A. The Maxwell Stress Tensor

The Maxwell equations of the electromagnetism provide a frame in which the interaction between radiation and matter can be studied from a classical point of view. It allow us to understand what are the mechanisms that make possible the optical trapping of particles. Here, we will develop the general theory of radiation-matter interaction starting from the principle of conservation of momentum combined with the Maxwell equations [159]. The derivation is done for radiation and matter that interact in the vacuum. The obtained expressions can be extended to other homogeneous and isotropic dielectric mediums by substituting the electric permittivity ( $ \epsilon _0$) and the magnetic permeability ($ \mu_0$) of the vacuum by the actual values of the medium ( $ \epsilon,\mu$).

The total electromagnetic force $ \vec{F}$ that acts on a charged particle is given by the Lorentz force [159]

$\displaystyle \vec{F} = \frac{d\vec{p}}{dt} = q\left(\vec{E}+\vec{v}\times\vec{B}\right)$ (A.1)

where $ q$ is the charge of the particle, $ \vec{v}$ is the velocity of the particle and $ \vec{E}$ and $ \vec{B}$ are the electric and magnetic fields.

If we sum the contribution of all the particles contained in the volume V we can write the total force ( $ \Delta \vec{F}$) acting on the volume,

$\displaystyle \Delta\vec{F}=\int_V\left(\rho\vec{E}+\rho\vec{v}\times\vec{B}\right) dV$ (A.2)

where $ \rho$ is the charge density and $ \vec{J}=\rho\vec{v}$ is the current density vector.

We can also define the force acting on a tiny volume,

$\displaystyle d\vec{F}=\left(\rho\vec{E}+\vec{J}\times\vec{B}\right) dV$ (A.3)

that allow us to define the force per unit volume as,

$\displaystyle \vec{f}=\frac{d\vec{F}}{dV}=\left(\rho\vec{E}+\vec{J}\times\vec{B}\right)$ (A.4)

At this point, we can use the two following Maxwell equations (Poisson and Maxwell-Ampère equations) to eliminate $ \rho$ and $ \vec{J}$,

$\displaystyle \rho$ $\displaystyle =$ $\displaystyle \epsilon_0\left(\vec{\nabla}\cdot\vec{E}\right)$  
$\displaystyle \vec{J}$ $\displaystyle =$ $\displaystyle \frac{1}{\mu_0}\vec{\nabla}\times\vec{B}-\epsilon_0\frac{\partial\vec{E}}{\partial t}$ (A.5)

and write the density of force in terms of the electromagnetic fields,
$\displaystyle \vec{f}$ $\displaystyle =$ $\displaystyle \epsilon_0\left(\vec{\nabla}\cdot\vec{E}\right)\vec{E}+\left(\fra...
...}\times\vec{B}-\epsilon_0\frac{\partial\vec{E}}{\partial t}\right)\times\vec{B}$  
  $\displaystyle =$ $\displaystyle \epsilon_0\left(\vec{\nabla}\cdot\vec{E}\right)\vec{E}+\frac{1}{\...
...right)\times\vec{B}-\epsilon_0\frac{\partial\vec{E}}{\partial t}\times\vec{B}~.$ (A.6)

We want to group the time dependent contributions. We note that using the product derivative rule and another Maxwell equation (Maxwell-Faraday equation) $ \left(-\frac{\partial\vec{B}}{\partial t}=\vec{\nabla}\times\vec{E}\right)$ we can write the last term as,

$\displaystyle \frac{\partial\vec{E}}{\partial t}\times\vec{B}$ $\displaystyle =$ $\displaystyle \frac{\partial\left(\vec{E}\times\vec{B}\right)}{\partial t}-\vec{E}\times\frac{\partial \vec{B}}{\partial t}$  
  $\displaystyle =$ $\displaystyle \frac{\partial\left(\vec{E}\times\vec{B}\right)}{\partial t}+\vec{E}\times\left(\vec{\nabla}\times\vec{E}\right)$  
  $\displaystyle =$ $\displaystyle \mu_0\frac{\partial\vec{S}}{\partial t}+\vec{E}\times\left(\vec{\nabla}\times\vec{E}\right)$  
  $\displaystyle =$ $\displaystyle \mu_0\frac{\partial\vec{S}}{\partial t}-\left(\vec{\nabla}\times\vec{E}\right)\times\vec{E}$ (A.7)

where we have used the definition of the Poynting vector $ \vec{S}=\frac{1}{\mu_0}\left(\vec{E}\times\vec{B}\right)$.

The second terms of equations A.6 and A.7 are the cross product of the curl of a vector with itself, which can be written as:

$\displaystyle \left(\vec{\nabla}\times\vec{B}\right)\times\vec{B}=\left(\vec{B}\cdot\vec{\nabla}\right)\vec{B}-\frac{1}{2}\vec{\nabla}\vec{B}^2$      
$\displaystyle \left(\vec{\nabla}\times\vec{E}\right)\times\vec{E}=\left(\vec{E}\cdot\vec{\nabla}\right)\vec{E}-\frac{1}{2}\vec{\nabla}\vec{E}^2~.$     (A.8)

Substituting A.8 into A.6 and A.7, and A.7 into A.6 we have
$\displaystyle \vec{f}$ $\displaystyle =$ $\displaystyle \epsilon_0\left[\vec{E}\left(\vec{\nabla}\cdot\vec{E}\right)+\left(\vec{E}\cdot\vec{\nabla}\right)\vec{E}\right]$  
    $\displaystyle +\frac{1}{\mu_0}\left[\vec{B}\left(\vec{\nabla}\cdot\vec{B}\right)+\left(\vec{B}\cdot\vec{\nabla}\right)\vec{B}\right]$  
    $\displaystyle -\vec{\nabla}\left(\frac{\epsilon_0}{2}\vec{E}^2+\frac{1}{2\mu_0}\vec{B}^2\right)-\frac{1}{c^2}\frac{\partial \vec{S}}{\partial t}~.$ (A.9)

An extra term proportional to $ \vec{\nabla}\cdot\vec{B}=0$ that vanishes has been added to make the expression look more symmetric. Here it is convenient to introduce the Maxwell Stress Tensor $ \overleftrightarrow{T}$,

$\displaystyle \overleftrightarrow{T}_{ij}=\epsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}E^2\right)+\frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}B^2\right)~.$ (A.10)

The component $ T_{ij}$ of the tensor is the flux (i.e., the time rate of change) of the $ i$ component of the electromagnetic momentum across the plane $ j$. The tensor has units of pressure. The diagonal elements $ \overleftrightarrow{T}_{ii}$ represent a force acting in a direction that is perpendicular to the surface. The off-diagonal elements $ \overleftrightarrow{T}_{ij} (i\neq j)$ represent a shear stress, acting in a direction that is parallel to the surface. All the terms in A.9 except the last one are exactly the divergence of the Maxwell Stress tensor. Using this notation, the force density can be written in a simpler manner,

$\displaystyle \vec{f}=\vec{\nabla}\cdot\overleftrightarrow{T}-\frac{1}{c^2}\frac{\partial\vec{S}}{\partial t}~.$ (A.11)

This equation is the local version of the conservation law of momentum and it holds in every region of the space. If we now want to calculate the total force acting on a material system of volume $ V$ enclosed in a boundary surface $ A$ that interacts with the electromagnetic field we can integrate equation A.11 and write,
$\displaystyle \vec{F}$ $\displaystyle =$ $\displaystyle \int_V\left(\vec{\nabla}\cdot\overleftrightarrow{T}-\frac{1}{c^2}\frac{\partial\vec{S}}{\partial t}\right)dV$  
$\displaystyle \vec{F}$ $\displaystyle =$ $\displaystyle \oint_A\overleftrightarrow{T}\cdot d\vec{a}-\frac{d}{dt}\frac{1}{c^2}\int_V\vec{S}dV$ (A.12)

where we have used the Gauss' theorem to convert a volume integral into a surface one ($ d\vec{a}$ is the normal element of area). Now we can identify the three terms of the previous equation and give physical meaning to them. The first term is the total force acting on volume $ V$ and can be written as $ \vec{F}=d\vec{P}_\mathrm{mech}/dt$, where $ \vec{P}_\mathrm{mech}$ is the mechanical linear momentum of the system enclosed in $ V$. On the other hand, the last term can be identified as the derivative of the total electromagnetic momentum $ \vec{P}_\mathrm{field}$ in the volume $ V$,

$\displaystyle \vec{P}_\mathrm{field}=\frac{1}{c^2}\int_V\vec{S}dV~.$ (A.13)

Putting everything together we can write,

$\displaystyle \frac{d\vec{P}_\mathrm{mech}}{dt}=\oint_A\overleftrightarrow{T}\cdot d\vec{a}-\frac{d\vec{P}_\mathrm{field}}{dt}$ (A.14)

and grouping the time-dependent terms it can be expressed as a conservation law

$\displaystyle \frac{d}{dt}\left(\vec{P}_\mathrm{mech}+\vec{P}_\mathrm{field}\right)=\oint_A\overleftrightarrow{T}\cdot d\vec{a}~.$ (A.15)

The right term of the equation accounts for the flux of momentum across the surface $ A$ into the volume $ V$. In other words, it is the time rate of change of electromagnetic momentum inside the volume $ V$. This flux acts on the combined system of particles and fields. The left hand side of the equation accounts for the changes in the linear momentum of the particles and the electromagnetic field. Thus it is a balance equation. It stands that what gets into volume $ V$ (i.e., right hand side) affects whatever was inside (i.e., left hand side).

The experimental setup described in section 2.2.1 was designed to measure forces using the conservation of linear momentum. The force exerted by a laser beam on a particle can be measured starting from equation A.15.

JM Huguet 2014-02-12