Application of the 2D etendue conservation to the design of achromatic aplanatic aspheric doublets

Conservation of etendue or phase space volume has been a useful tool in nonimaging design and analysis. It is one of the Poincare's invariants associated to any Hamiltonian system. It expresses that the phase space volume of a region representing a bundle of rays do not vary when the rays proceed along the optical system. Another of these invariants is the 2D etendue conservation in 3D optical systems. This invariant can be expressed as the conservation along the ray trajectories of the differential form: dxdp+dydq+dzdr where x, y, z are position coordinates and p, q, r are the conjugate variables in the Hamiltonian formulation. When the optical system is frequency dependent (through the dependence of the refractive index of omega) or it is time dependent, then the Hamiltonian formulation must include two new variables: t (time) and its conjugate variable -omega. The application of the 2D etendue conservation to this new set of variables allows formulating the conditions for achromatic designs in a simple way. The results are coincident with Conrady's formula and its simplicity permits a direct application to the design of achromatic lenses. We have applied these concepts to the design of achromatic aplanatic aspherical doublets, where the aplanatic condition means free of spherical aberration and circular coma of all orders and the achromatic condition means that the doublet is aplanatic for wavelengths in a neighborhood of the design wavelength. Several examples of these designs are given.