Single-slit diffraction Knowledge-Based Study

Study notes

1. What is Diffraction?

   Diffraction refers to the bending of light waves around the edges of an opening or obstacle and their subsequent spreading into the geometric shadow region. While geometric optics predicts a sharp shadow, the wave nature of light causes it to spread out when the dimensions of the aperture are comparable to the wavelength of the light.

   According to Huygens' Principle, every unobstructed point of a wavefront emerging from a slit acts as a secondary source of spherical wavelets. As these secondary wavelets propagate forward, they interfere with one another, producing an alternating pattern of bright and dark bands on a distant viewing screen.

   Deriving the Minima Condition

   Consider a monochromatic light wave of wavelength $\lambda$ incident normally on a slit of width $a$. The rays traveling at an angle $\theta$ relative to the original direction of propagation are focused on a screen located at a distance $L$.

   To locate the points of zero intensity (dark fringes or minima), the slit can be divided into equal segments:

   • For the first minimum, divide the slit into two halves of width $a/2$. A ray from the top edge and a ray from the midpoint have a path difference of $\frac{a}{2} \sin\theta$. Destructive interference occurs when this path difference equals $\frac{\lambda}{2}$, leading to $a \sin\theta = \lambda$.
   • For the $m$-th minimum, divide the slit into $2m$ equal zones. Adjacent zones cancel each other out in pairs.

   The general condition for minima is given by:

   $a \sin\theta = m\lambda \quad (m = \pm 1, \pm 2, \pm 3, \dots)$

   Note: The integer $m = 0$ is excluded because $\theta = 0$ corresponds to the central maximum, where all wavelets arrive in phase.

   Features of the Diffraction Pattern

   Unlike double-slit interference, where fringes have roughly equal brightness, single-slit diffraction exhibits a pronounced variation in intensity:

   • Central Maximum: Located at $\theta = 0$. It is the brightest and widest feature of the pattern, bounded by the first minima on either side at $a \sin\theta = \pm\lambda$.
   • Secondary Maxima: Weaker bright fringes located approximately halfway between consecutive minima. Their intensity decreases rapidly as $|m|$ increases because wavelets from only a fraction of the slit remain uncancelled.

   The angular half-width of the central maximum for small angles ($\sin\theta \approx \theta$) is $\theta_1 = \frac{\lambda}{a}$. Therefore, the full angular width is:

   $2\theta_1 = \frac{2\lambda}{a}$

   On a screen placed at distance $L$, the linear width $W$ of the central maximum is:

   $W = 2y_1 = \frac{2\lambda L}{a}$

   Factors Influencing the Pattern

   The spatial extent of the diffraction pattern depends directly on the relationship between slit width and wavelength:

   • Slit Width ($a$): Decreasing $a$ increases the angular spread $\theta$, resulting in a wider central maximum. Conversely, as $a \gg \lambda$, the diffraction angles become negligibly small, and the behavior approaches geometric optics.
   • Wavelength ($\lambda$): Longer wavelengths (such as red light) spread more broadly than shorter wavelengths (such as blue or violet light).
   • Screen Distance ($L$): Increasing $L$ scales up the linear separation between fringes linearly without altering their angular separation.