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1. ISSN 1063 7850, Technical Physics Letters, 2011, Vol. 37, No. 5, pp. 397–400. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © A.M. Kurbatov, R.A. Kurbatov, 2011, published in Pis’ma v Zhurnal Tekhnicheskoі Fiziki, 2011, Vol. 37, No. 9, pp. 14–21.
397
In recent years, for high grade navigation, fiber
optic gyroscopes (FOGs) are being used more and
more frequently. FOG has fiber ring interferometer
(FRI) and processing electronic unit. One of the
sources of FOG errors are the effects of polarization in
FRI elements.
In Fig. 1, the scheme of a fiber ring interferometer
(FRI) is shown that, in particular, contains an input
fiber with linear birefringence bin, length Lin, and
depolarization length Lγ, in; it also contains a sensing
coil of fiber with linear birefringence b, length L, and
depolarization length Lγ Ӷ L. In the absence of an
input fiber (FRI classical scheme [1–5]), the main
contributions to polarization error (PE) in the angle
rate are due to the polarization mode coupling in the
coil fiber (PE 1) and the misalignment of the
waveguide axes of the coil fiber and Y junction (PE 2).
For PE 1, three components of which correspond to
light Stokes parameters s1,2,3, we have [1]
(1)
where ε is the ratio of amplitude extinctions of channel
waveguides (5) of Y junction (3) (Fig. 1), p is the grade
of the residual polarization of the light, and h is
parameter h of the coil fiber. Error 1 is conditioned by
fiber sections with lengths Lγ, which are arranged sym
metrically along the entire length of the fiber relative
to its midpoint [4] and errors Φ2, 3 are conditioned by
the initial and final fiber sections with lengths Lγ [1–
5]. If sensing coil is made of polarizing (PZ) fiber with
y waves power suppressing coefficient α, then, as in
[1], calculations yield the following estimates:
(2)
Φ1 ε
2
ph LγL( )
1/2
, Φ2 3, εp hLγ( )
1/2
,∼ ∼
Φ1 ε
2
ph Lγ/α( )
1/2
,∼
Φ2 3, εp hLγ αLγ/2)
2
]erfc αLγ/2( )[exp{ }
1/2
.∼
Here, erfc(x) = 1 – erf(x) is the error function and h is
the h parameter at α = 0. Thus, even under a nearly
inapplicable condition αLγ ӷ 1, the influence of
dichroism on errors Φ2,3 is too small. Furthermore, the
influence of the dichroisn of the coil fiber on PE 2 is
even smaller [4], which ultimately seems to make it
useless. However, below, we will show that, in FRI with
highly anisotropic fiber at the input, the application of
a PZ fiber in the sensing coil is essentially useful.
Let the anisotropic fiber be at the input of the FRI
and let Lin ӷ Lγ, in (Fig. 1). This is similar to the aniso
tropic element in FRI [6], by which PE, which was
obtained in [7] in the form Φ = Φ2 + Φ3 + Φ1 (see (1)–
(2)), is suppressed to Φ = Φ1 (since, originally, it was
Φ1 Ӷ Φ2, 3) [6]. However, in [6] (and also in (1) and
(2)), only the coupling of the first order polarization
mode is considered. A numerical model of the polar
ization maintaining (??) coil fiber described in [5] in
the presence of an input fiber with length Lin Ӷ L led
us to the empirical formula
(3)
or rather, to a reduction in errors Φ2,3 by a factor of
only 1/(hL), instead of their full suppression, as in [6].
This is not sufficient for a navigation grade fiber optic
gyroscope (FOG). Now, we will show that, when
approaching condition Φ2,3 ≈ 0, one may apply a PZ
fiber for the coil.
For a PZ fiber, we generalized a known model [5].
The rest was made in full accordance with [5]. Other
FRI parameters include an operating wavelength λ0 =
1.55 μm, spectral width of 5 nm, ε = 0.01, p = 0.02,
coil radius of 40 mm, b = 6 · 10–4
, L = 100 m (for faster
calculations because the calculation time is ~L2 [5]).
We tested the model on the FRI scheme in the absence
of an input fiber and, with a correction by a coefficient
of ~1, we obtain good agreement with (2).
Φ2 3, εphL Lγ( )
1/2
,∼
Suppression of Polarization Errors
in Fiber Ring Interferometer by Polarizing Fibers¶
¶
A. M. Kurbatov* and R. A. Kurbatov
Kuznetsov Research Institute for Applied Mechanics, Moscow, 111123 Russia
*e mail: akurbatov54@mail.ru
Received October 4, 2010
Abstract—Polarization errors in a Sagnac fiber ring interferometer are considered. It is shown that the joint
application of polarizing fibers at the input of the interferometer and the sensing coil is able to radically sup
press these errors.
DOI: 10.1134/S1063785011050129
¶The article was translated by the authors.

2. 398
TECHNICAL PHYSICS LETTERS Vol. 37 No. 5 2011
A.M. KURBATOV, R.A. KURBATOV
Let us consider an input PM fiber (α = 0) with bire
fringence bin = b at two different lengths, i.e., Lin = 1
and 10 m. The angle rate errors Ω1,2,3, which corre
spond to Φ1,2,3 listed in Table 1 (values for Lin = 10 m
are shown in brackets). Here, we will not take into
account coil fibers, the misalignment of the waveguide
axes of the Y junction channel, or the coupling of the
polarization mode of the input fiber.
From Table 1, we have the following:
(1) errors Ω2,3 are reduced, even without dichroism
in the sensing coil fiber (see (3));
(2) errors Ω1,2,3 are further reduced by the dichro
ism of the coil fiber;
(3) for α > 1 dB/m and Lin = 10 m, errors Ω2,3 are
much smaller than for Lin = 1 m.
Let us give a physical interpretation of these results.
Due to the polarization mode coupling (PMC,) in
the sensing coil fiber, one may represent the direct
wave (running in clockwise direction) and the reverse
wave (counterclockwise direction) in the form E±
=
+ + . Here, is the x wave in the absence
of PMC, is the x wave that came from the input y
wave due to the first order power transfer, and
came from the input x wave due to the second order
power transfer.
In addition, in the absence of an input fiber, errors
Φ2,3 (1) are determined by waves and [1]. How
ever, when the input fiber is present, these waves are
incoherent and, therefore, yield Φ2,3 = 0 [6]. However
there is also interference of waves and , which,
E0
±
E1
±
E2
±
E0
±
E1
±
E2
±
E1
+−
E0
+−
E1
±
E2
±
unlike [6] (see Table 1 for Φ2, 3 = 0), yields Φ2, 3 ≠ 0.
This new PE generation is illustrated in Fig. 2.
Polarization mode e0, y enters the coil fiber at the
beginning and polarization mode e0, x at the end. The
first of these PMCs transfers to x wave and the sec
ond PMC transfers first to the y wave e1, y and then to
x wave . Due to input fiber, modes e0, y and e0, x have
an optical path difference that is incoherent at Lin ӷ
Lγ, in.
Now, assume that the mode e0, y partially trans
ferred to wave e0, x on the section dz at distance z from
the end of the. The wave e1, y will generate x wave
along all the remaining length L – z. It is clear that
components of wave that are coherent with wave
are generated at distances from the end of the coil
fiber exceeding z + z0 (z0 = Linbin/b) because this is the
only way that this path length difference is compen
sated for, which is initially due to the input fiber when
the wave is in the state of wave e1, y and passes the
necessary distance along the y axis of the coil fiber. In
this situation, the wave components further gener
ated by wave e1, y at some distance z + z0 + z1 from the
end of the coil fiber are coherent with wave com
ponents generated by e0, y in the region of distances of
z1 – Lγ to z1 + Iγ from the beginning of the fiber. This
scheme is half of the picture, though the other half is a
mirror image of these processes relative to the mid
point of the coil fiber.
E1
+
E2
–
E2
–
E1
+
E2
–
E2
–
E2
–
E1
+
Table 1. PE for FRI in Fig. 2 with input PM fiber and polarizing fiber of sensing coil, without misalignment between its
optical axes and axes of Y junction
α, dB/m Ω1, deg/h Ω2, deg/h Ω3, deg/h
0 8.5 × 10–4
(8.7 × 10–4
) 0.0055 (0.0051) 0.0052 (0.0054)
10–1 3.5 × 10–4 (2.6 × 10–4) 0.0022 (0.0023) 0.0024 (0.0026)
100
7 × 10–5
(6.4 × 10–5
) 0.0011 (2.9 × 10–4
) 0.0011 (3.1 × 10–4
)
101
2.3 × 10–5
(2 × 10–5
) 5.2 × 10–4
(2.7 × 10–9
) 5.3 × 10–4
(2.9 × 10–9
)
1
8
2
9
10
3 4 5 6
7
Fig. 1. FRI scheme: (1) light source; (2) isotropic coupler; (3) integrated optic Y junction; (4) electrodes for phase modulation
voltage; (5) channel waveguides; (6) jointing points of Y junction channel waveguides with sensing coil fiber; (7) sensing coil;
(8) photodetector; (9) input fiber; (10) joint of input fiber and Y junction.

3. TECHNICAL PHYSICS LETTERS Vol. 37 No. 5 2011
SUPPRESSION OF POLARIZATION ERRORS 399
Based on the above, for a coil PM fiber, at z0 > L, it
is theoretically possible to obtain Ω2,3 ≈ 0 because
there is no wave components that are coherent
with wave . However, here, in practice, the PMC of
the input fiber will manifest by limiting the suppres
sion of errors Ω2,3. Furthermore, it is also necessary to
fulfill the condition Linbin > Lb (usually when Lin ~ L).
How does the dichroism of the sensing coil fiber
will influence Ω2,3?
(1) The dichroism suppresses the wave e0,y (Fig. 2)
and, at z > 1/α and the waves ( ) are no longer
generated from the beginning (end) of the coil fiber.
(2) If αz0 ӷ 1, then wave e1, y (Fig. 2) will decay
before passing to distance z0. As a result, in Table 1, for
α = 10 dB/m and Lin = 10 m, errors Ω2,3 is much
smaller than for Lin = 1 m. Thus, we have a radical
mechanism for suppressing errors Ω2,3 (as in [6]).
Unfortunately, the second mechanism has almost
no influence on Ω1, which is mainly suppressed due to
the fact that deposits in it only remain from sections
with length Lγ situated at distances less than 1/α from
the beginning and end of the coil PZ fiber (2). The fur
ther suppression of Ω1 is only possible if the input fiber
is also polarizing.
Consider now an FRI with a coil fiber and the mis
alignment of the waveguide axes of the Y junction
within 2°. The dichroism of the input fiber is αin =
60 dB; furthermore, α = 10 dB/m, bin = 8 · 10–4
, b =
6 · 10–4. Consider the combinations with PMC in the
input fiber and without it, as well as with and without
the misalignment of the axes (four cases). Table 2 lists
PE values for Lin = 1 m; values for Lin = 20 m are
shown in brackets. Table 2 also presents PE values for
the PM fiber of the coil. Here, in order to identify the
E2
+−
E2
±
E1
+
E1
–
jointed action of input fiber and coil fiber, we assume
that the dichroism of the Y junction 5 waveguides 3
(Fig. 1) is absent (ε = 1).
Thus, for Lin = 20 m, errors Ω2,3 are not zero, but
only due to the PMC of the input fiber; for Lin = 1 m,
they are determined by the PMC in the coil fiber and
with the misalignment of the axes (the latter could be
treated as singular PMC centers at the beginning and
the end of the coil fiber, so it is possible to apply the
scheme in Fig. 2). Thus, the depolarization in the
input PZ fiber and coil PZ fiber may radically reduce
the errors Ω2,3 (if αz0 ӷ 1), or to achieve the results
derived in [6]. Note that here one may use Lin Ӷ L,
contrary to FRI with PM fibers (see above).
Jointed operation of depolarization and dichroism
in the input and coil fibers could be considered to be
part of the general principle of PE suppression in FRI
by depolarization and dichroism. Earlier [8], this prin
ciple manifests through the attenuation of polarizer
requirements due to depolarization in the coil fiber.
As for error Ω1, it is not suppressed to zero, even for
Lin = 20 m; only Ω1 now determines the whole PE.
However, the suppression of this error by the input and
the dichroism of the coil fibers is still essential. In
addition to this, we only considered a length L of 100
m, although Ω1 ~ 1/L (2). Furthermore, we did not
take into account the dichroism of the waveguides of
the Y junction; however, the PE is still small, even for
navigation grade FOG.
In the case of errors in the ?? fiber of the coil, the
difference for Lin = 1 m and 20 m is small and all of
these errors are due to PMC in the coil fiber and the
misalignment of the axes; furthermore, the PMC of
the input fiber does not influence the error. Here,
errors Ω2,3 are on the level of several orders of magni
tude and the error Ω1 is several times larger than cor
responding errors in the case of PZ fiber of the coil,
e0,y
3
E1
+
1
2
E2
−
e1,y
5
e0,y
z
z0
Fig. 2. New PE generation scheme. Arrow 1 unites waves that yield classical PE; arrow 2 unites waves that yield new PE; e0, x, y
are original x and y polarization modes entering the sensing coil fiber; arrow 3 shows generation of wave from input mode
e0, y; arrows 4 and 5 show generation of wave from input mode e0, x (z0 = Linbin/b).
E1
+
E2
–

4. 400
TECHNICAL PHYSICS LETTERS Vol. 37 No. 5 2011
A.M. KURBATOV, R.A. KURBATOV
which indicates that the use of the PZ fiber in the FRI
sensing coil may be very useful.
In our point of view, measurements of further PE
suppression are only reasonable in extremely accurate
unitary measurements where the requirements of FRI
compactness are absent. In [4], FRI was described for
detecting the effects of general relativity using a sens
ing coil made of single mode isotropic fiber with a
diameter of several kilometers. In our case, the coil
radius may be on the order of several meters or even
smaller.
Dichroism in the coil fiber may find practical
applications based on convenient anisotropic fibers
with so called small apertures [9] or in anisotropic W
fibers. The input PZ fiber could also be implemented
based on the W profile of the refractive index [10].
Consider the prototype of the W fiber of the sensing
coil described by us in [11]. Here, dichroism is sup
plied by the difference in the bending losses of the fun
damental polarization modes. If the bending radius is
not small (>50 mm), the spectral curves of these losses
grow fairly rapidly, so it is quite realistic from our point
of view to achieve the above mentioned values of
dichroism. As can be seen from Table 2, for naviga
tion grade FOG, lower values of the dichroism of the
sensing coil fiber are sufficient (~1 dB/m according to
calculations).
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(1986).
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Table 2. PE for FRI in Fig. 2 with polarizing input fiber for the cases of polarizing and polarization maintaining sensing
coil fiber
Polarizing sensing coil fiber
FRI Ω1, deg/h Ω2, deg/h Ω3, deg/h
Without PMC or misalignment 2 × 10–6
(1.9 × 10–6
) 2.2 × 106
(0) 2.3 × 10–6
(0)
Without PMC and with misalignment 3 × 10–6
(3.2 × 10–6
) 8.0 × 10–5
(0) 7.7 × 10–5
(0)
With PMC and without misalignment 2 × 10–6
(1.9 × 10–6
) 2.2 × 10–6
(9.6 × 10–9
) 2.3 × 10–6
(9.3 × 10–9
)
With PMC and misalignment 5.4 × 10–6
(3.3 × 10–6
) 8.0 × 10–5
(6 × 10–8
) 7.7 × 10–5
(5.8 × 10–8
)
Polarization maintaining sensing coil fiber
FRI Ω1, deg/h Ω2, deg/h Ω3, deg/h
Without PMC or misalignment 6.3 × 10–6
(5.4 × 10–6
) 2.7 × 10–4
(2.2 × 10–4
) 2.6 × 10–4
(2.1 × 10–4
)
Without PMC and with misalignment 7.9 × 10–6
(7 × 10–6
) 5.6 × 10–4
(4.8 × 10–4
) 5.3 × 10–4
(4.8 × 10–4
)
With PMC and without misalignment 6.3 × 10–6
(5.4 × 10–6
) 2.7 × 10–4
(2.2 × 10–4
) 2.6 × 10–4
(2.1 × 10–4
)
With PMC and misalignment 10–5
(8.5 × 10–6
) 5.6 × 10–4
(4.8 × 10–4
) 5.3 × 10–4
(4.8 × 10–4
)