1

Comparative theoretical study of polarising Panda-type and microstructured fibres for fibre-optic gyroscope

A. M. Kurbatov, R. A. Kurbatov

E-mail: romuald75@mail.ru (originally published in Optical and Quantum Electronics, September, 48, paper 439)

Abstract Comparative study is fulfilled for different polarising fibres: Panda fibres with match-clad and depressed-clad W-profiles, along with microstructured fibres. Comparison is made for their spectral width of single-polarisation windows taking into account bending, and for their splice losses with conventional fibres. A new type of match clad fibre is also proposed polarising even without bending. It is shown that the optimal is W-fibre Panda. At the same time, truly broadband polarising microstructured fibres demonstrate enormously large splice losses, whereas those of them with relatively low splice losses could not be wound into the coil of reasonable size due to large bend losses.

Keywords polarising fibre, W-profile, Panda fibre, microstructured fibre, bend loss, fibre optic gyroscope.

1. Introduction Polarisation non-reciprocity (PNR) is the fundamental limit of Sagnac fibre ring interferometer (FRI) accuracy (Kintner 1981). It could be strongly suppressed by sensing coil of highly birefringent (Hi-Bi) fibre together with broad-band optical source (Burns and Moeller 1983). Further suppression is available due to highly birefringent (Hi-Bi) optical element before or after FRI polariser (Jones and Parker 1986), which could be in the form of polarisation maintaining (PM) fibre section (Cordova et al. 1996). However, in Ref. (Carrara et al. 1987) it is mentioned that this suppression has its own essential limit, and in Ref. (Kurbatov 2011a) it is shown that this is due to high order polarisation mode coupling in coil fibre and to optical axes misalignments of spliced components. In this case, further dramatic PNR suppression could be reached by sensing coil of polarising (PZ) fibre together with the lightguide at the input of ring interferometer, when the following condition is fulfilled (Kurbatov 2011a)

𝛼𝐿𝑖𝑛𝐵𝑖𝑛 𝐵 ≫ 1⁄ (1) Here 𝛼 is the attenuation coefficient of rejected fundamental y-polarised mode (y-mode) intensity in PZ-fibre of the coil with birefringence 𝐵, while 𝐿𝑖𝑛 and 𝐵𝑖𝑛 are the length and birefringence of input lightguide. For 𝛼 ~ 30 dB/km, condition (1) means that 𝐿𝑖𝑛 value should be order of 𝐿 (length of the coil fibre), but for 𝛼 ~ 0.1-1.0 dB/m one may use 𝐿𝑖𝑛 ≪ 𝐿. Input lightguide may also be polarising with y-mode attenuation coefficient 𝛼𝑖𝑛. Note that term “fibre” is reserved for FOG coil fibre, while the term “lightguide” is used for input fibre, for their easy differing.

Basic parameter of PZ-fibre is the width of dichroism window, a spectral range, where the loss of fundamental x-polarised mode (x-mode) is below 0.1 dB, and y-mode loss is above 30 dB (rigid criterion of dichroism window). In the literature, х-mode loss is often set equal to 1 dB (soft criterion). Also, according to (1), the α-value at 1.55 μm is considered as one more dichroism criterion for coil fibre only.

Nowadays, FOG coils are usually wound of high-aperture match-clad (MC) fibre (Δn ~ 0.015). It cannot be a broadband polariser, but the low-aperture (Δn < 0.005) MC-fibre can be

2

(Varnham et al. 1983). The same is right for fibres having refractive index W-profile (Fig. 1), which becomes MC-profile at Δn- = 0. In this case, the value Δn+ could be done much smaller than 0.015.

Long polarising W-fibres (roughly speaking, of over 100 m length) are known with elliptical stress cladding (Wang et al. 2009; Jacobsen et al. 2013), having small Δn+ and Δn- (Fig. 1), and Panda fibre (Kurbatov et al. 2010 and 2013) with arbitrary Δn±. Shorter PZ-lightguides (less than 50 m) can be implemented at FRI input, they could be divided into bend-polarising MC-lightguides and W-lightguides which can be polarising even without bending. For the first case, bow-tie lightguides are known (Varnham et al. 1983, Hill et al. 2012), along with Panda lightguides (Okamoto 1984). As for W-lightguides, the ones are known with elliptical stress cladding (Simpson et al. 1983; Messerly et al. 1991, Wang et al. 2009; Jacobsen et al. 2013), along with Panda fibres (Kurbatov 1990, 2004, 2010, 2011b). Finally, present paper offers one more Panda-type MC-fibre which is polarising even without bending, where the reduced index stress applying parts (SAP) play the key role.

Also, microstructured (MS) fibres are known as promising ones for a lot of fields (Zheltikov 2000), including PZ-fibres applications (Kubota et al. 2004; Saitoh and Koshiba 2005; Chen and Shen 2007; Jian 2011; Lu 2013).

Below, three kinds of highly birefringent PZ-fibre are considered (straight and bent), having the following refractive index profiles: 1) low-aperture MC-profile; 2) W-profile; 3) MS-profile. For these kinds of fibres a comparative study is fulfilled, concerning the width of dichroism window and splice losses with conventional fibres (for example, SMF-28), or with the waveguides of integrated-optic chip (IOC), whose fields are approximated by Gaussian beams. For FOG coil fibre, splice loss should be doubled because it is spliced from both ends with two IOC waveguides.

Among the fibres with stress applying parts (SAP), present paper treats only Panda-type ones, which are simulated realistically enough, taking into account the smoothed and essentially asymmetric Panda-type refractive index profiles, along with their non-uniform material birefringence distribution within fibre cross section. This simulation is the essential development of that from Ref. (Kurbatov et al. 2013), and it is successfully tested experimentally (including numerous experimental data from literature). To the authors knowledge, no such realistic description has been published, and they believe that this is one of the reasons why PZ-fibres are still often considered as hardly available for manufacturing. Also, below presented consideration of bending properties of microstructured PZ-fibres is of primary importance for fibre gyroscopy which never deals with straight fibres. 2 Mathematical simulation procedures Rectangular configuration 1 at Fig. 2a illustrates the rectangular cross section of W-fibre with boron-doped SAP and absorbing complex-index layer (black ring). Configurations 2 and 3 at Fig. 2a illustrate the alternative geometries for absorbing (black) region. For light absorption, configurations 1 and 2 leads to the same result, while configuration 3 leads to the same result

Fig. 1 Refractive index W-profile, n1-3 are

refractive indices of the core, depressed cladding

and outer silica cladding, 2ρ and 2τ are diameters

of the core and depressed cladding.

3

as considerably more complicated PML-method (Berenger 1994). We assume the real part of absorbing region dielectric constant is equal to silica one, only correcting it as Re2𝑛 − Im2𝑛. The width of this region was chosen equal to 10 μm. The axes x and y are the fibre slow and fast optical axes, 𝐿𝑥,𝑦 are the lengths of simulation region. Fig. 2b illustrates the angle 𝛳 of SAP

orienting relative to bending plane for bent fibre (axes xbend and ybend are parallel and perpendicular to bending plane).

Fig. 2a W-fibre Panda cross section for simulation of mechanical stresses induced by boron-doped SAP, and

of the light absorption by different complex-index black regions (rectangular configurations 1-3).

Fig. 2b illustration of the angle 𝜭 of SAP orienting with respect to fibre bend plane (axes 𝒙𝒃𝒆𝒏𝒅 and 𝒚𝒃𝒆𝒏𝒅 are

parallel and perpendicular to bend plane), 𝑹 is the fibre bending radius.

In the similar manner, an absorption region and 𝛳-angle could be illustrated for MS-fibre.

Mechanical stresses are simulated by Goodier potential ψ(x,y) (Chu and Sammut 1984), which satisfies the equation

(𝜕2 𝜕𝑥2⁄ + 𝜕2 𝜕𝑦2⁄ )𝜓(𝑥, 𝑦) = [(1 + 𝑣) (1 − 𝑣)⁄ ]𝜅(𝑥, 𝑦)𝛥𝑇, Where 𝑣 is Poisson ratio, ΔТ is the difference of SAP melting temperature and fibre temperature, κ(x,y) is the thermal expansion coefficient distribution. Temperature, Young modulus Е and v are assumed to be the same everywhere. Stress components are calculated in the following form (Chu and Sammut 1984):

𝜎𝑥 =𝐸

(1+𝑣)(1−2𝑣)[(1 − 𝑣)

𝜕2

𝜕𝑥2 + 𝑣𝜕2

𝜕𝑦2] 𝜓, 𝜎𝑦 =𝐸

(1+𝑣)(1−2𝑣)[(1 − 𝑣)

𝜕2

𝜕𝑦2 + 𝑣𝜕2

𝜕𝑥2] 𝜓.

Goodier potential is represented in the form of Fourier decomposition:

𝜓(𝑥, 𝑦) = ∑ 𝜓𝑚,𝑛 cos[𝜋(𝑚 + 1 2⁄ ) 𝑥 𝐿𝑥⁄ ] cos[𝜋(𝑛 + 1 2⁄ ) 𝑦 𝐿𝑦⁄ ]𝑚,𝑛 .

Values σх and σy are zero at external boundary of layer 1 at Fig. 2, and the profiles for x- and y-polarised light could be written in the form (Okamoto et al. 1981)

𝑛𝑥(𝑥, 𝑦) = 𝑛0(𝑥, 𝑦) + 𝐶1𝜎𝑥(𝑥, 𝑦) + 𝐶2𝜎𝑦(𝑥, 𝑦),

𝑛𝑦(𝑥, 𝑦) = 𝑛0(𝑥, 𝑦) + 𝐶1𝜎𝑦(𝑥, 𝑦) + 𝐶2𝜎𝑥(𝑥, 𝑦).

Here 𝐶1,2 are elastooptic constants. Material birefringence field is defined as 𝐵(𝑥, 𝑦) =𝑛𝑥(𝑥, 𝑦) − 𝑛𝑦(𝑥, 𝑦), the difference Δ𝜅 for SAP and silica is chosen for desired 𝐵(0,0) value,

4

because for Panda fibres, 𝐵(0,0) is almost equal to modal birefringence 𝐵𝑚𝑜𝑑 = (𝛽𝑥 − 𝛽𝑦) 𝑘⁄

(Section 3), which determines the dichroism window width (here 𝛽𝑥,𝑦 are propagation

constants of х- and у-modes, 𝑘 is the vacuum wavenumber). This rough approach maintains the basic features of stress distributions yielded by others (Chu and Sammut 1984). Similar to this, one may also calculate the stress field in bow-tie and elliptical stress-clad fibres.

Panda fibres under consideration are weakly guiding, satisfying the scalar wave equation implemented individually for x- and y-modes (Snyder and Young 1978):

[𝜕2 𝜕𝑥2⁄ + 𝜕2 𝜕𝑦2⁄ − 𝑘2𝑛𝑥,𝑦2 (𝑥, 𝑦)]𝐸𝑥,𝑦(𝑥, 𝑦) = 𝛽𝑥,𝑦

2 𝐸𝑥,𝑦(𝑥, 𝑦), (2)

where 𝐸𝑥,𝑦 are electric fields of х- and у-modes. The solution of this equation is derived by

frequency domain finite difference method (FDFDM) (Stern 1988). For 125- and 80-μm diameter fibres, meshes 300×300 and 200×200 are enough. This is due to the fact that realistic profiles are smoothed.

For MS-fibres, FDFDM is implemented from Ref. (Zhu and Brown, 2002). Here the complex-index layer is added, as for configuration 1 at Fig. 2a, and for silica index a Sellmeier equation is used.

For all fibres, a supermode method is implemented, proposed in Ref. (Francois and Vassallo 1983), along with the procedure of necessary supermode searching (Kurbatov et al. 2013). For bent fibre profile 𝑛𝑏𝑒𝑛𝑡(𝑥, 𝑦), and for straight fibre profile 𝑛𝑠𝑡𝑟(𝑥, 𝑦), one may wright the following interrelation, starting from the result of Ref. (Heilblum and Harris 1975)

𝑛𝑏𝑒𝑛𝑡(𝑥, 𝑦) ≈ 𝑛𝑠𝑡𝑟(𝑥, 𝑦)[1 + 𝑥𝑏𝑒𝑛𝑑(𝑥, 𝑦) 𝑅⁄ ] = 𝑛𝑠𝑡𝑟(𝑥, 𝑦)[1 + (𝑥 cos 𝜃 − 𝑦 sin 𝜃) 𝑅⁄ ]. During the coil winding, the angle 𝛳 is out of control, so their bend loss calculations are done for all 𝛳 from 00 to 900 with the step 2.50, and after that the averaged loss curve is calculated. This method of simulation was verified experimentally (Kurbatov et al. 2013). As for MS-fibres, only certain values of 𝛳 are considered, which fact is explained in Section 4 (however, 𝛳-averaged calculations could be done).

Finally, splice losses are calculated by the overlap integrals of mode fields. 3 Polarising Panda fibres for FOG coils In this section, comparative study is carried out of polarising fibres with low-aperture MC-profile and with W-profile. Consider two groups of such fibres having MFD = 8 μm (fibres MC-8 and W-8) and with MFD = 10 μm (fibres MC-10 and W-10), where MFD is fundamental mode field diameter.

MC-fibres are known as the basis for bend-type polarisers. However, in this case dichroism occurs even in the absence of bending, because the fundamental mode cutoff of practical finite clad fibre is also finite (Black and Bourbonnais 1986). Also, a pair of SAP with reduced index shifts the dichroism window towards smaller wavelengths. In W-fibres, the fundamental mode cutoff may be finite due to depressed clad only (even for infinite silica clad). In another cases, the depressed clad only is not enough, so cutoff is finite due to finite silica clad and reduced SAP index, similar to MC-fibres. Both situations occur for y- and x-modes of W-fibre from Ref. (Messerly et al. 1991), respectively.

Table 1 contains the parameters of fibres W-8, MC-8, W-10 and MC-10. Table 1

Geometrical parameters and calculated optical characteristics of fibres W-8, MC-8, W-10 and MC-10

5

Parameter MC-8 MC-10 W-8 W-10

Core diameter, 2ρ, μm 4.9 8.0 9.2 12.5 χ = τ/ρ (Fig. 2) - - 1.6 1.5 Material birefringence at fiber center B(0,0) 8×10-4 8×10-4 8×10-4 8×10-4

Δn+ 7.5×10-3 4.35×10-3 5×10-3 3.45×10-3

Δn- 0 0 9.0×10-3 9.0×10-3

Dichroism window (rigid criterion), μm 1.54-1.55 1.5-1.55 1.5-1.55 1.46-1.55 Dichroism window (soft criterion), μm 1.54-1.64 1.5-1.64 1.5-1.61 1.46-1.63 Attenuation of у-mode at 1.55 μm, dB/km 60 120 500 1000 Material loss in SAP, dB/km 0.44 0.6 0.09 0.28

Fig. 3а shows the spectral losses of x- and y-modes in straight 1000-m fibres MC-8 and W-8, Fig. 3b is the same for these fibres wound with diameter 100 mm.

Fig. 3а Spectral loss curves for straight fibres W-8 (black solid curves) and MC-8 (grey dashed curves), “y”

and “x” denote the graphs for y- and x-modes of corresponding fibres.

Fig. 3b The same as at Fig. 3(a) for fibres wound with 100-mm diameter.

Dichroism occurs in all cases, and the fibre W-8 has wider dichroism window than MC-8. What is more important, loss of y-mode at 1.55 μm in fibre W-8 is almost 10 times larger than loss in fibre MC-8, so only fibre W-8 satisfies the condition (1). This is due to sharper growth of loss curve of fibre W-8. Also, material loss in SAP for fibre W-8 is 4.5 times lower than for fibre MC-8. Loss in SAP was calculated taking into account their index reduced by 0.01 relative to silica. This decreases the loss in SAP by 3-5 times comparing to the case when SAP index is equal to silica one, as in Ref. (Tajima and Sasaki 1989).

Note, that in Ref. (Varnham et al. 1983) microbending losses of x- and y-modes are considered as a reason of dichroism window in straight fibres at long wavelengths. These losses also could be simulated by supermodes method taking into account the influence of SAP, as in Ref. (Kurbatov et al. 2011b). However, there is no data for microbends statistical properties.

Fig. 4 illustrates the spectral graphs of 𝐵𝑚𝑜𝑑 for fibres MC-8 and W-8. Up to 1.7 μm, they are almost the same as 𝐵(0,0), and for fibre W-8 this differing from 𝐵(0,0) is much less due to tight confinement of x- and y-modes within the core.

6

Consider the cutoff of x- and y-modes for fibre W-8. Fig. 5а shows the graphs of x- and y-modes effective indices.

Fig. 5а Spectral graphs of effective indices of x- and y-modes in fibre W-8.

Fig. 5b Spectral losses of x- and y-modes in fibre W-8 of length 1 m (dashed) and 1000 m (solid).

Their crossing points with silica index (it is set equal to 1.46) are the mathematical cutoffs of x- and y-modes. Do they mean the beginning of the growth of these modes losses (physical cutoff)? Arrows from Fig. 5а to Fig. 5b reveal that mathematical and physical cutoffs agree well for 1-m fibre, unlike the 1000-m one. Thus, mathematical cutoff which may be calculated in a simple manner (Kurbatov et al. 2011b) could not be considered for simulating the fibre for FOG coil.

Bend polarisers of anisotropic MC-fibres are always considered as low-aperture (Δn < 0.005, MFD > 10 μm). For this case, fibres MC-10 and W-10 were compared revealing the same advantages of fibre W-10, as those from comparing the fibres W-8 and MC-8 (see Table 1). Moreover, y-mode loss at 1.55 μm in fibre W-10 is even 2 times larger than in fibre W-8. However, material loss in SAP is increased to 0.3 dB/km, because the distance between SAP is the same as in W-8, while the core is larger. 4 Microstructured fibres for FOG coils MS-fibres have significant advantages over MC- and W-fibres for large number of situations, and often they have no alternatives (Zheltikov 2000). As for their application in FOG coils, their extremely high birefringence is mentioned in literature, indistinguishable for fibres with SAP. Consider the examples of three MS-fibres with fragments of cross section illustrated at Fig. 6.

Fig. 4 Dash-dotted horizontal line is the level of

material birefringence 𝑩(𝟎, 𝟎) at fibres center,

black solid curve and grey solid curve are the

spectral graphs of modal birefringence 𝑩𝒎𝒐𝒅 in

fibres W-8 и MC-8, respectively.

7

Fibre MS-I is the most known. The rest two MS-fibres, among other applications, are offered for FOG PZ-coils. However, in Ref. (Kubota et al. 2004; Saitoh and Koshiba 2005; Chen and Shen 2007) they are treated only being straight. Because of their extremely high birefringence, it is meant (although not stated explicitly) that dichroism window is also extremely wide. Let’s show that this is not always so. Table 2 presents the geometrical parameters and calculated optical characteristics of fibres considered here.

Table 2

Geometrical parameters and calculated optical characteristics of microstructured fibres

Parameter MC-I-1 MC-I-2 MC-II MC-III

Λ, μm 5.588 1.7 2.2 1.4 a/(2Λ) 0.35 0.5 0.5 0.48 b/(2Λ) 0.35 0.5 0.5 0.24 d1/(2Λ) 0.95 0.95 0.95 0.66 d2/(2Λ) 0.95 0.95 0.95 0.48 Splice loss with IOC (MFD = 8 μm), dB 1.65 8.5 11 12.5 Splice loss with IOC (MFD = 10 μm), dB 2.5 11.5 15 16 Dichroism window width (rigid criterion), nm 0 20 110 300 Dichroism window width (soft criterion), nm 0 60 160 > 400

4.1 Fibre MS-I For this fibre, birefringence is due to the pair of large air holes. The indices of x- and y-modes may be lower than that of the cladding of small holes (cutoff). In Ref. (Kubota et al. 2004) a first practical PZ-fibre is described with MS-I structure. Its x- and y-modes losses are 28 and 196 dB/km at 1.55 μm. However, this fibre structure is not optimal (Ju et al. 2006), where four different similar fibres are treated at 1.55 μm. Here we consider two kinds of fibre MS-I (MS-I-1 and MS-I-2, see Table 2), similar to fibres denoted as PCF-VI and PCF-VIII in Ref. (Ju et al. 2006).

Fig. 7 illustrates the spectral losses of x- and y-modes for 1000-m FOG coil of fibre MS-I-1, wound with 25-cm diameter. Only one 𝛳-angle orientation is considered for holes structure with respect to bend plane, because other orientations yield the same type of result. Clearly, dichroism window does not satisfy even the soft criterion. One reason is that the phase birefringence is only 2×10-4 at 1.55 μm, while its maximum (4.55×10-4) occurs at 2.3 μm, where this fibre acts as 1-m PZ-fibre, similar to fibre PCF-VI from Ref. (Ju et al. 2006). This two-fold reducing of birefringence is unavoidable for long fibres due to their operational wavelength shift towards smaller wavelengths comparing to short fibres (similar to Fig. 6(b)). Fig. 7a also illustrates the short-wavelength boundary 1 of bend loss, typical for MS-fibres (Nielsen et al. 2004), and long-wavelength boundary 2, typical to MC- and W-fibres. Oscillations in the region 1 are the resonances with modes of clad of small holes (Murao et al. 2009). Winding this fibre with 100-mm diameter leads to overlapping of boundaries 1 and 2, so the loss exceeds 100 dB/km at all wavelengths.

Fig. 6 Fragments of cross sections of fibres MS-I,

MS-II, and MS-III. Grey background is silica;

white regions are the air holes.

8

Fig. 7a Spectral losses of x- and y-modes of fibre MS-I-1, wound with 25-cm diameter (only for one value of angle 𝜭). Regions 1 and 2 are the short- and long-wavelength limits of bend losses. Fig. 7b Spectral graph of phase birefringence, determined as 𝑩𝒎𝒐𝒅 in section 2.

Situation is different for fibre MS-I-2. Even for winding with 20-mm (!) diameter its dichroism window parameters are almost the same as for straight fibre (Fig. 8a) due to larger size of small air holes.

Fig. 8a Spectral loss curves of x- and y-modes of fibre MS-I-2, straight (𝟐𝑹 = ∞, black solid lines) and winded with diameter 𝟐𝑹 = 20 mm at fixed 𝜭 (grey dashed lines) for mesh 500×500. Fig. 8b Spectral loss curves in straight fibre for meshes with 𝑵 = 𝟒𝟎𝟎 and 𝑵 = 𝟓𝟎𝟎. Fig. 8c Spectral graph of phase birefringence.

9

The worst case of holes structure orienting is considered relative to bend plane (upper inset at Fig. 8a). The lower inset informs that FDFDM mesh is used with 𝑁𝑥 = 𝑁𝑦 = 𝑁 = 500. Fig. 8b

illustrates the spectral loss curves in straight fibre for 𝑁 = 500 and 𝑁 = 400. In the literature, attempts are made to reduce the calculation error of such graphs below 1% (Guo et al. 2004). Fig. 8b reveals much larger difference of graphs, at least for х-mode. However, for dichroism window width, the difference is small due to sharp growth of loss curves, so there is no need in very large N. Fig. 8c illustrates the spectral dependence of phase birefringence. It is approximately 10 times larger than in all above considered fibres, but dichroism window is narrower than even in MC-fibres Panda. Also, for this MS-fibre splice loss is very large (see Table 2). 4.2 Fibres MS-II and MS-III These fibres are described in References (Saitoh and Koshiba 2005; Chen and Shen 2007) in the absence of bending. Here we’ll not present the graphs for bent fibres. Note instead that winding the fibre MS-II with 60-mm diameter does not influence the dichroism window, similar to Fig. 8a. This could be treated as extremely wide dichroism window, but splice loss is too large (Table 2). For fibre MS-III, calculation reveals that its extremely wide dichroism window is almost the same for winding diameters up to 20 mm, but with even larger splice loss (Table 2).

Another two extremely broad-band polarising MS-fibres are known (Jian et al. 2011; Lu et al. 2013), also with large splice loss. Thus, the wider the dichroism window in MS-fibre the larger are splice loss. On the contrary, for smaller splice loss (fibre MS-I-1), FOG coil with required size could not be wound. 5 Short polarising lightguides (~1 m) Short polarising lightguides (~1 m) could be used at FOG ring interferometer input, where they may act as polariser and depolariser (Kurbatov et al. 2011). Also, it effectively filters high-order modes, because their cutoffs are below 1.0 μm. Finally, such lightguide suppresses half of optical y-polarised intensity, which otherwise was suppressed by IOC PZ-waveguides, leading to undesired additional IOC heating.

Here we’ll compare 1-m length sections of MC- W- and MS-lightguides. For MC- and W-lightguides we assume that birefringence is 𝐵𝑖𝑛= 0.001, because their diameters are 125 μm instead of 80 μm for coil fibre, which simplifies the birefringence enlarging (Chu and Sammut 1984). We start from the point that all these lightguides should be polarising, both being straight or bent. 5.1 MC- and W-lightguides Anisotropic MC-lightguides are known as bend polarisers (Varnham et al. 1983). Here another kind of MC-polariser is considered with dichroism even without bending, due to the indices of x- and y-modes shifting below the silica index by depressed index SAP. Fig. 9 presents six loss graphs of х- and у-modes: 1) in straight lightguide (dashed lines); 2) in bent lightguide with 60-mm diameter when the graphs are 𝛳-averaged within 𝛳 = 0-300 (solid lines); 3) in bent lightguide with 95-mm diameter without any SAP orienting, so graphs are 𝛳-averaged within 𝛳 = 0-900 (dashed-dotted lines).

10

Lightguide parameters are listed in Table 3. Clearly, this is PZ-lightguide even being straight. For both bend diameters, x-mode loss is 0.1 dB/m. One may say that pair of depressed-index SAP form W-profile along x-axis leading to dichroism without bending. However, it is not so, because light tunnels basically along y-axis (Fig. 2a).

The advantage of this lightguide is that SAP are close to thin enough core, allowing enlarging the birefringence even above 0.001. However, x-mode is deformed by SAP, so the splice loss with SMF-28 fibre is 0.4 dB for double splice passing towards the coil and back. In Ref. (Kurbatov et al. 2011 and 2013), a polarising W-lightguide is treated theoretically and experimentally. Here we turn to some other W-profile (Table 3) for which Fig. 10 presents the same six loss graphs as at Fig. 9. One may see that W-lightguide characteristics are similar to those of MC-lightguide. However, х-mode of W-lightguide is almost circular, so splice loss with SMF-28 fibre are 0.07 dB for splice double passing. Here a slight х-mode field squeezing by SAP still occurs, but SAP could be surrounded by silica layer, or/and their index could be raised. This spreads х-mode between SAP, and according to calculations, allows decreasing of splice loss to 0.02 dB for splice double passing. Of course, the loss value 0.4 dB for MC-lightguide is not

Fig. 9 Spectral losses of y- and x-modes

of MC-lightguide. Dashed curves are

for straight fibre, solid curves are for

fibre wound with 60-mm diameter and

with SAP orienting within 𝜭 = 0-300,

dash-dotted curvesare for fibre

winded with diameter 95 mm without

SAP orienting.

Fig. 10 The same graphs as at

Fig. 9 for W-lightguide.

11

critical, but W-profile allows reducing the MFD at least up to 8 μm (Kurbatov et al. 2013) for polarising lightguides, which are still broadband, unlike the MC-fibres.

Table 3

Parameters of MC- and W-lightguides for mathematical simulation

Parameter MC-lightguide W-lightguide

Core diameter (μm) 8.4 9.5 χ = τ/ρ (Fig. 2) - 1.8 Δn+ 0.003 0.0031 Δn- 0 0.0033 Linear birefringence 0.001 0.001

5.2 MS-lightguides Basic advantage of MS-lightguides, according to the literature, is their extremely large birefringence, unavailable for fibre with SAP. However, this means small MFD and unavoidable large asymmetry of x-mode field, i.e. large splice loss. Thus, there is the reason to consider only the lightguide МS-1 (Fig. 6), similar to PCF-VI from Ref. (Ju 2006). In Ref. (Ju 2006) the optimal structure has the dichroism window width 0.0 and 103.5 nm, according to rigid and soft criteria, with 6-dB splice loss for double passing. 6 Conclusions Comparative study is fulfilled for different types of polarising fibres for fibre optic gyro (FOG) sensing coil and for input polarisation/modal filters for FOG ring interferometer. Three kinds of fibre are considered: Panda fibres with matched-clad (MC) and W-profile, along with microstructured (MS) fibres. It is shown that W-fibres Panda are the optimal for high-accuracy FOG, having broad-band dichroism and minimal splice losses with other fibres and IOC waveguides. Also, a new MC-profile fibre is proposed polarising even without bending. As for polarising MS-fibres, they may be extremely anisotropic. But they don’t demonstrate similar several times wider dichroism window. MS-fibres with widest dichroism window are extremely bend resistant, but have too large splice loss. This makes such fibres not to be so promising for FOG, at least with modern FOG element base. References Berenger J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J.

Computat. Phys. 114, 185–200 (1994). Black, R.J., Bourbonnais, R.: Core-mode cutoff for finite-cladding lightguides. IEE Proceedings.

133, 377-384 (1986). Burns, W.K., Chen, C.-L., Moeller, R.P.: Fiber-optic gyroscopes with broad-band sources. Journal

of Lightwave Technol. 1, 98-105 (1983). Carrara, S.L.A., Kim, B.Y., Shaw, H.J.: Bias drift reduction in polarization-maintaining fiber

gyroscope. Optics Letters. 12, 214-216 (1987). Chen D., Shen L.: Highly birefringent elliptical-hole photonic crystal fibres with double defect.

Journal of Lightwave Technol. 25, 2700-2705 (2007). Chu, P.L., Sammut, R.A.: Analytical method for calculation of stresses and material birefringence

in polarisation-maintaining optical fiber. Journal of Lightwave Technol. 2, 650-662 (1984).

12

Cordova, A., Patterson, R., Rahn, J., Lam, L., Rozelle, D.: Progress in navigation grade IFOG performance. Proc. of SPIE. 2837, 207-217 (1996).

Francois, P.L., Vassallo, C.: Finite cladding effects in W fibres: a new interpretation of leaky modes. Applied Optics. 22, 3109-3120 (1983).

Guo, S., Wu, F., Albin, S., Tai, H., Rogowski, R.: Loss and dispersion analysis of microstructured fibres by finite-difference method. Optics Express. 12, 3341-3352 (2004).

Heilblum, M., Harris, J. H.: Analysis of curved optical waveguides by conformal transformation. IEEE Journal of Quantum Electron. 11, 75-83 (1975).

Hill, M.D., Alvarez, J.M., Hart, T., Gillooly, A., Wooler, J.P.F.: HiBi single polarizing HB-Z ZingTM fibers for fiber optic gyroscopes. Inertial Sensors and Systems, Karlsruhe (2012).

Jacobsen, W., Mayfield, J., Fournier, P., Bolte, D., Elmaola, H., Wang, Ch.-H., Drenzek, H., Soufiane, A.: Single-polarization fiber. US Patent no. 8,369,672 B2 (2013).

Jian, L., Maojin, Y., Feng, X., Meiling, L.: A kind of single-polarisation single-mode photonic crystal fibre. Proc. SPIE. 8120, 81200Z-1-81200Z-7 (2011).

Jones, E., Parker, W.J.: Bias reduction by polarisation dispersion in the fibre-optic gyroscope. Electron. Lett. 22, 54-56 (1986).

Ju, J., Jin, W., Demokan, M.S.: Design of single-polarisation single-mode photonic crystal fibre at 1.30 and 1.55 μm. Journal of Lightwave Technol. 24, 825-830 (2006).

Kintner, E.C.: Polarization control in optical-fiber gyroscopes. Optics Letters. 6, 154-156 (1981). Kubota, H., Kawanishi, S., Koyanagi, S., Tanaka, M., Yamaguchi, S.: Absolutely single polarisation

photonic crystal fibre. IEEE Photon. Technol. Lett. 16, 182-184 (2004). Kurbatov, A.M.: Single mode single polarisation fibre for polarisation modal filter. RU Patent no.

2040493. filed 1990/04/09. Kurbatov, R.A.: Mathematical simulation of single-mode polarizing W-fibre (in Russian). PhD

thesis (2004). Kurbatov, A.M., Kurbatov, R.A.: New optical W-fibre Panda for fibre optic gyroscope sensitive

coil. Technical Physics Letters. 36, 789-791 (2010). Kurbatov, A.M., Kurbatov, R.A.: Suppression of polarisation errors in fibre-ring interferometer

by polarising fibres. Technical Physics Letters. 37, 397-400 (2011a). Kurbatov, A.M., Kurbatov, R.A.: Fibre polariser based on W-lightguide Panda. Technical Physics

Letters. 37, 626-629 (2011b). Kurbatov A.M., Kurbatov R.A.: Polarisation and modal filters based on W-fibres Panda for fibre-

optic gyroscopes and high-power fibre lasers. Optical Engineering. 52, 035006-1-035006-9 (2013).

Lu, D., Zhang, X., Chang, M., Wang, G., Pan, L., Zhuang, S.: Endlessly single-polarisation single-mode holey fibres with low confinement loss. Optics Letters. 38, 2915-2918 (2013).

Messerly, M.J., Onstott, J.R., Mikkelson, R.C.: A broad-band single polarisation optical fibre. Journal of Lightwave Technol. 9, 817-820 (1991).

Murao, T., Saitoh, K., Koshiba, M.: Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers. Optics Express. 17, 7615-7629 (2009).

Nielsen, M.D., Mortensen, N.A., Albertsen, M., Folkenberg, J.R., Bjarklev, A., Bonacinni, D.: Predicting macrobending loss for large-mode area photonic crystal fibres. Optics Express. 12, 1775-1779 (2004).

Okamoto, K., Hosaka, T., Edahiro, T.: Stress analysis of optical fibres by a finite element method. IEEE Journal of Quantum Electron. 17, 2123-2129 (1981).

Okamoto, K., Takada, K., Kawachi, M., Noda. J.: All-Panda-fibre gyroscope with long-term stability. Electron. Lett. 20, 429-430 (1984).

Saitoh, K., Koshiba, M.: Numerical modeling of photonic crystal fibres. Journal of Lightwave Technol. 23, 3580-3590 (2005).

13

Simpson, J.R., Stolen, R.H., Sears, F.M., Pleibel, W., MacChesney, J.B., Howard, R.E.: A single-polarisation fiber. Journal of Lightwave Technol. 1, 370-373 (1983).

Snyder, A.W., Young, W.R.: Modes of optical waveguides. Journal of Optical Society of Am. 68, 297-309 (1978).

Stern, M.S.: Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles. IEE Proceedings. 135, 56-63 (1988).

Tajima, K., Sasaki, Y.: Transmission loss of a 125-μm diameter PANDA fibre with circular stress-applying parts. Journal of Lightwave Technol. 7, 674-679 (1989).

Varnham, M.P., Payne, D.N., Birch, R.D., Tarbox, E.J.: Bend behaviour of polarising optical fibres. Electron. Lett. 19, 679-680 (1983).

Wang, C.-H., Jacobsen, B., Bolte, D., Fournier, P., Mayfield, J., Soufiane, A., Highly polarizing single-mode optical fiber for sensing applications. Proc. of SPIE. 7503, 75034F-1-75034F-4 (2009).

Zheltikov, A.M.: Holey fibres. Physics Uspekhi. 43, 1125–1136 (2000). Zhu, Z., Brown, T.G., Full-vectorial finite-difference analysis of microstructured optical fibres.

Optics Express. 10, 853-864 (2002).