The state-of-the-art infrared (IR) photodetectors are either thermal detectors (bolometers) or quantum detectors (photovoltaic and photoconductive detectors). Compared to quantum IR photodetectors, IR bolometers are slower and less sensitive but in turn, they offer lower cost without need for cooling and exotic materials (e.g. HgCdTe). Phonon/photon engineered materials offer interesting routes for enhancing room-temperature IR bolometers. We have recently demonstrated experimentally a nano-thermoelectric bolometer for long-wave IR detection. The technology utilizes efficient thermoelectric transducers based on silicon nanomembranes, which have an enhanced thermoelectric figure of merit arising from the low thermal conductivity stemming from the nano-scale thickness. For the absorption of the IR radiation the nano-thermoelectric bolometer utilizes a nanomembrane based quarter-wave resistive absorber, which is also known as the Salisbury screen. The use of nanomembranes in both the thermoelectric transducer and the absorber results in a very small thermal mass, and thereby high speed for the detector. In this article, we present an analytical model for quarter-wave resistive absorbers (i.e. Salisbury screens). It can be applied both in radio frequency (RF) and optical applications. The results of the analytical model are compared with the ones obtained with the transfer-matrix method using the optical material data available in the literature. We present also a device model of the nano-thermoelectric IR detector and estimate the full performance of this technology.