Here we calculate the differential scattering cross section, degree of circular polarization and principal angle of polarization for light scattered from PSL and Copper spheres on a Si substrate. Results are compared to those published by Kim et al.

## Simulation setup

The simulation technique is similar to that described in the Defect scattering and detection example. The challenge in this example is the very thin layer SIO2 of 1.5 nm. If we use a mesh override of two cells in resolving the thickness, the simulation time is much longer, but the results are not significantly different from without SiO2 layer. For fast result, we will ignore this layer.

## Analysis

### Differential scattering cross section

Kim et al. state "the differential scattering cross section relates the intensity on a single particle to the power scattered by it per solid angle" In FDTD, this is easily calculated with the standard far field projection.

### Degree of circular polarization Pc

The degree of circular polarization is defined as

$$P_{c}=\frac{I_{c+}-I_{c-}}{I_{c+}+I_{c-}}$$

where I_{c+} and I_{c-} are the intensities of the left and right handed circular polarization components. The far field projection function *farfieldpolar3d* returns the linear vector field components E_{θ} and E_{φ}. Before calculating P_{c}, we must convert E_{θ}, E_{φ} to E_{c+}, E_{c-} with the following relations.

$$\vec{E}=a \vec{u}_{ \theta }+b \vec{u}_{ \phi}$$

$$\vec{E}=\alpha \vec{u}_{c +}+\beta \vec{u}_{c-}$$

where

$$\vec{u}_{c+}=\frac{1}{\sqrt{2}}\left(\vec{u}_{\theta}+i \vec{u}_{ \phi}\right)$$

$$\vec{u}_{c-}=\frac{1}{\sqrt{2}}\left(\vec{u}_{\theta}-i \vec{u}_{ \phi}\right)$$

Solving for α and β, we get

$$\alpha=\frac{\sqrt{2}}{2}(a-i b)$$

$$\beta=\frac{\sqrt{2}}{2}(a+i b)$$

The formulas for alpha and beta allow us to convert the linear polarization component to their equivalent circular polarization components. In the far field, intensity is proportional to \(|E|^2\).

### Principal angle of polarization η

Kim et al. measure the polarization angle from S-polarization in a right-handed fashion (counterclockwise, looking into the beam).

## Results

The analysis script will produce the following figures, which can be compared to Figure 7 and 8 from Kim et al. To reproduce the results from figure 7, open *PSL_Cu_scattering.lsf* and set the variable *fig8* equal to 0. To reproduce figure 8, set this variable equal to 1.

The agreement is quite good, even though the simulation region is small and uses a coarse mesh. Because the sphere is metal, we use the mesh refinement option "Conformal Variant 1" to achieve a better approximation of its shape. The accuracy can further be improved by increasing the X and Y simulation span to 10 um, increasing the mesh accuracy slider to 3, and by setting the "def" mesh override region to use an effective index of 5, rather than 3.

92, 123 and 155 nm diameter Cu sphere scattering at 442 nm wavelength. Figure 7 of Kim et al.

155nm diameter radius PSL sphere scattering as a function of wavelength. Figure 8 of Kim et al.

For a comparison, we add the 1.5 nm SiO2 layer back to the simulation with 2 cells in z direction using override mesh, for the 155 nm PSL sphere with different illumination wavelengths, the results are shown below:

Please note that since the differential scattering cross sections are in logarithm scale, the difference between with and without the thin SiO2 layer is small. Therefore, for initial test, it is proper to ignore the very thin layer. However, for final simulations, the thin layer should be included.

### Related publications

J. H. Kim, S. H. Ehrman, G. W. Mulholland, and T. A. Germer, "Polarized Light Scattering by Dielectric and Metallic Spheres on Silicon Wafers", Appl. Opt. 41, 5405-5412 (2002)