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Analytic Hankel operators are generally defined in the time domain, and via the Fourier transform their frequency domain representation is obtained. We will skip this part and introduce Hankel operators directly as frequency-domain objects. Our choice is to develop the theory of continuous-time systems. This means that the relevant frequency domain spaces are the Hardy spaces of the left and right half-planes. Thus we will study Hankel operators defined on half-plane Hardy spaces rather than on those of the unit disk as was done by Adamjan, Arov, and Krein (1968a,b, 1971, 1978). In this, we follow the choice of Glover (1984). This choice seems to be a very convenient one, since all results on duality simplify significantly, due to the greater symmetry between the two half-planes in comparison to the unit disk and its exterior.