Total internal reflection for illumination and displays
Total internal reflection (TIR) can occur when a light ray traveling in a transparent material encounters an interface with another transparent, but less optically dense material. The phenomenon is interesting not only because the reflection can be close to perfect, but because the effect depends critically on the angle of incidence. Although TIR is used widely, in this article we discuss two less well known applications with which we have considerable experience. The first example is a device known as a prism light guide, a hollow dielectric structure that can transport a large luminous flux for illuminating engineering applications. The second example is a form of electronic paper. In both cases, it is the unique characteristics of TIR that make these practical devices possible.
TIR is commonly said to occur when the angle of incidence exceeds the critical angle. Yet this notion is based on the inherent approximation of the ray model of light, an idea that is physically incorrect for any system of finite extent. Rather, the finite wavefront is equivalent to a superposition of a distribution of plane waves, which in turn have a range of propagation directions. Another fundamental shortcoming is that in any real system there is some degree of absorption loss in the materials. As a result, in reality, TIR is never truly total and the critical angle is never truly critical. Nevertheless, there is a profound quantitative difference between TIR and metallic reflection. In addition, in the case of TIR, the reflection can be modified by changing the external medium, which creates interesting possibilities.
It is worth considering these ideas a bit more carefully and quantitatively. Light rays traveling from plastic (n1 = 1.5) to air (n2 = 1.0) will be transmitted through the interface for small angles of incidence, and the reflectance grows rapidly as a function of angle as the angle of incidence approaches the critical angle (see Figure 1).
It is important to note how tiny the angular range is in the plot in Figure 1. For all angles of incidence below the critical angle, an antireflective coating can be designed for the interface to completely eliminate reflection at that particular angle. Above the critical angle, the situation is fundamentally different. The reflection will be total, and a dielectric coating cannot change that situation at all. It is remarkable that such a radically different state of affairs could exist across an infinitely thin mathematical boundary, and intuition might suggest that this would be unphysical.
Indeed it would be, as shown by the three additional curves. The second curve shows that for an uncoated surface, the actual surface reflectance does rise as the angle of incidence approaches the critical angle. The next curve shows the result of building in a small amount of angular spread (in this case about 1 microradian, equivalent to the diffraction spread of a 0.5m telescope!). It is apparent that this angular spread smoothes out all of the derivatives at the critical angle. Even without the angular spread, another real effect—absorption in the external medium—also has a similar smoothing effect, as shown by the third curve. Thus, while the effect of TIR is critical and has a surprising onset, it is clear that it does not really represent a basic unphysical discontinuity. Rather, it is a physical effect with surprising strength and usefulness.
In the field of illumination engineering, it is sometimes desirable to transport light from a bright source and distribute it where needed. Optical fibers effectively guide light for small-scale applications. But they are unsuitable for large-scale applications since the cost of a large solid-core fiber would be prohibitive. Moreover, the weight of such a structure would be impractical. For these reasons, hollow light guides are preferred. Prism light guides are hollow structures that pipe light by means of TIR. They can achieve high efficiency and uniform illumination.1,2
As shown in Figure 2, the wall of the hollow guide has molded in its external surface a series of isosceles right-angle prisms, running parallel to the axis of the guide. These prisms reflect, with near perfect efficiency, light rays that travel within a range of directions, provided their angular deviation from the guide axial direction is less than ∼30%. Prism light guides are currently widely used, primarily in applications where one wishes to separate the source from the region being illuminated. Common installations include tunnel lighting, industrial spaces such as warehouses, high ceiling areas, and architectural highlights on buildings.
We have been focusing on a related research program with the goal of cost-effectively illuminating office buildings with sunlight to reduce the required electrical lighting load.3 Many experimental systems have successfully demonstrated the required optical performance,4,5 but the capital costs of those systems have been prohibitively high compared with their energy savings. This new optical and mechanical system for core daylighting will have a sufficiently low life-cycle cost that the technology can be adopted in standard building construction.
An important feature of our new system is a modified dual-function prism light guide that not only guides sunlight and releases it in a controlled fashion into the space below, but also operates as an efficient fluorescent luminaire.5 The illumination levels provided by the new solar canopy system are well above the recommended standards, so under direct sunlight there is no need for additional electrical lighting. In addition, because the color-rendering properties are high, the quality of illumination is very good. Although it is premature to precisely quantify the energy savings that will result, we are optimistic that this system will, for the first time, yield truly economical daylight-based energy savings in the core regions of standard office buildings.
The CLEAR approach uses microfabrication technology to texture the rear surface of a polymer sheet with tiny hemispheres,9 as shown in the SEM in Figure 4(a). Light rays that strike the outer region of the hemisphere will undergo a series of reflections by TIR until they exit from the flat surface. As a result, the hemisphere displays a bright ring with a dark central spot when it is viewed through the flat side, as shown in the optical microscope image: see Figure 4(b). But these individual rings are invisibly small in a device based on this principle. The high reflectance of the hemisphere occurs even when it is viewed at an oblique angle, resulting in a display with a wide viewing angle.
A primary advantage of this technique is that switching from the reflective state to the absorptive state requires only a microscopic movement (1μm) of the absorbing material, such as electrostatically charged pigment particles or dye molecules that are electrophoretically moved into and out of the evanescent region associated with TIR. The reflectance level can be modulated quickly and efficiently, enabling the display to have both low power requirements and fast response times. This approach has the potential to create a very high brightness, video-rate image display with paperlike appearance.
Despite the fact that total internal reflection is a well-known and well-studied phenomenon in optics, there remain some subtle and interesting characteristics relating the nature of the critical angle that are generally not well known. Investigations in this area have led to the development of several new and useful applications for TIR. We hope that further work will be equally productive.
Lorne Whitehead is a professor in the Department of Physics and Astronomy, and the Vice President Academic and Provost at the University of British Columbia.
