1.1.

Spectral Conversion Efficiency Analysis

When a PV cell is used in a spectrum splitting system (SSS), the spectral response of the filters used to separate different spectral bands must be included in the conversion efficiency expression. The filtered efficiency ${\eta }_k^{*}$ of the $k$’th PV cell of an SSS can be expressed as

The total efficiency of the SSS (${\eta }_{\mathrm{SSS}}$) is equivalent to the optical-to-electrical conversion efficiency reported in the literature for different SSSs (as summarized by Mojiri et al.⁷). The spectral conversion efficiency analysis for different spectrum splitting configurations is discussed in detail by Russo et al.⁸

2.1.

Dispersion and Geometrical Parameters

The spectral overlap function will depend on the dispersion and geometrical parameters of the SSS listed in Table 1 and shown in Fig. 1.

Table 1

Dispersion and geometrical parameters used to calculate the spectral overlap function.

2.1.3.

Receiver size and position

Although the PV cell size and position depends on fabrication and design constraints, the selection of semiconductor materials (their spectral responsivities) will dictate the optimum spectral range $\mathrm{\Delta }{\lambda }_k$ defined in Eq. (4) above. It is possible to calculate optimum size and position for each of the PV cells based on the SCE of the cells that comprise the SSS using Eq. (7) to convert the optimum spectral range from wavelengths to positions at the receiver axis using the following equations:

Eq. (9)

$$\begin{gathered} x_1^{(k)}=d·\tan [\theta ({\lambda }_1^{(k)})], \\ {x}_2^{(k)}=d·\tan [\theta ({\lambda }_2^{(k)})], \\ {a}_k=x_2^{(k)}-x_1^{(k)}, \\ {c}_k=x_1^{(k)}+\frac{a_k}{2}. \end{gathered}$$

The equations above were used to obtain the correspondence between horizontal axes of the two sections of Figs. 2(a) and 2(b). The equations above guarantee that the ${\lambda }_1^{(k)}$ and ${\lambda }_2^{(k)}$ wavelength components of the dispersed continuum align with the edges of the $k$’th PV cell at the receiver plane. The optimum size for a PV cell depends on the separation distance $d$ and the dispersion factor $D_{\mathrm{F}}$ as shown in Eq. (9) and Fig. 3. Also shown in Fig. 3 is the fact that as the separation distance, $d$, is reduced, the optimum PV cell aperture, $a_k$, is also reduced.

Fig. 3

PV aperture size ($a_k$) versus separation distance ($d$). Increasing the dispersion increases the slope of the graph.

If all dispersed wavelength components, $\mathrm{\Delta }{\lambda }_k$, are focused down to a perfect image point, all the wavelength components in the optimum spectral range are collected by an optimum sized PV cell, the overlap function ${\tau }_k$ is ideal (as shown in Fig. 2) and the dispersion losses are zero. However, since the wavelength components $\mathrm{\Delta }{\lambda }_k$ have a finite extent, they either overlap completely, partially, or miss the PV aperture giving rise to dispersion loss. This is explained in more detail in Sec. 3.3.

3.1.

Rectangular Function Representations at the Receiver Plane

As shown in Fig. 1 and by Videos 1 and 2, the continuum incident at the receiver plane is comprised of consecutive and overlapping wavelength components that have finite extent. The size and separation of the components will depend on the amount of focusing defined in Eq. (8) and the parameters listed in Table 1. It is possible to represent the spectral continuum as a set of rectangular functions $S(x)$ positioned along the receiver axis as

As shown in Fig. 2, the $k$’th PV cell can be represented as a rectangular function ${\mathrm{PV}}_k(x)$ along the receiver axis as

3.2.

Cross-Correlation Analysis to Obtain the Spectral Overlap Function

Dispersion will cause the wavelength components $S(x)$ to have varying degrees of overlap with different PV cell apertures on the receiver plane. A spectral overlap function can be obtained by computing the overlap between $S(x)$ functions and the PV cell aperture function ${\mathrm{PV}}_k(x)$ for different positions of the dispersed wavelengths $x(\lambda )$ as

The spectral overlap function ${\tau }_k(\lambda )$ can be obtained as a function of wavelength using the equation for $x(\lambda )$ from Eq. (7) in Eq. (13). The result of the cross-correlation operation is shown in Fig. 4 (and in Videos 1 and 2) for an equal spot size to PV cell aperture size ($A^{\prime }=a_k$) and for a spot size 20 times smaller ($A^{\prime }/20=a_k$). This figure shows, as expected, that reducing the RMS spot size relative to the PV cell aperture size results in a more square shape for the spectral overlap function.

Fig. 4

Overlap functions for $A^{\prime }=a_k$ (a) and for $A^{\prime }=a_k/20$ (b). An animation of the cross-correlation operation shown one wavelength at a time for $A^{\prime }=a_k$ and for $A^{\prime }=a_k/20$ in Videos 1 and 2, respectively (Video 1, MP4, 0.98 MB) [URI: http://dx.doi.org/10.1117/1.JPE.5.1.054599.1] and Video 2, MP4, 0.98 MB) [ http://dx.doi.org/10.1117/1.JPE.5.1.054599.2].

3.2.1.

Dispersion losses and aperture transfer efficiency

Incident light can be incident on one or more cells, or miss the apertures of all of the cells (be a loss). Since energy must be conserved, the summation of all spectral overlap functions for each wavelength is

Eq. (14)

$$\sum _1^K{\tau }_k(\lambda )\le 1.$$

Dispersed light that does not illuminate any of the PV cell apertures is a dispersion loss and can be calculated as

Eq. (15)

$$\begin{gathered} {\eta }_{\text{aperture loss}}=\frac{1}{P_{\mathrm{AM}1.5}}\int [1-\sum _1^K{\tau }_k(\lambda )]·\mathrm{MAX}[{\mathrm{SCE}}_k,{\mathrm{SCE}}_{k+1},\dots ,{\mathrm{SCE}}_K]·E_{\mathrm{AM}1.5}(\lambda )·T(\lambda )·\mathrm{d}\lambda , \\ {\eta }_{\text{aperture loss}}\propto 1-\sum _1^K{\tau }_k(\lambda ). \end{gathered}$$

For the case where the summation term in Eq. (15) is equal to zero (as in Fig. 5 where ${\tau }_{k-1}(\lambda )+{\tau }_k(\lambda )+{\tau }_{k+1}(\lambda )=0$ for $\lambda <0.3\mu \mathrm{m}$ and $\lambda >1.2\mu \mathrm{m}$), the dispersed light misses all the cells in the system at that wavelength range and is completely lost due to dispersion. If the summation term in Eq. (15) is less than one [as in Fig. 5 where ${\tau }_{k-1}(\lambda )+{\tau }_k(\lambda )+{\tau }_{k+1}(\lambda )<1$ for $\lambda =[0.3-0.565]\mu \mathrm{m}$ and $\lambda =[0.975-1.2]\mu \mathrm{m}$], some of the light at that wavelength is incident on one or more PV cells and some of it is lost due to dispersion. Finally, if the summation term is equal to one [as in Fig. 5 where ${\tau }_{k-1}(\lambda )+{\tau }_k(\lambda )+{\tau }_{k+1}(\lambda )=1$ for $\lambda =[0.565-0.975]\mu \mathrm{m}$], all of the light is incident on one or more PV cells and there is no aperture loss. Note that a value larger than one for the summation term would indicate more energy at the receiver axis than at the entrance aperture and is invalid since it violates energy conservation.

Fig. 5

Spectral overlap functions for a system using ${\mathrm{InGaP}}_2$ (blue), GaAs (red), and Si (black) with aperture and spatial crosstalk losses highlighted. The optimal spectral range is highlighted for GaAs.

In an SSS, PV cells that are not properly matched with the incident spectrum will incur thermalization losses below ${\lambda }_1^{(k)}$ and will not be absorbed above ${\lambda }_2^{(k)}$ similar to a broadband PV system. Figure 5 shows ideal and nonideal overlap functions for a GaAs PV cell (in red). The nonideal overlap function extends beyond and does not completely fill the $\mathrm{\Delta }{\lambda }_k$ optimal wavelength range (as shown in Fig. 5). The extension of the overlap function beyond its corresponding $\mathrm{\Delta }{\lambda }_k$ range is defined as a mismatch loss in this paper. These losses are proportional to the difference between the $k$’th (mismatch cell) SCE and the maximum possible SCE of the system and can be calculated as

Eq. (16)

$$\begin{gathered} {\eta }_{\text{mismatch loss}}=\frac{1}{P_{\mathrm{AM}1.5}}\int E_{\mathrm{AM}1.5}(\lambda )·(\mathrm{MAX}[{\mathrm{SCE}}_k,{\mathrm{SCE}}_{k+1},\dots ,{\mathrm{SCE}}_K]-{\mathrm{SCE}}_k)·{\tau }_k(\lambda )·T(\lambda )·\mathrm{d}\lambda \\ {\eta }_{\text{mismatch loss}}\propto (\mathrm{MAX}[{\mathrm{SCE}}_k,{\mathrm{SCE}}_{k+1},\dots ,{\mathrm{SCE}}_K]-{\mathrm{SCE}}_k)·{\tau }_k(\lambda ), \end{gathered}$$

where the SCEs are a functions of wavelength. The equation above is equal to zero in the optimal spectral range since ${\mathrm{SCE}}_k$ is matched and maximum. At wavelengths corresponding to energies below the bandgap energy, ${\mathrm{SCE}}_k=0$ and the mismatch loss is maximum. In Fig. 5, it can be seen that for the same range $\lambda =[0.565-0.975]\mu \mathrm{m}$ that has zero aperture loss $[{\tau }_{k-1}(\lambda )+{\tau }_k(\lambda )+{\tau }_{k+1}(\lambda )=1]$, some light is incident on the $k-1$ PV cell (highlighted in blue) and $k+1$ PV cell (highlighted in gray). The mismatch losses in this example are proportional to ${\mathrm{SCE}}_{k-1}-{\mathrm{SCE}}_k$ (where ${\mathrm{SCE}}_{k-1}=0$) and ${\mathrm{SCE}}_{k+1}-{\mathrm{SCE}}_k$ (where ${\mathrm{SCE}}_{k+1}<{\mathrm{SCE}}_k$), respectively, and are both negative.

The dispersion loss terms defined in this section [in Eqs. (15) and (16)] quantify the amount the overlap function deviates from ideal conditions and affects the overall efficiency of the SSS. Using these loss definitions, consider the efficiency of the system in the absence of dispersion losses as

Eq. (17)

$${\eta }_{\mathrm{SSS}}^{\mathrm{ideal}}={\eta }_{\mathrm{SSS}}+{\eta }_{\text{dispersion loss}}={\eta }_{\mathrm{SSS}}+{\eta }_{\text{aperture loss}}+{\eta }_{\text{missmatch loss}}.$$

3.3.

Form Factor: Spectral Shape of the Overlap Function

In Eq. (12) with $A^{\prime }=a_k$ (equal spot size and ${\mathrm{PV}}_k$ aperture size), only a narrow spectral band will be collected completely [${\tau }_k(\lambda )=1$] by the PV cell. This is indicated by the triangular spectral shape of the overlap functions in Fig. 5 where ${\tau }_k=1$ at the peaks of each function. In Fig. 5, ${\tau }_k<1$ for the rest of the wavelength components due to a mix of the aperture and mismatch losses defined in the previous section. As shown in Fig. 6(a), when the spot size $A^{\prime }$ and the PV aperture size $a_k$ are equal, all the wavelength components of the continuum preceding and subsequent to the peak only partially overlap with the PV cell aperture, hence the ${\tau }_k<1$ overlap function value and only a part of the matched spectrum is collected.

Fig. 6

Spectral overlap function for a form factor of (a) unity and (b) four. Above, overlapping projections $S(x)$ shown for (a) two and (b) three wavelengths. The spatial crosstalk due to incomplete separation is highlighted for both cases.

Changing the relationship between the spot size and the PV aperture to $A^{\prime }<a_k$ by incorporating focusing ($M$) in the collection aperture, the width of the spectral band captured by the cell increases. This case is shown in Fig. 6(b), where $A^{\prime }=a_k/4$ and the overlap function has a trapezoidal shape as a function of wavelength. Assuming that the geometry of the system allows for $A^{\prime }\ll a_k$, the spectral overlap function can approximate a square shape. Another useful parameter in specifying an SSS is the form factor $F_k$ defined as

The form factor is analogous to specifying a spatial frequency of the SSS in terms of how many of the dispersed spectral projections and what proportion fit in the aperture of each PV cell. In this context, the energy collection efficiency ${\eta }_{\tau k}$ can be considered as the transfer efficiency of the aperture of the $k$’th PV cell compared to a system with no dispersion losses (ideal spectral overlap function). The case where there are no dispersion losses corresponds to 100% transfer efficiency. The “transfer efficiency” can be used to determine the amount of focusing power ($M$) required to achieve a desired amount of energy collection efficiency for an SSS with a fixed collection aperture ($A$) and PV cell aperture size ($a_k$), for example, when $F_k=1/2$ results in an energy collection efficiency of ${\eta }_{\tau k}=87\%$ (for an optimally sized PV cell) as defined in Eq. (5). If the ratio of $A^{\prime }/a_k=1/5$, a focusing power of $M=10$ is required to achieve this collection efficiency. Reducing the form factor to values much less than $1/2$ increases the aperture transfer efficiency to values close to unity and reduces the dispersion losses to a minimum. This effect can be seen in Fig. 7, where the optical transfer efficiency of the PV apertures is near 100% with form factor values $F_k\ll 1/2$. Values of $F_k>1$ correspond to incomplete spectrum separation due to a spot size larger than the PV aperture ($A^{\prime }>a_k$). As seen in seen in Fig. 7, the mismatch loss (${\eta }_{\text{mismatch loss}}$) is much larger than the aperture loss due to the mismatched light that falls on neighboring PV cells. One important conclusion of this analysis is that dispersive spectrum splitting requires form factors of less than unity to have high aperture transfer efficiencies. This implies that focusing power (i.e., concentration) is necessary. This requirement increases the complexity of the system, reduces the acceptance angle,³ and has an effect on the angular distribution and hence the response of the PV cell.^20,21

Fig. 7

Aperture transfer efficiency (${\eta }_{\tau k}$), SSS efficiency (${\eta }_{\mathrm{SSS}}$) and dispersion loss (${\eta }_{\mathrm{disp}}={\eta }_{\text{aper. loss}}+{\eta }_{\text{mismatch loss}}$) as a function of form factor ($F$) for optimized cell sizes ($a_{k-1}\ne a_k\ne a_{k+1}$ since for the selected cells $\mathrm{\Delta }{\lambda }_{k-1}\ne \mathrm{\Delta }{\lambda }_k\ne \mathrm{\Delta }{\lambda }_{k+1}$). The case where the PV cell size is forced to be equal is expanded in the Appendix.

3.3.1.

Scaling effects

Since a form factor $F_k\ll 1/2$ (as shown in Fig. 7) is required for maximum optical transfer efficiency of the PV cell aperture, a spot size $A^{\prime }\ll a_k/2$ [following Eq. (18)] is also required. When the height of the system ($d$) is reduced (Fig. 2) greater focusing power $M$ (i.e., smaller $f\#$) is required to produce a smaller diffraction limited spot on the receiver plane. Figure 8 shows the required spot size when the separation height ($d$ in Fig. 1) is reduced from 100 to 0.001 mm for a system with a 10-mm aperture. This figure also shows that there is a critical separation distance for this system (near 10 mm at $f/1$ for $D_{\mathrm{F}}=10.25\mathrm{deg}/\mu \mathrm{m}$ and smaller for others) below which the optical transfer efficiency is low due to insufficient dispersion. As the system cannot maintain a spot size to sustain $F_k\ll 1/2$, the form factor increases and the transfer efficiency decreases. As shown in Fig. 8, increasing the dispersion increases the transfer efficiency.

Fig. 8

(a) Spot size ($A^{\prime }$) versus distance from filter ($d$) for $F_k=1/4<1/2$ and ${\eta }_{\tau k}=96\%$ shown for multiple values of the dispersion factor ($D_{\mathrm{F}}$). (b) The form factor increases as the optical system cannot achieve the optimum spot size and hence the optical transfer efficiency drops to zero close to the entrance aperture.