This document discusses reducing polarisation nonreciprocity (PNR) in small Sagnac fibre ring interferometers (FRIs) used in fibre optic gyroscopes (FOGs). It presents an analytical model for how PNR scales exponentially with the length of the input polarising lightguide. Unlike previous models, this shows that by making the lightguide sufficiently long, the required polarisation extinction ratio can be made arbitrarily small, improving FOG accuracy for applications requiring high precision. It also describes a numerical simulation approach to model PNR in FRIs considering the specific properties and interactions of components like the polarised light, proton-exchanged waveguides, fibre coil, and splices between optical axes.

1. Eliminating polarisation nonreciprocity of small size Sagnac fibre ring
interferometer: a realistic study
A. M. Kurbatov, R.A. Kurbatov
Kuznetsov Research Institute of Applied Mechanics, Moscow,
1. INTRODUCTION
Polarisation nonreciprocity (PNR) [1, 2] is the fundamental accuracy limit of Sagnac fibre ring
interferometer (FRI) playing the principal role in high-grade fibre optic gyro (FOG) [3]. Fig. 1 sketches
FRI scheme of commercial small size FOG of highly birefringent (Hi-Bi) optical components.
At splices, optical axes of components are slightly rotated with respect to each other due to technology
imperfection (imperfect splices, one of PNR sources). Input lightguide and coil fibre may be polarising
(PZ). Channel waveguides of integrated optic chip (IOC) may be done by proton-exchanged (PE)
technology being PZ, having the intensity polarisation extinction ratio (PER) ε2
=10–5
–10–8
, and
possessing the extremely large birefringence order of 0.1 [4]. Polarisation mode coupling (PMC) in coil
fibre, input lightguide, and IOC waveguides also yields PNR, along with mutual interferences of spurious
waves from all these PMC kinds and with those from imperfect splices.
Perfect polariser (ε2
= 0) cancels PNR [1]. For more realistic polariser (ε2
> 0), large amplitude
PNR ~ ε1
was established in Ref. [2], along with smaller intensity PNR ~ ε2
. This amplitude PNR leads to
unrealistic required polariser PER value 120-160 dB for 0.01 deg/h accuracy at 1-km fibre coil. Ref. [5-
10] describe the evolution of PNR reducing by employing the Hi-Bi components and broad band light
with low degree of polarisation (DOP), none of which was treated in Ref. [2]. Nowadays, PNR
estimations according to Ref. [10-12] (Eq. (8)-(11) below) reveal that these solutions made it small to
prevent FOG from high grade applications. On the other hand, they reveal why only giant FOG [13] is
treated as the project for general relativity effects detecting.
In Ref. [10], PNR reducing up to 10-9
deg/h is mentioned for several meters size FRI and the coil
of PZ-fibre, even at the input lightguide PER artificially limited by 60-dB regardless to its length Lin. This
limitation was made because input lightguide is assumed to have random twists along its length as a
source of its PMC [14], and twisted fibre has two elliptically polarised eigenmodes, each of which, in
turn, has two linearly x- and y-polarised fields (major and minor) [15]. Below, minor field is treated in
natural manner, unlike the Ref. [10]. Another limitation of input lightguide PER is due to its PMC leading
to PER rigid limitation, the same for all Lin > 1/ξin. This is because only y-wave survives under dichroism
which is cross coupled from initial x-wave within the last input lightguide section of the length ~ 1/ξin.
Generally, polariser PER is determined by intensities ratio of output co-propagated y- and x-
waves (“intensity” PER). The above unrealistic values 120-160 are ascribed to input PZ-lightguide
exactly in this sense, while its PMC yield much smaller PER values (50-60 dB), which could not be
overcome by further enlarging of Lin. However, PNR is interferometric phenomenon for counter-
propagating waves passing the input lightguide twice, so in this case, there is no reason to postulate a
priori this rigidly limited “intensity” PER. Below it is shown that for FRI with PMC in FRI components
and imperfect splices, input lightguide dichroism influences PNR in the following manner:
 ~ exp 2 ,in inPNR L (1)
where ξin ≡ ξin,y – ξin,x, with ξin,x and ξin,y as the x- and y-waves losses within input lightguide. Unlike the
Ref. [10], only the coil of РМ-fibre is treated because of greater simplicity of its manufacturing
comparing to PZ-coil.
Fig. 1. Sagnac FRI scheme
of commercial FOG

2. The exponent from Eq. (1) is the analog of ε from Ref. [2], but unlike the fixed ε, it could be
made arbitrarily small just by enlarging of Lin to overcome PER values 120-160 dB associated with Ref.
[2], making it in the sense of “interferometric” PNR (Eq. (1)). This is close to conclusion of Ref. [1] for
PNR cancelling by perfect polariser, revealing one more fibre technology benefit in addition to detecting
of Sagnac effect itself by long-length small size fibre coil.
PNR cancelling by input PZ-lightguide of enough length reminds the idea of several input
polarisers sequence [16]. The necessity of spatial (modal) filtering between them is one of the reasons
made this idea impractical. At the same time, input PZ-lightguide may be such a filter due to low cutoffs
of its high-order modes. This filtering occurs along the lightguide in parallel to its PMC and dichroism.
2. GENERAL RELATIONSHIPS FOR PNR
Formula for PNR from Ref. [3, 11] may be rewritten as
     *
arg , , .CW CCW
t d E t E t       (2)
Here CW
E and CCW
E are electric fields of clockwise and counter-clockwise waves, sign “•” denotes the
scalar product. Using FRI Jones matrix M(λ,t) for CW-waves, one may write the following [3]:
         , , , , , ,
T TCW CW
x y x yE t E t M t e t e t          
         , , , , , ,
T TCCW CCW T
x y x yE t E t M t e t e t          
where ex,y are the fields of waves entering FRI. For small ψ, one may yield PNR in the form of rotation
rate (instead of phase difference ψ), ΩPNR = Ω1+Ω2+Ω3, where
   
   
   
1
1 3 1 31
1 3 1 3 3
0
, Im ,
.
, Re ,k kk
SF d t A t
t t SF
d t A t
  

  

 
 


   


 
(3)
Here SF ≡ 4πRL/(λ0c) is FRI scale factor (λ0 and c are the light mean wavelength and speed in vacuum),
 2 2 *
0 11 22 12 212 2Re ,A M M M M    2 2 *
1 11 22 12 212 2 Im ,A M M i M M  
* * * *
2 11 12 11 21 12 22 22 212 ,A M M M M M M M M       * * * *
3 11 12 11 21 22 12 22 212 ,A i M M M M i M M M M   
* *
0 ,x x y ye e e e   * *
1 ,x x y ye e e e    *
2 2Re ,x ye e   *
3 2Im .x ye e  (4)
Here the dependence on λ and t is skipped for compactness. One may see that PNR time evolution is
determined by that of light electric fields and by time dependence of FRI parameters.
2.1. One illustrative analytical result
Values of ImA1-3 from Eq. (3)-(4) could be rewritten as
 *
1 12Im Im ,A M M    * *
2 11 222Im Im ,A M M M M     * *
3 11 222Im Re ,A M M M M    (5)
where ΔM ≡ M12 – M21. This means that PNR is zero if ΔM = 0 (diagonal FRI Jones matrix) [3].
Consider one important consequence of ΔM structure for lowest order PMC model in input
lightguide. FRI Jones matrix is T
in inM F UF , where U is the Jones matrix of FRI part located right with
respect to input lightguide (Fig. 1). The latter is described by Jones matrix Fin (generalization of PM-fibre
Jones matrix from Ref. [12]) 𝐹𝑖𝑛 = (
𝑎 𝑖𝑘1
𝑖𝑘2 𝐺𝑎∗), where G ≡ exp(–ξinLin/2) is “interferometric” PER of input
lightguide (it is referred below as G-multiplier), values k1,2 describe PMC, limiting its “intensity” PER by
spurious y-waves of the form ey,1-2 = ik2-1ex,0. These y-waves generated by initial x-wave ex,0 passed the

3. input lightguide in one direction (ey,1) or in another (ey,2). FRI Jones matrix off-diagonal elements have the
form (first PMC order):
*
12 12 2 22 1 11,M GU iGa k U iak U   *
21 21 2 22 1 11,M GU iGa k U iak U   (6)
so for PNR determining value M , one has
 12 21 .M G U U   (7)
Eq. (7) contains G-multiplier, leading to Eq. (1), in spite of the fact that both of M12 and M21 contain
spurious y-waves limiting rigidly the input lightguide “intensity” PER. This is because y-waves iGa*
k2U22
and iak1U11 are the same for both M12 and M21, because they are generated by equal CW and CCW x-
waves passing the input lightguide both times in the same direction (i.e., under the same conditions),
unlike the coil fibre. This means no parasitic phase shifts due to their cross-interferences. Note that matrix
U is of maximally general form, including any PMC kind (not limited by below described concrete
numerical models), all axes misalignments at splices (not only small) and so on.
A few words should be written about the influence of the first 50×50 fibre coupler (Fig. 1). In
Ref. [11], expression is presented for such parasitic phase shift which could be rewritten as Δφ = s1|txy|2
.
Here |txy|2
is the intensity of cross-coupled waves 𝑒 𝑦,0
𝑐𝑤
and 𝑒 𝑦,0
𝑐𝑐𝑤
within the coupler. This phase shift does
not depend on any dichroism being, thus, very large. However, waves 𝑒 𝑦,0
𝑐𝑤
and 𝑒 𝑦,0
𝑐𝑐𝑤
are generated by parts
of initial waves 𝑒 𝑥,0
𝑐𝑤
and 𝑒 𝑥,0
𝑐𝑐𝑤
passed the FRI without perturbations along x-axis only. These x-waves are
equal to each other, i.e., they did not get parasitic phase shifts within FRI, and they pass the coupler in the
same direction. Thus, spurious waves 𝑒 𝑦,0
𝑐𝑤
and 𝑒 𝑦,0
𝑐𝑐𝑤
are equal to each other too (no PNR), yielding
parasitic phase shift only due to their interference with other spurious y-waves generated within FRI and
enforced to pass the input lightguide along its rejecting y-axis (G-multiplier).
For taking into account of higher order PMC induced values, it’s better to turn to numerical
model, because there is no guarantee that higher order spurious waves are also suppressed by G-
multiplier. Below, numerical simulations reveal that this is the case.
3. NUMERICAL SIMULATION OF PNR
PNR numerical simulation includes the concretization of light and of components Jones matrices
properties.
3.1. Light properties
Values X0-3(λ,t) from Eq. (4) may be treated as the light instantaneous spectral Stokes parameters. Being
averaged over infinite time, they become usual spectral Stokes parameters from Ref. [17], where they are
dependent on cyclic optical frequency 𝜔, instead of λ. Here, the worst cases are considered, of
specifically broad-band polarised light with s1-3 = 1 (normalized Stokes parameters [17]), so X0-3(λ,t) ≈
S(λ) (the light spectral density).
3.2. FRI Jones matrix
Jones matrix M for Eq. (2) has the form 2, 2 1 1, ,T T T T T
in in in out out in in inM F R PEW PEW R FR PEW PEW R F where F is the
coil fibre Jones matrix, PEWin and PEW1-2,out are Jones matrices for input and output PE-waveguides
(PEW), Rin and R1,2 are rotation matrices for input (in) and output (1, 2) splices (Fig. 1).
3.3. PE-waveguide and PZ-lightguide Jones matrices
Proton-exchange waveguide (PEW) Jones matrix may be represented as a product of N Jones matrices for
its sections of lengths ln (n = 1…N) with discrete PMC centre at the end of each section:
 
   
   1
1 , ,
,
, 1 ,
N
n n n n
n n n n n n n
a l a l
PEW
a l a l
   

     
  
  
    


4. where a(λ,t) ≡ exp[–iΔβPEW(λ)ln/2], εn ≡ exp(–ξPEWln/2), ΔβPEW(λ) ≈ 2πBPEW/λ is PEW modal birefringence,
ξPEW ≡ ξPEW,y – ξPEW,x, where ξPEW,x and ξPEW,y are the losses of x- and y-waves in PEW. It is assumed that ln
<< 2πλ0/BPEW. In Ref. [18], PER increasing of IOC is described from 60 to 80 dB by trapping of leaky y-
wave energy. Thus, loss ξPEW,y is chosen to yield ε = ε1×ε2×…εN = 10–6
by fitting Py/Px = 10–6
at PEW
output for input х- and у-waves of equal amplitudes in the absence of PMC in PEW; for PMC, non-
negative random numbers n are selected to provide Py/Px = 10–6
at IOC output, when only х-wave
enters IOC (at ε = 10–6
). In this case, residual 80 dB are due to PMC in IOC. Numbers αn are uncorrelated,
because PEW are made by microscopic processes, so it is reasonable to assume zero PMC correlation
length. These numbers are provided by random numbers generator.
PZ-lightguide Jones matrix is based on that for PM-fibre [14], divided into sections of
exponentially distributed random lengths ln (mean value is 25 mm) and with non-correlating random
twists Ɵn. Jones matrix elements for such fibre section of length l and with twist  are calculated from
PM-fibre elements of Ref. [14], replacing βx,y → βx,y + iξx,y/2, where βx,y are x- and y-waves propagation
constants, ξx,y are their losses, yielding the following:
   1
1,1 1 2 1 20.5 cos sin sin cos ,F i   
            1
1,2 2,1 3 exp ,F F i
     
   1
2,2 4 5 4 50.5 cos sin sin cos ,F i   
          
where the following values are introduced (in the order of appearance in calculations programming): Δβ =
βx – βy, ξ = ξy – ξx, 2ξ+ = ξy + ξx, B = Δβ2
– 0.25ξ2
+ 4Θ2
, H = (B2
+ Δβ2
ξ2
)1/4
, 2θ = arctan(ξΔβ/B), O1 =
Hcosθ, O2 = Hsinθ, C = cos(O1l/2), S = sin(O1l/2), A1 = O1 + Δβ, A2 = O2 + ξ/2, B1 = O1 – Δβ, B2 = O2 –
ξ/2, E = exp(–ξ+l/2), T1 = Eexp(O2l/2), T1 = E2
/T1, and
   1 1 2A C A S T B C B S T        ,    2 1 2A C A S T B C B S T        ,
   3 1 2 1 2S T T iC T T     ,
   4 1 2B C B S T A C A S T        ,    5 1 2.B C B S T A C A S T       
Lengths l and twists Θ are provided by random numbers generators. The whole lightguide Jones matrix is
the product of such short sections Jones matrices.
3.4. Integration over wavelengths
For integration in Eq. (2), light spectrum is discretized with step [14] δλ ≤ 2λ2
/(2BinLin + 2BPEWLPEW + BL),
where Bin, BPEW, B are birefringences of input lightguide, PEW and coil fibre, LPEW is PEW total length.
4. FRI WITH INPUT PZ-LIGHTGUIDE
Fig. 2 shows the graphs for σ(Ω1-3) (Bin = 10×10-4
). Coil basic parameters are also presented at Fig. 2.
Fig. 2. PNR
dependence on Lin.

5. Approximation by exp(–ξinLin/2) occurs for σ(Ω2,3) at any Lin value, and for σ(Ω1) at Lin > 1 m, while for
σ(Ω1) at Lin < 1 m, this approximation is exp(–ξinLin). This transition at Lin = 1 m occurs due to the fact
that for Lin > 1 m, σ(Ω1) value is determined, basically, by input lightguide PMC. However, PNR < 10-9
deg/h is reached only for Lin > 1 m, so one may write the following for all σ(Ω1-3):
   1 3 1 3 1 3 exp 2 .in ins L       (8)
Values Λ1-3 depend on all PMC and decoherence kinds, splices imperfections, light properties, PEW
dichroism, so, generally, they could be calculated only numerically. Thus, PNR still could be cancelled
(at Lin = 6 m, according to Fig. 2, for PNR = 10-9
deg/h). This complete enough consideration is in
accordance with Eq. (7) for low-order PMC, revealing that higher-order processes do not break this
simple result. Obviously, light with low DOP yields even better result.
Note that for graphs at Fig. 2, input lightguide PER is the same 61 dB for all Lin > 1/ξin (see
Introduction). Note also that ten times larger twists of input lightguide lead to its PER much more severe
limitation by the value 41 dB, enlarging, however, only σ(Ω1) value, remaining it smaller than σ(Ω2,3), so
total PNR is almost the same. Consequently, authors believe that this result, in principle, could be
observed in Ref. [19], for FOG with several-meters bend-type input PZ-lightguide (and with PER
limitation similar to treated here), but other imperfections (of electronic processing scheme, for instance)
prevented this.
6. CONCLUSIONS
It is shown that complete polarisation nonreciprocity cancelling of small size Sagnac fibre ring
interferometer (FRI) could be reached with polarising (PZ) optical lightguide of the lengths order of 5 m,
in spite of severe limitation of polarisation extinction ratio (PER) of this PZ-lightguide, measured as a
standard intensities ratio of co-propagating y- and x-polarised waves. Instead, the effective
“interferometric” PER, specific for Sagnac FRI, could be made arbitrarily large just by increasing the
input PZ-lightguide length, because PNR is purely interferometric phenomenon for counter-propagating
waves, passing the input PZ-lightguide twice in both directions.
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