Hello.

If one consider the original deduction of the gradient forces that is applied to the trapped particle we can find the following:

$F_{grad}=\frac{1}{4} \alpha \nabla E_0^2(r)$, where $E_0(r)$ is an amplitude of electric wave:$E(r,t)=E_0(r) \cos(\omega t)$.

See e.g.: Harada et al., Opt.Comm.124, 529, (1996)

However, on the other hand the potential energy of such system (dipole - external EM field ) might be deducted as: $U=(\alpha E(r,t))\:E(r,t)$ what after averaging over the time brings us to $U=\frac{1}{2} \alpha E_0^2(r)$.

Since the transition from the potential energy to the force occurs by $\nabla$ operator, I still be confused with the missed factor of 1/2.

Can someone explain where i did a mistake ?

Thank you in advance.