Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 19, Number 1, 2023
EOSAM 2022



Article Number  1  
Number of page(s)  20  
DOI  https://doi.org/10.1051/jeos/2022014  
Published online  13 January 2023 
Review Article
Comments about birefringence dispersion, with group and phase birefringence measurements in polarizationmaintaining fibers
^{1}
iXblue Photonics, Rue Paul Sabatier, 22300 Lannion, France
^{2}
iXblue, 34 Rue de la Croix de Fer, 78100 SaintGermainenLaye, France
^{3}
Laboratoire Hubert Curien, UJMCNRSIOGS, 18 Rue Professeur Benoît Lauras, 42000 SaintEtienne, France
^{*} Corresponding author: herve.lefevre@ixblue.com
Received:
27
September
2022
Accepted:
27
November
2022
A recent JEOSRP publication proposed Comments about Dispersion of Light Waves, and we present here complementary comments for birefringence dispersion in polarizationmaintaining (PM) fibers, and for its measurement techniques based on channeled spectrum analysis. We start by a study of early seminal papers, and we propose additional explanations to get a simpler understanding of the subject. A geometrical construction is described to relate phase birefringence to group birefringence, and it is applied to the measurement of several kinds of PM fibers using stressinduced photoelasticity, or shape birefringence. These measurements confirm clearly that the difference between group birefringence and phase birefringence is limited to 15–20% in stressinduced PM fibers (bowtie, panda, or tigereye), but that it can get up to a 3fold factor with an ellipticalcore (Ecore) fiber. There are also surprising results with solidcore microstructured PM fibers, that are based on shape birefringence, as Ecore fibers.
Key words: Birefringence / Birefringence dispersion / Channeledspectrum analysis / Group birefringence / Phase birefringence / Polarizationmaintaining fiber / Polarizationmode dispersion
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
A recent JEOSRP publication proposed Comments about Dispersion of Light Waves, as well as simple geometrical constructions to relate group index n_{g} to phase index n, and it showed that these constructions can be extended to phase birefringence B and group birefringence B_{g} [1], that is noted as G, in some publications. Here, we propose a complementary paper to develop these comments and constructions for birefringence dispersion in polarization maintaining (PM) fibers.
We also present experimental measurements of group birefringence B_{g}(λ) of several types of PM fibers over a wide spectrum range, knowing that it is quite easy today using a supercontinuum fiber source and channeled spectrum analysis with an optical spectrum analyzer (OSA).
We also show how to recover, from them, the value of phase birefringence B(λ) over this wide spectrum, with a singlewavelength phase measurement B(λ_{0}), since as it is wellknown, phase birefringence measurements are not as easy to perform as group measurements. For these singlewavelength phase measurements, we use fiber Bragg gratings (FBG) and RayleighOFDR (optical frequency domain reflectometry), and we explain why they measure phase birefringence.
As it will be shown, unlike the geometrical construction proposed in [1] that relates group birefringence B_{g}(λ) to phase birefringence B(λ), this requires now to relate phase birefringence B(λ) to group birefringence B_{g}(λ), to retrieve B(λ) over a wide spectrum. A reversed geometrical construction is proposed for a simpler understanding … in addition of some math, we must admit, since derivative, used to relate group to phase as seen in [1], is simple, whilst integration, needed to relate phase to group, yields firstly a primitive function (or indefinite integral), and then requires a reference at one wavelength to get a definite integral.
The preparation of this paper was also the opportunity to go back to early seminal papers on the subject. We may say that everything, or almost everything, that we are going to say can be found in the 1983 letter of Scott Rashleigh [2]. However, as you will see, this great threepage/onefigure letter deserves some complementary explanations, to say the least.
As the previous one [1], the present paper proposes many clear figures to ease the understanding, and it tries to avoid, as much as possible, complicated mathematical equations that might be not easily understandable.
2 Comments about early seminal papers
The first publication about the fabrication of what is called today polarizationmaintaining (PM) fiber was by Stolen et al. in 1978 [3]. This fiber was based on stressinduced birefringence, and it was called a highbirefringence (HB, or also HiBi) fiber since, as it was understood very early by Snitzer and Ostenberg [4], birefringence induces phase mismatch between both eigen polarization modes, which limits polarization degradation, since the various parasitic crossed couplings distributed along the fiber require phase matching to build efficiently a cumulative effect. This mismatch maintains a linear polarization that is coupled along one of the two principal axes of the highbirefringence fiber. These fibers were then called polarizationpreserving [5], or polarizationholding [6]. It is amusing to see that the same coauthors of these two papers, Rashleigh and Stolen who are very important pioneers in the domain, oscillated between preserving and holding, in terms of vocabulary. In any case, they became subsequently polarizationmaintaining (PM) fibers [7], which is the established vocabulary today.
The birefringence strength was originally given with the “beat length” (noted Λ), over which the relative phase difference between the two orthogonal modes changes by 2π [3]. Even if it might sound obvious, it could be useful to remind that a classical bulkoptic halfwave plate has a thickness of half a beat length, and it is a quarter of beat length for a quarterwave plate. This beat length was observed with transverse Rayleigh scattering [3, 8], which is working on the principle of a dipole radio antenna that cannot emit in a direction parallel to its axis. The equivalent of the antenna axis is the axis of the linear state of polarization. Like in a bulk crystal, launching a linear polarization at 45° of the principal birefringence axes yields a linear polarization that has been rotated by 90° after half the beat length, and that is back in the original direction after propagating along the whole beat length. Looking transversely at the fiber in the direction of the input polarization, there is no visible scattered light at the beginning, while there is scattering after half the beat length, since the polarization is now perpendicular to the direction of observation. One is back to no visible scattering after a full beat length since the polarization is parallel again. The fiber looks like a glowing dashed line [3], with a lightpattern periodicity equal to the beat length Λ (Fig. 1).
Figure 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. 
It is interesting to notice that this method is still influencing the vocabulary that is used today in the specification sheets of PM fiber manufacturers. Since the most common visible laser, at that time, was the helium–neon (He–Ne) laser, birefringence specification was, and is still given, with the beat length @ 633 nm, even if the operating wavelength is 1300 nm or 1550 nm! It is clear that the λ ^{−4} dependency of Rayleigh scattering does not make this sideview observation as easy for nearinfrared telecom wavelengths, as outlined by Rashleigh [2]. We shall see that phase birefringence B and group birefringence B_{g}, at the operating wavelength, are more relevant.
The simple and classical method that is used today for measuring birefringence of PM fibers is derived from what was proposed independently by Rashleigh in 1982 [9] and by Kikuchi and Okoshi in 1983 [10]. It was called wavelength scanning [2], or wavelength sweeping [10], but it is actually based on what is known as interferometric channeled spectrum analysis, in courses of optics, like Born and Wolf [11]. A broadband light source is coupled at 45° of the principal axes of the birefringent PM fiber, and there is a second polarizer at the output, with the same 45° orientation. The output channeled spectrum is then measured with an optical spectrum analyzer, OSA (Fig. 2).
Figure 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). 
This forms a polarimeter that is equivalent to an unbalanced twowave interferometer, considering the fast polarization mode as the short path, and the slow polarization mode as the long path. The polarizers at 45° are equivalent to the 50–50 spittersrecombiners of an interferometer, and the interferences are perfectly contrasted. The optical path length difference ∆L_{opt} is then:$$\Delta {L}_{\mathrm{opt}}=({n}_{\mathrm{eff}\mathrm{s}}{n}_{\mathrm{eff}\mathrm{f}})\cdot {L}_{f,}$$(1)where L_{f} is the PM fiber length, n_{eff−s} is the effective index of the slow (higherindex) mode, and n_{eff−f} is the one of the fast (lowerindex) mode. The phase birefringence B, also called modal birefringence, is given today with the difference between these effective phase indexes, as it is done in optics with birefringent crystals, such as quartz or lithium niobate:$$B=\Delta {n}_{\mathrm{eff}}={n}_{\mathrm{eff}\mathrm{s}}{n}_{\mathrm{eff}\mathrm{f}.}$$(2)There is a bright channel when ∆L_{opt} is an integer number m of the freespace wavelength λ(m) that corresponds to this mth channel, i.e., when:$$\Delta {L}_{\mathrm{opt}}=\left(m1\right)\cdot \lambda \left(m1\right),\hspace{0.5em}\mathrm{or}m\cdot \lambda \left(m\right),\hspace{0.5em}\mathrm{or}\left(m+1\right)\cdot \lambda \left(m+1\right).$$(3)The integer m − 1, or m, or m + 1 is called the channel order. These bright channels are equally spaced in inverse of wavelength, i.e., in spatial frequency σ(m) = 1/λ(m). The period of the spatial frequency spacing between adjacent bright channels is called the free spectral range, FSR_{σ} (Fig. 3). If one assumes that the birefringence B is not dispersive, i.e., that it does not vary with wavelength (or frequency), the result is very simple since, from (1) and (2), there is:$$\Delta {L}_{\mathrm{opt}}=B{\cdot L}_{f}=\Delta {n}_{\mathrm{eff}}\cdot {L}_{f}=m\cdot \lambda \left(m\right)=m/\sigma \left(m\right),\hspace{0.5em}\mathrm{and}\mathrm{then}:\sigma \left(m\right)=m/(B\cdot {L}_{f}).$$(4)Therefore, the free spectral range FSR_{σ} is:$${\mathrm{FSR}}_{\sigma}=\sigma \left(m\right)\sigma (m1)=[m/(B\cdot {L}_{f}\left)\right]\left[\right(m1)/(B\cdot {L}_{f}\left)\right]=1/B\cdot {L}_{f}$$(5)and it yields a simple measurement of the birefringence B (Fig. 3). This birefringence B is related to the beat length Λ by:$$B=\lambda /\mathrm{\Lambda}\hspace{0.5em}\mathrm{and}\hspace{0.5em}\mathrm{\Lambda}=\lambda \cdot {\mathrm{FSR}}_{\sigma}\cdot {L}_{f}.$$(6)This assumption of dispersionfree birefringence was done by Kikuchi and Okoshi in [10], as well as in an earlier Okoshi’s paper of 1981 [12] and, very often, it is still considered today.
Figure 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 quasiinfinite wavelength. 
Notice that is also possible to perform this channeled spectrum measurement with a tunable laser and a power meter (Fig. 4).
Figure 4 Principle of measurement of birefringence with channeled spectrum analysis, using a tunable laser and a power meter, instead of a broadspectrum source and an OSA. 
Rashleigh made the same assumption in his 1982 letter [9], but only for PM fibers based on stressinduced photoelasticity. He was clear about the fact that ellipticalcore (or Ecore) PM fibers are very dispersive, as we will see later in this paper. In his great 1983 letter [2], he corrected reference [10], saying that channeled spectrum analysis inherently measures group delay and not phase delay, even for PM fibers based on stressinduced birefringence. However, he did not explain very clearly why. It was obvious to him … but not obvious to us! As we will see in the following section, it is because the free spectral range, FSR, measured with channeled spectrum analysis, depends in fact on group birefringence B_{g}, and not on phase birefringence B. Group birefringence B_{g} is the difference between the effective group indexes of the two eigen modes [1]:$${B}_{g}=\Delta {n}_{\mathrm{g}\mathrm{eff}}={n}_{\mathrm{g}\mathrm{eff}\mathrm{s}}\u2013{n}_{\mathrm{g}\mathrm{eff}\mathrm{f}}.$$(7)With the term delay used by Rashleigh, one can notice that the vocabulary of that time was influenced by telecoms. Telecom scientists use angular frequencies to avoid the use of 2π in propagation equations. With this influence, phase birefringence was given [2, 9, 10, 12] as the difference ∆β between the propagation constant β_{s} = 2π · n_{eff−s}/λ of the slow mode and β_{f} = 2π · n_{eff−f}/λ, the one of the fast mode. These propagation constants are the effective angular spatial frequencies (or wave numbers) of the two eigen polarization modes. In addition, freespace angular spatial frequency (or angular wave number), k = 2π/λ, was used, instead of the wavelength λ. If you look at the figure of Rashleigh’s paper of 1983 [2], you see an angular wavenumber of 7.39 μm^{−1}, which corresponds in fact to a wavelength of 0.85 μm, as said in the text, and the birefringence ∆β is given in cm^{−1}. It is not easily understandable since the radian unit is obviated. Radian being a dimensionless unit, it can be obviated mathematically, but this does not help the understanding, as it was shown in [1] for group velocity dispersion (GVD). A wavelength λ of 0.85 μm corresponds to a spatial frequency σ of 1.176 μm^{−1}, and it is clearer to say that the angular spatial frequency or wave number k is 7.39 rad·μm^{−1} or rad/μm, i.e., 2π radian times 1.176 μm^{−1}, rather than to have 7.39 μm^{−1} that has, then, the same unit as σ.
In addition, the use of ∆β to define phase birefringence as a function of k is not very convenient since:$$\Delta \beta \left(k\right)=\Delta {n}_{\mathrm{eff}}\cdot k=B\cdot k.$$(8)Even with dispersionfree birefringence, i.e., when B is constant, ∆β(k) is not constant; it is proportional to k. As already seen, today, phase birefringence B is defined by the effective phase index difference ∆n_{eff}. Without dispersion, this index difference is constant. The other advantage of the use of the index is that it is a dimensionless value, which avoids dealing with such strange units as cm^{−1} , obviating radian.
It is interesting to revisit the figure of Rashleigh’s letter [2] of 1983 (Fig. 5). Following equation (8), the slope of the secant line that connects the origin to a point of the curve ∆β(k) is in fact the phase birefringence B, as it is defined today. A slope is a ratio, and when it is the ratio between parameters using the same strange unit … strangeness disappears!
Figure 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 bowtie fibers. Fiber C (red) combines stressinduced birefringence with form birefringence. Freespace wavenumber k should be read 7.39 rad·μm^{−1}, and it corresponds to a wavelength λ of 0.85 μm. 
Fiber A is a bowtie fiber [13] that is based on stressinduced birefringence and has a phase birefringence B = 5.2 × 10^{−4}, which is a typical value for industrial PM fiber products, today. Fiber B is a bowtie fiber with lower birefringence (3 × 10^{−4}). Both have little dispersion since the slope varies only slightly when k is changing. Fiber C combines stressinduced birefringence with form birefringence, created by an elliptical core. It has a higher phase birefringence (7 × 10^{−4}) but, mainly, a much higher dispersion.
It is interesting to notice that equation (8) of Rashleigh’s paper of 1983 [2] gives the group delay time difference τ_{g} between the two polarization modes:$${\tau}_{g}=({L}_{f}/c)\cdot \mathrm{d\Delta}\beta /\mathrm{d}k,$$(9)where L_{f} is the length of the PM fiber. Today, τ_{g} is called differential group delay, or DGD. Since group birefringence B_{g} equates τ_{g} · c/L_{f}, there is:$${B}_{g}=\mathrm{d\Delta}\beta /\mathrm{d}k.$$(10)Group birefringence B_{g} is the derivative of ∆β(k) with respect to k, i.e., d∆β/dk in Leibniz’s notation, or ∆β ^{′}(k) in Lagrange’s notation. Therefore, B_{g} is the slope of the tangent to the curve ∆β(k). Rashleigh’s figure can be revisited again (Fig. 6). With fiber C, that is very dispersive because of its elliptical core, and choosing λ = 1.3 μm, i.e., k = 4.83 rad μm^{−1}, as in Table 1 of [2], group birefringence B_{g} is 3.4 times phase birefringence B: 16 × 10^{−4} instead of 4.7 × 10^{−4}! Such an important difference will be confirmed with measurements of an ellipticalcore (or Ecore) PM fiber, later in this paper.
Figure 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! 
As we just saw, using ∆β as the definition of the birefringence has some geometrical interest with the use of the various slopes, but using ∆n_{eff} is much simpler, and it is the common meaning of phase birefringence, today.
3 How to understand simply channeled spectrum analysis
As we just saw, an unbalanced interferometer yields a channeled spectrum with a bright channel when the optical path difference ∆L_{opt} is equal to an integer number m of freespace wavelength λ. In the case of a polarimeter, ∆L_{opt} = B · L_{f} = ∆n_{eff} · L_{f}, where L_{f} is the length of birefringent PM fiber under test. In the general case where there is a wavelength dependence of the phase birefringence B:$$\Delta {L}_{\mathrm{opt}}=B\left[\lambda \left(m\right)\right]\cdot {L}_{f}=m\cdot \lambda \left(m\right),$$(11) $$m=B\left[\lambda \right(m\left)\right]\cdot [1/\lambda (m\left)\right]\cdot {L}_{f,}$$(12)where λ(m) is the freespace wavelength that corresponds to the mth channel, and B[λ(m)] is the phase birefringence of the PM fiber for this wavelength λ(m). Using now the corresponding spatial frequency σ(m) = 1/λ(m), there is:$$m=\Delta {L}_{\mathrm{opt}}\cdot \sigma \left(m\right)=B\left[\sigma \right(m\left)\right]\cdot \sigma \left(m\right)\cdot {L}_{f.}$$(13)Mathematically, one can consider that the spatial frequency σ(m) and the phase birefringence B[σ(m)] are continuous functions of the parameter m, a bright channel corresponding to an integer value of m. With continuous functions, one can differentiate, and:$$\mathrm{d}m=\mathrm{d}\left[B\left[\sigma \left(m\right)\right]\cdot \sigma \left(m\right)\right]\cdot {L}_{f}.$$(14)As seen in equation (4) of reference [1], the group index n_{g} is the derivative of the product of the phase (or refractive) index n by the frequency. With the angular temporal frequency ω = 2π · c/λ, there is:$${n}_{g}\left(\omega \right)=\mathrm{d}\left[n\right(\omega )\cdot \omega ]/\mathrm{d}\omega .$$(15)With the spatial frequency σ = ω/(2π · c), since dσ/σ = dω/ω, there is similarly:$${n}_{g}\left(\sigma \right)=\mathrm{d}\left[n\right(\sigma )\cdot \sigma ]/\mathrm{d}\sigma .$$(16)Since the birefringence is the difference between the indexes of the polarization modes, for phase as well as for group, it simply yields:$${B}_{g}\left(\sigma \right)=\mathrm{d}\left[B\right(\sigma )\cdot \sigma ]/\mathrm{d}\sigma .$$(17)Combining equations (14) and (17), one gets:$$\mathrm{d}m={B}_{g}\cdot \mathrm{d}\sigma \cdot {L}_{f}\hspace{0.5em}\mathrm{and}\hspace{0.5em}\mathrm{d}\sigma /\mathrm{d}m=1/({B}_{g}\cdot {L}_{f}).$$(18)The free spectral range in spatial frequency, FSR_{σ}, is the spatial frequency difference ∆σ corresponding to a parameter difference ∆m that equates 1. Since ∆σ = (dσ/dm) · ∆m, one gets:$${\mathrm{FSR}}_{\sigma}=\mathrm{d}\sigma /\mathrm{d}m=1/({B}_{g}\cdot {L}_{f}).$$(19)To summarize, the free spectral range FSR_{σ} depends on group birefringence B_{g}, and not on phase birefringence B, in the general case of birefringence dispersion. Theoretically, the position of the bright channels does depend on phase birefringence B, as seen in equation (13) but, in practice, there is no mean to know the precise value of the channel order m, since the broadband spectrum that is used remains limited. To have this precise value of m would require knowing all the channels down to zero frequency, i.e., an infinite wavelength, which is obviously not possible (Fig. 7). Notice that m is on the order of 10^{4} for 30 m of PM fiber. As said very early by Rashleigh [2], channeled spectrum analysis is inherently a group measurement!
Figure 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). 
Notice finally that there is a dual technique to measure a channeled spectrum: pathmatched white light interferometry. As it is wellknown, the interference power P, as a function of the path unbalance ∆L_{r} of an interferometer, is the Fourier transform (FT) of the power spectrum of the input light. With the channeled spectrum created by the PM fiber under test, it yields a central peak with the fringe amplitude following the coherence function of the original broadband source, and also two symmetrical secondary peaks with a contrast C of 0.5, for ∆L_{r} = ±B_{g} · L_{f} (Fig. 8). This technique was developed to make a distributed measurement of polarization crossed couplings along a PM fiber [14–18].
Figure 8 Principle of pathmatched whitelight interferometry: (a) setup with an unbalanced readout interferometer, using a beam splitter BS, and two mirrors M_{1} and M_{2}; (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 λ ^{2}/∆λ_{FWHM}. 
4 Measurement techniques of phase birefringence B
The first techniques to measure phase birefringence were based on creating a localized crossed polarization coupling in the PM fiber with the magnetooptic Faraday effect [19], or with the application of a transverse force that creates additional stresses in the fiber, which modifies the orientation of the principal birefringence axes [20–23]. The setups that were used are quite complicated, but they all rely on the same basic idea: when the localized crossed coupling is moved along the fiber, it yields a modulation of the processed signal that has a period equal to the beat length Λ, that is simply related to phase birefringence B with Λ = λ/B, as we saw in (6).
One can understand it easily by considering a single wavelength source, at λ_{0}, that is coupled along one of the principal axes of the PM fiber. A localized polarization crossed coupling is created, and there is an output polarizer at 45° of the principal axes (Fig. 9a). This yields a polarimeter and a channeled spectrum like what we just saw, considering that the fiber length L_{f} is now the length between the localized coupling and the fiber end. The only difference is that the contrast is not perfect anymore since the crossed coupling is equivalent to a low reflectivity splitter. Assuming a power splitting ratio of ɛ ^{2} and 1 − ɛ ^{2}, instead of 50–50, the contrast becomes 2ɛ. As wellknown in interferometry, the contrast depends on the amplitude ratio ɛ between the two interfering waves, and not on the power ratio ɛ ^{2}. It can be viewed simply, considering a main wave with a normalized amplitude of 1 and a low wave with a normalized amplitude of ɛ. Depending on their relative phase shift, the interference amplitude oscillates between 1 + ɛ and 1 − ɛ; the interference power is then (1 ± ɛ)^{2} = 1 ± 2ɛ + ɛ ^{2} ≈ 1 ± 2ɛ.
Figure 9 Measurement of phase birefringence B(λ_{0}) of a PM fiber: (a) setup with a moving crossedpolarization coupling; (b) resulting channeled spectrum (black sinusoidal curve) for a length L_{f} corresponding to a bright channel of order m for λ_{0}; 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). 
As seen in (11), there is a bright channel when the length L_{f} between the coupling point and the PM fiber end yields an optical path length difference ∆L_{opt} equal to an integer number m of wavelength λ_{0}:$$\Delta {L}_{\mathrm{opt}}=B\left({\lambda}_{0}\right)\cdot {L}_{f}=m\cdot {\lambda}_{0}.$$(20)If this length L_{f} is reduced by a quarter of a beat length Λ, the channeled spectrum becomes shifted by a quarter of the fringe period since B(λ_{0}) · Λ = λ_{0}, which reduces the output power. For half the beat length, the shift is half the period, and the power becomes minimum; for a full beat length, it is back to the maximum, but the channel order is then m − 1:$$B\left({\lambda}_{0}\right)\cdot \left({L}_{f}\mathrm{\Lambda}\right)=\left(m1\right)\cdot {\lambda}_{0}.$$(21)As we saw, channeled spectrum analysis does not allow one to get the precise value m of the channel order, but it is possible to know its relative variation by moving the crossed polarization coupling, which yields a measurement of the beat length Λ, and then of the phase birefringence B(λ_{0}).
Another technique of phase birefringence measurement is the use of a fiber Bragg grating (FBG) [24], with a very low reflectivity to ensure that it does not modify the birefringence. The reflected wavelength λ_{R} is given by:$${\lambda}_{R}=2{n}_{\mathrm{eff}}\cdot {\mathrm{\Lambda}}_{\mathrm{Bragg}},$$(22)where n_{eff} is the effective index of the fiber mode and Λ_{Bragg} is the period of the Bragg grating. A PM fiber being birefringent by principle, it yields two reflected wavelengths, λ_{s} and λ_{f}, that correspond respectively to the effective index n_{eff−s} of the slow mode, and n_{eff−f}, the one of the fast mode (Fig. 10), with:$${\lambda}_{s}=2{n}_{\mathrm{eff}\mathrm{s}}\left({\lambda}_{s}\right)\cdot {\mathrm{\Lambda}}_{\mathrm{Bragg}}\hspace{0.5em}\mathrm{and}{\lambda}_{f}=2{n}_{\mathrm{eff}\mathrm{f}}\left({\lambda}_{f}\right)\cdot {\mathrm{\Lambda}}_{\mathrm{Bragg}}.$$(23)The phase birefringence B is:$$B\left({\lambda}_{f}\right)={n}_{\mathrm{eff}\mathrm{s}}\left({\lambda}_{f}\right){n}_{\mathrm{eff}\mathrm{f}}\left({\lambda}_{f}\right)\hspace{0.5em}\mathrm{or}\hspace{0.5em}B\left({\lambda}_{s}\right)={n}_{\mathrm{eff}\mathrm{s}}\left({\lambda}_{s}\right){n}_{\mathrm{eff}\mathrm{f}}\left({\lambda}_{s}\right).$$(24)There are:$${n}_{\mathrm{eff}\mathrm{s}}\left({\lambda}_{f}\right)={n}_{\mathrm{eff}\mathrm{s}}\left({\lambda}_{s}\right)+\left[\right({\lambda}_{f}{\lambda}_{s})\cdot (\mathrm{d}{n}_{\mathrm{eff}\mathrm{s}}/\mathrm{d}\lambda \left)\right]\hspace{0.5em}\mathrm{and}\hspace{1em}{n}_{\mathrm{eff}\mathrm{f}}\left({\lambda}_{s}\right)={n}_{\mathrm{eff}\mathrm{f}}\left({\lambda}_{f}\right)+\left[\right({\lambda}_{s}{\lambda}_{f})\cdot (\mathrm{d}{n}_{\mathrm{eff}\mathrm{f}}/\mathrm{d}\lambda \left)\right].$$(25)Assuming that dn_{eff−s}/dλ ≈ dn_{eff−f}/dλ ≈ dn_{eff}/dλ, it yields:$$B\left({\lambda}_{f}\right)\approx B\left({\lambda}_{s}\right)\approx ({\lambda}_{s}{\lambda}_{f})\cdot \left[\right(1/\left(2{\mathrm{\Lambda}}_{\mathrm{Bragg}}\right))(\mathrm{d}{n}_{\mathrm{eff}}/\mathrm{d}\lambda \left)\right].$$(26)Phase dispersion ${\mathrm{d}n}_{\mathrm{Si}{\mathrm{O}}_{2}}/\mathrm{d}\lambda $ of silica is about −0.01 μm^{−1} in the 1200–1600 nm range [1], and waveguide phase dispersion is about −0.005 μm^{−1} for a single mode fiber with a numerical aperture NA of 0.19, i.e., a core index step of 0.0125 [1], whilst 1/(2Λ_{Bragg}) is about 1 μm^{−1} at 1550 nm. Therefore, within an approximation of 1.5%, phase birefringence B(λ) is simply:$$B\left(\lambda \right)\approx ({\lambda}_{s}{\lambda}_{f})/\left(2{\mathrm{\Lambda}}_{\mathrm{Bragg}}\right).$$(27)A last method is based on RayleighOFDR (Rayleigh opticalfrequencydomain reflectometry). This technique developed by Froggatt and coworkers [25] is impressive, and a commercial test instrument is available (OBR 4600 by Luna). OFDR was proposed very early by Eickhoff and Ulrich [26]. They explained it in terms of radar specialists with frequency chirp, temporal delay, and frequency beating, but it can be also viewed as channeled spectrum analysis, which is much easier to understand for opticsphotonics specialists.
Figure 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 (lowerindex) mode, while λ_{s} is the one of the slow (higherindex) mode. 
It is like what we saw earlier in Figure 9. Instead of a polarimeter, it is a twowave interferometer with a reference mirror on one arm and a small reflection (ɛ ^{2} in power) on the measurement arm. It yields a channeled spectrum that is measured with a tunable laser; the contrast is 2ɛ, and there are bright channels when the optical path length difference, ∆L_{opt} = 2n_{eff} . ∆L_{f}, is equal to an integer number of wavelength λ. ∆L_{f} is the fiber length difference between the reference mirror and the low reflection, and n_{eff} is the effective index of the mode (Fig. 11).
Figure 11 Principle of OFDR: (a) setup with a frequencyswept tunable laser, an unbalanced twowave interferometer with a reference mirror on one arm, and the low reflection point (ɛ ^{2} in power, and ɛ in amplitude) that is analyzed, on the other arm; (b) measured channeled spectrum with a contrast of 2ɛ. 
As in (13) for the polarimeter, the mth channel follows m = ∆L_{opt} · σ(m) and, for the interferometer, it yields:$$m=2{n}_{\mathrm{eff}}\cdot \Delta {L}_{f}\cdot \sigma \left(m\right).$$(28)With a PM fiber, one can select the slow mode or the fast one, and there is:$$m=2{n}_{\mathrm{eff}\mathrm{s}}\cdot \Delta {L}_{f}\cdot {\sigma}_{s}\left(m\right)=2{n}_{\mathrm{eff}\mathrm{f}}\cdot \Delta {L}_{f}\cdot {\sigma}_{f}\left(m\right).$$(29)There is a frequency shift ∆σ between the respective channeled spectra (Fig. 12a). The frequency of the mth channel of the slow mode is σ_{s}(m), and it is σ_{f}(m) for the fast mode. Since:$${n}_{\mathrm{eff}\mathrm{s}}\cdot {\sigma}_{s}\left(m\right)={n}_{\mathrm{eff}\mathrm{f}}\cdot {\sigma}_{f}\left(m\right).$$(30)There is:$$B={n}_{\mathrm{eff}\mathrm{s}}{n}_{\mathrm{eff}\mathrm{f}}={n}_{\mathrm{eff}\mathrm{f}}\cdot ({\sigma}_{f}{\sigma}_{s})/{\sigma}_{s}.$$(31)
Figure 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 crosscorrelated. 
Considering the mean effective index n_{eff}, and the mean spatial frequency σ, it yields:$$B\approx {n}_{\mathrm{eff}}\cdot ({\sigma}_{f}{\sigma}_{s})/\sigma ={n}_{\mathrm{eff}}\cdot \Delta \sigma /\sigma .$$(32)With (32), one could then think that it is possible to measure phase birefringence B, while we saw earlier that channeled spectrum analysis inherently measures group delay and not phase delay [2]. Actually, it is not possible because one can decide that a certain channel of the slowmode spectrum is the mth one, but there is no mean to know where is the corresponding mth channel of the fastmode spectrum: shifted sine functions cannot be correlated over more than one period (Fig. 12a).
The question is then why is it possible to measure phase birefringence B with RayleighOFDR, as proposed by Froggatt and coworkers [25]? As seen in Section 2, Rayleigh scattering is based on dipolar antenna emission of the numerous covalent Si–O bonds of silica, that are excited by the E field of the incoming light. Silica has a glassy (or amorphous) structure, and these bonds are randomly positioned; they are equivalent to many lowreflection points randomly distributed along the fiber. With channeledspectrum analysis, it yields a channeled spectrum that looks like a white noise, instead of the sine function obtained with a single reflector. Now, the randomness of the positions of these reflectors is fixed for a given fiber, and the Rayleigh channeled spectrum obtained with OFDR is an actual signature of the fiber [25]. Testing with the slow mode and the fast mode yields two identical noisy channeled spectra that are offset by the frequency shift ∆σ (Fig. 12b). This shift is identical to the first case with sine spectra but now, noisy channeled spectra can be crosscorrelated to know ∆σ, and then phase birefringence B, with (32), which is a great idea. However, this explanation is very simplified to allow the reader to grasp the principle of the method, but the signal processing of RayleighOFDR is actually very sophisticated [25] … and impressive!
We saw in Sections 2 and 3 that channel spectrum analysis of birefringence can use three equivalent methods: broadband source with an OSA, tunable laser, or pathmatched whitelight interferometry. It is the same for reflectometry but, if the three methods are equivalent, they do not have at all the same noise floor to detect very low reflections. With an OSA, the simplest technique, it is only −70 dB, which corresponds to a backreflection of 3 × 10^{−4} in amplitude, and 10^{−7} in power; with pathmatched white light interferometry, it is typically −90/−100 dB; with a tunable laser, as used in OFDR, it can go down to −130 dB [18]. Rayleigh backscattering signal is around −110 dB, and it requires OFDR to be measurable.
These three equivalent methods do not have at all the same spectral resolution either. For an OSA (Fig. 2), it is typically ∆λ_{OSA} ≈ 0.01 nm. The free spectral range in spatial frequency, FSR_{σ}, is 1/∆L_{optg}, where ∆L_{optg} is the group optical path length difference, i.e., (n_{g}/n) · ∆L_{opt}. In wavelength, it is:$${\mathrm{FSR}}_{\lambda}=\frac{{\lambda}^{2}}{\Delta {L}_{\mathrm{opt}\mathrm{g}}}.$$(33)Then, the maximum path length, ∆L_{max}, that can be explored is:$$\Delta {L}_{\mathrm{max}}={\lambda}^{2}/\Delta {\lambda}_{\mathrm{OSA}}.$$(34)For ∆λ_{OSA} = 0.01 nm at 1550 nm, ∆L_{max} is about 25 cm; it means about 8 cm of fiber in reflectometry, where n_{g} ≈ 1.45, and about 500 m in a polarimeter, where B_{g} ≈ 5 × 10^{−4}.
With pathmatched interferometry, it depends on the travel distance of the moving mirror M_{2} of the readout interferometer (Fig. 8). In practice, it is difficult to have more than few meters, even with a multifold path. With a tunable laser that has a very narrow linewidth, it yields a great spectral resolution, and the maximum path length, ∆L_{max}, can go up to few kilometers, as seen with the OBR 4600 of Luna. Impressive!
5 How to derive phase birefringence B(λ) from group birefringence B_{g}(λ)
As we saw, measurement of group birefringence B_{g}(λ) over a wide spectrum is very easy with channeled spectrum analysis using a broadband source and an OSA, while measurement of phase birefringence B is more complicated, even if it is not too difficult for a single wavelength. Reference [1] presented a geometrical construction to derive group birefringence B_{g}(λ) from phase birefringence B(λ), while one has now to derive B(λ) from B_{g}(λ) and a single measurement B(λ_{0}).
As seen in [1], group birefringence B_{g}(λ) is related to phase birefringence B(λ) with:$${B}_{g}\left(\lambda \right)=B\left(\lambda \right)\u2013\left[\lambda \cdot {B}^{\mathrm{\prime}}\left(\lambda \right)\right].$$(35)And the tangent T_{i}(λ) to the curve B(λ), at the wavelength λ_{i}, follows:$${T}_{i}\left(\lambda \right)=B\left({\lambda}_{i}\right)+\left[\left(\lambda {\lambda}_{i}\right)\cdot {B}^{\mathrm{\prime}}\left({\lambda}_{i}\right)\right].$$(36)Then:$${T}_{i}\left(0\right)=B\left({\lambda}_{i}\right)\left[\left({\lambda}_{i}\right)\cdot {B}^{\mathrm{\prime}}\left({\lambda}_{i}\right)\right]={B}_{g}\left({\lambda}_{i}\right).$$(37)Knowing phase birefringence B(λ) and its derivative B ^{′}(λ), over a wide spectrum, allows one to retrieve group birefringence B_{g}(λ) over this wide spectrum with a simple geometrical construction, using the tangent to B(λ) (Fig. 13).
Figure 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. 
Conversely, if one knows the whole curve B_{g}(λ) and a single value B(λ_{1}) for phase birefringence, one can define the tangent T_{1}(λ) and the derivative B ^{′}(λ_{1}). However, one cannot define yet the tangent T_{2}(λ), nor B(λ_{2}), for another wavelength λ_{2}. One must do some math to get it.
We saw in (35) that B_{g}(λ) = B(λ) − [λ · B′(λ)]. It yields for the derivatives:$${B}_{g}^{\mathrm{\prime}}\left(\lambda \right)=\lambda \cdot {B}^{\mathrm{\u2033}}\left(\lambda \right)\mathrm{and}{B}^{\mathrm{\u2033}}\left(\lambda \right)={B}_{g}^{\mathrm{\prime}}\left(\lambda \right)/\lambda $$(38)and there is also:$$B\left(\lambda \right)={B}_{g}\left(\mathrm{d}\lambda \right)+\left[\lambda \cdot B\mathrm{\prime}\left(\lambda \right)\right].$$(39)One can integrate B″(λ), seen in (38), to get B′(λ):$${B}^{\mathrm{\prime}}\left({\lambda}_{2}\right)=B\mathrm{\prime}\left({\lambda}_{1}\right){\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\left({B\mathrm{\prime}}_{g}^{}\right(\lambda )/\lambda )\cdot \mathrm{d}\lambda $$(40)and phase birefringence B(λ) for λ_{2} can be retrieved:$$B\left({\lambda}_{2}\right)={B}_{g}\left({\lambda}_{2}\right)+\left[{\lambda}_{2}\cdot {B}^{\mathrm{\prime}}\left({\lambda}_{1}\right)\right]\u2013\left[{\lambda}_{2}\cdot {\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\left(\frac{{B}_{g}^{\text{'}}\left(\lambda \right)}{\lambda}\right)\cdot \mathrm{d}\lambda \right].$$(41)With a small interval between λ_{1} and λ_{2}, one can consider that the slope ${B}_{g}^{\text{'}}$ of B_{g}(λ) is constant and:$$B\left({\lambda}_{2}\right)={B}_{g}\left({\lambda}_{2}\right)+\left[{\lambda}_{2}\cdot {B}^{\mathrm{\prime}}\left({\lambda}_{1}\right)\right]\u2013\left[{\lambda}_{2}\cdot {B}_{g}^{\mathrm{\prime}}\cdot {\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\frac{\mathrm{d}\lambda}{\lambda}\right],$$(42) $$B\left({\lambda}_{2}\right)={B}_{g}\left({\lambda}_{2}\right)+[{\lambda}_{2}\cdot {B}^{\mathrm{\prime}}({\lambda}_{1}\left)\right]\u2013[{\lambda}_{2}\cdot {B}_{g}^{\mathrm{\prime}}\cdot \mathrm{ln}({\lambda}_{2}/{\lambda}_{1}\left)\right].$$(43)A geometrical construction remains possible, using again the tangents T_{i}(λ). Equations (36) and (37) yield:$${T}_{i}\left(\lambda \right)={B}_{g}\left({\lambda}_{i}\right)+\left[\lambda \cdot \mathrm{B\text{'}}\left({\lambda}_{i}\right)\right].$$(44)They cross each other for the wavelength λ_{x} where T_{1}(λ_{x}) = T_{2}(λ_{x}), and then:$${\lambda}_{x}=\left[{B}_{g}\right({\lambda}_{2}){B}_{g}({\lambda}_{1}\left)\right]/\left[{B}^{\mathrm{\prime}}\right({\lambda}_{2})\u2013{B}^{\mathrm{\prime}}({\lambda}_{1}\left)\right].$$(45)
There is ${B}_{g}\left({\lambda}_{2}\right){B}_{g}\left({\lambda}_{1}\right)\approx {\mathrm{B\text{'}}}_{g}^{}\cdot ({\lambda}_{2}{\lambda}_{1})$ and, from (40):$${B}^{\mathrm{\prime}}\left({\lambda}_{2}\right)\u2013{B}^{\mathrm{\prime}}\left({\lambda}_{1}\right)\approx {\mathrm{B\text{'}}}_{g}^{}\cdot {\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\frac{\mathrm{d}\lambda}{\lambda}={\mathrm{B\text{'}}}_{g}^{}\cdot \mathrm{ln}({\lambda}_{2}/{\lambda}_{1}).$$(46)It yields:$${\lambda}_{x}=({\lambda}_{2}{\lambda}_{1})/\mathrm{ln}({\lambda}_{2}/{\lambda}_{1}).$$(47)The Taylor series of natural logarithm, ln, leads to:$$\mathrm{ln}(1+\delta )=\delta {\delta}^{2}/2+{\delta}^{3}/3{\delta}^{4}/4+\cdots $$(48)Therefore, when the interval δ = (λ_{2} − λ_{1})/λ_{1} is small, there is:$${\lambda}_{x}\approx {\lambda}_{1}(1+\delta /2)=({\lambda}_{2}+{\lambda}_{1})/2.$$(49)The wavelength λ_{x} is simply in the middle of the interval between λ_{1} and λ_{2}, which makes the geometrical construction easy (Fig. 14). We know B_{g}(λ) and B(λ_{1}); we can deduce T_{1}(λ) with B_{g}(λ_{1}) and B(λ_{1}); we can find T_{2}(λ) with B_{g}(λ_{2}), and also T_{2}(λ_{x}) = T_{1}(λ_{x}), with λ_{x} ≈ (λ_{2} + λ_{1})/2; finally, we can construct the curve B(λ) between λ_{1} and λ_{2}, with the two tangents T_{2}(λ) and T_{1}(λ).
Figure 14 Geometrical construction to derive phase birefringence B(λ) from group birefringence B_{g}(λ) (black solid line) and a singlewavelength phase measurement B(λ_{1}) (red circular dot); (a) construction of the tangent T_{2}(λ) that crosses tangent T_{1}(λ) at λ_{x} = (λ_{2} − λ_{1})/ln(λ_{2}/λ_{1}) (dashed blue lines), which allows one to find B(λ_{2});when the interval (λ_{2} − λ_{1})/λ_{1} is small, λ_{x} ≈ (λ_{2} + λ_{1})/2; (b) construction of B(λ) (dashed red curve) that must fit the tangents T_{1}(λ) and T_{2}(λ). 
6 Origin of birefringence dispersion in PM fibers using stressinduced birefringence
The method, that is now widely generalized to make PM fibers, is to use stressinduced linear birefringence with additional materials that have a thermal expansion coefficient larger than silica (several 10^{−6}/°C instead of 5 × 10^{−7}/°C for pure silica). The fiber preform is fabricated with two rods (sometimes called stressapplying parts, or SAPs) of highly doped silica (usually with boron and/or phosphorous) located on each side of the core region. Present PM products are based on three techniques that lead to similar results, as we shall see in the next section. They have nicknames based on obvious similarities with their look: bowtie, panda and tigereye (Fig. 15). After pulling the fiber at high temperature, these highly doped rods will tend to contract on cooling, but their thermal contraction is blocked by the surrounding silica, which has a much lower thermal contraction. This puts the rods under quasiisotropic tensile stress, and, by reaction, this also induces stresses in the core region where light propagates; there is a tensile stress in the xaxis of the rods and a compressive stress in the perpendicular yaxis. Because of the photoelastic effect, these anisotropic stresses yield anisotropic index changes, i.e., birefringence. These stresses are quite uniform along the xaxis between the rods (or SAPs), but they decrease rapidly along the perpendicular yaxis (Fig. 16). This creates birefringence dispersion since a longer wavelength yields a wider mode that sees lower stresses, which reduces the birefringence. The derivative B ^{′}(λ) of phase birefringence B(λ) is negative, and then, group birefringence B_{g}(λ) is larger than B(λ), since B_{g}(λ) = B(λ) – [λ · B ^{′}(λ)], as seen in (35).
Figure 15 Different kinds of PM fibers based on stressinduced linear birefringence: bowtie, panda and tigereye. The slow axis is along the axis of the stress structure, and the fast axis is perpendicular. 
Figure 16 Principle of PM fibers based on stressinduced linear birefringence: (a) SAPs (panda structure, here) are under quasiisotropic tensile stress, and they pull on the fiber cladding; (b) in the core region, this yields tensile stress (red arrows) in the xaxis, and a compressive stress (blue arrows) in the perpendicular yaxis; (c) these stresses decrease when one moves away from the core, along the perpendicular yaxis. 
We made a numerical model that confirms this behavior. In this model, we do not include the circular core to avoid being blinded by the significant isotropic stress that this one generates. Because of the high thermal expansion of germania (GeO_{2}) glass, the germanosilicate core behaves like the SAPs, but the additional isotropic stress that is created does not give any birefringence. To suppress the core in the model is easing the understanding: we clearly see that the compressive stress T_{y}(x,y) does decrease, in absolute value, along the yaxis to get to zero at the outer surface of the cladding (Fig. 17).
Figure 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 yaxis that decreases along this yaxis, and gets to zero at the outer limit of the cladding. 
The stress T_{x}(x,y) on the xaxis does also decrease along the yaxis; this is even faster than for T_{y}(0,y). The curve T_{x}(0,y) is positive (tensile stress) in the core region, becomes null and gets negative (compressive stress) (Fig. 18).
Figure 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 xaxis, along the yaxis: it is positive in the core region, and it becomes negative at a distance that corresponds to about the height of the SAPs. 
Note that the color code produces an artefact: when the stresses are at 45° of the axes, the color corresponds to the sum of the projections on these axes. Around the SAPs, the orthogonal stress is positive (tensile), while the tangential stress is negative (compressive); they have about the same absolute value and, at 45°, the sum of their projections on the x or y axes becomes null … but it does not mean that the stresses are null (Fig. 19)!
Figure 19 Stresses around the SAPs: (a) at 0° and 90°; (b) at 45°, where the sum of their projections on the x or yaxis becomes null … but it does not mean that the stresses are null! 
There is a second source of birefringence dispersion that is not often mentioned: dispersion of the photoelastic effect in silica. This effect is usually treated with two photoelastic coefficients: p_{11} and p_{12} that are dimensionless since they are related to elastic strains that are also dimensionless. They are assumed to be wavelength independent in most publications, with p_{11} = 0.121 and p_{12} = 0.270. To be honest, it is what we had in mind!
It is Rashleigh … again … who outlined in his great 1983 letter [2] that it is not completely negligible. In his references, he cited a publication of Sinha [27] about this dispersive effect in bulk silica. With Figure 4 of this Sinha’s paper, one can estimate that the relative variation of the photoelastic effect is about minus 0.05 μm^{−1} in the 1.2–1.6 μm range. Applying this on the geometrical construction that relates group birefringence to phase birefringence (Fig. 13), it yields a difference of 8% between them, at 1550 nm. The very interesting result of Sinha’s paper is that the relative dispersion of the photoelastic effect in silica, i.e., fused quartz, follows the same law as the relative dispersion of birefringence in crystallin quartz!
A similar relative variation of −0.05 μm^{−1} can be evaluated with Figure 6 of reference [28], that also cites Sinha [27]. Note that the title of this last reference of 1983 used modal birefringence for what is called phase birefringence today, and polarization mode dispersion (PMD) for what is called group birefringence, as outlined in Section 5 of [1]. There is the same use of vocabulary in reference [22] of 1993. We prefer to save the term PMD for coupledmode PMD in lowbirefringence telecom fibers, where it is a random process that grows as the square root of the length [29]. With highbirefringence PM fibers, it is actually intrinsic PMD that grows linearly with the fiber length. In this case, the concept of group birefringence looks more convenient to us [1].
7 Measurement of birefringence dispersion in PM fibers that use stress induced birefringence.
We performed measurements of birefringence of the three standard kinds of PM fibers based on stressinduced birefringence: bowtie, panda and tigereye (Fig. 15). Group birefringence B_{g} was measured with channeled spectrum analysis over a very wide spectrum (1250–1650 nm), using an inhouse supercontinuum fiber source and an OSA (Fig. 2). The three fibers have about the same characteristics: cutoff wavelength λ_{c} around 1250–1350 nm for an operating wavelength of 1550 nm, numerical aperture (NA) of 0.19, i.e., core index step of 0.0125, modefield diameter at 1/e (MFD) of 7 μm, and cladding diameter of 80 μm. Phase birefringence B was measured at 1550 nm with a Bragg grating, and also with RayleighOFDR (OBR 4600 of Luna), with similar results. B was then deduced over the 1250–1650 nm range with the geometrical construction of Figure 14.
Results presented in Figure 20 show that the three fibers have similar phase birefringence B(λ): around 4.2 × 10^{−4}; it corresponds to a beat length Λ of about 3.7 mm @ 1550 nm. As we already saw, the historical habit is to give the beat length @ 633 nm in commercial product specification sheets, but it is just calculated with the ratio of the wavelengths, which would yield here 3.7 × (633/1550) ≈ 1.5 mm. Because of birefringence dispersion, it is clear that this habit should be abandoned, or at least explained.
Figure 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) bowtie fiber; (b) panda fiber; (c) tigereye fiber. Photographs of the end face of the fibers are inserted; SAPs are dark, while the circular cores are illuminated. 
From (35), one sees that the slope of the relative variation of the phase birefringence B(λ) is:$${B}^{\mathrm{\prime}}/B=[1\u2013({B}_{g}/B\left)\right]/\lambda .$$(50)At 1550 nm, it is −0.11 μm^{−1} for the bowtie fiber, −0.09 μm^{−1} for the panda fiber, and −0.17 μm^{−1} for the tigereye fiber, when the sole relative dispersion of the photoelastic effect is −0.05 μm^{−1}, as we just saw. The breakdown between the dispersion due to the decay of the stresses, away from the core, and the pure photoelastic dispersion is about half and half for the bowtie and panda fibers, and it is about two thirdsone third for the tigereye fiber. This last higher value is due to the smaller height of the SAPs of the tigereye fiber, which yields a faster decay of the stresses, away from the core.
The three fibers have also a significant dispersion of group birefringence, with a relative variation ${B}_{g}^{\text{'}}/{B}_{g}$ of about +0.16 μm^{−1}, at 1550 nm. This must be considered with distributed measurement of polarization crossed couplings along a PM fiber [17, 18].
8 Case of an ellipticalcore (Ecore) PM fiber
The case of an ellipticalcore, or Ecore, PM fiber (Fig. 21) is much more complicated. There is shape birefringence induced by the elliptical core, but also stressinduced birefringence, and both are very dispersive. For shape birefringence, it involves difficult math. There are many publications on the subject, but, for us, the less complicated one is by Kumar and Ghatak [30] … even if they must solve two simple transcendental equations … which can be viewed as an oxymoron! A first simple important result is that shape phase birefringence depends on the square of the index step ∆n of the core: this requires a highindex step to get enough birefringence, which implies a very small core to remain single mode. The other simple important result is that this shape birefringence saturates when the ratio between the majoraxis width, 2a, and the minoraxis one, 2b, becomes higher than 2: it reaches the birefringence of a planar waveguide that has a thickness of 2b.
Figure 21 Ellipticalcore (Ecore) PM fiber, based on shape birefringence, with its slow and fast axes. 
This calls for a comment: the origin of shape birefringence can be grasped simply. As wellknown, or at least known, in bulk optics, there is a phase shift with total internal reflection (TIR) and it depends on the state of polarization, s (perpendicular to the plane of incidence; s comes from senkrech, i.e., perpendicular in German), or p (parallel to the plane of incidence) (Fig. 22). These states correspond respectively to the TE (transverse electric) mode or the TM (transverse magnetic) mode of a planar waveguide. Shape birefringence is, for us, simply due to this difference of phase shift. It is the opportunity to recall that TIR phase shift difference is the principle of the Fresnel rhomb that gives a quarterwave retarder. It is a very broadband retarder since the retardation depends on the index n that has only a small dispersion. A classical quarterwave (Λ/4) plate using a birefringent crystal, creates an optical path difference ∆L_{opt}, and not a phase retardation, strictly speaking. This path delay does yield a phase retardation ∆φ, but this retardation is inversely proportional to the wavelength λ, since ∆φ = 2π · ∆L_{opt}/λ, as it is wellknown.
Figure 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. 
The origin of stressinduced birefringence is complicated too. The highlydoped core is under tensile stress, as what we saw in the previous section for the SAPs of PM fibers (Fig. 19), while there is also some compressive stress in the surrounding cladding. With a circular core, the stresses are isotropic, but it is not the case anymore with an elliptical shape, which induces birefringence. To be honest, the theory of this effect is not clear to us … and it is not clear in the literature. It will require some more analysis.
In any case, the total phase birefringence is very dispersive as confirmed by measurements at various wavelengths by Urbanczyk and coworkers [31]. On our side, we performed measurement of group birefringence over a wide spectrum (1000–1650 nm), and we deduced phase birefringence with a single wavelength measurement, as what we did for the PM fibers with SAPs, seen in the previous section (Fig. 23). The elliptical core has a majoraxis 2a width of 4 μm and a minoraxis width 2b of 2 μm; the cutoff wavelength λ_{c} is 1050 nm for an operating wavelength of 1300 nm; the numerical aperture (NA) is 0.27, i.e., a core index step of 0.025; the cladding diameter is 80 μm. It is confirmed that phasebirefringence dispersion is very high, which yields a big difference between group and phase birefringences. At the operating wavelength of 1300 nm, B is only 2.3 × 10^{−4}, while B_{g} is 5.8 × 10^{−4}, i.e., 2.5 times bigger. We directly checked this big difference with an additional Bragg grating at 1300 nm to confirm it.
Figure 23 Measurement, by channeled spectrum analysis, of group birefringence B_{g}(λ) of an Ecore fiber, over 1000–1650 nm (black square dots); measurement of phase birefringence B at 1550 nm with Bragg grating and RayleighOFDR (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. 
Finally, as described by Mélin and coworkers [32], shape birefringence is also found in solidcore microstructured PM fiber, where there is a very strong phase birefringence: few 10^{−3}, when it is typically 5 × 10^{−4} in classical PM fibers. There is also an interesting effect: the value of phase birefringence dispersion dB/dλ is very high, and it is positive, i.e., anomalous [1]. With the geometrical construction that was discussed in Figure 13, it is easy to see that group birefringence can become negative! Slow and fast modes are inverted, between phase birefringence and group birefringence. Revisiting Figure 3 of [32], phase birefringence at 1550 nm is about 3.6 × 10^{−3}, and phasebirefringence dispersion dB/dλ is about 5 × 10^{−3} μm^{−1}; using (35), one finds a group birefringence B_{g} of minus 4.15 × 10^{−3} (Fig. 24).Worth to outline, isn’t it?
Figure 24 Revisited Figure 3 of [32]: the phasebirefringence 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 microstructured solidcore PM fiber described in [32]. 
It is also worth to outline that the shape birefringence of such solidcore microstructured PM fibers can be engineered to get cancellation of group birefringence at a specific wavelength, as described by Morin and coworkers [33], and Kibler and coworkers [34]. Their figures can be revisited as well, and this confirms that the tangent to the phase birefringence curves B(λ) does crosse at zero the ordinate axis corresponding to λ = 0, when group birefringence B_{g} is cancelled (Figs. 25 and 26). The slopes are inverted, because these two papers defined phase birefringence as negative, which is just a matter of convention.
Figure 25 Revisited Figure 3b of [33]: the tangent to phase birefringence curve B(λ) does cross at zero the zerowavelength 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. 
Figure 26 Revisited Figure 1b of [34]: the tangent to phase birefringence curve B(λ) does cross at zero the zerowavelength ordinate axis, for 1508 nm, where group birefringence is cancelled; the insert shows the shape of the microstructured solidcore PM fiber described in [34], that has less anisotropy than the one described in [32], and Figure 24. 
It is interesting to compare the result of [32], revisited in Figure 24, with the one of [34], revisited in Figure 26. As seen in the inserts, the anisotropy of the shape of the microstructured solidcore PM fiber of [34] is smaller than the one of [32], which yields less phase birefringence B (1.65 × 10^{−3} instead of 3.4 × 10^{−3} @ 1.5 μm, in absolute values), and less phase birefringence dispersion dB/dλ, or B ^{′} (1.05 × 10^{−3} μm^{−1} instead of 5 × 10^{−3} μm^{−1} @ 1.5 μm, in absolute values). According to equation (35), the condition for nulling out B_{g}(λ) is λ · B ^{′}(λ) = B(λ), i.e., the relative phase birefringence dispersion B ^{′}(λ)/B(λ) equates 1/λ. This relative phase birefringence dispersion B ^{′}(λ)/B(λ) is 1/(1.508 μm) = 0.663 μm^{−1} for [34], where B_{g}(1.508 μm) = 0, while it is 1.47 μm^{−1} for [32], which is more than twice higher. Then, group birefringence B_{g} goes beyond zero. It gets an opposite sign, and a larger absolute value than phase birefringence B. It is clear that obtaining the adequate design to null out group birefringence is complicated, but the visual geometrical construction with the tangent still applies and eases the understanding, even if you are not familiar, as us, with crossphasemodulationinstability band gap.
9 Conclusion
This paper is a complement to an earlier JEOSRP publication [1]. It presents comments and a reversed geometrical construction that should ease the understanding of birefringence dispersion in PM fibers. It also gives measurements of group birefringence B_{g}(λ) and phase birefringence B(λ) of several kinds of PM fibers. The points to remember are:

The easy technique of measurement of birefringence is channeled spectrum analysis with a broadband source and an OSA, but it is inherently a group measurement, as outlined very early by Rashleigh [2].

Phase birefringence B(λ) over a wide spectrum is more complicated, but it can be retrieved from group birefringence B_{g}(λ) over this wide spectrum, and a singlewavelength measurement B(λ_{0}), with a simple geometrical construction.

Birefringence dispersion is low in stressinduced PM fibers (bowtie, panda, or tigereye), but not completely negligible for certain applications: the difference between B_{g} and B is about 15–20%, and group birefringence dispersion dB_{g}/dλ is also significative. It is due, for a good part, to the decreasing of the stresses away from the core, but also to the intrinsic dispersion of photoelasticity [2], which is not often mentioned.

It is confirmed that birefringence dispersion is very high in ellipticalcore (Ecore) PM fibers: the difference between B_{g} and B can get up to a 3fold factor.

Shape birefringence in solidcore microstructured fibers yields a surprising result: phase birefringence dispersion dB/dλ is very high and positive, which can lead to a group birefringence B_{g} with an opposite sign! It is also possible to engineer the design to cancel out group birefringence at a specific wavelength.

The use of the beat length @ 633 nm, to specify the strength of the birefringence, should be avoided when 633 nm is not the operating wavelength.

To us, the concept of group birefringence is easier to grasp than the one of intrinsicPMD, even if they are obviously related.
We hope that this paper will be useful and help to simplify the subject, knowing that seminal publications are not always easy to understand, if only because of the evolution of vocabulary over four decades.
References
 Lefèvre H.C. (2022) Comments about dispersion of light waves, J. Eur. Opt. Soc.: Rapid Publ. 18, 1. https://doi.org/10.1051/jeos/2022001. [Google Scholar]
 Rashleigh S.C. (1983) Measurement of fiber birefringence by wavelength scanning, Opt. Lett. 8, 6, 336–338. [NASA ADS] [CrossRef] [Google Scholar]
 Stolen R.H., Ramaswamy V., Kaiser P., Pleibel W. (1978) Linear polarization in birefringent singlemode fibers, Appl. Phys. Lett. 33, 8, 669–701. [Google Scholar]
 Snitzer E., Ostenberg H. (1961) Observed dielectric waveguide modes in the visible spectrum, J. Opt. Soc. Am. 51, 499–505. [NASA ADS] [CrossRef] [Google Scholar]
 Rashleigh S.C., Stolen R.H. (1984) Status of polarizationpreserving fibers, in: Conference on Lasers and ElectroOptics, Anaheim, California, USA, 19–21 June 1984, OSA Technical Digest, Paper WH4. [Google Scholar]
 Stolen R.H., Rashleigh S.C. (1985) Polarizationholding fibers, in: Optical Fiber Sensors Conference, San Diego, California, USA, 13–14 February 1985, OSA Technical Digest, paper ThCC1. [Google Scholar]
 Noda J., Okamoto K., Sasaki Y. (1986) Polarizationmaintaining fibers and their applications, J. Lightwave Technol. LT4, 8, 1071–1089. [NASA ADS] [CrossRef] [Google Scholar]
 Papp A., Harms H. (1975) Polarization optics of indexgradient optical waveguide fibers, Appl. Opt. 14, 10, 2406–2411. [NASA ADS] [CrossRef] [Google Scholar]
 Rashleigh S.C. (1982) Wavelength dependence of birefringence in highly birefringent fibers, Opt. Lett. 7, 6, 294–296. [NASA ADS] [CrossRef] [Google Scholar]
 Kikuchi K., Okoshi T. (1983) Wavelengthsweeping technique for measuring the beat length of linearly birefringent fibers, Opt. Lett. 8, 2, 122–123. [NASA ADS] [CrossRef] [Google Scholar]
 Born M., Wolf E. (1999) Principles of Optics, 7th ed., Cambridge University Press, Subsection 7.3.3, pp. 295–296. [Google Scholar]
 Okoshi T. (1981) Singlepolarization singlemode optical fibers, IEEE J. Quantum Electron. QE17, 6, 879–884. [NASA ADS] [CrossRef] [Google Scholar]
 Birch R.D., Payne D.N., Varnham M.P. (1982) Fabrication of polarisationmaintaining fibres using gasphase etching, Electron. Lett. 18, 24, 1036–1038. [NASA ADS] [CrossRef] [Google Scholar]
 Takada K., Noda J., Okamoto K. (1986) Measurement of spatial distribution of mode coupling in birefringent polarizationmaintaining fiber with new detection scheme, Opt. Lett. 11, 10, 680–682. [NASA ADS] [CrossRef] [Google Scholar]
 Lefèvre H.C. (1987) Comments about the fiberoptic gyroscope, SPIE Proc. 838, 86–97. [Google Scholar]
 Martin P., Le Boudec G., Lefèvre H.C. (1991) Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry, SPIE Proc. 1585, 173–179. [Google Scholar]
 Yao X.S. (2019) Techniques to ensure highquality fiber optic gyro coil production, in: Udd E., Digonnet M. (eds), Design and Development of Fiber Optic Gyroscopes, Chapter 11, SPIE Press, pp. 217–261. [Google Scholar]
 Lefèvre H.C. (2022) Testing with optical coherence domain polarimetry (OCDP), or today, distributed polarization crosstalk analysis (DPXA), in: The FiberOptic Gyroscope, 3rd ed., Artech House, Subsection 5.4, pp. 110–123. [Google Scholar]
 Simon A., Ulrich R. (1977) Evolution of polarization along a singlemode fiber, Appl. Phys. Lett. 31, 8, 517–520. [NASA ADS] [CrossRef] [Google Scholar]
 Takada K., Noda J., Ulrich R. (1985) Precision measurement of modal birefringence of highly birefringent fibers, Appl. Opt. 24, 24, 4387–4391. [NASA ADS] [CrossRef] [Google Scholar]
 Calvani R., Caponi R., Cisternino, F., Coppa G. (1987) Fiber birefringence measurement with an external stress method and heterodyne polarization detection, J. Light. Technol. LT5, 9, 1176–1182. [NASA ADS] [CrossRef] [Google Scholar]
 Bock W.J., Urbanczyk W. (1993) Measurement of polarization mode dispersion and modal birefringence in highly birefringent fibers by means of electronically scanned shearingtype interferometry, Appl. Opt. 32, 30, 5841–5848. [NASA ADS] [CrossRef] [Google Scholar]
 Shlyagin M.G., Khomenko A.V., Tentori D. (1995) Birefringence dispersion measurement in optical fibers by wavelength scanning, Opt. Lett. 20, 8, 869–871. [NASA ADS] [CrossRef] [Google Scholar]
 Kashyap R. (2010) Fiber Bragg Gratings, 2nd ed., Academic Press. [Google Scholar]
 Froggatt M.E., Gifford D.K., Kreger S., Wolfe M., Soller B.J. (2006) Characterization of polarizationmaintaining fiber using highsensitivity opticalfrequencydomain reflectometry, J. Light. Technol. 24, 11, 4149–4154. [NASA ADS] [CrossRef] [Google Scholar]
 Eickhoff W., Ulrich R. (1981) Optical frequency domain reflectometry in singlemode fiber, App. Phys. Lett. 39, 9, 694–695. [Google Scholar]
 Sinha N.K. (1978) Normalized dispersion of birefringence of quartz and stress optical coefficient of fused silica and plate glass, Phys. Chem. Glass. 19, 4, 69–77. [Google Scholar]
 Shibata N., Okamoto K., Tateda M., Seikai S., Sasaki Y. (1983) Modal birefringence and polarization mode dispersion in singlemode fibers with stressinduced anisotropy, IEEE J. Quant. Electron. QE19, 6, 1110–1115. [NASA ADS] [CrossRef] [Google Scholar]
 Poole C.D. (1989) Measurement of polarizationmode dispersion in singlemode with random mode coupling, Opt. Lett. 14, 10, 523–525. [NASA ADS] [CrossRef] [Google Scholar]
 Kumar A., Ghatak A. (2011) Ellipticalcore fibers SPIE Tutorial, in: Polarization of light with applications in optical fibers, vol. TT90, SPIE Tutorial Text, Subsection 9.2.1.1, pp. 171–175. [Google Scholar]
 Urbanczyk W., Martynkien T., Bock W.J. (2001) Dispersion effects in ellipticalcore highly birefringent fibers, Appl. Opt. 40, 12, 1911–1920. [NASA ADS] [CrossRef] [Google Scholar]
 Mélin G., Cavani O., Gasca L., Peyrilloux A., Provost L., Rejeaunier X. (2003) Characterization of a polarization maintaining microstructured fiber, in: ECOCIOOC Proceedings, Rimini, Italy, AEI, September 21–25, vol. 3, Paper We1.7.4. [Google Scholar]
 Morin P., Kibler B., Fatome J., Finot C., Millot G. (2010) Group birefringence cancellation in highly birefringent photonic crystal fibre at telecommunication wavelengths, Electron. Lett. 46, 7, 525–526. [NASA ADS] [CrossRef] [Google Scholar]
 Kibler B., Amrani F., Morin P., Kudlinski A. (2016) Crossphasemodulationinstability band gap in a birefringenceengineered photoniccrystal fiber, Phys. Rev. A 93, 0138571–0138577. [NASA ADS] [CrossRef] [Google Scholar]
All Figures
Figure 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. 

In the text 
Figure 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). 

In the text 
Figure 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 quasiinfinite wavelength. 

In the text 
Figure 4 Principle of measurement of birefringence with channeled spectrum analysis, using a tunable laser and a power meter, instead of a broadspectrum source and an OSA. 

In the text 
Figure 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 bowtie fibers. Fiber C (red) combines stressinduced birefringence with form birefringence. Freespace wavenumber k should be read 7.39 rad·μm^{−1}, and it corresponds to a wavelength λ of 0.85 μm. 

In the text 
Figure 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! 

In the text 
Figure 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). 

In the text 
Figure 8 Principle of pathmatched whitelight interferometry: (a) setup with an unbalanced readout interferometer, using a beam splitter BS, and two mirrors M_{1} and M_{2}; (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 λ ^{2}/∆λ_{FWHM}. 

In the text 
Figure 9 Measurement of phase birefringence B(λ_{0}) of a PM fiber: (a) setup with a moving crossedpolarization coupling; (b) resulting channeled spectrum (black sinusoidal curve) for a length L_{f} corresponding to a bright channel of order m for λ_{0}; 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). 

In the text 
Figure 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 (lowerindex) mode, while λ_{s} is the one of the slow (higherindex) mode. 

In the text 
Figure 11 Principle of OFDR: (a) setup with a frequencyswept tunable laser, an unbalanced twowave interferometer with a reference mirror on one arm, and the low reflection point (ɛ ^{2} in power, and ɛ in amplitude) that is analyzed, on the other arm; (b) measured channeled spectrum with a contrast of 2ɛ. 

In the text 
Figure 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 crosscorrelated. 

In the text 
Figure 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. 

In the text 
Figure 14 Geometrical construction to derive phase birefringence B(λ) from group birefringence B_{g}(λ) (black solid line) and a singlewavelength phase measurement B(λ_{1}) (red circular dot); (a) construction of the tangent T_{2}(λ) that crosses tangent T_{1}(λ) at λ_{x} = (λ_{2} − λ_{1})/ln(λ_{2}/λ_{1}) (dashed blue lines), which allows one to find B(λ_{2});when the interval (λ_{2} − λ_{1})/λ_{1} is small, λ_{x} ≈ (λ_{2} + λ_{1})/2; (b) construction of B(λ) (dashed red curve) that must fit the tangents T_{1}(λ) and T_{2}(λ). 

In the text 
Figure 15 Different kinds of PM fibers based on stressinduced linear birefringence: bowtie, panda and tigereye. The slow axis is along the axis of the stress structure, and the fast axis is perpendicular. 

In the text 
Figure 16 Principle of PM fibers based on stressinduced linear birefringence: (a) SAPs (panda structure, here) are under quasiisotropic tensile stress, and they pull on the fiber cladding; (b) in the core region, this yields tensile stress (red arrows) in the xaxis, and a compressive stress (blue arrows) in the perpendicular yaxis; (c) these stresses decrease when one moves away from the core, along the perpendicular yaxis. 

In the text 
Figure 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 yaxis that decreases along this yaxis, and gets to zero at the outer limit of the cladding. 

In the text 
Figure 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 xaxis, along the yaxis: it is positive in the core region, and it becomes negative at a distance that corresponds to about the height of the SAPs. 

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

In the text 
Figure 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) bowtie fiber; (b) panda fiber; (c) tigereye fiber. Photographs of the end face of the fibers are inserted; SAPs are dark, while the circular cores are illuminated. 

In the text 
Figure 21 Ellipticalcore (Ecore) PM fiber, based on shape birefringence, with its slow and fast axes. 

In the text 
Figure 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. 

In the text 
Figure 23 Measurement, by channeled spectrum analysis, of group birefringence B_{g}(λ) of an Ecore fiber, over 1000–1650 nm (black square dots); measurement of phase birefringence B at 1550 nm with Bragg grating and RayleighOFDR (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. 

In the text 
Figure 24 Revisited Figure 3 of [32]: the phasebirefringence 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 microstructured solidcore PM fiber described in [32]. 

In the text 
Figure 25 Revisited Figure 3b of [33]: the tangent to phase birefringence curve B(λ) does cross at zero the zerowavelength 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. 

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

In the text 
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