Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems.

We propose a method for designing refractive optical elements for collimated beam shaping in the geometrical optics approximation. In this method, the problem of finding a ray mapping is formulated as a linear assignment problem, which is a discrete version of the corresponding mass transportation problem. A method for reconstructing optical surfaces from a computed discrete ray mapping is proposed. The method is suitable for designing continuous piecewise-smooth optical surfaces. The design of refractive optical elements transforming beams with circular cross-section to variously shaped (rectangular, triangular, and cross-shaped) beams with plane wavefront is discussed. The presented numerical simulation results confirm high efficiency of the designed optical elements.

Beam shaping with a plano-freeform lens pair.

It has been shown in [Arch. Rational Mechanics and Analysis 201, 1013 (2011)]. Such approach provides a rigorous methodology for designing freeform optics for irradiance redistribution. In this paper, this approach is applied to design of a laser beam shaping system with two lenses.

Controlling light with freeform multifocal lens designed with supporting quadric method(SQM).

The SQM is a versatile methodology for designing freeform optics for irradiance redistribution. Recently it has often been applied in design of freeform lenses, mirrors and diffractive optical elements. Still, many questions regarding theory and performance of optics designed with the SQM are open. Here we investigate theoretically plano-freeform refractive lenses designed with the SQM when an incident collimated beam must be transformed into a beam illuminating with prescribed irradiances a large number of pixels on a flat screen. It is shown that a lens designed for such task with the SQM operates as a multifocal lens segmented into subapertures with focal lengths providing accurate control of the irradiance distribution between pixels. These subapertures are patches of hyperboloids of revolution. Two different designs are possible, one of which defines a concave lens. Eikonal function for such lenses is also derived. As a proof of concept, we numerically analyze performance of a plano-freeform lens designed with the SQM for transforming a uniform circular parallel light into an image of A. Einstein represented by gray values at ≈ 38K pixels.

On the use of the supporting quadric method in the problem of the light field eikonal calculation.

A new iterative technique for calculating the eikonal function of a light field providing the focusing into a set of points is introduced. This technique is a modification of the supporting quadric method widely used for design of reflecting and refracting optical surfaces for generating prescribed illuminance distributions at given discrete set of points. As an example, we design a refractive optical element which focuses an incident beam into a set of points with energy pattern forming an image of a keyboard of a calculator. It is shown that the proposed technique is well-suited for the design of diffractive optical elements producing continuous intensity distributions within the scalar theory of diffraction. It is also shown that the calculated eikonal function is a good initial guess when designing diffractive optical elements using the iterative Gerchberg-Saxton algorithm.

On design of free-form refractive beam shapers, sensitivity to figure error, and convexity of lenses.

The problem of design of a two-lens optical system for reshaping the irradiance distribution of a laser beam in a prescribed manner is considered in the geometrical optics approximation. The presented design equations are derived in a rigorous manner and are applicable to free-form two-lens optical systems without any a priori symmetry assumptions on radiance profiles and beam cross sections. The obtained equations are applied to derive an equation defining sensitivity of the output radiation intensity to figure errors. This equation is applied to analyze sensitivity in several cases, including rotationally symmetric reshapers with nonrotationally symmetric figure error. The presented approach shows also that even in the general case two different designs are available for the same data. In one of these designs one lens is always concave or convex and the second is convex or concave, while in the second design the lenses may be neither convex nor concave. Since, in general, the surface lenses are aspherical, the availability of a design with convex/concave lenses is particularly important for fabrication.

Optical design of freeform two-mirror beam-shaping systems.

The problem of design of a two-mirror optical system for reshaping the irradiance distribution of a laser beam in a prescribed manner is considered in the geometrical optics approximation. The presented design equations are derived in a rigorous manner and are applicable to two-mirror optical systems not limited to radiance profiles and beam cross sections that are rotational or rectangular symmetric. The resulting mirrors are free-form surfaces not restricted by a priori constraints. Moreover, the presented approach shows also that even in the general case two different designs are available for the same data. In one of these designs the first mirror is always concave and the second is convex, while in the second design the resulting mirrors may be neither convex nor concave. Since, in general, the surface mirrors are aspherical, the availability of a design with convex and concave mirrors is particularly important for fabrication.