Subject of study. Possible methods for increasing the range of optical parameters of fabricated high-aperture non-classical ruled toroidal diffraction gratings and new mathematical relations for describing the equations of curvilinear projections of grooves, laws of variation of the curvature radius of the closest vertex circle, and variable pitch for these groove projections are presented. Aim of study. The study aimed to improve the principles of design-ing pendulum-type ruling engines ensuring the implementation of a new method for fabricating high-aperture non-classical ruled concave toroidal diffraction gratings and derive analytical expressions for determining geo-metric parameters of curvilinear projections of the grooves of these gratings. Method of study. The investigation method was based on the theory of designing precision mechanisms, analytic geometry, and calculus. Main results. The possibility of solving the relevant issue of increasing the size, relative aperture, and relation between the curva-ture radii in meridional and sagittal cross-sections of high-aperture non-classical ruled toroidal diffraction gratings considering the optimization of their aberrations was demonstrated using pendulum-type ruling engines. In con-trast to the available engines, pendulum-type ruling engines should be designed to ensure the intersection between the rotation axes of the diamond and blank carriages as well as alignment of the dimension chains of the structural parameters of the diamond and blank carriages according to the specified curvature radii in meridional and sagittal cross-sections of concave toroidal diffraction gratings. Considering that the cutting edge of the diamond tool must always be in contact with the concave toroidal surface of a substrate along the normal to the surface in all substrate points while ruling all grooves, we established that the system of curvilinear groove projections is described by a sys-tem of second-order curves, i.e., ellipse equations. With increasing discrete angle at which the grooves are ruled, the coordinates of the centers of the ellipses gradually shifted in the meridional cross-section from the origin of coordi-nates (grating vertex) in the direction of the edge light zone by a value directly proportional to the sine of the discrete angle and the difference between curvature radii in the meridional and sagittal cross-sections. Curvature radii of the closest vertex circles of the groove projections are variable parameters directly proportional to the product of curva-ture radii in the meridional and sagittal cross-sections of the grating and inversely proportional to the variable value of the specified coordinate. The variable pitch of the groove projections on the specified plane is a function that, to a first approximation, depends on the initial groove pitch at the grating vertex and the squared variable value of the specified coordinate and is inversely proportional to the squared curvature radius in the meridional cross-section of the grating. Practical significance. The proposed principles of shaping groove structures of the ruled toroidal diffraction gratings using pendulum-type ruling engines enable fabricating such gratings with extremely large relative aperture and optical size as well as increased range of ratios of their curvature radii in the meridional and sagittal cross-sections, which can be used to design novel advanced spectral instruments. The obtained analytical expressions describing the geometric parameters of the curvilinear groove projections enable correction of the first through third order aberrations of the gratings of the considered type. (c) 2023 Optica Publishing Group
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