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 lensmirror combination. A ray tracing of
threedimensional concentrators (with rotational symmetry) is also done, showing that the lensmirror
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 mirrornonimaging 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 threedimensional (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.
00036935/92/16305110$05.00/0.
C 1992 Optical Society of America.
tation of manifoldei in phase space xp (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 phasespace 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 edgeray 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 edgeray
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 CPClike 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 xp 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
thermalenergy 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 heighttoentry 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 heighttoaperture 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 xp.
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 pointbypoint 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 xnegative 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 thexp representation of ray ra at the exit aperture must be ra', i.e.,
the ray linking the xnegative edge of the exit aperture and the xpositive edge of the receiver.
Because of the symmetry of the lens, the conditions
stated are for the rays crossing the xpositive 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
pointbypoint 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 xpositive 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 xpositive side of the lens is designed such that the surfaces are normal to the z axis
at x = 0, the xnegative 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 xpositive 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 xnegative 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 raytracing 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 edgeray 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 xnegative side of the lens can also be generated by using the same procedure by which the
xpositive 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 xnegative 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.ThreeDimensional 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 raytracing
analysis (no optical losses have been considered).
The curves in this figure are the transmissionangle
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
6Oor'
OCD )
4
oM
8
0
CD
CD
CD
(P~~~~~(
20
0
4
8
12
16
20
incidence angle (degrees)
24
28
Fig. 8. Transmissionangle 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
transmissionangle 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 ///
edgeray 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 RayTracing 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 edgeray 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 transmissionangle
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 transmissionangle 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 LensMirror 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 imagingforming 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 lensmirror 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 flowline 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 transmissionangle 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
mirrornonimaging 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 lensmirror
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 lensmirror 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. Transmissionangle curves for several 3D nonimaging
lensmirror 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
abovementioned regions of the mirror and the lens.
V. ThreeDimensional Ray Tracing of the LensMirror
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 transmissionangle curves of
several lensmirror 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 lensmirror 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 lensmirror 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 RayTracing Results of Some selected LensMirror 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 dielectricfilled
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 lensmirror combinations.
The concentrators that are formed by a combination of a parabolic mirror with a CEC are also bigger
than the nonimaging lensmirror combinations and
their optical performance can be poorer. The theoretical limit of concentration can be attained with
these twostage 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 fnumbers much greater
than 0.287. For instance consider a combination of
an f/2 parabolic mirror with a dielectricfilled 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
secondstage 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
mirrorCEC combination is approximately 90%, i.e.,
4.6% less than the nonimaging lensmirror combination of Table II. As the fnumber increases the
transmissionangle curve of the mirrorCEC 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 quasispherical
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 lensmirror 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 lightemitting 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 lensmirror 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 lensmirror 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 lensmirror
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
lensmirror 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.
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