A portable imaging Mueller matrix polarimeter based

A portable imaging Mueller matrix polarimeter based
A portable imaging Mueller matrix polarimeter based on a
spatio-temporal modulation approach: theory and
implementation
Israel J. Vaughna , Oscar G. Rodrı́guez-Herrerab , Mohan Xua , J. Scott Tyoa,c
a College
b Centro
of Optical Sciences, University of Arizona, Tucson, USA;
de Ciencias Aplicadas y Desarrollo Tecnológico,Universidad Nacional Autónoma de
México, Cd. Universitaria, Mexico
c University of New South Wales @ ADFA, Canberra, ACT, Australia
ABSTRACT
Imaging polarimeters have been largely used for remote sensing tasks, and most imaging polarimeters are division
of time or division of space Stokes polarimeters. Imaging Mueller matrix polarimeters have just begun to be
constructed which can take data quickly enough to be useful. We have constructed a Mueller matrix (active)
polarimeter utilizing a hybrid modulation approach (modulated in both time and space) based on a micropolarizer array camera and rotating retarders. The hybrid approach allows for an increase in temporal bandwidth
(instrument speed) at the expense of spatial bandwidth (sensor resolution). We present the hybrid approach and
associated reconstruction schemes here. Additionally, we introduce the instrument design and some preliminary
results and data from the instrument.
Keywords: polarimetry, modulated polarimetry, linear systems, active polarimetry, Mueller matrix
1. INTRODUCTION
Recent advances in polarimetric instrument design1–5 have facilitated systematic instead of ad-hoc designs for
specific tasks using a linear systems framework. Particularly Alenin and Tyo4 have provided a general framework
to design any periodically modulated polarimetric instrument. We utilized this framework to design an active
polarimeter optimized for temporal bandwidth. The specific instrument presented here is designed optically for
general remote sensing purposes at the 10 − 200m range. One motivating example of the need for more temporal
bandwidth is skin cancer detection, in 2011 our group began evaluating combined hyperspectral and polarization
measurements for utility in skin cancer and skin interaction.6 A significant issue with our instrumentation was
the requirement for image registration due to patient movement, and a great deal of effort was expended on
improving image registration performance.
Faster instruments minimize the need for complex image registration algorithms and processing. In addition
to mitigating image registration, the linear systems framework allows designers to reduce channel crosstalk.
Channel crosstalk is often the cause of ”erroneous” polarization measurements and can be thought of as a
theoretical limitation instead of a systematic ”error.” We will address channel crosstalk in detail in another
communication. In this communication, we present a spatio-temporal system design as a linear system, we
develop an instrument design based on the theory, and finally we demonstrate the actual system and related
issues.
Send correspondence to IJV or JST
IJV: [email protected]
JST: [email protected]
2. FORMALISM AND CHANNELS
We use the Mueller-Stokes mathematical formalism here, as it is most commonly used in instrumental polarization
and polarimeter design. This analysis is, however, agnostic to the formalism used, a coherence formalism7 with
periodic modulators could also be used and would have similar results. In the next sections, it should be
kept in mind that modulations are done in some physical domain, they are periodic, i.e., a superposition of
sinusoidal functions, and the ”channels” are the resultant δ-functions which ensue from the Fourier transform of
the sinusoidal modulations.
2.1 Modulated Mueller formalism
The Stokes parameters are described by
  
 
h|Ex |2 i + h|Ey |2 i
s0
s1  h|Ex |2 i − h|Ey |2 i 
 
 
s=
s2  =  2 Re hEx Ey∗ i  = 
s3
2 Im hEx Ey∗ i


 , where s0 > 0, s20 ≥ s21 + s22 + s23

(1)
where h·i denotes the time average, s0 is proportional to the total irradiance, s1 is proportional to the prevalence of horizontal (0◦ ) over vertical (90◦ ) polarization, s2 is proportional to the prevalence of +45◦ over −45◦
polarization, and s3 is proportional to the prevalence of right circular over left circular polarization.8 Because
optical sensors measure a quantity proportional to the time averaged Poynting vector, the phase information is
lost, and only the incoherent time averaged polarization information can be obtained.7, 8
For materials which can be described via linear optical interactions, we can use the Mueller-Stokes formalism.
A Mueller matrix, M , is a matrix which linear transforms one set of Stokes parameters, sin , into another set of
Stokes parameters, sout :
sout = M · sin
(2)
Notice that M ∈ R4×4 but not every 4 × 4 real valued matrix is a Mueller matrix due to the constraints in
Eqn.1, see Gil9 for details.
With an active, or Mueller matrix, polarimeteric instrument, we must modulate in irradiance to infer the
T
Mueller matrix of an object, M (x), where x = x y z t σ . We can then rewrite Eqn. 2 to have Mueller
matrices and Stokes parameters be functions of space, time, and wavelength or wavenumber. Eqn. 2 then
becomes

 
 

s0,out (x)
m00 (x) m01 (x) m02 (x) m03 (x)
s0,in
s1,out (x) m10 (x) m11 (x) m12 (x) m13 (x) s1,in 

 
 

(3)
s2,out (x) = m20 (x) m21 (x) m22 (x) m23 (x) · s2,in 
s3,out (x)
m30 (x) m31 (x) m32 (x) m33 (x)
s3,in
where for simplicity we fix sin . Our detector then measures a quantity proportional to s0,out (x). For a Mueller
matrix measuring instrument, we have an unknown object Mueller matrix, Mobj (x), and we write down the
instrument equation which modulates Stokes parameters:10
sout (x) = A(x) · Mobj (x) · G(x) · sin
= A(x) · Mobj (x) · sG (x)
(4)
(5)
where G(x), A(x) are the generator and analyzer Mueller matrices respectively, known and modulated via the
physical instrument. The generator modulation can then be thought of as only a Stokes parameter modulation,
sG (x).
laser
object
lp
lr
lr
lr
lr
micropolarizer array
(a) layout
(b) micropolarizer tiling
Figure 1: Spatio-temporally modulated polarimeter schematic, lp=linear polarizer, lr=linear retarder, blue components denote the polarization state generator (PSG), green components denote the polarization state analyzer
(PSA). The micropolarizer array is the typical tiling, shown in (b). (a) was derived and modified from a figure
created by Andrey Alenin.
2.2 Channels
Eqn. 5 can be expanded to obtain a linear equation8 for s0,out (x).
s0,out (x) =
3 X
3
X
a0i (x)sj (x)mij (x)
(6)
i=0 j=0
where a0i (x) are the elements of the first row of A(x), sj (x) are elements of sG (x), and mij (x) are elements of
Mobj (x). We can then take the Fourier transform of s0,out (x) to obtain
S0,out (ρ) =
3 X
3
X
A0i (ρ) ∗ Sj (ρ) ∗ Mij (ρ)
(7)
i=0 j=0
Where x → ρ in the Fourier transform, ∗ denotes convolution, and the shift to capital letters indicates a function
has been Fourier transformed. If a0i (x) and sj (x) are superpositions of sinusoidal functions, then A0i (ρ) ∗ Sj (ρ)
is a set of δ-functions, and each Mij (ρ) is then convolved with each δ-function in the set. The complete set of
δ-functions for the system
3 X
3
X
A0i (ρ) ∗ Sj (ρ)
i=0 j=0
are defined as the channels of the system, or the system’s channel structure.4
3. SPATIO-TEMPORAL INSTRUMENT DESIGN
Our group has had a need for an active portable polarimeter for some time, with the following capabilities:
• General remote sensing tasks, including multi-class detection
(8)
(a) Spatio-temporal design
(b) Actual Instrument
Figure 2: Hybrid Domain Modulated Imaging Polarimeter (HyDMIP)
• Data acquisition for supervised learning in detection tasks
• Validation of channeled polarimeter theory
• General data collection, with capability to access any Stokes parameter state for both generator and
analyzer
• Fast and portable
The instrument presented here was not designed from the ground up as a channeled system, it began as a
portable system that could potentially test some aspects of channeled systems while still meeting the other
capabilities listed above as a general research instrument. This large set of general priorities resulted in an initial
design consisting of a micropolarizer array camera and 4 rotating retarders. This design would meet all of the
capabilities while still allowing us to validate channeled designs.
The initial design was specified to be spectrally broadband from 635−830nm, however cost constraints forced
us to design only the full Stokes polarization state analyzer (PSA) to be broadband, and the polarization state
generator (PSG) to be narrowband at 671nm. This allows for the PSA to be operated as a broadband portable
full Stokes polarimeter by itself, or as a narrowband active Mueller matrix polarimeter when combined with the
PSG. The PSG is a typical rotating retarder design,11 with the polarization components placed in a collimated
optical space, the only difference is that we use two rotating retarders instead of one. The PSA is a spatiotemporally modulated hybrid channeled system. The micropolarizer array provides spatial modulation,3, 5, 12
and two rotating retarders provide temporal modulation. The retarders are in a collimated optical space. We
call this polarimeter the Hybrid Domain Modulated Imaging Polarimeter (HyDMIP). See appendix A for more
details.
PSA PSA PSA PSA PSG PSG PSG PSG
Table 1: System specifications
Component
Type
Detail
source
671nm laser
200mW
retarder(s)
671nm
1in dia.
stages(s)
up to 1000rpm
< 15arc sec accuracy
optics
aplanatic @ 633nm
matched to PSA FOV
sensor
silicon 1M P array @ 28.4 fps
9µm pixels, micropolarizer array
retarder(s)
630 − 835nm
2in dia.
stages(s)
up to 1000rpm
< 15arc sec accuracy
optics
afocal section for retarders
f/2.4
3.1 System equation
Using the Mueller-Stokes formalism, the system designed above can be described by the following equation:
sout = P (x, y) · R(ν4 , 4 , δ4 ) · R(ν3 , 3 , δ3 ) · Mobj (x, y, t) · R(ν2 , 2 , δ2 ) · R(ν1 , 1 , δ1 )sin
(9)
where
P (x, y) = micropolarizer array Mueller matrix
(10)
R(νj , j , δj ) = retarder Mueller matrix
(11)
radians
s
j = retarder start position in 2π radians
νj = retarder frequency in 2π
(12)
(13)
δj = retarder retardance in radians
(14)
Mobj (x, y, t) = Mueller matrix of the object
(15)
The first element of sout , s0,out is then sampled by the camera’s focal plane array (FPA). Note that each νj is
implicitly dependent on time, t. We can then Fourier transform a 3-dimensional, (x, y, t) acquired data cube to
retrieve the convolved data. For details see Alenin and Tyo4 or Vaughn et al .13
4. CHANNEL DESIGN
Table 2: Notation for channels.
imag. real
positive
δ1 = π,
ν1 =
1
,
2
negative
Although HyDMIP is designed as a multipurpose instrument, in this communication we focus on using HyDMIP as a fast portable active polarimeter. This
requires a channeled system design, given the constraints of our quad-retarder
+ micropolarizer instrument. The channels for HyDMIP were optimized using the free parameters available; ν1 , · · · , ν4 , 1 , · · · , 1 , δ1 , · · · , δ4 . More details
about the optimization process are addressed in our other paper in this conference.13 The optimization process yielded
1
δ2 = π − arccos √ ,
3
1
δ3 = π − arccos √ ,
3
δ4 = π
ν2 = −1,
ν3 = 1,
ν4 =
1
2
(16)
(a) Spatio-temporal domain
(b) Fourier domain
Figure 3: Space-time (x, y, t) modulated data cube transformed into Fourier domain.
up to a scaling constant for the νj s, and irrespective of the j s. In the following analysis and results, however,
δ2 = π2 due to retarder availability. This corresponds to the set of retarders actually used in the instrument. Fig.
4 shows the optimized channel structure for m23 . Note the triangles, the notation of which is explained in table
2. The size of the triangles corresponds to the magnitude of the respective δ-function. Each δ-function has an
associated triangle which represents the complex scalar magnitude of that δ-function.
As Alenin and Tyo have shown4 (and Azzam11 for the specific case of a temporally modulated system which
assumes zero temporal bandwidth in the object), Mueller matrix reconstruction can be directly accomplished in
the Fourier domain using the pseudoinverse of the Q matrix, Q+ . The effect of changing the retardance of the
second retarder of the PSG, δ2 , to be π2 has no effect on the bandwidth of the system, but results in a Q matrix
which isn’t as well conditioned as in the optimal case. For our instrument this results in an increased effect of
noise on reconstruction when compared with the optimal case.
5. RECONSTRUCTION
Conceptually, in a channeled system with measurements of data that contain some information, with some
bandwidth content, we can think of each Mueller matrix element as a function over the domain of measurement,
T
mij (x), with the domain for our specific instrument being x = x y t . For the channeled measurements,
we can take the Fourier transform of each mij (x):
 
ξ
Mij (ρ) = F {mij (x, y, t)}x→ρ ; ρ = η 
(17)
ν
then the channeled system will convolve each Mij (ρ) with the set of δ-functions in the channel structure, resulting
in a mixture of data in the Fourier domain. The exact mixing is characterized by the Q matrix,4 and we can
unmix by using the pseudoinverse, Q+ . Note that due to differences between the ideal channel structure, and the
actual channel structure of an instrument which requires calibration, Q must be determined via a semi-empirical
process.
Additionally, the bandwidth extent of each Mij (ρ) is formally determined by where Mij (ρ) 6= 0, with most real
objects having infinite bandwidth. In practice, a filter can be determined or applied (hopefully) for |Mij (ρ)| > c
where c is some cut off threshold. The filter can then attenuate Mij (ρ) over some subdomain of ρ. If |Mij (ρ)| >
c when ρ is outside of the range surrounding a channel for bandlimited reconstruction, then we have channel
crosstalk.
5.1 Calibration
Actual instruments, of course, differ from ideal models. Our channel structure optimization assumed perfect
polarizers, retarders, etc. This gave us a good design to first order, but calibration is needed to validate
the bandwidth achievable and also to compute a proper Q matrix.4 The Q matrix is computed with the
semi-empirical model described below.
5.1.1 Semi-empirical model
The first step of calibration was calibrating the micropolarizer array. This calibration was accomplished in a
non-imaging setup by 1) placing white paper in front of a rotatable polarizer as a diffuser, which was then placed
in front of the micropolarizer array and masked off for stray light, 2) rotating the polarizer with 1◦ increments
and recording an image frame at each increment, 3) fitting a Malus’ like law to each micropolarizer pixel and
recording the parameters. The Malus’ like law is
f (a, b, θ0 ) = a + b · cos2 (θ + θ0 )
(18)
where θ is the rotation angle of the reference polarizer and θ0 is the angle of the micropolarizer of the pixel with
respect to the lab reference frame (set by the reference polarizer). The extinction ratio can then be estimated
ν
1
0
−1
.5
0
η
−.5 −.5
.5
0
ξ
Figure 4: Optimized channel structure showing δ-functions specific to m23
by either using the actual data for each pixel, or by
ER =
a
a+b
(19)
note that the above doesn’t account for dark current. If bad pixels are eliminated, our extinction ratios range
from about 1 : 10 to 1 : 50, with the mean at ∼ 1 : 17. These ratios were calculated directly from the data. The
Malus’ like law parameter fits for each pixel are then used as inputs for Mueller matrices of diattenuators.
The actual bulk retardances of each retarder are then used in the model, this is a valid assumption since
the retarders are in collimated optical spaces and therefore the retardance doesn’t have any significant spatial
variation at the object or image planes. Finally, sampling at the correct framerate is implemented into the model.
The model was designed to address Johnson (Gaussian like) detector noise and the remaining systematic error
of retarder position wander during rotation. The Stokes parameters at each pixel, (xk , yl ) are then computed at
each time step, tm as follows:
sout (xk , yl , tm ) = P (xk , yl ) · R(ν4 (tm ), 4 , δ4 ) · R(ν3 (tm ), 3 , δ3 )
· Mtest · R(ν2 (tm ), 2 , δ2 ) · R(ν1 (tm ), 1 , δ1 )sin
(20)
where


1
−1

sin = 
 0  , due to the laser polarization orientation
0
(21)
and Mtest is a matrix with one element at 0 ≤ i, j ≤ 3 equal to 1 and all other elements zero. This gives the
channel response of each specific Mueller matrix element. Each response for each time step and each pixel is
then assembled into a 3-dimensional data cube and Fourier transformed. The channel weights can then be read
off to assemble the Q matrix.
5.2 Inversion
After the Q matrix is computed from the semi-empirical model, then data can be acquired and inverted to obtain
Mueller matrix images. The inversion process is outlined below:
• Acquire data (image stack sampled at evenly spaced time points)
• Compute 3-dimensional Fourier transform of acquired data cube
• Filter data around each channel site (for our case we will get 23 smaller data cubes)
• Apply the pseudoinverse, Q+ in the Fourier domain elementwise to obtain a Fourier representation of each
Mueller matrix element.
• Inverse Fourier transform the Fourier representation of each element to obtain a spatio-temporally resolved
data cube for each Mueller matrix element.
• Display the Mueller matrix images (over time if needed).
There are some additional steps which are glossed over here, for example since we are sampling at the Nyquist
frequency in both time and space, we have to use Hermicity conditions to fully reconstruct filtered data on the
edges of the data cube, which is error prone to implement in 3-dimensions. Fig. 5 shows an example of filtered
data in the Fourier domain from the actual instrument.
Our specific system also has some constraints that weren’t apparent at the outset of the design. Since we are
discretely sampling, and we want to build a fast instrument, we must assume that we can know our channels,
and hence Q a priori. Due to spectral leakage (which we won’t delve into here) this system requires Fourier
Figure 5: Filtered data from HyDMIP in the Fourier domain.
transforms occur on a time window which is a multiple of 8 frames wide, and must be greater or equal to 16
frames wide. This forces a minimum time lag of
8
where R is the camera frame rate in frames per second.
R
(22)
Our system has a camera operating at 28.4f ps which corresponds to a minimal lag of 282ms. This implies that
whatever is happening in the scene now won’t show up in the Mueller matrix images until 282ms later. This
isn’t a huge lag, but could be an issue for some applications.
6. RESULTS
Our instrument was very recently completed, so the results presented here are preliminary. First, we will discuss
some issues that we could not overcome prior to publication, but we are confident that our design and analysis
are still valid. We shall then present some results which illustrate the channel crosstalk issue.
6.1 Validation
After initial data collection and inversion of that data using the Q+ derived from the semi-empirical model, it
was clear that the actual Qinst of our instrument and the derived Qse from our semi-empirical model did not
match. We noticed this because some measured Mueller matrix images appeared to physically incorrect. Fig. 6
elucidates the issue clearly, the left hand panel shows the channels from our semi-empirical model with a linear
polarizer as an object, and the right hand panel shows the channels from an actual polarizer measured with the
instrument. We have made some consistency checks and have validated most of the model assumptions, but the
following discrepancies could be causing the difference:
1
1
0.5
0.5
0
0
0.2
0.2
0.1
0
−0.1
0
−0.2
0.1
0.1
0
0
−0.1
−0.1
10
20
30
40
50
60
(a) Semi-empirical model polarizer channels
10
20
30
40
50
60
(b) Instrument polarizer channels
Figure 6: Difference between our semi-empirical model and measured data for the channel structure when a
linear polarizer oriented at 0◦ is present.
• Rotation of the focal plane array between the real instrument and the model.
• Misalignment between the in plane axis of the PSG and the focal plane array.
• Missed frame or trigger causing a time shift between our model and the actual instrument.
• Stage offsets are incorrect (although we have repeatedly verified that they are correct).
• Model coordinate frame is otherwise rotated from the actual instrument coordinate frame.
• sin is rotated from our assumptions in the actual instrument.
• Coordinate frame definitions are inconsistent between the model and the actual instrument.
• Gap between the micropolarizer array and FPA (this can be investigated by changing the f /#)14
Qualitatively, Fig. 6 indicates that we are close to the correct Q matrix, and even though the inversion Q+ isn’t
quite correct at the moment, the bandwidth and crosstalk effects are still relevant. The channels are all in the
right position, which means channel bandwidth and crosstalk can be demonstrated.
6.2 Bandwidth and crosstalk
The theory behind crosstalk and implications for this instrument are discussed elsewhere,13 but we show crosstalk
effects from actual data here. Briefly, channel crosstalk occurs when high frequency data from an one channel
”bleeds” into an adjacent channel. There is no way to ”unmix” the high frequency data which ”bled over” from
the data at the adjacent channel(s) to the current channel. This fact arises because for a general object, there
is no way to know, a priori, what the data frequency content of that object will be.
Fig. 7 shows an image of m01 reconstructed from actual instrument data (shown in Fig. 5). In this data
acquisition we manually rotated a glass plate at different speeds. The image shown is reconstructed from data
Figure 7: Reconstructed movement of hand and glass plate for m01 , notice the “artifacts” due to channel
crosstalk.
where the plate was moving quite fast, and ”artifacts” can be seen around the edges of the hand holding the
plate. These ”artifacts” are a result of channel crosstalk, and cannot be completely mitigated given a fixed
camera acquisition rate.
6.3 Performance
We believe that we have designed and built one of the fastest (if not the fastest) portable Mueller matrix
imaging polarimeters to date, and we hope to soon have a deployable instrument to collect data for remote
sensing detection tasks. Our polarimeter can reconstruct full Mueller matrix images at a rate of 1/8 of the
base camera rate, e.g., for a 30f ps camera, we can accomplish a Mueller matrix imaging rate of 3.75f ps. For
comparison, a typical dual rotating retarder Mueller matrix polarimeter (which is temporally modulated only)
can attain a rate of 1/24 of the base camera rate, resulting in a Mueller matrix image rate of 1.25f ps for a camera
operating at 30f ps. However, because our polarimeter exploits a tradeoff between spatial image resolution and
temporal bandwidth, we only attain Mueller matrix images with 1/4 of the resolution of the base camera.
7. CONCLUSION
We have demonstrated the design and implementation of a portable imaging Mueller matrix polarimeter, HyDMIP, and we believe that it is the fastest imaging Mueller matrix polarimeter built to date. The design relied on
the general linear systems framework introduced by Alenin and Tyo4 to optimize our instrument for temporal
bandwidth in a systematic way, given our constraints and specifications. Our actual instrument appears to perform close to what we would expect given the theory, but there are some discrepancies between our instrument
and our semi-empirical instrument model. These discrepancies will have to be addressed before the instrument
can be deployed to the field to collect data for detection tasks.
7.1 Future work
Finalizing the instrument design and implementation includes fixing the semi-empirical model, writing software
to reconstruct Mueller matrix images in “real time,” (re-)calibrating the entire system via known polarimetric
elements, ensuring camera triggers aren’t missed, and collecting data.
APPENDIX A. INSTRUMENT DETAILS
We include here some opto-mechanical details, the Source Assembly is the PSG, and the Receiver Assembly is
the PSA. Fig 8 shows a top view of the instrument design.
(a) Top view of PSG
(b) Top view of PSA
Figure 8: Renderings of the source and receiver for HyDMIP.
Beam collimator
671nm laser
Glan Thompson polarizer
Rotation stages
Exit lens assembly
A
Q.A
MFG
APPV'D
CHK'D
DRAWN
NAME
SECTION A-A
SIGNATURE
125
FINISH:
DATE
SCALE:1:5
Part #
TITLE:
REVISION
15
Quantity : 1
SHEET 1 OF 1
A22
Source Assembly
A3
University of Arizona, Optical Sciences
DO NOT SCALE DRAWING
0.01 A B C
ALL UNTOLERANCED BASIC
DIMESIONS DEFINED BY
DEBUR AND
BREAK SHARP
EDGES .005-.025,
MACHINED FILLET
RADII .015-.10
MIxed
Weight: ERROR!:Weight
MATERIAL:
ALL MACHNED SURFACES
UNLESS OTHERWISE SPECIFED
ALL ANGLES ±1° UNLESS
OTHERWISE SPECIFIED
Retarder #2
Retarder #1
A
Afocal lens assembly
Camera lens
4D Microgrid array camera
Rotation stages
A
Q.A
MFG
APPV'D
CHK'D
DRAWN
SECTION A-A
NAME
SIGNATURE
125
FINISH:
DATE
SCALE:1:5
Part #
TITLE:
REVISION
21
Quantity : 1
SHEET 1 OF 1
A23
Receiver Assembly
A3
University of Arizona, Optical Sciences
DO NOT SCALE DRAWING
0.01 A B C
ALL UNTOLERANCED BASIC
DIMESIONS DEFINED BY
DEBUR AND
BREAK SHARP
EDGES .005-.025,
MACHINED FILLET
RADII .015-.10
Mixed
Weight: ERROR!:Weight
MATERIAL:
ALL MACHNED SURFACES
UNLESS OTHERWISE SPECIFED
ALL ANGLES ±1° UNLESS
OTHERWISE SPECIFIED
Retarder #1
Retarder #2
A
ACKNOWLEDGMENTS
The authors would like to thank Andrey Alenin for his insight and discussion about the design of this polarimetric
instrument.
This work was supported by the Air Force Office of Scientific Research under award FA-9550-10-0114, the
National Science Foundation under award DGE-0841234, and the University of Arizona TRIF Imaging Fellowship
Program.
The hardware was procured through the Defense University Research Instrumentation Program under award
FA-9550-12-1-0014.
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