Bending-induced inter-core group delays in multicore fibers

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Introduction
In recent years challenges were tackled on the way towards robust fiber-based minimally invasive lensless endoscopes that would ultimately operate in a clinical setting. Both multicore [1,2] (MCF) and multimode fibers [3] (MMF) have been considered together with wavefront shaping devices providing the ability to control the relative phase between the different injected fiber modes. MMF are appealing for their commercial availability, their small diameter (∼100 µm) and their large number of modes that can be used for focusing [4] and imaging [5,6]. When dealing with flexible endoscopes phase control through MMF has been challenging because the fiber transmission matrix (TM) is strongly altered when the fiber is bent. Interestingly, recent works have identified parameters and regimes where deformation-induced changes in the TM can be minimized [7], measured and compensated for dynamically (in limited conditions) [8,9] or even predicted [10], which brings hope to build MMF-based flexible lensless endoscopes. MCF, in strong contrast with MMF, are made with a multitude of single mode fiber cores which show weak or zero coupling, they maintain and translate, to some extent, their output diffraction patterns when a simple phase tip and/or tilt is applied on the input wavefront. This so-called "memory effect" [11] has been extensively used for imaging using scanning [12][13][14] or wide field modalities [15][16][17]. Because a freely moving distal end is expected to deform and compress the MCF inner and outer curvature sides, respectively, it was noted that a slight angular bend (< 3°) causes tip and tilt to be added to the phase profile at the distal tip of the MCF [13,18], similar to the memory effect. Hereinafter we refer to this type of deformation as an L-type bend. When dealing with ultrashort pulses, the bending may also affect the group delays between pulses traveling in the different fiber cores such that they don't overlap and interfere anymore at the distal tip [19,20], precluding any focusing and imaging. This deformation-induced inter-core group delays are important in the context of 2-photon flexible lensless endoscope [14,21] that might ultimately require active phase [18] and group delay controls [20].
In the scope of designing and building flexible lensless 2-photon endoscopes, this paper investigates numerically and experimentally the deformation-induced inter-core group delays resulting from bending MCFs with large (up to 200°) angular bends and assesses the impact on imaging performances. We concentrate our investigation on MCFs showing virtually no core-to-core coupling and having an infinite memory effect [22].

Experiments
Our experimental set-up ( Fig. 1) utilizes phase-stepping spectral interferometry to measure delays between the ultrashort pulses, transmitted through the cores of interest, with respect to the reference core, as detailed in [20]. The output beam from a femtosecond laser source (Amplitude Systèmes t-Pulse, λ = 1030 nm, τ = 170 fs, repetition rate 50 MHz) is expanded with a telescope (not shown) to overfill the aperture of a spatial light modulator (Hamamatsu LCOS-SLM X10468-07). The SLM is used to shape the wavefront entering each of the MCF cores at its proximal facet, relayed via a lens L1 and a microscope objective MO1 (Olympus Plan N, 20x NA 0.40). Transmitted light is collected from the MCF distal end using a 10x NA 0.25 microscope objective MO2 (Nikon Plan) and filtered with a linear polarizer LP (Thorlabs LPNIR100) to ensure maximum contrast of the interference fringes, as the output states of polarization in the MCF under study are random [23]. Distal facet is imaged with a lens L2 onto a camera CCD1 (FLIR FL3-U3-32S2M-CS) in order to monitor the evolution of the transmitted z ' x z power during fiber bending. Far field of the distal end face (via a lens L3) is coupled into a multimode fiber (core diameter 62.5 µm), which relays it to an optical spectrum analyzer OSA (Yokogawa AQ-6315A). Magnification in this configuration is chosen so that only a part of the far field much narrower than the spatial interference fringe is spatially selected with the MMF probe (k x D > 2π, where k x corresponds to the fringe spatial frequency and D is the inner probe diameter). We use phase-shifting spectral interferometry technique, previously described in Ref. [20], to measure the inter-core group delays in the MCF. The entire far-field interference pattern is imaged onto a camera CCD2 (Thorlabs DCU223M) to record the point spread function (PSF) of the fiber imaging system for different bending conditions. The entire distal detection part, including the clamped MCF distal end, is mounted onto a portable unit allowing its translation in xz plane (see Figs. 1,6). Bent fiber geometry is recorded with a mobile camera (Appendix A provides an example of such recording).
Inset in Fig. 1 depicts the MCF under study, which was described in [14]. Note that this MCF exhibits very low cross talk between its cores (< -25 dB). The MCF was fabricated with the following parameters: Ge-doped single-mode cores with a parabolic refractive index profile (maximum difference of 0.0031 compared to silica) and mode field diameter of 3.6 µm, triangular lattice pitch 11.8 µm and the outer diameter of 357 µm (including the double cladding). The total length of the MCF used in the experiments is approximately 300 mm.

Simulations
We perform the bent fiber simulation using the curvature loss formula, typically employed to predict the bending losses in both single mode and multimode fibers [24]. We consider two main effects which induce additional group delays across the MCF face -the fiber elongation/compression in the circular bend and the local refractive index change due to stress-optic effect upon bending. A circularly curved segment of the fiber is transformed to an equivalent straight one via conformal mapping. The modified refractive index distribution across the fiber cross-section is given by the following [24]: where n (x, y) is the refractive index of the unperturbed waveguide structure, R is the radius of curvature, ν is the Poisson's ratio, and p 11 , p 12 are the photoelastic tensor components. The exponential term accounts for the change in optical path length, whereas the term in the square brackets describes the changes to the physical refractive index with the photoelastic effect. We consider a 300 mm long cylinder of 200 µm diameter made of silica with the following material properties: Poisson's ratio ν = 0.17, refractive index of stress-free material n = 1.45 (fused silica for λ = 1030 nm) and photoelastic tensor elements p 11 = 0.16, p 12 = 0.27 (the latter are also wavelength-dependent, the closest values we found in literature are for λ = 1150 nm [25]). For the given values of material properties the geometric change is more significant than the stress-optical contribution. Three characteristic types of bends were studied as the ones most commonly encountered in the experimental conditions. In this article they are referred to as L-, Uand S-type geometries and are produced by applying a force ì F as shown schematically in Fig. 2. For the sake of simplicity, only F x component was chosen to be non-zero, and adjusted to produce displacements of the same magnitude [Figs. 2(a)-2(c)], reflecting the experimentally achievable ones. Fixed constraints are defined at the input fiber facet (L-type bend) and at both end facets for U-and S-bend cases. For the forces applied to x y cross-sections at the corresponding z positions as shown in Fig. 2, we trace the resulting deformed geometries and calculate the total bend-induced group delays ∆[∆τ i ] using Eq. (1) and a least-squares curve fitting [26] to estimate the local radii of curvature along the fiber. ∆[∆τ i ] is calculated as (OPD i0 − OPD i ) /c, where c is the speed of light, OPD i is the optical path difference between the central and the ith core due to local length and refractive index changes upon bending, and OPD i0 is the intrinsic OPD for the ith core (equals to zero in simulations).
Substituting the values of material properties into the stress-optic term of Eq. (1) and replacing the exponential by its first-order Taylor expansion, we can estimate the bend-induced delays by integrating along the fiber: where s is a curvilinear abscissa (ds 2 = dx 2 + dy 2 + dz 2 in Cartesian coordinates). We can further substitute ds/ρ = dα, where α is the turning angle. Hence we obtain where ∆α is the variation of angular increase along the curve formed by the fiber, i.e. ∆α = ∆α n −π, where α n denotes an angle between outward-pointing normal vectors of proximal and distal MCF facets for a deformed fiber geometry without loops. ∆α is referred to as "bending angle" in this paper.

Simulations
Inter-core group delays due to bending are expected to arise from the refractive index inhomogeneity among the cores and change in physical length. Therefore, in our simulations we estimated the group index change across the MCF face, and calculated the total accumulated optical path difference (OPD) and the corresponding ∆ [∆τ i ] (since the group index only exhibits linear longitudinal variations). The model fiber diameter was chosen to be comparable with the circumference which includes the cores in our sample MCF (see inset in Fig. 1).
The simplest bending geometry, which we refer to as L-bend, is simulated by applying a force to the distal fiber end while the proximal end is fixed [ Fig. 2(a)]. OPD change along the arc length (z axis in the unperturbed geometry) for a given x, y [ Fig. 2(a)] represent the situation for one MCF core, taken at the same coordinates. Finally, we calculated the spatial distribution of ∆ [∆τ i ], allowing to retrieve the induced group delay for any given location (x, y) of the core within the MCF [ Fig. 2(g)] with respect to the central core.
The same type of simulations was next performed for the remaining two types of bending, whose denominations were inspired by the corresponding deformed geometry shapes [Figs. 2(b)-2(c)]. In the case of U-bend, the force is applied at z = 15 cm plane, producing a symmetric displacement and OPD distributions along the fiber length [Figs. 2(b), 2(e)]. When estimating the total ∆ [∆τ i ], it resulted in a cancellation of bending-related effects such that across the fiber face its maximum absolute change is less than 1 fs, most likely reflecting the numerical error of the curve fitting procedure. In the case of the S-bend, two forces of opposite directions and different magnitude were applied at 1/3 and 2/3 of the fiber length, which resulted in the asymmetric deformed geometry and OPD distributions along the fiber [Figs. 2(c), 2(f)]. Nevertheless, the overall ∆ [∆τ i ] along the total MCF length is very small (< 1 fs) and can be interpreted as the numerical error -as in the case of U-type deformation, see Fig. 2 All the above calculations were performed using Eqs. (1) and (2). Note that we obtain the same results as the ones displayed in Figs. 2(g)-2(f), using Eq. (3) and substituting ∆α = 22°for L-bend and ∆α = 0 for U-and S-type geometries, and this without the need to integrate over the changing radii of curvature along the deformed fiber.

Experiments
Measurements of induced inter-core group delays for each type of bending geometry are shown in Figure 3. To have a precise estimation of the bending-related inter-core group delays, we first perform a calibration of the intrinsic GDD (relative to the fabrication process and material imperfections), the MCF being kept straight in its reference geometry, then this intrinsic GDD ∆τ i is subtracted from the subsequent inter-core group delays measurements while bending the MCF to give ∆[∆τ i ]. Figure 3(b) displays an example of MCF inter-core group delays ∆τ i spatial distribution, measured for an MCF held relatively straight in its reference geometry [ Fig. 3(a)]. Measurements from the cores with low SNR were discarded; the displayed data covers 125 out of 169 cores. For the intrinsic GDD measurement we obtain a normal distribution ∆τ i with 2σ = 188 fs [ Fig. 3(c)], which is comparable to [20]. Subsequent measurement of the bending-induced inter-core group delays for the L-bend where the distal end was displaced by ∆x of about 7.  Fig. 3(e) show a clear trend spanning from negative added delays to positive ones in the range of about ±150 fs over the entire MCF facet. Force in this experiment was applied along the negative direction of x axis (as indicated by the white arrow), and a qualitative agreement with the stress-induced refractive index simulation can be seen when comparing Fig. 3(e) and Fig. 2(d).  Measurements for U-and S-types of bending geometries are displayed in Figs. 3(g)-3(i) and Figs. 3(j)-3(l) correspondingly. We aimed to achieve displacements along the x axis similar to the L-bend case for an easier comparison between the obtained values. For both of the double-clamped geometries, we found that ∆[∆τ i ] spreads show a marginal contribution with 2σ 20 fs, which we relate to the measurement uncertainty. Note that the error bars (three standard deviations) in presented GDD measurements using phase-stepping interferometry were below 20 fs for most of the cores. We scaled ∆[∆τ i ] colorbars according to Fig. 3(e) (L-bend) experiment for easier comparison of the obtained values given that ∆x fiber displacements are also comparable in all three cases. Here again the close to zero ∆[∆τ i ] trend is in good agreement with the simulated results Figs. 2(h), 2(l).
We focused next on L-type bending, where the proximal and distal MCF facets are not parallel anymore. This is the situation similar to the real endoscopic operation where the distal MCF facet is free to move. As a geometrical parameter of the bend which reflects the evolution of ∆[∆τ i ] spread magnitude we used ∆α, introduced earlier in Eq. (3). Therefore, our reference geometry measurement as well as U-and S-bends in Fig. 3 Fig. 4(g)] exhibit a gradient along the bending axis, whose magnitude is proportional to the bending angle ∆α. As opposed to the first set of measurements (Fig. 3), now the added inter-core group delays exhibit larger ∆[∆τ i ] distribution spreads, as can be observed in Figs. 4(d)-4(f).  Fig. 4(g). An example of such R fitting for ∆α = 180°bending geometry is given in Appendix B. As it can be seen from the plots, bending-induced inter-core group delays can span as far as almost ±1 ps for ∆α = 180°bend, which in turn would require a group delay compensating scheme such as a group delay controller (GDC) [20] to assure the temporal overlap of pulses at the distal MCF facet. For large bending angles one would expect to see some polarization mode dispersion [27]; therefore we performed all our measurements for a single polarization. Additionally, the bend radii in our experiments were kept sufficiently big in order to not induce a considerable bend loss (the transmitted intensity was continuously monitored with the CCD1 looking at the MCF distal facet).
The imaging performance of a 2-photon MCF lensless endoscope is directly related to the peak irradiance, requiring a spatially and temporally compact focal spot [14]. To assess the spatial aspect, we perform an investigation using CCD2 camera placed in the far field of the distal MCF facet (Fig. 1), and use the SLM to produce a distal focused spot [ Fig. 5(a)]. In pulsed laser mode, bending-induced inter-core group delays reduce the number of temporally overlapping beamlets thus reducing the delivered power at the focus. As a figure of merit, we used an L-bent fiber with ∆α = 180°together with the laser in CW operation regime so that individual beamlets from all the cores interfere and generate the brightest focal spot, with power P CW [ Fig. 5  which the coherent combining of the ultrashort pulse will be evaluated for various bending angles. We now move to pulse laser mode (τ = 170 fs) and investigate for various ∆α the focus power loss resulting from bending-induced inter-core group delays. Emergence of a spatiotemporal effect due to added inter-core group delays can be easily observed from Figs. 5(a)-5(c), where the PSF evolves from a circular to an elliptical shape. This can be intuitively inferred from the anisotropy of the OPD change due to bending along only one axis. In other words, the large bending-induced inter-core group delays focus the various frequencies within the spectral width of the pulse into different spatial spots; this effect is reminiscent of the time-space coupling as observed in temporal pulse shapers [28,29]. With this one observes a bending-induced distortion of the focal spot in Figs. 5(c), 5(e) where its full width at half maximum (FWHM) γ along the bending axis γ x is 3 times larger than along y axis for ∆α = 180°L-bend (Appendix A).
We illustrate this effect by calculating the far field distribution from the known intensity and group delay data at MCF distal facet. Figure 5(f) shows the complex field E distribution for the given laser bandwidth, simulated along the bending (x) axis in the CCD2 plane, with the experimentally measured ∆[∆τ i ] values taken into account. Note that different wavelengths get focused (come in phase) at different spatial coordinates. This is analogous to the broadening of an interference fringe due to partial coherence, in this case along the bending axis. Note that on the experimental data, along the orthogonal axis, the fringe width is still diffraction limited. Figure 5(g) represents a simulated case with no added inter-core group delays, resulting in a fully coherent state, thus retaining a symmetric and diffraction limited focus.

Discussion
The first important conclusion that can be drawn from the data presented in Figs. 2, 3 is that only the L-type bending geometry has a considerable impact on the inter-core group delays in the studied MCF. Both U-and S-bends exhibit a self-compensation of the bend-induced inter-core group delays on a scale of the entire fiber length, not affecting the total group delays spread. Second, in the case of L-bends, the added spatial inter-core group delays follow a linear law ∆[∆τ i ] = ax i , where the slope a depends on the bending angle ∆α, and x i is ith core coordinate along the bending axis. More generally, for a given ∆α and known MCF length one can have an estimate of ∆[∆τ i ] for an arbitrary location (x, y) whose precision depends mainly on the bending angle estimation error. ∆[∆τ i ] dependence was found to be linear versus bending angle ∆α, as shown in Fig. 4(h) and from Eq. (1) for a constant L and substituting R = L/∆α. One would expect the same behavior during fiber bending along y axis (generally speaking, along any direction in x y plane), assuming isotropic material properties. Additionally, experimental data in Figs. 4(g), 4(h) indicates that parameter a depends only on the angle between fiber end faces and not on the radius of curvature or relative position of input and output facets. For large angles of the L-bend and with femtosecond pulses, we found that spatial ∆[∆τ i ] distribution affects the imaging system PSF through the loss of coherence between some of the delivered pulses, precluding several cores from contributing to the generation of the interferometric focal spot and thus lowering the delivered peak power. Both these characteristics become crucial when such fibers are employed for nonlinear imaging (such as 2-photon for instance) and therefore a group delay controller [20] covering ∆[∆τ i ] range for the maximum bending angle would be required to ensure constant imaging performance in a flexible lensless endoscope set-up.

Conclusion
We experimentally studied multiple MCF bending geometries in terms of ultrashort (fs) pulse delivery and imaging performance. U-and S-type bends, even for large displacements, did not induce any significant additional inter-core group delays, whereas the added inter-core delays in L-type bending geometry depend linearly on the angle between distal and proximal fiber facets and the core coordinate along the bending axis. When using ultra-short pulses, imaging performance in terms of focal spot quality degrades for large bending angles when one fiber end is moving freely, requiring an active inter-core group delay compensation in the picosecond range in order to secure a stable and diffraction-limited imaging performance. These trends show a good agreement with a simple linear model (Eq. (3)) for the inter-core group delays as predicted by our simulations, where the bending-induced refractive index change for a single-clamped bending geometry (L-type) varies linearly along the bending axis with zero dispersion line crossing the fiber face at its middle, perpendicular to the applied stress. Simulations of the bending geometries where both ends are fixed display axial symmetry of bending-induced refractive index change, resulting in no significant dispersion of mean refractive index along the total fiber length. Our investigations highlight the suitability of MCFs for highly miniaturized, robust and flexible multi-photon endoscopes that could ultimately operate in clinical settings.