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Scattering of adiabatically aligned molecules by nonresonant optical standing waves

The state-dependent alignment of molecules affects their scattering from nonresonant standing waves.


INTRODUCTION
Molecular alignment by nonresonant laser fields has been studied extensively for the past three decades (1,2). Alignment or orientation of molecules is critical for investigations of structural aspects and has been exploited in various experiments including x-ray (3) or electron diffraction in the gas phase (4), generation of high harmonics (5), or tomographic reconstruction of molecular orbitals (6). Molecular alignment also plays a key role when controlling molecular translation with laser fields (7)(8)(9).
Alignment occurs through the state-dependent interaction of nonspherical molecules with laser fields, which creates directional states via hybridization of molecular rotation (10). As a result, for instance for linear molecules, the molecular axis becomes aligned along the laser polarization direction. The initially freely rotating molecules are confined and librate angularly around the laser polarization axis without dissipation of the rotational energy. Therefore, the molecules are more tightly angularly confined when their initial rotational energy is small or the laser intensity is large. Namely, the degree of alignment of a given directional state depends on the rotational state associated with it and on the laser intensity. The state-dependent molecular alignment and its coupling to translation are instrumental in determining the motion of molecules in laser fields (10)(11)(12)(13)(14)(15)(16). However, so far, experimental results have been interpreted without considering the effects of molecular alignment (17)(18)(19)(20)(21). Only recently have the state-averaged alignment effects and the state-dependent interaction been demonstrated for CS 2 molecules in propagating laser fields (7,8) and standing waves (22), respectively. A low rotational temperature T rot ~ 35 K of the molecules played a crucial role in both experiments. Nonetheless, state-dependent alignment was not considered in these experiments. A subsequent theoretical study predicted that scattering of CS 2 molecules at T rot ~ 1 K by an optical standing wave would exhibit strong state-dependent alignment effects on the molecular trajectories (23). The present study experimentally confirms this theoretical prediction.
An optical standing wave is created by two counter-propagating laser beams of wavelength  and amounts to a periodic structure with period /2. A matter-wave diffracted from the periodic structure exhibits diffraction peaks separated by integer multiples of 2ħk in momentum space, with k and -k the wave vectors of the two laser beams and k = 2/. This matter-wave diffraction is known as Kapitza-Dirac (KD) diffraction (24) and was first demonstrated for point-like particles such as atoms (25) and electrons (26). The scattering can also be understood as a deflection of particles by their interaction potential with an optical standing wave. The spatial gradient of the potential imparts a force to the particles, as a result of which their velocities vary after passing through the standing wave. For manipulating atoms, a dipolar interaction potential between the standing wave and the induced dipole moment of the atoms was applied and used to measure polarizabilities (27), and an accelerating, periodic, dipolar potential was used to accelerate metastable Ar atoms from 0 to 200 m/s (28).
Both the coherent and incoherent scattering has been extended to molecules. For large molecules such as C 60 and C 70 (29), the coherent KD effect has been explored in a Talbot-Lau interferometer (30). The Talbot-Lau interferometer with KD diffraction then served as an analytical tool to pinpoint the process where molecular fragmentation occurs in a mass spectrometer (31) and to measure the electronic properties of molecules (32). On the other hand, the classical dynamics of small molecules in standing waves has been studied to tailor the translational motions of molecules in a supersonic beam. For instance, standing waves were used to change the longitudinal or transverse velocity distributions of H 2 molecules (33) and CS 2 molecules (22), respectively, by aligning the standing wave parallel or perpendicular to the molecular beam axis.
When the frequencies of the two laser beams creating a standing wave differ by , the standing wave moves with velocity v sw = /k (34)(35)(36). Slow NO molecules (37) and stationary C 6 H 6 molecules (38) were produced by the incoherent molecular scattering from a standing wave moving at constant velocity along the molecular beam direction. However, in these extensions of coherent or incoherent KD scattering to molecules, the effects of molecular rotation remained largely unresolved. And yet, it is the molecular rotation, in addition to vibration, that distinguishes molecules from atoms or subatomic particles.
Here, we demonstrate the effect of state-dependent alignment on the scattering of CS 2 molecules by a standing wave formed by two counter-propagating pulsed infrared (IR) laser beams of identical properties: wavelength , waist radius (e −2 radius) w 0 , pulse duration [full width at half maximum (FWHM)] , polarization, and peak intensity I 0 . To investigate the alignment effect, we use rotationally cold molecules (T rot ~ 1 K) occupying only a few rotational states. Control experiments are performed with molecules at T rot ~ 35 K, where the alignment effect is predicted to be averaged out (23). The molecular beam of T rot ~ 1 or 35 K is transversely dispersed by the optical standing wave. We measure the resulting velocity distributions using a velocity map imaging technique (39). The shape of the distribution varies drastically with increasing I 0 . This effect is well reproduced by quantum classical simulations if these include the state-dependent alignment of the molecules. Our results therefore demonstrate how rotational state-specific alignment of molecules can be exploited to manipulate their translational motion. This illustrates that the rotational state-dependent force imparted by the nonresonant optical standing wave can be used as a new tool for rotational state selection of nonpolar molecules.
The interaction potential between the ground-state CS 2 molecules ( 1 Σ) and a laser field of intensity where  || and  ⊥ are the polarizability components parallel and perpendicular to the molecular axis,  is the polar angle between the molecular and laser polarization axes, and Z 0 is the vacuum impedance. Spherical harmonics Y j,M = |j,M> and their superpositions  J,M (I ) = ∑ j C j J,M (I ) | jM > (2) describe the rotational motion of the molecules in the absence (I = 0) and presence (I > 0) of the field, respectively. Here, j and M are the quantum numbers respectively associated with the rotational angular momentum and its projection onto the laser polarization axis with j restricted to even integers for 12 C 32 S 2 (40).  J,M evolves from  J,M (I = 0) = | j,M> to  J,M (I) when the field is turned on slowly over a time scale longer than the rotational period (41).
The expectation value < J,M |cos 2 | J,M > represents the statedependent directional properties of molecules in a laser field and is known as the alignment cosine. The alignment cosine denoted by <cos 2 > J,M = <cos 2 > J,M (I) is a rotational state-dependent function of the laser intensity. Thus, the polarizability component along the space-fixed laser polarization axis is  J,M (I) = ( || −  ⊥ ) <cos 2 > J,M +  ⊥ (12). Since the time scale of the intensity variation associated with molecular translation generally surpasses the rotational period by several orders of magnitude (14), the classical trajectories of molecules or the quantum phase accumulated along the beam path in slowly varying laser fields is determined by the approximate potential Fig. 1B, we plot the calculated alignment cosine <cos 2 > J,M (I) and the corresponding  J,M (I) for the lowest rotational states of J = 0 (black) and J = 2 (red, blue, and light orange) for an optical standing wave field characterized by the spatial intensity profile plotted in Fig. 1A. )| and |U 0,0 (x = 0)| and |U 2,0 (x = 0)| are, respectively, the largest with and without considering molecular alignment. Therefore, the state-dependent alignment needs to be considered for the proper analysis of experiments at low T rot . Figure 2 shows an illustration of our experiment [see (9) and Materials and Methods for details]. A rotationally cold molecular beam is collimated by two skimmers and a vertical slit before it interacts with the optical standing wave of intensity I(x,y,z,t) = 4I 0 exp[−2(y 2 + z 2 )/w 0 2 ]exp.[−4ln(2)(t/) 2 ]cos 2 (2x/), formed by two counterpropagating laser beams (IR1 and IR2). Their waist radius w 0 and pulse width  are 21.5 ± 1.0 m and 7.5 ± 0.1 ns, respectively. Only molecules that pass through the center of the standing wave and occupy the vibrational ground state are ionized by a probe dye laser beam (Probe) through resonance enhanced multiphoton ionization (REMPI) processes with the transition [ (42,43). An ion lens system consisting of three electrodes performs velocity map imaging, namely, ions with the same initial velocity are mapped onto the same position at the two-dimensional ion detector (39). The ion signal is amplified and converted into a light signal by a multichannel plate (MCP) and a phosphor screen (PS), respectively. The

RESULTS
is a convolution of the initial transverse velocity distribution function with the velocity changes during ionization v recoil and blurring at the detector. In our experiment, the detector blurring dominates f(v x ,v y ) off . The simulated and measured data agree well except that the measured image and profile have wider tails than simulated counterparts. This difference might be due to the high peak intensity (about 10 11 W/cm 2 ) of the probe laser. The detailed mechanisms of the dependence on the probe intensity are not yet fully understood.
The measured images and profiles vary drastically with increasing I 0 as shown in Figs. 3 (B to H) and 4 (B to H): Pairs of outer peaks appear sequentially, while the width and position of the central peak vary. The outer peaks are associated with rainbow-like singularities (15,22), which are substantially suppressed by the detector blurring. In Fig. 3H, there exist shoulders at v x = 12 and -29 m/s, which are manifestation of additional rainbow-like singularities and are absent in Fig. 4H. Also note that the dataset for I 0 = 2.4 × 10 10 W/cm 2 in Fig. 3D is similar to the one for I 0 = 3.2 × 10 10 W/cm 2 in Fig. 4E. This is well reproduced by our simulations including statedependent alignment. The deviation in Figs. 3 and 4 can be attributed to the same mechanisms related to the imperfect reproduction of the measured off images in Figs. 3A and 4A and to the lateral fluctuation of the IR foci (SD, 2.9 m). In the simulations, we use the <v 0x > value as a fitting parameter, determined as 10 ± 1 m/s, with other parameters predetermined in the previous experiment (T rot ) (9) or determined by deflection experiments with a single propagating laser beam (w 0 ). The details of the simulation can be found in Material and Methods.
In addition, simulated results using the alignment-ignored polarizability  J,M (0) are compared with the experimental data. Figure 5A shows the measured and the two simulated velocity profiles for I 0 = 3.2 × 10 10 W/cm 2 . Dissimilar from the simulated profile with  J,M (I), A molecular beam of CS 2 molecules is scattered by a pulsed optical standing wave. The rotationally cold molecular beam is produced by supersonic expansion from an Even-Lavie valve with stagnation pressure P 0 and temperature T 0 . The standing wave is formed by focusing two counter-propagating laser beams (IR1 and IR2). Their wavelengths  and waist radii w 0 are 1064 nm and 22 m, respectively. The velocity change due to the scattering is measured by the velocity map imaging technique. We ionize molecules passing through the center of the standing wave by a probe dye laser beam (Probe). An ion lens system consisting of three electrodes performs velocity map imaging. The ion signal is amplified and converted into a light signal by an MCP and a PS, respectively. The luminescence is recorded by an ICCD camera and a PMT. The accumulation time for the measured images is 1200 s, which corresponds to 12,000 laser shots. The images are normalized to the total signal intensity of each image. Gray profiles with shading and solid black curves correspond to the profiles of the measured and simulated images, respectively. The profile in (A) represents the initial transverse velocity distribution, which is centered at v x ≈ 10 m/s. As I 0 increases, the profile becomes asymmetric and its central peak position moves to v x ≈ −10 m/s. These asymmetries result from the phase-space rotation of the molecules during the laser pulse duration and the average initial velocity along the x axis, <v 0x > ≈ 10 m/s (see On the other hand, at the higher rotational temperature of T rot = 35 K, the two corresponding simulation results are almost identical and agree well with the measurements (Fig. 5C), thereby confirming previous experimental results (22) and theoretical predictions (23) quantitatively. The agreement also validates the determination of <v 0x >.   Fig. 3, a distinctive central peak around v x = -10 m/s appears at higher I 0 for T rot = 35 K. This peak turns up in (F) at I 0 = 3.9 × 10 10 W/cm 2 , while it is clearly visible at I 0 = 3.2 × 10 10 W/cm 2 for the lower rotational temperature (Fig. 3E). The shift of the peak arises from the half phase-space rotation of the initial distribution of the molecules. At T rot = 35 K, average values of <cos 2 > J,M and  J,M (I) are smaller than at T rot = 1 K, as the molecules are less aligned at the higher temperature. Since the degree of the rotation increases with the product of  J,M (I) and I, the half rotation requires higher I for T rot = 35 K.

DISCUSSION
The different dipolar potentials for the two cases shown in Fig. 1C are responsible for the dissimilar simulated profiles in Fig. 5A. The acceleration given by the optical dipole force F = −∇U determines molecular trajectories and final velocities. Figure 5D shows calculated molecular trajectories (solid curves) with  J,M (I) for I 0 = 3.2 × 10 10 W/cm 2 and initial conditions of v 0x = 10 m/s, v 0y = 0 m/s, v 0z = 1740 m/s, and J = 0. Two representative trajectories shown as thick solid curves (A and B) are compared with the corresponding ones simulated with  J,M (0) (thick dashed curves, A′ and B′). The alignment effect leads to a large variation in molecular positions. The slopes of the corresponding pairs of solid and dashed curves at t = 10 ns, representing the calculated final velocities, appear to be substantially different. The two aspects manifested in Fig. 5B can be understood qualitatively with the molecular dynamics in phase space. The two phasespace plots in Fig. 5 (E and F) show the calculated results at t = 5 ns with and without considering molecular alignment, respectively, together with the corresponding separatrices at t = 0 (dotted lines). Here, we neglect the effects of electron recoil and detector blurring. If the laser intensity I and the molecular polarizability  were constant, the time required for a half phase-space rotation, t half , would be reversely proportional to (4I) 1/2 for molecules near the center of the diagram where the potential can be assumed to be harmonic. In other words, the extent of phase-space rotation for a given time increases with (4I) 1/2 .
Although the standing wave is pulsed in our work, this proportionality still gives a qualitative understanding. Aspect (i), the decrease in I 0 * and I 0,min , is closely related to the condition at which t half =  (36,37). For example, t half = 7.1 ns at I 0 = 3.2 × 10 10 W/cm 2 , which is evaluated with I = I 0 and  = < J,M (4I 0 )>. Thus, the molecules near the center of the separatrix (blue trajectories in Fig. 5 (D and E) have performed nearly a half rotation at t = 5 ns, as shown in Fig. 5E corresponding to the minimum of W 0.5 . On the other hand, when the alignment effect is ignored, t half = 8.7 ns and the corresponding molecules (blue trajectories in Fig. 5F) rotate quite less than 180° at the same conditions. The half rotation might occur at I 0,min = 4.7 × 10 10 W/cm 2 , which is quite close to I 0,min of the dotted blue curve in Fig. 5B. Therefore, the half phase-space rotation during the laser pulse duration explains the rather distinctive peaks at v x ≈ -10 m/s of the profiles in Figs. 3 (E and F) and 4 (F to H). The underlying physics of the similar peak in Fig. 3 (G and H) is not clear at this moment. At least, these peaks are not caused by the detection of untrapped particles, to which strong peaks at an initial velocity or energy value was attributed in the previous scattering experiments (38,44), since they are different from the initial peak position in Fig. 3A.
Aspect (ii), the increase in W 0.1 , is associated with the molecules following anharmonic dynamics (45). The vertical width of the separatrix is proportional to (4I) 1/2 , which determines the maximum |v x | of molecules near the separatrix where the anharmonicity of the potential plays a important role. Since, generally,  J,M (4I 0 ) >  J,M (0), the alignment-considered analysis always shows a larger W 0.1 than the alignment-ignored analysis.
The rotational temperature dependence in the transverse velocity distributions affects the degree of phase-space rotation and the increase in the outer width, both of which are proportional to (4I) 1/2 . Since the state-averaged  J,M (4I) is bigger for T rot = 1 K than for T rot = 35 K, stronger I 0 is necessary for T rot = 35 K to attain the similar phase-space rotation and outer width for T rot = 1 K. This explains the resemblance between Figs. 3D and 4E. The distance between inner rainbow-like peaks also scales with (4I) 1/2 , which, together with the narrow distribution of rotational states at T rot = 1 K, accounts for the distinctive shoulders in Fig. 3H. At T rot = 35 K, the molecules occupy a large number of rotational states. The distance between the rainbowlike peak pair varies with the rotational state, and the maximum distance is smaller for the higher rotational temperature. Therefore, the weak rainbow-like feature spreads out, and it is not visible in Fig. 4H (22).
Overall, for T rot = 1 K, the simulation based on  J,M (I) shows much better agreement with the experimental data than the one based on  J,M (0). This agreement confirms the effect of state-dependent molecular alignment on molecular scattering by an optical standing wave.
Our demonstration has implications for various experimental studies on both coherent and incoherent scattering of molecules in directional states by optical standing waves. Our experiment can serve as a new tool for measuring the polarizability anisotropy  =  || −  ⊥ of small molecules. The alignment cosine depends on anisotropy, as does the interaction potential. Therefore, the molecular polarizability anisotropy could be measured by comparing widths of measured and simulated profiles as shown in Fig. 5 (B and C), with other parameters such as a rotational temperature of predetermined samples. Such a method for measuring the polarizability anisotropy is complementary to the techniques based solely on the molecular alignment by nonresonant propagating laser fields (46)(47)(48)(49).
Furthermore, scattering from a nonresonant optical standing wave can be applied to achieve rotational state selection of nonpolar molecules, complementing recently demonstrated techniques for state selection of polar molecules (50,51). According to our analysis considering the alignment effect, the velocity profile of Fig. 3B implies that the part of v x < −17.4 m/s (about 2.1% of the total area) contains molecules in the J = 0 state with a purity of 95%. The fraction and purity can be improved further by optimizing parameters such as I 0 , w 0 , , <v 0x >, and T rot . For example, the fraction and the purity for the part of v x < −29.9 m/s are 3.2 and 95.4%, respectively, in the simulation using I 0 = 1.16 × 10 10 W/cm 2 , w 0 = 80 m,  = 28 ns, and <v 0x > = 30 m/s (see fig. S2). These parameters are feasible with current technology (28). Furthermore, a larger w 0 makes this method more robust, because the lateral fluctuations of the IR foci (about 3 m) get less influential with increasing w 0 . Note that state-dependent alignment is critical not only in state-selection studies but also in state-dependent deceleration of molecules of low rotational temperature with moving lattice potentials (37,38), where an analysis neglecting the state-dependent alignment will lead to incorrect results.
Although our present analysis concerns CS 2 , the rotational state selection by an optical standing wave can be applied more broadly. Since the acceleration of two adjacent rotational states differs by the order of (B/I) 1/2 /m (10), the selection of the rotational ground state J = 0 can be easily implemented for all linear molecules with sufficiently large rotational constant B or small mass m. Further studies are required to test the applicability of this method to larger molecules, symmetric tops or asymmetric tops, and rotational states of J > 0. However, the effect of the state-dependent alignment will be critical for all these studies.
Last, we note that the degree of molecular alignment can be finetuned to some extent by varying the standing wave's polarization state. For instance, an optical standing wave formed by two counterrotating elliptically polarized light beams with ellipticity ϵ and intensity of (1 + ϵ 2 )I 0 exhibits linear polarization rotating periodically along the x axis, while its intensity profile is given by I(x) = 2I 0 [(1 + ϵ 2 ) + (1 -ϵ 2 )cos(4x/)] (52). Thus, varying ϵ makes it possible to adjust the minimum intensity 4I 0 ϵ 2 > 0, thereby allowing control of the degree of molecular alignment in the standing wave field.
The state-dependent directional properties of molecules are important in controlling translational molecular motions in various combinations of optical standing waves, electric, and magnetic fields. As well as the interaction of polarizable molecules with an optical field, it is known that interaction of polar molecules with combined optical and electric fields depends on their directional properties. Angular motions of molecules in such combined fields have been studied theoretically (53,54) and experimentally (55,56) for propagating laser fields. It will be intriguing to extend these studies toward controlling molecular translation using standing wave fields.

Experimental design
The apparatus operating at 10-Hz repetition rate is the same as the one used in our previous work (9), except optical standing waves replace propagating waves. A pulsed cold molecular beam is produced by expanding CS 2 molecules at 427 mbar, which is the CS 2 vapor pressure at 295 K, in 21 or 81 bar of He carrier gas through an Even-Lavie valve into high vacuum. Under these source conditions, the respective high-vacuum pressures in the source chamber are 3.9 × 10 −7 and 1.1 × 10 −6 mbar, respectively. The resulting rotational temperatures T rot were estimated to be 1 and 35 K, respectively, in the previous report (9). The molecular beam is collimated by two skimmers (Skimmer1 and Skimmer2) of 3-and 1-mm diameter, respectively. The second chamber is located between the two skimmers. Skimmer2 separates the second chamber and the detection chamber. The pressures of the two chambers are below 3.0 × 10 −8 mbar. A vertical slit of 200-m width and 3-mm height further collimates the molecular beam along the x axis. The collimated molecular beam is crossed at 89.7° by a pulsed standing wave. All the distances between the components are given in Fig. 2.
Two counter-propagating laser pulses (IR1 and IR2) with the same properties overlap at the center of the detection chamber to form the pulsed standing wave. A single pulse from an injection-seeded Nd:YAG laser, whose wavelength  and pulse width  are 1064 nm and 7.5 ± 0.1 ns (FWHM), respectively, passes through an optical isolator and is split into two by a beam splitter. After the beam splitter, the energy and polarization of the two pulses are adjusted by two sets of a zero-order half-wave plate and a Glan laser polarizer, respectively, to achieve the same peak intensity I 0 and linear polarization along the y axis. After propagating the same distance, they are focused by two lenses with focal lengths of 18.0 cm mounted on the opposite sides of the detection chamber. The waist radii w 0 of the focused laser beams are estimated as 21.5 ± 1.0 m.
The molecules affected by the pulsed standing wave are ionized by a linearly polarized probe pulse of  probe = 4 ns and  probe = 477.7 nm. At this wavelength, ionization occurs through a (3 + 1) REMPI process with the transition [ (42,43). The probe pulse is produced by a dye laser pumped by a third harmonic of another Nd:YAG laser, of which the energy and polarization are set to 1.0 mJ and vertical polarization by a zero-order half-wave plate and a Glan laser polarizer. After being expanded by a set of a concave and convex lenses, the beam is focused by the same plano-convex lens that is used to focus IR2, which results in its waist radius w 0,probe ~ 5 m. The probe pulse is delayed by 30 ns with respect to the IR pulses to avoid any two-color processes. The delay then requires a spatial shift of the dye focus along the z axis for the selected ionization of the dispersed molecules. To detect the most dispersed molecules, the y position of dye focal point is adjusted to y = 0.
The molecular ions are accelerated and focused by an electrostatic lens system onto a multi-channel plate (MCP) after flying 342 mm through a time-of-flight (TOF) tube. A repeller, an extractor, and ground electrodes made out of stainless steel constitute the ion lens system. They are 2-mm-thick disks of 90-mm diameter with holes at their center, whose diameters are 4 mm for the repeller and 20 mm for the others. The extractor is separated from the repeller and the ground by 16 mm. The voltages of the repeller and extractor are set to 900 and 644 V, respectively, which provide the velocity map imaging condition. Then, the ratio of the position at the detector to the TOF of an ion is its transverse velocity just after the ionization of a neutral molecule. This velocity is assumed to be the same as the velocity of the neutral molecule.
The ion signals are amplified and converted into luminescence by an MCP and a PS at its backside, respectively. The light signal from the PS is simultaneously detected and converted into electric signals by a PMT and an ICCD camera. The signals from the PMT and ICCD are transferred to a digital oscilloscope and a personal computer, respectively. We monitor the TOF spectrum on the scope to gate the intensifier, which allows us to record images of CS 2 + .

Velocity calculation
For the quantitative comparison of experimental and simulated data, we include the effect of the detector blurring in the calculation. The blurring at the MCP, the PS, and the intensifier constitute the total detector blurring function g(v x ,v y ) detector . To quantify it, we analyze the smallest ion images with the number and intensity of the laser pulses set to minimal values. We fit a two-dimensional Gaussian function to each of those images. The average FWHMs along the x and y axes of the resulting Gaussian functions are 12.9 ± 1.5 and 13.3 ± 1.5 m/s, respectively. We approximate g(v x ,v y ) detector to a twodimensional Gaussian function with these two FWHMs,  v x detector and  v y detector .
To simulate f(v x , v y ) off in the second panel in Figs. 3A and 4A, we approximate the initial velocity distribution function g(v 0x ,v 0y ) init to a two-dimensional Gaussian function centered at (<v 0x > = 10 m/s, <v 0y > = 0 m/s) with the FWHMs v 0x = 4.3 and v 0y = 3.6 m/s, assume isotropic electron recoil during ionization, and use g(v x , v y ) detector for the blurring at the detector. v 0x and v 0y are approximately determined from the line-of-sight argument. The given REMPI process results in v recoil = 2.7 m/s (8). Since the spreads of g(v x ,v y ) detector are much larger than the other two factors, the detector blurring governs f(v x , v y ) off .
The Monte Carlo sampling method is used to select the initial velocity (v 0x , v 0y , v 0z ), the initial position (x 0 , y 0 , z 0 ), and the initial rotational state |j,M> of each individual molecule. Initial transverse velocities are given by g(v 0x ,v 0y ) init . The probability function for v 0z is a Gaussian function with the most probable velocity v mp of 1740 m/s and an FWHM of 174 m/s. We determine z 0 from v 0z according to the following equation: z 0 = v mp t detectionv 0z t simul . Here, t detection and t simul are the detection and total simulation times, respectively. From this initial point, the individual molecule arrives at the detection plane z = v mp t detection at t = t detection . x 0 and y 0 are chosen randomly from a 200-m-long line and a Gaussian distribution with a 2.9-m-wide FWHM, respectively. The FWHM of the Gaussian function is decided by considering w 0,probe and the (3 + 1) REMPI process. The initial rotational state follows the Boltzmann distribution e −Bj(j + 1)/kT rot /q r , where k is the Boltzmann's constant and q r is the rotational partition function.
For the given standing wave intensity, the dipolar potential is  y, z, t ) dt(i = x, y, z) .
By collecting 20,000 trajectory calculation results, v i = v 0i + v i (i = x, y, z), we obtain the transverse velocity distribution function g(v x , v y ) on . In the velocity width calculations for T rot = 1 K (35 K), trajectories of 100,000 molecules are used for 1.5 × 10 10 W/cm 2 < I 0 < 3.0 × 10 10 W/cm 2 (2.0 × 10 10 W/cm 2 < I 0 < 2.8 × 10 10 W/cm 2 and 3.4 × 10 10 W/cm 2 < I 0 < 3.6 × 10 10 W/cm 2 ), the intensity ranges where the velocity widths change abruptly, to reduce the fluctuation in the values due to insufficient statistical average. With the standing wave field off, the corresponding distribution g(v x , v y ) off is equal to the initial transverse velocity distribution function g(v 0x , v 0y ) init .
The final velocity distribution functions f(v x , v y ) on and f(v x , v y ) off are the respective convoluted functions of g(v x , v y ) on and g(v x , v y ) off with g(v x ,v y ) ionization and g(v x ,v y ) detector . g(v x ,v y ) ionization originates from the velocity change v recoil induced by electron recoil during ionization. If spherical symmetric ionization is assumed, the resulting velocity distribution forms a thin spherical shell, v x 2 + v y 2 + v z 2 = v recoil 2 . Then, g(v x ,v y ) ionization is the projection of the sphere distribution on the detection plane.