Ray optics

Ray optics explanation (unfocused laser). When the bead is displaced from the beam center (right image), the larger momentum change of the more intense rays cause a net force to be applied back toward the center of the laser. When the bead is laterally centered on the beam (left image), the resulting lateral force is zero. But an unfocused laser still causes a force pointing away from the laser.

Ray optics explanation (focused laser). In addition to keeping the bead in the center of the laser, a focused laser also keeps the bead in a fixed axial position: The momentum change of the focused rays causes a force towards the laser focus, both when the bead is in front (left image) or behind (right image) the laser focus. So, the bead will stay slightly behind the focus, where this force compensates the scattering force.

In cases where the diameter of a trapped particle is significantly greater than the wavelength of light, the trapping phenomenon can be explained using ray optics. As shown in the figure, individual rays of light emitted from the laser will be refracted as it enters and exits the dielectric bead. As a result, the ray will exit in a direction different from which it originated. Since light has a momentum associated with it, this change in direction indicates that its momentum has changed. Due to Newton's third law, there should be an equal and opposite momentum change on the particle.

Most optical traps operate with a Gaussian beam (TEM₀₀ mode) profile intensity. In this case, if the particle is displaced from the center of the beam, as in the right part of the figure, the particle has a net force returning it to the center of the trap because more intense beams impart a larger momentum change towards the center of the trap than less intense beams, which impart a smaller momentum change away from the trap center. The net momentum change, or force, returns the particle to the trap center.

If the particle is located at the center of the beam, then individual rays of light are refracting through the particle symmetrically, resulting in no net lateral force. The net force in this case is along the axial direction of the trap, which cancels out the scattering force of the laser light. The cancellation of this axial gradient force with the scattering force is what causes the bead to be stably trapped slightly downstream of the beam waist.

The standard tweezers works with the trapping laser propagated in the direction of gravity^[30] and the inverted tweezers works against gravity.

Electric dipole approximation

In cases where the diameter of a trapped particle is significantly smaller than the wavelength of light, the conditions for Rayleigh scattering are satisfied and the particle can be treated as a point dipole in an inhomogeneous electromagnetic field. The force applied on a single charge in an electromagnetic field is known as the Lorentz force,

${\displaystyle \mathbf {F_{1}} =q\left(\mathbf {E} (\mathbf {x} _{1})+{\frac {d\mathbf {x_{1}} }{dt}}\times \mathbf {B} \right).}$

The force on the dipole can be calculated by substituting two terms for the electric field in the equation above, one for each charge. The polarization of a dipole is ${\displaystyle \mathbf {p} =q\mathbf {d} ,}$ where ${\displaystyle \mathbf {d} }$ is the distance between the two charges. For a point dipole, the distance is infinitesimal, ${\displaystyle \mathbf {x} _{1}-\mathbf {x} _{2}.}$ Taking into account that the two charges have opposite signs, the force takes the form

${\displaystyle {\begin{aligned}\mathbf {F} &=q\left(\mathbf {E} (\mathbf {x} _{1})-\mathbf {E} (\mathbf {x} _{2})+{\frac {d(\mathbf {x} _{1}-\mathbf {x} _{2})}{dt}}\times \mathbf {B} \right)\\&=q\left(\mathbf {E} (\mathbf {x} _{1})+\left((\mathbf {x} _{1}-\mathbf {x} _{2})\cdot \nabla \right)\mathbf {E} -\mathbf {E} (\mathbf {x} _{1})+{\frac {d(\mathbf {x} _{1}-\mathbf {x} _{2})}{dt}}\times \mathbf {B} \right).\\\end{aligned}}}$

Notice that the ${\displaystyle \mathbf {E_{1}} }$ cancel out. Multiplying through by the charge, ${\displaystyle q}$, converts position, ${\displaystyle \mathbf {x} }$, into polarization, ${\displaystyle \mathbf {p} }$,

${\displaystyle {\begin{aligned}\mathbf {F} &=\left(\mathbf {p} \cdot \nabla \right)\mathbf {E} +{\frac {d\mathbf {p} }{dt}}\times \mathbf {B} \\&=\alpha \left[\left(\mathbf {E} \cdot \nabla \right)\mathbf {E} +{\frac {d\mathbf {E} }{dt}}\times \mathbf {B} \right],\\\end{aligned}}}$

where in the second equality, it has been assumed that the dielectric particle is linear (i.e. ${\displaystyle \mathbf {p} =\alpha \mathbf {E} }$).

In the final steps, two equalities will be used: (1) a vector analysis equality, (2) Faraday's law of induction.

1. ${\displaystyle \left(\mathbf {E} \cdot \nabla \right)\mathbf {E} =\nabla \left({\frac {1}{2}}E^{2}\right)-\mathbf {E} \times \left(\nabla \times \mathbf {E} \right)}$
2. ${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$

First, the vector equality will be inserted for the first term in the force equation above. Maxwell's equation will be substituted in for the second term in the vector equality. Then the two terms which contain time derivatives can be combined into a single term.^[31]

${\displaystyle {\begin{aligned}\mathbf {F} &=\alpha \left[{\frac {1}{2}}\nabla E^{2}-\mathbf {E} \times \left(\nabla \times \mathbf {E} \right)+{\frac {d\mathbf {E} }{dt}}\times \mathbf {B} \right]\\&=\alpha \left[{\frac {1}{2}}\nabla E^{2}-\mathbf {E} \times \left(-{\frac {d\mathbf {B} }{dt}}\right)+{\frac {d\mathbf {E} }{dt}}\times \mathbf {B} \right]\\&=\alpha \left[{\frac {1}{2}}\nabla E^{2}+{\frac {d}{dt}}\left(\mathbf {E} \times \mathbf {B} \right)\right].\\\end{aligned}}}$

The second term in the last equality is the time derivative of a quantity that is related through a multiplicative constant to the Poynting vector, which describes the power per unit area passing through a surface. Since the power of the laser is constant when sampling over frequencies much longer than the frequency of the laser's light ~10¹⁴ Hz, the derivative of this term averages to zero and the force can be written as^[32]

${\displaystyle \mathbf {F} ={\frac {1}{2}}\alpha \nabla E^{2}={\frac {2\pi n_{0}a^{3}}{c}}\left({\frac {m^{2}-1}{m^{2}+2}}\right)\nabla I(\mathbf {r} ),}$

where in the second part we have included the induced dipole moment (in MKS units) of a spherical dielectric particle: ${\displaystyle \mathbf {p} =\alpha \mathbf {E} (\mathbf {r} ,t)=4\pi n_{1}^{2}\epsilon _{0}a^{3}(m^{2}-1)/(m^{2}+2)\mathbf {E} (\mathbf {r} ,t)}$, where ${\displaystyle a}$ is the particle radius, ${\displaystyle n_{0}}$ is the index of refraction of the particle and ${\displaystyle m=n_{0}/n_{1}}$ is the relative refractive index between the particle and the medium. The square of the magnitude of the electric field is equal to the intensity of the beam as a function of position. Therefore, the result indicates that the force on the dielectric particle, when treated as a point dipole, is proportional to the gradient along the intensity of the beam. In other words, the gradient force described here tends to attract the particle to the region of highest intensity. In reality, the scattering force of the light works against the gradient force in the axial direction of the trap, resulting in an equilibrium position that is displaced slightly downstream of the intensity maximum. Under the Rayleigh approximation, we can also write the scattering force as

${\displaystyle \mathbf {F} _{\text{scat}}(\mathbf {r} )={\frac {k^{4}\alpha ^{2}}{6\pi cn_{0}^{3}\epsilon _{0}^{2}}}I(\mathbf {r} ){\hat {z}}={\frac {8\pi n_{0}k^{4}a^{6}}{3c}}\left({\frac {m^{2}-1}{m^{2}+2}}\right)^{2}I(\mathbf {r} ){\hat {z}}.}$

Since the scattering is isotropic, the net momentum is transferred in the forward direction. On the quantum level, we picture the gradient force as forward Rayleigh scattering in which identical photons are created and annihilated concurrently, while in the scattering (radiation) force the incident photons travel in the same direction and ‘scatter’ isotropically. By conservation of momentum, the particle must accumulate the photons' original momenta, causing a forward force in the latter.^[33]

Harmonic potential approximation

A useful way to study the interaction of an atom in a Gaussian beam is to look at the harmonic potential approximation of the intensity profile the atom experiences. In the case of the two-level atom, the potential experienced is related to its AC Stark Shift,

${\displaystyle \mathbf {\Delta E} _{\text{AC Stark}}={\frac {3\pi c^{2}\Gamma \mu }{2\omega _{0}^{3}\delta }}\mathbf {I(r,z)} }$

where ${\displaystyle \Gamma }$ is the natural line width of the excited state, ${\displaystyle \mu }$ is the electric dipole coupling, ${\displaystyle \omega _{o}}$ is the frequency of the transition, and ${\displaystyle \delta }$ is the detuning or difference between the laser frequency and the transition frequency.

The intensity of a gaussian beam profile is characterized by the wavelength ${\displaystyle (\lambda )}$, minimum waist ${\displaystyle (w_{o})}$, and power of the beam ${\displaystyle (P_{o})}$. The following formulas define the beam profile:

${\displaystyle I(r,z)=I_{0}\left({\frac {w_{0}}{w(z)}}\right)^{2}e^{-{\frac {2r^{2}}{w^{2}(z)}}}}$

${\displaystyle w(z)=w_{0}{\sqrt {1+\left({\frac {z}{z_{R}}}\right)^{2}}}}$

${\displaystyle z_{R}={\frac {\pi w_{0}^{2}}{\lambda }}}$

${\displaystyle P_{0}={\frac {1}{2}}\pi I_{0}w_{0}^{2}}$

To approximate this Gaussian potential in both the radial and axial directions of the beam, the intensity profile must be expanded to second order in ${\displaystyle z}$ and ${\displaystyle r}$ for ${\displaystyle r=0}$ and ${\displaystyle z=0}$ respectively and equated to the harmonic potential ${\displaystyle {\frac {1}{2}}m(\omega _{z}^{2}z^{2}+\omega _{r}^{2}r^{2})}$. These expansions are evaluated assuming fixed power.

${\displaystyle {\frac {1}{2!}}{\frac {\partial ^{2}I}{\partial z^{2}}}{\Biggr |}_{r,z=0}z^{2}={\frac {2P_{0}\lambda ^{2}}{\pi ^{3}w_{0}^{6}}}z^{2}={\frac {1}{2}}m\omega _{z}^{2}z^{2}}$

${\displaystyle {\frac {1}{2!}}{\frac {\partial ^{2}I}{\partial r^{2}}}{\Biggr |}_{r,z=0}r^{2}={\frac {4P_{0}}{\pi w_{0}^{4}}}r^{2}={\frac {1}{2}}m\omega _{r}^{2}r^{2}}$

This means that when solving for the harmonic frequencies (or trap frequencies when considering optical traps for atoms), the frequencies are given as:

${\displaystyle \omega _{r}={\sqrt {\frac {8P_{0}}{\pi mw_{0}^{4}}}}}$

${\displaystyle \omega _{z}={\sqrt {\frac {4P_{0}\lambda ^{2}}{m\pi ^{3}w_{0}^{6}}}}}$

so that the relative trap frequencies for the radial and axial directions as a function of only beam waist scale as:

${\displaystyle {\frac {\omega _{r}}{\omega _{z}}}={\sqrt {2}}{\frac {w_{0}\pi }{\lambda }}}$