Polarized Light Scattering Biophysics More rigorous Treatment of

Polarized Light Scattering Biophysics

More rigorous Treatment of Light Scattering • This is not a solution • Polarizability per unt volume is a tensor (matrix) • Transversality is ensured by

Transversality condition is the transversality condition Example: If show E is perpendicular to

First Born and coupled dipole approximation • The first born approximation assumes the electric field inside the particle is equal to the incident electric field. • The coupled dipole approximation

Linearly Polarized Light

Circularly Polarized Light • E (x, t) = E 0 ( j ± i k) ei(kx-wt) – Ey = Re(E 0 ei(kx-wt)) = E 0 cos(kx-wt) – Ez = Re(± i E 0 ei(kx-wt)) =- ± E 0 sin(kx-wt) – At a particular position x (say x = 0) get • The electric field sweeps out a circle

Elliptically and Unpolarized Light • Elliptical – E (x, t) = (E 1 j +i E 2 k) ei(kx-wt), with E 1 not equal to E 2. • Unpolarized light • E always perpendicular to B but changes direction fast, on order of 10 ns or greater.

Polarizers • • Transmit light parallel to polarization axis Unpolarized: I Io/2 Circular: I Io/2 Linear: I = Iocos 2(q) – [Remember I a E 2] • Birefringence

Example Two light beams of intensities I 10 and I 20 are propagating in the z-direction. One is polarized along the x-direction and the other is circularly polarized. a) Write the electric field for both b) If they pass through a polarizer perpendicular to the z-axis, oriented at 45 o w/r the y-direction, what are the transmitted intensities? c) If they now go through a 3 rd polarizer oriented along the y-axis, what is the transmitted intensity d) Write the electric field for the transmitted fields.

Stokes Vector and Mueller Matrices Stokes vector completely describes intensity and polarization state of light. Mueller matrix is a transformation matrix that is a property of material object Check out examples of both

Example of Stokes Vectors • What do these (below) represent? How would we represent right circularly polarized light

Examples of Muller Matrices Linear Polarizer Circular polarizer Retarder at ± 45 o where d is the strain

Examples: Show that a vertical, horizontal, circular polarizer do what they are supposed to for incident un- and linearly polarized lights. Show that a vertical polarizer + 90 degree retarder at 45 o can act as a circular polarizer, with the polarizer first. Work out in class. (See Maple).

Mueller Matrix Elements Each element describes a change in polarization Eg. Ir = (M 11 + M 14) Io IL = (M 11 - M 14) Io M 14 = ½(Ir – IL)/Io

Significance of Muller matrix elements in absorption

Significance of Muller matrix elements in scattering • Upper case elements are non-zero even for small particles with low polarizability • Lower case very sensitive to polarizability • h, , f, j, k are zero in orientation average unless there is intrinsic chirality • M 11 is total intensity of scattered light

Measurement of Mueller matrix Example • How would you measure M 12? – Put in horizontal light and then vertical light – Take the difference and the sum. – Note only measure total intensity of stokes vector

Measurement of Mueller matrix Small elements • Use photoelasatic modulator taking advantage of modulation technique to get good signal to noise. d = A Sin(wt) PEM Quartz (strain) Apply ac voltage – piezoelectric crystal

Apparatus Light source 90 o Polarizer PEM at 45 o Sample Filter? detector

We can expand the cosine and sine terms as Bessel Functions: Cos(Asin(wt)) = J 0(A) + 2 J 2(A)cos(2 wt) + 2 J 4(A)cos(4 wt) + … Sin(Asin(wt))= 2 J 1(A)sin(wt) + 2 J 3(A)sin(3 wt) + … So if no final filter is used, The DC current is M 11 The 1 f signal is M 14/M 11 The 2 f signal is M 12/M 11

Example How would you measure M 14 from a sample using the Scanning polarization-modulation nephelometer starting with unpolarized light? Show the stokes vector you end up with, what the PMT will measure and how you extract M 14.