Operation of a Stark decelerator with optimum acceptance

With a Stark decelerator, beams of neutral polar molecules can be accelerated, guided at a constant velocity, or decelerated. The effectiveness of this process is determined by the six-dimensional (6D) volume in phase space from which molecules are accepted by the Stark decelerator. Couplings between the longitudinal and transverse motion of the molecules in the decelerator can reduce this acceptance. These couplings are nearly absent when the decelerator operates such that only every third electric-field stage is used for deceleration, while extra transverse focusing is provided by the intermediate stages. For many applications, the acceptance of a Stark decelerator in this so-called $s=3$ mode significantly exceeds that of a decelerator in the conventionally used $(s=1)$ mode. This has been experimentally verified by passing a beam of OH radicals through a $2.6\text{−}\mathrm{m}$-long Stark decelerator. The experiments are in quantitative agreement with the results of trajectory calculations, and can qualitatively be explained with a simple model for the 6D acceptance. These results imply that the 6D acceptance of a Stark decelerator in the $s=3$ mode of operation approaches the optimum value, i.e., the value that is obtained when any couplings are neglected.

Figure 1

(Color online) Scheme of the experimental setup. A pulsed beam of OH radicals is produced via photodissociation of $\mathrm{H}\mathrm{N}{\mathrm{O}}_3$ seeded in Xe, Kr, or Ar. The beam of OH radicals passes through a $2.6\text{−}\mathrm{m}$-long Stark decelerator that consists of three modules of $\sim 100$ stages each. The OH radicals can be state-selectively detected using a laser-induced fluorescence scheme at the end of the decelerator, and in the region between the first two modules. In the top inset, a photograph of a decelerator module is shown.

Figure 2

Schematic representation of the two electric-field configurations that are used in the deceleration process, together with the potential energy $W$ for an OH molecule along the molecular beam axis. By switching between the two field configurations when the molecules are at the positions indicated by the vertical dashed lines, an amount of kinetic energy $\Delta W$ is removed from the molecules. In the conventional $(s=1)$ mode of operation, each electric-field stage is used simultaneously for deceleration (D) and focusing in alternating transverse directions $(F_x,F_y)$. In the $s=3$ mode of operation, only every third stage is used for combined deceleration and transverse focusing, while the intermediate stages provide additional focusing.

Figure 3

Time-of-flight profiles of OH radicals, recorded at the exit of the 316 stage Stark decelerator. The Stark decelerator is operated at the $s=3$ mode, and accelerates [curve (a)], guides [curve (b)], or decelerates [curve (c)] a packet of OH radicals with an initial velocity of $350\mathrm{m}∕\mathrm{s}$. The TOF profiles that result from simulations of the experiment are shown underneath the experimental profiles.

Figure 4

Time-of-flight profiles of OH radicals that exit the Stark decelerator using the $s=1$ and 3 mode of operation. The OH radicals are detected after 103 and after 316 electric field stages for $s=1$ and 3, respectively. The measurements are recorded under otherwise identical conditions, and are shown on the same vertical scale. The beam of OH radicals has a mean initial velocity of $350\mathrm{m}∕\mathrm{s}$. The mean final velocity of the molecules and the phase angle used are indicated for selected profiles.

Figure 5

The ratio of the maximum signal intensities (squares, connected by straight line segments) for $s=3$ operation versus $s=1$ operation as a function of the final velocity. The two additional data points (stars) apply to a bimodal operation of the decelerator (see text for details).

Figure 6

Lower curve: The ratio of the signal intensity for $s=1$ operation of a decelerator with 316 vs 103 stages as a function of the final velocity. Upper curve: The ratio of the signal intensity using a 316 stage decelerator operating on the $s=3$ vs the $s=1$ mode.

Figure 7

Maximum signal intensity of decelerated packets of OH radicals as a function of the final velocity (upper panel) and as a function of ${\varphi }_0$ (lower panel), using a 316 stage decelerator operating at the $s=3$ mode. Beams of OH radicals with three different initial velocities, produced by seeding in Xe, Kr, or Ar, have been used. The intensities that are obtained from numerical simulations of the experiment are shown as solid lines.

Figure 8

6D phase-space acceptance of a decelerator as a function of the phase angle ${\varphi }_0$, resulting from numerical trajectory calculations (squares connected with solid lines for $s=3$; triangles connected with dashed lines for $s=1$), together with the model predictions (solid line for $s=3$; dashed line for $s=1$). In the inset, the $s=1$ data are shown enlarged. In the lower part, the longitudinal phase-space distributions that result from the simulations are shown for ${\varphi }_0=-10°$, both for $s=1$ and 3.

Figure 9

Natural transverse oscillation frequency ${\omega }_y∕2\pi$ for an OH ($X{}^2\Pi _{3∕2}$, $J=3∕2$, $M_J\Omega =-9∕4$) radical as a function of its phase $\varphi$ in a Stark decelerator, for the operation modes $s=1$ and 3.

Figure 10

Schematic representation of the method used to calculate the transverse phase-space acceptance. (a) The trajectory of a molecule in longitudinal phase space, shown as a dashed line. (b) The time dependence of the natural transverse oscillation frequency is constructed from the time dependence of the phase $\varphi \left(t\right)$ and the phase dependence of the natural transverse oscillation frequency ${\omega }_y\left(\varphi \right)∕2\pi$. (c) The time-averaged transverse focusing force results in elliptical orbits in transverse phase space.

Figure 11

Prediction for the longitudinal (a), transverse (b), and total 6D (c) phase-space acceptance of a Stark decelerator for the operation modes $s=1$ (dashed curves) and $s=3$ (solid curves) as a function of the phase angle ${\varphi }_0$. For the transverse and 6D acceptances, three curves are shown for each mode of operation that predict the phase-space acceptance in three limiting cases, as explained in the text. The center curves describe the best estimate for the acceptance in the limit that instabilities in the decelerator can be neglected, and have been used in Fig. 8 of Sec. 3.