New method of design of nonimaging concentrators Juan C. Miano and Juan C. Gonzalez A new method of designing nonimaging concentrators is presented and two new types of concentrators are developed. The first is an aspheric lens, and the second is a lens-mirror combination. A ray tracing of three-dimensional concentrators (with rotational symmetry) is also done, showing that the lens-mirror combination has a total transmission as high as that of the full compound parabolic concentrators, while their depth is much smaller than the classical parabolic mirror-nonimaging concentrator combinations. Another important feature of this concentrator is that the optically active surfaces are not in contact with the receiver, as occurs in other nonimaging concentrators in which the rim of the mirror coincides with the rim of the receiver. Key words: Nonimaging concentrators, optical design. I. Introduction A nonimaging concentrator is an optical device designed to transfer the incoherent radiation from a source to a receiver" 2 (see Fig. 1). Usually it is a requirement for the receiver to have the smallest possible area. In these cases the aim is for the nonimaging concentrator to reach the maximum concentration of irradiance on the receiver. In other cases the nonimaging concentrator has to couple two elements so that the transference of radiation between them is optimum. In either case, no imaging formation is needed, but only a transfer of radiant energy. The concentrator shown in Fig. 1 casts onto receiver R all the radiation received from source S. Let Ai be the set of rays connecting S and the concentrator's entry aperture 1i and letAY be the set of rays connecting R and the concentrator's exit aperture SO. If the concentrator is optimal then all the rays of Xi emerge from the exit aperture as rays of JO and vice versa; any ray of A/ comes from a ray of X, i.e., the manifolds Xi and XO are formed by the same rays. Henceforth the analysis is restricted to two dimensional (2D) geometry unless three-dimensional (3D) geometry is specified. Figure 2 shows the represenThe authors are with the Instituto de Energia Solar, Universidad Politdcnica de Madrid, E. T. S. I. Telecomunicaci6n, Ciudad Universitaria, 28040 Madrid, Spain. Received 25 June 1991. 0003-6935/92/163051-10$05.00/0. C 1992 Optical Society of America. tation of manifoldei in phase space x-p (x is the coordinate along the entry aperture and p is the optical direction cosine of the rays with respect to the x axis). Any point (x, p) of the dashed region represents a single ray of Xi. The representation of A/ is also shown in this figure. If the concentrator is optimal then, as we said above, and A are the same manifold and so the areas of their representation in the phase space is the same, i.e., the areas of the dashed regions must coincide.' This is because of the conservation of the phase-space volume theorem (which is also called the conservation of 6tendue).' Observe that the design of the nonimaging concentrator does not require an ordered transformation of the rays of Xi in the rays of A but only that the rays of X be transformed in rays of AO (no matter which one) and vice versa. Let da'i (and a.4) be the set of rays represented by the boundaries of the shaded region Xi (and ff,) in Fig. 2. Figure 1 shows the trajectory of four rays of adi and four rays of aA. These eight rays are those whose representation in the phase space are in the corners of the shaded regions (the rays have been designated by r, rb, r, rd, and ra', rb', r', rd'). The rays of dJ4 and a//4 are called the edge rays and they give the key to the design of the concentrators by means of the edge-ray theorem. 3 This theorem establishes that if an optical device is such that the rays of aXi are coupled with the rays of a4J, i.e., if the rays of adi and aA are the same, then the rays of Xi and 4 are also the same. Then we just have to couple the rays of afi and df to obtain the desired concentra1 June 1992 / Vol. 31, No. 16 / APPLIED OPTICS 3051 S s Fig. 1. General scheme of a concentrator showing the source of radiation SS', receiver RR', and entry (i) and exit (.) apertures of the concentrator. Several rays linking the source and the concentrator's entry aperture and linking the concentrator's exit aperture and the receiver are also shown (rays that are denoted r and r' are not necessarily the same ray even if the subscript is coincident). tor. This design principle is known as the edge-ray principle.' The design of the classical compound parabolic concentrator (CPC) and other similar concentrators such as the compound elliptical concentrator (CEC) does not need to take into account the edge rays impinging on the extreme points of the concentrator's entry (i) and exit (O) apertures. For example, a design of a CPC-like concentrator for the conditions of Fig. 1 needs to take into account only the rays of segments rard and rbr, at 1i and the rays of rarb' and r'rd' at Y, (see Fig. 2). The rays that correspond to segments rrb, rrd rrd', and rrb', which also belong to dai or to aX1, do not have any function in the design process. The most important advantage of the classical solution is its simplicity. Of the disadvantages we want to point out two: (a) when geometrical concentration Cg3D (ratio of the entry aperture area to the exit aperture area) is high the height of the concentrator turns out to be big when compared with the entry or exit aperture width. For instance, the ratio of the height to the entry aperture width for a 3D CPC with C 3D = 3283 (acceptance angle ±10) is 29.14. (b) Bhen exit aperture Y, coincides with receiver R the classical designs require that the reflector and reP ra rd Al rd Ao 'I C rb rb Fig. 2. Representation in phase space x-p of the set of rays4/ linking the source and the entry aperture (left) and the set of rays A linking the exit aperture and the receiver (right). The rays that are represented by the borders of the shaded regions are called edge rays. Some of these edge rays are drawn in Fig. 1. 3052 APPLIED OPTICS / Vol. 31, No. 16 / 1 June 1992 ceiver touch. This is a problem for practical purposes, such as the mechanical ensemble of the reflector and the receiver or the need to thermally isolate the absorber from its surroundings in a solar thermal-energy collection." 4 This problem can be solved by using virtual absorbers or by using cavities 5 if a small optical performance degradation is allowed. The first disadvantage is solved by truncation of the concentrator or by the combination of an imaging device (a lens or a mirror) with a secondary nonimaging concentrator. For instance, a concentrator with an acceptance angle of -- 0.24° and Cg3D 130 000 can have a ratio height-to-entry aperture diameter of 2.5 with a combination of a mirror and a secondary nonimaging concentrator. Such a concentrator has been built. 6 A solar irradiance as high as 72 W/mm 2 has been measured at its exit aperture (the world's record for solar flux concentration). The combinations of imaging and nonimaging devices are not theoretically perfect but they are close (in 2D geometry and also in 3D geometry) if the lens (or the mirror) forms a good image of the source at the entry aperture of the secondary concentrator. Here we propose a new 2D nonimaging concentrator design method that provides almost ideal nonimaging concentrators (in 2D geometry) whose height-toaperture ratios can be smaller than the combinations of imaging and nonimaging devices and whose refractive or reflective surfaces do not touch the receiver. Section II is devoted to the design of the two surfaces of an aspheric lens that fulfill the conditions of the nonimaging concentrator of Fig. 1. The same design method is used in Section IV to obtain ideal nonimaging concentrators with maximal concentration (the receiver and the concentrator exit aperture coincide) by using one refractive surface and one reflective surface. Ray tracing is done with the axisymmetrical 3D concentrators that are obtained from the two types of 2D concentrators designed in Sections II and IV. The results of these ray tracings are the subject of Sections III and V. Some of them give an optical performance that is even better than their equivalent 3D CPC. 11. Aspheric Nonimaging Lenses The purpose of this section is to design a nonimaging concentrator as an aspheric lens. In order to fix the conditions of the design let us assume that the receiver width is 2 (the receiver edges R and R' are at x = -1 and x = 1) and that the source is at the segment SS', as shown in Fig. 3. Figure 4 shows the representation of the edge rays (at the entry and exit apertures of the lens) in phase space x-p. It is well known that a single refractive surface can image sharply a bundle of rays into a point if each point of the surface is crossed by a single ray of the bundle. In general a single refractive surface can transform a given bundle of rays into another one that is predetermined if there is no more than one ray crossing each point of this surface. These refractive surfaces are called Cartesian ovals. 7 Our problem is S SI Fig. 3. Construction of an aspheric nonimaging lens begins at the extreme points of lenses X and N. Rays with the same subscript are the same ray. different: there are two surfaces to design and each point of the two surfaces is crossed by two edge rays (excepting the extreme points of these surfaces, which are crossed by a bundle of edge rays). The solution to this problem is also possible and can be obtained using a point-by-point method similar to the one used by Schulz in the design of aspheric lenses. 8 9 Before applying this method we impose certain conditions on the transformation of the rays of aXi into the rays of Ag,. These conditions derive from the statement of the problem. For instance, note that the rays reaching the extreme point N (or N') of the lens cannot be the same as the rays departing from the extreme point X (or X') of the lens unless the lens has zero thickness at its edges. Only ray ra (and its symmetric counterpart rd) crosses N' and X' (rd crosses N and X; see Fig. 3). The trajectory of ra reaches point N' of the entry aperture with the most negative value of p (this ray comes from S). Necessarily this ray must cross the most x-negative point of the exit aperture (point X') and the value of p of this ray at point X' must be the highest compared with the other rays of aXO crossingX'. Then thex-p representation of ray ra at the exit aperture must be ra', i.e., the ray linking the x-negative edge of the exit aperture and the x-positive edge of the receiver. Because of the symmetry of the lens, the conditions stated are for the rays crossing the x-positive side of the lens only (see Fig. 4; the notation r and r' means the same ray before and after crossing the lens): (a) ray rd (a corner of agi) is transformed into ray rd' (of 3J'); (b) the rays of the other corner of aXi, i.e., re, is S ra mlr rb transformed in a ray (re') that crosses the lens exit aperture at a point Y that is different from X; and (c) the other ray of the corner of aXO4 r,', comes from ray r, that crosses the entry aperture at a point M that is different from N. Similar conditions hold for rays r, rb, and rf. The above conditions determine portions MN (and M'N') and XY (and X'Y') of the two surfaces of the lens. The profile MN is a portion of a Cartesian oval that images the rays coming from S' (between r, and rd) at point X; profile YX is also a portion of a Cartesian oval that images N at R'. Points R and R' are assumed to have coordinates x = 1 and x = -1, respectively. The size and position of the source (relative to the receiver) are assumed to be known. The design procedure is as follows: first, 6tendue &' of manifoldAe7 (which coincides with the 6tendue ofA) is chosen. After this selection points N and X are chosen arbitrarily, taking into account that tendue 9' of manifolds/i andch' is the same. This means that point N must lie on the hyperbola INS' - INS = 9'/2 and point X must lie on the hyperbola defined by XR' | - XR I = 9'/2, where IXR Imeans the optical length from X to R (see Ref. 1 for the calculation of the 6tendues). The selection of N and X determines fully the Cartesian ovals MN and XY. The first Cartesian oval is the one crossing N and imaging S' and X. The second is a Cartesian oval crossing X and imaging N and R'. Point M is the intersection of ray r, coming from R and crossing X (note that the normal to refractive surface at X is known since the Cartesian oval crossing X is known and so it is possible to trace the trajectory of ray r, inside the lens and then calculate point M). In a similar way point Y can be calculated with ray r, coming from S and refracting at N. Now consider an arbitrary point 0 of the lens surface between M and N (see Fig. 5). Ray rg impinging on 0 from S must be directed to R'. The trajectory of rg inside the lens can be easily calculated (with the refraction law) since the portion MN is known. Then point P, where ray rg leaves the lens, can be calculated by requiring that the optical length 1 from S to R' be the same through re or through rg, i.e., ISNYR' = ISOPR'l (note that ISNYR'I can be rc rd A0 x -x ra"7f rd Fig. 4. Representation in the phase space ofei and o. Some special edge rays are marked with a solid circle and their trajectories can be seen in Fig. 3. Fig. 5. Remaining points of the lens are obtained with the point-by-point method departing from Cartesian ovals NM andXY. 1 June 1992 / Vol. 31, No. 16 / APPLIED OPTICS 3053 calculated because all the points S, N, Y, and R' are known). A new portion of the rightmost surface of the lens can be obtained by applying the above procedure to all points between N and M. The derivative of this new portion of surface at point P can be calculated by using its neighbor points or by application of the refraction law to ray rg at point P. Now consider the ray rh that links P and R. The trajectory of rh inside the lens can be calculated because the normal to the profile at P is known. This ray must impinge on the leftmost surface of the lens at a point Q in such a way that rh comes from S'. If the lens is symmetric (with respect to the z axis) the optical length along rh from S' to R must be 1, and so the point Q and the normal to the profile at Q can be calculated similarly as we did with point P. A new portion of the leftmost surface of the lens can be calculated by repeating the preceding procedure with the rays linking R with the calculated portion of the rightmost surface of the lens (i.e., the portion XY and the portion calculated during the first step of the procedure). The remaining points of the lens are calculated by repeating the procedure. The x-positive side of the two lens profiles is calculated according to the above procedure. The other side of the lens is obtained by symmetry. Generally the lens obtained with this procedure is not normal to the z axis at x = 0 and so a discontinuity of the derivative of the profile may exist. To get lens profiles that are normal to the z axis at x = 0 it is necessary to iterate the design procedure with different initial points N and X. First point N can be kept in its initial position and point X is moved along the hyperbola IXI? ' - XI? I = 9'/2 until the leftmost surface of the lens is normal to the z axis at x = 0 (more than a single solution can exist). Observe that the tendue of X,, is the same for different positions of X along the hyperbola. Second point X is kept constant and point N is moved along the hyperbola INS' - INS = /2 until the rightmost surface of the lens is normal to the z axis at x = 0. By iteration of this procedure it is possible to find a lens with both surfaces normal to the z axis at x = 0. Finally, when the x-positive side of the lens is designed such that the surfaces are normal to the z axis at x = 0, the x-negative side of the lens is obtained by symmetry, as we said above. Generally there is not a single solution. To choose the best solution it is possible to consider other features of the lenses, for instance, their performance as 3D lenses, or the thickness at the center of the lens, etc. The thinnest lens is that in which N coincides with X. This is not necessarily the best solution. This design procedure ensures that manifolds a.i and ad (see Fig. 4) are the same except for a small subset of rays crossing the central region of the lens. In order to study this small subset of rays assume that the x-positive side of the lens is designed so that both surfaces of the lens are normal to the z axis at x = 0. The design procedure ensures that all rays of Delhi impinging on the entry aperture from points N to 3054 APPLIED OPTICS / Vol. 31, No. 16 / 1 June 1992 S SI Fig. 6. Construction of the lens at the center, where the method requires an additional degree of freedom for obtaining an ideal nonimaging concentrator rigorously. In practice this additional degree of freedom is not needed and ray tracing shows that the resulting lenses cannot be distinguished from ideal concentrators. L (see Fig. 6) are rays of 4 after crossing the lens. The same can be said about the rays that cross the lens through the portion XZ of the rightmost surface of the lens. The design also ensures that the rays of aXi coming from S' and impinging on the portion CIL of the leftmost surface of the lens become rays of &//4 (in the case shown in Fig. 6 these rays are focused at R), and that the rays crossing CZ (the rightmost surface) and focused to R' are rays of aXJ coming from S. Let us construct the x-negative side of the lens by symmetry with respect to the z axis. The design method does not ensure that the rays from S impinging on the lens through C1L (these rays belong to Xi) will be imaged at R' with the exception of two of these rays: those crossing points C and L (see Fig. 7). The same fact can be viewed at the exit aperture of the lens. There is no evidence that rays focused to R from CrZ are rays coming from S', with the exception of the rays crossing points Cr and Z. The portions CIL and CrZ are determined fully by the design procedure so there are no more degrees of freedom to solve this problem unless we accept, for example, the existence of an additional small lens (with a different refractive index) between Q and Q'. IP -X I. -----XL *y - Fig. 7. Method ensures that the part of #i represented with a solid line in this figure is transformed by the lens in rays of w~. Ray tracing shows that, in the studied cases, the remaining rays of a1 are also transformed in rays of A~. XL is the x coordinate of point L. In practice this problem does not exist, and the rays impinging on the lens through CIL and coming from S result in being focused to point R' up to the accuracy of the ray-tracing program (a similar thing happens with the rays focused to R and crossing CrZ), so a81 = a//4 in 2D geometry. Nevertheless we cannot establish rigorously that this method ensures Since this fact has no practical that &1 = aRX. importance we shall proceed as if d1i and aWo were absolutely the same manifold of rays (in 2D geometry). The edge-ray theorem establishes that a.1 = &,'0 is a sufficient condition to obtain Xi =. In any case it would not be critical even if a,1 • aX, at the central regions of the profiles because these central regions do not generate an important amount of area of the 3D concentrators when these concentrators are generated by rotational symmetry around the z axis. The x-negative side of the lens can also be generated by using the same procedure by which the x-positive side is calculated, i.e., by following the procedure without stopping at x = 0. In this case the lens is, in general, asymmetric (except when the lens surfaces are normal to the z axis at x = 0) and so a 3D rotational symmetric lens (with rotational symmetry around the z axis) cannot be constructed. Another possibility is constructing the x-negative side of the lens by symmetry even if the surfaces are not normal to the z axis at x = 0, thereby accepting that the lenses have a kink at the center. This possibility has not been studied. When the source is placed at infinity then the rays of Xi are those that reach the entry aperture within a given angular field, i.e., with I I < sin (Ha ), where 20a is called the acceptance angle of the concentrator. In this case the 6tendue 9' is linked with the acceptance angle by a simple formula W= 4Cg sin(Oa), (1) where Cg is the x coordinate of point N (i.e., Cg is the 2D geometrical concentration). Hereafter we restrict our analysis of the nonimaging lenses to this case. 111.Three-Dimensional Ray Tracing of the Lenses As part of the 3D ray tracing of the lenses a 2D ray tracing is done to verify that Xi =XJ, in 2D geometry, i.e., to verify that: (a) any ray impinging on the entry aperture of lens NN' with -sin (0a) < p < sin(Oa) reaches receiver RR', and (b) any ray linking the exit aperture of lens XX' and receiver RR' crossed entry aperture NN' with -sin (0 ) ' p sin(Oa). Figure 8 shows the results of the 3D ray-tracing analysis (no optical losses have been considered). The curves in this figure are the transmission-angle curves T(O, Oa) for several nonimaging 3D lenses obtained by rotational symmetry (around the z axis) from 2D lenses with different acceptance angles Oa, which are designed as in Section II (other characteristics of these lenses are given in Table I). These transmission (%) 10 08 6O-or' OCD ) 4 oM 8 0 CD CD CD (P~~~~~( 20 0 4 8 12 16 20 incidence angle (degrees) 24 28 Fig. 8. Transmission-angle curves for several 3D aspheric nonimaging lenses with rotational symmetry. The lenses are designed for sources at infinity that subtend an angle called the acceptance angle of the concentrator. The number by each curve is the design acceptance angle of each lens. Other characteristics of the lenses can be seen in Table I. curves give the ratio of power transmitted to the receiver over the power transported by rays that impinge on the entry aperture at a chosen incidence angle 0 (assuming that these rays have a constant radiance). An amount of 9000 rays were traced for each incidence angle 0. The function T(0, a) was more deeply explored near the transition (around 0 = 0a) to ensure that T(0j, 0a) - T(Oi, 0a) < 0.1 (i and 0j~j are two consecutive values of 0 by which the function T(0, 0,a) is calculated). If meridian rays only were considered then the transmission-angle curves would be T(0, 03a) = 1 if 0 < 0 and T(H, 0a) = 0, if 0 > 0a, because the lenses are ideal in 2D geometry. Nevertheless the lenses are not ideal in 3D geometry (some skew rays with 0 < a are not sent to the receiver and other skew rays with 0 > a are sent to it) so the transition from T(0, 0a) = 1 to T(O, 0ta) = 0 is not abrupt in a 3D lens. In other words the set of rays impinging on the concentrator's entry aperture with 0 ' 0 a (the setA' ) does not coincide with the set of rays linking the concentrator exit aperture and the receiver (4') in 3D geometry. Nevertheless the etendue 9'3D of these two sets is the same because of the method used for the construction of the concentrator. This 6tendue can be calculated at the entry or exit aperture': 3D = rAe 2 2 sin O, = r2/4 (XR'I - IXR 1) , (2) where Ae is the area of the concentrator's entry aperture (Ae = 'MXN 2 , where XN is the coordinate x of point N). does not mean that the The fact that /// edge-ray theorem fails in 3D geometry'0 because the design method does not ensure that O1i = &(4 in 3D geometry (this is ensured in 2D geometry only). Let X9 be the set of rays that link the source and the receiver through the concentrator. Obviously ./ is a subset of/i and also a subset of A. Let 9ci be the representation of // at the entry aperture and let XcO be the representation of./c at the exit aperture. The inverse edge theorem in 3D geometry states that, 1 June 1992 / Vol. 31, No. 16 / APPLIED OPTICS 3055 Table I. Geometric Characteristics and 3D Ray-Tracing Results of Some Selected Nonimaging Lensesa Acceptance Angle, Oa (deg) 0.5 2 5 10 20 Geometric concentration, Cg3D Total transmission, T(e0 ) (%) Cutoff angular spread, AO) (deg) Estimation of error, Err (%) Thickness at the center of the lens Length/entry aperture diameter, f Exit aperture radius, R. Exit aperture to receiver distance, ZR - ZX Entry aperture to receiver distance, ZR - ZN 3600 97.5 0.04 0.21 101.5 1.161 20 32.53 72.35 225 98.0 0.065 -0.20 25.65 1.16 5 8.09 17.95 36 96.9 0.5 -0.19 9.5 1.161 2.5 3.98 7.84 12.25 96.6 0.6 0.47 3.99 1.036 2.5 3.16 4.28 2.25 95.9 2 -0.40 1.33 1.073 1.2 1.81 2.27 aLens refractive index 1.483, receiver radius 1. since Ci =C , i= CC Nevertheless, in gen- eral, daYi ali and a;,, • a4. So it is possible that some rays of da4i enter the concentrator and are turned back (as occurs in a CPC) but this does not invalidate the edge-ray theorem because the method of design does not ensure that &4' = a.4 in 3D geometry (neither the method of design presented here nor the method of design of a 3D CPC with rotational symmetry). An important figure characterizing the transmission of the concentrators is the total transmission T(Oa), which can be defined as the ratio of the 6tendue of 4; to 3D [therefore T(Oa) < 1]. It also can be defined as the total flux transmitted inside the design collecting angle.' Its expression is T(0a) =(Ae fOa T(e, O)a)sin 20d0] /'3 D The integral in Eq. (4) times Ae is the tendue of the set of rays that crosses the entry aperture and reaches the receiver. This manifold coincides withg4 (this coincidence does not occur if some reverse ray departing from the receiver suffers a total internal reflection at the entry aperture surface). Since the tendue of A; can be directly calculated [Eq. (2)] the comparison of both etendues can give a measure of the errors that are due to ray tracing and to numerical calculation. Figure 9 shows a cross section of the lens with a = 10° (see Table I for other data) and the position of points N, M, X, Y, L, and Z. Figure 10 shows a cross section of another lens with Oa = 10°. This lens has Cg3D = 6.25, T(Oa) = 97.0%, Err = 004%,f= 1.3, and R,= 1.2. (3) IV. T(Oa) is a quality factor of the concentrator [an ideal 3D concentrator has a rectangular cutoff at 0 = and so T(0a) = 1]. Another figure characterizing transmission-angle curves is the angle difference AO = 09 - 0, where 09 and 0 fulfill T(09 , Oa) = 0.9 and T(01, Oa) = 0.1. Table I shows the total transmission and AO obtained from the 3D ray tracing of the nonimaging lenses whose transmission-angle curve is shown in Fig. 8. The table also gives some other features of the lenses such as the geometric concentration Cg3D (which is the ratio of the entry and exit aperture areas), the ratio f of the length of the concentrator (which is the difference between the z coordinates of the receiver and the center of the leftmost surface of the lens) to the lens diameter, the thickness at the center of the lens, the exit aperture radius R, and the z coordinate of points X and N relative to the receiver plane RR' (ZR - ZX and ZR - ZN, respectively). The receiver radius is always 1 and the refractive index of the lens is 1.483. We have not done a systematic analysis of all the possible lenses so the lenses chosen in Table I are not necessarily the best. To evaluate the error carried during the calculation of T(Oa) we have also calculated the following variable Err: Err = 100[1 3056 A g 2 T(0, 0)sin 20do]. APPLIED OPTICS / Vol. 31, No. 16 / 1 June 1992 (4) Nonimaging Lens-Mirror Combination The most interestingproperties of nonimagingconcentrators appear when the size of the receiver has to be as small as possible or, in other words, when the receiver has to be illuminated isotropically by the rays of4/ (then the concentration on the receiver is the maximum that is possible). Practical concentrators based on imaging-forming designs fall a long way short of the maximum concentration.' For instance a parabolic mirror can provide only approximately 1/4 of the maximum possible concentration'," while nonimaging devices reach nearly the maximum concentration. 8FZ 6 4 2 -4 -3 -2 -1 0 1 2 3 4 Fig. 9. Aspheric nonimaging lens with an acceptance angle of 100 and a geometric concentration of 12.25. Cartesian ovals MN and XY and points L and Z are specified. 0 z -20 -40F -100 -80 -60 -40 -20 0 20 40 60 80 100 Fig. 11. Nonimaging lens-mirror combination for a source at infinity subtending an angle of 1° with respect to the z axis. This concentrator has maximal concentration. Cartesian oval XY and points Z and L are also shown. x~ -3 -2 -1 0 1 2 3 Fig. 10. Aspheric nonimaging lens with an acceptance angle of 100 and a geometric concentration of 6.25. The lens designed in the Section II and III is not one of these cases; the receiver is illuminated by the exit aperture of the lens that does not coincide with the receiver (and so the receiver is not isotropically illuminated). If the exit aperture of the lens is forced to coincide with receiver then the resulting lens is thick and, probably, impractical. A better solution for obtaining a high concentration is to combine the lens with a flow-line concentrator (FLC) (see Refs. 12 and 13), which is also called a trumpet concentrator. Such combinations already have been produced by using thin lenses.' 4 The lens proposed in Section III must provide advantages over thin lenses for such combinations when the acceptance angle Oa is large. The FLC is ideal in 3D geometry. Then transmission-angle curves of a combination of an aspheric nonimaging lens (ANL) and a FLC coincide with those shown in Fig. 8 (if we neglect the optical losses), as do the parameters T(Oa) and AO. Cg3D fulfills the equation Cg3D = 1/sin 2 (Oa) for such combinations. Another solution for obtaining an isotropic illumination on the receiver is to use the same design method as the ANL but, instead of designing with two refractive surfaces, we design a concentrator with reflective and refractive surfaces. The design of a 2D nonimaging concentrator with maximal concentration and with such a combination of refractive and reflective surfaces is the subject of this section. The result is a concentrator that, in some cases, has a better performance than that of the classical imaging mirror-nonimaging concentrator combinations and also better than that of the CPC. The design procedure is qualitatively identical to that of the lens. The only difference is that now there is one reflective surface and one refractive surface instead of the two refractive surfaces of the lens. Figure 11 shows one of these concentrators designed for maximal concentration and for a source placed at infinity. We describe the general procedure of design in which the source is placed at a finite distance of the concentrator and the receiver is not isotropically illuminated (see Fig. 12). The location of points N and X is done as before, i.e., taking into account that the conservation of the tendue theorem must be fulfilled. This means, in this case, that point X must I = and point N lay on the hyperbola IXIV'I - XI? I/2 on the hyperbola INS'I - INS = 9'I2 (points R and R' are assumed to be on the straight line z = 0 with x = +11). Note that optical path lengths IXR ' and IXR| are the lengths between the corresponding points times the refractive index n. Once points N and X have been chosen, taking into account the above condition, the design of the profiles can start with the portion XY of the lens. This portion is a Cartesian oval that images points N and R'. Since the position of points N, X, and R' is known such a Cartesian oval can be constructed easily. Point Y is obtained as the intersection of the Cartesian oval and the ray coming from S and reflecting at N. The portion NM of the mirror is obtained in a similar way: NM is part of an ellipse imaging the points S' and X. Point M is the intersection of this ellipse and the ray departing from R and crossing X. Observe the qualitative similarity between the portions XY and NM of the nonimaging lens-mirror combination and those portions of the ANL (see Figs. 3 and 12). The design can start now from the portion XY: first, rays crossingXY are traced from R and a portion of mirror is calculated requiring that these rays be the reflection of those reaching the mirror from point S'; second the rays coming from S are reflected in the N M S z mirrors Fig. 12. Construction of the nonimaging lens-mirror combination begins at the extreme points of lens X and of mirror N. 1 June 1992 / Vol. 31, No. 16 / APPLIED OPTICS 3057 an angle 0 with the z axis such that 0 < a. The calculations are also restricted to the case of isotropic illumination of the receiver, i.e., point X is aligned with R and R'. Points N and M of the mirror become the same point in this case. The 3D 6tendue of the manifold of rays crossing the entry aperture and reaching the receiver is 83D = Trn 2A, where A is 0 1 2 3 4 5 6 7 8 9 10 incidence angle (degrees) Fig. 13. Transmission-angle curves for several 3D nonimaging lens-mirror combinations with rotational symmetry. The concentrators are designed for sources at infinity that subtend an angle called the acceptance angle of the concentrator. The number attached to each curve is the design acceptance angle of each concentrator. Other characteristics of these concentrators can be seen in Table II. last calculated portion of the mirror and another portion of the lens surface is obtained by requiring that these rays be focused at R'. The procedure is repeated until the profiles at x = 0 are known. If it is desired to have surfaces normal to the z axis at x = 0 then usually it is needed to iterate all the procedure with different starting points X and N until the surfaces are normal to the z axis at x = 0. As in the case of the ANL there is a region around the center of the lens (from point L to the center of the mirror and from point Z to the center of the lens; see Fig. 11) where there are not enough degrees of freedom to ensure that all the edge rays are correctly directed. Again it is found that, up to the accuracy of the 2D ray tracing, these edge rays are correctly directed even though they lack these degrees of freedom, so afir = a., Nevertheless we cannot establish rigorously that this last condition is exactly fulfilled by a subset of the edge rays crossing the above-mentioned regions of the mirror and the lens. V. Three-Dimensional Ray Tracing of the Lens-Mirror Combination The 3D concentrators that are analyzed here are constructed by rotational symmetry around the z axis. As before we restrict our calculations to the case in which the source is at infinity. Then the rays of the source reach any point of the mirror that forms the receiver area. Figure 13 shows the transmission-angle curves of several lens-mirror combinations, one of which is the concentrator shown in Fig. 11 (the one that corresponds to a = 1). Other data of these concentrators appear in Table II. As in the case of the lenses, these lens-mirror combinations are not necessarily the best ones. The design method fails for large values of Oa, or the resulting lens is so big that its shadow on the mirror makes the concentrator useless. That is why the concentrators appearing in Table II have small a. Table II includes two values of the total transmission T(0a). The first takes into account the shadow made by the lens on the mirror, i.e., it is assumed that the light impinging on the entry aperture at the back side of the lens (or the back side of the receiver) is lost. The second does not consider the shadow. Since the 2D designs from which these concentrators derive do not consider the shadow this value of T(0a) gives us a measure of the results that can be expected with a 3D concentrator whose 2D design takes into account the shadow on the mirror. The ratio Err is calculated according to Eq. (4), taking into account the new expression of 83D. The lens thickness is measured at the center of the lens. ZN - ZR iS the distance from the plane of the entry aperture to the plane of the receiver. This distance can be negative when Oa is small, which means that the lens and the receiver are fully inside the volume enclosed by the mirror surface and the plane of the entry aperture. The nonimaging lens-mirror combination acts as an imaging device for incidence angles 0 close to a except for the rays reaching the neighborhood of the entry aperture border. When 0 is not close to a the system does not work, by far, as an imaging device. This can be verified, for instance, by inspecting the spot diagrams at the receiver of the rays impinging normally on the entry aperture (0 = 0). Such a spot Table II. Geometric Characteristics and 3D Ray-Tracing Results of Some selected Lens-Mirror Combinationsa 00 (deg) 0.5 1 3 5 7 Geometric concentration, Cg3D Total transmission, T(Oa) (%) Total transmission without shadow, T(00 ) Cutoff angular spread, AO (deg) Estimation of error, Err (%) Length/entry aperture diameter, f Lens thickness at the center Lens radius Entry aperture to receiver distance, N - R 28880 94.43 98.67 0.05 0.26 0.257 13.56 35 -5.42 7221 94.62 95.64 0.15 -0.10 0.287 4.56 8.5 10.08 802.9 90.22 92.38 0.65 0.27 0.313 2.57 4 5.88 289.5 90.03 93.9 1.0 0.20 0.356 2.57 3 5.91 148 80.74 88.18 1.5 0.20 0.348 2.33 3 3.77 aLens refractive index 1.483, receiver radius 1. 3058 APPLIED OPTICS / Vol. 31, No. 16 / 1 June 1992 diagram shows that the point of incidence at the receiver is distributed throughout the receiver surface. One of the most interesting features of these concentrators is their small size, which is especially important when 0 is small. For instance consider the case 0 = 1 of Table II. The concentrator thickness relative to the entry aperture diameter (the ratio f) is around 0.287. This ratio is 42.98 for an equivalent nontruncated CPC, i.e., a dielectric-filled CPC with 0 = 1 (refractive index 1.483), which obviously has the same geometric concentration Cg3D. This is achieved in combination with a total transmission factor (in which the shadow is considered) that is approximately equal to the one of the CPC's (Tcpc (0a) 95% when 0a = 1 (see Ref. 1). No optical losses are considered in these values of the total transmission for the CPC nor are they considered for the lens-mirror combinations. The concentrators that are formed by a combination of a parabolic mirror with a CEC are also bigger than the nonimaging lens-mirror combinations and their optical performance can be poorer. The theoretical limit of concentration can be attained with these two-stage systems if the imaging stage (parabolic mirror) is without aberrations, i.e., when the mirror forms the image of the source at the CEC entry aperture; this requires f-numbers much greater than 0.287. For instance consider a combination of an f/2 parabolic mirror with a dielectric-filled CEC (n = 1.483) for forming a concentrator with a design semiacceptance angle 0a = 10. The geometric concentration of the first stage is Cg9 = sin 2 + cos2 +/sin 2 0 = 181.76 (see, for instance Ref. 15), where + is the rim angle of the mirror, i.e., + = tan-(1/2f). The second-stage concentration is approximately Cg2 = n2 /sin 2 4 = 37.39 so Cg3D = CglCg2 = 6796, which is 94% of the maximum concentration (Cg3D < n 2/sin 2 00). If we consider this loss of concentration as a loss of total transmission and assume that the total transmission of the CEC is similar to that of a CPC 0.962 ( [i.e. TCEC with an acceptance angle 00 (see Ref. 1)], then the total transmission of the mirror-CEC combination is approximately 90%, i.e., 4.6% less than the nonimaging lens-mirror combination of Table II. As the f-number increases the transmission-angle curve of the mirror-CEC combination becomes more sharp but the concentration of the first stage decreases and then the concentration (and the size) of the second stage increases. This reduces the total transmission of the CEC. When f is high the shadow of the second stage on the mirror can be the cause of the decrease of the total transmission. The rays suffer a single metallic reflection in both systems (assuming that the CEC can work with total internal reflection). The Fresnel reflections at the CEC entry aperture are, approximately, those that correspond to a beam impinging on its entry aperture at angles (with the normal to the entry aperture) below + (+ = 14.04°). The angular distribution at the lens entry aperture of the nonimaging lens- mirror combination is less homogeneous. At the center of the lens the beam is impinging on the lens at angles below 18.52°. Because of the quasi-spherical shape of the lens and because the center of the lens is the point of the surface that is closest to the receiver center, it can be suspected that the rays will not form angles much greater than this with the normal to the lens surface, and so we can conclude that there will not be much differences in the Fresnel losses of both systems. Another important feature of the lens-mirror combination is that there are no optical surfaces in contact with the receiver as happens in the classical CPC (and CEO) designs, whose mirror rims touch the receiver. This feature should simplify the assembly of the concentrator and the receiver when this is small, for instance, when the device at the receiver place is a photodiode or a light-emitting diode (LED) (obviously the LED acts as an emitter but we prefer to use the same terminology as before and say that the LED is at the place of the receiver). The total transmission of the nonimaging lensmirror combinations decrease when the acceptance angle increases. This restricts the use of these concentrators to small values of 00, at which they compete advantageously with other nonimaging concentrators. VI. Conclusions We have presented a new method of designing nonimaging concentrators with two examples. The first is an aspheric nonimaging lens. The second, a nonimaging lens-mirror combination, has been studied more deeply and has been compared with other existing nonimaging concentrators, showing that it improves almost all the features of the concentrators when the acceptance angle is small. In particular the nonimaging lens-mirror combination has the following advantages over preceeding nonimaging designs: (1) its total transmission within the acceptance angle is one of the highest reported, (2) the size of the concentrator is minimal, (3) the amount of dielectric material is small, (4) no optical surface is in contact with the receiver, (5) the rays within the acceptance angle suffer a single reflection (nevertheless, this reflection cannot be a total internal reflection at a difference with the CPC, which, in some cases, can work by using total internal reflections only), and (6) it is simpler to manufacture than other compact concentrators.16 Their most important disadvantages are the complexity of the calculation of the profiles and the restriction of their use to small acceptance angles because the design method fails with large acceptance angles. This failure can probably be solved with a deeper study of the design method. We have studied only two possible cases of concentrators designed with this new method. There are probably many others, for instance: a lens-mirror combination such that the radiation from the source crosses the refractive surface first and then is reflected by the mirror; a combination of two mirrors; a 1 June 1992 / Vol. 31, No. 16 / APPLIED OPTICS 3059 lens mirror combination such that the radiation from the source crosses the refractive surface first, then is reflected by the mirror and finally crosses the refractive surface again before reaching the receiver; or a lens-mirror combination that is similar to those studied in Section IV, whose optically dense medium is between the reflective and refractive surfaces. This work was done under contract number 303/89 (PRONTIC program) with the Spanish agency Comisi6n Interministerial de Ciencia y Technologia. The authors acknowledge Philip Davies for his help in the preparation of the manuscript. References 1. W. T. Welford and R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989). 2. I. M. Bassett, W. T. Welford, and R. Winston, "Nonimaging optics for flux concentration," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1989), Vol. XXVII, pp. 161226. 3. J. C. Mifiano, "Synthesis of concentrators in two-dimensional geometry," in Solar Cells and Optics for Photovoltaic Concentration, A. Luque, ed. (Hilger, Bristol, UK, 1989), pp. 353-396. 4. R. Winston, "Ideal light concentrators with reflector gaps," Appl. Opt. 17, p. 1668 (1978). 5. R. Winston, "Cavity enhancement by controlled directional scattering," Appl. Opt. 19, 195-197 (1980). 6. D. Cooke, P. Gleckman, H. Krebs, J. O'Gallagher, D. Sagie, 3060 APPLIED OPTICS / Vol. 31, No. 16 / 1 June 1992 View publication stats and R. Winston, "Sunlight brighter than the Sun," Nature (London) 346, 802 (1990). 7. 0. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics, (Academic, New York, 1972). 8. G. Schulz, "Achromatic and sharp real imaging of a point by a single aspheric lens," Appl. Opt. 22, 3242-3248 (1983). 9. G. Schulz, "Aspheric surfaces, " in Progressin Optics, E. Wolf, ed. (North Holland, Amsterdam, 1988), Vol. XXV, pp. 351416. 10. J. C. Mifiano, "Design of three-dimensional concentrators," in Solar Cells and Optics for Photovoltaic Concentration, A. Luque, ed. (Hilger, Bristol, UK, 1989), pp. 397-429. 11. P. Gleckman, J. O'Gallagher and R. Winston, "Approaching the irradiance of the sun through nonimaging optics," Opt. News 15(5), pp. 33-36 (1989). 12. R. Winston and W. T. Welford, "Geometrical vector flux and some new nonimaging concentrators," J. Opt. Soc. Am. 69, 532-536 (1979). 13. R. Winston and W. T. Welford, "Ideal flux concentrators as shapes that do not disturb the geometrical vector flux field: a new derivation of the compound parabolic concentrator," J. Opt. Soc. Am. 69, 536-539 (1979). 14. W. T. Welford, J. O'Gallagher, and R. Winston, "Axially symmetric nonimaging flux concentrators with the maximum theoretical concentration ratio," J. Opt. Soc. Am. A 4, 66-68 (1987). 15. R. Winston, "Nonimaging Optics," Sci. Am. 264(3), 76-81, (1991). 16. G. W. Forbes and I. M. Basset, "An axially symmetric variableangle nonimaging transformer," Opt. Acta 29, 1283-1297 (1982).