We present a new theory for the description of detectors in terms of modes. Although the theory is very general, it is expected to be of particular use in the modelling of far-infrared and submillimeter instruments. Such a theory is needed because at far-infrared frequencies, both optical systems and detectors show partially coherent behaviour. That is to say, even when the instrument is illuminated by an incoherent source, the resultant field at the detector is partially coherent, and the detector itself is sensitive to the coherence properties of the field and not just the intensity. We have previously developed a modal description of optical systems at far-infrared wavelengths; here we describe a modal theory for the detectors themselves. The theories can be combined to provide a complete modal description of far-infrared instruments. The theory presented here applies equally well to pulsed or ergodic radiation, and incorporates polarisation effects. We also show how the statistics of the detector output can be determined from the theory. This is important for instruments such as bolometers, where the internal noise of the detector is very low, and either sky noise or radiation from the optical components dominate. We illustrate our work with a number of simulations, showing signal to noise ratios for different detector types, and we show how how the theory may be applied to the array packing density problem.