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Design of high birefringence stress-induced polarization-maintaining fiber based on utilizing geometrical birefringence
Optical Fiber Technology 53 (2019) 102065
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Design of high birefringence stress-induced polarization-maintaining fiber based on utilizing geometrical birefringence
T
⁎
Haoyu Lia, , Xuyou Lia, Yue Zhanga, Pan Liub, Hanrui Yangc a
College of Automation, Harbin Engineering University, Harbin 150001, China Beijing Institute of Control and Electronic Technology, Beijing 100032, China c School of Automation Engineering, Northeast Electric Power University, Jilin 132012, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Polarization-maintaining fiber Geometrical birefringence Fiber design Structural optimization
Considering the utilizing of geometrical birefringence, we propose a new type of stress-induced polarizationmaintaining fiber (PMF) with a ‘leaf-shaped’ core —a structure found by combining the merits of elliptical and rhombic core shapes. The finite element method (FEM) is used to compare and analyze the contribution of the elliptical, rhombic, ‘star-line shaped’ and ‘leaf-shaped’ core on geometrical birefringence. The results show that this kind of PMF improves the polarization-maintaining performance of the fiber significantly, and its birefringence value can reach 7.692 × 10−4 in Panda-Type PMF and 7.732 × 10−4 in Bow-Tie PMF which are doubled compare with the traditional Panda-Type PMF. Analysis of the mode field performance also indicates that the designed fiber is suitable for the sensing field. At the same time, this fine-diameter PMF conforms to the trend of miniaturization of current used fiber-optic sensors and can be applied to fiber optic gyroscopes.
1. Introduction High birefringence fibers have been widely used in many applications benefitting from their polarization-maintaining property. These fibers are capable of maintaining linear polarization along the birefringence axis over the entire length [1–3]. Applying two axially asymmetric stresses to the core of the fiber causes difference of the propagation constant between the fast and slow axis of the fiber at the same time increases the value of the birefringence. When a beam of polarized light emits along the slow axis of the fiber, the light is forced to propagate at a lower velocity than if it has been launched along the fast axis [4,5]. The modal birefringence can be improved in two ways: the first method is creating asymmetric core geometry in accordance with geometrical birefringence. Geometrical birefringence is essentially the anisotropy of the geometry of the dielectric material resulting in the dielectric constant of the material and the anisotropy of the permeability, causing the anisotropy of the refractive index of the material [6–9]. The other method is inducing stress birefringence, that is, adding symmetrically stress-applying zone (SAP) with a high thermal expansion coefficient on both sides of the core, or adding SAP around the elliptical core. The use of PCF as a representative of applied geometrical birefringence has received much attention in recent years. It is also easy to obtain high birefringence PCFs by designing the fiber structure [10]. ⁎
Researchers have designed many kinds of air hole shapes of PCF with high birefringence such as pentagons [11], rectangles [12,13], rhombus [14], and elliptical core shapes [15]. Unfortunately, PCFs are expensive to manufacture and have higher loss than the conventional PMFs. The refractive index and stress birefringence in the orthogonal direction of PMF depend on the thermal stress caused by the SAP. Therefore, the PMF utilizes the stress distribution to obtain the high birefringence magnitude of the core region. These fibers can be structurally optimized to achieve better polarization-maintaining performance to meet the process requirements [16]. Stress-induced PMF is still widely used in the field of optical fiber communication and optical fiber sensing [17]. However, such fibers can still increase their birefringence values by introducing geometrical birefringence [18]. As a way to increase the value of birefringence, the study of geometrical birefringence is not very comprehensive by now because the relevant research on geometrical birefringence mostly focuses on the elliptical core [19,20], furthermore, there is no relevant research on the influence of other special shapes on birefringence. In this paper, we discuss the influence of several special shape cores on the birefringence value. And we design a new ‘leaf-shaped’ core stress-induced PMF with high birefringence. The birefringence analysis and calculation method of stress-induced PMF is given by thermal stress analysis theory and fiber polarization theory. Then the birefringence of the circular and elliptical core fiber is studied numerically as the
Corresponding author. E-mail address: [email protected] (H. Li).
https://doi.org/10.1016/j.yofte.2019.102065 Received 14 May 2019; Received in revised form 16 August 2019; Accepted 21 October 2019 1068-5200/ © 2019 Elsevier Inc. All rights reserved.
Optical Fiber Technology 53 (2019) 102065
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comparison of subsequent part. Considering the effect of core curvature on birefringence, we study the birefringence values of the elliptical, rhombus and ‘star-line’ shaped cores. By combining the advantages of elliptical and rhombic shapes, we propose a ‘leaf-shaped’ core PMF and evaluate the birefringence performance. From the point of fabrication, we optimize the structure of this fiber and obtaine the birefringence value of 7.692 × 10−4 in Panda-Type PMF and 7.732 × 10−4 in Bow-Tie PMF. Finally, we study the mode field performance of this fiber and briefly explain the fabrication method. 2. Birefringence theory and analysis method As we all know, the formation of geometrical birefringence is due to the asymmetry of the core geometry. In order to improve this kind of birefringence, the influence of the core geometry on the magnitude of birefringence will be discussed in this paper. FEM has been widely used as an accurate and convenient method to analyze the birefringence characteristics of PMFs [21]. Especially in the analysis of arbitrary cross-section waveguides, nonlinear waveguides and anisotropic waveguides, this method is more suitable for analyzing the electric field and magnetic field distribution [22]. Due to the diversity and complexity of the waveguide problem, most problems are solved by the full vector FEM. The birefringence performance is extremely important for studying the polarization properties of the fibers. High values of birefringence provide superior polarization retention. Modal birefringence is related to many factors such as temperature, core ellipticity, core-cladding refractive index difference, thermal expansion coefficient and wavelength of light. The FEM analysis of the birefringence for stress-induced PMF can be divided into two steps: firstly, if the composite glass fibers are stretched at the high temperature of the preform, thermal stress is generated once the fibers are cooled to the temperature at which the glass solidifies [23], then the thermal stress analysis is used to find the direct stress of the core in both the x and y directions [24], after that the refractive index of the x and y directions is calculated by the photoelastic effect expression [25,26]:
Fig. 1. Cross-section of circular core Panda-Type PMF.
B=
3.1. Circular core PMF verification In order to verify the correctness of our model, we first calculate the birefringence value of the circular core PMF with 125 μm outer diameter to compare with the value in Ref. [16]. The schematic cross section of the typical circular core PMF is shown in Fig. 1, where the radius of the fiber cross section is W which is fixed at 62.5 μm and the gap between the core and SAP is d. The radius of the core and the SAPs are a and r respectively. The two B2O3-doped-silica SAPs are symmetrically placed beside a GeO2-doped-silica core [29]. The refractive index of the SAPs and core are 1.436 and 1.474 respectively. All simulations in this paper perform at the wavelength of 1550 nm. The thermal expansion coefficient α of the above doped material can be obtained by the expression shown below:
(1)
α = (1 − m) α 0 + mα1
(6)
where m is the mole percentage of the doped material. The cladding is made of silica. The used material parameters for the analysis model including Young’s modulus (E), Poisson radio (v), the density (ρ), the thermal expansion coefficient (α), the stress-optic coefficient (C1 and C2), the drawing temperature and operating temperature which are listed in Table 1. The above materials parameters applied have proven to be applicable to high birefringence fibers in [30]. The desired refractive index can be obtained by changing the doping percentage according to the hybrid Sellmeier equation. In order to compare the birefringence value with Ref. [16], we fix d = 6.6 μm and sweep the other two parameters r (from 7 μm to 17 μm with 1 μm spacing) and a to calculate the modal birefringence. As can be seen in Fig. 2, compared to Ref. [16], the birefringence magnitude changes in the same trend but has small deviations in value. The reason for the small deviation is that in the Ref. [16], while radius of the core and SAPs are changing, the gap between the SAPs and the core also varies. It can be seen that as the area of the SAPs increase, the amount of birefringence also increases. This conclusion is also the same as that obtained in Ref. [16]. We can conclude that our model can be used for further analysis of stress-induced PMF.
(2)
(3)
where β is the propagation constant of the fiber and ω is the angular frequency. The eigenvalue equation for the electric field can be obtained from the Helmholtz equation:
∇ × (n−2∇ × E ) − k 02 E = 0
(5)
3. Simulation and discussion
The modal birefringence is defined as the sum of geometrical and stress birefringence. Since the dimension in the z-direction of the fiber is much larger than the size of the cross-section, the load can be seen to an act on the cross-section but does not change with the length of the fiber, so the refractive index in the z direction is regarded as a constant. The electric field of the wave has the form:
E (x , y, z , t ) = E (x , y ) exp[j (ωt − βz )]
x = neff − neffy
x and are effective where βx and βy are the propagation constants; neff mode refractive indices in the x-axis and y-axis directions respectively. Thus the mode birefringence of the PMF is the differential value in the effective refractive index of the two refractive index principal axes of the core layer.
where C1 and C2 are respectively first and second stress-photoelastic coefficients; σx and σy are the direct stresses in the x-axis and y-axis directions; n0 is the refractive index of the stress-free material. Anisotropic changes in refractive index will result in higher values of birefringence due to stress optical effects in the fiber. Stress birefringence is defined as
Bs = n y − n x = (C2 − C1 )(σy −σx )
k0
neffy
n x = n 0 − (C1 σx +C2 σy ) n y = n 0 − (C2 σx +C1σy )
βx − βy
(4)
where k0 is the number of waves propagating in vacuum. Using the analysis method of wave optics to calculate the effective refractive index of the fiber [27,28], then the modal birefringence value of the stress-type PMF is calculated by: 2
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Table 1 Elastic Parameters Used for Modeling. Parameters
Core
Thermal expansion coefficient α (1/℃) Young’s modulus E (Pa)
5.4 × 10
Poisson ratio v Density ρ (kg / m3 ) First stress optical coefficient C1 (1/Pa) Second stress optical coefficient C2 (1/Pa) Drawing temperature (℃) Operating temperature (℃)
Cladding
-7
7.83 × 1010 0.186
5.4 × 10
-7
7.25 × 1010 0.186 2203
Stress-applying rod
3.38 × 10
-6
2.86 × 1010 0.265
− 6.9 × 10−13 − 4.19 × 10−12
Fig. 4. Colormap of modal birefringence as function of e and a (μm) for elliptical core fiber.
1320 20
optical fibers. Then we sweep the other two parameters a (from 1.5 μm to 3.5 μm with 0.2 μm spacing) and e (from 0.4 to 1.4 with 0.1 spacing) to analysis modal character in order to calculate the birefringence of the elliptical core fiber. Obviously, the elliptical core PMF can be changed as common circular core Panda-Type PMF with e = 1. Fig. 4 shows a colormap of the modal birefringence numerical value of all combinations of e and a. One can see that the birefringence values for the region of 0.4 < e < 1 are manifest larger than the value for the region of e greater than 1 and the birefringence value will increase with the elliptic rate e decreases. The birefringence magnitude range of an elliptical core Panda-Type PMF under selected material and structural parameters of e = 0.4 ~ 0.6 is4.805 × 10−4 ~ 6.611 × 10−4 . Fig. 5 shows the geometrical birefringence curve of an elliptical core fiber. From Fig. 5 we can see that as the core ellipticity e and the core short axis a increase, the geometrical birefringence shows a significant downward trend. This is because as e decreases, the geometric asymmetry of the core increases, resulting in an increase in the magnitude of the geometrical birefringence. But when e and a are excessively small, the area of the core will also decrease, so the appropriate value should be chosen to fabricate the fiber. In order to further explore the modal birefringence properties of the elliptical core PMF, we fix a = 1.9 μm, e = 0.5 and sweep the other two parameters d (from 3 μm to 6 μm with 0.5 μm spacing) and r (from 10 μm to 14 μm with 0.5 μm spacing) to calculate the birefringence value which is shown in Fig. 6. After further increasing the area of the SAPs and bringing the SAPs closer to the core, the elliptical core PMF obtains a modal birefringence value having a maximum value of 7.187 × 10−4 . This means that a larger area of the SAP will result in a significant increase in the amount of birefringence when it is close to the core.
Fig. 2. Modal birefringence value of the circular core PMF as a function of r (μm).
3.2. B. Elliptical core PMF The schematic cross section of the elliptical core PMF is shown in Fig. 3, where the radius of the fiber cross section is W which is fixed at 40 μm for the design of fine-diameter fiber and the gap between the elliptical core and SAP is d. The semi-minor, semi-major axis of the elliptical core are a and b respectively, the elliptic rate is defined as e = a/ b . The radius of the SAP is r. Other parameters are the same as the previous section. In order to determine the structure parameters of elliptical core PMF which can supply high birefringence, we fix d = 5 μm, r = 12 μm in this model which consider the technical requirements for the fabrication of
Fig. 5. Geometrical birefringence of elliptical core PMF versus e from 0.4 to 0.8 as a function of a (μm).
Fig. 3. Cross-section of elliptical core Panda-Type PMF. 3
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radius r of the stress region. As can be seen from the Fig. 9, the maximum birefringence of the diamond core can reach 7.089 × 10−4 . The amount of birefringence increases significantly, which is consistent with previous conclusions. Compared to the elliptical core PMF, the rhombus core can only achieve high birefringence values over a relatively several combination of values. However, as a fiber shape that relies on geometrical-induced polarization, rhombus core is still a desirable shape. 3.4. D. ‘leaf-shaped’ core PMF We decide to increase the geometrical asymmetry of the core in the y-axis direction and preserve the convex function curve in the x-axis direction, that is, the ‘leaf-shaped’ core Panda-Type PMF with a crosssection as shown in Fig. 10. The structure is obtained by filleting the two vertices of the rhombus in the x-axis direction. The semi-minor, semi-major axis of the core are defined as a and b respectively, the semiaxis rate here also defined as e = a/ b . The fillet radius is defined as t. In order to facilitate comparison with the elliptical core, we fix t = 5 μm. The material used in the core, cladding and stress-applying rods are consistent with the elliptical core PMF, the same as the gap size and the radius of SAP. Using the same research method, we sweep the two parameters a (from 1.5 μm to 2.5 μm with 0.1 μm spacing) and e (from 0.4 to 0.55 with 0.05 spacing), because we estimate that the fiber will get high birefringence performance from the previous results in this range of a and e. Fig. 11(a) shows the von Mises stress of the fiber obtained by thermal stress analysis, and (b) shows the stress birefringence distribution of the fiber. The direct stress of the fiber in both the x-axis and the y-axis can be obtained in Fig. 12, which will be used to calculate the effective mode index of the fiber and further analyze the birefringence performance. Fig. 13 shows the birefringence colormap calculated by sweeping two parameters e and a. It can be seen that the ‘leaf-shaped’ core can provide stably a high birefringence value in the range of 0.4 ⩽ e ⩽ 0.55 and 1.5 ⩽ a ⩽ 2.2 , which can only be in the ellipse when e and a are particularly small. It is obtained that the ‘leaf-shaped’ core can ensure a high birefringence while relatively increasing the core area, thanks to the geometrical birefringence provided by this particular asymmetric shape. The performance can also provide tolerance for the fabrication of optical fibers. One can see from the comparison of Fig. 14, the birefringence magnitude of the ‘leaf-shaped’ core is significantly improved compared to the elliptical and rhombic cores at the region of 1.7 < a. We can take the core structure of e = 0.5, a = 1.9 μm, the value of modal birefringence can achieve the value of 6.3904 × 10−4 at the same time the value of the geometrical birefringence is
Fig. 6. Colormap of modal birefringence as function of r (μm) and d (μm) for elliptical core fiber.
3.3. C. Rhombic and Star-line shaped core PMF Considering the effect of core curvature on birefringence, we study the birefringence properties of rhombic core fiber and star-line shaped core fiber, i,e the closed curve corresponding to the straight line edge, the concave function and the convex function are respectively mirrored twice. Fig. 7 shows the schematic cross section of the rhombic core and ‘star-line’ shaped core PMF, where the cladding radius, the gap between the elliptical core and stress-applying rods, the radius of the stress-applying rods are the same as the elliptical core PMF. For the purpose of comparison, the semi-minor, semi-major axis of the two cores are a and b respectively, the semi-axis rate here also defined as e = a/ b . The modal birefringence of the three shapes of core fiber are shown in Fig. 8. Compared to the ‘star-line’ shaped core, the elliptical and rhombic core can maintain a high amount of birefringence at the region of 0.4 ⩽ e ⩽ 0.7, which proves that these two shapes are capable of increasing geometrical birefringence. This means that the concave side of the ‘star-line’ shaped is not suitable for the core of a high birefringence fiber. Consistent with the conclusions described above, as the core asymmetry increases, the magnitude of birefringence also increases significantly. In order to design a higher birefringence core shape, it is necessary to retain the circular arc in the × direction similar to an elliptical shape and the sharp angle in the y-axis direction similar to a rhombic shape. Similar to the analysis process of the elliptical core PMF, we further study the maximum amount of modal birefringence that the rhombus core PMF can obtain after shortening the gap d and increasing the
Fig. 7. Cross-section of (a) rhombic core fiber and (b) ‘star-line’ shaped core fiber. 4
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Fig. 9. Colormap of modal birefringence as function of r (μm) and d (μm) for rhombus core fiber.
Fig. 10. Cross section of ‘leaf-shaped’ core fiber.
Fig. 11. (a) von Mises stress and (b) stress-induced birefringence distributions.
core fiber, we fix r = 12 μm and sweep the core fillet radius t (from 3 μm to 6 μm) to increase the geometrical birefringence while scanning the gap between the core and SAP d (from 2.5 μm to 6 μm) to change the stress birefringence. Fig. 15 shows the birefringence colormap calculated by sweeping the two parameters t and d. It can be seen that the magnitude of the modal birefringence increases as the distance between the stress-acting regions and the core decrease. This value can also be increased again by changing the fillet radius to provide a higher geometrical birefringence magnitude. In order to satisfy the manufacturing process, we fix d at 3 μm and study the relationship between the birefringence value and the fillet radius t (from 3 μm to 6 μm) when the SAP radius r (from 11 μm to 14 μm) of the stress region changes. As can be seen from Fig. 16, the increase of the radius of SAP provides greater stress birefringence resulting in improvement of modal birefringence. As the radius of SAP
Fig. 8. Modal birefringence contrast of ellipse, rhombus and ‘star-line’ versus (a) e = 0.4, (b) e = 0.5, (c) e = 0.6, (d) e = 0.7 as a function of a (μm) (with 0.2 μm spacing).
1.9727 × 10−4 ,which is 3.447 × 10−5 higher than that provided by the elliptical core. 3.5. E. Structural optimization of ‘leaf-shaped’ core fiber To further optimize the birefringence properties of the ‘leaf-shaped’ 5
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H. Li, et al.
Fig. 12. Direct stress in the (a) x directions and (b) y directions.
Fig. 13. Colormap of modal birefringence as function of e and a for leaf-shaped core fiber.
increases, the magnitude of the birefringence also increases. When t varies between 3 μm-4.7 μm, the maximum value of birefringence is 7.6921 × 10−4 , of which the geometrical birefringence is 3.2744 × 10−4 . While at the range of 4.7 μm < t < 5 μm, the magnitude of birefringence decreases; when t is greater than 5 μm, although the magnitude of birefringence increases, as the fillet radius increases, the area of the core decreases which is not conducive to fabricate fibers. The effective mode area is also an important indicator of the performance of PMFs. We usually use the following formula to calculate the effective mode area:
Fig. 14. Birefringence contrast of ellipse, rhombus and leaf-shaped core PMF versus (a) e = 0.4, (b) e = 0.45, (c) e = 0.5, (d) e = 0.55 as a function of a.
1.7 μm with 0.05 μm spacing). At a typical sensing wavelength of 1.55 μm, the mode field area of the x-polarization mode is 23.74 μm2, and the mode field area of the y-polarization mode is 24.49 μm2. We can see that the effective mode area of the designed “leaf-shaped” PMF is suitable for sensing fibers. Excessive effective mode area increases bending loss and cause the fiber susceptible to external interference. An appropriately sized mode field area ensures that energy propagates through the core region.
2
⎛⎜∬ |E|2 dxdy ⎞⎟ Aeff = ⎝
S
⎠
∬ |E|4 dxdy S
(7)
where E is the transverse electric field component, S is the integral area of core region. Fig. 17 shows the effective mode area for the x and y polarization as function of the operate wavelength λ (from 0.85 μm to 6
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Fig. 15. Colormap of modal birefringence as function of t and r of leaf-shaped core fiber for d = 3 μm.
Fig. 18. Cross-section of the Bow-Tie fiber with 'leaf-shaped' core.
3.6. F. ‘leaf-shaped’ core Bow-Tie fiber We explore the application of the proposed ‘leaf-shaped’ core in Bow-Tie fiber. The schematic cross section of the Bow-Tie fiber with ‘leaf-shaped’ core is shown in Fig. 18, where the cladding radius W is fixed at 40 μm, the semi-minor axis of the core a is fixed at 1.9 μm, e is 0.5. We sweep the inner and outer radius of the SAP which are defined as r1 (from 6 μm to 10 μm with 0.5 μm spacing) and r2 (from 23 μm to 33 μm with 0.5 μm spacing) respectively. The used material parameters for the model are the same as which are shown in Table 1. The simulation conditions are the same as the previous section. Fig. 19 shows the modal birefringence as function of r1(μm) and r2(μm) for 'leaf-shaped' core Bow-Tie fiber. We can see that the birefringence value increases significantly when the inner radius r1 decreases and the outer radius r2 increases of the SAPs. This is because the small r1 will make the stress region close to the core, resulting in the increase of birefringence and the increase of r2 means that the area of the stress region increases, which leads to the increase of birefringence value. This is consistent with the previous conclusion of Panda-Type PMF. The maximum value of birefringence can reach 8.124 × 10−4 as r1 = 6 μm and r2 = 33 μm. In order to compare with the ‘leaf-shaped’ core Panda-Type PMF in the previous section, we fix r1 = 5.7 μm and r2 = 28.6 μm to ensure the gap between the SAPs and the core as well as the area of the SAPs are close to the same to the Panda-Type PMF. The birefringence value is 7.732 × 10−4 . The birefringence value is slightly larger than that of Panda-Type PMF, which is due to the matching effect between the special structure of the Bow-Tie fiber’s stress region and the ‘leaf-
Fig. 16. Calculated birefringence as a function of t versus r ranging from 11 μm to 14 μm.
Fig. 17. Effective mode area (μm2) of 'leaf-shaped' core fiber (a) x; (b) y.
Fig. 19. Colormap of modal birefringence as function of r1(μm) and r2(μm) for 'leaf-shaped' core Bow-Tie fiber. 7
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References
shaped’ core structure. Considering the practical application of the proposed ‘leaf-shaped’ core PMF, we birefly state the fabrication process of Panda-Type PMF as an example. The mandrel is prepared by chemical vapor deposition (CVD) and the mandrel is flat-polished to form the ‘leaf-shaped’ mandrel. During the grinding process, the flat grinding of the long and short axes are performed according to the different grinding ratios of the core diameter, the long axis direction is shaped to ensure the distortion of the mandrel in the long axis direction. Then prepare two stress bars and place together with the grinded mandrel in the cladding. The mandrel, the stress rod and the cladding integrally form the optical fiber preform. The optical fiber preform is subjected to casing fabrication, and finally an external acrylate coating is disposed outside the optical fiber preform to form the ‘leaf-shaped’ core Panda-Type PMF. Another method can be used to achieve the tip-distorted core by the casing fabrication process, that is, to increase the area of the stress region and bring it closer to the core to squeeze it to this special shape. In order to further explore the effect of the outer coating on the birefringence of the proposed ‘leaf-shaped’ core PMF, we study the change of birefringence value when the temperature of the fiber was changed from 20℃ to 60℃ after the acrylate coating. We fix the fiber structure the same as the utilized parameters in the part E. The acrylate material parameters are as follows: the Young’s modulus (E) is 1.259 × 109 Pa, the Poisson radio (v) is 0.4, the thermal expansion coefficient (α) is 1.2 × 10−4 1/K, the density is 1190kg / m3 . The outer diameter of the acrylate coating layer is 245 μm. The value of birefringence after the temperature changed is 7.774 × 10−4 . The reason for the slight increase in the birefringence value is the change of stress caused by the SAPs on the core due to the increase of temperature. However, the amount of change in birefringence is small, which proves that the designed fiber has no significant influence on the amount of birefringence when it is used in a temperature-changing environment after adding an acrylate coating.
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Jian, Design and fabrication of Panda-type erbium-doped polarization-maintaining fibres, Chin. Phys. 16 (2) (2007) 478–484. [17] A.N. Trufanov, N.A. Trufanov, N.V. Semenov, Numerical analysis of residual stresses in preforms of stress applying part for PANDA-type polarization maintaining optical fibers in view of technological imperfections of the doped zone geometry, Opt. Fiber Technol. 31 (2016) 83–91. [18] J. Sakai, T. Kimura, Birefringence and polarization characteristics of single-mode optical fibers under elastic deformations, IEEE J. Quantum Electron. 17 (6) (1981) 1041–1051. [19] N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, Birefringence in single-mode optical fiber due to elliptical core deformation and stress anisotropy, IEEE J. Quantum Electron. 16 (11) (1980) 1267–1271. [20] W. J. Bock, T. Martynkien, W. Urbanczyk, “Influence of ellipticity and refractive index of the core on sensing characteristics of elliptical-core fibers,” in: 2002 15th Optical Fiber Sensors Conference Technical Digest. OFS 2002(Cat. No.02EX533), Portland, OR, USA, 2002, pp. 479-482 vol.1. [21] K. Okamoto, T. Hosaka, T. Edahiro, Stress analysis of optical fibers by a finite element method, IEEE J. Quantum Electron. 17 (10) (1981) 2123–2129. [22] R. M. Anwar, M. S. Alam, “Thermal stress effects on higher order modes in highly elliptical core optical fibers,” in: 2008 International Conference on Electrical and Computer Engineering, Dhaka, 2008, pp. 561-565. [23] K. Hayata, M. Koshiba, M. Suzuki, Stress-induced birefringence of side-tunnel type polarization-maintaining fibrers, J. Lightwave Technol. 4 (6) (1986) 601–607. [24] M. Fontaine, Computations of optical birefringence characteristics of highly eccentric elliptical core fibers under various thermal stress conditions, J. Appl. Phys. 75 (68) (1994) 68–73. [25] M. Ji, D. Chen, L. Huang, Integration method to calculate the stress field in the optical fiber, Opt. Commun. 403 (2017) 103–109. [26] K. Saitoh, M. Koshiba, Y. Tsuji, Stress analysis method for elastically anisotropic material based on optical waveguides and its application to strain-induced optical waveguides, J. Lightwave Technol. 17 (2) (1999) 255–259. [27] A. Antikainen, R. Essiambre, G.P. Agrawal, Determination of modes of elliptical waveguides with ellipse transformation perturbation theory, Optica 4 (12) (2017) 1510–1513. [28] R.H. Stolen, V. Ramaswamy, P. Kaiser, W. Pleibel, Linear polarization in birefringent single-mode fibers, Appl. Phys. Lett. 33 (8) (1978) 699–701. [29] S. Fu, Y. Wang, J. Cui, Q. Mo, X. Chen, B. Chen, M. Tang, D. Liu, Panda type fewmode fiber capable of both mode profile and polarization maintenance, J. Lightwave Technol. 36 (24) (2018) 5780–5785. [30] H. Yan, S. Li, Z. Xie, X. Zheng, H. Zhang, B. Zhou, Design of PANDA ring-core fiber with 10 polarization-maintaining modes, Photon. Res. 5 (1) (2017) 1–5. [31] J. Zhang, X. Qiao, T. Guo, Y. Weng, R. Wang, Y. Ma, Q. Rong, M. Hu, Z. Feng, Highly sensitive temperature sensor using PANDA fiber sagnac interferometer, J. Lightwave Technol. 29 (24) (2011) 3640–3644. [32] S. Chen, J. Wang, Design of PANDA-type elliptical-core multimode fiber supporting 24 fully lifted eigenmodes, Opt. Lett. 43 (15) (2018) 3718–3721.
4. Conclusion In this paper, we discuss the influence of several core geometry on the value of birefringence. Based on advantages of the rhombic and elliptical core shapes, we innovatively propose a ‘leaf-shape’ core Panda-type PMF, which the birefringence value can reach6.3904 × 10−4 , the geometrical birefringence magnitude is 1.9727 × 10−4 . The birefringence value is twice as high as that of the actual Panda-Type PMF. By optimizing the geometry parameters of this new shape core, the birefringence value can be increased to achieve 7.6921 × 10−4 while with the Bow-Tie fiber, the birefringence can achieve 7.732 × 10−4 , the polarization maintaining performance is significantly enhanced over conventional PMF simultaneously. The mode field performance of this core-shaped fiber has been verified to be suitable for sensing PMF. This kind of fine-diameter fiber can be applied to applications requiring miniaturization such as fiber optic gyroscopes, fiber optic hydrophones, etc[31]. Meanwhile, this special shape can also be used as a reference core shape in communication fibers [32]. Acknowledgment This work was supported by National Outstanding Youth Science Fund Project of National Natural Science Foundation of China (No. 61703090). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.yofte.2019.102065.
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Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
Design of high birefringence stress-induced polarization-maintaining fiber based on utilizing geometrical birefringence
T
⁎
Haoyu Lia, , Xuyou Lia, Yue Zhanga, Pan Liub, Hanrui Yangc a
College of Automation, Harbin Engineering University, Harbin 150001, China Beijing Institute of Control and Electronic Technology, Beijing 100032, China c School of Automation Engineering, Northeast Electric Power University, Jilin 132012, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Polarization-maintaining fiber Geometrical birefringence Fiber design Structural optimization
Considering the utilizing of geometrical birefringence, we propose a new type of stress-induced polarizationmaintaining fiber (PMF) with a ‘leaf-shaped’ core —a structure found by combining the merits of elliptical and rhombic core shapes. The finite element method (FEM) is used to compare and analyze the contribution of the elliptical, rhombic, ‘star-line shaped’ and ‘leaf-shaped’ core on geometrical birefringence. The results show that this kind of PMF improves the polarization-maintaining performance of the fiber significantly, and its birefringence value can reach 7.692 × 10−4 in Panda-Type PMF and 7.732 × 10−4 in Bow-Tie PMF which are doubled compare with the traditional Panda-Type PMF. Analysis of the mode field performance also indicates that the designed fiber is suitable for the sensing field. At the same time, this fine-diameter PMF conforms to the trend of miniaturization of current used fiber-optic sensors and can be applied to fiber optic gyroscopes.
1. Introduction High birefringence fibers have been widely used in many applications benefitting from their polarization-maintaining property. These fibers are capable of maintaining linear polarization along the birefringence axis over the entire length [1–3]. Applying two axially asymmetric stresses to the core of the fiber causes difference of the propagation constant between the fast and slow axis of the fiber at the same time increases the value of the birefringence. When a beam of polarized light emits along the slow axis of the fiber, the light is forced to propagate at a lower velocity than if it has been launched along the fast axis [4,5]. The modal birefringence can be improved in two ways: the first method is creating asymmetric core geometry in accordance with geometrical birefringence. Geometrical birefringence is essentially the anisotropy of the geometry of the dielectric material resulting in the dielectric constant of the material and the anisotropy of the permeability, causing the anisotropy of the refractive index of the material [6–9]. The other method is inducing stress birefringence, that is, adding symmetrically stress-applying zone (SAP) with a high thermal expansion coefficient on both sides of the core, or adding SAP around the elliptical core. The use of PCF as a representative of applied geometrical birefringence has received much attention in recent years. It is also easy to obtain high birefringence PCFs by designing the fiber structure [10]. ⁎
Researchers have designed many kinds of air hole shapes of PCF with high birefringence such as pentagons [11], rectangles [12,13], rhombus [14], and elliptical core shapes [15]. Unfortunately, PCFs are expensive to manufacture and have higher loss than the conventional PMFs. The refractive index and stress birefringence in the orthogonal direction of PMF depend on the thermal stress caused by the SAP. Therefore, the PMF utilizes the stress distribution to obtain the high birefringence magnitude of the core region. These fibers can be structurally optimized to achieve better polarization-maintaining performance to meet the process requirements [16]. Stress-induced PMF is still widely used in the field of optical fiber communication and optical fiber sensing [17]. However, such fibers can still increase their birefringence values by introducing geometrical birefringence [18]. As a way to increase the value of birefringence, the study of geometrical birefringence is not very comprehensive by now because the relevant research on geometrical birefringence mostly focuses on the elliptical core [19,20], furthermore, there is no relevant research on the influence of other special shapes on birefringence. In this paper, we discuss the influence of several special shape cores on the birefringence value. And we design a new ‘leaf-shaped’ core stress-induced PMF with high birefringence. The birefringence analysis and calculation method of stress-induced PMF is given by thermal stress analysis theory and fiber polarization theory. Then the birefringence of the circular and elliptical core fiber is studied numerically as the
Corresponding author. E-mail address: [email protected] (H. Li).
https://doi.org/10.1016/j.yofte.2019.102065 Received 14 May 2019; Received in revised form 16 August 2019; Accepted 21 October 2019 1068-5200/ © 2019 Elsevier Inc. All rights reserved.
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comparison of subsequent part. Considering the effect of core curvature on birefringence, we study the birefringence values of the elliptical, rhombus and ‘star-line’ shaped cores. By combining the advantages of elliptical and rhombic shapes, we propose a ‘leaf-shaped’ core PMF and evaluate the birefringence performance. From the point of fabrication, we optimize the structure of this fiber and obtaine the birefringence value of 7.692 × 10−4 in Panda-Type PMF and 7.732 × 10−4 in Bow-Tie PMF. Finally, we study the mode field performance of this fiber and briefly explain the fabrication method. 2. Birefringence theory and analysis method As we all know, the formation of geometrical birefringence is due to the asymmetry of the core geometry. In order to improve this kind of birefringence, the influence of the core geometry on the magnitude of birefringence will be discussed in this paper. FEM has been widely used as an accurate and convenient method to analyze the birefringence characteristics of PMFs [21]. Especially in the analysis of arbitrary cross-section waveguides, nonlinear waveguides and anisotropic waveguides, this method is more suitable for analyzing the electric field and magnetic field distribution [22]. Due to the diversity and complexity of the waveguide problem, most problems are solved by the full vector FEM. The birefringence performance is extremely important for studying the polarization properties of the fibers. High values of birefringence provide superior polarization retention. Modal birefringence is related to many factors such as temperature, core ellipticity, core-cladding refractive index difference, thermal expansion coefficient and wavelength of light. The FEM analysis of the birefringence for stress-induced PMF can be divided into two steps: firstly, if the composite glass fibers are stretched at the high temperature of the preform, thermal stress is generated once the fibers are cooled to the temperature at which the glass solidifies [23], then the thermal stress analysis is used to find the direct stress of the core in both the x and y directions [24], after that the refractive index of the x and y directions is calculated by the photoelastic effect expression [25,26]:
Fig. 1. Cross-section of circular core Panda-Type PMF.
B=
3.1. Circular core PMF verification In order to verify the correctness of our model, we first calculate the birefringence value of the circular core PMF with 125 μm outer diameter to compare with the value in Ref. [16]. The schematic cross section of the typical circular core PMF is shown in Fig. 1, where the radius of the fiber cross section is W which is fixed at 62.5 μm and the gap between the core and SAP is d. The radius of the core and the SAPs are a and r respectively. The two B2O3-doped-silica SAPs are symmetrically placed beside a GeO2-doped-silica core [29]. The refractive index of the SAPs and core are 1.436 and 1.474 respectively. All simulations in this paper perform at the wavelength of 1550 nm. The thermal expansion coefficient α of the above doped material can be obtained by the expression shown below:
(1)
α = (1 − m) α 0 + mα1
(6)
where m is the mole percentage of the doped material. The cladding is made of silica. The used material parameters for the analysis model including Young’s modulus (E), Poisson radio (v), the density (ρ), the thermal expansion coefficient (α), the stress-optic coefficient (C1 and C2), the drawing temperature and operating temperature which are listed in Table 1. The above materials parameters applied have proven to be applicable to high birefringence fibers in [30]. The desired refractive index can be obtained by changing the doping percentage according to the hybrid Sellmeier equation. In order to compare the birefringence value with Ref. [16], we fix d = 6.6 μm and sweep the other two parameters r (from 7 μm to 17 μm with 1 μm spacing) and a to calculate the modal birefringence. As can be seen in Fig. 2, compared to Ref. [16], the birefringence magnitude changes in the same trend but has small deviations in value. The reason for the small deviation is that in the Ref. [16], while radius of the core and SAPs are changing, the gap between the SAPs and the core also varies. It can be seen that as the area of the SAPs increase, the amount of birefringence also increases. This conclusion is also the same as that obtained in Ref. [16]. We can conclude that our model can be used for further analysis of stress-induced PMF.
(2)
(3)
where β is the propagation constant of the fiber and ω is the angular frequency. The eigenvalue equation for the electric field can be obtained from the Helmholtz equation:
∇ × (n−2∇ × E ) − k 02 E = 0
(5)
3. Simulation and discussion
The modal birefringence is defined as the sum of geometrical and stress birefringence. Since the dimension in the z-direction of the fiber is much larger than the size of the cross-section, the load can be seen to an act on the cross-section but does not change with the length of the fiber, so the refractive index in the z direction is regarded as a constant. The electric field of the wave has the form:
E (x , y, z , t ) = E (x , y ) exp[j (ωt − βz )]
x = neff − neffy
x and are effective where βx and βy are the propagation constants; neff mode refractive indices in the x-axis and y-axis directions respectively. Thus the mode birefringence of the PMF is the differential value in the effective refractive index of the two refractive index principal axes of the core layer.
where C1 and C2 are respectively first and second stress-photoelastic coefficients; σx and σy are the direct stresses in the x-axis and y-axis directions; n0 is the refractive index of the stress-free material. Anisotropic changes in refractive index will result in higher values of birefringence due to stress optical effects in the fiber. Stress birefringence is defined as
Bs = n y − n x = (C2 − C1 )(σy −σx )
k0
neffy
n x = n 0 − (C1 σx +C2 σy ) n y = n 0 − (C2 σx +C1σy )
βx − βy
(4)
where k0 is the number of waves propagating in vacuum. Using the analysis method of wave optics to calculate the effective refractive index of the fiber [27,28], then the modal birefringence value of the stress-type PMF is calculated by: 2
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Table 1 Elastic Parameters Used for Modeling. Parameters
Core
Thermal expansion coefficient α (1/℃) Young’s modulus E (Pa)
5.4 × 10
Poisson ratio v Density ρ (kg / m3 ) First stress optical coefficient C1 (1/Pa) Second stress optical coefficient C2 (1/Pa) Drawing temperature (℃) Operating temperature (℃)
Cladding
-7
7.83 × 1010 0.186
5.4 × 10
-7
7.25 × 1010 0.186 2203
Stress-applying rod
3.38 × 10
-6
2.86 × 1010 0.265
− 6.9 × 10−13 − 4.19 × 10−12
Fig. 4. Colormap of modal birefringence as function of e and a (μm) for elliptical core fiber.
1320 20
optical fibers. Then we sweep the other two parameters a (from 1.5 μm to 3.5 μm with 0.2 μm spacing) and e (from 0.4 to 1.4 with 0.1 spacing) to analysis modal character in order to calculate the birefringence of the elliptical core fiber. Obviously, the elliptical core PMF can be changed as common circular core Panda-Type PMF with e = 1. Fig. 4 shows a colormap of the modal birefringence numerical value of all combinations of e and a. One can see that the birefringence values for the region of 0.4 < e < 1 are manifest larger than the value for the region of e greater than 1 and the birefringence value will increase with the elliptic rate e decreases. The birefringence magnitude range of an elliptical core Panda-Type PMF under selected material and structural parameters of e = 0.4 ~ 0.6 is4.805 × 10−4 ~ 6.611 × 10−4 . Fig. 5 shows the geometrical birefringence curve of an elliptical core fiber. From Fig. 5 we can see that as the core ellipticity e and the core short axis a increase, the geometrical birefringence shows a significant downward trend. This is because as e decreases, the geometric asymmetry of the core increases, resulting in an increase in the magnitude of the geometrical birefringence. But when e and a are excessively small, the area of the core will also decrease, so the appropriate value should be chosen to fabricate the fiber. In order to further explore the modal birefringence properties of the elliptical core PMF, we fix a = 1.9 μm, e = 0.5 and sweep the other two parameters d (from 3 μm to 6 μm with 0.5 μm spacing) and r (from 10 μm to 14 μm with 0.5 μm spacing) to calculate the birefringence value which is shown in Fig. 6. After further increasing the area of the SAPs and bringing the SAPs closer to the core, the elliptical core PMF obtains a modal birefringence value having a maximum value of 7.187 × 10−4 . This means that a larger area of the SAP will result in a significant increase in the amount of birefringence when it is close to the core.
Fig. 2. Modal birefringence value of the circular core PMF as a function of r (μm).
3.2. B. Elliptical core PMF The schematic cross section of the elliptical core PMF is shown in Fig. 3, where the radius of the fiber cross section is W which is fixed at 40 μm for the design of fine-diameter fiber and the gap between the elliptical core and SAP is d. The semi-minor, semi-major axis of the elliptical core are a and b respectively, the elliptic rate is defined as e = a/ b . The radius of the SAP is r. Other parameters are the same as the previous section. In order to determine the structure parameters of elliptical core PMF which can supply high birefringence, we fix d = 5 μm, r = 12 μm in this model which consider the technical requirements for the fabrication of
Fig. 5. Geometrical birefringence of elliptical core PMF versus e from 0.4 to 0.8 as a function of a (μm).
Fig. 3. Cross-section of elliptical core Panda-Type PMF. 3
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radius r of the stress region. As can be seen from the Fig. 9, the maximum birefringence of the diamond core can reach 7.089 × 10−4 . The amount of birefringence increases significantly, which is consistent with previous conclusions. Compared to the elliptical core PMF, the rhombus core can only achieve high birefringence values over a relatively several combination of values. However, as a fiber shape that relies on geometrical-induced polarization, rhombus core is still a desirable shape. 3.4. D. ‘leaf-shaped’ core PMF We decide to increase the geometrical asymmetry of the core in the y-axis direction and preserve the convex function curve in the x-axis direction, that is, the ‘leaf-shaped’ core Panda-Type PMF with a crosssection as shown in Fig. 10. The structure is obtained by filleting the two vertices of the rhombus in the x-axis direction. The semi-minor, semi-major axis of the core are defined as a and b respectively, the semiaxis rate here also defined as e = a/ b . The fillet radius is defined as t. In order to facilitate comparison with the elliptical core, we fix t = 5 μm. The material used in the core, cladding and stress-applying rods are consistent with the elliptical core PMF, the same as the gap size and the radius of SAP. Using the same research method, we sweep the two parameters a (from 1.5 μm to 2.5 μm with 0.1 μm spacing) and e (from 0.4 to 0.55 with 0.05 spacing), because we estimate that the fiber will get high birefringence performance from the previous results in this range of a and e. Fig. 11(a) shows the von Mises stress of the fiber obtained by thermal stress analysis, and (b) shows the stress birefringence distribution of the fiber. The direct stress of the fiber in both the x-axis and the y-axis can be obtained in Fig. 12, which will be used to calculate the effective mode index of the fiber and further analyze the birefringence performance. Fig. 13 shows the birefringence colormap calculated by sweeping two parameters e and a. It can be seen that the ‘leaf-shaped’ core can provide stably a high birefringence value in the range of 0.4 ⩽ e ⩽ 0.55 and 1.5 ⩽ a ⩽ 2.2 , which can only be in the ellipse when e and a are particularly small. It is obtained that the ‘leaf-shaped’ core can ensure a high birefringence while relatively increasing the core area, thanks to the geometrical birefringence provided by this particular asymmetric shape. The performance can also provide tolerance for the fabrication of optical fibers. One can see from the comparison of Fig. 14, the birefringence magnitude of the ‘leaf-shaped’ core is significantly improved compared to the elliptical and rhombic cores at the region of 1.7 < a. We can take the core structure of e = 0.5, a = 1.9 μm, the value of modal birefringence can achieve the value of 6.3904 × 10−4 at the same time the value of the geometrical birefringence is
Fig. 6. Colormap of modal birefringence as function of r (μm) and d (μm) for elliptical core fiber.
3.3. C. Rhombic and Star-line shaped core PMF Considering the effect of core curvature on birefringence, we study the birefringence properties of rhombic core fiber and star-line shaped core fiber, i,e the closed curve corresponding to the straight line edge, the concave function and the convex function are respectively mirrored twice. Fig. 7 shows the schematic cross section of the rhombic core and ‘star-line’ shaped core PMF, where the cladding radius, the gap between the elliptical core and stress-applying rods, the radius of the stress-applying rods are the same as the elliptical core PMF. For the purpose of comparison, the semi-minor, semi-major axis of the two cores are a and b respectively, the semi-axis rate here also defined as e = a/ b . The modal birefringence of the three shapes of core fiber are shown in Fig. 8. Compared to the ‘star-line’ shaped core, the elliptical and rhombic core can maintain a high amount of birefringence at the region of 0.4 ⩽ e ⩽ 0.7, which proves that these two shapes are capable of increasing geometrical birefringence. This means that the concave side of the ‘star-line’ shaped is not suitable for the core of a high birefringence fiber. Consistent with the conclusions described above, as the core asymmetry increases, the magnitude of birefringence also increases significantly. In order to design a higher birefringence core shape, it is necessary to retain the circular arc in the × direction similar to an elliptical shape and the sharp angle in the y-axis direction similar to a rhombic shape. Similar to the analysis process of the elliptical core PMF, we further study the maximum amount of modal birefringence that the rhombus core PMF can obtain after shortening the gap d and increasing the
Fig. 7. Cross-section of (a) rhombic core fiber and (b) ‘star-line’ shaped core fiber. 4
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Fig. 9. Colormap of modal birefringence as function of r (μm) and d (μm) for rhombus core fiber.
Fig. 10. Cross section of ‘leaf-shaped’ core fiber.
Fig. 11. (a) von Mises stress and (b) stress-induced birefringence distributions.
core fiber, we fix r = 12 μm and sweep the core fillet radius t (from 3 μm to 6 μm) to increase the geometrical birefringence while scanning the gap between the core and SAP d (from 2.5 μm to 6 μm) to change the stress birefringence. Fig. 15 shows the birefringence colormap calculated by sweeping the two parameters t and d. It can be seen that the magnitude of the modal birefringence increases as the distance between the stress-acting regions and the core decrease. This value can also be increased again by changing the fillet radius to provide a higher geometrical birefringence magnitude. In order to satisfy the manufacturing process, we fix d at 3 μm and study the relationship between the birefringence value and the fillet radius t (from 3 μm to 6 μm) when the SAP radius r (from 11 μm to 14 μm) of the stress region changes. As can be seen from Fig. 16, the increase of the radius of SAP provides greater stress birefringence resulting in improvement of modal birefringence. As the radius of SAP
Fig. 8. Modal birefringence contrast of ellipse, rhombus and ‘star-line’ versus (a) e = 0.4, (b) e = 0.5, (c) e = 0.6, (d) e = 0.7 as a function of a (μm) (with 0.2 μm spacing).
1.9727 × 10−4 ,which is 3.447 × 10−5 higher than that provided by the elliptical core. 3.5. E. Structural optimization of ‘leaf-shaped’ core fiber To further optimize the birefringence properties of the ‘leaf-shaped’ 5
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Fig. 12. Direct stress in the (a) x directions and (b) y directions.
Fig. 13. Colormap of modal birefringence as function of e and a for leaf-shaped core fiber.
increases, the magnitude of the birefringence also increases. When t varies between 3 μm-4.7 μm, the maximum value of birefringence is 7.6921 × 10−4 , of which the geometrical birefringence is 3.2744 × 10−4 . While at the range of 4.7 μm < t < 5 μm, the magnitude of birefringence decreases; when t is greater than 5 μm, although the magnitude of birefringence increases, as the fillet radius increases, the area of the core decreases which is not conducive to fabricate fibers. The effective mode area is also an important indicator of the performance of PMFs. We usually use the following formula to calculate the effective mode area:
Fig. 14. Birefringence contrast of ellipse, rhombus and leaf-shaped core PMF versus (a) e = 0.4, (b) e = 0.45, (c) e = 0.5, (d) e = 0.55 as a function of a.
1.7 μm with 0.05 μm spacing). At a typical sensing wavelength of 1.55 μm, the mode field area of the x-polarization mode is 23.74 μm2, and the mode field area of the y-polarization mode is 24.49 μm2. We can see that the effective mode area of the designed “leaf-shaped” PMF is suitable for sensing fibers. Excessive effective mode area increases bending loss and cause the fiber susceptible to external interference. An appropriately sized mode field area ensures that energy propagates through the core region.
2
⎛⎜∬ |E|2 dxdy ⎞⎟ Aeff = ⎝
S
⎠
∬ |E|4 dxdy S
(7)
where E is the transverse electric field component, S is the integral area of core region. Fig. 17 shows the effective mode area for the x and y polarization as function of the operate wavelength λ (from 0.85 μm to 6
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Fig. 15. Colormap of modal birefringence as function of t and r of leaf-shaped core fiber for d = 3 μm.
Fig. 18. Cross-section of the Bow-Tie fiber with 'leaf-shaped' core.
3.6. F. ‘leaf-shaped’ core Bow-Tie fiber We explore the application of the proposed ‘leaf-shaped’ core in Bow-Tie fiber. The schematic cross section of the Bow-Tie fiber with ‘leaf-shaped’ core is shown in Fig. 18, where the cladding radius W is fixed at 40 μm, the semi-minor axis of the core a is fixed at 1.9 μm, e is 0.5. We sweep the inner and outer radius of the SAP which are defined as r1 (from 6 μm to 10 μm with 0.5 μm spacing) and r2 (from 23 μm to 33 μm with 0.5 μm spacing) respectively. The used material parameters for the model are the same as which are shown in Table 1. The simulation conditions are the same as the previous section. Fig. 19 shows the modal birefringence as function of r1(μm) and r2(μm) for 'leaf-shaped' core Bow-Tie fiber. We can see that the birefringence value increases significantly when the inner radius r1 decreases and the outer radius r2 increases of the SAPs. This is because the small r1 will make the stress region close to the core, resulting in the increase of birefringence and the increase of r2 means that the area of the stress region increases, which leads to the increase of birefringence value. This is consistent with the previous conclusion of Panda-Type PMF. The maximum value of birefringence can reach 8.124 × 10−4 as r1 = 6 μm and r2 = 33 μm. In order to compare with the ‘leaf-shaped’ core Panda-Type PMF in the previous section, we fix r1 = 5.7 μm and r2 = 28.6 μm to ensure the gap between the SAPs and the core as well as the area of the SAPs are close to the same to the Panda-Type PMF. The birefringence value is 7.732 × 10−4 . The birefringence value is slightly larger than that of Panda-Type PMF, which is due to the matching effect between the special structure of the Bow-Tie fiber’s stress region and the ‘leaf-
Fig. 16. Calculated birefringence as a function of t versus r ranging from 11 μm to 14 μm.
Fig. 17. Effective mode area (μm2) of 'leaf-shaped' core fiber (a) x; (b) y.
Fig. 19. Colormap of modal birefringence as function of r1(μm) and r2(μm) for 'leaf-shaped' core Bow-Tie fiber. 7
Optical Fiber Technology 53 (2019) 102065
H. Li, et al.
References
shaped’ core structure. Considering the practical application of the proposed ‘leaf-shaped’ core PMF, we birefly state the fabrication process of Panda-Type PMF as an example. The mandrel is prepared by chemical vapor deposition (CVD) and the mandrel is flat-polished to form the ‘leaf-shaped’ mandrel. During the grinding process, the flat grinding of the long and short axes are performed according to the different grinding ratios of the core diameter, the long axis direction is shaped to ensure the distortion of the mandrel in the long axis direction. Then prepare two stress bars and place together with the grinded mandrel in the cladding. The mandrel, the stress rod and the cladding integrally form the optical fiber preform. The optical fiber preform is subjected to casing fabrication, and finally an external acrylate coating is disposed outside the optical fiber preform to form the ‘leaf-shaped’ core Panda-Type PMF. Another method can be used to achieve the tip-distorted core by the casing fabrication process, that is, to increase the area of the stress region and bring it closer to the core to squeeze it to this special shape. In order to further explore the effect of the outer coating on the birefringence of the proposed ‘leaf-shaped’ core PMF, we study the change of birefringence value when the temperature of the fiber was changed from 20℃ to 60℃ after the acrylate coating. We fix the fiber structure the same as the utilized parameters in the part E. The acrylate material parameters are as follows: the Young’s modulus (E) is 1.259 × 109 Pa, the Poisson radio (v) is 0.4, the thermal expansion coefficient (α) is 1.2 × 10−4 1/K, the density is 1190kg / m3 . The outer diameter of the acrylate coating layer is 245 μm. The value of birefringence after the temperature changed is 7.774 × 10−4 . The reason for the slight increase in the birefringence value is the change of stress caused by the SAPs on the core due to the increase of temperature. However, the amount of change in birefringence is small, which proves that the designed fiber has no significant influence on the amount of birefringence when it is used in a temperature-changing environment after adding an acrylate coating.
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4. Conclusion In this paper, we discuss the influence of several core geometry on the value of birefringence. Based on advantages of the rhombic and elliptical core shapes, we innovatively propose a ‘leaf-shape’ core Panda-type PMF, which the birefringence value can reach6.3904 × 10−4 , the geometrical birefringence magnitude is 1.9727 × 10−4 . The birefringence value is twice as high as that of the actual Panda-Type PMF. By optimizing the geometry parameters of this new shape core, the birefringence value can be increased to achieve 7.6921 × 10−4 while with the Bow-Tie fiber, the birefringence can achieve 7.732 × 10−4 , the polarization maintaining performance is significantly enhanced over conventional PMF simultaneously. The mode field performance of this core-shaped fiber has been verified to be suitable for sensing PMF. This kind of fine-diameter fiber can be applied to applications requiring miniaturization such as fiber optic gyroscopes, fiber optic hydrophones, etc[31]. Meanwhile, this special shape can also be used as a reference core shape in communication fibers [32]. Acknowledgment This work was supported by National Outstanding Youth Science Fund Project of National Natural Science Foundation of China (No. 61703090). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.yofte.2019.102065.
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