Repeated kicks from a circular optical trap ($f_p=f_T\Delta x/(L+\Delta x)$)

For a passing optical trap on a circle of circumference $L$ with nonoverlapping support, a particle that is stationary between encounters receives displacement $\Delta x$ in each encounter. The trap revolution frequency $f_T$ and particle revolution frequency obey $L{f}_p=\Delta x(f_T-f_p)$. This corrects the naive independent-kick estimate when the particle motion changes the encounter rate.

The exact separated-kick response is $f_p=f_T\Delta x/(C+\Delta x)$ for circumference $C=2\pi R$. Its relative correction to the independent-kick estimate is controlled by $\Delta x/C$. For a localized well with force-induced speed bounded by $ϵv_T$, $\Delta x/C\le (w/C){ϵ}^2/(1-ϵ)$. Thus high speed is sufficient when the support width is below one circumference. The local straight-force approximation also requires a narrow trap on the curved track.

A triangular localized potential energy has forces $+F$ behind and $-F$ ahead of the trap centre. With $v_c=F/\zeta =L{f}_c$, $\alpha =2X_0/L$ and $\beta =f_T/f_c$, its circular response is $f_p/f_c=\alpha \beta /({\beta }^2-1+\alpha )$ for $\beta >1$. A trapped particle has $f_p/f_c=\beta$ for $0\le \beta \le 1$, interpreting the discontinuous force at the cusp by a sticking or rounded-well limit.