P G can now be expressed as a function of the parameters g i and the optimum is then found by setting its gradient to zero, i.e., equating the partial derivatives of P G with respect to all g i to zero, and solving the resulting n equations, which have the form: $$ I_\rm sol,i\cdot h\nu_i \cdot e^-\sigma_i= buy SIS3 \left[kT\cdot e^\mu/kT \cdot I_\rm bb,i\cdot h\nu_i+\fracP_\rm in\sum_i=1^n\sigma_i/h\nu_i \cdot \fracC_P_\rm inC_P_\rm in+C_\rm G \right]\cdot \frac1\mu+kT $$with the proviso that the transmittance e −σ ≤ 1. The term on the left-hand side is the transmitted
power spectrum. The σ i cannot be retrieved directly from this equation as they appear in summed form on the right-hand side as well. This fixed point equation can be solved by the method of iterative
mapping. The derivation of the equation and a description of the method for solving it is given in the S.M. The first term on the right-hand side of the equation is just the black body radiation at ambient temperature multiplied by a very large number (for μ values in the relevant range) and effectively causes an abrupt rise of the transmittance to 1 below a certain photon energy, a condition that is almost perfectly met by the bandgap in semiconductor photovoltaic cells. The second term on the right is spectrally constant, so at photon energies above the bandgap the dipoles should be distributed such that they absorb all power above a constant level that is MG-132 determined by their energy cost. This level is spectrally constant due to the diminishing returns caused by Beer’s law. It is constant transmitted power rather than intensity because the absorption cross-section of a dipole is proportional to its resonance
frequency, and does not indicate that photon energies in excess of the bandgap have been used. The cost of chemical storage of the absorbed power, \(C_P_\rm out\), has no influence (the equation implies that P sat is optimized accordingly) and the level depends only on the ratio between the cost of light harvesting, \(C_P_\rm in\), and that of “the rest of the cell”, C G. Results and discussion Figure 1 illustrates what fraction of the solar irradiance spectrum would be transmitted by a photosynthetic cell optimized for growth power, for a few values of the relative cost \(C_P_1/C_P_\rm in\) + C G). At zero cost, the second term in the transmitted power equation is zero and only the power at photon energies below about 1.14 eV is transmitted (shown in black). The corresponding absorptance (1 − e −σ) spectrum plotted on a wavelength scale is the outermost curve in Fig. 2, showing 50% cut-off at 1,090 nm. This is the supposedly ideal absorptance spectrum of a single-bandgap photovoltaic cell in full sunlight. Fig. 1 Solar irradiance transmitted by a photosynthetic cell optimized for growth power at different costs.