Q18Engineering Physics

When a plane wave of monochromatic light is forced to pass through a very narrow rectangular aperture (a slit), it does not travel straight through like a bullet. Instead, the light wave severely bends around the sharp edges of the slit and spreads out—a phenomenon universally known as diffraction. In the Fraunhofer class of diffraction, both the incident light source and the observation screen are placed at optical infinity, which is practically achieved in the laboratory using two converging convex lenses.

1. The Experimental Setup and Principles

Consider a perfectly monochromatic plane wavefront of wavelength $\lambda$ traveling perpendicularly toward a narrow rectangular slit of width $a$. According to Huygens' principle, the instant this plane wavefront hits the slit, every single microscopic point of exposed space within that width $a$ acts as a brand new, independent source of secondary spherical wavelets. These millions of secondary wavelets immediately begin expanding forward in all possible forward directions.

If we focus only on the wavelets traveling straight forward (undeviated, $\theta =0$), they all arrive at the exact center point (O) of the screen perfectly in phase. They undergo massive constructive interference, producing a brilliant, wide white band known as the Central Maximum.

However, to understand the surrounding dark and bright fringes, we must analyze the wavelets that are diffracted at an arbitrary angle $\theta$ from the normal axis. These rays are gathered by the converging lens and focused to a distinct off-center point $P$ on the screen. Because these rays travel at an angle, rays originating from the bottom of the slit must travel a physically longer distance to reach point $P$ than rays originating from the top of the slit.

2. Mathematical Derivation of Intensity

To calculate the resultant intensity at point $P$, we must integrate the complex amplitudes of all the individual wavelets across the entire width of the slit.

By dropping a perpendicular from the top edge of the slit (point A) to the ray originating from the bottom edge (point B), we form a right-angled triangle. Simple trigonometry reveals that the total extreme path difference ($\Delta x$) between the wavelet from the very top and the wavelet from the very bottom is: $\Delta x=a\sin \theta$

The corresponding total phase difference ($\varphi$) between these extreme rays is: $\varphi =\frac{2\pi }{\lambda }\Delta x=\frac{2\pi }{\lambda }a\sin \theta$

We now conceptually divide the slit into a massive number ($N$) of infinitesimally small parallel strips, each acting as a discrete source. The phase difference between adjacent strips is $\delta =\varphi /N$. Using the standard mathematical technique for summing $N$ identical wave vectors exhibiting a constant phase difference, the resultant amplitude $R$ at point $P$ is found to be:

Let us define a new convenient variable $\alpha$, which represents half the total phase difference: $\alpha =\frac{\varphi }{2}=\frac{\pi a\sin \theta }{\lambda }$

Substituting $\alpha$, the resultant amplitude equation simplifies elegantly to: $R=R_0\frac{\sin \alpha }{\alpha }$

Since physical intensity ($I$) is always directly proportional to the square of the amplitude ($I\propto R^2$), the definitive intensity distribution equation for Fraunhofer single-slit diffraction is:

Where $I_0$ is the absolute peak intensity at the central maximum.

3. Analysis of the Diffraction Pattern

This single intensity equation perfectly predicts every feature of the observed diffraction pattern on the screen:

• The Central Maximum: When the angle $\theta =0$, the variable $\alpha$ is exactly $0$. The limit of $(\frac{\sin \alpha }{\alpha })$ as $\alpha$ approaches $0$ is precisely $1$. Therefore, the intensity $I=I_0$. This mathematically proves the existence of the incredibly bright, central undeviated fringe.
• The Principal Minima (Dark Fringes): The intensity $I$ will mathematically plummet to zero whenever $\sin \alpha =0$ (excluding the case where $\alpha =0$). The sine function is zero at all integer multiples of $\pi$. Therefore, minima occur when $\alpha =\pm m\pi$ (where $m=1,2,\mathrm{3...}$). Substituting the definition of $\alpha$ back in: $\frac{\pi a\sin \theta }{\lambda }=\pm m\pi ⟹a\sin \theta =\pm m\lambda$ This provides the exact angular locations of every dark fringe flanking the central maximum.
• The Secondary Maxima (Weak Bright Fringes): The secondary bright fringes occur approximately halfway between the dark minima. We can find their exact positions by differentiating the intensity equation with respect to $\alpha$ and setting it to zero. This leads to the transcendental equation $\tan \alpha =\alpha$. The solutions are roughly $\alpha =1.43\pi ,2.46\pi ,3.47\pi$, etc. Substituting these values into the intensity equation reveals that these secondary peaks are incredibly dim. The first secondary peak is only about $4.5\%$ as bright as the central maximum ($I_1\approx 0.045I_0$), the second is $1.6\%$, and they rapidly fade into total darkness. This proves why diffraction patterns appear strictly as a single bright central band with faint ghostly "wings".