Design of axisymmetrical tailored concentrators for LED light source applications Bart Van Giel*, Youri Meuret and Hugo Thienpont Applied physics and photonics department (TONA), Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussel, Belgium ABSTRACT In our contribution we present a solution to an important question in the design of a LED-based illumination engine for projection systems: the collimation of the LED light. We tested the principle in the modification of a common device in non-imaging optics, the compound Parabolic Concentrator. This CPC-like achieves a collection of 72% (ideal reflective coating presumed). This CPC-like was tailored by numerically solving an differential equation. This approach has some serious drawbacks. For a compact collection device with high collimation, an other approach is required. A more elegant design strategy will rely fully on geometrical principles. The result of our work is a compound collection lens that achieves a collection efficiency of 87%, assuming an ideal reflective coating and neglecting Fresnel losses. We study the performance of this device in detail. Further enhancements are suggested. Keywords: light-emitting diode, projection display, edge ray theorem, etendue, collimation, illumination, optimization, optical design, tailored surfaces
1 INTRODUCTION The increase of the emitted light flux of light emitting diodes (LEDs) makes it interesting to use these devices as light sources for projector applications. LEDs are small light sources with a narrow spectral emission band and a low operating voltage, which makes them a ideal light source for compact, very light and inexpensive projector applications. [1] However, LEDs have also an important disadvantage. The optical power per unit of étendue is significantly lower than e.g. an UHP lamp (at least 20 times) [2]. This makes it difficult to achieve a high luminance on the screen. For this reason we need to design optics that collect the light of the LED source both with high optical efficiency and high collimation. The LED we use in our study is a Luxeon V High Power LED from Lumileds. This LED consists of a rectangular 2x2mm² die encapsulated in an epoxy dome. Our problem was to efficiently transfer light from the rectangular die of the
Figure 1: Compact illumination engine for a LED-based projector encapsulated LED to a circular target with a certain acceptance angle. This acceptance angle has a square distribution in *
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direction cosine space, due to the symmetry of the system and the square form of the die. The etendue of the resulting light beam should be approximately the same as the etendue of the LED and the optical efficiency should be as high as possible. Collimation devices for this type of LEDs were proposed earlier. Munoz and all proposed a high efficiency LED collection lens, tailored with the edge-ray principle of non-imaging optics (RXI) [3]. Shatz and Bortz proposed a lens doublet that was designed by searching an optimal set of parameters (65 parameters) of an axisymmetrical configuration [4]. Kudaev optimized CPC-like and RXI-like devices [5] [6]. In this contribution we propose two solutions to the problem. The first solution is a single surface reflector. It's design is inspired by the compound parabolic concentrator, a classical solution in non-imaging optics. We will call them CPC-like reflectors. The surface of the reflector is tailored using non-imaging techniques. Our second solution is an extension and simplification of these principles. Its a compound collection lens that is compact, efficient and it reaches a high collimation of the light bundle with small angular distribution at the exit of the lens (10 degrees). Our lens will be characterized by a small set of parameters. To achieve this non-imaging techniques are used to tailor surfaces that transform the light distribution of the LED. We provide a simple method to tailor surfaces fully relying on basic principles. The purpose of our collecting device was to use it in a compact LED-based projector system. Directly after the collecting device we place a fly's eye integrator. A relay lens system projects the uniform beam onto the light valve. This results in a compact illumination engine for our projector. [7] (figure 1) In this paper we will use two quantities that describe the performance of collecting devices; collection efficiency and collimation.
= C= C1
tar
LED
(1)
E
2
E , max
The collection efficiency is the flux on the target in relation to the total flux of the LED. The collimation is the flux in a certain etendue in relation to the maximal possible flux in that etendue, given by etendue conservation law. The ray-tracing code we use in this contribution is the Advanced Systems Analysis Program (ASAP) from Breault. [8] This is a very flexible and powerful non-sequentially ray-tracing software package. Its script-driven interface and batch mode permits collaboration with other software packages. In this contribution we tailor lens profiles in MATLAB. These profiles can be imported directly into a ASAP script. ASAP returns its results into Matlab using a textfile. Other ASAP features include targets and sources with practically any angular or spatial distribution and free-form surfaces. In section two we will explain the design of the classical CPC and the etendue concept. In section three we will develop our 'CPC-like' and in the fourth section we design a compound collection lens.
2 Etendue and the Compound Parabolic Concentrator 2.1 Etendue The etendue of a light bundle is its integral over angular and positional extent,
E =∫ dLdMdxdy
(2)
, where L and M are the directional cosines with respect to the x- and y-axis [9]. The etendue conservation law states that in an optical system without losses etendue is preserved. In an optical system with maximum collimation, the etendue of the source and the target are the same and all the light of the source is transferred to the target.
Figure 2: Calculating etendue with optical Path Length A more elegant geometrical definition of etendue is reported in the literature. We will use it further in this contribution. In two dimensions, the etendue of the light bundle in figure 3 is calculated,
= ∗[ S1T2 ]−[ S1T1 ]
e 2
(3)
, where the square brackets denote optical path lengths. In three dimension this becomes,
E = [ S1T2 ]−[ S1T1 ] 2
2
4
(4)
This equation yields also in more complex situations with refractive index variations or reflections in the optical path. 2.2
The compound parabolic concentrator
Figure 3: (a) the Compound Parabolic Concentrator (b) using a differential equation to tailor the CPC (c) using the string method to tailor the CPC The compound parabolic concentrator is a well known device for etendue limited collimation of light. It's design is based on the edge ray theorem. This theorem give a sufficient condition to have etendue limited collimation between a source and a target with the same etendue. The phase space boundaries of the source (spatial and angular) should be displayed on the phase space boundaries of the target . In figure 4, the rays that intersect the entrance of the reflector at the extreme angle of the incoming telecentric bundle are projected on the edge of the target. The entrance diameter a' and the target a are connected through etendue conservation, a'
=
a
sin
The CPC and CPC-like devices can be tailored by solving a differential equation, fig 3b, [10]
(5
− = ⋅
dR d
, where
, R
,R
R tan
(6)
2
is a polar coordinate system and
,R
is the direction of the ray in the point
, R . is the desired direction of the ray after reflection (with respect tot the vertical direction) For the classical CPC, there is an analytical solution and
= ,R
(7)
It is easy to solve this equation numerically for simple reflectors such as the CPC-like of the next section. However for more complex situations with multiple reflective surfaces or with refractive media variations this becomes too difficult. For our compound collection lens we use an elegant alternative approach, the so-called string method of non-imaging optics, that will overcome the disadvantages of using the differential equation. [11] It is based on Fermat's principle. In general when a wavefront is connected to a point,
∫incomming wavefront ndL=constant absorber edge
(8)
where n is the index of refraction of the media where the ray passes and dL is the corresponding path length increment. Figure 3c is an illustration of the string method for the CPC. [12] The points of the CPC reflector are determined by the condition that the optical path length from a plane perpendicular to the incoming extreme angle beam to the edge of the absorber is a constant. This constant can be calculated easily from the geometry of the CPC. The string method can be used in situations with multiple reflection or with refractions and refraction index variation.
3 The CPC-like tailored reflector As we mentioned in our introduction, we want to design a efficient collimator for a LED source that exists essentially of a die and a dome encapsulating it. To get some experience in such systems we designed a CPC-like reflector. In such an illumination system the terms 'target' and the 'source' have to be switched in the previous sections. The classical CPC is not a suitable collimator for this type of device: •
The edges of the CPC can not touch the edges of the LED source. This implies that it should be possible to collimate all LED light into a certain angle, but that this device would not be etendue limited.
•
The edge of the LED is not projected to the desired extreme angle due to refraction at the LED dome
In our CPC-like solution, we take the refraction of the edge rays at the dome of the LED into account. This lead to a smaller reflector and thus such a reflector should have a better etendue performance. The design and performance study of this collecting device, happens in two stages. In the first stage, we design the curve of the reflector in a meridian plane (2D) using a simple geometrical model of the LED. In the second stage, we evaluate the performance of this device in ASAP. This has an important advantage. The design only counts on a simple model of the LED, while the performance evaluation can be done with a more sophisticated model of the LED including e.g. the LED's spectrum, an angular radiation pattern that differs from a Lambertian pattern.
Figure 4: The CPC-like design
Applying the CPC design principle to our problem (Figure 4), we have a device that projects one edge-ray of our source onto the desired extreme angle at the target. We use a simple geometrical model for the geometry of the used LED, •
The dome of the LED has a diameter of 5.6mm and a refraction index of 1.58 (epoxy),
•
The die has a height of 0.1mm, we suppose that both the bottom of the hemispherical dome and the die are in the same plane,
•
In our 2D design model, the width of the 'die' is equal to the diameter of a the circle with the same area as the square die, 2.26mm. With this diameter we design a device for a LED with a circular die with the same etendue as the actual etendue of the LED (with a square die). This should ensure a proper match between the 2D design and the 3D performance of the device.
Figure 5: The CPC-like vs. the CPC
We designed a CPC-like for an exit angle of 20 degrees. The reflector was tailored by numerically solving equation (6). To calculate a new point from a previous one , our algorithm traces the edge ray to the old point. It calculates the direction of the edge ray incident to that point. From the old point and the calculated slope, a new point can be calculated.
This method uses an implicit calculation of in equation (6) using geometry. To trace a ray through the ,R LED dome to a point on the reflector we look for the minimum for the optical path length. Fermat principle says that the path with the minimal length is the actual path of the light. We found an important disadvantage of the use of a differential equation to tailor profiles with this method. Your start at a point but do not know where the curve will end. In our rather complex situation the explicit solution of the differential equation is not really an option.
Figure 6: Collection efficiency of an ideal concentrator, the CPC-like 20 degrees and the CPC 20 degrees
Starting from the bottom of the reflector we calculate the whole CPC-like reflector profile. This was done with a very fine grid, 2000 points, to avoid inaccuracies. To check the accuracy, we recalculated the profile using the ,R points in the calculation of the slope. In figure 5 see the new CPC-like profile in comparison with a classical CPC.
Such a device does not reach the maximum concentration ratio. Due to the dome which encapsulates the die, it is impossible to let coincide the sources edges and the bottom of the reflector. There is a hole in the light distribution at the target. We found that there was an optimal distance for this device to work. The collection efficiency reached a maximal value of 71% at a distance of 23.5mm from the LED. On figure 6 we see a improvement by 4% in comparison with a classical CPC. The working distance is 10% shorter than the CPC. We found that for LED types with a smaller die in the same cap (such as the Luxeon III) the benefits of our CPC-like are more pronounced. .
4 A compound collection lens CPC-like collectors have a major disadvantage, they tend to be very long for small angles. The fly's eye integrator in our illumination system needs smaller angles than the 20 degrees of our CPC-like device to ensure a uniform illumination on the imager.
Figure 7: The final design of our lens
From the CPC-like device we learned that geometrical methods are for more simple to use than a differential equation. We already used a geometrical method to trace the light rays trough the LED cap. For the design of the compound collection lens we rely fully on formulas (3) and (8). This permits us to calculate the position of the intersections between the surfaces and the dimensions of the lens before any surface is tailored. In what follows we will show the final design and explain how we tailored this design step by step. Again, the design of the lens was done in two stages: a 2D design step and a 3D ray-tracing step.. Given the free parameters of the design, a MATLAB routine calculates the tailored profiles B and D. Then the lens is evaluated in ASAP. These steps are repeated in a MATLAB optimization loop until a (global) maximum for a merit function is reached. For this optimization we use the Nelder-Mead algorithm as it is implemented in MATLAB [13]. The merit function for this optimization was the flux that gets through the lens in the right angle. We optimize for etendue transfer between the die and the target. We call our method a smart optimization method. We use tailored profiles to accurately transform the light of the source. But etendue efficiency is often not the only consideration in the design of a collection lens. The presence of TIR at the reflecting surface or the need for an anti-reflective coating can influence the usability of our device. Such bounding conditions can be taken into consideration in our second ray-tracing step. In the generated ASAP code, we can insert realistic coatings on the refractive surfaces (e.g. a bare coating satisfying Fresnel's law.). In this way such design problems are part of the global merit function of a particular lens. This lens has two collecting parts. Light falling on surface A is refracted by this surface and then refracted by surface B. Light falling on surface C will be refracted by this surface and subsequently reflected by surface D and refracted by surface E. Surfaces B and D are tailored to redirect the light in the right direction. In the presented design parts C and E are straight lines and surface A is a spherical surface. In our design we have just a few degrees of freedom. The slopes of parts C and E are varied during the optimization loop. An important degree of freedom is the width of the middle part of our lens (W2). It is a measure for the amount of light that is falling on the inner part. In the next paragraph we will use this parameter to calculate the etendue falling on this inner part. This is crucial for the etendue efficiency of our device.
Figure 8: a) calculating the etendue falling on the inner part of the lens b) Tailoring the lens profile
With figure 8 in mind we will explain our tailoring method into detail. Firstly we ensure that the etendue of the light falling on surface A is equal to the collimated etendue leaving surface B. The condition that both values of the etendue should be equal is necessary because we want a device with maximal collimation. Each of the two subsystems of our lens captures a certain etendue from the LED and transform it into a collimated beam with the same etendue. In our optimization routine we make sure that this condition is met, even for intermediate designs. The etendue falling on surface A is calculated easily with equation (3) and is a function of z-position and the width of the spherical surface A,
= ⋅ n1⋅s4 s3 − n1⋅s1 s2
e1 2
(9)
where n1 denotes the refracting index of the dome and s1, s2, s3 and s4 denote the lengths of the light rays in the different media. It is straightforward to calculate the collimated etendue of the light leaving surface B,
e
= ⋅W2⋅ 4
sin
(10)
, where denotes the angle to which we want to collimate the light. From the condition that both should be equal the z-position (height) of the spherical surface A can be determined as a function of it's width, the radius of curvature of the lower surface, which are a free parameter in our design, and the thickness of the middle part of the lens (L3). This thickness is needed to support this part of the lens.
Figure 9: Tailoring the reflective surface D of the collection lens
The fact that we are dealing with a square die and we want to design our device in 2D encounters a difficulty. Which width of the die should be taken to have a good performance of the axisymmetrical 3D device? We briefly discussed this topic already in the context of the CPC-like reflector. We chose the width D of the die such that the area of a circle with diameter D is the same as the area of our square die. This choice ensures a proper etendue conversion in our real device. The choice of the angle was made the same way. For our collecting device we want, for reasons of symmetry, a square directional output. The angle
was chosen such that the solid angle of an cone with the half opening angle
is the same as the solid angle of the square directional distribution. Other choices for the diameters of the die in the 2D design were found to deliver less efficient devices. Secondly we start to tailor the lens profile starting from the left. Given the thickness and the height of the lens, we know already one point of each surface A and B. We calculate the total optical path length of the light ray trough these two points (OPL1). The path of this ray is drawn on figure 8b. The other points of the upper surface are determined from the condition that the optical path length of all rays from the edge point of the die, passing surfaces A and B, to the depicted wavefront should be equal to OPL1. Fermat's principle ensures that the edge point is properly connected to the desired wavefront. With this method and given the start conditions, we tailor the unique continuous surface that connects the edge point to the wavefront. We start in the left corner of the lower spherical surface. We calculate points of the upper surface starting from a set of 20 points of the lower surface. For each point P n , A we can calculate a point of the upper lens surface such that the total optical path length of the resulting ray is equal to OPL1. At the end of this procedure, there is still a gap in the upper surface. We complete the upper lens with a spherical lens part which connects smooth to the already calculated lens profile. For the design of the reflective surface of our compound collection lens we use an analogue method as for the middle part of the lens (figure 9). The total light path is now divided in five parts. The total optical path length from the edge of the die to the desired wavefront is kept constant. In the design we present here, the surfaces A and C are simple surfaces. Our geometrical method permits the use of more complex surfaces in lens designs. The only thing we need for that is a description of the surface in terms of one parameter between the two endpoints of the surface. We designed a collection lens for an output angle of 10 degrees. Prior to the design of the full compound collection lens, we studied the performance of the inner section in detail. We found that this part achieves very high collimation. Up to 94% of the light falling on surface A was redirected into the right angular distribution. We saw indeed a square directional distribution at the output of this inner lens. To benchmark our geometrical method for the tailoring of lens profiles, we compared the performance of the inner section with the performance of a optimized lens with aspherical (conical and 1st aspherical term) profiles. The collimation of the light by our tailored lens was 5% better than that of the optimized lens. This is because much degrees of freedom would be required to describe our tailored lens profile in an analytic form. In figure 10, we see the performance of the optimized compound collection lens. Ideal reflective and AR coatings on all surfaces are assumed. The lens reaches a collection up to 87.3%. This light is collected on a target with an etendue that is a little larger than that of the LED. This high value for the collection proves that with our method to tailor optical surfaces, we can design etendue efficient optics. The resulting design is very compact in comparison to single surface reflectors. The height of the lens is 20mm. A CPC-like design for this type of LEDs and an output angle of 10 degrees would be 70mm long. We also optimized a lens with a bare (with Fresnel reflections) coating on all refractive surfaces and a 90% reflective coating on the reflective surface. This lens reached a collection efficiency of 70%. The losses are due to the non-ideal reflective coating on surface D and Fresnel losses, particularly on surface E. Further work should be done to improve the design of our collection lens. We already mentioned the possibility to add more complex curves on the non-tailored surfaces (in our design A, C and E). This would increase the number of degrees of freedom in the optimization loop. Our method should be extended to RXI-like designs. This can possibly eliminate the need for a reflective coating on one of the surfaces.
5 Conclusions This work resulted in the design of an efficient collection device for an LED based projection illumination engine. The design method was fully relying on geometrical principles. A smart combination of optimization of some crucial parameters and tailored profiles permits a simple design which has a good performance. Our design strategy makes it possible to take non-ideal behaviour of the source and the surfaces of the device into consideration. Further enhancements are still possible.
Acknowledgments The authors would like to thank the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT) and of the Fonds voor Wetenschappelijk Onderzoek – Vlaanderen for their kind financial support. This work was supported by FWO in the context of the project 'New optical architecture for LCOS projectors'.
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