*3.2.6 Graded index ring Core fibers (GI-RCFs)*

The ring notion touched the graded shape and a family of graded index ring core fibers (GI-RCF) has been proposed, designed and fabricated to support OAM mode group. **Figure 12** shows the refractive index of GI-RCF. In [66], Zhu and coworkers designed and fabricated the GI-RCF for OAM modes. The fiber supports 22 OAM modes with low insertion loss (less than 1 dB/km). The crosstalk between the highest order mode groups is less than 14 dB after10-km propagation. With such fiber, a successful transmission of 32 Gbaud QPSk-data overall 80 channels is experimentally demonstrated. A transmission capacity of 5.12 Tbits/s and a spectral efficiency of 9 bit/s/Hz, over 10 km propagation was reported [67].

The second demonstration was performed over 18-km propagation. Recently, the same group demonstrate the transmission of 12 Gbaud (8QAM) over 224 channels (2 OAM 112 wavelengths). A transmission capacity of 8.4 Tbits/s was achieved without MIMO DSP because of the large high-order mode group separation of the OAM fiber [68].

To increase even further the capacity of the fiber link, OAM transmission was reported over uncoupled multi core fibers. While a complete review on this topic exceed the scope of this chapter, we can nevertheless mention some contributions. Li and Wang designed seven-ring core fiber (MOMRF) supporting 154 datachannels in total (22 modes 7 rings) [69]. The proposed fiber featuring low-level inter-ring crosstalk (30 dB for a 100-km-long fiber) and intermodal crosstalk over a wide wavelength range (1520–1580 nm). Later, in [70], Li and Wang proposed a compact trench multi OAM ring fiber (TA-MOMRF) with 19 rings each supporting 22 modes (18 OAM states). The authors stated that such fiber is suitable for long distance OAM transmission enabling Pbit/s total transmission capacity and hundreds bit/s/Hz spectral efficiency. In [71], the authors proposed a coupled multi core fibers to support OAM modes (multi-orbital-angular-momentum (OAM)

**Figure 12.** *Design of the GIRCF.*

multicore supermode fiber (MOMCSF). The designed supermode fiber show favorable performance of low mode coupling, low nonlinearity, and low modal dependent loss.

#### *3.2.7 Inverse raised cosine few mode fibers (IRC-FMFs)*

Using IPGI fiber as a benchmark, we proposed a novel fiber that is based on inverse raised cosine function (IRCF). The standard raised cosine function (RCF) when applied to a wideband signal steeply removes the high out-of-band signals, making the filtered signal highly purified. Moreover, RCF is used in the same context because it eliminates intersymbol interference [72]. The IRCF profile is given by the following expression [73]:

$$m(r) = \begin{cases} n\_2 & \text{if } 0 \le r \le a \frac{1-a}{2} \quad (Core) \\ -\frac{1}{2(\mathbf{n}\_2 - \mathbf{n}\_3)} \left( 1 + \cos\left[ \left( \frac{\pi}{a \times a} \right) \left( \mathbf{r} - a \frac{\mathbf{1} - a}{2} \right) \right] \right) & \text{if } a \frac{\mathbf{1} - a}{2} \le r \le a \frac{\mathbf{1} + a}{2} \quad (Core) \\ n\_3 & \text{if } r \ge a \frac{\mathbf{1} + a}{2} \quad (GLadding) \end{cases} \tag{14}$$

where *a* is the core radius, n1 and n2 are respectively the maximum and the minimum refractive indices of the core, n3 is the refractive index of the cladding (r > *a*), and α is the profile shape. The refractive index of IRCF is shown in the **Figure 13**. The IRC profile is practically thinner (or more concentrated around the fiber axis) than the IPGI profile [73]. However, it is worthy to note that our profile becomes much smoother when reaching the maximum index value n1. When compared with IPGI-FMF, the inverse-raised-cosine function offers a large modal separation. The enhanced separation is likely to hinder mode coupling, reducing the system crosstalk and improving the transmission. Moreover, IRC-FMF has the potential to handle OAM modes with high purity hence low intrinsic crosstalk [73, 74].

**Figure 13.** *Index profiles of the IRC fiber (solid lines), with α ranging from 0 to 1 (reproduced from 73).*

*OAM Modes in Optical Fibers for Next Generation Space Division Multiplexing (SDM) Systems DOI: http://dx.doi.org/10.5772/intechopen.97773*

#### *3.2.8 Hyperbolic tangent few mode fibers (HTAN-FMFs)*

Based on hyperbolic tangent function (HTAN), we proposed and designed a ring core few mode fiber that we refer to as hyperbolic tangent few mode fiber (HTAN-FMF). The function HTAN was not common in optical fiber profiling. It is widely used in various fields/domains such as digital neural networks, image processing, digital filters, and decoding algorithms [75–77] but not common in waveguide and optical fiber designs. Intuitively, one of the most attractive criteria in hyperbolic tangent function, used as an activation function in neural network, is its strong gradient centered around the inflection point (switch point). This is the same criteria required from an optical fiber profile in order to enhance the intermodal separation. The refractive index of HTAN-FMF is given by the following expression [78]:

$$n(r) = \begin{cases} n\_2 \text{ if } 0 \le |r| \le a \frac{(1-a)}{2} \text{ (Core)}\\ \frac{n\_1 + n\_2}{2} + \frac{\Delta n}{2 \operatorname{Tanh}(\pi)} \times \left[ \operatorname{Tanh}\left(\frac{\pi \times (r - a\_1)}{a\_1 \times a}\right) \right] \text{ if } a \frac{(1-a)}{2} \le |r| \le a \frac{(1+a)}{2} \text{ (Core)}\\ n\_1 \text{ if } a \frac{(1+a)}{2} \le |r| \le a \text{ (Core)}\\ n\_3 \text{ if } |r| \ge a \text{ (GLadding)} \end{cases} \tag{15}$$

Where n1, n2, n3 are the refractive index at the core-cladding interface, at the core center, and at the cladding region, respectively. *a*, *a1* and α are the core radius, the half of core radius (*a1* = a/2) and the shape parameter respectively. *Δn* is the actual refractive index difference (i.e. *Δn=n1-n2*) which corresponds to the extent of hyperbolic tangent function inside the core. The shape parameter α controls the shape behavior of HTAN function. The refractive index of HTAN is illustrated in **Figure 14**. The proposed HTAN-FMF achieves a wide intermodal separation (between cylindrical vector modes) especially between TE0,1, HE2,1, and TM0,1

**Figure 14.** *Refractive index profile of HTAN fiber for different values of profile shape α [78].*

(≥<sup>3</sup> <sup>10</sup><sup>4</sup> ). This enables low-level crosstalk channels carrying data during propagation and outperforms what is existing in the literature [78]. On the other hand, even with an exterior abrupt variation, the inner smooth behavior of HTAN-FMF guarantees the enhancement of the obtained OAM mode purities (≥ 99.9%) leading to intrinsic crosstalk as minimum as 30 dB during propagation. Moreover, the obtained results in term of chromatic dispersion (max CD = 60 ps/(km.nm)), differential group delay (max DGD = 55 ps/m), and bending insensitivity, demonstrate that the HTAN-FMF could be a viable candidate for enhancing the transmission capacity and the spectral efficiency in next generation OAM mode division Multiplexing (OAM-MDM) systems [78].

#### **3.3 Photonic crystal fibers**

Photonic crystal fibers (PCF) has shown its design flexibility to guide appropriate OAM modes. With adjustable parameters, PCF can offer more flexible design structures to provide unique fiber properties. Due to that, several kinds of OAM-PCF with various structures (hexagonal, circular, kagome...) and materials (As2S3, SiO2, polymer … ), having promising features have been designed and even fabricated. PCF have been proposed and fabricated to ensure good transmission quality of OAM modes. While a review on this topic exceed the scope of this thesis, we can nevertheless mention some details and contributions. PCFs supporting one, 2, 10, 12, 14, 26, 34, 42 and 48, first order OAM modes have been proposed featuring good transmission properties [79–89].

The race is still ongoing to increase the number of OAM modes in PCF featuring good transmission proprieties. To the best of our knowledge, the most supporting OAM modes number in a circular PCF reaches 110 over C + L communication bands [90]. The designed fiber featured large effective indices separation (are at the order of 10<sup>3</sup> ), low nonlinear coefficient, low confinement loss (under 10<sup>7</sup> dB/m), and relatively flat chromatic dispersion. Such fiber could find potential application in high capacity OAM-MDM system. By analysis of these recent mosaic OAM-PCFs literature, we can come to the general requirements in PCF design that ensure good transmission quality of OAM modes in the following five points or guidelines [91–94].


*OAM Modes in Optical Fibers for Next Generation Space Division Multiplexing (SDM) Systems DOI: http://dx.doi.org/10.5772/intechopen.97773*

• The guided OAM modes would possess good transmission features such as low confinement loss, flat dispersion, large effective mode area, and low nonlinear coefficient over a large wavelength range (at least covering C + L bands defined by ITU-T).
