Nonimaging optics
Nonimaging optics (also called anidolic optics) is the branch of optics concerned with the optimal transfer of light radiation between a source and a target. Unlike traditional imaging optics, the techniques involved do not attempt to form an image of the source; instead an optimized optical system for optimal radiative transfer from a source to a target is desired. Applications. The two design problems that nonimaging optics solves better than imaging optics are: Typical variables to be optimized at the target include the total radiant flux, the angular distribution of optical radiation, and the spatial distribution of optical  radiation. These variables on the target side of the optical system often must be optimized while simultaneously considering the collection efficiency of the optical system at the source. Solar energy concentration. For a given concentration, nonimaging optics provide the widest possible acceptance angles and, therefore, are the most appropriate for use in solar concentration as, for example, in concentrated photovoltaics. When compared to "traditional" imaging optics (such as parabolic reflectors or fresnel lenses), the main advantages of nonimaging optics for concentrating solar energy are: Also, for low concentrations, the very wide acceptance angles of nonimaging optics can avoid solar tracking  altogether or limit it to a few positions a year. The main disadvantage of nonimaging optics when compared to parabolic reflectors or Fresnel lenses is that, for high concentrations, they typically have one more optical surface, slightly decreasing efficiency. That, however, is only noticeable when the optics are aiming perfectly towards the sun, which is typically not the case because of imperfections in practical systems. Illumination optics. Examples of nonimaging optical devices include optical light guides, nonimaging reflectors, nonimaging lenses or a combination of these devices. Common applications of nonimaging optics include many areas of illumination engineering (lighting). Examples of  modern implementations of nonimaging optical designs include automotive headlamps, LCD backlights, illuminated instrument panel displays, fiber optic illumination devices, LED lights, projection display systems and luminaires. When compared to "traditional" design techniques, nonimaging optics has the following advantages for illumination: Examples of nonimaging illumination optics using solar energy are anidolic lighting or solar pipes. Other applications. Modern portable and wearable optical devices, and systems of small sizes and low weights may require nanotechnology. This issue may be addressed by nonimaging metaoptics, which uses metalenses and metamirrors to deal with the optimal transfer of light energy. Collecting radiation emitted by high -energy particle collisions using the fewest photomultiplier tubes. Collecting luminescent radiation in photon upconversion devices with the compound parabolic concentrator being to-date the most promising geometrical optics collector. Some of the design methods for nonimaging optics are also finding application in imaging devices, for example some with ultra-high numerical aperture. Theory. Early academic research in nonimaging optical mathematics seeking closed form solutions was first published in textbook form in a 1978 book. A modern textbook illustrating the depth and breadth of research and engineering in this area was published in 2004. A thorough introduction to this field was  published in 2008. Special applications of nonimaging optics such as Fresnel lenses for solar concentration or solar concentration in general have also been published, although this last reference by O'Gallagher describes mostly the work developed some decades ago. Other publications include book chapters. Imaging optics can concentrate sunlight to, at most, the same flux found at the surface of the sun. Nonimaging optics have been demonstrated to concentrate sunlight to 84,000 times the ambient intensity of sunlight, exceeding the flux found at the surface of the sun, and approaching the theoretical (2nd law of thermodynamics) limit of heating objects to  the temperature of the sun's surface. The simplest way to design nonimaging optics is called "the method of strings", based on the edge ray principle. Other more advanced methods were developed starting in the early 1990s that can better handle extended light sources than the edge-ray method. These were developed primarily to solve the design problems related to solid state automobile headlamps and complex illumination systems. One of these advanced design methods is the "simultaneous multiple surface" design method (SMS). The 2D SMS design method () is described in detail in the aforementioned textbooks. The 3D SMS design method  () was developed in 2003 by a team of optical scientists at Light Prescriptions Innovators. Edge ray principle. In simple terms, the edge ray principle states that if the light rays coming from the edges of the source are redirected towards the edges of the receiver, this will ensure that all light rays coming from the inner points in the source will end up on the receiver. There is no condition on image formation, the only goal is to transfer the light from the source to the target. Figure "Edge ray principle" on the right illustrates this principle. A lens collects  light from a source SS and redirects it towards a receiver RR. The lens has two optical surfaces and, therefore, it is possible to design it (using the SMS design method) so that the light rays coming from the edge S of the source are redirected towards edge R of the receiver, as indicated by the blue rays. By symmetry, the rays coming from edge S of the source are redirected towards edge R of the receiver, as indicated by the red rays. The rays coming from an inner point S in the source are redirected towards the target, but  they are not concentrated onto a point and, therefore, no image is formed. Actually, if we consider a point P on the top surface of the lens, a ray coming from S through P will be redirected towards R. Also a ray coming from S through P will be redirected towards R. A ray coming through P from an inner point S in the source will be redirected towards an inner point of the receiver. This lens then guarantees that all light from the source crossing it will be redirected towards the receiver. However, no image of the source is  formed on the target. Imposing the condition of image formation on the receiver would imply using more optical surfaces, making the optic more complicated, but would not improve light transfer between source and target (since all light is already transferred). For that reason nonimaging optics are simpler and more efficient than imaging optics in transferring radiation from a source to a target. Design methods. Nonimaging optics devices are obtained using different methods. The most important are: the flow-line or Winston-Welford design method, the SMS or Miñano-Benitez design method and the Miñano design method using Poisson brackets. The  first (flow-line) is probably the most used, although the second (SMS) has proven very versatile, resulting in a wide variety of optics. The third has remained in the realm of theoretical optics and has not found real world application to date. Often optimization is also used. Typically optics have refractive and reflective surfaces and light travels through media of different refractive indices as it crosses the optic. In those cases a quantity called optical path length (OPL) may be defined as formula_1 where index "i" indicates different ray sections between successive deflections (refractions or reflections), "n" is the refractive  index and "d" the distance in each section "i" of the ray path. The OPL is constant between wavefronts. This can be seen for refraction in the figure "constant OPL" to the right. It shows a separation "c"("τ") between two media of refractive indices "n" and "n", where "c"("τ") is described by a parametric equation with parameter "τ". Also shown are a set of rays perpendicular to wavefront "w" and traveling in the medium of refractive index "n". These rays refract at "c"("τ") into the medium of refractive index "n" in directions perpendicular to wavefront "w". Ray "r" crosses "c  at point c"("τ") and, therefore, ray "r" is identified by parameter "τ" on "c". Likewise, ray "r" is identified by parameter "τ" on "c". Ray "r" has optical path length . Also, ray "r" has optical path length . The difference in optical path length for rays "r" and "r" is given by: In order to calculate the value of this integral, we evaluate , again with the help of the same figure. We have and . These expressions can be rewritten as and . From the law of refraction and therefore , leading to . Since these may be arbitrary rays crossing "c", it may be  concluded that the optical path length between "w" and "w" is the same for all rays perpendicular to incoming wavefront "w" and outgoing wavefront "w". Similar conclusions may be drawn for the case of reflection, only in this case . This relationship between rays and wavefronts is valid in general. Flow-line design method. The flow-line (or Winston-Welford) design method typically leads to optics which guide the light confining it between two reflective surfaces. The best known of these devices is the CPC (Compound Parabolic Concentrator). These types of optics may be obtained, for example, by applying the edge  ray of nonimaging optics to the design of mirrored optics, as shown in figure "CEC" on the right. It is composed of two elliptical mirrors "e" with foci S and R and its symmetrical "e" with foci S and R. Mirror "e" redirects the rays coming from the edge S of the source towards the edge R of the receiver and, by symmetry, mirror "e" redirects the rays coming from the edge S of the source towards the edge R of the receiver. This device does not form an image of the source SS on the receiver RR as indicated  by the green rays coming from a point S in the source that end up on the receiver but are not focused onto an image point. Mirror "e" starts at the edge R of the receiver since leaving a gap between mirror and receiver would allow light to escape between the two. Also, mirror "e" ends at ray r connecting S and R since cutting it short would prevent it from capturing as much light as possible, but extending it above r would shade light coming from S and its neighboring points of the source. The resulting device is called  a CEC (Compound Elliptical Concentrator). A particular case of this design happens when the source SS becomes infinitely large and moves to an infinite distance. Then the rays coming from S become parallel rays and the same for those coming from S and the elliptical mirrors "e" and "e" converge to parabolic mirrors "p" and "p". The resulting device is called a CPC (Compound Parabolic Concentrator), and shown in the "CPC" figure on the left. CPCs are the most common seen nonimaging optics. They are often used to demonstrate the difference between Imaging optics and nonimaging optics. When seen from  the CPC, the incoming radiation (emitted from the infinite source at an infinite distance) subtends an angle ±"θ" (total angle 2"θ"). This is called the acceptance angle of the CPC. The reason for this name can be appreciated in the figure "rays showing the acceptance angle" on the right. An incoming ray "r" at an angle "θ" to the vertical (coming from the edge of the infinite source) is redirected by the CPC towards the edge R of the receiver. Another ray "r" at an angle "α"<"θ" to the vertical (coming from an inner point of the infinite source) is  redirected towards an inner point of the receiver. However, a ray "r" at an angle "β">"θ" to the vertical (coming from a point outside the infinite source) bounces around inside the CPC until it is rejected by it. Therefore, only the light inside the acceptance angle ±"θ" is captured by the optic; light outside it is rejected. The ellipses of a CEC can be obtained by the (pins and) string method, as shown in the figure "string method" on the left. A string of constant length is attached to edge point S of the source and edge point R of  the receiver. The string is kept stretched while moving a pencil up and down, drawing the elliptical mirror "e". We can now consider a wavefront "w" as a circle centered at S. This wavefront is perpendicular to all rays coming out of S and the distance from S to "w" is constant for all its points. The same is valid for wavefront "w" centered at R. The distance from "w" to "w" is then constant for all light rays reflected at "e" and these light rays are perpendicular to both, incoming wavefront "w" and outgoing wavefront "w". Optical path length  (OPL) is constant between wavefronts. When applied to nonimaging optics, this result extends the string method to optics with both refractive and reflective surfaces. Figure "DTIRC" (Dielectric Total Internal Reflection Concentrator) on the left shows one such example. The shape of the top surface "s" is prescribed, for example, as a circle. Then the lateral wall "m" is calculated by the condition of constant optical path length "S"="d"+"n d"+"n d" where "d" is the distance between incoming wavefront "w" and point P on the top surface "s", "d" is the distance between P and Q and "d" the distance between  Q and outgoing wavefront "w", which is circular and centered at R. Lateral wall "m" is symmetrical to "m". The acceptance angle of the device is 2"θ". These optics are called flow-line optics and the reason for that is illustrated in figure "CPC flow-lines" on the right. It shows a CPC with an acceptance angle 2"θ", highlighting one of its inner points P. The light crossing this point is confined to a cone of angular aperture 2"α". A line "f" is also shown whose tangent at point P bisects this cone of light and, therefore, points in the  direction of the "light flow" at P. Several other such lines are also shown in the figure. They all bisect the edge rays at each point inside the CPC and, for that reason, their tangent at each point points in the direction of the flow of light. These are called flow-lines and the CPC itself is just a combination of flow line "p" starting at R and "p" starting at R. Variations to the flow-line design method. There are some variations to the flow-line design method. A variation are the multichannel or stepped flow-line optics in  which light is split into several "channels" and then recombined again into a single output. Aplanatic (a particular case of SMS) versions of these designs have also been developed. The main application of this method is in the design of ultra-compact optics. Another variation is the confinement of light by caustics. Instead of light being confined by two reflective surfaces, it is confined by a reflective surface and a caustic of the edge rays. This provides the possibility to add lossless non-optical surfaces to the optics. Simultaneous multiple surface (SMS) design method. This section describes The design procedure  The SMS (or Miñano-Benitez) design method is very versatile and many different types of optics have been designed using it. The 2D version allows the design of two (although more are also possible) aspheric surfaces simultaneously. The 3D version allows the design of optics with freeform surfaces (also called anamorphic) surfaces which may not have any kind of symmetry. SMS optics are also calculated by applying a constant optical path length between wavefronts. Figure "SMS chain" on the right illustrates how these optics are calculated. In general, the rays perpendicular to incoming wavefront "w" will be coupled to outgoing  wavefront "w" and the rays perpendicular to incoming wavefront "w" will be coupled to outgoing wavefront "w" and these wavefronts may be any shape. However, for the sake of simplicity, this figure shows a particular case or circular wavefronts. This example shows a lens of a given refractive index "n" designed for a source SS and a receiver RR. The rays emitted from edge S of the source are focused onto edge R of the receiver and those emitted from edge S of the source are focused onto edge R of the receiver. We first choose a point T and  its normal on the top surface of the lens. We can now take a ray "r" coming from S and refract it at T. Choosing now the optical path length "S" between S and R we have one condition that allows us to calculate point B on the bottom surface of the lens. The normal at B can also be calculated from the directions of the incoming and outgoing rays at this point and the refractive index of the lens. Now we can repeat the process taking a ray "r" coming from R and refracting it at B. Choosing now  the optical path length "S" between R and S we have one condition that allows us to calculate point T on the top surface of the lens. The normal at T can also be calculated from the directions of the incoming and outgoing rays at this point and the refractive index of the lens. Now, refracting at T a ray "r" coming from S we can calculate a new point B and corresponding normal on the bottom surface using the same optical path length "S" between S and R. Refracting at B a ray "r" coming from R we can  calculate a new point T and corresponding normal on the top surface using the same optical path length "S" between R and S. The process continues by calculating another point B on the bottom surface using another edge ray "r", and so on. The sequence of points T B T B T B is called an SMS chain. Another SMS chain can be constructed towards the right starting at point T. A ray from S refracted at T defines a point and normal B on the bottom surface, by using constant optical path length "S" between S and R. Now  a ray from R refracted at B defines a new point and normal T on the top surface, by using constant optical path length "S" between S and R. The process continues as more points are added to the SMS chain. In this example shown in the figure, the optic has a left-right symmetry and, therefore, points B T B T B can also be obtained by symmetry about the vertical axis of the lens. Now we have a sequence of spaced points on the plane. Figure "SMS skinning" on the left illustrates the process used to fill the  gaps between points, completely defining both optical surfaces. We pick two points, say B and B, with their corresponding normals and interpolate a curve "c" between them. Now we pick a point B and its normal on "c". A ray "r" coming from R and refracted at B defines a new point T and its normal between T and T on the top surface, by applying the same constant optical path length "S" between S and R. Now a ray "r" coming from S and refracted at T defines a new point and normal on the bottom surface, by applying  the same constant optical path length "S" between S and R. The process continues with rays "r" and "r" building a new SMS chain filling the gaps between points. Picking other points and corresponding normals on curve "c" gives us more points in between the other SMS points calculated originally. In general, the two SMS optical surfaces do not need to be refractive. Refractive surfaces are noted R (from Refraction) while reflective surfaces are noted X (from the Spanish word refleXión). Total Internal Reflection (TIR) is noted I. Therefore, a lens with two refractive surfaces is an RR optic, while  another configuration with a reflective and a refractive surface is an XR optic. Configurations with more optical surfaces are also possible and, for example, if light is first refracted (R), then reflected (X) then reflected again by TIR (I), the optic is called an RXI. The SMS 3D is similar to the SMS 2D, only now all calculations are done in 3D space. Figure "SMS 3D chain" on the right illustrates the algorithm of an SMS 3D calculation. The first step is to choose the incoming wavefronts "w" and "w" and outgoing wavefronts "w" and "w" and the optical path  length "S" between "w" and "w" and the optical path length "S" between "w" and "w". In this example the optic is a lens (an RR optic) with two refractive surfaces, so its refractive index must also be specified. One difference between the SMS 2D and the SMS 3D is on how to choose initial point T, which is now on a chosen 3D curve "a". The normal chosen for point T must be perpendicular to curve "a". The process now evolves similarly to the SMS 2D. A ray "r" coming from "w" is refracted at T and, with the  optical path length "S", a new point B and its normal is obtained on the bottom surface. Now ray "r" coming from "w" is refracted at B and, with the optical path length "S ", a new point T and its normal is obtained on the top surface. With ray "r" a new point B and its normal are obtained, with ray "r" a new point T and its normal are obtained, and so on. This process is performed in 3D space and the result is a 3D SMS chain. As with the SMS 2D, a set of points and normals  to the left of T can also be obtained using the same method. Now, choosing another point T on curve "a" the process can be repeated and more points obtained on the top and bottom surfaces of the lens. The power of the SMS method lies in the fact that the incoming and outgoing wavefronts can themselves be free-form, giving the method great flexibility. Also, by designing optics with reflective surfaces or combinations of reflective and refractive surfaces, different configurations are possible. Miñano design method using Poisson brackets. This design method was developed by Miñano and is based on  Hamiltonian optics, the Hamiltonian formulation of geometrical optics which shares much of the mathematical formulation with Hamiltonian mechanics. It allows the design of optics with variable refractive index, and therefore solves some nonimaging problems that are not solvable using other methods. However, manufacturing of variable refractive index optics is still not possible and this method, although potentially powerful, did not yet find a practical application. Conservation of etendue. Conservation of etendue is a central concept in nonimaging optics. In concentration optics, it relates the acceptance angle with the maximum concentration possible. Conservation of etendue may be seen as constant a  volume moving in phase space. Köhler integration. In some applications it is important to achieve a given irradiance (or illuminance) pattern on a target, while allowing for movements or inhomogeneities of the source. Figure "Köhler integrator" on the right illustrates this for the particular case of solar concentration. Here the light source is the sun moving in the sky. On the left this figure shows a lens L L capturing sunlight incident at an angle "α" to the optical axis and concentrating it onto a receiver L L. As seen, this light is concentrated onto a hotspot on the receiver . This may be a problem in some applications. One way around this is to add a new lens extending from L to L that captures the light from L L and redirects it onto a receiver R R, as shown in the middle of the figure. The situation in the middle of the figure shows a nonimaging lens L L is designed in such a way that sunlight (here considered as a set of parallel rays) incident at an angle "θ" to the optical axis will be concentrated to point L. On the other hand, nonimaging lens L L is  designed in such a way that light rays coming from L are focused on R and light rays coming from L are focused on R. Therefore, ray "r" incident on the first lens at an angle "θ" will be redirected towards L. When it hits the second lens, it is coming from point L and it is redirected by the second lens to R. On the other hand, ray "r" also incident on the first lens at an angle "θ" will also be redirected towards L. However, when it hits the second lens, it is coming from point L and  it is redirected by the second lens to R. Intermediate rays incident on the first lens at an angle "θ" will be redirected to points between R and R, fully illuminating the receiver. Something similar happens in the situation shown in the same figure, on the right. Ray "r" incident on the first lens at an angle "α"<"θ" will be redirected towards a point between L and L. When it hits the second lens, it is coming from point L and it is redirected by the second lens to R. Also, Ray "r" incident on the first lens at an  angle "α"<"θ" will be redirected towards a point between L and L. When it hits the second lens, it is coming from point L and it is redirected by the second lens to R. Intermediate rays incident on the first lens at an angle "α"<"θ" will be redirected to points between R and R, also fully illuminating the receiver. This combination of optical elements is called Köhler illumination. Although the example given here was for solar energy concentration, the same principles apply for illumination in general. In practice, Köhler optics are typically not designed as a combination of nonimaging optics , but they are simplified versions with a lower number of active optical surfaces. This decreases the effectiveness of the method, but allows for simpler optics. Also, Köhler optics are often divided into several sectors, each one of them channeling light separately and then combining all the light on the target. An example of one of these optics used for solar concentration is the Fresnel-R Köhler. Compound parabolic concentrator. In the drawing opposite there are two parabolic mirrors CC' (red) and DD' (blue). Both parabolas are cut at B and A respectively. A is the focal point of parabola CC ' and B is the focal point of the parabola DD' The area DC is the entrance aperture and the flat absorber is AB. This CPC has an acceptance angle of "θ". The parabolic concentrator has an entrance aperture of DC and a focal point F. The parabolic concentrator only accepts rays of light that are perpendicular to the entrance aperture DC. The tracking of this type of concentrator must be more exact and requires expensive equipment. The compound parabolic concentrator accepts a greater amount of light and needs less accurate tracking. For a 3-dimensional "nonimaging compound parabolic concentrator", the  maximum concentration possible in air or in vacuum (equal to the ratio of input and output aperture areas), is: formula_3 where formula_4 is the half-angle of the acceptance angle (of the larger aperture). History. The development started in the mid-1960s at three different locations by V. K. Baranov (USSR) with the study of the focons (focusing cones) Martin Ploke (Germany), and Roland Winston (United States), and led to the independent origin of the first nonimaging concentrators, later applied to solar energy concentration. Among these three earliest works, the one most developed was the American one, resulting in what nonimaging  optics is today. A good introduction was published by - Winston, Roland. “Nonimaging Optics.” Scientific American, vol. 264, no. 3, 1991, pp. 76–81. JSTOR, There are different commercial companies and universities working on nonimaging optics. Currently the largest research group in this subject is the Advanced Optics group at the CeDInt, part of the Technical University of Madrid (UPM).