## Abstract

This paper details the design of a ray-leakage-free sawtooth-shaped planar lightguide solar concentrator. The concentrator combines Unger’s dimpled planar lightguide solar concentrators [1] with a prism array dimpled planar lightguide solar concentrator. The use of a sawtooth-shaped boundary on the planar lightguide prevents leakages of the guiding ray after multiple reflections in the lightguide. That is, the proposed solar concentrator can achieve a higher geometrical concentration ratio, while maintaining a high optical efficiency at the same time. Numerical results show that the proposed sawtooth-shaped planar lightguide solar concentrator achieves 2300x geometrical concentration ratio without any guiding ray-leakages from the planar lightguide.

© 2013 OSA

## 1. Introduction

Green energies have become very important for the sake of environmental protection and the energy crisis. Solar energy is one kind of important green energy. Generally, sunlight is collected by a solar collector, and the solar energy is then transferred to heat energy or to electricity by the photovoltaic (PV) cell. Researchers use solar concentrators for achieving high light concentration ratios by collecting sunlight from a large area and directing it to a smaller light-receiving region of PV cells. Parabolic dish type [2] and trough type [3] sunlight collectors have previously been developed to collect sunlight. Recently, planar lightguide solar concentrators have attracted much attention for their promising thin form-factor, lightweight and inexpensive replacements for the current generation of refractive and reflective solar concentrators [4–11]. The general goal of designing a good planar lightguide solar concentrator is to have a high light concentration ratio and high optical efficiency at the same time.

Generally speaking, the planar lightguide solar concentrator could be classified into two types. One type is the active-type, the fluorescence-based concentrators [4]. Such active concentrators involve repeatedly nonlinear light absorption and light re-emission, where the large light re-emission angular spectrum and the repeatedly nonlinear light absorption lead to its low optical efficiency. The other type of planar lightguide solar concentrator is the passive-type, the non-ray-emitting concentrators [5–11]. The solar concentrator this paper proposed belongs to this type. A common design for non-ray-emitting-type planar lightguide concentrators usually consists of three parts: the lens array, secondary optical elements and the lightguide itself [6–8]. The lens array of a large sunlight collecting area focuses sunlight from a large area into a lightguide. The secondary optical elements then redirect the focusing light in the direction that the PV cell is situated. Finally, the lightguide of an index higher than the surrounding media guides the redirected sunlight to the PV cell of a small area by total internal reflection (TIR). For example, Karp *et al.* proposed planar micro-optic solar collectors [6, 7], which use round and small micro-structures as secondary optical elements. This kind of planar lightguide solar concentrator can stand a great concentration ratio by lowering its interaction probability between micro-structures. In this design, the sequential scattering by other micro-structures leads to guiding ray leakages. Such a decrease in optical efficiency could be reduced by narrowing down the micro-structures, but this might also introduce intrinsic lens coupling loss. Unger *et al.* proposed dimpled planar lightguide solar concentrators [1]. This planar solar concentrator uses a smart dimpled structure as the secondary optical elements. The dimpled structure can efficiently redirect the collecting sunlight in the direction of the light exit surface of a lightguide where the PV cell is situated. However, the reflections by the sequential dimpled structures also add an angular spectrum of guided rays which leads to the break in TIR and also leads to the limitation in concentrator optical efficiency and concentrator light concentration ratio. Similarly, reducing the depth of the dimpled structure can successfully decrease the possibility of sequential guiding ray reflections, but it will also narrow down the light-coupling area which introduces the intrinsic lens coupling loss. Increasing the lightguide depth with a fixed dimpled structure depth can also decrease the possibility of sequential guiding ray reflections, but it will also decrease the light concentration ratio of the solar concentrator.

Recently, Moore *et al.* proposed a kind of planar solar concentrator of “stepped-shape” [9]; the smart stepped lightguide design successfully avoids guiding-ray interaction with other injection facets, i.e. prevents the ray leakages from the lightguide. The concept of the “stepped-shape” was further explored by Selimoglu *et al.* [10]. With suitable lightguide end-cutting and the secondary compound parabolic concentrators (CPC) design at the lightguide exit port, Selimoglu’s solar concentrator can achieve a high concentration ratio without any ray-leakage from the planar lightguide. In this paper, we detail another design of a ray-leakage-free planar lightguide solar concentrator [11], i.e. sawtooth-shaped planar lightguide solar concentrators. This design adopts Unger’s smart dimpled structure as the secondary optical element of the lightguide. Besides, in our design, the boundary of the planar lightguide was shaped in a sawtooth-shape on one side. The combination of a sawtooth-shaped lightguide boundary and the dimpled structure horizontally constructs a “stepped-like” structure; thus, the lightguide structure successfully prevents light leakages from the planer lightguide. Moreover, the use of a sawtooth-shaped boundary for the planar lightguide substantially decreases the reflection times of guiding rays in the lightguide; thus, it can achieve higher light concentration ratios while maintaining a high optical efficiency. The numerical results show that while excluding the Fresnel loss at each interface, the proposed sawtooth-shaped planar lightguide solar concentrator design can achieve 2000x-, 735x- and 240x- geometrical concentration ratios with high optical efficiency of 96.80%, 97.93% and 98.99%, respectively, while their dimple structures have different length-width ratios: *r* = 25, *r* = 15 and *r* = 10, respectively. The minor energy loss is introduced by the intrinsic material absorption of the lightguide and microlens array. No guiding ray leakage happened in the lightguide.

This paper is organized as follows. Section 2 describes the design concept of the sawtooth-shaped planar lightguide solar concentrator. Section 3 details the solar concentrator design approach. Section 4 shows the numerical results of the optical properties of the proposed sawtooth-shaped solar concentrator and its comparison to Unger’s dimpled planar lightguide solar concentrator [1]. Finally, section 5 gives a brief summary of this study, discusses some concerns in regard to making a sawtooth-shaped planar lightguide solar concentrator, and provides a comparison between our concentrator designs and another ray-leakage-free planar solar concentrator.

## 2. Solar concentrator structure

Figure 1 illustrates the structure of the proposed sawtooth-shaped planar lightguide solar concentrator. The concentrator combines the Unger’s dimpled planar lightguide solar concentrators (on the left) [1] with a prism array dimpled planar lightguide solar concentrator (on the right). As the common design of a planar lightguide solar concentrator, the concentrator consists of three parts: a lens array, a secondary optical element and the lightguide itself. In this design, sunlight is collected by microlenses in a two-dimensional lens array and is coupled into the planar lightguide using localized secondary optical elements placed at each lens focus. The structure of the secondary optical element used in this design is Unger’s smart dimple structure [1]. Especially, this design modifies the lightguide’s one-side boundary using a sawtooth shape. The simpler modification substantially advances the planar lightguide property. Section 2.1 briefly reviews the smart dimpled structure function, and section 2.2 explains how a sawtooth-shaped lightguide boundary prevents guiding ray leakages. Section 2.3 explains why this design chooses a compound lightguide boundary. Referring to Fig. 1, this paper assumes that sunlight coming from the *z*-axis is vertically incident on the solar concentrator. The design goal is to collect and guide sunlight along the *y*-axis to the lightguide exit port where the solar cell is situated without any ray leakages. The wording “ray leakages” means only lightguide energy loss from the guiding rays which breaks the TIR-condition. In accordance with the arrangement of the proposed solar concentrator, as Fig. 1 shows, two important properties of the sawtooth-shaped planar lightguide solar concentrators, *geometric concentration ratio**C*_{geo} and *optical efficiency**η*, are given by:

#### 2.1 Dimpled air spaces

The secondary optical element used in this design is several micro air spaces situated at the bottom of the lightguide. The structure of the micro air spaces is Unger’s smart dimpled structure [1]. As Fig. 2 shows, the dimple structure comprises two air prisms, i.e. ** injection element** and

**. The length, width and depth of the dimple structure are denoted by symbols**

*bypass element**dy*,

*dx*, and

*dx*, respectively. This design takes the interior angle between the injection element tilted surface and x-y plane as 45 degrees, so the dimple width and depth are equal. The small

*expanding angle*of the bypass element is denoted by

*α*, where

*α*is related to the dimple

*length-width ratio*,

*r*=

*dy*/

*dx*, by

*α*= 2tan

^{−1}(1/2

*r*). As Fig. 2(b) shows, sunlight is collected by a lens array and is focused at the center of the tilted surface of the injection element. When the conical focusing sunlight from each microlens reaches the tilted surface of the injection element, the conical sunlight is redirected in the direction of the lightguide exit port along

*y*axis. The bypass element of the subsequent dimple structure prevents guiding ray leakages from the subsequent injection element scattering. In a perfect situation, the side surface of all the dimple structures and the lightguide itself should guide focusing sunlight to the lightguide exit port by TIR without any ray leakage.

#### 2.2 Preventing guiding ray leakages by the sawtooth-shaped lightguide boundary

To prevent guiding ray leakages, it is important to make sure that the incident angles of all guiding rays at the lightguide side surface and all dimple side surfaces exceed the *critical angle*. That is, the TIR condition is maintained until a guiding ray reaches the PV cell. Figure 3(a) illustrates the guiding ray paths in *Unger’s dimpled planar lightguide* [1]. The guiding ray will gradually leak out from the lightguide owing to the ray-incident-angle decreasing after several reflections. Referring to Fig. 1 and Fig. 3(b), to prevent the decreasing in incident angle of guiding rays, i.e. ray leakages, this design lets the right-side lightguide boundary be a sawtooth shape. The interior angles of each sharp tooth of the sawtooth boundary are the same as the *expanding angle* of all bypass elements. The side surfaces of the sawtooth boundary correct the change in ray-incident angle resulting from dimpled structure reflection. With the structural arrangement, the focusing conical sunlight from the microlens is directly redirected to the lightguide exit port. The usage of the sawtooth-shaped boundary of the planar lightguide substantially decreases both the number of guiding ray reflection times and the propagating distance of the guiding rays in the lightguide. Figures 4(a) and 4(b) plot the side view of the focusing ray paths from one microlens in Unger’s dimpled planar lightguide and the proposed sawtooth-shaped planar lightguide, respectively. Figure 4 shows that the design of the sawtooth boundary successfully prevents guiding ray leakages from the planar lightguide.

#### 2.3 Maximized solar concentrator’s geometrical concentration ratio

Referring to Fig. 5, we see that to maximize the solar concentrator’s geometrical concentration ratio, *C _{geo}*, the planar lightguide of the proposed solar concentrator is composed of two kinds of boundaries: the

*parallel part*and

*sawtooth part*. Considering the diffraction effect of a microlens, there is an intrinsic limitation on the dimple injection element sizes, which is related to the

*F*-number of microlenses. The Limitation restricts the length of the lightguide sawtooth part, and thus results in a limitation of

*C*. To maximize the

_{geo}*C*, the proposed planar concentrator includes a section of Unger’s dimpled planar lightguide of a length,

_{geo}*L*. After propagating through the planar lightguide parallel part of length

_{parallel}*L*, all guiding rays can still maintain a TIR condition without ray leakages. Section 3 will detail the calculations of parameter,

_{parallel}*L*. Adding the parallel part into the planar lightguide increases the light-collection area while maintaining the same optical efficiency, i.e. maximizes

_{parallel}*C*of solar concentrator while maintaining the same optical efficiency.

_{geo}## 3. Ray-leakage-free solar concentrator model

This section details approaches to find all parameters of the proposed ray-leakage-free sawtooth-shaped planar lightguide solar concentrator.

#### 3.1 Ray-leakage-free propagating length of incident rays in the meridional plane

This section finds the lightguide parallel part length, *L _{parallel}*, in which incident rays in the meridional plane can be guided without ray leakage. Here, the meridional plane means the y-z plane that contains the microlens optical axis. Figure 6 plots symbols used in this section. Referring to Fig. 6(a), the sunlight in the meridional plane coming from microlenses is redirected toward

*y*-axis by injection elements. After several reflections by the downstream dimple structures, the light rays are redirected in the direction with a

*leading angle δ*. Referring to Fig. 6(b), the

_{n}*leading angle δ*is the included angle between the guiding ray with y-z plane, where the suffix

_{n}*n*denotes the times that a guiding ray leaves from the dimple columns. The increase of the leading angle

*δ*of a guiding ray by multiple reflections is illustrated in Fig. 6(a), and the value of the leading angle

_{n}*δ*is given by:

_{n}*δ*of a guiding ray increases, twice the ray-traveling distance between the reflections by dimple structures will decrease, which is given by:where

_{n}*a*is the

*x*-distance between two columns of dimple structures, and Eqs. (3) and (4) hold while only the TIR condition is maintained.

Figure 6(b) depicts the relationship between the ray incident angle at the bypass element *ϕ _{in}* and the leading angle

*δ*, i.e. ${\varphi}_{in}=\pi /2-{\delta}_{n}-\alpha /2$. The TIR condition, ${\varphi}_{in}\ge {\theta}_{C}$, gives the relation of the

_{n}*maximum guiding ray reflection times N*with the

*critical angle θ*:

_{C}*leakage-free propagating length*,

*L*:However, the actual extended length of the lightguide parallel part,

_{l-f}*L*, is shorter than the leakage-free propagating length,

_{parallel}*L*. Referring to the green dotted rectangular area shown in Fig. 7, any reflected ray from bypass elements in this

_{l-f}*conjoint zone*(see dotted red arrow) will not fall on the sawtooth boundary. That is, the ray incident angle of all guiding rays from the

*conjoint zone*will be changed by the sequential dimple structures and parallel boundary. Thus,

*the length of the conjoint zone*,

*L*, should be included in the leakage-free propagating length,

_{overlapping}*L*. Simple geometrical calculations give the lengths of the conjoint zone, the sawtooth part and the parallel part, respectively. That is: and

_{l-f}The planar lightguide *overall length* is *L _{parallel}* +

*L*. However, it should be noted that Eqs. (4) to (10) assume that the

_{sawtooth}*x*-distance between twice the reflections from dimple structures is the same as the x-distance between two columns of dimpled structures

*a*. Actually, the value varies from (

*a*-

*dx*) to (

*a*+

*dx*). Replacing value

*a*of Eqs. (4) to (10) by (

*a*-

*dx*) gives a ray-leakage-free sawtooth-shaped planar lightguide solar concentrator with the shortest lightguide

*overall length*,

*L*+

_{parallel}*L*, i.e. gives the estimation of the smallest (or the worst) geometrical concentration ratio

_{sawtooth}*C*

_{geo}of solar concentrator.

#### 3.2 The increment of the leading angle of skewed incident rays on the sagittal plane

The “*skew incident rays*” in this paper denotes the focusing rays coming from microlenses with an oblique incident on the planar lightguide which were not contained in the meridional plane, y-z plane. The oblique incident skew rays result in the increase in the guiding ray incident angle at the first bypass element, and also, leading angle *δ _{n}*. This paper uses the symbol

*δ*’ with a comma superscript to denote

_{n}*the leading angle of skew incident rays*. Similar to the definition of

*n*in Eq. (3), here

*n*also denotes the times that the guiding ray leaves the dimple columns. Specifically, we use the symbol

*δ*

_{ini}to denote

*the initial leading angle of skew incident rays, i.e. δ*’

_{1}*= δ*

_{ini},

*i.e.,*the

*ray leading angle when it leaves the skew ray-incident dimple column*. Accordingly, the leading angle of a guiding skew ray that has left the neighbor column will be

*δ*’ =

_{2}*δ*+ α. The sequential skew ray leading angles

_{ini}*δ*’ will be:

_{n}*propagating length*,

*L*. The oblique incident skew rays result in the increase in the leading angle

_{l-f}*δ*’. Thus, in the lightguide parallel part, the skew incident ray will also be the crucial ray to be guided to the PV cell. In designing a leakage-free solar concentrator, this study uses this most crucial ray to find the lightguide parallel part length,

_{n}*L*. Referring to Fig. 8(a) regarding a focusing ray cone from a micro lens, the

_{parallel}*most crucial incident ray*to be guided in the lightguide is situated at the x-z plane that transverses in the lightguide ray-guiding direction which contains the microlens optical axis. This paper names the plane the

*sagittal plane*. The remainder of this section details how to find the initial leading angle

*δ*of the skew incident ray that is situated on the sagittal plane.

_{ini}Figure 8(b) plots the sagittal plane of the solar concentrator, which illustrates the parameters used in the following discussions. The *microlens’ thickness* is *d*_{1} and its *radius* is *ρ*. The *air space* between the microlens array and lightguide is *d*_{2} and half the *lightguide thickness* is *d*_{3}. The *refractive indexes* of the microlens array, air space and lightguide are denoted by symbols *n*_{1}, *n*_{0} and *n*_{2}, respectively. Symbols *θ*_{n} and *θ*_{n}’ denote the included angles between the surface normal with an incident ray and refracted ray, respectively. The *lateral distance of the ray intersect point* at each interface to the microlens optical axis is denoted by symbol *x _{n}*. The suffix of

*θ*

_{n},

*θ*

_{n}’ and

*x*indicates the interface where each interface is numbered in Fig. 8(b) by an orange rectangle:

_{n}The red arrows in Fig. 8 show the ray path of one focusing ray that is situated at the sagittal plane. The ray incident angle at the micro lens *θ* _{1} is related to the ray intersection’s lateral distance *x*_{1} and the microlens’ radius *ρ* by *θ* _{1} = sin^{−1}(*x*_{1}/ *ρ*). The refracted angle *θ* _{1}’ at interface 1 is easily calculated by Snell’s Law:

*exterior angle theorem*gives the incident angle at interface 2, i.e.

*θ*

_{2}=

*θ*

_{1}-

*θ*

_{1}’. With Snell’s Theorem, it is easy to find angles

*θ*

_{n}and

*θ*

_{n}’:

*d*

_{1}’ is the thickness of the lens at the ray intersect point at the microlens. The value is related to the microlens radius

*ρ*by:Referring to Figs. 8 and 9, the skew incident ray on the sagittal plane was finally reflected to the x-y plane by the injection element, where the included angle between the reflected guiding ray with y-z plane is denoted by the symbol Δ

*θ*. The value of Δ

*θ*will be the value of angle

*θ*

_{3}’:

Figure 9 illustrates the influence of two values, *x _{4}* and Δ

*θ*, on the guiding ray path in the planar lightguide of skew incident rays. As Fig. 9 shows, the guiding ray paths on the x-y plane are classified into two situations. In the first situation, the guiding ray forms the injection element that will pass in the space between the injection element and next bypass element (as shown by two green arrows in Fig. 9 plot). In this situation, the initial leading angle of these skew rays will be

*δ*= Δ

_{ini}*θ*. In the second situation, the guiding ray from the injection element will be reflected by the downstream bypass element before the guiding ray

*leaves the ray-incident dimple structure column*for the neighbor dimple column (as the red arrows in Fig. 9 indicate). In this situation, the initial leading angle of these skew rays will be

*δ*= Δ

_{ini}*θ + α*. The

*lateral ray intersect position at the injection elements*of the guiding rays

*x*and Δ

_{4}*θ*of the second situations satisfy the following condition:

#### 3.3 Finding the leakage-free sawtooth-shaped planar lightguide solar concentrator structure

This section addresses approaches to find a ray-leakage-free sawtooth-shaped planar lightguide solar concentrator’s parameters. Referring to Fig. 1, this study constructed the planar lightguide solar concentrator by combining several *square units* with the same lateral dimensions as microlens, *D*. Note, that this study lets the diameter of microlens *D* equal the dimple length *dy*, and lets the value of the lightguide thickness 2*d _{3}* equal the dimple depth

*dx*, to make a leakage-free sawtooth-shaped planar lightguide solar concentrator structure with the greatest geometrical concentration ratio

*C*. In real situations, the microlens diameter could be greater than the dimple length and the lightguide thickness could also be greater than the dimple depth. Besides, this study takes the lightguide thickness

_{geo}*dx*as the

*scaling variable*, since the geometrical concentration ratio

*C*

_{geo}in this design is related to the lightguide thickness.

The leakage-free sawtooth-shaped solar concentrator parameters are calculated by the following processes: First, with a given microlens diameter *D*, thickness *d*_{1}, index *n*_{1}, radius *ρ* (or focal length *f*) and dimple length-width ratio *r*, this study uses Eqs. (12) to (16) to find two values, *x _{4}* and Δ

*θ*, of all incident rays on the sagittal plane. The process finds the most critical ray to be guided by the planar lightguide, where the most crucial ray possesses the largest initial leading angle

*δ*. Second, with a given

_{ini}*x*-distance between two dimple columns

*a*, this study substitutes the

*δ*of the most crucial ray into Eq. (11) giving the largest leading angles

_{ini}*δ*’. Substituting

_{n}*δ*’ into Eqs. (9) and (10) gives us lightguide sawtooth part length,

_{n}*L*

_{sawtooth}and the estimation of the shortest leakage-free lightguide parallel part length,

*L*

_{parallel}. Third, dividing the solar concentrator’s overall length

*L*+

_{parallel}*L*by the lens diameter

_{sawtooth}*D*gives the overall longitudinal (i.e.

*y*-direction)

*square unit number*used in constructing the leakage-free solar concentrator, while there is no limits in the lateral (i.e.

*x*-direction)

*square unit number*. Finally, this study constructs the planar lightguide solar concentrator by combining multiple

*square units*of the estimated numbers.

In addition, the parameter-finding processes give the largest lateral ray intersect position at the injection elements of all incident rays, which is denoted by (*x*_{4})_{max}. The value provides the estimation of the minimum dimple width *dx*, and also, provides the limit of the lightguide thickness, dimple length-width ratio *r* and leakage-free lightguide geometrical concentration ratio *C _{geo}*.

## 4. Simulation results

This section provides a numerical simulation of the optical properties of the leakage-free sawtooth-shaped planar lightguide solar concentrator. All simulations in this paper were modeled by the commercial ray-tracing simulation package, LightTools [12]. First, we use the approaches addressed in section 3.3 to construct leakage-free solar concentrators. The solar concentrator parameters used in simulations are addressed as follows. This study constructs the planar lightguide solar concentrator by combining several square units with lateral dimensions of the microlenses, i.e. 5 mm × 5 mm. The material of the microlens and lightguide are both BK7, and the thickness of the microlens is 3 mm. Dimple structure of three different length-width ratios, of *r* = 25, *r* = 15, and *r* = 10, are constructed. Dividing the microlens diameter by the length-width ratio *r* gives the dimple structure thicknesses of three cases, i.e. 0.20 mm, 0.33 mm, and 0.50 mm, respectively. Since each injection element is situated at the focus of a microlens, the *x*-distance between two columns of dimpled structures *a* will be the same as the microlens diameter, naturally.

#### 4.1 Confirm ray-leakage-free property of the sawtooth-shaped planar lightguide solar concentrator

The major concern of this study is preventing the most crucial loss of the lightguide, i.e., the ray leakage from the planar lightguide. Thus, all simulation results shown in this section ignore the energy loss from the Fresnel loss at all surfaces and the material absorptions. This study includes discussions on the Fresnel loss and material absorptions in section 4.3. In all simulations of this section, sunlight of wavelength 0.55 μm is vertically incident on the solar concentrator. Moreover, to show the improvement of the proposed sawtooth-shaped concentrator on Unger’s dimpled lightguide concentrator [1], all simulation results shown in this sections include the results of Unger’s dimpled lightguide solar concentrator.

Two main optical parameters of the planar lightguide solar concentrators, *geometric concentration ratio C*_{geo} and *optical efficiency η* are calculated from simulations, where the definitions of two parameters are given by Eqs. (1) and (2). Figures 10(a) and 10(b) show the relation between the *C _{geo}* of solar concentrators with microlens focal lengths ranging from 24 mm to 50 mm, while the dimple structures possess three different length-width ratios,

*r*= 25,

*r*= 15, and

*r*= 10, respectively. The red-dotted line in Fig. 10 plots the smallest estimated

*C*

_{geo}for the ray-leakage-free sawtooth-shaped planar lightguide solar concentrators, i.e. calculating the leakage-free solar concentrator’s overall length by replacing value

*a*of Eqs. (4) to (10) by (

*a*-

*dx*). Since the real

*x*-distance between twice reflections from the two dimple column ranges between (

*a*-

*dx*) and (

*a*+

*dx*), the proposed ray-leakage-free sawtooth-shaped solar concentrators surely have a greater

*C*

_{geo}than estimated. This study finds the

*real ray-leakage-free overall length*of solar concentrators by adding more square units into the solar concentrators until the lightguide leaks any guiding rays. The

*real ray-leakage-free overall length*of solar concentrators is the length of total square units along y-axis before the ray leakage happens. The

*real C*

_{geo}of a ray-leakage-free sawtooth-shaped solar concentrator is calculated by dividing the

*real ray-leakage-free overall length*by the lightguide depth. The blue and black solid lines in Fig. 10 plot the real

*C*of the proposed sawtooth-shaped solar concentrators and Unger’s dimpled planar lightguide solar concentrators, respectively.

_{geo}Several things are illustrated by Fig. 10. First, it shows that the proposed dimpled planar lightguide solar concentrator achieves a higher *C _{geo}* than Unger’s dimpled planar lightguide solar concentrator in all situations. Second, using a dimple structure with a larger

*length-width ratio r*(i.e. a small expanding angle

*α*) can construct ray-leakage-free solar concentrators with a higher

*C*. At the same time, the increment in the

_{geo}*C*value compared with the previous Unger’s solar concentrators will also be more. Third, the focal length of the microlens directly influences the guiding ray leading angle

_{geo}*δ*, as to the geometrical concentration ratio

_{n}’*C*of a ray-leakage-free planar lightguide solar concentrator. Generally, increasing the focal length of the microlens increases the

_{geo}*C*of the ray-leakage-free planar lightguide solar concentrator (as Fig. 10(a) shows). It seems that Fig. 10(b) and Fig. 10(c) did not exhibit that effect; however, it did. The effect was not revealed in Fig. 10(b) and Fig. 10(c) because this study constructs a solar concentrator by combining the square unit of the same size one-by-one. Also, in Fig. 10(b) and Fig. 10(c), the increase in the solar concentrator overall length from the increase of the microlens focal length does not exceed the length of one square unit all the time. Fourth, some data points demonstrate unusually high

_{geo}*C*, i.e. (

_{geo}*r*= 25,

*f*= 30 mm) and (

*r*= 10,

*f*= 28 mm). We have closely checked the guiding ray tracks of these two situations further. Referring to Fig. 9; in these two situations, most of the reflected skew incident rays from the injection element pass through the space between the injection element and the next bypass element owing to the microlens aberration. Thus, the collected sunlight in these two situations can be guided by the planar lightguide in a longer length, which leads to the unusually high

*C*.

_{geo}As a comparison with Unger’s dimple lightguide solar concentrator, this study also discusses the non-ray-leakage-free situations of two kinds of solar concentrators. That is, this study simulates how the optical efficiency of these two kinds of solar concentrators decreases when the lightguide length exceeds their ray-leakage-free length. Figures 11(a), 11(b), and 11(c) show the relation between the optical efficiency with a *C _{geo}* of two kinds of solar concentrators when the dimple structures possess three different length-width ratios,

*r*= 25,

*r*= 15, and

*r*= 10, respectively. In this discussion, we choose the microlens focal lengths as 24 mm at random. The blue and red lines plot the situations of the proposed sawtooth-shaped solar concentrator and Unger’s dimpled solar concentrator, respectively. Numerical results show that while two kinds of solar concentrators were constructed with the same geometrical concentration ratio

*C*and with the same dimple structure, the proposed sawtooth-shaped solar concentrator can always achieve a better optical efficiency than Unger’s solar concentrator.

_{geo}#### 4.2 Concentration-acceptance product (CAP)

Geometric concentration ratio *C*_{geo} and acceptance angle *α* are two important characteristics of solar concentrators, where the acceptance angle *α* is defined as the ray-incident angle at which the optical efficiency of concentrator drops by 10%. It is known that these two parameters are not independent; thus, another parameter *CAP* (Concentration-acceptance product) is used to show the performance of concentrators [13]. The *CAP* value is defined as:

*CAP*value. Figure 12 shows the corresponding

*CAP*values of all ray-leakage-free sawtooth-shaped solar concentrators constructed in section 4.1. In Fig. 12, the black lines indicate the models of the ray-leakage-free sawtooth-shaped concentrators constructed in section 4.1, and the blue lines show the corresponding

*CAP*value of each concentrator. The

*CAP*values of all sawtooth-shaped concentrators are ranged between 0.093 and 0.189.

Figure 12 shows that using a lens array of a shorter focal length in the proposed sawtooth-shaped concentrator can make the proposed sawtooth-shaped concentrator a higher *CAP* value because, while using the microlens of a small focal length, the tilted incident ray will cause a less focal shift at the injection element. Also, a large focal shift at the injection element that departs from the injection element will lead to the decoupled loss (ray leakages). In the proposed sawtooth-shaped planar lightguide solar concentrator, the focal length of lens array is limited by the TIR-condition at the tilted surface of the injection element. The *CAP* value of the sawtooth-shaped concentrator can be further increased by using mirror coating at the tilted surface of the injection element, while also using a lens array of shorter focal length.

Figure 13 plots the curve “efficiency versus incident-ray tilted angle” of three ray-leakage-free planar lightguide solar concentrators that possess the largest acceptance angle *α*, while their dimple structures have different length-width ratios (*r* = 25, *r* = 15, and *r* = 10). The solid and dotted lines show the efficiencies of concentrators, while the incident rays are tilted about the y-axis and x-axis, respectively. Figure 13 shows that as the dimple length-width ratio *r* increases, the acceptance angle of the ray-leakage-free sawtooth-shaped solar concentrator decreases. The acceptance angle of three concentrators: (*r* = 10, *C*_{geo} = 240), (*r* = 15, *C*_{geo} = 735) and (*r* = 25, *C*_{geo} = 2000), are greater than ± 0.47°, ± 0.33° and ± 0.18°, respectively. Similarly, the acceptance angle of the sawtooth-shaped concentrator can be further increased by using mirror coating at the tilted surface of the injection element as well as using a lens array of shorter focal length.

#### 4.3 Optical efficiency and solar spectrum response

The major concern of this study is preventing the most crucial loss of the planar lightguide solar concentrator, i.e. the ray leakage from the planar lightguide. Thus, numerical simulations of section 4.1 and 4.2 considered only the energy loss from the ray leakage from the lightguide, without including the other two major kinds of energy losses, i.e. Fresnel loss at each interface and the material absorption. Figure 14 shows the real optical efficiency of the ray-leakage-free sawtooth-shaped planar lightguide solar concentrator, while including Fresnel loss and material absorption in simulations. In the simulations of Fig. 14, the lens array has MgF2 AR coating on both side, and both lens array and lightguide include the absorption coefficient of the BK7, 3 × 10^{−6} cm^{−1}. In Fig. 14, the black lines indicate the models of the ray-leakage-free sawtooth-shaped concentrators that were constructed in section 4.1. The blue and red lines show real optical efficiency of the sawtooth-shaped concentrators while the incident light are “vertically incident monochromatic light of wavelength 0.55 μm” and “light possessing ±0.26° divergence angle and the Air Mass 1.5 Solar Spectrum” [14], respectively.

The blue lines of Fig. 14 show the sum of the Fresnel loss and the material absorption of the sawtooth-shaped solar concentrator for a vertically incident monochromatic light. For vertically incident monochromatic light of wavelength 0.55 μm, the sawtooth-shaped solar concentrator can achieve concentration ratios: 2000x, 735x and 240x, without any ray leakage, while their dimple structures have different length-width ratios: *r* = 25, *r* = 15 and *r* = 10, respectively. The optical efficiencies of the three sawtooth-shaped concentrators are: 90.25%, 91.30% and 92.29%, respectively. The material absorption losses of the three sawtooth-shaped concentrators are only: 3.20%, 2.07% and 1.01%, respectively. The other energy losses are all coming from the Fresnel loss. The major Fresnel reflection loss occurs at the air/lightguide interface. Replacing the airspace between the lens array and the lightguide by a low-index guiding layer can reduce the Fresnel reflection loss at two air/material interfaces. The sum of the Fresnel reflection loss and material absorption of all sawtooth-shaped solar concentrators for a vertically incident light are all smaller than 10%, where the material absorption loss is around 1~3% and the Fresnel reflection loss is around 7%.

The red lines of Fig. 14 show the solar spectrum response of the sawtooth-shaped solar concentrator. The results show that for an on-axis incident sunlight (light possessing ± 0.26° divergence angle and the Air Mass 1.5 Solar Spectrum), the sawtooth-shaped solar concentrator can achieve concentration ratios: 2000x, 735x and 240x, and optical efficiencies: 69.6%, 85.9% and 90.7%, while their dimple structures have different length-width ratios: *r* = 25, *r* = 15, *r* = 10, respectively. The low optical efficiency of the (*r* = 25, C_{geo} = 2000) sawtooth-shaped concentrator results from its concentrator acceptance angle ( ± 0.18°) being smaller than the divergence angle of sunlight ( ± 0.26°). The sawtooth-shaped concentrators possessing length-width ratio between *r* = 25 and *r* = 15 can still achieve both high concentration ratio and high optical efficiency.

## 5. Discussions and summary

In summary, this study details the design of a ray-leakage-free sawtooth-shaped planar lightguide solar concentrator. In this design, the combination of a sawtooth-shaped lightguide boundary and the dimpled structure horizontally constructs a “step-like” structure; thus, the lightguide structure successfully prevents light leakages from the planer lightguide. Besides, the use of the sawtooth-shaped boundary of the planar lightguide substantially decreases both the guiding ray reflection times and the propagating distance of guiding rays in the lightguide. Thus, the proposed planar lightguide solar concentrator can achieve a higher light concentration ratio while maintaining high optical efficiency. Using the design approach shown in this paper, one can construct ray-leakage-free planar lightguide solar concentrators with a high geometrical concentration ratio, and without any ray-leakage energy from the lightguide. Numerical results of this study show that for vertically incident monochromatic light of wavelength 0.55 μm, the sawtooth-shaped solar concentrator can achieve concentration ratios of 2300x, 750x and 310x, without any ray leakage, while their dimple structures have different length-width ratios: *r* = 25, *r* = 15 and *r* = 10, respectively. The totals of the Fresnel reflection loss and material absorption of the sawtooth-shaped solar concentrator for a vertically incident light of all sawtooth-shaped concentrators are smaller than 10%. For an on-axis incident sunlight (light possessing ± 0.26° divergence angle and the Air Mass 1.5 Solar Spectrum), the sawtooth-shaped solar concentrator can achieve concentration ratios of 2000x, 735x and 240x, and optical efficiencies of 69.6%, 85.9% and 90.7%, while their dimple structures have different length-width ratios: *r* = 25, *r* = 15 and *r* = 10, respectively.

In comparison with Selimoglu’s ray-leakage-free planar lightguide solar concentrator (*C*_{geo} = 1000x, *α* = ± 1.0°, *η* = 96.5%) [10], the (*r* = 15, *f* = 24mm) sawtooth-shaped solar concentrator seems to possess worse characteristics (*C*_{geo} = 735x, *α* = ± 0.33°, *η* = 91.2%). However, the performance of the proposed sawtooth-shaped planar lightguide solar concentrator can easily be substantially improved. First, using secondary concentrator design (e.g. CPC) at the lightguide exit port can multiply the concentrator *C*_{geo} right away. Second, from the discussion in section 4.2, it is known that using mirror coating at the tilted surface of the injection element accompanying with using a lens array of shorter focal length can substantially increase the concentrator acceptance angle *α*. The achievement of the high acceptance angle of Selimoglu’s concentrator actually resulted from setting the reflective injection surfaces with “ideal reflectors”. Third, replacing the airspace between the lens array and lightguide by a low-index guiding layer can reduce the Fresnel reflection loss at two air/material interfaces.

The fabrication of the sawtooth-shaped dimpled lightguide should be the most important issue for the future realization of the sawtooth-shaped planar lightguide solar concentrators. Molding technique may be a practicable approach to make the sawtooth-shaped planar dimpled lightguide. Actually, the prototype of Unger’s dimpled lightguide has previously been successfully fabricated by molding [1]. For making a sawtooth-shaped dimpled lightguide by the molding technique, the material of the sawtooth-shaped dimpled lightguide should be chosen as polymer-like material. A manufacturer can make the mold containing dimple structures and sawtooth boundary first, and then make the sawtooth-shaped dimpled lightguide by injection molding [15, 16]. Concerning the fabrication of the sawtooth-shaped planar lightguide, two factors require attention. One is the “lightguide sidewall surface roughness”, and the other one is the “dimple acute angle tail.” In our lightguide structure design, the light ray is guided in the lightguide by TIR-reflection between the sidewall of lightguide boundary and the dimple structures. It should be noted that the “lightguide sidewall surface roughness” arising in making the dimple mold or in the injection molding process, may lead to an obvious decrease in the concentrator optical efficiency. Besides, the efficiency of the sawtooth-shaped concentrator is also closely related to the “dimple acute angle tail”. The acute tail of the dimple structure might be hard to make from the injection molding. Further simulations show that replacing the cross-section of dimple acute angle by a circular shape with the diameter of half-dimple width leads to about a 40% decrease in the concentrator optical efficiency. Except for the “sidewall surface roughness” and the “dimple acute angle tail”, other lightguide structure parameters (e.g., the tiled angle of the sidewall of both lightguide boundary and dimple structure, the acute tail of the sawtooth-shaped lightguide boundary, etc.) will not lead to large decrease in concentrator optical efficiency.

## Acknowledgment:

This work was supported by the Industrial Technology Research Institute and Advanced Optoelectronic Technology Center, National Cheng Kung University, under projects from the Ministry of Education and the National Science Council (NSC 99-2112-M-006-007-MY3) and (NSC 102-2112-M-006 −006 -MY3) of Taiwan.

## References and links

**1. **B. L. Unger, G. R. Schmidt, and D. T. Moore, “Dimpled planar lightguide solar concentrators,” in Proceedings of International Optical Design Conference, (OSA, 2010), ITuE5P. [CrossRef]

**2. **D. Feuermann and J. M. Gordon, “High-concentration photovoltaic designs based on miniature parabolic dishes,” Sol. Energy **70**(5), 423–430 (2001). [CrossRef]

**3. **Z. Hongfei, T. Tao, D. Jing, and K. Huifang, “Light tracing analysis of a new kind of trough solar concentrator,” Energy Convers. Manage. **52**(6), 2373–2377 (2011). [CrossRef]

**4. **V. Wittwer, W. Stahl, and A. Goetzberger, “Fluorescent Planar Concentrators,” Sol. Energy Mater. **11**(3), 187–197 (1984). [CrossRef]

**5. **J. C. Goldschmidt, M. Peters, A. Bosch, H. Helmers, F. Dimroth, S. W. Glunz, and G. Willeke, “Increasing the efficiency of fluorescent concentrator systems,” Sol. Energy Mater. Sol. Cells **93**(2), 176–182 (2009). [CrossRef]

**6. **J. H. Karp, E. J. Tremblay, J. M. Hallas, and J. E. Ford, “Orthogonal and secondary concentration in planar micro-optic solar collectors,” Opt. Express **19**(S4Suppl 4), A673–A685 (2011). [CrossRef] [PubMed]

**7. **J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express **18**(2), 1122–1133 (2010). [CrossRef] [PubMed]

**8. **M. Brown, D. Moore, G. Schmidt, and B. Unger, “Measurement and Characterization of Dimpled Planar Light Guide Prototypes,” in Proceedings of International Optical Design Conference, (OSA, 2010), JMB45P. [CrossRef]

**9. **D. Moore, G. R. Schmidt, and B. Unger, “Concentrated photovoltaic stepped planar light guide,” in Proceedings of International Optical Design Conference, (OSA, 2010), JMB46P. [CrossRef]

**10. **O. Selimoglu and R. Turan, “Exploration of the horizontally staggered light guides for high concentration CPV applications,” Opt. Express **20**(17), 19137–19147 (2012). [CrossRef] [PubMed]

**11. **S.-C. Chu, H.-Y. Wu, and H.-H. Lin, “Planar lightguide solar concentrator,” Proc. SPIE **8438**, 843810, 843810-7 (2012). [CrossRef]

**12. **Software LightTools, See website: http://www.opticalres.com/lt/ltprodds_f.html.

**13. **P. Benítez, J. C. Miñano, P. Zamora, R. Mohedano, A. Cvetkovic, M. Buljan, J. Chaves, and M. Hernández, “High performance Fresnel-based photovoltaic concentrator,” Opt. Express **18**(S1), A25–A40 (2010). [CrossRef]

**14. **See website: http://rredc.nrel.gov/solar/spectra/am1.5/

**15. **G. E. Engineering Thermoplastics Design Guide, See website: http://www.pdnotebook.com/wp-content/themes/thesis_18/custom/images/GE_plastic_design.pdf

**16. **Design Guidelines – DSM, See website: http://www.dsm.com/en_US/downloads/dep/designbroch05USweb.pdf