Cylindrical-Lens Light Sheets Generated from Circularly and Rectangularly Apertured Gaussian and Top-Hat Beams: A Unified Paraxial Diffraction Theory

Cylindrical lenses are widely used to generate static light sheets, yet practical estimates of sheet thickness and propagation length are commonly based either on ideal Gaussian-beam formulas or on a one-dimensional slit model. A finite circular beam and circular hard stop couple the focused and non-focused pupil coordinates through the circular boundary. We develop a unified scalar paraxial diffraction theory for circularly and rectangularly apertured Gaussian and top-hat fields focused by a thin cylindrical lens. The principal analytical contribution is a Cartesian aperture-slice decomposition (CASD) of the finite circular-pupil cylindrical-lens field. Analytical integration along the focused pupil coordinate reduces the two-dimensional Fresnel diffraction integral to one numerical quadrature at any observation point, while retaining the coupling between the focused and non-focused coordinates imposed by the circular boundary. This reduction is exact within the scalar paraxial model. CASD extends to other two-dimensional apertures with Gaussian or top-hat illumination and quadratic pupil phase. The same propagation model yields complete separable solutions for rectangular Gaussian and top-hat pupils. We derive nominal-focal-plane, axial-propagation-plane, and optical-axis forms, including an incomplete-gamma axial series and a polar Bessel–incomplete-gamma series for the full focal region. Cartesian hypergeometric and recurrence-based incomplete-gamma series provide complementary full-field representations, with the former also supplying analytical moment coefficients. The intensity field defines the principal peak position, nominal- and peak-plane transverse full widths at half maximum (FWHMs), connected axial intensity FWHM, local sheet thickness, and integrated illumination. A further contribution is a common explicit moment-based formula for the nominal transverse FWHM, with geometry-specific coefficients connecting top-hat, truncated-Gaussian and effectively untruncated-Gaussian illumination continuously. Maximum relative errors against complete-field numerical widths remain below 0.5% on the tested domains for both pupil geometries. Exact profile roots and strong-focus approximations provide complementary axial-width relations. The circular slice field agrees with independent polar quadrature to a representative relative complex-field error below 8.8e-9. The theory quantifies finite-Fresnel focal displacement and the transition from top-hat to effectively untruncated Gaussian illumination while retaining the physical aperture, beam radius, wavelength, and focal length. A low-numerical-aperture air–glass–sample example illustrates refractive focus displacement and the distinct transverse and axial scaling.