Diffraction of Light - The Single Slit

When light passes through a single narrow slit, it spreads out and produces a pattern of alternating bright and dark bands on a screen. This spreading of light is called diffraction.

Key elements of the setup:

• Slit LN of width a*, with midpoint M
• A screen placed at a large distance
• C = point on the screen along the normal to the slit
• P = any point on the screen at angle θ from the normal
• Light from every small part of the slit acts as a secondary source; all secondary sources within the slit are in phase with each other

The single-slit diffraction pattern has the following characteristics:

• A broad, bright central maximum at the centre of the screen (at point C, along the normal)
• Alternating dark and bright regions on either side of the central maximum
• The bright regions (secondary maxima) have progressively decreasing intensity as you move away from the centre
• Dark fringes (minima) separate the bright regions

Every point within the slit of width a acts as a secondary source of Huygens' wavelets. To find dark and bright fringes, the slit is divided into equal halves, and rays from corresponding points in each half are compared for their path difference.

Step-by-Step: Condition for Central Maximum

• At θ = 0 (point C on the normal), all secondary sources are at equal distances from C
• All path differences = 0
• All wavelets arrive in phase → constructive superposition
• Result: Central maximum at θ = 0 — this is the brightest point

Step-by-Step: Condition for Minima (Dark Fringes)

Divide the slit into two equal halves. For any ray from the top half, there is a corresponding ray from the bottom half. When the path difference between these paired rays equals λ/2, they cancel (destructive interference).

For the slit of width a, the path difference between rays from the top and middle of the slit:

For destructive interference: path difference = \[\frac {λ}{2}\]

Generalising by dividing the slit into 2n equal parts:

Condition for Secondary Maxima

Between consecutive minima, there are bright fringes (secondary maxima). These occur approximately at:

These secondary maxima have much lower intensity than the central maximum because a significant portion of the slit still cancels out.

• The intensity is maximum at the centre (θ = 0) and falls off on both sides
• Secondary maxima exist between the minima but are significantly weaker
• The pattern is symmetric about the central maximum
• As n increases, the intensity of the secondary maxima keeps decreasing.