Fourier moment analysis of velocity-map ion images

Article abstract of DOI:10.1063/1.1514978

A new method for extracting velocity and angular momentum distributions from velocity-map ion images was developed. The technique was demonstrated by using it to analyze sets of images from two different photolysis processes. The potential of the method for application to bimolecular reactions was illustrated using simulated images based on the results of recent quantum scattering calculations on the reaction H+D₂→HD(v=0,j=0,9)+D.

Full text of DOI:10.1063/1.1514978

Fourier moment analysis of velocity-map ion images
Mark J. Bass, Mark Brouard, Andrew P. Clark, and Claire Vallance
Citation: The Journal of Chemical Physics 117, 8723 (2002); doi: 10.1063/1.1514978
View online: http://dx.doi.org/10.1063/1.1514978
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/117/19?ver=pdfcov
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 117, NUMBER 19
15 NOVEMBER 2002
Fourier moment analysis of velocity-map ion images
Mark J. Bass, Mark Brouard,a) Andrew P. Clark, and Claire Vallance
Department of Chemistry, The Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford,
OX1 3QZ, United Kingdom
͑
Received 11 April 2002; accepted 27 August 2002͒
An alternative to inverse Abel transform and forward convolution methods is presented for
extracting dynamical information from velocity-map ion images. Unlike most competing methods,
that presented here does not require the probed three-dimensional distribution to possess cylindrical
symmetry. The new method involves analysis of the Fourier moments of images measured in
different experimental geometries, and allows speed distributions, angular differential cross
sections, and angular momentum alignment and orientation to be determined from raw images of the
products of photodissociation and photon-initiated bimolecular reactions. The methodology is
developed within the semiclassical framework of Dixon’s bipolar moment formalism ͓R. N. Dixon,
J. Chem. Phys. 85, 1866 ͑1986͔͒, although it is equally applicable to other common formulations of
the product scattering distribution. To allow a comparison of the method with the Abel inversion,
which requires that the velocity distribution of the probed product has an axis of cylindrical
symmetry, the method is applied to newly acquired experimental images of atomic chlorine
produced in the photolysis of NOCl. Extraction of product rotational alignment information is
illustrated using newly acquired images of rotationally aligned NO formed by NO photolysis.
2
Application of the Fourier moment methodology to studies of bimolecular reactions is also
demonstrated, using simulated images for the reaction HϩD →HD(vϭ0,jϭ0,9)ϩD. © 2002
2
American Institute of Physics. ͓DOI: 10.1063/1.1514978͔
I. INTRODUCTION
several limitations. A considerable number of researchers are
now using imaging techniques to investigate systems for
which the symmetry constraints of the inverse Abel trans-
form render it invalid, and alternatives must be found. The
transform is also highly sensitive to shot noise in the image.
This has become more of a problem with the advent of
Velocity-map ion imaging is a powerful experimental
technique that allows direct measurement of a two-
dimensional ͑2D͒ projection of the velocity distribution of a
photofragment or reaction product. Although experimental
procedures are now emerging which allow a 2D ‘‘slice’’
through the three-dimensional ͑3D͒ distribution to be ob-
tained directly ͑see, for example, Refs. 1 and 2͒, these slicing
methods involve an inherent loss of signal which may not be
affordable in all circumstances. For this reason, velocity-map
ion imaging, in its 2D projection guise, is likely to remain an
important experimental technique for some time. In order to
take full advantage of the technique, reliable methods are
required for reconstructing the 3D scattering distribution and
extracting the dynamical quantities of interest from the 2D
images. The two main approaches used to date have been the
inverse Abel transform and forward convolution methods.
The inverse Abel transform3 is a direct mathematical
transform that returns a slice through the 3D velocity distri-
bution for systems containing an axis of cylindrical symme-
try. The transform is ideally suited to studies of simple pho-
tolysis systems with no product angular momentum
alignment, and has been widely used in the analysis of im-
7
event-counting techniques, which provide a significant in-
crease in velocity resolution at the expense of a ‘‘grainy’’
appearance to the images. For the same reason, systems with
inherently low signal levels also present problems for the
inverse Abel transform.
An alternative approach involves forward convolution of
the image, whereby images are simulated based on a trial
scattering distribution and compared with the experimental
data. The trial distribution may then be adjusted iteratively
until satisfactory agreement is obtained. In a variation on the
forward convolution approach, a set of ‘‘basis images’’ is
simulated and fitted to the experimental images to extract the
parameters of interest. One clever example of an iterative
approach to generating the 3D scattering distribution, devel-
,4
8
oped by Vrakking, exploits the similarities that exist for
cylindrically symmetric systems between the form of the an-
gular and radial distributions of the 3D distribution and of its
2D projection. While, like the Abel inversion, the method is
limited to distributions with an axis of symmetry, the re-
turned 3D distributions are virtually noise-free. In general,
forward convolution approaches are not limited to cylindri-
cally symmetric distributions, and may be used to analyze
images both from imaging experiments using the conven-
tional single molecular beam and from crossed beam experi-
ments. Examples include applications as diverse as photo-
5
6
ages from such systems. Recently, Rakitzis has shown that
with careful choice of experimental geometries it is also pos-
sible to determine the photofragment alignment from a series
of Abel-invertible images. Even so, the inversion does have
a͒Author to whom correspondence should be addressed. Electronic mail:
mark.brouard@chemistry.ox.ac.uk
0021-9606/2002/117(19)/8723/13/$19.00
8723
© 2002 American Institute of Physics
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1

8724
J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Bass et al.
9
electron imaging, crossed beam studies of inelastic
1
0–12
scattering,
and determination of orbital alignment of
atomic photolysis products.1
3,14
Nestorov et al. were the first to apply Dixon’s semiclas-
1
5
sical bipolar moment formalism to the analysis of velocity-
1
6
map images. Their approach involves forward convolution
of basis image functions, which are then fitted to experimen-
tal images in order to extract the bipolar moments that char-
acterize vector correlations between the transition dipole mo-
ment of the parent molecule and the recoil velocity and
angular momentum distribution of the photofragment.
In the current work, a Fourier moment analysis for
velocity-map images is presented. The analysis is carried out
within the framework of the bipolar moment formalism,
though it is equally applicable to density matrix or other
representations of the product scattering. In a similar ap-
proach to that used in Doppler-resolved laser-induced fluo-
rescence dynamics studies,17 functions depending on indi-
vidual terms in the bipolar harmonic expansion of the
scattering distribution may be projected out from the images
and analyzed separately to obtain information on angular
correlations in both photodissociation and photon-initiated
bimolecular reactions. The Fourier moment method has sev-
eral major advantages over previous methods. Unlike the
Abel inversion, it does not require the 3D scattering distri-
bution to have an axis of cylindrical symmetry, and may
therefore be used to investigate angular momentum align-
ment and orientation in molecular systems, the presence of
which in general destroys the cylindrical symmetry of the
detected fragment distribution. Unlike many forward convo-
lution methods, no prior knowledge of the scattering distri-
bution is required. In this sense, the principle advantage of
the Abel transform, that it is an ‘‘analytic’’ solution to the
inversion problem, is retained. Also, the method is relatively
insensitive to noise in the image data. This is due to the fact
that every angle in the image contributes to each point in the
Fourier moments of the image—the functions from which
the dynamical parameters of interest are determined. Averag-
ing over the angular coordinate in this way means that the
signal-to-noise ratio of the Fourier moments is considerably
higher than that of the raw image.
To demonstrate the versatility of the Fourier moment
method, we have applied it to newly acquired images of
photofragments from two very different photolysis systems.
In the first, photolysis of NOCl near 240 nm, we have ig-
nored alignment effects and concentrated on extracting the
velocity distribution and velocity-dependent spatial anisot-
ropy distribution of the Cl fragment from the images. The
distributions are compared directly with those obtained from
an Abel inverted image. One of the principal advantages of
the Fourier moment analysis scheme is that it may be applied
to images of noncylindrically symmetric distributions. We
have demonstrated this using newly acquired images of
state-selected NO fragments from 308 nm photolysis of
FIG. 1. Schematic of the experimental apparatus.
from images of reactive scattering products. In a future paper
we will apply the method to the determination of electronic
orbital alignment in the photodissociation of Cl2 .18
II. EXPERIMENT
A. NOCl photolysis
NOCl photolysis experiments were carried out using a
standard velocity-map ion imaging setup, shown schemati-
cally in Fig. 1. A skimmed molecular beam of NOCl seeded
in helium passes through a 2 mm hole in the repeller plate
and is intersected by a laser beam from a frequency doubled
LPD series tunable dye laser, which serves both to photolyze
NOCl and to ionize the Cl photofragments via (2ϩ1) reso-
nantly enhanced multiphoton ionization ͑REMPI͒. Ionized
photofragments are extracted along a 40 cm flight tube
through 20 mm holes in the extraction plates to a 40 mm
chevron microchannel plate detector coupled to a P47 phos-
phor screen. The resulting images are captured by an inten-
sified charge coupled device camera ͑Photonic Science͒
gated to the time-of-flight of the ion of interest, then sent to
a PC for signal averaging and processing. Images are gener-
ally averaged over 10–20 000 laser shots. Use of velocity-
mapping ion optics introduces a calibration factor into the
conversion of the radial coordinate of the image into photo-
fragment velocity. This factor was determined as a function
NO . The complete set of alignment moments obtained are
2
compared with the results of Doppler resolved laser-induced
fluorescence experiments recently carried out in our labora-
tory. Finally, we also demonstrate the potential use of the
analysis scheme in determining differential cross sections
of extraction voltage using images of Cl from Cl photolysis
2
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J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Analysis of velocity-map ion images
8725
and NO from NO photolysis, for which the photofragment
2
velocities may be calculated exactly from known bond dis-
sociation energies.
NOCl was prepared by mixing small amounts of NO2
and Cl in a helium buffer gas ͑around 1 part NO to 4 parts
2
2
Cl up to a partial pressure of ϳ5% of the total gas pressure͒.
2
Images were obtained for both ground and spin–orbit excited
state Cl photolysis products. Ground state fragments were
4
2
4
2
detected via both the 4p P3 � P and 4p P � P
/2
3/2
1/2
3/2
REMPI transitions at 240.46 and 239.84 nm, respectively,
and excited state fragments were detected on the 4p D
P1/2 transition at 240.17 nm. Images were recorded us-
ing both horizontal and vertical laser polarizations.
4
FIG. 2. The time-of-flight frame of reference.
3
/2
2
19
�
k1
q
K
k2
K
K
1/2
B ͑k ,k ;⍀ ,⍀ ͒ϭ
͑Ϫ1͒ ͓K͔
ͩ
ͪ
0
1
2
v
j
͚
q
0
Ϫq
B. NO photolysis
2
ϫC ͑⌰ ,⌽ ͒C
͑⌰ ,⌽ ͒,
k q
v
v
k Ϫq
j
j
1
2
For the NO photolysis experiments, a beam of 2% NO
2
2
͑
2͒
in helium was intersected by counterpropagating pump and
probe laser beams, timed to fire simultaneously. The pump
beam was provided by the 308 nm output from a Lambda
Physik EMG103 excimer laser. The NO photofragments
were probed state selectively by (1ϩ1Ј)REMPI, with the
resonant photon provided by the doubled dye laser output
described previously, and the ionization photon provided by
the 308 nm photolysis pulse. Using this scheme it was pos-
sible to use an unfocused probe beam, eliminating probe-
only signal in which photolysis and one-color (1
ϩ1)REMPI are effected by the focused dye laser output. NO
where ͓K͔ϭ2Kϩ1, c(0)ϭ1, c(2)ϭ2/5 for a photolysis
process, and c(2)ϭ␤/5 for a photon-initiated reaction.17 The
angles ⍀ ϭ(⌰ ,⌽ ) and ⍀ ϭ(⌰ ,⌽ ) are the polar coor-
dinates of the product velocity and rotational angular mo-
mentum vectors, and the ͑body fixed͒ bipolar moments
v
v
v
j
j
j
K
0
b (k ,k ; ) are the ͑in general͒ speed-dependent coeffi-
cients of the expansion. The angular coordinates ⍀ and ⍀
v
1
2
v
j
are defined relative to the lab frame axis system, in which z
is parallel to the photolysis polarization vector and x lies
along the photolysis propagation vector. For clarity in the
following discussion, two further reference frames are now
defined. In the time-of-flight frame, shown in Fig. 2, z lies
along the time-of-flight axis and x along the photolysis
propagation vector; the polar coordinates of the product ve-
locity vector in this frame are denoted ⍀ . The detection
frame has z along the probe laser propagation axis and x
along the probe polarization vector.
fragments in (vϭ0,Nϭ30) were probed on the (0,0)Q (30)
1
2
ϩ
2
and (0,0)R (30) transitions of the A( ⌺ )� X( ⌸1/2) band
1
at 224.87 and 224.35 nm, respectively. Images were recorded
in two different experimental geometries, denoted HH and
HV, denoting the polarizations of the pump and probe lasers
as ‘‘horizontal’’ ͑H͒ or ‘‘vertical’’ ͑V͒ with respect to the
plane of the image.
T
The detection frame defined previously is the customary
frame in which to describe the intensity of the velocity-map
III. THEORY AND ANALYSIS
image arising from ions produced via a REMPI detection
scheme.1
5,20,21
However, the time-of-flight frame provides
The basis of the Fourier moment analysis scheme lies in
the fact that the intensity distribution in a velocity map im-
age ͑i.e., the projection of the 3D scattering distribution onto
the image plane͒ may be expressed as a Fourier series in the
angular coordinate of the image. Analysis of the Fourier mo-
ments of the image, and their dependence on pump–probe
geometry and probe transition, then allows information on
vector correlations between the electric vector of the pho-
tolysis light and product translational and rotational motions
to be extracted from the data.
the most natural coordinate system for describing velocity-
map ion images: ␪ is the angle the product velocity vector
T
makes with the time-of-flight axis, while ␾T is the polar
coordinate of the image, i.e., the angle made between the
projection of the product velocity onto the image plane and
the x axis. For this reason, it is convenient to express the
scattering distribution in time-of-flight-frame coordinates by
rewriting the lab-frame bipolar harmonics appearing in Eq.
͑1͒ as a linear combination of time-of-flight-frame bipolar
harmonics. The expansion coefficients are simply the rota-
A. Fourier moments of the image
k
1
tion matrix elements D (␣,␤,␥), where ͑␣,␤,␥͒ are the
q q
Ј
The normalized product scattering distribution following
polarized laser photolysis may be expressed as an expansion
in bipolar harmonics15
Euler angles describing the rotation
k1
q
K
k2
K
K
1/2
B ͑k ,k ;⍀ ,⍀ ͒ϭ
͑Ϫ1͒ ͓K͔
ͩ
ͪ
0
1
2
v
j
͚
q
0
Ϫq
1
P͑v,⍀ ,⍀ ͒ϭ
c͑K͓͒k1͔
v
j
6␲2 Kϭ0,2 k ,k
͚ ͚
1
ϫC
͑⌰ ,⌽ ͒
k Ϫq j j
1
2
2
K
K
ϫ͓k ͔b ͑k ,k ;v͒B ͑k ,k ;⍀ ,⍀ ͒ ͑1͒
2
0
1
2
0
1
2
v
j
k
1
q q
Ј
ϫ
D
͑␣,␤,␥͒C
͑␪ ,␾ ͒. ͑3͒
k q T T
Ј
͚
1
with
qЈ
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1

8726
J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Bass et al.
TABLE I. Euler angles for the lab to time-of-flight ͑␣, ␤, ␥͒ and lab to
detection frame (␣Ј,␤Ј,␥Ј) transformations for some commonly used ex-
perimental geometries. The notation HH, HV, etc., denotes the polarizations
of the pump and probe lasers ͑horizontal or vertical͒ relative to the image
plane.
Substituting Eq. ͑3͒ into Eq. ͑1͒, the lab-frame product scat-
tering distribution becomes
1
K
0
P͑v,⍀ ,⍀ ͒ϭ
c͑K͓͒k ͔͓k ͔b ͑k ,k ;v͒
T
j
6␲2 Kϭ0,2 k k
͚ ͚
1
2
1
2
1
1
2
Pump–probe
propagation
Pump–probe
polarization
͑␣, ␤, ␥͒
(␣Ј,␤Ј,␥Ј)
k1
q
K
k2
K
1/2
ϫ
͑Ϫ1͒ ͓K͔
ͩ
ͪ
Copropagating
Copropagating
Counterpropagating
Counterpropagating
Counterpropagating
Counterpropagating
HH
VV
HH
HV
VH
VV
͑␲/2, ␲/2, Ϫ␲/2͒
͑0, 0, 0͒
͑␲/2, ␲/2, Ϫ␲/2͒
͑␲/2, ␲/2, Ϫ␲/2͒
͑0, 0, 0͒
͑␲, Ϫ␲/2, 0͒
͑␲, ␲/2, 0͒
͑0, Ϫ␲/2, 0͒
͑0, Ϫ␲/2, ␲/2͒
͑0, Ϫ␲/2, Ϫ␲/2͒
͑0, Ϫ␲/2, 0͒
͚
q
0
Ϫq
k
1
ϫC
ϫC
͑⌰ ,⌽ ͒
D
͑␣,␤,␥͒
k Ϫq
j
j
͚
2
qЈq
qЈ
͑0, 0, 0͒
͑␪ ,␾ ͒.
Ј
͑4͒
k q
T
T
1
1
The lab-frame expression for the projection of the scattering
k
q
␳
͑v ,␾ ͒detϭ
cЈ͑K͒
͓k1͔
p
T
͚
͚
Љ
4␲
distribution onto the image plane, P (v ,␾ ,⍀ ), is ob-
Kϭ0,2
k
1
xy
p
T
j
tained by integrating Eq. ͑4͒ over the time-of-flight axis us-
2
2
2
p
1/2
k1
K
k
ing the Jacobian, dvTOF /dvϭ͓v /(v Ϫv )͔ , where v is
k
qq
p
ϫ
ͩ
ͪ
D
͑␣Ј,␤Ј,␥Ј͒
͚
Љ
the velocity projection onto the image plane
q
q
0
Ϫq
k
1
k1ϩqЈ
1
ϫ
D
͑␣,␤,␥͓͒1ϩ͑Ϫ1͒
͔
͚
q Ϫq
Ј
P ͑v ,␾ ,⍀ ͒ϭ
c͑K͓͒k ͔͓k ͔
xy
p
T
j
6␲2 Kϭ0,2 k k
͚ ͚
1
2
q
Ј
1
1
2
ϱ
ϱ
iqЈ␾T
K
ϫe
͵
b ͑k ,k;v͒C
͑␪ ,0͒
k q T
Ј
K
K
1/2
0
1
1
ϫ
͵
ͩ
b ͑k ,k ;v͒
͑Ϫ1͒ ͓K͔
͑⌰ ,⌽ ͒
v
0
1
2
͚
p
v
q
p
v2
1/2
k
K
k2
1
ϫ
vp dv.
͑7͒
ͩ
v2Ϫv2
ͪ
ϫ
ϫ
ͪ
C
k Ϫq
j
j
p
2
q
0
Ϫq
The resulting expression may be written more compactly as
k
1
D
͑␣,␤,␥͓͒C
͑␪ ,␾ ͒
1
͚
k q
T
T
q q
1
Ј
k
q
Ј
iq ␾
k ϩq
Ј
Ј
T
͚ ͚ ͓1ϩ͑Ϫ1͒ 1
qЈ
␳ ͑v ,␾ ͒detϭ
͚
4␲
qЈ
e
͔
p
T
Љ
Kϭ0,2
k
1
v2
1/2
K
0
K
0
ϩC
͑␲Ϫ␪ ,␾ ͔͒
vp dv.
k q
T
T
ͩ
v2Ϫv2
ͪ
ϫf ͑k ,k,qЈ,qЉ,R,RЈ͒F ͑k ,k,qЈ;v ͒,
1
Ј
1
1
p
p
͑
8͒
͑5͒
where
k1ϩqЈ
K
0
Using the fact that
C
(␲Ϫ␪ ,␾ )ϭ(Ϫ1)
k q T T
Ј
f ͑k ,k,qЈ,qЉ,R,RЈ͒
1
1
C
(␪ ,␾ ), the rotational moments of the projected dis-
k q
T T
1
Ј
k1
K
k
k
qq
k
1
tribution are then given by
ϭcЈ͑K͓͒k1͔
ͩ
ͪ
D
͑RЈ͒D
͑R͒
͚
Љ
qЈϪq
q
q
0
Ϫq
k
q
͑9͒
␳
͑v ,␾ ͒labϭ
͵
C ͑⌰ ,⌽ ͒P ͑v ,␾ ,⍀ ͒d⍀
p
T
kq
j
j
xy
p
T
j
j
and
1
k1
K
k
ϱ
ϭ
cЈ͑K͒
͓k1͔
ͩ
ͪ
K
0
K
0
͚
͚
4
␲ Kϭ0,2
k
q
0
Ϫq
F ͑k1 ,k,qЈ;vp͒ϭ
͵
b ͑k1 ,k;v͒Ck q ͑␪T,0͒
1
1
Ј
v
p
k
v2
v2Ϫv2
1/2
1
k ϩqЈ
͑␣,␤,␥͓͒1ϩ͑Ϫ1͒ 1
ϫ
D
͔
͚
q Ϫq
Ј
ϫ
vp dv
͑10͒
ͩ
ͪ
qЈ
p
ϱ
iqЈ␾T
K
and R and RЈ are the Euler angles for the rotations from the
time-of-flight frame to the lab frame, and from the lab frame
to the detection frame, respectively. The specific Euler angles
for typically used copropagating and counterpropagating
pump–probe geometries are given in Table I. While the deri-
vation of the above-given expression is somewhat complex,
the final result is fairly simple. Equation ͑8͒ represents a
ϫe
͵
b ͑k ,k;v͒C
͑␪ ,0͒
k1qЈ T
0
1
v
p
v2
v2Ϫv2
1/2
ϫ
vp dv,
͑6͒
ͩ
ͪ
p
K
1/2
where cЈ(K)ϭ(Ϫ1) ͓K͔ c(K).
The detection-frame moments are obtained by rotating
Eq. ͑6͒ into the detection frame ͑Euler angles ␣Ј, ␤Ј, ␥Ј),
Fourier expansion in the angle ␾ , with the Fourier coeffi-
T
K
cients given by sums of products of the factors f (¯) and
0
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J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Analysis of velocity-map ion images
8727
K
K
F (¯). The f (¯) parameters depend only on experimen-
k
ϭ͑Ϫ1͒ ͑2Ϫ␦ ͒1/2 Re͓␳ ͔, q у0,
qЉ
k
q
0
0
␳
␳
Љ
q ϩ
qЉ0
Љ
Љ
tal geometry and are straightforward to calculate, while the
K
͑11͒
F (¯) factors characterize the dynamics. Each of the
0
K
F (¯) factors depends on a single bipolar moment
k
qЉ 1/2
k
0
ϭ͑Ϫ1͒ 2 Im͓␳ ͔, qЉϾ0.
K
q Ϫ
q
Љ
Љ
b (k ,k;v). Note that the bipolar moments in Eq. ͑10͒ have
0
1
not been renormalized to the more commonly employed
K
␤
(k ,k;v) moments. The appropriate normalization con-
In general, for planar symmetric distributions, such as that
resulting from photodissociation of a nonchiral molecule,
0
1
stants may be found in Ref. 15.
2
2
k
k
The Hertel–Stoll scheme may be used to transform the
only the terms ␳
nonzero. Furthermore, for the present application, which em-
ploys linearly polarized pump and probe light, only the ␳
with even k, and ␳
with odd k, are
q ϩ
Љ
q Ϫ
Љ
k
rotational moments ␳q of Eq. ͑8͒, which in general have
both real and imaginary parts, into the real quantities ␳
and ␳
Љ
k
k
q ϩ
q ϩ
Љ
Љ
k
͑the subscript specifying the detection frame will
moments with even qЉ are nonzero. The rotational moments
of Eq. ͑8͒ then become
q Ϫ
Љ
now be dropped for ease of notation͒:
Ϫ1͒ ͑2Ϫ␦ ͒1/2
qЉ
͑
k
qЉ0
k
K
0
K
1
␳
͑v ,␾ ͒ϭ
͓1ϩ͑Ϫ1͒ ͔f ͑k ,k,0,qЉ;R,RЈ͒ϫF ͑k ,k,0;v ͒
q ϩ
p
T
͚ ͚
ͭ
1
0
1
p
Љ
4␲
Kϭ0,2
k
1
k ϩqЈ
K
qЈ K
ϩ
cos͑qЈ␾ ͓͒1ϩ͑Ϫ1͒ 1 ͔͓ f ͑k ,k,qЈ,qЉ;R,RЈ͒ϩ͑Ϫ1͒ f ͑k ,k,ϪqЈ,qЉ;R,RЈ͔͒
͚
T
0
1
0
1
qЈϾ0
K
0
ϫF ͑k ,k,qЈ;v ͒ , qЉу0.
͑12͒
1
p
ͮ
Equations ͑8͒–͑12͒ are equivalent to those published previ-
ously by Nestorov et al.16
the angular coordinate ␾T of the image I(v ,␾ )
p
T
ϵI(v ,v )v ,
x
y
p
2
␲
B. Image analysis
c ͑v ͒ϭN
͵
I͑v ,v ͒cos͑n␾ ͒v d␾ ,
͑14͒
n
p
x
y
T
p
T
0
1. Fourier analysis
where the normalization constant N is equal to 1 when n
ϭ0 and 2 when nϾ0. In obtaining these normalization con-
stants the Fourier series has been defined as
In systems for which there is no product angular mo-
0
mentum alignment, only the ␳ rotational moment is non-
0
zero, simplifying the analysis considerably. There are three
K
0
nonzero F (¯) functions, which may be extracted from two
images obtained using different experimental geometries.
Since in the one-laser experiments carried out on NOCl pho-
tolysis the only two geometries available were copropagating
pump and probe with the laser polarization vector either
horizontal or vertical with respect to the image plane, the
I͑v ,␾ ͒ϭc ͑v ͒ϩ2
c ͑v ͒cos n␾
p
T
0
p
ͭ
͚
n
p
T
nϾ0
ϩ
cЈ͑v ͒sin n␾
.
͑15͒
͚
n
p
T
ͮ
nϾ0
analysis was limited to obtaining the bipolar moments of the
With linearly polarized pump and probe radiation, n is re-
stricted to even terms ͓as seen from Eq. ͑12͔͒, and the cnЈ
0
0
␳
rotational moment. Higher moments were assumed to be
zero, an approximation that was justified by the insensitivity
of the images to the probe transition. For the two geometries
coefficients are all zero. The even cЈ moments only differ
n
k
from zero for the ␳
moments with k odd, and could be
q Ϫ
used in the one-color NOCl experiments, after substitution
Љ
K
0
probed with circularly polarized probe radiation. From Eq.
13͒ it is seen that for the horizontal geometry the zeroth and
for the geometrical factors f (¯), Eq. ͑12͒ becomes
͑
0
0
0
0
2
0
␳
͑v ,␾ ͒ ϭF ͑0,0,0,v ͒ϪF ͑2,0,0,v ͒
second Fourier moments are nonzero, while for the vertical
geometry only the zeroth moment is nonzero, i.e.,
p
T
H
p
p
2
Ϫ 6F ͑2,0,2,v ͒cos 2␾ ,
ͱ
0
p T
͑13͒
0
0
2
0
0
0
0
0
2
0
c ͑v ͒ ϭF ͑0,0,0,v ͒ϪF ͑2,0,0,v ͒,
␳
͑v ,␾ ͒ ϭF ͑0,0,0,v ͒ϩ2F ͑2,0,0,v ͒,
0
p
H
p
p
p
T
V
p
p
in which the subscripts H and V denote the laser polarization
with respect to the image plane.
2
0
c ͑v ͒ ϭϪ
ͱ
6F ͑2,0,2,v ͒,
͑16͒
2
p
H
p
The Fourier moments c (v ) are extracted from the
experimental data by carrying out a series of integrals over
n
p
0
0
2
0
c ͑v ͒ ϭF ͑0,0,0,v ͒ϩ2F ͑2,0,0,v ͒.
0
p
V
p
p
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8728
J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
TABLE II. Expressions for nonzero Fourier moments of experimentally accessible ␳ (v ,␾ ) rotational
Bass et al.
k
qЉϩ
p
T
moments for velocity-map images of NO photofragments from NO2 photolysis, obtained using (1
ϩ1Ј)REMPI.
Geometry
HH
Moment
Fourier moments
0
0
0
0
2
␳
␳
c (v )ϭF (0,0,0;v )ϪF (2,0,0;v )
0
p
p
0
p
2
c (v )ϭϪ2.449F (2,0,2;v )
2
p
0
p
2
0
2
2ϩ
0
2
Ϫ3␳
c (v )ϭ4.691F (2,2,0;v )Ϫ1.240F (0,2,0;v )
0 p p p
0
0
2
0
2
0
Ϫ1.069F (2,2,0;v )Ϫ2.676F (4,2,0;v )
p
p
0
0
2
0
c (v )ϭ7.482F (2,2,2;v )Ϫ4.536F (2,2,2;v )
2
p
p
p
2
Ϫ5.270F (4,2,2;v )
c (v )ϭϪ5.498F (4,2,4;v )
0
p
2
0
4
p
p
0
0
0
2
HV
␳
␳
c (v )ϭF (0,0,0;v )ϪF (2,0,0;v )
0 0
0 p p p
2
c (v )ϭϪ2.449F (2,0,2;v )
2
p
0
p
2
0
2
2ϩ
0
2
Ϫ3␳
c (v )ϭϪ5.928F (2,2,0;v )ϩ0.839F (0,2,0;v )
0 p p p
0
0
2
2
Ϫ1.069F (2,2,0;v )ϩ3.157F (4,2,0;v )
c (v )ϭϪ2.005F (2,2,2;v )ϩ4.536F (2,2,2;v )
ϩ5.270F (4,2,2;v )
c (v )ϭ1.473F (4,2,4;v )
0
p
0
p
0
2
2
p
0
p
0
p
2
0
p
2
0
4
p
p
For this system, in which there is no alignment, it is a simple
where h(2)(J ,J ) is a line strength factor depending on the
i f
matter to solve these equations to obtain expressions for the
initial and final states of the resonant transition, taking the
K
F (¯) functions:
value ϩ1 for a Q branch transition, ϪJ /(2J ϩ3) for a P
0
i
i
branch transition, and Ϫ(J ϩ1)/(2J Ϫ1) for an R branch
0
1
3
i
i
F ͑0,0,0,v ͒ϭ ͓c ͑v ͒ ϩ2c ͑v ͒ ͔,
0
p
0
p
V
0
p H
transition. Both of the transitions used to detect NO photo-
fragments in the present work have an intensity contribution
from overlapping satellite lines. The correct h( (J ,J ) fac-
2
0
1
3
F ͑2,0,0,v ͒ϭ ͓c ͑v ͒ Ϫc ͑v ͒ ͔,
͑17͒
p
0
p
V
0
p H
2)
i
f
1
tors in this case are therefore an average over the values for
the two overlapping lines, weighted by the transition prob-
ability for each line.
2
0
F ͑2,0,2,v ͒ϭϪ
c ͑v ͒ .
2 p H
p
ͱ
6
These functions are then fitted to a basis set, as described in
Before proceeding with the analysis, the image moments
must be properly normalized to take account of the total
rotational alignment. The appropriate normalization con-
stants are determined by integrating Eq. ͑19͒ over the angu-
lar coordinate of the image,
K
more detail later, to return the bipolar moments b (k ,k;v),
0
1
which contain all the required dynamical information.
0
0
In systems with alignment, ␳ and higher rotational mo-
ments may be projected out by taking the appropriate linear
combinations of the Fourier moments of images measured on
different REMPI transitions using the intensity expressions
1
2
h͑2͒͑Ji ,Jf ͓͒2 f ͑0,2,0,0͒
2
Iϰ1Ϫ
0
2
3
developed by Greene and Zare and Kummel, Sitz, and
2
0
2
0
k
Ϫ6&f ͑0,2,0,2͔͒b ͑0,2͒.
͑20͒
2
0,21
Zare.
In terms of the detection frame moments ␳
, the
q Ϯ
Љ
raw image may be written
2
0
The value of the b (0,2) bipolar moment has been measured
independently33 to be Ϫ0.23Ϯ0.02. Once the image mo-
ments have been normalized, Eq. ͑19͒ may be used to project
k
k
I͑v ,␾ ͒ϭC
P
͑i, f ͒␳ ͑v ,␾ ͒,
͑18͒
p
T
͚
q Ϯ
q Ϯ
p
T
Љ
Љ
k,qу0
0
2
out ␳ (v ,␾ ) and the combination ␳ (v ,␾ )
0
p
T
0
p
T
k
2
where P (i, f ) is a moment of the line strength for a tran-
Ϫ3␳ (v ,␾ ) by taking linear combinations of experi-
q Ϯ
Љ
2ϩ
p T
sition between states i and f and depends on the REMPI
scheme ͓e.g., the line strength moments for (1ϩ1)REMPI
will differ from those for a (2ϩ1) process͔, and C is a
constant, dependent ͑among other factors͒ on the populations
in initial state i. The values taken by the indices k and qЉ
depend on the number of photons and their polarization.
For a (1ϩ1Ј)REMPI detection scheme, as used in the
mentally determined Fourier moments for different rotational
branches involving the same lower state. For the Q and R
line transitions used in the present work, the appropriate lin-
ear combinations are
͑
R
2͒
͑2͒
h
IQϪh IR
R
Q
0
0
␳
͑v ,␾ ͒ϭ
,
p
T
͑2͒
͑2͒
Q
h
Ϫh
NO experiments, if we assume the ionization step is satu-
2
͑
21͒
rated, the intensity of the image is equivalent to that provided
by Dixon in his description of a chemiluminescence
2
͑I ϪI ͒
Q
͑2͒
R
͑2͒
2
0
2
␳ ͑v ,␾ ͒Ϫ3␳ ͑v ,␾ ͒ϭ
.
p
T
2ϩ
p
T
1
5
hR ϪhQ
experiment,
1
3
0
1
͑2͒
I͑v ,␾ ͒ϭ I ͓␳ ͑v ,␾ ͒Ϫ h ͑J ,J ͒
Expressions for the Fourier moments of these images ͓analo-
gous to those presented for NOCl photolysis in Eq. ͑13͔͒ are
given in Table II.
p
T
0
0
p
T
2
i
f
2
0
2
ϫ
͕
␳ ͑v ,␾ ͒Ϫ3␳ ͑v ,␾ ͒
͖
͔,
͑19͒
p
T
2ϩ
p
T
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1

J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Analysis of velocity-map ion images
8729
In the case of aligned systems, rather than extracting
K
0
individual F (¯) functions from the data, it is often simpler
to fit the basis functions directly to the Fourier moments of
the population and alignment-dependent images of Eq. ͑21͒
in order to extract the bipolar moments. In the present case,
the expressions appearing in Table II have been used to fit
the experimentally determined Fourier moments. Whether
K
0
fitting the F (¯) functions or fitting the Fourier moments
directly, the basis functions used to fit the data consist of a
K
0
set of simulated F (¯) functions. There are two cases to
consider.
i. Single recoil velocity. In systems for which the de-
tected product has a single, well-defined velocity ͑common
in photolysis of diatomics or photolysis of triatomics in
which the diatomic fragment is probed state-selectively, as in
the present experiments on the photolysis of NO ) the fitting
2
procedure becomes very simple. Equation ͑10͒ reduces to
ϱ
K
K
F ͑k ,k,qЈ;v ͒ϭ
͵
b ͑k ,k;v͒␦͑vϪv ͒C
͑␪ ,0͒
0
1
p
0
1
0
k1qЈ
T
0
FIG. 3. Fourier smoothing: Raw image ͑top left͒; inverse Abel transform of
raw image ͑top right͒; Fourier smoothed image ͑bottom left͒; and inverse
Abel transform of smoothed image ͑bottom right͒.
v2
v2Ϫv2
1/2
ϫ
vp dv
ͩ
ͪ
p
2
1/2
v0
K
in order to determine the expansion coefficients a . In the
ϭb ͑k ,k͒C
͑␪ ,0͒
vp .
n
0
1
k1qЈ
T
ͩ
2
2
p
ͪ
v Ϫv
0
above-given expressions, vЈϭ2(v/vmax)Ϫ1 is a reduced ve-
locity parameter lying between Ϫ1 and 1, the limits between
which the Legendre polynomials form an orthogonal basis.
In the present work, a genetic algorithm fitting program was
used to carry out the analysis.
͑22͒
The basis functions are then simply the analytical functions
2
2
2
1/2
C
(␪ ,0) ͓v /(v Ϫv )͔ vp and the fitting parameters
k q
T
0 0 p
1
Ј
K
are the bipolar moments b (k ,k). Note that the angle ␪
0
1
T
2. Inverse Abel transform analysis
appearing in the spherical harmonics is calculated using the
Ϫ1
equivalence ␪ ϭsin (v /v) ͑see Fig. 2͒. If fitting to experi-
In the next section, the results of the Fourier moment
T
p
K
mentally determined F (¯) functions, the bipolar moments
analysis on images of NOCl photolysis fragments are com-
pared with product scattering distributions determined using
the inverse Abel transform. As mentioned previously, one of
the problems with the inverse Abel transform is that it is
highly sensitive to experimental noise. The transform works
best on smooth functions, and does not cope well with the
‘‘graininess’’ intrinsic to event-counted velocity map images.
A common method for overcoming this problem is to carry
out a three-point Gaussian blur on the images before calcu-
lating their inverse Abel transform. This overcomes the noise
problem to some extent, but also leads to a small loss in
velocity resolution. In the present work, a procedure has
been developed for smoothing the images which does not
entail any loss in resolution. By taking advantage of the fact
that a velocity map image may always be expressed as a
Fourier expansion in the angular coordinate of the image, the
Fourier moments of the image may be extracted using Eq.
0
are simply the scaling factors between these functions and
the ͑properly normalized͒ experimental data. If fitting the
Fourier moments, the basis functions are used in conjunction
with the mathematical expressions for the image Fourier mo-
ments ͓e.g., those in Table II for (1ϩ1Ј)REMPI] and best fit
values for the bipolar moments determined from a numerical
fitting routine.
ii. Multiple recoil velocities. In systems for which the
probed product has a distribution of velocities, as is the case
for the Cl photofragments in the UV photolysis of NOCl, the
bipolar moments are velocity dependent. Each moment may
be expressed as an expression in Legendre polynomials ͑or
any other suitable set of functions͒
K
1
2
b ͑k ,k;vЈ͒ϭ
͑2nϩ1͒a P ͑vЈ͒
͑23͒
0
1
͚
n
n
n
͑
14͒ and used to reconstruct the image using Eq. ͑15͒. An
K
and the corresponding F (¯) function is fitted to a set of
0
example of this ‘‘Fourier smoothing’’ and its effect on the
inverse Abel transform is shown in Fig. 3. There is a marked
improvement in the inverted image following the smoothing
procedure, with a significant reduction in centerline noise.
The reduction of graininess in the Fourier-smoothed image
relative to the raw image is also reflected in the Abel inver-
sions of the two images. Similar results have been achieved
basis functions
ϱ
1
K
F ͑k ,k,qЈ;v ͒ ϭ
͵
͑2nϩ1͒P ͑vЈ͒C
͑␪ ,0͒
k1qЈ T
0
1
p
n
n
2
v
p
v2 1/2
ϫ
vp dv
ͩ
v2Ϫv2
ͪ
8
p
by Vrakking, although we believe the present method might
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1
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8730
J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Bass et al.
FIG. 4. Cl photofragment images following NOCl photolysis together with their zeroth- and second-order Fourier moments. The left-hand column shows
images and moments measured with horizontal laser polarization and the right-hand column with vertical polarization relative to the image plane. From top
2
4
2
4
2
4
to bottom, the images are: Cl( P ) detected via the 4p P state; Cl( P ) detected via the 4p P state; and Cl( P ) detected via the 4p D state. The
3
/2
1/2
3/2
3/2
1/2
3/2
Ϫ1
scale on the lower right-hand image denotes the velocity in km s
.
be simpler to implement. It should be emphasized at this
point that, unlike the Fourier moment fitting method de-
scribed previously, the Abel inversion ͑and the inversion
method of Vrakking ͒ may only be used for images of a
distribution containing an axis of cylindrical symmetry.
in the Legendre polynomial expansion of the bipolar mo-
ments. Using ten basis functions the general shape of the
distribution is reproduced, and as the number of basis func-
tions is increased, the finer details of the distribution emerge.
In the fits presented in Fig. 5, around 50 basis functions were
used, allowing most of the features present in the alternative
inverse Abel transform analysis to be reproduced. The fitting
8
IV. RESULTS AND DISCUSSION
A. NOCl photolysis
2
2
parameters for the F0(2,0,0;vp) and F0(2,0,2;vp) functions
were constrained to be the same, since these two functions
2
2
Velocity map images of the Cl( P ) and Cl( P1/2)
3
/2
2
depend on the same bipolar moment b (2,0;v).
0
products of NOCl photolysis near 240 nm are shown in Fig.
for the two experimental geometries used. Also shown are
the zeroth- and second-order Fourier moments of the images.
NOCl photolysis has been studied previously at a num-
4
ber of wavelengths in the range from 193 to 248 nm.2
4–29
In
K
0
this region, photolysis occurs via the A band, which is
though to consist of three partially overlapping electronic
transitions.25 The photodissociation dynamics are strongly
dependent on photolysis wavelength, and a wide range of ␤
parameters and photofragment velocity distributions have
been reported. These have been summarized previously by
Felder and Morley25 and by Matsumi and co-workers.
Figure 5 shows the F (¯) functions extracted from the
Fourier moments and the fits to these functions using the
basis set method described previously. The bipolar moments
returned from the analysis are also shown, and compared
with those returned from an inverse Abel transform analysis.
Note that the b (0,0;v) moment is simply the fragment
speed distribution, while the velocity-dependent spatial an-
isotropy parameter is given by ␤(v)ϭ2b (2,0;v)/
0
0
26
2
0
Measured photofragment velocity distributions are generally
strongly bimodal,2
4–26
with similar anisotropy for the two
0
0
b (0,0;v). In Fig. 6, the product translational energy distri-
velocity components, though some studies have indicated a
slightly lower anisotropy for the faster component, particu-
larly at shorter wavelengths. Previously measured ␤ param-
eters range from 0.45 for the fast component for photolysis at
193 nm24 to 1.8 near 235 nm. Measurements at 248 nm
butions are plotted.
The product distributions extracted from the images us-
ing the two different analyses are in good agreement with
one another. The velocity resolution of the Fourier moment
analysis is dependent on the number of basis functions used
26
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J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Analysis of velocity-map ion images
8731
FIG. 6. Translational energy distributions for the products of NOCl photoly-
sis. The corresponding vibrational energy levels of the NO photofragment
are marked along the top axis.
to transitions to the A electronic state, probably in conjunc-
a
tion with a contribution from the A state.
b
In the present work, the speed distributions for Cl
2
2
formed in the P3 and P states are markedly different
͑
/2
1/2
Ϫ1
see Fig. 5͒. Both display a large peak near 2500 m s ,
corresponding to formation of the NO cofragment in vϭ5.
However, the speed distribution of spin–orbit excited Cl is
Ϫ1
clearly bimodal, peaking again at 1870 m s ͑corresponding
to NO formation in vϭ9 or 10͒ while that of the ground
state photofragment is much broader, with two lower energy
peaks in the distribution corresponding to NO formation in
vϭ9 and vϭ11. The anisotropy is fairly independent of ve-
locity. The two sets of images collected for the ground state
K
FIG. 5. F (¯) functions extracted from the experimental data ͑solid lines͒
0
and fits to these functions using the basis set expansion method detailed in
the text ͑dotted lines͒. Also shown are the b (0,0;v) and b (2,0;v) bipolar
moments returned from the fit ͑solid lines͒, together with those obtained
from the images using the inverse Abel transform ͑dotted lines͒.
0
2
0
0
4
2
4
2
photofragment on the 4p P3 � P and 4p P � P
/2
3/2
1/2
3/2
transitions yield velocity-averaged ␤ parameters of 1.09 and
.94 ͑while the discrepancy in values is probably within ex-
0
perimental error, small variations may arise from the fact that
since these are single laser experiments, the photolysis en-
ergy is slightly different for the two detection schemes, at
5.157 and 5.171 eV, respectively͒. ␤ for the excited state is
0.90 at a photolysis energy of 5.164 eV. These values are
somewhat lower than those measured by Matsumi and co-
workers at a slightly shorter wavelengths, a discrepancy that
give a ␤ value of around 1.2.25 Apart from the work of Mat-
sumi and co-workers, these measurements were carried out
using techniques such as photofragment translational spec-
troscopy, in which it was not possible to resolve the two
spin–orbit states of the Cl photofragment. The resulting dis-
tributions therefore represent an average over the two spin–
orbit states. Matsumi and co-workers used photofragment
ion imaging ͑as opposed to the higher resolution velocity
map ion imaging used in the present study͒ to detect the Cl
products state selectively, and observed significant differ-
ences between the velocity distributions of the ground and
excited state fragments.26 He concluded that the observed
product distributions were consistent with initial excitation to
may be due in part to involvement of the A state, but is also
a
likely to reflect the higher rotational temperature of the mo-
lecular beam in the current experiments. Rotation of the par-
ent NOCl molecule during dissociation reduces the vector
correlation between the parent transition dipole moment and
the fragment velocity, leading to a reduced value of ␤. At-
tempts to treat the effects of molecular rotation quantitatively
have been made by several researchers.3
0–32
a common upper electronic state ͑labeled A ), followed by a
Though quantitative measurements on the spin–orbit ra-
b
sequence of strong nonadiabatic interactions as the photo-
fragments separate, both at small internuclear separation ͑due
to an avoided crossing͒ and at large internuclear distances.
Two other excited states, labeled A and A , were cited as
being important at short and long wavelengths, respectively.
It was suggested that the lower ␤ parameter and fragment
internal energy noted in the 248 nm experiments25 were due
tio of the products were not carried out, signal levels would
2
indicate that Cl is formed predominantly in the P1 excited
/2
spin–orbit state following 240 nm photolysis. This is consis-
tent with previous measurements at 248 nm25 and implies
that the dissociation dynamics are somewhat different from
those at 235 nm, for which branching ratios Cl*:Cl have
c
a
26
27
been reported as 0.3 and 0.31Ϯ0.02. Comparison of the
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8732
J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Bass et al.
FIG. 7. Photofragment images for the NO(vϭ0,Nϭ30) products of 308 nm
NO2 photolysis. Products were detected on the (0,0)Q (30) and
1
(
0,0)R (30) transitions in HH and HV counterpropagating geometries. For
1
the (0,0)Q (30) transition in the HH geometry the contribution of a fourth-
1
order Fourier moment c4 to the image, which can only arise because of
photofragment rotational alignment, is clearly visible.
0
2
0
FIG. 8. Experimentally determined Fourier moments of the ␳ and ␳
0
2
Ϫ3␳2 rotational moments of the NO product scattering distribution fol-
ϩ
lowing 308 nm photolysis of NO . Fits to the data using the methodology
2
detailed in the text are also shown.
translational energy distributions measured in the present
work with the data available in the literature reveals a sys-
tematic increase in the importance of the fast component of
the bimodal distribution relative to the slow component as
the wavelength increases. If the different peaks in the mea-
sured velocity distributions were associated with dissociation
that dissociation at 308 nm is rapid on the time scale of
molecular rotation. A value of Ϫ0.5Ϯ0.1 for ␤ (2,2) sug-
gests a near limiting alignment of jЌv. The bipolar moments
measured here are consistent with the prompt dissociation of
a cold triatomic molecule, which necessarily yields jЈ
aligned perpendicular to the molecular plane.
0
0
on different potential energy surfaces, as suggested by
Matsumi and co-workers26 and Felder et al.,
24,25
the data
would imply involvement of at least two surfaces. The lower
energy surface is associated with higher translational energy
release and hence lower NO vibrational excitation.
C. Application to bimolecular reactions
The Fourier analysis method has considerable potential
for use in ‘‘photoloc’’ experiments on photon-initiated bimo-
lecular reactions. In these experiments velocity-map ion im-
ages of the reaction products contain information about the
differential cross section and kinetic energy release, as well
as angular momentum alignment parameters. In general
these cannot be determined by direct inversion of the image,
and some form of data fitting is necessary.
B. NO photolysis
2
Velocity map images of the NO photofragments for the
HH and HV geometries and Q (30) and R (30) transitions
1
1
are shown in Fig. 7. Fourier moments and fits to the data for
0
0
2
0
2
2ϩ
the ␳ and ␳ Ϫ3␳ rotational moments ͑discussed in Sec.
III B 1͒ are shown in Fig. 8. The bipolar moments obtained
from the fitting procedure are given in Table III, together
with data from the only previous set of measurements at 308
nm, carried out by Brouard et al.33 for NO in Nϭ29 using
Doppler-resolved laser-induced fluorescence ͑LIF͒ using the
The general reaction scheme for a photoloc experiment
is
AMϩhv→AϩM,
P (29) transition. The results of the current measurements,
1
which assume that the two lambda doublet components are
identically polarized, are in excellent agreement with those
of the earlier experiments, and reinforces the reliability of
both the ion images and the data analysis procedures.
The fact that the bipolar moments are consistently of
somewhat greater magnitude than the previous study is ac-
counted for by the lower rotational temperature of the parent
molecule; the Doppler-resolved LIF experiments were car-
ried out at room temperature in a bulb, while the imaging
experiments utilize a molecular beam. The limiting value of
TABLE III. The even k bipolar moments of the NO product scattering
distribution following 308 nm NO2 photolysis. The moments shown have
K
0
been renormalized to the more commonly used ␤ (k ,k) ͑Ref. 15͒. Errors
1
in the final digit are given in parentheses. The data obtained in the present
experiments are compared with those obtained previously using Doppler-
resolved LIF techniques ͑Ref. 33͒.
Bipolar moment
Current measurement
Brouard et al. ͑Ref. 33͒
0
␤ (0,0)
1
1
0
2
␤
␤
␤
␤
␤
(2,0)
(2,2)
(0,2)
(2,2)
(4,2)
0.70͑5͒
Ϫ0.50͑11͒
Ϫ0.36͑2͒
0.30͑7͒
Ϫ0.08͑6͒
0.61͑2͒
Ϫ0.38͑4͒
Ϫ0.23͑2͒
0.27͑5͒
Ϫ0.27͑8͒
0
0
0
2
0
2
0
2
0
␤, based on the orientation of the transition dipole moment
relative to the N–O bond axis, is 1.54. The value of 1.4
Ϯ0.1 determined from the imaging experiments suggests
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J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Analysis of velocity-map ion images
8733
acterized, images of reaction products may be analyzed to
return the differential cross section, product speed distribu-
tion, and information on product angular momentum polar-
ization. Further details and applications of the photoloc tech-
nique may be found in Refs. 17, 34, and 35. An application
of the technique in an imaging experiment may be found in
3
6
FIG. 9. Imaging applied to reactions: when imaging a photofragment, the
radial coordinate provides a measure of the fragment velocity, and the scat-
tering distribution may be well characterized ͑left͒; forward-scattered reac-
tion products result in a larger Newton sphere in the lab frame ͑center͒;
backward scattered products result in a smaller Newton sphere in the lab
frame ͑right͒.
the work of Kitsopoulos et al. on the ClϩC H reaction.
2
6
As an example, simulated images for the HϩD reac-
2
tion, in which the HD product is ‘‘detected’’ quantum state
selectively ͑for example, using REMPI techniques͒, have
been fitted using the Fourier moment analysis. The simula-
tions are based on differential cross sections derived from
quantum scattering calculations carried out by Aoiz, Banares,
˜
and Castillo37 at a collision energy of 1.49 eV. The model
AϩBC→AB͑vЈ, jЈ͒ϩC,
differential cross sections are used as input to a Monte Carlo
In the proposed experiments, a photolysis precursor AM and
target molecule BC ͑not necessarily a diatomic͒ are coex-
panded in a molecular beam. Polarized laser photolysis of
the precursor produces a velocity-aligned photofragment A
that reacts with the target molecule, and the reaction product
AB(vЈ, jЈ) is ionized state-selectively via REMPI and de-
tected using velocity-map imaging. In simple photolysis
studies, the radial coordinate of a photofragment image sim-
ply provides information on the product velocity. However,
for a reaction of the kind described previously, the velocity
of the center-of-mass of the reacting system will lie approxi-
mately along the velocity vector of the fast reactant ͑i.e., the
reactant formed during photolysis of the precursor͒. When
reaction products are detected using velocity-map imaging,
the radial coordinate of the image is now sensitive to both
the product speed distribution and to the angular differential
cross section, with ‘‘rings’’ in the images reflecting peaks in
either or both of these distributions. This is illustrated sche-
matically in Fig. 9, and discussed in more detail in the fol-
lowing in relation to the current simulations. So long as the
scattering distribution of the fast reactant has been well char-
K
0
simulation of the bipolar moments b (k ,k;v) using equa-
1
tions developed in Ref. 17, which are then used to recon-
struct the 3D scattering distribution using Eq. ͑1͒. Images are
obtained by numerical integration of the 3D distribution over
the time-of-flight axis. Using this approach, the methods
used in generating the simulated images and in performing
the analysis are independent, and fitting the simulations pro-
vides a stringent test of the performance of the Fourier mo-
ment analysis method. Finally, random noise is added to the
images to assess the effects that it has on the derived center-
of-mass differential cross sections and energy disposals.
As in any three-atom reaction, when the molecular prod-
uct is detected in a specific quantum state, the total transla-
tional energy of the atom plus the detected product ͑and
therefore the velocity of the detected product͒ has a single
well-defined value. In this example, the simulations are car-
ried out at fixed collision energy, with the consequence that
the velocity-map images of the HD products are sensitive
only to the differential cross section of the reaction. The law
of cosines relates each scattering angle in the center-of-mass
FIG. 10. Simulated images for two HD product quantum states of the HϩD reaction at a collision energy of 1.49 eV, together with input differential cross
2
sections from quantum scattering theory, and the functions returned from a Fourier moment analysis.
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8734
J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
Bass et al.
frame to a unique velocity in the lab frame, and in this way,
fluctuations in the image intensity along the radial coordinate
of the image directly reflect oscillations in the differential
cross section.
has been illustrated using simulated images based on the
results of recent quantum scattering calculations on the re-
action HϩD →HD(vϭ0,jϭ0,9)ϩD. The differential cross
2
sections returned from the analysis procedure are in excellent
agreement with those input to the simulations, even when
realistic levels of noise are added to the simulated images.
For comparison two sets of simulations are performed
for HD products formed in the (vϭ0,jϭ0) and (vϭ0,
jϭ9) quantum states, chosen for their markedly different
differential cross sections, are shown in Figure 10. Also
shown in Fig. 10 are the differential cross sections used as
input to the simulations and the functions returned from the
fitting procedure. The peaks in the differential cross sections
are clearly visible as separate rings in the corresponding im-
ages. The simulations, and fits to them, demonstrate that us-
ing the Fourier moment analysis it is possible to extract even
fairly complicated differential cross sections from velocity
map images with a high degree of accuracy. The high angu-
lar resolution of the method is illustrated by its sensitivity to
even the very sharp forward and backward peaks in the j
ϭ0 differential cross section. The fitting method also ap-
pears to be remarkably robust to added noise on the simu-
lated data, with the returned differential cross section faith-
fully reproducing the input data for even very grainy images.
As discussed previously, for a three-atom reaction the
image reflects principally the differential cross section for the
chosen product channel of the reaction ͑in a real experiment,
the image will also be affected by the collision energy dis-
tribution, particularly if the excitation function varies signifi-
cantly over the range of collision energies sampled͒. How-
ever, for a reaction in which the coproduct is molecular, the
kinetic energy release for the probed channel does not take a
single value, and in general the detected product will have a
distribution of velocities. It has been shown in a previous
ACKNOWLEDGMENTS
The authors are very grateful to Javier Aoiz, Luis Ba-
nares, and Jesus Castillo for allowing us to use the results of
˜
their quantum scattering calculations on HϩD , and to
2
Javier Aoiz and Bruno Mart ´ı nez-Haya for helpful discus-
sions. We would like to acknowledge Hannah Reisler for
providing a preprint of Ref. 14. We would also like to thank
the Oxford Supercomputing Center for use of parallel com-
puting facilities, and the EPSRC, the Royal Society and the
EU ͑through Project No. HPRN-CT-1999-00007͒ for re-
search grants.
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1
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V. CONCLUSION
A new method has been presented for extracting velocity
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J. Chem. Phys., Vol. 117, No. 19, 15 November 2002
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1