New family of 2-D nonimaging concentrators: the compoundtriangular concentrator

Juan C. Mifiano

A new family of 2-D nonimaging concentrators is presented. The most significant characteristics of these

concentrators are their small size and the fact that they use straight, as opposed to curved, reflective, or

refractive interfaces. The concentrators are filled with a medium of refractive index n > 1 and use narrow

strips of refractive index n, < n. These strips are transparent, or act as mirrors, depending on the angle of in-

cidence of the rays. The direction of the collected rays (i.e., rays reaching the receiver) at they entry aperturecan be varied by modifying the refractive indices of the strips.

1. Introduction

In this paper we present a new family of 2-D idealand optimal concentrators. An ideal concentrator isone in which the collection of a ray depends only on itsdirection at the entry aperture, not on the point ofinterception with that surface. An optimal concentra-tor is one having the maximum geometric concentra-tion allowed by phase space conservation'. The bestknown family of ideal and optimal 2-D concentrators isthe CPC concentrators .

The new family considered in this paper is a subsetof a more general type of concentrator called com-pound triangular concentrator (CTC)2 . The CTCsdiscussed in Ref. 2 are composed of four triangularregions, two of which are symmetric. The three differ-ent triangular regions use, in general, media of differ-ent refractive indices.

The subset of concentrators described here also hasfour triangular regions but now introduces thin bound-aries between the regions that are characterized bytheir refractive indices. The concentrator studiedtakes one value of refractive index for all four triangu-lar regions and another value (in some cases two) forthe boundaries.

II. Description of the Concentrator

Figure 1 shows one of this class of concentrators.The semiangle of acceptance is 0' and the exit semian-

The author is with Universidad Politecnica de Madrid, Institutode Energia Solar, ETSI Telecomunicacion, 28040 Madrid, Spain.

Received 12 April 1985.0003-6935/85/223872-05$02.00/0.© 1985 Optical Society of America.

gle is 00. The concentrator is filled with a mediumhaving a refractive index n. Let 0 be the angle formedby any ray and the t axis (see Fig. 1). Then, thecollected rays have 1 01 • Oi (where n sin0k = sin0i') at thepoint at which they cross the entry aperture. Theheight of this concentrator, if its receiver (or collector)width is unity, is given by

sin00 _1

sin0

2tan( 0 M)(1)

Its geometrical concentration (ratio of the entry aper-ture width to the receiver width) is the maximumpermitted by its acceptance and exit semiangles, i.e.,'

C sin0g sin0i

(2)

These two equations then fix the geometry of theconcentrator; from them the angles a and # can bederived, as defined in Fig. 1:

o0 + Oj2

2

(3)

(4)

In this concentrator, the straight lines EC and E'C'are mirrors if n cosa < 1, otherwise the collected rayssuffer a total internal reflection at EC and at E'C' andno mirrors are required (assuming that the index ofrefraction of the medium outside the concentrator is1). Moreover, the straight lines EC' and E'C indicatethe use of a narrow strip of a medium with index ofrefraction n, given by

n= n cosf3. (5)

3872 APPLIED OPTICS / Vol. 24, No. 22 / 15 November 1985

1-

n-arcsin

2. Q a.csin. 0 arcsin a.n n

incidence angleFig. 3. Transmissivity and reflectivity of an ideal

sorption is assumed to be zero.

Fig. 1. CTC of' = 30'86°, Oi = 20',00 = 750 filled with amedium ofrefractive index 1.5. The refractive index of the strips is 1.33. Fromthe geometry of the CTCs we can derive that the angle CE'C' is equal

to As and the angle EC'E' is equal to Oo.

arcsin !3nn

na

Fig. 2. Behavior of an ideal strip of refractive index n immersed ina medium of greater refractive index n. When the angle formed by aray and the normal to the strip is greater than arcsin(n,/n), the stripacts as a reflector, otherwise the ray follows its rectilinear trajectory.

The thickness of each strip is assumed to be smallcompared to the concentrator height and to the widthsof the entry aperture and the receiver. When this stripis immersed in a medium of refractive index n, it acts asa reflector if the angle formed by a ray and the normalto the strip is greater than arcsin(n,/n). When thisangle is less, the ray crosses the strip without changingits rectilinear trajectory. These ideal properties of thestrip are represented in Fig. 2.

When a practical CTC concentrator is designedwith, for example, axial or linear symmetry, the stripbecomes a layer. The behavior of the reflectivity ortransmissivity vs the angle of incidence for this layerimmersed in the medium with greater refractive indexis not actually discontinuous. This behavior dependson the refractive indices of the different media andtheir thicknesses and on the wavelengths and polariza-tion of the radiation considered. Nevertheless, weassume that its behavior is the ideal one as shown inFig. 3, i.e., that predicted by geometric optics.

The refractive-index distribution as well as the ge-ometry of this concentrator can be derived from ageneral method of design of 2-D concentrators withinhomogeneous media 2 as shown in the Appendix.

If the rays of the source form at the entry aperture anangle 0 with the t axis so that -' < 0 < O', these samerays also form an angle 0 between -Oi and Oi when theyare in region 1 of the concentrator (see Fig. 4). The

-o

0>1

ID1

90

strip. The ab-

Fig. 4. Directions of the extreme rays in each region of the concen-trator. These directions are independent of the point considered in

each region.

angle 0 of the rays of the source is between 0 and -Oi inregion 2 and between 00 and -O0 in region 3. Note thatthe position of the strip EC' is such that the raysarriving from region 1 with 0 = i and the rays arrivingfrom region 2 with 0 = 00 form an angle with the normalto the strip equal to arcsin(n8/n) = 90 - Al, so they areat the limit angle of transmission of the strip. Whenthe rays reach EC with 0 = -0i, they are reflected with 0= 0. All the rays with 0o 0 <-0 in region 2 cross EC'to region 3 until the case is reached that the ray with 0= 00 is reflected. That is, all these rays cross fromregion 2 to region 3 provided that

Oo - (90. - a) < 900 - s,

which is equivalent to

00 < 900.

(6)

(7)

Obviously, rays having 0 > 90° cannot illuminate themonofacial receiver and so the cases excluded by theinequality (7) correspond to nonideal concentrators.

111. Several Properties fo the CTC Family

One of the most interesting properties of these con-centrators is their small size. We consider that twoconcentrators are equivalent if their acceptance andexit angles, their receiver characteristics, and the indi-ces of refraction of the media surrounding the receiversare the same. We use the height of the concentrator as

15 November 1985 / Vol. 24, No. 22 / APPLIED OPTICS 3873

>1.1

.9

C

12I

v 30

a parameter of comparison of the size of two equivalentconcentrators. To simplify the comparison, let us fo-cus our interest on concentrators without an exit anglerestriction, i.e., 00 = 900. In this case, Eq. (1) becomes

1h=2 tanOi (8)

from which it follows that the ratio of the entry aper-ture length Ae to h is

Ae 2

h cosi(9)

This ratio is always > 2.Comparing the height of one of these concentrators

with an equivalent CPC [i.e., a CPC filled by a mediumwith index of refraction n, acceptance semiangle O'i =arcsin (n sin0i), and exit semiangle 00 = 900] gives theresult that

hcpc =cc =(O ) Cg +1. (10)

hCTC (Sinai =

It should be noted that both concentrators (CTCand CPC) are equivalent in 2-D geometry. (Note that,in general, 3-D concentrators with axial or linear sym-metry are not equivalent even if their cross-sectionsare equivalent 2-D concentrators).

Another property of the CTCs is that all the lines atwhich refraction or reflection takes place are straight.This property is also found in the simple prism concen-trator (SPC) and its variations3 . In that reference itwas shown that the illumination produced on the col-lector of that concentrator was fairly uniform when theentry aperture was illuminated by parallel rays within.the acceptance angle. This is an interesting propertyin photovoltaic applications because nonuniform illu-mination reduces the efficiency of the cell4. The sim-ple prism concentrator is, in general, a 0 - 00 trans-former, and 00 can only be equal to 90° if 0 = 90°. Inthat case the ratio of the height of this concentrator tothat of an equivalent CTC (see Fig. 5) is

hspc 2 (11)

hCTc n

Now assume that the indices of refraction of thestrips EC' and E'C are not given by Eq. (5) but by (I -yJ <03:

nEC' = n cos(3 + -y), (12)

nE'C = n cos(3 - -y), (13)

where : = (900 - 0)/2. Then the collected rays arethose which, once they cross the entry aperture, havean angle fulfilling-O -7y< 0 Oi--y. The concentra-tor is no longer optimal if IyM > 0, i.e., its geometricalconcentration is not the maximum corresponding tothe bundle of collected rays. The ratio of Cg to themaximum geometrical concentration, also called thedegree of isotropy g5

,6 , is given by

mirror receiver irro

SPC CTC

Fig. 5. Simple prism concentrator of 0, = 900, = 900 and n = 1.5

compared with an equivalent CTC (n, = 1.37).

Ev

.

YFig. 6. Lines of the geometrical vector flux in a CTC filled with amedium of refractive index n = 1.6. The refractive index of the stripEC' is nEC' = 1.18 and the one of the strip E'C is nEC = 1.48. Thelines of the geometrical vector flux outside the concentrator form an

angle of 210 with the vertical; Oi = 250,00 = 10°, y = 100.

sin(0s - y)} for receiver length 1] to the etendue of therays reaching the receiver (2n for the same receiverlength).

Figure 6 shows the lines of the geometrical vectorflux when -y > 0. The lines of this flux represented inthe figure in the triangle CO'C' correspond to the casein which the receiver is removed and CO is a mirror.When the receiver is in position, the lines of the fluxchange in CO'C' but the collected beam is the same(independent of CO).

The concentrator is ideal for any value of y (I -Y </3),i.e., the collection of a ray depends only on its 0 at theentry aperture.

If Oo < 90° and IJy > 0, the concentrator is neitheroptimal nor ideal. In this case, all the rays having -O- y < < O - y at the entry aperture arrive at thereceiver, but they are not the only ones crossing theentry aperture and reaching the receiver.

Summarizing the results of this section we can saythat the CTC has a small size, that its refractive orreflective lines are all straight, and that the angles 0formed by the extreme rays at the entry aperture canbe varied by controlling the indices of refraction of thestrips EC' and E'C. This last property establishes thetheoretical possibility of having an ideal static concen-trator with tracking. Unfortunately, structures usingliquid crystal, in which the index of refraction can bevaried with the voltage applied, are anisotropic and theindex of refraction depends on the polarization of thelight and the angle of incidence.

g = cos-Y. (14)

This parameter g can be calculated as the ratio of theetendue of collected rays [which is nCg {sin (Oi + -y) +

IV. Combinations of CTCs and CPCs

The major advantages of the CTCs derive from theuse of the strips with indices of refraction lower than

3874 APPLIED OPTICS / Vol. 24, No. 22 / 15 November 1985

mrror -- \reivC.

Fig. 7. Combination of a CTC and a CPC for obtaining a 2-D idealand optimal concentrator for a bifacial collector.

those of the media in which they are immersed. Thesestrips can also be used in designs of 2-D ideal andoptimal concentrators having receiver shapes differentfrom a flat monofacial shape. A simple way to do thisis by combining parts of a CTC and of a CPC, as shownin Fig. 7 where the receiver is flat and bifacial. Thepart of the concentrator defined by the line EOCO'E'corresponds to a CTC of Oi = 20 (i = 13'18°), o =109'56° and the part OC'O'C corresponds to a CPC fora bifacial receiver with 'i = 200 that is filled with amedium of n = 1.5. The slope of the mirror of this newconcentrator is continuous at points 0 and O'. Theratio of the height of the concentrator of Fig. 7 to theone of an equivalent CPC is 0.17. The reduction ofsize is evident. These combinations should be doneusing the edge-ray principle'. For that it can be con-sidered that a ray inciding at the limit angle on thestrip follows its trajectory along the strip.

V. Conclusions

We have proposed a new family of 2-D optimal andideal concentrators that are small and do not usecurved reflective or refractive surfaces. They arebased on the use of narrow strips of a medium having arefractive index lower than that of the medium inwhich they are immersed. It was assumed in the con-centrator's design that such strips have a transmissivi-ty of 0 or 1 and a reflectivity of 0 or 1 depending on theangle formed by a ray and the normal to the strip. Inpractice, these ideal properties cannot be achieved andthis fact requires that the strips of a high-concentra-tion CTC be well designed to avoid low efficiency. Inthese high concentration CTC's all the collected rayshave an angle of incidence at the strip near the angle oftransition from high to low transmissivity.

We have also seen that the direction of the extremerays of the manifold of collected rays at the entryaperture can be varied by controlling the indices of

!refraction of the strips. This result establishes thetheoretical possibility of having an ideal static concen-trator with tracking if the indices of the strip can becontrolled.

Combinations of CTCs and CPCs can be producedto obtain 2-D concentrators for different receivershapes, thus profiting from the small size of the CTCand the simplicity of the CPC.

Appendix

Let G(x,t) be a continuous function so that the vec-tor field 2(-Gt,Gx) is the so-called geometrical vectorflux7 corresponding to the manifold of collected rays(see Fig.1 for the definition of the coordinates; thesubscripts denote partial differentiation). Let F(x,t)be a second continuous function such that 2(-Ft,Fx) isthe geometric vector flux of the manifold of noncollect-ed rays passing through the concentrator. It is under-stood that a collected ray is one that intercepts thereceiver at a nonrestricted angle. It has been shownelsewhere2 that F and G fulfill the conditions

FGX + FtG = 0,

F' + Ft + G + G = n

(Al)

(A2)

where n(x,t) is the refractive-index distribution in theconcentrator. The partial derivatives of F and G atthe entry aperture (t = 0, 1 xi • Ae/2) and at the receiver(t = h, I xI • AI2 with h being the height of the concen-trator, and A, the receiver length) define the manifoldof collected rays when this intercepts entry apertureand receiver segments.

The optical direction cosines with respect to the xand t axes of one of the extreme rays at a given point(x,t) are given by the functions Fx + Gx and Ft + Gt,respectively, and for the other extreme ray by Fx Gxand Ft - Gt.

In Ref. 2 several refractive-index distributions arestudied that, placed inside a cone concentrator, allow itto behave as a 2-D Oi - 0o transformer'. The functionsF and G chosen for these concentrators are linear withrespect to x and t but the multiplicative constants aredifferent in regions 1, 2, and 3 of the concentrator (seeFig. 4). Then F and G have a polyhedral aspect whenrepresented in an F-x-t space or in a G-x-t space.This implies that Fx, Ft, G, and Gt are not continuousat the boundaries of the different regions of the con-centrator. The refractive-index distribution n, de-rived from these functions [Eq. (A2)], generates theconcentrator that we want to design only when n is thelimit case of refractive-index distributions that areobtained with other F and G functions with continuouspartial derivatives that approach our F and G. Insome cases, that is, those analyzed in Ref. 2, we foundthat n is constant in each of the regions 1, 2, and 3 anddifferent from one region to another. In the othercases the refractive-index distribution obtained as alimit case of F and G with continuous partial deriva-tives requires, as we shall see in an example, that n hasa minimum value when crossing from one region toanother. If this minimum is not taken into account,the refractive-index distribution which allows as possi-ble rays those of one of the manifold of extreme rays isdifferent from the corresponding distribution for theother manifold. However, the square of both distribu-tions are equal.

Consider the concentrator of Fig. 1 and the transi-tion from regions 1 to 2 (see Fig. 4). Figure 8 repre-sents the values of the different partial derivatives in

15 November 1985 / Vol. 24, No. 22 / APPLIED OPTICS 3875

F1

minimum refractive

index 1.33

0FX, Gx

Fig. 8. Representation of the partial derivatives of the functions Fand G corresponding to the transition from region 1 to region 2 of theconcentrator of Fig. 1. The arrows indicate the movement directionof the end points of vectors (F,,Ft), (G.,Gt) in the transition from

region 1 to region 2.

regions 1 and 2. The vector F, is formed by the com-ponents Fx and Ft in the region 1. In a similar way F2 ,G,, and G2 represent the other partial derivatives.Equation (Al) implies that F1 and G,, as well as F2 andG2, must be orthogonal. The length of the vector F, +G, is the refractive index in region 1 [see Eq. (A2)] andIF 2 +Gd =n 2 -

As we have seen, the function F in regions 1 and 2that we want to approach is represented in the space F-x-t by two planes of different slope whose straight lineof intersection m is projected in the x-t plane at line Iseparating regions 1 and 2. A function Fwith continu-ous partial derivatives in the neighborhood of thatapproaches our F function would be that whose partialderivative FX and Ft take the values of the partialderivatives of the different planes that contain thestraight line m. These possible partial derivativesform a straight line when represented in the Fx-Ftplane (see Fig. 8). Obviously, this straight line linksthe end points of the vectors F1 and F2. Then thepartial derivatives of F in the continuous transitionfrom region 1 to 2 can be represented by a vector whoseend point is in the segment linking F, and F2. Asimilar reasoning holds for the function G.

Remember that Eq. (Al) implies that the vectorrepresenting the partial derivatives of F and the onerepresenting the partial derivatives of G are orthogo-

nal. Then n is determined once a given vector Fx, Ft(or G., Gt) is fixed. Obviously, n is continuous nowwhen crossing from region 1 to 2 and has a minimumvalue equal to n = 1.33 in the example studied. This isthe only required condition for the strip, i.e., that therefractive-index distribution has a specified minimumin the cross section of the strip.

References

1. W. T. Welford and R. Winston, The Optics of Nonimaging Con-centrators (Academic, New York, 1978).

2. J. C. Mihano, "Two-Dimensional NonImaging Concentratorswith Inhomogeneous Media: A new look," to be published in J.Opt. Soc. Am. A Jan. 1986 issue.

3. D. R. Mills and J. E. Giutronich, "Ideal Prism Solar Concentra-tors," Sol. Energy, 21, 423 (1978).

4. E. Lorenzo, E. Sanchez, and A. Luque, "Experimental Verifica-tion of the Illumination Profile Influence on the Series Resistanceof Concentrator Solar Cells," J. Appl. Phys. 51, 535 (1981).

5. J. C. Mifiano and A. Luque, "Limit of Concentration under Ex-tended Nonhomogeneous Light Sources," Appl. Opt. 22, 2751(1983).

6. J. C. Miffano, "Application of the Conservation of Etendue theo-rem for 2-D Subdomains of the Phase Space in NonimagingConcentrators," Appl. Opt. 23, 2021 (1984).

7. R. Winston and W. T. Welford, "Geometrical vector flux andsome new nonimaging concentrators," J. Opt. Soc. Am. 69, 532(1979).

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