Higher-order resonances in a Stark decelerator

The motion of polar molecules can be controlled by time-varying inhomogeneous electric fields. In a Stark decelerator, this is exploited to select a fraction of a molecular beam that is accelerated, transported, or decelerated. Phase stability ensures that the selected bunch of molecules is kept together throughout the deceleration process. In this paper an extended description of phase stability in a Stark decelerator is given, including higher-order effects. This analysis predicts a wide variety of resonances that originate from the spatial and temporal periodicity of the electric fields. These resonances are experimentally observed using a beam of OH $({}^2\Pi _{3∕2},v=0,J=3∕2)$ radicals passing through a Stark decelerator.

Figure 1

Scheme of the Stark decelerator, together with the Stark energy of a molecule as a function of position $z$ along the molecular beam axis. Adjacent electric field stages are spaced a distance $L=11\mathrm{mm}$ apart. Each electric field stage consists of two parallel 6-mm-diameter cylindrical rods with a center-to-center distance of 10 mm. A maximum voltage difference of 40 kV is applied to opposing rods in an electric field stage. The longitudinal phase space acceptance of the decelerator for $\mathrm{OH}(J=3∕2,M_J\Omega =-9∕4)$ is given for the resonances $s=1$, 3, and 19 (see text), together with the longitudinal emittance of the OH beam (shaded area).

Figure 2

Contour plot of the average velocity of $\mathrm{OH}(J=3∕2,M\Omega =-9∕4)$ radicals as a function of their initial position and velocity, that results from a one-dimensional numerical simulation of their trajectories through a 300-stage decelerator. The decelerator is operated at a phase angle ${\varphi }_0=0°$ and a “switching velocity” $v_{\mathrm{sw}}=150\mathrm{m}∕\mathrm{s}$. Resonances appear as regions in phase space where all the molecules have the same average velocity. The structure inside the separatrices is a result of the finite number of stages used in the simulation.

Figure 3

Scheme of the experimental setup. A pulsed beam of OH with a mean velocity of 450 m/s is produced via ArF-laser photodissociation of ${\mathrm{HNO}}_3$ seeded in Kr. The molecular beam passes through a skimmer, hexapole, and Stark decelerator into the detection region. State-selective LIF detection is used to measure the arrival time distribution of the $\mathrm{OH}(J=3∕2)$ radicals that exit the decelerator.

Figure 4

Observed TOF profile when the decelerator is operated at $v_{\mathrm{sw}}=450.0\mathrm{m}∕\mathrm{s}$ and phase angle ${\varphi }_0=0°$. The TOF profile that is obtained from a three-dimensional trajectory simulation of the experiment is shown underneath the experimental profile. The individual contributions of both $M_J\Omega$ components to the profile is indicated. In the inset, the longitudinal phase space distribution of $\mathrm{OH}(J=3∕2,M_J\Omega =-9∕4)$ radicals close to the end of the decelerator is shown, together with the separatrix, as it follows from the analytical model.

Figure 5

Observed and simulated TOF profiles showing a series of first-order resonances with $n=1$. The decelerator is operated at phase angle ${\varphi }_0=0°$ and a “switching velocity” $v_{\mathrm{sw}}=$ (a) 450.0, (b) 151.1, (c) 91.9, (d) 40.9, (e) 30.0, and (f) $23.6\mathrm{m}∕\mathrm{s}$. The TOF profiles that are obtained from three-dimensional trajectory simulations of the experiment are shown underneath the experimental profiles. In the insets, the longitudinal phase-space distributions for OH radicals in the $J=3∕2,M_J\Omega =-9∕4$ state, close to the end of the decelerator, are shown. In these phase-space plots the part of the beam that is responsible for the central feature in the observed TOF profiles is shown.

Figure 6

Observed and simulated TOF profiles when the decelerator is operated at values of $v_{\mathrm{sw}}$ such that $s=\ell ∕n$ ($\ell$ and $n$ odd; $n>1$) resonances appear at the center of the TOF distributions. The decelerator is operated with $v_{\mathrm{sw}}=$ (a) 270.1, (b) 71.1, and (c) $321.5\mathrm{m}∕\mathrm{s}$, giving rise to the resonances $s=5∕3$, $19∕3$, and $7∕5$ at the center of the TOF distributions, respectively.

Figure 7

Observed and simulated TOF profiles showing second-order resonances. The second-order resonances $s=2$ (a) and $3∕2$ (b) are observed at the center of the TOF distribution when the decelerator is operated with a “switching velocity” of 225.0 and $299.9\mathrm{m}∕\mathrm{s}$, respectively.

Figure 8

Observed (upper curves) and calculated (lower curves) TOF profiles when the molecular beam is modulated by the resonance $s=$ (a) 3 and (b) $5∕3$. The time sequences that are applied to the decelerator have been carefully selected to obtain a maximum contrast in the TOF profile. In the insets, the phase-space distributions of $\mathrm{OH}(J=3∕2,M_J\Omega =-9∕4)$ radicals are shown, when the most intense part of the beam is close to the exit of the decelerator.