Introduction

Orbital angulr momentum (OAM) beams characterized by a helical phase front of exp(ilφ)1, where l is the topological charge number, have been widely used in microscopy2, particle trapping3 and photon entanglement4. Recently, using OAM beams as carriers in a communication system has gained much interest due to intrinsic spatial orthogonality of OAM modes with different topological charge numbers. OAM mode multiplexing has been realized in both free space and fiber communication systems5,6,7,8. It is expected that space-division multiplexing (SDM) using OAM modes may offer an alternative to address the coming capacity crunch.

Most of previous reports of OAM mode multiplexing in a fiber adopt a single high-contrast-index ring-core structure7,8,9,10. Such structure can reduce the inter-mode crosstalk and stably support multiple OAM modes. Although two or more OAM modes have been explored in this kind of fiber, more comprehensive analyses are required to clearly show how many modes can be well supported with low-level inter-mode crosstalk for an optimized design. In addition, one would also expect to see its compatibility with existing SDM techniques and fiber designs. In view of well-known SDM techniques using multi-core fiber (MCF) and few-mode fiber (FMF)11,12, especially more recent works of few-mode multi-core fiber13, a laudable goal would be to combine OAM mode multiplexing and multi-core-like structure for high-capacity OAM communications14,15, where both inter-mode crosstalk and inter-ring crosstalk and their suppression should be considered.

In this paper, we present an improved compact design of trench-assisted multi-OAM multi-ring fiber (TA-MOMRF) for ultrahigh-density SDM. The designed TA-MOMRF has 19 rings with each ring supporting 22 modes (18 OAM ones), i.e. 418 channels in total. The large difference of effective refractive index of fiber eigenmodes (>10−4) in each ring enables low inter-mode crosstalk. The high-contrast-index ring and trench design ensure pretty low-level inter-ring crosstalk and give a compact structure. Wide-range operations covering the whole C and L bands are also available, which are compatible with well-established wavelength-division multiplexing (WDM) technique. The tolerance to birefringence induced by practical ellipticity of ring and fiber bending is studied.

Results

Design of trench-assisted multi-OAM multi-ring fiber (TA-MOMRF)

Figure 1(a) shows a schematic cross-section of the designed 19-ring TA-MOMRF. The ring pitch Λ and cladding diameter are 25 and 130 μm, respectively. Compared to previously reported 19-core fiber with the core pitch and cladding diameter of 35 and 200 μm16, our structure is more compact. Trench-assisted high-contrast-index ring is adopted to support multiple OAM modes in each ring and greatly reduce the inter-ring crosstalk among 19 rings. The index profile of a single ring is illustrated in Fig. 1(b). The glasses used in the TA-MOMRF are (i) ring: Schott SF4 with nr = 1.72, (ii) trench: Schott LLF1 with nt = 1.5, (iii) cladding: Schott SF2 with nc = 1.62 (at 1550 nm). Those materials of high-contrast-index structure have been utilized in previous fiber design for generating OAM modes9,10. In addition, practical fabrication of special fiber using similar type of materials has also been reported17,18. In the designed TA-MOMRF, the inner (r1) and outer (r2) radius of a single ring are 3 and 4 μm, respectively. The trench has an inner radius r3 of 6 μm and an outer radius r4 of 11 μm. As an example, the calculated intensity and phase distributions of HE31 related OAM mode in a single ring are depicted in Fig. 1(c).

Figure 1
figure 1

(a) Cross-section of a 19-ring trench-assisted multi-OAM multi-ring fiber (TA-MOMRF). (b) Index profile of a single ring. (c) Intensity and phase distributions of HE31 related OAM mode in a single ring.

Full size image

Mode properties of the designed TA-MOMRF

Table 1 presents calculated effective refractive index (neff) of 12 different order fiber eigenmdes at 1550 nm supported by a single ring. Owing to the high-contrast-index ring structure design, the maximum difference of neff can be as high as 0.01335 between EH31 and EH41 modes and even the minimum difference of neff can reach 0.00013 between EH41 and HE61 modes. Large difference of neff benefits efficient suppression of inter-mode crosstalk.

Table 1 neff of fiber eigenmdes in a high-contrast-index ring
Full size table

In fact, there are 22 fiber eigenmdes since each HEm1 or EHn1 mode also contains even and odd modes. One can obtain OAM modes by combine even and odd modes of HEm1 or EHn1 with a ±π/2 phase shift19, i.e. , . The topological charge number of and is ±(m − 1) and ±(n + 1), respectively. As an example of the phase distribution of shown in Fig. 1(c), the azimuthal phase variation is 4π corresponding to a topological charge number of 2. Hence, one can get 18 nonzero topological charge number OAM modes. Considering the fundamental mode HE11 (x and y polarization), TE01 and TM01 modes, 22 modes (18 OAM ones) in total are supported with low-level inter-mode crosstalk in a single ring.

Inter-ring crosstalk of the designed TA-MOMRF

In addition to inter-mode crosstalk, another important parameter is inter-ring crosstalk in the designed TA-MOMRF. Similar to a multi-core fiber, the number of rings in the TA-MOMRF is limited the by the ring pitch under a fixed cladding diameter. Small ring pitch gives more rings but leads to a large crosstalk between neighboring rings. To suppress the inter-ring crosstalk and increase the number of rings, trench design is employed.

In order to determine the ring pitch necessary for achieving negligible crosstalk, we employ coupled-power method to assess the crosstalk between two adjacent rings. The normalized power transfer between two identical rings is given by sin2[πz/(2Lc)] with Lc being the coupling length and z being the propagation length20. The coupled-mode theory method which has the ability to accurately evaluate ultra-long coupling length (~1010 m)21 has been used to calculate Lc.

Figure 2 shows Lc of different OAM modes in adjacent rings as a function of Λ. It is found that Lc increases with Λ, which is easy to understand that large separation between adjacent rings can reduce mode overlap and power transfer between adjacent rings, resulting in a long Lc. One may note the oscillation in Fig. 2, which can be explained with the fact that the adopted coupled-mode theory features a calculation precision of ~1010 m. However 1010 m coupling length is longer enough to guarantee a pretty low inter-ring crosstalk. When Λ is 25 μm and wavelength is 1550 nm, the maximum Lc has an order of 1010 m for the HE11 mode and the minimum Lc has an order of 107 m for EH41 related OAM mode.

Figure 2
figure 2

Coupling length (Lc) of OAM modes in adjacent rings versus ring pitch Λ.

Full size image

When the propagation length is 100 km, Figure 3 shows the inter-ring crosstalk of HE11 mode (best case) and EH41 related OAM mode (worst case) in adjacent rings as a function of Λ for TA-MOMRF with trench and multi-OAM multi-ring fiber (MOMRF) without trench. Under a ring pitch Λ of 25 μm, the crosstalk in adjacent rings of TA-MOMRF with trench is −99.3 dB for HE11 mode and −45.6 dB for EH41 related OAM mode, while −68.9 dB for HE11 mode and −0.7 dB for EH41 related OAM mode in MOMRF without trench. For MOMRF without trench, the ring pitch should be as large as 32 μm to achieve a −45 dB crosstalk. Hence, the trench design can greatly reduce the inter-ring crosstalk, especially for high-order OAM modes, which facilitates a compact fiber structure.

Figure 3
figure 3

Inter-ring crosstalk of HE11 mode and EH41 related OAM mode in adjacent rings with and without trench versus ring pitch Λ under a propagation length of 100 km.

Full size image

To further evaluate the performance of the designed TA-MOMRF, we calculate the inter-ring crosstalk with varied propagation length and wavelength. Figure 4 shows the inter-ring crosstalk of HE11 mode and EH41 related OAM mode in adjacent rings as a function of propagation length for TA-MOMRF with trench and MOMRF without trench under a ring pitch Λ of 25 μm. The summarized results under propagation lengths of 1 km, 10 km and 100 km are listed in Table 2. One can clearly see from Fig. 4 and Table 2 that the trench structure can effectively reduce the inter-ring crosstalk. Moreover, the pretty low inter-ring crosstalk of TA-MOMRF implies potential use in practical applications, especially for short-distance (e.g. 1 m ~ 1000 m) communications.

Table 2 Crosstalk corresponding to Fig. 4 at 1, 10, 100 km
Full size table
Figure 4
figure 4

Inter-ring crosstalk of HE11 mode and EH41 related OAM mode in adjacent rings with and without trench versus propagation length.

Full size image

Figure 5 shows the inter-ring crosstalk of OAM modes in adjacent rings as a function of wavelength under a propagation length of 100 km. Within a wide wavelength range from 1520 to 1630 nm which covers the whole C band and L band (1530 to 1625 nm), the inter-ring crosstalk in adjacent rings for all 22 modes is less than −26 dB. That is, the designed TA-MOMRF is also compatible with the existing WDM technique. For short-distance (e.g. 1 m ~ 1000 m) communications, similar broadband feature is also achievable with even wider wavelength range and much lower inter-ring crosstalk.

Figure 5
figure 5

Crosstalk of OAM modes in adjacent rings versus wavelength under a propagation length of 100 km.

Full size image

Tolerance to ellipticity and bending

Practical fibers might not be cylindrically symmetric because of the fabrication defects. The strain, stress, twist and other perturbations can also cause the nonperfect circularity (i.e. ellipticity) of fiber, giving rise to birefringence and polarization mode dispersion. Additionally, fiber bending can also induce birefringence. Considering that OAM modes are composed of fiber eigenmodes, i.e. the combination of even and odd modes of HEm1 or EHn1 with a ±π/2 phase shift, the ellipticity of ring and fiber bending can affect the mode profile and purity of OAM modes, resulting in the mode coupling and crosstalk.

We study the impacts of the ellipticity of ring and fiber bend radius on the performance of OAM modes. Figure 6(a) shows the neff difference between even and odd fiber eigenmodes as a function of the ellipticity of ring. One can clearly see that high-order OAM modes (EH31, HE41, EH41, HE51 and HE61 related OAM modes) feature stronger tolerance to the ellipticity variations compared to low-order modes (HE11 mode, EH11, HE21, EH21 and HE31 related OAM modes). For low-order modes, the neff difference between even and odd fiber eigenmodes increases with the ellipticity of ring. When the ellipticity is 2%, the maximum neff difference is 1.76 × 10−4 for the EH11 related OAM modes. For high-order OAM modes, the calculated neff difference between the even and odd fiber eigenmodes is around 1 × 10−9. The high-order OAM modes show enhanced tolerance to the ellipticity variations, which can be explained with the fact that high-order OAM modes have more azimuthal periods in their transverse field distribution, resulting in the mitigation of asymmetric effects from ellipticity. Figure 6(b) shows neff difference between even and odd fiber eigenmodes as a function of the fiber bend radius. When reducing the bend radius from 100 to 1 cm, high-order OAM modes having more azimuthal periods in their transverse field distribution feature stronger tolerance to the fiber bending effect, while the neff difference between even and odd fiber eigenmodes of low-order modes increases. The maximum neff difference is 1.84 × 10−4 for the EH11 related OAM modes under a fiber bend radius of 1 cm.

Figure 6
figure 6

Effective refractive index (neff) difference between even and odd fiber eigenmodes as functions of (a) ellipticity of ring and (b) fiber bend radius.

Full size image

The ellipticity of ring and fiber bending cause neff difference between even and odd fiber eigenmodes that form OAM modes. Such neff difference gives rise to temporal walk-off effect upon propagation. Two parameters named 2π walk-off length (L2π) and 10-ps walk-off length (L10 ps) are adopted to characterize the intra-mode walk-off effect of OAM modes22. Figure 7(a) and (b) plot 2π walk-off length and 10-ps walk-off length for different OAM modes as functions of ellipticity and bend radius, respectively. It can be clearly seen that high-order OAM modes with more azimuthal periods in their transverse field distribution feature longer 2π walk-off length and 10-ps walk-off length. For low-order OAM modes, both 2π walk-off length and 10-ps walk-off length decrease as increasing the ellipticity or reducing the bend radius.

Figure 7
figure 7

2π walk-off length and 10-ps walk-off length for different OAM modes as functions of (a) ellipticity of ring and (b) fiber bend radius.

Full size image

The neff difference between even and odd fiber eigenmodes due to ellipticity of ring and fiber bending can affect the profile and purity of OAM mode and cause OAM mode coupling and crosstalk. OAM channels with low crosstalk are highly desired for optical communications using OAM multiplexing5. In order to assess the crosstalk among different OAM modes, we study the OAM spectra which are also known as the charge weight distribution of OAM modes defined by the electric field overlap integral of the achieved modes and the corresponding pure eigenmodes of an ideal ring fiber23.

Figure 8(a) and (b) show two examples of calculated OAM spectra for the HE21 and HE61 related OAM modes under an ellipticity of 0.5%. One can clearly see that the HE21 and HE61 related OAM modes introduce low-level crosstalk less than −32 and −22 dB respectively to all other 21 modes (17 OAM ones).

Figure 8
figure 8

OAM spectra for (a) HE21 and (b) HE61 related OAM modes under an ellipticity of 0.5%.

The OAM numbers “−0” and “+0” denote modes .

Full size image

Figure 9(a)–(f) show another two examples of calculated OAM spectra for the HE21 and HE61 related OAM modes under different fiber bend radii of 5, 2 and 1 cm. It can be clearly seen that the HE21 and HE61 related OAM modes contribute low-level crosstalk less than −28 and −40 dB respectively to all other 21 modes (17 OAM ones) under a fiber bend radius of 5 cm. For a fiber bend radius of 2 cm, low-level crosstalk less than −20 and −40 dB from HE21 and HE61 related OAM modes are induced to all other 21 modes (17 OAM ones). When further reducing the fiber bend radius to 1 cm, the maximum crosstalk from HE21 and HE61 related OAM modes increases to −14.9 and −37.3 dB, respectively.

Figure 9
figure 9

OAM spectra for (a)(c)(e) HE21 and (b)(d)(f) HE61 related OAM modes under different fiber bend radii of (a)(b) 5 cm, (c)(d) 2 cm and (e)(f) 1 cm.

The OAM numbers “−0” and “+0” denote modes .

Full size image

The designed fiber features favorable tolerance to both small ellipticity of ring and small fiber bending. With the increase of ellipticity or the decrease of bend radius, the mode crosstalk also increases. In the case of relatively large mode crosstalk, multiple-input multiple-output (MIMO) technique24,25 and low-density parity-check (LDPC) coding26 might be employed to enable mitigation of crosstalk effects and improve the transmission performance.

Discussion

Benefiting from the high-contrast-index ring structure and trench design, the proposed TA-MOMRF offers simultaneous low-level inter-mode crosstalk and inter-ring crosstalk within a wide wavelength range (1520 to 1630 nm). For optical communications, it is possible to transmit 57266 channels in total (19 rings × 22 modes × 137 wavelengths from 1520 to 1630 nm with 100-GHz spacing) with negligible inter-channel crosstalk in a single TA-MOMRF, which could increase 57266 times total transmission capacity and 418 times aggregate spectral efficiency compared with a single channel transmission. For example, when using 42.8-Gbaud/s 16-ary quadrature amplitude modulation (16-QAM) signal for each channel5, one might achieve 9.8-Pbit/s total transmission capacity and 1337.6-bit/s/Hz aggregate spectral efficiency.

The designed TA-MOMRF, fully exploring the spatial domain and limited space resources, is not a simple combination of two concepts of single-ring OAM fiber and multi-core fiber which would not be able to simultaneously support multiple OAM modes (22 modes) in multiple rings (19 rings) in a compact fiber design (130 μm cladding diameter) with both low-level inter-mode crosstalk and inter-ring crosstalk over 110 nm wavelength range. These distinct features come from the optimized design of the geometric parameters of ring structure and trench structure, which also offers favorable tolerance to both small ellipticity of ring (<−22 dB crosstalk under an ellipticity of 0.5%) and small bend radius (<−20 dB crosstalk under a bend radius of 2 cm) of practical fibers.

The calculated results shown in Fig. 4 and Table 2 indicate that the inter-ring crosstalk is lower than −45 dB for all 22 modes in adjacent rings even under a 100-km propagation length. However, propagation loss is not considered here. For long-distance transmission, one of the most important limiting effects would be the loss. Fiber structure using high-doping materials might induce relatively large loss. The recent ring fiber OAM transmission showed a propagation loss of ~1.6 dB/km7,27, which is larger than the loss of typical single-mode fiber (<0.2 dB/km). The large loss of OAM ring fiber might be due to the high doping of the ring. The designed TA-MOMRF might show relatively larger loss because of the use of even higher-doping material. Actually, there is a trade-off between the tight light confinement or inter-mode crosstalk and the loss. High-contrast-index ring fiber design benefits tight mode confinement and increased effective refractive index difference among different OAM modes and resultant low-level inter-mode crosstalk, while the loss of the ring fiber is relatively large.

In view of relatively large loss, the most possible applications using the designed fiber could be short-distance (1 m ~ 1000 m) communications, e.g. broadband optical access network (FTTx) and indoor optical fiber communications to end users28. The combination of multiple OAM modes and multiple rings could offer large transmission capacity by ultrahigh-density SDM with tolerable loss for the short-distance communications. In addition to short-distance communications, the short-length OAM ring fiber might also find interesting applications in nonlinear optical signal processing29.

In order to reduce the loss of the designed fiber, possible solutions might be considered: 1) using low-loss materials while optimizing the geometric structure parameters7,27; 2) using similar photonic crystal fiber structure to confine light in the air gap region to reduce the loss30.

Methods

Mode analyses

We use 2D finite element analysis in the COMSOL software package to compute the effective refractive index (neff) of fiber eigenmdes as well as the electric field and phase distributions of OAM modes. A circular perfectly matched layer (PML) is used to absorb the incident electromagnetic waves at the outer boundary of cladding. In the ring core domain, the maximum size of grid is set to be 100 nm to ensure the precision of simulation.

Coupling coefficient

In order to analyze the coupling behavior and resultant inter-ring crosstalk in the designed fiber, the coupling coefficient Cpq between two adjacent rings (p, q) in a fiber, which determines the coupling length Lc = π/(2κ), can be calculated using the similar coupled more theories by Marcuse and Snyder31,32

where and are normalization terms. () and ( ) represent the transverse electric (magnetic) fields of the guided modes of each independent single-ring fiber. k is the wave number in vacuum. ε0 and μ0 denote the permittivity and permeability of vacuum, respectively. is the unit vector along the z-axis direction. n(x, y) and are the refractive index profiles of the two-ring fiber and the single-ring fiber, respectively. Note that the integral of Eq. (1) has actually only to be calculated over the region where , i.e. non-null only in the one of two rings.

Considering two identical rings with a spacing of d, we have approximate expressions of and Np = Nq. Consequently, Eq. (1) can be simplified and written by21

Therefore, two field calculations (, ) are required when using Eq. (1) while only one field calculation () is needed by applying Eq. (2). Similar computing precisions are provided by Eq. (1) and Eq. (2) giving reliable results for the coupling coefficients.

Walk-off length

The neff difference between even and odd fiber eigenmodes that compose OAM modes leads to temporal walk-off effects upon propagation.

2π walk-off length (L2π) represents the propagation length when the even and odd fiber eigenmodes walk off to each other with a relative phase shift of 2π, which is written by22

where λ is the wavelength in vacuum. and are effective refractive index of even and odd fiber eigenmodes, respectively.

The walk-off effect between the even and odd fiber eigenmodes distorts the profile of the OAM mode and resultant mode purity, resulting in the mode coupling and crosstalk. It is noted that such behavior features periodic and the period is equal to L2π.

10-ps walk-off length (L10 ps) denotes the propagation length when the even and odd fiber eigenmodes have a 10-ps temporal walk off, which is expressed by22

where c is the light velocity in vacuum and Δt = 10 ps is the temporal walk off time.

L10 ps characterizes the impact of temporal walk-off on the signal quality. For 10-Gbaud/s signals, 10-ps temporal walk-off corresponds to one-tens of the symbol period. Consequently, tolerable signal qualities of 10-Gbaud/s signals are achievable after the L10 ps propagation.

OAM spectra

OAM spectra, also known as OAM charge weight distributions, are written by23

where is the OAM charge weight and . e(x, y) is the electric field of the achieved mode in a perturbed fiber (e.g. in presence of ellipticity of ring or fiber bending). ψi(x, y) is the electric field of eigenmodes in an ideal ring fiber.

Eq. (5) can be used to estimate the mode coupling and crosstalk in presence of perturbations such as the ellipticity of ring and fiber bending.