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Abstract
By solving the time-independent power flow equation (TI PFE), we study mode coupling in a multimode W-type microstructured polymer optical fiber (mPOF) with a solid-core. The multimode W-type mPOF is created by modifying the cladding layer and reducing the core of a multimode singly clad (SC) mPOF. For such optical fiber, the angular power distributions, the length Lc at which an equilibrium mode distribution (EMD) is achieved, and the length zs for establishing a steady state distribution (SSD) are determined for various arrangements of the inner cladding’s air-holes and different launch excitations. This information is useful for the implement of multimode W-type mPOFs in telecommunications and optical fiber sensors.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Multimode POFs are frequently used for short-range communication links. POFs are simple to treat due to their ductility, light weight and large core diameters (up to 1 mm). POFs with a large core diameter can be easily combined with VCSELs, LEDs and lasers, to create low-cost and efficient communication systems. They have also used illumination, sensing, and data processing in their applications. Microstructured optical fibers (MOFs) have demonstrated performance unrivaled by conventional optical fibers because the optical fiber's micro-structured patterning provides flexibility to affect the sectional profile at the design stage [1–7]. As we know, “endlessly single-mode” MOF that operates over a wide range of wavelengths has been realized [2]. Also, because of photonic bandgap guidance, the hollow core MOF is defined [8–13]. MOFs have been used in light dispersion [14–16], birefringence [17], supercontinuum light generation [18–20], light wavelength conversion [21,22], optical fluids [23], and sensing [24]. The typical numerical aperture (NA) of the MOF is NA≃0.5-0.6 [25–27]. High NA MOFs [28] have been used to achieve lensless focusing with excellent resolution. In a typical MOF design, as shown in Fig. 1, the cladding-layer can be effected by changing the pattern and/or size (d) of air-holes within a concentric ring-like region. A MOF with a varying pattern in the cladding imitating a doubly clad W-type optical fiber is shown in Fig. 1. The advantage of W-type MOF over conventional W-type optical fiber is greater flexibility in adjusting the geometric parameters: air-hole diameters dq, dp and pitch Λ.
Fig. 1. (a) The cross section of the multimode doubly clad W-type MOF. Λ is the pitch, dq and dp are the diameters of inner outer cladding air-holes, respectively. (b) The RI performance of the referent multimode W-type MOF.
The propagation characteristics of a multimode optical fibers are influenced by mode coupling, modal attenuation, and modal dispersion. Light scattering in multimode optical fibers, which is due to fiber’s intrinsic perturbations, is the primary cause of mode coupling. Light scattering can be caused by a variety of irregularities, including voids, cracks, microscopic bends, and density variations. Only short fibers will produce a highly defined ring radiation pattern when light is launched at a given angle θ0>0 with respect to the fiber axis. The boundaries of such a ring grow fuzzy at the end of longer fibers due to mode coupling. At length Lc, where the highest-order mode ring-pattern evolves into a disk, an EMD is established.
Until recently, commercial simulation software packages were not designed for investigation of transmission characteristics of multimode microstructured optical fibers. This deficiency is addressed in this paper by numerically solving the time-independent power flow equation in the case of W-type mPOF. This enabled us to investigate the transmission performance of the multimode W-type mPOF, which is affected by the width of the inner cladding, spacing and size of air-holes, and the type of launch condition. For multimode W-type mPOF with a solid-core, we calculated characteristic lengths for achieving the EMD and SSD for launch beam distribution with different widths, for two different diameters of air-holes in the inner cladding. We assumed that cladding air-holes form a triangular pattern with a constant pitch Λ (see Fig. 1). In our best knowledge for the first time in this paper we investigate the transmission performance of mPOF with W-type refractive index distribution.
2. W-type mPOF design
Cladding air-holes of conventional mPOFs are typically arranged as a regular triangular lattice. The effective refractive index (RI) profile for a selected optical fiber layer can be adjusted by changing the geometric parameters dq, dp andΛ, as Fig. 1-(a) shows. The air-holes in the outer cladding are smaller than those in the inner cladding (dp vs dq in Fig. 1-(a)). The RI of the core n0 (central part) is higher than those in two claddings. The lowest effective RI nq is found in the inner cladding, and the outer cladding's effective RI is np (nq<np<n0). We used the TI PFE to simulate such a system.
3. Time-independent power-flow equation
The light transmission in a multimode optical fiber can be treated by Gloge’s TI PFE [29]:
The W-type optical fiber with the index profile depicted in Fig. 1-(b) can be thought as a system of a SCq optical fiber and cladding [31], in which the angle ${\theta _q} \cong {({2{\Delta _q}} )^{1/2}}$ is the critical angle for the guided modes – where ${\Delta _q} = ({{n_0} - {n_q}} )/{n_0}$. Likewise, the angle ${\theta _p} \cong {({2{\Delta _p}} )^{1/2}}$ is the critical angle of a singly clad SCp optical fiber. For the complete analyzed W-type fiber, the modes are guided if they propagate with angles $\theta \le {\theta _p}$, while modes with angles between ${\theta _p}$ and ${\theta _q}$ are lossy leaky modes:
4. Numerical results
We investigated a light transmission in a multimode W-type mPOF with a solid-core (Fig. 1). The effective V parameter for such fiber is given as:
where ${a_{eff}} = \Lambda /\sqrt 3 $ [33], and ${n_{fsm}}$ is cladding's effective RI, which can be obtained from Eq. (4), with the effective V parameter [33]:
Table 1. The fitting coefficients in Eq. (6).
Significant values of the effective RI of the inner cladding ${n_q}$, relative RI difference ${\Delta _q}$, critical angles ${\theta _p}$, and inner cladding air-holes diameters dq are shown in Table 2, for λ=645 nm and $\Lambda = 3\mathrm{\mu m}.$ The critical angle ${\theta _p}$ for ${d_p} = 1\mathrm{\mu m}$ is θp = 5.79°, and the effective RI of the outer cladding is np = 1.4844 (Table 2).
Table 2. For two air-hole diameters dq in the inner cladding, the data show the effective refractive index ${n_q},$ relative refractive index difference Δq and the corresponding critical angle ${\theta _q}$ - all for the inner cladding and at λ=645 nm.
The explicit finite differences method was used to solve the TI PFE (1) for the multimode W-type mPOF with a solid-core [29,34], for a Gaussian launch-beam distribution of the form:
The W-type mPOF was designed from the SC mPOF, which we theoretically investigated in our recently published work [35]. The characteristics of the SC mPOF were: n0 = 1.492, optical fiber diameter b = 1 mm, and coupling coefficient D = 1.649${\times} {10^{ - 4}}$ rad2/m (typical value of D for conventional POFs and mPOFs) [34,35]. It is worth noting that when modeling the W-type mPOF, the typical values of D that characterize a conventional POF can be used, because the strength of mode coupling both in conventional POFs and mPOFs is related to the polymer core material. In modeling a silica W-type MOF, a similar assumption was made [36].
To examine the effect of air-holes diameter in the inner cladding on the angular power distribution, the cases with dq = 1.5 and 2 µm were simulated, for a fixed diameter of the outer cladding air-holes dp = 1 µm, for two widths of the inner cladding δa = 2.4 µm (δ=0.008) and δa = 7.2 µm (δ=0.024), where core diameter 2a = 600 µm of W-type mPOF is assumed in the calculations. We examined cases of launch beam distributions with (FWHM)z = 0 = 1°, 5° and 10°. Figure 2 shows the output angular power distribution at different optical fiber lengths, in the case of dp = 1 µm, dq = 2 µm, and δ=0.008, calculated for Gaussian launch beam distributions with input angles θ0 = 0°, 6°, and 12° with (FWHM)z = 0 = 5°. Figure 2 shows that when the Gaussian launch beam distribution at the input fiber end is centered at θ0 = 0°, with increasing the optical fiber length, its width increases as a result of mode coupling. As the optical fiber length increases, low-order modes coupling becomes stronger: the distributions shift more towards θ=0°. Higher-order modes significantly couple only after longer optical fiber lengths. At the optical fiber's coupling length z = Lc =34.5 m, the EMD is established. The optical fiber length zs≡zs = 89 m denotes the point at which the mode- distribution becomes completely independent of the launch beam distribution, i.e. it becomes a SSD.
Fig. 2. The evolution of the normalized output angular power distribution with fiber length for the case with dq =1 µm (np = 1.4844), dq =2 µm (nq = 1.4458), δ=0.008, calculated for Gaussian launch distribution with input angles θ0 = 0° (solid line), 6° (dashed line), and 12° (dotted line) with (FWHM)z = 0 = 5° for: (a) z = 2 m; (b) z = 15 m; (c) z ${\equiv} $ Lc = 34.5 m and (d) z ${\equiv} $ zs =89 m.
We also solved the TI PFE (1) for different excitations to determine the optical fiber lengths Lc and zs. The resulting graphs in Figs. 3 and 4 show the lengths Lc and zs as a function of the (FWHM)z = 0 of the launch beam, for two inner cladding’s air-holes sizes dq (Fig. 3(a)) and two widths δa of the inner cladding (Fig. 3(b)). Figures 3 and 4 show that the wider the launch beam distribution results in the earlier onset of the EMD and SSD, respectively. This is a consequence of more uniform energy distribution among guided modes, so the EMD and SSD are established at shorter distances. The effect of launch excitation on lengths Lc and zs is also affected by fiber structural parameters (Table 3). With large leaky mode losses, the influence of launch excitation is less pronounced for smaller inner cladding’s air-holes diameter dq. Leaky mode losses decrease with increasing dq, that leads to a more pronounced influence of the launch excitation.
Fig. 3. Length Lc as a function of the launch beam’s angular distribution that is Gaussian with (FWHM)z = 0 = 1°, 5° and 10°, for dp = 1 µm (np = 1.4844) and (a) dq = 1.5 µm (nq = 1.4757) and 2 µm (nq = 1.4458), where δ=0.008, and (b) δ=0.008 and 0.024, where dq = 1.5 µm (nq = 1.4757).
Fig. 4. Length zs as a function of the launch beam’s angular distribution that is Gaussian with (FWHM)z = 0 = 1°, 5° and 10°, for dp = 1 µm (np = 1.4844) and (a) dq = 1.5 µm (nq = 1.4757) and 2 µm (nq = 1.4458), where δ=0.008, and (b) δ=0.008 and 0.024, where dq = 1.5 µm (nq = 1.4757).
Table 3. Coupling length Lc (for EMD) and length ${\textrm{z}_\textrm{s}}$ (for SSD) in mPOF with dp = 1 µm, ${\boldsymbol \varLambda } = 3\mathrm{\mu m}$, for different δ, dq, and (FWHM)z = 0 of the incident Gaussian launch beam distribution.
Figures 3(a) and 4(a) show that the lengths Lc and zs increase with the size of the inner cladding’s air-holes (diameter dq). This is due to a decrease in leaky mode losses for larger air-holes; power then remains in higher-order leaky modes over longer transmission lengths, postponing the onset of the EMD and SSD (longer Lc and zs).
The lengths Lc and zs are shorter for the optical fiber with a narrower inner cladding (Figs. 3(b) and 4(b)). Because leaky mode losses are lower in optical fibers with wider inner cladding, power remains in leaky modes for a longer light transmission length. The lengths Lc and zs for W-type mPOF (Figs. 3 and 4) are shorter if compared to these lengths for two SC mPOFs, shown in Fig. 5: SCq(1) fiber with n0 = 1.492 and nq = 1.4757, and SCq (2) fiber with n0 = 1.492 and nq = 1.4458. In particular, a portion of the guiding modes in W-type mPOFs become leaky modes, reducing the number of actually guided modes and, as a result, the lengths required to establish an EMD and SSD are shorter compared to the SC mPOFs.
Fig. 5. (a) Length Lc as a function of the launch beam’s angular distribution that is Gaussian with (FWHM)z = 0 = 1°, 5° and 10°, for SCq(1) fiber with n0 = 1.492 and nq = 1.4757 (dq = 1.5 µm) and SCq(2) fiber with n0 = 1.492 and nq = 1.4458 (dq = 2 µm); (b) Length zs as a function of the launch beam’s angular distribution that is Gaussian with (FWHM)z = 0 = 1°, 5° and 10°, for SCq(1) fiber with n0 = 1.492 and nq = 1.4757 (dq = 1.5 µm) and SCq(2) fiber with n0 = 1.492 and nq = 1.4458 (dq = 2 µm).
Compared with mPOF investigated in this work, silica microstructured optical fibers show much weaker mode coupling (D≃10−6 rad2/m) and therefore much longer lengths at which SSD is achieved (zs≃1 to 10 km) [36].
It is worth noting that the length dependence of the bandwidth of W-type MOF is determined by mode coupling behavior. The bandwidth is inverse linear function of length below the coupling length Lc. However, it has a $z^{-1/2}$ dependence beyond this equilibrium length. Thus, the shorter Lc leads to the faster transition to the regime of slower bandwidth decrease [37,38]. Since the lengths required to establish an EMD and SSD in W-type mPOFs are shorter compared to the SC mPOFs (Figs. 3 and 5(a)), a faster bandwidth improvement in mPOFs is expected than in SC mPOFs, which prove that W-type mPOFs are better choice for short-range telecommunication links. It is also important to be able to determine a modal distribution at a certain length of the W-type mPOF employed as a part of optical fiber sensory system.
Finally, it is interesting to note that the theoretical approach of modal diffusion in microstructured optical fibers employed in this work can be used for calculation of fiber’s bandwidth, but instead of time-independent power flow Eq. (1) which is solved in this work, one has to solve the time-dependent power flow equation [38].
5. Conclusion
We investigated how the transmission of the multimode W-type mPOFs is affected by the width of the inner cladding, the size and spacing of inner cladding’s air-holes, and the type of launch excitation. We demonstrated that the greater the width of the launch beam distribution, the shorter the lengths at which an EMD and SSD are established. Due to large leaky mode losses, the influence of launch excitation is less pronounced for smaller air-holes diameters of the inner cladding. The lengths required to achieve the EMD and SSD are shorter for the fiber with a narrower inner cladding. Because leaky mode losses are lower in fibers with wider inner cladding, power remains in leaky modes for a longer fiber length. Finally, because of these adjustable parameters, W-type mPOFs can be designed with greater versatility.
Funding
National Natural Science Foundation of China (62003046, 6211101138); Ministry of Education, Science and Technological Development (451-03-68/2020-14/200122); Science Fund of the Republic of Serbia (CTPCF-6379382); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515011997); City University of Hong Kong (CityU 7004600); The Innovation Team Project of Guangdong Provincial Department of Education (2021KCXTD014); Special project in key field of Guangdong Provincial Department of Education (2021ZDZX1050).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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