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 guides all critical angles like a normal straight lightpipe. Second, for an outer surface, BD, we chose the fixed angle of equiangular spiral surface to be greater than π/2 + θc, so that the incident angle of all critical rays will be greater than ∑c. For the third part, the inner surface of the bend, we also choose an equiangular shape, OE.
We verified the lossloss properties of the proposed lightpipe bend via the simulation shown in Figure 3. We assumed that the incoming light was from a point-like Lambertian source that emits 50,000 rays with angular distribution between -90° to +90°. Our simulation demonstrated that these surfaces provided lossless characteristics.
Optical elements similar to light pipes can be used to mix or split light fluxes. We illustrate two conceptual designs of the equiangular-spiral no-loss bent lightpipe in Figure 4: a no-loss light-splitting element could split light flux in any specified flux ratio without any light leakage; while a no-loss light-mixing element could mix light flux from several ports without any light leakage. In both designs, we used an equiangular-spiral bent lightpipe with a bent angle of 30° as the basic unit. We numerically verified that the flux ratios and number of entrance ports for both elements can vary without losing the lossless performance.
In conclusion, we developed a novel approach to bent lightpipes, based on the design of a nautilus, that eliminates light leakage.We used numerical simulations to explore their characteristics as well as the possibility of creating optical splitters and mixers using this basic design. Such elements are necessary for a variety of practical applications including concentrator designs6 and illuminators for projectors.7,8 Highly efficient lightpipes are an important development for the field of illumination and the approach based on equiangular spirals provides designers with a useful tool.
Shu-Chun Chu received her BS in physics from the National Kaohsiung Normal University, Kaohsiung, Taiwan, and her MS and PhD from the National Chiao Tung University, Hsinchu, Taiwan. She is currently an assistant professor at the Department of Physics at National Cheng Kung University.
Jyh-Long Chern received his BS and MS, both in physics, from the National Tsing Hua University, Hsinchu, Taiwan, and his PhD in Optical Science from the University of New Mexico. He is currently a professor at the National Chiao Tung University.
