## Abstract

During the past few years there has been a significant interest in spectrum splitting systems to increase the overall efficiency of photovoltaic solar energy systems. However, methods for comparing the performance of spectrum splitting systems and the effects of optical spectral filter design on system performance are not well developed. This paper addresses these two areas. The system conversion efficiency is examined in detail and the role of optical spectral filters with respect to the efficiency is developed. A new metric termed the Improvement over Best Bandgap is defined which expresses the efficiency gain of the spectrum splitting system with respect to a similar system that contains the highest constituent single bandgap photovoltaic cell. This parameter indicates the benefit of using the more complex spectrum splitting system with respect to a single bandgap photovoltaic system. Metrics are also provided to assess the performance of experimental spectral filters in different spectrum splitting configurations. The paper concludes by using the methodology to evaluate spectrum splitting systems with different filter configurations and indicates the overall efficiency improvement that is possible with ideal and experimental designs.

© 2014 Optical Society of America

## 1. Introduction

Considerable research is being devoted to increasing the conversion efficiency of photovoltaic (PV) systems. Increased efficiency will lower the cost per energy output of solar energy systems and make them a more viable renewable energy alternative. Shockley and Queisser have shown that systems based on single junction PV cells are limited to a system efficiency of 33% [1]. This restriction results from the mismatch between the photon energy of the incident solar illumination and the inability of a single junction device to optimally convert the broad incident spectrum.

One approach to overcome this difficulty is to incorporate multiple bandgap PV cells that are optimized to convert different parts of the incident spectrum to electrical power. Two approaches are being investigated to implement this concept: integrated multi-junction PV cells in tandem [2], and a spatially separated collection of single junction cells [3–5]. Tandem multi-junction cells must be made from similar materials that can be latticed matched and the cell interfaces. In addition the different junction cells are connected in series and therefore the output current is limited by the cell with the lowest output current [6]. Finally tandem cell systems are typically used at high efficiency to increase cell conversion efficiency and to reduce the cost of the system by incorporating a large collection optic with a lower cost/area than the multi-junction PV cell.

Spectrum splitting configurations distribute incident photons onto several single bandgap PV cells that are spatially separated [3]. This allows PV cells with dissimilar materials to be used and is not restricted by current matching effects. In addition the system can be operated at lower concentration ratios thereby requiring less stringent tracking and cooling techniques. However, the spectrum splitting optical system must be designed to spatially separate spectral bands that optimally match the spectral responsivity of the PV cells in the system. The optical system must also provide high transfer efficiency and negligible overlap between spectral bands.

In this paper, the conversion efficiency of each PV cell will be defined as a spectral quantity that is modified by the filter spectral transmittance (and different system configurations) and then integrated to determine the overall spectrum splitting system (SSS) optical to electrical conversion efficiency.

Although there have been different spectrum splitting geometries proposed to date, they have not been presented with a uniform and detailed evaluation method [4,5]. In particular, the overall advantage (or disadvantage) of using a particular SSS design has not been quantified equivalently across different systems [5]. In this paper, the important metrics for spectrum splitting systems are developed to provide an assessment of conversion efficiency, the influence of spectral filters on system performance, and guidance on system design. It is shown that careful consideration of the wavelength dependence of the spectral filter and PV cell must be made to achieve high system conversion efficiency. A metric that compares the efficiency of the spectrum splitting system with respect to a similar system with the highest efficiency single bandgap cell from the system is also introduced. This parameter called the Improvement over Best Bandgap (IoBB) provides an indication of the benefit of the added complexity of the spectrum splitting system with respect to a simpler single cell system. Metrics are also defined that indicate the relative performance of experimental to ideal spectral filters and their impact on overall system efficiency.

The properties of different types of filters for use in spectrum splitting systems are also examined. These include spectral band separation and dispersive optical filters. The effect of the linear dependence of the spectral responsivity of PV cells on the overlap characteristics of spectrum splitting filters is also examined and is found to limit system conversion efficiency.

The paper concludes with an evaluation of ideal and experimental filters as well as ideal and experimental PV cell characteristics for 4- and 2-cell spectrum splitting systems. The metrics illustrate the conversion efficiency that is possible with ideal components and the sensitivity to different component properties.

## 2. Efficiency metrics for spectrum splitting systems

#### 2.1 Spectral conversion efficiency (SCE)

The overall conversion efficiency of a PV cell or system is the ratio of electrical output power to the optical power of the incident solar illumination. For a single junction PV device, the optical-to-electrical conversion efficiency can be expressed as [2,7]:

_{SC}in [A/m

^{2}], the open circuit voltage V

_{OC}in [V], and the fill factor (FF) of the cell. The incident optical power P

_{IN-optical}[W/m

^{2}] is obtained by:where E

_{IN}in [W/m

^{2}-μm] is the spectral irradiance incident on the cell. PV cells are typically characterized at standard testing conditions (STC) with an incident spectrum corresponding to the Air Mass 1.5 standard spectrum (E

_{AM1.5}) [8]. In this case P

_{IN-optical}= P

_{AM15}.

The current density, J_{SC}, is a function of both the incident spectrum and the spectral responsivity, SR(λ) [A/W], of the PV cell according to [2]:

Using Eqs. (2) and (3) we can rewrite (1) as:

_{OC}⋅FF⋅SR(λ). The SCE is a useful PV cell parameter because it expresses the absorption properties of the cell (SR(λ)), the bandgap properties (V

_{OC}), and the circuit properties of the PV cell (FF).

When a PV cell is used in a spectrum splitting system the transmittance of the spectral filters used to separate different spectral bands must be included in the conversion efficiency expression. The efficiency for the k^{th} PV cell in the system is expressed as:

_{k}* is the filtered efficiency and T(λ) is the transmittance of the filter (incident on the k

^{th}PV cell). The wavelength dependent filter transmittance function, T(λ), allows for ideal and non-ideal filter characteristics to be used in determining the efficiency of the multiple bandgap PV system.

For a system with K different PV cells, the total overall system efficiency is given by the following expression:

where η_{SSS}is the total efficiency of the spectrum splitting system. The total efficiency of the spectrum splitting system (η

_{SSS}) is the optical-to-electrical conversion efficiency reported in the literature for any spectrum splitting system (for example in Table 1 of [5]).

#### 2.2 Optical efficiency

The optical efficiency of a spectrum splitting system is a function of the spectral
transmittance and the ability of the optical system to deliver different spectral components to
their respective PV cells. However the definition is complicated by the overlap of the spectral
responsivities of the different PV cells used in the system. Figure 1 above shows the response for three ideal PV cells with different bandgaps and
absorption properties. Since the ideal PV cell responsivity is a linear function of wavelength
there is overlap at short wavelengths and as a result it is not possible to isolate a distinct
spectral band with an optical filter. In addition it is not possible to separate the optical
efficiency (η^{oe}) as a spectrally distinct factor affecting the total
efficiency as in:

_{SSS}metric.

#### 2.3 Filter performance

Spectrum splitting filters separate wavelength bands for use by different PV cells with
matching spectral responsivities. The process of specifying a filter is complicated by the
overlap and lack of sharp transitions of PV cell spectral responsivity characteristics. As a
result the filter set must be optimized for the particular set of PV cells used in the system.
An ideal spectral filter will have a transmittance or reflectance value of 100% within the
selected spectral band and zero outside the band. Figure
2 shows the spectral responsivities for three different PV cells: an InGaP_{2}-
Emcore [9,10], an
GaAs-Alta [9] and Si-PERL [11] PV cells. The figure also shows the filter transmittance for an ideal and
experimental filter implemented with a volume hologram that are designed to select the
k^{th} (GaAs) spectral band. In addition, if more than two PV cells are considered,
then significant spectral overlap of the cell SCEs can occur making optimization of the filter
the transmittance function more complex typically reducing the IoBB of the system.

It is possible to consider the filter performance within (In-Band) and outside (Out-of-Band) the designed spectral band of the filter. These two regions have the following characteristics:

- In-Band Filter Region: This region contains spectral components that are in the primary transmission band of the filter and overlap with the spectral range where the maximum SCE of the k
^{th}PV cell occurs. There is no appreciable overlap with adjacent PV_{k ± 1}cell spectral response. This is the optimized spectral range for the bandgap energy. This filter region corresponds to the maximum output from k^{th}filter-PV cell combination. - Out-of-Band Region: This filter region consists of two areas. The first corresponds to long wavelengths that are not absorbed by the k
^{th}PV cell. As shown in Fig. 2, SCE = 0 in this range for the k^{th}PV cell, changes to the filter do not have any effect on the performance of the k^{th}PV cell but affect the output of the k^{th}+ 1 PV cell. Wavelengths shorter than the in-band spectral range have significant response from both the k^{th}and (k^{th}-1) PV cells. Therefore in this region, the spectral range and transmittance of the filter must be optimized in conjunction with consideration of the in-band performance of the k^{th}-1 filter/PV cell combination.

#### 2.4 Improvement over best single bandgap (IoBB)

The total efficiency is frequently stated as a performance metric for a variety of PV systems including spectrum splitting [4,5,12]. However this parameter does not provide insight into how much of an improvement the spectrum splitting system provides with respect to using a single type of PV cell with one bandgap over the same system aperture. Comparing the performance of a spectrum splitting systems with the output when a single cell with the highest conversion efficiency is used indicates the performance gain that will be obtained with the more complex system. This metric is called the Improvement over Best Bandgap (IoBB) and is defined as:

where η_{1}, η

_{2}… η

_{k}are the full-spectrum efficiencies of the individual PV cells that are part of the system as reported by the manufacturer. If this ratio is less than one, the system performs worse than one (or more) of its PV cells. Conversely, if this ratio is larger than one, the efficiency of the system is larger than any single PV cell performance. If the improvement is marginal or equal to the single junction system then the value of implementing the spectrum splitting system is questionable and may not be worthwhile.

#### 2.5 Filter overlap metric

In order to evaluate SSS configurations and compare them with ideal filter performance, we define the filter overlap O as,

where the IoBB is calculated using Eq. (8) with practical filter performance data (measured, simulated, etc.) and IoBB_{Ideal}is calculated using ideal filter parameters (i.e. a filter with 100% in-band performance and no spectral components in the out-of-band regions as shown in Fig. 2) and is the maximum improvement that can be obtained for a specific combination of PV cells. The overlap metric (O) indicates how close particular filter is of achieving the maximum ideal improvement.

Deviation of actual filter performance with respect to ideal filter characteristics can be quantified with:

where O_{filter loss}is the total loss caused by the non-ideal filter. Since the filter overlap metric includes the IoBB, filter crosstalk and overlap with adjacent PV cell spectral responses are included in this quantity.

## 3. Spectrum splitting system configurations

The metrics from the previous section provide a methodology to evaluate and compare different spectrum splitting systems. For clarity, the expressions below only include PV cell and filter spectral characteristics of the components used in the spectrum splitting system. Other factors (such as shadowing, reflectance, tracking error, etc.) have been omitted but can be easily included in the analysis if necessary [13].

#### 3.1 Tandem spectrum splitting PV systems

In a tandem spectrum splitting configuration, the absorption of each junction effectively filters the incident spectrum for subsequent PV cells. Multi-junction PV cells are typically designed so that the top layer completely absorbs shorter wavelength bands allowing longer wavelengths to penetrate to the lower energy bandgap junction that is near the bottom of the multi-junction stack. For a three bandgap system, the total efficiency can be calculated as:

_{i}is the transmittance of each of the junctions above. Since all photons with energy higher than the bandgap of a junction are absorbed, the multi-junction stack performs as a cascade of low-pass filters (in frequency, red-pass in wavelength). The transmittance T

_{i}of a bandgap as a red-pass filter is given by following expression:

_{BGi}) is a step function with cut-off wavelength corresponding to the i

^{th}bandgap wavelength λ

_{BGi}.

#### 3.2 Lateral spectrum splitting PV systems

Lateral spectrum splitting can be realized in a variety of configurations. In each case spectral bands are spatially separated outside of the PV cell layer with external optical components.

### 3.2.1 Band pass filter spectrum splitting

Reflection type band pass filters can be formed using dielectric coating (dichroics) or
with holographic reflection gratings. The m^{th} reflected spectral band,
R_{m}(λ), is laterally separated as the transmitted band
T_{m}(λ) continues down the optical axis as shown in Fig. 3(a). In order to realize a multi-PV cell system the filters can be
cascaded as shown in Fig. 3(b). In this case, the total
efficiency of a system with M cascaded band pass filters and K PV cells can be calculated
as:

_{i}(λ) is the transmittance of each band pass filters expressed as:In band pass spectrum splitting, each PV cell is affected by filter performance, including the first one. This is in contrast to tandem spectrum splitting, where the first junction in the system receives direct unfiltered illumination while subsequent junctions receive filtered sunlight. Note that the effective collection area is greater than the incident beam diameter (D

_{A}) and should be taken into account when computing the system efficiency.

### 3.2.2 Spatially separated spectral band systems

In a spatially separated spectral band configuration, the filters are arranged next to each other instead in a cascade. The separated band is directed to an adjacent spectrally matched PV cell as shown in Fig. 4.In this case the unfiltered spectrum is matched to the PV cell directly below the filter.

The PV cell below a filter receives the spectrally matched pass band of the filter above it and the separated band of the adjacent filter. In this case, the total system efficiency can be calculated as:

_{m}(λ) and the pass band of the m

^{th}filter is 1-T

_{m}(λ), as defined in Fig. 4 above.

### 3.2.3 Dispersive spectrum splitting

Spectral separation can also be achieved using the dispersion of a filter caused by diffraction or refraction. The magnitude of filter dispersion depends on the type of component used and can be expressed as a dispersion factor given by:

When the filter is illuminated with the solar spectrum (i.e. AM1.5), each wavelength is dispersed at a different angle forming a continuum of wavelengths spatially separated along the receiver plane. The spatial extent of the dispersed wavelengths along the receiver plane depends on the separation distance between the filter and the receiver plane (d) and the dispersion factor (D_{F}) as defined in Eq. (16). If the dispersion factor is uniform across the filter aperture then the situation shown in Fig. 5(a) results in which the filter aperture with a mixture of wavelengths is projected onto the receiver plane. This configuration by itself is not useful for spectrum splitting since the incident wavelengths are not completely separated at the receiver plane except at the edges of the projected aperture. In order to realize a useful spectrum splitting element, the dispersion factor must be varied across the aperture (with an element like a focusing hologram as shown in Fig. 5(b)) or a separate focusing element must be used as in [13].

The degree to which the spatially dispersed spectrum on the receiver plane is matched to
the responsivity of a PV cell can be quantified using a spectral overlap parameter
τ_{k}(λ). The spectral overlap parameter will be maximum when the PV
cell aperture, spectral responsivity and the dispersed spectrum match in size and location,
and decreases with mismatch of these parameters. A diagram of this function is shown in Fig. 6 and illustrates the effect of focusing on the overlap
parameter as a function of wavelength.

With the above definitions, the total efficiency of a spectrum splitting system using a dispersive optical element with K different PV cells can be calculated using the following expression:

_{k}(λ) is the overlap parameter and depends on the position of the PV cell on the receiver plane.

## 4. Examples of spectrum splitting systems

In this section the metrics and methodology discussed in the previous sections are applied to systems using ideal and experimental PV cell data and filter characteristics.

#### 4.1 Ideal filter band pass spectrum splitting system with ideal PV cell SCE performance

Figure 7 shows the results of modeling a band pass
configuration as shown in Fig. 1 with K = 4 and the
highest efficiency PV cell as PV_{4}. Each curve was obtained using: the SCE definition
in Eq. (4), the bandgap wavelength [10,14] indicated in
the graph for each cell, and the Current-Voltage (IV) characteristics were obtained with
detailed balance analysis [1,15]. The filters were modeled with ideal spectral response (with 100% in-band
and 0% out-of-band transmittance as shown overlaid on the SCE curves). The filter parameters
(bandwidths: short and long wavelength cutoffs, and central wavelengths) were obtained by
maximizing the total system efficiency η_{SSS}. The filtered efficiencies are
less than the reported STC full spectrum efficiencies but the overall SSS efficiency is larger
than any of the PV cells that comprise the system.

For this case, all components of the SSS have ideal characteristics and the efficiency is maximum at η_{SSS} = 51.89% (for the four PV cells selected). The system’s efficiency corresponds to an IoBB of 67.02% over the InGaP_{2}-Emcore PV cell (best single junction efficiency). Since the filters are ideal, the IoBB = IoBB_{Ideal} and the filter overlap O is equal to unity.

#### 4.2 Ideal filter band pass spectrum splitting system with reported PV cell SCE characteristics

Figure 8 shows modeling results for the same
system as the previous section using experimental PV cell performance data and an ideal filter
spectral response. The IoBB shows that the SSS performance can be expected to be 47.78% more
efficient that the GaAs-Alta PV cell (at 28.19%) for a total system efficiency of
η_{SSS} = 41.65%. The filter overlap (O) is still equal to unity since ideal
filters were used in the analysis.

#### 4.2 Holographic filter spectrum splitting systems with reported PV cell SCE characteristics

In the following system experimental values for a holographic reflection filter are used
with two different PV cells (GaAs-Alta [9] and Si-PERL
[11] PV cells). The resulting system performance is
shown in Fig. 9 as well as results for the same PV cells
with ideal spectral filters. With ideal filters, a maximum system efficiency
(η_{SSS}) of 34.38% can be achieved and the improvement over best single cell
efficiency is (IoBB = IoBB_{Ideal}) 21.99%. The overall system efficiency and the IoBB
are reduced to 31.88% and 13.09% respectively when experimental holographic filter performance
[17] is considered. From examination of Fig. 9 it can be seen that out-of-band loss is primarily due
to secondary Bragg matching at a wavelength near 0.5μm. Light diffracted away from the
GaAs cell by the secondary order is incident on the Si-PERL cell with lower SCE than GaAs and
therefore reduces the overall system performance. The in-band losses are caused by the
non-square spectral shape and limited bandwidth of the holographic filter reflectance spectrum.
As a result the most efficient part of the Si-PERL cell is not utilized effectively.

The holographic filter overlap (O) for this system is 59.53% with a corresponding filter loss (O_{filter loss}) of 40.47% (Eqs. (9) and (10) respectively). The in-band losses account for 96.22% of the losses while the out-of-band regions account for an additional 3.78% of the losses. Therefore the system efficiency can be significantly increased with improved hologram diffraction efficiency within the in-band spectral range at the expense of increasing the efficiency of the secondary order of the hologram and its effect on the out of band loss.

Results for a dispersive spectrum splitting system with three PV cells implemented with a
holographic transmission filter and a lens [13] are
shown in Fig. 10. In this analysis, the reported data for three PV cells (GaAs-Alta [9], Si-PERL [11], and a GaSb [14,16]) are used. When
ideal filters are used, a maximum system efficiency (η_{SSS}) of 38.24% is
achieved. The IoBB improvement of this system is 35.66%. When experimental data for holographic
spectral filters are used, the overall system efficiency and the IoBB are reduced to 33.43% and
18.61% respectively. The corresponding holographic filter overlap and filter loss are O =
52.18% and O_{filter loss} = 47.81% respectively. The spectral overlap parameter
τ_{k}(λ) with the geometry in [9]
has a spectral response close to ideal. However, since the diffraction efficiency of the
hologram is less than 100% the in-band performance is reduced. The in-band losses are then
7.45%, 23.90%, and 16.50% for PV cells k = 1 (GaSb), 2 (Si-PERL) and 3 (GaAs-Alta)
respectively. The filter overlap analysis shows that improving the filter shape for k = 2
followed by reducing the loss for k = 3 can significantly improve the performance of this
configuration.

The overlap metric (O) can be used to compare systems that have different configurations and different numbers of PV cells, (i.e. the systems shown in Figs. 9 and 10). The dispersive system in Fig. 10 has a better IoBB and system efficiency than the band pass system of Fig. 9. But the larger filter overlap metric of the band pass system indicates a filter that is closer to ideal. The improved performance of the dispersive system is due to increased number of PV cells with different bandgaps offsetting the losses of the dispersive transmission grating.

## 5. Conclusions

In this paper a set of metrics were presented for evaluating spectrum splitting systems. These metrics were then used to evaluate the specific issues related to different types of spectral filters used in spectrum splitting systems. The system efficiency (η_{sss}) incorporates the spectral conversion efficiency (SCE(λ)) of the cell and the filtered Air Mass (T(λ)⋅E_{AM1.5}(λ)) spectrum and provides a measure for comparing different configurations. Another important metric that was defined is the Improvement over Best Bandgap (IoBB) and indicates the degree to which the efficiency increases with respect to a similar system that uses a single bandgap PV cell. This parameter provides an indication of whether the improved performance is worth the greater complexity of the spectrum splitting system. Parameters specific to the spectral filter performance were also introduced and include the spectral overlap with adjacent spectral bands in the filter system and loss with respect to an ideal filter.

A description of different types of spectral filters and spectrum separation mechanisms was provided giving an overview of the capabilities and limitations of each approach. These include: absorption filtering in tandem PV cell systems, band pass filters, spatially separated spectral band filters, and dispersive filters.

An analysis was then performed of two spectrum splitting configurations: a system with ideal spectral filters and 4 ideal PV cells compared to a similar system with ideal spectral filters and 4 experimental PV cells; and a system with ideal spectral filters and two experimental PV cells compared to a similar system with experimental holographic spectral filters. The results showed: η_{SSS} = 51.89% and IoBB of 67.02% for the ideal filter with four ideal PV cell system compared to η_{SSS} = 41.65% and IoBB of 47.78% for the ideal filter with 4 experimental PV cells. For the two cell system η_{SSS} = 34.38% and IoBB = 21.99% for the ideal filter system and η_{SSS} = 31.88% and IoBB = 13.09% for the holographic filter case. Analysis of the holographic filter characteristics shows that the primary loss with respect to an ideal filter occurs due to lower in-band efficiency rather spectral overlap with adjacent PV cell spectral bands. This characteristic can be improved using different hologram exposure and processing methods.

Future improvements to the system metrics and analysis methods will incorporate the PV cell efficiency change with concentration ratio. In ideal spectrum splitting systems the concentration ratio increases with the number of PV cells. Systems with greater numbers of PV cells therefore require using cells that are more efficient with higher concentration. This will also require optical systems that can readily incorporate different focusing powers to realize correspondingly different concentration ratios.

## Acknowledgments

The authors wish to acknowledge support from the NSF/DOE ERC cooperative agreement No. EEC-1041895, NSF Grant No. 0925085, State of Arizona TRIF(WEES) program and the Research Corporation.

## References and links

**1. **W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. **32**(3), 510 (1961). [CrossRef]

**2. **A. Luque and S. Hegedus, *Handbook of Photovoltaic Science*, 1st ed. (John Wiley & Sons Ltd., 2003), Chap. 3, 9.

**3. **A. Barnett, D. Kirkpatrick, C. Honsberg, D. Moore, M. Wanlass, K. Emery, R. Schwartz, D. Carlson, S. Bowden, D. Aiken, A. Gray, S. Kurtz, L. Kazmerski, M. Steiner, J. Gray, T. Davenport, R. Buelow, L. Takacs, N. Shatz, J. Bortz, O. Jani, K. Goossen, F. Kiamilev, A. Doolittle, I. Ferguson, B. Unger, G. Schmidt, E. Christensen, and D. Salzman, “Very high efficiency solar cell modules,” Prog. Photovolt. Res. Appl. **17**(1), 75–83 (2009). [CrossRef]

**4. **A. G. Imenes and D. R. Mills, “Spectral beam splitting technology for increased conversion efficiency in solar concentrating systems: a review,” Sol. Energy Mater. Sol. Cells **84**(1-4), 19–69 (2004). [CrossRef]

**5. **A. Mojiri, R. Taylor, E. Thomsen, and G. Rosengarten, “Spectral beam splitting for efficient conversion of solar energy—A review,” Renew. Sustain. Energy Rev. **28**, 654–663 (2013). [CrossRef]

**6. **M. Yamaguchi, K.-I. Nishimura, T. Sasaki, H. Suzuki, K. Arafune, N. Kojima, Y. Ohsita, Y. Okada, A. Yamamoto, T. Takamoto, and K. Araki, “Novel materials for high-efficiency III–V multi-junction solar cells,” Sol. Energy **82**(2), 173–180 (2008). [CrossRef]

**7. **M. Hamdy, F. Luttmann, and D. Osborn, “Model of a spectrally selective decoupled photovoltaic/thermal concentrating system,” Appl. Energy **30**(3), 209–225 (1988). [CrossRef]

**8. **D. R. Myers and C. Gueymard, “Description and availability of the SMARTS spectral model for photovoltaic applications,” Proc. SPIE **5520**, 56–67 (2004). [CrossRef]

**9. **M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl. **20**(1), 12–20 (2012). [CrossRef]

**10. **A. L. Gray, M. Stan, T. Varghese, A. Korostyshevsky, J. Doman, A. Sandoval, J. Hills, C. Griego, M. Turner, P. Sharps, A. Haas, J. Wilcox, J. Gray, and R. Schwartz, “Multi-terminal dual junction InGaP_{2}/GaAs solar cells for hybrid system,” in Proceedings of IEEE Photovolatic Specialists Conference (Institute of Electrical and Electronics Engineers, San Diego, California, 2008), pp. 1–4.

**11. **J. Zhao, A. Wang, M. A. Green, and F. Ferrazza, “19.8% efficient “honeycomb” textured multicrystalline and 24.4% monocrystalline silicon solar cells,” Appl. Phys. Lett. **73**(14), 1991 (1998). [CrossRef]

**12. **A. Barnett and X. Wang, “High Efficiency, Spectrum Splitting Solar Cell Assemblies: Design, Measurement and Analysis,” in Imaging and Applied Optics Congress, OSA Technical Digest (CD) (Optical Society of America, 2010), paper SWB1. [CrossRef]

**13. **J. M. Russo, D. Zhang, M. Gordon, S. D. Vorndran, Y. Wu, and R. K. Kostuk, “Grating-over-lens concentrating photovoltaic spectrum splitting systems with volume holographic optical elements,” Proc. SPIE **8821**, 882106 (2013). [CrossRef]

**14. **L. M. Fraas, J. E. Avery, J. Martin, V. S. Sundaram, G. Girard, V. T. Dinh, T. M. Davenport, J. W. Yerkes, and M. J. O’Neil, “Over 35-percent efficient GaAs/GaSb tandem solar cells,” IEEE Trans. Electron. Dev. **37**(2), 443–449 (1990). [CrossRef]

**15. **T. O. M. Tiedje, E. L. I. Yablonovitch, G. D. Cody, and B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Dev. **31**(5), 711–716 (1984). [CrossRef]

**16. **N. Rahimi, A. A. Aragon, O. S. Romero, D. M. Kim, N. B. J. Traynor, T. J. Rotter, G. Balakrishnan, S. D. Mukherjee, and L. F. Lester, “Ohmic contacts to n-type GaSb grown on GaAs by the interfacial misfit dislocation technique,” Proc. SPIE **8620**, 86201K (2013). [CrossRef]

**17. **D. Zhang, M. Gordon, J. M. Russo, S. Vorndran, M. Escarra, H. Atwater, and R. K. Kostuk, “Reflection hologram solar spectrum-splitting filters,” Proc. SPIE **8468**, 846807 (2012). [CrossRef]