214
OPTICS LETTERS / Vol. 12, No. 3 / March 1987
Bias drift reduction in polarization-maintaining fiber gyroscope
S. L. A. Carrara,
B. Y. Kim, and H. J. Shaw
Edward L. Ginzton Laboratory, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California 94305
Received September 18, 1986; accepted December 5, 1986
The output signal of a birefringent
fiber gyroscope is analyzed, and sources of error due to polarization cross
coupling in the fiber and fiber components are identified. Techniques to suppress such errors by modulating the
fiber birefringence and balancing the optical power in the two polarization modes are proposed and demonstrated.
An ideal polarizer located at the common input/output port of a fiber-optic gyroscope eliminates the nonreciprocal phase error due to polarization cross coupling in the fiber sensing loop.1' 2 In practice, however,
polarizers have a finite polarization-extinction ratio,
leading to a bias drift in rotation measurement. The
use of a broadband optical source3 in conjunction with
highly birefringent fiber4- 6 suppresses this phase error
by reducing the polarization mode coupling and by
making the primary optical waves that travel reciprocal optical paths incoherent with the cross-coupled
waves. A depolarizer in an ordinary, low-birefringence fiber loop also provides a means of reducing
nonreciprocal phase errors.7 However, in order to
suppress phase errors to inertial navigation standards
without imposing unrealistic requirements on the performance of individual components such as polarizers,
optical fibers, and directional couplers, it is desirable
propagation in the single-mode, high-birefringence fiber, including the effects of the directional coupler
that forms the loop; 2k5 is the Sagnac phase shift; and
the subscripts 1 and 2 refer to the fiber-polarization
eigenmodes. By reciprocity, it can be shown that, in
the absence of external magnetic fields and time-varying perturbations,
G.CW(w)= G'WT(W).8With zero ro-
tation, the phase error is given by
Aikerr = arglEintGecwtGcwEin}
where the dagger indicates the Hermitian conjugate
and arg~z1is the phase of the complex number z.
Consider the input optical power divided between
the two polarization eigenmodes, where we define tan 0
= la 2/ay1. Since the off-diagonal elements of the G
matrices are much smaller in magnitude than the diagonal terms for a high-birefringence fiber gyro, Eq. (2)
yieldsAq0err= A/qamp+ Afint, with
, ImI(g11 *g12 + g12*g2 2)al*a 2 + (g11g21 * + g21g22*)ala2*C
2
2
Atkamp ;:1~
-
Ig1 la 11
to implement additional means of suppressing these
errors. In this Letter we characterize the main
sources of phase error in a highly birefringent fiber
gyroscope and experimentally demonstrate techniques to suppress the corresponding errors.
In Fig. 1, consider the optical fields in the common
input/output fiber, disregarding the birefringence
modulators (B-mod's) for the moment. The fields
leaving the interferometer after traversing the loop in
the clockwise and counterclockwise directions are
Ecw(w) = Gcw(co)e+i0sEin,
Ercw(w) = Gccw(,)e,
S,
(1,
where
E [al(w)1
[a2(wM
is the input optical field to the sensing loop;
Fgii(w) g 12 (CW) 1
GiW(s) te trsW)
g22(W)J
2
is the transfer matrix corresponding to the clockwise
-
0146-9592/87/030214-03$2.00/0
(2)
+ 1g2 2a 21
Im1g 12 g2 21 (la
2
112 -
Ia21)1
2
Ignlata1+ 1g22 a2 1
c
sin 20,
(3a)
(3b)
where the proportionalities on the right-hand sides
hold for Ig111 1g2 2 1. These expressions are general in
the sense that all possible combinations of cross-coupled waves causing nonreciprocal phase error are covered. It can be seen that two different types of error
term with distinct characteristics are present. Amplitude-type error (A'Pamp) depends on the relative
phases of the input field components a, and a2 , whereas intensity-type error (Aki\it)depends on the optical
power difference between the two polarization modes,
regardless of their relative phases. It should be mentioned that insertion of a polarizer in the common
input/output path having an amplitude-extinction ratio Eand its transmission axis parallel to a principal
axis of the fiber would reduce the amplitude-type error by E,as pointed out by Kintner,2 and the intensitytype error would be suppressed by e2.
Amplitude-type error arises from the coherent interference of field components that were orthogonally
polarized at the input of the loop and were brought
into the same polarization mode by cross-coupling
©1987, Optical Society of America
March 1987 / Vol. 12, No. 3 / OPTICS LETTERS
Ein
B-mod
B-mod
Eout
LOO
4-,
Fig. 1.
Schematic of a fiber Sagnac interferometer
with
birefringence modulators (B-mod's).
the coherence between a1 and a2. However, error con-
tributions from multiple coupling centers in the fiber
loop6 and from residual coherence between the two
polarization modes will remain. It has also been suggested8 that reduction of the coherence between a, and
their relative
phases right after the polarizer. This could be done
by modulating the fiber birefringence in a section of
the input fiber. We point out that one can achieve
smaller phase errors without requiring high-performance components by combining these two approaches to reduction of amplitude-type error.
Intensity-type error is due to interference between
waves originally in the same polarization mode that
cross coupled into the other mode. Therefore this
error is not affected by either of the techniques mentioned above. Contributions to Akint come mostly
from pairs of scattering centers symmetrically located,
to within a depolarization length, with respect to the
midpoint of the fiber in the loop6' 10 and are limited by
1a 2 12)/(1a 112 +
(1a1 12 -
1a 212), where
the
h pa-
rameter of the fiber describes how well the fiber eigenmodes are isolated, L is the loop length, and LD is the
depolarization length of the source in the fiber. We
note that a way of suppressing this intensity-type
error is to balance the input optical power in the two
polarization modes. We also point out that if a birefringence modulator is placed at one end of the fiber
loop, as depicted in Fig. 1, error contributions
from
such symmetrical pairs of coupling centers located in
the sensing fiber coil will be time averaged through
the modulation of g12*g21 in expressions (3). In this
case, the birefringence modulation in the loop also
introduces nonreciprocal phase modulation for the
two counterpropagating
gent-fiber gyroscope was constructed as shown in Fig.
was about
centers. It has been
that a dispersive
birefringent element in the input lead of the Sagnac
loop would reduce amplitude-type error by decreasing
h LLD72
gent fiber directional coupler. Therefore, if the directional coupler is the dominant error source, a better choice would be the combination of a birefringence
modulator outside the loop and equal excitation of
optical power in both polarization modes.
To verify the foregoing derivations, an all-birefrin2. The beat length between the two polarization
modes was 3.3 mm at 820 nm, and the h parameter
suggested6' 9
a 2 could be achieved by modulating
215
waves. However, if the fre-
quency of the birefringence modulation is much lower
than that of the bias phase modulation, although
higher than the detection bandwidth, this effect can
10-5
m- 1 . The sensing fiber was 50 m long,
wrapped on a 12.5-cm-diameter spool. The net area
enclosed by the loop was made equal to zero by reversing the direction of winding for the second half of the
fiber length in order to avoid the presence of both
rotation-induced offset and scale-factor instabilities
in the drift measurements. A polarizer was used to
control the amount of optical power launched in each
polarization mode. The optical source was a superluminescent diode emitting at 820 nm, with a bandwidth of 10 nm (FWHM). Assuming a Gaussian
spectrum, the depolarization length LD in the fiber
was 0.3 m.
Polarization-preserving
directional cou-
plers of the polished type were constructed after the
birefringent axes of the fiber were carefully aligned in
the interaction region.' Polarization cross coupling
It12was of the order of 10-3. Birefringence modulators were constructed separately and spliced into the
system. The principal axes of the fiber were aligned
in the middle of a 2-m-long piece, as described in Ref.
11, and the fiber was glued between two quartz slabs,
with the fast axis of the fiber perpendicular to the
faces of the plates. This orientation minimizes polarization coupling caused by small misalignment.1
This fiber sandwich was then clamped against a piezoelectric transducer. An electrical signal applied
to the transducer produces a modulation in the fiber
birefringence through the elasto-optic effect. A deterministic signal was applied to the modulator, with
optimum amplitude dependent on the particular
waveform. We point out that birefringence modulation can easily be achieved in integrated-optics form
and can be incorporated into the same substrate as
the directional coupler and the phase modulator.
Figure 3(a) shows the behavior of the amplitude
error as a function of the fractional optical power in
one polarization mode with and without birefringence modulation. Reduction of phase error due to
birefringence modulation by a factor of 50 is observed
with optical power balance. The phase error is measured by heating the input fiber section, causing the
relative phase between orthogonal input field compo-
be made negligible.
Therefore by combining the two birefringence modulators (inside and outside the loop), or one birefringence modulator in the common input/output lead
and the optical power balance, we can suppress all
nonreciprocal phase errors mentioned in Ref. 6, regardless of the nature of the polarization cross-coupling mechanisms involved. When two birefringence
modulators are used, however, intensity-type error
due to polarization cross coupling in the directional
coupler still remains. This error will be limited by
HAint
_ It12(1a
1 12
-
la 2 12)/(1a112 + 1a 2 12), with
1t12 being
the power cross-coupling coefficient in the birefrin-
POLARIZER
SLID
Ein
B-mod
DC
/ a SPLICE
Fig. 2.
Experimental
setup.
ode; PM, phase modulator.
SLD, superluminescent
di-
OPTICS LETTERS / Vol. 12, No. 3 / March 1987
216
sensing loop.
No appreciable improvement was no-
ticed, which can be explained by the fact that the
error expected from symmetrical coupling points in
the fiber, for our setup, is only 3% of that caused by
the loop directional coupler. The remaining phase
2 1000 _
error was not affected by temperature-induced
-Gs 500
<1
0
1.0
FFRACTIONOF OPTICAL POWER IN SLOW MODE
(a)
1.04
0.
E 0.8
-
* EXPERIMENT
THEORY
changes in the fiber birefringence at any location in
the optical circuit, except for the directional coupler,
and it swept through +Alpintwhen the coupler was
heated. These observations confirmed that polarization cross coupling in the directional coupler is the
main source of intensity-dependent drift in our gyroscope. As mentioned earlier, careful balance of input optical power in the two polarization modes
should be used in order to suppress this error. In Fig.
-0-
4 the intensity-type
<10.6
a
W
the fraction of optical power in one polarization
mode. The error for maximum power imbalance is
dependent on environmental conditions. A residual
drift of 20 grad was observed when the input polarizer
was adjusted to 450 with the principal axes, which
seemed to be due to imperfect optical power balance.
In summary, we have presented time-averaging
techniques to reduce output signal drift in fiber gyroscopes, by employing birefringence modulation at the
common input/output port and at one end of the fiber
coil, solving all problems caused by imperfections in
the fiber. Imperfections in the birefringent fiber directional coupler were seen to be the main source of
error left. This error was reduced by exciting both
polarization modes with equal power. We have also
noted that, when they are used together with a polarization filter, these techniques can reduce errors even
further, relieving the requirements on the performance of individual components.
N
j
0.4
o~ 0.2
I
ol
0
27r
677
47r
AMPLITUDE OF TRIANGULAR-WAVE MODULATION
(b)
Fig. 3. (a) Behavior of amplitude-type error with and without birefringence modulation. (b) Reduction of drift observed with birefringence modulation.
* EXPERIMENT
-THEORY
0
0
0
This research was supported by Litton Systems, Inc.
0
1._
error is shown as a function of
S. L. A. Carrara is supported
by the Brazilian Air
Force.
0
0.2
0.4
0.6
0.8
1.0
FRACTION OF OPTICAL POWER IN SLOW MODE
Fig. 4. Intensity-type error as a function of the fraction of
the input optical power in one polarization mode.
References
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Lett. 17, 352 (1981).
nents to vary over several times 27r. The maximum
peak-to-peak variation is measured. Birefringence
modulation with a triangular waveform at fb of 100 Hz
and peak-to-peak amplitude of 27r was used, while
the phase modulation for the dynamically biased detection system employed was at fm = 25 kHz. Essentially, birefringence modulation shifts the error signals present at fm to fm i nfb, i.e., outside the detection bandwidth centered at fi. Figure 3(b) shows the
characteristic drift reduction as a function of the
modulation signal amplitude for a triangular waveform.
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integral multiple of 2r.
To investigate the reduction of intensity-type error
terms, a second B-mod was spliced at one end of the
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