MP Board · Class 12 · Physics · Wave OpticsWhat is diffraction of light? Explain diffraction on a single slit and derive the condition for the formation of secondary minima. Also, draw the intensity distribution curve for single-slit diffraction.

Definition of Diffraction\nDiffraction of light is defined as the phenomenon of bending of light waves around the corners of an obstacle or an aperture into the region of geometrical shadow, provided the size of the obstacle or aperture is comparable to the wavelength of the light.

Diffraction at a Single Slit\nWhen a parallel beam of monochromatic light is incident normally on a single slit $AB$ of width $a$, a diffraction pattern consisting of a central bright maximum flanked by alternate dark and bright fringes of decreasing intensity is obtained on a screen placed at the focal plane of a convex lens.

Theory and Derivation for Secondary Minima\nLet:

• $a$ be the width of the slit.
• $\lambda$ be the wavelength of the incident light.
• $\theta$ be the angle of diffraction for a particular point on the screen. \nConsider a wave front incident on the slit. Every point in the slit acts as a source of secondary wavelets. To find the intensity at a point $P$ on the screen where light is diffracted at an angle $\theta$:

1. Drop a perpendicular $AN$ from $A$ to the extreme ray starting from $B$.
2. The path difference between the wavelets originating from the extreme points $A$ and $B$ of the slit is given by: $$\Delta = $BN = a \sin \theta$$

Condition for Secondary Minima:\nTo obtain minima (dark fringes) at point $P$, the path difference $a \sin \theta$ must be an integral multiple of the wavelength $\lambda$. $$a \sin \theta = n\lambda \quad \text{where } n = \pm 1, \pm 2, \pm 3, \dots$$

Reasoning: \nWe can conceptually divide the slit into two equal halves of width $a/2$. For every point in the upper half, there is a corresponding point in the lower half at a distance of $a/2$ such that the path difference between their secondary wavelets is $\lambda/2$. They interfere destructively and cancel each other's effect, resulting in a minimum of intensity.

Intensity Distribution Curve\nThe intensity distribution curve shows a very sharp and intense central maximum flanked by secondary maxima of rapidly decreasing intensity on both sides. The width of the central maximum is given by:

$$\beta_0 = \frac{2\lambda D}{a}$$