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According to Huygens’s principle, each portion of the slit acts as a source of light waves. Hence, light from one portion of the slit can interfere with light from another portion, and the resultant light intensity on a viewing screen depends on the direction to analyze the diffraction pattern, it is convenient to divide the slit into two halves, as shown in above figure. Keeping in mind that all the waves are in phase as they leave the slit, consider rays 1 and 3. As these two rays travel toward a viewing screen far to the right of the figure, ray 1 travels farther than ray 3 by an amount equal to the path difference a/2 .Sin θ, where a is the width of the slit. Similarly, the path difference between rays 2 and 4 is also a/2 .Sin θ. If this path difference is exactly half a wavelength (corresponding to a phase difference of 180°), then the two waves cancel each other and destructive interference results. This is true for any two rays that originate at points separated by half the slit width because the phase difference between two such points is 180°. Therefore, waves from the upper half of the slit interfere destructively with waves from the lower half when;
a/2 .Sin θ = λ/2 or when, Sin θ = λ/a
The general condition for destructive interference is;
Sin θ = m. λ / a
m= ±1, ±2, … …
The above equation gives the values of θ for which the diffraction pattern has zero light intensity i.e., when a dark fringe is formed. However, it tells us nothing about the variation in light intensity along the screen.
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Waves and Optics