Nonimaging optics

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:[5]

• wider acceptance angles resulting in higher tolerances (and therefore higher efficiencies) for:
  • less precise tracking
  • imperfectly manufactured optics
  • imperfectly assembled components
  • movements of the system due to wind
  • finite stiffness of the supporting structure
  • deformation due to aging
  • capture of circumsolar radiation
  • other imperfections in the system
• higher solar concentrations
  • smaller solar cells (in concentrated photovoltaics)
  • higher temperatures (in concentrated solar thermal)
  • lower thermal losses (in concentrated solar thermal)
  • widen the applications of concentrated solar power, for example to solar lasers
• possibility of a uniform illumination of the receiver
  • improve reliability and efficiency of the solar cells (in concentrated photovoltaics)
  • improve heat transfer (in concentrated solar thermal)
• design flexibility: different kinds of optics with different geometries can be tailored for different applications

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.

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:

• better handling of extended sources
• more compact optics
• color mixing capabilities
• combination of light sources and light distribution to different places
• well suited to be used with increasingly popular LED light sources
• tolerance to variations in the relative position of light source and optic

Examples of nonimaging illumination optics using solar energy are anidolic lighting or solar pipes.

Collecting radiation emitted by high-energy particle collisions using the fewest photomultiplier tubes.[6]

Some of the design methods for nonimaging optics are also finding application in imaging devices, for example some with ultra-high numerical aperture.[7]

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 S₁S₂ and redirects it towards a receiver R₁R₂.

Edge ray principle

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 S₁ of the source are redirected towards edge R₁ of the receiver, as indicated by the blue rays. By symmetry, the rays coming from edge S₂ of the source are redirected towards edge R₂ 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 S₁ through P will be redirected towards R₁. Also a ray coming from S₂ through P will be redirected towards R₂. 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.

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 ${\displaystyle S=\textstyle \sum _{i}n_{i}d_{i}}$ where index i indicates different ray sections between successive deflections (refractions or reflections), n_i is the refractive index and d_i the distance in each section i of the ray path.

Constant OPL

The OPL is constant between wavefronts.[1] 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 n₁ and n₂, where c(τ) is described by a parametric equation with parameter τ. Also shown are a set of rays perpendicular to wavefront w₁ and traveling in the medium of refractive index n₁. These rays refract at c(τ) into the medium of refractive index n₂ in directions perpendicular to wavefront w₂. Ray r_A crosses c at point c(τ_A) and, therefore, ray r_A is identified by parameter τ_A on c. Likewise, ray r_B is identified by parameter τ_B on c. Ray r_A has optical path length S(τ_A) = n₁d₅ + n₂d₆. Also, ray r_B has optical path length S(τ_B) =n₁d₇ + n₂d₈. The difference in optical path length for rays r_A and r_B is given by:

${\displaystyle S(\tau _{B})-S(\tau _{A})=\int _{A}^{B}dS=\int _{\tau _{A}}^{\tau _{B}}{\frac {dS}{d\tau }}d\tau =\int _{\tau _{A}}^{\tau _{B}}{\frac {S(\tau +d\tau )-S(\tau )}{(\tau +d\tau )-\tau }}d\tau }$

In order to calculate the value of this integral, we evaluate S(τ+dτ)-S(τ), again with the help of the same figure. We have S(τ) = n₁d₁+n₂(d₃+d₄) and S(τ+dτ) = n₁(d₁+d₂)+n₂d₄. These expressions can be rewritten as S(τ) = n₁d₁+n₂dc sinθ₂+n₂d₄ and S(τ+dτ) = n₁d₁+n₁dc sinθ₁+n₂d₄. From the law of refraction n₁sinθ₁=n₂sinθ₂ and therefore S(τ+dτ) = S(τ), leading to S(τ_A)=S(τ_B). Since these may be arbitrary rays crossing c, it may be concluded that the optical path length between w₁ and w₂ is the same for all rays perpendicular to incoming wavefront w₁ and outgoing wavefront w₂.

Similar conclusions may be drawn for the case of reflection, only in this case n₁=n₂. This relationship between rays and wavefronts is valid in general.

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 e₁ with foci S₁ and R₁ and its symmetrical e₂ with foci S₂ and R₂.

CEC

Mirror e₁ redirects the rays coming from the edge S₁ of the source towards the edge R₁ of the receiver and, by symmetry, mirror e₂ redirects the rays coming from the edge S₂ of the source towards the edge R₂ of the receiver. This device does not form an image of the source S₁S₂ on the receiver R₁R₂ 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 e₂ starts at the edge R₁ of the receiver since leaving a gap between mirror and receiver would allow light to escape between the two. Also, mirror e₂ ends at ray r connecting S₁ and R₂ since cutting it short would prevent it from capturing as much light as possible, but extending it above r would shade light coming from S₁ and its neighboring points of the source. The resulting device is called a CEC (Compound Elliptical Concentrator).

CPC

A particular case of this design happens when the source S₁S₂ becomes infinitely large and moves to an infinite distance. Then the rays coming from S₁ become parallel rays and the same for those coming from S₂ and the elliptical mirrors e₁ and e₂ converge to parabolic mirrors p₁ and p₂. 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 2θ). 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 r₁ at an angle θ to the vertical (coming from the edge of the infinite source) is redirected by the CPC towards the edge R₁ of the receiver.

Rays showing the acceptance angle

Another ray r₂ 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 r₃ 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 S₁ of the source and edge point R₁ of the receiver.

String method

The string is kept stretched while moving a pencil up and down, drawing the elliptical mirror e₁. We can now consider a wavefront w₁ as a circle centered at S₁. This wavefront is perpendicular to all rays coming out of S₁ and the distance from S₁ to w₁ is constant for all its points. The same is valid for wavefront w₂ centered at R₁. The distance from w₁ to w₂ is then constant for all light rays reflected at e₁ and these light rays are perpendicular to both, incoming wavefront w₁ and outgoing wavefront w₂.

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.

DTIRC

The shape of the top surface s is prescribed, for example, as a circle. Then the lateral wall m₁ is calculated by the condition of constant optical path length S=d₁+n d₂+n d₃ where d₁ is the distance between incoming wavefront w₁ and point P on the top surface s, d₂ is the distance between P and Q and d₃ the distance between Q and outgoing wavefront w₂, which is circular and centered at R₁. Lateral wall m₂ is symmetrical to m₁. The acceptance angle of the device is 2θ.

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 2θ, highlighting one of its inner points P.

CPC flow-lines

The light crossing this point is confined to a cone of angular aperture 2α. 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 p₁ starting at R₂ and p₂ starting at R₁.

There are some variations to the flow-line design method.[1]

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.[15] 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.

This section describes

a nonimaging optics design method known in the field as the simultaneous multiple surface (SMS) or the Miñano-Benitez design method. The abbreviation SMS comes from the fact that it enables the simultaneous design of multiple optical surfaces. The original idea came from Miñano. The design method itself was initially developed in 2-D by Miñano and later also by Benítez. The first generalization to 3-D geometry came from Benítez. It was then much further developed by contributions of Miñano and Benítez. Other people have worked initially with Miñano and later with Miñano and Benítez on programming the method.[1]

The design procedure

is related to the algorithm used by Schulz[16][17] in the design of aspheric imaging lenses.[1]

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 w₁ will be coupled to outgoing wavefront w₄ and the rays perpendicular to incoming wavefront w₂ will be coupled to outgoing wavefront w₃ 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 S₁S₂ and a receiver R₁R₂.

SMS chain

The rays emitted from edge S₁ of the source are focused onto edge R₁ of the receiver and those emitted from edge S₂ of the source are focused onto edge R₂ of the receiver. We first choose a point T₀ and its normal on the top surface of the lens. We can now take a ray r₁ coming from S₂ and refract it at T₀. Choosing now the optical path length S₂₂ between S₂ and R₂ we have one condition that allows us to calculate point B₁ on the bottom surface of the lens. The normal at B₁ 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 r₂ coming from R₁ and refracting it at B₁. Choosing now the optical path length S₁₁ between R₁ and S₁ we have one condition that allows us to calculate point T₁ on the top surface of the lens. The normal at T₁ 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 T₁ a ray r₃ coming from S₂ we can calculate a new point B₃ and corresponding normal on the bottom surface using the same optical path length S₂₂ between S₂ and R₂. Refracting at B₃ a ray r₄ coming from R₁ we can calculate a new point T₃ and corresponding normal on the top surface using the same optical path length S₁₁ between R₁ and S₁. The process continues by calculating another point B₅ on the bottom surface using another edge ray r₅, and so on. The sequence of points T₀ B₁ T₁ B₃ T₃ B₅ is called an SMS chain.

Another SMS chain can be constructed towards the right starting at point T₀. A ray from S₁ refracted at T₀ defines a point and normal B₂ on the bottom surface, by using constant optical path length S₁₁ between S₁ and R₁. Now a ray from R₂ refracted at B₂ defines a new point and normal T₂ on the top surface, by using constant optical path length S₂₂ between S₂ and R₂. 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 B₂ T₂ B₄ T₄ B₆ 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.

SMS skinning

We pick two points, say B₁ and B₂, with their corresponding normals and interpolate a curve c between them. Now we pick a point B₁₂ and its normal on c. A ray r₁ coming from R₁ and refracted at B₁₂ defines a new point T₀₁ and its normal between T₀ and T₁ on the top surface, by applying the same constant optical path length S₁₁ between S₁ and R₁. Now a ray r₂ coming from S₂ and refracted at T₀₁ defines a new point and normal on the bottom surface, by applying the same constant optical path length S₂₂ between S₂ and R₂. The process continues with rays r₃ and r₄ 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.

SMS 3D chain

The first step is to choose the incoming wavefronts w₁ and w₂ and outgoing wavefronts w₃ and w₄ and the optical path length S₁₄ between w₁ and w₄ and the optical path length S₂₃ between w₂ and w₃. 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 T₀, which is now on a chosen 3D curve a. The normal chosen for point T₀ must be perpendicular to curve a. The process now evolves similarly to the SMS 2D. A ray r₁ coming from w₁ is refracted at T₀ and, with the optical path length S₁₄, a new point B₂ and its normal is obtained on the bottom surface. Now ray r₂ coming from w₃ is refracted at B₂ and, with the optical path length S ₂₃, a new point T₂ and its normal is obtained on the top surface. With ray r₃ a new point B₂ and its normal are obtained, with ray r₄ a new point T₄ 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 T₀ can also be obtained using the same method. Now, choosing another point T₀ 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.

This design method was developed by Miñano and is based on Hamiltonian optics, the Hamiltonian formulation of geometrical optics[1][2] 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 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.