Compact freeform illumination system design for pattern generation with extended light sources.

One of the major problems in freeform illumination design in a geometrical optics approximation is picture generation with extended light sources. In contrast to the freeform design with zero-étendue sources, the extension of the light source leads to the typical blurring effect and a contrast reduction of the required irradiance. This effect can be minimized by increasing the distance between the freeform surface and the light source, which according to étendue conservation, results in an impractically large projection optic. To tackle this problem, we propose a design concept consisting of the combination of a pattern-generating double freeform surface for collimated beam shaping, which is calculated for a zero-étendue light source, and an imaging projection system with a telecentric object space. The design concept works independently of the shape of the emission area of the light source and does not require a representation of the extended light source by several individual wavefronts. By interpreting the pattern blurring effect as a composition of a shift contribution and a distortion contribution, we show that both can be minimized simultaneously by an appropriate placement of the object plane of the imaging optics and by making the distance between both freeform surfaces as small as possible. This allows the calculation of compact, energy-efficient freeform illumination systems for picture generation with real extended light sources. We demonstrate the significant blurring reduction by designing a simple illumination system consisting of a collimation optic, a (zero-étendue) double freeform lens for collimated beam shaping, and a projection lens for the generation of the target distribution "Elaine" with an extended Lambertian emitter of 3 mm×3 mm extension and ±42 deg maximum opening angle. For a working distance to the projection system of 500 mm and a target area of 300 mm×300 mm, a relative blurring extension of 2% is estimated, compared to 23% for a single freeform projector with the same energy throughput and similar lateral extension. The influence of the doublefreeform thickness on the blurring reduction is demonstrated, and a summary of the design procedure for the developed design concept is given.

Double freeform illumination design for prescribed wavefronts and irradiances.

A mathematical model in terms of partial differential equations (PDE) for the calculation of double freeform surfaces for irradiance and phase control with predefined input and output wavefronts is presented. It extends the results of Bösel and Gross [J. Opt. Soc. Am. A34, 1490 (2017)JOAOD60740-323210.1364/JOSAA.34.001490] for the illumination design of single freeform surfaces for zero-étendue light sources to double freeform lenses and mirrors. The PDE model thereby overcomes the restriction to paraxiality or the requirement of at least one planar wavefront of the current design models in the literature. In contrast with the single freeform illumination design, the PDE system does not reduce to a Monge-Ampère type equation for the unknown freeform surfaces, if nonplanar input and output wavefronts are assumed. Additionally, a numerical solving strategy for the PDE model is presented. To show its efficiency, the algorithm is applied to the design of a double freeform mirror system and double freeform lens system.

Single freeform surface design for prescribed input wavefront and target irradiance.

In beam shaping applications, the minimization of the number of necessary optical elements for the beam shaping process can benefit the compactness of the optical system and reduce its cost. The single freeform surface design for input wavefronts, which are neither planar nor spherical, is therefore of interest. In this work, the design of single freeform surfaces for a given zero-étendue source and complex target irradiances is investigated. Hence, not only collimated input beams or point sources are assumed. Instead, a predefined input ray direction vector field and irradiance distribution on a source plane, which has to be redistributed by a single freeform surface to give the predefined target irradiance, is considered. To solve this design problem, a partial differential equation (PDE) or PDE system, respectively, for the unknown surface and its corresponding ray mapping is derived from energy conservation and the ray-tracing equations. In contrast to former PDE formulations of the single freeform design problem, the derived PDE of Monge-Ampère type is formulated for general zero-étendue sources in Cartesian coordinates. The PDE system is discretized with finite differences, and the resulting nonlinear equation system is solved by a root-finding algorithm. The basis of the efficient solution of the PDE system builds the introduction of an initial iterate construction approach for a given input direction vector field, which uses optimal mass transport with a quadratic cost function. After a detailed description of the numerical algorithm, the efficiency of the design method is demonstrated by applying it to several design examples. This includes the redistribution of a collimated input beam beyond the paraxial approximation, the shaping of point source radiation, and the shaping of an astigmatic input wavefront into a complex target irradiance distribution.

Ray-mapping approach in double freeform surface design for collimated beam shaping beyond the paraxial approximation.

Numerous applications require the simultaneous redistribution of the irradiance and phase of a laser beam. The beam shape is thereby determined by the respective application. An elegant way to control the irradiance and phase at the same time is from double freeform surfaces. In this work, the numerical design of continuous double freeform surfaces from ray-mapping methods for collimated beam shaping with arbitrary irradiances is considered. These methods consist of the calculation of a proper ray mapping between the source and the target irradiance and the subsequent construction of the freeform surfaces. By combining the law of refraction, the constant optical path length, and the surface continuity condition, a partial differential equation (PDE) for the ray mapping is derived. It is shown that the PDE can be fulfilled in a small-angle approximation by a mapping derived from optimal mass transport with a quadratic cost function. To overcome the restriction to the paraxial regime, we use this mapping as an initial iterate for the simultaneous solution of the Jacobian equation and the ray mapping PDE by a root-finding algorithm. The presented approach enables the efficient calculation of double freeform lenses with small distances between the freeform surfaces for complex target irradiances. This is demonstrated by applying it to the design of a single-lens and a two-lens system.

Ray mapping approach for the efficient design of continuous freeform surfaces.

The efficient design of continuous freeform surfaces, which maps a given light source to an arbitrary target illumination pattern, remains a challenging problem and is considered here for collimated input beams. A common approach are ray-mapping methods, where first a ray mapping between the source and the irradiance distribution on the target plane is calculated and in a subsequent step the surface is constructed. The challenging aspect of this approach is to find an integrable mapping ensuring a continuous surface. Based on the law of reflection/refraction and an integrability condition, we derive a general condition for the surface and ray mapping for a collimated input beam. It is shown that in a small-angle approximation a proper mapping can be calculated via optimal mass transport - a mathematical framework for the calculation of a mapping between two positive density functions. We show that the surface can be constructed by solving a linear advection Eq. with appropriate boundary conditions. The results imply that the optimal mass transport mapping is approximately integrable over a wide range of distances between the freeform and the target plane and offer an efficient way to construct the surface by solving standard integrals. The efficiency is demonstrated by applying it to two challenging design examples, which shows the ability of the presented approach to handle target illumination patterns with steep irradiance gradients and numerous gray levels.

Nonlocal effects: relevance for the spontaneous emission rates of quantum emitters coupled to plasmonic structures.

The spontaneous emission rate of dipole emitters close to plasmonic dimers are theoretically studied within a nonlocal hydrodynamic model. A nonlocal model has to be used since quantum emitters in the immediate environment of a metallic nanoparticle probe its electronic structure. Compared to local calculations, the emission rate is significantly reduced. The influence is mostly pronounced if the emitter is located close to sharp edges. We suggest to use quantum emitters to test nonlocal effects in experimentally feasible configurations.