Question 1 a) State and prove the convolution theorem. b) The process of sampling a function 𝑓 may be represented by multiplication of the function by a Dirac comb (an infinite set of regularly spaced Dirac distributions). Prove that in the Fourier domain the sampling in the object (time / space) domain is equivalent to replication in the Fourier domain of 𝐹 = ℱ(𝑓), the Fourier transform of 𝑓 (i.e. replication of the spectrum). c) What is the equivalent operation in the object space of sampling in the Fourier domain? d) Show the relationship between the sequences of samples transformed by the Discrete Fourier Transform (DFT) and the continuous Fourier transform with appropriate sampling in both object and Fourier domain. Question 4* A planar Computer Generated Hologram (CGH) consists of a square aperture tessellated by 𝑀𝑁 × 𝑀𝑁 square pixels of dimension 𝑤 × 𝑤 organised as a 𝑀 × 𝑀 array of identical subarrays of 𝑁 × 𝑁 pixels. When the CGH is illuminated by a uniform plane optical wave, each pixel acts as a source of a scattered wave with overall amplitude equal to a sample at its mid-point of a complex function ℎ defined over the plane of the CGH. These scattered waves diffract and interfere on propagation to form the desired farfield diffracted field as described by Kirchhoff (or Rayleigh-Sommerfeld) diffraction in the Fraunhoffer regime. The far-field amplitude is consequently described essentially by the Fourier transform of the object field (the collection of sources corresponding to each pixel). Using arguments based on Fourier analysis supported by appropriate sketches: a) Describe the form of the far-field you would expect to observe in the case where the source associated with a pixel may be approximated as a point source and 𝑀 is sufficiently large that it may be taken to be infinite. Explain how the DFT may be used to accurate numerically evaluate this field (see Question 1). b) Describe the effect on the expected far-field if the each pixel may be approximated as a source by an aperture of the same dimensions as the pixel uniformly illuminated by a plane wave of amplitude equal to the associated sample of ℎ. c) Describe the effect on the expected far-field if 𝑀 ≫ 1 is finite. d) In applications often it is only the intensity of the far-field that can be observed (measured) or is of significance. The phase of the far-field is then a free variable for design purposes. To make best use of a hologram with limited dynamic range, the free-phase may be essentially randomized by some means (i.e. the action of a ‘diffuser’ is emulated numerically and often optimised† ). Explain why the free-phase has no observable effect on the far-field intensity under the circumstances of part (a) but can have a profound effect on the appearance of the far-field intensity under the circumstances of part (c) , especially if 𝑀 = 1. What is this phenomenon called? * This question could have as easily concerned any of a variety of ‘phased-arrays’ such as the spatial light modulator (SLM) used in Reconfigurable Optical Add-Drop Modules (ROADM); RF phased array radar (PAR) antennas or the optical phased array (OPA) emerging in optical wireless applications. † In the interests of energy efficiency a CGH is most often designed to have a phase-only transmittance and the design problem recast as an optimisation problem to find the best approximation of a scaled desired far-field by the predicted far-field provided by the CGH. The effect of the approximation error at the hologram may be rendered harmless by ensuring it is scattered outside some measurement aperture in the far-field, i.e. one finds a hologram that when spatially low-pass filtered best approximates the desired hologram structure. This is the same principle as half-tone printing; pulse position / width modulation; and Σ− Δ modulation / ADC. The effect computationally is to introduce an interaction between pixels similar to that found in the Ising model of spin-glasses. It is also desired that the hologram is as efficient as possible; solutions are found that maximise the aforementioned scaling factor subject to meeting a specified measure of the quality of the reconstruction. This is the basis of the Generalised Error Diffusion design algorithm and its phase-optimised variant (POGED). .