Introduction

There has been growing interest, recently, in multiphoton nonsequential ionization and dissociative-ionization precesses. This interest is due to a desire to understand the physics associated with collective phenomena driven by manybody wavefunctions leading to results that cannot be explained by single-particle wavefunctions. Imaging, combined with correlation techniques, have proven to be extremely valuable for such studies [1],[2].

In a previous paper [1], we introduced an image spectrometer that is capable of collecting and analyzing charges ejected over 4π sr with high energy and angular resolution. We demonstrated that spatial imaging makes it possible to identify ejection partners unambiguously and determine if a specific charge is ejected by itself or in conjunction with another charge. This ability allows us to distinguish collective and sequential processes. The focus of our earlier work was on two-atom systems, H₂ and ${\mathrm{H}}_2^{+}$ in particular, and one-electron ionization in which only two entities needed to be correlated. Our current focus is on more complicated systems – three-atom systems and two-electron atoms – in which three or more entities must be correlated. To study such systems, we extended our approach developed for two-component correlation. The extension we describe in this paper involves a new way to visualize correlation in the ejection dynamics; we call this approach image labeling. Image labeling not only allows correlated partners to be viewed graphically, it allows the degree of correlation to be measured and the initial geometry prior to the multiphoton-induced atomic or molecular fragmentation to be determined. We will demonstrate the utility of image labeling on the Coulomb explosion [3] spectrum of 3-atom systems.

Imaging 3-atom ejection dynamics

Linearly polarized, diffraction limited 100 fs pulses at 800 nm from a mode-locked Ti:Sapphire laser were used to induce the Coulomb explosions. The pulses were focused into the 4π image detector with a 150 mm radius of curvature, 38 mm diameter spherical mirror to intensities that could be varied between 10¹⁴ and 10¹⁶ W/cm².

The detector’s design and principle of operation have been described in detail else-where [1],[4]. Briefly, it consists of an image quality microchannel plate (MCP) with a phosphor screen at the back end. Laser pulses are brought to a focus a distance l from the center of the MCP with their polarization axis oriented parallel to the MCP. Charges are swept toward the MCP with a uniform static electric field (~250 V/cm) as they are ejected from the focal point. The center of mass of the ejection dynamics coincides with the lab frame with its origin at the center of the image. The momentum distribution of the ejected charges is proportional to the spatial intensity distribution of the light emitted by the phosphor. The light is digitized by a digital CCD camera and recorded to disk in real time at up to 730 Hz. The laser repetition rate is set to match the camera frequency so that each frame corresponds to one and only one laser pulse. The camera exposure time and the MCP/phosphor gain are adjusted to produce a zero background while allowing near single-charge detection. Measurements are made at gas pressures between 1×10^-9 and 3×10^-8 Torr to minimize space charge effects. We exploited the different arrival times for ions of different mass and charge state to isolate specific charges for detailed study by gating the MCP. A temporal resolution of 100 ns is sufficient to separate low mass charges (H, C, N, O, etc.) and their various charge states. Larger time windows can be used to isolate groups of ions (N⁺ and O⁺, for example).

 

Fig. 1. The Coulomb explosion image of NO₂ (inset) and surface plot showing the momentum distribution of N^q+ and O^q+ ions (75% q=2 and 25% q=3). This distribution contains 500,000 laser pulses, each centered at 800 nm, with a pulse width of 100 fs and linearly polarized (horizontal in the inset) with a peak intensity of 10¹⁵ W/cm². The vertical distribution is composed of N^q+ ions while the distribution parallel to the polarization axis is a mixture of both N^q+ and O^q+ ions.

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Figure 1 shows a typical 3-atom Coulomb explosion image. This particular image displays the momentum distribution of the doubly and triply charged N and O ions originating from the explosion of NO₂. This total momentum distribution image is a composite of 500,000 camera frames (i.e., 500,000 laser shots). Along the polarization axis in Fig. 1, energetic ions appear in the image at a distance from the center of the image given by

where ε is the energy of the ion, q is the charge on the ion, F is the static electric field pushing the ions toward the detector and l is the distance between the focal point and the detector. Since the angular distribution is independent of the azimuthal angle about the polarization axis, ϕ, Eq. 1 holds absolutely only for polar angles θ=0 and π. It is possible to deconvolve the images so that Eq. 1 holds for all θ (see Ref. [4]). Although deconvolved images can be used to determine energy and angular distributions, they cannot be used to identify decay partners uniquely when three or more charges are ejected together. In this case, a unique identification is only possible with correlation techniques, such as the one we describe next.

Image labeling

Image labeling is the 2-dimensional analog of covariance mapping that has been employed so successfully to analyze 2-atom Coulomb explosions [5]. The technique allows all the correlated partners to be visualized with a family of momentum images similar to the total momentum distribution image but with only the correlated partners highlighted. We call each of these images a correlation image and the family of images a correlation map.

 

Fig. 2. This figure explains how to read a correlation image. The concentric circles and spokes in the upper left panel divide the momentum distribution image (the same as that displayed in the inset of Fig. 1) into sectors. The grey arrow indicates labeled ions (i.e., a subset of ions with a narrow momentum distribution moving downward, 6 o’clock); the white arrows indicate the correlated sectors (ions moving toward 2 and 10 o’clock). The right panel shows the correlation image for the Coulomb explosion where all three atomic ions are ejected simultaneously. This image shows the momenta of the charges ejected simultaneously. The grey (white) arrows indicate the final momenta of the labeled (correlated) charges. The correlation image in the lower left panel is the difference between averaging only those frames that have a nonzero count in the labeled sector and the average of all 500,000 frames.

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To produce a correlation image, we divide the momentum distribution image in Fig. 1 into sectors, S (r, θ). The size of each sector is chosen such that Δε (∝rΔr, see Eq. 1) and Δθ are the same for all sectors. We show this in Fig. 2 for the image in Fig. 1. Next, we calculate the statistical correlation, C_ij , between sectors i and j, which is given by

where 〈〉 means we average over all laser shots and ${\sigma }_{{\mathit{S}}_{\mathit{i}}}$ is the square root of the variance in the counts for sector i with laser shots. For C_ij to be meaningful, it is critical that the average number of ions generated per sector per laser shot be less than one and that the momentum distribution for each laser shot be stored separately. The i^th correlation image, which labels the i^th sector, is assembled by replacing the ion counts for the j^th sector in the composite image (i.e., the upper left panel in Fig. 2) with the correlation value C_ij . The right panel in Fig. 2 shows one correlation image of a 3-atom Coulomb explosion in NO₂. There are as many correlation images as there are sectors. Typically, we employ about 500 sectors.

 

Fig. 3. Two correlation images for the Coulomb explosion of CO₂ taken under the same conditions as Fig. 1. We label O ion moving toward 3 o’clock in the left image and those moving toward 6 o’clock in the right image. We isolate linear explosion events on the left and bent events on the right.

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The correlation images not only tell us which ions are ejected simultaneously, they give us a clearer picture of the explosion dynamics. Specifically, they make it possible to extract the final momentum of each of the correlated charges and the total energy for the event. The length and direction of the arrows in the right panel in Fig. 2 indicated the momentum of each charge relative to the center of mass of the dynamics, the center of the image. Imposing energy and momentum conservation allows us to determine the initial configuration (e.g., internuclear separation and bond angle in the case of molecular Coulomb explosion) of the system just prior to the explosion. For example, the correlation image in Fig. 2 corresponds to the case where six electrons are removed from NO₂ primarily (six-electron channel [6]), leading to N²⁺ ions moving downward correlated with O²⁺ ions moving toward the upper left and right corners of the image. In this particular case, the explosion occurred from a geometry where the bonds between the N and O nuclei are stretched about a factor of 2.5 compared with the equilibrium bond lengths (0.12 nm) while the bond angle is approximately the same as the equilibrium bond angle (134°). The correlation map, the family of correlation images, provides a way to study the ejection dynamics as a function of the momentum of a particular partner. Each correlation image corresponds to labeling a different region of the momentum distribution image [7].

We are using image labeling to explore the details of strong-field dynamics of linear and bent systems. Figure 3 shows how we can isolate linear and bent channels for Coulomb explosions of CO₂. In distinction with NO₂, the equilibrium geometry of CO₂ is linear. We can study the linear channel by labeling ions along the polarization axis as done in the left panel. In this case we label an O ion on the righthand side of the system and the correlation image highlights the companion O ion on the opposite side of the systems as well as the C ion in the center. The bent channels are studied by labeling the C ions that are ejected along an axis perpendicular to the polarization axis.

Given the energy and momentum of each partner, it is possible to test models of the Coulomb explosion quantitatively. Consider CO₂ again. From Fig. 3, we can extract an initial potential energy, the classical electrostatic potential, of about 58 eV prior to the explosion of the linear system. This implies a C-O separation prior to the explosion of about 0.25 nm. Ionization at extended lengths in diatomic systems at 100 fs has been well documented [8]–[10] and apparently is active in 3-atom systems as well [11],[12]. When we label bent events, we extract a minimum initial bond angle of about 170°. This minimum angle can be understood in terms of the bending vibrational frequency. For the ground state of CO₂, this frequency is 667 cm^-1; the corresponding force constant is about 57 N/m. Classically, these parameters will produce a minimum angle in the neighborhood of 168°, in good agreement with what we extract.

The family of correlation images provide a quantitative measure of the correlation as well. The degree of correlation, for example, can be determined by comparing the C_ij values of the correlated area with that of the labeled area. As an example, the degree of correlation between the labeled and the correlated areas in Fig. 2 is about 0.5, which means these ions participate in multiple dissociation channels. This is typical for most NO₂ correlation images. In contrast, we frequently observe a degree of correlation near unity for CO₂. This is the case for the left correlation image in Fig. 3; the degree of correlation for the right image is about 0.7. The uncertainty in these values is between ±0.1 and ±0.2.

Although the C_ij values for the sectors can be used to determine the degree of correlation, we find it more accurate to work directly with the pixels, as shown in the lower left panel of Fig. 2. This correlation image was generated by taking the difference between the average distribution shown in the upper left panel in Fig. 2 (all 500,000 frames) and an average distribution composed only of frames that have nonzero counts in the i^th sector – a selective average. The two approaches give about the same result but the selective average approach provides higher resolution and more direct control over the size of the labeled area.

Isolating and controlling dynamics

In closing we point out that image labeling should be a useful tool for quantum control studies. The 3-atom strong-field dissociative-ionization process on which we have been focusing serves to illustrate this point as well. The complete breakup of a system XYZ into X, Y and Z ions can proceed through several possible channels. Two possibilities are a sequential dissociation-channel, involving an intermediate state such as XY+Z followed by the breakup of XY, and a simultaneous dissociation-channel where X+Y+Z ions are produced in one step. Figure 4 shows these channels not only exist in NO₂, they can be distinguished unambiguously. An ability to isolate multiple channels to essentially the same final state (complete decomposition in this case) provides an opportunity to focus on the Lagrangian of the system, which determines which path the system takes not just the final products. Furthermore, since labeling isolates, and hence, simplifies the momentum distribution image, this approach might possibly provide a clearer picture of the physics underlying optimal control algorithms.

 

Fig. 4. Two correlation images for NO₂ taken from the same data set as Fig. 1 showing a sequential dissociation-channel (left) and a simultaneous dissociation-channel. The sequential channel involves an explosion of NO₂ into NO+O ions followed by the explosion of the NO ion. Pictured on the left is the explosion of NO. The explosion dynamics is asymmetric, the center of mass of the two correlated charges is not the center of the image.

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