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) = E0 ( j ± i k) ei(kx-wt) Ey = Re(E0 ei(kx-wt)) = E0cos(kx-wt) Ez = Re(± i E0 ei(kx-wt)) =- ± E0sin(kx-wt) At a particular position x (say x = 0) get The electric field sweeps out a circle

Elliptically and Unpolarized Light E (x,t) = (E1 j +i E2 k) ei(kx-wt), with E1 not equal to E2. 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 = Iocos2(q) [Remember I a E2] Birefringence

Example Two light beams of intensities I10 and I20 are propagating in the z-direction. One is polarized along the x-direction and the other is circularly polarized. Write the electric field for both If they pass through a polarizer perpendicular to the z-axis, oriented at 45o w/r the y-direction, what are the transmitted intensities? If they now go through a 3rd polarizer oriented along the y-axis, what is the transmitted intensity 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 ±45o where d is the strain

Work out in class. (See Maple). 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 45o 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 = (M11 + M14) Io IL = (M11 - M14) Io  M14 = 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 M11 is total intensity of scattered light

Measurement of Mueller matrix Example How would you measure M12? 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 90o Polarizer Sample Filter? detector PEM at 45o

We can expand the cosine and sine terms as Bessel Functions: Cos(Asin(wt)) = J0(A) + 2J2(A)cos(2wt) + 2J4(A)cos(4wt) + … Sin(Asin(wt))= 2J1(A)sin(wt) + 2J3(A)sin(3wt) + … So if no final filter is used, The DC current is M11 The 1f signal is M14/M11 The 2f signal is M12/M11

Example How would you measure M14 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 M14.

Other Examples (for home?) How would you measure M13 in a simple system (no PEM, just polarizers)? How would you measure M12 and M22?

Higher Order DNA Structure www.library.csi.cuny.edu

Polarizability modeling

Possible Models