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Mie Scattering (DGTD)

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cover_picture_mie_3d_dgtd_zoom33Calculate scattering and absorption cross sections, local field enhancements and far field scattering distributions from a nano particle excited by a planewave (mie scattering). The cross section and farfield results are compared with the analytic solution to validate the accuracy of the simulation.

Related: Mie Scattering (FDTD)

Files and required solvers

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Contents

Overview

Run and results

Important model settings

Updating the model with your parameters

Taking the model further

Convergence

Additional resources

Overview

Understand the simulation workflow and key results

mie_3d_dgtd_workflow_zoom104

The scattering properties of a nano-particle are generally described in terms of field enhancement, cross sections, and farfield distribution. This example shows how these results can be obtained from a single DGTD simulation.

Run and results

Instructions for running the model and discussion of key results

1.Open the simulation file and click the Run button.

2.Results can be explored manually by right-clicking the monitors and visualizing the quantity of interest.

3.The associated script file can be used to plot the representative results (cross-sections and far-fields) shown below.

Local field enhancement

Interactions of the electromagnetic field with the nanoparticle 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 the wavelength of about 502 nm. To visualize these results, right-click on the 'field_XY', 'field_XZ' and 'field_YZ' monitors, select 'field.' and then choose 'Abs^2' in the 'Scalar operation.'  Notice that the edge of the TFSF source is clearly visible in the plots. The inside of the spherical region with the radius of 100 nm corresponds to the the 'total' field and the outside to the 'scattered' field.

mie3d_dgtd_e2_xy_zoom50mie3d_dgtd_e2_yz_zoom50mie3d_dgtd_e2_xz_zoom50

Absorption and scattering cross sections

The absorption cross-section (the rate at which energy is removed from the incident plane wave by absorption) is automatically calculated by the 'scat' frequency monitor surrounding the gold particle since it is placed on the same closed surface as the plane wave source. The same monitor also calculates the scattering cross-section of the gold particle.  The 'sigma_front' flux result corresponds to the scattering cross-section and the ' sigma_back' to the absorption cross-section.

 

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, \( \pi r^2 \); and the size parameter is  \( \frac{2 \pi n_1}{\lambda}  \), where \( n_1 \) is the background index of FDTD region and is 1 for air.  

mie3d_dgtd_sigma_scat_zoom40mie3d_dgtd_sigma_abs_zoom40

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 associated script file computes the scattered far field over a sphere using the createsphericalsurface and near2far scripting commands. The scattered far field is compared with the analytic result given by the mie3ds12 command.

The far-field figures show the electric field intensity at points on a sphere with 1 m radius. The radiation patterns uses the same sets of far-feild data and plots the intensity in terms of the polar and azimuthal angles, providing a better means of visualizing the directionality of the radiating field. Both the farfields and the radiation patterns from the theory and simulation show a good agreement.

mie3d_dgtd_farfield_xyz_plot_zoom30 mie3d_theory_farfield_xyz_plot_zoom30

mie3d_dgtd_farfield_radiation_plot_zoom30 mie3d_theory_farfield_radiation_plot_zoom30

Important model settings

Description of important objects and settings used in this model

TFSF source

The TFSF source is specifically designed for this type of situation, where a non-periodic object is illuminated by a planewave.  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. In this example, the source boundary is defined over a spherical surface formed by the "source" object. The TFSF source boundary can have an arbitrary shape in DGTD, in contrast to FDTD where is must have a rectangular shape. For further information about this versatile feature, see the Reference geometries page.  

mie3d_dgtd_tf_sf_schematic_zoom25

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 (\( m^{-2} \) for a 3d simulation and \( m^{-1} \) for 2d).  For more information, see the Power normalization section of the TFSF sources page.

Scattering and absorption measurements

As with sources, monitors in the DGTD solver can be defined over a curved surface. It is important to define the 'scat' monitor over the closed spherical surface of the "source" object, where the TFSF source is also defined. As a result, the 'sigma_front' (outward) flux result returned by the monitor is calculated in the 'scattered' field, hence its direct correspondence to the scattering cross section. Likewise, the 'sigma_back' (inward) flux results is calculated in the 'total' field, hence its equivalence to the absorption cross-section.

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 defined on the surface of the "sphere" object, corresponding to the interface of the scatterer and the surrounding medium.

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.

1.Set the source wavelength range and polarization.

2.Set the material or index of the nanoparticle.

3.Set the spans of the source and the simulation region. The source span in the current simulation file corresponds to diameter of the "source" object and should be large enough to enclose the scatterer.

4.Symmetry boundary conditions can be utilized to reduce the memory and the simulation time when there are symmetries in both the field and the structure. To take advantage of the symmetries in the current simulation file, the simulation settings need to be modified as follows:
- Disable the "outer" geometry
- In the general tab of the simulation region object, change all the boundaries to "closed"
- Set x_min =y_min = 0 and x_max = y_max = 0.4 um for the simulation region.
- Add PEC and PMC boundary objects from DGTD tab. Select "x min" for the PEC and "y min" for the PMC.

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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. The technique used in this example to calculate the far field (projecting from a closed spherical monitor) only works when the monitor is 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 a monitor located above or below the particle (depending if you want scattering in the forward or backwards direction). You can then use the standard near2far command for the farfield projection.

When using an open-surface monitor such as a 2d planar or a half-sphere monitor, as opposed to the closed-surface in this example, it's important to set the monitor such that it captures most of the scattered light.  For a half-sphere monitor, this can be achieved by placing the base of the monitor close to the interface of the particle and the substrate. For a 2d planar monitor, you might need to increase the simulation span as well as the monitor span and place the monitor close to the particle or the particle/substrate interface depending on the farfield projection direction you want. See the Nanoparticle scattering (DGTD) example for further information.

mie3d_dgtd_particle_on_sub_zoom20

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. For further information, see the Unpolarized beam page.

Convergence

Tips for ensuring that your model is giving accurate results

The default settings for this simulation are designed to give reasonably accurate results while minimizing the simulation time.  If higher accuracy is desired, uses a finer mesh over the gold particle.  The following figures show the cross sections from the higher accuracy simulation with 'max edge length' set to 0.003 um in the Mesh constraint. Agreement between the DGTD and theoretical results is clearly much better.  

mie3d_dgtd_sigma_scat_high_accuracy_zoom40mie3d_dgtd_sigma_abs_high_accuracy_zoom40

 

The following figures show the electric field intensities at the XY, XZ and YZ planes from the higher accuracy simulation.

mie3d_dgtd_e2_yz_high_accuracy_zoom50mie3d_dgtd_e2_xz_high_accuracy_zoom50mie3d_dgtd_e2_xy_high_accuracy_zoom50

Additional resources

Additional documentation, examples and training material

Mie scattering (FDTD) example

Plasmonic Resonator

Related publication: Bohren, C.F., and D.R. Huffman, 1983: Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York).

Related publication: H C van de Hulst, "Light Scattering by Small Particles", John Wiley, (1957). The 1981 edition is available through Google Books.

Lumerical University courses:

Course_DGTD100

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