The Single Slit

The figure shows a single slit of width $a$, labelled LN, with its midpoint M. A parallel beam of monochromatic light (plane wavefront) falls normally on the slit from the left. The slit opening is vertical, and the light spreads out after passing through it.

The screen is placed to the right, perpendicular to the original direction of the beam. The point C on the screen is directly opposite the midpoint M — it lies on the straight line through M perpendicular to the slit plane. The point P is any other point on the screen, at an angle $\theta$ measured from the line MC.

Rays from different points across the slit (L, M, N) travel to P. Because P is far away, these rays are treated as parallel, all making the same angle $\theta$ with the normal MC. The key geometric feature is the path difference between a ray from the top edge L and a ray from the bottom edge N. This path difference is labelled as $a \sin \theta$ in the diagram.

Physical idea: The slit is divided into many tiny segments, each acting as a secondary source of light (Huygens’ principle). All these sources are in phase because the incident wavefront is plane and parallel to the slit. However, at an angle $\theta$, the waves from different parts of the slit travel different distances to reach P, creating a phase difference. The net amplitude at P is the sum of contributions from all these sources, taking into account their relative phases.

Key formula(s) developed from this figure:

The condition for minima (zero intensity) on the screen is:

where:

• $a$ = width of the slit,
• $\theta$ = angle of the point P from the central line MC,
• $\lambda$ = wavelength of the monochromatic light,
• $n$ = order of the minimum (non-zero integer). …

The figure presents two complementary views of the single-slit diffraction pattern. The upper part is a graph plotting intensity (vertical axis) against angle $\theta$ (horizontal axis, measured from the central normal line $MC$). The lower part is a schematic band strip representing the actual photograph of the fringes: a broad, bright central band flanked by much fainter side bands.

What the graph shows:

• A single, tall central maximum at $\theta = 0$. This is the brightest and widest feature.
• On either side, minima (zero intensity) occur at angles $\theta = \pm \frac{\lambda}{a},\ \pm \frac{2\lambda}{a},\ \pm \frac{3\lambda}{a},\ \dots$ where $\lambda$ is the wavelength of light and $a$ is the slit width.
• Between successive minima lie secondary maxima — much weaker and narrower than the central maximum. Their positions are approximately at $\theta = \pm \frac{3\lambda}{2a},\ \pm \frac{5\lambda}{2a},\ \dots$ (i.e., $\theta \approx \pm (n+\frac12)\frac{\lambda}{a}$ for $n=1,2,3,\dots$).
• The intensity of these secondary maxima falls off rapidly as $|\theta|$ increases.

What the band strip shows:

• A broad bright central region corresponding to the central maximum.
• On each side, faint narrow bands (the secondary maxima) separated by dark gaps (the minima). The bands become progressively dimmer away from the centre.

Physical idea taught by the figure:

The pattern arises because each point across the slit acts as a coherent secondary source (Huygens’ principle). Light from different parts of the slit travels different distances to a point $P$ on the screen, creating a phase difference. The resultant amplitude at $P$ is the sum of contributions from all these sources. When the path difference between the two edges of the slit equals an integer multiple of $\lambda$, destructive interference produces a minimum. The central maximum is brightest because all contributions arrive nearly in phase.

Key formula developed with this figure:

The intensity $I(\theta)$ for a single slit of width $a$ is given by

where

• $I_0$ is the intensity at the centre ($\theta = 0$),
• $\beta = \frac{\pi a \sin \theta}{\lambda}$,
• $\lambda$ is the wavelength of the monochromatic light,
• $a$ is the slit width. …