Eigenvalue calibration method for 3 × 3 Mueller polarimeters

Abstract

3 × 3 Mueller polarimetry has shown potential for tissue characterization applications, however, calibration has not been fully addressed. We demonstrate a 3 × 3 Mueller polarimeter eigenvalue calibration method, inspired by those for full Mueller polarimeters. We also investigate the optimal combination of calibration measurements. Our method does not rely on modeling the polarization state generator, polarization state analyzer, or precise knowledge of calibration sample properties or orientations. It is therefore easy to implement, and the experimental results of a linear polarizer test sample, as well as a biological specimen, are presented. 3 × 3 Mueller polarimetry measures the top left 3 × 3 sub-matrix of a 4 × 4 Mueller matrix, conveying a substantial proportion of the sample polarization properties [1-3]. Partial Mueller matrices can be determined without using phase retarders, simplifying the system and measurement procedures by only using linear polarizers. 3 × 3 Mueller polarimetry has been demonstrated as feasible in several potential applications, including endoscopy [4-9]. The polarization state generator and analyzer (PSG/PSA) of a 3 × 3 Mueller polarimeter only involve linear polarizers (LPs). The 4 × 4 Mueller matrix of a general LP is given by M LP ˆ Rotθ† 0 0 1 ! 2 6 6 4 q ‡ r q − r 0 0 q − r q‡ r 0 0 0 0 1 0 0 0 0 1 3 7 7 5 Rot−θ† 0 0 1 ! Rotθ† ˆ 2 6 4 1 0 0 0 cos θ sin θ 0 − sin θ cos θ 3 7 5, (1) where θ is the orientation angle, and q and r are the maximum and minimum attenuations along two principal axes of the LP. In practice, θ, q, and r of the LPs used in the PSG and PSA of 3 × 3 Mueller polarimeters may deviate from their nominal values. The light source (considered as a part of the PSG) may also not be perfectly unpolarized. Therefore, it is important to develop a calibration method to obtain the actual PSG and PSA instrumental matrices. A traditional calibration method uses an additional LP with the orientation precisely controlled, an unpolarized light source, and a detector, thereby calibrating the orientation of the LPs within the PSG and PSA, based on the established null intensity calibration method (NICM) [4,5,10]. This emphasizes the calibration of θ for the LPs, rather than q and r. Ignoring the calibration of q is problematic, especially for PSGs/PSAs with multiple LPs such as division-of-focal-plane PSAs or division-of-amplitude PSAs [4,6] that usually have different q values. Ignoring the calibration of r might lead to errors , e.g., in multispectral polarimetry, where r may not be 0 for all wavelengths. Obtaining an additional unpolarized light source and precisely controlling the orientation of the additional LP also requires extra time and effort. Another calibration method [6] uses nine gain coefficients obtained by fitting data for an additional rotating LP to correct radiometric measurements. This method assumes that the transformation matrix between the actual and the nominal PSG/PSA matrix is diagonal, which is normally not the case. Here we demonstrate a calibration method for 3 × 3 Mueller polarimeters inspired by that for complete Mueller polarimeters [11]. This method does not require to (1) model the PSG/PSA, so θ, q, and r of the PSG/PSA LPs, and unpolarized light sources are calibrated altogether; or (2) precisely know the properties and orientations of the calibration samples (CSs). This method is easy, quick, and convenient to implement. Eigenvalues of calibration measurements. In complete Mueller polarimetry, the Mueller matrix is solved from [2]: P ˆ M PSA M 4×4† M PSG , (2) 2362 Vol. 44, No. 9 /