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- Things to remember

When light passes through a narrow aperture or obstacle, it spread out into the geometric shadow of the aperture or the obstacle. This spreading of light around the edge of an aperture or obstacle is called the diffraction. Diffraction of a sound wave is larger than light waves as a wavelength of sound is larger than a wavelength of light.

The light passing through a narrow slit produces a diffraction pattern consisting of a broad, intense central band called the central maximum and a series of narrower less intense bands called secondary maxima.

Diffraction pattern associates with light passing through a sharp edge of n object are shown in the figure. A similar pattern is observed when the light passes through the edges at both inside and outside the edge. As the light passes the vertical edge at left, it flares left and right and undergoes interference producing a pattern along the left edge. Actually that pattern lies within what would have been a shadow of the blade of geometric optics prevailed. The diffraction pattern of a disc in which a bridge spot called ‘poison’s spot’ is at the center and circular fringes extend outward from shadow’s edge the occurrence of the ‘poison’s spot’ is called “ Freshnel bright spot”.

The diffraction patterns, as discussed above, produce due to wave nature of light. It can be explained in terms of Huygens’ Principle which states that every point of a wavefront can be considered as a source of secondary wavelets, that spread out in all directions with a speed equal to the speed of propagation of the wave. The intensity of light at any point on the screen is obtained by superposing the individual displacements produced by these secondary wavelets, arising from the aperture or obstacle. So, diffraction is the interference produced by the secondary waves from different parts of the same wavefront.

Diffraction patterns are classified into two categories on which source and screen are placed. When either the source or the screen is near the aperture or obstacle, the wavefronts are spherical and the pattern is quite complex. This is called the Fresnel diffraction. When both source and screen are placed at a greater distance from the aperture, the incident light planes waves and the rays leaving the opening are parallel. This is called the Fraunhofer diffraction.

A narrow parallel beam of light from a source is incident normally on a rectangular, vertical slit of width a. the waves propagating out of the slit diffract and produce a diffraction pattern on the screen with a central bright fringe and a number of fainter fringes on both sides of the central fringe. These fringes are images of the single slit.

**Theory**

Suppose a plane wave of wavelength λ, falls normally on a narrow rectangular slit of width a. now, divide the slit into two equal halves as shown in the figure. All the waves are in phase at the slit. Consider two rays 1 and 3 travelling toward the screen at an angle Ï´. Ray 1 travels farther than ray 3 by a path difference between rays 2 and 4 are also (a/2) sin Ï´, as between rays 3 and 5. If this path difference is exactly half a wavelength, two waves cancel each other and produce destructive interference. So all such pairs of rays from two halves interfere destructively and in condition,

\begin{align*} \frac a2\sin \theta &= \pm \frac {\lambda }{2} \\ \text {or,} \: \sin \theta &= \pm \frac {\lambda }{a} \\ \begin{align*}

If the slit is divided into four equal parts, dark images are obtained on the screen and we have \begin{align*} \\ \sin \theta &= \pm \frac {2\lambda }{a} \\ \end{align*}

So, by using similar pairing process, the destructive interference of higher order is obtained and for this,

$$\sin \theta = m \frac {\lambda }{a} \dots (v) m = \pm 1, \pm 2, \pm 3, \dots$$

The \( \pm \) sign indicate the destructive interference occur in both sides of the central maxima on the screen. Above equation gives the value of Ï´ for diffraction pattern of zero intensity. The first maximum is formed when the slit is divided into three equal parts as shown in the figure. And a direction is considered in which the path difference between their ends are λ/2. Wavelets from strips in two adjacent parts cancel, and only one part remains that gives a much less bright band.

Through, the equation (v) does not give the variation of intensity, the general features of intensity distribution is shown in figure. At both sides of central maximum, secondary maxima of lower intensity lie between the minima at angle such that \( \theta = \pm \frac {3\lambda }{2a}, \pm \frac {5\lambda }{2a} \).

**Width of Central Maximum**

In figure, two minima lie on two sides of the sides of the central maximum. So the width of the maximum is the distance between first minimum on its both sides.

\begin{align*} \text {So, far} \: 1^{st} \: \text {minimum, we have} \\ \sin \theta &= \pm \frac {2\lambda }{a} \\ \text {For small angle,} \: \sin \theta = \theta \: \text {and so} \\ \theta &= \pm \frac {\lambda }{a} \\ \text {For small angle,} \ \sin \theta = \theta \: \text {and so} \\ \theta &= \pm \frac {\lambda }{a} \\ \end{align*}

The angular width of the central maximum

\begin{align*} \\ 2\theta &= \frac {2\lambda }{a} \: \left ( i.e. \frac {\lambda }{a} + \frac {\lambda }{a} = 2 \frac {\lambda }{a} \right ) \end{align*}

If y is the distance of the first minimum from the central maximum, the width of central maximum is 2y. If D is the distance between the slit and screen which is very large compared to the width of slit,

\begin{align*}\ \theta &= \frac yD = \frac {\lambda }{a} \\ \text {or,} \: y &= \frac {\lambda D}{a} \\ \text {And width of the maximum,} \\ 2y &= \frac {2\lambda D}{a} \\ \end{align*}

References

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. *Principle of Physics*. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. *A text Book of Physics*. Kathmandu: Surya Publication, 2003.

The spreading of light around the edge of an aperture or obstacle is called the diffraction.

When either the source or the screen is near the aperture or obstacle, the wavefronts are spherical and the pattern is quite complex. This is called the Fresnel diffraction.

When both source and screen are placed at a greater distance from the aperture, the incident light planes waves and the rays leaving the opening are parallel. This is called the Fraunhofer diffraction.

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