We use angle-dependent Mueller-matrix spectroscopic ellipsometry (MMSE) to determine Mueller matrices of Scarabaeidae beetles which show fascinating reflection properties due to structural phenomena in the exocuticle which are often depolarizing. It has been shown by Cloude  that a depolarizing matrix can be decomposed into a sum of up to four non-depolarizing matrices according to M= aM1+bM2+cM3+dM4, where a, b, c and d are eigenvalues of the covariance matrix of M. Using the same eigenvalues the matrices Mi can be calculated. This method provides the full solution to the decomposition with both the non-depolarizing matrices and the weight of each of them in the sum.
An alternative to Cloude decomposition is regression decomposition. Here any Mueller matrix can be decomposed into a set of matrices Mi which are specified beforehand. Whereas in Cloude decomposition the only constraint on the matrices is that they are physically realizable non-depolarizing Mueller matrices, we can now limit the constraint and only use Mueller matrices representing pure optical devices having direct physical meaning, such as polarizers, retarders, etc. This leaves a, b, c, d as fit parameters to minimize the Frobenius norm Mexp -Mreg where Mexp is the experimentally determined Mueller matrix to be decomposed and Mreg is the sum of all Mi. Depending on Mexp an appropriate choice of Mreg matrices has to be made and different values of a, b, c and d are obtained through regression analysis.
We have previously shown that regression decomposition can be used to show that the Mueller matrix of Cetonia aurata can be decomposed into a sum of a circular polarizer and a mirror . Here we expand the analysis to include angle-resolved spectral Mueller matrices, and also include more species of Scarabaeidae beetles.
One effect of the decomposition is that when depolarization is caused by an inhomogeneous sample with regions of different optical properties the Mueller matrices of the different regions can be retrieved under certain conditions. Regression decomposition also has potential to be a classification tool for biological samples where a set of standard matrices are used in the decomposition and the parameters a, b, c, d are used to quantify the polarizing properties of the sample.
 Cloude S.R. 1989. Conditions for the physical realisability of matrix operators in polarimetry. Proc. SPIE 1166, Polarization Considerations for Optical Systems II, pp. 177-185
 Arwin H, Magnusson R, Garcia-Caurel E, Fallet C, Järrendahl K, De Martino A, Ossikovski R, 2015. Sum decomposition of Mueller-matrix images and spectra of beetle cuticles. Opt. Express, vol. 23, no. 3, pp. 1951–1966
American Vacuum Society (AVS) , 2015.
Mueller-matrix spectroscopic ellipsometry, matrix decomposition, Scarabaeidae beetles
AVS 62nd International Symposium & Exhibition, San Jose, CA, USA, October 18-23 2015