Criteria for correction of all aberrations with quadratic field dependence

Aberrations of imaging systems can be described using a polynomial expansion of the dependence on field position. Aberrations on axis and those with linear field dependence can be calculated and controlled using Fermat's principle and the Abbe Sine Condition. We now present a powerful new set of relationships that fully describe the aberrations that depend on the second power of the field. A simple set of equations, derived using Hamilton's characteristic functions, which we call the Pupil Astigmatism Criteria, use on-axis behavior to evaluate and control all aberrations with quadratic field dependence and arbitrary dependence on the pupil. These relations are explained, validated, and applied to design optical systems that are free of all quadratic field dependent aberrations.