Coherence refers to the phase relationship between different points along a wavefront. In single-slit diffraction, spatial coherence is particularly important. It ensures that the light waves passing through different parts of the slit maintain a fixed phase relationship, leading to the observable diffraction pattern. The source of light needs to be spatially coherent to some extent; otherwise, the superposition of waves from different parts of the slit would be random, and a clear diffraction pattern wouldn’t form. The coherence length should be comparable to or larger than the slit width to observe a distinct diffraction pattern.

In single-slit diffraction, the fringe visibility and intensity distribution are characterised by a broad central maximum followed by rapidly diminishing intensities of subsequent maxima. In contrast, double-slit interference typically exhibits more evenly spaced and similarly intense fringes. The addition of single-slit diffraction to double-slit setups leads to the modulation of the interference pattern, creating a combined pattern where the intensity of the interference fringes is not uniform. This modulation is attributed to the overlapping of the diffraction pattern from each slit on the interference pattern, creating variations in the intensity and visibility of the fringes.

The material of the slit can impact the diffraction pattern in a number of ways. Different materials can introduce variables such as absorption and reflection of light, impacting the intensity and clarity of the diffraction pattern. Additionally, the edges of the slit can also play a role. If the edges are not defined sharply, it can lead to irregularities in the diffraction pattern. Moreover, the mechanical properties of the material, including its stability and ability to maintain a precise slit width under varying environmental conditions, are essential to ensure the consistency of the observed diffraction patterns in experimental setups.

Practical applications of single-slit diffraction are extensive in optics and photonics. For instance, in imaging systems like microscopes and telescopes, understanding diffraction is crucial to comprehend and mitigate the resolution limits. The formation of diffraction patterns and the spreading of light impacts the clarity and sharpness of images. By understanding the principles of single-slit diffraction, enhancements in the design, such as the incorporation of apodization filters or computational methods to mitigate diffraction effects, can be employed to improve image quality and resolution. This knowledge is fundamental in pushing the boundaries of what can be resolved by optical systems.

The intensity distribution in the single-slit diffraction pattern can be calculated using the formula for intensity I(θ) as a function of the angle θ. It incorporates factors like the slit width and the wavelength of light. The intensity is highest at the central maximum, where constructive interference is at its peak. Mathematical methods involving integrals and complex numbers can be used to detail the light's wave function across the slit, accounting for every point source of light within the slit and calculating the superposition of waves at a particular point on the screen.