Diaboloidal mirrors: algebraic solution and surface shape approximations

Area of Science:

• Optics and Photonics
• X-ray Optics
• Optical Engineering

Background:

• Cylindrical waves are utilized in various beamline applications, often requiring precise focusing.
• Current optical elements, like toroidal mirrors, can introduce aberrations in focusing applications.
• Existing systems often involve multiple components for collimation and focusing.

Purpose of the Study:

• To analytically describe a new optical element capable of focusing cylindrical waves to a point focus.
• To provide an aberration-free solution for point-to-point imaging in beamline systems.
• To present fabrication-friendly solutions for the novel optical element.

Main Methods:

• Analytical derivation of the optical element's shape in a laboratory coordinate system.
• Development of approximate solutions based on bending geometric shapes (oblique circular cone, bent cylinder).
• Focus on achieving zero slope at the center for the ideal solution.

Main Results:

• An exact analytical solution for an optical element that focuses cylindrical waves to a point is presented.
• The proposed element provides completely aberration-free, point-to-point imaging.
• Approximate solutions are offered to aid in the fabrication and metrology of the optical element.

Conclusions:

• The new optical element offers a superior alternative to traditional toroidal mirrors for specific beamline configurations.
• This development enables precise, aberration-free imaging critical for advanced scientific instrumentation.
• The proposed fabrication methods facilitate the practical implementation of this advanced optical component.