The measurement of linear polarization is one of the hot topics of high-energy astrophysics. Gas detectors based on the photoelectric effect have paved the way for the design of sensitive instruments, and mission proposals based on them have been presented in the last few years in the energy range from about 2?keV to a few tens of keV. In addition, a number of polarimeters based on Compton scattering are approved or being discussed for launch on board balloons or space satellites at higher energies. These instruments are typically dedicated to pointed observations with narrow field of view telescopes or collimators, but there are also projects aimed at the polarimetry of bright transient sources such as soft gamma repeaters or the prompt emission of gamma-ray bursts. Given the erratic appearance of such events in the sky, these polarimeters have large fields of view to catch a reasonable number of them, and as a result, photons may impinge on the detector off-axis. This dramatically changes the response of the instrument to polarization, regardless of whether photoabsorption or Compton scattering is involved. Instead of the simple cosine-squared dependence expected for polarized photons that are incident on-axis, the response is never purely cosinusoidal, and a systematic modulation also appears for unpolarized radiation. We investigate the origin of these differences and present an analytical treatment that proves that such systematic effects are actually a natural consequence of how current instruments operate. Our analysis provides the expected response of photoelectric or Compton polarimeters to photons impinging with any inclination and state of polarization.
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