Enhancing brightness of Lambertian light sources with luminescent concentrators: the light extraction issue

: Luminescent concentrators (LC) enable breaking the limit of geometrical concentration imposed by the brightness theorem. They enable increasing the brightness of Lambertian light sources such as (organic) light-emitting diodes. However, for illumination applications, light emitted in the high-index material needs to be outcoupled to free space, raising important light extraction issues. Supported by an intuitive graphical representation, we propose a simple design for light extraction: a wedged output side facet, breaking the symmetry of the traditional rectangular slab design. Angular emission patterns as well as ray-tracing simulations are reported on Ce:YAG single crystal concentrators cut with different wedge angles, and are compared with devices having flat or roughened exit facets. The wedge output provides a simple and versatile way to simultaneously enhance the extracted power (up to a factor of 2) and the light directivity (radiant intensity increased by up to 2.2.)


Introduction
Light-emitting diodes (LEDs) are not only the new reference for general lighting, they have also proved to be reliable and economically-viable alternatives to halogen or arc lamps for more specific, bright illumination sources useful for e.g. automotive [1] or medical applications [2]. LEDs have also recently shown their potential as lower-cost alternatives to lasers or laser diodes for laser pumping [3][4][5]. More recently, Organic Light-emitting diodes have also emerged as flat, potentially flexible, large-area lighting devices with high potential in lighting, optical communications [6] or medical care [7]. Many of those novel applications triggered by the availability of these sources require the beam to be not only intense but also directional, which enables for example the light beam to be tightly focused. This is challenging with LED chips or OLED panels as their emission is not directional but quasi-Lambertian: their radiance -also referred to as "brightness", that is power per unit apparent area and solid angle -is independent of observation angle. The brightness theorem [8] indeed states that the irradiance (in W/m 2 ) produced at some remote location by a Lambertian source, will be always lower than the irradiance measured directly onto the exit surface of the emitter (i.e., the source power density). This is important pointing out that this result remains true whatever the nature (imaging or non-imaging) of the optical system between the source and detector, and no matter the number of individual emitters that one could possibly imagine to combine alongside. However, if light is not just refracted or reflected but instead absorbed and reemitted at a lower energy, through a luminescence process, overcoming the brightness theorem becomes possible: this is the key idea behind Luminescent Concentrators (LC). LCs have been proposed in 1976 in the context of photovoltaics to concentrate sunlight with a wide field of view and no need for tracking [9][10][11]. They consist in a slab of a solid-state fluorescent material surrounded by air (or more generally a lower index material) in which luminescence is guided by Total Internal Reflection (TIR) towards the edges, where the irradiance can reach values higher than the irradiance of incoming radiation by typically one order of magnitude in most practical cases. More recently, LCs have been used in association with LEDs disposed on the top largest surface to produce irradiances that could not be accessible with LEDs alone: this for instance enabled Ti-Sapphire lasers to be pumped for the first time by LEDs [12]. This example shows that the association of planar Lambertian LED light sources with LCs opens a novel route for high-brightness illumination sources that furthermore extends the wavelength capabilities of available LEDs. However, using LED + LC combinations to build bright directional sources requires that the radiation emitted in a high-index material is eventually outcoupled to air, which raises the important issue of light extraction, a long-studied issue in other contexts like device design of LEDs [13], OLEDs [14] or scintillators [15].
In planar light-emitting devices, classical strategies for light extraction include e.g. surface texturing [16], or surface modification with microlens arrays [17], networks of inverted micropyramids [18] or photonic crystal nanostructures [19]. For LCs based on plastic or crystalline slabs, the realization of such micro or nanostructures is both complicated and costly. The idea to change the geometry of the LC on the macroscopic scale has been investigated by De Boer et al. [20] who proposed to use a Compound Parabolic Concentrator (CPC) attached to the edge: the CPC both increases the emitting area and decreases the emission solid angle (as the étendue is conserved in a passive optical device) but it offers a way out for trapped rays (defined in following section): in the end, the outgoing radiation has a reported brightness that is 4.5 times higher than the brightness of LED illumination, and is more directional.
In this paper we study how more cost-effective and universal techniques can be used to improve the extraction capabilities and beam characteristics of concentrators that are used in the context of illumination. Firstly (section 2), we present a simple graphical representation in k space that enables an easy classification of rays inside a concentrator, offering a convenient tool for making rapid estimates but also understanding what happens in situations where the concentrator loses some of its symmetries. We then present the experimental methods and compare the measurements of concentration factors of a simple rectangular polished slab concentrator with a concentrator in which the output facet is simply frosted (section 3). In section 4, we investigate an original and simple way to improve light extraction from luminescent concentrators, applicable to any material, consisting in cutting the exit facet with a wedge. We studied Ce:YAG single crystals, chosen for their attested very good efficiency as luminescent concentrators under blue LED excitation [21]. The impact of the design on the achievable gain in extracted power, intensity and radiance (brightness) will be finally discussed.

Light extraction in luminescent concentrators: representation of internal rays in k space
It is first instructive to gain some insights in the light extraction issue in the "classical" symmetric rectangular slab design, considered polished on its 6 orthogonal faces, to investigate how this design can be further modified to improve extraction. In this respect, an elegant and insightful picture consists in representing rays only by their direction (in k space). Usually the problem of extraction is considered in planar light sources (LEDs, OLEDs…), where one dimension is much smaller than the others: in this case, all internal rays that fall within the escape cones (and only 2 cones are considered, with axes normal to the source plane) are directly outcoupled without undergoing any total internal reflection before exiting, while all the others are referred to as "guided rays", and are subsequently either absorbed, scattered, or coupled to other surface modes. In a 3D macroscopic geometry all 6 faces can contribute to TIR, and escape cones have to be considered on every face of the concentrator.
for YAG, η trapped = 52%. Each Total Internal Reflection (TIR) changes sign of k x , k y or k z : the trajectory of a "trapped ray" in k-space is therefore figured by the 8 apices (blue dots in Fig. 1) of a parallelepiped inscribed in the sphere. When considering a ray that belongs to an escape cone (let's say the cone with k y <0, see red dots in Fig. 1), the k x and k z components can change sign many times when the ray is travelling but k y is conserved. The trajectory is then represented in k-space by a dot bouncing between 4 points with the same k y component, always remaining within the same escape cap, before the ray can eventually exit -the number of reflections undergone by a ray will obviously depend on the position of the luminophore with respect to the exit surface.
In practice, scattering and reabsorption complicates this simple sketch: rays that experience propagation paths that are longer than the reabsorption length will not necessarily escape by the facet defined by the initial ray direction. Trapped rays will acquire a finite lifetime and will not bounce "forever". In high-quality crystals with large Stokes shift, the reabsorption length can however be quite high. In the Ce:YAG samples used in our study (see next section), the reabsorption length, which is simply estimated here from the inverse of the average absorption coefficient weighted by the fluorescence spectrum, is 3.4 cm, while the typical slab dimensions are in the mm to cm range. As a general rule of thumb, this framework for representing concentrators is useful in cases where the reabsorption length is longer than the typical dimensions of the concentrators.
The concept is here introduced in the simplest case of a symmetric (3 pairs of mutually parallel facets) optically-isotropic medium surrounded by a homogeneous isotropic medium. It can be especially useful however in more complex situations: for instance, the sphere becomes ellipsoids in anisotropic media; the caps may also have different dimensions if the surrounding medium is not homogeneous, which is the case if the concentrator slab sits on top of a substrate, for example.
In these non-trivial cases, the "trapped ray" area is determined by mapping the zones that are not covered by any cap and by any cap mirror-image: indeed, one has also to remove from the trapped ray region all the rays that will find themselves into an escape cap after one or several reflections. This means that one has to pave the sphere with all the mirror images of the caps through all the symmetry planes (where TIR can occur) to determine what is finally the 'trapped ray' (uncovered) region.
From this example, one can see that breaking the symmetry is a simple and efficient strategy to eliminate the existence of trapped rays, and hence improve the fraction of rays that can potentially be outcoupled.
In this paper, we investigated how a simple wedge in one direction, appended to the small exit facet, can significantly improve light extraction.
In this graphical representation in k space, tilting the output facet by an angle β consists in sliding the escape "cap" of the side edge towards the "pole" (top face escape cone), as represented in Fig. 2. Not only will the rays belonging to this tilted escape cap emerge, but also all those which find themselves in this cap after TIR. As the faces perpendicular to z k are the largest -so that most of TIR events occur on those facets -the new escape cone has to be completed by its mirror image across z axis: tilting the facet therefore increases notably the surface of the sphere that is paved by escape cones. Note that the tilted facet is also still a possible surfa emerge any m These rays wo to the plane o several exit fa In the end images of the within a thin would be stra  " that exit has to be d reaches th. In the previous section, we estimated the fraction of photons outcoupled by the side facet to be 8% of the total number of photons generated inside a perfect lossless concentrator. As all incident photons are not absorbed and as all absorbed photons are not reemitted as fluorescence (i.e. pump absorption efficiency and fluorescence Quantum yield are below unity), we can expect a theoretical ratio of outcoupled photons over incident photons slightly lower than 8%: in fact, the quantum yield is as high as 97% in Ce:YAG and absorption is 97.5%, which does not make so much difference. However, the experimental value is almost twice higher than the expectation from these simple considerations. This can be attributed for some part to reabsorption, which reduces the number of trapped rays and redirects them in all extraction cones. It is also due for another part to scattering on defects on the surfaces and to diffraction around sharp corners and edges. It is possible to evaluate the relative importance of reabsorption and diffraction/scattering effects, by resorting to ray-tracing simulations, which can include the effect of reabsorption but do not go beyond ray propagation considered within the limits of geometrical optics. We used Light Tools software to model a perfectly polished LC with a refractive index n = 1.84, having passive losses corresponding to a measured 97% transmission over 100 mm of propagation in Ce:YAG. Reabsorption was taken into account by considering the medium as homogeneously doped with an isotropic emitter whose absorption and emission spectra were those experimentally measured, while the luminophore concentration was calculated from the experimental absorption at 450 nm. The simulation yields a photonic extraction efficiency of 11.3% (9.6% in power ratio), meaning that starting from the rough estimate of 8% based only on geometrical considerations, reabsorption accounts for an additional 3.3%. The remaining difference of 2.8% between the experimental 14.1% efficiency and the simulated value of 11.3% is accounted by all phenomena that are not taken into account by ray tracing simulations, i.e. scattering and diffraction. Adding some further scattering seems to be a straightforward and simple idea to increase the light extraction, an idea that we explored in order to compare it with the wedge structure presented in the next section. The symmetric polished slab of 49 × 9 × 1 mm dimensions was left polished on 5 sides and frosted on one of the smallest facets of dimension 9 × 1 mm. The exit facet was roughened using abrasive grinding over a brass tool in order to obtain a RMS roughness of 430 nm, measured using Atomic Force Microscopy. An increase in the extracted power of up to 50% was observed, corresponding to a concentration factor of 8. The intensity indicatrix was measured in one angular dimension with a simplified version of the experiment described by Parel et al. [23] that is shown in Fig. 3a.

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We then inve paths that are significant pe In order to the reference 20°, 40°, and    (Fig. 3), mulated and e s for allegedly ts having been hrough the top but also shows ased.
ch wedge angle (le ns correspond to o he wedged sam an the average LC shows ripp d by taking th only enhances ace, which can ngular width a ngles, twice as trator volume e TIR limit ang s the relative in s of the wedg s wedged con . 3 left) are sh gly angle-depe especially for experimental v y the same reas n ignored in surface is bot angle concent eft) and calculated one angular coord mple with β = e intensity mea ples around γ = he average val s intensity in p n be a useful fe at half maximu s much as the r will hit the to gle θ lim = Arcsi ncrease of down P for the reference down to 52° for the LC with β = 60°. Table 1 summarizes the gain in terms of total extracted power and maximum intensity with respect to the reference unwedged concentrator.   In Table 1 is also reported the 'Brightness concentration factor', calculated as follows. The standard 'concentration factor' C defined as a ratio of irradiances at the output and input area is not an adequate metric when radiation is directional and detected after free-space propagation. It should be replaced by a brightness (or radiance) ratio of output and input radiation fields, or C B = B out /B in . In general, this brightness enhancement factor will depend on the direction of observation, except of course in the case of an angle-independent brightness, that is for a Lambertian source. When both input and output fields are Lambertian, the brightness concentration factor simply equals the classical concentration factor C. It is the case for our reference sample, so that where C ref is the usual concentration factor for the unaltered symmetric slab, g(γ) the gain in intensity at observation angle γ compared to the reference rectangular slab at normal incidence, and the last term a tilting term that takes into account the modification of apparent area. Table 1 reveals that highest brightness is obtained for β = 40° as a compromise between gain in intensity and increase of apparent area. It however remains slightly lower than the brightness achievable with the frosted device of maximal RMS roughness. Hence, depending on the parameter that one wishes to optimize (total extracted power, intensity in a specific direction or brightness), the optimal design will not be the same. It has also to be noticed that a simple wedge in one direction obviously sharpens the beam in only one direction while keeping the emission pattern in the perpendicular direction coarsely Lambertian. It is illustrated by the simulation of the 3D emission pattern for the 60°wedged LC reported in Fig. 6.
It is interesting to note that the apparently complex 3D intensity profile is indeed straightforward to interpret when looking back at the angular 3D representation. As soon as the wedge angle is higher than the critical angle, escape cones overlap. Rays that belong to the intersection of two different cones will statistically preferentially exit through the largest surface. Here, the small tilted edge facet and the large top surface respective escape cones have a strong overlap (for α > −3° using notations of Fig. 6 as β = 60° > θ crit = 33°) whose shape is easily bottom of the flat facet opp reflections on the visual rep at least quali resorting to so   Fig. 7(a)), we the middle of e) TIR on the nts (relatively t line). If we ex observe in the IR overlap wh of the energy in Fig. 3