Q18Engineering Physics

In Fraunhofer diffraction, the incident light is strictly a plane wavefront (the source is effectively at optical infinity), and the physically diffracted light is systematically focused directly onto a viewing screen using a highly precise convex lens. When this plane wavefront strikes a narrow rectangular slit of absolute width '$e$', every single mathematical point strictly within the slit explicitly acts as a completely independent source of secondary Huygens wavelets.

Optical Path Difference

Consider secondary wavelets physically diffracting specifically at an angle $\theta$. The absolute total optical path difference ($\Delta$) between the exact wavelets originating entirely from the absolute top edge and the absolute bottom edge of the slit is strictly determined purely by geometry:

This absolute path difference mathematically corresponds to an exact total phase difference ($\varphi$) of:

Intensity Distribution

By rigorously integrating the complex electric field amplitudes of all these infinite infinitesimal secondary wavelets entirely across the full width of the slit, the absolute total resultant amplitude ($R$) and subsequent intensity ($I=R^2$) mathematically emerge as:

Analysis of Maxima and Minima

• Central Maximum ($\theta =0$): Here, $\alpha =0$. By rigorous application of L'Hopital's rule, $(\sin \alpha /\alpha )\to 1$. The intensity physically reaches its absolute global maximum, $I_0$. All secondary wavelets travel completely straight and interfere perfectly constructively.
• Condition for Minima (Zero Intensity): Absolute destructive interference strictly occurs when the intensity perfectly drops to zero. This happens when $\sin \alpha =0$ (but $\alpha \ne 0$). Thus, $\alpha =n\pi$, where $n=\pm 1,\pm 2,\dots$
• * Substituting $\alpha$: $\frac{\pi e\sin \theta }{\lambda }=n\pi ⟹e\sin \theta =n\lambda$.
• Secondary Maxima: Highly subdued, drastically weaker bright fringes physically appear roughly halfway strictly between the absolute minima. Their intensities mathematically diminish rapidly ($I_1\approx I_0/22$, $I_2\approx I_0/61$).