Final answer:
The first dark band in a single-slit diffraction pattern arises because the path difference between pairs of wavelets is one-half of a wavelength, leading to destructive interference. This foundational concept in wave optics follows from Huygens's principle and explains the characteristic single-slit diffraction pattern.
Step-by-step explanation:
At the first dark band in a single-slit diffraction pattern, the path lengths of selected pairs of wavelets differ by one-half of a wavelength (λ/2). Huygens's principle explains this phenomenon, stating that every point on a wave front can be considered as a source of secondary wavelets that spread out in all directions. When light passes through a single slit, these wavelets overlap and interfere with each other. If two wavelets arrive at a screen and their path length difference is λ/2, then they will be exactly out of phase, causing destructive interference and producing a dark band.
The first minimum in the diffraction pattern occurs because all the rays from the wavelets across the slit can be paired in such a way that each pair cancels out due to destructive interference. This concept is critical in understanding the single-slit diffraction pattern and the precise locations of dark and bright fringes produced on the screen. Thus, the pattern can be predicted using the analysis of the wavelet paths and their interference.