How do you derive the formula for single-slit diffraction?

The formula for single-slit diffraction is derived using Huygens' principle and the concept of path difference, closely related to the understanding of wavefronts and rays.

To derive the formula for single-slit diffraction, we start with Huygens' principle, which states that every point on a wavefront can be considered as a source of secondary wavelets. These wavelets spread out in the forward direction, at the same speed as the wave. The new wavefront is the tangential surface that can be drawn to all of these secondary wavelets, an idea that ties back to the superposition principle.

Consider a single slit of width 'a' and a screen at a distance 'D' from the slit. When a monochromatic light of wavelength 'λ' is incident on the slit, diffraction occurs and a pattern of bright and dark fringes is observed on the screen. The central maximum is the brightest and as we move away from the centre, the intensity of light decreases. This phenomenon is a practical example of how diffraction patterns are formed.

To find the condition for minima (dark fringes), consider two rays, one from the top and one from the middle of the slit. The path difference between these two rays is 'a/2 sinθ', where 'θ' is the angle the ray makes with the line perpendicular to the slit. For destructive interference to occur, this path difference must be equal to half the wavelength, i.e., 'λ/2'. Therefore, the condition for minima is 'a sinθ = mλ', where 'm' is an integer (1, 2, 3, ...). This derivation highlights the concept of interference, a crucial aspect of the polarization of light, although in a different context.

The angle 'θ' can be related to the position 'y' on the screen using the small angle approximation 'tanθ ≈ sinθ ≈ y/D'. Therefore, the position of the mth minima on the screen is given by 'y = mλD/a'.

This derivation assumes that the width of the slit 'a' is of the order of the wavelength of light 'λ', and the distance to the screen 'D' is much larger than 'a'. It also assumes that the light is incident normally on the slit and the screen is perpendicular to the direction of the incident light.

IB Physics Tutor Summary: The formula for single-slit diffraction is found using Huygens' principle, which tells us every point of a wavefront acts like a new wave source. By looking at light going through a slit and hitting a screen, we see a pattern because of differences in path length. The key formula 'a sinθ = mλ' helps predict where dark spots appear, depending on the slit size, light wavelength, and screen distance.