(Solved): Explain what is meant by the convolution of two functions. State the convolution theorem and explai ...

Explain what is meant by the convolution of two functions. State the convolution theorem and explain how it may be used to determine the diffraction patterns of regular arrays of a basic pattern. The diffraction pattern of an infinite regular array of narrow slits, separated by a distance \( d \), is given by: \[ a(\theta)=a_{0} \sum_{n=0}^{\infty} \delta\left(\sin \theta-n \frac{\lambda}{d}\right) \] where \( \theta \) is the angle through which the incident beam is diffracted and \( \delta(x) \) is the Dirac \( \delta \)-function. By first writing the transmission of a real diffraction grating as a combination of products and convolutions of functions, determine the diffraction pattern of a grating of width \( b \), composed of slits of width \( c \) spaced by a distance \( d \), and sketch the pattern. Hence estimate the resolution of a spectrograph for use at \( \lambda=589 \mathrm{~nm} \) when the grating parameters are \( b=25 \mathrm{~mm}, d=1 /\left(450 \mathrm{~mm}^{-1}\right), c=0.5 \mu \mathrm{m} \).