in the grating to resolve \text{D}_1 and \text{D}_2 lines of sodium in the first order, assuming \lambda and \Delta\lambda respectively as 600\text{ nm} and 0\cdot6\text{ nm}?
With a planoconvex lens of radius of curvature R resting on a plane glass surface, Newton's rings are observed with incident light of wavelength \lambda. Find the radii of m\text{th} order dark rings. What will be the colour of the innermost ring with white light illumination?
In Young's double-slit experiment, a thin mica sheet (refractive index = 1 \cdot 5) is introduced in the path of one of the beams. If the central fringe gets shifted by 0 \cdot 2\text{ cm}, calculate the thickness of the mica sheet. Assume separation between the two slits (d) and between slit and source (D) as d = 0 \cdot 1\text{ cm} and D = 50\text{ cm}.
Explain how you can construct a zone plate from Fresnel's half period zone. Show that a zone plate has multiple foci.
Starting with the rate equations for matter-radiation interaction, show that the ratio of the Einstein's A and B coefficients is proportional to the third power of the frequency of radiation.
Draw a neat diagram to show how light propagates through a graded index optical fiber from the entrance to the fiber end and explain.
What are the step index and graded index optical fibers ? What are the conventional optical windows for light propagation through optical fibers ?
How does the population inversion in an active medium lead to the amplification of light in a laser ? Explain in details.
A left circularly polarized beam (\lambda = 5893^\circ\text{A}) is incident normally on a calcite crystal (with its optic axis cut parallel to the surface) of thickness 0.005141\text{ mm}. What will be the state of polarization of the emergent beam ? Justify your answer. Here the refractive indices for the ordinary and the extraordinary rays are 1.65836 and 1.48641 respectively.
Write down the mathematical representation of Fermat's principle and explain all the notations used. With the help of a neat diagram, show that this principle can be used to obtain the law of refraction : n_1 \sin \theta_1 = n_2 \sin \theta_2 , where n_1 and n_2 are the refractive indices of the two media while \theta_1 and \theta_2 are the angles of incidence and refraction of the light beam.
If the system matrix for a thin lens of focal length f is given by : S = \begin{bmatrix} 1 & -1/f \\ 0 & 1 \end{bmatrix} , show that the system matrix for a combination of two thin lenses of focal lengths f_1 and f_2 separated by a distance d can be obtained as S_{12} = \begin{bmatrix} 1 - d/f_2 & -(1/f_1 + 1/f_2 - d/f_1 f_2) \\ d & 1 - d/f_1 \end{bmatrix}
Discuss the applications of lasers in communication and medicine.
Classify fibers based on the number of modes that can propagate through them. Depict step-index and graded-index multimode fibers pictorially.
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.
Discuss spatial evolution of Fresnel diffraction into Fraunhofer diffraction.
Consider a zone plate with radii r_n = 0\cdot 11 \sqrt{n}\text{ cm} illuminated by a monochromatic light of wavelength 589\text{ nm}. Calculate the positions of first two foci.
In a double slit interference experiment, show that the fringe shape is a hyperbola.
If a thin sheet of glass of thickness t and refractive index \mu is placed in the path of one of the interfering waves, show that the distance through which a fringe is displaced is a function of t.
