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In this example, we look at the scattering spectrum of light from an electron beam.

 

Solvers

FDTD

Associated files

sp_ebeam.fsp

sp_ebeam.lsf

Related publications

Pratik Chaturvedi et al, "Imaging of Plasmonic Modes of Silver Nanoparticles Using High-Resolution Cathodoluminescence Spectroscopy", ACS Nano 3 (10), 2965-2974 (2009)

 

Pabitra Das and Tapas Kumar Chini, "Spectroscopy and Imaging of Plasmonic Modes Over a Single Decahedron Gold Nanoparticle: A Combined Experimental and Numerical Study", The Journal of Physical Chemistry C 116 (49), 25969-25976 (2012)

 

Pabitra Das, Tapas Kumar Chini, and James Pond, "Probing Higher Order Surface Plasmon Modes on Individual Truncated Tetrahedral Gold Nanoparticle Using Cathodoluminescence Imaging and Spectroscopy Combined with FDTD Simulations", The Journal of Physical Chemistry C 116 (29), 15610-15619 (2012)

See also

nonorm

 

sp_ebeam_screenshot1_zoom64

Creating the ebeam source

A screenshot of sp_ebeam.fsp is shown above. The simulation consists of a gold particle on a glass surface. The electron beam is represented by a series of dipoles with phase delay that is related to the electron velocity. Note that, we are simulating a single electron (as you can see in the movie). The actual experiment is a continuous beam, our results can be scaled by the beam intensity. The current density due to the electron is given by

$$ \vec{J}(t, \vec{r})=-e v \hat{u}_{z} \delta(z-v t) \delta\left(x-x_{0}\right) \delta\left(y-y_{0}\right) $$

 

where e is the electron charge and v is the velocity of the electron. In the frequency domain, this corresponds to a current density of

 

$$ \vec{J}(\omega, \vec{r})=e \hat{u}_{z} \exp \left(\frac{-i \omega z}{v}\right) \delta\left(x-x_{0}\right) \delta\left(y-y_{0}\right) $$

 

This current density can be simulated by using a large number of closely space dipoles along the electron trajectory, of the form

 

$$ \vec{p}(\omega, \vec{r})=\frac{p_{0}}{i \omega} \hat{u}_{z} \exp \left(\frac{i \omega z}{v}\right) $$

 

Since the system is linear, we can study a system of dipoles of the form

 

$$ \vec{p}(\omega, \vec{r})=p_{0} \hat{u}_{z} \exp \left(\frac{i \omega z}{v}\right) $$

 

and multiply the electromagnetic fields by a factor of 1/iw during the post processing phase. We will not be concerned with the overall normalization factor p0. In the time domain, we can create the desired sources by delaying the source pulses by z/v.

 

In the absence of any structure, similarly to how a constant DC current through a wire will not produce any radiation, the electron beam will not generate any radiation because it is moving at a constant velocity. In FDTD, we are obliged to simulate only a finite portion of the electron path and the sudden appearance and disappearance of the electron will generate radiation. To minimize this problem, a raised-cosine filter is introduced to turn on and off the dipoles' amplitude gradually. We are also running a reference simulation (without the nanoparticle and substrate) to subtract anything that can potential obscure the signal from the nanoparticle. Although this makes the analysis slightly more complex, we can calculate the electromagnetic fields at frequency w by taking the difference in fields between the simulations, using sp_ebeam.lsf. To get an accurate difference, we must force the simulation mesh to be exactly the same with and without the structure. For this reason, we use mesh override regions over all the structures.

Setup scripts

We use a setup script contained in the ebeam group to create all the sources with the correct pulse delays and positions. In this example, we use an electron velocity of 0.2*c. A raised-cosine filter is also introduce to gradually turn on and off the dipoles.

Results

To reproduce the plots below, open the sp_ebeam.fsp simulation and run the sp_ebeam.lsf script file. The script file will run two simulations, take the difference in electromagnetic fields and calculate the scattering spectrum. This means that it will calculate the total power scattered into the upper z half space due to the presence of the gold particle and substrate.

 

Note that the CW normalization with a large number of sources leads to a variety of complications. Instead, we use the no normalization state, and remove the spectrum of the source pulse by calculating the sourcenorm for a single source pulse, rather than a sum of all the dephased source pulses.

 

sp_ebeam_power_upper_zoom58

The total scattering into the upper half-space. Transmission has been normalized by the maximum value.

 

sp_ebeam_nearfield_pz_zoom63

Poynting vector at the first peak wavelength.

 

sp_ebeam_farfield_e2_zoom63

Angular distribution of the electric field intensity in the far field calculated at the first peak wavelength.

 

Movie of simulation. A number of interesting phenomena are visible, including the strong influence of the nanoparticle on the scattering direction as well as on the scattering spectrum, the width of the pulse gives some indication about the spatial extent of the e-beam field profile and injection errors that occur at the beginning and end of the simulation. These errors exist because we only inject a finite length portion of the electron beam and are already minimized using a filter to control the dipoles' amplitude.

 

Convergence testing

This example was run with a mesh size of only 10nm. Some convergence testing should be done. Typically, a mesh size of 1-5 nm is required to obtain acceptable accuracy for gold nanoparticles in this wavelength range. The z span is also important in convergence testing where the injection error can be further reduced for longer z span.

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