sinc2mønster

Sinc2mønster, or the sinc squared pattern, refers to the pattern produced by the squared sinc function, often written as sinc^2(x). The normalized sinc function is defined as sinc(x) = sin(πx)/(πx), making sinc^2(x) = [sinc(x)]^2. The function is nonnegative, symmetric, and has a single peak at x = 0 with zeros at all nonzero integer values of x.

In optics, sinc2mønster describes the intensity distribution of diffraction from a rectangular aperture. For a slit of width a observed in the far field, the intensity is proportional to sinc^2(π a sin θ / λ). The main maximum occurs at θ = 0, and zeros appear when a sin θ = mλ for m = ±1, ±2, … The width of the central lobe is inversely related to the aperture width a.

In signal processing, the Fourier transform of a rectangular time-domain pulse of width T yields a frequency-domain pattern proportional to sinc(fT), and the power spectrum is proportional to sinc^2(fT). Thus the sinc2mønster describes how energy distributes in frequency, with a dominant central lobe and side lobes that decay roughly as 1/f^2. In two dimensions, a rectangular aperture with width a and height b produces a pattern I(x,y) ∝ sinc^2(π a x / λ) · sinc^2(π b y / λ).

Applications of sinc2mønster include diffraction analysis, optical imaging resolution assessment, antenna radiation patterns, and certain digital filtering and sampling scenarios. The pattern highlights a trade-off between main-lobe width and side-lobe levels, influencing resolution and spectral leakage.