1 Introduction

In the twenty-first century, communication technology is developed rapidly by photonic crystal fiber (PCF). In this era of communication technology, time-division multiplexing (TDM), dense mode-division multiplexing (DMDM), and space division multiplexing (SDM) are developed and enriched with increasing the capacity of transmission. But the SDM is a popular and attractive method to overcome the capacity limit of optical communication [1, 2]. Orbital angular momentum (OAM) is a promising candidate for the SDM [3]. It provides some important features to improve fiber communication strategy. In 1992, the OAM techniques of optics were widely recognized by Allen via Lager Gaussian beams at first [4]. The OAM beams have two ordinary characteristics of eddy wavefront and rounded field power distribution. It is expressed by \(e^{{\left( {il\varphi } \right)}}\). Where, l (….., − 3, − 2, − 1, 0, + 1, + 2, + 3,.….) is the topological accusation and φ is the angle of azimuth [5]. Recently, the OAM beams are generated by some strategy. This method uses cylindrical lenses [6], spiral plates [7], diffractive optical elements [8, 9]. The number of OAM modes is an important property for optical communication. Thus, we need to compose new structures of fiber to increase transmission capacity and support higher OAM modes.

The act of OAM transporting is sustained by circular PCF [1], dual guided ring-core PCF [2], spiral PCF [10], circular PCF with a square hole [11], etc. These are separately good for confinement loss, nonlinearity, numerical aperture (NA) and, dispersion. The robust transmission and communication capacity are increased by a higher number of OAM modes. Zhang et al. [3] proposed a C-PCF supporting 26 OAM mode. Jia et al. [5] proposed a hollow-core ring PCF supporting 38 OAM mode. Nevertheless, it was not enough for an efficient communication system. Thus, it is imperative to design a special PCF that supports more OAM mode. It also supports the robust transmission with low confinement loss, low dispersion variation, high refractive index differences, better nonlinearity, and numerical aperture.

This article proposed a circular photonic crystal fiber with a special design of air holes in the cladding region. This PCF supports up to 38 OAM modes with 1000 nm bandwidth. The highest refractive index difference is 4.899 × 10–2 at 2000 nm wavelength for |HE11,1–EH9,1|. Besides, low confinement loss, smooth dispersion curve, convenient nonlinear coefficients, and relatively better numerical aperture are obtained from the proposed PCF. Benefitting from these properties, the raised PCF could be a strong candidate for the transmission scheme.

2 Design strategy and theoretical analysis

The basic structure of the proposed PCF is shown clearly in Fig. 1 with a little physical explanation. The proposed PCF is composed of a large center air-hole and the three-layer well-shaped circular air holes in the cladding region which enhanced the performance of the PCF. In the designed PCF, the radius of the center air-hole is r3 = 4.00 μm. The large center air-hole is favorable for the high number of OAM modes. Also, the radius of the inner and outer circle is r1 = 17.5 μm and r2 = 16 μm, respectively. Thus, the width of the perfectly matched layer (PML) is 1.5 μm or 8.5714% which should maintain the manner of the boundary condition (less or equal 10%). The PML reduces the undesirable immaterial scattering. The diameter of the air holes in the cladding region is 2 μm and the pitch (distance between adjacent circles, Λ) is 2.5 μm. The air holes of the raised PCF are built by the circle through the COMSOL Multiphysics 4.3b software.

Fig. 1
figure 1

The cross-sectional view of the proposed C-PCF

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The light leakage of the PCF is reduced by the well-structured layer of air holes. The design and simulation of the proposed PCF are done by the COMSOL Multiphysics software. All numerical analyses and simulated calculations are performed by the finite element method (FEM). By performing the simulation on the proposed PCF, some PCF properties are calculated. These are an effective refractive index difference (ERID), dispersion variation, effective area, confinement loss, nonlinearity, and numerical aperture.

The fused silica is used as the background material. The refractive index (RI) of fused silica is 1.446 at the wavelength of 1.55 μm [12]. The flat dispersion is influenced by the lower effective refractive index. Thus, the fused silica is the best for this parameter. The RI of the background material is evaluated by the Sellmeier equation Eq. (1) [13]:

$$ n^{2} \left( \lambda \right) = 1 + \mathop \sum \limits_{i = 1}^{3} \frac{{A_{i} \lambda^{2} }}{{\lambda^{2} - \lambda_{i}^{2} }}, $$
(1)

where \( \lambda_{i}^{2}\) and \(A_{i}\) are the Sellmeier coefficient and n \(\left( \lambda \right)\) is the RI that varies with the wavelength.

The ERID of the OAM mode can be calculated through Eq. (2) and it is symbolized by \(\Delta n_{{{\text{eff}}}}\):

$$ \Delta n_{{{\text{eff}}}} = \left| {n_{{{\text{eff}}_{{{\text{HE}}_{l + 1,1} }} }} - n_{{{\text{eff}}_{{{\text{EH}}_{l - 1,1} }} }} } \right| > 10^{ - 4} . $$
(2)

The waveguide dispersion is expressed by Eq. (3) [10]:

$$ D = - \frac{1}{\lambda c}\left( {\lambda^{2} \frac{{d^{2} {\text{Re}} \left[ {n_{{{\text{eff}}}} } \right]}}{{{\text{d}}\lambda^{2} }}} \right), $$
(3)

where \(\lambda\) is the wavelength, c is the velocity of light in the free space (\(3 \times 10^{8}\) m s−1) and \({\text{Re}} \left[ {n_{{{\text{eff}}}} } \right]\) is the real part of the effective RI.

The confinement loss (\(C_{{{\text{loss}}}}\)) is induced by internal material absorption of light and waveguide geometry. The \(C_{{{\text{loss}}}}\) can be expressed by Eq. (4) [11]:

$$ C_{{{\text{loss}}}} = \frac{40\pi }{{\ln \left( {10} \right)\lambda }}{\text{Im}} \left( {n_{{{\text{eff}}}} } \right)\left( {{\text{dB}}/{\text{m}}} \right), $$
(4)

where λ is the operating wavelength and \({\text{Im}} \left( {n_{{{\text{eff}}}} } \right)\) is the imaginary part of the RI.

The effective area of the OAM mode is characterized by \(A_{{{\text{eff}}}}\) and can be figured by the mathematical Eq. (5) [14]:

$$ A_{{{\text{eff}}}} = \frac{{\left\{ {\left( {\iint {\left| {E\left( {x,y} \right)} \right|^{2} {\text{d}}x{\text{d}}y}} \right)} \right\}^{2} }}{{\iint {\left| {E\left( {x,y} \right)} \right|^{4} {\text{d}}x{\text{d}}y}}}, $$
(5)

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

Numerical aperture is the entire capacity of gathering beam in optical fiber. It is evaluated by Eq. (6) [14]:

$$ {\text{NA}} = \frac{1}{{\sqrt {1 + \frac{{\pi A_{{{\text{eff}}}} }}{{\lambda^{2} }}} }}. $$
(6)

The nonlinearity of the PCF is symbolized by the γ and it is expressed by the Eq. (7) [15]:

$$ \gamma = \frac{{2\pi n_{2} }}{{\lambda A_{{{\text{eff}}}} }}, $$
(7)

where \(n_{2}\) = 2.3 × 10–20 m2 W−1 [16] denotes the nonlinear index of fused silica.

3 Results, discussion and analysis

In this section, we have widely described some OAM parameters such as the supported OAM modes for the proposed PCF, ERID, dispersion profile, confinement loss, effective refractive area, numerical aperture, and nonlinearity with proper diagram, equation, and discussion.

3.1 The supported OAM modes for the proposed PCF

The proposed PCF supports up to 38 OAM modes with the wavelength ranging from 1000 to 2000 nm. In addition, a center air-hole area, air-hole pattern in the cladding region also effects to produces the OAM mode. The large center air-hole, ring number, and well-organized air holes in the cladding of the PCF help to increase the OAM mode number because it acquired adequate space to produces the OAM mode as well as it reduces the light leakage in the cladding region. Besides, from the experiment, the raised PCF supports up to 38 OAM modes within the 1000 to 2000 nm wavelength. The intensity and phase are used in multiplexing. It also ensures the robust transmission of data. The number of OAM modes is calculated through Eqs. (8) and (9) [15]:

$$ \left\{ {\begin{array}{*{20}l} {{\text{OAM}}_{{ \pm l,1}}^{ \pm } = {\text{HE}}_{{l + 1,1}}^{{{\text{even}}}} \pm j{\text{HE}}_{{l + 1,1}}^{{{\text{odd}}}} } \hfill \\ {{\text{OAM}}_{{ \pm l,1}}^{ \mp } = {\text{EH}}_{{l - 1,1}}^{{{\text{even}}}} \pm j{\text{EH}}_{{l - 1,1}}^{{{\text{odd}}}} } \hfill \\ \end{array} } \right\}\quad ;(l > 1) $$
(8)
$$ \left\{ {\begin{array}{*{20}l} {{\text{OAM}}_{{ \pm 1,1}}^{ \pm } = {\text{HE}}_{{2,1}}^{{{\text{even}}}} \pm j{\text{HE}}_{{2,1}}^{{{\text{odd}}}} } \hfill \\ {{\text{OAM}}_{{ \pm 1,1}}^{ \mp } = {\text{TM}}_{{0,1}} \pm j{\text{TE}}_{{0,1}} } \hfill \\ \end{array} } \right\}\quad ;(l = 1), $$
(9)

where l is the topological charge and radial order is equal to 1 for the OAM mode calculation. Accidental degeneracies are occurred by the higher-order [17]. It is a challenge for the multiplexing scheme. All of the OAM modes for the designed PCF are given bellow: \({\text{OAM}}_{ \pm 1,1 }^{ \pm } ({\text{HE}}_{2,1} )\), \({\text{OAM}}_{ \pm 2,1 }^{ \pm } ({\text{HE}}_{3,1} ,{\text{EH}}_{1,1} )\), \({\text{OAM}}_{ \pm 3,1 }^{ \pm } ({\text{HE}}_{4,1} ,{\text{EH}}_{2,1} )\), \({\text{OAM}}_{ \pm 4,1 }^{ \pm } ({\text{HE}}_{5,1} ,{\text{EH}}_{3,1} )\), \({\text{OAM}}_{ \pm 5,1 }^{ \pm } ({\text{HE}}_{6,1} ,{\text{EH}}_{4,1} )\), \({\text{OAM}}_{ \pm 6,1 }^{ \pm } ({\text{HE}}_{7,1} ,{\text{EH}}_{5,1} )\), \({\text{OAM}}_{ \pm 7,1 }^{ \pm } ({\text{HE}}_{8,1} ,{\text{EH}}_{6,1} )\), \({\text{OAM}}_{ \pm 8,1 }^{ \pm } ({\text{HE}}_{9,1} ,{\text{EH}}_{7,1} )\), \({\text{OAM}}_{ \pm 9,1 }^{ \pm } ({\text{HE}}_{10,1} ,{\text{EH}}_{8,1} )\), \({\text{OAM}}_{ \pm 10,1 }^{ \pm } ({\text{HE}}_{11,1} ,{\text{EH}}_{9,1} )\).

The intensity of the OAM mode is obtained by the electromagnetic wave of x component and phase for including argument function. Some of them are shown in Fig. 2. The intensity and phase of the proposed PCF are achieved by choosing a rainbow color from the color table in the simulation software (COMSOL Multiphysics). We know that the effective refractive index (RI) of HE mode is lower than EH modes for the same order OAM modes at the same wavelength. So, discrimination between HE and EH modes appeared at the time of simulation of the raised PCF in the COMSOL Multiphysics software. Since the refractive index (RI) of the background material (silica) is higher than the center large air-hole, therefore OAM modes will be propagated between the center air-hole and air-holes pattern of the cladding region. Therefore, the gap between center air-hole and air-holes pattern of the cladding region is also called the light-guiding area or core of the raised PCF. The same phenomenon was displayed through recently published articles [1, 14, 15].

Fig. 2
figure 2

a–i The intensity of some OAM modes and j–o phase of \({\text{HE}}_{2,1}\), \({\text{HE}}_{3,1}\), \({\text{EH}}_{4,1}\), \({\text{HE}}_{5,1}\),\({\text{EH}}_{6,1}\), and \({\text{EH}}_{8,1}\) OAM modes

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The intensity of Fig. 2a–g shows the different OAM modes with two different colors. The OAM as a multiplexing system in digital communication is transmitted maximum (1) or minimum (0) types of multiple data at a time. For intensity, two colors are indicator and phase separator of two types of data. Two colors are separated by an angle. This angle is changed by varying the number of OAM modes. For example, in mode of EH6,1, the angle between one color to another color is 30° because of 12 (6 + 6) data are transmitted. For mode of HE8,1, the angle between one color to another color is 22.5° because of 16 (8 + 8) data are transmitted here. Besides, the phase distribution is separated by one color. This color is not same as the color of intensity; it is a random selection from color table at the time of simulation. Both TE0,1 and TM0,1 given in Fig. 2i, j are the types of two vector modes that approve us to construct OAM modes with topological charge l = 1. The red arrows in Fig. 2 indicate the vector notations of the OAM mode [10].

3.2 Effective refractive index differences (ERIDs)

The ERID is the absolute value of differences between HE and EH mode. It is calculated mathematically through Eq. (2). It is shown in Fig. 3 and the pinkie straight line \(10^{ - 4}\) is used as a reference. In Fig. 3, the ERID of any OAM is greater than \(10^{ - 4}\) over the wavelength from 1000 to 2000 nm. It can be achieved for a big index difference between silica cladding and hollow air-hole. Thus, each mode propagates independently in an effective way and it also prevents a couple of linear polarization (LP) of the contiguous vector modes. Here, it is also observed that \(\Delta n_{{{\text{eff}}}}\) is higher for the higher group of OAM modes except for OAM3,1 mode and as well as \(\Delta n_{{{\text{eff}}}}\) increases with raising the wavelength.

Fig. 3
figure 3

ERID of the different OAM modes versus operating wavelength

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3.3 Dispersion profile

The dispersion and ERID are affected by the real value of RI. There are two types of dispersion: material dispersion and waveguide dispersion. The material dispersion has little effect on total dispersion. So, we focused only on the waveguide dispersion. The dispersion is calculated according to Eq. (3) and it is displayed in Fig. 4. It is noted that the dispersion of the operating last two-wavelength is discarded. It has occurred for the second-order derivation as the dispersion equation. In the simulation of proposed PCF, it was verified that the effective refractive index is decreased by increasing operating wavelength and the identity relation between wavelength and dispersion is shown in Eq. (3). In Fig. 4a, b it is precisely displayed that dispersion is increased by raising the wavelength and the lower order OAM modes have a lower dispersion. Thus, the smooth dispersion variation is obtained for the use of fused silica as a cladding region because the fused silica has a small RI.

Fig. 4
figure 4

Dispersion of different a HE modes, b EH, TE, and TM modes as a function of wavelength

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The dispersion variation of HE1,1, HE2,1, HE3,1, TM0,1, TE0,1, EH1,1, EH2,1, and EH3,1 are 23.049 ps/km nm, 18.9937 ps/km nm, 28.2265 ps/km nm, 15.2512 ps/km nm, 0.916 ps/km nm, 20.0409 ps/km nm, 10.8982 ps/km nm, and 22.1766 ps/km nm, respectively. As far we know, the dispersion variation 0.916 ps/km nm for TE0,1 is the lower from the previous OAM related articles. Besides, the lower groups of the OAM modes obtain flat dispersion. Moreover, we also noticed in Fig. 4a, b that the dispersion variations of the HE mode is larger than the EH mode at the same order of OAM modes. Nonetheless, the higher groups of OAM modes are insecure in optical transmission. In this manner, the flat dispersion of a lower group of OAM modes is a strong candidate for stable fiber communication.

3.4 Confinement loss (CL)

Fiber losses will be produced at the time of transmitting light due to the internal absorption element properties, micro bending, and PCF design imperfection [18]. The confinement losses are exterminatory for optical communication. It is preceded by periodic cladding and well-organized air holes. The proposed PCF is constructed with a regular shape of cladding with the special size of air holes. Thus, the mediocre CL is obtained for the PCF. In the proposed PCF, the average CL is gained about \(10^{ - 7}\) dB/m for wavelength ranging from 1000 to 2000 nm and the lowest CL is 1.534 × 10–9 dB/m at operating wavelength 1400 nm for the EH11,1 mode.

The CL is figured out by Eq. (4). It is directly correlated to the imaginary part of effective refractive indices and the obtained CL is plotted in Fig. 5. In the raised PCF, the CL variation of the different OAM modes is not linear compared to other parameters such as ERID, dispersion, numerical aperture, and nonlinearity, etc. because the values of CL is changed for a large wavelength difference (which is 0.1 μm in Fig. 5). For this reason, the graph displays larger zigzag trends, and also the same phenomenon was shown by the recently published articles [14, 15, 19]. The small confinement losses ensure the robust and efficient transmission of light in the fiber.

Fig. 5
figure 5

Confinement loss of different a HE modes, b EH modes with respect to the wavelength

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3.5 Effective refractive area and numerical aperture

The effective refractive area of the proposed PCF is mathematically calculated by Eq. (5). The calculation of the effective area is done in the simulation of the PCF. The effective area as a function of wavelength ranging from 1000 to 2000 nm is plotted in Fig. 6. Moreover, Fig. 6a shows all the HE modes and Fig. 6b shows all the EH modes and other Eigenmodes.

Fig. 6
figure 6

The effective area of different a HE modes, b EH, TE, TM modes with respect to the wavelength

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Another optical parameter, numerical aperture (NA) is reclined in an effective area. The NA is specialized by the total amount of optical capacity. It is a unit-less parameter of PCF. The NA is calculated through Eq. (6) The NA for all OAM modes is depicted in Fig. 7. According to the equation and figure, the NA is increased by increasing the wavelength. By comparing Figs. 6 and 7, it is found out that the high profile of an effective area shows the low NA because both are inversely correlated. The proposed PCF shows the highest effective area for TM0,1 mode and this area, the designed PCF has obtained the lowest NA. The fused silica is used as the background material of the proposed PCF. Silica is a material with a low refractive index. Thus, the PCF supports a low numerical aperture.

Fig. 7
figure 7

The numerical aperture of different a HE modes, b EH, TE, TM modes with respect to the wavelength

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3.6 Nonlinearity

Nonlinearity is one of the most important parameters of the PCF. The nonlinearity of a PCF depends on the core size of the design. The nonlinearity is established a reverse relation with the core size and effective area of the PCF. The proposed PCF has a larger center air-hole. The core size of the raised PCF provides a lower nonlinearity. The nonlinearity is mathematically calculated through Eq. (7) The nonlinearity of the proposed PCF is graphically shown in Fig. 8. The lowest nonlinearity of the PCF is 0.997 W−1 km−1 at wavelength 2000 nm for TM0,1 mode. It can be achieved for the highest effective area 7.25 × 10–11 m2 in the same situation as this mode. Moreover, the low nonlinear coefficient is advantageous for transmitting the OAM modes into optical fibers.

Fig. 8
figure 8

Nonlinearity of different a HE modes, b EH, TE, and TM modes concerning the wavelength

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Table 1 shows the different parameters of PCF with a comparison. This table shows the performance between previous OAM related published articles and our proposed PCF. Thus, the proposed PCF is a strong candidate for OAM mode transmission and other related optical communications.

Table 1 A comparison chart of structure type, bandwidth, dispersion variation, and confinement losses with the raised PCF and the recently published works
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4 Conclusion

This article introduces a hollow larger center air-hole C-PCF with a new well-organized shape of air holes in the cladding region. Silica is used as a background element of this PCF. It can transmit data within a greater bandwidth (1000 nm) and supports up to 38 OAM modes with ensuring efficient transmission. All the propagation characteristics of the designed PCF are achieved by applying the FEM method and PML as a boundary condition through the COMSOL Multiphysics 4.3b simulator. It also prevents the linear polarization of adjacent vector modes. Additionally, this PCF has produced some special OAM features. In concise, the lowest dispersion variation and confinement loss is 0.916 ps/km nm for TE0,1 mode and 1.534 × 10–9 dB/m for EH11,1 mode at the operating wavelength of 1400 nm, respectively. Therefore, from the above analysis, this PCF ensures robust OAM mode transmission, and also other relevant optical communications with high capacity.