A Novel Hyperbolic Tangent Profile for Optical Fiber for Next Generation OAM-MDM Systems

In this work, we propose and theoretically/numerically investigate a novel parametric refractive index profile for few mode fibers, referred to as a hyperbolic tangent few mode fiber (HTAN-FMF). To the best of our knowledge, the HTAN-FMF is adopted for the first time, to simultaneously maximize the inter-channels separation and OAM purity. The proposed FMF achieves a wide separation between cylindrical vector modes ( $\geq 3\times 10\,\,^{\mathrm{ -4}}$ between TE0,1, HE2,1, and TM0,1). This enables low-level crosstalk and clearly overcomes recently reported levels in literature. On the other hand, simulations show that the generated and evaluated OAM modes are mostly high pure (≥ 99.9%) leading to intrinsic crosstalk as minimum as -30 dB during propagation through HTAN-FMF. Other propagation performance metrics of the proposed fiber including chromatic dispersion and differential mode delay are also evaluated over the C ITU-T band showing that HTAN-FMF has the potential to achieve stable OAM modes transmission. Bending insensitivity is numerically proved showing that the proposed FMF has the potential to enhance the transmission capacity and the spectral efficiency in next generation OAM mode division Multiplexing (OAM-MDM) systems.


I. INTRODUCTION
During last years, the unprecedented explosion in data traffic caused by the colossal progress in social networking, the cloud computing, and big data applications, put pressure on telecommunications community to develop higher capacity optical networks [1], [2].
Space division multiplexing (SDM) has gained a widespread popularity due to its capability to overcome the capacity crunch of already deployed single mode fiber (SMF) [3]. SDM has the advantage of increasing the number of data channels available inside an optical fiber. An attractive embodiment of SDM is the mode division multiplexing (MDM) based on excitation and propagation of several optical modes within few mode fibers (FMF). Often, FMF uses linear polarized modes (LP) as a mode basis carrying The associate editor coordinating the review of this manuscript and approving it for publication was Md Selim Habib . data. Heavy and complex ''multiple input multiple output'' digital signal processing (MIMO DSP) is required at the fiber end (receiver) to lift the degeneracy of propagating LP modes since they are formed by a superposition of several degenerate and polarization vector modes. The complexity of MIMO DSP increases as the number of propagating LP modes increases [4], [5].
Orbital angular momentum (OAM), also known as twisted light, is arguably one of the most promising approaches that has recently received special attention in both: free space and optical fiber networks. In fact, OAM is known by the unboundedness and the orthogonality among the co-propagating modes (channels) [6]. This will make a breakthrough in optical communication.
interestingly, OAM modes have proved its effectiveness in Terabit to Petabit scale capacity transmission and unleashed from dozen to hundred (bit/s/Hz) spectral efficiency over free space communication link [7]. Recalling here that FSO links are much appropriate when the optical fiber deployment is costly or not applicable. It has been demonstrated that carrying data on Laguerre-Gaussian beams (LG), incorporating two degrees of freedom (i.e. with various azimuthal indices and nonzero radial indices), has shown a great capability to enhance free-space system capacity and spectral efficiency. On the other hand, FSO based OAM modes faces several challenges where atmospheric turbulence (AT) is the main of them. AT leads to wavefront distortion and beam spreading & wandering where several techniques have been performed to mitigate AT effects including adaptive optics and low-density parity-check coding [8].
Considering the future of optical fiber communication, OAM modes may possibly look to replace the classic LP modes in new generation optical fibers. Special fibers proposed, designed, and manufactured, to support OAM modes transmission are referred to as OAM-fibers. Three main common features between these special optical fibers are identified. The first feature is the high contrast between core and cladding refractive indices, which increases the mode effective indices separation ( n eff ), hence enables low induced crosstalk. The second feature is pertaining to the refractive index that matches the donut shape of OAM mode intensity (Ring shape). Finally, the interfaces (core/cladding) are smoothed (instead of being in a step form) in order to eliminate the spin-orbit-coupling inducing mode purity impairment [9]- [21].
Targeting these design recommendations, Ramachandran et al. have reported a vortex fiber (VF) having n eff = 1.8 × 10 -4 between cylindrical vector modes [9]. Subsequently, the same team reported the transmission of two OAM states over 1.1 km vortex fiber, reaching terabit scale in capacity transmission [10].
Hollow (air) core fiber has been reported in [11] and [12]. In [11], the fiber (supporting 12 OAM states) is characterized by a good separation among its supported vector modes ( n eff = 1 × 10 -4 ). In [12], 17 OAM states equivalent to 36 information channels, have been demonstrated over air core fiber (ACF). Recently, a free MIMO 10.56 Tbit/s has been demonstrated over 1.2 km ACF by carrying data over 12 OAM modes combined with wavelength division multiplexing (WDM) [13]. Latest air core ring fiber is designed to support more than 1000 OAM modes (using As 2 S 3 as ring material ) across wide wavelength band covering S, O, E, S, C, and L Bands [14].
On the other hand, the general ring shape has been applied considering both step and graded refractive index profiles. In one side, step ring core fiber (RCF) has been investigated in the context of OAM transmission demonstrating good features in terms of effective index separation and dispersion [15]. Recently, 50 OAM states divided into 13 mode groups (MGs) have been numerically investigated in RCF using small MIMO DSP blocks [16]. Experimentally, a 50 km ring core fiber transmission over two OAM mode-group is demonstrated without MIMO DSP [17].
On the other side, graded index ring core fiber (GIRCF) is a nascent kind of OAM fiber that has been reported and shown to have capability to alleviate the spin-orbit coupling [18].Recently, using two OAM modes and 112 Wavelength channels (WDM), 8.4 Tbit/s has been reached using GIRCF [19].
Recently, we have proposed inverse raised cosine fiber (IRC-FMF) for OAM mode transmission that has shown great capability to lift modal degeneracies [20]. One particularly attractive profile is known as inverse parabolic graded index fiber IPGIF [21], characterized by an inner ''smooth'' gradient profile controlled by shape parameter N and outer Sharpe index behavior. with this heterogeneous structuremixture, this profile ensures large effective indices separation among its supported cylindrical vector modes (>2.1 × 10 -4 ). The transmission of eight OAM states over 1 m and two OAM states over more than 1 km have been experimentally demonstrated; thus, making the IPGI fiber suitable for both short and long-distance OAM based multiplexing system. In 2017, the multiplexing/transmission and demultiplexing of 3.36 Tbit/s have been demonstrated over 10-meters inverse parabolic graded index fiber using four OAM modes and 15 wavelengths [22].
The race is ongoing and it is still quite challenging to find the appropriate refractive index profile for optical fiber that both supports high number of OAM states while keeping the inter channels crosstalk very low. other performance metrics are also critical to ensure a robust OAM mode transmission including chromatic dispersion, differential group delay, OAM mode purity and intrinsic crosstalk. Moreover, given the huge demonstrated data capacity in these OAM-fibers categories, it is essential to examine the fibers under practical environment (bending, twist, and ellipticity, etc) which could both direct the design (as well as the optimization) process and contribute to progress from prototyping (in labs) to commercialization and then to deployment (FTTH: Fiber To The Home).
Motivated by the promising results above, we propose and theoretically/numerically examine a novel parametric refractive index profile for FMF (referred to Hyperbolic Tangent FMF: HTAN−FMF). This aims to ensure wider effective indices separation among the supported cylindrical vector modes. To the best of our knowledge, the hyperbolic tangent function has never been used as an optical fiber profile. Similar to graded index function, HTAN has a shape factor (α) that controls the shape of the index variation.
After analyzing the generic parameters of HTAN fiber (Section II), in Section III, we investigate the impact of the profile parameters on intermodal separation. Aiming to find the optimum parameters for wider separation, the investigation was carried out using a theoretical approach validated by a numerical simulation using finite element method (FEM). Once achieved, the OAM modes are generated and assessed in terms of purity and intrinsic crosstalk (Section IV). A full OAM mode analysis was performed over wavelength including the chromatic dispersion and the differential group delay. Finally, the effects of tight bend on the intermodal separation, on OAM quality, and on modal confinement are investigated (Section V). Section VI provides conclusions and some future research perspectives.

II. HYPERBOLIC TANGENT PROFILE DESCRIPTION AND DESIGN GUIDELINES
As reported in [21], the idea of designing a refractive index profile that is adapted to OAM transmission is driven by the need to achieve a wide separation between supported cylindrical vector modes. In principle, this is achievable when the index profile fulfills both criteria: (1) high-refractive-index contrast between the core and the cladding, and (2) sharp index change at the core-cladding boundary. Fortunately, both criteria are intrinsic characteristics of our proposed profile. We refer to the later as modified hyperbolic tangent profile (HTAN). This is widely used in various fields/domains such as digital neural networks [23], image processing [24], digital filters [25], and decoding algorithms [26] but not common in waveguide and optical fiber designs.
The standard hyperbolic tangent function ( Fig. 1 (a)) is an odd function (tanh(−r) = − tanh(r)) given by tanh(r) = (e r −e -r )/(e r +e -r ), defined from −∞ to +∞, which monotonically increases from −1 to 1 ( Fig. 1(a)). Since the standard tanh function is defined over an infinite real interval, it could not be used directly for fiber index profile because of its limited/realistic extent. This leads to a truncation operation of tanh in a way that keeps only its central part where the most of the change occurs between both levels (minimum and maximum). This truncation limits the extent of the function to an interval that is shorter than [−1, 1].
Moving to transpose the function into the fiber dimension, where r is the distance from core center to the cladding and a is the core radius, we make a variable change where The resulting function is written as tanh [(πx)/(αa/2)]/tanh(π).
We then shift the function from the origin to r = a/2. The shifted function becomes: n(r) = tanh[ π α ( r−a/2 a/2 )]/tanh(π) for a(1 − α)/2 ≤ r ≤ a(1 + α)/2, (see Fig 1(e)) where a(1 + α)/2 is the truncated inner core radius. Considering the full space [−∞ + ∞], the final function is written as: Similarly, the next step is to scale the function in order to limit its extent between n 1 and n 2 (maximum refractive index at the core cladding interface, and minimum refractive index at the core center, respectively). We shift and scale the function in the form of A + B × tanh[ π α ((r − a 2 )/ a 2 )]/tanh(π), where A = (n 1 + n 2 )/2, and B = n/2. The obtained HTAN distribution n(r) is as follows: where n 1 , n 2 , n 3 are the three levels of refractive index as illustrated in Fig. 2, a and α are the core radius and the shape parameter, respectively, and r denotes the distance from core center. n = n 1 − n 2 corresponds to the extent of hyperbolic tangent function, and a 1 = a/2. At core subbounds: a(1 − α)/2 and a(1 + α)/2, n(r) = n 2 and n(r) = n 1 , respectively. The parameter α, where 0 ≤ α ≤ 1, controls the shape behavior of HTAN function and will also be referred to as smoothness parameter. Considering the special case of α = 0, HTAN corresponds to the known step index inner & outer ring core fiber. From Figure 2, one can see that the proposed HTAN-fiber has quite similar profile complexity (i.e. inner graded & outer Sharpe variation) like IPGI VOLUME 8, 2020 fiber [21] but simpler than VF [9] and GIRC fiber [18] where both fibers observe an interior and outer graded refractive index evolution''.

III. FIBER KEY PARAMETERS AND INTER-CHANNELS SEPARATION
We theoretically investigate and numerically validate the modal separation between supported vector modes in the proposed HTAN fiber. In this section, we investigate the fiber parameters that directly influence the intermodal separation. However, this helps to optimize the fiber-improved separation between propagating modes. In [21], [27] the authors proposed/introduced a theoretical analysis of intermodal separation inside optical fiber by applying a first order perturbation. We exploited their analysis and applied it for our proposed profile index. Ung and co-workers stated that wide effective index separation between vector modes require: (1) Large refractive index gradient expressed as − → ∇ (ln(n 2 )) (e.g. Sharp core-cladding boundary together with large refractive index separation (contrast, jump)), (2) high transverse field amplitudes ( − → e ), and (3) large field variation − → ∇. − → e . 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). In Appendix I, we provide the theoretical approach used for identifying the fiber parameters that control the effective index separation behavior. In Appendix I, by applying the expression of gradient (A.12) on HTAN refractive index distribution (core region), and evaluating it at the sub-bound a(1 + α)/2, we find the following proportionality relation: Theoretically, for HTAN profile, the effective index separation is proportional to the relative refractive index difference inside the core n/n 1 while it is inversely proportional to the shape parameter α and to the core radius a. Hence, any variation in either of these fiber parameters changes automatically the intermodal separation. The mode field and the mode field gradient have been numerically computed using FEM (A.8). Fig. 3 (a and c) depicts the normalized transverse electric fields TE 0,1 and HE 2,1 together with simulated refractive index profile of HTAN profile with α = 1 and α = 0.5, respectively, while Fig. 3(b and d) shows its corresponding gradients in cylindrical coordinates. From Fig. 3(b and d), one can see that both maxima of gradients superpose at the same point (core-cladding boundary (r = a). Moreover, quantities have the same sign (negative here). This means that high effective index separation between vector modes is accomplishable in the proposed HTAN fiber. In order to both prove (validate) the obtained relation (3) and optimize the fiber parameters (improve n eff ), we investigate the intermodal separation between TE 0,1 , HE 2,1 , and TM 0,1 that forms the mode group G 2 . The selection of G 2 (known as LP 1,1 ) comes from the fact that TE 0,1 and TM 0,1 are parasitic modes that do not provide an OAM mode and induce crosstalk to the other OAM modes OAM ±1,1 if they are not enough spaced. In addition, as revealed from [21], the higher order modes (HE l,m /EH l,m ) with the same radial number (m = 1) share the same intermodal separation behavior. We aim to validate the theoretical investigation highlighted above, and optimize α, n max , n and a for the fiber design in order to achieve wide effective index separation among vector modes belonging to G 2 .

A. INTER-CHANNELS SEPARATION VERSUS THE SHAPE α AND REFRACTIVE INDEX CONTRASTS
We investigate the minimum intermodal separation n eff among (TE 0,1 , HE 2,1 , and TM 0,1 ). Fig. 4 (a) depicts the minimum n eff inside G 2 versus the shape parameter (α) and the maximum contrast between core and cladding indexes ( n max ). In addition, Fig. 4(b) shows the minimum n eff versus α for selected values of n max . It is clear that for a given value of α, n eff increases with the increase of n max . This confirms that a high contrast refractive index between fiber core and cladding favors large n eff , enabling low intermodal crosstalk. Furthermore, from Fig. 4(a) a very large n eff of around 7.5 × 10 -4 is achievable with ultra large n max = 0.08. This value is experimentally achievable with modified chemical vapor deposition (MCVD) process with Germania doped silica [28]. On the other hand, Fig. 4(b) reveals that for selected values of n max , n eff inside G 2 slowly decreases with α. The impact of actual refractive FIGURE 5. Minimum effective index separation between (TE 0,1 , HE 2,1 , and TM 0,1 ) as a function of: (a) the relative difference = n/n 1 and for various smoothness parameter α, collocated with those calculated in GIF (for different α G ) (b) the smoothness parameter and for various values n, a = 3µm, λ = 1.55µm, and n max = 0.05. FIGURE 6. Minimum effective index separation between (TE 0,1 , HE 2,1 , and TM 0,1 ) as a function of smoothness parameter α and the core radius a, n 3 = 1.444, n max = 0.05, n 2 = 1.454 and λ = 1.55 µm.
index difference ( n = n 1 − n 2 ) on n eff is investigated. In Fig. 5(a), the minimum n eff inside G 2 is depicted versus the relative refractive index difference inside the core ( n/n 1 ) for a given shape parameter α. In order to evaluate the effects of α on n eff , we add the n eff in standard graded index fiber (GIF) simulated with the same parameters as in HTAN fiber for different values of shape parameter (α G ). In Fig. 5(b), we depict minimum n eff versus α for given values of n. Figure 5(a) confirms that n eff increases with n as predicted by expression (3). Furthermore, HTAN improves the intermodal separation compared to the standard GIF. Fig. 6 highlights the minimum n eff inside G 2 versus a and α. We observe that small core (at the vicinity of 3µm) enhances the intermodal separation (a ≤ 3.5µm, n eff ≥ 3.5 × 10 -4 ). In order to investigate and validate the theoretical studies above more, we calculate n eff : TE 0,1 − HE 2,1 and HE 2,1 − TM 0,1 versus the core radius for selected partical values of maximum contrast n max : [0.04, 0.05, and 0.06], for selected values of the difference n:[0.03, 0.04, and 0.05] and for different values of shape parameter α (Fig 7 (a-d)).

B. INTER-CHANNELS SEPARATION VERSUS CORE RADIUS
One can see that the modal separation increases with core radius until it reaches its maximum (peak) at an optimal core radius a opt . Interestingly, when r ≤ a opt , we find n eff (TE 0,1 , HE 2,1 ) ≥ n eff (HE 2,1 , TM 0,1 ). On the other hand, when r ≥ a opt , we find n eff (HE 2,1 , TM 0,1 ) ≥ n eff (TE 0,1 , HE 2,1 ). This indicates that the vector modes (TE 0,1 , HE 2,1 , and TM 0,1 ) formed LP 1,1 are spaced regularly at a opt . Hence, the located a opt is useful for directing the design ( n max & n) of HTAN fiber towards the improvement of intermodal separation.

IV. OAM MODES ANALYSIS
In this section, we select the fiber parameters that enhance the separation among first radial OAM modes. We then generate the OAM modes and assess its purity and the induced intrinsic crosstalk at fixed wavelength λ = 1.55 µm. Subsequently, we expand the wavelength band to cover the entire C (ITU-T) band ranging from 1525 nm to 1625 nm, where we investigate the impact of the shape parameter ''α'' on various performance metrics including chromatic dispersion, and differential group delay. We ignored the part when n eff <10 -4 . Each sub-figure (map) could be considered as ID card for fiber (for each α) since it provides number & order of modes, separation among them, and the optimum fiber parameter values (cutoff radii, n max , and n). From those figures, it is confirmed that n eff between first radial modes (m = 1) decreases as ''a'' increases while it increases for higher radial modes, (e.g. HE 1,2 lines without symbols). Interestingly, n intervenes in the order of excited mode: when n decreases; it tends to delay the excitation of higher radial order modes (e.g. HE 1,2 , comparison between n = 0.04 and n = 0.05) while it decreases n eff among modes. Thus, a trade-off-design should be considered between mode number, n eff between modes as well as the mode order (i.e. m).
Since higher radial orders OAM modes cause trouble in multiplexing and demultiplexing operations due to the intensity and phase variety distribution (complex modal distribution), we limited our study to the first radial OAM modes that fulfil the criteria n eff > 10 -4 . Hence, there are 10 possible eigenmodes. Moreover, the HTAN-FMF will perform comparably to several other OAM-fibers by supporting higher order OAM modes packaging into mode groups (MG) where the inter-MG separation should be maintained with more tradeoffs.
At last, we denote OAM modes as OAM ± ±l,m where ± superscript describes the spin angular momentum SAM (circular polarization). l and m subscript denote the azimuthal and radial indices, respectively, and m describes the number of nulls radially (rings) in the intensity profile of the OAM mode. Fig. 9 recapitalizes the guided vector modes supported in HTAN fiber with the parameters cited under the graph. In addition, we depict the corresponding OAM modes generated by combining the even and odd modes of HE l,m or EH l,m modes with ± (π /2) phase shift calculated using finite element method (FEM) [29]. The combination between TE 0,1 and TM 0,1 do not provide OAM states due to the large separation between its propagation constants.

B. OAM MODE QUALITY
In order to qualitatively assess (evaluate) the obtained OAM modes in the proposed HTAN-FMF and evaluate the distribution impact, we investigate the OAM purity in the designed fiber. Interestingly, it has been demonstrated in [24] that the refractive index profile structure intervenes in enhancing the OAM purity enabling low induced intrinsic crosstalk. The attention of authors was concentrated on the evolution of refractive index from the core center till the cladding region (graded versus step). They demonstrated theoretically that outer graded index structure enhances OAM purity in contrast to step index structures (straight evolution). They draw our attention to the effects of the shape factor α (in HTAN-FMF) on the synthetized OAM's purity.
Investigating the purity versus the parameter α seems to be justifiable since the only fiber parameter that controls the shape profile in HTAN fiber is the parameter α. In addition, this choice helps to evaluate the impact of the only inner smooth behavior of hyperbolic tangent function on the fiber performance, which is one of the main purposes of this research. Considering all the OAM states (OAM ± ±l,m ), we calculate the power weight of major and minor OAM light beams through processing the integral of right part constituting the transverse field of OAM modes versus the shape factor α expressed as [30]: where k, n eff and n HTAN are the wave number in vacuum, mode effective index, and the refractive index of HTAN profile at given α, respectively. − → x and − → y are two orthogonal unit vectors of cartesian coordinates. s, P(r) and Q(r) are given as follows: s ≡ ωµ 0 C kA , P(r) ≡ 1 2 [ ∂F lz (r) ∂r + 1 r F lz (r)] and Q(r) ≡ 1 2 [ ∂F lz (r) ∂r − 1 r F lz (r)]. A and C are undetermined coefficients and F lz (r) is the radial dependance of the field profile. The OAM purity are depicted in Fig. 10 (a). In addition, we calculate the corresponding induced intrinsic crosstalk. The purity and the intrinsic crosstalk of OAM light beam are respectively calculated as [30]: Purity = OAM major OAM major + OAM minor (5) and IntrinsicCrosstalk = OAM minor OAM major + OAM minor where OAM major/minor is the power weight of the major/minor OAM modes. It is clear that the OAM purity increases with the increase of α. Hence, as the profile interiorly smooths, the OAM purity improves. This is confirmed by the increase of OAM purity of OAM ± 0,1 , OAM ± ±1,1 , OAM ∓ ±2,1 and OAM ± ±2,1 versus α. The minimum OAM purity is found at α = 0 which corresponds to step ring core structure. In step ring core structure, the refractive index presents two discontinuity areas with large refractive index contrast. The first is located at r = a/2 (inner core) when the refractive index ''jumps'' from n 2 to n 1 , while the second is located at r = a (outer core), the area at which the refractive index 'falls' from n 1 to n 3 . Except OAM 0,1 , all the OAMs' purity are below 99.9%. Recalling here that an OAM purity of 99.9% is equivalent to an intrinsic crosstalk of −30dB. This key value is a theoretical limit of the practical crosstalk of the OAM optical fiber communication networks [30]. The worst purity is for OAM ∓ ±2,1 valuing 69.907% (intrinsic crosstalk = −5,215dB). Moreover, given the fact that α = 1 corresponds to the slow changes of refractive index, it presents the highest OAM purity compared to other α values. The OAM purity of OAM ± 0,1 , OAM ± ±1,1 ,OAM ∓ ±2,1 and OAM ± ±2,1 (at α = 1) are 99.960%, 99.971%, 99.814% and 99.965%, which correspond to intrinsic crosstalk of −33.97 dB, −35.37 dB, −27.30 dB and −34.55 dB, respectively. The lower value of OAM purity of OAM ∓ ±2,1 can be attributed to the high contrast between core and cladding refractive indexes ( n max ) and/or the step index interface at the vicinity between the core and the cladding (r = a), hence presenting discontinuity area that enhances the anti-aligned spin orbit coupling that deteriorates the synthetized OAM purity. Even though, this value is larger than those in vortex fiber [9] (99.8%), and comparable to those in designed known graded index FMF in [30] and in IRCF [20].

V. TRANSMISSION PERFORMANCE METRICS ACROSS TELECOMMUNICATION BANDS
The wide separation at fixed wavelength (λ = 1.55µm) has been analyzed and discussed in previous sections. Here, we extend the wavelength to cover the entire C&L (ITU-T) bands, ranging from 1525 nm to 1565 nm. We assessed the OAM modes separation over C&L, the chromatic dispersion (CD) and differential group delay (DGD) both over C band.

A. INTER-CHANNELS CROSSTALK ACROSS C+L BAND
We investigate the effective indices separation between all supported modes along the selected bands. Using FEM, Fig. 11 (a) depicts the mode effective indices versus the wavelength. The proposed HTAN-FMF supports eight OAM modes in the entire C+L band. These OAM modes are divided into 3 groups: G 1 (OAM ± 0,1 ), G 2 (TE 01 , OAM ± ±1,1 , TM 01 ) and G 3 (OAM ∓ ±2,1 ,OAM ± ±2,1 ). On the other hand, we investigate the effective indices differences between successive modes (inter and intra-group separation) across the wavelength. The results are shown in Fig. 11 (b). The n eff between adjacent modes slowly increases with wavelength. All the modes are sufficiently spaced (above 10 -4 ) enabling low crosstalk between different channels.

B. CHROMATIC DISPERSION ACROSS C BAND
The chromatic dispersion (CD) of supported modes is evaluated through the following expression [20]: where λ and c are the wavelength and the light velocity both in vacuum, respectively. n eff is the mode effective index. the results are depicted in Fig. 12 for different values of α= [0, 0.2, 0.8, and 1]. One can clearly see that the chromatic dispersion curves are flat, negative or slowly decreases with wavelength for G 1 (OAM ± 0,1 ) and G 2 (TE 0,1 / OAM ± ±1,1 /TM 01 ), while it decreases significantly versus wavelength for G 3 (OAM ∓ ±2,1 ,OAM ± ±2,1 ). The effective indices of OAM ∓ ±2,1 and OAM ± ±2,1 are close to the cutoff index (refractive index of cladding), hence placing the dispersion in high value range around −60 (ps/(km.nm)) compared to the chromatic dispersion of G 1 and G 2 .
Considering the effect of α on mode chromatic dispersion, Fig. 12 provides that the chromatic dispersion of G 1 and G 2 slowly decreases with α (the chromatic dispersion get close to zero as α increases). On the other hand, the chromatic dispersion of G 3 increases with α. This behavior may be explained by the impact of the shape parameter α on the profile structure. When the shape parameter α increases, the effective difference between the radii of the inner and outer cores reduce, and the higher mode effective indices approach the cutoff index (i.e. the cladding (n 3 )).
Considering the case when α = 1, over the whole C band, the G 1 and G 2 dispersions are less than 30 (ps/(km.nm)) while larger for the third group (about 60 (ps/(km.nm))). These dispersion values are close to those obtained for air core fiber reported in [31] (> 62 (ps/(km.nm))) or in ring core fiber [31] (> 39.7 (ps/(km.nm)). The dispersion properties of designed fiber are promising since they are lower than those in the reported IPGI-FMF (around −100 (ps/(km.nm))) for G 3 ) [21] and air core fiber [14]. Hence, HTAN-FMF is quite promising for long distance fiber communication using OAM modes carrying data.

VI. FIBER TOLERANCE TO BENDING
In view of practical uses (environment) of the optical fibers, the fiber is susceptible to perturbations such as bending and twisting. Bent fibers can affect seriously the transmission operation including the intermodal separation, the purity of OAM modes and power loss [34], [35]. After analyzing the unperturbed HTAN fiber, in this section, we examine (investigate) the immunity of the proposed fiber when it is subjected to bend. We study the impacts of fiber bends on the performance of OAM modes. The latter includes the intermodal separation (i.e. Crosstalk), the intra-mode walk-off effect of OAM modes, and the confinement losses of supported modes induced by tighter bending radius (Rb) ranging from 2 mm to 25 mm. Other fiber parameters are kept as: a = 3µm, n 2 = 1.444, n max =0.05, n 2 = 1.454 and λ = 1.55µm. According to the standard conformal mapping approach [35], the straight fiber and bent fiber refractive indexes are linked by the following expression: where n(x, y) and n eq (x, y) are the straight fiber refractive index and the equivalent bent fiber refractive index, respectively. Rb is the bend radius applied along the transverse x-axis direction. The additional term 1.40, located in denominator, denotes the photoelastic effects on the refractive index induced by the local strain in bent fused silica fibers [36].
By implementing the expression (6) into FEM, we numerically compute the bending effects on modes and address the robustness of such fiber against fiber bend. Figure 14 illustrates the intermodal separation ( n eff ) between successive modes as a function of bending radius and for different values of shape parameter α. Note here that the supported modes number is the first parameter that is affected by the bending. The higher modes tend to leak out from the core into the cladding (unguided). In HTAN, only at Rb = 8 mm, the fiber supports all the 10 modes presented in straight case. From Figure 14 (a-d), one can see that even under tight bend radius, the intermodal separation remains > 10 -4 enabling low-level induced channels crosstalk. Hence, HTAN features stronger resilience (tolerance) to fiber bending effect.

B. MODES WALK OFFS
In addition to the effects on the intermodal separation, fiber bending affects the separation between even and odd modes that form OAM modes, which gives mode walk-off upon propagation. In order to characterize the intra-mode walk-off effect of OAM modes against the fiber bend radius, we calculate the 2π and 10-ps walk offs lengths (L 2π , L 10ps ) for all the OAM modes in HTAN fiber (for different values of α). L 2π is defined as the propagation length when the even and odd fiber eigenmodes walk off to each other with a relative phase shift of 2π, while L 10ps presents the propagation length after the two eigenmodes has a 10-ps temporal walk off.it characterizes the impact of temporal walk-off on the signal quality. Note here that, for 10-Gbaud/s signals, 10-ps temporal walk-off corresponds to 10% of the symbol period [34]. L 2π , L 10ps are given as follows:  L 10ps = c × t n even eff − n odd eff (11) where λ, c, and t are the wavelength, speed of light in a vacuum, and walk-off time ( t = 10 ps), respectively. The results are shown in Figure 15 (a-d). One can see that L 2π and L 10ps of OAM modes increase exponentially with bending radius. HE 3,1 related OAM mode has longer L 2π and L 10ps , hence smaller effective index separation between even and odd modes, which means that it has more azimuthal periods in its transverse field distribution. Regarding the effect of α, (L 2π , L 10ps ) slowly decrease with α. The minimum L 10ps at α = 1 is around 200 km even under tight bend Rb = 1 cm.These values are reasonable since they are consistent with those recently reported in OAM-fibers in [14], [34].

C. BENDING LOSS
At last, the bend induces confinement losses (CL) of supported modes. The results are shown in Figure 16 (a-d), where the confinement losses are computed versus Rb and for different values of α. We add also the CL of both higher modes at Rb = 1 cm for n max = 0.048. From that figure, one can see that higher modes (EH 1,1 and HE 3,1 related OAM modes) achieve larger confinement losses due to the proximity to the cut-off (high susceptibility to leak out from the core into the cladding). In contrast, lower modes observe strong modal confinement within the core even under tight bending radius (below 10 -10 (consistent with [38]) (Rb = 1cm). Even though it is well known that high index contrast at the core cladding boundary raises the fiber loss, HTAN with the selected parameters shows great bending insensitivity that prevents the addressing of additional complex structure (e.g. by adding trench [39]) to original profile structure. The obtained results are reasonable and in full agreement with those in IPGI [21]. Regarding the effect of bend with parameter α, the CL slowly decreases as α increases.

VII. CONCLUSION
In this work, we have proposed a novel kind of refractive index profile for few mode fiber referred to as a hyperbolic tangent FMF (HTAN-FMF) that could be a good candidate for next generation cylindrical vector modes as well as OAM modes based MDM systems. By performing a theoretical/numerical analysis, we identify the design (key parameters & values) that provide optimized n eff among vector modes. These parameters include core radius (a = 3µm), index contrasts ( n max = 0.05, n = 0.01) and shape parameter (α = 1). The results show that the optimized n eff outperforms what is existing in the literature and enables low crosstalk among channels carrying data during propagation. In addition, the simulations shows that the HTAN-FMF structure complexity (inner smooth & outer sharpe) enhances the purity of synthesized OAM leading to reduced intrinsic crosstalk. Hence, this makes the proposed fiber suitable for next generation long distance OAM fiber. Furthermore, the propagation performance metrics in terms of chromatic dispersion (CD) and differential mode delay (DGD) are examined (across the C ITU-T band) and discussed. The obtained results (max CD = -60 ps/(km.nm), max DGD=55 ps/m) show that HTAN-FMF can be a viable candidate for long distance OAM-MDM based communication. Even under practical condition (bending), the proposed HTAN-FMF shows great insensitivity to bending effects. In future work, we aim to deeply examine the new profile including the sensitivity to further fiber perturbations such as core ellipticity and fiber twisting and the inter group/core crosstalk in multimode/multicore fiber scenario involving HTAN. The results of such investigation may encourage the industry to prototype the proposed HTAN design.

APPENDIX I: INTERMODAL SEPARATION: ANALYTICAL APPROACH
The effective index n eff and the propagation constant β of a given vector mode are linked with its scalar degenerate counterpart ( n eff ) and ( β) by the following expressions [21], [27]: where δβ 2 expresses the applied correction factor lifting the degeneracy of vector modes, k 0 is the wave number given by 2π/λ. If we neglect the longitudinal component of mode field (e zẑ ), the transverse field of a given vector mode is presented as − → e = [ê rr ,ê ϕφ ] in cylindrical coordinates. The correction factor δβ 2 is linked with − → e by [21], [27]: − → ∇ is the gradient operator, and n is the refractive index function. As reported in [21], the expression A.13 is general and exact for any refractive index profile. The intermodal separation is proportional to δβ 2 [15]. Since the proposed HTAN−FMF presents a circular symmetric (the index depends only on r), the gradient expressions (of refractive index and mode field are expressed in cylindrical coordinates as [27]: Applying the expression (A.14) on HTAN refractive index distribution (core region), we have: 2tanh(π) tanh( π(r−a 1 ) Tanh( π(r − a 1 ) a 1 α )) 2 (A.16) VOLUME 8, 2020 In order to facilitate the analysis, we define X = [π(r − a 1 )]/(a 1 α), and since dtanh(X ) dr = π a 1 ×α (1 − tanh(X ) 2 ), the expression (A.16) is written as: − → ∇.(ln(n 2 )) = n ( n 1 +n 2 2 + n1−n2 2×tanh(π) × tanh( π(r−a 1 ) a 1 ×α )) 2 which is then simplified and expressed as: Evaluating the resulting expression at the bound a (1+α) 2 , we find the following proportionality relation:  He has also an active involvement in the review process of a number of research journals, KACST general directorate grants programs, and national and international symposiums and conferences. VOLUME 8, 2020