Equiangular spirals provide lossless bends for lightpipes

High efficiency optical throughput is necessary for optical instruments and equipment, including lightpipes: non-imaging optical elements that guide light from one place to another. Typical applications of lightpipes include projector-engine illumination,1 LCD backlight systems,2 and automobile dashboard lighting.3 The shape of these elements vary but the design usually requires bends. Lightpipes can be optimized in a number of ways, e.g., controlling the light angular distribution or spatial distribution. One long-standing issue for lightpipes involves optimizing efficiency by reducing light leakage, especially at bends. Our recent work shows that by using equiangular-shaped spirals, lightpipes can guide light flux with arbitrary bend angles and no leakage, and this design method can also be extended to optical elements that split and mix light.4, 5 A lightpipe uses total internal reflection to guide light. Therefore, the incident angle of a light ray at the lightpipe’s guiding surface should be greater than or equal to the total internal reflection angle, θc , to keep light from leaking out. Hence, the angular distribution of guided rays inside the lightpipe is between -θc to θc . Thus, we can limit our discussion to identify a basic no-loss bent lightpipe geometric unit, which transfers incoming light with an angular distribution between -θc to θc to the other end of a lightpipe without leaks. The most crucial ray to be guided in the bent lightpipe is the ray incident from an inner bend point to the outer surface on the principle section. The key idea of our design is to choose an outer surface that ensures that all critical rays have an incident angle greater than critical angle, θc , so that they can be guided without any light loss. The natural geometric form of the nautilus, a type of mollusk, inspired our choice of shape for our no-loss bent lightpipe (see Figure 1). The shape of the nautilus’s shell is an equiangular spiral, which has a fixed angle between the ray from the center and the surface tangent. As shown in Figure 2, the proposed lossless bent lightpipe contains three parts. First, the straight AB part Figure 1. The general shape of a nautilus is an equiangular spiral.