Light recycling can increase the brightness of light sources in apparent violation of the brightness theorem, which states that no image can appear brighter than its source. We explain this phenomenon and present an application that employs, as a physical model, the standard halogen lamp.1 Although not designed to facilitate light recycling, the lamp's coiled filament makes it a prime example with practical potential.
The size of optical illumination systems scales with that of the light source. The smaller the source, the smaller the system. For the future, this will mean brighter back-lit flat displays, for example, and pen-size beamers. From lithography to medical instrumentation, smaller sources will make possible optics of reduced size and simpler design. All that is needed are light sources of smaller dimensions yet with brightness comparable to present devices, or else brighter light sources of unchanged dimensions.
Simply increasing wattage to achieve greater brightness is limited by cooling requirements of the LED and material properties used. Wire temperature defines the brightness of any thermal light source, and tungsten, of which the wire filament is made, has the highest melting point (3410°C) of any metal. In view of these limitations, recycling is a viable option for improving the quality of emitted light. It increases source brightness without consuming more power.
The brightness of a light source is its radiance measured in watts per area within a given solid angle. Thermal sources are Lambertian emitters. The emitted solid angle 2π is a sphere surrounding the light source. Imagine one part of the solid angle reflected through the source as shown in Figure 1. Direct light and recycled light are superposed in the same phase space, because they come from the same location. The lamp appears to shine more brightly.
Figure 1. The principle of light recycling: recycled light is superposed onto direct light in the same phase space.
The total radiance of the lamp, however, has not changed. In fact, the amount of light emitted is reduced by the quantity absorbed by the source during the transit of the reflected light. We can also see that it is not possible to increase the brightness of the source ad infinitum, because more reflections and transits will cause greater absorption.
The property of light recycling in the coiled filament is largely geometrical. Consider a filament to be a physical model of the spectral radiance distribution of a light source consisting of geometric modules of the filament and its material characteristics. These two submodules, integrated with the thermal model, yield the radiation model. Viewed in geometric detail, location along the filament coil is parameterized by location in space. Recycling light from facing portions of the filament wire is important for calculating emitted radiance. We include polarization and specularly dependent reflection in metals. The material model deals with the thermal conductivity, electrical resistivity, heat capacity, and spectral emissivity of tungsten. The thermal model gives the temperature distribution along the filament as calculated by an instationary heat balance equation, taking into account ohmic heat and thermal conduction. Finally, for the radiation model, results of the submodules are integrated within Planck's law of spectral radiance. The result is dynamic spectral radiance L(x,φ,λ,I,t), where x is the location on the filament length, φ is the angle on the circumference of the wire, λ is the wavelength of the radiation emitted, I is the driving current, and t is time.
Inasmuch as parts of the geometry of the coil are concave, a fraction of the rays emitted from a surface element on the inside of the coil will strike the coil again. Indeed, this effect is the source of light recycling. We define the geometrical recycling factor ξ as that fraction of the emitted light from a surface element that returns to the source, with the assumption of Lambertian emittance. In this sense, light recycling is a purely geometrical phenomenon.
The recycling factor is determined via Monte Carlo ray tracing. Consider a surface element on the filament as a Lambertian source and the entire filament as the receiver. Radiated light from any surface element will either leave the configuration directly as emitted light, or strike other parts of the filament. The recycling factor ξ is this latter fraction. A continuous factor function is obtained via interpolation. Absorption limits brightness enhancement to a factor of about 2. The effects of factor ξ, and material emissivity Ε (equal to absorptivity, or 1 minus reflectivity for mirrors, as stated by Kirchhoff's law) on brightness enhancement are given in Figure 2. Typical reflectance of the wire at operating temperatures around 2900K is 0.55.
Figure 2. Brightness enhancement increases with the geometrical light recycling factor ξ, but decreases with the material emissivity Ε.
The brightness distribution of the filament in operation is shown in a photographic negative in Figure 3. Dark gray means high brightness. Clearly, the brighter parts of the coil, which lie within, are subject to geometrical light recycling. Note, however, that temperatures outside and inside the wire are nearly equal due to high conductivity thinness (0.1mm). Intensified brightness could not result from temperature differences.
Figure 3. In this photographic negative of the filament of a 20W halogen lamp, the darker parts are brighter.
Our model predicts spectral radiance in good agreement with measurements, as indicated in Figure 4 The light originating from inside the coil is approximately 30% brighter than the light from the surfaces on the outside of the coil. There is a significant dependence of the material properties on wavelength.
Figure 4. Modeled radiance vs. experimental spectroscopic measurement at the filament lamp indicates intrinsic light recycling.
Light recycling contributes significantly to the brightness of the coiled filament lamp. Any model of an incandescent lamp must consider light recycling in order to correctly predict spectral irradiance. Light recycling depends on the geometry of the coil and reflectance of the tungsten wire, with the laws of Planck and Kirchhoff central to our calculations. Our model can reproduce the dynamic behavior of radiance distribution with high accuracy and in positive agreement with a real source.
Apart from explaining the nature of light recycling in filament lamps, our model has potential practical applications. For example, a simulated source could be employed in ray-tracing packages. It would require small computational load or data storage space.
We wish to thank the German Federal Ministry of Education and Research (BMBF) for supporting this work in the framework of the RIOS collaboration under contract FKZ 13N8419. We would also like to thank the editors of SPIE, who made this article possible.
Ralf Leutz, Ling Fu, Harald Ries
Physics Department, Philipps-University
Ralf Leutz is scientific assistant in the Optics Workgroup, part of the Physics Department at Philipps University in Marburg, Germany. He has written a book on nonimaging Fresnel lenses, and is most interested in tailored optics for solar concentration, including fundamentals based on the thermodynamics of light.
Ling Fu has just finished her PhD on increasing the brightness of light sources in the Optics Workgroup. She intends to teach at the Institute of Technology in Harbin, China.
Harald Ries is professor of experimental physics at Philipps University. He is also president of OEC AG in Munich, Germany (www.oec.net).