## Abstract

The understanding of how bending modifies the dispersion of optical fibers, in particular, the zero-dispersion wavelength (*λ*_{0}), is essential in the development of compact nonlinear optical devices such as parametric amplifiers, wavelength converters, soliton lasers and frequency comb generators. Typically, substantial variations in the parametric gain and/or conversion efficiency are significant for changes in *λ*_{0} of ~0.1 nm, which occur for variations on the bending radius (*R _{b}*) of 1 cm or less. Measuring

*λ*

_{0}as a function of bending radius (

*R*) is challenging, as it requires detecting changes < 0.1 nm and in short fibers. By using a method based on four-wave mixing (FWM) generated by an incoherent-pump with relatively broad spectrum and a weak laser, we report measurements of

_{b}*λ*

_{0}as a function of

*R*in a dispersion-shifted fiber with <0.1 nm accuracy on

_{b}*λ*

_{0}. This method is sensitive enough to measure small variations in

*λ*

_{0}of ~0.04 nm in very short fibers (~20 m). We observe that

*λ*

_{0}increases by 12 nm when

*R*is decreased from 10 cm to 1 cm, and a change of 1 nm is obtained for

_{b}*R*= 3 cm. We also present numerical simulations of the bent fiber that are in good agreement with our measurements, and help us to explain the observations and to predict how high-order dispersion is modified with bending. This study can provide insights for dispersion engineering, in which bending could be used as a tuning, equalization, or tailoring mechanism for

_{b}*λ*

_{0}, which can be used in the development of compact nonlinear optical devices based on fibers or other bent-waveguide structures.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Bending modifies the guiding characteristics of an optical fiber by inducing changes in the loss and dispersion parameters. These changes are important in the design of compact devices where bending is unavoidable to save space. Dispersion induced changes are of key relevance, for example, for fiber optical parametric amplifiers (FOPAs), wavelength converters, and other compact fiber devices based on phase-matched nonlinear processes [1,2]. In such devices, one convenient way of taking into account the effect of bending is through the induced shift of the zero-dispersion wavelength (*λ*_{0}) of the fiber. In a typical fiber-optic parametric amplifier, for example, a variation of ~0.1 nm in *λ*_{0} ~1500 nm (or less than 1 part in 10^{4}) produces a sizable reduction of the parametric gain. Bending induced changes in *λ*_{0} were far less studied than bending induced loss, most likely due to the lack of sensitive methods to measure *λ*_{0} in short fibers (and better than 0.1 nm accuracy on *λ*_{0}). The shortness of the fiber is required for two reasons. The first is that, in most cases, *λ*_{0} varies randomly along the length of the fiber, with a typical correlation length (*ℓ _{c}*) of few tens to few hundreds of meters [3–5]; thus, because in long fibers one measures an average value of

*λ*

_{0}, we should be able to measure fibers with a uniform

*λ*

_{0}, that is, fibers shorter than

*ℓ*. Note however that a short fiber (<

_{c}*ℓ*) does not warrant fiber uniformity. For example, in the highly nonlinear fibers analyzed in [6], fibers as short as 5 m exhibited detectable variations in the CD. The second reason is that in order to have a well-defined and constant bending radius (Rb) along the whole length of fiber, we should be able to wind the total fiber length in a single layer (or few layers) around a cylinder of well-defined radius. This was evidenced in the measurement of Δ

_{c}*λ*

_{0}=

*λ*

_{0}(

*R*) –

_{b}*λ*

_{0}(

*R*= ∞) made in [7] in a dispersion-shifted fiber (DSF) of 1.5 km. The author of [7], who used a commercial chromatic dispersion (CD) measuring instrument, called attention to the fact that more sensitive methods for CD characterization would be important to measure Δ

_{b}*λ*

_{0}(

*R*) in shorter fibers, that could be wound (in a cylinder of reasonable length) in one or few layers with a precise value of

_{b}*R*.

_{b}It is well known that phase-matched four wave mixing (FWM) is sensitive to minute changes in CD, a fact that is employed in several methods to measure *λ*_{0} [8–10], as is the case of our method explained below. Even though, these methods have shown minimum uncertainty on *λ*_{0} ~0.1 nm [8] or meter-scale resolution [10], they have not been applied to the study of *λ*_{0}(*R _{b}*) dependence.

In this paper, we report on measurements of *λ*_{0} in a 20 m long fiber (DSF Corning), wound in a single layer with a good control of *R _{b}*. Our measuring method is based on FWM of an incoherent pump and a weak laser [8], that allow us to measure Δ

*λ*

_{0}as small as 0.04 nm, corresponding to a change in

*R*from 10 to 8 cm. In long fibers, this method permits an easy way to see if the fiber is uniform or not, since each distinct value of

_{b}*λ*

_{0}produces a FWM peak. In a short fiber, however, each FWM peak is broad and a single peak is not signature of fiber uniformity. In our measurements, we are able to identify if the fiber is uniform by using a different approach, namely by the observation of the spectral sinc

^{2}function predicted by the theory of FWM, as explained in the next section. To the best of our knowledge, other groups that used FWM driven by an incoherent pump have not reported a clear sinc

^{2}spectrum with well-defined zeros in their measurements. We have achieved this by several experimental improvements of the method [8], as described below and in the Appendices.

We present also a theoretical analysis of the dependence of *λ*_{0} on bending in a numerical model of the DSF. The numerical simulations, based on conformal mapping [11,12] and the finite-elements method (FEM), are in good agreement with our experimental results. Our calculations also allow us to explain qualitatively the variation of *λ*_{0} with bending in terms of the mode profile deformation and fiber geometry. Previous theoretical studies in single mode fibers (SMFs) of several index profiles have found that *λ*_{0} increases as *R _{b}* is decreased [7,13]. Based on our simulations, however, we predict that this behavior can be reversed in fibers with very small core. The ideas in the present study can provide insights and design rules for the dispersion engineering of future compact parametric devices [14,15].

This paper is organized as follows: in Section 2 we describe our measuring method; in section 3 we present our experiments; in section 4 we show our measured and calculated results. Finally, we discuss the effect of bending on fiber dispersion, summarize the main implications of our study and draw our conclusions in Section 5. Details of the experiments and the simulation methodology, as well as the effect of bending on high order-dispersion can be found in the Appendices.

## 2. Measuring method

The interaction of a weak laser at angular frequency *ω*_{ℓ} with a relatively strong incoherent pump with a spectrum centered at a frequency *ω*_{p} close to the zero-dispersion frequency of the fiber, *ω*_{0} = 2*πc*/*λ*_{0} produces, in general, a FWM field with a power spectral density *S*(*ω*) in the region covered by *ω*_{1} + *ω*_{2} – *ω _{ℓ}*, where

*ω*

_{1}and

*ω*

_{2}are any two frequencies within the pump spectrum. It turns out that

*S*(

*ω*) exhibits a peak at some frequency

*ω*. If the pump spectrum can be regarded as flat-top and with spectral width Δ

_{FWM}*ω*satisfying Δ

_{p}*ω*<< |

_{p}*ω*–

_{p}*ω*|, then the power spectral density of the FWM field may be approximated by:

_{ℓ}*ϒ*is the nonlinearity coefficient,

*P*is the laser power,

_{ℓ}*S*the power spectral density of the incoherent pump,

_{p}*L*is the fiber length and Δ

*β*(

*ω*) is the wavevector mismatch of the FWM process given by:

*ω*is the mid frequency between the laser and the FWM peak i.e.

_{c}*ω*= (

_{c}*ω*+

_{ℓ}*ω*)/2. In Eq. (1), the central frequency of pump was assumed to coincide with

_{FWM}*ω*, a condition that maximizes the FWM peak power. In Fig. 1 we plot

_{c}*S*(

*ω*) as given by Eq. (1), normalized to the peak

*S*(

*ω*).

_{FWM}Note that the FWM spectrum is not symmetric with respect to *ω _{FWM}* and this asymmetry is due to the fact that |Δ

*β*| is not a symmetric function of Ω =

*ω*–

*ω*.

_{FWM}By making a Taylor expansion of each term in Eq. (2) around *ω _{c}* up to the fourth order, the phase-matching condition Δ

*β*= 0 can be written as:

*β*=

_{nc}*d*/

^{n}β*dω*are evaluated at

^{n}*ω*=

*ω*. In addition, by expressing these coefficients in terms of those evaluated at

_{c}*ω*

_{0}:

We can combine Eqs. (3) and (4) and express the FWM peak frequency as a function of the laser frequency, namely:

where*ω*is an independent variable, while

_{ℓ}*ω*

_{0}and Ω

_{34}=

*β*

_{3}/

*β*

_{4}are fixed parameters. Then, by tuning

*ω*and measuring

_{ℓ}*ω*, it is possible to fit the results with Eq. (5), and retrieve from this process the fitting parameters

_{FWM}*ω*

_{0}and Ω

_{34}. In order to illustrate how the FWM spectrum shifts as a function of the laser wavelength (

*λ*), we show in Fig. 2 measured FWM spectra generated in 20 m of DSF coiled with an

_{ℓ}*R*≈8.17 cm, for nine values of

_{b}*λ*tuned in the 1604.8 - 1652.2 nm range. Note that the sinc

_{ℓ}^{2}function structure predicted by Eq. (2) and shown in Fig. 1 can be clearly observed in Fig. 2, warranting fiber uniformity.

Throughout this paper, we call *R _{b}* the radius of the cylinder (

*R*) where the fiber was wound plus the radius of the fiber with the protective coating (i.e.

_{cylinder}*R*=

_{b}*R*+ 125 μm).

_{cylinder}One aspect that influences the accuracy of our measurements is the width of the FWM central peak. As seen in Fig. 2, as the laser is tuned closer to *λ*_{0} (which in this fiber is near 1550 nm) the central peak becomes broader. This may be explained by expanding Δ*β*(*ω*) around the peak in Eq. (2) as:

*β*´ = dΔ

*β*(

*ω*)/d

_{FWM}*ω*. The width of the central peak of the FWM spectrum can be taken as the first zero of the function sinc

^{2}(ΩΔ

*β*´L/2), i.e. ΔΩ = 2

*π*/

*L*Δ

*β*´. From Eq. (2) we have that Δ

*β*´ ≈(

*β*

_{3}/8)(

*ω*–

_{FWM}*ω*)

_{ℓ}^{2}, thus:

Considering that the FWM broadening affects the accuracy of our measurements and that the value of (*ω _{FWM}* –

*ω*)

_{ℓ}^{2}is limited by how far we can tune the laser away from the field generated by FWM, one can infer from Eq. (7) that the spectral resolution in our method is limited by

*L*and the third-order dispersion (

*β*

_{3}). Furthermore, the value of Δ

*Ω*also gives the period of the spectral oscillations (at least for the first few side lobes around the main peak) observed in Fig. 1. This may be used to measure

*β*

_{3}directly (and not just the ratio

*β*

_{3}/

*β*

_{4}); however, a large signal to noise ratio (SNR) is required to clearly identify the minima in the measured spectrum. This is not easily obtained as it requires, in short fibers, the use of high power for both the incoherent pump and laser, and the use of optical filters to suppress the pump and laser entering the OSA, thus reducing stray light, as described in Section 3 and the Appendices.

## 3. Experiments

In the experimental setup shown in Fig. 3(a), the fiber under test (FUT) consisted of 20 m of DSF (Corning), the pulsed incoherent pump (see Appendix A) was centered around 1550 nm with ~32 nm bandwidth and the tunable laser (derived from a pulsed tunable coherent source, see Appendix B) was tuned in the range of 1600 −1650 nm. The pump and the laser had ~3 W and ~200 mW of peak power respectively, both with pulse duration of 20 ns and repetition rate of 100 kHz. This pulse duration was short enough to avoid Stimulated Brillouin Scattering (SBS) from the laser, inside the erbium-doped fiber amplifiers (EDFAs) or in the FUT, but it was limited by the response time of the (acousto-optic) modulator to 20 ns. The low repetition rate allows for measurements with good SNR, and the low duty cycle allows to extract more peak power from the EDFAs. Also, the pulses must have very low (ideally zero) continuous wave (CW) baseline. This was achieved by using acousto-optic modulators (extinction ratio ≳ 40 dB). By decreasing the repetition rate one can extract even higher peak power from the EDFAs, but at the cost of lower SNR. The values of 100 kHz and – 27 dB duty cycle employed in our experiments represent good values in this trade-off. The requirement of high-peak power was due to the very short length of the FUT and the low nonlinear coefficient of the DSF (i.e. *γ* ≈2.3 W^{−1} km^{−1} [16]). However, these peak powers were well below that required for the parametric gain regime (*γPL* ≳ 1) in order to keep the phase matching condition independent of the pump power. We verified in the experiments that the normalized FWM spectrum did not change when reducing the power of the pump nor that of the laser.

The pump and laser were combined by means of a wavelength division multiplexer (WDM) coupler. After the FUT, we used three concatenated high-pass optical filters (HPFs) to suppress the pump and laser. This allowed us to measure details of the FWM spectrum at very low power levels (< – 60 dBm) with good spectral resolution (RBW = 100 pm), as it is shown in the inset of Fig. 3(b). These HPFs were commercial 1480/1550 nm WDMs couplers (AC Photonics MWDM 54), each with a cutoff wavelength of ~1510 nm and an out-of-band rejection of ~25 dB. An extra benefit of suppressing both the incoherent pump and the laser, was avoiding the need to use attenuators to protect the optical spectrum analyzer (OSA).

We measured the FWM peak for different values of *λ*_{ℓ} in order to determine *λ*_{0} as described previously (Section 2). We then repeated the measurements for the same FUT coiled in cylinders of different radii and with smooth surfaces, in a single layer, in order to have a constant *R _{b}*. We used glass cylinders since in this material the surface is quite smooth and thus little or no microbending effects take place. In Fig. 4(a) we compare the FWM spectra generated in the FUT for different

*R*and at the same laser wavelength,

_{b}*λ*

_{ℓ}≈1640.4 nm. We see a clear shift of the FWM peak position as

*R*is varied, which is related to (but not exactly proportional to) Δ

_{b}*λ*

_{0}. For small values of

*R*we observe that the spectral width depends on the laser state of polarization (SOP), and eventually, for very small values of

_{b}*R*, the spectrum shows a double peak structure, as shown, for example, in Fig. 4(a) for

_{b}*R*≤ 1.5 cm. We attribute these effects to bending-induced birefringence, which from our calculation increases the total birefringence from Δ

_{b}*n*= |

*n*–

_{x}*n*| ≈10

_{y}^{−9}for

*R*= 10 cm to 10

_{b}^{−7}for

*R*= 1 cm. we assumed that the larger variation in

_{b}*λ*

_{0}(

*R*) corresponds to the $x$ polarization of the laser field with the cylinder axis aligned to $y$ [see Fig. 3(a)], which is what we observe in our numerical calculations. Note that the incoherent pump is unpolarized in our experiments, so the polarization dependence is defined by the SOP of the laser. We identified the FWM spectrum peaks in the OSA trace for certain values of

_{b}*R*by modifying the SOP of the laser via a polarization controller (PC). This is illustrated in Fig. 4(b) for the case

_{b}*R*= 4 cm. In contrast, for the smallest values of

_{b}*R*it was difficult to identify the FWM spectrum peaks, as discussed in Section 4 and illustrated in the case of

_{b}*R*= 1.2 cm in Fig. 4(c). We emphasize that all the spectra shown in Fig. 4, as well as those used for our measurements, were obtained with RBW = 10 pm.

_{b}The accuracy of the measurement of the FWM peak frequency is also affected by the presence of spectral dips due to water absorption within the OSA [see Figs. 4(a) and 4(c)] for normal relative humidity conditions (i.e. ~40%). We have overcome this problem by placing the OSA inside a low humidity chamber, as described in Appendix C.

## 4. Results

Figure 5 shows the measured *λ*_{0}(*R _{b}*) together with theoretical curves detailed in Appendix D. The measured values of

*λ*

_{0}for each

*R*[circles in Fig. 5] were obtained by performing a nonlinear fitting routine of Eq. (5) using the measured values of

_{b}*ω*and

_{ℓ}*ω*, as described in Section 2. Since the FWM spectrum was distorted due to the bending induced birefringence [see Figs. 4(a) and 4(c)], we were able to measure

_{FWM}*λ*

_{0}(

*R*) for the $x$ polarization in 9 different values of

_{b}*R*ranging from 10 to 1 cm. For the $y$ polarization [triangles in Fig. 5], we could only measure

_{b}*λ*

_{0}for

*R*≈3 and 4 cm. For smaller values of

_{b}*R*, the FWM spectra for the two polarizations are indistinguishable due to the significant broadening [see Figs. 4(a) and 4(c)] of the FWM spectrum. That is because

_{b}*λ*

_{0}is shifted towards

*λ*, which implies a reduction of (

_{ℓ}*ω*–

_{FWM}*ω*)

_{ℓ}^{2}in Eq. (7). However, there is a good agreement between the measured and calculated

*λ*

_{0}(

*R*) for both polarizations in Fig. 5.

_{b}We also plot in the right axis of Fig. 5 the calculated total loss for the 20 m of DSF and the loss for *R _{b}* = 1 cm, measured with a power meter. For larger

*R*the loss was within the error of this instrument (0.4 dB). One can notice from Fig. 5 that the bending induced loss appears in very tight curvature while Δ

_{b}*λ*

_{0}is noticeable even for values of

*R*as large as 8 cm.

_{b}As described in the Section 2, besides *λ*_{0} were able to find information about high-order dispersion. We found that the ratio *β*_{3}/*β*_{4} or Ω_{34} ≈– 213 ps^{−1} remained unchanged for *R _{b}* = 10 to 4 cm. In addition, by measuring the width of the FWM spectrum and using Eq. (7) we were also able to independently measure the value of

*β*

_{3}≈0.114 ps

^{3}/km for

*R*= 10 to 4 cm, that also remained unchanged. By combining Ω

_{b}_{34}and

*β*

_{3}we obtain a value of

*β*

_{4}≈– 5.3x10

^{−4}ps

^{4}/km within this range of

*R*. Our calculations show that these values of high order dispersion are in agreement with our measurements at least for

_{b}*R*= 10 cm, and that these quantities vary slightly from

_{b}*R*= 10 to 4 cm [see Appendix E].

_{b}In the case of tight bending (*R _{b}* < 4 cm) we were not able to measure

*Ω*

_{34}nor

*β*

_{3}. This was because in that case the spectrum becomes very broad and exhibits secondary peaks due to birefringence as illustrated in Figs. 4(a) and 4(c). However, our calculations also show that high-order dispersion changes significantly for

*R*< 4 cm.

_{b}## 5. Discussion

#### 5.1 Effect of bending on fiber dispersion

In order to obtain a deeper understanding of the relationship between the change of the mode profile due to bending with Δ*λ*_{0}, we modeled a step-index profile with a raised-index ring. The raised-index ring is one of the multi-cladding options, commonly adopted in the design of DSFs, that allows low bending loss and low splicing loss with SMFs [17,18]. The specification of the refractive index profile of the modeled DSF that is in agreement with our measurements of Δ*λ*_{0}(*R _{b}*) is illustrated in Fig. 6(a), where the parameters

*n*,

_{core}*n*and

_{cladd}*n*are the refractive indices of the core, cladding and raised-index ring, respectively, and the geometrical parameters

_{ring}*r*,

_{c}*r*and

_{i}*r*are the core, inner, and outer ring radius (see Appendix D for more details of the numerical simulations).

_{o}In Figs. 6(a) and 6(b) we show the refractive index profiles for the bent fiber obtained by conformal mapping, for *R _{b}* = 10 and 0.5 cm respectively, both as a function of the horizontal coordinate of the transformation plane $u={R}_{b}ln\left(x/{R}_{b}\right)$. In addition, the mode normalized intensities corresponding to each refractive index profile, obtained from FEM numerical calculations, are illustrated in Figs. 6(c) and 6(d). Note that in the case of the tighter bend, the mode distribution becomes significantly asymmetric, with substantial energy shifted away from the center of curvature, but still inside the core, thus the mode is more confined. This is illustrated by the reduction of the shadowed area in Fig. 6(b) relative to that in Fig. 6(a). As shown below, in this fiber

*λ*

_{0}increases when

*r*is reduced.

_{c}In order to further understand the dispersion properties of this type of fiber, Fig. 7(a) shows *λ*_{0} as a function of *r _{c}*, obtained from our numerical DSF model (in straight condition,

*R*= ∞). The fiber model that matches the measured

_{b}*λ*

_{0}= 1546.48 nm corresponds to

*r*= 2.1 µm [point B in Fig. 7(a)]. We note from this point that a small decrease of

_{c}*r*leads to a positive Δ

_{c}*λ*

_{0}over the range of our measurements presented in Fig. 5. On the other hand, there is another value of

*r*that gives a the same

_{c}*λ*

_{0}= 1546.48 nm, that is point A in Fig. 7(a) (

*r*= 1.7 µm). We have also simulated the effect of bending for this case and observed that

_{c}*λ*

_{0}is shifted to lower wavelengths as

*r*decreases.

_{c}The effects of mode deformation with bending mentioned above have been treated in different theoretical studies on single-mode [19] and multimode [11,20,21] optical fibers, as well as on high-index contrast waveguides [14]. These studies have shown that, for moderate values of *R _{b}*, the mode profile tends to contract and moves towards a region closer to the outer core/cladding interface but still within the core, while for very small values of

*R*a larger portion of the mode moves into the cladding [14,19–21]. In particular, in [14] the dependence of Δ

_{b}*λ*

_{0}with bending is analyzed in terms of this mode deformation. The authors of [14] explain the observed Δ

*λ*

_{0}(

*R*) as an interplay between the increase of anomalous dispersion due to “mode squeezing,” and the increase of normal dispersion promoted by a larger amount of energy in the cladding for very tight bending. However, in contrast to the results of the waveguides in [14] where

_{b}*λ*

_{0}is always shifted to longer wavelengths, we observe from Fig. 7(a) that in optical fibers

*λ*

_{0}may be shifted to longer or shorter wavelengths, depending on the precise value of

*r*.

_{c}We point out that, in the case of a DSF and the range of *R _{b}* used in our experiments, the contraction of the mode profile [see Fig. 6(d)] or the change in dispersion is not as evident as in the cases of single-mode fibers bent with smaller values of

*R*[19,21], multimode fibers [11,21], or high-index contrast waveguides [14]. In addition, the change on dispersion with bending in DSFs is not as drastic as, for example dual-core photonic-crystal dispersion compensating fibers [22] that exhibit a narrow band of high negative CD due to the coupling between the modes of the inner and outer cores. The theoretical study in [22] shows that bending influences significantly this coupling, causing a shift of the central wavelength of the negative CD narrow band up to 128 nm for

_{b}*R*= 1 cm.

_{b}We should note that many choices of the fiber profile parameters give a reasonable agreement with the measured dispersion parameters of our fiber for large *R _{b}*. In Fig. 7(b) we show how

*λ*

_{0}varies as we modify

*r*(maintaining

_{i}*r*= 2.1 µm and

_{c}*r*–

_{o}*r*= 2 µm). The calculations were repeated for three values of

_{i}*R*(10, 1.5 and 1.0 cm). We can see that only for

_{b}*r*= 4.6 µm, the variation of

_{i}*λ*

_{0}matches that observed in our experiments [denoted by the vertical line in Fig. 7(b)]. In fact, this figure was used as a guideline to choose the parameters of our simulated DSF. However, for a different choice of GeO

_{2}concentration in the core and/or in the ring, one should find another profile matching the measurements of

*λ*

_{0}(

*R*). Our intention was not to perform reverse engineering, but only have a simulation model at hand to understand the experimental results.

_{b}#### 5.2 Summary and conclusions

Thanks to the sensitivity of FWM to minute changes in the phase-matching condition, we have been able to measure the change of *λ*_{0} as a function of *R _{b}* with <0.1 nm accuracy in

*λ*

_{0}. We have shown that

*λ*

_{0}may be shifted by ~12 nm in a DSF when it is wound with

*R*= 1 cm, which is a large Δ

_{b}*λ*

_{0}compared to those obtained from other environmental modifications [23], such as temperature (30 pm/ºC), pressure (7.6 pm/MPa) or strain (1.7 nm/%).

We have also shown that in a DSF, as we gradually reduce *R _{b}*, changes in dispersion manifests more quickly than bending induced loss. For example, in our DSF, we were able to measure the bend induced loss only for

*R*= 1 cm (

_{b}*α*~ 1.2 ± 0.4 dB in 320 turns), while Δ

*λ*

_{0}≈ 12 nm, which is large and easier to measure.

According to our measurements, the method based on FWM between and incoherent pump and a weak laser is sensitive enough to measure values of Δ*λ*_{0} as small as 0.04 nm. In addition, we have measured high-order dispersion such as the ratio *β*_{3}/*β*_{4} ≈ – 213 ps^{–1}, *β*_{3} ≈ 0.114 ps^{3}/km and *β*_{4} ≈ – 5.4x10^{–4} ps^{4}/km within *R _{b}* = 10 to 4 cm. Furthermore, our numerical model for the DSF that is in good agreement with our measurements of Δ

*λ*

_{0}with bending shows that high-order dispersion should change significantly with bending for

*R*< 4 cm (see Appendix E).

_{b}Our measuring method is, in principle, very simple to implement and requires a minimum of instrumentation. In practice, however, one needs a laser and an incoherent-pump source operating in the required wavelength ranges for a given FUT. These can be obtained by using wavelength conversion and supercontinuum generation, as we did in this work. Moreover, several experimental issues have to be handled carefully. First, especially for small *R _{b}*, the fiber should be wound in a single layer (with a precisely defined

*R*) around a cylinder with a smooth surface (thus minimizing microbending effects). Of course, this requires using short fibers and a sensitive method. Second, since detecting the weak FWM in short fibers requires high pump power, one should filter out the pump, thus reducing the stray light (which is always present in any OSA). The stray light level usually prevents the observation of a clear sinc

_{b}^{2}spectrum with sharply defined zeros (which is a good evidence that the fiber is uniform). Third, to determine the weak FWM spectral peaks with good accuracy, one has to reduce the spectral dips due to water vapor absorption within the OSA. In our case, we used glass cylinders with smooth surfaces, reduced the stray light level by ~25 dB, and enclosed the OSA in a low humidity chamber (~7%).

Our observations may impact the design of photonic devices based on FWM or other nonlinear effects that depend on *λ*_{0}, such as parametric amplifiers [24], wavelength converters [1,2], soliton lasers [25], and frequency comb generators [26]. In these devices, the fiber should be wound with a uniform *R _{b}* to prevent variations of

*λ*

_{0}that may reduce the efficiency of the device. If Δ

*λ*

_{0}induced by bending can be accomplished in a controlled manner, it could be used as a mechanism for

*λ*

_{0}tuning [27] to obtain high parametric efficiency in a desired spectral region, or to obtain a fiber with a gradual variation

*λ*

_{0}(

*z*) to maintain phase matching in the presence of loss [28], or even for gain equalization to compensate fluctuations of the fiber geometry [29].

Our measuring method can be applied for fibers with positive *β*_{4} (see for example [6]), the only difference is that the center wavelength of the incoherent-pump spectrum should be in the anomalous dispersion region of the fiber. In the case of a fiber with significant spatial variation of *λ*_{0}, the FWM spectrum should exhibit several peaks (one for each value of *λ*_{0}) and with bending all these peaks should shift (of course, each piece of fiber with a constant *λ*_{0} must be long enough to display a large and sufficiently narrow peak to be detectable). Furthermore, our method can be scaled to the cases of short, high-index contrast waveguides and ring resonators [14,15], if the dispersion parameters and the nonlinearity coefficient are large enough to compensate for the very small length.

Finally, it is interesting to estimate how *λ*_{0} varies in a typical spool that most industries use to wound a 25 km fiber. These spools have an inner radius of 7.96 cm and are 9.5 cm high. The conventional fibers with primary polymer protection have an outer diameter of 250 µm. Thus, the bending radius varies between 7.97 cm for the innermost layer and 10.80 cm for the outermost layer. Interpolating our data, one could expect that *λ*_{0} will decrease from the inner to the outer layers by 0.04 nm.

## Appendix A Pulsed incoherent pump

An EDFA was used as an amplified spontaneous emission (ASE) source, which was modulated by an acousto-optic modulator (AOM) with ≳ 40 dB of extinction ratio, driven by 20 ns pulses at 100 kHz repetition rate. The optical pulses at the output of the AOM have ~10 mW peak power. A second EDFA was used as a booster to increase the peak power of the incoherent optical pulses up to ~100 W. The obtained ASE spectrum (between 1546 and 1564 nm) was not quite adequate to measure our FUT that has *λ*_{0} ≈ 1546. In order to obtain a spectrum centered around this value of *λ*_{0}, we broadened this source in a 2 m highly non-linear fiber (HNLF) with a nonlinear coefficient *γ* ≈ 14 W^{−1} km^{−1} and *λ*_{0} ≈ 1560 nm. The resultant supercontinuum was then spectrally sliced by means of an optical tunable filter. The pulsed incoherent pump was nearly 32 nm of width [see Fig. 3(b)] and had a pulse peak power of approximately ~3 W. All measurements of peak power of the incoherent pulses were done with a photodetector of 1 GHz of bandwidth and a 4 GHz bandwidth oscilloscope.

## Appendix B Pulsed tunable coherent source

The pulsed tunable coherent source (used as a tunable laser) was derived from a FWM-based wavelength converter (WC). The choice of setting up the WC was due to the wavelength and power limitation of the tunable lasers available in the laboratory. The WC was constructed by combining a pump near 1550 nm and a signal tuned around 1464 nm and 1510 nm in 100 m of HNLF with a nonlinear coefficient *γ* ≈ 6 W^{−1} km^{−1} and *λ*_{0} ≈ 1554 nm, by which we were able to obtain a pulse peak power of ~200 mW around 1600 – 1650 nm. For the pump, a tunable laser was modulated by an AOM, driven by 20 ns pulses at 100 kHz repetition rate, and then amplified by an EDFA, resulting in a peak power of ~10 W. The pump was combined with a CW signal derived from the second tunable laser using a WDM coupler. At the output of the HNLF, an optical tunable filter was used to select only the generated idler wave.

## Appendix C Low humidity chamber

Due to the presence of water vapor inside the OSA, we decided to place it inside a low humidity chamber [See Fig. 3(a)]. This chamber was sealed and allowed us to maintain the OSA inside a low humidity environment (RH < 10%) by putting 4 kg of silica gel inside the chamber and injecting nitrogen gas.

## Appendix D Numerical simulations

The modified refractive index profile due to bending [see in Fig. 6(b)] was obtain by means of the conformal mapping method [11,12,19,21], which is widely used for bending analysis of waveguides. In this method, the modal analysis is performed in a coordinate system where the bent fiber of homogeneous (by parts) refractive index profile *n _{b}* is transformed into an equivalent straight fiber with an inhomogeneous refractive index profile given by [11,12]:

*n*

_{s}is the refractive index of the unstressed fiber,

*ν*= 0.20 is the Poisson's ratio coefficient, and

*P*

_{11}= 0.121 and

*P*

_{12}= 0.270 are the components of the photo-elastic tensor of SiO

_{2}[11,30].

The refractive indices of our numerical model of the DSF *n _{core}*,

*n*and

_{ring}*n*were obtained from the mixed model for GeO

_{cladd}_{2}-SiO

_{2}glasses proposed in [31], which takes into account the concentration of GeO

_{2}. We started from a step-index fiber model with a Δ

*n*= (

_{core}*n*–

_{core}*n*)/

_{cladd}*n*= 0.5% and

_{core}*r*≈ 2 µm, that shows more stability of

_{c}*λ*

_{0}near 1550 nm to variations of

*r*. Subsequently, we have included the raised-index ring with Δ

_{c}*n*= 0.05% and obtain a DSF fiber model that matches our experimental results with

_{ring}*r*= 2.1 µm and

_{c}*r*= 4.6 µm as shown in Fig. 6(a).

_{i}## Appendix E Effect of bending on high-order dispersion

In Fig. 8 we present the percentage variation obtained from our numerical calculations for *λ*_{0} and high-order dispersion (i.e. *β*_{3}, *β*_{4} and the ratio *β*_{3}/*β*_{4} = Ω_{34}) as a function of *R _{b}* for the $x$–polarization. In this figure, the percentage variation of the high-order dispersion coefficients is larger than that for λ

_{0}, a fact that becomes more evident for tight bending (i.e.

*R*< 4 cm).

_{b}## Funding

Sao Paulo Research Foundation (FAPESP) (2008/57857-2, 2012/50259-8, 2013/20180-3, 2015/11779-4); National Council for Scientific and Technological Development (CNPq) (574017/2008-9); Coordination for the Improvement of Higher Education Personal (CAPES); Administrative Department of Science, Technology and Innovation (Colciencias).

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