Complete polarization characterization of single plasmonic nanoparticle enabled by a novel Dark-field Mueller matrix spectroscopy system

Information on the polarization properties of scattered light from plasmonic systems are of paramount importance due to fundamental interest and potential applications. However, such studies are severely compromised due to the experimental difficulties in recording full polarization response of plasmonic nanostructures. Here, we report on a novel Mueller matrix spectroscopic system capable of acquiring complete polarization information from single isolated plasmonic nanoparticle/nanostructure. The outstanding issues pertaining to reliable measurements of full 4 × 4 spectroscopic scattering Mueller matrices from single nanoparticle/nanostructures are overcome by integrating an efficient Mueller matrix measurement scheme and a robust eigenvalue calibration method with a dark-field microscopic spectroscopy arrangement. Feasibility of quantitative Mueller matrix polarimetry and its potential utility is illustrated on a simple plasmonic system, that of gold nanorods. The demonstrated ability to record full polarization information over a broad wavelength range and to quantify the intrinsic plasmon polarimetry characteristics via Mueller matrix inverse analysis should lead to a novel route towards quantitative understanding, analysis/interpretation of a number of intricate plasmonic effects and may also prove useful towards development of polarization-controlled novel sensing schemes.

The steps of the eigenvalue calibration method which is used for the calibration of the developed Dark-field Mueller matrix spectroscopic microscopy system are outlined in this supporting information file.

Eigenvalue Calibration method
The actual experimental polarization state generator and analyzer matrices are determined using measurements on a set of ideal calibrating samples (pure diattenuators (polarizers) and retarders (waveplates)), as follows. Sixteen (4×4) set of spectral measurements are performed separately with the calibrating sample (s) in place ( and without any sample (blank) ( . These are related as Here, is the unknown Mueller matrix of the calibrating sample. Two set of matrices and are then constructed such that the former is independent of and the latter is independent Using equation (2) Where  and  are the conventional ellipsometric parameters for the diattenuating retarder with  being its transmittance. This Mueller matrix M has two real and two complex The transmittance  and the ellipsometric parameters ,  can thus be obtained from the eigenvalues of M (as determined from the experimental or matrices, Eq. 2) as The Mueller matrix M of the reference sample can then be constructed using Eqs. 3 and 5. [1,3] Once the Mueller matrix is determined, the generator and the analyzer matrices can be determined using Eq. 2. The matrix is determined by solving the following equation.
In order to solve the above equation, a linear operator K is formed such a way that is the only eigenvector associated with the null eigenvalue (satisfying K 16 × 1 = 0). [1,3] Note that K has all different eigenvalues from zero except 1 , which is supposed to be null and practically as close as zero (0= 1 << 2 < 3... 16 ). The smallest eigenvalue of the matrix (corresponding to the obtained eigenvector (W 16 × 1 )) K is reshaped in 4 × 4 matrix to obtain the generator matrix . With the W matrix in hand, the analyzer matrix A can be determined The exact nature of the system and analyzer matrices and their wavelength dependence was determined by performing measurements on two different types of calibrating reference samples. We used linear polarizer (pure diattenuator) and broadband quarter waveplate (pure retarder over = 400 -700 nm) as reference samples. Once, the experimental and matrices are determined, they can be used to determine Mueller matrices of any unknown sample using Equation.
2 of the manuscript.

Scattering angle dependence of diattenuation and retardance
In the Fig. S1 below, the dependence of the Mueller matrix-derived (using Eq. As expected from the predictions of Eq. 5, in the dipolar scattering approximation, the parameter is relatively insensitive to the scattering angle. In contrast, the -parameter varies with , due to the presence of the he factor associated with dipolar scattering. Figure S1: The (theoretically computed) Mueller matrix-derived scattering angle ( ) variation of the linear retardance (right axis, red dotted line) and linear diattenuation (left axis, blue solid line) parameters for the preferentially oriented Au-nanorods (40×14 nm). While for the parameter, the wavelength is chosen to be = 650 nm (corresponding to the peak of the longitudinal dipolar plasmon resonance plasmon), for the parameter, is chosen to be 550 nm (corresponding to the spectral overlap region of the two orthogonal dipolar plasmon resonance).