A method developed by Lansraux for calculating diffraction images when the pupil function of the objective has radial symmetry is applied to finding the coefficients of frequency response of the optical system when the object is periodic in one dimension and emits or scatters incoherently. A series solution is obtained in which each member is a product of a Hankel transform <i>∫</i><sup>∞</sup><sub>0</sub><i>J <sub>n</sub></i> (<i>z</i>) <i>J <sub>m</sub></i> (<i>z</i>) <i>J</i><sub>0</sub> (<i>sz</i>)<i>z</i><sup>1-<i>n</i>-<i>m</i></sup><i>dz</i> and a term depending on the optical properties of the system. The first few transforms are evaluated from a general trigonometric integral.
LUTHER W. SMITH and HAROLD OSTERBERG, "Modulation Transfer from Lansraux’s Series," J. Opt. Soc. Am. 54, 523-524 (1964)