Using the method of Laplace transform the field amplitude in the paraxial
approximation is found in the two-dimensional free space using initial values
of the amplitude specified on an arbitrary shaped monotonic curve. The obtained
amplitude depends on one a priori unknown function, which can be found
from a Volterra first kind integral equation. In a special case of field
amplitude specified on a concave parabolic curve the exact solution is derived.
Both solutions can be used to study the light propagation from arbitrary
surfaces including grazing incidence X-ray mirrors. They can find applications
in the analysis of coherent imaging problems of X-ray optics, in phase
retrieval algorithms as well as in inverse problems in the cases when the
initial field amplitude is sought on a curved surface.
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