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Abstract
Helical core fibers (HCFs) suffer from low coupling efficiency and unavoidable excitation of higher order modes below the cutoff wavelength because of a core tilt with respect to the symmetry axis of the cladding. We propose an effective way of increasing the coupling efficiency to a HCF by untwisting its beginning section in a hydrogen flame. The proposed solution provides also a control over the excitation of higher order modes in HCFs and can be applied in splicing as well as in a free-space launching configuration. We experimentally demonstrate that by using the proposed method, the coupling efficiency between a SMF-28 and HCF can be increased to the level reachable for straight fibers, which is limited only by a modal fields mismatch. We also present detailed numerical and analytical studies of the coupling efficiency between a HCF and SMF versus the pitch distance in the partially untwisted input section of the HCF, which takes into account fundamental and higher-order modes.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spun fibers with a centrally located core and helical core fibers (HCFs) with a decentered core attracted much attention in recent years. Several unique features of such fibers have been already identified, including high circular birefringence useful in magnetic field and electric current sensing [1,2], single-mode propagation even for high normalized frequency- beneficial in laser applications [3,4], generation of modes with orbital angular momentum [5–8] or reduced polarization mode dispersion [9]. Twist-induced resonant couplings between the core and cladding modes in standard and microstructured fibers have been also observed experimentally for appropriate combination of wavelengths and helix pitches [10–12] and studied theoretically with the use of numerical methods, specially developed for modelling twisted waveguides [13,14]. These specific coupling effects can be potentially exploited for building novel fiber optic devices, such as couplers, filters, polarizers, polarization converters or sensors [10–12,15–20]. Moreover, light guidance in a decentered core created by an arrangement of air holes making a partially open ring has been recently experimentally demonstrated for a possible application as a bent sensor [21].
Although the helical twist provides an additional degree of freedom in controlling the propagation characteristics of optical fibers, the large lateral offset (tens of µm) - creating an angular tilt of the core with respect to the cladding symmetry axis - limits the efficiency of free space light coupling [22,23] and increases losses on splices with conventional fibers [24]. Moreover, because of the core tilt, it is practically impossible to purely excite the fundamental mode below the cut-off wavelength in a helical core fiber. Additionally, the relative power division between the fundamental and higher order modes cannot be easily controlled by the cores offset as it is at splices of conventional fibers with a straight core. All these coupling issues limit practical applications of HCFs.
As a possible solution to this problem, an adiabatically tapered splice technique was proposed in [24]. The authors demonstrated that the guided modes in the tapered splicing region are mainly confined by the cladding/air boundary. In consequence, the original helical core has a negligible effect on their guidance as long as the waist region is thin enough (about 30 µm). Although the possibility of achieving the coupling efficiency between SMF and HCF up to 98% was declared based on the simulations, in experiment only 26% efficiency was demonstrated which was associated with the limited length of the tapered splice. This figure may be insufficient in many practical applications like nonlinear optics and lasers, in which low loss coupling is of great importance.
In this paper, we propose a more effective solution for increasing coupling efficiency between HCFs and SMFs and for controlling the excitation of higher order modes in HCFs. It relies on untwisting the input section of the HCF in a hydrogen flame, which makes the core almost straight at the fiber end-face. If the twist gradient in a few millimeters long transition section is not too high, the modes excited at the beginning of the fiber (where the core is locally straight) are adiabatically transformed to the modes of a highly twisted fiber without a significant loss. The proposed solution provides also a control over the excitation of higher order modes in the HCF as they are adiabatically transformed in the transition fiber section without any coupling to the modes with different orbital angular momentum. The proposed solution can be applied to reduce the coupling loss at splices between standard SMFs and HCFs as well as in a free-space launching configuration. We experimentally demonstrate that our approach increases the coupling efficiency between the SMF-28 and the highly twisted HCF from initial 6–7% to the level of about 60%, which is limited only by modal fields mismatch in the straight cores. We also report detailed numerical and analytical studies of the coupling efficiency between the HCF and SMF versus the initial helix pitch in the partially untwisted input section of the HCF, which takes into account fundamental and higher-order modes. We point to the possibility of controlling the relative excitation of the first order and the fundamental modes without the significant loss increase by properly adjusting the helix pitch at the interface of the two-mode HCF and a single mode fiber. The results of simulations conducted for the SMF-28 and the highly twisted HCF are in good agreement with the experimental observations.
2. Experimental studies of the coupling efficiency to a helical core fiber with gradient twist
To conduct experimental studies of the coupling efficiency, we have used a specially fabricated helical core fiber with a helix pitch (Λ) equal to 1.5 mm. The scanning electron microscope (SEM) image of the fiber cross section is shown in Figs. 1(a),(b). The fiber cladding has a hexagonal shape with a diagonal of approximately 115 µm. Because of imperfections of the technological process, the core shape is not perfectly cylindrical and depending on the direction its size varies from 5.1 to 5.6 µm. The core is laterally shifted from the symmetry axis of the cladding by Qc=39.5 µm. We have measured the transmission loss in this fiber by the cutback method, Fig. 1(c). The fiber has been fabricated by the stack and draw technology using silica and germanium doped rods made of Heralux WG glass with relatively high OH contamination. As a result, the OH absorption peak at 1400 nm is clearly visible in the loss characteristic even for short fiber samples (loss of about 2 dB/m), Fig. 1(c). In the spectral range from 600 to 1300 nm, the loss of the fabricated HCF is about 0.5 dB/m and increases to about 1 dB/m at 1550 nm. Moreover, we have experimentally verified that the HCF fiber is single mode for wavelength greater than 1530 nm.
Fig. 1. Scanning electron microscope images of the helical core fiber used in the experiments (a, b) and measured loss characteristic (c).
The propagation characteristics of the HCF used in the experiment were calculated using the finite element method combined with the transformation optics formalism [10,11] and implemented in COMSOL Multiphysics software. In the simulations of the fabricated HCF we have used the geometrical parameters reproduced from the SEM image, Fig. 1, while the refractive index contrast was Δnc=0.0329 at λ=1550 nm. For comparison, we have modeled the fiber with idealized geometry (perfectly circular core) to find out which of the experimentally observed effects is caused by technological imperfections of the fabricated fiber. We have assumed that the idealized HCF has the circular core of diameter Dc=5.3 µm, while the other parameters are the same as in the fabricated fiber.
The calculated intensity profiles for the fundamental and the first order modes in the idealized and real helical core for λ=1200 nm are shown in Fig. 2. In the idealized structure, the circular shape of the fundamental and the first-order modes is only weakly perturbed by fiber twisting.
Fig. 2. Calculated intensity profiles of the fundamental and first-order modes in the fabricated helical core fiber with Λ=1.5 mm at λ=1200 nm (a,b,c) and in the idealized fiber with a circular core (d,e). In both fibers, modes are circularly polarized and their intensity profiles are independent of polarization handedness.
In the real structure, the fundamental mode is nearly circular but the first-order modes split into two spatial modes with tangentially and radially oriented intensity maxima because of a flattened shape of the core. In both cores, the initial linear polarization of the fundamental and first order modes in the non-twisted fiber gradually evolves towards left- or right-handed circular polarization with decreasing helix pitch. The cut-off of the first order modes occurs at λc=1530 nm and 1370 nm, respectively for the mode with tangentially and radially oriented intensity maxima, while the next higher order mode is cut-off at λc=1100 nm.
Large lateral offset of the core and short helix pitch result in the significant tilt of the core with respect to the symmetry axis of the cladding equal to ϑco=9.5° (ϑco=atan(2πQc/Λ). This makes a free space light coupling into the fiber core difficult because of the necessity of alignment of the focused input beam in the horizontal, vertical and longitudinal directions as well as tilting the beam with respect to the fiber symmetry axis by an angle ϑ (Fig. 3(a)). One should note the angles ϑ and ϑco are related by Snell’s law (sinϑ=ncosinϑco), where nco is the refractive index of the core [24]. To facilitate the alignment of the input/output beam, the fiber input/output was angularly adjusted in such a way that the core at the fiber input/output is inclined with respect to the fiber symmetry axis only in the horizontal direction (Fig. 3(a)). For the alignment without a beam tilt (ϑ=0°, Fig. 3(b)), a pure excitation of the fundamental mode was impossible and the higher order modes were clearly visible at the fiber output for any lateral alignment of the focused beam on the core surface. As an example, we show in Fig. 4(a) the intensity profiles registered at the output of the HCF for the input beam tilt ϑ=0° and wavelengths much below the cut-off (λ=800 and 1200 nm) using a supercontinuum light source SC (NKT, SuperK COMPACT) and band pass interference filters. The transmitted power registered with an optical spectrum analyzer OSA (Yokogawa, AQ6370C) in the single mode range was low for such alignment and showed significant step-like jumps at cut-off wavelengths of successive higher order modes (λc=1530 nm and 1370 nm), Fig. 4(d). For beam tilt in the range 10°<ϑ<14°, only the fundamental and the first order modes were excited (Fig. 4(b)) and the power coupled to the fundamental mode was higher than in the previous case, Fig. 4(d). Apparently, for the incidence angle 10°<ϑ<14°, the beam inclination with respect to the core axis is too small to excite higher order modes. For pure excitation of the fundamental mode having a slightly disturbed Gaussian profile (due to core shape and twist), the beam tilt ϑ needed to be about 14° (Fig. 4(c)). In this case, the power transmitted in the fundamental mode of the HCF was the greatest, although slightly lower than for the optimal alignment of the standard SMF-28 fiber with the same excitation conditions. This is related to the difference in modal field diameters in the two fibers, which are equal to 9.4 µm and 5.1 µm at 1550 nm, respectively for the SMF-28 and the HCF.
Fig. 3. Scheme of the setup for light coupling into the HCF using a free space configuration (a). The fiber input and output are angularly aligned in such a way that the input/output beams are inclined with respect to the fiber symmetry axis only in the horizontal direction. In (b) we show the scheme of the experimental setup for measuring coupling efficiency versus wavelength (b); MO-microscope objective, F-bandpass filter.
Fig. 4. Registered intensity distributions at the output of the HCF for λ=800 and 1200 nm showing the presence of higher order modes depending on the input beam inclination ϑ=0° (a), ϑ=10° (b), ϑ=14° (c), and registered output power for different input beam inclinations (d): ϑ=0° (black curve), ϑ=10° (red), ϑ=14 ° (green) and for the SMF-28 (cyan). The cut-off wavelengths of the first-order modes with tangentially (λc=1530 nm) and radially (λc=1370 nm) oriented intensity maxima are marked with dashed lines.
To improve the coupling efficiency to the helical core fiber at splices with a standard fiber having a straight core or in a free-space launching configuration, we propose a simple method based on untwisting the input section of the HCF. To untwist the fiber in a controlled way, we have used the Lightel CW-5000 workstation containing computer controlled translational platforms with two rotating fiber holders and a hydrogen burner. To ensure effectiveness and repeatability of the untwisting process, we have experimentally selected the following processing parameters: flow of the hydrogen - 120 ml/min, platform speed - 0.004 mm/s, holders rotation speed - 180°/s, torch height - 12.3 mm, pulling length (indirectly untwisting time) from 0.004 to 0.012 mm.
The dependence of the pitch length Λ upon the fiber length in a partially untwisted fiber section was determined using the microscope fiber image. In Fig. 5 we show pictures of the twisted fiber with constant pitch length Λ=1.5 mm (Fig. 5(a)) and the partially untwisted fiber sections with different maximum pitch length (Fig. 5(b)-(g)). Microscope photos of the fibers presented in Fig. 5(a)-(h) were taken with the same magnification (28×). The fibers were placed on the translation stage to set its most untwisted part in the center of the microscope view field. Based on these pictures, we have measured the distances between characteristic black points (intensity minima) clearly visible in the fibers images because of a hexagonal shape of the cladding. Taking into account the fact that due to the six fold symmetry of the fiber cladding, the distance between successive characteristic features in its image (black points) corresponds to the twist angle equal to π/3, we were able to determine the function ϕ(z), representing the dependence of the total twist angle expressed in radians upon the propagation distance z. This function was then approximated with the 4-th degree polynomial and differentiated versus z, which yields the local value of the twist period:
Fig. 5. Pictures of the non-processed helical core fiber with the constant helix pitch Λ=1.5 mm (a) and partially untwisted fibers with different helix pitch in the center of the view field: (b) Λ=1.9 mm, (c) Λ=2.16 mm, (d) Λ=3.88 mm, (e) Λ=5.47 mm, (f) Λ=11.09 mm, (g) Λ=24.5 mm. In (h) we show an example of ready to splice partially untwisted input fiber section with the initial pitch length Λ0=24.5 mm. The splices between the SMF-28 fiber and the non-processed (i) and partially untwisted HCF with helix pitch Λ0=24.5 mm (j).
In Fig. 6 we show the functions ϕ(z) and Λ(z) determined for the all partially untwisted fiber sections presented in Fig. 5(a)-(g). Prior to splicing, the processed fiber was cut in the center of the partially untwisted section by using the fiber optic cleaver under the microscope inspection. An example of the ready to splice partially untwisted input section of the HCF with initial pitch length Λ0=24.5 mm is shown in Fig. 5(h). In Fig. 5(i),(j) we show the splices between the SMF-28 fiber and the twisted fiber with constant pitch length Λ=1.5 mm (i) and the partially untwisted fiber with the initial pitch length Λ0=24.5 (j). In order to see better the splices, microscope photos shown in Fig. 5(i),(j) were taken with the greater magnification of 65×.
Fig. 6. Total twist angle (a) and local twist period (b) vs. propagation distance determined for all the partially untwisted fiber sections shown in Fig. 5.
The coupling efficiency between the fundamental modes in the HCF and the SMF-28 was first determined for a butt-coupled configuration based on the formula:
where α is the attenuation in the HCF, L is the fiber length, PHCF is the power measured at the end of the HCF and PSMF is the power measured at the output of the leading-in fiber SMF-28. The measurements of the output power were performed in the system schematically shown in Fig. 3(b) using a stable laser emitting 0.3 mW power at the wavelength of 1550 nm (Grandway FHS1D02). For this wavelength, the leading-in SMF-28 fiber and the HCF are single-mode. The output power was measured using the Thorlabs PM100USB, S122C power meter operating at the wavelength range 700–1800 nm. The leading-in SMF-28 fiber coupled with light source was positioned with respect to the HCF using the fusion splicer (Ericsson FSU-975) to obtain the maximum output power. The measurements were repeated several times to eliminate accidental alignment errors.The coupling efficiency η00 between the fundamental modes in the SMF-28 and in the HCF measured versus the initial helix pitch Λ0 at λ=1550 nm is shown in Fig. 7. It asymptotically increases with Λ0 from initial 6% for Λ0=1.5 mm up to 60% for the partially untwisted fiber with the initial pitch length Λ0 greater than 8 mm. It is worth to mention that the HCF with Λ=1.5 mm is single mode at 1550 nm but in the partially untwisted input section with greater pitch, the first order mode is cut-off at longer wavelengths. This effect is related to the decrease of bending loss introduced by the curvature of the helical core, which is smaller for greater helix pitch. For example, for the HCF with constant pitch Λ=1.9 mm, which was the shortest initial helix pitch, the single mode propagation is for λ>1675 nm, while for Λ=24.5 mm the cut-off occurs at λc=1950 nm. One should note that even though the higher order modes can propagate in the partially untwisted beginning section of the HCF, they do not contribute to the transmitted power for λ>1530 nm because of high attenuation in the rest of the HCF with stronger twist, Λ=1.5 mm. The contribution of higher order modes is only observed in the spectral range below 1530 nm. Indeed, as it is shown in Fig. 8, for the butt-coupled HCF with constant pitch Λ=1.5 mm and the SMF-28, regardless of the relative offset of the two fibers, we always observed at the output of the HCF the superposition of the fundamental and the first order modes for λ<1530 nm. In contrast, for the HCF with untwisted input section (Λ0=24.5 mm), we could easily selectively excite the fundamental mode and both first order modes by vertically and horizontally shifting the leading-in SMF-28 with respect to the HCF. Finally, the optimally aligned fibers (for maximum transmission) fibers were spliced. Because of a large core offset in the HCF fiber, only about 80% of splicing attempts were successful. Moreover, we have observed a systematic loss increase after splicing the fibers by about 0.1–0.5 dB compared to butt-coupling configuration.
Fig. 7. Comparison of the coupling efficiency between the fundamental modes in the SMF-28 fiber and the helical core fibers with different initial helix pitch Λ0 for λ=1550 nm obtained by measurement (black dots) and numerical simulations (red line).
Fig. 8. Registered intensity distributions for λ=1200 nm at the output of the HCF with the untwisted input section with Λ0=24.5 mm proving the possibility of controlled excitation of the first order modes by vertical (a) and horizontal (b) displacement of the HCF with respect to the SMF-28. In (c) we show the intensity profiles obtained at the output of the butt-coupled HCF with constant pitch length Λ=1.5 mm and the SMF-28 for the same horizontal offsets of both fibers proving unavoidable excitation of the first order mode.
In the next step, the transmission was measured in a wide spectral range (1200–1700 nm), for which the leading-in SMF-28 fiber is still single-mode while the HCF can propagate fundamental and first order modes. Measurements were carried out using a broadband supercontinuum light source (NKT, SuperK COMPACT) and a monochromator for two HCFs with partially untwisted input sections with Λ0=2.92 mm and Λ0=10.44 mm and for the HCF with constant pitch length Λ=1.5 mm, Fig. 9. Due to the presence of higher order modes in the HCF below 1530 nm, the transmission coefficient experiences clear step-like change close to the cut-off wavelength, which is especially well pronounced for shorter initial pitches.
Fig. 9. Comparison of the transmission between the output of the SMF-28 fiber guiding the fundamental mode and the output of the helical core fiber with constant pitch Λ=1.5 mm and with partially untwisted beginning sections with Λ0=2.92 mm and Λ0=10.44 mm obtained by measurement (dots) and numerical simulations (lines).
This effect is related to the presence of higher order modes below 1530 nm which are strongly excited for small initial pitches (Λ0=1.5 and Λ0=2.92 mm) and carry a significant fraction of the total power. For longer initial pitches (Λ0=10.44 mm), the excitation of the first order mode and in consequence the step-like transmission change at around 1530 nm gradually weakens and completely vanishes for Λ0=∞. As it is shown in Fig. 7 and 9, the obtained measurement results are in qualitative agreement with the results of numerical calculations discussed in the next section.
3. Numerical simulations
In this section we present numerical simulations providing a detailed physical justification for the experimental results and discussing the influence of non-circular shape of the core in the investigated HCF on the coupling efficiency. The coupling efficiency ηimkn, understood as the ratio of the power coupled from m-th polarization mode within i-th spatial mode guided in the SMF-28 fiber (limited to the left- and to the right-handed fundamental mode in the analyzed case) to n-th polarization mode within k-th spatial mode characterized by a distinct intensity profile (see Fig. 2) guided in the helical core fiber, can be expressed as the normalized modes overlap:
To compare the simulation results with the polarization insensitive measurements, we have used the above formulas to calculate the coupling coefficients between different spatial modes. Moreover, we have numerically verified that the gradual decrease in the pitch length in the partially untwisted transition fiber section does not cause any couplings between the spatial modes of different orders. Therefore, the coupling efficiency at the interface between the two fibers characterized by initial pitch length Λ0 can be directly related to the power carried in the respective spatial modes at the output of the HCF, which was measured in the experiment.
In Fig. 10(a) we show the coupling efficiency between the fundamental mode in the SMF-28 and the fundamental mode (η00) and first order modes (η01) in the idealized HCF calculated versus Λ0 for λ=1200 nm. For real HCF, the coupling efficiencies η01r and η01t were calculated separately for two first order spatial modes, respectively with the radially and tangentially oriented intensity maxima, Fig. 10(b). The cores in both fibers were centrally aligned to maximize η00 between the fundamental modes.
Fig. 10. Polarization independent coupling efficiency between the fundamental mode in the SMF-28 fiber and the core modes in the idealized (a) and real (b) helical core fiber calculated versus Λ0 for λ=1200 nm; solid black lines represent the coupling efficiency to the fundamental mode of the HCF (η00), dotted red line to circular first-order modes of the idealized fiber (η01), dashed red line to the first-order mode with tangentially oriented maxima in the real fiber (η01t), dash-dotted blue line to the first-order mode with radially oriented maxima in the real fiber (η01r).
The obtained results are very similar for the idealized and the real fiber geometries. The calculated coupling efficiencies between the fundamental modes are almost equal in both cases and monotonically increase with Λ0, reaching the maximum value of η00=0.66 for the fiber completely untwisted at the beginning (Λ0=∞) for λ=1200 nm. One should note that this value is slightly greater than the calculated coupling efficiency η00=0.64 between straight fibers at λ=1550 nm shown in Fig. 7.
The coupling between the fundamental and the first order modes does not occur for the non-twisted centrally aligned cores. However, the coupling becomes possible if one of the cores is non-parallel to the fiber axis due to the twist and for the considered fibers geometry reaches the maximum efficiency equal to η01=η01r+η01t=0.37 for Λ≈2.5 mm. Sum of the coupling efficiencies for all the first order modes is very similar in the idealized and the real fiber. However, if we consider each of four first order modes individually then every polarization mode in the idealized fiber contributes to approximately ¼ of the total coupling efficiency η01. In the real fiber, each of the modes with tangentially oriented maxima contribute almost twice as much, while the contribution related to the modes with perpendicularly oriented maxima is smaller by approximately one order of magnitude.
In Fig. 9 we show the total transmission calculated versus wavelength for the experimentally investigated HCF of length 50 cm and Λ=1.5 mm with the partially untwisted beginning section. The simulation results shown in this figure take into account the coupling efficiency at the interface between the SMF-28 fiber. Both the fundamental and the higher order modes are excited in the partially untwisted beginning section of the fiber with the efficiency which does not change much within the analyzed wavelength range from 1200 to 1700 nm, however, the total power transmitted by the strongly twisted part of the fiber (Λ=1.5 mm) experiences an abrupt change at around 1530 nm because of the cut-off of the first order modes at this wavelength. In consequence, regardless of the initial pitch length Λ0, the transmission coefficient experiences a clear step-like change between 1350 and 1500 nm. This is because the transmission characteristics of the twisted fiber with the partially untwisted beginning section not only depends on the coupling coefficients to the fundamental and to the first-order modes at the interface but also on the attenuation of the first order modes in the strongly twisted fiber part (Λ=1.5 mm). As a result the total transmission below 1350 nm is almost equal to the sum of the coupling coefficients η00 and η01 for the fundamental and the first order modes (in this spectral range the loss of the first order modes is small) while above 1530 nm the transmitted power is approximately equal to η00 because the first order modes are highly attenuated in the strongly twisted fiber section. Such behavior of the transmission is almost identical for the real and the idealized HCF and shows qualitative agreement with the experimental results also displayed in Fig. 9.
Finally, we demonstrate in Fig.11a that one can easily control the relative excitation of the fundamental and the first order modes in the HCF by adjusting Λ0 at the interface between the SMF-28 and the HCF, without any drop in the total coupled power (actually a small increase in the total power is observed when the coupling efficiency to the first order mode becomes greater). In particular, for Λ=2.8 mm, the coupling efficiencies to the fundamental and the first order modes are equal, while the total coupled power remains on a high level η00+η01=70%. This is in contrast to the wide spread method of exciting the first order modes by lateral shift of the butt-coupled cores. In Fig. 11(b) we show the calculated coupling coefficients η00 and η01 between the fundamental mode of the SMF-28 and the first order modes of the HCF versus the lateral shift. In this method, the maximum coupling efficiency to the first order mode reaches only η01=5%, while the total coupled power drops to η01+η01=25% for such lateral alignment of the cores.
Fig. 11. Calculated coupling efficiency between the fundamental mode of the SMF-28 and the fundamental and the first order modes of the HCF calculated versus the helix pitch of the HCF at the interface (a) and versus the lateral offset between the two straight cores, for the initial pitch length of the helical core fiber equal to infinity Λ0=∞ (b), λ=1200 nm.
4. Conclusions
We have proposed an effective method for increasing the coupling efficiency between HCFs and SMFs by untwisting the beginning section of the HCF in a hydrogen flame. If the twist gradient in a few millimeters long transition section is not too high, the modes excited at the beginning of the fiber (where the core is locally almost straight) are adiabatically transformed into the modes of highly twisted fiber without significant loss and without any conversion to the modes of different orders (with different orbital angular momentum). The proposed solution can be applied to increase the coupling efficiency at splices between standard SMFs and HCFs as well as in a free-space launching configuration. The experimental results are in good agreement with the results of numerical simulations conducted for SMF-28 and for the fabricated highly twisted HCF with the helix pitch Λ=1.5 mm. Our approach allowed to increase the coupling efficiency between the two fibers from an initial 6% for constant Λ=1.5 mm up to 60% for the partially untwisted HCF with the initial pitch length Λ0 greater than 8 mm. We have shown that the coupling efficiency increases asymptotically with the initial pitch length Λ0 to the maximum value which is limited only by the modal fields mismatch in the straight cores. Moreover, the detailed numerical studies as well as the experimental results show that due to the unavoidable excitation of higher-order modes in the HCF, its transmission experiences a clear step-like change close to the cut-off wavelength, which is especially well pronounced for shorter initial pitches.
Our solution also provides a control over excitation of higher order modes in the HCF. We show that in contrast to the widespread method of exciting the first order modes by the lateral shift of the butt-coupled cores, one can easily control the relative excitation of the fundamental and the first order modes in HCF by adjusting the helix pitch at the SMF-28/HCF interface while keeping the total coupled power at almost constant level.
Appendix - analytical model
The relations between the coupling coefficients and the initial pitch length Λ0 obtained in the numerical simulations for straight conventional fiber and the helical core fiber can be also derived using an analytical approach. The first attempt of this type was reported in [23] to determine the coupling coefficient between the fundamental modes in the standard SM and HCF by using the modified expression for normalized modes overlap derived in [24,25] for tilted fibers:
The corrected formula, which can be used to accurately describe the coupling between the modes of different orders, may be obtained by taking into account the twist-induced change in the phase profile of the mode guided in the helical core in the plane perpendicular to the fiber axis. Such correction differs from the cosine factor used in Eq. (5). In [26] it was shown that the inclination of the core modes’ phase fronts with respect to the cladding symmetry axis introduces the tangential component of the propagation constant βθ, which can be expressed as:
The above equation can be further approximated by substituting the field in the plane transverse to the helical core $({{\textbf e}_i^{HC}} )$ with the field of untwisted fibers (not deformed by the twist). In the case of the idealized helical core fiber, the relative difference between the results obtained without mode deformation and results presented in Fig. 10(a) is lower than 4% for the fundamental modes and lower than 1% for the first order modes. Furthermore, the relative difference increases with decreasing helix pitch, for which the coupling efficiency is the smallest. Therefore, in the analyzed case, the results obtained with formulas derived in this section are almost indistinguishable from those presented in Fig. 10(a). Such small difference is related to the fact that in the analyzed case, the fields in the helical core fiber do not change significantly due to a twist, as can be seen in Fig. 2.
The coupling efficiency between individual modes of different orders can be analyzed more directly if we use a weak-guidance approximation for a circular core, which leads to degeneration of the first order modes and allows to represent ${\textbf e}_i^{HC}$ and ${\textbf e}_k^{SM}$ by the circularly polarized field profiles expressed by a single angular harmonic:
Modulation of the field profile related to the tangential component of the propagation constant can be expressed in the cylindrical coordinates as:
According to the above equation, a significant coupling occurs only between the modes of the same polarization handedness ($\sigma _k^{SM} = \sigma _i^{HC}$) and the coupling efficiency between fundamental modes, for which 0-th order Bessel function (J0) will be used, decreases with increasing Q/Λ0. Furthermore, Eq. (10) also predicts that at the interface between the single mode and the helical core fibers there will be non-zero coupling between modes of different orders with maximum efficiency for a certain value of Q/Λ0, which depends on the difference between angular momenta of the modes and their radial profiles. It should be noted that if we used cos(βϑcoy) given in Eq. (5) instead of exp(iAQβzy) to represent the modulation of the phase profile, we would obtain the same efficiency for coupling between fundamental modes. However, in such a case, the coupling between fundamental and first order modes would be forbidden because the amplitude of the harmonics with odd q in Eq. (9) would be equal to zero.
The following superposition of two angular harmonics for which Mi=σi and Mi=−σi can be used to represent the first order modes with two maxima (Fig. 2) guided in the deformed core of the fabricated helical core fiber:
For [exp $({j2\sigma_i^{HC}\theta } )$+1] the intensity maxima in the modal field distribution will be oriented along the x-axis while for [exp $({j2\sigma_i^{HC}\theta } )$−1] they will be oriented along the y-axis. The coupling efficiency between such modes of the real HCF and the fundamental mode of the single mode fiber of the same polarization handedness ($\sigma _k^{SM} = \sigma _i^{HC} = \sigma$), for which the angular dependence of the field is represented by $\textrm{exp}({j\sigma \theta } )$, can be expressed in the following way:
Funding
Narodowe Centrum Nauki (Maestro 8, DEC-2016/22/A/ST7/00089).
Disclosures
The authors declare no conflicts of interest.
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