This example has been updated. Find the latest version at Mie Scattering (FDTD).
We address Mie scattering in three dimensions from a gold nano-particle and compare the scattering and absorption cross sections to the analytic solution. We also show how to calculate the angular scattering in a plane around the particle.
The file mie_example_3d.fsp contains an example of the Mie problem in 3D, using a total-field scattered-field source(TFSF) that surrounds a gold particle. There are two analysis groups, each of which consist of a box of power monitors: one in the total field region and one in the scattered field region. These analysis groups can be used to calculate the absorption and scattering cross sections, as well as the angular distribution of scattered radiation. In addition, 3 frequency profile (normal profile) monitors are added in the total field region to calculate the electric field enhancement. The total scattered field source covers a wavelength range of 300 to 600 nm.
The gold material is a copy of the "Au (Gold) - Johnson and Christy" material that is contained in the default material database. Since the default number of coefficients is not sufficient to provide a good fit to the sampled data over the source bandwidth, the maximum coefficients is set to 10. The theoretical Mie scattering cross sections (absorption and scattering) have been calculated for material fit, and mie3d command recalls this data.
With the default settings (simulation span is 1x1x1 um3 and a mesh accuracy of 3, plus an override region of 5nm at the sphere) the simulation will require about 250 MB of memory.
Once the simulation is finished, mie_analysis_3d_modified.lsf will perform the following analysis.
Note: Mesh override region
For simulations with metals, the mesh override region is often used to more accurately resolve the locations of the metal interface, especially with curved surfaces. In this simulation, the mesh override region is large enough to encompass not only the gold sphere, but also the entire TFSF region. This was done intentionally; the TFSF sources work best in uniformly meshed regions.
Also note that the mesh spacing of the override region effects the location of the analysis boxes. Sources require a certain amount of space to inject the fields. The amount of space required is ~2 mesh cells and is depicted graphically by light white shading that surrounds the source. Within this region, the fields are not physically meaningful. Hence monitors should not be placed in this region.
The absorption cross section (the rate at which energy is removed from the incident plane wave by absorption) is calculated by an analysis group located inside the TFSF source. The analysis group calculates the net power flow into the particle and hence the absorption cross section using the optical theorem. Similarly, the scattering cross section is calculated by an analysis group located outside the TFSF source. This group measures the net power scattered from the particle. To obtain these plots, increase the number of frequency points from global monitor settings to 50 from the Edit tab of x-normal-profile.
The MIe efficiency is defined as the ratio of cross_section of scattering/absorption and the geometrical area pi*r^2; and the size parameter is 2*pi*r/lambda*n1, where n1 is the background index of FDTD region and is 1 for air.
The above plots compare the FDTD results with the results from theory which were computed using the Gold data from the material fit. The script uses mie3d, available in 2017 R5 and higher, and material fit index data to calculate the values for mie theory.
Note: Power normalization with the TFSF source
Power normalization with the TFSF source can be slightly confusing. Rather than normalizing results to the source power (which is infinite for an ideal plane wave since it has infinite extent), it is best to normalize it by the source intensity. This leads to power measurements being returned in cross section type units. For more information, see the Power normalization page of the TFSF sources section.
The default settings for this simulation are designed to give reasonably accurate results while minimizing the simulation time. If higher accuracy is desired, make the following modifications. These modifications will increase the memory requirements to around 1.7 GB.
Mesh refinement: Set the mesh refinement to 'conformal variant 1' to achieve sub-cell resolution for the gold particle boundary. Care must be taken when selecting this setting if the mesh is coarse and there is a large difference in permittivity between the metal and surrounding medium at the frequencies of interest. It is best to perform some convergence testing. A more detailed discussion on convergence testing can be found on the convergence testing page.
Mesh size: Set the mesh override mesh size to 0.8nm
Simulation span: Set the simulation span to 2um in all directions.
When the simulation region is too small, evanescent tails of the resonant surface plasmon modes will interact with the PML boundary conditions.
PML reflections: Any light reflecting from the PML boundary conditions may affect the results. More PML layers will reduce reflections. However, if you are using the "stretched coordinate pml" with its default 8 layers, no need to change it, except you need much higher accuracy.
Symmetry: Set the X min boundary condition to Symmetric. Set the Z min boundary condition to Anti-Symmetric. We can take advantage of the symmetry of the simulation to reduce the simulation memory and time by a factor of 4.
The following figures show the cross sections from the higher accuracy simulation. Agreement between the FDTD and theoretical results is clearly much better.
In the majority of scattering experiments, measurements of the scattered field (radiation pattern) are made far away from the scatterer. We will, therefore, examine the behavior of the scattered field in the far field in the X-Y, X-Z, and the Y-Z plane. The image on the top-right shows how the numerical angles are oriented in 3D space.
Each of these polar plots contain two graphs: the blue graph uses scat_ff analysis group, and green graphs use mie3ds12 script command, for each and every wavelength recorded by the field and power monitors if the option "do polar plot" is enabled. Alternatively, it is possible to open the results (XY, XZ or YZ) from scat_ff in the Visualizer using the polar plot option and sweep over wavelength with the parameter slider, as shown below. For more information about polar plots in the Visualizer please visit this page.
Note: Particles on a substrate
This example studies a particle surrounded by a homogeneous material. If the particle is on a substrate, the far field part of the analysis must be modified. The technique used in this example (projecting from a closed box of monitors) only works when all of the monitors are in a homogeneous material. When a substrate is present, the best way to calculate the far field scattering pattern is to use one monitor, located above or below the particle (depending if you want scattering in the forward or backwards direction). You can then use the standard farfield3d function. When using a single monitor, it's important to make the simulation span large enough that most of the scattered light will pass through the monitor before hitting the PML absorbing boundary conditions.
This issue only applies to the farfield analysis. It is not necessary to change the analysis for the power absorption and scattering measurements, because the technique used to calculate power flow is valid even when the monitor box spans several materials.
We use a series of profile monitors to image the field profile |E|^2 in the YZ, XZ and XY planes through the center of the particle. Notice that the edge of the TFSF source is visible in the plots. These figures were created with the higher accuracy settings.
The field enhancement (excitation rate enhancement) calculation is usually the first step for fluorescence enhancement simulations. For the more information, please visit the methodology page.