Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Theoretical Investigation of the Capacity of Space Division Multiplexing with Multimode Step-Index Air-Clad Silica Optical Fibers

Theoretical Investigation of the Capacity of Space Division Multiplexing with Multimode... hv photonics Communication Theoretical Investigation of the Capacity of Space Division Multiplexing with Multimode Step-Index Air-Clad Silica Optical Fibers 1 , 2 3 4 5 2 Svetislav Savovic ´ , Alexandar Djordjevich , Isidora Savovic ´ , Branko Drljaca ˇ , Ana Simovic ´ 1 , and Rui Min * Center for Cognition and Neuroergonomics, State Key Laboratory of Cognitive Neuroscience and Learning, Beijing Normal University at Zhuhai, Zhuhai 519087, China; savovic@kg.ac.rs Faculty of Science, University of Kragujevac, R. Domanovica ´ 12, 34000 Kragujevac, Serbia; asimovic@kg.ac.rs Department of Mechanical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China; mealex@cityu.edu.hk Laboratory of Neurodegenerative Disease, School of Biomedical Sciences, LKS Faculty of Medicine, The University of Hong Kong, 21 Sassoon Road, Pokfulam, Hong Kong, China; u3008169@connect.hku.hk Faculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, L. Ribara 29, 38220 Kosovska Mitrovica, Serbia; branko.drljaca@pr.ac.rs * Correspondence: rumi@doctor.upv.es Abstract: We studied the effect of mode coupling on the space division multiplexing (SDM) capabili- ties of multimode step-index (SI) air-clad silica optical fibers by numerically solving the power flow equation. Mode coupling considerably reduces the length of these fibers at which space division mul- tiplexing may be achieved with minimal crosstalk between neighboring optical channels, according to the findings. Up to 120 m and 30 m, respectively, the two and three spatially multiplexed channels in the investigated multimode step-index silica optical fibers can be used with low crosstalk. When building a space division multiplexing-based optical fiber transmission system, such characterization Citation: Savovic, ´ S.; Djordjevich, A.; of optical fibers should be taken into account. Savovic, ´ I.; Drljaca, ˇ B.; Simovic, ´ A.; Min, R. Theoretical Investigation of Keywords: air-clad silica optical fibers; microbends; mode coupling; space division multiplexing the Capacity of Space Division Multiplexing with Multimode Step-Index Air-Clad Silica Optical Fibers. Photonics 2022, 9, 127. 1. Introduction https://doi.org/10.3390/ Global network traffic has expanded dramatically in recent decades, owing primarily photonics9030127 to the rapid expansion of the Internet [1]. Optical fiber systems now support the majority Received: 25 January 2022 of this data flow. Multiplexing, fiber amplifiers, and high-efficiency spectral coding have Accepted: 22 February 2022 all contributed to this capacity gain [1]. Optical data multiplexing is possible not just in Published: 23 February 2022 wavelength, but also in space, time, polarization, and phase. Optical fiber transmission Publisher’s Note: MDPI stays neutral systems can benefit from the multiplexing technique [1,2]. SDM, which includes mode divi- with regard to jurisdictional claims in sion multiplexing using few-mode fibers or multimode and/or core multiplexing utilizing published maps and institutional affil- multicore fibers, has gotten a lot of attention as a way to increase optical communication’s iations. multiplicative capacity [1,3–8]. SDM can operate at the same or separate wavelengths [9]. If the SDM channels inside the carrier fiber are assigned radially distributed optical signals at the same wavelength, the central channel is launched along the fiber axis in the form of a disk, whereas all other channels are in the form of concentric rings. Thus, one increases the Copyright: © 2022 by the authors. capacity of the optical fiber link (Figure 1). Licensee MDPI, Basel, Switzerland. Silica optical fibers (SOFs) are suited to long-distance signal communication [10], while This article is an open access article plastic optical fibers (POFs) with a large core are extensively used in short-distance (less than distributed under the terms and 100 m) networks [11]. Optical measurements, telecommunications, and sensor applications conditions of the Creative Commons all use plastic-clad silica fibers (PCSFs) [12,13]. Currently, the only technology capable Attribution (CC BY) license (https:// of reaching exceptionally high NAs is PCSFs and air-clad silica fibers. In commercially creativecommons.org/licenses/by/ marketed PCSFs, NA is limited to ~0.46. An all-silica fiber is promising for SDM because 4.0/). Photonics 2022, 9, 127. https://doi.org/10.3390/photonics9030127 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 7 applications all use plastic-clad silica fibers (PCSFs) [12,13]. Currently, the only technol- ogy capable of reaching exceptionally high NAs is PCSFs and air-clad silica fibers. In com- mercially marketed PCSFs, NA is limited to ~0.46. An all-silica fiber is promising for SDM because of the material durability and sufficient silica purity for strong transmission per- formance in the large wavelength range of 300 nm to 2.4 µm [14–16]. In contrast, air-clad fibers can have an extraordinarily high numerical aperture (NAmeas > 0.9) [17]. Figure 2 is a cross-sectional sketch of an all-silica air-clad fiber with an annulus made up of a single ring of holes, produced at the University of Sydney [15]. The ring, or rings, of air holes, whose surface roughness has been measured to be 0.5 nm [18], are solely responsible for Photonics 2022, 9, x FOR PEER REVIEW the NA of air-clad fibers. The thin bridges are typically between 100 and 400 nm th 2 ick, of 7 with the ring of holes forming a corrugated surface at the core. To decrease light leakage, the thickness was chosen. applications all use plastic-clad silica fibers (PCSFs) [12,13]. Currently, the only technol- Photonics 2022, 9, 127 2 of 7 ogy capable of reaching exceptionally high NAs is PCSFs and air-clad silica fibers. In com- mercially marketed PCSFs, NA is limited to ~0.46. An all-silica fiber is promising for SDM because of the material durability and sufficient silica purity for strong transmission per- of the material durability and sufficient silica purity for strong transmission performance formance in the large wavelength range of 300 nm to 2.4 µm [14–16]. In contrast, air-clad in the large wavelength range of 300 nm to 2.4 m [14–16]. In contrast, air-clad fibers fibers can have an extraordinarily high numerical aperture (NAmeas > 0.9) [17]. Figure 2 is can have an extraordinarily high numerical aperture (NA > 0.9) [17]. Figure 2 is a meas a cross-sectional sketch of an all-silica air-clad fiber with an annulus made up of a single cross-sectional sketch of an all-silica air-clad fiber with an annulus made up of a single ring ring of holes, produced at the University of Sydney [15]. The ring, or rings, of air holes, Fig of ure holes, 1. Apr sch oduced ematic at of a the thrUniversity ee-channel Sof DM Sydney system, [15 con ].si The sting ring, of a cor ent rings, er diskof an air d tw holes, o conc whose entric whose surface roughness has been measured to be 0.5 nm [18], are solely responsible for rsurface ings. roughness has been measured to be 0.5 nm [18], are solely responsible for the NA of the NA of air-clad fibers. The thin bridges are typically between 100 and 400 nm thick, air-clad fibers. The thin bridges are typically between 100 and 400 nm thick, with the ring with the ring of holes forming a corrugated surface at the core. To decrease light leakage, Fiber imperfections and inhomogeneities introduced during the optical fiber fabrica- of holes forming a corrugated surface at the core. To decrease light leakage, the thickness the thickness was chosen. tion process cause power transfer between adjacent modes [9,19,20]. Because SDM entails was chosen. densely packed spatial channels in a fiber, mode coupling is critical since it allows cross- talk between channels. As a result, the expected beam parameters are altered. The optical power distribution of an optical fiber determines its far-field pattern, which is influenced by launch conditions, fiber parameters, and fiber length. Only short fibers will produce a highly defined ring radiation pattern when light is launched at a given angle > 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 “coupling length” Lc, where the highest-order mode ring- pattern evolves into a disk, an equilibrium mode distribution (EMD) is established. Fig Figure ure 1 1. . A A sch schema emat tiic c o of f a a tt h h rr ee ee -c -h ch an an ne nleS l D SM DM sysy ste st m em, , con cs on istsi in sg tin ofga of cea ntc een r d tier sk d ais nd k t aw no d c to w no ce cn otn ric cen rin tr g ic s. rings. Fiber imperfections and inhomogeneities introduced during the optical fiber fabrica- tion process cause power transfer between adjacent modes [9,19,20]. Because SDM entails densely packed spatial channels in a fiber, mode coupling is critical since it allows cross- talk between channels. As a result, the expected beam parameters are altered. The optical power distribution of an optical fiber determines its far-field pattern, which is influenced by launch conditions, fiber parameters, and fiber length. Only short fibers will produce a highly defined ring radiation pattern when light is launched at a given angle > 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 “coupling length” Lc, where the highest-order mode ring- Figure 2. Sketch of cross section of air-clad silica fiber based on the design used in the experiments of [15]. Figure 2. Sketch of cross section of air-clad silica fiber based on the design used in the experiments pattern evolves into a disk, an equilibrium mode distribution (EMD) is established. of [15]. Fiber imperfections and inhomogeneities introduced during the optical fiber fabrica- tion process cause power transfer between adjacent modes [9,19,20]. Because SDM entails Because information on SDM in multimode air-clad silica optical fibers is lacking in densely packed spatial channels in a fiber, mode coupling is critical since it allows crosstalk the literature, we investigated mode coupling in multimode SI air-clad silica optical fibers between channels. As a result, the expected beam parameters are altered. The optical in this work by numerically solving the power flow equation. This fiber was previously power distribution of an optical fiber determines its far-field pattern, which is influenced investigated experimentally by Åslund et al. [15]. This allows one to determine the maxi- by launch conditions, fiber parameters, and fiber length. Only short fibers will produce a mum fiber lengths for an SDM system that employs multimode SI air-clad silica optical highly defined ring radiation pattern when light is launched at a given angle q > 0 with fibers. respect to the fiber axis. The boundaries of such a ring grow fuzzy at the end of longer fibers due to mode coupling. At “coupling length” L , where the highest-order mode ring-pattern evolves into a disk, an equilibrium mode distribution (EMD) is established. Because information on SDM in multimode air-clad silica optical fibers is lacking in Fig theure literatur 2. Skete, chwe of cinvestigated ross section of mode air-clacoupling d silica fibin er mult based imode on the SI deair sig-clad n used silica in the optical experimen fibers ts of in[1 this 5]. work by numerically solving the power flow equation. This fiber was previously in- vestigated experimentally by Åslund et al. [15]. This allows one to determine the maximum Because information on SDM in multimode air-clad silica optical fibers is lacking in fiber lengths for an SDM system that employs multimode SI air-clad silica optical fibers. the literature, we investigated mode coupling in multimode SI air-clad silica optical fibers 2. Power Flow Equation in this work by numerically solving the power flow equation. This fiber was previously investigated experimentally by Åslund et al. [15]. This allows one to determine the maxi- Gloge’s power flow equation is [19]: mum fiber lengths for an SDM system that employs multimode SI air-clad silica optical ¶P(q, z) D ¶ ¶P(q, z) fibers. = a(q)P(q, z) + q , (1) ¶z q ¶q ¶q where P(q,z) is the angular power distribution, q is the propagation angle in respect to the core axis, z is the distance from the input end of the optical fiber, D is the constant coupling Photonics 2022, 9, 127 3 of 7 coefficient [19,21–23], and a(q) is the modal attenuation. Since a(q) need not be accounted for in solving Equation (1) for mode coupling [22,23], Equation (1) reduces to [15]: ¶P(q, z) D ¶P(q, z) ¶ P(q, z) = + D . (2) ¶z q ¶q ¶q The explicit finite-difference approach [24] was used to derive a numerical solution of the power flow Equation (2) for a Gaussian launch-beam distribution of the form: " # 1 (q q ) P(q, z) = exp , (3) 2s s 2p with 0  q  q , where q is the mean value of the launch angular distribution, s is the standard deviation, and FWHM = 2s 2 ln 2 = 2.355s. We discretized Equation (2) using explicit finite difference method, so Equation (2) now reads [24]: ! ! DzD DzD 2DzD DzD DzD P = P + 1 P + + P , (4) i,j+1 i1,j i,j i+1,j 2 2 2 2q Dq 2q Dq Dq Dq Dq i,j i,j where indexes i and j refer to the discretization step lengths Dq and Dz for the angle q and length z, respectively. This is a simple formula for P at the (i, j + 1)th mesh point in terms i,j+1 of the known values along the jth distance row. The truncation error for the difference in Equation (4) is O(Dz,Dq ). The grid dimension in the q direction is N = q /Dq and the grid dimension in the z direction is M = L/Dz, where q is the critical angle and L is the fiber length. 3. Results and Discussion In this study, we theoretically investigated the effect of mode coupling on SDM capabilities in a multimode SI air-clad silica optical fiber employed in a prior experiment by Åslund et al. [15]. The single-material air-clad fiber was made of low-grade natural silica and had material losses of less than 10 dB/km at 1550 nm. The fiber had NA = 0.54, a core diameter of d = 180 m, and 59 bridges with a length of l  26 m to sustain core it. The maximum thickness of the bridge was 340 nm. The constant coupling coefficient –5 2 for the fiber was D = 3.5  10 rad /m at  = 1550 nm [15,25], which we adopted in this work. The numerical solution to Equation (4) was obtained using discretization step lengths Dq = 0.1 and Dz = 0.001 m. Figure 3 illustrates the normalized output angular power distribution at two lengths of the multimode SI air-clad optical fiber obtained as a solution of Equation (2). The three launch beams in the Gaussian form with (FWHM) = 3.9 and different input z=0 angles q = 0 , 13 , and 26 represent three optical channels [15,25]. In the short SI air-clad optical fiber in Figure 3a, the mode coupling is minimal, and as a result there is no crosstalk between optical channels. Due to mode coupling, spatial power distributions broaden with increasing fiber length, so the three-channel SDM can be realized in this fiber up to a fiber length of z = 30 m (Figure 3b). In the case of two co-propagating optical SDM channels, the two Gaussian launch distributions with input angles q = 0 and 26 , and (FWHM) = 3.9 , are investigated. One can see from Figure 4b that practical realization z=0 of two-channel SDM can be done up to a fiber length of z = 120 m. As can be seen, SDM mode coupling severely restricts the length of multimode SI air-clad silica optical fibers that can be used for SDM (Table 1). Ph Pho otto on niic cs s 2 20 02 22 2,, 9 9,, x x F FO OR R P PEE EER R R RE EV VIIEW EW 4 4 o off 7 7 a are re in inve vest stiiga gated ted.. On One e ca can n se see e f from rom Fi Figure gure 4 4b b th tha at t pr pra acti ctica cal l re rea ali liz za atio tion n o of f t two wo- -cha chan nn nel el S SDM DM ca can n be be d do on ne e up up to to a a f fiber iber leng lengt th h of of z zSD SDM M = = 1 12 20 0 m m.. A As s c ca an n be be see seen n,, m mo od de e cou coupli plin ng g s sev ever erely ely Photonics 2022, 9, 127 4 of 7 re rest stricts ricts th the e le len ngt gth h o of f m mult ultiim mo od de e S SI I a air ir- -cla clad d si sili lica ca o optic ptica al l f fiber ibers s th tha at t ca can n be be used used f fo or r S SDM DM (Table 1). (Table 1). Figure 3. Normalized output intensity at different lengths of the multimode SI air-clad silica optical Fig Figure ure 3 3.. N Nor orm ma ali liz ze ed d ou outtpu putt iin ntten ensi sitty y a att di diffe ffer ren entt le len ng gtth hs s of of tth he e m mult ultimode imode S SI I a air ir- -c cla lad d s sil ilic ica a op opttica icall 5 2 −5 −5 2 2 fiber obtained as solutions of Equation (2) for D = 3.5 × 10 rad /m, three launch beams in the Gauss- fiber fiberobtained obtained as as solutions solutions of of E Equation quation ((2) 2) for for D D = = 3.5 3.5 × 1 0 10 rad /m rad , t/m, hree thr launch ee launch beams beams in the G inath uss- e ia ian n for form m,, a an nd d in inpu putt a an ng gle les s = = 0 0 (s (so oli lid d li lin ne) e),, 1 13 3° ° ( (d da as sh he ed d li lin ne e) ) a an nd d 2 26 6° ° ( (do dotttted ed li lin ne e) ),, w wit ith h Gaussian form, and input angles q = 0 (solid line), 13 (dashed line) and 26 (dotted line), with (FWHM) = 3.9 for: (a) z = 10 m and (b) z = 30 m. (F (FWH WHM) M)z=0 z=0 = = 3 3..9 9° ° for for:: ( (a a) ) z z = = 1 10 0 m m a an nd d ( (b b) ) z z = = 3 30 0 m m.. z=0 Figure 4. Normalized output intensity at different lengths of the multimode SI air-clad silica optical Figure 4. Normalized output intensity at different lengths of the multimode SI air-clad silica optical Figure 4. Normalized output intensity at different lengths of the multimode SI air-clad silica optical −5 25 2 −5 2 fiber fib fiber er obtained ob obtta ain ined ed a as as s so so solutions lut lutio ion ns s of of ofE EEquation qua quattio ion n ( (2 2(2) ) ) fo fofor r r D D =D = 3 3= .5 .5 3.5 × × 10 10 r r 10 a ad d /m /m rad ,, ttw w/m, o o la launch unch two b b launch ea eam ms s in in beams tth he e G Ga ain ussi ussi the a an n o o form, and input angles = 0 (solid line) and 26° (dotted line), with (FWHM)z=0 = 3.9° for: (a) z = 40 form, and input angles = 0 (solid line) and 26° (dotted line), with (FWHM)z=0 = 3.9° for: (a) z = 40 Gaussian form, and input angles q = 0 (solid line) and 26 (dotted line), with (FWHM) = 3.9 for: 0 0 z=0 (a m ) z an =d 40 (bm ) z and = 12(0 b m ) z. = 120 m. m and (b) z = 120 m. Ta Table ble 1 1.. Le Len ng gtth h z zS SD DM M for for ttw wo o a an nd d tth hr ree ee spa spattia iall lly y m mult ultip iple lex xed ed c ch ha an nn nel els s w wit ith h m min inim ima al l c cr ros oss stta alk lk iin n S SI I Table 1. Length z for two and three spatially multiplexed channels with minimal crosstalk in SDM air-clad silica optical fibers and PCSFs, with different numerical apertures NA and coupling coeffi- air-clad silica optical fibers and PCSFs, with different numerical apertures NA and coupling coeffi- SI air-clad silica optical fibers and PCSFs, with different numerical apertures NA and coupling c cie ien ntts s D D.. coefficients D. D D z zS SDM DM ( (m) m) S SDM DM ( (m) m) D z (m) (m) Fiber type NA Fiber type NA SDM SDM Fiber Type NA 2 2 (rad /m) 2 (2-channel) (3-channel) (rad /m) (2-channel) (3-channel) (rad /m) (2-Channel) (3-Channel) Air-clad silica fiber (this Air-clad silica fiber (this −5 Air-clad silica fiber (this work) 0.54 3.5 −5 10 120 30 0 0..5 54 4 3 3..5 5 × × 10 10 120 120 30 30 wo work) rk) PCSF (1) [26] 0.4 1.28  10 50 14 −5 −5 5 PCSF (1) [26] 0.4 1.28 × 10 50 14 PCS PCSF F (1(2) ) [2 [26 6] ] 0.4 0.37 1.28 × 10 4.5  10 50 25 14 7 −5 −5 PCS PCSF ( F (2 2) ) [ [2 26 6] ] 0 0..3 37 7 4 4..5 5 × × 10 10 25 25 7 7 The coupling coefficient D of the multimode SI air-clad silica fiber investigated in this The The coupl couplin ing g coef coeff ficient icient D D o of f th the e m mult ultim imo od de e S SI I a air ir- -cla clad d si sili lica ca f fiber iber in inve ves stig tiga ated ted iin n t th his is study is of the same order of magnitude as that of the previously investigated multimode study is of the same order of magnitude as that of the previously investigated multimode study is of the same order of magnitude as that of the previously investigated multimode SI PCSFs [26] (Table 1). SI PCSFs have a lesser capacity for SDM due to their lower NA of SI PCSFs [26] (Table 1). SI PCSFs have a lesser capacity for SDM due to their lower NA of SI PCSFs [26] (Table 1). SI PCSFs have a lesser capacity for SDM due to their lower NA of 0.37 to 0.4. For example, the capacity for three-channel SDM in the SI air-clad silica fiber investigated in this work, compared to PCSF (2) [26], is about four times greater, which is mainly due to the much higher NA of the SI air-clad silica fiber (these two fibers have a similar coupling coefficient D). On the other hand, despite the higher coupling coefficient D (which restricts the capacity for SDM) of the SI air-clad silica fiber compared to that of PCSF (1) [26], the capacity for three-channel SDM in the SI air-clad silica fiber compared to PCSF Photonics 2022, 9, 127 5 of 7 (1) is about two times greater, due to the higher NA of the SI air-clad silica fiber. Therefore, in practical realization of SDM in PCSFs and air-clad silica fibers, the coupling coefficient is the dominant factor for lower NA PCSFs compared to high NA air-clad silica fibers. As a result, two- and three-channel SDM can be achieved in SI air-clad silica optical fibers with lengths longer than PCSFs (Table 1). Air-clad silica fibers are, hence, more suitable for SDM applications. It is worth noting that, in our previous works, we have shown that this kind of SDM can be employed in standard multimode SI plastic optical fibers at lengths of up to few meters [27–29] and multimode SI silica optical fibers at lengths of several hundreds of meters [30]. In order to investigate the influence of (FWHM) of launch beam distribution on z=0 the capacity of SDM in the SI air-clad silica fiber, we solved Equation (1) for four different Gaussian beams with (FWHM) = 0.5, 1, 2, and 3 . The numerical results are summarized z=0 in Table 2. One can see that, by increasing the width of the launch beam, the length of two- and three-channel SDM decreases. One can conclude that a narrower launch beam distribution is more desirable in practical realization of SDM in the investigated SI air-clad silica fiber. Table 2. Length z for two and three spatially multiplexed channels with minimal crosstalk in SI air- SDM clad silica optical fibers for different widths ((FWHM) ) of the Gaussian launch beam distributions. z=0 (FWHM) z (m) z (m) z=0 SDM SDM (deg) (2-Channel) (3-Channel) 3.9 120 30 3 130 33 2 142 36 1 151 39 0.5 158 41 To conclude, an optical fiber with weaker mode coupling and higher NA should be employed in order to achieve a higher capacity of SDM. A further improvement of the capacity of SDM can be achieved by choosing a narrower launch beam. In general, for spatially multiplexed channels, it is difficult to precisely anticipate the amount of crosstalk that will prevent the system from operating by computing the normalized output power distribution for different fiber lengths. In practice, a transmission matrix should be employed for a more precise assessment of the SDM capacity of a specific fiber, taking into consideration the noise factors that are dependent on the receiver ’s individual implementation [27]. Our numerical results provide a good estimate of the optical fiber length, where SDM with three and two channels might be implemented with low crosstalk in the analyzed SI air-clad silica fibers. 4. Conclusions The power flow equation was used to explore the effect of mode coupling on SDM in multimode SI air-clad silica optical fibers. We have shown that mode coupling limits the length at which the SDM can be realized in high-NA air-clad silica optical fiber. When compared to SI PCSFs with smaller NA, this constraint is less noticeable. For SDM, an optical fiber with a lower mode coupling and a higher NA is preferable. Furthermore, a narrower launch beam results in longer fiber lengths at which SDM can be realized. When building an optical fiber transmission system with space division multiplexing, such characterization of optical fibers should be taken into account. Author Contributions: Methodology and software, S.S. and B.D.; conceptualization, S.S.; writing— original draft preparation, A.S. and B.D.; writing—review and editing, A.D., I.S. and R.M.; super- vision, S.S.; funding acquisition, S.S. and R.M. All authors have read and agreed to the published version of the manuscript. Photonics 2022, 9, 127 6 of 7 Funding: This research was funded by the National Natural Science Foundation of China (62003046, 6211101138); the Guangdong Provincial Department of Science and Technology (2021A1313030055); the Serbian Ministry of Education, Science, and Technological Development (Agreement No. 451- 03-68/2020-14/200122); a Strategic Research Grant of the City University of Hong Kong (Project No. CityU 7004600); the Science Fund of the Republic of Serbia (Agreement No. CTPCF-6379382); the Innovation Team Project of Guangdong Provincial Department of Education (2021KCXTD014); a Special project in key field of the Guangdong Provincial Department of Education (2021ZDZX1050); and the Guangdong Basic and Applied Basic Research Foundation (2021A1515011997). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conflicts of Interest: The authors declare no conflict of interest. References 1. Richardson, D.; Fini, J.; Nelson, L. Space-division multiplexing in optical fibres. Nat. Photonics 2013, 7, 354–362. [CrossRef] 2. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. Influence of mode coupling on angular division multiplexing in seven-core plastic optical fibers. Laser Phys. 2020, 30, 065103. [CrossRef] 3. Winzer, P.J. Optical networking beyond WDM. IEEE Photonics J. 2012, 4, 647–651. [CrossRef] 4. Li, G.; Bai, N.; Zhao, N.; Xia, C. Space-division multiplexing: The next frontier in optical communication. Adv. Opt. Photonics 2014, 6, 413–487. [CrossRef] 5. Brunet, C.; Ung, B.; Belanger, P.-A.; Messaddeq, Y.; LaRochelle, S.; Rusch, L.A. Vector mode analysis of ring-core fibers: Design tools for spatial division multiplexing. J. Lightwave Technol. 2014, 32, 4046–4057. [CrossRef] 6. Uden, R.G.H.; Huijskens, R.C.E.L.F.; Xia, C.; Li, G.; Schulzgen, A.; Waardt, H.; Koonen, A.; Okonkwo, C. Ultra-high-density spatial division multiplexing with a few-mode multicore fibre. Nat. Photonics 2014, 8, 865–870. [CrossRef] 7. Zhang, L.; Chen, J.; Agrell, E.; Lin, R.; Wosinska, L. Enabling Technologies for Optical Data Center Networks: Spatial Division Multiplexing. J. Lightwave Technol. 2020, 38, 18–30. [CrossRef] 8. Rommel, S.; Dodane, D.; Grivas, E.; Cimoli, B.; Bourderionnet, J.; Feugnet, G.; Morales, A.; Pikasis, E.; Roeloffzen, C.; Dijk, P.; et al. Towards a Scaleable 5G Fronthaul: Analog Radio-over-Fiber and Space Division Multiplexing. J. Lightwave Technol. 2020, 38, 5412–5422. [CrossRef] 9. Murshid, S.H.; Chakravarty, A.; Biswas, R. Attenuation and modal dispersion models for spatially multiplexed co-propagating helical optical channels in step index fibers. Opt. Laser Technol. 2011, 43, 430–436. [CrossRef] 10. Zhou, J.; Yang, C.; Sui, Q.; Wang, H.; Gao, S.; Feng, Y.; Liu, W.; Yan, Y.; Li, J.; Yu, C.; et al. Burst-Error-Propagation Suppression for Decision-Feedback Equalizer in Field-Trial Submarine Fiber-Optic Communications. J. Lightwave Technol. 2021, 39, 4601–4606. [CrossRef] 11. Apolo, J.; Ortega, B.; Almenar, V. Hybrid POF/VLC Links Based on a Single LED for Indoor Communications. Photonics 2021, 8, 254. [CrossRef] 12. Drljaca, ˇ B.; Simovic, ´ A.; Djordjevich, A.; Savovic, ´ S. Wavelength dependence of equilibrium mode distribution and steady state distribution in W-type plastic-clad silica fibers. Opt. Fiber. Technol. 2020, 54, 102077. [CrossRef] 13. Savovic, ´ S.; Djordjevich, A. Mode coupling in multimode step-index plastic-clad silica fibers with corrugated surfaces. Opt. Laser Technol. 2017, 97, 400–404. [CrossRef] 14. Wang, W.; Wang, N.; Li, K.; Geng, Z.; Jia, H. A novel dual guided modes regions photonic crystal fiber with low crosstalk supporting 56 OAM modes and 4 LP modes. Opt. Fiber. Technol. 2020, 57, 102213. [CrossRef] 15. Åslund, M.L.; Canning, J. Air-clad fibres for astronomical instrumentation: Focal-ratio degradation. Exp. Astron. 2009, 24, 1–7. [CrossRef] 16. Corbett, J.C.W. A brief introduction to photonic crystal fibres for astronomical instrumentalists. New Astron. Rev. 2006, 50, 305–312. [CrossRef] 17. Wadsworth, W.J.; Percival, R.M.; Bouwmans, G.; Knight, J.C.; Birks, T.A.; Hedley, T.D.; Russel, P.S.J. Very high numerical aperture fibers. IEEE Photonics Technol. Lett. 2004, 16, 843–845. [CrossRef] 18. Canning, J.; Buckley, E.; Huntington, S.; Lyytikäinen, K. Using multi-micro channel capillaries for determination of the zeta potential of a microfluidic channel. Electrochim. Acta 2004, 49, 3581–3586. [CrossRef] 19. Gloge, D. Optical power flow in multimode fibers. Bell Syst. Tech. J. 1972, 51, 1767–1783. [CrossRef] 20. Garito, A.F.; Wang, J.; Gao, R. Effects of random perturbations in plastic optical fibers. Science 1998, 281, 962–967. [CrossRef] 21. Djordjevich, A.; Savovic, ´ S. Numerical solution of the power flow equation in step index plastic optical fibers. J. Opt. Soc. Am. B 2004, 21, 1437–1442. [CrossRef] 22. Gambling, W.A.; Payne, D.N.; Matsumura, H. Mode conversion coefficients in optical fibers. Appl. Opt. 1975, 14, 1538–1542. [CrossRef] [PubMed] Photonics 2022, 9, 127 7 of 7 23. Rousseau, M.; Jeunhomme, L. Numerical solution of the coupled-power equation in step index optical fibers. IEEE Trans. Microwave Theory Tech. 1977, 25, 577–585. [CrossRef] 24. Djordjevich, A.; Savovic, ´ S. Investigation of mode coupling in step index plastic optical fibers using the power flow equation. IEEE Photonics Technol. Lett. 2000, 12, 1489–1491. [CrossRef] 25. Savovic, ´ S.; Djordjevich, A.; Min, R. Investigation of mode coupling in step-index air-clad silica optical fibers. Opt. Fiber Technol. 2022, submitted for publication. 26. Savovic, ´ S.; Djordjevich, A. Mode coupling and its influence on space division multiplexing in step-index plastic-clad silica fibers. Opt. Fiber Technol. 2018, 46, 192–197. [CrossRef] 27. Tsekrekos, C.P.; Martinez, A.; Huijskens, F.M.; Koonen, A.M.J. Design considerations for a transparent mode group diversity multiplexing link. IEEE Photonics Technol. Lett. 2006, 18, 2359–2361. [CrossRef] 28. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. Influence of mode coupling on three spatially multiplexed channels in multimode graded index plastic optical fibers. Laser Phys. 2020, 30, 115102. [CrossRef] 29. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. Influence of mode coupling on three, four and five spatially multiplexed channels in multimode step-index plastic optical fibers. Opt. Laser Technol. 2018, 106, 18–21. [CrossRef] 30. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. A transmission length limit for space division multiplexing in step-index silica optical fibers. J. Mod. Opt. 2019, 66, 1695–1700. [CrossRef] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Photonics Multidisciplinary Digital Publishing Institute

Theoretical Investigation of the Capacity of Space Division Multiplexing with Multimode Step-Index Air-Clad Silica Optical Fibers

Loading next page...
 
/lp/multidisciplinary-digital-publishing-institute/theoretical-investigation-of-the-capacity-of-space-division-4pFyRx011z
Publisher
Multidisciplinary Digital Publishing Institute
Copyright
© 1996-2022 MDPI (Basel, Switzerland) unless otherwise stated Disclaimer The statements, opinions and data contained in the journals are solely those of the individual authors and contributors and not of the publisher and the editor(s). MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Terms and Conditions Privacy Policy
ISSN
2304-6732
DOI
10.3390/photonics9030127
Publisher site
See Article on Publisher Site

Abstract

hv photonics Communication Theoretical Investigation of the Capacity of Space Division Multiplexing with Multimode Step-Index Air-Clad Silica Optical Fibers 1 , 2 3 4 5 2 Svetislav Savovic ´ , Alexandar Djordjevich , Isidora Savovic ´ , Branko Drljaca ˇ , Ana Simovic ´ 1 , and Rui Min * Center for Cognition and Neuroergonomics, State Key Laboratory of Cognitive Neuroscience and Learning, Beijing Normal University at Zhuhai, Zhuhai 519087, China; savovic@kg.ac.rs Faculty of Science, University of Kragujevac, R. Domanovica ´ 12, 34000 Kragujevac, Serbia; asimovic@kg.ac.rs Department of Mechanical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China; mealex@cityu.edu.hk Laboratory of Neurodegenerative Disease, School of Biomedical Sciences, LKS Faculty of Medicine, The University of Hong Kong, 21 Sassoon Road, Pokfulam, Hong Kong, China; u3008169@connect.hku.hk Faculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, L. Ribara 29, 38220 Kosovska Mitrovica, Serbia; branko.drljaca@pr.ac.rs * Correspondence: rumi@doctor.upv.es Abstract: We studied the effect of mode coupling on the space division multiplexing (SDM) capabili- ties of multimode step-index (SI) air-clad silica optical fibers by numerically solving the power flow equation. Mode coupling considerably reduces the length of these fibers at which space division mul- tiplexing may be achieved with minimal crosstalk between neighboring optical channels, according to the findings. Up to 120 m and 30 m, respectively, the two and three spatially multiplexed channels in the investigated multimode step-index silica optical fibers can be used with low crosstalk. When building a space division multiplexing-based optical fiber transmission system, such characterization Citation: Savovic, ´ S.; Djordjevich, A.; of optical fibers should be taken into account. Savovic, ´ I.; Drljaca, ˇ B.; Simovic, ´ A.; Min, R. Theoretical Investigation of Keywords: air-clad silica optical fibers; microbends; mode coupling; space division multiplexing the Capacity of Space Division Multiplexing with Multimode Step-Index Air-Clad Silica Optical Fibers. Photonics 2022, 9, 127. 1. Introduction https://doi.org/10.3390/ Global network traffic has expanded dramatically in recent decades, owing primarily photonics9030127 to the rapid expansion of the Internet [1]. Optical fiber systems now support the majority Received: 25 January 2022 of this data flow. Multiplexing, fiber amplifiers, and high-efficiency spectral coding have Accepted: 22 February 2022 all contributed to this capacity gain [1]. Optical data multiplexing is possible not just in Published: 23 February 2022 wavelength, but also in space, time, polarization, and phase. Optical fiber transmission Publisher’s Note: MDPI stays neutral systems can benefit from the multiplexing technique [1,2]. SDM, which includes mode divi- with regard to jurisdictional claims in sion multiplexing using few-mode fibers or multimode and/or core multiplexing utilizing published maps and institutional affil- multicore fibers, has gotten a lot of attention as a way to increase optical communication’s iations. multiplicative capacity [1,3–8]. SDM can operate at the same or separate wavelengths [9]. If the SDM channels inside the carrier fiber are assigned radially distributed optical signals at the same wavelength, the central channel is launched along the fiber axis in the form of a disk, whereas all other channels are in the form of concentric rings. Thus, one increases the Copyright: © 2022 by the authors. capacity of the optical fiber link (Figure 1). Licensee MDPI, Basel, Switzerland. Silica optical fibers (SOFs) are suited to long-distance signal communication [10], while This article is an open access article plastic optical fibers (POFs) with a large core are extensively used in short-distance (less than distributed under the terms and 100 m) networks [11]. Optical measurements, telecommunications, and sensor applications conditions of the Creative Commons all use plastic-clad silica fibers (PCSFs) [12,13]. Currently, the only technology capable Attribution (CC BY) license (https:// of reaching exceptionally high NAs is PCSFs and air-clad silica fibers. In commercially creativecommons.org/licenses/by/ marketed PCSFs, NA is limited to ~0.46. An all-silica fiber is promising for SDM because 4.0/). Photonics 2022, 9, 127. https://doi.org/10.3390/photonics9030127 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 7 applications all use plastic-clad silica fibers (PCSFs) [12,13]. Currently, the only technol- ogy capable of reaching exceptionally high NAs is PCSFs and air-clad silica fibers. In com- mercially marketed PCSFs, NA is limited to ~0.46. An all-silica fiber is promising for SDM because of the material durability and sufficient silica purity for strong transmission per- formance in the large wavelength range of 300 nm to 2.4 µm [14–16]. In contrast, air-clad fibers can have an extraordinarily high numerical aperture (NAmeas > 0.9) [17]. Figure 2 is a cross-sectional sketch of an all-silica air-clad fiber with an annulus made up of a single ring of holes, produced at the University of Sydney [15]. The ring, or rings, of air holes, whose surface roughness has been measured to be 0.5 nm [18], are solely responsible for Photonics 2022, 9, x FOR PEER REVIEW the NA of air-clad fibers. The thin bridges are typically between 100 and 400 nm th 2 ick, of 7 with the ring of holes forming a corrugated surface at the core. To decrease light leakage, the thickness was chosen. applications all use plastic-clad silica fibers (PCSFs) [12,13]. Currently, the only technol- Photonics 2022, 9, 127 2 of 7 ogy capable of reaching exceptionally high NAs is PCSFs and air-clad silica fibers. In com- mercially marketed PCSFs, NA is limited to ~0.46. An all-silica fiber is promising for SDM because of the material durability and sufficient silica purity for strong transmission per- of the material durability and sufficient silica purity for strong transmission performance formance in the large wavelength range of 300 nm to 2.4 µm [14–16]. In contrast, air-clad in the large wavelength range of 300 nm to 2.4 m [14–16]. In contrast, air-clad fibers fibers can have an extraordinarily high numerical aperture (NAmeas > 0.9) [17]. Figure 2 is can have an extraordinarily high numerical aperture (NA > 0.9) [17]. Figure 2 is a meas a cross-sectional sketch of an all-silica air-clad fiber with an annulus made up of a single cross-sectional sketch of an all-silica air-clad fiber with an annulus made up of a single ring ring of holes, produced at the University of Sydney [15]. The ring, or rings, of air holes, Fig of ure holes, 1. Apr sch oduced ematic at of a the thrUniversity ee-channel Sof DM Sydney system, [15 con ].si The sting ring, of a cor ent rings, er diskof an air d tw holes, o conc whose entric whose surface roughness has been measured to be 0.5 nm [18], are solely responsible for rsurface ings. roughness has been measured to be 0.5 nm [18], are solely responsible for the NA of the NA of air-clad fibers. The thin bridges are typically between 100 and 400 nm thick, air-clad fibers. The thin bridges are typically between 100 and 400 nm thick, with the ring with the ring of holes forming a corrugated surface at the core. To decrease light leakage, Fiber imperfections and inhomogeneities introduced during the optical fiber fabrica- of holes forming a corrugated surface at the core. To decrease light leakage, the thickness the thickness was chosen. tion process cause power transfer between adjacent modes [9,19,20]. Because SDM entails was chosen. densely packed spatial channels in a fiber, mode coupling is critical since it allows cross- talk between channels. As a result, the expected beam parameters are altered. The optical power distribution of an optical fiber determines its far-field pattern, which is influenced by launch conditions, fiber parameters, and fiber length. Only short fibers will produce a highly defined ring radiation pattern when light is launched at a given angle > 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 “coupling length” Lc, where the highest-order mode ring- pattern evolves into a disk, an equilibrium mode distribution (EMD) is established. Fig Figure ure 1 1. . A A sch schema emat tiic c o of f a a tt h h rr ee ee -c -h ch an an ne nleS l D SM DM sysy ste st m em, , con cs on istsi in sg tin ofga of cea ntc een r d tier sk d ais nd k t aw no d c to w no ce cn otn ric cen rin tr g ic s. rings. Fiber imperfections and inhomogeneities introduced during the optical fiber fabrica- tion process cause power transfer between adjacent modes [9,19,20]. Because SDM entails densely packed spatial channels in a fiber, mode coupling is critical since it allows cross- talk between channels. As a result, the expected beam parameters are altered. The optical power distribution of an optical fiber determines its far-field pattern, which is influenced by launch conditions, fiber parameters, and fiber length. Only short fibers will produce a highly defined ring radiation pattern when light is launched at a given angle > 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 “coupling length” Lc, where the highest-order mode ring- Figure 2. Sketch of cross section of air-clad silica fiber based on the design used in the experiments of [15]. Figure 2. Sketch of cross section of air-clad silica fiber based on the design used in the experiments pattern evolves into a disk, an equilibrium mode distribution (EMD) is established. of [15]. Fiber imperfections and inhomogeneities introduced during the optical fiber fabrica- tion process cause power transfer between adjacent modes [9,19,20]. Because SDM entails Because information on SDM in multimode air-clad silica optical fibers is lacking in densely packed spatial channels in a fiber, mode coupling is critical since it allows crosstalk the literature, we investigated mode coupling in multimode SI air-clad silica optical fibers between channels. As a result, the expected beam parameters are altered. The optical in this work by numerically solving the power flow equation. This fiber was previously power distribution of an optical fiber determines its far-field pattern, which is influenced investigated experimentally by Åslund et al. [15]. This allows one to determine the maxi- by launch conditions, fiber parameters, and fiber length. Only short fibers will produce a mum fiber lengths for an SDM system that employs multimode SI air-clad silica optical highly defined ring radiation pattern when light is launched at a given angle q > 0 with fibers. respect to the fiber axis. The boundaries of such a ring grow fuzzy at the end of longer fibers due to mode coupling. At “coupling length” L , where the highest-order mode ring-pattern evolves into a disk, an equilibrium mode distribution (EMD) is established. Because information on SDM in multimode air-clad silica optical fibers is lacking in Fig theure literatur 2. Skete, chwe of cinvestigated ross section of mode air-clacoupling d silica fibin er mult based imode on the SI deair sig-clad n used silica in the optical experimen fibers ts of in[1 this 5]. work by numerically solving the power flow equation. This fiber was previously in- vestigated experimentally by Åslund et al. [15]. This allows one to determine the maximum Because information on SDM in multimode air-clad silica optical fibers is lacking in fiber lengths for an SDM system that employs multimode SI air-clad silica optical fibers. the literature, we investigated mode coupling in multimode SI air-clad silica optical fibers 2. Power Flow Equation in this work by numerically solving the power flow equation. This fiber was previously investigated experimentally by Åslund et al. [15]. This allows one to determine the maxi- Gloge’s power flow equation is [19]: mum fiber lengths for an SDM system that employs multimode SI air-clad silica optical ¶P(q, z) D ¶ ¶P(q, z) fibers. = a(q)P(q, z) + q , (1) ¶z q ¶q ¶q where P(q,z) is the angular power distribution, q is the propagation angle in respect to the core axis, z is the distance from the input end of the optical fiber, D is the constant coupling Photonics 2022, 9, 127 3 of 7 coefficient [19,21–23], and a(q) is the modal attenuation. Since a(q) need not be accounted for in solving Equation (1) for mode coupling [22,23], Equation (1) reduces to [15]: ¶P(q, z) D ¶P(q, z) ¶ P(q, z) = + D . (2) ¶z q ¶q ¶q The explicit finite-difference approach [24] was used to derive a numerical solution of the power flow Equation (2) for a Gaussian launch-beam distribution of the form: " # 1 (q q ) P(q, z) = exp , (3) 2s s 2p with 0  q  q , where q is the mean value of the launch angular distribution, s is the standard deviation, and FWHM = 2s 2 ln 2 = 2.355s. We discretized Equation (2) using explicit finite difference method, so Equation (2) now reads [24]: ! ! DzD DzD 2DzD DzD DzD P = P + 1 P + + P , (4) i,j+1 i1,j i,j i+1,j 2 2 2 2q Dq 2q Dq Dq Dq Dq i,j i,j where indexes i and j refer to the discretization step lengths Dq and Dz for the angle q and length z, respectively. This is a simple formula for P at the (i, j + 1)th mesh point in terms i,j+1 of the known values along the jth distance row. The truncation error for the difference in Equation (4) is O(Dz,Dq ). The grid dimension in the q direction is N = q /Dq and the grid dimension in the z direction is M = L/Dz, where q is the critical angle and L is the fiber length. 3. Results and Discussion In this study, we theoretically investigated the effect of mode coupling on SDM capabilities in a multimode SI air-clad silica optical fiber employed in a prior experiment by Åslund et al. [15]. The single-material air-clad fiber was made of low-grade natural silica and had material losses of less than 10 dB/km at 1550 nm. The fiber had NA = 0.54, a core diameter of d = 180 m, and 59 bridges with a length of l  26 m to sustain core it. The maximum thickness of the bridge was 340 nm. The constant coupling coefficient –5 2 for the fiber was D = 3.5  10 rad /m at  = 1550 nm [15,25], which we adopted in this work. The numerical solution to Equation (4) was obtained using discretization step lengths Dq = 0.1 and Dz = 0.001 m. Figure 3 illustrates the normalized output angular power distribution at two lengths of the multimode SI air-clad optical fiber obtained as a solution of Equation (2). The three launch beams in the Gaussian form with (FWHM) = 3.9 and different input z=0 angles q = 0 , 13 , and 26 represent three optical channels [15,25]. In the short SI air-clad optical fiber in Figure 3a, the mode coupling is minimal, and as a result there is no crosstalk between optical channels. Due to mode coupling, spatial power distributions broaden with increasing fiber length, so the three-channel SDM can be realized in this fiber up to a fiber length of z = 30 m (Figure 3b). In the case of two co-propagating optical SDM channels, the two Gaussian launch distributions with input angles q = 0 and 26 , and (FWHM) = 3.9 , are investigated. One can see from Figure 4b that practical realization z=0 of two-channel SDM can be done up to a fiber length of z = 120 m. As can be seen, SDM mode coupling severely restricts the length of multimode SI air-clad silica optical fibers that can be used for SDM (Table 1). Ph Pho otto on niic cs s 2 20 02 22 2,, 9 9,, x x F FO OR R P PEE EER R R RE EV VIIEW EW 4 4 o off 7 7 a are re in inve vest stiiga gated ted.. On One e ca can n se see e f from rom Fi Figure gure 4 4b b th tha at t pr pra acti ctica cal l re rea ali liz za atio tion n o of f t two wo- -cha chan nn nel el S SDM DM ca can n be be d do on ne e up up to to a a f fiber iber leng lengt th h of of z zSD SDM M = = 1 12 20 0 m m.. A As s c ca an n be be see seen n,, m mo od de e cou coupli plin ng g s sev ever erely ely Photonics 2022, 9, 127 4 of 7 re rest stricts ricts th the e le len ngt gth h o of f m mult ultiim mo od de e S SI I a air ir- -cla clad d si sili lica ca o optic ptica al l f fiber ibers s th tha at t ca can n be be used used f fo or r S SDM DM (Table 1). (Table 1). Figure 3. Normalized output intensity at different lengths of the multimode SI air-clad silica optical Fig Figure ure 3 3.. N Nor orm ma ali liz ze ed d ou outtpu putt iin ntten ensi sitty y a att di diffe ffer ren entt le len ng gtth hs s of of tth he e m mult ultimode imode S SI I a air ir- -c cla lad d s sil ilic ica a op opttica icall 5 2 −5 −5 2 2 fiber obtained as solutions of Equation (2) for D = 3.5 × 10 rad /m, three launch beams in the Gauss- fiber fiberobtained obtained as as solutions solutions of of E Equation quation ((2) 2) for for D D = = 3.5 3.5 × 1 0 10 rad /m rad , t/m, hree thr launch ee launch beams beams in the G inath uss- e ia ian n for form m,, a an nd d in inpu putt a an ng gle les s = = 0 0 (s (so oli lid d li lin ne) e),, 1 13 3° ° ( (d da as sh he ed d li lin ne e) ) a an nd d 2 26 6° ° ( (do dotttted ed li lin ne e) ),, w wit ith h Gaussian form, and input angles q = 0 (solid line), 13 (dashed line) and 26 (dotted line), with (FWHM) = 3.9 for: (a) z = 10 m and (b) z = 30 m. (F (FWH WHM) M)z=0 z=0 = = 3 3..9 9° ° for for:: ( (a a) ) z z = = 1 10 0 m m a an nd d ( (b b) ) z z = = 3 30 0 m m.. z=0 Figure 4. Normalized output intensity at different lengths of the multimode SI air-clad silica optical Figure 4. Normalized output intensity at different lengths of the multimode SI air-clad silica optical Figure 4. Normalized output intensity at different lengths of the multimode SI air-clad silica optical −5 25 2 −5 2 fiber fib fiber er obtained ob obtta ain ined ed a as as s so so solutions lut lutio ion ns s of of ofE EEquation qua quattio ion n ( (2 2(2) ) ) fo fofor r r D D =D = 3 3= .5 .5 3.5 × × 10 10 r r 10 a ad d /m /m rad ,, ttw w/m, o o la launch unch two b b launch ea eam ms s in in beams tth he e G Ga ain ussi ussi the a an n o o form, and input angles = 0 (solid line) and 26° (dotted line), with (FWHM)z=0 = 3.9° for: (a) z = 40 form, and input angles = 0 (solid line) and 26° (dotted line), with (FWHM)z=0 = 3.9° for: (a) z = 40 Gaussian form, and input angles q = 0 (solid line) and 26 (dotted line), with (FWHM) = 3.9 for: 0 0 z=0 (a m ) z an =d 40 (bm ) z and = 12(0 b m ) z. = 120 m. m and (b) z = 120 m. Ta Table ble 1 1.. Le Len ng gtth h z zS SD DM M for for ttw wo o a an nd d tth hr ree ee spa spattia iall lly y m mult ultip iple lex xed ed c ch ha an nn nel els s w wit ith h m min inim ima al l c cr ros oss stta alk lk iin n S SI I Table 1. Length z for two and three spatially multiplexed channels with minimal crosstalk in SDM air-clad silica optical fibers and PCSFs, with different numerical apertures NA and coupling coeffi- air-clad silica optical fibers and PCSFs, with different numerical apertures NA and coupling coeffi- SI air-clad silica optical fibers and PCSFs, with different numerical apertures NA and coupling c cie ien ntts s D D.. coefficients D. D D z zS SDM DM ( (m) m) S SDM DM ( (m) m) D z (m) (m) Fiber type NA Fiber type NA SDM SDM Fiber Type NA 2 2 (rad /m) 2 (2-channel) (3-channel) (rad /m) (2-channel) (3-channel) (rad /m) (2-Channel) (3-Channel) Air-clad silica fiber (this Air-clad silica fiber (this −5 Air-clad silica fiber (this work) 0.54 3.5 −5 10 120 30 0 0..5 54 4 3 3..5 5 × × 10 10 120 120 30 30 wo work) rk) PCSF (1) [26] 0.4 1.28  10 50 14 −5 −5 5 PCSF (1) [26] 0.4 1.28 × 10 50 14 PCS PCSF F (1(2) ) [2 [26 6] ] 0.4 0.37 1.28 × 10 4.5  10 50 25 14 7 −5 −5 PCS PCSF ( F (2 2) ) [ [2 26 6] ] 0 0..3 37 7 4 4..5 5 × × 10 10 25 25 7 7 The coupling coefficient D of the multimode SI air-clad silica fiber investigated in this The The coupl couplin ing g coef coeff ficient icient D D o of f th the e m mult ultim imo od de e S SI I a air ir- -cla clad d si sili lica ca f fiber iber in inve ves stig tiga ated ted iin n t th his is study is of the same order of magnitude as that of the previously investigated multimode study is of the same order of magnitude as that of the previously investigated multimode study is of the same order of magnitude as that of the previously investigated multimode SI PCSFs [26] (Table 1). SI PCSFs have a lesser capacity for SDM due to their lower NA of SI PCSFs [26] (Table 1). SI PCSFs have a lesser capacity for SDM due to their lower NA of SI PCSFs [26] (Table 1). SI PCSFs have a lesser capacity for SDM due to their lower NA of 0.37 to 0.4. For example, the capacity for three-channel SDM in the SI air-clad silica fiber investigated in this work, compared to PCSF (2) [26], is about four times greater, which is mainly due to the much higher NA of the SI air-clad silica fiber (these two fibers have a similar coupling coefficient D). On the other hand, despite the higher coupling coefficient D (which restricts the capacity for SDM) of the SI air-clad silica fiber compared to that of PCSF (1) [26], the capacity for three-channel SDM in the SI air-clad silica fiber compared to PCSF Photonics 2022, 9, 127 5 of 7 (1) is about two times greater, due to the higher NA of the SI air-clad silica fiber. Therefore, in practical realization of SDM in PCSFs and air-clad silica fibers, the coupling coefficient is the dominant factor for lower NA PCSFs compared to high NA air-clad silica fibers. As a result, two- and three-channel SDM can be achieved in SI air-clad silica optical fibers with lengths longer than PCSFs (Table 1). Air-clad silica fibers are, hence, more suitable for SDM applications. It is worth noting that, in our previous works, we have shown that this kind of SDM can be employed in standard multimode SI plastic optical fibers at lengths of up to few meters [27–29] and multimode SI silica optical fibers at lengths of several hundreds of meters [30]. In order to investigate the influence of (FWHM) of launch beam distribution on z=0 the capacity of SDM in the SI air-clad silica fiber, we solved Equation (1) for four different Gaussian beams with (FWHM) = 0.5, 1, 2, and 3 . The numerical results are summarized z=0 in Table 2. One can see that, by increasing the width of the launch beam, the length of two- and three-channel SDM decreases. One can conclude that a narrower launch beam distribution is more desirable in practical realization of SDM in the investigated SI air-clad silica fiber. Table 2. Length z for two and three spatially multiplexed channels with minimal crosstalk in SI air- SDM clad silica optical fibers for different widths ((FWHM) ) of the Gaussian launch beam distributions. z=0 (FWHM) z (m) z (m) z=0 SDM SDM (deg) (2-Channel) (3-Channel) 3.9 120 30 3 130 33 2 142 36 1 151 39 0.5 158 41 To conclude, an optical fiber with weaker mode coupling and higher NA should be employed in order to achieve a higher capacity of SDM. A further improvement of the capacity of SDM can be achieved by choosing a narrower launch beam. In general, for spatially multiplexed channels, it is difficult to precisely anticipate the amount of crosstalk that will prevent the system from operating by computing the normalized output power distribution for different fiber lengths. In practice, a transmission matrix should be employed for a more precise assessment of the SDM capacity of a specific fiber, taking into consideration the noise factors that are dependent on the receiver ’s individual implementation [27]. Our numerical results provide a good estimate of the optical fiber length, where SDM with three and two channels might be implemented with low crosstalk in the analyzed SI air-clad silica fibers. 4. Conclusions The power flow equation was used to explore the effect of mode coupling on SDM in multimode SI air-clad silica optical fibers. We have shown that mode coupling limits the length at which the SDM can be realized in high-NA air-clad silica optical fiber. When compared to SI PCSFs with smaller NA, this constraint is less noticeable. For SDM, an optical fiber with a lower mode coupling and a higher NA is preferable. Furthermore, a narrower launch beam results in longer fiber lengths at which SDM can be realized. When building an optical fiber transmission system with space division multiplexing, such characterization of optical fibers should be taken into account. Author Contributions: Methodology and software, S.S. and B.D.; conceptualization, S.S.; writing— original draft preparation, A.S. and B.D.; writing—review and editing, A.D., I.S. and R.M.; super- vision, S.S.; funding acquisition, S.S. and R.M. All authors have read and agreed to the published version of the manuscript. Photonics 2022, 9, 127 6 of 7 Funding: This research was funded by the National Natural Science Foundation of China (62003046, 6211101138); the Guangdong Provincial Department of Science and Technology (2021A1313030055); the Serbian Ministry of Education, Science, and Technological Development (Agreement No. 451- 03-68/2020-14/200122); a Strategic Research Grant of the City University of Hong Kong (Project No. CityU 7004600); the Science Fund of the Republic of Serbia (Agreement No. CTPCF-6379382); the Innovation Team Project of Guangdong Provincial Department of Education (2021KCXTD014); a Special project in key field of the Guangdong Provincial Department of Education (2021ZDZX1050); and the Guangdong Basic and Applied Basic Research Foundation (2021A1515011997). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conflicts of Interest: The authors declare no conflict of interest. References 1. Richardson, D.; Fini, J.; Nelson, L. Space-division multiplexing in optical fibres. Nat. Photonics 2013, 7, 354–362. [CrossRef] 2. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. Influence of mode coupling on angular division multiplexing in seven-core plastic optical fibers. Laser Phys. 2020, 30, 065103. [CrossRef] 3. Winzer, P.J. Optical networking beyond WDM. IEEE Photonics J. 2012, 4, 647–651. [CrossRef] 4. Li, G.; Bai, N.; Zhao, N.; Xia, C. Space-division multiplexing: The next frontier in optical communication. Adv. Opt. Photonics 2014, 6, 413–487. [CrossRef] 5. Brunet, C.; Ung, B.; Belanger, P.-A.; Messaddeq, Y.; LaRochelle, S.; Rusch, L.A. Vector mode analysis of ring-core fibers: Design tools for spatial division multiplexing. J. Lightwave Technol. 2014, 32, 4046–4057. [CrossRef] 6. Uden, R.G.H.; Huijskens, R.C.E.L.F.; Xia, C.; Li, G.; Schulzgen, A.; Waardt, H.; Koonen, A.; Okonkwo, C. Ultra-high-density spatial division multiplexing with a few-mode multicore fibre. Nat. Photonics 2014, 8, 865–870. [CrossRef] 7. Zhang, L.; Chen, J.; Agrell, E.; Lin, R.; Wosinska, L. Enabling Technologies for Optical Data Center Networks: Spatial Division Multiplexing. J. Lightwave Technol. 2020, 38, 18–30. [CrossRef] 8. Rommel, S.; Dodane, D.; Grivas, E.; Cimoli, B.; Bourderionnet, J.; Feugnet, G.; Morales, A.; Pikasis, E.; Roeloffzen, C.; Dijk, P.; et al. Towards a Scaleable 5G Fronthaul: Analog Radio-over-Fiber and Space Division Multiplexing. J. Lightwave Technol. 2020, 38, 5412–5422. [CrossRef] 9. Murshid, S.H.; Chakravarty, A.; Biswas, R. Attenuation and modal dispersion models for spatially multiplexed co-propagating helical optical channels in step index fibers. Opt. Laser Technol. 2011, 43, 430–436. [CrossRef] 10. Zhou, J.; Yang, C.; Sui, Q.; Wang, H.; Gao, S.; Feng, Y.; Liu, W.; Yan, Y.; Li, J.; Yu, C.; et al. Burst-Error-Propagation Suppression for Decision-Feedback Equalizer in Field-Trial Submarine Fiber-Optic Communications. J. Lightwave Technol. 2021, 39, 4601–4606. [CrossRef] 11. Apolo, J.; Ortega, B.; Almenar, V. Hybrid POF/VLC Links Based on a Single LED for Indoor Communications. Photonics 2021, 8, 254. [CrossRef] 12. Drljaca, ˇ B.; Simovic, ´ A.; Djordjevich, A.; Savovic, ´ S. Wavelength dependence of equilibrium mode distribution and steady state distribution in W-type plastic-clad silica fibers. Opt. Fiber. Technol. 2020, 54, 102077. [CrossRef] 13. Savovic, ´ S.; Djordjevich, A. Mode coupling in multimode step-index plastic-clad silica fibers with corrugated surfaces. Opt. Laser Technol. 2017, 97, 400–404. [CrossRef] 14. Wang, W.; Wang, N.; Li, K.; Geng, Z.; Jia, H. A novel dual guided modes regions photonic crystal fiber with low crosstalk supporting 56 OAM modes and 4 LP modes. Opt. Fiber. Technol. 2020, 57, 102213. [CrossRef] 15. Åslund, M.L.; Canning, J. Air-clad fibres for astronomical instrumentation: Focal-ratio degradation. Exp. Astron. 2009, 24, 1–7. [CrossRef] 16. Corbett, J.C.W. A brief introduction to photonic crystal fibres for astronomical instrumentalists. New Astron. Rev. 2006, 50, 305–312. [CrossRef] 17. Wadsworth, W.J.; Percival, R.M.; Bouwmans, G.; Knight, J.C.; Birks, T.A.; Hedley, T.D.; Russel, P.S.J. Very high numerical aperture fibers. IEEE Photonics Technol. Lett. 2004, 16, 843–845. [CrossRef] 18. Canning, J.; Buckley, E.; Huntington, S.; Lyytikäinen, K. Using multi-micro channel capillaries for determination of the zeta potential of a microfluidic channel. Electrochim. Acta 2004, 49, 3581–3586. [CrossRef] 19. Gloge, D. Optical power flow in multimode fibers. Bell Syst. Tech. J. 1972, 51, 1767–1783. [CrossRef] 20. Garito, A.F.; Wang, J.; Gao, R. Effects of random perturbations in plastic optical fibers. Science 1998, 281, 962–967. [CrossRef] 21. Djordjevich, A.; Savovic, ´ S. Numerical solution of the power flow equation in step index plastic optical fibers. J. Opt. Soc. Am. B 2004, 21, 1437–1442. [CrossRef] 22. Gambling, W.A.; Payne, D.N.; Matsumura, H. Mode conversion coefficients in optical fibers. Appl. Opt. 1975, 14, 1538–1542. [CrossRef] [PubMed] Photonics 2022, 9, 127 7 of 7 23. Rousseau, M.; Jeunhomme, L. Numerical solution of the coupled-power equation in step index optical fibers. IEEE Trans. Microwave Theory Tech. 1977, 25, 577–585. [CrossRef] 24. Djordjevich, A.; Savovic, ´ S. Investigation of mode coupling in step index plastic optical fibers using the power flow equation. IEEE Photonics Technol. Lett. 2000, 12, 1489–1491. [CrossRef] 25. Savovic, ´ S.; Djordjevich, A.; Min, R. Investigation of mode coupling in step-index air-clad silica optical fibers. Opt. Fiber Technol. 2022, submitted for publication. 26. Savovic, ´ S.; Djordjevich, A. Mode coupling and its influence on space division multiplexing in step-index plastic-clad silica fibers. Opt. Fiber Technol. 2018, 46, 192–197. [CrossRef] 27. Tsekrekos, C.P.; Martinez, A.; Huijskens, F.M.; Koonen, A.M.J. Design considerations for a transparent mode group diversity multiplexing link. IEEE Photonics Technol. Lett. 2006, 18, 2359–2361. [CrossRef] 28. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. Influence of mode coupling on three spatially multiplexed channels in multimode graded index plastic optical fibers. Laser Phys. 2020, 30, 115102. [CrossRef] 29. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. Influence of mode coupling on three, four and five spatially multiplexed channels in multimode step-index plastic optical fibers. Opt. Laser Technol. 2018, 106, 18–21. [CrossRef] 30. Savovic, ´ S.; Djordjevich, A.; Simovic, ´ A.; Drljaca, ˇ B. A transmission length limit for space division multiplexing in step-index silica optical fibers. J. Mod. Opt. 2019, 66, 1695–1700. [CrossRef]

Journal

PhotonicsMultidisciplinary Digital Publishing Institute

Published: Feb 23, 2022

Keywords: air-clad silica optical fibers; microbends; mode coupling; space division multiplexing

There are no references for this article.