Second-order semi-aplanatic freeform optics as a limit case of the SMS 3D design method.

Second-order semi-aplanatism provides better imaging quality along a line of the object plane close to a point than conventional aplanatic optics, which is of interest in applications with high aspect ratio sensors. Designing an optic with second-order semi-aplanatism requires the use of freeform surfaces, and can be done as a limit case of the SMS 3D design method applied to stigmatically image 3 collinear object points. The algorithm for this specific design problem is described and a lens example with 3 freeform surfaces is designed and analyzed.

Line-focus solar concentration 10 times higher than the 2D thermodynamic limit.

Line-focus solar concentrators have traditionally been limited by the 2D concentration limit due to the continuous translational symmetry in these systems. This limit is orders of magnitude lower than the 3D limit, severely limiting the achievable concentration ratio compared to point-focus systems. We propose a design principle for line-focus solar concentrators that bypasses this 2D limit, while maintaining a trough-like configuration and only requiring single-axis external solar tracking. This is achieved by combining the concept of étendue squeezing with the concept of tracking integration. To demonstrate the principle, we present a design example that achieves a simulated average yearly efficiency of 80% at a geometric concentration of 335x under light with a ±9mrad angular distribution and horizontal single-axis external tracking. We also show how the same design principle can achieve a line-focus with 1563x geometric concentration at 90% efficiency if design constraints are relaxed by foregoing tracking-integration and assuming two-axis external solar tracking. This design principle opens up the design space for high-concentration line-focus solar concentrators, and may contribute to a reconsideration of the trade-off between concentration and acceptance angle in such systems.

Three surface freeform aplanatic systems.

We address, in detail, the system of differential equations determining a freeform aplanatic system with illustrative examples. We also demonstrate how two optical surfaces, in general, are insufficient in achieving freeform aplanatism through the use of integrability condition for a given reflective freeform aplanatic configuration. This result also alludes to the fact that a freeform aplanatic system fulfills a broader set of conditions than its rotationally symmetric counterpart. We also elaborate on the above results with two illustrative examples (1) A semi aplanatic system which satisfies the generalized sine condition in only one direction and (2) A fully freeform aplanatic reflective system.

Analytical solution of an afocal two freeform mirror design problem.

We investigate a new afocal two freeform mirror design problem in first order optics. The resulting first-order partial differential equations for the freeform two mirror system have an analytic solution with the sole condition that the x-y and x'-y' axes are parallel. Two selected solutions are presented. One of them is semiaplanatic (fulfilling the aplanatic condition only for the x-coordinates), while the other is, to our knowledge, the first example of an aplanatic two-mirror system without rotational symmetry.

Freeform aplanatic systems as a limiting case of SMS.

We present here aplanatic systems in 3D geometry as a limiting case of a SMS 3D design. We extend the basic formulations governing rotationally symmetric aplanatic systems to freeform aplanatic systems and provide a formal proof that a SMS 3D design in the limiting case of 3 coincident points leads to a freeform aplanatic system.

Multifield direct design method for ultrashort throw ratio projection optics with two tailored mirrors.

In this work, we present a multifield direct design method for ultrashort throw ratio projection optics. The multifield design method allows us to directly calculate two freeform mirror profiles, which are fitted by odd polynomials and imported into an optical design program as an excellent starting point. Afterward, these two mirrors are represented by XY polynomial freeform surfaces for further optimization. The final configuration consists of an off-the-shelf projection lens and two XY polynomial freeform mirrors to greatly shorten the regular projection distance from 2 m to 48 cm for a 78.3 inch diagonal screen. The values of the modulation transfer function for the optimized freeform mirror system are improved to over 0.6 at 0.5 lp/mm, in comparison with its rotationally symmetric counterpart's 0.43, and the final distortion is less than 1.5%, showing a very good and well-tailored imaging performance over the entire field of view.

Design of compact and ultra efficient aspherical lenses for extended Lambertian sources in two-dimensional geometry.

The energy efficiency and compactness of an illumination system are two main concerns in illumination design for extended sources. In this paper, we present two methods to design compact, ultra efficient aspherical lenses for extended Lambertian sources in two-dimensional geometry. The light rays are directed by using two aspherical surfaces in the first method and one aspherical surface along with an optimized parabola in the second method. The principles and procedures of each design method are introduced in detail. Three examples are presented to demonstrate the effectiveness of these two methods in terms of performance and capacity in designing compact, ultra efficient aspherical lenses. The comparisons made between the two proposed methods indicate that the second method is much simpler and easier to be implemented, and has an excellent extensibility to three-dimensional designs.

Simultaneous multiple surface design method for diffractive surfaces.

The simultaneous multiple surface (SMS) design method is extended to include design of diffractive optical surfaces besides refractive and reflective ones. This method involves the simultaneous and direct (no optimization) calculation of diffractive and refractive/reflective surfaces. Using the phase-shift properties of diffractive elements as an extra degree of freedom, two rays for each point on each diffractive surface are controlled. Representative diffractive systems designed by the SMS method are shown.

Modal wavefront estimation from its slopes by numerical orthogonal transformation method over general shaped aperture.

Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.

Direct design of aspherical lenses for extended non-Lambertian sources in two-dimensional geometry.

Illumination design for extended sources is very important for practical applications. The existing direct methods that are all developed for extended Lambertian sources are not applicable to extended non-Lambertian sources whose luminance is a function of position and direction. What we present in this Letter is to our knowledge the first direct method for extended non-Lambertian sources. In this method, the edge rays and the interior rays are both used, and the output intensity at a given direction is calculated to be the integral of the luminance function of all the outgoing rays at this direction. No cumbersome iterative illuminance compensation is needed. Two examples are presented to demonstrate the elegance of this method in prescribed intensity design for extended non-Lambertian sources in two-dimensional geometry.

Double aspheric imaging design with unconstrained object to image mapping.

An imaging design approach for optical systems consisting of two aspheres which is free of astigmatism is presented in this paper. A set of implicit differential equations is derived from generalized ray tracing. The solution of the derived equations provides the profiles of the two aspheres as well as the object to image mapping. The obtained design can be used as a good starting point for optimization. Particular examples are given.

Prescribed intensity design for extended sources in three-dimensional rotational geometry.

Regulating the intensity distribution of an extended source to produce a prescribed illumination in three-dimensional (3D) rotationally symmetric geometry remains a challenging issue in illumination design. In this Letter, we present an effective method focusing on creating prescribed intensity designs for extended sources. By this method, a prescribed 3D intensity design is first converted into a two-dimensional intensity design for the extended source, a new approach is used to calculate the initial patch to generate a more stable design, and then a feedback strategy is employed to improve the performance of the aspherical lens in 3D rotational geometry. Three examples are presented to demonstrate the effectiveness of the proposed method in terms of performance and capacity for tackling complex designs.

Initial design with L(2) Monge-Kantorovich theory for the Monge-Ampère equation method in freeform surface illumination design.

The Monge-Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L(2)-Kantorovich (LMK) theory. An efficient approach is proposed to find the optimal mapping of the LMK problem. The characteristics of the new approach are introduced and the limitations of the LMK theory in illumination design are presented. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.

9-fold Fresnel-Köhler concentrator with Fresnel lens of variable focal point.

Non-uniform irradiance patterns over Multi-Junction Cells gives rise to power losses, especially when considering spectral irradiance distributions over different junctions. Thermal effects on Silicone-on-Glass lenses affect spectral irradiance distributions. A new Photovoltaic Concentrator (CPV), formed by nine optical channels, each one with a Köhler configuration, has been designed to overcome these effects at high concentrations for a large acceptance angle. A Fresnel Lens with a Variable Focal Point is proposed to prevent optical crosstalk in multichannel systems. When integrated into the concentrator, improves the acceptance angle. These designs are designed to fulfill the expected requirements of four junction CPV systems.

Improving performances of Fresnel CPV systems: Fresnel-RXI Köhler concentrator.

The optical design presented here has been done in order to achieve superior optical performance in comparison with the state-of-the-art Fresnel CPV systems. The design consists of a Photovoltaic Concentrator (CPV) comprising a Fresnel lens (F) as a Primary Optical Element (POE) and a dielectric solid RXI as a Secondary Optical Element (SOE), both with free-form surfaces (i.e. neither rotational nor linearly symmetric). It is the first time the RXI-type geometry has been applied to a CPV secondary. This concentrator has ultra-high CAP value ready to accommodate more efficient cells eventually to be developed and used commercially in future.

Improving performances of Fresnel CPV systems: Fresnel-RXI Köhler concentrator.

The optical design presented here has been done in order to achieve superior optical performance in comparison with the state-of-the-art Fresnel CPV systems. The design consists of a Photovoltaic Concentrator (CPV) comprising a Fresnel lens (F) as a Primary Optical Element (POE) and a dielectric solid RXI as a Secondary Optical Element (SOE), both with free-form surfaces (i.e. neither rotational nor linearly symmetric). It is the first time the RXI-type geometry has been applied to a CPV secondary. This concentrator has ultra-high CAP value ready to accommodate more efficient cells eventually to be developed and used commercially in future.

Influence of the characteristics of a light source and target on the Monge-Ampére equation method in freeform optics design.

It was previously demonstrated in [Opt. Lett.38, 229 (2013)] that the problem of freeform surface illumination design can be converted into a nonlinear boundary problem for the elliptic Monge-Ampére equation based on the ideal source assumption. But how the Monge-Ampére equation method is affected by the characteristics of the light source and target was not discussed there. This Letter systematically analyzes the influence of discontinuity, nonconvexity, and connectivity of light source and target on the Monge-Ampére equation method and presents some intrinsic features of this design method. These features are applied in practical examples in freeform optics design.

Single freeform surface imaging design with unconstrained object to image mapping.

An imaging design approach which is free of third-order astigmatism for one freeform optical surface and the image is presented in this paper. A set of differential equations is derived from generalized ray tracing. The solution of the above derived equations provides the anastigmatic freeform optical surface, the image surface and the object to image mapping. The obtained design can be used as a good starting point for optimization. As an example, a reflective freeform surface is designed for a single reflective Head Mounted Display (HMD). This example has a 3 mm pupil, 15mm eye clearance, 24-degree diagonal full field of view, and the final design yields an average MTF of 62.6% across 17 field points.

Free-form optics for Fresnel-lens-based photovoltaic concentrators.

The Concentrated Photovoltaics (CPV) promise relies upon the use of high-efficiency triple-junction solar cells (with proven efficiencies of over 44%) and upon high-performance optics that allow for high concentration concurrent with relaxed manufacturing tolerances (all key elements for low-cost mass production). Additionally, uniform illumination is highly desirable for efficiency and reliability reasons. All of these features have to be achieved with inexpensive optics containing only a few (in general no more than 2) optical elements. In this paper we show that the degrees of freedom using free-forms allow the introduction of multiple functionalities required for CPV with just 2 optical elements, one of which is a Fresnel lens.

Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces.

The two-dimensional analytic optics design method presented in a previous paper [Opt. Express 20, 5576-5585 (2012)] is extended in this work to the three-dimensional case, enabling the coupling of three ray sets with two free-form lens surfaces. Fermat's principle is used to deduce additional sets of functional differential equations which make it possible to calculate the lens surfaces. Ray tracing simulations demonstrate the excellent imaging performance of the resulting free-form lenses described by more than 100 coefficients.