- 1Faculty of Science, University of Kragujevac, Kragujevac, Serbia
- 2Department of Mechanial Enginering, City University of Hong Kong, Hong Kong, China
- 3Center for Cognition and Neuroergonomics, State Key Laboratory of Cognitive Neuroscience and Learning, Beijing Normal University at Zhuhai, Zhuhai, China
- 4Faculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, Kosovska Mitrovica, Serbia
- 5Department of Electrical Engineering, Ajman University, Ajman, United Arab Emirates
- 6Center of Medical and Bio-Allied Health Sciences Research, Ajman University, Ajman, United Arab Emirates
- 7School of Microelectronics and Communication Engineering, Chongqing University, Chongqing, China
Up to now, there have been no commercial simulation tools accessible for researching the transmission properties of multimode microstructured optical fibers (MOFs). In order to avoid this problem, this study uses the time-independent power flow equation (TI PFE) numerical solution to examine the wavelength dependency of the equilibrium mode distribution (EMD) and steady state distribution (SSD) in multimode graded-index microstructured polymer optical fibers (GI mPOF) with a solid core. We showed that the lengths zs at which an SSD is obtained in GI mPOF and the coupling length Lc necessary to create an EMD are shorter at λ = 568 nm than they are found to be at λ = 633 nm. The lengths Lc and zs stay constant when the wavelength decreases further from λ = 568 to 522 and then to 476 nm. As a result, it is anticipated that a faster bandwidth enhancement in the tested GI mPOF will take place at wavelengths around λ = 568 nm as opposed to 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.
1 Introduction
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].
FIGURE 1. (A) The cross-section of the multimode GI MOF. The pitch Λ determines the position of air holes in a triangular lattice. The air holes that make the four rings of the core have the following diameters: d1, d2, d3, and d4. Two rings of air holes of the same diameter as those in the outermost core ring form the cladding (d4 = d5 = d6). (B) The dashed blue line shows the referent multimode GI MOF’s RI distribution. The solid black line represents the RI distribution in the core based on Eq. 1, when g = 2.0, and at λ = 633 nm.
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 zs. 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 Lc). 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/z1/2 (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.
2 GI mPOF design
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 (d4 = d5 = d6). This system was simulated using the TI PFE.
3 Time-independent power flow equation
The GI optical fibers have the following RI profile:
Here nco(λ) is the core’s highest index (measured at the fiber axis), ncl(λ) is the cladding’s index,
The TI PFE for GI optical fiber is [32]:
where
where k = 2π/λ.
The principal mode m excited at the input fiber end is [32]:
where
4 Numerical simulation results
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:
where aeff =
The fitting parameters Ai (i = 1–4) are given as:
The coefficients ai0 to ai3 and bi1 to bi3 (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 nco 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 d1 = 0.6 μm, d2 = 0.7 μm, d3 = 1.3 μm, and d4 = 3.1 μm, the refractive indices n1, n2, n3, and n4, 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 d4 = d5 = d6, and therefore the refractive index of the cladding is n4 = n5 = n6 = ncl. 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].
TABLE 1. Refractive index n0, n1, n2, n3, n4, n5, and n6, the relative index difference Δ, the core index exponent g, and the maximum principal mode number M at different wavelengths.
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
FIGURE 2. Normalized output modal power distribution P (m,λ,z) acquired by numerically solving the TI PFE (2) over a range of radial offsets
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 Lc has an inversely linear effect on the bandwidth. Beyond this equilibrium length LC, 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 Lc between 1.45 and 1.65 km at which an EMD is achieved, and a length zs between 3.30 and 3.80 km for the establishment of an SSD [34].
5 Conclusion
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 Lc and zs 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 Lc and zs stay constant when the wavelength decreases further from λ = 568 to 522 and then to 476 nm. Therefore, the shorter Lc 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.
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
AS: Conceptualization, Methodology, Software, Writing–original draft. SS: Conceptualization, Methodology, Software, Supervision, Writing–original draft, Writing–review and editing. ZW: Funding acquisition, Project administration, Writing–review and editing. BD: Methodology, Software, Writing–original draft. MK: Conceptualization, Methodology, Software, Writing–original draft. LK: Methodology, Software, Writing–original draft. AD: Writing–original draft, Writing–review and editing. KA: Writing–original draft, Writing–review and editing. CC: Writing–original draft, Writing–review and editing, Conceptualization.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This research was funded by the National Natural Science Foundation of China (62003046, 6211101138); a grant from Ajman University (Grant No. 2023-IRG-ENIT-14); a grant from City University of Hong Kong (Project No. CityU 7004600); a grant from the Serbian Ministry of Science, Technological Development, and Innovations (Agreement No. 451-03-47/2023-01/200122); and a grant from Guangdong Basic and Applied Basic Research Foundation (2021A1515011997).
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.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
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Keywords: polymer optical fiber, graded-index optical fiber, microstructured optical fiber, power flow equation, wavelength dependent transmission
Citation: Simović A, Savović S, Wang Z, Drljača B, Kovačević MS, Kuzmanović L, Djordjevich A, Aidinis K and Chen C (2024) Wavelength dependent transmission in multimode graded-index microstructured polymer optical fibers. Front. Phys. 12:1340505. doi: 10.3389/fphy.2024.1340505
Received: 18 November 2023; Accepted: 26 January 2024;
Published: 09 February 2024.
Edited by:
Jitendra Bahadur Maurya, National Institute of Technology Patna, IndiaReviewed by:
Satyendra Kumar Mishra, Centre Tecnologic De Telecomunicacions De Catalunya, SpainSushank Chaudhary, Chulalongkorn University, Thailand
Copyright © 2024 Simović, Savović, Wang, Drljača, Kovačević, Kuzmanović, Djordjevich, Aidinis and Chen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Zhuo Wang, emh1b3dhbmdAYm51LmVkdS5jbg==