from obvious geometrical considerations q¢ cannot exceed p/2, this suggests (p/2q) 2 as a theoretical upper limit to the concentration. Unfortunately, this argument is invalid because the étendue as we have defined it is essentially a paraxial quantity. Thus, it is not necessarily an invari- ant for angles as large as p/2. In fact, the effect of aberrations in the optical system is to ensure that the paraxial étendue is not an invariant outside the paraxial region so that we have not found the correct upper limit to the concentration. There is, as it turns out, a suitable generalization of the étendue to rays at finite angles to the axis, and we will now explain this. The concept has been known for some time, but it has not been used to any extent in classical optical design, so it is not described in many texts. It applies to optical systems of any or no sym- metry and of any structure—refracting, reflecting, or with continuously varying refractive index. Let the system be bounded by homogeneous media of refractive indices n and n¢ as in Figure 2.16, and suppose we have a ray traced exactly between the points P and P¢ in the respective input and output media. We wish to consider the effect of small displacements of P and of small changes in direction of the ray segment through P on the emergent ray so that these changes define a beam of rays of a certain cross section and angular extent. In order to do this we set up a Cartesian coordinate system Oxyz in the input medium and another, O¢x¢y¢z¢, in the output 2.7 The Generalized E ´ tendue or Lagrange Invariant 19 Figure 2.15 The étendue for a multielement optical system with an internal aperture stop. Figure 2.16 The generalized étendue. 6 It is necessary to note that the increments dL and dM are in direction cosines, not angles. Thus, in Figure 2.17 the notation on the figure should be taken to mean not that dM is the angle indicated, but merely that it is a measure of this angle. 20 Chapter 2 Some Basic Ideas in Geometrical Optics medium. The positions of the origins of these coordinate systems and the direc- tions of their axes are quite arbitrary with respect to each other, to the directions of the ray segments, and, of course, to the optical system. We specify the input ray segment by the coordinates of P(x, y, z), and by the direction cosines of the ray (L, M, N). The output segment is similarly specified. We can now represent small displacements of P by increments dx and dy to its x and y coordinates, and we can represent small changes in the direction of the ray by increments dL and dM to the direction cosines for the x and y axes. Thus, we have generated a beam of area dxdy and angular extent defined by dLdM. This is indicated in Figure 2.17 for the y section. 6 Corresponding increments dx¢, dy¢, dL¢, and dM¢ will occur in the output ray position and direction. Then the invariant quantity turns out to be n 2 dx dy dL dM—that is, we have (2.8) The proof of this theorem depends on other concepts in geometrical optics that we do not need in this book. We have therefore given proof in Appendix A, where references to other proofs of it can also be found. The physical meaning of Eq. (2.8) is that it gives the changes in the rays of a beam of a certain size and angular extent as it passes through the system. If there are apertures in the input medium that produce this limited étendue, and if there are no apertures elsewhere to cut off the beam, then the accepted light power emerges in the output medium so that the étendue as defined is a correct measure of the power transmitted along the beam. It may seem at first remarkable that the choice of origin and direction of the coordinate systems is quite arbitrary. However, it is not very difficult to show that the generalized étendue or Lagrange invariant as calculated in one medium is independent of coordinate translations and rotations. This, of course, must be so if it is to be a meaningful physical quantity. The generalized étendue is sometimes written in terms of the optical direction cosines p = nL, q = nM, when it takes the form (2.9) dx dy dpdq n dx dy dL dM n dx dy dLdM¢¢¢¢¢= 22 Figure 2.17 The generalized étendue in the y section. An étendue value is associated to any 4-parameter bundle of rays. Each combina- tion of the four parameters defines one single ray. In the example of Figure 2.16, the four parameters are x, y, L, M (or x¢, y¢, L¢, M¢), but there are many other pos- sible sets of 4 parameters describing the same bundle. For the cases in which the rays are not described at a z = constant (or z¢=constant planes), then the follow- ing generalized expression can be used to calculate the differential of étendue of the bundle dE: (2.10) The total étendue is obtained by integration of all the rays of the bundles. In what follows we will assume that the bundle can be described at a z = constant plane. In 2D geometry, when we only consider the rays contained in a plane, we can also define an étendue for any 2-parameter bundle of rays. If the plane in which all the rays are contained is a x = constant plane, then the differential of étendue can be written as dE = ndydM. As in the 3D case, the étendue is an invariant of the bundle, and the same result is obtained no matter where it is calculated. For instance, it can be calculated at z¢=constant, and the result should be the same: n¢ dy¢ dM¢=ndydM, or, in terms of the optical direction cosines, dy¢ dq¢=dy dq. We can now use the étendue invariant to calculate the theoretical maximum concentration ratios of concentrators. Consider first a 2D design, as in Figure 2.18. We have for any ray bundle that transverses the system (2.11) and integrating over y and M we obtain (2.12) so that the concentration ratio is (2.13) In this result a¢ is a dimension of the exit aperture large enough to permit any ray that reaches it to pass, and q¢ is the largest angle of all the emergent a a n n¢ = ¢¢sin sin q q 44na n asin sinqq=¢¢ ¢ ndydM n dydM= ¢¢ ¢ dE dx dy dpdq dydzdqdr dzdx dr dp=++ 2.7 The Generalized E ´ tendue or Lagrange Invariant 21 Figure 2.18 The theoretical maximum concentration ratio for a 2D optical system. rays. Clearly q¢ cannot exceed p/2, so the theoretical maximum concentration ratio is (2.14) Similarly, for the 3D case we can show that for an axisymmetric concentrator the theoretical maximum is (2.15) where again q is the input semiangle. The results in Eqs. (2.14) and (2.15) are maximum values, which may or may not be attained. We find in practice that if the exit aperture has the diameter given by Eq. (2.15), some of the rays within the incident collecting angle and aperture do not pass it. We sometimes also find in a number of the systems to be described that some of the incident rays are actually turned back by internal reflections and never reach the exit aperture. In addition, there are losses due to absorption, imperfect reflectivity, and so forth, but these do not represent fundamental limi- tations. Thus, Eqs. (2.14) and (2.15) give theoretical upper bounds on performance of concentrators. Our results so far apply to linear concentrators [Eq. (2.14)] with rectangular entrance and exit apertures and to rotational concentrators with circular entrance and exit apertures [Eq. (2.15)]. We ought, for completeness, to discuss briefly what happens if the entrance aperture is not circular but the concentrator itself still has an axis of symmetry. The difficulty with this case is that it depends on the details of the internal optics of the concentrator. It may happen that the internal optical system forms an image of the entrance aperture on the exit aperture—in which case it would be correct to make them similar in shape. For an entry aper- ture of arbitrary shape but uniform entry angle ±q i all that can be said in general is that for an ideal concentrator the area of the exit aperture must equal that of the entry aperture multiplied by sin 2 q i . We will see in Chapter 6 that such con- centrators can be designed. 2.8 THE SKEW INVARIANT There is an invariant associated with the path of a skew ray through an axisym- metric optical system. Let S be the shortest distance between the ray and the axis—that is, the length of the common perpendicular—and let g be the angle between the ray and the axis. Then the quantity (2.16) is an invariant through the whole system. If the medium has a continuously varying refractive index, the invariant for a ray at any coordinate z 1 along the axis is obtained by treating the tangent of the ray at the z value as the ray and using the refractive index value at the point where the ray cuts the transverse plane z 1 . The skew-invariant formula will be proved in Appendix C. If we use the dynamical analogy described in Appendix A, then h corresponds to the angular momentum of a particle following the ray path, and the skew- hnS= sing C a a n n max sin = ¢ Ê Ë ˆ ¯ = ¢ Ê Ë ˆ ¯ 22 q C n n max sin = ¢ q 22 Chapter 2 Some Basic Ideas in Geometrical Optics invariant theorem corresponds to conservation of angular momentum. In terms of the Hamilton’s equations, the skew invariant is just a first integral that derives from the symmetry condition. 2.9 DIFFERENT VERSIONS OF THE CONCENTRATION RATIO We now have some different definitions of concentration ratio. It is desirable to clarify them by using different names. First, in Section 2.7 we established upper limits for the concentration ratio in 2D and 3D systems, given respectively by Eqs. (2.14) and (2.15). These upper limits depend only on the input angle and the input and output refractive indices. Clearly we can call either expression the theoreti- cal maximum concentration ratio. Second, an actual system will have entry and exit apertures of dimensions 2a and 2a¢. These can be width or diameter for linear or rotational systems, respec- tively. The exit aperture may or may not transmit all rays that reach it, but in any case the ratios (a/a¢) or (a/a¢) 2 define a geometrical concentration ratio. Third, given an actual system, we can trace rays through it and determine the proportion of incident rays within the collecting angle that emerge from the exit aperture. This process will yield an optical concentration ratio. Finally, we could make allowances for attenuation in the concentrator by reflection losses, scattering, manufacturing errors, and absorption in calculating the optical concentration ratio. We could call the result the optical concentration ratio with allowance for losses. The optical concentration ratio will always be less than or equal to the theoretical maximum concentration ratio. The geometrical concentration ratio can, of course, have any value. REFERENCE Welford, W. T. (1986). “Aberrations of Optical Systems.” Hilger, Bristol, England. Reference 23 [...]... 4.1 The flux from a spherically symmetric sun falls off (r/R )2 = (1/sin q )2 Concave Focusing Mirror qs qs qs D /2 r F rqs F d /2 D /2 D = 2 r sin f C = (D/d )2 = (1/4) sin2 2f / sin2 qs £ (1/4)1/ sin2 qs £ (1/4) Cmax d = 2 r sin q / cos f D/d = sin f cos f /sin q = sin 2f / 2 sin q Figure 4 .2 Image-forming solar concentrator for planar absorber 4 .2 IMAGING DEVICES AND THEIR LIMITATIONS If one were to ask... clearly attained when f = p /2 At this point the optical concentration ratio is, allowing for the obstruction caused by the absorber, 2 1 cos2 2q max Ê a¢ ˆ - 1 = ◊ Ë a¯ 4 sin 2 q max cos2 q max (3.13) 40 Chapter 3 Some Designs of Image-Forming Concentrators Figure 3 .20 The Schmidt camera This optical system has no spherical aberration or coma, so, in principle, it could be a good concentrator for small... the circle, it subtends 2qmax at the ends of the mirror, so the collecting angle is 2qmax The mirror is not specified to be of any particular shape except that it must reflect all inner rays to the inside of the exit aperture Then if the mirror subtends 2f at the center of the circle, we find a¢ sin 2q max = a sin f (3. 12) and the minimum value of a¢ is clearly attained when f = p /2 At this point the optical... qs qs qs D /2 r F F ª r qs D /2 C = (D /2 p r sin qs = sin f / sin qs £ 1/p sin qs £ (1/p) Cmax Figure 4.3 Image-forming solar concentrator for tubular absorber desire to bridge the gap between the levels of concentration achieved by common imaging devices and the sine law of concentration limit that motivated the invention of nonimaging optics Entirely similar considerations can be applied to 2D or trough... of mimeographed notes and the book is a word-for-word transcription Maxwell, J C (1958) On the general laws of optical instruments Q J Pure Appl Math 2, 23 3 24 7 Morgan, S P (1958) General solution of the Luneburg lens problem J Appl Phys 29 , 1358–1368 4 NONIMAGING OPTICAL SYSTEMS 4.1 LIMITS TO CONCENTRATION The relationship between the concentration ratio and the angular field of view is a fundamental... the conservation laws in a natural way—has to work Perhaps a more serious “proof” is to notice that the 2D CPC rejects all stray radi- 48 Chapter 4 Nonimaging Optical Systems Imaging Optics ͵D dl = constant [Fermat 1601-1665], where P¢ P ray n = index of refraction I = path length a) P¢ P Nonimaging Optics: b) P ͵n dl = constant P¢ P string P¢ Figure 4.5 Fermat’s principle for rays and strings String... “Principles of Optics, ” 5th Ed Pergamon, Oxford, England Cornbleet, S (1976) “Microwave Optics. ” Academic Press, New York Harper, D A., Hildebrand, R H., Pernic, R., and Platt, S R (1976) Heat trap: An optimised far infrared field optics system Appl Opt 15, 53–60 Krenz, J H (1976) “Energy Conversion and Utilization.” Allyn & Bacon, Rockleigh, New Jersey Luneburg, R K (1964) “Mathematical Theory of Optics. ”... the 20 0inch telescope on Mt Palomar or any preferred telescope Of course, such an experiment had better remain in the realm of imagination only, since beginning astronomers are admonished never to point their telescopes at the sun at the risk of catastrophic damage to the instrument But to continue this train of thought, the concentration limit of a telescope is readily shown to be sin2 (2f)/4 sin2 q... refractive index n = 2; the rays shown emerge in air as the extreme rays in a solid angle 2p aplanatic imaging system of Figure 3.10 could be used by itself as a concentrator, as in Figure 3.17 The surface of radius a/n forms the spherical exit surface, and the internal angle 2a of the cone meeting this face is such that sin a = 1/n Thus, the emerging rays cover a solid angle 2p, as with the Luneburg... blocks without regard to the message, the task is much easier This is the key idea of nonimaging optics 4.4 THE EDGE-RAY PRINCIPLE OR “STRING” METHOD Figure 4.4c suggests the notion of transporting the boundary or edge of the container of rays in phase space This leads to one of the most useful algorithms of nonimaging optics We shall see that transporting the edges only, without regard to interior order, . Consider first a 2D design, as in Figure 2. 18. We have for any ray bundle that transverses the system (2. 11) and integrating over y and M we obtain (2. 12) so that the concentration ratio is (2. 13) In. the skew- hnS= sing C a a n n max sin = ¢ Ê Ë ˆ ¯ = ¢ Ê Ë ˆ ¯ 22 q C n n max sin = ¢ q 22 Chapter 2 Some Basic Ideas in Geometrical Optics invariant theorem corresponds to conservation of angular. dzdx dr dp=++ 2. 7 The Generalized E ´ tendue or Lagrange Invariant 21 Figure 2. 18 The theoretical maximum concentration ratio for a 2D optical system. rays. Clearly q¢ cannot exceed p /2, so the theoretical