## Abstract

The spatially resolved spectral (S^{2}) imaging method is applied on an active microstructured fiber, with a multi-filament core (MFC). This type of fiber has been designed to be the last amplifying stage of a source for a long range coherent lidar. Studying the influence of the bending radius on the modal content with or without gain, we demonstrate that an upper-bound of the high-order modes content can be found by performing the S^{2} imaging on the bleached fiber. S^{2} imaging is then used to verify that the output beam of the MFC fiber can be made effectively single-mode. We also show that it can be simply adapted for measuring the fiber birefringence. Finally, a comparison of the MFC fiber mode area with that of a standard large mode area Erbium doped step index fiber illustrates the interest of the MFC structure for high power amplifiers.

© 2012 OSA

## 1. Introduction

Laser applications such as coherent detection lidars require the emission of light pulses with excellent spectral and spatial coherence, together with a high peak power, in a portable system. Rare-earth ion doped optical fibers are well-known for their ability to amplify quasi-gaussian spatial modes, in robust compact setups. For long range systems, the 1.5 µm transition of Erbium is widely preferred, because it offers an excellent trade-off between eye safety, high efficiency detectors, and atmosphere transparency.

However, the high peak power pulses propagating in small core fibers induce high intensities that generate non-linear effects. Single frequency lasers used in coherent detection lidars are first limited by stimulated Brillouin scattering: the propagating light wave generates acoustic waves, with a periodic pattern that scatters back a fraction of the input light wave. This backscattered radiation reduces the amplifier efficiency, and more important, it can destroy input or output components (isolators, couplers, pump diodes…).

There are basically three ways to reduce the Brillouin effect, either by hindering the propagation of the acoustic wave (antiguiding) [1], or by reducing the effective length of amplification [2], or finally by increasing the optical mode size to reduce the peak intensity. This last approach has led to the development of Large Mode Area (LMA) fibers with low numerical aperture (NA), where single-mode propagation is now achieved for mode areas of about 10.000 µm^{2} in passive fibers [3]. However, the task is more challenging for Er^{3+} doped silica fibers, since the necessary high Aluminium co-doping of the core increases the NA.

Peak powers of several kW have been obtained recently for Yb doped fibers thanks to strong doping and high amplification efficiency [4]. Yet doping standard silica fibers with Er^{3+} with the same concentrations is not possible, for the ions clusters induce inter-ions energy transfers and reduce the pump efficiency. One original approach to alleviate this limit with Er^{3+} is to use phosphate as the host matrix, to dilute the ions. This allowed reaching peak powers at the kW level [2]. The same performance has been demonstrated by our group in a silica based fiber, thanks to the multi-filament core (MFC) original design [5]. This Er:Yb doped fiber amplifier has delivered 1 µs duration pulses, with a repetition rate of 5 kHz, and a beam quality parameter M^{2} = 1.3. Reducing the pulse duration to 95 ns, we could increase the peak power and reach the Brillouin threshold for a signal output of about 2 kW [6]. The mode effective area in the MFC was then 880 µm^{2}, which is very large for Er:Yb doped silica fibers.

The basic principle of the MFC fiber is to synthesize a large core with low numerical aperture by assembling several high NA filaments, doped with Erbium together with Ytterbium and Phosphorus, and assembled in a hexagonal lattice [5]. The material in between the filaments is made of fluorine doped silica, to control the NA. The filaments are separated from each other by a distance of a few wavelengths, thus reducing the filling factor of the low index core, but still allowing a strong coupling between each other. The light field thus propagates as a super-mode in the artificially low NA fiber. We developed a numerical model based on fundamental space filling mode method, and showed that the high order modes losses are the same as in an equivalent step index (SI) fiber with NA = 0.04 and core diameter 32 µm [7].

Here we report on the measurement of the actual high order modes (HOM) content in an MFC fiber sample. Various methods have been proposed to determine the relative power of the various eigenmodes of a slightly multimode fiber, and are generally based on the measurement of inter-modal interference or projection of the output beam on the eigenmode basis [8–14]. We choose the spatially resolved spectral interference imaging (so-called S^{2} imaging) proposed by Nicholson et *al.* [10], because it is the simplest way to yield all following information for a few modes fiber: HOM amplitude, relative group delay, and mode profile. We then draw conclusions on the ability to make the MFC fiber amplifier perfectly single-mode, and show how the S^{2} imaging can be simply adapted to measure the fiber birefringence. We finally test an Er^{3+} doped step index fiber with the largest core commercially available for a relatively low NA, and compare its modal content with that of the MFC fiber.

## 2. Spatially resolved spectroscopy of the propagating modes

#### 2.1 Principle of the measurement

The S^{2} imaging consists in the spectral analysis of the near-field image of the interference pattern between the fundamental mode and the various HOMs guided in the fiber [8]. For an optical angular frequency *ω*, the various guided modes have different group velocities. Since the equations of propagation in fibers are generally solved in terms of phase indices of the eigenmodes, we will consider the group indices, instead of the group velocities. By definition, the group index results from the contributions of phase index *n _{φ}^{(i)}* of mode (

*i*) and of the mode dispersion:

For a given point (*x,y*) of the near field image, where the modes (*i)* and (*j)* overlap, their phase difference after a propagation over a fiber length *L* can be written $\Delta {\phi}^{i,j}=\frac{\omega}{c}L\left({n}_{g}^{(i)}-{n}_{g}^{(j)}\right)$. Therefore, scanning the optical frequency *ω* of the input light around a central frequency *ω*_{0} will translate into a modulation of the output intensity *I(x,y,ω)*, with a period that is directly related to the group index difference:

Calculating the Fourier transform *Ĩ*(*x,y,τ*) of the signal modulus at one point of the near field, we obtain the spectrum of the interference between the fundamental and higher order modes at this point, as a function of their propagation group delay. The map of the intensity spectra *Ĩ*(*x,y,*Δ*τ _{0}*) for one particular group delay Δ

*τ*gives the image of the mode that propagate with the corresponding delay compared to the fundamental mode. The sum $\sum _{x,y}\tilde{I}(x,y,\Delta \tau )$ of all the points of the map as a function of Δ

_{0}*τ*eventually yields the intermodal interference spectrum.

A relevant parameter to define the ratio between the fundamental mode power (*P _{0}*) and the HOM power (

*P*) is the so-called multipath interference (MPI) amplitude, defined as MPI = 10 Log (

_{HOM}*P*/

_{HOM}*P*). Throughout the article, the MPI is calculated following the method described in [10], using the same approximation that the contribution of the higher order modes is neglected in the slowly varying component of the interference signal (i.e. power of the fundamental mode as the frequency is scanned). As will be shown later, the MPI values inferred from our measurements in the MFC are lower than −12 dB for the LP

_{0}_{11}mode. A simple simulation of the interference between LP

_{01}and LP

_{11}modes shows that in this case, the error induced by the approximation is lower than 0.7 dB.

#### 2.2 Experimental setup

There are two ways to perform the S^{2} imaging of the inter-modal interference. One can inject a broadband source in the fiber and analyze the spectrum after the propagation, for each point of the near file. The second method, suggested in [10], consists in injecting a source with tunable frequency, and recording the interference images on a CCD as the frequency is scanned. We chose the latter solution, as the time requested is on the order of a minute, instead of one hour for the first one [11], and because the setup is more simple.

The 6 m long MFC fiber that is analyzed has a core made of 37 filaments doped with Er:Yb (Fig. 1
). The filling ratio of the hexagonal core by the active filaments is 13%, and the effective core absorption is 13 dB/m at 1530 nm. The equivalent SI fiber has a numerical aperture NA = 0.04, a core radius of 16 µm [7], yielding a normalized frequency V = 2.6 and a mode field diameter MFD = 34 µm. The MFC fiber is thus slightly multimode, and a propagation simulation by fundamental space filling mode method shows that 3 types of super-modes should be guided, namely LP_{01}, LP_{11} and LP_{21}. One could object that according to the equivalent SI fiber model, only the two first mode groups should propagate since V_{model} = 2.6<V_{cut}(LP_{21}) = 3.83. This is a limit for our use of this model, which assumes that the mode is a plane wave propagating in a core made of an infinite periodic lattice [15]. Since the real core has a finite width, the periodicity condition breaks down as the electric field starts entering the clad, which is the case for the HOM modes.

As illustrated on Fig. 1, our seed source is an extended cavity diode laser (ECDL), with high wavelength stability and resolution (Δλ=10^{−3} nm). It is tunable from 1456 to 1584 nm, and delivers a maximum power of about 4 mW. The fiber coupled output of the ECDL is injected, via a fiber isolator (not represented), into a fiber pre-amplifier that can deliver up to 1 W between 1535 and 1584 nm. The output of the pre-amplifier is a polarization maintaining single mode fiber, with MFD=10 µm. The polarization of the input field can be adjusted with a half wavelength plate (HWP). A polarizing beam splitter (PBS) at the fiber output allows selecting the projection of the field analyzed on the CCD (128x128 InGaAs pixels, 12 bits). Finally, the fiber can be pumped in the inner clad by a diode laser that delivers up to 60 W at 976 nm.

## 3. Influence of pumping and curvature on the modal content

As a first step, we performed the S^{2} imaging without pumping the fiber, taking advantage of the saturable absorption at the signal wavelength in a simple setup. This preliminary measurement gives an estimate of the modal content in the fiber. It also serves as a reference when we test how the pumping improves the fundamental mode content to the detriment of the higher order modes.

#### 3.1 S^{2} measurement without pumping

In order to see the modes interference at the output of the doped fiber, the input signal power is set high enough to bleach the Er^{3+} ions. The S^{2} spectra are displayed on Fig. 2
, for increasing values of the input power, with a wavelength span from 1544 to 1576 nm. The fiber alignment was not optimized, to make the various HOM appear more clearly.

For each power, the same four peaks can be identified. Mapping the amplitudes of the interference spectrum for every pixel, we can identify the peaks at 0, 0.4, 3.7 and 4.3 ps/m as respectively the LP_{01}, LP_{11}, and LP_{21} (odd and even) super-modes. The near field images of the super-modes clearly reproduce the multi-filament structure, but this does not appear in the far-field (see [5]). The three lobes aspect of the LP_{21} image at 3.7 ps/m is simply due to the bending of the fiber, which induces an overlap between the two lobes that are aligned with the curvature plane [16]. As for the orientation of the LP_{11} lobes, it is essentially dictated by the direction of the fiber misalignment.

As expected, as the injected power increases, the noise level decreases whereas the signal (LP_{11} peak amplitude) remains constant. Since the noise floor does not vary significantly for input power between 470 mW and 680 mW, we find that 500 mW is enough to saturate the active core, and to reach the best SNR. The possible noise sources in the measurement are: amplified spontaneous emission in the core, spontaneous emission guided by the inner clad, signal coupled in the clad, and electronic noise. By comparing results with various commercial single and double clad fibers, we checked that the main noise contribution is the spontaneous emission collected by the clad. For the highest value of the injected power, the MPI values are −12.5 dB for the LP_{11} mode and −26 dB for the highest LP_{21} mode. We remind here that the fiber coupling was purposely not optimal.

Note that although both odd and even LP_{11} modes are able to propagate in the fiber, their relative power in the modal content depends strongly on the injection, and one of them is generally much higher than the other. This explains why only one LP_{11} mode can be identified from our images (see [10]). As for the increase of the amplitude of the LP_{21} peaks with decreasing input power, while the LP_{11} peak remains constant, it is not clear whether it is due to an increase of the LP_{21} MPI, or to an increase of the noise floor due to the higher amplitude of the modes of cladding guided spontaneous emission for these group delays.

In order to check the ultimate sensitivity of our measurement, we measured the S2 spectrum of a single mode fiber (SMF28) that was directly connected to the fiber-coupled output of the ECDL. The noise floor of this spectrum stands at about −33 dB, about 10 dB lower than the best noise floor of the MFC measurements. This limit corresponds to the electronic noise of the CCD camera, as illustrated by the measurement performed with the laser turned off. Since the power received by the pixels is digitally coded on values between 1 and 2^{11} = 2048, their dynamic is limited to 33 dB. The residual peaks appearing on the single-mode fiber spectrum (notably at 1.4 ps/m) cannot be due to HOM interference. They result from the various fiber inter-connections before the SMF coupling (ECDL-isolator-amplifier-collimator) that create parasitic micro-reflections, each inducing a phase delay.

#### 3.2 Influence of the bending on the HOM amplitude

We then use the S^{2} measurement to study the damping of the intermodal interference as we spool the MFC fiber with decreasing bending radius R_{c}. The measurement is first performed for the saturated fiber, optimally injected, without pumping. As expected, the amplitude of the interference decreases with R_{c} (Fig. 3
, left), because of the stronger coupling of the HOM with the fiber clad [16]. For R_{c}<8 cm, the peak even disappears in the measurement noise that increases itself, since the distortion of the modes induces a leakage of the fluorescence in the clad. No reduction of the output power has been noticed for any value of the curvature.

The same measurement is performed while pumping the fiber at 976 nm in the clad, looking for a typical gain of about 10 dB. Injecting a seed power of 115 mW at 1545 nm, for an input pump power of 10 W, we obtain a gain of 12 dB. The pump absorption is 93%, so the optical-optical efficiency is about 20%. In this case, the measurement noise is lower by about 5 dB (Fig. 3, right), thanks to the much better contrast between the amplified core signal and the fluorescence guided in the inner clad. As in the saturated regime, the amplitude of the interference peak decreases as R_{c} decreases, without any output power reduction either.

We quantify the difference between the two regimes by calculating the MPI for each of these spectra as a function of R_{c}. According to the results plotted on Fig. 4
, the amplitude of the interference in the active regime (with gain) is systematically lower than in the passive regime (bleached fiber). This corresponds to a lower amplification for the LP_{11} mode than for the fundamental mode when the signal is amplified. Calculations using a model of the MFC fiber by the finite element method (FEM) show that for any bend radius, the overlap between the doped filament structure and the LP_{01} mode is only 10% higher than for the LP_{11} mode. Therefore, the overlap between modes and gain medium does not explain the MPI improvement. A strong depletion of the population inversion by the LP_{01} mode could be a good candidate. This effect is the object of ongoing work. We note that it confirms the tendency that was observed in [17] for a similar fiber structure.

## 4. Measure of the birefringence

The same S^{2} method can be applied to measure the group birefringence induced by the Panda structure inserted in the MFC inner cladding. The setup is modified by rotating the polarization of the input field at 45° of the axis of the stress, in order to inject both H and V polarizations. At the output of the fiber, the PBS polarizer is also set at 45° of the birefringence axis, in order to recombine the orthogonal transmitted fields and to collect their beat-note. In this setup, the input power fluctuations are suppressed by collecting a fraction of the injected power to normalize the output images. The output polarizer may be set parallel (maximum transmission) or perpendicular (minimum transmission) to the input polarization.

Given its principle, the S^{2} method can only yield the group birefringence. However, it is known that in the case of stress induced anisotropy, the difference between group and phase birefringence is on the order of 10% [18,19]. Indeed, the anisotropy caused by Panda rods consists of a refractive index offset between the orthogonal axes, while the cylindrical symmetry of the core is kept intact. The modal dispersion is thus identical for the two axes, and only the material dispersion is slightly different because of the index offset.

For an anisotropic waveguide, on the other hand (e.g. highly elliptic core), one can intuitively picture that the two orthogonal axis have different guiding properties, which modify notably their respective group indices. Although the hexagonal shape of the MFC core is slightly elliptic, we can check through FEM calculation that the expected difference between phase and group birefringence is still below 10%.

Eventually, this approximation allows relating the phase birefringence *B = n _{H} – n_{V}* to the group delay given in the S

^{2}spectrum by the position of the interference peak between the orthogonal polarizations

*H*and

*V*of the fundamental mode:

For each polarization H or V, the two relevant modes guided in the MFC fiber are LP_{01} and LP_{11}. Given these four coherent fields, and neglecting the inter-LP_{11} interference, four peaks should appear in the spectrum in addition of the bias term, as explained in Table 1
. The first peak corresponds to the interference between the crossed polarized fundamental modes, and is the one that allows measuring the birefringence, appearing at delay *LB/c*. Then, the interference peaks between LP_{01} and LP_{11} can occur for either parallel or orthogonal polarisations. In the first case, the peak appears at the usual group delay Δτ_{g} between LP_{01} and LP_{11}. In the second case, since the birefringence induces a positive or negative delay Δτ_{B} between the two orthogonal modes, it shifts the group delay in one way or in the other, leading to one peak at Δτ_{g} + Δτ_{B} and the other at Δτ_{g}-Δτ_{B}.

The resulting spectrum is plotted on Fig. 5
, along with the images of the power and phase of the various modes. The measurement was performed without pumping the fiber. The peak corresponding to the interference between the crossed fundamental modes appears at 0.21 ± 0.02 ps/m, which yields a group birefringence of (6.3±0.6) 10^{−5}. This also corresponds to the phase birefringence, with a slightly higher error: *B* = (6.3±0.9) 10^{−5}. This value is consistent with the value of (7 ± 1) 10^{−5} that can be expected from the geometry of the stress-applying rods in the clad [5,20]. As a comparison, the birefringence for a polarization maintaining single-mode fiber is typically one order of magnitude higher.

The nature of the interfering modes is inferred from their power images, and confirmed by the phase images, which clearly show the *π* shift between the two lobes of the LP_{11} modes. In our particular case, only three peaks appear instead of four. This is due to the coincidence that makes the group delay Δ*τ _{g}* = 0.38 ps/m very close to twice the birefringence delay Δτ

_{B}. Therefore, the peak that should appear at Δ

*τ*-Δ

_{g}*τ*is hidden by the birefringence peak I, which is about 10 dB higher (since the fourth peak involves the same amplitudes as peak III).

_{B}## 5. Comparison with a step index LMA

In order to evaluate the potential advantage of the MFC structure in an active fiber, we compare it to an Er-doped step index (SI) fiber, for the relevant parameters that are modal content and mode effective area. The 9 m long SI fiber under study has a core diameter 20 µm and NA = 0.09, doped with Er only (Liekki Er60-20/125 DC). It is slightly more multimode (V = 3.65) than the MFC fiber, but for proper injection, the HOM beyond LP_{21} still have negligible amplitudes. Therefore, this LMA fiber stands for one of the best compromise between large mode area and low HOM content that is available on the erbium fiber market.

As previously for the MFC, we look over the influence of the curvature on the amplitude of the inter-modal interference (Fig. 6
left). The first observation is that, unlike for the MFC, the HOM amplitude does not decrease significantly as we reduce R_{c}. This is due to the higher NA of the step index LMA fiber that confines more the HOM in the core, thus reducing their bending losses. In addition, we find a notable shift of the group delay of the two modes (LP_{01} and LP_{11}). This is also due to the higher NA of the LMA fiber: the group velocities are initially more separated, so this difference is more rapidly affected by the bend induced distortion of the index profile. This influence of the NA on the group delays reduction has been reproduced with FEM calculations.

We then compare the S^{2} spectra obtained with SI and MFC fibers for the lowest radius of curvature that we applied (Fig. 6 right). The effective areas of the fundamental modes propagating in these fibers are respectively 250 µm^{2} and 880 µm^{2}, and the corresponding MPI are −13 dB and −29 dB. In both case, no reduction of the output power is detected. Furthermore, we notice that the noise floors are identical for S^{2} measurement on the SI and MFC fibers (about −23 dB for the largest bending radius). We conclude that the possible scattering losses induced by the microstructured core do not affect the modal content, in the limit of the measurement sensitivity.

Finally, given that the MFC fiber allows reaching a much larger single mode area than the largest mode area commercial SI fiber, the comparison illustrates the interest of the core micro-assembling to obtain a large core Er^{3+} doped fiber with high beam quality.

## 6. Conclusion

The S^{2} imaging has been applied to determine the modal content of an active multi-filament core fiber, doped with Er:Yb. We first demonstrate that applying this method to a saturated fiber, without the complexity and precautions of pumping, allows obtaining an upper bound of the MPI amplitude. We then show that for radius of curvature lower than about 10 cm, with gain, the MPI is lower than −25 dB. This means that the interfering higher-order modes represents 0.3% of the total power, and even less when the fiber is used as an amplifier (for a gain of more than 10 dB). We believe that this is a fair clue that the fiber amplifier presented in [5] was effectively single-mode. The comparison with the largest mode area commercial SI fiber confirms the interest of the multifilament core structure for amplifying a single fundamental mode with a very high beam quality.

In addition, we demonstrate that the S^{2} imaging can be used to infer the group birefringence induced by a Panda structure in the cladding, and thus the phase birefringence with a relative accuracy of 10%. We find a value of (6.3±0.9) 10^{−5} in good agreement with the value that can be calculated from the distribution of the stress rods. Although this last figure is not optimal, we conclude that the MFC fiber is a very good candidate to improve peak powers in lidar sources.

## Acknowledgment

We wish to thank Sylvia Jetschke, Sonja Unger and Johan Kirchhof from the Institute of Photonic Technology of Jena (IPHT) for providing the MFC fiber.

## References and links

**1. **M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express **15**(13), 8290–8299 (2007). [CrossRef]

**2. **W. Shi, E. B. Petersen, Z. Yao, D. T. Nguyen, J. Zong, M. A. Stephen, A. Chavez-Pirson, and N. Peyghambarian, “Kilowatt-level stimulated-Brillouin-scattering-threshold monolithic transform-limited 100 ns pulsed fiber laser at 1530 nm,” Opt. Lett. **35**(14), 2418–2420 (2010). [CrossRef]

**3. **S. Ramachandran, J. Fini, M. Mermelstein, J. Nicholson, S. Ghalmi, and M. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev. **2**(6), 429–448 (2008). [CrossRef]

**4. **V. Fomin, M. Abramov, A. Ferin, A. Abramov, D. Mochalov, N. Platonov, and V. Gapontsev, “10 kW single-mode fiber laser,” presented at 5th International Symposium on High-Power Fiber Lasers and Their Applications (2010).

**5. **G. Canat, S. Jetschke, S. Unger, L. Lombard, P. Bourdon, J. Kirchhof, V. Jolivet, A. Dolfi, and O. Vasseur, “Multifilament-core fibers for high energy pulse amplification at 1.5 microm with excellent beam quality,” Opt. Lett. **33**(22), 2701–2703 (2008). [CrossRef]

**6. **G. Canat, L. Lombard, P. Bourdon, V. Jolivet, O. Vasseur, S. Jetschke, S. Unger, and J. Kirchhof, “Measurement and modeling of Brillouin scattering in a multifilament core fiber,” in CLEO 2009, JTuB3.

**7. **G. Canat, R. Spittel, S. Jetschke, L. Lombard, and P. Bourdon, “Analysis of the multifilament core fiber using the effective index theory,” Opt. Express **18**(5), 4644–4654 (2010). [CrossRef]

**8. **A. Barthelemy, P. Facq, C. Froehly, and J. Arnaud, “New method for precise characterisation of multimode optical fibres,” Electron. Lett. **18**(6), 247–249 (1982). [CrossRef]

**9. **G. Brun, I. Verrier, M. Ramos, J.-P. Goure, P. Ottavi, and A.-M. Lambert, “Measurement of mode times of flight in multimode fibers by an interferometric method using polychromatic light: theoretical approach and experimental results,” Appl. Opt. **35**(7), 1129–1134 (1996). [CrossRef]

**10. **J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. **15**(1), 61–70 (2009). [CrossRef]

**11. **J. Nicholson, J. Jasapara, A. Desantolo, E. Monberg, and F. Dimarcello, “Characterizing the modes of a core-pumped, large-mode area Er fiber using spatially and spectrally resolved imaging,” in CLEO 2009, CWD4.

**12. **T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express **17**(11), 9347–9356 (2009). [CrossRef]

**13. **D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C^{2}) imaging of fiber and waveguide modes,” Opt. Express **19**(14), 13008–13019 (2011). [CrossRef]

**14. **D. M. Nguyen, S. Blin, T. N. Nguyen, S. D. Le, L. Provino, M. Thual, and T. Chartier, “Modal decomposition technique for multimode fibers,” Appl. Opt. , in press.

**15. **M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. **18**(7), 1031–1037 (2000). [CrossRef]

**16. **R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. **43**(10), 899–909 (2007). [CrossRef]

**17. **V. Kuhn, S. Unger, S. Jetschke, D. Kracht, J. Neumann, J. Kirchhof, and P. Weßels, “Experimental comparison of fundamental mode content in Er:Yb-codoped LMA fibers with multifilament- and pedestal-design cores,” J. Lightwave Technol. **28**, 3212–3219 (2010).

**18. **S. C. Rashleigh, “Measurement of fiber birefringence by wavelength scanning: effect of dispersion,” Opt. Lett. **8**(6), 336–338 (1983). [CrossRef]

**19. **M. Legre, M. Wegmuller, and N. Gisin, “Investigation of the ratio between phase and group birefringence in optical single-mode fibers,” J. Lightwave Technol. **21**(12), 3374–3378 (2003). [CrossRef]

**20. **P. L. Chu and R. A. Sammut, “Analytical method for calculation of stresses and material birefringence in polarization-maintaining optical fiber,” J. Lightwave Technol. **LT-2**, 650–662 (1984).