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Backward Secondary-Wave Coherence Errors in
Photonic Bandgap Fiber Optic Gyroscopes
Xiaobin Xu *, Ningfang Song, Zuchen Zhang and Jing Jin
Department of Opto-electronics Engineering, Beihang University, Beijing 100191, China;
[email protected] (N.S.); [email protected] (Z.Z.); [email protected] (J.J.)
* Correspondence: [email protected]; Tel.: +86-10-8231-6887
Academic Editor: Jörg F. Wagner
Received: 25 April 2016; Accepted: 3 June 2016; Published: 8 June 2016
Abstract: Photonic bandgap fiber optic gyroscope (PBFOG) is a novel fiber optic gyroscope (FOG)
with excellent environment adaptability performance compared to a conventional FOG. In this
work we find and investigate the backward secondary-wave coherence (BSC) error, which is a bias
error unique to the PBFOG and caused by the interference between back-reflection-induced and
backscatter-induced secondary waves. Our theoretical and experimental results show a maximum
BSC error of ~4.7˝ /h for a 300-m PBF coil with a diameter of 10 cm. The BSC error is an important
error source contributing to bias instability in the PBFOG and has to be addressed before practical
applications of the PBFOG can be implemented.
Keywords: photonic bandgap fiber; fiber optic gyroscope; bias error
1. Introduction
Current fiber optic gyroscopes (FOGs) have very good performance [1], but inevitably they still
have large Shupe effects, Faraday effects and radiation effects due to the fact that conventional silica
fiber is very sensitive to these environments. Photonic bandgap fibers (PBFs), a novel kind of fiber
in which the light travels not in a conventional silica core but in an air core [2–4], have attracted
considerable interest owing to their unique optical properties such as significantly lower nonlinear
coefficients and especially, their improved adaptability to temperature, magnetic fields and radiation
in comparison with the properties of silica fibers [5,6].
A photonic bandgap fiber optic gyroscope (PBFOG), as illustrated in Figure 1, consists of a
broad-spectrum source, a coupler, an integrated optic chip (IOC), and a PBF coil [7,8]. A previous
study [9] has reported reductions in the errors induced by the Kerr effect (>170), Shupe effect (~6.5), and
Faraday effect (>20) in a PBFOG when compared with the corresponding error values of a conventional
FOG. Therefore, the PBFOG represents a substantially improvement of these key properties compared
to a FOG, and it has excellent prospects, but as a new kind of FOG, it definitely exhibits some different
errors and noises from those which are normally observed in a traditional FOG owing to the use
of the PBF. References [7,8] have investigated shot noise, backscattering noise which is defined as
the error due to coherence interference between the backscattered waves and primary waves and
which also exists in resonant fiber optic gyroscopes using an air-core fiber [10,11], electronic noise,
and reflection-induced intensity noise; further, the polarization non-reciprocity error has also been
investigated [12]. In this study, for the first time to the best of our knowledge, we find and investigate
the backward secondary-wave coherence (BSC) error, which is defined as a bias error caused in the
PBFOG by interference between back-reflection-induced and backscatter-induced secondary waves.
Sensors 2016, 16, 851; doi:10.3390/s16060851
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Figure 1. Diagram of a photonic bandgap fibe r optic gyroscope .
Figure 1. Diagram of a photonic bandgap fiber optic gyroscope.
2. Theoretical Analysis
2. Theoretical Analysis
In the PBFOG, the pigtails of the IOC are conventional fibers that have a Ge-doped SiO2 core,
In the PBFOG, the pigtails of the IOC are conventional fibers that have a Ge-doped SiO2 core, but
but the coil comprises a commercially available 7-cell air-core PBF with hexagonal air holes
the coil comprises a commercially available 7-cell air-core PBF with hexagonal air holes triangularly
triangularly arranged in the cladding [12]. According to the measurement results in our previous
arranged in the cladding [12]. According to the measurement results in our previous study [13],
study [13], a strong reflection inevitably occurs at the two fusion splicing points between the PBF
a strong reflection inevitably occurs at the two fusion splicing points between the PBF coil and the
coil and the IOC pigtails. Both back-reflection-induced secondary waves have very large
IOC pigtails. Both back-reflection-induced secondary waves have very large magnitudes (only about
magnitudes (only about 3–10 times smaller than that of primary waves for a 300-m coil composed of
3–10 times smaller than that of primary waves for a 300-m coil composed of a PBF with a loss of
a PBF with a loss of 20 dB/km). Obviously, those back-reflection-induced secondary waves produce
20 dB/km). Obviously, those back-reflection-induced secondary waves produce an offset at the detector
an offset at the detector and add intensity-type noise because the coherence length of the
and add intensity-type noise because the coherence length of the broad-spectrum source is always
broad-spectrum source is always considerably less than the path mismatch length between the two
considerably less than the path mismatch length between the two IOC pigtails, and thus, there is no
IOC pigtails, and thus, there is no interference between these two sets of secondary w aves [7,9].
interference between these two sets of secondary waves [7,9]. However, we found that the strong
However, we found that the strong back-reflection-induced secondary waves at one fusion splicing
back-reflection-induced secondary waves at one fusion splicing point inevitably interfere with the
point inevitably interfere with the backscatter-induced secondary waves within the other pigtail,
backscatter-induced secondary waves within the other pigtail, thereby causing a BSC error. This
thereby causing a BSC error. This BSC error is a phase-type error that does not exist in a
BSC error is a phase-type error that does not exist in a conventional FOG, and it seriously affects the
conventional FOG, and it seriously affects the PBFOG performance.
PBFOG performance.
As illustrated in Figure 2a, before the primary-wave (black arrows) output from the IOC enters
As illustrated in Figure 2a, before the primary-wave (black arrows) output from the IOC enters
the PBF coil, secondary waves (W A and W B) are generated owing to the strong back-reflection at
the PBF coil, secondary waves (WA and WB ) are generated owing to the strong back-reflection at
fusion splicing points A and B. W A has an optical path of L(W A) = nIOC |OO1| + n SiO2 |O1 A| from the
fusion splicing points A and B. WA has an optical path of L(WA ) = nIOC |OO1 | + nSiO2 |O1 A| from
common input/output end O. nIOC and nSiO2 is, respectively, the refractive index of the IOC and
the common input/output end O. nIOC and nSiO2 is, respectively, the refractive index of the IOC and
conventional fiber. It is well known that the backscattering is randomly distributed along the fiber,
conventional fiber. It is well known that the backscattering is randomly distributed along the fiber,
and each backscattering point causes a secondary wave, but not all of those backscattering-induced
and each backscattering point causes a secondary wave, but not all of those backscattering-induced
secondary waves are able to interfere with W A, because it is the broad-spectrum light in the optical
secondary waves are able to interfere with WA , because it is the broad-spectrum light in the optical
circuit and the interference intensity is maximum when the optical path difference (OPD) is 0, but
circuit and the interference intensity is maximum when the optical path difference (OPD) is 0, but
decreases dramatically and rapidly as the OPD increases [14]. Therefore, i f a secondary wave W A1
decreases dramatically and rapidly as the OPD increases [14]. Therefore, if a secondary wave WA1
induced by the backscattering at point A1 has an optical path of L(W A1)= nIOC|OO2 | + nSiO2|O2B| +
induced by the backscattering at point A1 has an optical path of L(WA1 )= nIOC |OO2 | + nSiO2 |O2 B| +
nair |BA1 |from the common input/output end O and L(W A1) = L(W A), then the interference occurs
nair |BA1 |from the common input/output end O and L(WA1 ) = L(WA ), then the interference occurs
between W A and W A1 and the intensity is maximum that is given by the first term in:
between WA and WA1 and the intensity is maximum that is given by the first term in:
2 I in  splicing
cos(m`
)  2 I in pR
mm`2ϕ)2 q
 RBB RRBB11 q1{2cos(
BSC 
AR
A1  cospΦ
m 
“I2I
pR ARRA1
q1{2
ϕ
cospΦ
1 q1 ` 2Iininαin
in αin αinsplicing
1/ 2
IBSC
1/ 2
1{2
2
3
22
`2Iin α
α2coilpR
R A RqA1{2
cospΦ
ϕ3 q3 )`2I
α
cospΦ
ϕ q
in2α
I in2splicing
in  2splicing
cos(
2 Iininαinin α
3splicing
 m m`
m`
BBR
B1 q cos(
 RpR
coil
coilA RA1
1
m 
splicing  coil
BR
1
4) 4
1/ 2
1/ 2
(1)
(1)
which equation describes the summed intensity (as detected by the detector) of all the interference
which equation describes the summed intensity (as detected by the detector) of all the interference
phenomena involved (that is, IBSC ). Here nair is the refractive index of the PBF. In Equation (1), Iin
phenomena involved (that is, IBSC). Here nair is the refractive index of the PBF. In Equation (1), Iin
denotes the light intensity output from the IOC. Further, αin , αsplicing , and αcoil represent the losses at
denotes the light intensity output from the IOC. Further, α in, α splicing, and α coil represent the losses at
the input channel (from the IOC pigtail to detector), fusion splicing at points A and B, and PBF coil,
the input channel (from the IOC pigtail to detector), fusion splicing at points A and B, and PBF coil,
respectively. The typical values of αin , αsplicing , and αcoil are ~8 dB, ~1.5 dB and ~6.6 dB for a 300-m
respectively. The typical values of α in, α splicing, and α coil are~8 dB, ~1.5 dB and ~6.6 dB for a 300-m PBF
PBF coil, respectively. Parameters RA and RB represent the back-reflection coefficients at points A and
coil, respectively. Parameters RA and RB represent the back-reflection coefficients at points A and B,
B, respectively. Similarly, RA1 and RB1 represent the backscattering coefficients at points A1 and B1 ,
respectively. Similarly, RA1 and RB1 represent the backscattering coefficients at points A1 and B1,
respectively. Parameters ϕ1 , ϕ2 , ϕ3 , and ϕ4 denote the random phases between the two secondary
respectively. Parameters φ1, φ2, φ3 , and φ4 denote the random phases between the two secondary
waves involved in the interference. Φm is the modulation phase.
waves involved in the interference. Фm is the modulation phase.
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(a)
(b)
Figure 2. Se condary waves induce d by back-re flection and backscatte r of primary waves . (a) Be fore
Figure 2. Secondary waves induced by back-reflection and backscatter of primary waves. (a) Before
the primary waves e nter the PBF coil; (b) Afte r the primary waves have propagate d through 1 loop
the primary waves enter the PBF coil; (b) After the primary waves have propagated through 1 loop of
of the PBF coil.
the PBF coil.
Interference between W B and W B1 can similarly be examined, and the resulting interference
Interference
between
WB and
WB1
similarly
bethe
examined,
and
resulting
interference
intensity
is given
by the second
term
in can
Equation
(1 ). On
other hand,
asthe
illustrated
in Figure
2b,
when the
primary-wave
outputterm
fromin
the
PBF coil after
propagating
coil once, in
backward
intensity
is given
by the second
Equation
(1). On
the other through
hand, asthe
illustrated
Figure 2b,
secondary
waves (W′ A,output
W′ B, W′from
A1, W′the
B1) arise,
and after
interference
can also
occur between
W′ A and
W′ A1,
when
the primary-wave
PBF coil
propagating
through
the coil once,
backward
1
1
1
1
1 and
W′
B
and
W′
B1
with
the
intensities
given
by
the
last
two
terms
in
Equation
(1).
In
fact,
there
also
secondary waves (W A , W B , W A1 , W B1 ) arise, and interference can also occur between Wexist
A
1
1
1
other
backward
secondary
waves
that
propagate
over
multiple
loops;
however,
their
contribution
W A1 , W B and W B1 with the intensities given by the last two terms in Equation (1). In fact, there to
also
the
BSC backward
error is very
small, and
the relevant
terms have
been
neglected
in Equation
exist
other
secondary
waves
that propagate
over
multiple
loops;
however,(1).
their contribution
In the
standard
operation,
a square terms
wave with
τ and amplitude of ±π/8,
to the BSC
error
is veryFOG
small,
and the relevant
have eigenfrequency
been neglectedfin
Equation (1).
and a sawtooth wave with amplitude π are added and applied to the IOC to modulate primary
In the standard FOG operation, a square wave with eigenfrequency f τ and amplitude of ˘π/8,
waves for coherence detection and closing-loop operation [14]. The demodulation value of the
and a sawtooth wave with amplitude π are added and applied to the IOC to modulate primary
interference intensity between the primary waves indicates the angle velocity of the FOG. The
waves for coherence detection and closing-loop operation [14]. The demodulation value of the
back-reflection-induced and backscatter-induced secondary waves also propagate through the IOC,
interference intensity between the primary waves indicates the angle velocity of the FOG. The
and therefore, these waves are also modulated by the square and sawtooth waves; thus, the
back-reflection-induced
backscatter-induced
secondary
waves
also propagate through
the
IOC,
modulation phase Фm and
in Equation
(1) includes two
parts: the
square-wave-induced
phase (Ф
m_SQ)
andand
therefore,
these
waves
are
also
modulated
by
the
square
and
sawtooth
waves;
thus,
the
modulation
the sawtooth-wave-induced phase (Фm_SA). Because the secondary waves propagate twice
phase
Φm in
(1) of
includes
parts:
the
square-wave-induced
phase of
(Φfm_SQ
)Ф
and
through
theEquation
same branch
the IOC,two
Фm_SQ
has an
amplitude
of ±π/2 and frequency
τ, and
m_ SAthe
sawtooth-wave-induced
has an amplitude of 4π.phase (Φm_SA ). Because the secondary waves propagate twice through the
same branch
of thetoIOC,
an amplitude
of ˘π/2
and frequency
, and Φ
According
the Φ
theory
of coherence
detection,
because
Фm_SQ has of
thef τsame
modulation
m_SQ has
m_SA has an
amplitude
of (f4π.
frequency
τ) as that of the primary waves, the demodulation value of IBSC cannot be separated from
the
angle velocity
the FOG
which is the
demodulation
ofhas
thethe
primary
interference,
According
to theof
theory
of coherence
detection,
becauseresult
Φm_SQ
same waves’
modulation
frequency
leading
a bias error
(namely,
the BSC error)
[14].ofOn
the
other be
hand,
Фm_SA leads
(f τ )thus
as that
of thetoprimary
waves,
the demodulation
value
IBSC
cannot
separated
from to
theBSC
angle
error’s
variation
with
the
sawtooth
wave,
and
there
are
two
complete
periods
when
the
sawtooth
velocity of the FOG which is the demodulation result of the primary waves’ interference, thus leading
changes
from 0 the
(rad)
to π
(rad).[14].
Under
conditions,
the maximum
BSCerror’s
error isvariation
given
to awave
bias error
(namely,
BSC
error)
Onextreme
the other
hand, Φm_SA
leads to BSC
by:
with
the sawtooth wave, and there are two complete periods when the sawtooth wave changes from 0
2 maximum
2 error is given by:
(rad) toBSC
π (rad).
Under
extreme
the
BSC
Errormax
  splicing
RA RA1 conditions,
RB RB1  2splicing
coil
RA RA1 3splicing
coil
RB RB1   2LD / c   2splicing coil 
a
a
a
(2)
a
2
3
BSCError
“ pαLsplicing
R A R A1
` diameter
R B R B1 ` α
α2coil length
R A R A1of
`α
α2coil
R B R B1 q{p2πLD{λc
α2splicing αcoil q (2)
splicing
splicing
wheremax
D and
represent
the
and
fiber
the
coil,
respectively,
λ the¨wavelength,
and c the velocity of light.
where D and L represent the diameter and fiber length of the coil, respectively, λ the wavelength, and c
the velocity of light.
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3. Experimental Results
3. Experimental
To verify theResults
existence of the BSC error and measure its magnitude in a practical FOG, we
promoted
a
method
and established
experimental
as illustrated
in
To verify the existence
of the BSC the
errorcorresponding
and measure its
magnitude setup,
in a practical
FOG, we
Figure
3,
where
the
light
launched
from
an
amplified
spontaneous
emission
(ASE)
source
enters
the
promoted a method and established the corresponding experimental setup, as illustrated in Figure 3,
IOC through
single-mode
fiber
coupler,
and it travels
backward
the enters
detector
to the
where
the light alaunched
from an
amplified
spontaneous
emission
(ASE) to
source
theowing
IOC through
back-reflection
and
backscattering.
The
ASE
source
has
a
power
of
~5
mW
and
the
IOC
a single-mode fiber coupler, and it travels backward to the detector owing to the back-reflection is
anda
proton-exchanged
3 circuit
with
the half-wave
voltage
of IOC
~5 V.isIn
fact, the setup mainly
aims
backscattering.
TheLiNbO
ASE source
has
a power
of ~5 mW
and the
a proton-exchanged
LiNbO
3
at
determining
the
first
two
terms
in
Equations
(1)
or
(2),
which
are
the
dominant
BSC
error
terms.
circuit with the half-wave voltage of ~5 V. In fact, the setup mainly aims at determining the first two
Moreover,
our setup
a configuration
that
is nearlyBSC
identical
to the Moreover,
FOG configuration;
terms
in Equations
(1)has
or (2),
which are the
dominant
error terms.
our setupsquare
has a
and sawtooththat
waves
are applied
IOC, configuration;
and a lock-insquare
amplifier
is used towaves
demodulate
the
configuration
is nearly
identicalto
to the
the FOG
and sawtooth
are applied
interference
intensity
(I
BSC
)
at
the
detector
so
as
to
better
simulate
the
actual
FOG
operation.
The
one
to the IOC, and a lock-in amplifier is used to demodulate the interference intensity (IBSC ) at the detector
difference
in our
case isthe
that
two lengths
of PBFs (and
not adifference
PBF coil)in
are
connected
to the
IOC
so
as to better
simulate
actual
FOG operation.
The one
our
case is that
twotwo
lengths
pigtails
to
eliminate
interference
between
the
primary
waves
whose
demodulation
value
cannot
be
of PBFs (and not a PBF coil) are connected to the two IOC pigtails to eliminate interference between
separated
from
the
measurement
results.
From
the
aspect
of
BSC
error
determined
by
the
first
two
the primary waves whose demodulation value cannot be separated from the measurement results.
termsthe
in Equation
(1 ), this
does
notfirst
matter,
because
this part (1),
of the
error does
From
aspect of BSC
errordifference
determined
by the
two terms
in Equation
thisBSC
difference
does not
not
depend
on
the
subsequent
PBF
coil.
Considering
the
IOC’s
half-wave
voltage
and
the
signal
matter, because this part of the BSC error does not depend on the subsequent PBF coil. Considering
generator’s
single-end
output,
wethe
setsignal
the square
wave frequency
500 kHz
an square
amplitude
of
the
IOC’s half-wave
voltage
and
generator’s
single-endtooutput,
wewith
set the
wave
±1.25 V, and
the kHz
sawtooth
wave
is set toof
vary
from
0 V the
to 10
V. Based
on the
analysis
frequency
to 500
with an
amplitude
˘1.25
V, and
sawtooth
wave
is set
to varymentioned
from 0 V
above,
the
demodulation
result
of
I
BSC
should
have
two
sinusoidal
periods
when
the
sawtooth
to 10 V. Based on the analysis mentioned above, the demodulation result of IBSC should havewave
two
changes from
0 V when
to 10 V,
its peak-to-peak
value
can0 V
betoeasily
which value
directly
sinusoidal
periods
theand
sawtooth
wave changes
from
10 V, resolved,
and its peak-to-peak
value
can
1/2 or I in (R BR B1)1/2 or their sum according to which
1/2
1/2
reflects
I
in
α
splicing
(R
A
R
A1
)
splicing
point
exists
in
the
be easily resolved, which value directly reflects Iin αsplicing (RA RA1 )
or Iin (RB RB1 )
or their sum
system.
according to which splicing point exists in the system.
Figure 3. Expe rime ntal se tup for me asuring the BSC e rror.
Figure 3. Experimental setup for measuring the BSC error.
First, a length of PBF is normally connected to the IOC pigtail at point B through fusion
First, a length of PBF is normally connected to the IOC pigtail at point B through fusion splicing.
splicing. The micrograph of the splicing between the PBF and IOC pigtail is shown in the inset in
The micrograph of the splicing between the PBF and IOC pigtail is shown in the inset in Figure 4a.
Figure 4a. Since it is 0° splicing, the secondary wave W B is gen erated and its interference with W B1
Since it is 0˝ splicing, the secondary wave WB is generated and its interference with WB1 occurs
occurs simultaneously. The lock-in amplifier’s output after demodulation of the interference
simultaneously. The lock-in amplifier’s output after demodulation of the interference intensity IBSC
intensity IBSC is described by the dashed curve in Figure 4a where the fixed bias induced by such
is described by the dashed curve in Figure 4a where the fixed bias induced by such phenomena as
phenomena as Earth rotation has been taken out. Obviously, the output varies with the voltage of
Earth rotation has been taken out. Obviously, the output varies with the voltage of the sawtooth wave,
the sawtooth wave, and there are two periods when the voltage changes from 0 V to 10 V, a result
and there are two periods when the voltage changes from 0 V to 10 V, a result that agrees well with
that agrees well with the abovementioned analysis. The peak-to-peak value of the output is ~130 μV,
the abovementioned analysis.
The peak-to-peak value of the output is ~130 µV, which implies that
which implies that (RBRB1)1/2 ~3.5 × 10 −7, and therefore, the maximum BSC error in this case is
(RB RB1 )1/2 ~3.5 ˆ 10´ 7 , and therefore, the maximum BSC error in this case is ~1.36˝ /h according
~1.36°/h according to Equation (2) for a 300-m PBF coil with a diameter of 10 cm. Next, a second PBF
to Equation (2) for a 300-m PBF coil with a diameter of 10 cm. Next, a second PBF is also similarly
is also similarly connected to the other IOC pigtail at point A by fusion splicing. Th e resulting
connected to the other IOC pigtail at point A by fusion splicing. The resulting output is described by
output is described by the solid line in Figure 4a. The peak-to-peak value of the output is ~450 μV,
the solid line in Figure 4a. The peak-to-peak value of the output is ~450 µV, and consequently, the
and consequently, the total maximum BSC error is ~4.7°/h according to Equation (2). The
total maximum BSC error is ~4.7˝ /h according to Equation (2). The experimental results indicate that
experimental results indicate that interference between W A and W A1 contributes more to the BSC
interference between WA and WA1 contributes more to the BSC error than that between WB and WB1 ,
error than that between W B and W B1, which can be explained by the fact that the backscatter
which can be explained by the fact that the backscatter coefficient in the PBF is higher than that in
coefficient in the PBF is higher than that in conventional fiber s [15].
conventional fibers [15].
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(a)
(b)
Figure 4. Test results of (a) the BSC e rror and (b) bias stability in the PBFOG whe n the PBFs are
Figure 4. Test results of (a) the BSC error and (b) bias stability in the PBFOG when the PBFs are
connecte d to
to the
the IOC
IOC pigtails
pigtails through
throughnormal
normalfusion
fusionsplicing.
splicing.Inset:
Inset: micrograph
micrograph showing
showing the
the fusion
fusion
connected
splicing
be
twe
e
n
PBF
and
conve
ntional
fibe
r
at
point
A
or
B.
splicing between PBF and conventional fiber at point A or B.
In a real closed-loop PBFOG, the BSC error can seriously affect the bias stability. Bias stability
In a real closed-loop PBFOG, the BSC error can seriously affect the bias stability. Bias stability is
is defined as the random variation in bias as computed over specified finite sample time and
defined
as the random variation in bias as computed over specified finite sample time and averaging
averaging time intervals according to IEEE standard [16]. The secondary waves W A and W A1, W B
time
intervals according to IEEE standard [16]. The secondary waves WA and
W (see
, WFigure
are
B and W
B1 The
and W B1 are not reciprocal, because they propagate along different optical
pathsA1
2a).
not
reciprocal,
because
they
propagate
along
different
optical
paths
(see
Figure
2a).
The
environment
environment (temperature, vibration, and so on) definitely has different influences on different
(temperature,
vibration, and so on) definitely has different influences on different optical paths, as a
optical paths, as a result, their phase differences (φ1, φ2 ) are random environment-dependent
result,
their
phase
differences (ϕ , ϕ2 ) are
random environment-dependent variables. When those
variables. When those secondary1waves
interfere, their interference intensities and the induced BSC
secondary
waves
interfere,
their
interference
intensities and the induced BSC error would also vary
error would also vary randomly, so the PBFOG bias becomes unstable if it includes this error. In
randomly,
so the PBFOG bias becomes unstable if it includes this error. In order to verify the BSC error’s
order to verify the BSC error’s influence on PBFOG bias stability, a real PBF coil, instead of the two
influence
on
PBFOG bias stability, a real PBF coil, instead of the two lengths of PBFs, is connected to
lengths of PBFs, is connected to the IOC pigtails with normal fusion splicing on the basis of the
the
IOC
pigtails
with normal fusion splicing on the basis of the experimental setup in Figure 3, and
experimental setup in Figure 3, and forms a complete PBFOG. A signal-processing electronic circuit
forms
a complete PBFOG. A signal-processing electronic circuit substitutes the lock-in amplifier and
substitutes the lock-in amplifier and signal generator to implement the digital closing-loop
signal
generator to implement the digital closing-loop operation in the PBFOG [14]. The test result of
operation in the PBFOG [14]. The test result of the PBFOG bias is shown in Figure 4b, where the
the
PBFOG bias is shown in Figure 4b, where the fixed bias induced by such as earth rotation has also
fixed bias induced by such as earth rotation has also been taken out. The test result indicates that
been
taken out. The test result indicates that the bias remarkably fluctuates and the corresponding bias
the bias remarkably fluctuates and the corresponding bias stability is ~0.75°/h (standard deviation,
stability
is ~0.75˝ /h (standard deviation, 10 s integration time).
10 s integration time).
Although
backscatter within the fiber cannot be eliminated, back-reflection can be reduced
Although backscatter within the fiber cannot be eliminated, back-reflection can be reduced
through
an
angled
fiber end face in fiber termination in order to suppress the BSC error. A tilted angle
through an angled fiber end face in fiber termination in order to suppress the BSC error . A tilted
˝
of
8 is a commonly used value for conventional fiber and it is often applied in fiber connectors to
angle of 8° is a commonly used value for conventional fiber and it is often applied in fiber
suppress
back-reflection [17]. Although a larger cleavage angle can further reduce the back reflection,
connectors to suppress back-reflection [17]. Although a larger cleavage angle can further reduce the
this
will
come
at the cost of increased splicing loss, so in the experimental FOG, both the pigtails
back reflection, this will come at the cost of increased splicing loss, so in the experimental FOG, both
of
the IOC and PBF coil are cleaved at a tilted angle of ~8˝ to guarantee both the suppression of
the pigtails of the IOC and PBF coil are cleaved at a tilted angle of ~8° to guarantee both the
back-reflection
at the interface and the proper fusion splicing loss, as shown in Figure 5a [7,9–11,18].
suppression of back-reflection at the interface and the proper fusion splicing loss, as shown in
After
fusion
splicing,
according to the test results, the reflectance at the interface (see Figure 5b) is
Figure 5a [7,9–11,18]. After fusion splicing, according to the test results, the reflectance at the
reduced
from greater than ´20 dB to ~´54 dB, so the intensity of both WA and
WB decreases
to be
interface (see Figure 5b) is reduced from greater than −20 dB to ~−54 dB, so
the intensity
of both W A
negligible
compared
to
the
primary
waves.
Under
this
condition,
the
BSC
error
is
also
negligible
and W B decreases to be negligible compared to the primary waves. Under this condition, the BSC
according
to the experimental result, as illustrated in Figure 5c that gives the test result of the BSC
error is also negligible according to the experimental result, as illustrated in Figure 5c that gives the
error;
the
corresponding
PBFOG performance is shown in Figure 5d. Obviously, when the BSC error is
test result of the BSC error; the corresponding PBFOG performance is shown in Figure 5d.
suppressed,
the
bias
stability
is dramatically improved to ~0.4˝ /h (standard deviation, 10 s integration
Obviously, when the BSC error is suppressed, the bias stability is dramatically improved to ~0.4 °/h
time).
Therefore, we can conclude that the BSC error is an important bias error source that seriously
(standard deviation, 10 s integration time). Therefore, we can conclude that the BSC error is an
affects the PBFOG performance, and it has to be addressed before any practical application of the
important bias error source that seriously affects the PBFOG performance, and it has to be addressed
PBFOG
is considered.
before any practical application of the PBFOG is considered.
Sensors 2016, 16, 851
Sensors 2016, 16, 851
6 of 7
6 of 7
(a)
(b)
(c)
(d)
Figure
5. Micrographs
Micrographsshowing
showing(a)
(a)the
thecleaved
cleaveangles
d angles
of pigtails
of PBF
the PBF
coil IOC
and and
IOC(b)
and
(b)
Figure 5.
of pigtails
of the
coil and
cross
cross
section
of
the
corresponding
point
after
fusion
splicing;
(c)
Test
results
of
the
BSC
error
and
(d)
section of the corresponding point after fusion splicing; (c) Test results of the BSC error and (d) bias
bias
stability
in
the
PBFOG
whe
n
the
PBF
coil
is
connecte
d
to
the
IOC
pigtails
through
~8°
fusion
˝
stability in the PBFOG when the PBF coil is connected to the IOC pigtails through ~8 fusion splicing.
splicing.
4. Conclusions
4. Conclusions
In summary, we found and investigated a bias error (namely, the BSC error) in the PBFOG that is
In
we found
andback-reflection-induced
investigated a bias errorand
(namely,
the BSC error) in
the PBFOG
that
caused summary,
by interference
between
backscatter-induced
secondary
waves.
is
caused
by
interference
between
back-reflection-induced
and
backscatter-induced
secondary
The BSC error was theoretically analyzed and experimentally verified. Our experimental results
waves. The
BSC
error
was
theoretically
and aexperimentally
verified.
Our the
experimental
˝ /h
showed
a BSC
error
of ~4.7
for a 300-m analyzed
PBF coil with
diameter of 10 cm.
Therefore,
BSC error
results
show
ed
a
BSC
error
of
~4.7°/h
for
a
300-m
PBF
coil
with
a
diameter
of
10
cm.
Therefore,
the
is an important error source in the PBFOG, and it seriously affects the long-term stability of PBFOG.
BSC
error
is
an
important
error
source
in
the
PBFOG,
and
it
seriously
affects
the
long-term
stability
This error has to be addressed before the practical application of the PBFOG is considered, and our
of PBFOG.
Thisfocus
error
to be
addressed
before
the practical
application
of the
PBFOGthe
is
future
work will
on has
further
studying
possible
measures
to suppress
the BSC error
to improve
considered,
ourPBFOG.
future work will focus on further studying possible measures to suppress the
bias
stabilityand
of the
BSC error to improve the bias stability of the PBFOG.
Acknowledgments: This work was supported by National Natural Science Foundation of China (NSFC) under
Grants
61575012 andThis
61575013.
Ackno wledgments:
work was supporte d by National Natural Scie nce Foundation of China (NSFC) unde r
Grants
61575012
and
61575013.
Author Contributions: Xiaobin Xu conceived and theoretically analyzed the SIM error and wrote the paper;
Ningfang Song and Xiaobin Xu promoted the experimental methods and setup; Zuchen Zhang performed the
Author Contributions: Xiaobin Xu conce ive d and theore tically analyze d the SIM e rror and wrote the pape r;
experiments; Jing Jin involved in the analysis of the data.
Ningfang Song and Xiaobin Xu promote d the e xpe rime ntal me thods and se tup; Zuchen Zhang pe rforme d the
Conflicts
The authors
declare
no conflict
of interest.
e xpe rimeof
nts;Interest:
Jing Jin involve
d in the
analysis
of the data.
Conflicts of Interest: The authors de clare no conflict of inte re st.
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© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access
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