Nonimaging Optics Winston Episode 5 pot

If the inner side of the hyperboloid of revolution is mirrored, then it becomes an ideal nonimaging concentrator with the followingdefinitions of the input and output bundles: The input

are symmetrical transverse to the optic axis As already noted in Section 6.4, the

étendue H generalizes to the difference of optical path lengths (up to an overall

constant) This remains true even in the presence of refractive media, providedthe optical path lengths are measured along rays These rays need not be straight

lines Thus, in Figure 6.14, the étendue H from Lambertian source AA¢ to section PP¢ is proportional to [A¢P] - [AP], where the brackets indicate optical path

lengths It follows that the lines of flow (indicated by arrows in the figure) lie along

contours of H = constant Since the detailed balance condition holds in 2D, we may

construct concentrators by placing mirrors along the flow lines However, it doesnot follow that the 3D construction obtained by rotating the 2D flow line about theoptic axis will automatically satisfy detailed balance Specific cases will have to bechecked with respect to detailed balance before the usefulness of the 3D designscan be evaluated

6.12.1 Introduction

The principles of Geometrical Optics can be formulated in several ways, all of thembeing equivalent in the sense that they can provide the same information Never-theless, there are some particular problems for which one formulation is betterthan the others—for example, the problem is more easily stated and sometimesmore easily solved using one of the formulations This is common to disciplineshaving more than one mathematical model Probably, the most well-known formulation of Geometrical Optics is the variational one (Fermat’s principle) InSection 6.12.2 we will see another well-known formulation: the Hamiltonian equa-tions This formulation will be useful for stating and solving some nonimagingdesign problems both in 2D and 3D geometry with the Poisson Brackets method.This method is unique in the sense that it is able to give ideal designs in 3D geom-etry in some cases Unfortunately, the 3D designs obtained with this methodrequire graded refractive index materials, which limits its practical use

The Hamiltonian formulation has been widely used in imaging optics Themost important results are the characteristic functions and the simplicity withwhich some optical invariants are recognized (see, for instance, Luneburg, 1964)

A

P

A¢

P¢

Figure 6.14 Flow lines with refractive components AA¢ are a Lambertian source The

arrows indicate row lines; the plain lines, rays.

One of these invariants is a common tool in nonimaging optics, the conservation

of étendue Within the Hamiltonian formulation, this invariant is one of the Poincaré’s invariants Although Hamilton originally developed his equations foroptics, their applications in mechanics developed faster, so some of the results ofthe theory may sound as if they belong to mechanics more than to optics This may

be the case of the Poisson brackets In other cases, the same result has two ferent names: one for optics and one for mechanics For instance, in mechanicsFermat’s principle is known as the principle of least action or the principle of Maupertuis The Hamiltonian formulation, when applied to nonimaging optics,makes little use of the results for imaging optics, and because of this, its resultsmay appear more mechanic than optic

dif-6.12.2 Hamilton Equations and Poisson Bracket

As we will see, the Hamiltonian formulation is not unique We start with thedescription of the Hamilton equations that we will use in the most general form

we need Let x1 = x1(s), x2 = x2(s), x3 = x3(s), t = t(s) be the equations of a ray trajectory in parametric form (s is the parameter) in the space x1- x2- x3- t (x1

- x2 - x3are the Cartesian coordinates, and t is the time) For each point of the trajectory of a ray—that is, for each value of s, we have a value of the wave vector

k = (k1, k2, k3) and a value of the angular frequency w Let k1 = k1(s), k2= k2(s),

k3= k3(s), w = w(s) be the values of the three components of the wave vector and the angular frequency, respectively The set of eight functions x1= x1(s), x2= x2(s),

x3= x3(s), t = t(s), k1= k1(s), k2= k2(s), k3= k3(s), w = w(s) define a ray trajectory in the phase space x1- x2- x3- t - k1- k2- k3- w In general we are only interested

in the trajectory of the ray in the space x1- x2- x3, sometimes also including t.

The introduction of the other variables in this case is still interesting because they

simplify the formulation of the equations The variables k1, k2, k3, - w are called

the conjugate variables of x1, x2, x3, t in the Hamiltonian formulation.

A key point of the Hamiltonian formulation is the so-called Hamiltonian

func-tion In the case of optics, K(x1, x2, x3, t, k1, k2, k3, w) is a function such that K = 0 defines the surface of the wave vector k (Arnaud, 1974) The equation K = 0 is also

called Fresnel’s surface of wave normals, and it is directly related to the Fresnel’s

Differential Equation (Kline and Kay, 1965) The function K can be determined by

the properties of the medium where the rays are evolving

The Hamiltonian equations can be written as

K k

dk ds

K x dx

ds

K k

dk ds

K x dx

ds

K k

dk ds

K x dt

ds

K d ds

K t

1 1 1 1 2

2 2 2 3

3 3 3

surfaces K = constant of the phase space x1- x2- x3- t - k1- k2- k3- w—that is,

the function K is a first integral of the system (Arnold, 1976) A function F(x1, x2,

x3, t, k1, k2, k3, w) is a first integral of the system of Eq (6.4) if F is constant along any ray trajectory—that is, dF/ds = 0 F is said to be a “constant of motion” in

mechanics (Abraham and Marsden, 1978; Leech, 1958) This can be written as

of Eq (6.4) are contained in hypersurfaces K = constant.

Not all the solutions of Eq (6.4) represent ray trajectories The ray

trajecto-ries in the phase space x1- x2- x3- t - k1- k2- k3- w are only the solutions of

this equation system that are consistent with K = 0—that is, the curves contained

in the hypersurface K = 0.

The equation of this hypersurface K = 0 can be expressed in different ways For instance, let f (x) be any function such that f (x) = 0 only if x = 0 Then f(K) =

0 represents the same surface as K = 0 It can be easily seen that if f(K) is used

as the Hamiltonian function instead of K in Eq (6.4), then the same ray ries are obtained (with different parameterization) provided that df/dx π 0 when

trajecto-x = 0 In particular, if we multiply the Hamiltonian function by a nonzero

func-tion, the solutions of the Hamiltonian system remain the same but with different

parameterization, that is, instead of getting x1= x1(s), x2= x2(s), x3= x3(s), t = t(s),

k1= k1(s), k2= k2(s), k3= k3(s), w = w(s), we would get another set x1 = x1(s¢), x2=

x2(s¢), x3= x3(s¢), t = t(s¢), k1= k1(s¢), k2= k2(s¢), k3= k3(s¢), w = w(s¢) but still giving

the same phase space trajectories

One useful property fulfilled by the solutions of the Hamiltonian system isgiven by the Maupertius principle (in mechanics), also known as the least actionprinciple, which corresponds to the Fermat’s principle of optics (Arnold, 1974) Interms of the Hamiltonian system of Eq (6.4) this principle says that the integral

(6.7)

along the ray trajectories in the space x1, x2, x3, t is an extremal among all the curves connecting point A and point B that also fulfill K = 0—that is, among the curves whose trajectory in the phase space x1- x2- x3- t - k1- k2- k3- w is con-

tained in the hypersurface K = 0 A and B are two points of the space x , x , x , t.

F x

K k

F k

dK x

F x

K k

F k

K x

F x

K k F

k

K x

F t

K d

F K t

F x

dx ds

F k

dk ds

F x

dx ds

F k

dk ds

F x

dx ds F

k

dk ds

F t

dt ds

F d ds

1 1 1 1 2 2 2 2 3 3

3 3

0

In other words, choose any curve of the space x1, x2, x3, t connecting A and B Now choose arbitrary functions k1= k1(x1, x2, x3, t), k2= k2(x1, x2, x3, t), k3= k3(x1, x2, x3,

t), w = w(x1, x2, x3, t) such that the Hamiltonian K vanishes along the curve If these

functions are compatible with the solution of the Hamiltonian system of Eq (6.4),then the integral in Eq (1.7) is an extremal among the other possible choices

(Arnold, 1974) Observe that there is no restriction on the relationship of k jand

dx j /dt in this way to establish Fermat’s principle in contrast with the usual way

to present it Nevertheless, it can be proved that both ways to present the ple are equivalent (Arnold, 1974)

princi-We shall restrict the analysis to time-invariant isotropic media In this case,the surface of the wave vectors is a simple equation

(6.8)

where co is the light velocity in vacuum and n(x1, x2, x3, w) is the refractive index

at the point x1, x2, x3for the angular frequency w (see Arnaud, 1976, and Kline andKay, 1965, for obtaining the Hamiltonian function in other cases) Because the

media is time-invariant, the Hamiltonian function does not depend on t and thus

the last equation of the Hamiltonian system Eq (6.4) expresses that w is

inva-riant along any ray trajectory (dw /ds = 0) Thus, w is a first integral of the

Hamiltonian system in this case

If w = constant and we are not interested in the dependence of t with the meter s, then we only need the first six equations of the system Furthermore, if

para-we make the change of variables pj = kj· co/w a new Hamiltonian system is obtained

for each value of w The variables p1, p2, p3are called the optical direction cosines

of a ray—that is, p1 is n(x1, x2, x3) times the cosine of the angle formed by the

tangent to the ray trajectory with respect to the x1axis (p2and p3are defined in

a similar way with respect to the x2axis and the x3axis)

The Poisson bracket is defined in a similar way as before and the total

derivative of a function F(x1, x2, x3, p1, p2, p3) along the trajectories can also bewritten as

P∫ p1+p2+p3-n x x x2( 1, 2, 3,w)

dx ds

P p

dp ds

P x dx

ds

P p

dp ds

P x dx

ds

P p

dp ds

P x

1 1 1 1 2

2 2 2 3

3 3 3

The ray trajectories are now the solutions of the system, without restriction to

H = 0 The parameter of these ray trajectories is x3—that is, the conjugate

vari-able of p3in the system of Eq (1.9) The function H is H = -p3when solved from

the equation P = 0—that is,

(6.13)Eqs (6.12) and (6.13) are the usual way in which Hamiltonian equations are intro-duced in optics (Luneburg, 1964) Nevertheless, we won’t use it For our purposes,

Eq (6.9) with the condition P = 0 is a more convenient way to set the basic

equa-tions of Geometrical Optics

Before going further, we still need a last system of Hamilton equations This

is the one obtained when a change of variables from x1, x2, x3to a new set of

orthog-onal coordinates i1, i2, i3 is done This transformation belongs to a class of able transformations called canonical (Leech, 1958), and owing to this fact, theHamilton equations remain very similar (Leech, 1958; Miñano, 1986) Canonical

vari-transformations are characterized by a “generating function” G For our purposes the expression of G is

(6.14)

where the functions i1, i2, i3 in Eq (6.14) give the values of the coordinates i1, i2, i3

for a point x1, x2, x3· u1, u2, u3are the conjugate variables of i1, i2, i3 According tothe canonical transformation theory, the new conjugate variables can be expressedas

(6.15)

The resulting Hamiltonian system is

p p p

i x

i x

i x i

x

i x

i x i

x

i x

i x

u u u

1

2

3

1 1 2 1 3 1 1

2 2 2 3 2 1

3 2 3 3 3

1

2

3

ÊË

ÁÁ

ˆ

¯

˜

˜ =Ê

Ë

ÁÁÁÁÁÁ

ÁÁ

H p

dp dx

H x dx

dx

H p

dp dx

H x

P p

F p

P x

F x

P p

F k

P p

F x

P p

F p

P x

and the Hamiltonian function is

(6.17)

where a1, a2, and a3 are, respectively, the modulus of the gradient of i1, i2, and

i3over the refractive index n (i.e., a j = |—ij |/n) Remembering the expressions of the scale factors h j (Weisstein, 1999) of Differential Geometry, we can write a j=

1/(h j n) The refractive index n is in general a function of i1, i2, i3

With the aid of Eq (6.15) it is easy to find the physical meaning of the

conju-gate variables u i : A point i1, i2, i3, u1, u2, u3of the new phase space represents a

ray passing by the point i1, i2, i3with optical direction cosines a1u1, a2u2, a3u3with

respect to the three orthogonal vectors —i1, —i2, —i3 Figure 6.15 shows these three

orthogonal vectors and an arbitrary ray The i1lines are given by equations i2=

constant, i3= constant The i2, i3lines are defined in a similar way

6.12.3 Optical Path Length

With the information provided in Figure 6.15 it is easy to see that the differential

of path length dL can be written as

H u

du ds

H i di

ds

H u

du ds

H i di

ds

H u

du ds

H i

1 1 1 1 2

2 2 2 3

3 3 3

tra-function we are using.

The Poisson bracket design method is, as yet, one of the few known 3D ing concentrator design methods In general, this method provides concentratorsrequiring variable refractive index media, which is impractical in most of the cases.The main interest of the Poisson bracket method is that it provides ideal 3D con-centrators, and thus it proved that such ideal concentrators exist In particular,

nonimag-we will design a 3D maximal concentrator illuminated by a bundle of rays having

an angular spread q with respect the entry aperture’s normal, that is, the set ofrays that are concentrated are formed by all the rays that impinge a flat entryaperture forming an angle smaller than a certain value q with the normal to thisaperture The concentrator has maximal concentration, and thus the ratio of entry

to exit apertures areas is n2/sin2q, where n is the refractive index of the points of

the exit aperture, which is the same for all of them Figure 6.16 shows a scheme

of such a concentrator

The work presented here was developed some years ago (Miñano, 1985b;1985c; Miñano, 1993a; 1993b; Miñano and Benítez, 1999) Some nontrivial ideal3D nonimaging concentrators were already known when the Poisson bracketsmethod was developed Among these, the most important is the hyperboloid of rev-olution (Winston and Welford, 1979) Figure 6.17 shows one of these concentra-tors A reflector whose cross-section is a hyperboloid forms it The foci of this

hyperboloid generate the circumference C when the cross-section is rotated around

the axis of revolution symmetry If the inner side of the hyperboloid of revolution

is mirrored, then it becomes an ideal nonimaging concentrator with the followingdefinitions of the input and output bundles: The input bundle is formed by all the

rays crossing the entry aperture that would reach any point of the circle C (virtual

receiver) if there was no mirror The set of rays crossing the exit aperture formsthe output bundle The concentrator is ideal in the sense that any ray of the input

L AB H ds ds

B A

1 1 2 2 3 3

bundle is transformed in a ray of the output bundle by the concentrator, and anyray of the output bundle comes from a ray of the input bundle Thus, the samerays form both bundles The only difference is that the input bundle describes thetransmitted bundle at the entry aperture and the output bundle describes it atthe exit aperture Additionally, the concentrator has maximal concentrationbecause the output bundle comprises all the rays crossing the exit aperture, andthus the exit aperture has the minimum possible area such that all the rays of thetransmitted bundle cross it.

From the preceding definition of ideal concentrator we can conclude that anydevice may be an ideal concentrator with a proper definition of the input andoutput bundles Nevertheless, the name “ideal” used to be restricted to cases inwhich both input and output bundles have a practical interest There are two types

of bundle that deserve special attention

1 Finite source The rays of this bundle are those linking any point of a given

surface with any point of another given source (see Figure 6.18)

2 Infinite source This bundle can be described as formed by all the rays that

meet (real or virtually) a given surface forming an angle smaller than or equal

q with a given reference direction Then, this bundle is fully characterized bythe surface (also called aperture), by the angle q, and by the reference direc-tion This bundle is a typical input bundle for solar applications: The rays to

be collected are those reaching the concentrator aperture forming an anglewith the normal to this aperture smaller than the acceptance angle of thesystem (see Figure 6.19)

The input bundle of the hyperboloid of revolution of Figure 6.17 is a finite source

where C1is the entry aperture and C2is the virtual receiver The output bundle

is an infinite source of the type shown in Figure 6.19 with q = 90°

A thin lens with focal length f can be considered as a concentrator whose input

bundle is an infinite source of angle q and whose output bundle is a finite source

of the type shown in Figure 6.18, C1being the lens aperture and C2being a circlelocated at the focal plane with radius equal to f · tan(q) For a real lens this descrip-tion is approximate The approximation is better for smaller since q is smaller.Therefore, a combination of a hyperboloid of revolution reflector and a thin lens

C2

C1

aperture

q

Figure 6.18 Example of finite source The rays of this bundle are those linking any point

of the circle C1with any point of the circle C2

Figure 6.19 Example of infinite source of angle q It can also be considered as a lar case of the bundle shown in Figure 6.18 when one of the circles is infinitely far from the other and of infinite radius.

particu-is, approximately, an ideal concentrator of the type shown in Figure 6.16 (at leastfor small values of q) (Welford, O’Gallagher, and Winston, 1987), if the combina-tion is done in such a way that the output bundle of the thin lens, which is the

finite source defined by the circles C1and C2, is made to coincide with the inputbundle of the hyperboloid (see Figure 6.20)

A characteristic of the hyperboloid of revolution as a nonimaging tor is that its transmitted bundle is what we call an elliptic bundle An elliptic

concentra-bundle is defined as one whose edge rays cross any point of the x1- x2- x3spaceform—in this space, a cone with an elliptic basis Figure 6.21 shows one of these

cones corresponding to the bundle of rays illuminating the circle C This bundle

is of the elliptic type, and thus the rays form an elliptic cone at any point of thespace The figure shows also two flow lines of this bundle

If the elliptic bundle is such that its flow lines are the coordinate lines of a

three-orthogonal coordinate system (i1, i2, i3), then it can be easily proved that the

edge rays conjugate variables u , u , u fulfill an equation like

(6.22)where the functions a1, a2, a3are arbitrary functions of i1, i2, i3 This equation,together with Eq (6.17) defines a conic curve (ellipse, parabola, or hyperbole) in a

u n - um plane (n, m = 1, 2, 3, n π m) Note that it is necessary that Eq (6.17) be

fulfilled because the rays are the solutions of the Hamiltonian system that are

con-sistent with H = 0, which is Eq (6.17).

6.13.1 Statement of the Problem

The ray trajectories in the phase space x1- x2- x3- p1- p2- p3or (i1- i2- i3- u1

- u2- u3) do not cross between them This property, which derives from the ness of the solution of a system of first order differential equations passing through

unique-a given point of the phunique-ase spunique-ace is punique-articulunique-arly useful for describing visuunique-ally theproblem that we want to solve and comparing it with the typical synthesis problem

in imaging optics For the purpose of describing qualitatively both problems, weare going to consider a simplified case This is when the rays are contained in a

plane (for instance the x1- x2) We call this case a 2D system, and it can be derived

from the general case by establishing ∂ n/∂ x3= 0 and p3= 0 In this case the phase

space can be limited to four variables x1- x2- p1- p2 Moreover, since the ray

tra-jectories are restricted to P = 0 (P is defined in Eq (6.10)), then p2= ±[n2(x1, x2)

-p1]1/2—that is, for each point x1, x2, p1there are only two possible values of p2such

that x1, x2, p1, p2 describes a ray Both values of p2 give the same ray path (x2

increases with the parameter s for one value of p2 and for the other value x2

decreases with increasing s) Thus, if we forget one of the two possible directions

of the ray, we can say that each ray can be fully characterized by a point x1, x2, p1.Fortunately, these are only three variables, and the trajectories can be easily rep-resented For instance, Figure 6.22 shows the trajectory of a ray in the phase space

x1- x2- p1and its projection on the x1- x2plane This projection has the equation

(6.23)This ray trajectory is the one obtained for meridian rays in a fiber whose square

of the refractive index has a parabolic profile versus x1(see Miñano, 1985b)

A one-parameter family of ray trajectories in the phase space forms, in general,

a surface For instance, consider the family of rays derived from Eq (6.23) taken

C as the parameter of the family (A and B are kept constants) The

representa-tion of this family in the phase space are the cylinders shown in Figure 6.23 The

ray trajectories in this phase space are wrapped around the cylinder, and they

don’t cross Different cylinders correspond to different values of A and B.

Now let us consider the problem of designing a 2D nonimaging concentrator

in the phase space In general the problem involves determining the optical systemsuch that a given bundle of rays described at a line called the entry aperture istransformed by the optical system in another prescribed bundle of rays at the exit

aperture Assume for simplicity that the entry aperture is at x2= 0 and that the

exit aperture is at x2= 1 The bundle at the entry aperture can be defined by a

region of the plane x2= 0, as well as the bundle at the exit aperture is defined by

another region of the plane x2= 1 (see Figure 6.24 and Miñano, 1993a) Because

of the conservation of étendue, both regions must have the same area

The edge-ray principle simplifies the problem of design: To get the tioned goal we must design an optical system that transforms the edge rays of thebundle at the entry aperture in the edge rays of the bundle at the exit aperture.This is equivalent to stating that the edge rays’ trajectories will form a tubelike

aforemen-surface in the phase space that cuts the x2= 0 plane and the x2= 1 plane at thecontours of the regions defining the bundles at the entry and at the exit

Figure 6.24 The nonimaging design problem: The bundles of rays at the entry aperture

(x2= 0) and at the exit aperture (x2 = 1) are prescribed (left side) An optical system has to

be designed such that the edge rays at the entry are the same as the edge rays at the exit; that is, the edge ray trajectories in the phase space must form a surface connecting the edge rays’ representations at both apertures.

In general the imaging problem has less degrees of freedom Figure 6.25 showsthe phase space representation for this case At the object and at the image planethere is a prescribed family of one-parameter bundles of rays to be coupled Therays issuing or reaching a point of the object or imaging planes form each one ofthese one-parameter bundles From the mathematical point of view, in the non-imaging problem we have to find an optical system that admits a given particularintegral of the Hamilton equations, whereas in the image problem we have to find

an optical system that admits a given first integral of the Hamilton equations.Now let us go back to the 3D case Let us call restricted entry phase space to

the points of the phase space whose spatial coordinates x1, x2, x3belong to the entryaperture The restricted exit phase space is defined in a similar way In the non-imaging design the edge rays have prescribed descriptions in the restricted entryand exit phase spaces The edge rays in the restricted phase spaces form a curvethat encloses the set of points representing the rays of the transmitted bundle.Note that in the 3D case, the edge rays form a three-parameter bundle of rays,

and thus their trajectories in the six-dimensional phase space (x1- x2- x3- p1

-p2- p3) form a four-dimensional subset that must be contained in the subset P(x1,

x2, x3, p1, p2, p3) = 0 The subset P = 0 is then five-dimensional, and thus a dimensional subset can be characterized by an additional equation of the type w (x1,

four-x2, x3, p1, p2, p3) = 0 (w here has no relation with the angular frequency, which isnot considered in this analysis) The surface of the edge rays’ trajectories is then

defined by P = 0 together with w = 0 The function w(x1, x2, x3, p1, p2, p3) is notuniquely determined except when w = 0

The question now is to find the conditions on the function w so w is a surface

formed by ray trajectories The answer is that the Poisson bracket of w and P should be zero, when P = 0 and when w = 0—that is,

(6.24)Remember that the Poisson bracket is defined as

Figure 6.25 The imaging problem: A family of one-parameter bundles of rays at the object

plane (x2= 0) and another one at the image plane (x2 = 1) are prescribed (left side) An optical system has to be designed such that each bundle at the object plane is imaged to its corre- sponding bundle at the image plane.

Since the variable transformation from x1- x2- x3- p1- p2- p3to i1- i2- i3- u1

- u2- u3is canonical, the problem can be easily established in the new variables:

Eq (6.24) becomes (now w is a function of the variables i1, i2, i3, u1, u2, u3obtained

with the preceding transformation from the function w (x1, x2, x3, p1, p2, p3))

Hamiltonian function H (see Eq (6.17)) and thus the conditions on the refractive

index distribution and on the modulus of the gradients of the coordinate variables

(more precisely, the conditions on the functions aj = |—ij|/n).

Because we are not completely free to choose the Hamiltonian function(because, for instance, its dependence with the squares of the conjugate variablesmust be linear if a three-orthogonal coordinate system is used), then choosing thefunction w is not completely free either In order to find the restrictions we mustimpose to the function w, we must expand Eq (6.27)

The problem is further simplified if we restrict the analysis to elliptic edge ray

bundles that can be defined by a couple of equations H = 0 (H is defined in Eq.

(6.17)) together with

(6.28)

Observe that with this restriction, w and H are both linear functions of the squares

of the conjugate variables u1, u2, u3 The definition of Eq (6.28) implies that the

cone formed by the rays of the bundle passing through any point i1, i2, i3has threeplanes of symmetry Therefore, the flow lines of the bundle are one of the threecoordinate lines Thus, we are restricting elliptic bundles to those whose flow linesmay be coordinate lines of a three-orthogonal system This restriction implies, forinstance, that the flow lines are orthogonal to a family of surfaces and thus that

J · — ¥ J = 0 (J is the geometrical vector flux) which is not a necessary condition for J This restriction is not imposed in the analysis done further in this chapter

using the Lorenz geometry tools, where, nevertheless, the refractive index isassumed to be constant

The type of bundle given by Eq (6.28) can be used as an edge ray bundle inthe flow-line design method where the existence of a reflector surface connectingentry and exit apertures borders permits the edge ray bundle to be unbounded in

the spatial variables i1, i2, i3 (see Appendix B) This won’t to be the case of theSimultaneous Multiple Surface design method described in Chapter 8

Owing to the symmetries of the elliptic bundles, if we find a refractive indexdistribution that has as a solution a prescribed elliptic bundle, then we will easily

be able to design a concentrator with the flow-line method All we will have to do

is choose a surface formed by flow lines of the bundle as reflector In the definition

of elliptic bundles given by Eq (6.28) it is implicitly established that one of the

three coordinate lines are flow lines of the bundle and that the plane tangents to

the surfaces i j = constant ( j = 1, 2, 3) at each point i1, i2, i3are planes of

symme-try of the bundle at this point Assume, for instance, that the coordinate lines i1(i.e., the lines i2= constant i3= constant) are the flow lines of interest Then, any

surface i2 = constant will contain flow lines and its tangents are planes of

sym-metry of the bundle Then, if the surface i2= constant is mirrored, the bundle result

is unaffected (from the collection point of view), and thus we will obtain the desired

concentrator provided the surface i2= constant defines the entry and exit aperture

of the concentrator according to our requirements The concentrator will be formed

by a mirror with the i2= constant surface shape and the refractive index tion The mirror edges will define the shape of the entry and exit apertures Addi-tionally we know that the rotational hyperbolic concentrator is a solution of thistype, and thus we know that there is at least one solution for this problem.The analysis of elliptic bundles may appear too restrictive, and that may be

distribu-so Nevertheless, we have to take into account that the edge-ray bundle of anynontrivial ideal 3D concentrator known at present is an elliptic bundle Moreover,the concept of an elliptic bundle can also be applied in 2D geometry (see Section6.16), and in this case any edge-ray bundle can be viewed as an elliptic bundle.Since the majority of design methods of nonimaging concentrators (other than thePoisson brackets and the numerical methods) are actually 2D methods, we canconclude that the concept of elliptic bundle is not so restrictive

6.13.2 Elliptic Edge-Ray Bundle Analysis

Let us define the following vectors

(6.29)(6.30)(6.31)

The vectors a and a depend solely on i1, i2, i3 Using this notation, equations

H = 0, w = 0 remain as

(6.32)(6.33)

Note that for a given edge-ray bundle—that is, for a given surface (H = 0, w = 0)—

the vector a is not uniquely determined: Any vector

(6.34)where m is an arbitrary parameter m π 1 defines the same edge-ray bundle Eqs

(6.32) and (6.33) must be independent—that is, a ¥ a π 0.

The equation {w, H} = 0 can be written as