Instrument matrix of the four-detector photopolarimeter: physical meaning of its rows and columns and constraints on its elements

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Abstract
The four-detector photopolarimeter (FDP) is an arrangement of four photodetectors for measuring the state of polarization of light. The output current vector I of the FDP is related to the input Stokes vector S by I = AS, where A is the instrument matrix. The rows of A can be viewed as projection operators that determine the output currents of the detectors. This leads to the recognition of four special totally polarized input states, each of which maximizes the output of one detector. The associated four orthogonal states produce minimum signals. Because each detector is absorptive and its output is nonnegative, eight inequalities must be satisfied by the elements of A. For optically isotropic detectors, one element is identically zero and another can be made zero by an appropriate coordinate rotation. Three additional inequalities that are likely to apply are also stated. The columns of A have the following interesting meaning. The first column represents the normalized response of the FDP for incident unpolarized light or its average response to incident light polarized in any pair of orthogonal states. The second, third, and fourth columns represent the differential normalized responses of the FDP for incident light polarized in the following pairs of orthogonal states: (1) linear polarizations at 0 and 90° azimuths, (2) linear polarizations at 45° and 135° azimuths, and (3) the right and left circular polarizations, respectively. These are the same pairs of orthogonal states that are used in the phenomenological definitions of the Stokes parameters.