Calculate scattering and absorption cross-sections, local field enhancements and far-field scattering distributions from a nanoparticle excited by a planewave (Mie scattering). The cross-section and far-field results are compared with the analytic solution to validate the accuracy of the simulation.
Overview
Understand the simulation workflow and key results
The scattering properties of a nano-particle are generally described in terms of field enhancement, cross-sections, and far-field distribution. This example shows how these results can be obtained from a single FDTD simulation.
Run and results
Instructions for running the model and discussion of key results
- Open the simulation file and click the "Run" button.
- Results can be explored manually by right-clicking the monitors or analysis group and selecting the quantity of interest.
- The associated script file can be used to plot the representative results shown below.
Local field enhancement
Interactions of the electromagnetic field with the nanoparticle can create strong field enhancements at the particle surface. Frequency domain field monitors directly measure the local field enhancement. The following figures show \( |E|^2 \) in the XY, XZ and YZ planes through the center of the particle at a wavelength point closest to the 'target wavelength' specified in the script.
It can be noted that the edge of the TFSF source is visible in the plots as abrupt changes in the image color. Fields inside the source are the 'total' field (i.e. incident fields + scattered fields) while only the 'scattered' fields are visible outside the source.
Absorption and scattering cross-sections
The absorption cross-section (the total absorbed power divided by the power per unit area of the incident beam) is calculated by an analysis group located inside the TFSF source. The analysis group measures the net power flowing into the particle and, by normalizing it to the source intensity, returns the absorption cross-section. Similarly, the scattering cross-section is calculated by an analysis group located outside the TFSF source.
The cross-section, by definition, takes the unit of \( m^{2} \) for a 3D simulation and \( m\) for a 2D simulation.
Cross-section measurements are often normalized to the size of the scattering object, as shown in the following figures. The Mie efficiency is defined as the ratio of cross-section to the geometrical area \( \pi r^2 \) for a sphere (3D) and \(2r\) for a circle (2D) and generally plotted against the size parameter ( \( \frac{2 \pi n_1}{\lambda}\)), where \( n_1 \) is the background index of FDTD region and is 1 for air.
The FDTD results are compared with the analytic solution obtained from the mie3d script. The differences between the two results are noticeable, desiring some improvement in the simulation settings. This will be the subject of the following section on the convergence test.
Far-field angular scattering
In most scattering experiments, measurements of the scattered field (radiation pattern) are made far away from the scatterer with respect to the scale of the wavelength in consideration. The "scat_ff" monitor returns the scattered field distribution in the far-field. The following polar plots show the scattered field in the far-field in the X-Y, X-Z, and the Y-Z planes. Each plot contains lines in two colors: blue for the FDTD simulation results and green for the analytic results from the mie3ds12 script command. The first figure shows how the polar angle is defined in each plane.
Important model settings
Description of important objects and settings used in this model
Model setup script
A setup script in the model object is used to set the mesh size, simulation span, and particle location. The script is a convenient way to ensure the position of the simulation region, mesh override region, source, scat, and absorption monitors are correct. For example, it's important that the TFSF source is between the scat and absorption monitors, with at least two mesh cells of space between the objects. The position of these objects must be set via the setup script. Other properties, such as the simulation time, can be modified directly in the objects.
TFSF source
The TFSF source is specifically designed for this type of situation, where a non-periodic object is illuminated by a plane-wave. It makes the scattering analysis of nano-particles straightforward by separating the scattered field from the incident field. For the scattering analysis to work properly, it is crucial to make sure the scatterer is completely within the TFSF source.
Power normalization with the TFSF source
Power normalization with the TFSF source can be 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 by the source intensity. This leads to power measurements being returned in cross-section type units. For more information, see the Power normalization section of the TFSF source page.
'abs' and 'scat' analysis groups
In 3D simulations, the cross-section analysis group is composed of six 2D monitors, forming a closed box that measures the net power flowing in/out of the box. The location of these analysis groups is very important. The 'abs' analysis group, which measures the absorbed power, must be completely inside the TFSF source but outside of the particle. The 'scat' monitor must be completely outside the TFSF source.
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 set large enough to encompass not only the gold sphere, but also the entire TFSF region. This was done intentionally as the TFSF source works best in uniformly meshed regions.
Also, note that the mesh size affects how close the total and scat monitors can be to the source. It is best to keep at least two mesh cells spacing between the source and monitors to avoid unphysical results being returned by monitors placed in the grey-shaded source injection area. Note that these conditions are enforced by the 'model' setup script.
For more information, see the page about how close monitors can be to other objects .
Symmetry
This simulation has a plane of symmetry in both the X and Z dimensions. To reduce the simulation time and memory by a factor of 4, the X min boundary condition was set to Symmetric and the Z min boundary condition to Anti-Symmetric. Note that symmetry can only be used when both the particle and source have the necessary symmetry.
Updating the model with your parameters
Instructions for updating the model based on your device parameters
The simulation file is parametrized to make setting up the simulation easier. The template currently uses a spherical particle, but it can be used with an arbitrarily shaped particle or multiple particles. Once you specify the parameters in the 'model,' the size of the rest of the simulation objects will be automatically adjusted.
- Set the source wavelength range and polarization.
- Set the material or index of the nanoparticle.
- Set the spans and position of the nanoparticle, mesh size of mesh override, and simulation span in the 'model.' The source and 'abs'/'scat' analysis groups will be automatically separated by a maximum of two mesh cells, with the nanoparticle being completely enclosed by the 'abs' analysis group.
- When simulating a non-spherical particle or multiple particles, the boundary conditions might need to be updated to match the symmetry properties of the new structures. The associated script file also needs to be modified to correct the geometrical area and the size parameter of the scatterer.
Taking the model further
Information and tips for users that want to further customize the model
Particles on a substrate
This example uses 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 single homogeneous material that extends to outward to infinity. 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 on the major direction of the scattering). 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 can pass through the monitor before reaching the PML absorbing boundaries. This issue only applies to the far-field analysis. It is not necessary to change the analysis for the cross-section and near-field measurements. See the PSL and Cu Sphere Scattering example for further information.
Unpolarized illumination
For systems with incoherent unpolarized illumination, run a second simulation with the source polarization rotated 90 degrees and then average the results. This is easily accomplished with a 2 point parameter sweep over the source polarization angle. For further information, see the unpolarized beam page.
Convergence
Tips for ensuring that your model is giving accurate results
With the current settings (simulation span of 1x1x1 um 3 , mesh accuracy 3, 5nm mesh near the particle) the simulation requires about 150 MB of memory and runs in roughly 1 minute. These settings provide a reasonable level of accuracy while minimizing the simulation time. The following changes will provide higher accuracy.
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.
DGTD solver
Consider using the Mie scattering (DGTD) example for high accuracy results with metal nanoparticles. The nature of the finite element mesh in the DGTD solver can allow a better convergence and less prone to the staircasing and hotspots issues.
The following figures show the cross-sections from the higher accuracy FDTD simulation. Agreement between the FDTD and theoretical results is clearly much better. Additionally, the smaller mesh results in much higher resolution field profiles that better resolve the fields near the metal interface.
Additional resources
Additional documentation, examples and training material
Related publications
- Bohren, C.F., and D.R. Huffman, 1983: Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York).
- H C van de Hulst, "Light Scattering by Small Particles", John Wiley, (1957). The 1981 edition is available through Google Books .
See also
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