Heterodyne diffracted beam photonic Doppler velocimeter (DPDV) for measurement of
transverse and normal particle velocities in pressure-shear plate impact experiments
Michael Mello
, Christian Kettenbeil
, Moriah Bischann
, and
Guruswami Ravichandran
Citation:
AIP Conference Proceedings
1979
, 160017 (2018); doi: 10.1063/1.5045016
View online:
https://doi.org/10.1063/1.5045016
View Table of Contents:
http://aip.scitation.org/toc/apc/1979/1
Published by the
American Institute of Physics
Heterodyne
Di
ff
racted
Beam
Photonic
Doppler
Velocimeter
(DPDV)
for
Measurement
of
Transverse
and
Normal
Particle
Velocities
in
Pressure-Shear
Plate
Impact
Experiments
Michael
Mello
1,a)
,
Christian
Kettenbeil
1,b)
,
Moriah
Bischann
1,c)
and
Guruswami
Ravichandran
1,d)
1
California
Institute
of
Technology,
M
/
C
104-44,
1200
E
California
Blvd
Pasadena,
CA
91125, USA
a)
Corresponding author: mello@caltech.edu
b)
ckb@caltech.edu
c)
mbischan@caltech.edu
d)
ravi@caltech.edu
Abstract.
Pressure-shear plate impact (PSPI) experiments have traditionally relied on free space beam interferometers such as
the transverse displacement interferometer (TDI) and normal displacement interferometer (NDI) or normal velocity interferometer
(NVI), to measure transverse and normal velocities at the rear surface of the target plate [1]. Alternative interferometer schemes
feature a dual beam VISAR arrangement [2] and a recently developed all fiber-optic TDI-NDI
/
PDV configuration [3]. Here, we
present a heterodyne di
ff
racted beam PDV (DPDV) which interferes a pair of symmetrically di
ff
racted 1
st
order beams produced
by a thin, specular, metallic grating deposited on the rear surface of the target plate. Each beam is collected by a fiber-optic probe
and directed to interfere with a reference beam of a slightly increased wavelength to create an upshifted carrier signal frequency
at zero particle velocity. Signal frequencies are extracted from the two fringe records using a moving-window DFT algorithm and
then linearly combined in a post processing step to decouple the normal and transverse velocities. The 0
th
order beam can also be
interfered in a heterodyne PDV to obtain an additional independent measurement of the normal particle velocity [4]. An overview
of the DPDV configuration is presented along with a derivation of the interferometer sensitivities to transverse and normal particle
velocities. Results from a normal impact experiment conducted on y-cut
α
-quartz are presented as experimental validation.
INTRODUCTION
The present paper addresses the development, and experimental validation of a heterodyne di
ff
racted beam PDV
(DPDV) system for the measurement of normal and transverse particle velocity components in pressure-shear plate
impact (PSPI) experiments. Three fiber-optic probes collect the reflected (0
th
order) beam and symmetric
̆
1
st
order
beams produced by a 400 lines
/
mm metallic di
ff
raction grating deposited onto the polished rear surface of the target
plate. Fiber-optic probes are designed to collect light even as the di
ff
racted beams are slightly rotated due to tilt
between impact faces and de-centered due to the normal displacement of the target plate rear surface. The 0
th
and
̆
1
st
order beams are each passed through a fiber-optic circulator and combined with a reference beam of a slightly
higher wavelength to create a heterodyne signal with an upshifted carrier frequency at zero particle velocity. The
frequency content encoded within the recorded DPDV fringe records corresponds to a scaled linear combination
(sum or di
ff
erence) of the normal and transverse particle velocity components. A moving-window discrete Fourier
transform (DFT) algorithm is applied to extract the DPDV signal frequencies
f
`
p
t
q
and
f
́
p
t
q
, which also contains
the constant user-selected carrier frequency. Normal and transverse particle velocity components are subsequently
decoupled through addition or subtraction and appropriate scaling of the extracted signal frequencies. The normally
reflected (0
th
order) beam can also be interfered in a heterodyne PDV arrangement to obtain an additional independent
measurement of the normal particle velocity. The fiber-optic DPDV system has been configured with a powder gun
capable of achieving impact velocities of 1
.
8km
/
s and will enable future PSPI shock wave experiments designed to
investigate strength and failure properties of novel materials at shear strain rates approaching
9
γ
“
10
8
s
́
1
[5].
Shock Compression of Condensed Matter - 2017
AIP Conf. Proc. 1979, 160017-1–160017-6; https://doi.org/10.1063/1.5045016
Published by AIP Publishing. 978-0-7354-1693-2/$30.00
160017-1
X
X
Circulators
Single Mode
Fiber-Optic Cables
1
2
3
1
2
3
1
2
3
1
2
3
Fiber-Optic
Patches
Polarization-maintaining
Fiber-Optic Cables
1 x 4 coupler (splitter)
1 x 2 coupler
(splitter)
Reference
Laser (
h
R
)
Attenuators
Impact Chamber
DD
4km
Delay
1 x 2
coupler
1 x 2
coupler
1 x 2
coupler
1 x 2
coupler
A
A
Power Monitors
Oscilloscope
-n
th
order
flyer
target
reflected beam
(0
th
order)
+n
th
order
x
1
,
Ú
u
1
x
2
,
Ú
u
2
P
P
Photodetector
Photodetector
V
0
AAAA
probe
probe
“down barrel”
PDV probe
Drive Group
Reference Group
Sensing Group
Reference
beams
Source
Laser (
h
S
)
(2 beams supressed)
probe
FIGURE 1.
DPDV system for combined measurement of normal and transverse particle velocities in PSPI experiments
DPDV FIBER-OPTIC CONFIGURATION
The fiber-optic DPDV arrangement is comprised of the drive, reference, and sensing groups as depicted in Fig. 1.
The source light is produced by a KOHERAS BOOSTIK model E15 fiber laser by NKT Photonics with a Lorentzian
linewidth of 0.1kHz (which translates to a very long coherence length), adjustable center wavelength between 1550
nm and 1570 nm, and a maximum output power of 2W. The drive group hardware elements include a 1
ˆ
4 fiber-optic
splitter, fiber-optic circulators, attenuators, and fiber-optic probes. The “down barrel” probe is a collimating PDV
probe which collects normally scattered light from the roughened surfaced of the sabot. The center probe adjacent
to the target rear surface produces an incident,
«
0
.
3 mm diameter, collimated beam and collects the 0
th
order (col-
limated) beam reflected from the rear surface of the target. Symmetrically di
ff
racted (collimated)
̆
1
st
order beams
are received by pigtail style collimators with an f
=
6
.
2 mm aspheric lens with a 5 mm clear aperture. All four light
beams are directed back toward their respective circulators which re-direct the light through SM fiber and FC
/
APCs to
Variable Fiber Optic Attenuators as depicted in Fig 1. Each of the four collected light beams is passed through a fiber-
optic circulator and redirected to variable fiber-optic attenuators before propagating to the sensing group elements as
depicted in Fig. 1.
Reference group hardware elements depicted in Fig. 1 generate the reference beams, which are interfered with
the four collected drive group source beams. The reference light is produced by a KOHERAS ADJUSTIK fiber laser
by NKT Photonics, with a Lorentzian linewidth of 0.1kHz, adjustable center wavelength between 1535 nm and 1580
nm and a maximum power output of 40 mW. A reference beam wavelength of 1550
.
017 nm was selected for the
validation experiment, which generated a constant carrier signal frequency of 0
.
66 GHz at zero particle velocity when
mixed with the source light with a wavelength of 1550
.
012 nm.
Sensing group hardware elements depicted in Fig. 1 interfere the collected source light with the reference light
and convert the resulting interference signals into digitized fringe records for analysis. The four Drive Group source
beams are combined into PDV
/
DPDV beam pairs using two 1
ˆ
2 Single Mode (SM) fiber-optic couplers as shown.
The di
ff
racted light beams are each initially sent through a 4 km
ˆ
2 Network Simulation Module, which delays each
beam by
«
20
μ
s with respect to its PDV counterpart before the beams are combined. Each DPDV
/
PDV beam pair is
then combined with a reference beam using a 1
ˆ
2 SM fiber-optic coupler with a 10
{
90 coupling ratio. Interfering
beam trains terminate at a Photodetector where the modulating light fields are converted into electrical signals. The
time multiplexed PDV and DPDV fringe records are digitized at 20 GS
/
s and recorded using only 2-channels of a
4-channel oscilloscope wit
h a 4 GHz analog bandwidth.
160017-2
-n
th
order
flyer
target
A
A’
O
O
b
OPL
+
= -
b
u
1
(1+cos
O
n
) -
b
u
2
sin
O
n
b
OPL
-
= -
b
u
1
(1+cos
O
n
) +
b
u
2
sin
O
n
+n
th
order
-n
th
order
incident beam
reflected beam
(0
th
order)
B
+n
th
order
thin
specimen
O
n
O
n
grating
b
OPL
0
= -2
b
u
1
0
th
order
V
0
x
2
x’
2
x
1
x’
1
b
u
1
b
u
2
x
1
,
Ú
u
1
x
2
,
Ú
u
2
FIGURE 2.
Changes in optical path length (OPL) of the 0
th
order (reflected) and
̆
n
th
order di
ff
racted beams due to small (positive),
normal and transverse displacements (
δ
u
1
,δ
u
2
) of the target rear surface as point A displaces to A’.
INTERFEROMETER SENSITIVITY AND DATA ANALYSIS
DPDV measurement sensitivity is derived by invoking the two beam, time-averaged intensity formula
I
9
`
E
S
`
E
R
̆
̈
`
E
S
`
E
R
̆
̊
(1)
where
E
S
and
E
R
are the electric field plane wave representations of the source and reference beams and the asterisk
symbol denotes the complex conjugate operation [1, 6]. Substituting for all time varying phase terms including those
corresponding to changes in optical path length (OPL) of the
̆
n
th
order di
ff
racted beams, as shown in Fig. 2, leads to
an expression for the time-averaged intensity of the heterodyned DPDV interference signals given by
I
̆
p
t
q“
I
̆
S
`
I
̆
R
`
2
b
I
̆
S
I
̆
R
cos
„
2
π
λ
S
`
u
1
p
t
qp
1
́
cos
θ
n
q ̆
u
2
p
t
q
sin
θ
n
̆
`
2
π
p
ν
S
́
ν
R
q
t
́
φ
̆
j
.
(2)
Here
I
S
and
I
R
represent the time-averaged steady state intensity of the source and reference beams,
λ
S
represents the
wavelength of the source light,
ν
S
and
ν
R
are the optical frequencies of the source and reference light fields, and
φ
̆
is a arbitrary constant phase in each interference signal. Application of a moving-window discrete Fourier transform
(DFT) algorithm to the recorded DPDV fringe records extracts the encoded signal frequencies corresponding to
f
`
p
t
q“
1
λ
S
`
9
u
1
p
t
qp
1
`
cos
θ
n
q`
9
u
2
sin
θ
n
̆
`p
ν
S
́
ν
R
q
(3)
f
́
p
t
q“
1
λ
S
`
9
u
1
p
t
qp
1
`
cos
θ
n
q ́
9
u
2
sin
θ
n
̆
`p
ν
S
́
ν
R
q
(4)
where
9
u
1
p
t
q
and
9
u
2
p
t
q
correspond to the normal and transverse particle velocity components.
Subtracting the two DPDV signal frequencies given by Eqs. 3,4 and substituting for sin
θ
n
from the grating
equation
d
sin
θ
n
“
n
λ
s
n
“ ̆
1
,
2
,
3
...
(5)
yields an expression for the transverse particle velocity in terms of the grating pitch (
d
), di
ff
raction order (
n
), and the
extracted signal frequencies given by
9
u
2
“
d
2
n
`
f
`
p
t
q ́
f
́
p
t
q
̆
.
(6)
The frequency scaling factor
d
{
2
n
e
ff
ectively represents the fundamental measurement sensitivity of the DPDV to
changes in transverse velocity and is equivalent to the sensitivity of a TDI [1]. Using the
̆
1
st
order beams from a 400
lines
/
mm grating in the current DPDV configuration results in a transverse velocity measurement sensitivity of 1
.
25
m
/
s
/
MHz.
Addition of the two DPDV signal frequencies given by Eqs. 3 and 4 yields an expression for the normal particle
velocity in terms of the measured signal frequencies and the independently measured carrier frequency (
ν
C
“
ν
S
́
ν
R
)
160017-3
0 20 40 60 80 100 120 140 160 180 200
longitudinal velocity (m/s)
transverse velocity (m/s)
80
70
60
50
40
30
20
10
0
-20
Velocity Diagram (V
0
= 212m/s)
x
2
(mm)
Time (
+
s)
x-t Diagram
longitudinal
transverse
quasi-transverse
quasi-longitudinal
-6 -4 -2 0 2 4
2.0
1.5
1.0
0.5
0
(a)
(b)
-10
.
FIGURE 3.
(a) x-t diagram displays wave characteristics in the flyer plate (
́
6
.
5mm
ď
x
2
ď
0 mm) and target plate (0 mm
ď
x
2
ď
5 mm) (b) Predicted target plate particle velocities for an impact velocity of
V
0
“
212 m
/
s measured by the “down barrel”
PDV
given by
9
u
1
“
λ
S
2
p
1
`
cos
θ
n
q
`
f
`
p
t
q`
f
́
p
t
q ́
2
ν
C
̆
.
(7)
The scaling factor
λ
S
{p
2
p
1
`
cos
θ
m
qq
represents the fundamental measurement sensitivity of the DPDV to changes
in normal velocity. DPDV is evidently (1
`
cos
θ
n
qˆ
more sensitive to changes in normal velocity compared to a
standard PDV, which has a sensitivity of
λ
S
{
2 [4]. Using the
̆
1
st
order beams of a 400 lines
/
mm grating in the
current DPDV configuration results in a normal velocity measurement sensitivity of 0
.
434 m
/
s
/
MHz, which represents
a1
.
78
ˆ
increase over the sensitivity of the PDV when using the 0
th
order beam at the same source wavelength.
EXPERIMENTAL VALIDATION - NORMAL IMPACT OF Y-CUT
α
-QUARTZ
A normal plate impact experiment was conducted using a borosilicate glass flyer plate and a single crystalline y-cut
α
-quartz target plate as a means of validating the new DPDV-PDV system. Single crystalline y-cut
α
-quartz was
selected as a target material because of the strong anisotropic coupling exhibited between longitudinal and transverse
particle motion when impacted along its
x
2
axis. Quasi-longitudinal (QL) and quasi-transverse (QT) waves with wave
speeds,
c
QL
“
6015
.
64 m
/
s, and
c
QT
“
4318
.
39 m
/
s emerge as the only nonzero eigenvalues of the elasto-acoustic
tensor which governs the problem [1]. The sharp velocity jumps registered in the normal and transverse directions
upon arrival of the QL and QT waves at the rear surface of the target present an ideal scenario for evaluating various
attributes of the DPDV system such as predicted velocity measurement sensitivities and the optical heterodyne feature
for automatic, accurate detection of sharp velocity reversals.
The x-t diagram in Fig. 3(a) depicts the QL and QT wave characteristics in the quartz target plate and the
transverse and longitudinal wave characteristics in the isotropic, borosilicate flyer plate. Six unique states of constant
particle velocity and stress are identified and labele
dA-Finthex-t
diagram. The particle velocity components for
each state were computed based on an impact velocity of
V
0
“
212 m
/
s as measured by the “down barrel” PDV and
plotted as coordinate pairs (
9
u
1
,
9
u
2
) in the velocity diagram in Fig. 3(b). Sharp step-like velocity jumps of
9
u
C
1
“
150
.
79
m
/
s and
9
u
C
2
“
72
.
51 m
/
s are predicted upon arrival of the QL wave at the rear surface of the target plate. The
subsequent arrival of the QT wave induces a pronounced reversal of the transverse velocity from
9
u
C
2
“
72
.
51 m
/
sto
9
u
F
2
“ ́
17
.
02 m
/
s and a second positive jump in the normal particle velocity from
9
u
C
1
“
150
.
79 m
/
sto
9
u
F
1
“
193
.
84
m
/
s corresponding to the path C
Ñ
F in the velocity diagram (Fig. 3(b)).
The choice of borosilicate for the flyer plate material was based on three considerations: (a) choice of an isotropic
material to simplify material alignment requirements, (b) a low acoustic impedance to generate relatively low normal
stresses in both plates, and (c) su
ffi
ciently high yield strength in compression and shear to prevent premature failure of
the flyer plate. Cylindrical plates of Borofloat 33 glass were prepared with the following specifications: (a) diameter
=
33
.
909
̆
0
.
1 mm, (b) thickness
=
6
.
5
̆
0
.
1 mm, and (c) RMS surface roughness
ă
1 nm. The borosilicate plate
160017-4
Time (
+
s)
transverse velocity (m/s)
free surface velocity (m/s)
100
0
200
100
200
0
50
150
0 0.5 1.0 1.5 2.0
0 20 40 60 80 100 120 140 160 180 200
-20
0
20
40
60
80
transverse
longitudinal
longitudinal
transverse
predicted
predicted
experiment
(a)
(b)
normal velocity (m/s)
FIGURE 4.
(a) Measured longitudinal and transverse velocity profiles compared to predicted values based on the measured impact
speed of
V
0
“
212 m
/
s (b) Orthogonality of the measured velocity jumps during the arrival of the QL wave, and later the QT wave
thickness was chosen to prevent wave reflections from the backside of the flyer plate from reaching the impact face
before separation. Values for borosilicate material constants were obtained from Schott North America and are given
as given as follows: (a) density
ρ
“
2
.
23 g
/
cm
3
, (b) elastic modulus
E
“
64 GPa, (c) Poisson’s ratio
ν
“
0
.
2.
Cylindrical y-cut
α
-quartz plates were prepared with the following specifications: (a) diameter
=
30
̆
0
.
1 mm,
(b) thickness
=
5 mm, and (c) RMS surface roughness of 8
́
10 Å. The target plate diameter provided an observation
window of more than 2
μ
s before lateral release waves from the plate boundary reached the center of the target. The
z-axis of the y-cut substrate was found using a petrographic microscope and interference figures and marked with an
etched line on the plate’s circumference. Values for density and elastic-acoustic tensor constants were obtained from
literature and are given as follows: (a) density
ρ
“
2
.
65 g
/
cm
3
, (b)
C
22
“
87
.
16 GPa, (c)
C
24
“
18
.
15 GPa, (d)
C
44
“
58
.
14 GPa, (e)
C
66
“
40
.
26 GPa.
A 300 nm thick gold di
ff
raction grating was fabricated onto the backside of the polished y-cut
α
-quartz target
plate using standard photolithography procedures and metal vapor deposition techniques in a cleanroom environment.
A 400 lines
/
mm (d
“
2
.
5
μ
m) di
ff
raction grating was produced with its grating lines aligned parallel to the
x
1
-axis
of the y-cut quartz substrate. The grating thus produces sharp, specular 0
th
order and
̆
1
st
order di
ff
raction beams
along precisely known di
ff
raction angles
θ
̆
1
“ ̆
38
.
32
̋
per Eq. 5. No additional di
ff
raction orders are produced by
the 400 lines
/
mm grating at the source wavelength (
λ
“
1550 nm) used in the current DPDV system. DPDV probes
are aligned to the
̆
1
st
order di
ff
raction angles by optimizing the light return from each di
ff
racted beam until the
maximum light returns are achieved.
Normal and transverse velocity profiles measured at the rear surface of the y-cut
α
-quartz target plate using the
DPDV system are plotted in Fig. 4(a). The velocity profiles were obtained from the acquired fringe records using a
Hamming window of 50 ns with the window shifted by a 50 ps time step for every analysis. The dashed lines represent
the predicted velocity jumps of each respective motion component. The measured steady initial and final transverse
particle velocity levels are in excellent agreement with theory while the measured normal velocity jumps are within
6
́
7% of their respective predicted values. The observed deviation is attributed to a small tilt angle
φ
«
1mrad
between impact faces, which causes the QL wave to deviate from the
x
2
axis (quartz crystal y-axis) by an amount
consistent with the observed deviation. The observed deviation is also be partly attributed to uncertainties in the
values of the elastic constants for
α
-cut quartz and borosilicate glass.
Since the target plate’s free surface velocity is proportional to the orthogonal QL and QT wave polarization
vectors, the transient velocity jump is ideally orthogonal to the final velocity jump as depicted Fig. 4(b). The measured
velocity jumps during the arrival of the QL wave, and later the QT wave, show a slight deviation from predicted values.
Specifically, the QL wave’s polarization deviated by 1
.
7
̋
and the QT wave’s polarization deviated by 0
.
1
̋
. Measured
values are respectively quite comparable and even slightly lower than the jump o
ff
sets of 2
.
15
̋
and 1
.
7
̋
reported by
[1] in their original TDI validation experiments on y-cut quartz which had a measured impact tilt angle of 0
.
4 mrad.
ACKNOWLEDGMENTS
The authors are grateful for support from the O
ffi
ce of Naval Research (Award No. N00014-16-1-2839) for the devel-
opment of the PSPI capability at high pressures and the Air Force O
ffi
ce of Scientific Research (Award No. FA9550-
12-1-0091) for the development of the heterodyne di
ff
racted beam photonic Doppler velocimeter (DPDV) system.
160017-5
REFERENCES
[1] K. Kim, R. Clifton, and P. Kumar,
J. Appl. Phys.
48,10
, 4132–4139 (1977).
[2] L. Chhabildas, H. Sutherland, and J. Asay,
J. Appl. Phys.
50,8
, 5196–5201 (1979).
[3] B. Zuanetti, T. Wang, and V. Prakash,
Rev. Sci. Instrum.
88, 3
, p. 033108 (2017).
[4] D. Dolan,
Rev. Sci. Instrum.
81,5
, p. 053905 (2010).
[5] C. Kettenbeil, M. Mello, T. Jiao, R. Clifton, and G. Ravichandran, in
Shock Compression of Condensed Matter
- 2017
, edited by R. Chau, T. Germann, and M. Lane. (American Institute of Physics, Melville NY, 2018).
[6] H. Espinosa, M. Mello, and Y. Xu,
J. Appl. Mech.
64,1
, 123–131 (1997).
160017-6