(here the total number of photons is expressed in terms of equivalent current density assuming one electron-hole pair per photon). The total energy available is the area under the curve, while the useful energy absorbed by a single junction cell can be no greater than the area of a box touching the curve with the right edge at the material's bandgap, at maximum slightly less than half the total area. In a cascade cell, photons not absorbed by the top cell can pass through to a second cell. As shown in Fig. 4, a higher fraction of the total energy is available to a two bandgap cascade, where the fraction of energy used can be as large as the sum of the two box areas shown. In most of the regions of interest, the integrated solar spectrum of Fig. 4 can be approximated by a straight line. The photon utilization fraction can then be directly seen to be approximately n/(n+l) for a cascade of n elements. For one and two element cascades, this approximation yields photon utilization fractions of 50% and 67%, very close to the actual utilization fractions of 45% and 69%. (In practice, incomplete photon utilization is not the only source of loss in a solar cell, and actual efficiencies are about half this.) In principle, cascades could consist of an arbitrary number of elements, which would approach complete utilization of the solar spectrum. The largest jump in photon utilization comes from the increase from one bandgap to two. In practice, it is unlikely that thin-film materials will be made with more than two cascaded elements, at least in the reasonable future. In an optimum current-matched two-element cascade, the efficiency can be approximately calculated as equal to the top cell efficiency plus half the bottom cell efficiency. If current matching is not required, the efficiency is approximately equal to the top cell efficiency plus 1-Jsc(top cell)/Jsc(bottom cell) times the bottom cell efficiency.
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