Design of a lens-to-channel waveguide system as a solar concentrator structure

Yuxiao Liu; Ran Huang; Christi K. Madsen

doi:10.1364/OE.22.00A198

1. Introduction

A solar concentrator typically uses a large area optical structure to focus sunlight onto a small area so that high output irradiance can be generated. Ever since the III-V multijunction solar cells were developed, intense investigation has been done regarding to photovoltaic concentrators (CPVs). A CPV projects the concentrated sunlight onto small photovoltaic (PV) cells to reduce the total cost by replacing expensive PV cells with cheap concentrator materials, as well as to get a very high efficiency. III–V multijunction PV cells with efficiencies in excess of 40% under concentrated sunlight have been reported [1]. The final goal of a CPV system is to provide sustainable, highly efficient and cost effective electricity.

A fundamental CPV design is to use a lens array, each concentrating the sunlight directly onto a separated PV cell. Such designs may bring non-uniformity, cooling and connection issues. A new approach of CPV systems was proposed by Karp et al. in 2010 for a planar concentrating structure [2], where those PV cells are replaced by a single slab waveguide. Sunlight is coupled into the waveguide by a series of microstructure at each focal point of the lenses. Then the coupled light travels inside the slab waveguide by total internal reflections (TIRs) and finally exits from both edges directly onto PV cells. 82% theoretical optical efficiency is achieved at 300x concentration under $0.26 °$ incidence angle [2]. Several other designs use similar ideas [3–7]. However, there is an inherent loss mechanism by which light already in the waveguide is decoupled by subsequent couplers, leading to decreased optical efficiency. This decoupling loss is inevitable and a higher concentration ratio or larger tolerance angle will further enhance the loss mechanism. The three-dimensional concentration from the lens array is also lost due to the large thickness of the slab waveguide. Arizono et al. proposed an alternative design where the input end is replaced by multiple branches [8,9]. Ray tracing results show 86.4% efficiency under 108x concentration. Their structure employs much more complex designs comparing to Karp’s due to the use of double TIR surfaces and therefore leads to reduced efficiency in fabrication. In this paper, we propose a novel lens-to-channel waveguide system as a solar concentrator structure. The lens and the tapered waveguide act as the primary and secondary concentrator, respectively. Couplers are placed only at one end of the waveguide to eliminate any decoupling loss in the waveguide. High concentration, efficiency and uniformity at the output end are realized by this simple structure. We study the design parameters and present a detail math model to explore the tradeoffs in this configuration.

2. The lens-to-channel waveguide system

Figure 1 illustrates the top view of the proposed lens-to-channel waveguide system. It consists of a $M \times N$ square lens array and a corresponding waveguide. Each lens has a coupler at its focal point redirecting the light into the waveguide. Lenses are tiled at an angle $Θ$ and the waveguide unit in each row becomes wider along the z-axis to avoid light in the waveguide hitting subsequent couplers. Light first travels inside each waveguide unit, then combines into a tapered common waveguide and finally exits from the end where it couples directly to a PV cell. Assuming the side length of each lens is $D$ with a spot size of $d$ , we design the coupler size $d_{C}$ to be the diameter of the spot size’s circumcircle in order to guide all the light into the waveguide. To achieve the maximum fill factor, each waveguide unit width $w_{N}$ is set as the product of $d_{C}$ and the number of the lenses in a row $N$ . We divide the structure into two parts. One is the fundamental design of waveguide units. Assuming the waveguide thickness $t$ , the geometric concentration ratio $C_{1}$ for this fundamental part is

C_{1} = \frac{M \times N \times D^{2}}{t \times M \times N \times d_{C}} = \frac{D^{2}}{\sqrt{2} d \times t} .

On the other hand, the tapered waveguide part has a tapering angle

σ

and length

l

. Its concentration ability is expressed as

C_{2} = \frac{W_{0}}{W_{0} - 2 l \tan σ} = \frac{1}{1 - 2 \frac{l}{W_{0}} \tan σ} ≜ \frac{1}{1 - 2 l_{N} \tan σ},

and the total concentration is

C = C_{1} \times C_{2}

.

l_{N}

is designated as the normalized waveguide length. As can be seen from section 3, the waveguide thickness

t

can be as small as

d_{C}

; thus

C_{1}

will equal to half of the concentration ability of the lens

C_{l}

. Therefore we can always achieve higher concentration ratio than previous designs due to the smaller waveguide thickness without any decoupling losses [2–7]. We explore in detail several important parameters of the lenses, couplers and waveguides. Our goal is to design a feasible, easily fabricated and efficient structure with high concentration.

Fig. 1 A top schematic view of the lens-waveguide system. Light collected by the lens array is redirected by the couplers placed at each focus. Then it travels inside the waveguide unit and finally exits from the tapered common waveguide directly onto a PV cell. The lens array and the tapered waveguide act as the primary and the secondary concentrators. Since lenses are tilted and the waveguide becomes wider along the z-axis, any decoupling loss is eliminated.

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3. Parameter designs and performance estimation

Assuming the lens array consists of only thin lenses (paraxial approximation), a spot size $d = 2 f \tan δ_{M}$ will be formed at each lens focal point if the incoming light field has a half angle $δ_{M}$ (Fig. 2). The geometric concentration ratio $C_{l}$ of the lens is

C_{l} = {(\frac{D}{d})}^{2} = \frac{1}{{[2 (f / D) \tan δ_{M}]}^{2}};

and the marginal ray angle

θ_{M}

after the lens,

θ_{M} = arc \tan [\tan δ_{M} + \frac{1}{2 (f / D)}] .

We define the f-number (

F / #

) as the ratio between the lens paraxial focal length

f

and the lens diameter

D

. The output of the lens array is therefore totally determined by

δ_{M}

and

F / #

. As previously stated,

C_{1} = 0.5 C_{l}

is the fundamental concentration factor of the structure.

Fig. 2 The aberration-free lens focuses incoming light onto its paraxial image plane.

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The main loss mechanisms in our structure include coupling loss, propagation loss and Fresnel reflection loss. We begin our discussion by evaluating the coupler design. An angled waveguide/air TIR coupler interface is placed at each focal point of the lens array, which acts as a turning mirror redirecting light after the lens into the waveguide. Since TIR is not wavelength sensitive, the coupler can cover a broad range of the solar spectrum. The coupling efficiency $η_{C}$ depends on the marginal ray angle $θ_{M}$ and the waveguide core material refractive index $n_{w}$ . Consider a beam of light $θ$ after the lens ( $- θ_{M} \leq θ \leq θ_{M}$ ). It will be first refracted at the air/cladding/waveguide interface before hitting the coupler. According to Snell’s law, $\sin θ = n_{w} \sin γ .$ We express the light ${\vec{k}}_{i} = k (\sin γ \cos Ω, \cos γ, \sin γ \sin Ω)$ using angle definitions in Fig. 3(a).Only light with incidence angle larger than the critical angle can be coupled into the waveguide. Meanwhile, the reflected light ${\vec{k}}_{r} = (k_{x 0}, k_{y 0}, k_{z 0})$ is

{\vec{k}}_{r} = k (\sin γ \cos Ω, \cos θ_{y} \sqrt{1 - \sin^{2} γ \cos^{2} Ω}, \sin θ_{y} \sqrt{1 - \sin^{2} γ \cos^{2} Ω}),

where

θ_{y} = 2 β - arc \tan (\tan γ \sin Ω)

. Since both

k_{y 0}

and

k_{z 0}

are a function of

2 β

,

β = 45 °

is a unique angle leading to a symmetrical profile as shown in Fig. 3(b). Both of the reflection angles in XZ and YZ planes reach the minimum value. We stress that any other angles will require a larger waveguide numerical aperture (NA) to confine the reflected light. Consequently a coupler angle

β = 45 °

is always desired. Assuming the change of the spot size in the waveguide material (comparing to air) is negligible, the minimum waveguide thickness is

t = d_{C}

.

Fig. 3 (a) Each coupler is a tilted waveguide/air interface. (b) An illustrative plot of the reflection angles in XZ plane $ϕ_{x 0} = arc \tan (k_{x 0} / k_{z 0})$ and YZ plane $ϕ_{y 0} = arc \tan (k_{y 0} / k_{z 0})$ . Each ellipse represents a reflection angle range for one particular coupler angle $β$ . It is clear that the $45 °$ coupler yields the minimum angles in both planes.

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We evaluate the waveguide propagation loss in the fundamental part by inspecting one waveguide unit (Fig. 4). Consider a light ray with an angle $δ$ entering at lens $P$ ( $- δ_{M} \leq δ \leq δ_{M}$ , $1 \leq P \leq N$ ). Assuming the reflected light is ${\vec{k}}_{r} = (k_{x 0}, k_{y 0}, k_{z 0})$ , the total distance for a specific light ray travels in the fundamental waveguide can be expressed as

L_{P} (P, Ω, δ) = \frac{[(P - 1) + 1 / 2] D \times \cos Θ}{k_{z 0} / | {\vec{k}}_{r} |} .

The propagation loss is estimated as the exponential decay of the distance multiplied by the waveguide material absorption coefficient

α

,

η_{P} = < \sum_{p} \int_{- δ_{M}}^{δ_{M}} \int_{0}^{2 π} \exp (- α L_{P}) d Ω d δ >

. After entering the tapered waveguide structure, however, the light

{\vec{k}}_{r} = (k_{x 0}, k_{y 0}, k_{z 0})

gains

2 σ

in XZ plane propagation angle every time it hits the sidewalls. In other words, the light angular space becomes larger as additional concentration is achieved. The total distance

L_{T}

in the tapered waveguide the light travels can be expressed as

L_{T} (Ω, δ) = W_{0} \times \frac{\cos (ϕ_{0} - σ) - \sqrt{\cos^{2} (ϕ_{0} - σ) - 4 l_{N} \tan σ + 4 l_{N}^{2} \tan^{2} σ}}{2 \sin σ \times (k_{x 0}^{2} + k_{z 0}^{2}) / {| k |}^{2}},

where

ϕ_{0}

is the reflected light initial angle in the XZ plane. Similarly, the propagation loss in the tapered waveguide is estimated as

η_{T} = 〈 \int_{- δ_{M}}^{δ_{M}} \int_{0}^{2 π} \exp (- α L_{T}) d Ω d δ 〉

. Combining all the losses discussed above, we express the total optical efficiency as

η = η_{R} \times η_{C} \times η_{P} \times η_{T},

where

η_{R}

represents the total Fresnel reflection losses.

Fig. 4 One waveguide unit cell of the concentrator.

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4. Simulation results and discussion

As an example, we simulate an 800x concentration system performance under different tolerance angles and f-numbers. The waveguide material is assumed to be F2 glass ( $n_{w} = 1.64$ , $〈 α 〉 = 1.8 \times 10^{- 4} c m^{- 1}$ ), with an air cladding and a silica substrate ( $n_{s} = 1.46$ , $n_{c} = 1$ ). Each of the lenses has a diameter of $1 m m$ . An index-matching detector space is assumed and Fresnel reflection is intentionally neglected. Figure 5(a) illustrates the optical efficiency dependence on f-numbers under different maximum incident angles. The left portion of the increasing efficiency is coupler TIR limited in that the marginal ray angle is large under small f-numbers and there is light with angles exceeding the critical angle at the coupler surface. The coupler surface is the limiting factor in these scenarios. If these surfaces are replaced by reflective metal layers, the limiting factor would become the waveguide/substrate interface, i.e. the waveguide NA, instead. The right portion of the decreasing efficiency, in contrast, is determined by the tapered waveguide concentration ability. Since larger f-numbers bring smaller marginal ray angles as well as small concentration, it requires more concentration from the tapered waveguide part. When it reaches the maximum waveguide concentration, the efficiency begins to drop. There exists an optimum f-number for a particular system. As can be read from Fig. 5(a), for all tolerance angles smaller than $0.55 °$ (note the solar angular width is $\pm 0.26 °$ ), there is no inherent loss in the structure, which is much better than the 60% (or 84% for the orthogonal structure) in Karp’s designs [2,3].

Fig. 5 (a) An exemplar plot of optical efficiency under different f-numbers and tolerance angles, where $α = 1.8 \times 10^{- 4} c m^{- 1}$ , $n_{w} = n_{d} = 1.64$ , $n_{s} = 1.46$ and $n_{c} = 1$ . (b) The performance of the waveguide concentrator. The efficiency remains the same until $C_{2} ~ 7.9$ .

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It is worth noting that the incident angle in the YZ plane $δ_{M Y Z}$ is always the limiting factor comparing to that in the XZ plane $δ_{M X Z}$ since the maximum incident angle at the coupler interface is determined by $δ_{M Y Z}$ (the normal of the coupler surface is in the YZ plane) and the critical angle at the waveguide/substrate interface is much smaller than those at other interfaces. It indicates the proposed structure is promising for one-axis tracking. Active tracking may be implemented for the YZ plane while passive tracking can be used for the XZ plane. The same tracking configuration using a micro-tracking lateral translation stage can be adopted from [10]. The detailed discussion is beyond the scope of this paper and will be presented in the future.

In order to explore the maximum concentration can be generated by the tapered waveguide, Fig. 5(b) shows the efficiency plot based on a given f-number ( $F / # = 3$ ) and incident angle ( $δ_{M} = \pm 0.7 °$ ). It turns out that the efficiency begins to drop at around 7.9x concentration, which is approaching to the 2D etendue limit 9.3x. The maximum waveguide concentration would be achieved when the output light angle is the critical angle $θ_{c}$ , which can be expressed as

C_{2 \max} = \frac{1}{1 - 2 l_{N} \tan σ} = \frac{\cos θ_{c}}{\sin (ϕ_{0 M} + σ)} \approx \frac{\cos θ_{c}}{\sin ϕ_{0 M}},

where

ϕ_{0 M}

is the maximum reflected half angle. Recall the ideal 2D concentration

C_{2 D i d e a l} = 1 / \sin ϕ_{0 M}

[11]. Equation (8) indicates that the tapered waveguide can be viewed as a near-ideal concentrator. If the critical angle is small, its concentration ability would approach to the 2D etendue limit.

Using Eqs. (3), (4) and (8), we can estimate the upper limit of the concentration ratio without considering any coupling losses. The two concentration stages, $C_{1 \max}$ and $C_{2 \max}$ can be respectively expressed as

C_{1 \max} = \frac{1}{2 {[2 (f / D) \tan δ_{M}]}^{2}},

C_{2 \max} \approx \frac{\cos θ_{c}}{\sin ϕ_{0 M}} \approx \frac{n_{w} \cos θ_{c}}{\sin {arc \tan [\tan δ_{M} + \frac{1}{2 (f / D)}]}} .

The total concentration

C_{\max} = C_{1 \max} \times C_{2 \max}

is therefore a function of

δ_{M}

and

F / # = f / D

. Given the waveguide material

n_{w}

, the maximum achievable concentration is calculated using Eqs. (9) and (10). An exemplar plot using parameters in Fig. 5(a) is shown in Fig. 6, which corresponds to the right part (waveguide limited) in Fig. 5(a). Note the small

F / #

region is hard to realize since its large angle will finally be limited by the TIR conditions at the coupler interface (left portion in Fig. 5(a)).

Fig. 6 An exemplar plot of the estimated maximum concentration as a function of f-numbers and tolerance angles, where $n_{w} = 1.64$ .

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We also perform a ray tracing simulation using ZEMAX to explore the practical performance of the system at 800x. An ideal blackbody source from 400nm to 1900nm at 5777K with $\pm 0.7 °$ incidence angle is set as the light source to simulate the incoming useful sunlight. The $3 \times 12, D = 1 m m$ lens array is made of BK7 glass and each lens is optimized to yield the minimum spot size. An anti-reflection layer MgF₂ is deposited on the top surface of the lens array. The input surface area is $3 m m * 12 m m$ , while the output area is $0.083 m m * 0.54 m m$ ( $C_{1} = 145 x, C_{2} = 5.6 x$ ). All the losses including scattering, Fresnel reflections and material absorptions are accounted for. Since the Fresnel reflection loss at the air/waveguide surface is a severe problem resulting from the big refractive index difference between F2 and air, a layer of PDMS is inserted between the waveguide and the lens array to eliminate the air gap. It acts as both an index matching medium and a supporting structure for the lens array. We simulate this optimized structure in ZEMAX. The Fresnel reflection loss is reduced while the propagation loss increases by ~2% due to the absorption peaks of PDMS [12]. A total optical efficiency of 89.1% is achieved for this 800x system with uniform output. Table 1 shows the loss chart of the optimized practical system.

Table 1. Efficiency Chart for the Optimized Practical Setup

View Table

5. Conclusion

We propose a lens-to-channel waveguide concentrator system which avoids any inherent decoupling loss in the waveguide. Detailed math models both for concentration calculation and optical efficiency estimation are provided using geometric optics analysis. Simulation results indicate that a coupler angle $β = 45 °$ is the best to achieve high efficiency as well as maximum waveguide concentration. ZEMAX simulations provide an optimized structure with an index matching layer, showing 89.1% optical efficiency at 800x for a practical setup.

References and links

1. A. Luque and S. Hegedus, Handbook of Photovoltaic Science and Engineering, 2nd ed. (John Wiley, 2011), Chap. 8.

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

3. 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(S4), A673–A685 (2011). [CrossRef] [PubMed]

4. W. C. Shieh and G. D. Su, “Compact solar concentrator designed by minilens and slab waveguide,” Proc. SPIE 8108, 81080H (2011). [CrossRef]

5. S. Bouchard and S. Thibault, “Planar waveguide concentrator used with a seasonal tracker,” Appl. Opt. 51(28), 6848–6854 (2012). [CrossRef] [PubMed]

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

7. J. H. Karp and J. E. Ford, “Planar micro-optic solar concentration using multiple imaging lenses into a common slab waveguide,” Proc. SPIE 7407, 74070D (2009). [CrossRef]

8. K. Arizono, R. Amano, Y. Okuda, and I. Fujieda, “A concentrator photovoltaic system based on branched planar waveguides,” Proc. SPIE 8468, 84680K (2012).

9. I. Fujieda, K. Arizono, and Y. Okuda, “Design considerations for a concentrator photovoltaic system based on a branched planar waveguide,” J. Photonics Energy 2(1), 021807 (2012). [CrossRef]

10. K. Baker, J. Karp, J. Hallas, and J. Ford, “Practical implementation of a planar micro-optic solar concentrator,” Proc. SPIE 8485, 848504 (2012).

11. W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, 1978), Chap. 2.

12. S. Kopetz, D. Cai, E. Rabe, and A. Neyer, “PDMS-based optical waveguide layer for integration in electrical–optical circuit boards,” AEU Int. J. Electron. Commun. 61(3), 163–167 (2007). [CrossRef]

Design of a lens-to-channel waveguide system as a solar concentrator structure

Abstract

1. Introduction

2. The lens-to-channel waveguide system

3. Parameter designs and performance estimation

4. Simulation results and discussion

5. Conclusion

References and links

Cited By

Figures (6)

Tables (1)

Equations (10)

Optics Express