Fig. 1. Principle of observation of the beat length, using side viewing of transverse Rayleigh scattering. Linearly polarized light is launched at 45° of the principal axes (blue lines) of the PM fiber, that appears as a glowing dashed line [3], with a periodicity equal to the beat length Λ.

Fig. 2. Principle of measurement of birefringence with channeled spectrum analysis: a broadband source light is sent into the PM fiber that is between polarizers at 45°, and this generates a channeled spectrum that is measured with an optical spectrum analyzer (OSA).

Fig. 3. Principle of measurement of birefringence with channeled spectrum analysis: the measured free spectral range (FSR_σ) between two successive channels is equal to 1/(B · L_f), in absence of dispersion. The first channel, displayed with a grey line, is theoretical since it corresponds to a quasi-infinite wavelength.

Fig. 4. Principle of measurement of birefringence with channeled spectrum analysis, using a tunable laser and a power meter, instead of a broad-spectrum source and an OSA.

Fig. 5. Revisited figure of Rashleigh’s letter [2] of 1983: the slope of the various dashed secant lines in color is the phase birefringence B = ∆n_eff, as it is defined today. Fibers A (green) and B (cyan) are bow-tie fibers. Fiber C (red) combines stress-induced birefringence with form birefringence. Free-space wavenumber k should be read 7.39 rad·μm⁻¹, and it corresponds to a wavelength λ of 0.85 μm.

Fig. 6. Revisited figure of Rashleigh’s letter [2] of 1983: the slope of the tangent (black dashed line) to the curve ∆β(k) is the group birefringence B_g, as it is defined today. Fiber C is very dispersive: at λ = 1.3 μm, i.e., k = 4.83 rad·μm⁻¹, group birefringence B_g is 3.4 times higher than phase birefringence B!

Fig. 7. Principle of measurement of birefringence with channeled spectrum analysis, in the general case of birefringence dispersion. The free spectral range (FSR_σ), i.e., the spatial frequency difference between two successive bright channels, depends on group birefringence B_g: it is equal to 1/(B_g · L_f). Theoretically, the position of the bright channels does depend on phase birefringence B, but there is no mean to measure it, since the channel order m cannot be known precisely without access to all orders down to zero frequency (grey line channel).

Fig. 8. Principle of path-matched white-light interferometry: (a) set-up with an unbalanced readout interferometer, using a beam splitter BS, and two mirrors M₁ and M₂; (b) interferogram as a function of the path unbalance ∆L_r, with the secondary peaks yielding a measurement of group birefringence B_g; this interferogram is the Fourier transform (FT) of the channeled spectrum; and the coherence length L_c of the original broadband source is equal to λ²/∆λ_FWHM.

Fig. 9. Measurement of phase birefringence B(λ₀) of a PM fiber: (a) set-up with a moving crossed-polarization coupling; (b) resulting channeled spectrum (black sinusoidal curve) for a length L_f corresponding to a bright channel of order m for λ₀; and shifted channeled spectrum (red sinusoidal curve) when the length is reduced by a quarter of the beat length Λ, i.e., is equal to L_f − (Λ/4).

Fig. 10. Measurement of phase birefringence B(λ) of a PM fiber with a fiber Bragg grating that has a period Λ_Bragg; λ_f is the reflected wavelength of the fast (lower-index) mode, while λ_s is the one of the slow (higher-index) mode.

Fig. 11. Principle of OFDR: (a) set-up with a frequency-swept tunable laser, an unbalanced two-wave interferometer with a reference mirror on one arm, and the low reflection point (ɛ² in power, and ɛ in amplitude) that is analyzed, on the other arm; (b) measured channeled spectrum with a contrast of 2ɛ.

Fig. 12. Shifted channeled spectra, using the slow mode (green curves) and the fast mode (red curves) of a PM fiber: (a) case of a single reflector with sine spectra that cannot be correlated over more than one period; (b) case of Rayleigh backscattering with noisy spectra that can be cross-correlated.

Fig. 13. Geometrical construction to derive group birefringence B_g(λ) (dashed black curve) from phase birefringence B(λ) (red solid curve), as seen in [1]; the tangents T_i(λ) (blue dashed lines) cross the ordinate axis, i.e., where λ = 0, at B_g(λ_i); cases with i = 1 or 2.

Fig. 14. Geometrical construction to derive phase birefringence B(λ) from group birefringence B_g(λ) (black solid line) and a single-wavelength phase measurement B(λ₁) (red circular dot); (a) construction of the tangent T₂(λ) that crosses tangent T₁(λ) at λ_x = (λ₂ − λ₁)/ln(λ₂/λ₁) (dashed blue lines), which allows one to find B(λ₂);when the interval (λ₂ − λ₁)/λ₁ is small, λ_x ≈ (λ₂ + λ₁)/2; (b) construction of B(λ) (dashed red curve) that must fit the tangents T₁(λ) and T₂(λ).

Fig. 15. Different kinds of PM fibers based on stress-induced linear birefringence: bow-tie, panda and tiger-eye. The slow axis is along the axis of the stress structure, and the fast axis is perpendicular.

Fig. 16. Principle of PM fibers based on stress-induced linear birefringence: (a) SAPs (panda structure, here) are under quasi-isotropic tensile stress, and they pull on the fiber cladding; (b) in the core region, this yields tensile stress (red arrows) in the x-axis, and a compressive stress (blue arrows) in the perpendicular y-axis; (c) these stresses decrease when one moves away from the core, along the perpendicular y-axis.

Fig. 17. Numerical model of the compressive stress T_y(x,y), in a panda fiber, without the circular core, not to be blinded by the significant isotropic stress that this one generates: (a) color code with, in addition, double arrows to better visualize the stresses; (b) curve T_y(0,y) of the compressive stress in the y-axis that decreases along this y-axis, and gets to zero at the outer limit of the cladding.

Fig. 18. Numerical model of the stress T_x(x,y) in a panda fiber, without the circular core, not to be blinded by the significant isotropic stress that this one generates: (a) color code with, in addition, double arrows to better visualize the stresses; (b) curve T_x(0,y) of the variation of the stress on the x-axis, along the y-axis: it is positive in the core region, and it becomes negative at a distance that corresponds to about the height of the SAPs.

Fig. 19. Stresses around the SAPs: (a) at 0° and 90°; (b) at 45°, where the sum of their projections on the x or y-axis becomes null … but it does not mean that the stresses are null!

Fig. 20. Measured group birefringence B_g(λ) over 1250–1650 nm (black solid line, with square dots for the measurements), measured phase birefringence B at 1550 nm (red circular dot), and deduced phase birefringence B(λ) over 1250–1650 nm (red dashed line): (a) bow-tie fiber; (b) panda fiber; (c) tiger-eye fiber. Photographs of the end face of the fibers are inserted; SAPs are dark, while the circular cores are illuminated.

Fig. 21. Elliptical-core (E-core) PM fiber, based on shape birefringence, with its slow and fast axes.

Fig. 22. Phase shift induced by total internal reflection (TIR): p state of polarization is in red line, and s state is in green line. θ_c is the critical angle where TIR starts. A grazing incidence (angle of 90°) yields a π phase shift.

Fig. 23. Measurement, by channeled spectrum analysis, of group birefringence B_g(λ) of an E-core fiber, over 1000–1650 nm (black square dots); measurement of phase birefringence B at 1550 nm with Bragg grating and Rayleigh-OFDR (red circular dot); phase birefringence B(λ) over 1000–1650 nm (dash red line) deduced by geometrical construction; confirmation of the value of B at 1300 nm with an additional Bragg grating (green circular dot); a photograph of the end face of the fiber, with its illuminated elliptical core, is inserted.

Fig. 24. Revisited Figure 3 of [32]: the phase-birefringence dispersion dB/dλ is very high, and it has a positive slope; this yields a group birefringence B_g with an opposite sign. This figure shows the interest of the geometrical construction with the tangent, that allows one to visualize simply the relationship between phase and group birefringences. The insert shows the shape of the micro-structured solid-core PM fiber described in [32].

Fig. 25. Revisited Figure 3b of [33]: the tangent to phase birefringence curve B(λ) does cross at zero the zero-wavelength ordinate axis, for 1.55 μm, where group birefringence is cancelled; the insert shows Figure 3c, with group birefringence curve B_g(λ), noted as G(λ), in this paper.

Fig. 26. Revisited Figure 1b of [34]: the tangent to phase birefringence curve B(λ) does cross at zero the zero-wavelength ordinate axis, for 1508 nm, where group birefringence is cancelled; the insert shows the shape of the micro-structured solid-core PM fiber described in [34], that has less anisotropy than the one described in [32], and Figure 24.