In designing imaging optical systems, the primary task is to correct aberrations. Aberrations are deviations from perfect imagery. They depend on both the filed size and pupil position. When the Constant Optical Path Length (OPL) condition is satisfied, an optical system is free of all orders of spherical aberrations, which have zero field dependence. When the Abbe Sine condition is satisfied, all the aberrations with linear filed dependence are corrected. The Abbe Sine condition does not involve any off-axis ray properties, but it predicts the correction of off-axis aberrations.; We go one step beyond the Constant OPL condition and the Abbe Sine condition. By using Hamilton's characteristic functions, we developed a set of criteria for correcting the aberrations with quadratic field dependence and all orders of pupil dependence. These criteria involve only properties of the rays originating from the on-axis object point as the Abbe Sine condition does. Using these criteria, we analyzed some known designs and obtained new information about these designs. We also developed an algorithm to implement the criteria in designing well-corrected novel optical systems. Even when the criteria are not exactly satisfied, we now have a way to predict the residual quadratic field-dependent aberrations without tracing rays from any off-axis object point. We extended the Hamiltonian treatment to bilateral systems and developed similar criteria for correcting the quadratic field-dependent aberrations for this type of system.
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