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Comparative theoretical study of polarising Panda-type and microstructured fibres for fibre-optic gyroscope
A. M. Kurbatov, R. A. Kurbatov
E-mail: [email protected] (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)
πΌπΏπππ΅ππ π΅ β« 1β (1) Here πΌ is the attenuation coefficient of rejected fundamental y-polarised mode (y-mode) intensity in PZ-fibre of the coil with birefringence π΅, while πΏππ and π΅ππ are the length and birefringence of input lightguide. For πΌ ~ 30 dB/km, condition (1) means that πΏππ value should be order of πΏ (length of the coil fibre), but for πΌ ~ 0.1-1.0 dB/m one may use πΏππ βͺ πΏ. Input lightguide may also be polarising with y-mode attenuation coefficient πΌππ. Note that term βfibreβ is reserved for FOG coil fibre, while the term βlightguideβ 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, Ρ -mode loss is often set equal to 1 dB (soft criterion). Also, according to (1), the Ξ±-value at 1.55 ΞΌm is considered as one more dichroism criterion for coil fibre only.
Nowadays, FOG coils are usually wound of high-aperture match-clad (MC) fibre (Ξn ~ 0.015). It cannot be a broadband polariser, but the low-aperture (Ξn < 0.005) MC-fibre can be
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(Varnham et al. 1983). The same is right for fibres having refractive index W-profile (Fig. 1), which becomes MC-profile at Ξn- = 0. In this case, the value Ξn+ 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 Ξn+ and Ξn- (Fig. 1), and Panda fibre (Kurbatov et al. 2010 and 2013) with arbitrary ΞnΒ±. 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, 2Ο and 2Ο are diameters
of the core and depressed cladding.
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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 Re2π β Im2π. The width of this region was chosen equal to 10 ΞΌm. The axes x and y are the fibre slow and fast optical axes, πΏπ₯,π¦ are the lengths of simulation region. Fig. 2b illustrates the angle π³ 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 π of SAP orienting with respect to fibre bend plane (axes πππππ and πππππ are
parallel and perpendicular to bend plane), πΉ is the fibre bending radius.
In the similar manner, an absorption region and π³-angle could be illustrated for MS-fibre.
Mechanical stresses are simulated by Goodier potential Ο(x,y) (Chu and Sammut 1984), which satisfies the equation
(π2 ππ₯2β + π2 ππ¦2β )π(π₯, π¦) = [(1 + π£) (1 β π£)β ]π (π₯, π¦)π₯π, Where π£ is Poisson ratio, ΞΠ’ is the difference of SAP melting temperature and fibre temperature, ΞΊ(x,y) is the thermal expansion coefficient distribution. Temperature, Young modulus Π and v are assumed to be the same everywhere. Stress components are calculated in the following form (Chu and Sammut 1984):
ππ₯ =πΈ
(1+π£)(1β2π£)[(1 β π£)
π2
ππ₯2 + π£π2
ππ¦2] π, ππ¦ =πΈ
(1+π£)(1β2π£)[(1 β π£)
π2
ππ¦2 + π£π2
ππ₯2] π.
Goodier potential is represented in the form of Fourier decomposition:
π(π₯, π¦) = β ππ,π cos[π(π + 1 2β ) π₯ πΏπ₯β ] cos[π(π + 1 2β ) π¦ πΏπ¦β ]π,π .
Values ΟΡ and Οy 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)
ππ₯(π₯, π¦) = π0(π₯, π¦) + πΆ1ππ₯(π₯, π¦) + πΆ2ππ¦(π₯, π¦),
ππ¦(π₯, π¦) = π0(π₯, π¦) + πΆ1ππ¦(π₯, π¦) + πΆ2ππ₯(π₯, π¦).
Here πΆ1,2 are elastooptic constants. Material birefringence field is defined as π΅(π₯, π¦) =ππ₯(π₯, π¦) β ππ¦(π₯, π¦), the difference Ξπ for SAP and silica is chosen for desired π΅(0,0) value,
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because for Panda fibres, π΅(0,0) is almost equal to modal birefringence π΅πππ = (π½π₯ β π½π¦) πβ
(Section 3), which determines the dichroism window width (here π½π₯,π¦ are propagation
constants of Ρ - and Ρ-modes, π 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):
[π2 ππ₯2β + π2 ππ¦2β β π2ππ₯,π¦2 (π₯, π¦)]πΈπ₯,π¦(π₯, π¦) = π½π₯,π¦
2 πΈπ₯,π¦(π₯, π¦), (2)
where πΈπ₯,π¦ are electric fields of Ρ - and Ρ-modes. The solution of this equation is derived by
frequency domain finite difference method (FDFDM) (Stern 1988). For 125- and 80-ΞΌm diameter fibres, meshes 300Γ300 and 200Γ200 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 πππππ‘(π₯, π¦), and for straight fibre profile ππ π‘π(π₯, π¦), one may wright the following interrelation, starting from the result of Ref. (Heilblum and Harris 1975)
πππππ‘(π₯, π¦) β ππ π‘π(π₯, π¦)[1 + π₯ππππ(π₯, π¦) π β ] = ππ π‘π(π₯, π¦)[1 + (π₯ cos π β π¦ sin π) π β ]. During the coil winding, the angle π³ is out of control, so their bend loss calculations are done for all π³ 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 π³ are considered, which fact is explained in Section 4 (however, π³-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 ΞΌm (fibres MC-8 and W-8) and with MFD = 10 ΞΌm (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
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Parameter MC-8 MC-10 W-8 W-10
Core diameter, 2Ο, ΞΌm 4.9 8.0 9.2 12.5 Ο = Ο/Ο (Fig. 2) - - 1.6 1.5 Material birefringence at fiber center B(0,0) 8Γ10-4 8Γ10-4 8Γ10-4 8Γ10-4
Ξn+ 7.5Γ10-3 4.35Γ10-3 5Γ10-3 3.45Γ10-3
Ξn- 0 0 9.0Γ10-3 9.0Γ10-3
Dichroism window (rigid criterion), ΞΌm 1.54-1.55 1.5-1.55 1.5-1.55 1.46-1.55 Dichroism window (soft criterion), ΞΌm 1.54-1.64 1.5-1.64 1.5-1.61 1.46-1.63 Attenuation of Ρ-mode at 1.55 ΞΌm, dB/km 60 120 500 1000 Material loss in SAP, dB/km 0.44 0.6 0.09 0.28
Fig. 3Π° 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. 3Π° Spectral loss curves for straight fibres W-8 (black solid curves) and MC-8 (grey dashed curves), βyβ
and βxβ 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 ΞΌm 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 π΅πππ for fibres MC-8 and W-8. Up to 1.7 ΞΌm, they are almost the same as π΅(0,0), and for fibre W-8 this differing from π΅(0,0) is much less due to tight confinement of x- and y-modes within the core.
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Consider the cutoff of x- and y-modes for fibre W-8. Fig. 5Π° shows the graphs of x- and y-modes effective indices.
Fig. 5Π° 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. 5Π° 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 (Ξn < 0.005, MFD > 10 ΞΌm). 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 ΞΌm 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 π©(π, π) at fibres center,
black solid curve and grey solid curve are the
spectral graphs of modal birefringence π©πππ in
fibres W-8 ΠΈ MC-8, respectively.
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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. Letβs 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
Ξ, ΞΌm 5.588 1.7 2.2 1.4 a/(2Ξ) 0.35 0.5 0.5 0.48 b/(2Ξ) 0.35 0.5 0.5 0.24 d1/(2Ξ) 0.95 0.95 0.95 0.66 d2/(2Ξ) 0.95 0.95 0.95 0.48 Splice loss with IOC (MFD = 8 ΞΌm), dB 1.65 8.5 11 12.5 Splice loss with IOC (MFD = 10 ΞΌm), 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 ΞΌm. However, this fibre structure is not optimal (Ju et al. 2006), where four different similar fibres are treated at 1.55 ΞΌm. 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 π³-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 2Γ10-4 at 1.55 ΞΌm, while its maximum (4.55Γ10-4) occurs at 2.3 ΞΌm, 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.
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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 π). Regions 1 and 2 are the short- and long-wavelength limits of bend losses. Fig. 7b Spectral graph of phase birefringence, determined as π©πππ 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 (ππΉ = β, black solid lines) and winded with diameter ππΉ = 20 mm at fixed π (grey dashed lines) for mesh 500Γ500. Fig. 8b Spectral loss curves in straight fibre for meshes with π΅ = πππ and π΅ = πππ. Fig. 8c Spectral graph of phase birefringence.
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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 ππ₯ = ππ¦ = π = 500. Fig. 8b
illustrates the spectral loss curves in straight fibre for π = 500 and π = 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 Ρ -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 weβll 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 ΞΌm. 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 weβll compare 1-m length sections of MC- W- and MS-lightguides. For MC- and W-lightguides we assume that birefringence is π΅ππ= 0.001, because their diameters are 125 ΞΌm instead of 80 ΞΌm 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 Ρ - and Ρ-modes: 1) in straight lightguide (dashed lines); 2) in bent lightguide with 60-mm diameter when the graphs are π³-averaged within π³ = 0-300 (solid lines); 3) in bent lightguide with 95-mm diameter without any SAP orienting, so graphs are π³-averaged within π³ = 0-900 (dashed-dotted lines).
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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, Ρ -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 Ρ -mode field squeezing by SAP still occurs, but SAP could be surrounded by silica layer, or/and their index could be raised. This spreads Ρ -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 π = 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.
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critical, but W-profile allows reducing the MFD at least up to 8 ΞΌm (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 (ΞΌm) 8.4 9.5 Ο = Ο/Ο (Fig. 2) - 1.8 Ξn+ 0.003 0.0031 Ξn- 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 ΠS-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 donβt 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, 185β200 (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).
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