Comments about birefringence dispersion, with group and phase birefringence measurements in polarization-maintaining ﬁ bers

. A recent JEOS-RP publication proposed Comments about Dispersion of Light Waves , and we present here complementary comments for birefringence dispersion in polarization-maintaining (PM) ﬁ bers, 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 ﬁ bers using stress-induced photo-elasticity, or shape birefringence. These measurements con ﬁ rm clearly that the difference between group birefringence and phase birefringence is limited to 15 – 20% in stress-induced PM ﬁ bers (bow-tie, panda, or tiger-eye), but that it can get up to a 3-fold factor with an elliptical-core (E-core) ﬁ ber. There are also surprising results with solid-core micro-structured PM ﬁ bers, that are based on shape birefringence, as E-core ﬁ bers.


Introduction
A recent JEOS-RP 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 (k) 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(k) over this wide spectrum, with a single-wavelength phase measurement B(k 0 ), since as it is well-known, phase birefringence measurements are not as easy to perform as group measurements. For these single-wavelength phase measurements, we use fiber Bragg gratings (FBG) and Rayleigh-OFDR (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 (k) to phase birefringence B(k), this requires now to relate phase birefringence B(k) to group birefringence B g (k), to retrieve B(k) 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 three-page/one-figure 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 EOSAM 2022 Guest editors: Patricia Segonds, Gilles Pauliat and Emiliano Descrovi to avoid, as much as possible, complicated mathematical equations that might be not easily understandable.

Comments about early seminal papers
The first publication about the fabrication of what is called today polarization-maintaining (PM) fiber was by Stolen et al. in 1978 [3]. This fiber was based on stress-induced birefringence, and it was called a high-birefringence (HB, or also Hi-Bi) 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 high-birefringence fiber. These fibers were then called polarization-preserving [5], or polarization-holding [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 polarization-maintaining (PM) fibers [7], which is the established vocabulary today.
The birefringence strength was originally given with the "beat length" (noted K), over which the relative phase difference between the two orthogonal modes changes by 2p [3]. Even if it might sound obvious, it could be useful to remind that a classical bulk-optic half-wave plate has a thickness of half a beat length, and it is a quarter of beat length for a quarter-wave 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 light-pattern periodicity equal to the beat length K (Fig. 1).
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 k À4 dependency of Rayleigh scattering does not make this side-view observation as easy for near-infrared 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).
This forms a polarimeter that is equivalent to an unbalanced two-wave 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 spitters-recombiners of an interferometer, and the interferences are perfectly contrasted. The optical path length difference DL opt is then: where L f is the PM fiber length, n eff-s is the effective index of the slow (higher-index) mode, and n eff-f is the one of the fast (lower-index) 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: There is a bright channel when DL opt is an integer number m of the free-space wavelength k(m) that corresponds to this mth channel, i.e., when: The integer m À 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 r(m) = 1/k(m). The period of the spatial frequency spacing between adjacent bright channels is called the free spectral range, FSR r (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: Therefore, the free spectral range FSR r is: and it yields a simple measurement of the birefringence B (Fig. 3). This birefringence B is related to the beat length K by: This assumption of dispersion-free 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.
Notice that is also possible to perform this channeled spectrum measurement with a tunable laser and a power meter (Fig. 4).
Rashleigh made the same assumption in his 1982 letter [9], but only for PM fibers based on stress-induced photoelasticity. He was clear about the fact that elliptical-core (or E-core) 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 stress-induced 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]: 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 2p in propagation equations. With this influence, phase birefringence was given [2,9,10,12] as the difference Db between the propagation constant b s = 2p Á n eff-s /k of the slow mode and b f = 2p Á n eff-f /k, 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, free-space angular spatial frequency (or angular wave number), k = 2p/k, was used, instead of the wavelength k. If you look at the figure of Rashleigh's paper of 1983 [2], you see an angular wavenumber of 7.39 lm À1 , which corresponds in fact to a wavelength of 0.85 lm, as said  in the text, and the birefringence Db is given in cm À1 . 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 k of 0.85 lm corresponds to a spatial frequency r of 1.176 lm À1 , and it is clearer to say that the angular spatial frequency or wave number k is 7.39 radÁlm À1 or rad/lm, i.e., 2p radian times 1.176 lm À1 , rather than to have 7.39 lm À1 that has, then, the same unit as r.
In addition, the use of Db to define phase birefringence as a function of k is not very convenient since: Even with dispersion-free birefringence, i.e., when B is constant, Db(k) is not constant; it is proportional to k. As already seen, today, phase birefringence B is defined by the effective phase index difference Dn 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 À1 , 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 Db(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! Fiber A is a bow-tie fiber [13] that is based on stressinduced birefringence and has a phase birefringence B = 5.2 Â 10 À4 , which is a typical value for industrial PM fiber products, today. Fiber B is a bow-tie fiber with lower birefringence (3 Â 10 À4 ). Both have little dispersion since the slope varies only slightly when k is changing. Fiber C combines stress-induced birefringence with form birefringence, created by an elliptical core. It has a higher phase birefringence (7 Â 10 À4 ) 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 s g between the two polarization modes: where L f is the length of the PM fiber. Today, s g is called differential group delay, or DGD. Since group birefringence B g equates s g Á c/L f , there is: Group birefringence B g is the derivative of Db(k) with respect to k, i.e., dDb/dk in Leibniz's notation, or Db 0 (k) in Lagrange's notation. Therefore, B g is the slope of the tangent to the curve Db(k). Rashleigh's figure can be revisited again (Fig. 6). With fiber C, that is very dispersive because of its elliptical core, and choosing k = 1.3 lm, i.e., k = 4.83 rad lm À1 , as in Table 1 of [2], group birefringence B g is 3.4 times phase birefringence B: 16 Â 10 À4 instead of 4.7 Â 10 À4 ! Such an important difference will be confirmed with measurements of an elliptical-core (or E-core) PM fiber, later in this paper. As we just saw, using Db as the definition of the birefringence has some geometrical interest with the use of the various slopes, but using Dn 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 DL opt is equal to an integer number m of free-space wavelength k. In the case of a polarimeter, In the general case where there is a wavelength dependence of the phase birefringence B: where k(m) is the free-space wavelength that corresponds to the mth channel, and B[k(m)] is the phase birefringence J. Eur. Opt. Society-Rapid Publ. 19, 1 (2023) of the PM fiber for this wavelength k(m). Using now the corresponding spatial frequency r(m) = 1/k(m), there is: Mathematically, one can consider that the spatial frequency r(m) and the phase birefringence B[r(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: 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 x = 2p Á c/k, there is:   [2] of 1983: the slope of the tangent (black dashed line) to the curve Db(k) is the group birefringence B g , as it is defined today. Fiber C is very dispersive: at k = 1.3 lm, i.e., k = 4.83 radÁlm À1 , group birefringence B g is 3.4 times higher than phase birefringence B! n g ðxÞ ¼ d½nðxÞ Á x=dx: With the spatial frequency r = x/(2p Á c), since dr/r = dx/x, there is similarly: Since the birefringence is the difference between the indexes of the polarization modes, for phase as well as for group, it simply yields: Combining equations (14) and (17), one gets: The free spectral range in spatial frequency, FSR r , is the spatial frequency difference Dr corresponding to a parameter difference Dm that equates 1. Since Dr = (dr/dm) Á Dm, one gets: To summarize, the free spectral range FSR r 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! Notice finally that there is a dual technique to measure a channeled spectrum: path-matched white light interferometry. As it is well-known, the interference power P, as a function of the path unbalance DL 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 DL r = ±B g Á L f (Fig. 8). This technique was developed to make a distributed measurement of polarization crossed couplings along a PM fiber [14][15][16][17][18].

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 magneto-optic 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][21][22][23]. The set-ups 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 K, that is simply related to phase birefringence B with K = k/B, as we saw in (6).
One can understand it easily by considering a single wavelength source, at k 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, Figure 7. Principle of measurement of birefringence with channeled spectrum analysis, in the general case of birefringence dispersion. The free spectral range (FSR r ), i.e., the spatial frequency difference between two successive bright channels, depends on group birefringence B g : it is equal to 1/(B g Á 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).
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 e 2 and 1 À e 2 , instead of 50-50, the contrast becomes 2e. As well-known in interferometry, the contrast depends on the amplitude ratio e between the two interfering waves, and not on the power ratio e 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 e. Depending on their relative phase shift, the interference amplitude oscillates between 1 + e and 1 À e; the interference power is then (1 ± e) 2 = 1 ± 2e + e 2 % 1 ± 2e.
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 DL opt equal to an integer number m of wavelength k 0 : Figure 8. Principle of path-matched white-light interferometry: (a) set-up 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 DL 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 k 2 /Dk FWHM .
If this length L f is reduced by a quarter of a beat length K, the channeled spectrum becomes shifted by a quarter of the fringe period since B(k 0 ) Á K = k 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 À 1: 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 K, and then of the phase birefringence B(k 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 k R is given by: where n eff is the effective index of the fiber mode and K Bragg is the period of the Bragg grating. A PM fiber being birefringent by principle, it yields two reflected (b) resulting channeled spectrum (black sinusoidal curve) for a length L f corresponding to a bright channel of order m for k 0 ; and shifted channeled spectrum (red sinusoidal curve) when the length is reduced by a quarter of the beat length K, i.e., is equal to wavelengths, k s and k f , that correspond respectively to the effective index n effÀs of the slow mode, and n effÀf , the one of the fast mode (Fig. 10), with: The phase birefringence B is: There are: n effÀs ðk f Þ ¼ n effÀs ðk s Þ þ ½ðk f À k s Þ Á ðdn effÀs =dkÞ and Assuming that dn effÀs /dk % dn effÀf /dk % dn eff /dk, it yields: Bðk f Þ % Bðk s Þ % ðk s À k f Þ Á ½ð1=ð2K Bragg ÞÞ À ðdn eff =dkÞ: Phase dispersion dn SiO 2 =dk of silica is about À0.01 lm À1 in the 1200-1600 nm range [1], and waveguide phase dispersion is about À0.005 lm À1 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/(2K Bragg ) is about 1 lm À1 at 1550 nm. Therefore, within an approximation of 1.5%, phase birefringence B(k) is simply: A last method is based on Rayleigh-OFDR (Rayleigh optical-frequency-domain reflectometry). This technique developed by Froggatt and co-workers [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 optics-photonics specialists.
It is like what we saw earlier in Figure 9. Instead of a polarimeter, it is a two-wave interferometer with a reference mirror on one arm and a small reflection (e 2 in power) on the measurement arm. It yields a channeled spectrum that is measured with a tunable laser; the contrast is 2e, and there are bright channels when the optical path length difference, DL opt = 2n eff . DL f , is equal to an integer number of wavelength k. DL 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).
As in (13) for the polarimeter, the mth channel follows m = DL opt Á r(m) and, for the interferometer, it yields: With a PM fiber, one can select the slow mode or the fast one, and there is: There is a frequency shift Dr between the respective channeled spectra (Fig. 12a). The frequency of the mth channel of the slow mode is r s (m), and it is r f (m) for the fast mode. Since: There is: Considering the mean effective index n eff , and the mean spatial frequency r, it yields: 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 slow-mode spectrum is the mth one, but there is no mean to know where is the corresponding mth channel of the fast-mode 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 Rayleigh-OFDR, as proposed by Froggatt and co-workers [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 low-reflection 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 Dr (Fig. 12b). This shift is identical to the first case with sine spectra but now, noisy channeled spectra can be cross-correlated to know Dr, 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 Rayleigh-OFDR 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: Figure 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 (e 2 in power, and e in amplitude) that is analyzed, on the other arm; (b) measured channeled spectrum with a contrast of 2e.
broadband source with an OSA, tunable laser, or path-matched white-light 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 À70 dB, which corresponds to a back-reflection of 3 Â 10 À4 in amplitude, and 10 À7 in power; with pathmatched white light interferometry, it is typically À90/À100 dB; with a tunable laser, as used in OFDR, it can go down to À130 dB [18]. Rayleigh backscattering signal is around À110 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 Dk OSA % 0.01 nm. The free spectral range in spatial frequency, FSR r , is 1/DL optÀg , where DL optÀg is the group optical path length difference, i.e., (n g /n) Á DL opt . In wavelength, it is: Then, the maximum path length, DL max , that can be explored is: For Dk OSA = 0.01 nm at 1550 nm, DL 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 Â 10 À4 . With path-matched interferometry, it depends on the travel distance of the moving mirror M 2 of the read-out interferometer (Fig. 8). In practice, it is difficult to have more than few meters, even with a multi-fold path. With a tunable laser that has a very narrow linewidth, it yields a great spectral resolution, and the maximum path length, 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 cross-correlated. DL max , can go up to few kilometers, as seen with the OBR 4600 of Luna. Impressive! 5 How to derive phase birefringence B(k) from group birefringence B g (k) As we saw, measurement of group birefringence B g (k) 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 (k) from phase birefringence B(k), while one has now to derive B(k) from B g (k) and a single measurement B(k 0 ). As seen in [1], group birefringence B g (k) is related to phase birefringence B(k) with: And the tangent T i (k) to the curve B(k), at the wavelength k i , follows: Then: Knowing phase birefringence B(k) and its derivative B 0 (k), over a wide spectrum, allows one to retrieve group birefringence B g (k) over this wide spectrum with a simple geometrical construction, using the tangent to B(k) (Fig. 13).
Conversely, if one knows the whole curve B g (k) and a single value B(k 1 ) for phase birefringence, one can define the tangent T 1 (k) and the derivative B 0 (k 1 ). However, one cannot define yet the tangent T 2 (k), nor B(k 2 ), for another wavelength k 2 . One must do some math to get it.
We saw in (35) that B g (k) = B(k) À [k Á B 0 (k)]. It yields for the derivatives: and there is also: One can integrate B 00 (k), seen in (38), to get B 0 (k): and phase birefringence B(k) for k 2 can be retrieved: Figure 14. Geometrical construction to derive phase birefringence B(k) from group birefringence B g (k) (black solid line) and a single-wavelength phase measurement B(k 1 ) (red circular dot); (a) construction of the tangent T 2 (k) that crosses tangent T 1 (k) at k x = (k 2 À k 1 )/ln(k 2 /k 1 ) (dashed blue lines), which allows one to find B(k 2 );when the interval (k 2 À k 1 )/k 1 is small, k x % (k 2 + k 1 )/2; (b) construction of B(k) (dashed red curve) that must fit the tangents T 1 (k) and T 2 (k). Figure 13. Geometrical construction to derive group birefringence B g (k) (dashed black curve) from phase birefringence B(k) (red solid curve), as seen in [1]; the tangents T i (k) (blue dashed lines) cross the ordinate axis, i.e., where k = 0, at B g (k i ); cases with i = 1 or 2.
With a small interval between k 1 and k 2 , one can consider that the slope B 0 g of B g (k) is constant and: A geometrical construction remains possible, using again the tangents T i (k). Equations (36) and (37) yield: They cross each other for the wavelength k x where T 1 (k x ) = T 2 (k x ), and then: There is B g ðk 2 Þ À B g ðk 1 Þ % B 0 g Á ðk 2 À k 1 Þ and, from (40): It yields: The Taylor series of natural logarithm, ln, leads to: Therefore, when the interval d = (k 2 À k 1 )/k 1 is small, there is: The wavelength k x is simply in the middle of the interval between k 1 and k 2 , which makes the geometrical construction easy (Fig. 14). We know B g (k) and B(k 1 ); we can deduce T 1 (k) with B g (k 1 ) and B(k 1 ); we can find T 2 (k) with B g (k 2 ), and also T 2 (k x ) = T 1 (k x ), with k x % (k 2 + k 1 )/2; finally, we can construct the curve B(k) between k 1 and k 2 , with the two tangents T 2 (k) and T 1 (k).

Origin of birefringence dispersion in PM fibers using stress-induced birefringence
The method, that is now widely generalized to make PM fibers, is to use stress-induced linear birefringence with  additional materials that have a thermal expansion coefficient larger than silica (several 10 À6 /°C instead of 5 Â 10 À7 /°C for pure silica). The fiber preform is fabricated with two rods (sometimes called stress-applying 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: bow-tie, panda and tiger-eye (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 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 y-axis that decreases along this y-axis, and gets to zero at the outer limit of the cladding. 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 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.
quasi-isotropic tensile stress, and, by reaction, this also induces stresses in the core region where light propagates; there is a tensile stress in the x-axis of the rods and a compressive stress in the perpendicular y-axis. Because of the photo-elastic effect, these anisotropic stresses yield anisotropic index changes, i.e., birefringence. These stresses are quite uniform along the x-axis between the rods (or SAPs), but they decrease rapidly along the perpendicular y-axis (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 0 (k) of phase birefringence B(k) is negative, and then, group bire- , as seen in (35).
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 germano-silicate 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 y-axis to get to zero at the outer surface of the cladding (Fig. 17).
The stress T x (x,y) on the x-axis does also decrease along the y-axis; 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).
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)! There is a second source of birefringence dispersion that is not often mentioned: dispersion of the photo-elastic effect in silica. This effect is usually treated with two photo-elastic 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 photo-elastic effect is about minus 0.05 lm À1 in the 1.2-1.6 lm 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 photo-elastic 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 À0.05 lm À1 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 coupled-mode PMD in low-birefringence telecom fibers, where it is a random process that grows as the square root of the length [29]. With high-birefringence 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].

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 stress-induced birefringence: bow-tie, panda and tiger-eye (Fig. 15). Group birefringence B g was measured with channeled spectrum analysis over a very wide spectrum (1250-1650 nm), using an in-house supercontinuum fiber source and an OSA (Fig. 2). The three fibers have about the same characteristics: cut-off wavelength k 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, mode-field diameter at 1/e (MFD) of 7 lm, and cladding diameter of 80 lm. Phase birefringence B was measured at 1550 nm with a Bragg grating, and also with Rayleigh-OFDR (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(k): around 4.2 Â 10 À4 ; it corresponds to a beat length K 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 Â (633/1550) % 1.5 mm. Because of birefringence dispersion, it is clear that this habit should be abandoned, or at least explained.
From (35), one sees that the slope of the relative variation of the phase birefringence B(k) is: At 1550 nm, it is À0.11 lm À1 for the bow-tie fiber, À0.09 lm À1 for the panda fiber, and À0.17 lm À1 for the tiger-eye fiber, when the sole relative dispersion of the photo-elastic effect is À0.05 lm À1 , 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 bow-tie and panda fibers, and it is about two thirds-one third for the tiger-eye fiber. This last higher value is due to the smaller height of the SAPs of the tiger-eye 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 0 g =B g of about +0.16 lm À1 , at 1550 nm. This must be considered with distributed measurement of polarization crossed couplings along a PM fiber [17,18].
8 Case of an elliptical-core (E-core) PM fiber The case of an elliptical-core, or E-core, PM fiber (Fig. 21) is much more complicated. There is shape birefringence induced by the elliptical core, but also stress-induced 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 Dn of the core: this requires a high-index 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 major-axis width, 2a, and the minor-axis one, 2b, becomes higher than 2: it reaches the birefringence of a planar waveguide that has a thickness of 2b.
This calls for a comment: the origin of shape birefringence can be grasped simply. As well-known, 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 quarter-wave retarder. It is a very broadband retarder since the retardation depends on the index n that has only a small dispersion. A classical quarter-wave (K/4) plate using a birefringent crystal, creates an optical path difference DL opt , and not a phase retardation, strictly speaking. This path delay does yield a phase retardation Du, but this retardation is inversely proportional to the wavelength k, since Du = 2p Á DL opt /k, as it is well-known.
The origin of stress-induced birefringence is complicated too. The highly-doped 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 stres-ses 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 co-workers [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 major-axis 2a width of 4 lm and a minor-axis width 2b of 2 lm; the cut-off : the phase-birefringence dispersion dB/dk 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].  [33]: the tangent to phase birefringence curve B(k) does cross at zero the zero-wavelength ordinate axis, for 1.55 lm, where group birefringence is cancelled; the insert shows Figure 3c, with group birefringence curve B g (k), noted as G(k), in this paper.  . Measurement, by channeled spectrum analysis, of group birefringence B g (k) 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(k) 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. wavelength k 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 lm. It is confirmed that phase-birefringence 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 Â 10 À4 , while B g is 5.8 Â 10 À4 , i.e., 2.5 times bigger. We directly checked this big difference with an additional Bragg grating at 1300 nm to confirm it.
Finally, as described by Mélin and co-workers [32], shape birefringence is also found in solid-core microstructured PM fiber, where there is a very strong phase birefringence: few 10 À3 , when it is typically 5 Â 10 À4 in classical PM fibers. There is also an interesting effect: the value of phase birefringence dispersion dB/dk 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 Â 10 À3 , and phasebirefringence dispersion dB/dk is about 5 Â 10 À3 lm À1 ; using (35), one finds a group birefringence B g of minus 4.15 Â 10 À3 (Fig. 24).Worth to outline, isn't it?
It is also worth to outline that the shape birefringence of such solid-core micro-structured PM fibers can be engineered to get cancellation of group birefringence at a specific wavelength, as described by Morin and co-workers [33], and Kibler and co-workers [34]. Their figures can be revisited as well, and this confirms that the tangent to the phase birefringence curves B(k) does crosse at zero the ordinate axis corresponding to k = 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.
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 micro-structured solid-core PM fiber of [34] is smaller than the one of [32], which yields less phase birefringence B (1.65 Â 10 À3 instead of 3.4 Â 10 À3 @ 1.5 lm, in absolute values), and less phase birefringence dispersion dB/dk, or B 0 (1.05 Â 10 À3 lm À1 instead of 5 Â 10 À3 lm À1 @ 1.5 lm, in absolute values). According to equation (35), the condition for nulling out B g (k) is k Á B 0 (k) = B(k), i.e., the relative phase birefringence dispersion B 0 (k)/B(k) equates 1/k. This relative phase birefringence dispersion B 0 (k)/B(k) is 1/(1.508 lm) = 0.663 lm À1 for [34], where B g (1.508 lm) = 0, while it is 1.47 lm À1 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 cross-phase-modulationinstability band gap.

Conclusion
This paper is a complement to an earlier JEOS-RP 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 (k) and phase birefringence B(k) 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(k) over a wide spectrum is more complicated, but it can be retrieved from group birefringence B g (k) over this wide spectrum, and a singlewavelength measurement B(k 0 ), with a simple geometrical construction. Figure 26. Revisited Figure 1b of [34]: the tangent to phase birefringence curve B(k) 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.
-Birefringence dispersion is low in stress-induced PM fibers (bow-tie, panda, or tiger-eye), but not completely negligible for certain applications: the difference between B g and B is about 15-20%, and group birefringence dispersion dB g /dk 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 photo-elasticity [2], which is not often mentioned. -It is confirmed that birefringence dispersion is very high in elliptical-core (E-core) PM fibers: the difference between B g and B can get up to a 3-fold factor. -Shape birefringence in solid-core micro-structured fibers yields a surprising result: phase birefringence dispersion dB/dk 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 intrinsic-PMD, 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.