Thin-film interference filters illuminated by tilted apertures

Thin-film interference filters can be illuminated by a circular aperture at different angles. Each situation produces a different transmittance spectrum. We present an analytical model that, for small tilt angles, predicts the change in transmittance for an arbitrary position of the filter in three-dimensional space. The model is extended to take into account higher-order harmonics. We also derive a formula to predict the change in central wavelength, and we validate our results by comparison with thin-film transfer-matrix calculations. A key property of our approach is that the model can be combined with empirical data to predict the transmittance without knowing the filter design.