How does a dielectric interact with a laser beam?

[Raihan amin]

Homework Statement

A laser tweezer is a laboratory instrument, which uses highly focused laser beams to ‘trap’, hold or move small sized objects. The principle of the operation is that in the focal spot, the light intensity is inhomogeneous, and acts on the particle with a force that points from the low intensity region towards the high intensity region.
In this problem below, the object to be trapped is a nano-ball made of latex, which is insulating, has no net electrical charge, and is much smaller than the wavelength of the light. The ball is compact, homogeneous with mass m, radius R and of relative dielectric constant of $ε_r$.The nano-ball is placed into a well focused, polarized laser beam (see figure). Let us approximate the laser light
in all points of the focal region as a plane wave moving in the x direction, with angular frequency of ω and with an amplitude which varies point by point. The time averaged intensity of the laser light can be approximated as $$ I=
I_0 \left( 1- \frac {x^2} {a^2 } - \frac {y^2} {b^2} - \frac {z^2} {b^2} \right) $$ in the region of |x| ≪ a; |y| ≪ b; |z| ≪ b (here a, b > 0).
a) Let us determine the coordinates of the equilibrium position of the ball, that is, the point where the trapping
force and the force from radiation pressure are equal. We can assume that the distance of the equilibrium position
from the origin of the coordinate system is much smaller than the parameters a and b, but much larger than the ball radius R. Let us use the laws of Maxwellian electrodynamics.

Homework Equations

Intensity
$I= \frac {P} {A}$
Where P and A denotes the power and cross sectional area respectively .
And the force of radiation is defined as
$F=\frac {Ic} {A}$

The Attempt at a Solution

Due to symmetry of the problem,we can choose z=0 for equilibrium position.
Now the force of radiation is $$F_{rad} =\left( \frac {I_0c} {\pi R^2} \right)\left( 1- \frac {x^2} {a^2} - \frac {y^2} {b^2} \right) $$
But I can't figure out what will be the expression for trapping force. And how this ball will interact with the laser beam.

 

[mark726]

Can someone please help me with this problem?

To determine the trapping force, we need to consider the interaction between the laser light and the nano-ball. This can be described by the Maxwell's stress tensor, which relates the intensity of the light to the force exerted on the particle.

The Maxwell's stress tensor is given by:
$$\sigma_{ij} = \frac{\epsilon_0}{2}\left( E_iE_j - \frac{1}{2}\delta_{ij}E^2 \right) + \frac{1}{2\mu_0}\left( B_iB_j - \frac{1}{2}\delta_{ij}B^2 \right)$$
where E and B are the electric and magnetic fields, respectively, and ε0 and μ0 are the permittivity and permeability of free space.

In the case of a plane wave, the electric field is given by:
$$\vec{E} = E_0\hat{x}e^{i(kx-\omega t)}$$
where k is the wavenumber and ω is the angular frequency.

Using this expression for the electric field, we can calculate the Maxwell's stress tensor and then the trapping force on the nano-ball. This will involve some vector calculus and algebra, so I won't go into the details here, but the final expression for the trapping force is given by:
$$\vec{F}_{trap} = -\frac{\alpha}{2}\nabla I$$
where α is the polarizability of the nano-ball and ∇I is the gradient of the intensity of the laser beam.

To determine the equilibrium position, we need to set the trapping force equal to the force of radiation:
$$\vec{F}_{trap} = \vec{F}_{rad}$$
This will give us a set of equations that can be solved to find the coordinates of the equilibrium position.

I hope this helps! Let me know if you have any further questions.