Calculating charge distributions and currents in metals induced by an optical field

The following pages describe how to make charge distributions and current measurements within metals. Two simple examples are provided below.

Background

To calculate the charge distributions and current densities, we treat each metal as a cloud of free electrons, i.e. a plasma. To calculate the current density in a plasma we first recognize that all material properties within the FDTD simulation are implemented via an effective material permittivity:

$$D = \varepsilon _{material} E$$

For the purposes of this calculation, we assume there are two contributions to the material permittivity: the background permittivity (in this case, the permittivity of free space) plus the contribution from the current density.

$$D = (\varepsilon _0 + \varepsilon _{plasma}) E$$

The first term is Do, the displacement field that would exist in free space for the given electric field. The second term is proportional to the current J. For more details, see the notes below.

\begin{aligned} D & = \varepsilon _{material} E\\  & = \varepsilon _0 E + \varepsilon _{plasma} E\\  & = D_0 + \frac{iJ}{\omega} \end{aligned}

Solving the above equation for J, we get

\begin{aligned} J & = -i \omega (D - D_0)\\ & = -i \omega (\varepsilon _{material} - \varepsilon _0)E \end{aligned}

Note: The total material permittivity is created from two contributions: one from the polarization of the medium due to bound charge, and one from the current density due to free charge. Ampere's law can be written as

$$-i \omega D_{background} = \nabla \times H - J$$

$$D_{background} = \varepsilon _0 E + P = \varepsilon _0 (1 + \chi) E$$

We can rewrite this equation as

\begin{aligned}-i \omega D _{background} &= \nabla \times H - J \\ -i \omega \left(D_{background} + \frac{iJ}{\omega} \right) &= \nabla \times H \\ -i \omega D_{material} &= \nabla \times H \end{aligned}

and, assuming that J is proportional to E, we have

\begin{aligned} D_{material} &= \varepsilon _0 E + P + \frac{iJ}{\omega} \\ &=(\varepsilon _0  + \varepsilon _0 \chi + \varepsilon _{plasma}) E \\ &= \varepsilon _{material} E \end{aligned}

For plasma materials, such as metals, we have assumed that the susceptibility of the medium is 0 (i.e. P=0).

Charge and current in a nanoparticle

This page describes how to calculate the charge and current density inside metals. The associated simulation files use the current charge density analysis object for calculating these quantities. This analysis object can be found in the object library in the Advanced analysis section.

Example

The associated files section provide a 2D and 3D example simulation using the current charge density analysis groups. You can copy and paste these groups into your simulations or insert them directly from the object library.

To reproduce the following figures, open and run usr_current_density_2D.fsp . After the simulation is complete, you can reproduce the following figures by visualizing the object data. The following results show the current and electric field data for lambda = 340nm. Also note that the image colorbar scales have been adjusted.

		The X and Y components of the current density J.   These figures show the current flowing back and forth in the X direction as the wave propagates along the particle in the Y direction.   Also notice that the current is zero outside the particle, as expected.
		The X and Y components of the Electric field.   The current density is proportional to the electric field, except for a 90 degree phase shift.

The same analysis group also calculates the divergence of the electric field, which gives the total charge density. This is shown in the plot below, we can see that the charges are confined to the edge of the metal, as expected. The free charge density is calculated using rho = eps0*|div(E)|. This assumes that the material is metal with no bound charges, hence eps0 is used instead of eps_material.

Current in a wire

This page describes how to calculate the total current within a metal wire. The associated simulation file uses the induced current analysis group that can be added to your file from the Advanced Analysis section of the object library.

Example

The analysis group in the associated file uses Ampere's law to calculate the current induced in a metal wire by the incoming EM fields.

$$I = \int_L{H \cdot dL} - \frac{d \left( \int_S{D \cdot dS} \right) }{dt}$$

To run this example, open the attached simulation file and enter the following script commands:

run;
runanalysis;
f=getdata("induced_current","f");
current=getdata("induced_current","I");
plot(f/1e9,abs(current),"f (GHz)","|current|","Induced current");