Order-of-magnitude signal gain in magnetic sector mass
spectrometry via aperture coding
Evan X. Chen *1, Zachary E. Russell *1, Jason J. Amsden 1, Scott D. Wolter 1,2, Ryan M.
Danell 3, Charles B. Parker 1, Brian R. Stoner 4, Michael E. Gehm 1, Jeffrey T. Glass 1,
David J. Brady 1
Author affiliations
1
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
2
Department of Physics, Elon University, Elon, NC 27278
3
Danell Consulting, Winterville, NC 28590
4
Discovery-Science-Technology Division, RTI International, Research Triangle Park, NC 27709
Corresponding Author:
Prof. Jeffrey Glass
Department of Electrical and Computer Engineering
Duke University
Durham, NC 27708
(919) 660-5431
jeff.glass@duke.edu
*These authors contributed equally
Supporting Information
Supporting Information
Generalized Forward Model Derivation. The forward model is a mathematical representation
of the mass spectrometer system that can be used to calculate the pattern of ions on the detector
plane from the pattern of the aperture and the physical characteristics of the system. For
notational simplicity, in the forward model we use

to represent m/z, the ratio of the ion mass
(in amu) to the net number of elementary charges. The coordinate system used in this derivation
defines the x and y coordinates as the non-mass-dispersive direction and the mass-dispersive
direction in the detector plane, respectively, and the x' and y' coordinates are the spatial
dimensions of the coded aperture, as shown in Figure 1b. The ion intensity at a point ( x, y ) on
the detector plane, is g ( x, y ) :
g ( x, y ) = S ( x, y,  )t ( x, y)h  x, x ', y, y',   dxdy d . (1)
where S ( x ', y ',  ) is the ion spectral intensity at a point ( x ', y ') in the coded aperture plane,
t ( x, y)  {0,1} is the coded aperture transmission function describing the shape of the aperture,
and h( x, x ', y, y ',  ) is a kernel describing propagation through the spectrometer for an ion of
specific
.
Assuming that the ion beam spatial intenstiy in the coded aperture plane is independent of
,
S ( x ', y ',  ) can be seperated into f ( ) I ( x ', y ') where f ( ) is the mass spectrum and
I ( x ', y ') is the ion beam intensity spatial profile at ( x ', y ') in the coded aperture plane.
Therefore, we can describe the ion intensity at a point ( x, y ) on the detector as the following:
g ( x, y ) =  f ( ) I ( x, y)t ( x, y)h  x, x ', y, y ,   dxdy d . (2)
Grouping terms in Equation (2), we define a system transfer function, T:
T ( x, y,  ) = I ( x, y)t ( x, y)h  x, x ', y, y ',   dxdy. (3)
This system transfer function describes how ions at each
 travel from the coded aperture to the
detector. Substituting T into Equation (2) gives our generalized forward model:
g ( x, y ) =  f ( )T  x, y,   d . (4)
To enable reconstruction of the mass spectrum from experimental data, the analytical
forward model (Equation (4)) derived above must be discretized to be consistent with the
pixelated experimental measurements. The pixelation of the intensity pattern at the detector
plane can be represented by sampling with rect functions in both the x and y directions with Δ
representing the effective pixel sampling pitch size. The rect function is defined as the following:

2.
(5)

if x 
2

0
 x  
rect    
   1

if x 
Using the rect functions, the discrete model for the detector intensity is then:
 x  xi
g ( x, y ) = g i rect 
 
i

 y  yi
 rect 

 

.

(6)
Since the mass spectrum has a discrete nature, we can represent f ( ) by a series of Dirac delta
functions:
f ( ) =  f k     k  . (7)
k
Combining Equation (6) and (7) with Equation (4), leads to the discretized forward model:
gi =  f k 
k
xi  /2
xi  /2

yi  /2
yi  /2
T  x, y,  k  dydx. (8)
Furthermore, this discretized system forward model can be written in matrix form as:
gi = H i ,k f ( k ), (9)
where H is:
H i ,k = 
xi  /2
xi  /2

yi  /2
yi  /2
T  x, y,  k  dydx, (10)
and we assume the Einstein summation convention for repeated indices. This allows expressing
the discrete forward model as the linear system:
g  Hf .
(11)
where g is a vector of experimentally measured data and H is the forward matrix which maps the
mass spectrum f to the measurement data vector. Using Equation (11) combined with numerical
inversion algorithms we can estimate the desired mass spectrum
f
from measurements g.
Explicit Forward Model for the 90-degree MS. Reconstructing a mass spectrum, f ( ) , from
g ( x, y ) requires determining the system transfer function, generating the forward matrix H, and
using a numerical inversion algorithm. The system transfer function (Equation (3)) contains
information about the entire mass spectrometer system, including the ion beam properties, the
coded aperture geometry, and the mass analyzer geometry. The propagation kernel
h( x, x ', y, y',  ) describes how an ion at a point in the aperture plane traverses through the sector
to a point on the detector plane and therefore is different for various sector types. Figure 1b
shows a schematic of the magnetic sector mass spectrometer used in this study. The instrument is
composed of an electron ionization ion source, a coded aperture spatial filter, a 90-degree
magnetic sector (90 degrees being the angle between the aperture plane and the detector plane),
and an ion detector composed of a multichannel plate and phosphor screen assembly. Additional
details on the experimental apparatus appear in the Methods section. For the purpose of
demonstrating the concept, we selected a 90-degree sector geometry because it is the simplest
system to confirm the benefits of spatial aperture coding in mass spectrometry. Additional
spectrograph geometries and their compatibility with these techniques will be the subject of a
future manuscript.
For the 90-degree system geometry, the propagation kernel h can be approximated as:

h( x, x ', y, y',  )    x  x ', y 



8U  (u / e)
y ' y '2  . (12)

B

where B is the magnitude of the applied magnetic field, U is the applied ion acceleration voltage,
and (u/e) is the ratio of an amu to an elementary charge in the desired set of physical units.
Further, the system transfer function is:


8U  (u / e)
T ( x, y,  )  I ( x, y)t ( x, y)  x  x ', y 
y ' y '2  dxdy


B



 

2U  (u / e) 
B2
2U  (u / e) 
B2
2
2


 I  x,
1

1

y
t
x
,
1

1

y



 .
2

 
B2
2
U

(
u
/
e
)
B
2
U

(
u
/
e
)



 

 


(13)
Indicating that system response for a single  is a scaled and shifted version of the aperture
pattern t ( x, y ) product with ion spatial intensity I ( x, y ) . Detailed derivations for the propagation
kernel for the 90-degree MS as well as for arbitrary sector angles are in the following
“Propagation Kernel Derivations” section.
Of course, the transfer function in Equation (13) is an ideality that does not actually occur
and cannot be directly used as part of an inversion strategy. We can understand and address the
nature of these idealities by examining actual experimental data and calibration of the forward
model as described in the main text.
Propagation Kernel Derivations. In this section we derive an equation for the propagation
kernel h for a sector of arbitrary angle and then simplify to the specific 90-degree MS used in
the experiments. Supporting Figure 1 is a schematic of the general magnetic sector instrument.
The aperture plane and detector plane intersect at an angle
1 at the point O. O is the origin of
two local coordinate systems—(x’, y’) on the aperture plane and (x, y) on the detector plane. A
collimated ion beam (created via accelerating voltage U) is normally-incident on the aperture
plane at point A. After passing through the aperture, the ions interact with the magnetic field B,
and travel in a circular path of radius r centered at the point C. The ions then strike the detector
plane at the point D. Below we relate the impact location D to the aperture location A, the system
parameters U, B, and 1 , and the mass-to-charge ratio m / z (the ratio of the ion mass (in amu) to
the net number elementary charges ) of the ions (which we represent as

for notational
simplicity).
From a simple consideration of motion under a centripital force, we can write r, the radius of
curvature of the ion path, as
r
2U  (u / e)
, (14)
B
With u one amu expressed in kg and e the fundamental charge in C. From Supporting Figure 1,
we see that AO  y ' , CD  y , and OC  r  y ' . Applying the Law of Sines to the triangle
OCD , we write
r
r  y
y


, (15)
o
sin(180  1 ) sin  2 sin 3
which we can rewrite as
 (r  y)sin(180o  1 ) 
2  arcsin 
 , (16)
r


and
 (r  y) sin(180o  1 ) 
 . (17)
r


3  1  2  1  arcsin 
Combining Equation (16) and (17) with (15), and solving for the ion strike position y yields

 (r  y) sin(180o  1 )  
r
y
sin  1  arcsin 
  . (18)
sin(180o  1 )
r



Combining this with Equation (14) yields the final general result
y


2U  (u / e)
By
sin  1  arcsin  1 
o
Bsin(180  1 )
2U 


o
For the 1  90 case, Equation (19) can be simplified as,


o
 sin(180  1 )   . (19)


y
8U (u / e)
y ' y '2 . (20)
B
The propagation kernel used in the experiment can be written as:

h( x, x ', y, y',  )    x  x ', y 



8U  (u / e)
y ' y '2  . (21)

B

Path-length Dependent Magnification. In the derivation of the propagation kernel, we
assumed that the ion beam was normally-incident on the aperture plane. In actuality, however,
the beam has some angular spread and is therefore only normally incident near the axis. In this
section, we derive a result that shows how this effect impacts the propagation kernel in x.
We begin by analyzing the expected variation of the source cone angle with  . We
consider ions in the region between the accelerating grid and the aperture plane. The velocity of
an ion can be decomposed into radial and tangential components. Similarly, the kinetic energy of
the ion can divided into contributions arising from each of these components. The radial
contribution to kinetic energy is unaffected by the longitudinal acceleration provided by the grid,
and therefore maintains the distribution of radial kinetic energies present in the gas prior to
acceleration. Assuming the ions are originally thermally-distributed (Maxwell-Boltzmann), the
resulting distribution of radial kinetic energy is independent of
 (or m). The distribution of the
radial velocity, however, is proportional to 1/ m (and hence 1/  ) as a result of the fact that
2
the kinetic energy K  (1/ 2)mv . A similar argument holds in the longitudinal direction, except
that the longitudinal kinetic energy is dominated by effect of the accelerating grid
K  qU  (me /  u )U . Converting to the longitudinal velocity distribution we again find that
the distribution is proportional to 1/  . Given the distributions for the velocities, we can
compute the angular distribution of the ions by relating the tangent of the angle to the ratio of the
radial velocity to the longitudinal velocity. As both distributions are proportional to 1/  , this
cancels in the ratio and we conclude the angular distribution is independent of
 . The spatial
distribution of the ions at the aperture is governed by the angular distribution via propagation,
and thus the total ion distribution (spatial and angular) at the aperture plane must be

independent.
We next consider how the ion distribution at the aperture plane impacts the final size of
the detected aperture pattern. As the aperture can effectively be viewed as casting a shadow on
the detector plane, we expect the aperture pattern to be magnified by the joint action of the
angular spread of the source and the propagation distance to the detector plane. We have shown
above that the angular spread of the source is
 independent. Here we determine the  scaling
of the propagation distance. From Supporting Figure 1 we see that the propagation path length
can be written as
 
l  r
o
 180
 
 r
o
 180


 3

 (r  y) sin(180o  1 )  



arcsin


 1
r



2U  (u / e)  

o
B
 180
 (r  y) sin(180o  1 )  

 .
  1  arcsin 
r



o
For the 1  90 case, this reduces to
l


2U  (u / e)  
By '
  arcsin 1 
  . (23)

2

B
2
U

(
u
/
e
)



(22)
We have previously shown that for reasonable assumptions, the spatial distribution at the
aperture plane is independent of
 (i.e. that
y ' contains no implicit
 dependence). For the
conditions of the experiment, the argument of the arcsin is approximately 1. Taylor expansion of
the arcsin about this point scales as 1/  1/ 4 to the lowest order. Combining this with the

prefactor, we see the overall scaling of l is  1/4 to lowest order. Thus we make the substitution
x  x '  x  (1  M x 1/4 ) x ' in the propagation kernel to account for this scaling.
Supporting Figure 1. Generalized magnetic sector geometric model. Two local coordinate
systems were used here, one on the coded aperture plane, and one on the detector plane. 1 is the
angle between coded aperture plane and detector plane. Set the intersection of aperture and
detector plane as origin zero point, y ' is the distance from aperture opening to origin, and y is
the strike distance on the detector. r is the radius of curvature of ion beam.
Methods
As this work builds upon that of [1], much of the following section contains similar information
to that found in the previous work, but are included here for convenience. Additional details are
provided on the topic of ion source design.
90-degree Magnetic Sector Mass Spectrometer. In order to test the concept of onedimensional aperture coding for mass spectrometry we constructed a test system consisting of an
electron ionization (EI) ion source [2], a magnet with a maximum field of 0.45 T, and a 40 mm
diameter 10 µm pitch micro-channel plate (MCP) array imaging ion detector (see Figure 1b).
A 90-degree magnetic sector geometry was chosen for this work due to its simplicity for
proof-of-concept testing. The magnet was purchased from Dexter Magnetics (Elk Grove Village,
IL) and consisted of two 25 by 25 by 100 mm NdFeB bar magnets spaced 25 mm apart and
supported by a low-carbon stainless steel yoke. The maximum field in the gap was 0.45 T.
A single stack 40 mm diameter circular MCP array imaging ion detector with 10 µm
channel pitch spacing coupled to a phosphor screen was used (Beam Imaging Solutions, BOS40; Longmont, CO). The MCP and phosphor were biased at 1 kV and 3 kV respectively, with the
detector plane of the MCP at system ground. After losses are considered, the effective spatial
resolution of the detector system is approximately 50 µm. The coded spectra produced at the
detector were recorded using a 10-bit black and white camera (Sony XCD U100) with an 8.5-90
mm focal length manual zoom video lens (Edmund Optics part #68-679; Nether Poppleton,
York, UK) outside of the vacuum chamber.
Ion Source Design. In this section we elaborate on the design of the ion source used for this
work (shown in Supporting Figure 2). The EI ion source was constructed using a Kimball
Physics (Wilton, NH) eV Parts kit and a commercial tungsten filament assembly (Extrel EX100)
from Scientific Instrument Services (Ringoes, NJ). This electron ionization ion source design is
optimized to illuminate spatially coded apertrure pattenrs of just over 1cm2 with minimial
angular and energy dispersion. Minimizing source dispersion was of particular importance due to
the simplicity of the magnetic sector used for these experiments, as the 90 degree magnetic
sector geometry does not provide any correction for angular or energy dispersion from the ion
source.
To create a beam of ions capable of illuminating a 1cm2 area coded aperture, electrons
thermally generated using a commercial tungeston spiral filament with repeller plate were
directed towards a slit aperture between the electorn filament and the ionization region. Electrons
passing through this slit form a thin “sheet” of electrons from which ionization events can occur.
The electrons were created near the ground potential and experienced a slight repelling voltage
(under 5 V) from the filament repeller before being accelerated up to the potential of the
ioniation region. The regions where ions can be formed is large in the expanse of the dimensions
we are spatially coding in, but quite narrow in the dimenion of the potential gradient for the
extration of the subsequently formed ions.
To minimize energy dispersion in the ion beam, the potential gradient in the ion beam
was kept as small as possible. This narrow potential gradient across the ionization region reduces
the energy spread of the ion beam as shown in [3]. Here, this was accomplished by keeping the
entire ionization region (electron slit, anode, side panels, and first ion extraction aperture) at the
same potential as the desired ion beam potential (2 kV), placing a slight retarting potential on the
repeller electrode (5-10 V above ionization region) and then using a potential of about ½ that of
the ionization region on the second ion extraction aperture. The extraction aperture both extracts
ions and creates an initial ion crossover that can be used with subsequent focusing options. The
energy spread of this ion source was tested via SIMION 8.1 [4]simulation and then verified
experimentally by measuring the stopping potential, as shown from Supporting Figure 2.
Supporting Figure 2. Ion Source Design. The performance of the ion source used in this work
was simulated and optimized using SIMION 8.1 [4] as shown in part (a) with electrode values as
shown in Supporting Table 1. Equipotentials shown in the ionization region are in 1V steps from
2000V to 2005V. The focusing lenses were not used in this work. The energy dispersion of the
source was determined experimentally to be no more than 0.5% of the beam energy across beam
energies ranging from 250 to 1000 eV. Data above this energy was not acquired due to
limitations of the experimental setup. Stopping potential data for a 1000eV beam is shown in (bc).
Supporting Table 1:
Electrode Name
Reference Variable
Applied Potential
Filament
F
0-2V 2A
Filament Housing
H
0V
Electron Slit
S
2000V
Anode
A
2000V
Ion Repeller
R
2005V
Extraction Lens
E
1000V
Ion Aperture
I
2000V
Focusing Lenses
L
0V
Coded Aperture
CA
0V
Coded Aperture Fabrication. The coded apertures for the system were fabricated using UV
lithography patterning and deep reactive ion etching (DRIE) (SPTS Pegasus Deep Silicon
Etcher; San Jose, CA) of silicon and coated with Au to prevent charging from the ion beam. This
process enabled arbitrary geometries of coded patterns to be created in volume for pattern
optimization. For this study we used 250 µm thick silicon wafers and a minimum feature size of
125 µm. Coded aperture patterns were derived from an elongated Cyclic-S Matrix pattern with a
13X elongation in the non-mass-dispersive dimension. For example, an S-7 coded aperture of the
pattern {0101110} would consist of one 125 µm slit and one 375 µm slit spaced 125 µm apart.
Data Collection Procedure. A LabVIEW program (National Instruments Software; Austin, TX)
was developed to automate coded aperture image capture across multiple camera exposure times.
All the data presented in this paper used the same exposure time (100 ms). A bleed valve
regulated analyte flow into the system from a gas reservoir at approximately atmospheric
pressure. For each aperture and each analyte, spectral images were taken across a range of
pressures spanning from 1.0–10.0 µTorr in steps of 2.5 µTorr. System base pressures were held
constant across the data series to ensure quantitative results for aperture gain could be acquired.
Apertures were tested across multiple days and cycles of system venting/pump-down in a
random order. Several of the coded patterns were tested multiple times during this process with
consistent intensity results, verifying the reproducibility of the intensity data.
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Spectrometry. J Am Soc Mass Spectrom 2015, 26 (2), 248-256.
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Nier, A. O., A Mass Spectrometer for Isotope and Gas Analysis. Review of Scientific
Instruments 1947, 18 (6), 398-411.
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Ahn, J. R.; Lee, S.-G.; Park, C. J., Measurement of Ion Kinetic-energy Distributions in
Electron-impact Ion Sources of a Quadrupole Mass Spectrometer. J. Korean Phys.Soc 2011, 59,
2670.
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