Application of the conservation of etendue theorem for2-D subdomains of the phase space in nonimagingconcentrators

Juan C. Minano

The conservation of etendue for general 2-D bundles of rays (not necessarily coplanar) is examined (a 2-D

bundle of rays is that whose rays are distinguishable by giving each one two parameters). This is one of the

integral invariants of Poincar6 and it is directly related to the Lagrange invariant. The application of thistheorem to selected 2-D bundles of rays crossing an arbitrary cylindrical concentrator gives us a relationship

between the maximum geometrical concentration of a cylindrical concentrator and the angular field of viewwhich is more restrictive than the general one (i.e., the relationship is valid for an arbitrary concentrator)

when the collector is surrounded by a refractive medium.

1. Theoretical Background

In this paper we shall follow the Hamiltonian de-scription of geometrical optics, which has been veryuseful for treating problems of nonimaging optics. Inparticular, the Liouville theorem, which can be derivedfrom this description, has given a simple answer to afundamental problem of nonimaging concentrators: tocalculate the smallest exit aperture area of a concen-trator that will collect a given beam of rays.1 Thistheorem is also known as the etendue invariant 2 and theconservation of phase space volume; it is one of the in-tegral invariants of Poincar6.3

Let x (z),y (z) specify a light ray in a set of Cartesiancoordinates. In terms of these functions, the Fermatprinciple can be expressed as

6 fz n(x,y,z)(1 + x2 + 9 2 )dz = 0, (1)Zl

where the dot denotes differentiation with respect to z,and n (xy,z) is the index of refraction corresponding topoint (x,y,z) (it will be simply noted by n). We canidentify the integrand as the Lagrangian L(x,y,z,i,5)of the light ray. Then, the conjugate momenta corre-sponding to x and y are

aL _ _ _ _ _ _ _OL- Xi (2)

ax (1+x2+2)1/2(

The author is with Universidad Politecnica de Madrid, ETSITelecomunicacion, Instituto de Energia Solar, Madrid 3, Spain.

Received 20 July 1983.0003-6935/85/122021-05$02.00/0.© 1984 Optical Society of America.

OL _ _ _ _ _ _ _ _a== * ~ (3)a.y (1 + 2 + y2)1/2

A look at Eqs. (2) and (3) reveals that p and q are, re-spectively, n times the cosine of the angle formed be-tween the ray and a parallel to the x axis and betweenthe ray and a parallel to the y axis at point (x,y). Ob-serve that a ray leaving the x-y plane through the sem-ispace z positive is fully defined by the set of variablesx,y,p,q (Fig. 1).

When a 4-D region of the phase space x,y,p,q istransformed point to point into a 4-D region of otherphase space x',y',p',q' by a canonical transformation, 4

the Liouville theorem states2 3 56

dxdydpdq = dx'dy'dp'dq'. (4)

Let us consider an entry aperture x-y plane and an exitaperture plane of an optical system (Fig. 1). Each rayleaving the first plane and reaching the last one estab-lishes a correspondence point to point between (xyp,q)and (x',y',p',q'). This transformation is canonical andinvolves a special generating function called Hamilton'scharacteristic function.7 2 Then, Eq. (4) holds for thetransformation of the bundle of rays entering the entryaperture of the optical system into the bundle of raysleaving the exit aperture.

There is another canonical transformation that willbe in our interest: the transformation of the set ofvariables (x,y,p,q) into (p,6,j,h) where p and 0 are thepolar coordinates of a point in the x-y plane and j,h are,respectively, their conjugate momenta8

j = p cosO + q sinO,

h = p(q cosO - p sinO).

(5)

(6)

From these equations we can verify that j is n times the

15 June 1984 / Vol. 23, No. 12 / APPLIED OPTICS 2021

rays. Since the sum of the square of the three directioncosines of a ray (with respect to three orthogonal axes)is one, the field of variability of P/n(X,Y),Q/n(X,Y) [orP'/n(X',Y'),Q'/n(X',Y')] is restricted to the circle ofradius unity. We assume that the index of refractionoutside the concentrator is 1 and the one of the mediumsurrounding the collector is n,. Then we can express

E = Aed

Fig. 1. Coordinate systems used for the rays at the entry aperturex,y,p,q and the exit aperture x',y',p',q' of an arbitrary optical system.The two rays drawn have coordinates x = 0, y = 0 and x' = 0,

Y = 0.

cosine of the angle formed between the ray and thestraight line of the x-y plane defined by 0 = constant,and h is np times the cosine of the angle formed betweenthe ray and the tangent to the circumference p = con-stant at point p,O; h is an invariant (skew invariant) ina ray traversing an axisymmetrical system.2 TheLiouville theorem applied to the two aforementionedcanonical transformations states that

dxdydpdq = dpdOdjdh = dx'dy'dp'dq' = dp'dO'dj'dh', (7)

where p',0'j',h' are obtained from x',y',p',q' in the sameway as p,Oj,h derive from x,y,p,q.

Equation (4) is valid even if the x-y and x'-y' planesare not parallel. Then we can derive an equation sim-ilar to (4) for the sets of variables (X,YP,Q) and (X',Y',P',Q'), where X and Y are the coordinates of a pointin a given curved entry aperture surface, and P,Q andn (X, Y) times the cosine of the angle formed betweenthe ray and the line Y = constant and between the rayand the line X = constant, respectively. The lines X= constant and Y = constant are assumed to be or-thogonal. Note xlydXdYdPdQ is equal to a differ-ential dxdydpdq where the x axis is the straight linetangent to the line Y = constant and the y axis is tan-gent to the line X = constant; Ix and ly are two func-tions of X, Y that make lXdX and lyd Y be differentialsof length. Similar considerations can be done with re-spect to X',Y',P',Q' and so lx,ly',dX'dY'dP'dQ =dx'dy'dp'dq'. Then, using Eq. (4) we have thatlxlydXdYdPdQ = lxly'dX'dY'dP'dQ'. 9

Let us assume that X,Y are the surface coordinatesof a point of the entry aperture 2e of an arbitrary con-centrator and X',Y' are the coordinates of a point of itscollector 1, The concentrator casts on its collector agiven bundle of rays arriving at its entry aperture. Therays of this bundle occupy a region D of the X- Y-P-Qspace and a region D' of the X'-Y'-P'-Q' space. Ob-viously, if a ray (X,Y,P,Q) belongs to D, it is trans-formed by the concentrator into a point of the region D'.The conservation of etendue applied to the transfor-mation generated by each collected ray when linking aset (X,Y,P,Q) to another set (X',Y',P',Q') statesthat

E = fD IxlydXdYdPdQ = f lx1ly'dX'dY'dP'dQ', (8)

where E is called the etendue of the bundle of collected

(9)

and affirm that

E Arn., (10)

where Ae is the area of 1eX Ac is the area of 1c, and iscalled "average acceptance area."'10

Let us consider the region of the P-Q plane obtainedby the intersection of D and a domain of constant X, Y(i.e., the collected rays that cross 2e at X,Y), a is theaveraged area of all these regions corresponding to thepoints of 1e,. Since P,Q is restricted to a circle of radius1,i < •-r.

Combining (9) and (10) we obtain

Cg < 2nC = Cgm(d),ela)

(11)

where Cg is the geometrical concentration. Equation(11) expresses the maximum Cg achievable with a con-centrator of given d. The quotient Cg/Cg () is calledthe "degree of isotropy" g.11 For a concentrator of flatentry aperture which collects every ray forming, withthe normal to e, an angle lower than q5, Eq. (11) takesits more usual form (observe that in this case p2 + Q2

< sin2, ) 1:

Cg n, (12)

11. Conservation of Etendue for 2-D Subdomains ofthe Phase Space

Another integral invariant of Poincare3 will interestus now. This integral invariant refers to a 2-D regionof the phase space x,y,p,q and to the 2-D region of thephase space x',y',p',q' obtained by a canonical trans-formation of the first one. It is not necessary that thetrajectories of the rays represented by these 2-D regionsare contained in a plane and, in general, this will notoccur. The differential form of this invariant is

dx dp + dy dq = dx'dp'+ dy' dq'. (13)

A simple proof of the validity of (13) can be obtainedfrom the analogy between Hamiltonian optics andmechanics. 5 We can also see that (13) is only the La-grange invariant. Let us assume that u and v are twoparameters that determine each ray of the bundle de-fined by either of these 2-D regions:

x = x(u,v), x' = x'(u,

y = y(u,v), y' = y(u,v),

p = p(u,v) p' = P,(uv),

q = q(u,v), q' = q'(u,v).

What we should show is that the Jacobians fulfill

(14)

2022 APPLIED OPTICS / Vol. 23, No. 12 / 15 June 1984

OPT ICA LSYSTEM

car-, (_ q�nnHX.i'yYzI ,z

yy.

Ill. 2-D Bundles in Cylindrical Concentrators

tconst.

7aC CosI

Fig. 2. Coordinate system used for the rays at the entry aperture 2e

and at the exit aperture Z, of an arbitrary cylindrical concentrator.

D(x,p) D(y,q) D(x',p') D(y',q')+ = + (15)

D(u,v) D(u,v) D(u,v) D(u,v)

Equation (15) can also be written as (the dot denotesscalar product)(ax y\ ap aql - (ap aq x ay

\alu O uJ d av ai v dv OF. du d vav avi

au TU) ka -v TV) auaTO avavJ1

which is the Lagrange invariant. 1 2 Note that we haveimplicitly assumed that the 2-D bundle of rays iscrossing a plane z = constant (and z' = constant). Theetendue of the 2-D bundle is obtained by adding (inte-grating) each differential contribution dxdp + dydq (ordx'dp' + dy'dq') of the bundle. Considering that eachof these contributions is positive, the integral will alsobe positive. The value of this integral is sometimescalled the geometrical energy flux.13 As was shown byLuneburg, a 2-D bundle whose etendue is zero is anormal system of rays, i.e., there is a family of surfacesnormal to the rays (the wave fronts). The conversetheorem is also true: a normal system of rays has zeroetendue. From here it follows that the theorem ofMalus-Dupin is equivalent to the conservation of thezero etendue of the normal system of rays.

The usual expression of the Lagrange invariant is aline integral on a closed curve and it refers only to nor-mal systems of rays.14 This integral invariant relativeto closed curves can be transformed into the integralinvariant obtained from Eq. (13) with the Stokes the-orem.3

The integral invariants of Poincar6 have also beenused to describe the flow of phase space through non-imaging concentrators by means of the geometricalvector flux.1 5 The invariant was expressed there as theconservation (zero divergence) of this vector.

By similar reasoning to that in sec. I we can obtain forthe sets of variables X,Y,P,Q and X',Y',P',Q' thatlxdXdP + IydYdQ = lx'dX'dP' + ly'dY'dQ'. Usingthe sets p,Oj,h and p',0',j',h' the equivalent of Eq. (13)is

dp dj + dO dh = dp' dj' + dO' dh'. (17)

Let us assume that X,Y are the surface coordinatesof a point of the entry aperture of an arbitrary cylin-drical concentrator and X',Y' are the coordinates of apoint of 1c, We can choose the coordinate systems sothat Ix = ly = Ix' = ly' = 1 and that the lines Y = const,Y' = const are straight lines parallel to the axis oftranslational symmetry, and the lines X = const, X' =const are transverse to it. With this set of coordinateswe can say that P is an invariant of the ray in a cylin-drical concentrator and so P = P' (Fig. 2). Let us con-sider the 2-D bundle of rays which has, when crossing1, a constant X and a constant P. Since the concen-trator is cylindrical, we know that the rays of this bundlewill have, in 1, a constant P' = P. Then, applying Eq.(13) we have

dYdQ = dY' dQ'. (18)

Observe that this equation holds not only for the rayscontained in a plane normal to the axis of translationalsymmetry (rays of P = 0) but also for any 2-D bundle ofconstant P (it is not necessary to have a constant X).Let us call E2-D(X,P) the etendue of the 2-D bundle ofcollected rays crossing the entry aperture at a given Xand at a given P:

E 2 D(X,P) = Sfls(xP) dYdQ = S.(X P,)dY'dQ', (19)

where S(XP) is the 2-D region of the X- Y-P-Q spaceof X = const and P = const and D 2-D(X,P') is the 2-Dregion obtained by transforming D fl S(X,P). Theregion D2 D(X,P') has the properties that all its rayshave a constant P' = P and that there are not two rayswith equal Y',P',Q' and different X'. Note for this thatX can be expressed (because it is a cylindrical system)as -. _ . - no -11 I-

A = ' + [ Y's's').

Then we can say that

E2-D(X,P) < 2Ae(1 - p2)1 /2 (21)

[the index of refraction at Xe is assumed to be 1 and soI Q I < (1 - P 2 )1 /2 ] because of the first equality of (19),and that

E 2 D(X,P) < 2A,(n2 - p 2 )/2 (22)

because of the second equality of (19). A and Ac are,respectively, the areas per unit of length of 2e and 1,n, is the index of refraction of the medium surroundingX, so I Q' < (n2

- P/2 )1/2. E 2 -D(X,P) is, in fact, only

a function of P, because the coordinate X of a ray at Iedoes not care whether the ray is collected.

As a result of the first equality of (8), E can be ex-pressed as

X=1 P 1x= o P=1 E2-D(P)dX dP (23)

By combining inequalities (21) and (22) with Eqs. (23)and (10) and the definition of the degree of isotropy g,we can obtain that g for a cylindrical concentratorcannot be greater than the expression

15 June 1984 / Vol. 23, No. 12 / APPLIED OPTICS 2023

(ZU)

.78

I.76

3 5Gemetrical Con tmttlon

7 9It

Fig. 3. Upper bound of degree of isotropy for cylindrical concen-trators whose collector is surrounded by a medium of index of re-

fraction 1.5.

10

5

1 2 3

Fig. 4. Upper bound of geometrical concentration vs average ac-ceptance area a for cylindrical concentrators whose collector is sur-rounded by a medium of index of refraction n = 1.5. The dashedline shows the general upper bound for arbitrary concentrators of nc

= 1.5.

2(n,2-1 1/2g gM(Cgnc) = 2Cg sin' C 1)

rn2in'1 (C,2 - n 1/21l 24+ n2 sin1 |-(I, n) ]i ' (24)ncg

where Cg is the geometrical concentration. For detailsof these calculations see Ref. 16. In Fig. 3, gM(Cg,1.5)is plotted. The proof of Eq. (24) can be done withoutreferring explicitly to Eq. (13) (as is done in Ref. 16).

The degree of isotropy is an important parameter ofthe concentrators which, as we have seen in Sec. I, givesthe ratio of the Cg of a concentrator to the maximum Cgcorresponding to its average acceptance area . Thisparameter also appears when studying the concentra-tors illuminated by a given source. Then Cg and a be-come parameters less useful than the coupled opticalconcentration C, intercept factor I (CO is the ratio ofthe power collected to the maximum power on the col-lector when this is without concentrator and I is theratio of the power collected to the maximum powerreaching the entry aperture surface). It has been shownelsewhere10 that the source illuminating the concen-trator, which in Ref. 10 is assumed to have a flat entryaperture, sets an upper bound of the C achievable bythe concentrators that have a given I,COM(I). The ratioof the C achieved by a concentrator to the upper boundcorresponding to its I is gy, where -y is called the ad-aptation factor. Summarizing, we have that

Cg = gCgm (), (25)

C. = gyCM(I). (26)

Then, using (24) and (25), we have that (if n > 1) the

upper bound of Cg vs a for cylindrical concentrators islower than the general one Cgm (6):

Cg • gM(Cgnc)Cgm(i). (27)

The equality of Eq. (27) can be solved by iterations toobtain the upper bound of Cg for cylindrical concen-trators as a function of 6 (see Ref. 16). Figure 4 showsthis upper bound compared with the general one, whenn = 1.5. This upper bound is achieved by the cylin-drical CPCs for any n, The combination of Eqs. (24)and (26) is analyzed in Ref. 16. We have been able touse Eq. (13) for studying the degree of isotropy of cy-lindrical concentrators because the conservation of Phas given us information about the region transformedof D n S(X,P). In the concentrators of axial symmetrythe rays conserve their value of h (skew invariant).This suggests that we can obtain similar information inan axisymmetrical concentrator. Let us consider oneof these concentrators of flat 1e and 1, The coordi-nates of a point of 2e are p and 0 and those of a point of1, are p' and 0'. The study of the etendue of the 2-Dbundle of rays crossing z2 e at a constant 0 and at a con-stant h does not give us more information than thatderived from Eq. (11). This is because the upper boundof this E2-D(h) obtained from the conditions at 1e [i.e.,the equivalent inequality to (21)] and the upper boundobtained from the conditions at 1c [the equivalent in-equality to (22)] cannot be applied together to a singleconcentrator, i.e., if Ae > ncAc, the conditions at c aremore restrictive than the conditions at 1e for any valueof h, and if A < n2Ac, the conditions at ye are morerestrictive than the conditions at 1, These inequali-ties are

E 2 -D(h) p=R j=[-(h/p)2]1/2Jp=h j=[l-(h/p)21/2

r2Dh p- r rj dp=n2-d/p2]E2Dh)SJPp=h/n J j=-[n2-(h/p')21/2 d j

(28)

(29)

R is the radius of 2e and r is the radius of I. Note thatthe field of variability of j (and j') is given by the rela-tion between the direction cosines j2 + (h/p)2 < 1 [andj]2 + (h/p) 2 < n2] and that p cannot be smaller than h(and p' cannot be smaller than h/nc); this would requirehaving a direction cosine >1. If we change the variablep' in (29) to a new variable p = p'nc, we obtain the in-equality (28) with the only difference being that the firstintegral of (29) is extended from p = h to p = mc.Then, if R < rn, (29) is a more restrictive condition forany value of h, and if R < rnc, (28) is the most restrictivecondition. None of these inequalities gives us a newinformation by itself.

IV. Conclusions

We have presented an application to nonimagingconcentrators of one of Poincar's integral invariantsreferring to a two-parameter bundle of rays (2-D bun-dles). This integral invariant is directly related to theLagrange invariant. It permits us to assign a value

2024 APPLIED OPTICS / Vol. 23, No. 12 / 15 June 1984

I

I

aa�2i.Ico

(called etendue) to the 2-D bundle. This etendue canbe calculated on any surface intercepting the wholebundle and it is zero if the bundle is a normal system ofrays, i.e., if there is a family of surfaces (wave fronts)normal to all the rays of the bundle.

The application of this invariant to selected 2-Dbundles of rays crossing an arbitrary cylindrical con-centrator has given us two useful inequalities with whichwe have been able to obtain the particular upper boundof geometrical concentration vs average acceptance areafor cylindrical concentrators.

References1. R. Winston, "Light Collection Within the Framework of Geo-

metrical Optics," J. Opt. Soc. Am. 60, 245 (1970).2. W. T. Welford and R. Winston, The Optics of Nonimaging

Concentrators (Academic, New York, 1978).3. H. Poincar6, Les methodes nouvelles de la mecanique celeste

(Dover, New York, 1957), Vol. 3.4. R. Weinstock, Calculus of Variations (McGraw-Hill, New York,

1952).5. L. D. Landau and E. M. Lifshitz, Spanish translation: Mecdnica

(Revert6, Barcelona, 1965).6. W. T. Welford, Aberrations of the Symmetrical Optical System

(Academic, London, 1974).7. R. K. Luneburg, Mathematical Theory of Optics (U. California

Press, Berkeley, 1964).8. A generating function of this transformation can be G(p,q,p,O)

= -(pp cosO + qp sinO).9. The Hamiltonian characteristic function can also be defined for

the cases where the terminal points of the rays lie on a curvedsurface. See H. A. Buchdall, An Introduction to HamiltonianOptics (Cambridge U.P., London, 1970); G. W. Forbes, "NewClass of Characteristic Functions in Hamiltonian Optics," J. Opt.Soc. Am. 72, 1698 (1982).

10. J. C. Mifiano and A. Luque, "Limit of concentration underNonhomogeneous Extended Light Sources," Appl. Opt. 22, 2751(1983).

11. For the sake of simplicity we have assumed in this work that thecollector is monofacial (only collects rays coming from one faceof the collector surface). If it is assumed that the collector isbifacial, as is done in Ref. 10, the right-hand side of Eq. (10)should be multiplied by 2 and the definitions of Cgm (a) and gsuffer the corresponding changes. Note that, with the treatmentused here, a concentrator with a monofacial collector can haveg = 1, while in the treatment of Ref. 10 g cannot, in this case, be>1/2. From an optical point of view, a bifacial collector can beconsidered as having two monofacial faces.

12. See, for example, M. Herzberger, "Mathematics and GeometricalOptics," Supplementary Note III in Ref. 7.

13. M. Herzberger, Modern Geometrical Optics (Wiley Interscience,New York, 1958).

14. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1970).

15. R. Winston and W. T. Welford, "Geometrical Vector Flux andSome New Nonimaging Concentrators," J. Opt. Soc. Am. 69, 532(1979).

16. J. C. Mifiano and A. Luque, "Limit of Concentration for Cylin-drical Concentrators Under Extended Light Sources," Appl. Opt.22, 2437 (1983).

Richard Wyatt

British Telecom Research Labs.

Photo: F. S. Harris, Jr.

15 June 1984 / Vol. 23, No. 12 / APPLIED OPTICS 2025