Reprints Available Directly from the Publisher Photocopying Permitted by License Only Modeling Light-extraction Characteristics of Packaged Light-emitting Diodes

We employ a Monte Carlo ray-tracing technique to model light-extraction characteristics of light-emitting diodes. By relaxing restrictive assumptions on photon traversal history, our method improves upon available analytical models for estimating light-extraction efficiencies from bare LED chips, and enhances modeling capabilities by realistically treating the various processes which photons can encounter in a packaged LED. Our method is not only capable of calculating extraction efficiencies, but can also provide extensive statistical information on photon extraction processes, and predict LED spatial emission characteristics.


I. INTRODUCTION
Analytical models[I,2] developed in the past have been very useful for considering photon extraction from semiconductor LED chips in limiting cases.Taking advantage of high-speed computation platforms available today, we approach the problem with a numerical Monte Carlo ray-tracing technique which promises greater range of validity.Photon extraction from LED chips is difficult for several reasons.Absorption by the medium and contacts prevent many of the internally generated photons from ever reach- ing exposed chip surfaces in the first place.A sub- stantial fraction of the photons that do reach the surface end up being internally reflected because crit- ical angles at chip surfaces are typically small due to the large index of refraction mismatch between the semiconductor LED chip and its surrounding medium.Although each photon may encounter a number of these processes, it is straight-forward to keep track of photon traversal history numerically.Therefore Monte Carlo techniques well-suited for simulating photon extraction.

II. MODEL
Figure illustrates the T-1 3/4 LED lamp, which is used as the prototype structure in our Monte Carlo particle simulation.The particles used are classical photons (corpuscles) which obey the rules of geomet- rical optics.A simulation is performed with an ensemble of randomly generated photons, and the processes which each photon encounters (e.g., medium absorption, transmission/reflection at inter- faces) are governed by probabilistic models.We assume that in the LED chip photons are generated uniformly within a thin planar region at a height of z hjunc from the base of the chip, and are emitted in an isotropic spatial pattern.[ 1 We also assume mono- chromatic photon emission for simplicity.A trivial extension to our model would allow for the treatment of more general emission spectra.For each photon, a position vector on the emission plane and a direction vector are randomly generated.A ray-polygon inter- section algorithm [3] is used to compute the intersec- tion of photon trajectory with one of the six chip surfaces.Since a photon might be absorbed inside the chip before reaching the destination surface, we use a Monte Carlo routine to determine whether absorption takes place en route.Let the path length between the initial and final positions be A. The probability of a photon being absorbed in a distance A is given by where o is the medium absorption coefficient.We generate a random number R between 0 and 1; absorption takes place if R _< P(A).If a photon reaches a destination chip surface without being absorbed by the semiconductor, we check to see if it strikes either the top or the bottom contact, where contact absorption processes can occur.For simplicity, we assume that the contacts are totally absorbing, although par- tially absorbing contacts can be modeled quite easily.If the photon is not absorbed, we determine whether it is transmitted or reflected by generating a random number between 0 and 1, and comparing it to the transmission probability given by the FresneI equa- tions.[4] An internally reflected .photoncontinues on its path within the LED chip until it is eventually absorbed or transmitted.A photon emitted by the LED chip is then ray-traced to the reflector dish, and then to the epoxy lens.For speed and simplicity, we assume that the three stages of the simulation occur sequentially (chip, then reflector, then lens).There- fore photon re-entrance into the LED chip is not con- sidered, nor the possibility of internally reflected photons in the lens re-striking the reflector dish.The error associated with the sequential approximations has been estimated to be a few percent.Throughout the different stages of the simulation, extensive statis- tics are gathered for subsequent analyses.

III. RESULTS AND DISCUSSION
We demonstrate the potential usefulness of the Monte Carlo method by varying the junction height hjunc to examine the effect it has on light-emission characteristics.We also consider the effect of varying the value of epoxy index of refraction.We assume that the absorption coefficient for the semiconductor chip is a 10 cm-1.This corresponds to a weakly absorbing medium, with a characteristic absorption length 1/c of 1000 gm, or 3.75 times the width of the cubic chip.
Note that simulation of a realistic device would require that we use experimentally measured absorption coefficient spectrum o(E) for the LED material.
The index of refraction for the semiconductor chip is taken to be 3.1, and for the epoxy, both 1.5 and 1.7 are considered.Figure 2(a) shows the light-extraction efficiency for the LED as functions of the light-emit- ting junction height, which ranges from 0 to 265 gm (the height of the chip is 266.7 gm, or 1.05 rail.), for two different values of epoxy index of refraction.
Note that the efficiency for the n 1.5 case is consist- ently lower than the n 1.7 case by a significant amount.This is because the critical angle at the chip- epoxy interface is 29 for n 1.5, and 33 for n 1.7.(In practice, epoxy indices do not exceed 1.51.)In both cases, efficiency is low when the junction is near the bottom of the chip where most of the downward pointing photons generated at the junction are absorbed by the bottom contact.As the junction is raised, the efficiency increases until reaching a maxi- mum at around hjunc 190 tm, and then decreases by a small amount.At its maximum, the efficiency is nearly twice the hjunc 0 value.Figure 2(b) shows the emission patterns as functions of azimuthal angle 0 for a particular LED structure after each of the three simulation stages (chip, dish, and lens).The chip emission tends to be broadly distributed, the reflector dish brings all emission to 0 < 90 , and the lens tightly focuses a portion of the emission into 0 < 20 , while scattering the remainder to a much broader dis- tribution.
In summary, we employ a Monte Carlo ray tracing technique to model light-extraction characteristics of light-emitting diodes.By effectively utilizing readily available computational resources and relaxing restrictive assumptions on photon traversal history, our method improves upon available analytical mod- els for estimating light-extraction efficiencies from bare LED chips, and enhances modeling capabilities by realistically treating the various processes which photons can encounter in a packaged LED.Our method is not only capable of calculating extraction efficiencies, but can also provide extensive statistical information on photon extraction processes, and predict LED spatial emission characteristics.Simulations using our method can be performed very rapidly on modern workstations, making it a good candidate for an effective design tool.Several areas of our model could be improved to make it more realistic and ver- satile; a more detailed account of our model is given elsewhere [5].We believe that, with some refinement, our method could become a valuable LED design tool.