The document discusses Mie scattering from a sphere and the challenges involved in computing the electric field distribution behind the sphere. It describes how only the front part of an incoming Gaussian beam interacts with the sphere. It also discusses how the computation region behind the sphere is divided into plane wave components using a discrete Fourier transform, but that the plane waves may be skewed in different directions, making the computation difficult. The document emphasizes that to fully account for polarization effects, it is important to compute the electric field distribution rather than just the intensity, as the polarization information is lost when only considering intensity.

1. Mie scattering from a sphere
Challenges and local minima
I’m Stuck

2. Overview
• Region of interest in the beam
•Computation region behind the sphere
•Electric field vs Intensity

3. Region of interest in the beam
First slice of the matrix
Only the front part of the beam interacts with the sphere

4. Region of interest in the beam
Fourier Transform
Gaussian beam
Plane wave
components

5. Region of interest in the beam
Plane wave
components
If we add the scattering due to all the plane waves we
are looking at the scattering due to the whole
Gaussian beam. It is just a time averaging.

6. Region of interest in the beam
First slice of the matrix
2d Gaussian matrix

7. Region of interest in the beam
Fourier Transform
…
But these are not 3d plane waves!

8. Discrete Inverse Fourier Transform in 3D
Computation region behind the sphere
Where the matrix F is of dimension NXMXP
The propagation direction of the plane waves are perpendicular to the plane:

9. A 2D example
Computation region behind the sphere
u=1
v=1
u=1
v=2
u=2
v=1
u=2
v=2
M=2
N=2
Wavefront : xu/N + yv/M = 0
Propagation direction : xv/M – yu/N = 0
x/2+y/2=0
x+y/2=0
x/2+y=0
x+y=0
Propagation direction of wave
Wavefront

10. Extrapolating to 3D by analogy
Computation region behind the sphere
0
We see that the plane waves propagate in different skewed
directions

11. Computation region behind the sphere
Component plane waves
Plane of
computation
Resultant
intensity
The two planes cut and
we can compute the
resultant intensity

12. Computation region behind the sphere
Component plane waves
Plane of
computation
Here the two computation
planes do not cut and so we
are doomed!
The component plane
waves can be very skewed
in general.

13. Electric field vs Intensity
If the beam is not linearly polarized, we need to break the
beam into components and carry out the analysis separately
for each polarization.
E
Ex
Ez
Ey x
z
y
But if we proceed with only intensity, we cannot decompose
the beam into different polarizations

14. Electric field vs Intensity

15. Electric field vs Intensity
On Fourier Transform the constant vector or polarization
remains intact.
3D Discrete Fourier Transform

16. Electric field vs Intensity
Fourier Transform
Plane wave
components
Gaussian beam

17. Electric field vs Intensity
90o
Physically possible polarization
Computed polarization
For electromagnetic waves the
electric field should be
perpendicular to the direction
of propagation.

18. Thank You
Wake up! Its over!