## Abstract

Conventional condensers using rotational symmetric devices perform far from their theoretical limits when transferring optical power from sources such as arc lamps or halogen bulbs to the rectangular entrance of homogenizing prisms (target). We present a free-form condenser design (calculated with the SMS method) that overcomes the limitations inherent to rotational devices and can send to the target 1.8 times the power sent by an equivalent elliptical condenser for a 4:1 target aspect ratio and 1.5 times for 16:9 target and for practical values of target etendue.

©2008 Optical Society of America

## 1. Introduction

The main function of the condenser optics of a projection system is to collect the light from the source and transmit it to a homogenizing prism with a high average luminance. This prism will pass the light to the spatially modulating microdisplay, from which it goes towards the projection optics. A new conceptual design for condenser optics is presented herein. This device has been designed with the 3D SMS method [1–7], a detailed explanation of which has been recently given in [8]. The connection of the SMS method with other design methods can be found in references [4, 5].

Most conventional condensers use elliptic or parabolic mirrors (see Fig. 1). They perform far from the theoretical limits (calculated by etendue invariance) for sources such as arc lamps or halogen bulbs. Typical small displays in the 5–15 mm^{2}srad etendue range have collection efficiencies about 40–50% for the best condensers, although theory allows about 100% (collection efficiency defined as the ratio of power sent to the target to total source power).

To understand the limitations of conventional elliptic or parabolic condensers it is useful to consider the concept of the projection of source-images. A pinhole at the exit optical surface of the condenser will let pass a bundle of rays that bear the local image of the source (see Fig. 2). The limited collection efficiency for small etendues of a conventional elliptical condenser can be understood by examining its projected source images, which have two notable characteristics:

1. The size of the projected image varies (in meridian length m and sagittal width s) from point to point along the profile of the condenser exit aperture, not only because its rectangular shape (replicating the 4:1 elongated shape of the UHP arc) varies in apparent aspect ratio, but also because of the great variation in source-distance (*i.e.*, coma).

2. The elongated source-images rotate, due to the rotational symmetry of the mirror and the fixed orientation of the light source. These variously rotated bar-like arc images do not collectively fit well onto the usual rectangular targets.

Figure 2 shows the arc as a surface-emitting cylinder, which is a schematic simplification of the actual non-homogeneous luminance distribution of an arc Fig. 2(b). This simplification helps understand the inherent limitations of the elliptic condenser. Moreover, the etendue of the simplified arc is a better defined parameter than that of the real arc.

When the etendue of the target (i.e., of the homogenizing prism and consequently the etendue of the microdisplay) is much greater than the that of the source, the two aspects mentioned above (projected image size variation and images rotation) do not limit the collection efficiency, since the homogenizing prism aperture (shown as a dashed-line rectangle in Fig. 2(d)) will be much larger than all the projected arc images. This, however, implies a reduction of the average brightness of the source when seen from the target. The consequences of a low average brightness are the use of a large (and expensive) microdisplay and/or the use of an expensive optical projector. In the most cost-demanding applications the microdisplay etendue will be smaller and closer to that of the arc, and thus more vulnerable to the inefficiency shown in Fig. 2.

An interesting clarifying case is that in which both microdisplay and arc etendues are equal. There is no theoretical obstacle preventing a 100% collection efficiency although no such design is known. It can be easily seen that in this equi-etendue case the homogenizing prism entry aperture will have the same area as the average projected image (any ray of the target must come from the source). This makes clear that the variable size and the rotation of the images of the elliptical condenser do prohibit 100% collection efficiency. A condenser achieving such 100% value would necessarily meet the condition that all projected images are not only rectangular but also exactly match the contour of the prism entry aperture. This condition is general, i.e., it does not depend on the particular optical elements used in any such (hypothetical) 100% collection condenser.

A recent trend in improving collection efficiency for small etendues has been to reduce the arc etendue by reducing the gap between the electrodes. Furthermore, some optical designs have been developed to improve the collection efficiency by:

1. Reducing the arc etendue via a hemispherical mirror (concentric with the arc) which reflects half of the emitted light back through the arc (which is partially absorbent) and thus boosting its luminance [9, 10]. The half-etendue light from the hemisphere not covered by the mirror arc is then collected by a conventional elliptical reflector.

2. Creating a side-by-side image of the arc with a decentered hemispherical mirror, and creating a composite 1:1 image of that via a dual parabolic reflector [11].

3. The equalization of the meridian length of the projected arc images [12, 13] by correcting the elliptical mirror coma, using an aspheric reflector profile and an aspheric lens, both surfaces still being rotationally symmetric.

These optical approaches, apart from their complexity and technological challenges, have limited gain capability because their condenser optics is still restricted to rotational symmetry.

Other approaches to improve system efficiency include color recapture [14] and color scrolling [15] (which try to recover the losses produced by the color filtering in single microdisplay projectors), or polarization-recovery techniques [16] (which try to recover the 50% losses produced by the need for polarized light in LCD and LCoS systems). In these approaches, however, the resulting lamp etendue is accordingly increased (doubled and tripled in polarization and color recovery systems, respectively), further limiting the performance of small microdisplays that use classical condensers.

Regarding the state of the art of manufacturing condenser optics, all present systems are based exclusively on rotationally-symmetric surfaces. These are manufactured mainly from glass (due to its low cost) or by glass-ceramic (for higher thermal stability). The accuracy of both techniques is limited, so the manufactured profiles can differ substantially from those intended.

## 2. Design of the XX condenser

The simplest application of the SMS3D design method provides full control of two couples of wavefronts and the partial control of a third couple with just two optical surfaces (mirrors or dioptrics). Thus the size, position, and orientation of the projected source-images can be controlled to an unprecedented degree. SMS-designed dual-reflector devices are herein termed XX. In order to clarify the various XX condenser families, this section considers the design of an XX that must collect the light emitted by a rectangular flat source and transfer it to a rectangular flat target, in the geometric configuration shown in Fig. 3. The light source is placed at the y=0 plane and emits in the y>0 hemisphere of directions. The target is placed at a z=constant plane and will receive the emitted light in z>0 hemisphere of directions.

Assume for a while that the source size is small enough compared to the condenser size, and that the XX is able to image the source on the target so the following linear mapping holds (note that the SMS can guarantee a sharp image of 2 points, for instance A and B into A’ and B’, and a partial image of the remaining ones C and D):

where (x’,y’) is a point on the target, (x,z) is a point on the source (the same global coordinate system x-y-z for source and target is being used), and c_{1} and c_{2} are constants that define the mapping of the center of the source to the center of the target. This mapping implies that to a first-order approximation (valid for a small source) the SMS method provides an image-forming design whereby the light source is placed on the object plane and the target on the image plane. In Fig. 3, points A, B, C and D are object points while A’, B’, C’ and D’ are their corresponding image points. The diagonal of the matrix in the previous equation defines the magnifications of the optical system. Constants M and N, are defined as: magnification M: ratio between the segment of the target C’D’, and the segment CD of the source; magnification N: ratio between the segment of the target A’B’, and the segment AB of the source. Parameters M and N can be either positive or negative, so four families of XX can be considered.

For the calculation of the initial curve (*i.e.*, the SMS seed rib, [7]) two point sources (A and B in Fig. 3) are placed such that the line joining them is parallel to the x axis of the coordinate system. The line that joins the target points A’ and B’ will be parallel to the x axis of the coordinate system as well, see Fig. 4.

The SMS method induces a particular parameterization on the surfaces creating two types of curves: ribs and spines [7, 8]. For the SMS-ribs calculation, two point sources (C and D in Fig. 3) are placed such that the line joining them is parallel to the z axis of the coordinate system. The line that joins the target points C’ and D’ will be parallel to the y axis of the coordinate system. Fig. 5 shows some ribs on an XX.

## 2.1 The XX design for a cylindrical source

The XX configuration and geometry introduced in the previous section can also be applied for the problem of coupling the rays issuing a cylindrical source into a rectangular target. This is of practical interest in condenser applications, because the resulting design, as shown below, can efficiently couple the light from an arc into a rectangular aperture. The SMS3D’s control of the projected source images (via the control of selected wavefronts) allows the non-rotational projection of them, as well as a constancy of projected size that is completely constant in at least one dimension, and guaranteed around a curve of the mirrors (called seed rib) in the perpendicular dimension (see Fig. 6). This perfect control of one dimension of the projected size and partial control of the other dimension suffices for our purposes.

The formal definition of this problem is shown in Fig. 7: The rays of the source have all the same radiance and are those emitted from a cylinder surface with directions forming an angle greater than *β*
_{MIN} with the cylinder’s axis. The rays accepted by the target are those reaching a rectangle forming an angle smaller than *ϕ*
_{MAX} with the normal to the rectangle. The condenser has to maximize the power transferred from the source to the target.

Figure 8 shows the input and output wavefronts for the SMS 3D design process. All of them belong to the edge rays of the bundles defined for the source and the target. All of the output wavefronts WF_{ox} are spherical and centered at the midpoints of the sides of the rectangular target. Two of the wavefront couples WF_{i3}-WF_{o3} and WF_{i4}-WF_{o4} are used for the calculations of the SMS chains while the other two couples WF_{i1}-WF_{o1} and WF_{i2}-WF_{o2} are only used only for initial curve calculation. This is why there is only a partial control of one dimension of the source images. WF_{i3} and WF_{i4} are orthonormal rays issuing form the cylinder edges. WF_{i1} and WF_{i2} are formed by tangent rays to the cylinder.

Four families of solutions can be defined according to the signs of the magnifications. Since the input source is no longer a plane, the classical definition of magnification does not apply, but the four families still appear as a consequence of the wavefront-pair assignment, and for simplification of the families nomenclature the terms M and N magnification signs will remain. To illustrate this, Fig. 9 shows the spines contained on the plane x=0 for families with the two possible signs of magnification equivalent to M. Note that in this 2D section, when M<0, the rays of any of the two input wavefronts, after being reflected at the primary optical element (POE), form a real caustic before reaching the secondary optical element (SOE). On the other hand, when M>0, the caustics will be virtual, which implies that, for the optics at y>0, the rays emitted from the source towards the points of the POE with high/low z-values will be reflected towards the points of the SOE with low/high y-values in the case M>0, and towards the high/low y-valued SOE points in the case M<0 (see Fig. 9).

The sign of the magnification equivalent to N affects the seed rib calculation. The case showed in Fig. 4 has N<0. The four XX families generated by the two possible signs of M and N can be equivalently described by the real or virtual nature of their 2 caustic surfaces in 3D.

Since the cylindrical source emits light towards both the y>0 and y<0 half-spaces, there is still another Boolean variable to be added to the signs of magnifications M and N, raising the four families of XX solutions to eight. This third Boolean variable arises from the additional possibility of choosing that the half of the POE mirror at y>0 reflects the light towards the half of the SOE at y>0 (as shown in all previous figures) or towards the half of the SOE at y<0, as shown in Fig. 10.

## 3. Ray tracing results.

In this section, only the case with the best performance (among the aforementioned families of this XX condensers) is shown. This case has M<0, N<0, with non-adjacent POE and SOE paired halves (as in Fig. 10). The distance from source center to target plane was fixed at 30 mm. The input parameters of this design are:

1. Cylindrical source: Length L=1.2 mm; Diameter D=0.3 mm; *β*
_{MIN}=45°.

2. Rectangular flat target: Aspect ratio=4:1; *ϕ*
_{MAX}=19°.

Figure 11 shows the POE and SOE mirrors of this XX design. Fig. 12 shows the standard top, side and front views, indicating the dimensions.

In order to evaluate performance, the collection efficiency versus target etendue has been calculated by ray-tracing (using the commercial ray-tracing package LightTools® by Optical Research Associates [17]). The set of optical surface points obtained from the SMS design method were used to create the free-form surface patches with Rhino3D [18]. These surfaces were later exported to LightTools in SAT format. The source and the target were created in LightTools with its own CAD capability.

Fig. 13 shows efficiency versus target etendue given by *E*
_{target}=*A*
_{target} π sin^{2}(*ϕ*
_{MAX}), where *A*
_{target} is the target area and *ϕ*
_{MAX}=±19° is the target acceptance angle. The etendue of the target was varied by varying the target area *A*
_{target} while freezing the target aspect ratio (4:1) and its circular field of view *ϕ*
_{MAX}.

For comparison purposes, Fig. 13 shows two further curves. One of them shows collection efficiency versus target etendue for a conventional elliptical reflector working for the same source and a target with the same aspect ratio (4:1). The eccentricity of the ellipsoid has been set equal to 0.8 (which is the optimized standard) and the target field of view of *ϕ*
_{MAX}=±30° (which is also the standard value in the market). The third curve in Fig. 13 corresponds to the theoretical limit, which is imposed by etendue constraints: an ideal condenser achieving it (which may not exist) would transfer all the source power to the target if the target etendue is greater than the source etendue (i.e., it will have a 100% collection efficiency in this case), and would fully fill the target etendue with light from the source if the source etendue is larger than the target etendue. Then the ideal condenser will have a collection efficiency equal to the ratio of target to source etendue when the target etendue is smaller than the source etendue.

The etendue of the source can be calculated for a cylindrical source geometry (the two circular cylinder bases are included):

For the above input data E_{source}=3.13 mm^{2}. Fig. 13 shows that the XX performs much better than the elliptical reflector (for all mirrors, specular reflectivity has been set equal to 1), getting close to the theoretical limit. There are three factors, however, that prevent the XX from reaching the theoretical limit:

1. When target etendue is large some rays that reflect off the POE will miss the SOE, so that XX curve cannot reach 100% collection efficiency.

2. The “shoulder” of the XX efficiency curve is rounded (in contrast to the theoretical one, which shows a slope discontinuity). This is due to the non-stepped transitions of the illuminance distribution on the target plane and also to the rounded contour lines of that illuminance distribution (shown in Fig. 14).

3. When target etendue is small, the slope of the XX curve in Fig. 13 is less than the theoretical slope, because the XX does not fill completely and uniformly fill the circular field of view (see the intensity distribution in Fig. 14).

Fig. 15 shows another way to represent the data of Fig. 13, which is more interesting, at least from the academic point of view. In this figure the collection efficiency is represented vs the normalized brightness which is defined as the ratio of the average brightness on the target and the source brightness. The average brightness on the target is simply the ratio of the power on the target over the target’s etendue. The source brightness is the ratio of the source’s power and the source’s etendue. This calculation assumes that the brightness of the source is constant (as it is in our model). The points with the same target etendue in this representation form a straight line crossing the origin. The dashed diagonal line in Fig. 15 is the line where the etendue of the target equates the etendue of the source. Collection efficiency is a parameter affecting the total power on the screen of the projection system while the normalized brightness is related to the cost and complexity of the optics and microdisplay.

The 4:1 aspect ratio of the target is not seen in projection display applications. On the contrary, the 16:9 format can be considered the present standard. An XX condenser similar to that in Fig. 11, designed for a cylindrical source with diameter 0.62 mm (source etendue=6.96 mm^{2}) and keeping the rest of parameters unchanged, will produce a 16:9 irradiance distribution on the target plane. Fig. 16 shows the ray tracing results. The XX with target’s circular field of view of *ϕ*
_{MAX}=19° still performs better than elliptical reflector, although the gain is reduced to 1.5, due to the lower aspect ratio of the target (again, for all mirrors, specular reflectivity has been set equal to 1). The theoretical limit is also reduced to 2.

It is interesting to note that these XX condensers perform closer to the theoretical limit if a square field of view for the target is considered. Fig. 16 also shows the collection efficiency and target etendue using a square field of view of 28°×28° for the XX at the target (this square field of view is almost inscribed in the preceding 19° radius circular filed of view). This result is more than just an academic consideration if we take into account that the angular acceptance field of the dielectric filled mixing prism working with total internal reflection has a square shape. The theoretical limit stays unchanged but the XX performs much better because its intensity distribution matches better with the square field of view. Note that the slope of the efficiency curve of the XX with square field of view becomes now very close to the theoretical limit near the origin, indicating a uniform and well-filled field of view.

## 4. Demonstrator prototype

A demonstrator prototype has been fabricated to validate these design principles. For easier handling, instead of a projector arc lamp (whose high flux cannot be dimmed), an automotive H7 halogen lamp was selected. The H7 filament is a spiral enclosed by a cylinder of length L=4.3 mm and diameter D=1.55 mm. The lamp geometry constrained the choice to the XX configuration, so as to avoid shading and mirror to lamp interferences.

The design selected is an XX of the family N<0, M<0 with adjacent POE and SOE halves. Moreover each half has a separate rectangular target, as shown in Fig. 17. The 3D drawing in Fig. 17(b) shows the device rotated 90 degrees around the cylindrical source axis, in contrast to the device in Fig. 11.

The prototype, manufactured by LPI [19] using nickel electroforming for the reflectors, is shown in Fig. 18. The mirror coating is made of evaporated aluminium. The y>0 and y<0 halves of the SOE mirror were made as identical replicas of the mold. Similarly, the POE was made of two halves, but in this case the POE was split into x>0 and x<0 halves, for easier mold release. Fig. 18(a) shows the entire condenser. The upper POE has been taken out in Fig. 18(b) showing the lamp and the SOE. The photography of Fig. 18(c) was taken within the angular field of the target. It shows the two images of the filament formed at the exit plane. Fig. 18(d) shows the two spots formed by the condenser on a paper placed at the condenser target plane. For a clearer understanding of this arrangement, the upper POE has been taken out from Fig. 18(b), 18(c) and 18(d). Radiometric measurements on this prototype have yet to be done.

A detailed analysis of the optical tolerances is outside the scope of this paper. Nevertheless, some qualitative considerations can be established. Regarding the effect of tolerances on the manufacturing of the XX condenser, we distinguish between the source tolerances (which include the variability of the arc electrode, bulb shape, time-varying arc position, etc) and the optics tolerances. The source tolerances can be seen as a variation of the apparent source position and size, which will cause a corresponding modification of pin-hole projected images. The source tolerances will then affect the XX in a different way compared to a conventional elliptical reflector, as the pin-hole projected images of both devices differ significantly. Because the projected images of the XX have all similar sizes, we foresee a more relaxed tolerance for it than for the elliptic reflector.

The novelty of the free-form surface shapes makes that no effective standard of specification and testing is available yet for the optics tolerances. We should expect the XX to be less tolerant because the rays suffer two reflections instead of the single reflection of the conventional elliptic reflector. The optics accuracy of manufactured prototype is still under evaluation.

## 6. Conclusions

After introducing the efficiency limitations of some conventional condensers, we have presented a new free-form condenser (called XX) which has been designed with the SMS3D method. This condenser has capabilities unattainable by rotationally symmetric optics, particularly the control of the rotation of the projected source images. Ray traces simulated a cylindrical source emitting with constant radiance within a limited angular field and a rectangular target with a circular cone angular field. Using the same source and target we have also ray traced a conventional elliptical condenser. A comparison of the power transmitted to the target vs. the target etendue (done under equal conditions) shows that the XX condenser can send to the target up to 1.8 times the power sent by the elliptical condenser (for a 4:1 aspect ratio target rectangular aperture) and up to 1.5 times when the target aperture aspect ratio is 16:9.

Although it has long been known that rotational symmetry limits the capabilities of condensers for cylindrical sources and rectangular targets (see for instance [20]), this is the first free-form condenser design outperforming a rotational symmetric one. This is an important theoretical achievement particularly in showing how the extra degrees of freedom provided by free-form surfaces can be used to improve the performance of the design. Nevertheless much practical work remains. The main drawback of the XX condenser is its use of two mirrors instead of the single one used by a conventional elliptical condenser. The extra reflection makes the XX condenser less tolerant to surface errors in the mirrors although the XX design can be more tolerant to errors of position of the source (which could be the dominant ones in a mass production assembling procedure using arc lamps).

An XX condenser demonstrator has been built in electroformed nickel to validate the concept. For easy prototype handling, a lamp filament was made the source. As yet no radiometric measurement has been made.

## Acknowledgment

The authors thank the Spanish Ministries Mcei (Consolider program CSD2008-00066), Mityc (Foco 1000x: FIT-330100-2007-49 and PRV: FIT-330101-2007-3), and the Madrid Regional Government (CCG07-UPM/000-1602 and CCG07-UPM/ENE-1603) for the support given in the preparation of the present work. The authors from the Universidad Politécnica de Madrid also thank Optical Research Associates for the educational license of LightTools software.

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