Nonimaging Optics Winston Episode 6 ppt

case in which the flow lines are “reflected” by the flat mirror. For our purposes, we do not need to go further in these considerations (which also would require a more detailed definition of the flow lines). When the flow lines find the reflector surface such that the geometrical vector flux is contained in the tangent to the surface, then it may happen that the new flow lines are contained in the reflector surface. This property has been used to design a multifoliate ideal 3D concentrator by Forbes and Basset (1982), which is shown in insert Figure 6.34. This concentrator is formed by many (infinite in the theoretical limit case) two-sided mirrors with a spherical shape. The center of the spheres is at the center of the entry aperture. The reflectors’ surfaces are extended in the right hemisphere defined by the entry aperture plane, excluding the points 6.15 Multifoliate-Reflector-Based Concentrators 139 reflector entry aperture exit aperture two-sided reflector 2f 2f S Figure 6.33 Flow lines reflected by a flat mirror. Figure 6.34 Multifoliate ideal 3D nonimaging concentrator that collects with maximal con- centration all the rays impinging its entry aperture within an angular cone a = 20°. inside the sphere S. In this region, the meridian rays form a constant angle equal to f with respect to the tangents to the circumferences centered at the entry aper- ture center. At the entry aperture, not only the meridian rays but the remaining ones form an angle f with these tangents. The meridian section of S is the cir- cumference whose points subtend an angle 2f with the exit aperture diameter. Because of the multiple reflections suffered by the rays in their passage through the multifoliate mirrors, Forbes and Basset suggested total internal reflec- tion. Simply using a set of transparent solid thin spherical shells can get the desired performance. This 3D ideal concentrator can be very well approximated in the real world using nonzero thickness dielectric layers. Both articles by Bassett and Forbes (1982) examine other cases in which the desired output bundle is not the one illuminating the exit aperture isotropically but illuminating it with a restricted angle, called angle transformers. Again, the design of Forbes and Basset gives ideal concentrators in 3D geometry. It can be proved that the bundle of collected rays of the multifoliate concen- trator is also an elliptic bundle. For this reason, we can use the Poisson bracket method to find out the refractive index distribution that can substitute the multi- foliated region. For this, it is convenient to choose a triorthogonal coordinate system so the flow lines are one of the coordinate lines. These coordinates are 140 Chapter 6 The Flow-line Method for Designing Nonimaging Optical Systems x 3 x 1 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 –3.0 –2.5 –2.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 Figure 6.35 Flow lines in the multifoliate concentrator of Figure 6.34. spherical. The coordinate change from cylindrical coordinates (r, q, z) to the new ones is given by (6.110) Vector a is thus (6.111) The rotational symmetry of the bundle we are considering implies that the second component of the vectors V and N should be a constant. Application of Eqs. (6.54) and (6.55) to vector a, taking into account this consideration, gives (6.112) (6.113) We are looking for a refractive index distribution such that the meridian rays form an angle f with the tangents to the circumferences centered at r = 0 and the bundle is circular in the plane z = 0—that is, when i 3 = 0. With this description we can calculate vector a. We can choose vector a so its third component is zero because the third com- ponent of vector a is zero only when i 1 Æ•(see Section 6.13.2, Eq. (6.34)). Then a 1 = 1/(u 1 ) 2 when u 2 = 0, (Eq. (6.33))—and u 1 is the optical direction cosine of merid- ian edge rays with respect the i 1 lines. Thus, we can write (6.114) From Eq. (6.54), and taking into account that the edge rays form a circular bundle at the plane i 3 = 0, we derive (6.115) From Eq. (6.55) it is concluded that (6.116) Where C is an arbitrary constant (C 2 = (C 2 - v 2 )/cos 2 f). What is important in this result for our purposes is (1) there is an arbitrary multiplicative constant, and (2) the flow lines are contained in surfaces of the same refractive index (surfaces i 1 = constant). Let us take a thin layer between the surfaces i 1 = const. and i 1 +Di 1 = const. Since the flow lines are contained in these surfaces, we can isolate this region by mirroring the two surfaces. The bundle of edge rays is not disturbed by this action, and, therefore, it is still the same one we used for the preceding calculations. n C i 2 2 1 2 = a = Ê Ë ˆ ¯ 1 1 1 0 22 1 2 nisin ,, f a = ( ) = Ê Ë Á ˆ ¯ ˜ aa f a 12 22 2 0 1 0,, sin ,, n V =- - Ê Ë Á ˆ ¯ ˜ n C i vC v i 2 2 1 2 22 2 2 3 ,, cos N =- - Ê Ë Á ˆ ¯ ˜ C i nC n i 1 1 2 21 2 2 3 ,, cos a n iii n iii=— ( ) — ( ) — ( ) ( ) = ( ) ( ) - - 11 1 2 1 2 2 2 3 2 2 13 2 1 2 , , , cos , iz i i z 1 22 2 3 1 =+ = =- Ê Ë Á ˆ ¯ ˜ - r q r tan 6.15 Multifoliate-Reflector-Based Concentrators 141 Assume that Di 1 is thin enough to consider that the refractive index is constant in this layer. We can still choose C so the refractive index of this layer is a given one—for instance, n = 1. This reasoning is independent of the value of i 1 , and thus it can be applied to any thin layer in between two i 1 = constant. In this way we get that multifoliate region of the concentrator of Figure 6.34 can also be derived from the Poisson bracket method. In the Forbes and Bassett (1988) articles it is explained in detail that the bundle exiting the multifoliate region is coincident, at the sphere S, with the bundle of rays reaching the receiver disk. As also noted in these articles, a practical way to manufacture an efficient multifoliate concentrator should rely on total internal reflection. The geometry of the concentrator of Figure 6.34 permits the total internal reflection condition with practical materials of refractive index around 1.5. 6.16 THE POISSON BRACKET METHOD IN 2D GEOMETRY When the spatial variables are only 2, i 1 , and i 2 , it is called a 2D geometry problem. Consequently, their conjugate variables are also 2, u 1 , and u 2 . The formulation of the problem is identical to the 3D case, except i 3 and u 3 do not appear. The equa- tions H = 0 and w = 0 can be written as in Eqs. (6.32) and (6.33), but now, u, a, and a are two-dimensional vectors. (6.117) (6.118) Since it is assumed that a and a are independent, then these two equations define uniquely a vector u. This means that for any point i 1 , i 2 of the concentrator there is a single vector u associated to it—that is, there are four possible combinations of values of the conjugate variables (±u 1 , ±u 2 ), which represent the two possible directions of the edge rays passing through the point i 1 , i 2 . The result of the elliptic bundle analysis in 2D geometry is the equivalent to Eq. (6.55). (6.119) Where now V is a two-dimensional vector that fulfills (6.120) Vector u can be written as (equivalent to Eq. (6.56)) (6.121) In the 3D problem we restricted the analysis to a certain type of bundles (the ellip- tic bundles). In the 2D problem, the description of the edge-ray bundle given by Eqs. (6.117) and (6.118) is more general. In fact, it covers all the cases in which there are a couple of edge rays at every point i 1 , i 2 of the concentrator. This case includes all 2D concentrators considered in this book. The design procedure is much simpler in the 2D case, and the number of solu- tions is much broader than in the 3D case. One possible design procedure is to fix uV= V = ( ) ( ) [] vi vi 11 22 , aV V◊=◊=a 1 w ∫◊-=u a 10 H ∫◊-=ua 10 142 Chapter 6 The Flow-line Method for Designing Nonimaging Optical Systems the refractive index distribution and then find the vector V and the coordinates i 1 and i 2 that fulfill the contour conditions. This procedure is essentially the string method (see Section 4.4). The way in which we fix the string is what we call here the contour conditions. The fact that the string is straight between fixing points is equivalent to establish the refractive index distribution (rays are straight lines in a constant refractive index distribution). The movement of the pencil along the string is equivalent here to the calculation of the flow lines—one of the coordinate lines. In order to clarify this equivalence, let us introduce Eq. (6.121) in Eq. (6.119) using Eq. (6.120), and using the expression of a j in terms of |—i j | and n. (6.122) This equation together with the condition of orthogonality of the coordinate lines (—i 1 -—i 2 = 0) and noting that the components of vector u—that is, v 1 and v 2 must be positive lead to (6.123) where (6.124) The functions m 1 and m 2 are the optical path lengths of the two sets of edge rays (remember that there were two edge rays at every point) and Eq. (6.123) is simply their eikonal equation. 6.16.1 Example: The Compound Triangular Concentrator Another possible design procedure is to propose a vector V that fulfills the contour condition, select a set of orthogonal coordinates i 1 and i 2 , and then calculate the refractive index distribution in a way quite similar to the one shown in Section 6.14.2. For instance, let’s choose the set of orthogonal coordinates show in Figure 6.36. These coordinates can be chosen such that |—i j | has a constant value in each of the four regions defined by the straight-lines AE and A¢E¢. Let v 1 (i 1 ) and v 2 (i 2 ) be linear functions of i 1 and i 2 , respectively. These functions and this coordinate system define what is called the Compound Triangular Concentrator (CTC) (Miñano, 1985a, 1985c). Figure 6.36 shows the i 1 and i 2 lines of a CTC concentrator. These lines have three straight segments. Consequently, there is a discontinuity on —i 1 and —i 2 at the points where these straight segments connect (lines AE¢ and A¢E). The design is done in such a way that the refractive index is the same for all the points out of the lines where the discontinuities are (n b ). The refractive index at the discon- tinuities was studied as if the CTC was a limited case of concentrators in which —i 1 and —i 2 vary sharply but continuously at lines AE¢ and A¢E. It was concluded that the refractive index has a minimum in these lines, whose value is called n l . This refractive index minimum causes the lines AE¢ and A¢E 1 behave as non- existent layers for rays forming an angle with the normal smaller than sin -1 (n 1 /n b ) and behaves like a mirror (by total internal reflection) otherwise. Figure 6.37 mvidividi mvidividi 1111222 2111222 = ( ) + ( ) = ( ) - ( ) ÚÚ ÚÚ — ( ) =— ( ) =mmn 1 2 2 2 2 vi i vi i n 11 1 2 22 2 2 2 ( ) — ( ) + ( ) — ( ) = 6.16 The Poisson Bracket Method in 2D Geometry 143 shows the refractive index distribution. As in any other concentrator designed with the flow line method, the final step is to choose two flow lines as reflectors. In the case of Figure 6.37 the lines chosen are i 1 =±1. These flow lines define the edges of the entry (AA¢) and exit (EE¢) apertures. It should be emphasized that with our definitions, a 2D concentrator is not the same as a 3D concentrator with linear symmetry. An ideal 2D concentrator will not in general generate an ideal 3D concentrator by linear symmetry. The con- fusion arises from the fact that when the refractive index is constant (for instance, in the CPC), then the linear concentrator derived from the 2D concentrator is ideal with respect to the input and output bundles that are generated from the 2D case by the same linear symmetry. This is not the case of the CTC because its refrac- tive index is not constant. 6.17 ELLIPTIC BUNDLES IN HOMOGENEOUS MEDIA Gutiérrez et al. (1996) applied the Lorentz geometry to the problem of finding ellip- tic bundles in a medium of constant refractive index. The rapid development of Lorentz geometry is due to Einstein’s general relativity theory of gravitation. It provides a cone structure on an open subset ¬ of the space—that is, a cone at each point X of ¬ (formally, in the tangent space T x ¬ of X, which is isomorphic to ޒ 3 ). 144 Chapter 6 The Flow-line Method for Designing Nonimaging Optical Systems A¢ E A E ¢ i 1 = 1 i 1 = 0 i 2 = 0 i 2 = 1 Figure 6.36 i 1 and i 2 lines of a CTC concentrator. A ¢ E A E ¢ n b reflector n 1 Figure 6.37 Refractive index distribution of a CTC. In this example there are two differ- ent refractive indices for the materials within the concentrator, n b , which is the refractive index of the bulk material, and n 1 , which is the refractive index of the lines AE¢ and A¢E. This cone is called the light cone in Einstein’s theory because it is formed by the directions of the light rays in the continuous space-time. In nonimaging optics, Lorentz metric is used in a quite different interpretation. This cone is formed by the directions of the edge rays passing through a point in the conventional 3D space (see Figure 6.38a). We’ll briefly discuss Lorentz geometry here, but the interested reader can consult O’Neill (1983) for some background on differential geometry. Let B = {e 1 , e 2 , e 3 } be the canonical global basis vector field on ¬, which is ortho- normal with respect to the usual Euclidean metric. For each point X of ¬, B x is the canonical basis of the tangent space T x ¬. A Lorentz metric g on ¬ is a map that assigns to every point X of ¬ a bilinear map g x : T x ¬¥T x ¬Æޒ, with some additional properties. These properties may be expressed through the matrix G associated with the map g in the basis B. Using this matrix, one can formulate the map as g x (Y, Y¢) = Y t G(X)Y¢, where Y and Y¢ are two vectors of T x ¬ (the superscript t denotes transposition). The matrix G must verify the following three properties: 1. G is symmetric on every point X of ¬. 2. |G| π 0 on every point of ¬. 3. G has one negative and two positive eigenvalues on every point of ¬. This is expressed by saying that g has the signature (-, +, +). The elements of G(X) are functions on ¬ defined by g if (X) = g x (e i , e j ). If they are differentiable of class C • , the above properties characterize a C • Lorentz metric on ¬. Similarly, a C r Lorentz metric can be introduced (r Œ ގ). Three different types of vector Y are considered: spacelike if Y t GY > 0 or Y = 0, timelike if Y t GY < 0, and lightlike if Y t GY = 0 and Y π 0. The set of all lightlike vectors forms the light cone, whose equation is Y t GY = 0. If g : ޒ Ƭis a C 1 curve and g . is its derivative, then g is called spacelike if g . (t) is spacelike for all t. The timelike and the lightlike curves are defined in a similar way. Let us look at the role of the eigenvalues and eigenvectors of the matrix G of a Lorentz metric g. Because G is a symmetric matrix, it can be diagonalized, 6.17 Elliptic Bundles in Homogeneous Media 145 lightcone lightlike vector (a) (b) x 1 x 2 x 3 Figure 6.38 (a) The Lorentz geometry provides a cone structure in ¬, formed by the set of lightlike vectors at each point X. (b) The vectors {J, U, V} define the three planes of sym- metry of the cone of edge rays of the elliptical bundle at the point X. leading to a matrix G¢=diag(l 1 , l 2 , l 3 ), where l 1 , l 2 , l 3 Œ ޒ. This transformation from G to G¢ is just a change of basis, and thus G¢ may be expressed as G¢= A -1 GA, where A is the matrix changing between the orthonormal canonical basis B x and a new one B x ¢, which is also orthonormal (and thus A -1 = A t ). The new basis B x ¢ is formed by three eigenvectors of G. The columns of A are the components of these eigenvectors in the canonical basis B x , and the elements l 1 , l 2 , l 3 of the diag- onal of G¢ are the eigenvalues of G associated with the corresponding eigenvec- tors. Because g is Lorentzian, the signature is (-, +, +); thus only one eigenvalue is negative, and the other two are positive. There is no loss of generality if we assume that l 1 < 0 and |A| = 1 (changing the order of the columns of A if neces- sary). Thus, the basis B x ¢ in which G¢ is a diagonal matrix has a nice geometric interpretation, since it can be obtained from the canonical basis B x by a rotation. Now the study of the cone structure in the basis B x ¢ is trivial. B x ¢={J, U, V}, where J, U, and V are the eigenvectors of G associated with l 1 , l 2 , and l 3 , respec- tively. Let Y π 0 be a lightlike vector. If the components of Y in the basis B x ¢ are (y 1 , y 2 , y 3 ), the equation of the lightlike vectors in this basis is Y t G¢Y = 0—that is, (6.125) which is a cone whose axis is parallel to the vector J (remember that l 1 < 0, and l 2 , l 3 > 0). The intersection of the cone and the plane y 1 = 1 is the ellipse (6.126) The ellipse axes are parallel to the vectors U and V (see Figure 6.38b), and their respective semiaxis lengths are (-l 1 /l 2 ) 1/2 and (-l 1 /l 3 ) 1/2 . To summarize: 1. The light cone at X is elliptic, "X Œ¬. 2. The direction of the elliptic cone axis is the direction of J, the eigenvector of G associated with the negative eigenvalue of G. 3. The directions of the principal axis of the elliptic cone are the directions of U and V, the eigenvectors of G associated with the positive eigenvalues of G. Sections 6.17.1 and 6.17.2 are devoted to the “optical condition,” which is the con- dition that the Lorentz geometry must fulfill to make the lightlike curves define the edge rays of the bundle. In Section 6.17.2 we examine the system of partial differential equations, which are derived from the general optical condition. We also discuss a specific set of solutions of said system, which coincide with the already identified bundles found by other methods. Section 6.17.4 describes some examples of elliptic bundles, and the conclusions are given in Section 6.18. 6.17.1 The Restricted Optical Condition Let us consider a medium with constant refractive index n = 1. Thus, the ray tra- jectories will be straight lines. In the Lorentz geometry approach of Gutierrez et al. (1996), a “restricted optical condition” was used, which stated that every lightlike geodesic of the Lorentz metric must be a geodesic of the Euclidean metric. The application of this condition forces the lightlike curves to be geodesics of both the Lorentz metric and the Euclidean metric but provided tools to find new ellip- tic bundles. ll l 22 2 33 2 1 yy+=- lll 11 2 22 2 33 2 0yyy++= 146 Chapter 6 The Flow-line Method for Designing Nonimaging Optical Systems Let us eliminate requirement of the lightlike curves being geodesics of the Euclidean metric from restricted optical condition. Then every lightlike geodesic of the Lorentz metric must be a straight line. This condition will be called “refor- mulated restricted optical condition”. Thus, we can reformulate the restricted optical condition of Gutiérrez et al. (1996) as follows: If g is a C 2 curve in ¬ such that g . (t) π 0, " t Œ ޒ, then, for k Œ {1, 2, 3}, (6.127) where the symbol Ÿ denotes the vector product. The upper left equation establishes that g is a geodesic curve in Lorentz geom- etry. The lower left one implies that g is a lightlike curve. The equation on the right-hand side states that g ¨ (t) is parallel to g . (t), which is the general condition for g to be a straight line. G k is a matrix formed by the elements G k ij , which are the Christoffel symbols of the Lorentz metric (O’Neill, 1983). The expression of these symbols using Einstein’s notation for the summation (in which the summation symbol is omitted) is (6.128) where G = (g ij ); G -1 = (g ij ). Let us see that the restricted optical condition Eq. (6.127) is fulfilled if (6.129) where f k are functions on ¬, and the matrices A k are formed by the elements , a k are also functions on ¬ and d ki is the Kronecker delta function. By substitution of G k in the upper-left equation in Eq. (6.127), and taking into account the lower-left equation in Eq. (6.127), we obtain: (6.130) where a denotes the vector a = (a 1 , a 2 , a 3 ). Equations Eq. (6.129) are the reformulated restricted optical equations. Let us prove that Eq. (6.129) is not only sufficient but necessary for the restricted optical condition Eq. (6.127) to be fulfilled. Consider a point X in ¬ and a lightlike vector Y at X, which must fulfill (6.131) Let the straight line g (t) be the lightlike curve fulfilling g (t o ) = X and g . (t o ) = Y. If G denotes the vector whose components are the matrices G k , from (1): (6.132) where the right-hand side equation is obtained by substitution of the upper left equation in the lower left one. Setting t = t o in the equation on the right, the fol- lowing two independent equations are obtained: ˙˙ ˙ ˙ ˙˙ ˙ ˙˙ ˙ gggg gg gg gg tttt tt tt tt t t ( ) =- ( ) [] ( ) [] ( ) ( ) Ÿ ( ) = ¸ ˝ ˛ fi ( ) Ÿ ( ) [] ( ) [] ( ) = G G 0 0 YG Y t X ( ) = 0 ggg g gg g g ggg ktk k iik k k a+=fi+ ( ) =fi =-◊fiŸ= ˙˙ ˙˙ ˙˙ ˙ ˙˙˙˙ Aa00 0 A aa ij k i kj j k i =+ 22 dd G kk k fk=+ Œ {} GA,,,123 G ij kkm jm i mi j ij m gg g g=+- ( ) 1 2 ,,, ˙˙ ˙˙ ˙˙ ˙˙ ˙ yt t t t ttt tt k t k t ( ) + ( ) [] ( ) [] ( ) = ( ) [] ( ) [] ( ) = ¸ ˝ ˛ fi ( ) Ÿ ( ) = ggg ggg gg G 0 0 0 G 6.17 Elliptic Bundles in Homogeneous Media 147 (6.133) In Eq. (6.132) we indicate explicitly that G k is not a function of Y. As the equations Eq. (6.132) are third-order polynomials in the components of vector Y and must vanish for all the vectors Y defined by the quadric Eq. (6.131), then the third-order polynomials must be decomposed as: (6.134) where D = (d 1 (X), d 2 (X), d 3 (X)) and E = (e 1 (X), e 2 (X), e 3 (X)) are vectors whose coef- ficients are functions on ¬. Developing Eq. (6.134) and identifying the coefficients of the powers of Y, it is found that D and E fulfill (6.135) being f k functions on ¬, and the following system of differential equations is obtained: (6.136) It is straightforward to see that the 15 equations Eq. (6.136) and the 18 equations Eq. (6.129) are equivalent: Eq. (6.136) can be obtained from Eq. (6.129) by elimi- nation of the 3 unknown functions a k . Let us solve the restricted optical equations Eq. (6.129). Using the expressions for the Christoffel symbols G k ij , we can write the restricted optical equations as (6.137) Calling F k = f k + g mk a m /2, this system of equations can be rewritten as (6.138) Assume that the metric is C 2 and that the functional coefficients F k and a k are C 1 . The integrability conditions of system Eq. (6.138), given by forcing the equality of the second-order cross derivatives (g ij,mk = g ij,km ), lead to the following expressions for the functions F k and a k : (6.139) The upper equations say that the vector a = (a 1 , a 2 , a 3 ) is irrotational, and thus the must be a potential function V(X) on ¬ fulfilling: (6.140) Taking into account Eq. (6.140), the lower equations in Eq. (6.139) can be easily integrated, and the solution can be written in terms of V as: (6.141) Fex k kV kk =+ ( ) Œ {} - qm 123,, aV=— aa FaF ijij j i i j i j i j = -= πŒ {} 0 123,, ,, ggFggFgagijm im j mk k ij ik k jm j im, ,,, ,,=++ Œ {} 123 ggfggfg a g a gag ijm im j mk k ij ik k jm i mj m ij j im, ,,, ,,= + +++ Œ {} 22 123 G G GG ij kk ii ij kk ij ii i ik ki ii k ik fg i k fg i j i k j k fg fg i k =π =πππ -= - π ,, 22 D E =- ( ) =- ( ) 0 0 32 31 ,, ,, ff ff YY G YY G ttt ttt 2 3 3 2 3 1 1 3 YY YY YYDY YY YY YYEY GG GG [] - [] ∫ [] ( ) [] - [] ∫ [] ( ) YXYX YXYX t t 2 3 3 12 3 1 1 13 0 0 YYYY YYYY GG GG ( ) [] - ( ) [] = ( ) [] - ( ) [] = 148 Chapter 6 The Flow-line Method for Designing Nonimaging Optical Systems [...]... Winston, R (1989) High Collection Nonimaging Optics Academic Press, New York Welford, W T., OGallagher, J., and Winston, R (1987) Axially symmetric nonimaging ux concentrators with the maximum theoretical concentration ratio J Opt Soc Am A 4, 66 68 Winston, R., and Welford, W T (1979) Geometrical Vector Flux and some new imaging concentrators J Opt Soc Am A 69 , 5325 36 7 CONCENTRATORS FOR PRESCRIBED... calculations Figure 6. 40c shows this for the particular case p = f of bundle B2 6. 18 CONCLUSION Using poisson brackets is a smart way to establish the general nonimaging design problem Nevertheless it is not by itself a useful tool unless the bundle to work 1 56 Chapter 6 The Flow-line Method for Designing Nonimaging Optical Systems B1 flow line flow line B2 ( p = f ) (a) 90 (b) p Dr 2 Dr 45 (c) Figure 6. 40 In... Eq (6. 158), for the parameter values of this case, takes the form ẩ f G x3 =0 = 0 ẻg x2 whose eigenvalues are 0 g x2 p 0 0 p (6. 160 ) 155 6. 18 Conclusion Table 6. 1 Some characteristics of the elliptic bundles with 2 3 g 0, s = 0, a = 0, m = 0, pf 0 ( r = x2 + x3 ) Case p f ơ Negative Eigenvalue B1 >0 0 fp r> g l3 B2 >0 l1 = 2 ( f + p) + ( f - p)2 + 4g 2 + x2 2 l2 = p l3 = (6. 161 )... design of nonimaging concentrators: A review In Nonimaging Optics: Maximum-Efciency Light Transfer II (Winston, R., and Holman, R L., eds.) Proc SPIE 20 16, 98108 Miủano, J C (1993b) The Poisson Brackets method of design of nonimaging concentrators Optical Society of America Annual Meeting Optical Society of America, Washington, DC Postconference Edition Vol 16, 315 Miủano, J C., and Benớtez, P (1999) Poisson... variable-angle nonimaging transformer Optica Acta 29, 12831297 Garwin, R L (1952) The design of liquid scintillation cells Rev Sci Instrum 23, 755757 Gutiộrrez, M., Miủano, J C., Vega, C., and Benớtez, P (19 96) Applications of the Lorentz geometry to nonimaging optics: New three-dimensional ideal concentrators J Opt Soc Am A 13(3), 532540 Kline, M., and Kay, I W (1 965 ) Electromagnetic Theory and Geometrical Optics. .. f 2 h hậ hk k k a = a + h k {1, 2, 3} (6. 143) Let us select the function h = e-V, where V is the potential function associated to the vector a = (a1, a2, a3) By the denition of V in Eq (6. 140) and from Eq (6. 143), we get that for this function h, G fullls the restricted optical equations Eq (6. 129) with fÂk = eVFk and aÂk = 0 This implies, again by Eq (6. 140) that the potential function for GÂ... Arnaud, J A (19 76) Beam and Fiber Optics Academic Press, New York Arnold, V I (1974) Les Mộthodes Mathematiques de la Mộcanique Classique (French translation) Mir, Moscow Bassett, I M., and Forbes, G W (1982) A new class of ideal non-imaging transformers Optica Acta 29, 12711282 Benớtez, P (1999) Elliptic bundles in three-dimensional geometry for nonimaging optics: A new approach J Opt Soc Am A 16( 9), 22452252... G k = f  k G k {1, 2, 3} (6. 144) which differs from Eq (6. 129) in that aÂk = 0 This condition, when introduced in Eq (6. 130), provides that g (t) = 0, which means that the edge rays of G are not only geodesics of the Lorentz metric but also geodesics of the Euclidean metric These equations Eq (6. 144) are the restricted optical equations dened in Gutiộrrez et al (19 96) , where the restricted optical... Y = aP for certain a From the cone structure equation and Eq (6. 147), the lightlike vectors Y fulll Y tGY = 0 Y[ Y tGY ] = 0 (6. 148) Note that the second equation in Eq (6. 148) can also be obtained by deriving the left-hand side of Eq (6. 145) with respect to the parameter t and setting t = 0 for all X ơ As the second equation in Eq (6. 148) is a third-order polynomial in the components of vector Y... bundles In Nonimaging Optics: Maximum-Efciency Light Transfer V (Winston, R., ed.) Proceedings of SPIE Vol 3781, 211 ONeill, B (1983) Semi-Reimannian Geometry with Applications to Relativity Academic, New York Stavroudis, O N (1972) The Optics of Rays, Wavefronts, and Caustics Academic, New York Weisstein, E W (1999) CRC Concise Encyclopedia of Mathematics CRC Press, Boca Raton Welford, W T., and Winston, . 4, 66 68 . Winston, R., and Welford, W. T. (1979). Geometrical Vector Flux and some new imaging concentrators. J. Opt. Soc. Am. A. 69 , 532–5 36. 158 Chapter 6 The Flow-line Method for Designing Nonimaging. is obtained: (6. 1 36) It is straightforward to see that the 15 equations Eq. (6. 1 36) and the 18 equations Eq. (6. 129) are equivalent: Eq. (6. 1 36) can be obtained from Eq. (6. 129) by elimi- nation. geometry is the equivalent to Eq. (6. 55). (6. 119) Where now V is a two-dimensional vector that fulfills (6. 120) Vector u can be written as (equivalent to Eq. (6. 56) ) (6. 121) In the 3D problem we restricted