## Abstract

The optical thermodynamic efficiency is a comprehensive metric that takes into account all loss mechanisms associated with transferring flux from the source to the target phase space, which may include losses due to inadequate design, non-ideal materials, fabrication errors, and less than maximal concentration. We discuss consequences of Fermat’s principle of geometrical optics and review étendue dilution and optical loss mechanisms associated with nonimaging concentrators. We develop an expression for the optical thermodynamic efficiency which combines the first and second laws of thermodynamics. As such, this metric is a gold standard for evaluating the performance of nonimaging concentrators. We provide examples illustrating the use of this new metric for concentrating photovoltaic systems for solar power applications, and in particular show how skewness mismatch limits the attainable optical thermodynamic efficiency.

©2010 Optical Society of America

## 1. Introduction

In recent years there has been a considerable interest in developing efficient concentrating optics for photovoltaic (PV) solar energy applications. Establishing appropriate design metrics for these optics has proven to be a challenging problem that requires a judicious balance between radiometric measures of performance and other considerations, primarily related to costs of manufacture and operation. The concentrating PV design problem is furthermore special in that the optics are not essential to the proper function of the solar cell; rather, they are introduced primarily as a cost savings measure, enabling reduction of the cell area. The concentrating PV design problem motivates the discussion in this paper; however, the performance metric that we develop is, as we shall show, applicable to all types of nonimaging devices.

Spirkl et al[1] have derived an upper limit on the concentration achievable by non-tracking solar concentrators, thereby providing a standard with which non-ideal non-tracking solar concentrators can be compared. However, this standard is not applicable to tracking solar concentrators or to concentrators used in non-solar-energy-related applications.

A metric often used in characterizing tracking solar PV concentrators is the concentration-acceptance product (CAP), defined as

where ${C}_{g}$is the geometrical concentration and the acceptance angle*α*is defined as the incidence angle on the entrance pupil for which the concentrator collects 90% of the on-axis power.[2] Although it is a useful merit function in many cases of interest, CAP has a number of limitations. One drawback is that it is only applicable in cases where the desired acceptance solid-angular region is axisymmetric. Another problem is that it is not normalized to unity for ideal concentrators operating under étendue-matched conditions. In fact, it can be shown that the CAP value of such an ideal concentrator is given by the formulawhere ${n}_{src}$and ${n}_{trg}$are the indices of refraction in which the source and target are immersed, and ${\theta}_{trg}^{max}$ is the maximum allowed incidence angle on the target surface. It is apparent from Eq. (2) that different ideal étendue-matched concentrators will not even necessarily have the same CAP values. A more serious drawback of CAP as a metric is that it does not account for the absolute flux transfer from the source to the target. The ${C}_{g}$-term in Eq. (1) is a purely geometrical quantity containing no information about flux transfer. The term involving

*α*provides a measure of the off-axis flux-transfer relative to the on-axis flux transfer, but does not provide an absolute measure of flux transfer. Consider, for example, two concentrators that are identical except that one has optical coatings with significantly higher efficiency. The two concentrators could have the same CAP value, even though the concentrator with the better coatings transfers significantly more flux from the source to the target.

The purpose of a nonimaging concentrator is to transfer maximal flux from the phase space of the source to that of a target. A concentrator’s performance can be expressed relative to a thermodynamic reference. We discuss consequences of Fermat’s principle of geometrical optics. We review étendue dilution and optical loss mechanisms associated with nonimaging concentrators. We introduce the concept of optical thermodynamic efficiency which is a performance metric combining the first and second laws of thermodynamics. Our optical thermodynamic efficiency is a comprehensive metric that takes into account étendue dilution as well as all loss mechanisms associated with transferring flux from the source to the target phase space, including losses due to inadequate design, non-ideal materials, fabrication errors, and less than maximal concentration. As such, this metric is a gold standard for evaluating the performance of nonimaging concentrators.

## 2. Geometrical optics and thermodynamics

We are interested in applying the laws of thermodynamics to determine performance limits for nonimaging concentrators in the geometrical optics approximation. We first briefly review the principles underlying geometrical optics, with particular emphasis on how they relate to thermodynamics. We consider the geometrical-optics point of view only and show that the first and second laws of thermodynamics, which are independent postulates, are manifested through Fermat’s principle of geometrical optics, in surrogate form.

In the geometrical-optics approximation, the behavior of a nonimaging optical system can be formulated and studied as a mapping from input phase space to output phase space, where *S* is an even-dimensional piecewise differentiable manifold and *n* ( = 2) is the number of generalized coordinates. The starting point for this formulation is the generalization of Fermat's variational principle, which states that a ray of light propagates through an optical system in such a manner that the time required for it to travel from one point to another is stationary. This mapping is purely geometrical and is independent of thermodynamic quantities such as heat or temperature.

Let *g* be a differentiable mapping. The mapping *g* is called canonical[3,4] if *g* preserves the differential 2-form *w*^{2} = *dp _{i}* ^

*dq*

_{i}*i*=1..

*n*, where

*q*is the generalized coordinate and

*p*is the generalized momentum. Applying the Euler-Lagrange necessary condition to Fermat's principle and then the Legendre transformation, we obtain a canonical Hamiltonian system, which defines a vector field on a symplectic manifold (a closed nondegenerate differential 2-form). Now, a vector field on a manifold determines a phase flow, i.e., a one-parameter group of diffeomorphisms (transformations which are differentiable and also possess a differentiable inverse). The phase flow of a Hamiltonian vector field on a symplectic manifold preserves the symplectic structure of phase space and consequently is canonical. The properties of these mappings can be summarized as follows:

- 1) The mappings from input phase space to output phase space are piecewise diffeomorphic. Consequently they are one-to-one and onto.
- 2) The transformation of phase space induced by the phase flow is canonical, i.e., it preserves the differential 2-form.
- 3) The mappings preserve the integral invariants, known as the Poincaré-Cartan invariants. Geometrically, these invariants are the sums of the oriented volumes of the projections onto the coordinate planes.
- 4) One of the Poincaré-Cartan invariants preserved by the mappings is the phase-space volume element (i.e., étendue). The volume of
*gD*is equal to the volume of*D*, for any region*D*.

To begin, we need conservation of energy. For this, we invoke Noether’s theorem[3], which relates symmetry to conservation laws. It states that to every one-parameter group of diffeomorphisms of the configuration manifold of a Lagrangian system that preserves the Lagrangian function, there corresponds a first integral of the equations of motion. In the case of geometrical optics, symmetries in time, rotation and translation result, respectively, in the conservation of energy and the rotational and translational skewness[4-6], which are analogous to angular and linear momentum. The conservation of energy provided by Noether’s theorem applies to propagation in nondissipating media. Therefore if energy is applied at one end of the nonimaging mapping it will be transported by the phase space through the system. This result is purely geometrical and is a surrogate for general conservation of energy.

We also require the second law of thermodynamics. From statistical mechanics[7,8] we have the following relationship between entropy, *S*, and the étendue, *E*, which contains both geometrical and thermal terms:

To conclude, Fermat’s principle of geometrical optics generates surrogates (i.e. quantities that exhibit consistent behavior) for the first and second laws of thermodynamics. Note that Fermat’s principle is observed experimentally; consequently it is a postulate. Our point of view is that Fermat’s principle would not appear in nature unless it complied with the first and second laws of thermodynamics. The conservation of energy and entropy in surrogate form enable us to establish performance limits for nonimaging devices.

## 3. Optical thermodynamic loss mechanisms in PV systems

A photovoltaic (PV) system is comprised of collection/concentration optics and a solar cell that converts the incident light into electrical power. The performance of a PV system can be evaluated in terms of a thermodynamic reference. This is a strict reference that is useful both in the absolute sense and also in the relative sense since it provides an unambiguous method for comparing the performance of PV systems either within the same design category or across different types of design categories.

Since the incident flux must propagate through the optical system to reach the solar cell, the system efficiency can be expressed as the product of the optical efficiency and the cell efficiency:

where the use of the asterisk denotes a thermodynamic quantity. This formula shows that: a) all the efficiencies contributing to the system thermodynamic efficiency must be*thermodynamic*efficiencies, and b) the optical thermodynamic efficiency is just as influential as the cell thermodynamic efficiency when it comes to achieving total system thermodynamic efficiency. There are three primary mechanisms that cause loss of optical thermodynamic performance:

- 1. Losses that occur along the ray paths: coatings, bulk attenuation, scattering, etc. These are energy losses.
- 2. Losses that occur due to ray rejection. This is light that could have been collected by the solar cell but, due to an inadequate optical design, ends up either rejected outside the optical system or absorbed within the optical system at a location other than the solar cell. This type of loss is often caused by inappropriate use of optical components (e.g., utilizing imaging spherical optical surfaces).
- 3. Losses due to étendue dilution. Given the required active area of the solar cell and acceptance angle at the entrance aperture of the optical system, dilution provides a measure of how much additional flux could have been transferred to the cell by utilizing optics of larger entrance-aperture area than the candidate optics. This loss is an indirect loss since it represents the lost opportunity of failing to use a larger entrance aperture and/or a smaller target (i.e., a higher concentration ratio). An associated loss here is that lower concentration ratios are generally associated with lower cell efficiencies. However, it should be noted that cell efficiency does not increase indefinitely with concentration; at present, the maximum conversion efficiencies typically occur for concentration ratios between 300 and 600.

One particularly challenging form of étendue dilution has to do with the shapes of the entrance aperture and solar cell. Entrance apertures are typically square because of the requirement for tiling of modules; similarly solar cells tend to be square for economy of manufacture. Assuming that we start out with an étendue-matched axisymmetric optical system having a circular aperture and a circular solar cell, we may convert the entrance aperture to a square by cropping the circular aperture to the shape of the inscribed square. The circular cell shape is also often converted to a square by using a cell having the shape of the circumscribing square. The result of all of this is étendue dilution: the target-to-source étendue ratio is now two, rather than the original ratio of unity. Note that this particular problem does not occur with Köhler systems in which the entrance pupil is imaged onto a solar cell having the same shape.

## 4. The optical thermodynamic efficiency

We now define the optical thermodynamic efficiency and provide a convenient formula for computing it. We consider an optical source, an optical target, and a nonimaging optical system. The source and the target are assumed to occupy separate (i.e., non-overlapping) finite phase-space volumes. The source étendue ε* _{src}* is not necessarily equal to the target étendue ε

*. The source radiance is a function of position within the source’s phase-space volume and is zero outside this volume. For simplicity, we define radiance as the flux per unit phase-space volume, which is sometimes referred to as the generalized radiance. The advantage of using generalized radiance is that it is conserved along any loss-free ray path, even when the refractive index varies along the ray path.*

_{trg}Our definition of optical thermodynamic efficiency is dependent on two quantities: the source-to-target flux-transfer efficiency and the source-to-target étendue ratio. When the source étendue is greater than or equal to the target étendue, we define optical thermodynamic efficiency simply as the flux-transfer efficiency. When the source étendue is less than that of the target we define the optical thermodynamic efficiency as the flux-transfer efficiency times the source-to-target étendue ratio, where the étendue ratio is included in the definition in order to provide an appropriate penalty for étendue dilution, i.e., the inherent inability to completely fill the target’s phase space when the target’s phase-space volume is greater than that of the source. By defining the optical thermodynamic efficiency in this way, we obtain a metric that evaluates an actual concentrator’s performance relative to that of a hypothetical ideal concentrator operating under étendue-matched conditions.

We define the source flux Φ* _{src}* as the total flux emitted by the source. Similarly, we define the total flux transferred by the optical system from the source to the target as Φ

*. We also define the maximum dilution-free target flux Φ*

_{trg}*as the total flux that would be contained by the target phase space if it were to be completely filled with a constant radiance equal to the average source radiance. From the definition of generalized radiance, we find that the average source radiance equals the source flux divided by the source étendue:*

^{max}_{trg}Combining Eqs. (8) and (10) into a single equation, we obtain

*a*,

*b*) is equal to

*a*when

*a*≤

*b*and

*b*when

*b*<

*a*. Equation (11) provides a convenient formula for computing the optical thermodynamic efficiency.

The optical thermodynamic efficiency provides a single figure of merit for evaluating the performance of a nonimaging optical system when used with a specific source and target. This metric takes into account étendue dilution as well as all loss mechanisms associated with transferring flux through a nonimaging system, including losses due to inadequate design, materials, and less than maximal concentration. It is applicable to both symmetrical and nonsymmetrical optical systems.

## 5. Examples

We now show how the optical thermodynamic efficiency is calculated for a specific optical system. In addition, we compute the upper limit on the optical thermodynamic efficiency of East–west-oriented non-tracking translationally-symmetric solar concentrators for which the upper limit on the flux-transfer efficiency is 100%.

As a first example, we consider the refractive solar concentrator depicted in Fig. 1
. This is a plano-convex singlet lens having a 40-mm-diameter circular entrance aperture and a refractive index of 1.5. The convex lens surface is spherical, with a 28-mm radius of curvature. The convex and planar lens surfaces intersect the optical axis at x = −6 mm and 6 mm, respectively. Each lens surface has a transmittance of 96%, independent of incidence angle. Propagation within the lens material is assumed to be loss-free. The target is a 6-mm-square non-immersed solar cell centered on the optical axis at x = 46.5 mm. The cell is assumed to absorb flux at all incidence angles relative to its surface normal. The required acceptance angular region for the concentrator is defined as a circular cone of half angle θ* _{src}* = 5° centered on the optical axis. This type of acceptance-angle requirement could be derived, for example, from known alignment and tracking errors. The source and target étendue are

*A*is the source (i.e., entrance-aperture) area and

_{src}*A*is the target area. We also consider a hypothetical ideal concentrator that transfers 100% of the flux from the 40-mm-diameter, 5°-half-angle source to a flat on-axis étendue-matched target. Like the actual target, the étendue-matched target used in defining the ideal concentrator is assumed to absorb flux at all incidence angles relative to its surface normal.

_{trg}The source-to-target flux-transfer efficiency as a function of the source’s angular half width is plotted in Fig. 2 for the actual and ideal concentrators. We find that for the desired 5°-half-angle source, the flux-transfer efficiency for the plano-convex lens is

The ideal concentrator has a flux-transfer efficiency of 100% for the same source. Since it operates under étendue-matched conditions, the ideal concentrator has a thermodynamic efficiency of 100%. By substituting the values from the right hand sides of Eqs. (13) - (15) into Eq. (11), we find that the thermodynamic efficiency of the refractive concentrator isThe refractive concentrator suffers from étendue dilution, which results in its optical thermodynamic efficiency being reduced by a factor of ε*/ε*

_{src}*relative to the source-to-target flux-transfer efficiency.*

_{trg}As our second and final example, we now consider the problem of calculating the upper limit on optical thermodynamic efficiency imposed by conservation of translational skewness^{6} on East–west-oriented non-tracking translationally-symmetric solar concentrators. It should be emphasized that we are not here analyzing the performance of any specific concentrator. Instead, we are computing an upper limit on performance for a specific class of concentrator.

Translationally-symmetric concentrators are concentrators for which the surface normals of all refractive and reflective optical surfaces are perpendicular to a single Cartesian coordinate axis, referred to as the symmetry axis. In the case of concentrators that include one or more gradient-index (GRIN) components, the normals to all surfaces of constant refractive index must all be perpendicular to the symmetry axis. An East–west-oriented translationally-symmetric concentrator is one having a symmetry axis that is oriented parallel to the local latitude line. We assume the entrance pupil is rectangular, oriented such that it is parallel to the symmetry axis. The surface normal to the entrance pupil is tilted toward the equator, with tilt angle equal to the local latitude, measured relative to the local vertical. We assume year-round operation with a daily operation time of *T* (≤ 12 hr), where the time interval *T* is centered on solar noon. We neglect latitude-dependent shadowing effects by the earth’s horizon, which will in practice limit the operation time at certain times of the year for concentrators operating at high latitudes.

Flux incident on the entrance pupil is to be transferred by the concentrator to a translationally-symmetric target having the same symmetry axis as the concentrator. We assume the target is in air and that the maximum allowable incidence angle for rays incident on the target surface is 60° relative to the local surface normal. We also require that the upper limit on flux-transfer efficiency from the source to the target be 100%. Due to translational-skewness mismatch between the source and the target^{6}, it can be shown that 100% flux transfer using a translationally-symmetric concentrator is impossible when the source and target have equal étendue. Instead, the source-to-target étendue ratio has to be less than unity to meet the 100%-flux-transfer-limit requirement. This, in turn, means the optical thermodynamic efficiency must be less than 100%, as is apparent from Eq. (11).

From the above problem definition as well as the apparent motion and angular size of the sun in the earth’s frame of reference, it can be shown that the direction cosines of rays incident on the entrance pupil satisfy the following inequalities:

andwhere*k*and

_{y}*k*are referred to as the vertical and horizontal direction cosines, respectively. The

_{z}*k*direction cosine is parallel to the symmetry axis. Both of these direction-cosine coordinates are parallel to the plane containing the entrance pupil. The quantities ${\theta}_{y}$ and ${\theta}_{z}(T)$ in the above equations are the vertical and horizontal angular widths of the source, given by the formulasandwhere ${\theta}_{earth}=23.45\xb0$ is the tilt of the earth’s axis and ${\theta}_{sun}=0.25\xb0$ is the half width of the sun. Similarly, the direction cosines for the target satisfy the inequalitywhere ${\theta}_{trg}$ is maximum allowed incidence angle on the target surface relative to the local incidence angle. For this example, we’ll set ${\theta}_{trg}$ equal to 60°. Figure 3 depicts the boundaries of the direction-cosine regions corresponding to the source and target, for the case of

_{z}*T*= 6 hr. The projected solid angle of the source equals the area of its direction-cosine region. Similarly, the projected solid angle of the target equals the area of its direction-cosine region. To compute the étendue we multiply the projected solid angle by the surface area. For the source, the result is

*A*is the source area (i.e., the entrance-pupil area). The target étendue iswhere

_{src}*A*is the target area. Using the methods described in Ref. 6, we can show that the translational-skewness distribution of the source is given by the formula

_{trg}*S*is the translational skew-invariant. Similarly, the translational-skewness distribution of the target is

_{z}*A*we can rescale the target’s skewness distribution of Eq. (25) until it completely contains the skewness distribution of the source, as shown in Fig. 5 . In this case, the translational-skewness limit on the flux-transfer efficiency becomes 100%. Examination of Fig. 5 shows that the desired condition occurs when the two corners (i.e., slope discontinuities) of the source’s skewness distribution touch the target’s distribution. From Eqs. (24) and (25) we find that this condition is expressed by the formula

_{trg}*A*is the minimum required target area for which the translational-symmetry limit on the flux-transfer efficiency is 100%.

_{trg,req}We also must require that the maximum *S _{z}*-value for which the source’s skewness distribution is non-zero be less than or equal to the maximum

*S*-value for which the target’s distribution is non-zero. Using Eqs. (24) and (25), this leads to the requirement that

_{z}Combining Eqs. (11), (22), and (23) with Eq. (27), we obtain the following formula for the upper limit on the optical thermodynamic efficiency:${\eta}_{optics,max}^{*}(T)=\frac{2\mathrm{sin}[{\theta}_{z}(T)]\left[\mathrm{sin}({\theta}_{y})\mathrm{cos}({\theta}_{y})+{\theta}_{y}\right]\sqrt{{\mathrm{sin}}^{2}({\theta}_{trg})-{\mathrm{sin}}^{2}[{\theta}_{z}(T)]{\mathrm{cos}}^{2}({\theta}_{y})}}{\pi \mathrm{sin}({\theta}_{y}){\mathrm{sin}}^{2}({\theta}_{trg})}\text{.}$ (29)

This is the maximum possible thermodynamic-efficiency achievable by East–west-oriented translationally-symmetric concentrators under the stated conditions as a function of the daily operation time. A plot of this function is shown in Fig. 6 . The peak value of the upper limit is 67.605%, which occurs for an operation time of 5.563 hr. To achieve an optical thermodynamic efficiency above this value would require a non-translationally-symmetric optical system.

## 6. Conclusions

Fermat’s principle of geometrical optics provides surrogates for both the first and second laws of thermodynamics. We have introduced an optical thermodynamic efficiency metric, which combines both laws of thermodynamics. This is a comprehensive metric which takes into account étendue dilution as well as all loss mechanisms associated with transferring flux through a nonimaging system, which may include losses due to inadequate design, materials, and less than maximal concentration. This metric is applicable to both symmetrical and nonsymmetrical systems. It is therefore a gold standard for evaluating the performance of nonimaging optics for both projection and collection systems and has been put to use in evaluating concentration optics for PV systems in solar energy applications.

## References and links

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