Geometrical optics and optimal transport.

The Fermat principle is generalized to a system of rays. It is shown that all the ray mappings that are compatible with two given intensities of a monochromatic wave, measured at two planes, are stationary points of a canonical functional, which is the weighted average of the actions of all the rays. It is further shown that there exist at least two stationary points for this functional, implying that in the geometrical optics regime the phase from intensity problem has inherently more than one solution. The caustic structures of all the possible ray mappings are analyzed. A number of simulations illustrate the theoretical considerations.

Variational principle in optics II: Dissipative wave equations.

The problem of phase retrieval from intensity measurements is examined for the case of dissipative wave equations. Unlike the conservative case, it is not clear if and when the problem is solvable at all. We provide two solutions. First, it is shown that, for a certain class of dissipating potentials, the problem can be fully solved by converting it through a simple transformation to the framework of the weighted least action principle. Second, for all other dissipating potentials, a deep result from the theory of elliptic partial differential equations is used to show that the problem is always solvable up to a scaling of one of the measured intensities. Moreover, the solution in this general case can be obtained by solving a Monge-Ampere type differential equation. Two numerical examples are given to illustrate some of the theoretical considerations.

Intensity control with a free-form lens.

A family of free-form lenses for intensity control is designed. The lens can shape an incident collimated beam with a given intensity distribution I1 into a new collimated beam with intensity distribution I2. No symmetry is assumed for the two intensity profiles. The key idea is that the lens design problem can be formulated and solved in terms of an optimization process involving a specific action functional. It is further shown that the free-form lens can be manufactured by a surfacing process using a convex tool.

A variational principle in optics.

We derive a new variational principle in optics. We first formulate the principle for paraxial waves and then generalize it to arbitrary waves. The new principle, unlike the Fermat principle, concerns both the phase and the intensity of the wave. In particular, the principle provides a method for finding the ray mapping between two surfaces in space from information on the wave's intensity there. We show how to apply the new principle to the problem of phase reconstruction from intensity measurements.

Differential relations for the imaging coefficients of asymmetric systems.

We consider imaging properties embedded in the point eikonal for first-order asymmetric optical systems. We provide geometrical interpretations for the coefficients of the eikonal functions and proceed to show that there exist differential relations between them. The differentials are computed with respect to the position of the reference planes in the object or image spaces.