1. Introduction

Measurements of phase (modal) birefringence, B_phase, and group birefringence, B_group, are very important when characterizing high birefringence (HiBi) fibers, and especially elliptical core fibers (ECF) [1,2]. The difference between phase and group birefringence is often neglected, and very seldom they are reported for the same fiber [3–5]. For HiBi fibers dominated by stress-induced birefringence, it may be justified not to distinguish between them [6–8]. However, in ECFs it is not uncommon that B_phase and B_group differ quite significantly and thus they both have to be measured in order to characterize the fiber [6–8]. For some applications like polarimetric sensors, a large phase birefringence is of importance, while for others the group birefringence or the differential group delay (DGD) is the key parameter [9]. Knowledge about the dispersion of the phase and group birefringence may be important. ECFs have shown to be less sensitive to temperature, pressure and strain compared to fibers with birefringence dominated by stress [10,11]. This can be advantageous for some applications. Previous experimental studies on the birefringence of ECFs have mainly been at non-telecom wavelengths and only for a very limited range of ellipticities [3,4,11].

This paper presents an extensive study of B_phase and B_group for ECFs at telecom wavelengths around 1550 nm: The semi-minor, b, and semi-major, a, axis ratios, b/a ranging from 1 down to below 0.1, or ellipticity,$\epsilon =1-b/a$, in the range from 0 to 0.9. This is the first report on both types of birefringence and birefringence dispersion for ECFs covering such an extensive range of ellipticities. The core area is shown to impact the ratio between the group and phase birefringence, and the dispersion of the group birefringence or DGD slope is shown to be very dependent on the ellipticity of the core.

2. Phase and Group Birefringence

The phase (modal) birefringence

The group birefringence is given by

3. Experiments and Results

Fibers with elliptical core drawn from a preform with semi-minor to semi-major ratio b/a ranging from close to 1 down to below 0.1 were investigated. Fabrication details and cut-off properties have previously been reported [12]. The area of the elliptical core was on average 12.7 μm² (preform 1) and a maximum index difference of 0.026. In addition, ECFs originating from two other preforms with somewhat lower core areas, 10.8 μm² (preform 2) and 9.4 μm² (preform 3) respectively, and ellipticities up to 0.5 were investigated. The normalized cutoff frequencies, $V_b=(2\pi b/\lambda ){(n_{co}^2-n_{cl}^2)}^{1/2}$of the measured fibers were in the range 0.77-2.23 (Table 1 ).

Table 1. Elliptical core fibers investigated

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The phase birefringence was measured by a slightly modified version of the external stress method originally proposed by Calvani et al [15]. The fiber is laid flat on a hard smooth surface. By applying a lateral force at a point that moves along the fiber axis and simultaneous measuring the output state of polarization (SOP), the L_B,phase can be obtained by counting the periodic variation in the output SOP with a polarimeter (PAN9300 FIR). Unlike the classical lateral force method proposed by Takada et al [16], this method does not require a specific or even known input polarisation, but it relies only on the measurement of the output SOP as function of the displacement of the point like stress. The technique is non-destructive, and by measuring the output SOP, the beat-length could be determined at a number of wavelengths in the range from 1520 – 1590 nm provided by a tuneable laser diode source. An example of SOP variation and associated beat-length is illustrated in Fig. 1 . For short beat-lengths the main uncertainty comes from the reading of the displacement which in the applied setup had an accuracy of 0.5 mm. In order to reduce the uncertainty, the L_B,phase(λ) was determined by averaging over 2-20 periods depending on beat-length (4.5 – 110 mm). This reduced the uncertainty to less than 1%. Using the relation $L_{B,phase}\left(\lambda \right)=\lambda /B_{phase}\left(\lambda \right)$, the B_phase was calculated.

 

Fig. 1 a) Example of SOP evolution on the Poincare sphere. b) The three normalized Stokes vector components as function of displacement length from which the polarization beat-length is determined.

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B_group was determined from Eq. (3), where $d{B}_{phase}/d\lambda$ was found by making a linear fit to B_phase(λ) over the wavelength range. As illustrated in Fig. 2a , the results were in excellent agreement with the B_group determined from DGD measurements in the wavelength range from 1440 – 1590 nm obtained by the Jones-Matrix eigenanalysis (JME) measurement technique with an HP-8509 polarization analyzer. The slope of the DGD was found assuming a linear dependence on wavelength. Justification of this is evident from Fig. 2b which shows examples of DGD as function of wavelength.

 

Fig. 2 a) Group birefringence determined from the phase birefringence dependence on wavelength (in blue) and from DGD measurements (in green). b) Examples of DGD as function of wavelength, all well approximated by a linear dependence in the range covered.

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The measured B_phase and B_group at 1550 nm are displayed as function of ellipticity in Fig. 3 . Both are increasing with ellipticity going through a maximum and finally decreasing at very large ellipticities. They show a maximum of approximately $3.5\times {10}^{-4}$ and $6.5\times {10}^{-4}$corresponding to beat-lengths of 4.5 mm and 2.4 mm, respectively. Both B_stress and B_form and their sum, the total birefringence, B_phase, are also included in Fig. 3. The form-induced birefringence is calculated using the actual measured 2D-index profiles. In order to estimate the stress-induced birefringence, we have assumed a Δα of $2\times {10}^{-6}$K⁻¹ due to the Al doping and ΔT = 1000 °C. These are somewhat speculative. However, an excellent agreement is observed between the experimentally measured phase birefringence and the calculated total birefringence. For comparison the Fig. 3b shows the birefringence for a generic step index profile with equivalent flat top distribution [12]. There is close agreement with the experimental data even though the group birefringence is a little underestimated. This may be due to the fact that only the dispersion of the stress optic coefficient was included by assuming $\left(\lambda /C\right)\left(dC/d\lambda \right)=-0.1$ [6,17], and that the wavelength dependence of the factor describing the portion of the mode propagating in the core in the stress term had been neglected.

 

Fig. 3 a) Measured phase (blue) and group (black) birefringence for fiber 1. Calculated stress (green) and form birefringence (red) and the calculated total phase birefringence (magenta). b) Calculated stress, form, total phase and group birefringence for a generic elliptic step index profile with equivalent flat top distribution [6].

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The ratio $B_{group}/B_{phase}$ between the group and phase birefringence is shown in Fig. 4 . For preform 1 it is observed that the ratio is less than 1 at moderate ellipticities (~0.2) and then it increases to values larger than 2 with increasing ellipticities. The birefringence ratios for the fibers with smaller core areas are also included in Fig. 4. It is observed that the ratio increases with decreasing core area and thus effective index, n_eff, for the same ellipticity. This confirms that the stress birefringence (phase and group) varies considerably with the portion propagating in the core [7,8,13,18].

 

Fig. 4 a) Ratio of measured group and phase birefringence for fiber from preform 1, 2 and 3 with decreasing core area. b) Calculated ratio from simulated generic elliptic step index profiles with similar core areas and equivalent flat top distribution.

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The DGD slope at 1550 nm, found by a linear fit to the measured DGD, is shown in Fig. 5 . The slope starts at almost zero going through a maximum of approximately $1.5\cdot {10}^{-3}$ps/nm-m at an ellipticity of 0.7 and then it drops to negative values at very high ellipticity. The calculated slope values are found assuming $\left({\lambda }^2/C\right)\left(d^{2}C/d{\lambda }^2\right)=-0.06$ at 1550 nm [19]. Again an excellent agreement is found. The DGD slope of $1.5\cdot {10}^{-3}$ps/nm-m corresponds to a group birefringence dispersion of 4.5×10⁻⁷ nm⁻¹. This is more than one order of magnitude less than reported for highly birefringent microfiber [20].

 

Fig. 5 a) Measured DGD slope at 1550 nm for fiber from preform 1 (blue points) together with the calculated slope (bright blue). Also shown are the DGD slopes for fiber from preforms 2 and 3. b) Calculated DGD slope from simulated generic elliptic step index profiles with similar core areas and equivalent flat top distribution.

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4. Conclusion

Experimental measurements of phase and group birefringence and their wavelength slopes in elliptical core fibers, with ellipticities systematically varying from 0 to more than 0.9, were made at 1550 nm. Phase and group beat-lengths down to approximately 4.5 mm and 2.5 mm were obtained, respectively. However, the minimum beat-length was obtained at intermediate axis ratios for both the phase and group beat-length. The ratio$B_{group}/B_{phase}$was shown to be strongly dependent on both the ellipticity and the core area, and the DGD slope was shown to be positive and increasing to about $1.5\cdot {10}^{-3}$ps/nm-m at an ellipticity of 0.7 and then change to negative values at very high ellipticities.