A combsampling method for enhanced mass analysis in linear electrostatic ion traps
ABSTRACT In this paper an algorithm for extracting spectral information from signals containing a series of narrow periodic impulses is presented. Such signals can typically be acquired by pickup detectors from the imagecharge of ion bunches oscillating in a linear electrostatic ion trap, where frequency analysis provides a scheme for highresolution mass spectrometry. To provide an improved technique for such frequency analysis, we introduce the CHIMERA algorithm (Combsampling for Highresolution IMpulsetrain frequency ExtRAaction). This algorithm utilizes a comb function to generate frequency coefficients, rather than using sinusoids via a Fourier transform, since the comb provides a superior match to the data. This new technique is developed theoretically, applied to synthetic data, and then used to perform high resolution mass spectrometry on real data from an ion trap. If the ions are generated at a localized point in time and space, and the data is simultaneously acquired with multiple pickup rings, the method is shown to be a significant improvement on Fourier analysis. The mass spectra generated typically have an order of magnitude higher resolution compared with that obtained from fundamental Fourier frequencies, and are absent of large contributions from harmonic frequency components.
 R. N. Wolf, F. Wienholtz, D. Atanasov, D. Beck, K. Blaum, Ch. Borgmann, F. Herfurth, M. Kowalska, S. Kreim, Yu. A. Litvinov, D. Lunney, V. Manea, D. Neidherr, M. Rosenbusch, L. Schweikhard, J. Stanja, Z. ZuberInternational Journal of Mass Spectrometry 09/2013; · 2.23 Impact Factor

Article: LIADfs: A novel method for studies of ultrafast processes in gas phase neutral biomolecules
[Show abstract] [Hide abstract]
ABSTRACT: A new experimental technique for femtosecond (fs) pulse studies of gas phase biomolecules is reported. Using LaserInduced Acoustic Desorption (LIAD) to produce a plume of neutral molecules, a timedelayed fs pulse is employed for ionisation/fragmentation, with subsequent products extracted and mass analysed electrostatically. By varying critical laser pulse parameters, this technique can be used to implement control over molecular fragmentation for a range of small biomolecules, with specific studies of amino acids demonstrated.Journal of Physics Conference Series 11/2012; 388(1):2032.  [Show abstract] [Hide abstract]
ABSTRACT: Sidebands are observed in the Fourier transform of the pickup signal of an RFbunched beam oscillating in an electrostatic ion beam trap. We show that the sidebands are dominated by a collective longitudinal oscillation of the center of the bunch around the synchronous ion as the bunch travels within the storage device. We present evidence for a linear dependence of the sideband frequency and for a seemingly chaotic behavior of the amplitude of the sideband peaks on the number of stored ions.Journal of Instrumentation 04/2014; 9(04):P04008. · 1.53 Impact Factor
Page 1
REVIEW OF SCIENTIFIC INSTRUMENTS 82, 043103 (2011)
A combsampling method for enhanced mass analysis in linear
electrostatic ion traps
J. B. Greenwood,1,a)O. Kelly,1C. R. Calvert,1M. J. Duffy,1R. B. King,1L. Belshaw,1
L. Graham,1J. D. Alexander,1I. D. Williams,1W. A. Bryan,2I. C. E. Turcu,3C. M. Cacho,3
and E. Springate3
1Centre for Plasma Physics, School of Mathematics and Physics, Queen’s University Belfast,
Belfast BT7 1NN, United Kingdom
2Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
3Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot, Oxfordshire OX11 0QX,
United Kingdom
(Received 16 December 2010; accepted 9 March 2011; published online 8 April 2011)
In this paper an algorithm for extracting spectral information from signals containing a series of nar
row periodic impulses is presented. Such signals can typically be acquired by pickup detectors from
the imagecharge of ion bunches oscillating in a linear electrostatic ion trap, where frequency analy
sis provides a scheme for highresolution mass spectrometry. To provide an improved technique for
suchfrequencyanalysis,weintroducetheCHIMERAalgorithm(CombsamplingforHighresolution
IMpulsetrain frequency ExtRAaction). This algorithm utilizes a comb function to generate frequency
coefficients, rather than using sinusoids via a Fourier transform, since the comb provides a superior
match to the data. This new technique is developed theoretically, applied to synthetic data, and then
used to perform high resolution mass spectrometry on real data from an ion trap. If the ions are
generated at a localized point in time and space, and the data is simultaneously acquired with mul
tiple pickup rings, the method is shown to be a significant improvement on Fourier analysis. The
mass spectra generated typically have an order of magnitude higher resolution compared with that
obtained from fundamental Fourier frequencies, and are absent of large contributions from harmonic
frequency components. © 2011 American Institute of Physics. [doi:10.1063/1.3572331]
I. INTRODUCTION
Electrostatic devices are increasingly being used to
store ions for scientific studies and for mass spectrometry
applications.1Unlike ion traps which use magnetic or time
varying electric fields, the storage conditions in an electro
static trap are independent of mass for ions of the same en
ergy per charge; this allows molecules as large as proteins to
be studied. While the electrostatic Kingdon trap2has been in
use for nearly 90 years, it is only in the last 20 years that a
new range of electrostatic devices have been developed.
The electrostatic ion storage ring in Aarhus (ELISA)3
was one of the first of these instruments to be developed; it is
much smaller than the magnetic equivalent, and has inspired
the further development of cryogenic4,5and miniature6stor
age rings. Around the same time, the Kingdon trap was mod
ified to create the Orbitrap,7for which the mass resolution
reduces more slowly as a function of mass than with other
high performance mass spectrometers, such as ion cyclotron
resonance (ICR) devices.
The linear electrostatic ion trap (LEIT), which acts as
an analogue to a laser cavity, is another innovation in elec
trostatic storage,8–10with useful applications for ion tar
get preparation11and the potential for mass spectrometry
schemes.12Ions are stored along a linear trajectory, oscillat
ing back and forth within an electrostatic cavity. Unlike sig
nals acquired from ICRs or Orbitraps, the signal from LEITs
a)Electronic mail: j.greenwood@qub.ac.uk.
is nonsinusoidal in the time domain. Due to this, the corre
sponding Fourier transform inherently contains a rich spec
trum of harmonics. Through the analysis of these harmonics13
LEITs have demonstrated a higher resolution capability than
is achievable for a similar trapping time in an Orbitrap.7
However, mass analysis of LEIT data is complicated by
this existence of multiple harmonics in the Fourier frequency
spectra, such that the generation of a mass spectrum is non
trivial, even where there are relatively few ion species.
In this paper an alternative method is proposed for the
frequency analysis of anharmonic data acquired from a LEIT.
By using comb functions with different time offsets to sam
ple data obtained from several different pickup detectors, and
by comparing with a standard fast Fourier transform, a spec
trum free of harmonics can be obtained. This CHIMERA
(combsampling for highresolution impulsetrain frequency
extraction) algorithm allows frequencies to be easily con
verted into a mass spectrum with substantially higher res
olution than can be achieved through Fourier analysis. The
implementation of this technique is demonstrated using ex
perimental data obtained from the electrostatic trapping of
charged species created by an intense femtosecond laser in
teraction.
To provide the context for our new analysis approach, we
describe our ion trap in Sec. II. A discussion of the Fourier
treatment of the signal from the instrument and its draw
backs are given in Sec. III. In Sec. IV the CHIMERA al
gorithm is presented, while its application to synthetically
generated datasets is detailed in Sec. V. The application of
00346748/2011/82(4)/043103/12/$30.00© 2011 American Institute of Physics
82, 0431031
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0431032Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)
the CHIMERA algorithm to experimental results is reported
in Sec. VI, for both isotope identification in atoms and for
molecular fragmentation spectra. In Sec. VII the results are
summarized, the performance of the algorithm relative to
Fourier analysis is discussed, and suggestions for applications
in other devices are made. Appendices are used to detail some
of the mathematical derivations.
II. THE KEIRA ELECTROSTATIC ION TRAP
The potential energy surface for our kilovolt electrostatic
ion reflection analyzer (KEIRA) device is shown in Fig. 1.
The electrostatic mirrors at each end of the trap provide axial
confinement (z) of the ion motion. The mirror regions consist
of six 3 mm thick plates (R1–R6), with apertures of 16 mm
diameter, separated by 7 mm, and one end plate (R7) situated
a further 14 mm from R6. The potentials that are applied to
these electrodes increase incrementally up to a maximum of
5 kV.
Radial confinement (r) is provided by two sets of four
plates, which are also separated by 7 mm with 16 mm diame
ter apertures. In each set, the outer plates are earthed and the
two central plates are held at a negative potential to form an
electrostatic lens. Ions are generated by laser ionization of a
gasjet,whichemergescolinearwiththetrapaxisfroma1mm
hole in one end plate. The laser is aligned perpendicular to the
z axis, to enter the 14 mm gap between R6 and R7 and is fo
cused into the gas jet. The ions are generated in a well defined
spatial region within a couple of mm of the trap z axis, with
effectively zero initial velocity.
They are immediately accelerated by the electrostatic
field into the field free region at the center of the trap. With the
appropriate electrode potentials, a large fraction of these ions
oscillate between the mirrors on stable trajectories. The ions
travel a total distance of approximately L = 410 mm from
mirror to mirror.
KEIRA has three imagecharge detectors (pickup rings),
one at the geometric center of the trap and one either side,
separated by 48 mm centertocenter. The pickup itself is a
cylinder of 16 mm length and 9 mm inner diameter (shown
in more detail elsewhere15). It is situated within an earthed
housing which has 8 mm entrance and exit apertures which
are separated from the pickup by 1 mm. An ion bunch at the
center will have 97% of its charge imaged on the pickup.15
FIG. 1. (Color online) Potential energy surface14for the KEIRA ion trap,
with the mirror regions defined by the electrodes R7–R1, and two sets of four
electrodes (the inner two held at a negative potential) defining the lenses.
Three isolated pickup rings are located in the central, field free region of the
trap. The trap is cylindrically symmetric, with ion propagation along the z
axis direction.
0 1.02.0 3.04.05.0
0
0.2
0.4
time (ms)
amplitude
00.0050.0100.015
0
0.5
1.0
00.0050.0100.015
22.005
time (ms)
2.0102.015
0
0.04
0.08
2 2.005
time (ms)
2.010 2.015
Offset PickupCentre Pickup
(a)
(b)
(c)
(e)
(d)
FIG.2. (Coloronline)(a)Temporalspectraacquiredusingthecentralpickup
from ionization of H2O present in background gas. (b)–(d) show 15 μs win
dows at different trapping times for the central pickup ring (b, d) and a pickup
ring closer to the ionization region (c, e).
Figure 2 shows time signal obtained from photoioniza
tion of background gas at a pressure of 4 × 10−8mbar ob
tained from the central and one offset pickup ring in KEIRA.
The ion signal is almost exclusively due to H2O+, and was
produced using the Artemis16femtosecond laser at the Cen
tral Laser Facility. The laser pulse had a central wavelength
of 790 nm, pulse length of 15 fs, and a maximum intensity
of about 5 × 1014W cm−2. The trap potentials (in kV) were
set at R1 = 0.5, R2 = 1.0, R3 = 1.5, R4 = 2.0, R5 = 2.5,
R6 = 3.0, R7 = 3.55, Lens = −2.8. From a previous cal
ibration of the pickup ring,15we estimate that about 3.2 ×
104ions were initially produced by the laser, with 1.5 × 104
being captured on stable trajectories. The data was acquired
for 5 ms and, to improve the statistics, was averaged over 104
laser shots. The decay in the amplitude of the signal seen in
Fig. 2(a) is due to ion losses from collisions with background
gas, and an increase in the bunch length due to slight differ
ences in oscillation times for the ions. In Figs. 2(b) and 2(c)
the peak widths correspond to the time taken to traverse the
pickup, while the signals at 2 ms [Figs. 2(d) and 2(e)] indicate
that the bunch length now exceeds that of the pickup.
The evolution of ion bunches in LEITs has been exten
sively studied by the Weizmann group.17They identified three
types of behavior corresponding to
coherent diffusion : t2
w= t2
w= t2
w= constant,
0+ n2?T2,
0+ n?T2,
noncoherent diffusion : t2
(1)
selfbunching : t2
where tw is the bunch length in time, t0is the initial bunch
length, n is the number of full oscillations, and ?T is the
Page 3
0431033Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)
increase in bunch length per oscillation. In coherent diffusion,
the ions are weakly perturbed by external or internal (ion–ion)
interactions, and differences in ion oscillation times are solely
determined by the differences in the initial ion conditions. For
noncoherent diffusion, the ion trajectories are disturbed at a
rate greater than the oscillation frequency, resulting in the ion
having no “memory” of the previous oscillation.
For the data in Fig. 2, the ion density is too low to sup
port the selfbunching phenomenon, but we have observed the
selfbunching in KEIRA for larger ion numbers. The present
data set exhibits coherent diffusion with ?T ≈ 1 ns, a value
which can be reduced by tuning the trap potentials.
III. LEIT MASS RESOLUTION: FOURIER ANALYSIS
A. Constant ion bunch length
The mass resolution is defined as R = m0/?m, where
m0is the ion mass and ?m is the minimum resolvable mass
difference. We define the following quantities: f0is the ion
oscillation frequency, U is the total energy of the ion, ?v? is
the ion velocity averaged over one oscillation,and L is the
length of the trap. The frequency of oscillation can be given
by
f0=?v?
2L=
β
2L
?2U
m0
?1
2
,
(2)
where β is a constant that depends on the trap geometry and
applied potentials (for KEIRA β ≈ 0.75). In terms of fre
quency, the resolution is given by
R =
m0
?m≈
hf0
2?f,
(3)
where ?f is the full width half maximum (FWHM) of the hth
harmonic of the fundamental frequency f0.
The signal shown in Fig. 2 can be modeled by a comb
function convolved (⊗) with a distribution function and mul
tiplied by an exponential decay term of lifetime τ. The time
distribution generated from one pass of the ion bunch is it
self a convolution of the instrumental response of the pickup
with the ion density per unit length, which we represent by
a Lorentzian with FWHM tw, so that the signal in time (t) is
given by
s (t) ∝ e−t/τ
tw
t2
w/4 + t2⊗
?n +1
?
∞
?
??
n=−∞
δ
?
t −(n + c)
f0
?
+δ
?
t −
2− c?
f0
,
(4)
where c/f0is the time taken for the ions to travel from the
geometric center of the trap to the center of the pickup used
to acquire the data. We formally consider a signal symmetric
in time, for mathematical convenience. The values of c for the
three pickups in KEIRA are approximately 0 and ±0.038, and
change only slightly for different potential energy surfaces. If
?T ≈ 0, twremains constant and the amplitude of the Fourier
0.5
1.0
amplitude
0 0.51.01.5 2.02.53.03.5
0
0.5
0
1.0
0.3340.3360.3380.340
frequency (MHz)
0
0.5
1.0
3.3683.370 3.372
0
0.04
0.08
(a)
(b)
(c) (d)
FIG. 3. (Color online) FFT with a Welch window of the data in Fig. 2 for the
(a) central and (b) offset pickups, with closer inspection of (c) the 2nd and
(d) 20th harmonics for the central pickup only.
transform for positive frequencies ( f ) is
S ( f ) ∝ e−π f tw
?????cosπf?2c −1
τ−2+ (2π ( f − hf0))2
2
?
f0
?????
×
∞
?
h=1
τ−1
(5)
(see Appendix A). Figure 3 shows the Fourier transforms of
the two pickup signals from Fig. 2. For harmonic frequencies
f ≈ hf0and a central pickup with c = 0, Eq. (5) reduces to
S ( f ) ∝ τ?e−πhf0tw??1 + (−1)h?1/2.
Therefore, in Fig. 3(a) for the central pickup, the odd harmon
ics are suppressed, while the even ones reduce exponentially
at a rate determined by tw. For the offset pickup [Fig. 3(b)],
odd harmonics are included and the magnitudes follow the
more complex formula given in Eq. (5). Note that the 13th
harmonic is absent since for c = −0.038, 2πf (1/2 − 2c)/f0
≈ 15π.
From Eq. (6), the FWHM of the harmonic peaks (?fτ
= 1/τπ) is independent of h, hence the resolution increases
linearly with the harmonic order
(6)
R =π
2hf0τ.
(7)
This has been exploited to achieve the very high resolutions
previously attained for a LEIT in selfbunching mode.12
B. Coherent bunch diffusion
Figures 3(c) and 3(d) show the 2nd and 20th harmonics
from the center pickup data. The second harmonic represents
Page 4
0431034Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)
a resolution of R ≈ 500. According to Eq. (3), ?f should be
the same for both harmonics, but due to bunch diffusion the
20th harmonic is broader. If the ion bunch diffuses coherently
[Eq. (1)], and we assume that the initial bunch width t0is
negligible, then tw= n?T in Eq. (4). The Fourier transform
for this signal is derived in Appendix B. For frequencies close
to the hth harmonic f = hf0+ δf and for c = 0, τ → ∞, h
even, we obtain
?
S (δf ) ∝
1
?1 − e−π f ?T?2+ e−π f ?T(2πδf/f0)2
?1/2
,
(8)
which for low bunch spread per oscillation, i.e., f0?T ? 1,
yields
?
In this case, the contribution to the FWHM of the harmonic
due to coherent bunch dispersion is
√3 hf2
S (δf ) ∝
1
(πhf0?T)2+(2πδf/f0)2
?1/2
.
(9)
?f?T=
0?T.
(10)
Therefore if the ion lifetime is long, the peak width increases
linearly with h and hence there is no improvement in resolu
tion for higher harmonics. In Figs. 3(c) and 3(d) it can be seen
that the 20th harmonic is wider than that of the 2nd. It is also
evident from Fig. 2 that a proportion of the ions disperse more
slowly than the rest, resulting in the multiple component peak
in Fig. 3(d).
C. Windowing and discretization
In practice, the acquisition time TS is finite (5 ms in
Fig. 2) and the signal is multiplied by an apodization or win
dowing function a (t/TS) to reduce spectral leakage in the
Fourier transform. Therefore S ( f ) is also convolved with
the Fourier transform of a (t/TS), which introduces additional
broadening to harmonic peaks so that
(?f )2≈ (?fτ)2+ (?f?T)2+ (?fA)2,
where ?fA= A/TS and A is a constant specific to the
apodization function, e.g., for a rectangle window A = 1.21;
Welch A = 1.59; Blackman A = 2.3. With a Welch window
it is only for TS> 5τ that ?fA< ?fτ. Therefore, the peak
width of the 2nd harmonic in Fig. 3(c) is dominated by the
apodization ?fA= 318 Hz.
As the data is also discrete, the Fast Fourier Transform
(FFT) calculates the frequency spectrum at T−1
(200 Hz), which is often similar to ?f [Fig. 3(c)]. This can be
mitigated by zero padding the data, but the discontinuity this
introduces at t = TScan result in additional spectral leakage.
(11)
S
intervals
D. Harmonic phase
Since the ion bunch has a well defined starting point in
time,inprinciplethere isaclear phase associated withFourier
components from ion trapping signal, which should allow dis
crimination against uncorrelated electronic noise. If γ/f0is
the time taken by the ion bunch to reach the geometric center
of the trap after its initial formation (γ = 0.25 for KEIRA),
the phase of the Fourier frequencies is given by
φ ( f ) = −2πf
f0
?
γ +1
4
?
(12)
(see Appendix C). In principle, there is additional informa
tion encoded in the phase which could be used to identify the
harmonic order and to distinguish signal from external noise
sources. However, this is very difficult to achieve in practice
as the phase changes rapidly across the finite width of a har
monic peak.
E. Comparison with a time of flight mass
spectrometer
Some LEITs have been employed as multireflectron
time of flight mass spectrometers, where ions are trapped for
long enough to achieve mass separation and then extracted.18
By significantly increasing the effective time of flight length,
large enhancements in resolution capabilities can be achieved
compared to a standard linear or singlereflection device.
However, such a device is limited to studying similar mass
ions to ensure different ion bunches undergo the same num
ber of oscillations.
If an ion bunch of width tw is extracted from a LEIT
after a time TS, the time of flight resolution is TS/2tw. To
compare with the resolution achievable using Fourier anal
ysis, consider ions trapped for the same time with twf0
= 0.05 and τ → ∞. Using a Welch window, an equivalent
resolution would only be possible from the 32nd harmonic or
greater.
F. Drawbacks of Fourier analysis
The signal obtained from LEITs is nonsinusoidal un
like those obtained from other mass spectrometers such as an
ICR or an Orbitrap. While higher harmonic components give
better resolution, there are a number of drawbacks in using
Fourier transforms to analyze this signal;
(1)The spectral amplitude of the signal is spread out among
many harmonics, reducing the signaltonoise ratio, par
ticularly for higher harmonics.
The presence of multiple harmonics makes conversion
of frequencies into a mass spectrum difficult, especially
if many different ions are present.
As it is possible to have many pickup detectors acquir
ing data simultaneously, one would expect that these dif
ferent channels could be combined to yield more infor
mation and better signaltonoise, but the spectra from
pickups offset from the trap center generate even more
complicated harmonic spectra.
There is rich phase information present in the signal
which could be used to discriminate against electronic
noise and improve resolution, but this is very difficult to
extract from the Fourier transform.
(2)
(3)
(4)
Page 5
0431035Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)
(5)Apodization of the data is necessary to reduce spectral
leakage, but this reduces the achievable resolution and
the signaltonoise ratio.
IV. CHIMERA COMBSAMPLING THEORY
A. Determining the frequency of a comb
Consider a time signal s (t) which consists of a series of
impulses at regular intervals,
s (t) =
N
?
n=0
δ
?
t −n
f0
?
.
(13)
In this case, we use δ to represent a narrow function of fi
nite height. The frequency of such a distribution can be deter
mined froma simple measurement, that is,to simply count the
time between one impulse and the next. For such a scheme,
the resolution increases as the number of impulses measured
is increased. However, executing this measurement becomes
nontrivial if the signal contains additional frequencies. Alter
natively, using a comb function g (t), a spectral function S ( f )
may be generated, which gives the desired frequency f0,
g (t) =
m2
?
m=m1
δ
?
t −m
f
?
,
(14)
S ( f ) =
1
M
?∞
0
s (t)g (t)dt,
(15)
where M = m2− m1+ 1. However, S ( f ) would also con
tain an infinite number of harmonics and “fractional
harmonics.”
Introducing an offset from t = 0, such that both s (t) and
g (t) are temporally offset by an amount γ/f0, gives
?∞
n=0
m2
?
In this case, more than one coincidence of the impulses only
occurs if
f
f0
Therefore if γ is chosen to be irrational, S ( f ) only has a sig
nificant value when f = f0. LEITs are normally filled with
ions by injection of a bunch from outside the trap, where
bunches can be generated by, for example, chopping a contin
uous beam, pulsing ions out of a pretrap, or ionization with a
short pulse laser. For ions which emanate from such a pulsed
ion source and subsequently execute trajectories dictated by
electrostatic optics, the value of γ will be constant, irrespec
tive of the ion mass.
In practice, the measured impulses from an ion bunch
passing through the pickup are finite in time, so that there
will still be contributions from fractional harmonics even if γ
is an irrational number. If a second pickup is used to acquire
data simultaneously, it will have a different value of γ as the
S ( f ) =
1
M
0
?
N
?
?
δ
?
t −(n + γ)
f0
??
?
×
m=m1
δ
t −(m + γ)
f
dt.
(16)
=m + γ
n + γ.
(17)
distance between ion source/pulsar and pickup has changed.
Analysis of this data for the new γ will also have fractional
harmonics but at different values to the signal from the first
pickup. Therefore, by multiplying the two spectra and taking
the square root, the fractional harmonics will be suppressed,
producing a frequency spectrum with the main contribution
being from the fundamental frequencies only.
B. Resolution
As a comb function is a better match to the pickup signal
than a sinusoid, one would expect the frequency resolution to
be significantly improved. To simulate a more realistic sig
nal, we convolve the comb function with a pickup response
functionq (t).Formathematicalconvenience,wewillapprox
imate q (t) by a triangle function of full width in time 2twand
amplitude t−1
?
q (t) = 0 , t > tw.
w, such that
q (t) =
1
tw
1 −t
tw
?
, t ≤ tw,
(18)
Convolving this with the comb function in Eq. (13), we
may express the signal as
s (t) =
N
?
n=0
q
?
t −n
f0
?
(19)
(where we have set γ = 0). The corresponding frequency
function [cf. Eq. (16)] is thus
S ( f ) =
1
M
?∞
0
N
?
m2
?
n=0
q
?
?m
t −n
f0
?
m2
?
?
m=m1
δ
?
t −m
f
?
dt,
=
1
M
N
?
n=0
m=m1
q
f
−n
f0
.
(20)
Now, with twas the FWHM of q (t), where tw? 1/f0, and if
we let f = f0+ δf whereδf
to S ( f ) only when m = n, giving
1
M
m=m1
To model the effect of coherent diffusion, we let
f0< 1/m, there are contributions
S (δf ) =
m2
?
q
?mδf
f2
0
?
.
(21)
tw= (k + mp)/f0,
(22)
where k/f0is the initial FWHM, and the increase in width per
oscillation is ?T = p/f0, giving
f0
M
m=m1
For no bunch diffusion, p = 0, and for small values of δf this
yields
?
S (δf ) =
m2
?
?k + pm − mδf 
(k + pm)2
f0
?
.
(23)
S (δf ) =
f0
k
1 −δf 
2 f0k(m2+ m1)
?
.
(24)
Page 6
0431036Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
frequency (kHz)
amplitude
100.0
99.9
(a)
(b)
(c) (d)
99.9
100.1
100.0
100.1
FIG. 4. (Color online) Combsampling frequency analysis for simulated data
with a frequency of 100 kHz and FWHM of 200 ns for analysis windows now
extending from (a) 0–5, (b) 0.5–5, (c) 2.5–5, and (d) 4.5–5 ms.
(see Appendix D). This is also a triangle function, giving res
olution R,
R =
f0
2?f
=m2+ m1
4k
.
(25)
It is worth noting that as the sampling window nar
rows toward the end of the time data, i.e., m2= N, m1
→ m2, where N = f0TSis the number of oscillations within
the sample time, the resolution for a time of flight instrument
is recovered,
R →m2
2k=
TS
2tw.
(26)
At the other extreme, if the sampling window spans the whole
data set (m1= 0, m2= N), the resolution is a factor of two
lower. In Fig. 4, synthetic data with parameters f0= 100 kHz,
TS= 5 ms, N = 1000, tw= 200 ns, k = 0.04, and p = 0 has
been analyzed using combsampling for four sample windows
with different starting points (m1/f0= 0, 0.5, 2.5, 4.5 ms) and
the same end point (m2/f0= 5 ms). Note that the mean value
of the whole synthetic data set has been subtracted from each
data point prior to analysis.
As predicted from Eq. (25), delaying the start of the sam
pling increases the resolution, so that in Fig. 4(d) the reso
lution is close to that expected for a conventional TOF spec
trometer with a 5 ms flight time. Figure 5 shows the same data
but focusing on amplitudes close to zero and over a greater
range of frequencies. On either side of the main peak the
frequency function dips below zero. This corresponds to fre
quencies where the comb “misses” the pickup impulses and
samples only the baseline (= −mean). There is also a ring
ing artifact which begins when the sampling frequency is far
enough away from f0to start sampling the next impulse at the
end of the time window when
N f0= m2f = (N ± 1) f,
(27)
−0.1
0
0.1
0.2
0.3
amplitude
99100 101
−0.1
0
0.1
0.2
0.3
frequency (kHz)
99100 101
(c)
(a) (b)
(d)
FIG. 5. (Color online) As for Fig. 4 but with the frequency range increased
and a reduced amplitude range.
f − f0= ±f
N= ±100 Hz.
(28)
In Fig. 5(d) the ringing artifact is unacceptably high since
a number of different frequencies are capable of sampling
most of the impulses in this small sample window. The best
outcome can be found for Fig. 5(b) where the start time of the
window is 10% of the end time. Here, nearly all of the ring
ing is confined to negative values which can be eliminated if
values less than zero are rejected. Delaying the start of the
sampling also ensures that for real data, no ions on unstable
trajectories contribute to the frequency spectrum.
Alternatively, coherent bunch diffusion with k = 0,
yields
?
S (δf ) =
f0
Mp
1 −δf 
pf0
?
ln
?m2
m1
?
,
if m1,m2? 1
(29)
(see Appendix D), giving a resolution of R = 1/2p. In this
case, the resolution is solely determined by the rate at which
the bunch diffuses and not upon the length of the acquisition
time.
C. Influence of the ion lifetime and windowing
To simulate the influence of the ion lifetime τ, the data
is given a weighting which linearly decreases in time with a
gradient of τ−1, so that q (t) now becomes
1
tw
tw
effectively creating a trapezoidal window for the data. Tak
ing the case of zero bunch diffusion (p = 0) and m1= 0, the
resolution becomes
R =m2(3τ f0− 2m2)
6k (2τ f0− m2)
q (t) =
?
1 −t
??
1 −
m
τ f0
?
, t ≤ tw,
(30)
(31)
Page 7
0431037Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)
(see Appendix E). This reduces to Eq. (25) as τ → ∞ and
if the lifetime is equal to the width of the analysis window (τ
= m2/f0),theresolutionisafactorof1.5lower(R = m2/6k).
Another advantage of this technique, compared to FFT, is
that any shape of window can be employed. Since the analysis
is more sensitive to impulse peaks near the end of the time
spectrum, their relative weight can be enhanced. For instance,
a simple linear weighting increases the resolution by a factor
of 4/3 (R = m2/3k, Appendix E).
V. IMPLEMENTATION OF CHIMERA ALGORITHM
In this section, we extend the analysis to the general case
and apply it for the setup currently employed in KEIRA. If
we include the pickup offset, c, then Eqs. (13) and (14) now
become
?
?
s (t) =
N
?
n=0
δ
?
t −(n + γ + c)
?n + γ +1
f0
?
+δ
t −
2− c?
f0
??
,
(32)
g (t) =
m2
?
m=m1
?
δ
?
t −(m + γ + c)
f
?
??
+δ
?
t −
?m + γ +1
2− c?
f
.
(33)
For a proportion of the teeth in these two combs to coincide
periodically, i.e., contribute to S ( f ) more than once, one of
the following conditions must be met:
f
f0
=m + γ + c
n + γ + c,
=m + γ +1
n + γ +1
f
f0
=m + γ +1
n + γ + c
m + γ + c
n + γ +1
2− c
,
f
f0
2− c
2− c,
f
f0
=
2− c,
(34)
where m,n are integers in the specified ranges. The mag
nitude of S ( f ) is determined by the fraction of the sample
comb data which yields coincidences normalized to the value
at f = f0. Theoretical spectra for a pure comb signal with
similar conditions to those used in KEIRA (γ = 1/4,c = 0
and γ = 1/4,c = −1/28) are shown in Fig. 6. For the cen
ter pickup (c = 0), it can be seen that the relative contribu
tion of some fractional harmonics is high, particularly for
f/f0= 1/h where h is odd. The even harmonics are absent
while odd harmonics are reduced in amplitude by a factor of
h. In contrast, for the offset pickup fractional harmonics are
significantly reduced in magnitude, with no contribution at in
teger values.
As the frequency peaks in Figs. 6(a) and 6(b) only coin
cide for f = f0, when the two spectra are multiplied together
only the fundamental remains. For real data, the finite width
of the image charge impulses does result in some contribu
tions at other frequencies giving an incomplete suppression
of fractional harmonics. In this case, acquiring an additional
0
0.5
1.0
01.02.0
relative frequency
3.04.05.0
0
0.5
1.0
amplitude
(a)
(b)
FIG. 6. (Color online) Spectrum from analysis of a pure data comb at fre
quency f0for (a) central pickup ring with γ = 1/4,c = 0, and (b) offset
pickup ring with γ = 1/4,c = −1/28 (approximately equal to the experi
mental value of −0.038).
spectrum from a third pickup enables fractional harmonics to
be further suppressed by multiplying all three together and
taking the cube root.
In practice, the teeth of the comb are given a finite width
so that more than one data point is sampled per peak. The
average of all these data points is then normalized by the
number of teeth in the comb M. Increasing the width of the
teeth reduces the achievable resolution, but the change is neg
ligible provided that the width remains less than tw. If ions
of different mass have the same bunch length in space, then
tw∝ m1/2
cies f , the algorithm changes the teeth width in proportion
to f−1.
Figure 7 shows CHIMERA sampling analysis for sim
ulated data generated using the same parameters as those
used in Fig. 4, for the frequency range 0.1 f0–5.1 f0. Data
for three pickups (c = 0,−0.038, +0.06) has been gener
ated [Figs. 7(a), 7(b), and 7(c), respectively]. Also shown are
the results of combining two of the spectra [c = 0,−0.038,
Fig. 7(d)] and all three [Fig. 7(e)] together. It can be seen that
there is a steady improvement in the purity of the spectrum
as the number of combined data sets increases. In principle
this will improve further if data from additional pickups is
included.
Toachieveastillpurerspectrum,theFFTcanbeusedasa
discriminator.Comparingacombsamplingspectrumwiththe
FFT for c = 0 (at double the frequency since odd harmonics
aresuppressed),iftheFFTliesbelowausersetdiscrimination
level, the contribution to the combsampling spectra may be
set to zero at that frequency.
0
∝ f−1
0. Therefore, for different sampling frequen
VI. ANALYSIS OF KEIRA RESULTS
So far we have applied the CHIMERA algorithm only to
synthetic data. Figure 8 shows the analysis of real data ac
quired for TS= 8 ms from ionization of Xe gas by 40 fs laser
pulses with a maximum intensity of 2.5 × 1013W cm−2, av
eraged over 4000 shots. In Fig. 8(a), the FFT of data from
the center pickup shows 14 even harmonics above the gen
eral noise level. In Fig. 8(b), the combsampling method has
Page 8
0431038 Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)
0 0.1 0.2 0.3 0.4 0.5
0
0.5
1.0
amplitude
0 0.1 0.20.30.4 0.5
0 0.1 0.2 0.3 0.40.5
00.10.2 0.3 0.40.5
0
0.5
1.0
frequency (MHz)
00.10.20.3 0.40.5
(a)
(d)
(b) (c)
(e)
FIG. 7. (Color online) Spectra from analysis of simulated data for a single ion species oscillating at a frequency of 100 kHz with γ = 1/4 and (a) c = 0, (b)
c = −0.038, (c) c = +0.06, (d) c = 0,−0.038 combined, (e) all combined.
been applied using a window from 0.8 – 8 ms for both center
and offset pickup data, with a discriminator set just above the
general FFT noise in Fig. 8(a). Apart from a few minor contri
butions, harmonics are absent from the comb analyzed spec
trum. The frequency spectrum for isotopes of Xe+is shown in
Fig. 8(c), where the second harmonic of the FFT is compared
with CHIMERA analysis, which gives a resolving power of
5000, a factor 15 greater than the second harmonic in the FFT.
Whilst an equivalent FFT resolution can be obtained from the
30th harmonic as predicted by Eq. (3), this peak is barely dis
cernable above the noise level.
It can be seen that the relative amplitudes of the xenon
isotopes obtained from FFT and CHIMERA are different.
While the results of the CHIMERA analysis are in good
agreement with natural abundances of these isotopes, the
agreement for the FFT is poor. This is probably due to the
limitation on the FFT frequency step size as discussed in
Sec. III C.
As CHIMERA removes harmonics, it is also straightfor
ward to generate a mass spectrum. To demonstrate this, an
experiment was carried out on a gaseous target of αCyano
4hydroxycinnamic acid (CHCA), which has a formula of
C10H7NO3and a molecular mass of 189.17. The neutral tar
get of this molecule was produced by depositing a sample of
CHCA onto the surface of the final electrode, which was then
desorbed off the surface using a 4 ns, 355 nm laser pulse.
Subsequent ionization/fragmentation was enforced by a 40 fs
laser pulse of intensity 1013W cm−2. Data was acquired for
3 ms using the central pickup and one offset pickup detec
tor. Figure 9 compares mass spectra produced from a FFT of
the central pickup signal and from the combsampling anal
ysis using data from both pickups. As well as the improved
0
0.5
1.0
amplitude
0 0.2 0.4 0.60.81.01.21.4
0
0.5
1.0
0.10150.10200.10250.1030
frequency (MHz)
0.10350.1040 0.10450.1050
0
0.5
1.0
136
132
129
128
130
131
134
(b)
(c)
(a)
FIG. 8. (Color online) Frequency analysis of Xe+isotopes trapped for 8 ms using (a) FFT with a Welch window and (b) combsampling with a window from
0.8–8 ms. Plot (c) shows the fundamental frequency from combsampling (narrow peaks) compared to the second harmonic of the FFT. Corresponding masses
of Xe isotopes are indicated.
Page 9
0431039Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)
0
0.5
1.0
amplitude
020 4060 80 100120140160180 200
0
0.5
1.0
mass (atomic units)
C2Hn
+
C3Hn
+
C4Hn
+
C5Hn
+
C6Hn
+
C7Hn
+
–COOHCN
–COOH
–OH
M+(b)
(a)
FIG. 9. (Color online) Frequency analysis of ions generated from ionization of CHCA trapped for 3 ms from (a) FFT of the center pickup data with a Welch
window, and (b) combsampling of data from the center pickup and the offset pickup closest to the femtosecond laser focus. M+is the parent ion, with other
fragments corresponding to the hydrocarbons shown, or to the loss (−) of a particular chemical group.
resolution of the combsampling method, it can be seen that
multiple harmonics in Fig. 9(a) make analysis of masses less
than 30 very difficult with FFT. For low masses in Fig. 9(b)
there are some fractional harmonics present, but these are at
a low level and could be further suppressed if data from addi
tional pickups was available.
VII. SUMMARY
In this paper, a new algorithm (CHIMERA) for analyz
ing the oscillation frequencies of ions in a linear electrostatic
trap has been described. By using a comb function to sam
ple the data at different frequencies, it has been shown that
more complete utilization of the information contained in the
ion signal is possible. The key to extracting only the funda
mental frequency is for the initial ion bunch generation to be
temporally and spatially well defined. By generating multiple
spectra using data acquired from more than one pickup de
tector, integer and fractional harmonics arising from the finite
width of the pickup impulses can be suppressed.
We have tested CHIMERA on simulated data and real
data acquired from a linear electrostatic ion trap using in situ
ion generation with a femtosecond laser. As the sampling of
the narrow pickup peaks is very sensitive to the frequency
of the comb, mass resolutions obtained from relatively short
trapping periods (<10 ms) are shown to approach those which
could be obtained for a linear time of flight device of equiv
alent length (about 100 m). For instance, a mass resolution
of 5000 was obtained for Xe+isotopes for a trapping time
of 8 ms, which was a 15 fold improvement over the second
harmonic obtained from a Fourier transform. CHIMERA also
benefits in that it does not suffer from spectral leakage, which
is a feature of the windowing process in a Fourier transform.
And unlike FFT algorithms, there are no limitations on the
frequency steps used.
The value of the initial offset (γ) of the comb data and
that of the pickup electrodes from the trap center (c), in the
present setup have been fixed by the existing geometry and
operational mode (γ=1/4, c=0, −0.038). While ideally these
values should be irrational, this is not possible in practice due
to the finite width of the recorded impulses. However, if γ and
c are chosen to avoid values close to factorized fractions with
small denominators, significant contributions from fractional
harmonics can be avoided.
As such our present value of γ=1/4 is not ideal, but could
be changed by pulsing the potential of the end plate R7 once
all the ions have first left the ion generation region. For in
stance, if R7 = 3.2 kV when the ions are created and is raised
to 3.55 kV once the ions reach the field free region, a new γ
value of 0.283 would be generated. With appropriate choices
of γ and c, CHIMERA could also be easily applied to other
electrostatictrapsandringsforwhichionbunchesareinjected
from an external source. For instance, ion bunches from a
matrixassisted laser desorption and ionization source19and a
pulsed beam20have previously been injected into this type of
trap using electrostatic optics. Another possible implementa
tion would be for the case where ions are confined and cooled
in a radiofrequency trap prior to injection into a storage ring,
provided the ions are injected directly rather than being mass
selected by a magnet.21
Application to other mass spectrometers, such as the Or
bitrap and ICR may be possible if highly anharmonic signals
are acquired. The potential for generating higher mass resolu
tion from anharmonic signals in an ICR has previously been
recognised.22,23For some pickup electrode arrangements and
ion excitation schemes, signals from multiple pickups or the
differentialsignalfrompairscouldexhibitasignalresembling
the pulse trains analyzed in this paper.24
In conclusion, the ability to extract high resolution mass
spectra from a linear electrostatic ion trap has been enhanced
by employing a new CHIMERA analysis method, which sam
ples ion oscillation data with a comb function. Compared with
Fourier analysis, the CHIMERA algorithm makes better use
of the phase and frequency information present in the data.
This enables the fundamental ion oscillation frequencies to
be extracted with much higher resolution, while also sup
pressing uncorrelated electronic noise. When this algorithm is
combined with the mass independent trapping and ion detec
tion characteristics of the electrostatic trap, very complex ion
Page 10
04310310Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)
mixtures with an enormous mass range can be simultaneously
analyzed with very high resolution.
ACKNOWLEDGMENTS
The authors wish to note with appreciation, insightful
discussions with Gleb Gribakin who provided valuable input
to this paper. This work was supported by Leverhulme Trust
and utilized the Artemis Laser at the STFC Central Laser Fa
cility. C.R.C. and O.K. acknowledge funding from the Lev
erhulme Trust. L.B., M.J.D., and R.B.K. acknowledge fund
ing from Department of Employment and Learning (Northern
Ireland) and J.D.A. acknowledges funding from the European
Social Fund.
APPENDIX A: FOURIER TRANSFORM FOR AN ION
BUNCH OF CONSTANT LENGTH
s (t) ∝ e−t/τ
?
tw
t2
w/4 + t2⊗
2− c?
?
∞
?
n=−∞
δ
?
t −(n + c)
f0
?
+δ
t −
?n +1
f0
??
,
(A1)
for this signal with Lorentzian peaks and constant width tw,
the Fourier transform is
τ−1
τ−2+ (2πf )2
?
n=−∞
S ( f ) ∝
⊗
e−π f tw
?
∞
?
e−
2πif (n+c)
f0
+e−
2πif(n+1
2−c)
f0
??
,
(A2)
S ( f ) ∝
τ−1
τ−2+ (2πf )2
?
⊗
e−π f tw
?
1+e
2πif (2c−1
f0
2)
?
e−
2πif c
f0
∞
?
n=−∞
e−
2πif n
f0
?
,
(A3)
S ( f ) ∝
τ−1
τ−2+ (2πf )2
?
⊗
e−π f tw
?
1+e
2πif(2c−1
2)
f0
?
e−
2πif c
f0
∞
?
h=−∞
δ ( f −h f0)
?
.
(A4)
If the width of the Lorentzian is much less than the fun
damental frequency (i.e, τ−1? f0), only one delta impulse in
the comb makes a significant contribution to the convolution
at each frequency, yielding
?
S ( f ) ∝ e−π f tw
1 + e
2πif(2c−1
2)
f0
?
e−2πif c
f0
×
∞
?
h=−∞
τ−1
τ−2+ (2π ( f − hf0))2,
(A5)
S ( f ) ∝ e−π f twcosπf?2c −1
∞
?
and the amplitude for positive frequencies is
2
?
f0
e−πif
2 f0
×
h=−∞
τ−1
τ−2+ (2π ( f − hf0))2,
(A6)
S ( f ) ∝ e−π f tw
?????cosπf?2c −1
τ−2+ (2π ( f − hf0))2.
2
?
f0
?????
×
∞
?
h=1
τ−1
(A7)
APPENDIX B: FOURIER TRANSFORM FOR A
COHERENTLY DIFFUSING ION BUNCH
s (t) ∝ e−t/τ
∞
?
t −(n + c)
n=0
n?T
(n?T)2
4
+ t2
⊗
?
δ
?
f0
?
+ δ
?
t −
?n +1
2− c?
f0
??
.
(B1)
With the ion bunch lengthening linearly in time, the
Fourier transform for positive frequencies is
S ( f ) ∝
τ−1
τ−2+ (2πf )2
∞
?
⊗
n=0
e−π f n?T
?
e−
2πif (n+c)
f0
+ e−
2πif(n+1
2−c)
f0
?
.(B2)
Using the identity?∞
S ( f ) ∝
n=0pneinx=
τ−1
τ−2+ (2πf )2
e−2πif c/f0
1 − e−π f ( f0?T+2i)/f0
1
1−peix,
⊗
?
1 + e
2πif(2c−1
2)
f0
?
.
(B3)
For f = hf0+ δf , where δf ? f0, c = 0, τ → ∞ and h
even
S (δf ) ∝ (1 − e−π f ?Te−2πiδf/f0)−1,
S (δf ) ∝ ((1 − e−π f ?T)2+ e−π f ?T(2πδf/f0)2)−1
(B4)
2.
(B5)
APPENDIX C: FREQUENCY DEPENDENCE OF THE
FOURIER PHASE
If we include additional time delay γ/f0, and consider
only imaginary multiplicative terms in Eq. (A3) we obtain
?
S ( f ) ∝ e−
2πif (c+γ)
f0
1+e
2πif(2c−1
2)
f0
?
,
(C1)
Page 11
04310311Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)
S ( f ) ∝
?
?
cos2πf (c + γ)
f0
− i sin2πf (c + γ)
+ isin2πf?2c −1
f0
?
1+cos2πf?2c −1
2
?
f0
2
?
f0
?
, (C2)
S ( f ) ∝ cos2πf?c −1
?
4
?
f0
×
cos2πf?γ +1
4
?
f0
− i sin2πf?γ +1
?
4
?
f0
?
, (C3)
φ ( f ) = −2πf
f0
γ +1
4
?
.
(C4)
APPENDIX D: COMBSAMPLING THEORETICAL
RESOLUTION
S (δf ) =
f0
M
m2
?
m=m1
?k + pm − mδf 
(k + pm)2
f0
?
.
(D1)
For no bunch diffusion (p = 0),
f0
Mk
m=m1
S (δf ) =
m2
?
?
1 − mδf 
f0k
?
=
f0
k
−δf 
Mk2
m2
?
m=m1
m ,
(D2)
S (δf ) =
f0
k
−
δf 
2Mk2(m2− m1+ 1)(m2+ m1),
(D3)
S (δf ) =
f0
k
?
1 −δf 
2 f0k(m2+ m1)
?
.
(D4)
This expression [in Eq. (D4)] is only valid for
m2δf 
f0k
≤ 1.
(D5)
For coherent bunch diffusion (k = 0)
S (δf ) =
f0
Mp2
?
p −δf 
f0
?
m2
?
m=m1
1
m.
(D6)
Using the following identity for an infinite harmonic series:
lim
m2→∞
m2
?
?
m=1
1
m= lnm2 + 0.5772 ,
(D7)
S (δf ) =
f0
Mp
1 −δf 
pf0
?
ln
?m2
m1
?
,
if m1,m2? 1,
(D8)
S (δf ) =
if m1= 1,m2? 1 .
f0
Mp
?
1 −δf 
pf0
?
(lnm2+ 0.5772 ),
(D9)
APPENDIX E: RESOLUTION OBTAINED WITH A
LINEAR TIME WEIGHTING ON THE DATA
For a signal decreasing linearly with a gradient of τ−1,
m1=0, and p = 0,
f0
Mk
m=0
S (δf ) =
m2
?
?
1 −m
k
δf 
f0
??
1 −
m
τ f0
?
,
(E1)
S (δf ) ∝
m2
?
m=0
1 −
1
τ f0
m2
?
m=0
m−δf 
kf0
?m2
m=0
?
m −
1
τ f0
m2
?
m=0
m2
?
,
(E2)
S (δf ) ∝ m2+ 1 −
?
1
2τ f0m2(m2+ 1)
1
3τ f0m2(m2+ 1)(2m2+ 1)
−δf 
2kf0
m2(m2+ 1)−
?
(E3)
,
S (δf ) ∝ 1 −
m2
2τ f0
−δf m2
2kf0
?
1 −(2m2+ 1)
3τ f0
?
,
(E4)
S (δf ) ∝ 1 −δf m2(3τ f0− 2m2− 1)
3kf0(2τ f0− m2)
Since m2? 1, this gives a resolution of
R =m2(3τ f0− 2m2)
With a linearly increasing weighting ,
.
(E5)
6k (2τ f0− m2).
(E6)
S (δf ) ∝
m2
?
m=0
m −δf 
kf0
m2
?
m=0
m2,
(E7)
S (δf ) ∝ m2(m2+ 1) −δf 
3kf0
m2(m2+ 1)(2m2+ 1),
(E8)
S (δf ) ∝ 1 −δf 
3kf0
(2m2+ 1).
(E9)
Giving a resolution of R = m2/3k if m2? 1.
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