1 Introduction

The orbital angular momentum (OAM) is the recently used consolidated multiplexing process concerning light for securing information. It is also a space division multiplexing (SDM) technology that is used in free space for optical communication (Huang et al. 2014). For the optical transmissions, the OAM introduced with vacuum communication (Brunet et al. 2014). Nowadays, optical communication confronts different types of challenges such as the transmission of big data, telecommunication, broadcast, community antenna television coaxial cable (CATV), high-definition television (HDTV) for medium blocking and insufficient capacity of data transmission. Recently, for multimode fiber, the SDM technology provides the OAM method that enhances the capacity and efficiency of optical communication (Zhang et al. 2018; Richardson et al. 2013; Zhu et al. 2016; Li et al. 2018).

The beams of OAM individualized by \(e^{{\left( {il\varphi } \right)}}\) (Chen et al. 2016), where the topological charge is expressed by l and it is an integer number (…., − 3, − 2, − 1, 0, + 1, + 2, + 3,….) and azimuthal angle is expressed by φ. The azimuthal angle (φ) of a point in cylindrical or spherical coordinates is the positive angle between the projection of the vector to the XY-plane and positive X-axis (Zhu et al. 2016; Allen et al. 1992). Every positive (+) and negative (−) topological charges are spatially orthogonal to each and another (Jiao et al. 2019). There are several methods to achieve an OAM beam like twisted fiber (Wong et al. 2012), helical lattice (Fang and Wang 2015), etc. There are various applications such as optical communication (Wang et al. 2012), particle trapping (Padgett and Bowman 2011), high-resolution imaging (Tamburini et al. 2006) and many more (Heng et al. 2017) of the OAM beams.

For improving the number of modes, Liu et al. (2019a) designed a dual-cladding photonic quasi-crystal fiber with 6 OAM modes for the OAM optical communication system. The quasi-periodic structure used in the photonic quasi-crystal fiber (Liu et al. 2019b, c; Li et al. 2019) such as sensing (Shi et al. 2019; Liu et al. 2019d), filters (Liu et al. 2018; Yan et al. 2018), nanocavity (Ren et al. 2018), metasurface (Tang et al. 2019), lens (Liu and Fan 2018), etc. Zhang et al. (2015) proposed a C-PCF with silica as the background material supporting 10 OAM modes to transmit in one fiber. A hollow-core fiber was proposed by Li et al. (2016) that supports 16 OAM modes with high confinement loss. A C-PCF proposed by Hu et al. (2016) that supports 26 OAM modes with As2S3 as background material. Wang et al. (2018) is also designed a PCF with 26 OAM modes. Xu et al. (2017) proposed a C-PCF supporting 30 OAM modes with negative dispersion. The proposed PCF of Lei et al. (2018) supports more OAM modes (76 OAM modes). But that was not enough for stable transmission. So, there is still room to design a novel PCF with larger OAM modes based on PCF with better output.

The high number of OAM mode is good for the encoding of data transmission. The same number of the OAM mode is useful for detecting receiving information with respect to sending information. When the sender sends data to the receiver, then it has to find out the similarities between sending and receiving data. By checking the mode properties of two transmission sides, the receiver ensures require data which transmit by the sender. In the OAM modulation process, the sender sends a carrier signal with the message signal; where the carrier signal carries the OAM mode properties. So, the OAM mode plays an important act in optical communication. The traditional optical fiber fails to transmit the OAM mode thus we have to design a special PCF for supporting multimode and also to achieve good parameters of a fiber. The difference between the two adjacent effective refractive indexes of the OAM HE and EH modes must be greater than \(10^{ - 4}\) (Δneff > \(10^{ - 4}\)) (Zhang et al. 2017a). The OAM transmission is supported by many unique PCF like spiral PCF (Nandam and Shin 2018), C-PCF with a square hole (Bai et al. 2018), dual guided ring-core PCF (Xu et al. 2018), C-PCF (Zhang et al. 2018). These are separately good for confinement loss, dispersion, nonlinearity, and numerical aperture, but not better for the robust transmission.

This paper proposed a chain shaped photonic crystal fiber that is composed of the three well-structured chain type air holes ring with a large air hole and the background material is composed of fused silica. This structure of PCF is novel for OAM transmission. This novel design of PCF supports 26 modes with 750 nm bandwidth. This PCF supports a low dispersion variation of 3.8684 ps/km-nm for the \(HE_{2,1}\) mode, and 5.1421 ps/km-nm for the \(TE_{0,1}\) mode with low confinement loss that is 3.19 × \(10^{ - 10}\) dB/m at wavelength 1.80 µm.

2 Model and algorithm

As mention before, the proposed three-layer chain shaped air holes with a hollow-core C- PCF is designed and simulated by COMSOL Multiphysics 4.3b software. The hollow-core PCF is the part of PCFs that guide light through the core (Monfared 2018). The three well-patterned circular ring type chain shaped air holes would decrease the light outpouring of optical fiber. The radius of the core of the proposed model is 3.7 μm. The large core increases the number of OAM modes (Zhang et al. 2017a). The air holes are made by the Bezier Polygons. Seven Cartesian points are needed to build this type of curve. The width of each polygon is 1.5 μm (\(w_{2}\) = \(w_{3}\) = \(w_{4}\) = 1.5 μm) and each co-ordinate has two levels that are totally eight levels. The area between the outer circle and the inner circle is called PML (Perfectly Matched Layer) and the width between them (\(w_{1}\)) is also 1.5 μm. It is added here to truncate unwanted nonphysical scattering. All the numerical analysis are implemented by using the PML and the FEM. And the background material is fused silica with a refractive index of 1.446 (Monfared et al. 2013). The refractive index (RI) is calculated by the Sellmeier Eq. (1) (Biswas et al. 2019).

$${\text{n}}\left( \lambda \right) = \sqrt {1 + \mathop \sum \limits_{i = 1}^{m} \frac{{A_{i} \lambda^{2} }}{{\lambda^{2} - \lambda_{i}^{2} }}}$$
(1)

where m,\(\lambda_{i}^{2}\) and \(A_{i}\) are the Sellmeier coefficients and \({\text{n}}\left( \lambda \right)\) represents the RI that varies with wavelength. The cladding of the proposed PCF is constructed by fused silica. For fused silica, m = 3. The complete design of the PCF is shown in Fig. 1.

Fig. 1
figure 1

The cross-section view of the proposed PCF

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The OAM modes in the PCF can be expressed by the combinations of even and odd modes of EH and HE. It can be defined by Eqs. (2) and (3) (Pakarzadeh and Sharif 2019; Zhang et al. 2017b).

$$\left\{ {\begin{array}{*{20}c} {OAM _{ \pm l,m}^{ \pm } = HE_{l + 1,m}^{even} \pm jHE _{l + 1,m}^{odd} } \\ {OAM _{ \pm l,m}^{ \mp } = EH_{l - 1,m}^{even} \pm jEH _{l - 1,m}^{odd} } \\ \end{array} } \right\};\quad ({\text{l}} > 1)$$
(2)
$$\left\{ {\begin{array}{*{20}c} {OAM _{ \pm 1,m}^{ \pm } = HE_{2,m}^{even} \pm jHE _{2,m}^{odd} } \\ {OAM _{ \pm 1,m}^{ \mp } = TM_{0,m} \pm jTE _{0,m} } \\ \end{array} } \right\};\quad (l = 1)$$
(3)

where HE and EH are the composite mode of the OAM beam. The two main types of each mode are even and odd that obtains after π/2 phase difference of the modes. The sign of “± l” defines the right or left wave front orbit direction. Where l is a topological charge, j denotes a π/2 phase shift, and m denotes the radial order for the OAM mode, and equal to 1. The OAM mode does not support m > 1 because it causes “accidental degeneracies” (Pakarzadeh and Sharif 2019). Because it is a problem for the encoding, multiplexing and demultiplexing of the OAM modes. So, we use a lower-order radial mode (m = 1) (Zhang et al. 2018).

By applying the Eqs. (2) and (3), the proposed structure is supported 26 modes, these are \(OAM_{ \pm 1,1 }^{ \pm } \left\{ {HE_{2,1} } \right\}\), \(OAM_{ \pm 2, 1 }^{ \pm } \left\{ {HE_{3, 1} , EH_{1,1} } \right\}\), \(OAM_{ \pm 3, 1 }^{ \pm } \left\{ {HE_{4, 1} , EH_{2,1} } \right\}\),\(OAM_{ \pm 4, 1 }^{ \pm } \left\{ {HE_{5, 1} , EH_{3,1} } \right\}\),\(OAM_{ \pm 5, 1 }^{ \pm } \left\{ {HE_{6, 1} , EH_{4,1} } \right\}\), \(OAM_{ \pm 6, 1 }^{ \pm } \left\{ {HE_{7, 1} , EH_{5,1} } \right\}\),\(OAM_{ \pm 7, 1 }^{ \pm } \left\{ {HE_{8, 1} , EH_{6,1} } \right\}\).

The proposed PCF is supported by all the selected modes from the wavelength the ranges from 1.2 to 1.95 μm and simulations are done by those modes. It also supports some eigenmodes such as \(TE_{0,1}\) and \(TM_{0,1} ,\) but they can’t combine. Because the propagation constants of \(TE_{0,1}\) and \(TM_{0,1}\) modes are different.

The quality of the OAM based PCF depends on more numbers of the OAM modes. Because, it increases the capacity of transmission and also ensures the robust transmission. The encoding and multiplexing of lights are developed by the good quality of OAM mode. Therefore, the light intensity imbricate factor to qualitatively calculate the mode quality through the Eq. (4) (Zhang et al. 2017b).

$$\eta = \frac{{I_{r} }}{{I_{c} }} = \frac{{{\iint }_{rings} \left| {\vec{E}} \right|^{2} dxdy}}{{{\iint }_{cross - section} \left| {\vec{E}} \right|^{2} dxdy}}$$
(4)

where \(I_{r}\) defines the normal mode concentration of the ring area where all the OAM modes will be confined and \(I_{c}\) denotes the average mode intensity in full exploded view area of the C-PCF. In Fig. 2, different OAM mode intensity order of eigenmodes and the wave front phase are exhibited.

Fig. 2
figure 2

a The intensity ordinations of \(EH_{1,1}\), \(EH_{2,1}\), \(HE_{3,1}\), \(HE_{4,1}\), \(EH_{5,1}\), \(EH_{6,1}\), \(HE_{7,1}\), \(HE_{8,1}\), \(TE_{0,1}\) and \(TM_{0,1}\) modes, b the phase ordinations of \(EH_{2,1} HE_{3,1}\), \(EH_{4,1} , HE_{5,1}\) and \(HE_{8,1}\) modes

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For all modes, proposed PCF gains flat dispersion, less nonlinearity, better numerical aperture, less confinement loss and other OAM properties.

Dispersion is evaluated by Eq. (5) (Zhang et al. 2018).

$$D = - \frac{\lambda }{c}\frac{{d^{2} Re\left[ {n_{eff} } \right]}}{{d\lambda^{2} }}$$
(5)

where \(Re\left[ {n_{eff} } \right]\) denotes the real values of the effective refractive index and c defines the velocity of light in the vacuum.

The differences of the refractive index can be evaluated through Eq. (6) (Pakarzadeh and Sharif 2019). This difference is performed on two adjacent vector mode.

$$\Delta n_{eff} = \left| {n_{{eff_{{HE_{l + 1,m} }} }} - n_{{eff_{{EH_{l - 1,m} }} }} } \right| > 10^{ - 4}$$
(6)

So, \(\Delta n_{eff}\) measures, the absolute value within the HE and EH OAM modes. It must be larger than \(10^{ - 4}\).

The effective OAM mode area is symbolized by \(A_{eff}\) and can be evaluated through Eq. (7) (Jia et al. 2018).

$$A_{eff} = \frac{{\left( {{\iint }\left| {E\left( {x,y} \right)} \right|^{2} dxdy} \right)^{2} }}{{{\iint }\left| {E\left( {x,y} \right)} \right|^{4} dxdy}}$$
(7)

where \(E\left( {x,y} \right)\) is an electric field allocation of the electromagnetic (EM) transverse field.

Numerical aperture (NA) can be calculated by Eq. (8) (Biswas et al. 2019).

$${\text{NA}} = \left[ {1 + \frac{{\pi A_{eff} }}{{\lambda^{2} }}} \right]^{{ - \frac{1}{2}}}$$
(8)

The nonlinear property of PCF is expressed through Eq. (9) and it is denoted by the \(\upgamma\) (Jia et al. 2018).

$$\upgamma = \frac{{2\pi n_{2} }}{{\lambda A_{eff} }}$$
(9)

where \(n_{2}\) defines the nonlinear coefficient of fused silica and \(n_{2}\) = 2.6 × 10−20 m2/W.

Confinement loss (\(C_{loss}\)) can be evaluated by Eq. (10) (Xu et al. 2017).

$$C_{loss} = \frac{40\pi }{{\ln \left( {10} \right)\lambda }}Im\left( {n_{eff} } \right)\left( {{\text{dB}}/{\text{m}}} \right)$$
(10)

where \(Im\left( {n_{eff} } \right)\) is the unreal part of the refractive index of the OAM transmission mode and λ is the wavelength of the OAM mode.

3 Results and discussion

The proposed PCF supports 26 OAM modes with low confinement loss, flat dispersion, better nonlinearity, low numerical aperture and the refractive index difference greater than \(10^{ - 4}\) between two adjacent OAM modes. Now, here is described the OAM properties such as an effective refractive index, dispersion, effective refractive index difference, effective mode area, numerical aperture, nonlinearity, and confinement loss with appropriate discussion, equation, and the necessary figure.

3.1 The effective refractive index of eigenmodes

The refractive index of fused silica is calculated by Eq. (1). Higher modes are obtained by decreasing the search around of the refractive index at a fixed wavelength.

Figure 3 shows the relationship between all selected OAM modes effective refractive index (\(n_{eff}\)) and wavelength within the range from 1.2 to 1.95 µm in the proposed PCF. The changing rate of the refractive index is downward with respect to the increment of wavelength. The effective refractive index is related to the dispersion. The flat dispersion is obtained for the small differences of the refractive index at any specific mode.

Fig. 3
figure 3

The effective refractive index of different eigenmodes with respect to the wavelength

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3.2 Dispersion

The dispersion parameter is one of the most important parameters of PCF. Dispersion is the expansion of the light pulse at transmitting time. It decreases the performance of an optical fiber. It contains material and waveguide dispersion (Zhang et al. 2018). Material dispersion has little effect on the total dispersion. So, the calculation is only for the waveguide dispersion.

The dispersion is inversely proportional to the wavelength. The \(TM_{0,1}\), \(TE_{0,1}\), \(EH_{1,1}\), \(HE_{1,1}\), \(EH_{2,1}\), \(HE_{2,1}\) and others OAM modes are exhibited in Fig. 4.

Fig. 4
figure 4

Dispersion of different OAM modes with respect to the wavelength

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The dispersion of \(TE_{0,1}\), \(EH_{1,1}\), \(HE_{1,1}\), \(EH_{2,1}\), \(HE_{2,1}\), and \(TM_{0,1}\) modes are nearly horizontal while other OAM modes show flat dispersion. The high dispersion is shown in HE6,1 and EH6,1 modes of the OAM. The proposed PCF shows the dispersion variation of different OAM modes. For example, the dispersion variation for the TE0,1 mode is 5.1421 ps/km-nm, the HE2,1 mode is 3.8684 ps/km-nm, the HE3,1 mode is 18.0606 ps/km-nm, the HE4,1 mode is 39.5561 ps/km-nm, the HE6,1 mode is 64.8221 ps/km-nm, the HE7,1 mode is 47.2286 ps/km-nm, the EH2,1 mode is 3.8566 ps/km-nm, the EH3,1 mode is 11.6646 ps/km-nm, the EH4,1 mode is 26.7569 ps/km-nm, the EH5,1 mode is 43.4371 ps/km-nm, the EH6,1 mode is 81.6529 ps/km-nm, the EH7,1 mode is 56.7001 ps/km-nm within the wavelength from 1.2 to 1.85 μm.

Especially, for the TE0,1 and HE2,1 mode, the variation of dispersion is much better than indicated values (Nandam and Shin 2018; Zhang et al. 2016; Yue et al. 2012).

The dispersion of the HE mode is larger than the EH mode. Because, the effective index of the HE mode is smaller than the EH mode in the same order of the OAM modes (Bai et al. 2018). In Fig. 4, the dispersion of HE4,1 and the EH4,1 mode is 142.6587 ps/km-nm and 121.1491 ps/km-nm respectively. So, the higher orders of the OAM modes have a higher dispersion.

The dispersion of the proposed PCF is improved by modifying the basic element of the PCF. The improvement of dispersion is also dependent on the small refractive index differences.

3.3 The effective refractive index difference of OAM modes

Figure 5 shows the relationship between the \(\Delta n_{eff}\) and wavelength. For the disparity, in Fig. 5, a blue straight line is used to indicate for the \(10^{ - 4}\). If differences of OAM modes (HE and EH) are not greater than \(10^{ - 4}\), then this mode will be coupled with one another, and also command to linearly polarized (LP). The \(\Delta n_{eff}\) is increased by increasing the wavelength. The higher order of the OAM modes will be highly effective refractive index differences (\(\Delta n_{eff}\)). The highest \(\Delta n_{eff}\) is 4.4302 × 10−2 at 1950 nm wavelength for |HE8,1 − EH6,1|.

Fig. 5
figure 5

Effective refractive index differences of different OAM modes with respect to the wavelength

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Therefore, the PCF has the benefits of decrement of the OAM mode coupling as well as ensuring the robust transmission of OAM modes.

3.4 The effective mode area

The \(A_{eff}\) of the PCF has a significant response to the nonlinearity parameter. The effective area is increased by increasing with the wavelength. The effective area of different OAM modes is shown in Fig. 6.

Fig. 6
figure 6

The effective area of different OAM modes with respect to the wavelength

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Here, \(TM_{0,1}\) and \(TE_{0,1}\) achieves the lowest and the highest effective mode area respectively. The indicated part of Fig. 6 is the zoom view of the dashed shaped area that clarifies the effective mode area of different OAM modes. Low numerical aperture and nonlinearity are obtained for the flat effective area.

3.5 Numerical aperture (NA)

NA is acknowledged by the total amount of optical power. The higher values of NA have various applications in the optical field. Besides, the low values of NA provide depth of field (DOF).

From the Eq. (8), NA is dependent on the effective area and the wavelength. The NA is a unit less parameter. It is increased by increasing the wavelength as well as decreasing the effective area.

The indicated part of Fig. 7 is the zoom view of the dashed shaped area that clarifies the numerical aperture of different OAM modes.

Fig. 7
figure 7

The numerical aperture of different OAM modes with respect to the wavelength

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The low numerical aperture of the proposed PCF is obtained for the random variations of the refractive index.

3.6 Nonlinearity

Figure 8 displays the relationship between the nonlinearity and wavelength of all selected OAM modes. Nonlinearity is inversely proportional to the effective mode area and wavelength. In the proposed PCF, all selected OAM modes are low nonlinearity within the range from 1.2 to 1.95 μm. Especially, the nonlinearity of most OAM modes are below 4 W−1/km and particularly, the lowest nonlinearity of \(TE_{0,1}\) OAM mode is 1.58 W−1/km at the wavelength 1.8 μm.

Fig. 8
figure 8

Nonlinearity of different OAM modes with respect to the wavelength

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The indicated part of Fig. 8 is the zoom view of the dashed shaped area that clarifies the nonlinearity of different OAM modes.

The proposed PCF is achieved lower nonlinearity. Because, lower nonlinearity is obtained for the large core (Hu et al. 2016).

3.7 Confinement loss (CL)

The confinement loss is caused by structural imperfection, micro bending, and intrinsic material absorption characteristics when light is transmitted over the PCF (Shi et al. 2019). The CL is determined for the different modes of the OAM. The CL for the TM0,1, TE0,1, EH1,1, HE1,1, EH2,1, HE2,1, and other modes are represented in the Fig. 9.

Fig. 9
figure 9

Confinement loss of different OAM modes with respect to the wavelength

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In the scenario of Fig. 9, the inconstancy of CL of all different OAM modes with respect to the wavelength does not display significant regularity. It is mainly adequate to the high probability of leakage to cladding for the larger wavelength. The indicated part of Fig. 9 is the zoom view of the dashed shaped area that clarifies the CL of different OAM modes from the wavelength 1.5 to 1.6 μm of step size 1 nm.

In this proposed PCF design, the average CL of the most OAM modes is varied between 4 × 10−10 and 5 × 10−6 dB/m with the wavelength from 1.2 to 1.95 μm. Such as, The HE4,1 mode has the lowest confinement loss that is numerically 3.19 × 10−10 dB/m at wavelength 1.8 μm. In Fig. 9, the confinement loss of the HE1,1 mode is 1.01 × 10−9 dB/m, the EH1,1 mode is 6.62 × 10−9 dB/m, the HE2,1 mode is 5 × 10−8 dB/m, the EH2,1 mode is 3.46 × 10−7 dB/m, the EH3,1 mode is 5.28 × 10−7 dB/m, the HE4,1 mode is 3.22 × 10−7 dB/m, the EH4,1 mode is 4.57 × 10−8 dB/m, the EH6,1 mode is 3.39 × 10−7 dB/m, the HE7,1 mode is 7.86 × 10−8 dB/m, the EH7,1 mode is 6.15 × 10−9 dB/m and the TE0,1 mode is 1.5 × 10−7 dB/m at the wavelength of 1.45 μm.

The CL is destructive for the optical communication system. The CL is generated for structural imperfection and internal absorption. The CL of the proposed PCF is decreased for periodic cladding. For all OAM modes confinement loss does not show the regularity. It is caused for a greater chance of leakage to the cladding (fused silica) for higher wavelength.

The comparison of confinement loss and dispersion variation of OAM modes with some previous article of OAM based PCF is shown in Table 1.

Table 1 The comparison chart of dispersion variations for TE, HE modes and confinement loss of the proposed article and the recent published articles
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By comparing with some OAM modes PCF parameters that are previously reported and it can be investigated that our proposed PCF gives much better optical characteristics than other OAM based PCF articles.

4 Conclusion

In this article, a novel chain-shape PCF design is proposed with low confinement loss, flat dispersion, better nonlinearity and high refractive index differences between the wide bandwidth the range of 750 nm (from 1200 to 1950 nm). The proposed fiber parameters are numerically calculated by applying FEM and PML as a boundary condition. The design can support up to 26 OAM modes. The numerical exploration shows that the effective refractive index differences of all selected OAM modes are above than \(10^{ - 4} .\) The maximum \(\Delta n_{eff}\) is 4.4302 × 10−2 at 1950 nm wavelength for HE8,1 to EH6,1 mode. The low confinement loss is 3.19 × 10−10 dB/m at optical wavelength 1800 nm. The dispersion variation for the HE2,1 mode is 3.8684 ps/km nm and 5.1421 ps/km nm for TE0,1 mode. So, the proposed PCF will be a strong candidate in stable and high capacity transmissions, encoding and MDM applications.