## Abstract

A novel graphical approach for probing multicomponent collective ejection
processes is described. *Image labeling* provides a visual means
for identifying ejection partners and their relative momenta, isolating specific
decay channels for detailed study and determining initial electronic and/or
molecular geometries prior to ejection. The power of the technique is
demonstrated by looking at strong-fieldmultiphoton-induce 3-atom Coulomb
explosion spectra.

© 2001 Optical Society of America

## 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 4*π* 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, H_{2} 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
10^{14} and 10^{16} W/cm^{2}.

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).

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 NO_{2}. 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.

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
Δ*ε*
(∝*r*Δ*r*, see Eq. 1) and Δ*θ* 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 NO_{2}. There are as many correlation images as there are
sectors. Typically, we employ about 500 sectors.

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 NO_{2} primarily (six-electron channel [6]), leading to N^{2+} ions moving downward
correlated with O^{2+} 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 CO_{2}. In distinction with NO_{2}, the
equilibrium geometry of CO_{2} 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 CO_{2} 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 CO_{2}, 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 NO_{2} correlation
images. In contrast, we frequently observe a degree of correlation near unity for
CO_{2}. 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
NO_{2}, 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.

## Acknowledgements

We thank B. Yorgey, T. Colvin, Jr. and F. Ademeitz for technical assistance with computer codes This work was supported by the National Science Foundation through grant PHY 9876976.

## References and links

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**3. **By Coulomb explosion, we mean the energetic dissociation of a multiply ionized molecule into atomic ions. The final kinetic energies of the atomic ions are determined mostly by the classical electric potential associated with the internuclear separation of the ionized nuclei just prior to the explosion.

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