Irradiance in Mixed Coherent/Incoherent Structures: An Analytical Approach

We propose a new method for a light energy flux density (or irradiance) calculation in an arbitrary multilayer stack containing coherent and incoherent layers. Although the well known General Transfer-Matrix Method (GTMM) can be successfully used for the overall reflectance and transmittance calculation, it does not allow us to obtain the corresponding irradiance depth profile straightforwardly. We show in this paper that subsequent phase-shift integrations over the incoherent layers result in the reflectance and transmittance expressions identical to those of the GTMM formulation. However, the alternative mathematical approach allows us to derive an analytical expression for irradiance at an arbitrary depth of the multilayer stack, thus making it possible to calculate the absorptance depth profile. In fact, the GTMM expressions for the overall reflectance and transmittance turn out to be special cases of the irradiance calculation at the incident and emergent surface of the multilayer stack. Consequently, the proposed Phase-shift Integration Method (PIM) represents a continuous irradiance calculation model without any energy imbalances on layer interfaces. In addition, since we are able to obtain analytical layer thickness derivatives, the PIM is suitable for use with gradient optimization methods. We verify the method on three cases of an encapsulated bifacial heterojunction silicon (HJ Si) solar cell, a perovskite solar cell, and a perovskite/silicon tandem solar cell, which all consist of thin and thick layers.