Recent years have seen a significant increase in research interest in high-speed short-range signal transmission across polymer optical fibers (POFs) [1–3]. The assets of POFs, such as a large core and simple connection, may offer a cost-effective solution for the in-home network. Polymethyl methacrylate (PMMA) [4, 5], polydimethylsiloxane (PDMS) [6, 7], polystyrene (PS) [8, 9], polycarbonate (PC) [10, 11], perfluorinated polymer (CYTOP^®) [12, 13], cycloolefin polymer (ZEONEX^®) [14, 15], and cycloolefin copolymer (TOPAS^®) [16, 17] are just a few of the materials used to fabricate POFs. Due to the flexibility of POF material, it is feasible to produce POFs that meet the requirements of various applications by using alternative specifications or materials. PMMA is the material that has been used to make POFs the most frequently up until this time [18–22].

The RI distribution of GI multimode POF gradually decreases from the core axis to the cladding. The POF’s bandwidth and transmission distance can both be increased using this type of RI distribution. To create GI POF, however, advanced doping techniques are required. MOF, often referred to as photonic crystal fiber, was successfully proposed in the 1990s [23]. The flexibility of the optical fiber is considerably increased by the microstructure of MOFs. Numerous relevant MOF features have been investigated by changing the microstructure [24–27]. Eijkelenborg and associates created the first PMMA mPOF in 2001 [28]. The various applications of mPOF then attracted scientific attention [29, 30]. The core and/or cladding layer of a typical mPOF design can be changed by altering the placement and/or size (d) of air holes within a concentric ring-like region, as shown in Figure 1. In Figure 1, an mPOF that mimics a GI optical fiber features a core with different-sized air holes. GI mPOF offers more latitude in changing the air-hole diameters and pitch than typical GI POF, which calls for complex doping methods. Additionally, for communication purposes, it has been found that GI mPOF has a wider bandwidth and less loss than traditional GI POF [31].

Mode coupling is primarily caused by light scattering, which takes place when transient abnormalities in multimode optical fibers transmit power from one mode to another. Modal dispersion can be decreased and transmission bandwidth increased by using mode coupling [30]. Mode coupling prevents measurements of an optical fiber’s fundamental optical characteristics, such as attenuation and bandwidth, from being made until the steady state distribution (SSD) has not yet been fully obtained at length z _s . Thus, it is essential to comprehend the fiber lengths at which an equilibrium mode distribution (EMD) and SSD are established (EMD is achieved at length L _c ). It is of particular interest to explore how wavelength affects GI mPOF’s structural and physical parameters and therefore power flow at different fiber lengths.

Up to now, there have been no commercial simulation tools available for studying the transmission characteristics of multimode MOFs. To circumvent this problem, the time-independent power flow equation (TI PFE) is numerically solved in this study to investigate the wavelength dependent light transmission in GI mPOF. We calculated the lengths for achieving the EMD and SSD for multimode GI mPOF with a solid core using launch beam distributions with different radial offsets Δr at different wavelengths λ (the low attenuation windows of POFs). As shown in Figure 1, we proposed that the air holes in the core and cladding be arranged in a grid of triangles with regular pitch Λ. The shorter the GI mPOF’s length at which EMD is attained, the sooner the functional dependency of bandwidth changes from of 1/z to of 1/z^1/2 (slower bandwidth decline) [16]. This study is the first to examine how wavelength affects power flow in GI mPOF to the best of our knowledge. The numerical results reported in this work are very useful in communication and sensory systems that use multimode GI mPOFs at different wavelengths.

The GI mPOF that was examined in this study is depicted in Figure 1. This GI mPOF is made up of six air-hole rings, numbered 1, 2, ... , 6, respectively.

A triangular lattice with pitch Λ holds the air holes in the studied polymer fiber. The parabolic RI distribution in the core is a result of the appropriate choice of the air-hole diameters in the four inner air-hole rings. The air-hole diameters in rings 5 and 6 are equal to those in ring 4 (d ₄ = d ₅ = d ₆). This system was simulated using the TI PFE.

The GI optical fibers have the following RI profile: n r , λ = n c o λ   1 − 2 Δ λ r a g 1 / 2   0 ≤ r ≤ a n c o λ   1 − 2 Δ λ 1 / 2 = n c l λ   r > a (1)

Here n _co (λ) is the core’s highest index (measured at the fiber axis), n _cl (λ) is the cladding’s index, Δ = n c o λ − n c l λ / n c o λ is the relative index difference, g is the core index exponent, and a is the core radius.

The TI PFE for GI optical fiber is [32]: ∂ P m , λ , z ∂ z = D m ∂ P m , λ , z ∂ m + D ∂ P 2 m , λ , z ∂ m 2 (2) where P m , λ , z is power in the m-th principal mode (modal group), z is the coordinate along the fiber axis, and D is a constant mode coupling coefficient. The maximum principal mode number M(λ) can be calculated as [32]: M λ = g Δ λ g + 2 a k n c o λ (3) where k = 2π/λ.

The principal mode m excited at the input fiber end is [32]: m M = Δ r a g + θ 2 2 Δ g + 2 / 2 g (4) where ∆ r is the radial offset of the launch beam and θ is the launch beam angle. In this work, Equation 2 is solved using the explicit finite difference method [32].

Light transmission was examined in a multimode GI mPOF with a solid core (Figure 1). The effective V parameter for such a fiber is given as: V = 2 π λ a e f f n 0 2 − n f s m 2 (5) where a_eff = Λ / 3   [33, 34], and n _fsm is the effective RI for various core and cladding layers, as determined by combining the Equation 5 with the effective V parameter [33, 34]: V λ Λ , d Λ = A 1 + A 2 1 + A 3 ⁡ exp A 4 λ / Λ (6)

The fitting parameters A_i (i = 1–4) are given as: A i = a i 0 + a i 1 d Λ b i 1 + a i 2 d Λ b i 2 + a i 3 d Λ b i 3 (7)

The coefficients a _i0 to a _i3 and b _i1 to b _i3 (i = 1–4) are given in our previous work [34].

We employed our approach Eq. 2 on the GI mPOF with the core radius a = 4Λ = 16 μm, where Λ = 4 μm, and the diameter of the fiber b = 1 mm. Table 1 displays the core’s refractive index n _co at different wavelengths when measured along the fiber axis. For Λ = 4 μm and air-hole diameters of the four air-hole rings in the core d ₁ = 0.6 μm, d ₂ = 0.7 μm, d ₃ = 1.3 μm, and d ₄ = 3.1 μm, the refractive indices n ₁, n ₂, n ₃, and n ₄, respectively, calculated using Eqs 6, 7 for different wavelengths, are given in Table 1. Parabolic RI distribution Eq. 1 in the core with g = 2.0, 4.5, 4.7, and 5.0 is achieved at λ = 633, 568, 522, and 476 nm, respectively. The air-hole diameter in the cladding rings 5 and 6 is d ₄ = d ₅ = d ₆, and therefore the refractive index of the cladding is n ₄ = n ₅ = n ₆ = n _cl . Table 1, for the GI mPOF under investigation, at various wavelengths, provides the maximum principal mode number M (Eq. 3). The coupling coefficient is D = 1482 1/m [29]. The typical values of D that define a standard GI POF can be used when modeling the GI mPOF due to the fact that the intensity of mode coupling in all types of POFs is correlated with the polymer core material. An analogous foundation was used to model a silica MOF [31].

As an illustration, for λ = 568 nm, Figure 2 shows the development of the normalized output modal power distribution P (m,λ,z), which depends on the length of the fiber. Eq. 2 assumes a Gaussian beam P (θ,z) launched with θ = 0^o for numerical calculations. Results are displayed for radial offsets of ∆ r = 0, 4, 8, and 12 µm. It can be seen from Figure 2A that at short fiber lengths, due to mode coupling, only lower-order modes shift their midpoints of the power distributions to zero (m = 0). With increasing fiber length, higher order modes start to couple, shifting their distributions to m = 0 (Figure 2B). The EMD is obtained by shifting the midpoints of the power distributions of all modes to m = 0 at the coupling length of L _c = 6 m (Figure 2C). Figure 2D shows that SSD is established at z≡z _s = 30 m. The lengths L _c and z _s at various wavelengths are displayed in Table 2. It can be seen that the maximum principal mode number M drops with decreasing wavelength from λ = 633 to 568 nm, which causes the lengths L _c and z _s to decrease. Shorter lengths are required to accomplish EMD and SSD due to the smaller wavelength and fewer propagating modes. The maximum principal mode number maintains the constant value M = 22 with subsequent wavelength reduction from λ = 568 to 522 and finally to 476 nm, leading to the same lengths L _c = 6 m and z _s = 30 m. It is also worth noting that increasing the parameter g with decreasing the wavelength λ, i.e., modification of the GI distribution toward a step-index distribution, does not lead to longer lengths L _c and z _s . This is a consequence of the larger influence of wavelength λ and maximum principal number M on these two characteristic fiber lengths.

It is important to notice that mode coupling behavior controls how the GI MOF bandwidth varies with length. A length less than the coupling length L _c has an inversely linear effect on the bandwidth. Beyond this equilibrium length L _C , it has a _Z ^−1/2 dependence, though. As a result, a shorter LC would lead to a more rapid transition to a slower bandwidth drop phase [30, 35, 36]. The investigated GI mPOF is predicted to experience a faster bandwidth enhancement at a wavelength of λ = 568 nm than at λ = 633 nm. Such an improvement in bandwidth is not achieved by further reducing the wavelength from λ = 568 to 522 and subsequently to 476 nm.

In contrast to the GI mPOF that we focused on in this investigation, silica MOFs have much weaker mode coupling, resulting in a length L _c between 1.45 and 1.65 km at which an EMD is achieved, and a length z _s between 3.30 and 3.80 km for the establishment of an SSD [34].

In this study, the power flow along a GI mPOF at various wavelengths is examined using the TI PFE. We have demonstrated that the lengths L _c and z_s needed to achieve an EMD and an SSD, respectively, in GI mPOF are shorter at λ = 568 nm than they are at λ = 633 nm. The lengths L _c and z _s stay constant when the wavelength decreases further from λ = 568 to 522 and then to 476 nm. Therefore, the shorter L _c causes a quicker changeover to the slower bandwidth decrease regime. As a result, a faster bandwidth enhancement in the tested GI mPOF is only anticipated to take place at wavelengths λ = 568 nm as opposed to that at λ = 633 nm. Such a bandwidth improvement is not brought about by additional wavelength reduction. The study’s findings can be used in communication and sensory systems that use multimode GI mPOFs at different wavelengths, i.e., at different low attenuation windows. Calculating the modal distribution of the GI mPOF used as a component of the optical fiber sensory system at a specific length at different wavelengths is also important. The future research on this type of optical fiber should be calculations of bandwidth at different wavelengths, which can be realized by numerically solving the time-dependent power flow equation.