TN05 What is the Mueller matrix and why is it useful? The Mueller matrix provides the most general and complete description of the response of a medium to excitation by polarized light in either reflection or transmission configurations. The Mueller matrix totally characterizes the optical properties of the sample by the interaction of polarized light with matter in the absence of non-linear effects. M = ( m ij ) is a 4x4 real matrix which char1 ≤ ij ≤ 4 The Stokes-Mueller formalism acterizes the sample. Definition Lamp 1 – cos ( 2Ψ ) 0 0 – cos ( 2Ψ ) 1 0 0 M = R 0 0 sin ( 2Ψ ) cos ( ∆ ) sin ( 2Ψ ) sin ( ∆ ) 0 0 – sin ( 2Ψ ) sin ( ∆ ) sin ( 2Ψ ) cos ( ∆ ) Sr Si P Input head Output head S Sample Stokes vector The Stokes vector S represents the most general state of polarization of a light wave. t S = ( I, 〈 I 0 – I 90〉 , 〈 I ( 45 ) – I ( – 45 )〉 , 〈 I R – I L〉 ) Why is the Mueller matrix useful? What can be extracted from the Mueller matrix In reflection mode • Isotropic ellipsometry parameters: Ψ and ∆ the classical ellipsometric angles • Anisotropic ellipsometry parameters • Reflectance • s- and p- reflectance Where I is the total intensity, and Iα represents the intensity transmitted by a linear polarizer set at an angle α with respect to the P axis in the plane perpendicular to the propagation direction. IR and IL are the intensities transmitted by a right or a left circular polarizer. The superscript T stands for transpose. The brackets mean that these intensities are temporally and spatially averaged. • Depolarization effects Mueller matrix • s- and p- transmittance The Mueller matrix M of a sample is defined by the linear relationship: Sr=MSi Where Sr and Si are the Stokes vectors of the incident and the reflected beam. In transmission mode • Retardance magnitude and orientation • Optical rotation / circular retardance • Polarizer transmission axis orientation • Polarization dependent loss • Circular dichroism TN05 Additional information • Percent transmittance/Insertion loss • Depolarization effects Jones matrix vs. Mueller matrix All Jones matrices may be expressed as Mueller matrices but the reverse is not true. For the following conditions it is impossible to express a Mueller matrix as a Jones matrix: Depolarization (for example due to incoherent reflection, roughness, scattering) Because the Mueller matrix describes perfectly the polarization change due to a sample, it has been proved that the additional information provided greatly reduces the correlations observed between measured parameters using classical spectroscopic ellipsometry (SE). Examples of the benefits gained by the Mueller matrix formalism are found when characterising gratings2 and anisotropic samples. • Inadequate spectral resolution • Inhomogeneity Mueller matrices and their properties have been extensively investigated1. One of the properties is that: 4 ∑ P gen = m ij 2 i---------------------,j = 1 2 4m 11 ≤1 If Pgen<1 then the Mueller matrix is not expressible as a Jones matrix. Sample analysis Isotropic nondepolarizing sample Isotropic depolarizing sample Anisotropic nondepolarizing sample* Anisotropic depolarizing sample* P 1 P=Pgen&<1 <1 P≠Pgen&<1 Pgen 1 P=Pgen&<1 1 P≠Pgen&<1 *:For the anisotropic sample, two measurements in two different azimuths could be needed for having the necessary condition. If the anisotropy is along the normal axis (type n) then P&Pgen are equivalent to the isotropic sample case. 1. Chipman R.A, in Handbook of Optics, Ch22: Polarimetry, OSA 2. A.D.Martino, T.Novikova, A.BenHatit, B.Drévillon, D.Cattelan "Characterization of gratings by Mueller Polarimetry in Conical diffraction", conference, 2005 This document is not contractually binding under any circumstances - Printed in France - 04/2005 Access to full Mueller matrix allows us to have a complete picture of the sample, and for many cases this is vital for the correct characterisation of a sample.