1. Introduction

Polarization sensitive reflectometry (PSR) is the technique of choice for the measurement of local polarization properties of single-mode optical fibers. Since its first proposal [1], PSR has been deeply investigated and successfully demonstrated in many experiments [2–11]. Among the many applications, characterization of spin profile in spun fibers [12] has recently attracted attention, for PSR offers the most viable approach [8, 10, 13]. To achieve this kind of characterization, the angle of rotation of birefringence is calculated from the measured state of polarization (SOP) of the backscattered field. The practical implementation of this task, however, may be hindered by the limited spatial resolution and accuracy of the reflectometer.

In this paper, first we theoretically analyze the limits due to spatial resolution and measurement accuracy, providing a map of constraints that graphically shows the range of measurability as a function of birefringence and spin rate. Second, we describe and experimentally validate a technique to characterize the spin profile of fibers that fall outside the measurable range of the reflectometer. The proposed method consists in twisting the fiber at different and known rates, so that the spin profile is locally unwound.

2. Analysis of the constraints

It is known that the SOP of the field backscattered by a periodically spun fiber is characterized by the alternation of small loops and wide arcs [3, 14]. The former correspond to fiber sections where the spin rate is maximum, whereas the latter correspond to sections where the spin rate is close to zero. An accurate characterization of the spin profile depends on the ability to precisely track those fast and small oscillations. Therefore, the measurement device should provide both a sufficiently short spatial resolution and a sufficiently high SOP accuracy.

According to the above description, to quantify the constraints of a reflectometer we may limit the analysis to fiber sections where the spin rate is highest. By definition, this occurs at a local maximum of the spin rate and, around this point, the spin rate itself can be considered almost constant. Moreover, since we are focusing the attention on a rather short fiber span, we may neglect for simplicity the effect of birefringence randomness. Following this argumentation, we may restrict the analysis to the case of constantly birefringent and constantly spun fibers, which reasonably represents the worst case scenario.

With respect to a reference frame that rotates at the same rate of the spin profile, the 3 × 3 Mueller matrix R(z) representing forward propagation as a function of z is described by the differential equation ∂R _z = b × R, where b = (β, 0, −2α)^T is the equivalent birefringence vector in the rotated frame, β is the constant birefringence and α is the constant spin rate (i.e. physical angle of rotation per unit length). Accordingly, R(z) can be written as R(z) = I + (b×)(sinγz)/γ + (b×)²(1 − cosγz)/γ ², where γ = (β ² + 4α ²)^1/2 is the equivalent birefringence strength. The Mueller matrix that represents round-trip propagation is invariant with respect to a rotation of the reference frame around the fiber longitudinal axis [9]. As a consequence, with respect to the laboratory frame the round-trip Mueller matrix reads B(z) = MR(z)^T MR(z), where M = diag(1, 1, −1) is a diagonal matrix. Explicit calculations reveal that B(z) can be written as B(z) = C ₀(ε) + C ₁(ε) cosγz + C ₂(ε) cos 2γz + S ₁(ε) sinγz + S ₂(ε) sin 2γz, where ε = arctan(2α/β) is the equivalent birefringence ellipticity, C ₀ = I – C ₁ – C ₂, I is the 3 × 3 identity matrix and

2.1. SOP accuracy

Note that as ε → π/2, C ₁, C ₂, S ₁ and S ₂ approach zero, hence B(z) → I. This means that as the spin rate increases with respect to the linear birefringence, the amplitude of the oscillation of the backscattered SOP decreases and the measurement becomes more critical. To quantify this effect, we should compare the amplitude of the SOP oscillation with the uncertainty of SOP measurement. This amplitude may be estimated as 2〈|[B(z) – C ₀]s ₀|〉, which is twice the mean modulus of the oscillating component of the backscattered SOP, s _B(z) = B(z)s ₀, where s ₀ is the input SOP. The average 〈·〉 is evaluated with respect to s ₀, which we assume to be uniformly distributed on the Poincaré sphere. Indeed, this assumption is justified by the fact that, in practice, the actual SOP at the input of each highly spun fiber section is randomly oriented. As a result, the mean amplitude reads Δ = 2(5/3 – cos 2ε)^1/2 cosε, hence we may conclude that the oscillations of the SOP can be accurately detected only if Δ > ρσ, where σ is the SOP uncertainty and ρ > 1 is an arbitrary constant representing the least acceptable SNR. Typically, σ is less than 10% and an SNR of 10 dB is more than enough; hence, ρσ ≲ 1. Recalling that tanε = 2α/β, the constraint Δ > ρσ may finally be expressed as

2.2. Spatial resolution

The spatial resolution should cope with the highest spatial frequency of s _B(z). Since this frequency is equal to 2γ/(2π), the spatial sampling frequency should be at least the double and hence the spatial resolution δz must respect the constraint δz < π/(2γ), which yields

To legitimately apply this relaxed constraint, we require that the amplitude of the terms at angular frequency γ be at least κ times larger that the amplitude of the terms at 2γ. As we have done for Δ, we may estimate these amplitudes as Δ_n = 〈|C _n s ₀ cos nγz + S _n s ₀ sin nγz|〉, (n = 1, 2), which yields ${\Delta }_1\hspace{0.17em}=\hspace{0.17em}\sqrt{2/3}\hspace{0.17em}\text{sin}\hspace{0.17em}2\epsilon$ and ${\Delta }_2\hspace{0.17em}=\hspace{0.17em}\sqrt{2/3}\hspace{0.17em}{\text{cos}}^2\hspace{0.17em}\epsilon$. Therefore, we may relax the constraint (2) by a factor 2 only if α > κβ/4, where κ ≃ 10 is a reasonable choice.

2.3. Range of applicability

The analysis performed so far is grounded on two simplifying assumptions: namely, that the birefringence is deterministic and that the spin rate is constant. This, of course, does not occur in practice, where the birefringence varies at random and the spin rate is (typically) periodic. Nevertheless, the above theory can be applied as long as we may isolate a sufficiently short fiber span where the two assumptions hold. The length L of such a section, however, should be at least 2π/γ, so to enable the observation of a complete period of oscillation of s _B(z). Following these considerations, we may show that the constraints calculated above may be applied to a randomly birefringent, periodically spun fiber if

We finally remark that also the differential group delay of the fiber, Δτ, sets a limit to PSR, because it may distort the probe signal and hinder the measurement [15]. As a rule of thumb, it should be ΔτΔf ≪ 1, where Δf is the bandwidth of the probe signal. However, this limit has to be assessed for each specific fiber sample, because Δτ is not a local parameter and depends on fiber length, on birefringence and (in a non trivial way) on spin profile [12].

2.4. The map of constraints

The constraints calculated above can be conveniently represented in the graphical map shown in Fig. 1(a). Curve 𝒞₁ represents the constraint (2) given by the spatial resolution, whereas line ℒ₁ represents the constraint (1) given by the SOP accuracy. Curve 𝒞₃ represents the tightest of the constraints imposed by (3). All these constraints define the area of measurable parameters, represented in yellow. Also shown in the figure is the area of the relaxed constraints, colored in orange. This area is defined by the curve 𝒞₂, which is the spatial resolution constraint relaxed by a factor 2, and by the line ℒ₂, which represents the lower range, α > κβ/4, for the application of the relaxed constraint. Note that this area exists if and only if ℒ₁ is steeper than ℒ₂, which means σ ≲ 6.5/(ρκ) — i.e. the SOP uncertainty should be sufficiently small.

 

Fig. 1 (a) Map of constraints of PSR. (b) Map of constraints calculated for δz = 1 cm, σ = 1.2%, ρ = 6 dB and κ = 10. Markers indicate values measured on different fibers.

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The map clearly shows that if both the birefringence and the spin rate are too large, the measurement may fail for insufficient spatial resolution. Furthermore, the measurement may fail also when a low birefringence fiber is strongly spun, for in this case the oscillations of the backscattered SOP may be too small compared with the measurement accuracy. Note also that both the spatial resolution and SOP uncertainty may be varied by applying a low-pass filter to the data, to increase the SNR. Clearly, as the band of the filter is reduced, 𝒞₁ and 𝒞₂ tend to shrink, but at the same time ℒ₁ becomes steeper.

At a first glance, the area above ℒ₁ may seem of poor interest; but actually this is the range of medium-to-low birefringence, typically populated by telecommunication fibers. Therefore, it is convenient to represent the map of constraints as a function of beat length, L_B = 2π/β, as shown in Fig. 1(b). The markers represent a broad selection of various types of fibers measured with the POFDR described in refs. [10, 11]. Clearly the area of relaxed constraints plays an important role in these measurements. For those fibers p and L_F were both larger than 8 m, therefore the constraints (3) fall outside the shown axes. Note also that some results lay outside the achievable area; how those measurements have been possible is described in the next section.

3. Measurements beyond the measurable range

Whenever PSR is applied to a spun fiber which lays outside the measurable range, the result looks like in Figs. 2(a) and 2(b), where the physical angle of the birefringence and its z-derivative are shown. The curves clearly have two portions, highlighted in orange, that are much smoother than the rest and correspond to the sites where the local spin rate falls within the measurable range. On the contrary, the rest of the curve corresponds to fiber sections where the spin rate is too large for the SOP to be accurately measured. We remark that these differences in the portions are due to the limits of the PSR. Data refer to the G.655 fiber represented by the star in Fig. 1(b); the fiber is sinusoidally spun with period 8.85 m and amplitude 20 turns (which corresponds to a maximum spin rate of 89.2 rad/m) and its mean birefringence is about 1.2 rad/m. This measurement (as well as the other ones described below) has been performed with the POFDR described in refs. [10, 11]; the spatial resolution was 1 cm and the SOP uncertainty was about 1.2%. Despite the spatial resolution is adequate, the measurement of this fiber fails due to the lack in SOP accuracy.

 

Fig. 2 (a) Angle of birefringence measured on a spun fiber and (b) its z-derivative. In both figures the orange curves represent the good portion of the experimental data.

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In order to recover information about the spin also in the rough portions of the curve, one may unwind the spin by twisting the fiber at a known and constant rate, so to lower to a measurable level the local spin-rate in some portion of the fiber. This procedure alone, however, does not solve the problem, because the continuity of the curve is compromised by its rough portions. As a result, each good portion shown in Fig. 2(a) is affected by an unknown offset. A possible method to recover these offsets has been described in [13]. Here we propose a new approach that achieves the same accuracy while being more straightforward.

The fiber under test should be laid straight on the floor and twisted at different constant rates by simply rotating its farthest end, while keeping the other fixed. The physical angle of rotation of the birefringence is then measured for each twist rate and the good portions of the gathered curves are selected. Figure 3(a) shows the result of the procedure applied on the 8.53-m-long piece of fiber used for Fig. 2(a). For convenience, let ψ_{i , m}(z) be the mth good portion of the physical angle of the birefringence measured when the fiber is twisted at a constant rate τ_i. Then we may write ψ_{i , m}(z) = A(z) + (1 − g/2)τ_iz + ϕ_{i , m} + η(z), where A(z) is the spin profile, g ≃ 0.147 is the elasto-optic rotation coefficient [11], ϕ_{i , m} is the unknown offset that affects the portion and η(z) accounts for both the intrinsic random rotation of birefringence and the measurement noise. We may get rid of the offsets ϕ_{i , m} simply by taking the z-derivative of ψ_{i , m}(z). Note, however, that this is basically equivalent to perform a second order differentiation on experimental data—a task that requires some care. Therefore, we have calculated dψ_{i , m}/dz by applying a Savitzky-Golay filter of forth degree, on an 11-point-long (5 cm) sliding window [16]. The results are shown in Fig. 3(b), where the effects of the applied twist are visible. These effects can be compensated by subtracting (1 − g/2)τ_i to each curve, using the corresponding value of τ_i. In this way the spin rate of the fiber is reconstructed as shown in Fig. 3(c).

 

Fig. 3 (a) Good portions of the birefringence angle measured for different numbers of turns. (b) z-derivative of the same portions. (c) Complete spin rate obtained by compensating the applied twist. Graphs share the same z-axis. Same colors correspond to the same number of turns, from −105 [lowest curve in (b)] to +70 [upper curve in (b)], in steps of 35. (d) Estimate of the spin rate (black, solid) and of the spin profile (black, dashed); in orange, the best fitting sinusoid; in green, the result of the procedure described in [13].

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A better estimate of the spin rate, dA/dz, can be obtained upon averaging the overlapping portions of the curves dψ_{i , m}/dz; the result is shown in Fig. 3(d) (black, solid curve). This curve can be numerically integrated to achieve the estimate of the spin profile, A(z), which is shown on the same figure (black, dashed curve). Note that this is in very good agreement with the estimate provided by the method described in [13] and shown by the green curve. Finally, the orange curve in Fig. 3(d) represents the best fitting sinusoid; the resulting period and amplitude are 8.9 m and 21.6 turns, respectively, in good agreement with the nominal values.

4. Conclusion

In this paper we have presented a detailed analysis of the limits of PSR due to spatial resolution and SOP accuracy. Results are summarized in a map of constraints, which helps in understanding why some specific measurements may fail. Along with this analysis, we have also proposed and implemented a method to overcome these limits. The technique, based on the local unwinding of the spin by the application of external twist, enables the accurate characterization of the spin profile also outside the measurable range.