1. Polarized Light Scattering
Biophysics

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

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

4. 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

5. Linearly Polarized Light

6. 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

7. 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.

8. Polarizers Transmit light parallel to polarization axis
Unpolarized: I  Io/2
Circular: I  Io/2
Linear: I = Iocos2(q)
[Remember I a E2]
Birefringence

9. 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.

10. 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

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

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

13. 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).

14. Mueller Matrix Elements
Each element describes a change in polarization
Eg. Ir = (M11 + M14) Io
IL = (M11 - M14) Io
 M14 = ½(Ir – IL)/Io

15. Significance of Muller matrix elements in absorption

16. 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

17. 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

18. 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

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

20. 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

21. 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.

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

23. Higher Order DNA Structure

26. Polarizability modeling

28. Possible Models