This page shows two techniques of calculating optical force on a particle:
- Maxwell Stress Tensor (MST) technique
- Volumetric technic
Maxwell stress tensor technique
The time averaged force F on a particle due to harmonic fields may be calculated from the Maxwell Stress Tensor (MST). The net force can be found by integrating the MST over a closed surface surrounding the particle. We use a box of power monitors (four 1D monitors in 2D simulations, six 2D monitors in 3D simulations) to record the fields on this surface.
$$ F=\oint_{S} \sum_{\beta} \frac{1}{2} \operatorname{Re}\left(T_{\alpha \beta} \hat{n}_{\beta}\right)$$
where the elements of the Maxwell stress tensor for harmonic fields are given by
$$T_{\alpha \beta}=\varepsilon E_{\alpha} E_{\beta}^{*}+\mu H_{\alpha} H_{\beta}^{*} - \frac{1}{2} \delta_{\alpha \beta}(\varepsilon|\vec{E}|^{2}+\mu |\vec{H}|^2)$$
and \(\hat{n}\), is the unit normal of S.
The tweezer.fsp simulation contains an analysis group named optical_force_2D. This analysis group can be inserted from the Object library in the Advanced analysis category. The analysis group contains four identical subgroups, whose analysis scripts calculate the components of the stress tensor from the simulation. The analysis script of the main group integrates the stress tensor to provide the total force on the particle.
Volumetric technique
The force on a charged particle in the presence of an electric and magnetic field is given by
$$\vec{F}=q \vec{E}+q \vec{\nu} \times \vec{B}$$
where q, v, E, and B are, respectively, the charge, velocity, electric field and magnetic field.
In a medium with charge per unit volume and current density, the force per unit volume is given by
$$\vec{F}_{v}=\rho \vec{E}+\rho \vec{\nu} \times \vec{B}=\rho \vec{E}+\vec{J} \times \vec{B}$$
where ρ is the total charge per unit volume and J is the total current density. These quantities are given by
$$\begin{array}{l}{\rho=\varepsilon_{0} \vec{\nabla} \cdot \vec{E}} \\ {\vec J=-\frac{\partial \vec P}{\partial t}}\end{array}$$
where P is the polarization.
In Lumerical's FDTD solver, all the material properties are included in the permittivity. As a result there is no free current density and no free charge, therefore \(\vec{\nabla} \cdot \vec{D}=0\). This is an arbitrary choice without physical consequence. For example, in a conductor with conductivity σ, it is possible to solve Maxwell's equations using a relative permittivity of 1 and a free current density of \(\vec{J}=\sigma\vec{E}\). It is physically equivalent, however, to solve a system with no free current and relative permittivity of \(\varepsilon_{r}=1+i \sigma / \omega \varepsilon_{0}\). Both methods will give the same physical results.
In the frequency domain, and with Lumerical's sign convention of \(\vec P(\omega)=\int e^{i \omega t} \vec P(t) d t\), we have \(\vec{J}(\omega)=-i \omega \vec P(\omega)\) the final expression for the force per unit volume is therefore given by
$$\begin{aligned} \vec{F}_{v} &=\varepsilon_{0}(\vec{\nabla} \cdot \vec{E}) \vec{E}-i \omega \vec{P} \times \vec{B} \\ &=\varepsilon_{0}(\vec{\nabla} \cdot \vec{E}) \vec{E}-i \omega\left(\varepsilon-\varepsilon_{0}\right) \vec{E} \times \vec{B} \end{aligned}$$
In a medium with a background index that is not 1, it is most numerically efficient and accurate to get the net optical force that will result in motion of the particle by rescaling the background permittivity, yielding a final equation
$$\vec{F}_{v}=\varepsilon_{b} \varepsilon_{0}(\vec{\nabla} \cdot \vec{E}) \vec{E}-i \omega \varepsilon_{0}\left(\varepsilon_{r}-\varepsilon_{b}\right) \vec{E} \times \vec{B}$$
where εb is the background relative permittivity, and εr is the relative background permittivity throughout the volume.
Note that the equation for net force without rescaling with the background permittivity will give the total force on the volume including force on the background material which does not result in motion of the particle.
The volumetric technique is typically more accurate because many interpolation errors can be avoided. However, it can require a significant amount of memory because the electromagnetic fields and the permittivity must be recorded throughout the volume.
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Note: MST vs Volumetric technique The optical force can be calculated with either the MST or Volumetric analysis groups. The two techniques give the same result within numerical error, but each has its own strengths and weaknesses.
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