length.

For larger nanospheres, the measured size-dependent corrections must be applied. In the experiments described below, we find that the mechanical oscillation frequency is modulated by a finite size correction $\omega_M(r) = \omega_M(r\simeq 0) f(r)$ (see Fig. \[exp\_trap\_freq\] and description of the measurement of $f(r)$ in section 5 below). Thus, since: $$\begin{aligned}
\omega_M^2 \propto \frac{A(r)}{m},
\label{freq2}\end{aligned}$$

then $A(r)\equiv A_0(r) f^2(r)$ and the coupling is in turn modulated by the finite size correction. The experimental results suggest that for $r\lesssim 200$nm, then $f(r)\sim 1$ and $A(r) \simeq A_0(r)$.

For example, for $r=150$nm, $L=1$cm and $w=40\,\mu$m, then $A_0\simeq 8
\times 10^5$Hz. For reasonable values of cavity decay constants $\kappa \simeq 2-8 \times
10^5$Hz, then for $n_1 \sim 10^{9}$, $$\begin{aligned}
{\tilde g}_{1,2} \sim 5.4 \times 10^{-6} \left({\frac{A^3
n_{1,2}}{m}}\right)^{1/4} \simeq 10^6\,\mathrm{Hz} \gg \kappa\end{aligned}$$

For $r \lesssim 200$nm, ${\tilde g}_{1,2} \propto r^{3/2}$. Thus a 200nm sphere provides an optomechanical coupling about an order of magnitude larger than a 40-50nm sphere. To achieve a comparable increase in coupling by photon number enhancement would require increasing the driving power by a factor of order $10^4$.

A more careful analysis, including the effects of the finite-size correction function $f^2(r)$ is shown in Fig. \[beadg\]. We see that ${\tilde 