Extracting the smaller quantities requires first determining the 4 × 4 polarization transfer or Mueller matrix (𝐌) using an instrument that modulates the input and the output polarization. M is reconstructed pixel by pixel from a light intensity signal captured with a camera, and the raw data of a complete imaging polarimeter, normalized to the magnitude of the first element M11 as given in Fig. . The sixteen elements that construct 𝐌 encode the linear optical anisotropies of the medium including the following: the linear extinction (LE), the linear retardance (LR), the circular extinction (CE), and circular retardance (CR). For the transparent crystals, we are foremost interested in separating the dispersive quantities that introduce phase shifts, the retardances, and therefore we will not concern ourselves with extinctions that modulate amplitudes. Retardances are phase shifts that are linearly proportion to the intrinsic birefringences and the optical path length z: LR = 2p(n//-n^)z/l, and CR = 2p(nL-nR)z/l where l is the monochromatic wavelength. The polarization dependent optical properties, |LB| and CB, can in principle be recovered through inversion of the Mueller matrix that reduces the convoluted optical properties to the differential properties. This was achieved analytically. However, ambiguities arise as the pathlength increases; the LB oscillates between ±𝜋, in principle, because whole wave phase shifts of n2𝜋, where n is the integral order of the retardance, are not manifest, and in practice oscillate only between 0 and 𝜋, as a triangular waveform, (Fig. ) since |LB| is encoded as an absolute distance. Since the integral order is not known from polarimetry in a first analysis, we will maintain to the phenomenological language of retardance, |LR| and CR.