Nonlinear optomechanical pressure

A transparent material exhibits ultrafast optical nonlinearity and is subject to optical pressure if irradiated by a laser beam. However, the effect of nonlinearity on optical pressure is often overlooked, even if a nonlinear optical pressure may be potentially employed in many applications, such as optical manipulation, biophysics, cavity optomechanics, quantum optics, and optical tractors, and is relevant in fundamental problems such as the Abraham-Minkoswky dilemma or the Casimir effect. Here, we show that an ultrafast nonlinear polarization gives indeed a contribution to the optical pressure that also is negative in certain spectral ranges; the theoretical analysis is confirmed by first-principles simulations. An order-of-magnitude estimate shows that the effect can be observable by measuring the deflection of a membrane made by graphene.

Figure 1

(a) Sketch of the simulated structure, the input field, and the block are indicated; (b) snapshot from one typical FDTD simulation in cw of the field $E_x$ during propagation in the ($y$,$z$) section at $t=65$ fs and $P_0=95$ kW. The BB position is indicated by the black thick line in the horizontal plane.

Figure 2

(a) Dispersion of the linear refractive index $n_0\left(\lambda \right)$ (left scale) and the nonlinear Kerr coefficient $n_2\left(\lambda \right)$ (right scale) used for the theoretical analysis and in the numerical simulations (the model is detailed in Appendix pp5); (b) pressure per unit of intensity (force per Watt) for various $n_2$ in units of ${10}^{-22}{\text{m}}^2/\text{W}$; (c) nonlinear pressure coefficient versus wavelength, with $n_2$ as in panel (a); (d) nonlinear pressure coefficient versus pulse duration ($L=2\mu$m, $\lambda =800$ nm).

Figure 3

(a) Transmitted instantaneous power in the cw case for input peak power $P_0=15$ kW (thick red line) and $P_0=95$ kW (blue line); (b) as in (a) with pulsed excitation with pulse duration $T_0=10$ fs; (c) $\Delta I$ after Eq. (11) for $P_0=95$ kW and $n_2\cong {10}^{-22}$ m${}^2$/W (thick red line) and $n_2\cong 2\times {10}^{-22}$ m${}^2$/W (blue thin line); (d) as in (c) for pulsed excitation with $T_0=10$ fs.

Figure 4

(a) Calculated force $F_z\left(t\right)$ for two input peak powers $P_0=25$ kW (thick green line) and $P_0=95$ kW (blue thin line). Note the initial transient regime for $t<15$ fs needed for the wave to travel within the block ($n_2\cong {10}^{-22}$ m${}^2$/W); (b) the pressure per unit intensity calculated for various values of $n_2$ indicated in the panel and given in units of ${10}^{-22}$ m${}^2$/W (c); the pressure is calculated as the time average in the stationary regime $t>40$ fs; (c) nonlinear pressure coefficient $p_2$ as determined from data in panel (b) for various $n_2$.

Figure 5

Left panel, $p_2$ coefficient versus input wavelength for a cw excitation ($n_2\cong {10}^{-22}$ m${}^2$/W); note the region of negative $p_2$. Panels on the right show the trend of the calculated force $F_z\left(t\right)$ for specific wavelengths, as indicated.

Figure 6

(a) Calculated force $F_z\left(t\right)$ versus time in the presence of a pulsed excitation ($T_0=10$ fs, $P_0=70$ kW); note that the force is at a maximum in correspondence of the entrance and of the exit of the pulse from the BB; (b) pressure $p$ versus input intensity $I_0$ for the various pulse duration corresponding to the dots in panel (c); (c) nonlinear pressure $p_2$ versus pulse duration $T_0$ at fixed power $P_0=70$ kW and $n_2\cong {10}^{-22}$ m${}^2$/W.