1. Introduction

The difference between imaging optics and nonimaging optics depends on whether ray sources are treated as points or extended objects. The latter have temperature and entropy and therefore we can apply thermodynamics, while point sources do not because they do not possess any internal degrees of freedom.

Conventional imaging optics is most concerned with the idea of point to point mapping of an optical system. A point from the object space is ideally mapped via an optical device to a corresponding point in the image space. For example, a Gaussian optical system starts with mapping the center point of the object to its image on the optical axis, and then the aberrations of the off-axis points are minimized by optimization. Figure 1 shows a comparison between imaging (left) and nonimaging (right) optics. A “perfect” concentrating imaging system free from aberration cannot be achieved without infinite degrees of design freedom ([1], Appendix B).

 

Figure 1 Comparison between imaging (left) and nonimaging (right) optics.

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Nonimaging optics, however, starts with the idea of energy transfer. For this purpose, only the boundary of the object must be considered for the designing process. The mapping of points inside such a boundary is not required. In a sense, nonimaging optics design only guarantees that the edge rays, or the boundary consisting of extreme positions or directions, is carried over. The contents within the edge rays, or the rays of the positions and directions in between, can be scrambled.

1.1. Sphere Ellipse Paradox

Because imaging optics treats the source and its destination (sink) of light rays as points, it processes the sources and sinks by simplifying them. In other words, the analysis of the optical system is based on point to point mapping through rays. Although such an underlying assumption works very well, it fails to address the thermodynamic rules that the optical sources or sinks with extended surfaces are required to observe. This can lead to paradoxes.

One classical example [2] is the paradox of an elliptical spherical chamber, as shown in Fig. 2, where the assumption that radiation source and sink can be simple points will create a dilemma that violates the second law of thermodynamics. Consider a spherical-ellipsoid chamber with perfect reflectivity for its inner walls. The point object $A$ is positioned at the center of a spherical reflecting cavity, and it is also at the focus of an elliptical reflecting cavity; the point object $B$ is at the other focus. If we start $A$ and $B$ at the same temperature, the probability of radiation from $B$ reaching $A$ is clearly higher than $A$ reaching $B$, as shown by the arrows. One can easily validate such a result with ray tracing. So we conclude that $A$ warms up while $B$ cools off, despite starting at the same temperature, resulting in a violation of the second law of thermodynamics (heat only flows from higher temperature to lower temperature). The paradox is resolved by making $A$ and $B$ extended objects, no matter how small. In fact, a physical object with temperature has many internal degrees of freedom and cannot be point-like.

 

Figure 2 Elliptical paradox where the imaging optics fails.

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Obviously in such an example the geometrical principle of rays being reflected from one focus to the other focus for ellipses, and from the central point of a spherical mirror and back for spheres, is not at fault. Rather, it is the assumption that we can treat optical objects as simple points, producing a result that does not exist in reality. Simple points can neither produce rays nor receive rays in radiation heat transfer. Their usage can help to conceptualize and analyze an optical system. However, in many cases, the over-simplification of point-like sources and sinks for radiation heat transfer produces self-conflicting results, such as those of the sphere-ellipse paradox. The solution is not to fully give up on ray tracing. Instead, in addition to using rays and points to help guide the optical designing process, we must also consider thermodynamics as one of the fundamental principles that cannot be violated through such a designing process. Nonimaging optics at its core addresses such a challenge.

2. Thermodynamic Origin of Nonimaging Optics

To connect thermodynamics with real-world applications, we demonstrate the underlying thought process of nonimaging optics with a realistic problem: concentrator designs. In order to simplify the model, we assume that the designing process is pure geometric. In other words, effects such as wavelength shifts and polarization, which will have impacts on the thermodynamics of the system, are not our concern. For the readers who are interested in a more general analysis for effects such as luminescent concentrators, please refer to [3]. Compared to imaging optics, a nonimaging concentrator is concerned with transferring energy from the source to the absorber (sink) within the theoretical limit of thermodynamics. Instead of designing an optical system based on fine tuning and engineering light rays, we first need to look at the maximum concentration ratio allowed by the fundamental laws of physics. To do so, we must enter the regime of geometrical optics and probability.

The general setup of a concentrating system requires three components, as shown in Fig. 3: the source of radiation, the aperture of the concentrator, and the absorber (sink) of the radiation. We define the geometric concentration ratio as $C=\frac{\text{Area of the aperture}}{\text{Area of the absorber}}$.

 

Figure 3 Typical setup of a concentrator.

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To explore the thermodynamic limitations of a concentrator design using geometrical optics, we assume that the source and absorber are perfect Lambertian surfaces (blackbody).

The probability of radiation leaving one surface and arriving at another surface is defined as

Starting from this simple concept, we will describe nonimaging optics with two thermodynamic arguments.

2.1. Maximize Radiation Flux at the Absorber

Now we shall discuss the first thermodynamic argument, which can be used independent of the second argument. The probability defined in Eq. (1) can be readily utilized within the context of thermodynamics to provide the maximum concentration ratio of an optical device. For a configuration described in Fig. 3, according to the Stefan–Boltzmann law, the radiation from a blackbody source to a blackbody absorber is

Equation (3) represents a fundamental concept: given two blackbodies at the same temperature, the radiative power from one to the other is equivalent.

From here, we derive the radiative heat flux at the surface of the absorber:

Because $P_{31}\le 1$, $q_{13}$ reaches a maximum radiative flux equal to that of the source, which is consistent with the second law of thermodynamics:

2.2. Maximize the Optical Efficiency

Now we can discuss the second thermodynamic argument. Ideally, a concentrator design has a geometrical optical efficiency equal to one. In other words, for a configuration shown in Fig. 3 the energy from the radiation source passing through the aperture will all arrive at the absorber (for nonideal system, refer to [4])

Equation (6) provides the limit of concentration given the additional requirement that the optical efficiency of the device is also ideal. Such a requirement is not always enforced in the real-world application; theoretically, however, providing a device with the ideal design is at least an option to be considered.

2.3. Maximum Concentration Ratio

Using the principle of reciprocity once more, we have

Combining Eqs. (3), (6), and (7), we conclude that

Equation (8) demonstrates the design principle of an ideal nonimaging concentrator. Impose the following requirements: (1) maximize the concentrated flux at the absorber, and (2) maximize the optical efficiency of the concentrator. Then the concentration ratio of such a device is $1/P_{21}$. Equation (8) also shows that such a design is achieved when $P_{13}=1$, according to Eq. (5). The thermodynamic approach of maximizing the absorber temperature such that it is equivalent to the radiation source, or, $P_{31}=1$, serves as the underlying principle for most nonimaging designs.

2.4. Nonimaging Concentrator for Infinitely Far Away Source

Hoyt Hottel, an MIT engineer working on the theory of furnaces [5,6], showed a convenient method for calculating radiation transfer between walls in a furnace using “strings.” In order to calculate the $P_{21}$ from Eq. (8), we use the Hottel’s strings on the radiation source 1 and aperture 2, as shown in Fig. 4:

 

Figure 4 Hottel’s strings can be used to solve for $P_{21}$.

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As the source $AB$ approaches infinity, $\mathrm{\Delta }\theta$ approaches 0 and $\overline{AC}=\overline{AE}$. If the setup is kept symmetric, i.e., $\overline{AD}=\overline{BC}$ and $\overline{AC}=\overline{BD}$, then

Equation (10) gives the maximum ratio of concentration within the framework of thermodynamics, for an infinitely far away radiation source in 2D. One can easily generalize this limit to 3D:

2.5. Discussion of Étendue within Thermodynamic Framework

In this section we explain the concept of étendue based on conventional radiative heat transfer. This might be particularly helpful for readers who are familiar with such a background. And it can provide the reader with a clearer view of the connections between heat transfer and nonimaging optics. In the context of radiative heat transfer, the geometric setup is described using view factors [5]. Similar to [7], a probability concept can be established to describe the geometric capability for a radiative heat transfer device, even though the source and absorber may not have a direct view of each other:

Here $d{A}_1,d{A}_2$ are two infinitely small areas on the surfaces of two objects undergoing radiative heat transfer (Fig. 5). Variables ${\beta }_1$ and ${\beta }_2$ are the angles formed between the direction connecting them and the norms of the specific surface, and $r$ is the distance between them. Because we are in a geometric optical setup, some of the rays from the source of radiation to the sink can be reflected or refracted. Probability $P$ instead of view factor $F$ is used to include these rays. In other words, compared to the definition of view factor $F$, which only describes geometric configurations without considering optical devices, probability $P$ is a more general description of radiative heat transfer which includes ray paths that connect the source and the absorber through optical devices such as mirrors and lenses.

 

Figure 5 Radiative heat transfer between areas 1 and 2.

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From Eq. (12), the étendue ([8, 9], Eq. (3.11)) is

Compare Eqs. (12) and (13), we conclude

This result connects the étendue with view factor analysis, which is an important concept in radiative heat transfer studies, and its tabulated analytical results are widely available in the literature [5]. Notice that for nonimaging optical system designed in 2D, $E_{12}=2A_1{P}_{12}$ due to the integral in Eqs. (12) and (13) being limited in 2D.

The second law of thermodynamics forbids a colder body to transfer heat to a hotter body. Therefore [also from Eq. (12)], this can be described from the étendue perspective as

Or, for any two surfaces we can conclude that

We can implement Eq. (16) with Eqs. (6) and (7) and conclude that the shorthand expression of the two important thermodynamic arguments for nonimaging optics is [10]

Equation (17) shows that the ideal nonimaging optical designs are geometrically étendue matching devices. If we treat étendue as the ensemble of all the rays connecting two objects (such as radiation source, aperture, or absorber), then we can observe that for the nonimaging geometric optical design that achieved maximum concentration ratio with theoretical 100% optical efficiency, Eq. (17) is required from a thermodynamic point of view.

3. Phase Space Representation of Nonimaging Optics

Now we will introduce the second perspective of designing and understanding nonimaging optics, i.e., interpreting nonimaging optics under the phase space framework.

To understand the phase space of geometric optics, we use the simple example shown in Fig. 6. A source at $z=0$ emits light rays which subtend a half angle of 30 deg. Here we define the optical axis as the $z$ axis. Figure 6(a) shows how we can discretize both the position $x$ and the angle of such rays [11]. We pick four equally spaced $x$ positions and use seven equally spaced angles to represent all the rays from the screen. In Fig. 6(b) we plot these rays as points according to their $x$ positions and their directions as they intercept a screen at various $z$ positions, which are represented as the vertical lines in Fig. 6(a). However, instead of plotting the ray’s direction as angles, we plot the direction according to the sine of the incident angle, or $p_x=n\sin (\text{angle})$. $p_x$ is also called optical momentum under Hamiltonian optics [12], which is equal to directional cosine when the refractive index is 1. Because such angles are between $+30$ and $-30\mathrm{deg}$, the representations of the rays are also between 0.5 and $-0.5$ on the $p_x$ axis. The screen starts at $z=0$, or when the screen is right against the source, the “$+$” markers occupy a rectangular shape. The four corners of this rectangle represent the four extreme rays $(-5,-\sin (30))$, $(-5,\sin (30))$, $(5,-\sin (30))$, and $(5,\sin (30))$. Next, we move the screen to position $z=5$ to intercept these rays, and we plot the rays again in phase space with the “o” markers. Now the extreme rays with positive directions drift toward the $+x$ direction, and the extreme rays with negative directions drift toward the $-x$ direction. The rays incident on the screen occupy the shape of a parallelogram. As the screen takes on further positions such as $z=6,7,8\dots 10$, we see that the parallelogram becomes more skewed. This can be seen more clearly as we increase the density of the rays (Fig. 7). In Fig. 7 it is shown that such parallelograms occupy the same area. Without giving further detailed proof ([5]), we state that without any loss or generation of rays, the phase space volume is always conserved.

 

Figure 6 Discrete representation of the phase space.

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Figure 7 Dense representation of the phase space.

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3.1. Definition of Étendue within the Phase Space Framework

Here we reintroduce the concept of étendue under the phase space framework. We observe in Fig. 7(b) that the area in the phase space occupied by the rays remains the same. This area corresponds to the value of étendue ([7], page 75), or the “spread” of the rays in both positions and directions. We can express this as

The conservation of étendue can be derived from Hamiltonian optics; other forms of proofs have also been offered ([8], Appendix A.2, A.3). In Appendix A we also offer a proof due to Fermi (Appendix A, or [13], page 34). For a system described in Fig. 7, it can be visualized as follows. First, we choose an arbitrary position on the $z$ axis. The rays of an optical system that intersect a screen at this position [Fig. 7(a)] can be represented, one to one, as points in a phase space with $(x,p_x)$ coordinates [Fig. 7(b)]. As we move the screen along the $z$ axis, the boundary of such a phase space representation of all the rays in the optical system will also shift its shape. However, the area, or étendue, within this boundary will remain constant. In an analogy, if we treat $z$, or the position of the screen as time, then as we move away from the screen (changing the time $z$), the étendue of the system is like the volume of a noncompressible fluid. As the fluid takes on different shapes, its volume will remain the same.

3.2. Maximum Concentration Ratio under Phase Space Representation

If we are given a concentrator design task in air ($n=1$), with a specified profile of aperture phase space as shown in Fig. 8(a). All the rays coming into the aperture ($\overline{-aa}$) subtending half acceptance angle (notice Fig. 6 simply consists of the reversed rays, with $\theta =30$, $a=5$), which is typical for a far-away radiation source, then we can draw its phase space representation in Fig. 8(b). To fit the phase space volume (area in 2D), or étendue, into the smallest absorber area, the following limitation exists: the phase space volume cannot extend above $L=1$ or below $L=-1$. This limitation exists because directional cosine exceeding 1 is nonphysical. Thus, we arrive at a rectangular phase space volume, as shown in Fig. 9(a).

 

Figure 8 (a) Configuration of an aperture of a concentrator. (b) The phase space representation of an aperture.

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Figure 9 (a) Phase space representation of the absorber achieving maximum concentration. (b) The phase space example that does not achieve maximum concentration.

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Consider the following proof. If the phase space volume of the absorber takes on a shape other than the rectangular area with vertical boundaries, as shown in Fig. 9(b), the absorber phase space area is the parallelogram outlined by the black dotted–dashed lines. Then we can always find a new boundary with vertical line $x=x_0$ (blue dashed line) that allows part of the phase space area outside the new boundary to be folded inside, resulting in a smaller absorber area, where the new physical size of the absorber starts with $x=x_0$. In other words, having a nonvertical boundary for phase space volume means that certain small areas on the absorber are under-utilized. Or, some of the angles for such small areas are not being used to receive incoming radiation. In such a situation, one ends up “wasting” the phase space volume that the absorber has for accepting incoming light.

A quick calculation shows that the size of the new absorber is $2a\sin (\theta )$; therefore, the ideal concentration ratio is $1/\sin (\theta )$, which agrees with Eq. (10):

3.3. Flux Density of Nonimaging Optics in Directional Cosine Space

It is easy to pictorially describe the phase space concept of a 2D system; however, this is not the case for a 3D system. In a 2D system we can use $x$ and directional $x(p_x)$ to uniquely identify a ray, as we have shown in Fig. 6. In order to describe the 3D rays in the phase space geometrically, we need four unique parameters instead of two: $x$, $y$, directional $x(p_x)$, and directional $y(p_y)$. This four-dimensional space is no longer readily intuitive for us but we can still observe its properties in its projections. When we project it into the $x$, $y$ space it takes on the unit of $\text{watts}/{\mathrm{m}}^2$, which is represented as the typical radiation intensity for a screen; but if we instead project into the directional cosine spaces $(p_x,p_y)$ it becomes watts/sr. Later on we will discuss how an ideal nonimaging system behaves under such a projection.

First, we can expand the same phase space representation of étendue into 3D configurations. The étendue is defined as

Here the phase space “volume” is calculated based on the four-dimensional coordinates $(x,y,p_x,p_y)$. For any point $(x,y)$, the set of rays coming from the source looks the same:

The full étendue is

When the aperture is illuminated with identical pencil rays, in other words, $p_x,p_y$ are independent according to $x$, $y$ [e.g., Fig. 10(a)],

 

Figure 10 (a) 3D configuration of light coming into an aperture A subtending angle $\theta$. (b) The $L/M$ space representation of the incoming rays.

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Here A is the area of the aperture. $L$ and $M$ are directional cosine $x$ and directional cosine $y$, respectively.

As we mentioned before, although it is hard to visualize such a four-dimensional volume, for certain configurations, such a calculation can be greatly simplified. For example, given an aperture receiving light from an infinitely far away source, here we can pick any point ($x$, $y$) on the receiver, and the set of rays going through a small area $dxdy$ will not be dependent on the ($x$, $y$) position [Fig. 10(a)]. We can visualize the 4D phase space of such an aperture by projecting it on the 2D $LM$ (or, $p_x{p}_y$) space. The phase space projection into $LM$ space in such an example will be of a circular area of radius $\sin (\theta )$, as shown in Fig. 10(b). The phase space volume is therefore $\pi A{\sin }^2(\theta )$.

We can also calculate the $\int \mathrm{d}L\mathrm{d}M$ using the unit sphere method [5]. It is crucially important to understand that the directional cosine space is different from the 2D phase space. Although they are both two-dimensional, one represents the directional $x$ and directional $y$ ($L$ and $M$, or $P_x{P}_y$), which is a projection of the full four-dimensional phase space. The other is the full representation of $x$ and $P_x$ under a 2D configuration.

4. Constructing a Nonimaging Optics System

We have provided two ways of arriving at the same limit of concentration. With the thermodynamic argument Eq. (10) and the phase space argument Eq. (19), we reached the same conclusion. Equation (10) provides the limit of concentration under simple but fundamental thermodynamic assumptions. With Eq. (19) we see how étendue conservation in phase space will provide clues for the designs for edge rays (a more robust argument can be found in [14]). Although such thermodynamic understanding of nonimaging optics does not offer design methods to directly generate a nonimaging concentrator, it does provide a theoretical limit and some intuitions for the designing process. It also serves as a guideline on how nonimaging optical systems can be designed, as shown below.

4.1. Compound Parabolic Concentrator

4.1a. Basic Nonimaging Optics Design

One of the first nonimaging optical designs to be invented was the compound parabolic concentrator (CPC), which we look at now. Here the source is between two wavefronts (Fig. 11), or two groups of “edge rays” that represent the two extreme angles that the source subtends. The most rudimentary way of designing a concentrator for such a setup is to use two straight line reflectors, as shown in Fig. 12. This, indeed, was the first effort employed that led to the design of the CPC. However, it will result in a poor concentration ratio. The evolution of such a design into the CPC has been well documented ([9], Fig. 2.8). Here instead, we use the insights provided by Eq. (8) to review the result of this designing process to give another perspective of how nonimaging optics is related to thermodynamics. We require that (1) $P_{31}=1$, or all rays from the absorber back trace to the angle between the two wavefronts, and (2) $P_{12}=P_{13}$, or all rays between the two extreme directions will arrive at the absorber. Then the straightforward attempt is to map wavefront 1 to absorber edge $B^{\prime }$ and wavefront 2 to absorber edge $B$. This can be shown as the following. (1) To understand $P_{31}$, we look at the extreme rays, namely, rays from points $B$ and $B^{\prime }$. To prove that for $P_{31}=1$ requires that all rays from $B$ and $B^{\prime }$ should be mapped into extreme ray directions $\theta$ and $-\theta$, we use this following proof of contradiction. If some of the rays from $B$ or $B^{\prime }$ are mapped into angles larger than $\theta$, say, at ${\theta }_0$, then we can find their neighboring point $B_0$, where the small ray cluster emitted by $B{B}_0$ at ${\theta }_0$ will be outside of the source 1. (2) Similarly, if some of the rays from $B$ or $B^{\prime }$ are mapped into angles smaller than $\theta$, then some rays coming into the aperture will not be mapped between $B$ and $B^{\prime }$. In such a case we would not be able to guarantee $P_{12}=P_{13}$. By limiting all the rays from points $B$ and $B^{\prime }$ to be mapped to extreme angles, we ensure both conditions that are implied by Eq. (8).

 

Figure 11 Setup of edge rays or wavefronts.

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Figure 12 Using simple straight line reflectors to form the concentrator.

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Mapping these two groups of parallel rays into two points will result in two parabolic curves (Fig. 13). We determine the aperture $A{A}^{\prime }$ when the tangent of parabola becomes vertical. Because under this condition the aperture of the concentrator will be at a maximum. This, however, is only one way of designing optics that satisfy both thermodynamic requirements of Eq. (7).

 

Figure 13 Mapping points to edge rays.

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4.2. Concentration Ratio of the CPC

Here we prove that the concentrator constructed this way does achieve the ideal concentration ratio.

Because of the geometric property of the parabola,

Therefore,

Utilizing the property of right triangle $A{A}^{\prime }C$,

4.3. Phase Space Analysis of the CPC

Phase space of a plane in the optical device can be plotted based on the position and direction of a light ray. Here we use the CPC design from the previous section as an example. First, we examine the phase space representation of the aperture plane $A^{\prime }A$. At point $A^{\prime }$, the two extreme rays both intersect with the aperture at the same $x$ coordinate; therefore, their representation on the phase space coordinate has the same value on the $x$ axis. In phase space, instead of plotting the ray’s direction as the incident angle, we plot the direction according to the sine of the incident angle, or $p_x=n\sin (\text{in}\text{cident angle})$, where $n$ is the index of refraction. We also call $p_x$ the optical momentum or, when the refractive index is 1, directional cosine. The incident angle of ray 1 is $\theta$, while as ray 2 is at $-\theta$. Therefore, they are plotted as points 1 and 2 in the phase space. The same can be done for rays 3 and 4, which are represented as points 3 and 4 in the phase space plot. All the other rays incident at point $A^{\prime }$ have the same $x$ coordinates, with different values for $p_x$ between $\sin (\theta )$ and $-\sin (\theta )$. As a result, they will form a line between points 1 and 2. As we plot these phase space points one by one based on their position of $x$ and directional cosine $p_x$, we find that this “cloud” of points occupies an area shaped as a rectangle. The same plot can be created for the absorber. In this case the extreme rays will intersect with the absorber at angles of 90 deg; therefore, their $p_x$ coordinates correspond to 1 and $-1$, corresponding to the limits we saw for $L$ in Fig. 8(b).

4.4. Understanding the Maximum Concentration Based on Phase Space

For an optical device without light generation or loss, its phase space volume is conserved [15]. In a 2D configuration, as shown in Fig. 14, the phase space volume, represented by the area of the rectangles, is conserved between the aperture and the absorber plane:

 

Figure 14 Phase space of a CPC.

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Equation (24) shows that, from the phase space conservation perspective, the CPC is achieving the ideal concentration limit.

4.5. Edge Ray Principle

As an example shown in Fig. 15, an imaging system maps the rays from boundary points $a$ and $b$ to $a^{\prime }$ and $b^{\prime }$. It also requires the precise mapping of any point in between, such as $c$, to its corresponding point $c^{\prime }$. A nonimaging system, however, does not require the point $c$ to be mapped to $c^{\prime }$, or to be mapped at all. The edge ray principle requires that the rays traveling at extreme directions and or emanating from extreme positions at the aperture, must be directed to the extreme directions and or extreme positions at the absorber. Extreme is meant to refer to the largest angles accepted by a system and or the edge-most positions located on an aperture or absorber. We call such rays the “edge rays.” When plotted in phase space, they are represented by the boundaries of the phase space volume (area for 2D).

 

Figure 15 Difference between nonimaging and imaging optics.

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We use the phase space representations of an optical device to prove this concept, which is similar to the Liouville theorem in classical mechanics [16]. As shown in Fig. 16, it is impossible for a ray to retire from its position as a boundary point in its phase space representation. For example, if ray 1 at the boundary (red square) wants to become a ray that occupies a position in phase space located internally, the following must happen: ray 1 would need to select one of its neighboring points (for example, the blue square) to exchange positions. However, the Liouville theorem requires that the phase space volume occupied by all the rays of the system to have no source or sink. In other words, there is no generation, vanishing, or merging of the little squares representing each ray.

 

Figure 16 Ray 1 cannot (red square) interchange its position with a ray (blue square) that is not originally inside the boundary.

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In order for blue and red squares to interchange their positions in phase space at one point, they must be compressed, as in Fig. 17, and eventually merged (the green square in Fig. 18). However, merging the area of the phase space representations of any two rays will result in these rays then processing at the same direction and position (green square), which renders them indistinguishable from then on. This makes it clear that two separate rays in a system cannot during their propagation within the optical system acquire both the same position and same direction, or occupy the same phase space area. This proves that the edge rays cannot interchange their positions with internal rays, they are stuck as edge rays. Any edge rays from the absorber must always be mapped to the edge rays of the radiation source, and vice versa.

 

Figure 17 In order for the ray 1 to interchange its phase space position, it will have to merge with a small phase space that was originally not on the boundary. This violates the Liouville theorem.

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Figure 18 Merging of the edge ray with an internal ray in their phase space representations. Similarly, it violates the Liouville theorem.

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4.6. Nonimaging Concentrator with Convex Shaped Absorber

For a convex absorber, such as the tubular absorber shown in Fig. 19, we can again use the thermodynamic intuition proposed in Eq. (8). To design for a wavefront tilted at an angle of $\theta$, we use the principle of redirecting the edge ray to the absorber tangent. We require that the absorber can only see within the designed angle $\theta$, and we design reversely, evaluating the rays that come from the absorber. Starting from the bottom of the absorber $B$, we require that the unwinding string TP be the same length as the absorber section length $\stackrel{͡}{BT}$, effectively forming an involute part of the concentrator $\stackrel{͡}{BQ}$. In this part, the involute only redirects the rays from the absorber based on 1:1 concentration ratio. Past this involute part we require the tangent ray TP of the absorber to be reflected into the extreme ray direction, forming the curve $\stackrel{͡}{Q{A}^{\prime }}$, until the tangent of the reflector becomes vertical, resulting at the concentrator to be at its maximum aperture. Another way of thinking of this design is to use the string method. We assign the point $S$ to be on the slider which can move without friction, and it has a string attached to it. By tying this string to point $P$ and around the absorber shape to point $B$, we require that the string BTPS be always taught and have the same length. Now as we trace the point $P$ along the curve, the point $S$ slides freely, causing the string to be always perpendicular to the slider bar. This method also guarantees the constant optical path length between the wavefront and the tangent points of the absorber.

 

Figure 19 CPC design for a tubular absorber.

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An alternative way of designing the nonimaging concentrator is by replacing the idea of constant string length with correct ray angles. One starts from an arbitrary point at the bottom of the absorber, such as $B$. The curve of the reflector starts by reflecting the tangent ray of the absorber to its original direction to form the involute section $\stackrel{͡}{BQ}$ (TP is the same as PT). Then as soon as the reflector tangent can reflect the absorber tangent ray into the wavefront $S$ without being obstructed by the absorber, the tangent of the reflector should be constructed to do so [8] (PS is perpendicular to the bar AC). The source code as an example for this process can be acquired from this public repository [17]. A shape generated with such a procedure is effectively the same as that generated by the method previously described.

4.7. Asymmetric CEC

The asymmetric compound elliptical concentrator (CEC) was first introduced in [18] as a secondary concentrator used for imaging primary concentrator designs. Here we use the thermodynamic understanding to perform the design procedure in reverse, starting from the absorber $c{c}^{\prime }$, then naturally determine the aperture by using the strings. This design achieves the thermodynamic limit of concentration, or $1/P_{21}$.

The proceeding steps are followed to design the CEC with the string method (Fig. 20):

 

Figure 20 Asymmetric CEC setup.

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To prove that such a design can reach the thermodynamic limit, we use the properties of ellipses:

Therefore, we conclude that

We can derive the concentration ratio as

This method is not limited to $c{c}^{\prime }$ being a straight line; with $c{c}^{\prime }$ being any convex shape, a similar process will also produce ideal nonimaging optics [19]. An example can be found here [20].

5. Flow Line Design

The flow line design has been introduced as an alternative way of generating nonimaging optical shape.

5.1. Flow Line in the Context of Radiative Heat Transfer

Geometric vector flux, or flow line [8], has been used in illumination as a photic field [21], and in computer graphic rendering as a light field [22,23]. It was first introduced in 1893 [24,25] as a light vector. The definition of flow line is

Therefore, we can arrive at the relationship between étendue and flow line vector as [9] prescribes (eq 5.24),

For a two-dimensional configuration as shown in Fig. 21, AB is a blackbody radiation source. The $\overrightarrow{J}$ is a vector with a magnitude $|J|=\sin (\frac{{\theta }_1-{\theta }_2}{2})$, and a direction of $\frac{{\theta }_1+{\theta }_2}{2}$, or the direction bisecting the two extreme rays.

 

Figure 21 Flow line bisects the two extreme angles.

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5.2. Using Flow Line Design to Construct the CPC

Because the flow line always bisects the two extreme angles of the rays from the source, a mirror placed along the flow line will not disturb the flow line field. This can be shown in a 2D scenario; for 3D setups it is also true for Lambertian sources [8,27].

5.2a. Flow Line Examples

To visualize this, we use the example of a spherical source. Because the flow lines point away from the center of the solid angle subtended by the source, all the flow lines are simply straight lines from the center of the sphere (Fig. 22). One can take any surface that consists of flow lines and put a mirror along it, without disturbing the flow line field. In other words, one would see exactly the same thing even with the flow line mirrors in place.

 

Figure 22 Flow line of a spherical source.

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As an example, we put a cone that follows the flow line $AB$, $DC$ in Fig. 22, as shown in Fig. 23, and from all the angles we can see that the orange remains the same.

 

Figure 23 Flow line example of an orange cone.

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5.2b. Flow Line Design as an Alternative Way of Generating Nonimaging Optics

To use the flow line method for generating a CPC shape, we must include both the radiation source and radiation absorber information to come up with the flow line generator.

For a CPC, this means that the flow line generator must combine the information from both the acceptance angle and the radiation absorber shape. Such a setup is shown in Fig. 24. The flat radiation absorber is extended with two “shadow lines” from the source, which extend to infinity at the acceptance angle (e.g., line PR). Such a shape for the flow line generator can be shown to work as the following:

 

Figure 24 “Mountain top” shape as the source for flow line that generates CPC.

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As we observe such a process, we can see that the flow line design is capable of transforming the view of any point within the concentrator to be seeing the whole “mountain top” even if we only keep the flow line source $AP$. The phase space that $AP$ occupies for any point is effectively the same as the phase space the mountain top shape occupies. This setup does not, however, work for all 3D configurations [8].

5.2c. Flow Line as a Tool to Calculate Concentration Ratio

For the asymmetric setup, as shown in Figs. 25 and 26, the étendue at the aperture is (notice that the flow line is at the angle $\frac{{\theta }_1+{\theta }_2}{2}$ from the aperture)

 

Figure 25 Flow line plot of an asymmetric compound elliptical concentrator.

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Figure 26 Close-up flow line plot of the asymmetric CEC.

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The étendue at the absorber is

Because the étendue is conserved,

5.3. Generalizing the Flow Line Design

Although we can see from the previous sections that the flow line design is capable of producing same design for certain configurations, the very nature of the design method still remains to be explained. Here we try to generalize the flow line design and look at another motivation of nonimaging designs, the Hilbert integral.

The Hilbert integral has provided an understanding of nonimaging optics at a deeper level ([27], Section 6.6). In this section we summarize the Hilbert integral previously introduced to form a very general treatment of the 2D concentrator. In the example shown by Fig. 27, an input and an output surface are postulated enclosing and surrounded by regions of given refractive index distributions.

 

Figure 27 Hilbert integral as a high-level understanding of nonimaging optics.

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On each of these surfaces a distribution of extreme rays is given. Then we develop a procedure that enables us to design a 2D concentrator that will ensure that all rays between the extreme incoming rays and none outside are transmitted so that the concentrator is optimal. Suppose we have, as in Fig. 27, two surfaces $AB$ and $A^{\prime }{B}^{\prime }$, and let $AB$ be illuminated in such a way that the extreme angle rays at each point form pencils belonging respectively to wavefronts ${\mathrm{\Sigma }}_{\alpha }$ and ${\mathrm{\Sigma }}_{\beta }$. Similarly, rays at intermediate angles belong to other wavefronts, so the whole ensemble of rays comes ultimately from a line of point sources and is transformed by a possibly inhomogeneous medium in such a way that the rays just fill the aperture $AB$. These rays then have a certain étendue $E$, and we shall see following how to calculate it. Similarly, we draw rays and wavefronts emerging from $A^{\prime }{B}^{\prime }$ as indicated, and we postulate that these shall have the same étendue $E$. Now we want to know how we can design a concentrator system between the surfaces $AB$ and $A^{\prime }{B}^{\prime }$, possibly containing an inhomogeneous medium, that shall transform the incoming beam into the emergent beam without loss of étendue. To solve this problem, we postulate a new principle (we shall see that our edge ray principle can be regarded as derived from it). The optical system between $AB$ and $A^{\prime }{B}^{\prime }$ must be such as to exactly image the pencil from the wavefront ${\mathrm{\Sigma }}_{\alpha }$ into one of the emergent wavefronts and ${\mathrm{\Sigma }}_{\beta }$ into the other.

To see how this principle leads to a solution of the general problem, we must first show how to calculate the étendue of an arbitrary beam of rays at a curved aperture, as in Fig. 27. We use the Hilbert integral, a concept from the calculus of variations. In the optics context [9] the Hilbert integral for a path from $P_1$ to $P_2$ across a pencil of rays that originated in a single point is

Now let there be some kind of system constructed that achieves the desired transformation of incident extreme pencils with emergent extreme pencils, as in Fig. 27. The system takes ${\mathrm{\Sigma }}_{\alpha }$ into ${\mathrm{\Sigma }}_{\alpha }^{\prime }$ and ${\mathrm{\Sigma }}_{\beta }$ into ${\mathrm{\Sigma }}_{\beta }^{\prime }$, and we want it to do so without loss of étendue. We write down the optical path length from $P_{\overline{\alpha }}$ to $P_{\overline{\alpha }}^{\prime }$ and equate it to that from $P_{\alpha }$ to $P_{\alpha }^{\prime }$. Similarly, for the other pencil

The left-hand side of this equation can be treated as the difference between the étendues at the entry and exit apertures. Since we require this difference to vanish, we have to make the right-hand side of Eq. (41) vanish. A simple way to do this would be to ensure that the optical system of the concentrator is such that the $\alpha$ and $\beta$ ray paths from $A$ to $B^{\prime }$ coincide, and the same for those from $B$ to $A^{\prime }$. We can do this by starting segments of mirror surfaces at $A$ and $A^{\prime }$ in such directions as to bisect the angles between the incoming $\alpha$ and $\beta$ rays. In this way we effectively made the $P_{\alpha }A$ and $P_{\beta }A$ the same through the optical system between $AB$ and $A^{\prime }{B}^{\prime }$. In other words, the two extreme rays became one. We then continue the mirror surfaces in such a way as to make all $\beta$ rays join up with the corresponding emerging rays; in other words, we image the ${\beta }^{\prime }$ pencil exactly into the ${\beta }^{\prime }$ pencil, and similarly for the other mirror surface connecting $B$ and $B^{\prime }$. We have $E_{12}=E_{34}$ thus completed the construction and used up all degrees of freedom in doing so.

In Fig. 28 we construct a simplified configuration of Fig. 27 with only mirrors $A{A}^{\prime }$ and $B{B}^{\prime }$. The result of satisfying Eq. (41) is $E_{12}=E_{34}$, and if 3 and 4 overlaps, $E_{34}=\pi A_3$. This brings us back to Eq. (17). Notice that Eq. (41) is not the only way to enable $E_{12}=E_{34}$. However, $A^{\prime }{B}^{\prime }$ overlapping with ${\mathrm{\Sigma }}_{\alpha }^{\prime }{\mathrm{\Sigma }}_{\beta }^{\prime }$, being either the same or part of it, is essential to meet the condition $E_{34}=\pi A_3$.

 

Figure 28 Hilbert integral configured without refractive optics.

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5.4. Flow Line Method and Its Potential

As we can see, this particular generalized method is not limited by the refractive index. Such a general method provides one particular way of how to make a nonimaging optics system, and it also hints at an alternative explanation of what nonimaging optics is and how it works. However, this is not the only way to explain nonimaging optics and we are continuously exploring the fundamental reasoning and methodologies of creating nonimaging systems with newer tools. Thermodynamics has given us what can be done; the means to achieve the theoretical goals remains an open discussion.

6. Conclusion

Other designing methods include the simultaneous multiple surface, and tailored edge ray concentrators have been built based on the generalized method presented in Subsection 5.3. These methods either directly utilize the edge ray principle as described previously, or use a variation of the edge ray concept to form the optical device. The breakthrough of nonimaging optics compared to the traditional imaging optical design, however, lies within its capability of connecting the thermodynamic principles with the designing process through edge rays. This tutorial aims to show that the intuition of the multitude of nonimaging optics designing methods originates from a thermodynamic understanding of the optical system. This concept continues to fuel the development of this field. Therefore, to understand both how a nonimaging optical design works, and how to develop new ways of such optical designs, one could rely on the guidance of the thermodynamic understanding of the optical designing process.

Appendix A: Fermi Proof of Étendue Conservation

“An important result deserves more than one proof.”

Consider the Legendre transformation of the optical Lagrangian:

(A1)$$\begin{gathered} \mathrm{Construct}{p}_x\dot{x}+p_y\dot{y}-\mathcal{L}(x,y,\dot{x},\dot{y})=\mathcal{H} \\ d\mathcal{H}=p_{x}d\dot{x}+d{p}_x\dot{x}+p_{y}d\dot{y}+d{p}_y\dot{y}-[\frac{\partial \mathcal{L}}{\partial x}dx+\frac{\partial \mathcal{L}}{\partial y}dy+\frac{\partial \mathcal{L}}{\partial \dot{x}}d\dot{x}+\frac{\partial \mathcal{L}}{\partial \dot{y}}d\dot{y}]=-\dot{p_x}dx-\dot{p_y}dy+\dot{x}d{p}_x+\dot{y}d{p}_y. \end{gathered}$$

This results in $\frac{\partial H}{\partial x}=-\dot{p_x},\frac{\partial H}{\partial y}=-\dot{p_y},\frac{\partial H}{\partial p_x}=\dot{x},\frac{\partial H}{\partial p_y}=\dot{y}$, which are familiar to Hamiltonian equations.

Notice that $\frac{\partial \dot{p}x}{\partial p_x}=-\frac{\partial }{\partial p_x}\left(\frac{\partial H}{\partial x}\right)=-\frac{\partial }{\partial x}\left(\frac{\partial H}{\partial p_x}\right)=\frac{\partial \dot{x}}{\partial x}$, etc, which are the Hamilton’s equations of motion.

Now construct a vector $\overrightarrow{W}=(\dot{x},\dot{y},{\dot{p}}_x,{\dot{p}}_y)$, notice that $\nabla ·\overrightarrow{W}=\frac{\partial W}{\partial \dot{x}}+\frac{\partial W}{\partial \dot{y}}+\frac{\partial W}{\partial {\dot{p}}_x}+\frac{\partial W}{\partial {\dot{p}}_y}=\frac{\partial {\dot{p}}_x}{\partial p_x}+\frac{\partial \dot{x}}{\partial x}+\frac{\partial {\dot{p}}_y}{\partial p_y}+\frac{\partial \dot{y}}{\partial y}=0$. This means the field of four-dimensional vector $\overrightarrow{W}$ has the important property of divergence being zero. In other words, the four-dimensional hyperspace of $(x,y,p_x,p_y)$ has the property of conservation of volume as all the light rays evolve in an optical system.

Here we will also offer an intuitive analogy using an incompressible fluid (Fig. 29).

 

Figure 29 Phase space volume remains constant.

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Let $\mathcal{V}$ be the volume surrounded by a closed Surface, where $\overrightarrow{v}$ is the velocity field of the small elements within. As the fluid starts to flow according to the change of $t$, and the $\text{Surface}(t)$ starts to evolve into shape $\text{Surface}(t+\mathrm{\Delta }t)$, the enclosed volume of $\mathcal{V}(t+\mathrm{\Delta }t)$ will also change as

(A2)$$\mathcal{V}(t+\mathrm{\Delta }t)-\mathcal{V}(t)={∯}_{\text{Surface}(t)}\overrightarrow{v}\mathrm{\Delta }t·\mathrm{d}\overrightarrow{s},$$

where $d\overrightarrow{s}$ is the surface vector pointing outward along the normal direction.

Using Gauss’s theorem,

(A3)$${∯}_{\text{Surface}(t)}\overrightarrow{v}\mathrm{\Delta }t·\mathrm{d}\overrightarrow{s}=\iiint \nabla ·\overrightarrow{v}\mathrm{d}\tau ,$$

where $d\tau$ is the volume element.

If $\nabla ·\overrightarrow{v}=0$ everywhere, then obviously,

(A4)$$\mathcal{V}(t+\mathrm{\Delta }t)-\mathcal{V}(t)=0;\mathrm{or},\mathcal{V}(t)=\text{constant}.$$

Applying this to the étendue conservation, we find that the four-dimensional volume $(x,y,p_x,p_y)$ remains constant as it evolves with time. The vector field $\overrightarrow{W}=(\dot{x},\dot{y},{\dot{p}}_x,{\dot{p}}_y)$ replaces $\overrightarrow{v}$. ($\nabla ·\overrightarrow{W}=0$ just as $\nabla ·\overrightarrow{v}=0$.) This not only implies that the étendue is conserved, but also shows that there is no source or sink of field $\overrightarrow{W}$ as the light propogates.

Funding

University of California (UC) Advanced Solar Technologies Institute.

Acknowledgment

The authors want to thank Aaron Brinkerhoff for the flow line orange cone photo. We are grateful to our colleagues Bennett Widyolar, Jonathan Ferry, and Jordyn Brinkley, who provided expertise assisting the research. We also want to thank the anonymous reviewers who greatly helped to improve the quality of the paper. Last but not the least, we want to thank Sarah Boyd, who moderated this paper. The research was supported by University of California Advanced Solar Technologies Institute (ucsolar.org).

References

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6. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, 1972), pp. 204–207.

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11. L. Jiang, “2D Phase space representation example,” https://github.com/wormite/PhaseSpace/.

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Dr. Roland Winston is a leading figure in the field of nonimaging optics and its applications to solar energy. He is the inventor of the compound parabolic concentrator (CPC), used in solar energy, astronomy, and illumination. He is also a Guggenheim Fellow, a Franklin Institute medalist, past head of the University of Chicago Department of Physics, and a member of the founding faculty of University of California Merced, and he is currently the head of UC Solar.

Dr. Lun Jiang is a researching scientist at UC Solar. His expertise is with vacuum devices, nonimaging optics and solar thermal and hybrid systems, solar cooling, and solar desalination. In his Ph.D. thesis he demonstrated two novel solar collectors that reach a working temperature above 200°C, without tracking. He led the receiver designing team for a vacuum hybrid receiver that generates both electricity and heat under 70× concentration, commission by Arpa-E.

Dr. Melissa Ricketts is an illumination engineer at Acuity Brands Lighting. She received her Ph.D. (2017) from the University of California, Merced in physics with a concentration in nonimaging optics and a B.S. (2012) from the University of California, Merced in physics. Her research interests lie in nonimaging optics for illumination. She currently uses the nonimaging optics theory to design optics for illumination in both indoor and outdoor lighting.