Introduction

In order to facilitate a wide range of services, including the Internet of Things, augmented reality, virtual reality, video streaming, and online gaming with very high data and low latency, fifth-generation (5G) and sixth-generation (6G) communication are needed. Nevertheless, lower-band communication finds it difficult to accommodate the high data rate transmission due to the already congested current band. POFs have a variety of applications across different industries due to their unique properties, such as short-range data communication, illumination, sensing, medical imaging, automotive applications, high-power delivery, consumer electronics and safety equipment. Overall, the use of POFs continues to grow as technology advances and new applications are discovered. Their flexibility, durability, and resistance to electromagnetic interference make them an attractive choice for various industries. The advantage of microstructured plastic optical fiber (mPOF) over traditional POF is the ability to adjust air-hole diameters and pitches with greater flexibility, without the need for complex doping procedures as with regular POF.

Optical fiber bandwidth can be increased by different multiplexing, more sophisticated techniques, and appropriate refractive index design1,2,3,4,5. Because of this, it is critical to ascertain the circumstances in which an optical fiber has the largest bandwidth, i.e., to develop models that can predict and ascertain how fiber design characteristics affect bandwidth. Previous work on modeling and experimental investigation of the transmission characteristics of various multimode optical fiber types by a number of research teams demonstrated that the transmission characteristics of the fibers can be greatly enhanced by appropriately adjusting the fiber’s parameters6,7,8,9,10,11,12,13,14,15,16,17,18. The obtained results demonstrated that singly-clad (SC) graded-index (GI) POFs have a substantially lower modal dispersion than SC step-index (SI) POFs because of their gradually decreasing core refractive index with a radial distance from the fiber axis19. One anticipates that the proposed double clad W-type GI mPOF will perform better in terms of fiber bandwidth than all other currently available conventional POFs and mPOFs as the modal dispersion of double-clad W-type POF is less than that of SC POF20. Moreover, an advantage of a mPOF is its increased flexibility in modifying its geometric parameters, in contrast to conventional POF, which is based on core and cladding(s) with varying doping levels19.

The optical power distribution launched into a multimode optical fiber evolves gradually along the fiber length due to mode coupling effects. This process alters the expected beam characteristics, including the far-field radiation pattern. The far-field pattern of a fiber depends on the initial launch conditions, the fiber’s physical properties, and its length. When light is launched at a specific angle or radial offset relative to the fiber axis, it produces a well-defined ring-shaped radiation pattern at the output of short fibers. However, due to mode coupling, the edges of this ring become increasingly blurred with longer fiber lengths. Up to a certain point, known as the “coupling length” Lc​, this blurring intensifies with distance, and the ring pattern gradually transitions Lc, the fiber reaches an equilibrium mode distribution (EMD), where the output no longer exhibits ring patterns regardless of the launch angle. Although the disk-shaped distribution may still depend on the launch conditions, EMD signifies that mode coupling is nearly complete. At a distance zs>Lc from the fiber input, the disk patterns resulting from various launch angles converge into a single, uniform distribution across the fiber cross-section. This state, known as steady-state distribution (SSD), represents the full completion of mode coupling, rendering the output light distribution independent of the input launch conditions.”

There is no commercial modeling tool available for researching highly multimode microstructured optical fibers (MOFs) transmission characteristics. This study investigates the transmission characteristics of W-type GI mPOF by numerically solving the time-independent power flow equation (TI PFE) in order to get over this issue. More precisely, we looked into how different configurations of W-type mPOFs with GI distribution of the core affect bandwidth as well as the length at which an EMD and SSD are obtained. We proposed that the air holes in this optical fiber’s core and cladding be placed in a grid of triangles with regular pitch Λ (see Fig. 1).

Fig. 1
figure 1

(a) A quarter cross-section of the multimode W-type GI mPOF. The pitch Λ determines the position of air-holes with diameters d1, d2, d3, dp and dq in a triangular lattice, (b) The solid black line represents the RI distribution of W-type GI mPOF, for g = 2.0, at λ = 645 nm.

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Time-independent PFE

The TI PFE for multimode GI fibers is given as13:

$$\frac{{\partial P(m,\lambda ,z)}}{{\partial z}}= - \alpha (m,\lambda )P(m,\lambda ,z)+\frac{1}{m}\frac{\partial }{{\partial m}}\left( {md(m,\lambda )\frac{{\partial P(m,\lambda ,z)}}{{\partial m}}} \right)$$
(1)

where P(m,\(\lambda ,\)z) is power in the m-th principal mode (modal group), z is coordinate along the fiber axis, \(d(m,\lambda )\) is mode coupling coefficient (assumed constant D)14, \(\alpha (m,\lambda )={\alpha _0}(\lambda )+{\alpha _{\text{d}}}(m,\lambda )\) is modal attenuation, where \({\alpha _0}\) represents conventional losses (absorption and scattering). The term α0 leads only to a multiplier exp(−α0z) in the solution and can be neglected.

The RI profile of W-type optical fibers with GI distribution of the core (see Fig. 1(b)) can be written as follows:

$$n(r,\lambda )=\left\{ \begin{gathered} {n_{co}}(\lambda ){\left[ {1 - {\Delta _q}(\lambda ){{\left( {\frac{r}{a}} \right)}^g}} \right]^{1/2}}\,\,\,\,\,(0 \leqslant r \leqslant a) \hfill \\ {n_q}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(a<r \leqslant a+\delta a) \hfill \\ {n_p}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(a+\delta a<r \leqslant \frac{b}{2}) \hfill \\ \end{gathered} \right\}$$
(2)

where g is core index exponent, a is core radius, δa is intermediate layer width, b is fiber diameter, nco(λ) is the maximum RI of the core (measured at the fiber axis), nq and np are RI of intermediate layer and cladding, respectively, \({\Delta _{\text{q}}}=({n_{co}} - {n_{\text{q}}})/{n_{co}}\) is relative index difference between core and intermediate layer. The maximum principal mode number M(λ) given as21:

$$M(\lambda ) \equiv {m_q}=\sqrt {\frac{{g{\Delta _q}(\lambda )}}{{g+2}}} ak{n_{co}}(\lambda )$$
(3)

where \(\,k=2\pi /\lambda\) is the free-space wave number. Gaussian launch-beam distribution (LBD) P0(θ,\(\lambda\),z = 0) can be transformed into P0(m,\(\lambda\),z = 0) (one needs P0(m,\(\lambda\),z = 0) to numerically solve the PFE (1)), using the following relationship22:

$$\frac{m}{{M(\lambda )}}={\left[ {{{\left( {\frac{{{\:\varDelta\:r}}}{a}} \right)}^g}+\frac{{{\theta^2}}}{{2{\Delta _q}(\lambda )}}} \right]^{(g+2)/2g}}$$
(4)

where Δ r is radial distance (radial offset) between the position of the maximum of the LBD and the core center, θ is the propagation angle with respect to the core axis and \({\Delta _q}=({n_{co}} - {n_q})/{n_{co}}\) is the relative RI difference between the core and intermediate layer. Attenuation constant of leaky modes \({m_p}<m<{m_q}\) is given as23:

$${\alpha _{\text{L}}}(m,\lambda )=\frac{{4\sqrt {2{\Delta _q}} {{\left( {{{\left( {\frac{m}{{{m_q}}}} \right)}^{\frac{{2g}}{{g+2}}}} - {{\left( {\frac{{{m_p}}}{{{m_q}}}} \right)}^{\frac{{2g}}{{g+2}}}}} \right)}^{1/2}}}}{{a{{\left( {1 - 2{\Delta _q}{{\left( {\frac{m}{{{m_q}}}} \right)}^{\frac{{2g}}{{g+2}}}}} \right)}^{1/2}}}}\frac{{{{\left( {\frac{m}{{{m_q}}}} \right)}^{\frac{{2g}}{{g+2}}}}\left( {1 - {{\left( {\frac{m}{{{m_q}}}} \right)}^{\frac{{2g}}{{g+2}}}}} \right)}}{{\left( {1 - \frac{{{\Delta _p}}}{{{\Delta _q}}}{{\left( {\frac{{{m_p}}}{{{m_q}}}} \right)}^{\frac{{2g}}{{g+2}}}}} \right)}}\exp \left[ { - 2\delta a{n_{co}}k{{\left( {2{\Delta _q}\left( {1 - {{\left( {\frac{m}{{{m_q}}}} \right)}^{\frac{{2g}}{{g+2}}}}} \right)} \right)}^{1/2}}} \right]$$
(5)

where mp is given as \({m_p}=\sqrt {g{\Delta _p}(\lambda )/(g+2)} ak{n_{co}}(\lambda )\). The modal attenuation in a W-type GI optical fiber can be expressed as:

$${\alpha _d}(m,\lambda )=\left\{ \begin{gathered} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m \leqslant {m_{\text{p}}} \hfill \\ {\alpha _{\text{L}}}(m,\lambda )\,\,\,\,\,\,\,\,\,{m_{\text{p}}}<m<{m_{\text{q}}}\,\, \hfill \\ \infty \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m \geqslant {m_{\text{q}}} \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\,$$
(6)

Because of the strong dependence of αL(m) on the intermediate layer width δa (Eq. 5), it is expected that characteristics of a W-type GI mPOF also strongly depend on δa9,24,25. We solved Eq. (1) using the explicit finite difference method (EFDM). As first in the authors’ best knowledge, numerical solution of the TI PFE (1) is reported in this work for investigation of the transmission along a newly designed W-type GI mPOF.

Time-dependent PFE

Time-dependent (TD) PFE for multimode optical fibers with GI core distribution is:

$$\begin{gathered} \frac{{\partial P(m,\lambda ,z,\omega )}}{{\partial z}}+j\omega \tau (m,\lambda )P(m,\lambda ,z,\omega )= - \alpha (m,\lambda )P(m,\lambda ,z,\omega ) \hfill \\ +\frac{{\partial P(m,\lambda ,z,\omega )}}{{\partial m}}\frac{{\partial d(m,\lambda )}}{{\partial m}}+d(m,\lambda )\frac{1}{m}\frac{{\partial P(m,\lambda ,z,\omega )}}{{\partial m}}+d(m,\lambda )\frac{{\partial {P^2}(m,\lambda ,z,\omega )}}{{\partial {m^2}}} \hfill \\ \end{gathered}$$
(7)

where P(m,λ,z,ω) is power in the m-the principal mode, z is coordinate along the fiber axis from the input fiber end, α(m,λ) is attenuation of the mode m (Eq. (6)), d(m,λ) is the coupling coefficient of the mode m (assumed constant D), ω=2πf is the baseband angular frequency, τ(m,λ) is delay time per unit length of mode m, given as:

$$\tau (m,\lambda ) \cong \frac{{{n_{co}}(\lambda )}}{c}\left[ {1+\frac{{g - 2}}{{g+2}}{\Delta _q}(\lambda ){{\left( {\frac{m}{{M(\lambda )}}} \right)}^{2g/(g+2)}}+\frac{1}{2}\frac{{3g-2}}{{g+2}}{\Delta _q}{{(\lambda )}^2}{{\left( {\frac{m}{{M(\lambda )}}} \right)}^{4g/(g+2)}}} \right]$$
(8)

where c is the free-space velocity of light. We solved Eq. (8) using the EFDM.

Numerical results and discussion

Light transmission was examined in a multimode W-type GI mPOF (Fig. 1), for which the effective V parameter can be written as:

$$V=\frac{{2\pi }}{\lambda }{a_{eff}}\sqrt {n_{co}^{2} - n_{{fsm}}^{2}}$$
(9)

where aeff = \(\:\Lambda/\sqrt{3\:}\) is the effective core radius26, and nfsm is the effective RI for various core and cladding layers, as determined by combining the Eq. (9) with the effective V parameter26:

$$V\left( {\frac{\lambda }{\Lambda },\frac{d}{\Lambda }} \right)={A_1}+\frac{{{A_2}}}{{1+{A_3}\exp \left( {{A_4}\lambda /\Lambda } \right)}}$$
(10)

The fitting parameters Ai (i = 1 to 4) are given as:

$${A_i}={a_{i0}}+{a_{i1}}{\left( {\frac{d}{\Lambda }} \right)^{{b_{i1}}}}+{a_{i2}}{\left( {\frac{d}{\Lambda }} \right)^{{b_{i2}}}}+{a_{i3}}{\left( {\frac{d}{\Lambda }} \right)^{{b_{i3}}}}$$
(11)

The coefficients ai0 to ai3 and bi1 to bi3 (i = 1 to 4) are given in our previous work16.

The characteristics of W-type GI mPOF analyzed in this work are: The core radius of the fiber is a = 300 μm, the RI of the core at the fiber axis was nco = 1.4920 at wavelength λ = 645 nm. For pitch Λ = 3 μm and air-hole diameters of the five air-hole rings d1 = 0.3 μm, d2 = 0.5 μm, d3 = 1.0 μm, dp=1.0 μm and dq=1.5 μm, using Eqs. (10) and (11), we obtain the refractive indices n1 = 1.4907, n2 = 1.4887, n3 = 1.4884 and np =1.4844 and nq=1.4757, respectively (material dispersion has also been taken into account). For the investigated fiber, the maximum principal mode number is M = 322, g = 2.0 and \(\:{\varDelta\:}_{q}=({n}_{co}-{n}_{q})/{n}_{co}\)=0.0109. The normalized intermediate layer widths δ = 0.008 and δ = 0.024 were employed (actual width is δ·a [µm]). The constant coupling coefficient \(D=1482\,\,\,1{\text{/m}}\)14 and wavelength λ = 645 nm were used in the calculations. By numerical solving the TI PFE (1), we calculated the length Lc at which the EMD is achieved and length zs at which the SSD is established in W-type GI mPOF. Figure 2 shows the output modal power distribution in W-type GI mPOF with the width of the intermediate layer δ = 0.008 at different lengths. In the numerical calculations, a Gaussian LBD P(θ,z) is assumed to be launched with \( <\theta >\) = 0° and full width at half maximum (FWHM)0 = 3°. Results are shown for five different radial offsets \(\:\varDelta\:r\) = 0, 100, 150, 200 and 250 μm. As additional illustration, Fig. 3 shows calculated 3-dim output modal power distribution with radial offsets \(\:\varDelta\:r=0\:\mu\:\)m at different fiber lengths. One can see from Fig. 2 that for δ = 0.008, the EMD and SSD in W-type GI mPOF are achieved at fiber length Lc=6 m and zs=15 m, respectively. For δ = 0.024, these lengths are Lc=7 m and zs=18 m, respectively (Table 1). For comparison, the EMD in SC GI POF experimentally investigated in our earlier work was achieved at Lc = 31 m (for SC GI POF, M = 656 and \(D=1482\,\,\,1{\text{/m}}\))14. The length Lc is lowered in the case of W-type GI mPOF due to a decrease in the number of higher guided modes engaged in the coupling process. With increasing the width of the inner cladding of W-type GI mPOF, leaky mode losses reduce (number of guided modes increases), leading to longer lengths Lc and zs in the case δ = 0.024.

Fig. 2
figure 2

Calculated output modal power distribution P(m) in W-type GI mPOF with δ = 0.008, over a range of radial offsets \(\:\varDelta\:r\), at fiber lengths (a) z = 0.1 m, (b) z = 1 m, (c) z = 3 m, (d) z = 6 m and (e) z = 15 m, for Gaussian launch beam distribution with \(( \theta )\) = 0° and (FWHM)0 = 3°.

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Fig. 3
figure 3

Calculated 3-dim output modal power distribution P(m) in W-type GI mPOF with δ = 0.008, for Gaussian launch beam distribution with radial offsets \(\:\varDelta\:r=0\:\mu\:\)m, \(< \theta >\) = 0° and (FWHM)0 = 3°, at different fiber lengths.

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Table 1 Coupling length Lc (for EMD) and length zs (for SSD) in W-type GI mPOF with different widths of the intermediate layer δ, for the launch beam distribution with (FWHM)0 = 3°.
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We determined the bandwidth for the investigated W-type GI mPOF using the TD PFE. Figure 4 shows that, for short lengths, bandwidth drops linearly before switching to the 1/z1/2 functional dependence. With thinner intermediary layers, this changeover and the ensuing EMD happen at shorter fiber lengths. The faster bandwidth increase happens, the shorter Lc. It is also important to note that the larger leaky mode losses (fewer higher guided modes) in the case of the thinner intermediary layer leads to smaller modal dispersion and larger bandwidth. When compared to the bandwidth of the conventional SC GI POF (Fig. 5)18, the W-type GI mPOF examined in this work has a substantially larger bandwidth. Namely, in the proposed W-type GI mPOF, higher guided modes are filtered out, resulting in diminished modal dispersion. This is a result of the proposed W-type GI mPOF in which a higher guided modes are attenuated, leading to reduced modal dispersion. As shown in Fig. 4, at short fiber lengths, the bandwidth decreases with increasing radial offset. This is due to greater modal dispersion resulting from the initial excitation of higher-order guided modes in the case of larger radial offset. However, this effect diminishes over longer fiber lengths as a result of mode coupling. Such newly designed W-type GI mPOF is a promising candidate for a high bandwidth performance of short distance communication links.

Fig. 4
figure 4

Bandwidth versus transmission length in the W-type GI mPOF with (a) δ = 0.008 and (b) δ = 0.024, for a various radial offsets \(\:\varDelta\:r\) and Gaussian launch beam distribution with \(< \theta >\) = 0° and (FWHM)0 = 3°.

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Fig. 5
figure 5

Measured bandwidth versus transmission length in the conventional SC GI POF for a various radial offsets \(\:\varDelta\:r\) and Gaussian launch beam distribution with \( <\theta >\) = 0° and (FWHM)0 = 3° (lines are drawn to guide the eye)18.

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It is important to note here that, in general, mode coupling arises due to random perturbations and the propagation constant difference (Δβ) between modes, with Δβ determining the coupling strength. The smaller Δβ, the stronger the coupling. More modes mean that the effective refractive index difference (Δβ) of adjacent modes may decrease, resulting in enhanced coupling. In fact, for multimode fiber, the average coupling length decreases as the number of modes increases. Therefore, only by reducing the number of modes can the length (perturbation period) lc ∼1/ (∆β) be increased. For example, in the weak coupling regime (few-mode fiber), lc can reach several kilometers. In the strong coupling regime (multimode fiber), lc may drop to the centimeter level. For space division multiplexing (SDM) systems, there is a trade-off between transmission capacity and crosstalk. A high number of modes can improve the spatial multiplexing rate, but shortening lc will increase the crosstalk between modes. Similarly, increasing lc will reduce the crosstalk between modes, but will reduce the spatial multiplexing rate. Low crosstalk requires L≪lc (L is the transmission distance). This has been demonstrated in our previous works for POFs, where L is only few meters for three spatially multiplexed channels and 10 m for two spatially multiplexed channels27,28. Otherwise digital signal processing compensation is required, which increases the complexity. Another, and potentially superior, SDM solution is the design of a multicore optical fiber, but this requires careful consideration of inter-core crosstalk29,30,31.

Conclusion

A novel multimode W-type GI mPOF was proposed, and we presented a methodology for examining its transmission. Our findings indicate that the coupling length Lc required to generate an EMD in W-type GI mPOF is less than the length found in the experiment with conventional SC GI POF. The reduction in the number of higher guided modes involved in the coupling process, as a result of leaky mode losses, lowers the length Lc. In W-type GI mPOF, fewer higher guided modes result in a lower modal dispersion and a higher bandwidth. Thinner intermediate layers are proven to further boost bandwidth. This study’s analysis of the W-type GI mPOF indicates a much higher bandwidth when compared to commercially available conventional POFs. Bandwidth decreases as the radial offset increases, primarily due to increased modal dispersion caused by the initial excitation of higher-order guided modes. However, this effect becomes negligible over longer fiber lengths due to mode coupling. Consequently, a large gain in bandwidth performance in short-distance communication lines may be possible with such a W-type GI mPOF architecture.