Theoretical analyses on orbital angular momentum modes in conventional graded-index multimode fibre

Orbital angular momentum (OAM) modes are another mode basis to represent spatial modes. There have been increasing interests in exploiting OAM modes in specialty fibres. In this paper, we present a comprehensive characterisation of OAM modes in conventional graded-index multimode fibre (MMF). 1) We synthesise the circularly polarized OAM modes by properly combining two fold degenerate cylindrical vector modes (eigenmodes) and analyse the total angular momentum, i.e. spin angular momentum and orbital angular momentum. 2) We divide all the OAM modes of the conventional graded-index MMF into 10 OAM mode groups with effective refractive index differences between different mode groups above 10−4 enabling low-level inter-group crosstalk. 3) We study the chromatic dispersion, differential group delay, effective mode area, and nonlinearity for each OAM mode group over a wide wavelength ranging from 1520 to 1630 nm covering the whole C band and L band. 4) We discuss the performance tolerance to fibre ellipticity and bending. 5) We further address the robustness of performance against fibre perturbations including the core size, index contrast and the imperfect index profile of the practically fabricated MMFs. The obtained results may provide theoretical basis for further space-division multiplexing applications employing OAM modes in conventional graded-index MMF.


Cylindrical vector modes in conventional graded-index MMF.
In this work the MMF we study is an idealized conventional circular core optical fibre with a graded-index profile, e.g. an OM3 fibre. The radii of the fibre core and cladding are r core = 25 μm and r cladding = 62.5 μm, respectively. The relative refractive index difference (Δ = (n 1 − n 2 )/n 2 ) measured between the fibre core center (n 1 ) and cladding (n 2 ) is Δ = 1% at 632.8 nm. According to the three-term Sellmeier equation 44 , we can theoretically get the fibre index profile at 1550 nm with GeO 2 -doped core index n 1 = 1.458369 and pure-SiO 2 cladding index n 2 = 1.4440, as shown in Fig. 1(a). Figure 1(b) shows the n eff of all 110 fibre cylindrical vector modes at 1550 nm calculated by using a full-vector finite-element mode solver. Obviously, these 110 eigenmodes can be divided into 10 mode groups according to the different n eff values. Specifically, the n eff values in the same mode group are approximate equal, while different mode groups possess relatively large n eff difference. Each mode group contains a set of two fold strictly degenerate modes as listed in Fig. 1(c). That is, each HE mn e o / or EH mn e o / comprises even and odd modes, hence 110 fibre eigenmodes are supported in total. Figure 1(d) shows the n eff differences in arbitrary two adjacent vector modes. Remarkably, n eff differences can be divided into three types as marked with black elliptical circles. One can see that n eff differences between different mode groups in the top elliptical circle remain above 10 −4 , which benefits negligible inter-group crosstalk. n eff differences between the two fold degenerate HE mn OAM modes in conventional graded-index MMF. Remarkably, OAM-carrying light beam possesses a total angular momentum (AM) of (l + s) ħ per photon, where the OAM values lħ, and the spin angular momentum (SAM) valuing sħ is associated with circular polarization with s = ±1 corresponding to left-or right-circular polarization, respectively 45 . In this work, OAM modes are denoted by ± OAM l m L R , / . l and m in the subscript refer to azimuthal and radial indices, where l is the topological charge number, and m is the number of nulls radially in the intensity profile of the OAM mode. R and L in the superscript represent right and left handed circular polarization with SAM of −ħ and ħ, respectively. The total number of supported circular OAM modes contains all degeneracies in circular polarizations and in rotation directions of the phase front of the fields. The circularly polarized OAM modes in the fibre can be obtained by properly combining even and odd modes of HE nm or EH mn modes with a ±π/2 phase shift, i.e. ± * HE i HE lm even lm odd and ± * EH i EH lm even lm odd and the corresponding topological charge number is ±(l − 1) and ±(l + 1), respectively. For ± * HE i HE lm even lm odd combination, the circular polarization and phase front of the OAM mode rotate in the same directions, while ± * EH i EH lm even lm odd with a circular polarization in the opposite direction with respect to the field phase front rotation. In particular, the sum of TE 0m and TM 0m modes with a ±π/2 phase shift carries the same magnitude of SAM and OAM, while the rotation direction of the circular polarization and phase front is opposite, thus having opposite signs in the SAM and OAM, making the total AM equal to zero 46 .   In Fig. 3, we list all mode groups formed by 30 degenerate OAM modes at 1550 nm that are supported in the conventional graded-index MMF. Actually, the OAM modes in the same group have the same 2 m + l values. Specifically, the p th mode group include 2p modes with 2 m + l equal to p + 1.
is four fold degenerate including polarization and rotation, while OAM m L R 0, / is two fold degenerate including only polarization, thus the total number of supported circularly polarized OAM modes is 100, ignoring the other ten OAM modes with zero topological charge number. Those 110 modes are orthogonal with each other and the n eff differences between different groups can be maintained above 10 −4 , which enables low inter-group crosstalk. Chromatic dispersion and differential group delay. Conventional graded-index MMF supports hundreds of eigenmodes. Figure 4 shows the calculated guided eigenmode number sweeping wavelength from 1520 to 1630 nm. One can see that the mode number decreases with the increase of the wavelength. In the following calculations, we only present calculated results of the first and last synthesised OAM modes from each group of eigenmodes (typical thirteen OAM modes: OAM 0,1 , OAM 1,1 , OAM 2,1 , OAM 3,1 , OAM 4,1 , OAM 5,1 , OAM 6,1 , OAM 1,4 , OAM 7,1 , OAM 0,5 , OAM 8,1 , OAM 1,5 , OAM 3,4 ), e.g. OAM 0,1 in the 1 st mode group synthesized by HE 1,1 odd/even modes, OAM 0,5 and OAM 8,1 in the 9 th mode group synthesized by HE 1,5 odd/even and HE 9,1 odd/even modes, respectively. Here each OAM l,m mode denotes one of the degenerate ± OAM l m L R , / modes as the degenerate OAM modes have similar dispersion and DGD. Moreover, degenerate OAM modes also have similar effective mode area and nonlinearity, thus we only provide effective mode area and nonlinearity for one of the degenerate ± OAM l m L R , / modes below. Note that the conventional graded-index MMF supports at least 98 modes over 1520 nm to 1630 nm, thus we only take the first 98 modes into consideration. Figure 5 shows the chromatic dispersion and DGD of typical thirteen OAM modes over ten OAM mode groups with the wavelength ranging from 1524 nm to 1626 nm. DGD of OAM l,m modes is the mode delay between OAM 0,1 mode and OAM l,m mode, respectively. From Fig. 5(a), one can see that the chromatic dispersion of radially fundamental OAM modes OAM l,1 in the first nine OAM mode groups increases with l. The radially fundamental modes in the first nine OAM mode groups have positive chromatic dispersion slightly increasing with the wavelength, while the 10 th OAM mode group has negative chromatic dispersion rapidly increasing with the wavelength. In particular, as shown in the inset of Fig. 5(a), one can clearly see that a low and flat chromatic dispersion within (8.9 × 10 −2 , 25.9) ps/nm/km is achieved for the first nine OAM mode groups as the wavelength changes from 1524 to 1626 nm covering the whole C band and L band (1530 to 1625 nm). On the contrary, the 10 th OAM mode group has a relatively large and tilted chromatic dispersion within (−312.4, −21.4) ps/nm/km. From Fig. 5(b), one can see that the DGD of radially fundamental OAM modes OAM l,1 in the first nine OAM mode groups increases with l. The radially fundamental modes in the first nine OAM mode groups have positive DGD increasing with the wavelength. The DGD of the radially higher-order modes in the first nine OAM mode    groups and the 10 th OAM mode group decreases with the increase of the wavelength. The radially higher-order mode in the 9 th OAM mode group and the 10 th OAM mode group have negative DGD. Similarly, a relatively flat DGD within (122.5, 1289.4) ps/km is achieved for the radially fundamental modes in the first nine OAM mode groups, while the 10 th OAM mode group has a relatively titled DGD within (−2523.3, −18017.1) ps/km.
Additionally, Fig. 6 shows DGD of all supported degenerate fibre OAM l,m modes (see also Fig. 3) at 1550 nm. One can clearly see that in the lower OAM mode groups, DGD slightly increases with m and decreases with l, while in the higher OAM mode groups, DGD rapidly increases with l and decreases with m. Note that degenerate fibre OAM modes have similar dispersion and DGD. Actually, through similar analyses, we can conclude that the chromatic dispersion of OAM l,m modes in each mode group increases with l, while decreases with m.
Effective mode area and nonlinearity. Figure 7(a) and (b) show the effective mode area and nonlinearity of typical thirteen OAM modes over the 10 OAM mode groups with the wavelength ranging from 1520 nm to 1630 nm. Note that the fibre nonlinearity is inversely proportional to the effective mode area. One can see that the effective mode area of radially fundamental OAM modes OAM l,1 in the first nine OAM mode groups increases   with l and wavelength, and the fibre nonlinearity decreases with l and wavelength accordingly. Within the whole C band and L band, the conventional graded-index MMF keeps lower nonlinearity. In detail, the first nine OAM mode groups feature flat effective mode area sitting in (186.4546, 1.0816e3) μm 2 , while the 10 th OAM mode group has a rapid change of effective mode area sitting in (1.3675e3, 2.7224e3) μm 2 .
Additionally, Fig. 8(a) shows the outer 1/e 2 modal field radius (in blue) and inner radius (in red) of all 30 degenerate OAM modes at 1550 nm. The outer modal radius for radially higher-order OAM modes is as large as 22.75 µm which easily leaks to the cladding. Figure 8(b) shows the effective mode area of all supported degenerate fibre OAM l,m modes at 1550 nm. One can see that the effective mode area of OAM l,m modes increases with l and m, that is, for OAM l,m modes with same l (m) values, the effective mode area increases with m (l) values. Generally, the higher OAM mode groups have relatively larger outer modal radius and effective mode area. Note that degenerate OAM modes have similar effective mode area and nonlinearity.
Tolerance to ellipticity and bending. The intrinsic defects and external perturbations (e.g. stress/strain, squeezing, twisting, bending, heating/temperature) in practical fabrication process and deployment can cause non-ideal fibres. For instance, the stress/strain and squeezing can cause imperfect circularity (i.e. ellipticity) of fibre, which gives rise to the birefringence and resultant polarization mode dispersion of the degenerate fibre eigenmodes. Fibre twisting and bending can also cause birefringence. Those imperfections of non-ideal fibres may cause the difference of propagation constants between the two fibre eigenmodes forming the OAM modes, affect the mode profile and purity of the synthesised OAM modes, and induce OAM modes coupling and crosstalk. Hence, it is of importance to study the effects of non-ideal fibres (ellipticity by stress, fibre bend radius by bending) on the performance of OAM modes.
We first study the effects of ellipticity on the performance of OAM modes in a conventional graded-index MMF. Figure 9(a) shows the n eff difference between the even and odd fibre eigenmodes which form the OAM modes as a function of the fibre ellipticity. One can see that n eff differences increase with the ellipticity and the growth rate becomes relatively slow with larger ellipticity. When the fibre ellipticity takes 5%, the maximum n eff difference is 4.54 × 10 −7 for the HE 10,1 related OAM modes. Additionally, we adopt two parameters 2π walk-off length (L 2π ) and 10-ps walk-off length (L 10ps ) to characterise the intra-mode walk-off effect upon propagation of the OAM modes resulting from the fibre ellipticity. Figure 9(b) plots the 2π walk-off length and 10-ps walk-off length between the even and odd fibre eigenmodes which form the OAM modes as a function of the fibre ellipticity. Note that the 2π walk-off length and 10-ps walk-off length are inversely proportional to the n eff differences, and the 10-ps walk-off length is around 2000 times longer than the 2π walk-off length at 1550 nm. One can see that the propagation length keeps longer than 6.6 km when the decomposed two fibre eigenmodes have a 10-ps temporal walk off under an ellipticity of 5%. Furthermore, we study the crosstalks between different OAM modes caused by fibre ellipticity. Figure 9(c) presents the calculated OAM crosstalks for the lowest-order − OAM 1,1 R mode synthesized by HE 21 modes and highest-order + OAM 9,1 L mode synthesized by HE 10,1 modes under an ellipticity of 0.5%. Remarkably, the OAM modes coupling and resultant crosstalk only occurs within the same mode group, that is, the crosstalks between different mode groups are negligible. As listed in Fig. 9(c), for the lowest-order  mode groups. In particular, we calculate the minimum n eff differences between different mode groups with ellipticity varying from 0 to 5%, as listed in Fig. 9(d). Although the minimum n eff difference between different mode groups decreases with the increase of ellipticity, it still keeps larger than 7.67 × 10 −4 even under an ellipticity of 5%, indicating low-level crosstalk between different mode groups.
We then discuss the characteristics of OAM modes under different fibre bend radius. Note that the higher-order modes leak out from the core into the cladding with the decrease of bend radius. As a result, the number of supported eigenmodes reduces to 90 and 72 under fibre bend radii of 2 cm and 1 cm, respectively. Figure 10(a) shows the n eff difference between the even and odd fibre eigenmodes which form the OAM modes as a function of the fibre bend radius. One can see that n eff differences increase with the decrease of fibre bend radius. The azimuthally higher-order radially fundamental OAM modes show stronger tolerance to the bend radius variation, while the radially higher-order OAM modes show larger n eff differences. When the bend radius is 1 cm, the maximum n eff difference is 1.11 × 10 −7 for the EH 23 related OAM modes. Figure 10(b) plots the π 2 walk-off length and 10-ps walk-off length between the even and odd fibre eigenmodes which form the OAM modes as a function of fibre bend radius. The minimum 10-ps walk-off length is 26.9 km with 1 cm bend radius. Figure 10 When the bend radius takes 2 cm, the crosstalks for HE 21 and HE 91 related OAM modes are less than −16.7 dB and −10.6 dB, respectively, while the crosstalks for HE 21 and HE 10,1 related OAM modes with 10 cm bend radius are less than −30.7 dB and −24.2 dB, respectively. It is found that the lower-order OAM modes show relatively low-level crosstalk. Moreover, we calculate the minimum n eff differences between different mode groups with bend radius varying from 1 cm to 100 cm, as listed in Fig. 10(d). It is noted that the minimum n eff difference between different mode groups keeps larger than 1.25 × 10 −3 when decreasing the bend radius.
Robustness against fibre perturbations. In addition to ellipticity and bending, the heating or temperature variation can cause slight change of geometric parameters and material properties of fibre, i.e. the practically fabricated MMF might vary slightly in the core size, index contrast between the core and cladding, the graded index profile and so on. Thus we further address the robustness against fibre perturbations including the fibre core size, the core index and the distorted fibre index profile.
The fibre core diameter of the actual drawn MMF is assumed to have a deviation of 25 ± 2.5 µm. By sweeping the fibre core radius from 23.5 µm to 26.5 µm (with 0.5 µm spacing) at 1550 nm, the calculated guided eigenmode number and n eff distribution are shown in Fig. 11(a) and (b). With the increase of fibre core radius, the n eff of the same order fibre modes increase accordingly and more fibre eigenmodes are guided in the MMF. Moreover, within a small radius deviation of 25 ± 0.5 µm, the MMF keeps supporting 110 eigenmodes. When the fibre core radius becomes 26.5 µm, Fig. 11(c) lists all 124 fibre cylindrical vector modes supported including additional 14  eigenmodes (or OAM modes) in the last group compared with a 25-µm core radius MMF. Note that in the same mode groups, the mode order may change with different fibre parameters, such as the fibre core size, the core index and the perturbed fibre profiles. Figure 12 shows the calculated chromatic dispersion, DGD, effective mode area, and nonlinearity coefficient for the typical fourteen OAM modes discussed above versus fibre core radius. One can clearly see that the lower-order OAM modes in radial direction feature stronger tolerance to fibre core radius variation. For the higher-order modes in radial direction (OAM 0,5 , OAM 1,5 and OAM 0,6 ), the chromatic dispersion, DGD and nonlinearity coefficient increase with fibre core size, while the effective mode area decreases with the increase of fibre core size.
The fibre numerical aperture of the actual drawn MMF is assumed to have a deviation of 0.2 ± 0.015 (with fibre relative refractive index difference (Δn) changing from 0.82% to 1.1%). Similarly, we sweep the fibre relative refractive index difference from 0.8% to 1.2% (with 0.05% spacing) with fibre cladding index fixed to 1.4440 at 1550 nm. Figure 13(a) shows the calculated guided eigenmode number which keeps supporting 110 eigenmodes with the core index difference deviation of 1% ± 0.5%. The detailed n eff distributions are shown in Fig. 13(b). When the fibre relative refractive index difference becomes 1.2%, 132 modes including additional 22 eigenmodes (or OAM modes) in the 11 th group compared with a 1% relative refractive index difference MMF are supported as shown in Fig. 13(c). Figure 14 shows the calculated chromatic dispersion, DGD, effective mode area, and nonlinearity coefficient for the typical fourteen OAM modes as a function of the fibre relative refractive index difference. Similar to the core size perturbation effects, the lower-order OAM modes in radial direction feature larger tolerance to fibre core index variation, while the chromatic dispersion, DGD and nonlinearity coefficient of the higher-order modes in radial direction increase with fibre relative refractive index difference.
Due to the limitations of modern fibre manufacture technology, the actual drawn MMF might have slightly distorted square law profile, typically with a central index dip and the refractive index ripple near the core-cladding boundary 47 . Here we accordingly simulate a refractive index profile that is distorted by random ripples in the fibre core center and core-cladding boundary as shown in Fig. 15(a). By sweeping the ripple amplitude from 0 to 1‰ (with 0.1‰ spacing), we analyse the fibre robustness against the distorted fibre index profile. Note that in Fig. 15(a), the ripple amplitude is 0.5‰, that is, the maximum index perturbation has 0.5‰ index difference with the idealized index at the same position. The supported eigenmodes number keeps 110 and the n eff distributions are almost the same. Figure 15(b) lists all the 110 eigenmodes when the ripple amplitude is 0.5‰. Figure 16 shows the calculated chromatic dispersion, DGD, effective mode area, and nonlinearity coefficient for the typical thirteen OAM modes as a function of the ripple amplitude. One can see that with the increase of ripple amplitude, the chromatic dispersion of most of OAM modes keeps almost constant, while OAM 0,1 , OAM 1,4 ,

Discussions
OAM modes featuring helical phase fronts have recently seen emerging applications of optical communications in free space and specially designed fibres. It is always believed that specialty fibres are preferable to support OAM  modes. However, the most widely deployed or commercially available fibres are SMF and MMF. Conventional MMFs are widespread for short-reach optical interconnect applications because of their relaxed connector tolerances and efficient coupling to low-cost laser sources. Many present and future optical communication links for enterprise and data centre applications are based on MMFs. Fundamentally, the standard SMF does not support OAM modes at 1550 nm. It is valuable to exploit conventional MMF for OAM communications.
In summary, we give a comprehensive characterisation of OAM modes in the conventional MMF. We study the synthesis of OAM modes, effective refractive index, number of OAM modes, division of OAM mode groups, chromatic dispersion, DGD, effective mode area, and nonlinearity. We discuss in detail the effects of non-ideal fibres (variation of core size, refractive index and index profile by heating/temperature, ellipticity by stress/strain, fibre bend radius by bending) on the performance of OAM modes. The n eff differences, 2π walk-off length, 10-ps walk-off length, OAM crosstalks, and minimum n eff differences between different mode groups are analysed. The conventional MMF shows favorable tolerances to heating/temperature/stress/strain/bending induced fibre perturbations, indicating its potential robustness even used in practical applications.
The obtained results provide detailed theoretical understanding on OAM modes supported in the conventional MMF and show the feasibility to use the conventional MMF for OAM multiplexing communications. The OAM modes guided in the conventional MMF can be divided into multiple groups. Two OAM multiplexing approaches could be considered. First, one can employ different OAM mode groups to carry different data information channels. Different OAM mode groups with relatively large effective refractive index difference have low-level inter-group crosstalk, showing possible OAM mode group multiplexing communications. Second, one can also use different OAM modes to carry different data information. In the same OAM mode group with relatively large intra-group crosstalk, small-scale multiple-input multiple-output digital signal processing (MIMO-DSP) technique could be used to mitigate the mode crosstalk. Hence, small-scale MIMO-DSP assisted intra-group OAM modes multiplexing combined with crosstalk-free inter-group OAM modes multiplexing can be considered to facilitate the OAM multiplexing communications using a conventional MMF. With future improvement, experimental demonstrations on OAM modes multiplexing communications in the conventional MMF are under way. One challenge is the perfect excitation of OAM modes in the conventional MMF. Previous work has shown the excitation of OAM modes in specially designed fibres via perfect OAM beams 48 , which could provide an effective approach to facilitating perfect excitation of OAM modes in the conventional MMF.
It is of great interest to directly employ commercially available MMF but not specially designed and fabricated fibre with increased complexity for OAM communications. The presented work may open up new perspectives to more extensive OAM related applications using already existing conventional MMF. 633 nm to 79 nm and the integral radius remaining 45 µm. Figure 18(d-f) sweep the integral radius changing from 25 µm to 55 µm and the number of mesh element remains 1.1e4. According to the results shown in Fig. 18, we eventually employ the average mesh size of 158 nm with 1.4e4 mesh element and integral radius of 45 µm for simulations of effective mode area and nonlinearity.
Walk-off length. 2π walk-off length (L 2π ) and 10-ps walk-off length (L 10ps ) denote the propagation length when the even and odd fibre eigenmodes walk off to each other with a 2π relative phase shift or have a 10-ps temporal walk-off.  where λ and c are the wavelength and light velocity in vacuum, n eff even and n eff odd are effective refractive index of the even and odd fibre eigenmodes, and Δt = 10 ps is the temporal walk off time.
OAM crosstalks. OAM crosstalks, also characterised by OAM spectra or OAM charge weight distributions, are expressed by 10log 10  where ψ i (x, y) and e(x, y) are the electric field distributions of the synthesised circular OAM modes in the ideal fibre and the perturbed fibre with ellipticity or bending, respectively.