Improving the bandwidth of microstructured multimode W-type POFs in the visible light range

The bandwidth of multimode W-type microstructured plastic optical fibers (mPOFs) is analyzed using the time-dependent power flow equation (TD PFE). The results demonstrate that increasing the wavelength enhances the bandwidth in W-type mPOFs, depending on the inner cladding width and the launch beam distribution width. We observed that bandwidth improves with thinner and shallower inner cladding, as well as a narrower centrally launched beam. This characterization aligns with the fibers’ effectiveness in increasing bandwidth, allowing for the customization of various W-type optical fibers for specific applications at different wavelengths.

1 Introduction

Optical fiber communication systems offer greater reliability and flexibility compared to wireless communication, serving as the backbone of modern telecommunication networks [1]. Microstructured optical fibers (MOFs) or photonic crystal fibers (PCFs), represent a specialized optical fiber technology designed for light guiding [2]. In certain types of PCFs, a high refractive index (RI) material, such as silica or polymer, is used as the base, with periodically distributed air holes. This hole pattern lowers the effective refractive index (RI) of the fiber cladding, allowing the optical fibers to guide light [2–7]. The cladding hole pattern design enables modifications to the refractive index (RI) profile of the optical fiber. PCFs demonstrate exceptional performance, as their microstructure offers increased flexibility for adjusting the cross-section during the design process. Various types of PCFs can be utilized for different applications [8–24]. The spacing between the cladding holes determines the numerical aperture (NA) of the fiber, typically around 0.5 to 0.6 [25–27]. Additionally, certain PCFs incorporate heavy metal oxide glass fibers [28] and liquid-filled hollow-core fibers for specialized applications [29]. High NA PCFs have also exhibited excellent resolution in lensless focusing [30].

PCFs offer outstanding bandwidth performance and versatility, making them ideal for sensing and transmission applications. The propagation characteristics of PCFs are essential for their practical applications and are affected by modal dispersion, mode attenuation, mode coupling. Mode coupling primarily arises from light scattering caused by intrinsic perturbations within the fiber. One of the most effective methods for predicting mode propagation characteristics of multimode optical fibers involves using the PFE [31–39]. This study seeks to investigate the impact of wavelength on the fiber’s bandwidth across different configurations of multimode W-type mPOFs, taking into account variations in the intermediate layer width and the distribution of the centrally launched beam.

2 W-type mPOF

The effective refractive index (RI) profile of a chosen optical fiber layer can be modified by altering the geometric parameters d_q, d_p and Λ (Figures 1A). We employed the TD PFE to simulate this system (Figure 1B).

Figure 1

Figure 1. (A) The cross section of the W-type mPOF. Λ is the pitch, d_q and d_p are the diameters of inner and outer cladding air-holes, respectively. (B) The RI of the referent W-type mPOF.

3 TD PFE

The TD PFE for multimode optical fibers is given as [31, 35]:

$\frac{\mathrm{\partial }p\left(\theta ,z,t\right)}{\mathrm{\partial }z}+\tau \left(\theta \right)\frac{\mathrm{\partial }p\left(\theta ,z,t\right)}{\mathrm{\partial }t}=-\alpha \left(\theta \right)p\left(\theta ,z,t\right)+\frac{1}{\theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left[D\left(\theta \right)\frac{\mathrm{\partial }p\left(\theta ,z,t\right)}{\mathrm{\partial }\theta }\right](1)$

where t is time; $p\left(\theta ,z,t\right)$ is power distribution over propagation angle, length, and time; $\tau \left(\theta \right)$ is modal delay; $D\left(\theta \right)$ is the coupling coefficient; and $\alpha \left(\theta \right)\approx {\alpha }_d\left(\theta \right)$ is the attenuation [31, 35]. One should note that the condition of validity of the model proposed in this work is that guiding modes can be treated as a modal continuum. This is the case with all types of multimode optical fibers, such as a W-type mPOF investigated in this work. A more detailed explanation of the method for solving Equation 1 and calculating the bandwidth of W-type optical fibers can be found in our earlier publication [38].

W-type optical fiber (Figure 1) can be considered as a system of a SC_q optical fiber and cladding, in which the angle ${\theta }_q\cong {\left(2{\Delta }_q\right)}^{1/2}$ is the critical angle for the guided modes–where ${\Delta }_q=\left(n_0-n_q\right)/n_0$. Similarly, the angle ${\theta }_{\mathrm{p}}\cong {\left(2{\Delta }_{\mathrm{p}}\right)}^{1/2}$ is the critical angle for the guided modes of a SC_p optical fiber, where ${\Delta }_{\mathrm{p}}=\left(n_0-n_{\mathrm{p}}\right)/n_0$. The modes whose propagation angles are between ${\theta }_{\mathrm{p}}\cong {\left(2{\Delta }_{\mathrm{p}}\right)}^{1/2}$ and ${\theta }_{\mathrm{q}}\cong {\left(2{\Delta }_{\mathrm{q}}\right)}^{1/2}$, where ${\Delta }_{\mathrm{q}}=\left(n_0-n_{\mathrm{q}}\right)/n_0$ and ${\Delta }_{\mathrm{p}}=\left(n_0-n_{\mathrm{p}}\right)/n_0$, are leaky modes. Attenuation constants of leaky modes (Equation 2) are given as [35]:

${\alpha }_{\mathrm{L}}\left(\theta \right)=\frac{4{\left({\theta }^2-{\theta }_{\mathrm{p}}^2\right)}^{1/2}}{a{\left(1-{\theta }^2\right)}^{1/2}}\frac{{\theta }^2\left({\theta }_{\mathrm{q}}^2-{\theta }^2\right)}{{\theta }_{\mathrm{q}}^2\left({\theta }_{\mathrm{q}}^2-{\theta }_{\mathrm{p}}^2\right)}\exp \left[-2\delta a{n}_0{k}_0{\left({\theta }_{\mathrm{q}}^2-{\theta }^2\right)}^{1/2}\right](2)$

where $k_0$ = 2π/λ, a is the core radius and da is the width of the intermediate layer (inner cladding). The attenuation in this fiber can be expressed as [35]:

${\alpha }_{\mathrm{d}}\left(\theta \right)=\left\{\begin{array}{ll}0& ;\theta \le {\theta }_{\mathrm{p}}\\ {\alpha }_{\mathrm{L}}\left(\theta \right)& ;{\theta }_{\mathrm{p}}<\theta <{\theta }_{\mathrm{q}}\\ \infty & ;\theta \ge {\theta }_{\mathrm{q}}\end{array}\right.(3)$

In this study, to the best of our knowledge, we are the first to explore how bandwidth in multimode W-type mPOFs can be improved by transitioning from smaller to larger wavelengths, taking into account various widths of the intermediate layer (Equation 3) and different FWHM of the centrally launched beam distribution. The findings could be useful in the design of W-type mPOFs for communication and sensing applications.

4 Numerical results and discussion

We examined the bandwidth of a multimode W-type mPOF with a solid core (Figure 1) at various wavelengths. Its effective V parameter is given as:

$V=\frac{2\pi }{\lambda }{a}_{eff}\sqrt{n_0^2-n_{fsm}^2}(4)$

where n₀ is the RI of the core, n_fsm is the effective RI of the cladding and $a_{eff}=\mathrm{\Lambda }/\sqrt{3}$ [33]. The effective RI of the cladding n₁≡n_fsm, can be obtained from Equation 4, using the following equation [33]:

$V\left(\frac{\lambda }{\mathrm{\Lambda }},\frac{d}{\mathrm{\Lambda }}\right)=A_1+\frac{A_2}{1+A_3\exp \left(A_4\lambda /\mathrm{\Lambda }\right)}(5)$

with the fitting parameters A_i (i = 1–4) given as:

$A_i=a_{i0}+a_{i1}{\left(\frac{d}{\mathrm{\Lambda }}\right)}^{b_{i1}}+a_{i2}{\left(\frac{d}{\mathrm{\Lambda }}\right)}^{b_{i2}}+a_{i3}{\left(\frac{d}{\mathrm{\Lambda }}\right)}^{b_{i3}}(6)$

where the coefficients a_i0 to a_i3 and b_i1 to b_i3 (i = 1–4) in Equations 5 and 6 are provided in our earlier works [36, 37].

The W-type mPOF was developed from the singly-clad (SC) mPOF, which we analyzed theoretically in our recent publication [39]. The SC mPOF had n₀ = 1.492, diameter b = 1 mm, and coupling coefficient D = 1.649 × 10⁻⁴ rad²/m (typical value of $D$ for conventional multimode POFs and multimode mPOFs). In the calculations, the air-holes diameters d_q = 1.5 and 2 μm, and d_p = 1 μm, and pitch Λ = 3 μm, are assumed. The two widths of the inner cladding δa = 2.4 µm (δ = 0.008) and δa = 7.2 µm (δ = 0.024), where 2a = 600 μm, are assumed in the calculations. We examined cases of launch beam distributions with FWHM = 1° and 10^o. The EFDM was used to solve the TD PFE (1) for this type of mPOF [34, 36–39].

Our numerical solution of the TD PFE is shown in Figures 2, 3, which illustrate the evolution of the W-type mPOF’s bandwidth at 30 m across different visible wavelengths, considering various widths of the intermediate layer δ and different diameters of the inner cladding air holes d_q. These figures depict Gaussian launch excitation with FWHM values of 1° and 10°, respectively. It is apparent in Figures 2, 3 that the fiber bandwidth increases with increasing wavelength for both analyzed widths of the intermediate layer. This is explained by the rise of the leaky mode losses with increasing wavelength as fewer leaky modes remain guided along the fiber; which decreases modal dispersion and increases bandwidth. Notably, Figure 3 indicates lower bandwidth compared to Figure 2, as the wider launch beam distributes energy more uniformly among guided modes, which increases modal dispersion and reduces bandwidth. As the width of the inner cladding decreases (smaller d_q), the bandwidth improves because fewer leaky modes persist along the fiber, thereby reducing modal dispersion. Additionally, Figures 2, 3 demonstrate that a thinner inner cladding (smaller δ) results in increased bandwidth, as leaky modes are filtered out sooner along the fiber, leading to a quicker reduction in modal dispersion. Although the W-type optical fibers investigated in this work have a step-index distribution of the fiber core (Figure 1), which is simpler than the graded index distribution of a fiber core, a significant enhancement of the bandwidth of such W-type optical fibers could be achieved by increasing the operating wavelength, with appropriate choice of the excitation type and the width of intermediate layer of the W-fiber. It is worth noting that in a recently reported work [40], the effect of the structural parameters on the dispersion and attenuation of the MOFs has also been demonstrated. Furthermore, a realization of attenuator and amplifier using MOFs as well as a proposal for vacuum ultraviolet torch using plasmonic-based MOFs have been reported [41, 42].

Figure 2

Figure 2. Calculated bandwidth at 30 m vs. wavelength for W-type mPOF with δ = 0.008 and 0.024, and d_p = 1 μm, for Gaussian beam excitation with FWHM = 1^o, (A) d_q = 1.5 μm and (B) d_q = 2 μm.

Figure 3

Figure 3. Calculated bandwidth at 30 m vs. wavelength for W-type mPOF with δ = 0.008 and 0.024, and d_p = 1 μm, for Gaussian beam excitation with FWHM = 10^o, (A) d_q = 1.5 μm and (B) d_q = 2 μm.

The results presented in this work can be useful for employment the investigated W-type mPOF for data communication in short-range communication systems, such as those within buildings. The visible spectrum is the most commonly utilized wavelength range for consumer electronics. W-type POFs can also be used for decorative and automotive lighting, especially in applications where flexibility is important (e.g., car interiors or illuminated signs). Also, the obtained results can be useful for employment of the W-type POFs as a part of various sensory systems which operate at different visible wavelengths.

5 Conclusion

Bandwidth is numerically determined across a range of visible light beam wavelengths for W-type mPOFs with varying widths of the intermediate layer and different FWHM values of the launch beam. We showed that bandwidth increases with larger wavelengths, as well as with thinner and shallower inner cladding and narrower centrally launched beams. This characterization aligns with the observed effectiveness of these fibers in minimizing modal dispersion and enhancing bandwidth. Such findings enable the customization of various W-type optical fibers for specific applications at different wavelengths. In our future work, the W-type mPOF with graded index core distribution will be investigated, which is a promising fiber design for further enhancing the bandwidth.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

BD: Conceptualization, Formal Analysis, Methodology, Software, Visualization, Writing–original draft. AS: Conceptualization, Formal Analysis, Investigation, Methodology, Software, Visualization, Writing–review and editing. AD: Conceptualization, Formal Analysis, Funding acquisition, Methodology, Validation, Writing–review and editing. KA: Conceptualization, Formal Analysis, Investigation, Methodology, Validation, Visualization, Writing–review and editing. SS: Conceptualization, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Writing–original draft, Writing–review and editing.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The Serbian Ministry of Science, Technological Development, and Innovations (Agreement Nos 451-03-65/2024-03/200122; 451-03-65/2024-03/200123) and Ajman University (Grant ID: 2023-IRG-ENIT-14; 2024-IRG-ENIT-10) provided funding for this study.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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