Optical tweezers use a strongly focused laser beam to trap and manipulate microscopic particles.
The key idea is that a dielectric particle placed in a nonuniform optical field experiences forces caused by the transfer of momentum from light.
In a tightly focused beam, two force contributions are especially important:
the gradient force, which tends to pull the particle toward the region of highest intensity, and the scattering force, which pushes the particle along the beam direction because of radiation pressure.
In a simplified picture, the gradient force is associated with the spatial intensity variation near the focus, so one often writes schematically
\[
F_g \propto \nabla I.
\]
The scattering force instead scales more directly with optical power and forward momentum flow, so it is often written in the form
\[
F_s \propto \frac{n_m P}{c},
\]
where \(P\) is the beam power, \(n_m\) is the refractive index of the surrounding medium, and \(c\) is the speed of light.
In real trapping theory the exact coefficients depend on the particle size, refractive-index contrast, polarization, and whether the particle lies in the Rayleigh or Mie regime.
This calculator uses an order-of-magnitude estimate intended for intuition rather than a full electromagnetic solution.
First it estimates the focused-beam waist as
\[
w_0 \approx \frac{0.61\lambda}{NA},
\]
where \(\lambda\) is the trapping wavelength and \(NA\) is the numerical aperture of the objective.
A smaller waist means a stronger intensity gradient, which improves trapping.
The peak focal intensity is then estimated by the Gaussian-beam relation
\[
I_0 \approx \frac{2P}{\pi w_0^2}.
\]
To describe how strongly the particle polarizes in the field, the calculator uses the relative-index parameter
\(m=n_p/n_m\),
where \(n_p\) is the particle index and \(n_m\) is the medium index.
From this, it forms the usual Clausius-Mossotti-like contrast factor
\[
\chi = \frac{m^2-1}{m^2+2}.
\]
If the particle index is larger than the medium index, then \(\chi\) is positive and inward trapping is favored.
If the particle index is not larger, then ordinary gradient trapping becomes much weaker or can even reverse in sign.
The simplified force estimates are written as
\[
F_g \approx Q_g\frac{n_m P}{c},
\qquad
F_s \approx Q_s\frac{n_m P}{c},
\]
where \(Q_g\) and \(Q_s\) are dimensionless trapping efficiencies.
In this calculator they are constructed from the contrast factor, the beam waist, the particle size, and the numerical aperture so that the results stay in a realistic optical-tweezers range for common biological and colloidal trapping conditions.
This means the calculator is not claiming an exact first-principles result; it is providing a calibrated estimate suitable for learning and rapid comparison.
The trap stiffness is then estimated from a local harmonic approximation:
\[
F \approx -kx.
\]
If one takes a characteristic trapping displacement of about \(w_0/2\), then the stiffness scale becomes
\[
k \approx \frac{|F_g|}{w_0/2}.
\]
This gives a quick way to compare how “stiff” the trap is when power, wavelength, numerical aperture, or bead properties are changed.
A stiffer trap corresponds to a larger restoring force per unit displacement.
For the sample case of a
\(100\,\text{mW}\)
trap,
\(NA=1.2\),
water with
\(n_m=1.33\),
and a default high-index micrometer-scale bead,
the calculator gives a gradient force in the rough
\(10\)–\(20\,\text{pN}\)
range.
That is the same order of magnitude often discussed for practical optical tweezers in soft-matter and biological experiments.
In more advanced theory, the force model depends strongly on regime.
In the Rayleigh limit, the particle behaves like an induced dipole and the gradient force scales roughly with particle volume, while the scattering force rises with higher powers of size and depends on the wavelength more strongly.
In the Mie regime, full electromagnetic scattering becomes necessary and the force can no longer be captured accurately by simple dipole reasoning.
Aberrations, spherical interfaces, beam shape, polarization, and axial instability also become important.
Even with those limitations, the simplified relations are still very useful because they capture the most important experimental trends:
stronger power increases both gradient and scattering forces,
a higher numerical aperture makes the focus tighter and usually improves trapping,
and a larger refractive-index contrast strengthens the restoring force.
\[
w_0 \approx \frac{0.61\lambda}{NA},
\qquad
F_g \approx Q_g\frac{n_m P}{c},
\qquad
F_s \approx Q_s\frac{n_m P}{c},
\qquad
k \approx \frac{|F_g|}{w_0/2}.
\]
These formulas provide a practical first look at whether a laser trap is likely to be gradient-dominated, scattering-dominated, weak, or reasonably stiff for a chosen particle and optical setup.
