The mathematical process of combining the characteristic functions of two optical systems to obtain the characteristic function of the system created by adjoining these component systems is shown to be identical for restricted point, mixed, and angle characteristics. Consequently, the applicability of the technique of concatenation is significantly extended. By way of example, the angle characteristic of a refracting surface is considered; the explicit power series for the angle characteristic of a symmetric refracting surface is given to within terms of degree ten.
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