Calculation of the interaction between an overlapping spherical lens and a pin-type second optical element for spherical lens microtracking concentrator photovoltaic with a wide angle of incidence

Fossil fuels will be exhausted within 200 years; however, the Earth receives 170 billion MW of solar energy in every year. Photovoltaics can convert this enormous amount of solar energy into usable electrical energy. ¹⁾ Single-junction solar cells approach the Shockley–Kuisser limit of 32%–33%, but multijunction solar cells can exceed this limit because of the stacking of photovoltaic materials with an appropriate bandgap. ²⁾ Concentrator photovoltaics (CPVs) increase the conversion efficiency by several hundred times with a metal grid geometry ³⁾ and reduce the cost of smaller solar cells. A cell efficiency of 47.1% has been achieved. ⁴⁾ Using lens focusing and field measurements, a module efficiency of 36.7% has been achieved. ⁵⁾ High concentrations can achieve a high conversion efficiency.

However, three problems are associated with high-concentrator photovoltaics: (i) High-concentrator photovoltaics can only convert direct light, (ii) solar tracking is required to concentrate the light, and (iii) an acceptably wide angle of incidence (AOI) is required. Two solutions have been proposed to solve this problem, in which only direct light can be converted: (1) diffuse light, which is not direct light, is transmitted to the concentrator photovoltaic (transmitted CPV) and used for other purposes, such as agriculture. ⁶⁾ (2) Hybrid multijunction solar cells (hybrid CPV) are hybrids of multi-junction solar cells with concentrator and Si solar cells without concentrator. Direct light is concentrated and converted using the multijunction solar cells. Diffuse light is converted using Si solar cells. ⁷⁾ Solutions using (1) transmitted CPVs and (2) hybrid-type CPVs require a structure in which light travels from top to bottom within the CPV.

Conventional solar-tracking systems are large and heavy. Several microtracking CPV (MTCPV) systems have been proposed to address this significant problem. ⁸⁾ Previous MTCPV studies have investigated the following: (A) waveguide and lens structures on top of multijunction solar cells; (B) upper and lower mirrors on a multijunction solar cell structure; ^9,10) and (C) a three-dimensional controlled structure of a solar cell stage and an acceptably wide AOI lens.

Type (A) structures were studied in the following ways: (a) modifying the lower waveguide shape (e.g. horizontally tapered waveguide, ¹¹⁾ branched planar waveguide, ¹²⁾ horizontally staggered light guide, ¹³⁾ lens-channel waveguide, ¹⁴⁾ staggered-tapered light guide ¹⁵⁾); (b) modifying the waveguide reflector (e.g. small lateral shifts of the waveguide, ¹⁶⁾ horizontally spectrally separated waveguides ¹⁷⁾); (c) combining an external seasonal tracker; ¹⁸⁾ (d) adding optical elements between the lens and the waveguide (e.g. bio-inspired curved light guide ¹⁹⁾); and (e) adding two-lens structures in the upper structure of a multijunction solar cell. ^20,21) As light travels from top to bottom in the (A)-type tracker, this tracker can be applied as (1) a transmitted CPV and (2) a hybrid-type CPV. However, the AOI is restricted by the waveguide and lens conditions.

Type (B) structures have an acceptably wide AOI. However, this type of structure cannot be applied as a (1) transmitted CPV or (2) hybrid-type CPV because it uses an undermirror and light does not travel from top to bottom.

Type (C) structures use acceptably wide AOI lenses and movement mechanisms (e.g. horizontal and vertical displacement, ²²⁾ molded glass array lens, ²³⁾ ball guide support microtracking ²⁴⁾). Theoretically, a spherical gradient index lens ²⁵⁾ and solar cell stage ²⁶⁾ have been proposed. Hybrid-type CPVs ²⁷⁾ and transmitted CPVs ²⁸⁾ have been proposed because of their structure, in which light travels from top to bottom. However, the AOI is limited by the interference between the spherical lens and the solar cells.

In this study, a structure consisting of a spherical lens MTCPV (SMTCPV) and pin-type second optical element (SOE) was adopted as Type (C). The spherical lens had a focal point from any AOI without rotation ²⁹⁾ and an acceptably wide AOI lens. Our SMTCPV is applicable to transmitted and hybrid CPVs, where light travels from top to bottom. Solar tracking can only be performed with horizontal and vertical movements without rotation using a spherical lens. The overlap radius of the spherical lenses can be adjusted to achieve a wider maximum AOI and higher optical efficiency. The distance between the spherical lenses was varied and analyzed to resolve the interaction between the spherical lenses and the pin-type SOEs.

Figures 1(a) and 1(b) show the light incident on the SMTCPV without pin-type SOEs in the vertical direction (AOI = 0°) and inclined direction, respectively. Figures 1(c) and 1(d) show light incident on the SMTCPV with pin-type SOEs in the vertical direction (AOI = 0°) and inclined direction, respectively. A spherical lens has a focal point at all AOIs. The solar cells were placed at the focal point below the lens when the Sun was directly above it (AOI = 0°). When light was incident from an inclined direction, as shown in Fig. 1(b), the lens moved vertically and the solar cells moved horizontally, placing the solar cells at the focal point. These relative movements enabled the SMTCPV to perform solar tracking without lens rotation. However, interference between the spherical lens and the solar cell stage restricts the maximum AOI ${\theta }_{\max {AOI}}.$ We added pin-type SOEs to the SMTCPV as shown in Figs. 1(c) and 1(d). Solar cells were placed at the bottom of the SOEs. The SOEs were placed at the focal point below the lens when the Sun was directly above the lens (AOI = 0°). When light was incident from an inclination angle, the cones of the SOEs were placed at a focal point. In particular, when the tilt angle was large, the upper SOEs were positioned between the lenses.

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Two methods were proposed for three-dimensional (3D) mechanical control. (1) The solar cell stage moves laterally (XY-direction) and the spherical lens array moves vertically (Z-direction). ²⁹⁾ (2) Long-hole guides/pins on gears enable the lateral (XY-direction) movement of solar cells, whereas ball feet/guides passively enable the vertical (Z-direction) movement of solar cells. ²⁴⁾

Thus, even when the focal point is between the spherical lenses, the maximum AOI can be increased by guiding Sunlight to the solar cells via the pin-type SOEs.

Figure 1(e) shows the top view of the lens array showing the closest packed state. Figure 1(f) shows the top view of the lens array with less overlap. Figure 1(g) shows a side view of the lens array showing the closest packed state. Figure 1(h) shows a side view of the lens array with minimal overlap. An overlap radius ${r}_{\mathrm{over}}$ was introduced to explain the overlap of the spherical lenses. The distance between the center of the spherical lens and the adjacent center is the overlap radius ${r}_{\mathrm{over}},$ normalized by the spherical lens radius. $\phi$ is the director angle. $\phi =0^\circ$ is the direction from the spherical lens center to the adjacent spherical lens center, and $\phi =30^\circ$ is the direction from the spherical lens center to the gap between the three spherical lenses. ${A}_{\mathrm{gap}}$ is the gap between the three spherical lenses. The overlap radius at the closest packing is ${r}_{\mathrm{over}}=\sqrt{3}$ in Figs. 1(e) and 1(g), and the nonoverlapping but contacting condition is ${r}_{\mathrm{over}}=2.$ The overlap radii in Figs. 1(g) and 1(h) are ${r}_{\mathrm{over}}=1.9.$ Figure 1(g) shows that the maximum AOI is reduced when the overlap radius is small because pin-type SOEs are not inserted between the spherical lenses. However, as shown in Fig. 1(h), with an overlap radius ${r}_{\mathrm{over}}=1.9,$ pin-type SOEs can be inserted between spherical lenses, resulting in fewer overlapping spherical lenses and a wider maximum AOI.

Figure 2(a) illustrates the spherical lens and the upper part of the pin-type SOE at the top of the simulation model. Table I presents the simulation conditions. A spherical lens with a lens radius ${r}_{\mathrm{lens}}=5$ mm is on the upper side, and a receiver and SOE are on the lower side. The SOE is cone-shaped with a height ${h}_{\mathrm{soe}}=0.75$ mm and the receiver bottom radius is ${r}_{\mathrm{rec}}=0.5$ mm. The material used is high RI epoxy considering a previous study. ³⁰⁾ The receiver varied the focusing distance along the z-axis, the x-axis was perpendicular to the focusing distance direction, and the rotation angle was perpendicular to the y-axis. A 3D ray-tracing analysis was performed using the commercial software from BestMedia Inc. The 3D ray-tracing analysis was performed by considering the Fresnel reflection losses and neglecting the volume absorption of the lens.

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Table I. Simulation conditions I.

Spherical lens RI (PMMA)	1.491 @ 587.6 nm 31)
SOE RI (Epoxy)	1.589 @ 589 nm 32)
Lens radius rlens	5 mm
Receiver bottom radius rrec	0.5 mm
Receiver bottom shape	Rectangle 1 × 1 mm
Focusing distance f	5–8 mm
SOE height hsoe	0.75 mm, no SOE
AOI ${\theta }_{\mathrm{AOI}}$	0°–90°
Geometrical concentration ratio C	78.5×
Ray spectrum	750–949 nm (middle junction wavelength of the triple-junction solar cells)

As pin-type SOEs are inserted between spherical lenses, SMTCPVs with pin-type SOEs are preferred because of their short focal distances. However, the spherical lens RI should consider the interference between the spherical lenses and the pin-type SOEs. In other simulations, The spherical lens in $\mathrm{RI}=1.49$ which similar to RI of polymethyl methacrylate (PMMA) has the highest optical efficiency at $5.8$ mm which means ${r}_{\mathrm{over}}+{h}_{\mathrm{soe}}.$ Thus the PMMA lens is not interfere with the SOE. Therefore, PMMA was used for the spherical lens.

Figure 2(b) illustrates the thickness ${t}_{\mathrm{soe}}$ of the cone SOE. The cone SOE has two ends: the cone SOE top end ${P}_{\mathrm{top}}$ and the cone SOE bottom end ${P}_{\mathrm{bottom}}.$ The top end of the cone SOE is rotated from the receiver center as follows: ${t}_{\mathrm{soe}-\mathrm{top}}={h}_{\mathrm{soe}}\cos {\theta }_{\mathrm{AOI}}.$ The bottom end of the cone SOE is rotated from the receiver center as follows: ${t}_{\mathrm{soe}-\mathrm{bottom}}={r}_{\mathrm{rec}}\sin {\theta }_{\mathrm{AOI}}.$ The larger value between ${t}_{\mathrm{soe}-\mathrm{top}}$ and ${t}_{\mathrm{soe}-\mathrm{bottom}}$ is the thickness of the cone SOE ${t}_{\mathrm{soe}}.$ Figure 2(c) shows the relationship between ${t}_{\mathrm{soe}}$ and ${\theta }_{\mathrm{AOI}},$ with ${h}_{\mathrm{soe}}=0.75$ mm and ${r}_{\mathrm{rec}}=0.5$ mm.

Figure 3 shows the optical efficiency maps with a cone SOE with ${h}_{\mathrm{soe}}=0.75$ mm and without an SOE at each AOI; Z = 5 mm indicates that the receiver is placed on the surface of a spherical lens. The yellow line indicates the closest line that allows a cone SOE considering the cone SOE thickness ${t}_{\mathrm{soe}}$ as shown in Fig. 2(c).

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Table II lists the optical efficiency of the point with the highest optical efficiency with an SOE and the optical efficiency of the same point without an SOE. The optical efficiency with an SOE is higher than that without an SOE; for $\mathrm{AOI}=50^\circ$ and $60^\circ ,$ as the AOI is increased, the optical efficiency with the SOE is higher than that without the SOE. As shown in the optical efficiency map in Fig. 3, the white area, which indicates a high optical efficiency, is reduced under high-AOI conditions without the SOE.

Figure 4 shows the ray distribution points with and without the SOE in Table II. Figures 4(a) and 4(b) show the ray distributions at $\mathrm{AOI}=0^\circ .$ Figures 4(c) and 4(d) shows the ray distributions at $\mathrm{AOI}=30^\circ .$ Figures 4(e) and 4(f) show the ray distributions at $\mathrm{AOI}=60^\circ .$ Figures 4(a), 4(c), and 4(e) show the ray distributions with the SOE. Figures 4(b), 4(d), and 4(f) shows the ray distributions without the SOE. Figures 4(a) and 4(b) show the circular ray distributions. Figure 4(d) shows the ray distribution at $\mathrm{AOI}=30^\circ$ without the SOE. The ray distribution is elliptical owing to spherical aberration. Figure 4(c) shows the ray distribution at $\mathrm{AOI}=30^\circ$ with the SOE. The ray distribution is also elliptical, and the light ray distribution is caused by reflections from the SOE walls. Figure 4(f) shows the ray distribution for $\mathrm{AOI}=60^\circ$ without the SOE. The optical efficiency was reduced because the elliptical rays exceeded the measurement area. Figure 4(e) shows the ray distribution at $\mathrm{AOI}=60^\circ$ with the SOE. The number of reflected rays increased, covering the area. Hence, the optical efficiency increased compared with that without the SOE. Simulations were performed at the bottom of the cone SOE. When a pole was added, the ray was dispersed at the bottom of the pole.

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Table III lists the simulation condition considering the light ray spectrum of the multijunction solar cell. The energy generation by multijunction solar cells requires the consideration of the spectrum match between each subcell. The optical efficiency maps of each subcell at $\mathrm{AOI}=30^\circ$ and $60^\circ$ are shown in Fig. 5. Figures 5(a) and 5(b) show the optical efficiency maps of the top subcell. Figures 5(c) and 5(d) show the optical efficiency maps of the middle subcell. Figures 5(e) and 5(f) shows the optical efficiency maps of the bottom subcell. Figures 5(a), 5(c), and 5(e) show the optical efficiency maps for $\mathrm{AOI}=30^\circ .$ Figures 5(b), 5(d), and 5(f) show the optical efficiency maps for $\mathrm{AOI}=60^\circ .$ Table IV presents the optical efficiency at the point with the highest optical efficiency for the middle subcell. Figure 5 shows that the focusing distance difference between the top, middle, bottom subcells is less than 0.2 mm. Table IV shows that the difference in the optical efficiency at the highest optical efficiency point in middle sub-cell is less than 1%. In this study, spherical lenses were used and the focusing distance of the spherical lenses was short. Therefore, the effect of chromatic aberration was smaller.

Table III. Simulation conditions II.

Spherical lens RI (PMMA)	1.491 @ 587.6 nm 31)
SOE RI (Epoxy)	1.589 @ 589 nm 32)
Lens radius rlens	5 mm
Receiver bottom radius rrec	0.5 mm
Receiver bottom shape	Rectangle 1 × 1 mm
Focusing distance f	5–8 mm
SOE height hsoe	0.75 mm
AOI ${\theta }_{\mathrm{AOI}}$	0°–90°
Geometrical concentration ratio C	78.5×
Ray spectrum	280–749 nm (top junction)
	750–949 nm (middle junction)
	950–2000 nm (bottom junction)
	(In triple-junction solar cells)

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We introduce numerical equations for the maximum AOI. Figures 6(a)–6(c) show a schematic of a spherical lens and a pin-type SOE when expanded to an angle other than $\phi =0^\circ .$ Figure 6(a) presents an overview. Figure 6(b) illustrates the angular schematic in the horizontal plane through the center ${P}_{S1}$ of spherical lens S1 and the center ${P}_{S2}$ of spherical lens S2. Figure 6(c) shows the angular schematic in the vertical plane between ${P}_{S1}$ and the $\phi$ direction. The center of the spherical lens S2, when cut in the $\phi$ direction plane, is ${P}_{S2-\mathrm{slice}}.$ The slice overlap radius of S2 is ${r}_{\mathrm{over}-\mathrm{slice}}.$ The distance ${r}_{\mathrm{over}-\mathrm{slice}}$ between ${P}_{S2}$ and ${P}_{S2-\mathrm{slice}}$ is given by

$$\begin{eqnarray}&&{r}_{\mathrm{over}-\mathrm{slice}}={r}_{\mathrm{over}}\cos \left(\phi \right).\end{eqnarray} \tag{ 1 }$$

The vertical distance ${d}_{v-\mathrm{slice}}$ from ${P}_{S2}$ to ${P}_{S2-\mathrm{slice}}$ is given by

$$\begin{eqnarray}&&{d}_{v\mathrm{\_slice}}={r}_{\mathrm{over}}\sin \left(\phi \right).\end{eqnarray} \tag{ 2 }$$

The sliced circle radius ${r}_{S2-\mathrm{slice}}$ is the radius cut in the $\phi$ direction of the spherical lens S2, and is given as follows. The radius of the spherical lens $r$ was 1.

$$\begin{eqnarray}&&{r}_{S2\mathrm{\_slice}}=\sqrt{{r}^{2}-{\left({r}_{\mathrm{over}}\sin \phi \right)}^{2}}=\sqrt{1-{\left({r}_{\mathrm{over}}\sin \phi \right)}^{2}}\end{eqnarray} \tag{ 3 }$$

[In Fig. 6(c), the focusing distance projection, where ${f}_{\mathrm{project}}$ is projected onto the line connecting ${P}_{S1}$ and ${P}_{S2-\mathrm{slice}}$ is as follows.

$$\begin{eqnarray}&&{f}_{\mathrm{project}}={r}_{\mathrm{over}-\mathrm{slice}}\displaystyle \frac{{f}_{n}}{{f}_{n}+{r}_{S2-\mathrm{slice}}}\end{eqnarray} \tag{ 4 }$$

These formulas use the normalized focal length ${f}_{n},$ which is the focal length $f$ normalized by the lens radius $r.$ The complementary maximum AOI ${\theta }_{\mathrm{maxAOI}-c}$ of the sliced circle cut in the $\phi$ direction is given by

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}{\theta }_{\max \,\mathrm{AOI}-c} & = & {\cos }^{-1}\left(\displaystyle \frac{{f}_{\mathrm{project}}}{{f}_{n}}\right)={\cos }^{-1}\left(\displaystyle \frac{{r}_{\mathrm{over}-\mathrm{slice}}\tfrac{{f}_{n}}{{f}_{n}+{r}_{S2-\mathrm{slice}}}}{{f}_{n}}\right)\\ & = & {\cos }^{-1}\left(\displaystyle \frac{{r}_{\mathrm{over}-\mathrm{slice}}}{{f}_{n}+{r}_{S2-\mathrm{slice}}}\right)={\cos }^{-1}\left(\displaystyle \frac{{r}_{\mathrm{over}}\,\cos \,\phi }{{f}_{n}+\sqrt{1-{\left({r}_{\mathrm{over}}\,\sin \,\phi \right)}^{2}}}\right).\end{array}\end{eqnarray} \tag{ 5 }$$

The maximum AOI ${\theta }_{\mathrm{maxAOI}}\,$ when the sliced circle is cut in the $\phi$ direction is given by

$$\begin{eqnarray}&&{\theta }_{\mathrm{maxAOI}}=\displaystyle \frac{\pi }{2}-{\cos }^{-1}\left(\displaystyle \frac{{r}_{\mathrm{over}}\cos \phi }{{f}_{n}+\sqrt{1-{\left({r}_{\mathrm{over}}\sin \phi \right)}^{2}}}\right).\end{eqnarray} \tag{ 6 }$$

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Figure 6(d) shows the relationship between the overlap radius and the maximum AOI at $\phi =0^\circ .$ The maximum AOI ${\theta }_{\mathrm{AOI}}$ was $51.9^\circ$ at the overlap radius ${r}_{\mathrm{over}}=\sqrt{3}$ at ${f}_{n}=1.2.$ However, at ${f}_{n}$ = 1.1, the maximum AOI was $55.6^\circ .$ The maximum AOI can be increased by shortening the focusing distance. When the spherical lens overlap was reduced and the overlap radius was set to ${r}_{\mathrm{over}}=1.9$ at ${f}_{n}$ = 1.1, the maximum AOI was $64.8^\circ .$ Figure 4(e) shows the maximum AOI ${\theta }_{\mathrm{maxAOI}}\,$ when the directional angle $\phi$ and the overlap radius ${r}_{\mathrm{over}}$ are varied. The closest focusing distance $f=5.75$ mm for a spherical lens radius $r=5$ mm and SOE thickness ${t}_{\mathrm{soe}}=0.75$ mm, which gives the closest normalized focusing distance ${f}_{n}=1.15.$ There was interference between the spherical lenses and the pin-type SOEs, limiting the maximum AOI with the closest packing to ${r}_{\mathrm{over}}=\sqrt{3}:$ ${\theta }_{\mathrm{maxAOI}}=53.6^\circ$ at $\phi =0^\circ$ and ${\theta }_{\mathrm{maxAOI}}=65.4^\circ$ at $\phi =30^\circ .$

The shorter the overlap between the spherical lenses, the larger the ${r}_{\mathrm{over}}$ and the wider the distance between the spherical lenses. A gap occurred between the three spherical lenses centered at $\phi =30^\circ .$ As the distance increased and gaps appeared, the maximum AOI also increased. At overlap radius ${r}_{\mathrm{over}}$ = 1.9, ${\theta }_{\mathrm{AOI}}=62.1^\circ$ at $\phi =0^\circ$ and ${\theta }_{\mathrm{maxAOI}}=90^\circ$ at $\phi =26^\circ .$ ${\theta }_{\mathrm{AOI}}=90^\circ$ indicates that a pin-type SOE can be inserted into the horizontal plane of the lens array. Therefore, the maximum AOI can be increased by reducing the spherical lens overlap and using a pin-type SOE. However, the shortening of the overlap of the

The gap area ${A}_{\mathrm{gap}}$ and cover area ratio ${A}_{\mathrm{cover}}$ in the horizontal plane of the lens array are illustrated in Figs. 4(a)–4(c).

Figure 7(a) shows a schematic of the three spherical lenses. The three spherical lenses are denoted as S1, S2, and S3. S1, S2, and S3 are named Si or Sj or Sk using iterators i = 1, 2, 3 or j = 1, 2, 3 or k = 1, 2, 3. The centers of these lenses are ${P}_{S1},$ ${P}_{S2},$ and ${P}_{S3},$ respectively. The horizontal plane of the lens array was rotated about the ${P}_{S1}-{P}_{S2}$ axis, and the state of the lens array was calculated with varying AOI.

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The distance between ${P}_{S1}$ and ${P}_{S2}$ is ${d}_{12},$ which is given as follows:

$$\begin{eqnarray}&&{d}_{12}={r}_{\mathrm{over}}.\end{eqnarray} \tag{ 7 }$$

The distance between ${P}_{S1}$ and ${P}_{S3}$ is ${d}_{13},$ and that between ${P}_{S2}$ and ${P}_{S3}$ is ${d}_{23},$ which are given as follows:

$$\begin{eqnarray}&&{d}_{13}={d}_{23}={r}_{\mathrm{over}}\cos {\theta }_{\mathrm{AOI}}.\end{eqnarray} \tag{ 8 }$$

The distance between other centers of these lense are named ${d}_{ij}$ using iterators i = 1, 2, 3 or j = 1, 2, 3. The same also applied to ${d}_{ik}.$ The Si full angle ${\theta }_{Si-\mathrm{full}}$ consists of the lines ${P}_{Si}-{P}_{Sj}$ and ${P}_{\mathrm{Si}}-{P}_{\mathrm{Sk}}$ as follows.

$$\begin{eqnarray}&&{\theta }_{Si-\mathrm{full}}={\cos }^{-1}\left(\displaystyle \frac{{d}_{ij}}{2{d}_{ik}}\right)\left(i,j,k=1,2,3\right).\end{eqnarray} \tag{ 9 }$$

The area of the triangle with ${P}_{S1}-{P}_{S2}-{P}_{S3}$ connected to ${A}_{\mathrm{out}}$ is expressed below.

$$\begin{eqnarray}&&{A}_{\mathrm{out}}=\displaystyle \frac{1}{2}{\rm{\cdot }}{d}_{12}{\rm{\cdot }}{d}_{13}{\rm{\cdot }}\sin {\theta }_{S1-\mathrm{full}}.\end{eqnarray} \tag{ 10 }$$

Figure 7(b) shows a schematic of the spherical lens S1 and the gap. This figure illustrates S1 as i = 1; however, S2 can also be discussed with i = 2 and S3 with i = 3. ${P}_{{Gij}}$ is the gap vertex point between the spherical ${S}_{i}$ and ${S}_{j}$ sides. The Si side angle ${\theta }_{{Sij}}$ consisting of the lines ${P}_{Si}-{P}_{Sj}$ and ${P}_{{Si}}-{P}_{{Gij}}$ is given as follows.

$$\begin{eqnarray}&&{\theta }_{{Sij}}={\cos }^{-1}\left(\frac{{d}_{{ij}}}{2r}\right)(i,j=1,2,3)\end{eqnarray} \tag{ 11 }$$

The gap target angle ${\theta }_{{SiG}}$ consists of ${P}_{{Si}}-{P}_{{Gij}}$ and ${P}_{{Si}}-{P}_{{Gik}}$ as follows.

$$\begin{eqnarray}&&{\theta }_{{SiG}}={\theta }_{{Si}-{full}}-{\theta }_{{Sij}}-{\theta }_{{Sik}}(i,j,k=1,2,3)\end{eqnarray} \tag{ 12 }$$

Figure 7(c) shows a schematic of the gap between the three spherical lenses. The gap outside angle ${\theta }_{G-12-\mathrm{out}}$ consists of ${P}_{{Gij}}-{P}_{{Si}}$ and ${P}_{{Gij}}-{P}_{{Sj}}$ as follows.

$$\begin{eqnarray}&&{\theta }_{G-12-\mathrm{out}}=2\pi -2{\theta }_{S12}\end{eqnarray} \tag{ 13 }$$

The gap side angle ${\theta }_{{Gi}}$ consists of ${P}_{{Gij}}-{P}_{{Si}}$ and ${P}_{{Gij}}-{P}_{{Gik}}$ as follows.

$$\begin{eqnarray}&&{\theta }_{{Gi}}=\frac{\pi }{2}-\frac{{\theta }_{{SiG}}}{2}(i=1,2,3)\end{eqnarray} \tag{ 14 }$$

The gap inside angle ${\theta }_{G-12-{out}}$ consists of ${P}_{{Gij}}-{P}_{{Si}}$ and ${P}_{{Gij}}-{P}_{{Sj}}$ as follows.

$$\begin{eqnarray}&&{\theta }_{G-12-\mathrm{in}}=\pi -{\theta }_{G-12-\mathrm{out}}-{\theta }_{G1}-{\theta }_{G2}\end{eqnarray} \tag{ 15 }$$

Next, we describe this area. Subtracting the three fan sections ${A}_{{Gi}}$ from the gap triangle area ${A}_{\mathrm{tri}},$ which consists of ${P}_{G12}-{P}_{G13}-{P}_{G23},$ yields ${A}_{\mathrm{gap}.}$

The fan section area ${A}_{{Gi}}$ is given as follows.

$$\begin{eqnarray}&&{A}_{{Gi}}=\pi {r}^{2}\frac{{\theta }_{{SiG}}}{2\pi }-2{r}^{2}\sin \left(\frac{1}{2}{\theta }_{{SiG}}\right)(i=1,2,3)\end{eqnarray} \tag{ 16 }$$

The gap triangle area ${A}_{\mathrm{tri}}$ is given as follows.

$$\begin{eqnarray}&&{A}_{\mathrm{tri}}=\displaystyle \frac{1}{2}{\rm{\cdot }}2r{\rm{\cdot }}\sin \left(\displaystyle \frac{1}{2}{\theta }_{S1G}\right){\rm{\cdot }}\sin \left(\displaystyle \frac{1}{2}{\theta }_{S2G}\right){\rm{\cdot }}\sin {\rm{}}({\theta }_{G-12-\mathrm{in}})\end{eqnarray} \tag{ 17 }$$

The gap area ${A}_{\mathrm{gap}}$ is given as follows.

$$\begin{eqnarray}&&{A}_{\mathrm{gap}}={A}_{\mathrm{tri}}-{A}_{G1}-{A}_{G2}-{A}_{G3}\end{eqnarray} \tag{ 18 }$$

The cover area ratio ${A}_{\mathrm{cover}}$ is given as follows.

$$\begin{eqnarray}&&{A}_{\mathrm{cover}}=\displaystyle \frac{{A}_{\mathrm{out}}-{A}_{\mathrm{gap}}}{{A}_{\mathrm{out}}}.\end{eqnarray} \tag{ 19 }$$

Figure 7(d) shows the cover area ratio ${A}_{\mathrm{cover}}$ for varying overlap radius ${r}_{\mathrm{over}}$ and AOI ${\theta }_{\mathrm{AOI}}.$ ${A}_{\mathrm{cover}}$ is smallest at ${r}_{\mathrm{over}}$ = 2 when the spherical lenses do not overlap. On the other hand, ${A}_{\mathrm{cover}}$ is 1 at ${r}_{\mathrm{over}}=\sqrt{3}$ when the lens is the closest packed.

The modified optical efficiency ${\eta }_{\mathrm{modify}}$ is calculated using the optical efficiency ${\eta }_{\mathrm{opt}}$ and cover area ratio ${A}_{\mathrm{cover}},$ as follows.

$$\begin{eqnarray}&&{\eta }_{\mathrm{modify}}={\eta }_{\mathrm{opt}}\times {A}_{\mathrm{cover}}\end{eqnarray} \tag{ 20 }$$

The optical efficiency ${\eta }_{\mathrm{opt}}$ is obtained from Table II. The cover area ratio ${A}_{\mathrm{cover}}$ was selected from the overlap radius ${r}_{\mathrm{over}}$ = 1.9 in Fig. 7(d).

The cover area ratio ${A}_{\mathrm{cover}}$ was obtained from Fig. 7(d) for the overlap radius ${r}_{\mathrm{over}}$ = 1.9.

Figure 8 shows the optical efficiency with varying AOI for the overlap radius ${r}_{\mathrm{over}}=1.9$ and ${h}_{\mathrm{soe}}=0.75.$ The optical efficiency was 59.8% at the lowest ${\theta }_{\mathrm{AOI}}=70^\circ ,$ indicating that the SMTCPV maintains a high efficiency over a wide range of AOI. Thus, a wide AOI ${\theta }_{\mathrm{AOI}}$ and high optical efficiency can be achieved by introducing pin-type SOEs and adjusting the overlap radius ${r}_{\mathrm{over}}.$ After preparing the suitable samples, the actual conversion efficiencies were measured.

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A spherical lens can focus light from any AOI to any point and can contribute to an SMTCPV system. We introduced a pin-type SOE and adjusted the spherical lens overlap radius to achieve a maximum AOI and high optical efficiency. Formulas were developed to calculate the maximum AOI considering the interference between the pin-type SOE and the solar cell stage. We also calculated the loss equation owing to the gap when the overlap radius of the spherical lens was varied to increase the maximum AOI. When the spherical lens overlap was adjusted to the overlap radius ${r}_{\mathrm{over}}$ = 1.9, the optical efficiency ${\eta }_{\mathrm{opt}}$ was maintained at 62.2% at ${\theta }_{\mathrm{AOI}}=70^\circ .$ Therefore, a larger maximum AOI and higher optical efficiency at wide angles can be achieved by introducing pin-type SOEs and varying the spherical lens overlap. In this study, the concentration ratio was maintained constant at 78.5 times. Previous studies have shown that core–shell spherical lenses have acceptable focusing distances and high optical efficiency. ²⁹⁾ Changing the light concentration ratio and using core–shell spherical lenses may improve the optical efficiency at all angles. In the future, all energy flow rates and conversion efficiencies will be calculated and realized as modules.

The authors would like to thank Professor Hideo Fujikake of Tohoku University for insightful discussions.