Effect of rotational-state-dependent molecular alignment on the optical dipole force

The properties of molecule-optical elements such as lenses or prisms based on the interaction of molecules with optical fields depend in a crucial way on the molecular quantum state and its alignment created by the optical field. Herein, we consider the effects of state-dependent alignment in estimating the optical dipole force acting on the molecules and, to this end, introduce an effective polarizability which takes proper account of molecular alignment and is directly related to the alignment-dependent optical dipole force. We illustrate the significance of including molecular alignment in the optical dipole force by a trajectory study that compares previously used approximations with the present approach. The trajectory simulations were carried out for an ensemble of linear molecules subject to either propagating or standing-wave optical fields for a range of temperatures and laser intensities. The results demonstrate that the alignment-dependent effective polarizability can serve to provide correct estimates of the optical dipole force, on which a state-selection method applicable to nonpolar molecules could be based. We note that an analogous analysis of the forces acting on polar molecules subject to an inhomogeneous static electric field reveals a similarly strong dependence on molecular orientation.

Figure 1

Two schemes for the transverse manipulation of molecules. (a) A single propagating infrared (IR) laser beam and (b) a pulsed optical standing wave are used for the deflection and dispersion of molecular beams, respectively. The $\mathrm{C}{\mathrm{S}}_2$ molecular beam propagates along the $z$ axis, while the IR laser beams propagate along the $x$ axis. The coordinate origin is the focal point of the IR laser beams; the time zero is the moment when the intensity of the pulsed laser beam has its maximum value. The gray transparent beams indicate the molecular beam with the laser field off, while the pink beams indicate the molecular beam with the laser field on. The black dashed line represents the initial $y$ position of the molecular beam.

Figure 2

${\alpha }_{J,M}^U\left(I\right), {\beta }_J{,}_M\left(I\right)$, and ${\alpha }_{J,M}^{\mathbf{F}}\left(I\right)$ as a function of intensity for the selected rotational states. The first, second, and third rows are for $J\le 4, J=10$, and $J=20$, respectively. For the same $J$ value, curves for different $\left|M\right|$ values are distinguished by different line brightnesses (i.e., darker curves represent larger $\left|M\right|$ values). The solid and dashed curves distinguish the states with even and odd ($J-\left|M\right|$) values, respectively.

Figure 3

Three characteristic intensities $I^{(+)}$ (square and solid line), $I^{\left(\min \right)}$ (circle and dashed line), and $I^{(-)}$ (triangle and dashed-dotted line) for different rotational states. The variations of $I^{(+)},I^{\left(\min \right)}$, and $I^{(-)}$ are shown as a function of $J$ for (a) $\left|M\right|=0$ and $J$, as well as (b) $\left|M\right|=1$ and $J-1$. For $J=16$ and $J=10$, the three characteristic intensities are plotted against (c) even and (d) odd $\left|M\right|$ values, which correspond to even and odd ($J-\left|M\right|$) values, respectively.

Figure 4

The molecular interaction potential and the resulting dipole force given by the propagating wave of $I_0=100\times {10}^{10}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. Black and red curves are the variation of the negative potential and the corresponding optical dipole force, respectively, along the approximated path of the molecules occupying a specific rotational state. The potential and the force are calculated with (solid curves) and without (dashed curves) considering the molecular alignment effect. The numbers at the top left in each graph denote the quantum numbers $J$ and $\left|M\right|$ of the rotational state. The first column contains graphs for states of different $J$ values and same $\left|M\right|$ value (namely, $\left|M\right|=0$). The second and the third columns contain graphs for rotational states with various $\left|M\right|$ values with $J$ fixed at 10 and 20, respectively.

Figure 5

The molecular interaction potential with the optical standing wave and the resulting optical dipole force. The peak intensity $I_0$ of IR1 and IR2 is set to $5.0\times {10}^{10},10\times {10}^{10}$, and $20\times {10}^{10}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ for the first, the second, and the third columns, respectively. The format of the graph is the same as that in Fig. 4, except that the potential and the force are plotted along the $x$ axis. In each column, $J$ varies from 0 to 20 while $\left|M\right|=0$.

Figure 6

The molecular interaction potential with the optical standing wave and the resulting optical dipole force for the same $J$ value (namely, $J=10$) and different $\left|M\right|$ values (namely, $\left|M\right|=1$, 2, 9, and 10). The format of the figure is the same as that of Fig. 5.

Figure 7

Simulated velocity profiles $h\left(v_y\right)$ of the deflected molecules by the propagating laser beams with $I_0={10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The calculation was performed for the rotational temperature (a) $T=35$ and (b) 1 K. The three profiles were calculated using the alignment-neglecting state-dependent polarizability ${{\alpha }_J}_{,M}$ (black dotted curve), the alignment-including state-averaged polarizability $\langle {\alpha }_{J,M}^{\mathbf{F}}\left(I\right)\rangle$ (blue dashed curve), and the alignment-including state-dependent polarizability ${\alpha }_{J,M}^{\mathbf{F}}\left(I\right)$ (red solid curve). The velocity distribution in the absence of the laser field is represented by a gray solid curve.

Figure 8

Velocity $v_y$ of the $\mathrm{C}{\mathrm{S}}_2$ molecules under the influence of the propagating laser beams with $I_0={10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ for (a) $J\le 4$, (b) $J=10$, and (c) $J=20$. The initial conditions of the molecule—namely, $v_{0y},v_{0z}$, and $y_0$—are set to $0,v_{\mathrm{mp}}$, and ${\omega }_0/2$, respectively. The color and style of the curves representing different rotational states is the same as that in Fig. 2.

Figure 9

Simulated velocity profiles $h\left(v_x\right)$ of molecules dispersed by the optical standing wave with $I_0=1.0\times {10}^{10},5.0\times {10}^{10},10\times {10}^{10}$, and $20\times {10}^{10}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The upper and lower panels show the velocity profiles calculated for $T=35$ and 1 K, respectively. The coding of the curves is the same as in Fig. 7. The initial velocity distribution multiplied by 1/3 is shown by the gray solid curve in (a).