**2. (a) For Newton’s ring, prove that diameters of n ^{th} dark ring is directly proportional IR to the square root of natural number. In Newton’s ring experiment the diameter of n^{th} and (n+8)^{th }bright rings ar 4.2mm and 7mm respectively. Radius of curvature of lower surface of lens is 2m. Determine the wavelength of light used. [8M]**

Let R be the radius of curvature of the lens. Let a dark ring be located at the point B. The thickness

of the air film at B is t and AE = r, the radius of the ring at thickness t. By using the Pythagoras theorem

to triangle AEO we get,

AO^{2 }=AE^{2}+EO^{2}

R^{2 }= r^{2}+ (R-t)^{2} = r^{2} + R^{2} – 2Rt + t^{2}

r^{2} = 2Rt

or

t = r^{2}/2R ,As t is very small, t^{2} can be neglected

Using relation 2t = nλ, for dark rings.

r^{2} = nλR

$latex r=\sqrt{n\lambda R}$ Dark ring.

The Diameter of dark ring is given by

$latex D_n=\sqrt{4n\lambda R}$

Since, λ and R are constants

$latex D_n\alpha\sqrt n$

i.e. the diameter of the Newton’s n^{th} dark ring is directly proportional to square root of the ring number.

we have,

$latex \left(D^2\;_{n+p}\right)\;-\left(D^2\;_n\right)=\;4p\lambda R$

$latex \left(7\;\times\;10^{-3}\right)^2-\left ( 4.2\times 10^{-3} \right )^{2}=4\times8\times\lambda \times2$

$latex 31.36\times10^{-6}=64\lambda$

$latex \lambda=0.49\times10-6\;m$

$latex =4900\AA$

Optical fibers in general are of two types namely single mode and multimode fiber. A single mode fiber has a smaller core diameter and support only one mode of propagation. Whereas a multimode fiber has larger core diameter and support number of modes of propagation. The multimode fibers are further distinguished by its index profile as Step Index fiber and Graded Index fiber.

**1.Step index fiber**

These fibers has, core of uniform refractive index and the cladding. The RI changes abruptly at the core- cladding boundary.Entering at the different angles of incidence with the axis, they travel different paths and emerge out of fibre at different time and also the input pulse gets widened.

There are two types of step index fiber.

i. Single mode step index fiber

A single mode step index fiber has a very fine thin core of uniform refractive index of a higher

value which is surrounded by a cladding of lower refractive index. The refractive index changes abruptly at the core-cladding boundary, as illustrated in Figure, because of which it is known as a step index fiber

The fiber is surrounded by an opaque protective sheath. A typical SMF has a core diameter of 4 um, which is of the order of a few wavelengths of light. Light travels in SMF along a single path that is along the axis. Obviously, it is the zero order modes that are supported by a SMF. A SMF is characterized by a very small value of. It is of the order of 0.002

ii. Multimode step index fiber

A multimode step index fiber is very much similar to the single mode step index fiber except that its core is of larger diameter. A typical fiber has a core diameter of 100 pm which is very large compared to the wavelength of light being transmitted. Light follows zigzag paths inside the fiber. Many such zigzag paths of propagation are permitted in a MMF. A typical structure and index profile of a step index MMF are shown in figure.

The NA of a MMF is large as the core diameter of the fiber is large. It is of the order of 0.3.

**2. Graded index (GRIN) fiber**

A graded index fiber is a multimode fiber with a core consisting of concentric layers of different refractive indices. Therefore, the refractive index of the core varies with distance from the fiber axis. It has a high value at the center and the falls off with increasing radial distance from the axis. A typical structure and its index profile are shown in figure. Such a profile causes a periodic focusing of the light propagating through the fiber.

V-number or Normalized frequency: The number of modes that can propagate through an optical fibre is given by a parameter called V-number. It is given by the expression,

$latex V=\frac{2\pi a}\lambda\left(\sqrt{n_1^2-n_2^2}\right)—–\left(i\right)$

Normalized frequency is (V) is a relation among the fibre size, the refractive indices $latex \left(\frac{n_1}{n_2}\right)$ and the wavelength λ_{0}. In equation (1), a is the radius of the core, λ_{0} is the free space wavelength; n_{1} and n_{2} are refractive indices of the core and cladding respectively.

$latex \begin{array}{l}V=\frac{2\pi a}\lambda\left(NA\right)\\\\V=\frac{2\pi a}\lambda\left(n_1\sqrt{2\triangle}\right)—–\left(ii\right)\end{array}$

$latex V=\frac{2\pi a}\lambda\left(n_1\sqrt{2\triangle}\right)$

V-number determines the no. of modes Nm that can be propagated through a fibre. According to equation (2), the no. of modes that propagate through a fibre increases with increase in numerical aperture.

The maximum no. of modes Nm supported by S.I. fibre is determined by,

$latex Nm=\frac12\left(V^2\right).$

While the no. of modes in the GRIN fibre is about half that in a similar S.I. Fibre.

$latex Nm=\frac14\left(V^2\right).$

We have,

$latex \begin{array}{l}V=\frac{\pi a}\lambda\left(\sqrt{n_1^2-n_2^2}\right)\\\\V=\frac{3.14\times40\times10^{-6}}{1.5\times10^{-6}}\left(\sqrt{1.5^2-1.46^2}\right)\\\\V=9.914\end{array}$

The maximum no. of modes,

$latex \begin{array}{l}Nm=\frac12\left(V^2\right)\\\\\;\;\;\;\;\;=\frac{9.914^2}2\\\\Nm=49.14\;modes\end{array}$

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