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 andmore frequently. FOG has fiber�ring interferometer(FRI) and processing electronic unit. One of thesources of FOG errors are the effects of polarization inFRI elements.

In Fig. 1, the scheme of a fiber�ring interferometer(FRI) is shown that, in particular, contains an inputfiber with linear birefringence bin, length Lin, anddepolarization length Lγ, in; it also contains a sensingcoil of fiber with linear birefringence b, length L, anddepolarization length Lγ � L. In the absence of aninput fiber (FRI classical scheme [1–5]), the maincontributions to polarization error (PE) in the anglerate are due to the polarization mode coupling in thecoil fiber (PE�1) and the misalignment of thewaveguide axes of the coil fiber and Y junction (PE�2).For PE�1, three components of which correspond tolight Stokes parameters s1,2,3, we have [1]

(1)

where ε is the ratio of amplitude extinctions of channelwaveguides (5) of Y junction (3) (Fig. 1), p is the gradeof the residual polarization of the light, and h isparameter h of the coil fiber. Error 1 is conditioned byfiber sections with lengths Lγ, which are arranged sym�metrically along the entire length of the fiber relativeto its midpoint [4] and errors Φ2, 3 are conditioned bythe initial and final fiber sections with lengths Lγ [1–5]. If sensing coil is made of polarizing (PZ) fiber withy waves power suppressing coefficient α, then, as in[1], calculations yield the following estimates:

(2)

Φ1 ε2ph LγL( )

1/2, Φ2 3, εp hLγ( )

1/2,∼ ∼

Φ1 ε2ph 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 isthe h parameter at α = 0. Thus, even under a nearlyinapplicable condition αLγ � 1, the influence ofdichroism on errors Φ2,3 is too small. Furthermore, theinfluence of the dichroisn of the coil fiber on PE�2 iseven smaller [4], which ultimately seems to make ituseless. However, below, we will show that, in FRI withhighly anisotropic fiber at the input, the application ofa PZ fiber in the sensing coil is essentially useful.

Let the anisotropic fiber be at the input of the FRIand let Lin � Lγ, in (Fig. 1). This is similar to the aniso�tropic element in FRI [6], by which PE, which wasobtained in [7] in the form Φ = Φ2 + Φ3 + Φ1 (see (1)–(2)), is suppressed to Φ = Φ1 (since, originally, it wasΦ1 � Φ2, 3) [6]. However, in [6] (and also in (1) and(2)), only the coupling of the first�order polarizationmode is considered. A numerical model of the polar�ization�maintaining (??) coil fiber described in [5] inthe presence of an input fiber with length Lin � L ledus to the empirical formula

(3)

or rather, to a reduction in errors Φ2,3 by a factor ofonly 1/(hL), instead of their full suppression, as in [6].This is not sufficient for a navigation�grade fiber�opticgyroscope (FOG). Now, we will show that, whenapproaching condition Φ2,3 ≈ 0, one may apply a PZfiber for the coil.

For a PZ fiber, we generalized a known model [5].The rest was made in full accordance with [5]. OtherFRI 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 fastercalculations because the calculation time is ~L2 [5]).We tested the model on the FRI scheme in the absenceof an input fiber and, with a correction by a coefficientof ~1, we obtain good agreement with (2).

Φ2 3, εphL Lγ( )1/2

,∼

Suppression of Polarization Errorsin 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.ruReceived October 4, 2010

Abstract—Polarization errors in a Sagnac fiber�ring interferometer are considered. It is shown that the jointapplication 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.

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 = 1and 10 m. The angle�rate errors Ω1,2,3, which corre�spond to Φ1,2,3 listed in Table 1 (values for Lin = 10 mare shown in brackets). Here, we will not take intoaccount coil fibers, the misalignment of the waveguideaxes of the Y junction channel, or the coupling of thepolarization mode of the input fiber.

From Table 1, we have the following:

(1) errors Ω2,3 are reduced, even without dichroismin 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 aremuch smaller than for Lin = 1 m.

Let us give a physical interpretation of these results.

Due to the polarization�mode coupling (PMC,) inthe sensing�coil fiber, one may represent the directwave (running in clockwise direction) and the reversewave (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�orderpower 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 areincoherent 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 thebeginning 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 havean optical path difference that is incoherent at Lin �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 coilfiber exceeding z + z0 (z0 = Linbin/b) because this is theonly 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 thenecessary 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 ofz1 – Lγ to z1 + Iγ from the beginning of the fiber. Thisscheme is half of the picture, though the other half is amirror 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 itsoptical 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

29

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 modulationvoltage; (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.

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, itis theoretically possible to obtain Ω2,3 ≈ 0 because

there is no wave components that are coherent

with wave . However, here, in practice, the PMC ofthe input fiber will manifest by limiting the suppres�sion of errors Ω2,3. Furthermore, it is also necessary tofulfill the condition Linbin > Lb (usually when Lin ~ L).

How does the dichroism of the sensing�coil fiberwill influence Ω2,3?

(1) The dichroism suppresses the wave e0,y (Fig. 2)

and, at z > 1/α and the waves ( ) are no longergenerated from the beginning (end) of the coil fiber.

(2) If αz0 � 1, then wave e1, y (Fig. 2) will decaybefore passing to distance z0. As a result, in Table 1, forα = 10 dB/m and Lin = 10 m, errors Ω2,3 is muchsmaller than for Lin = 1 m. Thus, we have a radicalmechanism for suppressing errors Ω2,3 (as in [6]).

Unfortunately, the second mechanism has almostno influence on Ω1, which is mainly suppressed due tothe fact that deposits in it only remain from sectionswith length Lγ situated at distances less than 1/α fromthe beginning and end of the coil PZ fiber (2). The fur�ther suppression of Ω1 is only possible if the input fiberis also polarizing.

Consider now an FRI with a coil fiber and the mis�alignment of the waveguide axes of the Y junctionwithin 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 theinput fiber and without it, as well as with and withoutthe misalignment of the axes (four cases). Table 2 listsPE values for Lin = 1 m; values for Lin = 20 m areshown in brackets. Table 2 also presents PE values forthe PM fiber of the coil. Here, in order to identify the

E2+−

E2±

E1+ E1

–

jointed action of input fiber and coil fiber, we assumethat 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, butonly due to the PMC of the input fiber; for Lin = 1 m,they are determined by the PMC in the coil fiber andwith the misalignment of the axes (the latter could betreated as singular PMC centers at the beginning andthe end of the coil fiber, so it is possible to apply thescheme in Fig. 2). Thus, the depolarization in theinput PZ fiber and coil PZ fiber may radically reducethe errors Ω2,3 (if αz0 � 1), or to achieve the resultsderived in [6]. Note that here one may use Lin � L,contrary to FRI with PM�fibers (see above).

Jointed operation of depolarization and dichroismin the input and coil fibers could be considered to bepart of the general principle of PE suppression in FRIby depolarization and dichroism. Earlier [8], this prin�ciple manifests through the attenuation of polarizerrequirements due to depolarization in the coil fiber.

As for error Ω1, it is not suppressed to zero, even forLin = 20 m; only Ω1 now determines the whole PE.However, the suppression of this error by the input andthe dichroism of the coil fibers is still essential. Inaddition to this, we only considered a length L of 100m, although Ω1 ~ 1/L (2). Furthermore, we did nottake into account the dichroism of the waveguides ofthe Y junction; however, the PE is still small, even fornavigation�grade FOG.

In the case of errors in the ?? fiber of the coil, thedifference for Lin = 1 m and 20 m is small and all ofthese errors are due to PMC in the coil fiber and themisalignment of the axes; furthermore, the PMC ofthe 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–

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 FRIsensing coil may be very useful.

In our point of view, measurements of further PEsuppression are only reasonable in extremely accurateunitary measurements where the requirements of FRIcompactness are absent. In [4], FRI was described fordetecting the effects of general relativity using a sens�ing coil made of single�mode isotropic fiber with adiameter of several kilometers. In our case, the coilradius may be on the order of several meters or evensmaller.

Dichroism in the coil fiber may find practicalapplications based on convenient anisotropic fiberswith so�called small apertures [9] or in anisotropic Wfibers. The input PZ fiber could also be implementedbased on the W profile of the refractive index [10].Consider the prototype of the W fiber of the sensingcoil 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 isnot small (>50 mm), the spectral curves of these lossesgrow fairly rapidly, so it is quite realistic from our pointof view to achieve the above�mentioned values ofdichroism. 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 tocalculations).

REFERENCES

1. S.M. Kozel et al. , Opt. Spektrosk. 61 (6), 1295 (1986).2. W.K. Burns et al., J. Lightwave Technol. LT�1 (1), 98

(1983).3. A. M. Kurbatov, Report “Development of Fiber�Optic

Gyroscope” (OKB „Impul’s“ MAP, Arzamas, 1984) [inRussian].

4. I. A. Andronova and G. B. Malykin, Usp. Fiz. Nauk172 (8.), 849 (2002).

5. G. B. Malykin and V. I. Pozdnyakova, Opt. Spektrosk.95 (4), 646 (2003).

6. E. Jones and J. W. Parker, Electron. Lett. 22 (1), 54(1986).

7. E. C. Kintner, Opt. Lett. 6 (3), 154 (1981).8. W. Burns and R. Moeller, J. Lightwave Technol. LT�2

(4), 430 (1984).9. M. P. Varnham et al., Electron. Lett. 19 (7), 246 (1983).

10. M. Messerly et al., J. Lightwave Technol. 9 (7), 817(1991).

11. A. M. Kurbatov and R. A. Kurbatov, Pis’ma Zh. Tekh.Fiz. 36 (17), 23 (2010) [Tech. Phys. Lett 36, 789(2010)].

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)